Analysis of Shells and Plates
Phillip L. Gould
Analysis of Shells
and Plates
With 164 Illustrations in 237 Parts
Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo
Phillip L. Gould
Department of Civil Engineering
Washington University
St. Louis, MO 63130, USA
Cataloging-in-Publication Data
Gould, Phillip L.
Analysis of shells and plates / Phillip L. Gould.
p.
cm.
Includes bibliographies.
ISBN-13:978-1-4612- 8340·9
1. Shells (Engineering) 2. Plates (Engineering)
TA660.S5G644 1987
624.1'776-dc 1987-21011
I. Title.
Previous edition: Phillip L. Gould, Static Analysis of Shells. © 1977 by D.C. Heath Company
©1988 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1st edition 1988
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, New York
10010, lJSA), except for brief excerpts in connection with reviews or scholarly analysis. Use in
connection with any form of information storage and retrieval, electronic adaptation, computer
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the
former are not especially identified, is not to be taken as a sign that such names, as under~tood by
the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Typeset by Asco Trade Typesetting Ltd., Hong Kong.
9 8 7 654 3 2 1
ISBN-13:978·1-4612-8340·9
DOl: 10.1007/978-1-4612-3764·8
e-ISBN-13:978-1-4612-3764·8
To David and Belle Gould,
my parents, and
Deborah Gould, my wife
Preface
The study ofthree-dimensional continua has been a traditional part of graduate
education in solid mechanics for some time. With rational simplifications to the
three-dimensional theory of elasticity, the engineering theories of medium-thin
plates and of thin shells may be derived and applied to a large class of engineering structures distinguished by a characteristically small dimension in one
direction. Often, these theories are developed somewhat independently due to
their distinctive geometrical and load-resistance characteristics. On the other
hand, the two systems share a common basis and might be unified under the
classification of Surface Structures after the German term Fliichentragwerke.
This common basis is fully exploited in this book.
A substantial portion of many traditional approaches to this subject has been
devoted to constructing classical and approximate solutions to the governing
equations of the system in order to proceed with applications. Within the
context of analytical, as opposed to numerical, approaches, the limited generality of many such solutions has been a formidable obstacle to applications
involving complex geometry, material properties, and/or loading. It is now
relatively routine to obtain computer-based solutions to quite complicated
situations. However, the choice of the proper problem to solve through the
selection of the mathematical model remains a human rather than a machine
task and requires a basis in the theory of the subject.
With the requirement of a strong grounding in the engineering theories of
shells and plates remaining firm, this book presents a unified development
with emphasis on the fundamental engineering aspects. The basic material is
designed to be covered in a single semester graduate course or through equivalent self-study; also, ample enrichment is provided for further independent
study. Initially, the geometrical relationships are developed on a somewhat
general level, with specific applications to frequently encountered forms. Following the geometric description of the surface and the consideration of equilibrium, we introduce a first logical simplification, the membrane theory of
shells. After a further theoretical exposition of linear deformations, constitutive
relationships, and energy principles, we present the flexural theory of plates
and the bending theory of shells, including elastic instability. Although this
sequence postpones the introduction of the complete theory of plates until much
of the theory of shells is covered, we believe it is logical and consistent with the
vii
viii
Preface
unifying objective of the text. Additionally, instructive exercises are provided
at the end each chapter.
The fundamental geometric and static relationships are considered from an
integrated mathematical and physical point of view. Orthogonal curvilinear
coordinates and vector calculus are used to provide a concise general derivation
of the field equations. Early on, however, we present the physical resolution
and interpretation of forces and deformations for specialized geometric forms
such as rotational shells and flat plates. We believe that the physical notions
are more meaningful in the specialized geometric context, whereas the mathematical formulation yields a conciseness not attainable with a strictly physical
viewpoint. Also, we stress the energy aspects of the formulations because of the
importance of energy methods in modern computational techniques.
With regard to applications, our focus is on classical examples that illustrate
the basic resistance mechanisms of selected configurations without undue
mathematical complications. The availability of numerically based, computerimplemented solution algorithms has diminished the need to rely on cumbersome, sometimes oversimplified, analytical solutions for complex problems;
hence, they are paid scant attention in this text. Rather, we emphasize the
essential aspects of equilibrium and compatibility, as they apply to the resistance
of surface loading and the satisfaction of boundary constraints. We believe that
a firm grasp of these principles is necessary to perform the vital critical interpretation required of the analyst when computer-based solutions are employed.
This book is dedicated to developing in the engineer the physical and mathematical understanding required to perform analysis and design in an interactive
computer-assisted environment. The book is also designed to provide a foundation of the subject which will enable the interested reader to progress to more
advanced texts and technical papers.
Of special interest in the book is a thorough treatment of hyperboloidal shells
of revolution. This topic is of current interest because of the wide use of this
form in cooling tower applications.
Regarding the background of the reader, this book is written primarily for
advanced students and practitioners in Civil, Mechanical, and Aeronautical
Engineering and Engineering Mechanics. We presume that the reader has an
elementary knowledge of vector calculus and matrix algebra, and some familiarity with the linear theory of elasticity.
Phillip L. Gould
Acknowledgments
Many contributors, both direct and indirect, to an earlier volume Static Analysis of Shells (D. C. Heath, 1974), were acknowledged then and their contribution to this book is likewise appreciated.
The interest of the Springer-Verlag publishing program in allowing the
author to present a more complete, more accurate, and, hopefully, more incisive
treatment of the topic is greatly appreciated. The present work incorporates
the results of an additional decade of criticism, reflection and study, and,
hopefully, represents a significant improvement over the earlier volume.
In the past decade, the author has been influenced by a number of contemporary engineers and scientists. Among those whose contributions directly impacted this book are-in alphabetical order-Dr. P. Bergan, Veritec, Norway;
Dr. D. Bushnell, Lockheed-Palo Alto; Dr. J. Bobrowski, Consulting Engineer,
UK.; Prof. C. R. Calladine, University of Cambridge; Prof. J. G. A. Croll,
University College, London; Prof. K. J. Han, University of Houston; Prof. S.
Kato, Toyohashi Institute of Technology; Prof. M. Ketchum, University of
Connecticut; Prof. W. Kratzig, Ruhr-Universitat-Bochum; Prof. D. Pecknold,
University of Illinois at Urbana-Champaign; Prof. E. Reissner, University of
California, San Diego; Dr. J. M. Rotter, University of Sydney; Prof. W.
Schnobrich, University of Illinois at Urbana-Champaign; Prof. S. Simmonds,
University of Alberta; Prof. U. Wittek, University of Kaiserlautern; Prof. J. K.
Wu, Peking University.
The author is also indebted to current and recent students B. J. Lee, J. S. Lin,
Robert Elkin, Michael Williams, and Hidajat Harintho for careful proofreading
and suggestions. Also, Mr. Sakul Pochanart assisted in improving the illustrations and prepared many of the drawings. Ms. Kathryn Schallert carefully
revised and retyped much of the manuscript.
Finally, the author is appreciative of the academic atmosphere provided at
Washington University, and ofthe patience and cooperation of his colleagues
throughout the manuscript preparation process.
IX
Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
ix
Chapter 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1 Role of the Theory of Elasttcity . . . . . . . . . . . . . . . . . . . . . . .
Engineering Theories. . . . . . .
Load Resistance Mechanisms
References. . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . .
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14
Chapter 2. Geometry... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Curvilinear Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . .
Middle Surface Geometry. . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Unit Tangent Vectors and Principal Directions. . . . . . . . . .
2.4 Second Quadratic Form of the Theory of Surfaces. . . . . . . .
2.5 Principal Radii of Curvature. . . . . . . . . . . . . . . . . . . . . . . .
2.6 Gauss-Codazzi Relations . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Gaussian Curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Specialization of Shell Geometry. . . . . . . . . . . . . . . . . . . . .
2.9 References......................................
2.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
16
1.2
1.3
1.4
1.5
2.1
2.2
Chapter 3. Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Stress Resultants and Couples. . . . . . . . . . . . . . . .
3.2 Equilibrium of the Shell Element. . . . . . . . . . . . . .
3.3 Equilibrium Equations for Shells of Revolution. . .
3.4 Equilibrium Equations for Plates. . . . . . . . . . . . . .
3.5 Nature of the Applied Loading. . . . . . . . . . . . . . .
3.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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15
21
26
28
29
30
32
53
54
55
55
60
63
67
67
69
69
Chapter 4. Membrane Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.1 Simplification of the Equilibrium Equations. . . . . . . . . . . . .
70
4.2 Applicability of Membrane Theory . . . . . . . . . . . . . . . . . . . .
71
xi
xii
Contents
73
4.3 Shells of Revolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Shells of Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
4.5 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Appendix 4A. Summary of Surface Loading and Stress Resultants
for Quasistatic Seismic Loading on Hyperboloidal Shells of
Revolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
195
197
Chapter 5. Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
206
151
191
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206
206
213
218
224
226
226
Chapter 6. Constitutive Laws, Boundary Conditions, and Displacements
227
5.1
5.2
5.3
5.4
5.5
5.6
5.7
General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Strain-Displacement Relations for Shells of Revolution.
Strain-Displacement Relations for Plates. . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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227
237
245
258
Chapter 7. Energy and Approximate Methods. . . . . . . . . . . . . . . . . . . ..
261
6.1
6.2
6.3
6.4
6.5
Constitutive Laws. . . . . . . . . . . . .
Boundary Conditions. . . . . . . . . .
Membrane Theory Displacements.
References. . . . . . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . . . . . .
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259
General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Strain Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Potential Energy of the Applied Loads . . . . . . . . . . . . . . . ..
Energy Principles and Rayleigh-Ritz Methods. . . . . . . . . ..
Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
261
261
263
264
Chapter 8. Bending of Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
272
7.1
7.2
7.3
7.4
7.5
7.6
7.7
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
Governing Equations. . . . . . . . . . .
Rectangular Plates ............ .
Circular Plates .. . . . . . . . . . . . . . .
Plates of Other Shapes . . . . . . . . . .
Energy Method Solutions. . . . . . . .
Extensions of the Theory of Plates. .
Instability and Finite Deformation.
References.. . . . . . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . . . . . . .
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269
270
270
272
289
313
329
332
342
349
364
366
Contents
xiii
Chapter 9. Shell Bending and Instability. . . . . . . . . . . . . . . . . . . . . . . . ..
9.1
9.2
9.3
904
9.5
9.6
9.7
General. . . . . . . . . . . . . . . . . . . . . . . .
Circular Cylindrical Shells. . . . . . . . . .
Shells of Revolution. . . . . . . . . . . . . . .
Shells of Translation. . . . . . . . . . . . . .
Instability and Finite Deformations. . .
References. . . . . . . . . . . . . . . . . . . . . .
Exercises. . . . . . . . . . . . . . . . . . . . . . .
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372
373
413
443
451
466
469
Chapter 10. Conclusion... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
472
10.1 General.. . . . . . . . . . . . . . . . . . . .
10.2 Proportioning. . . . . . . . . . . . . . . .
10.3 Future Applications of Thin Shells.
lOA References. . . . . . . . . . . . . . . . . . .
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472
472
474
475
Index....... ......................................
477
CHAPTER
1
Introduction
1.1 Role of the Theory of Elasticity
The theory of elasticity is the basis for several engineering theories which, in
turn, are applied to mechanical and structural design. The basic components
of the elasticity problem are often designated as equilibrium, compatibility, and
a constitutive law. The equilibrium equations represent a statement of Newton's
laws, which are restricted here to the static case. The compatibility conditions
express the kinematic relationships between strains and displacements, and the
constitutive law embodies the stress-strain behavior of the material which is
presumed here to be linear elastic. In general, this set of basic components may
be collected as a set of differential equations or as an energy principle. The
simultaneous satisfaction of each component of the elasticity problem often
is foreboding from the mathematical standpoint; therefore, engineers have
naturally looked to simplifications and approximations.
One common simplification often imposed on elasticity problems has been
to formulate less restrictive theories based on distinctive geometric characteristics. Among the relaxed theories, we find (1) the theory of beams, which is
concerned with flexural members having one dimension characteristically far
greater than the other two; (2) the theory of plates, which treats initially flat
components having two dimensions far greater than the third; and (3) the theory
of shells, which deals with curved bodies having one small dimension. Within
these three theories, there are a variety of subtheories. For the purposes of this
introduction it is sufficient to refer to the most common subtheory for each
case; e.g., (1) shallow beam theory; (2) medium-thin plate theory, and (3) thin shell
theory.
Although a beam may be considered as a one-dimensional member, whereas
plates and shells are two-dimensional, the similarities in the theories are numerous, and the beam serves as a useful analogue for the exposition of the higher
theories.
1.2 Engineering Theories
Sections of a beam, a plate, and a shell are shown in figure 1-1. On each figure,
the characteristic dimensions in the transverse and lateral directions are de1
2
1 Introduction
~ middle
axis
... .. ,:~::~::;.:-: :~::::: ..;.:.:
",
1.
surface
Fig. 1-1
Characteristic Dimensions of Structural Forms
noted as h and I, respectively. Also, a reference position is identified in the
transverse direction, midway between the boundaries for symmetric cross sections. We call this reference position the neutral or middle axis for a beam, the
middle plane for a plate, and the middle surface for a shell. In the ensuing
treatment, the terms plane and plate, and surface and shell, are used synonymously and interchangeably to avoid repetitive distinction between the mathematical and physical objects.
The initial step in the derivation of each of the simplified theories usually
consists of a set of assumptions with respect to the ratio of the characteristic
dimensions, the relative magnitude of the deflection under the applied loading,
the rotation of a normal to the undeformed reference position, and the stresses
1.3 Load Resistance Mechanisms
3
in the transverse directions. The statement and justification of these assumptions are largely attributable to several distinguished mathematicians and
scientists of the eighteenth and nineteenth centuries. Thus, we have the Navier
hypothesis and the Bernoulli-Euler theory for beams, the Kirchhoff theory of
plates, and Love's first approximation to the theory of shells. a.1
These assumptions are collected in table 1-1. A number of comments with
respect to the assumptions and justifications are appropriate:
[lJ This assumption may be regarded as the most fundamental, since it clearly
delineates the class of problems with which we are concerned from a
physical standpoint. Also, [lJ is the justification for [3J and [4]. The
bounds on h/l are only approximate and are subject to considerable
latitude depending on loading, geometry, etc.
[2J This assumption is independent, although the cases for which it would be
violated might well coincide with the lower range of h/l stated in [1]. If
[2J is not justified, geometrically nonlinear theories can be formulated
retaining the remaining assumptions. (An illustration of such a formulation is presented in section 8.7.) We may also accommodate material
nonlinearities within the range of [2]. However, as the magnitude of the
admissible displacements increases, the possibility of exceeding the limits
of linear elastic material behavior increases proportionally.
[3J This assumption is developed in detail within the treatment of deformations in chapter 5. When only classical solution techniques were available,
the suppression of transverse shearing strains permitted many otherwise
intractable problems to be approached. With powerful numerical procedures now well developed, this necessity has diminished, although it is
still popular. It has been suggested in an independent observation that this
strategy, originally conceived to facilitate analytical solutions, indeed often
complicates numerically based solutions. 2 Relaxation of [3J enables the
upper limit on h/l in [1] to be extended in many cases.
[4J Situations for which assumption would not be justified, apart from the
immediate vicinity of concentrated loads, would coincide with the upper
limits of [1].
The assumptions and consequences collected in table 1-1 are of utmost
importance in what follows and are referred to frequently.
1.3 Load Resistance Mechanisms
The common basis of shallow beam, medium-thin plate, and thin shell theories
is illustrated by the unified set of underlying assumptions. However, the means
• Readers who are interested in the historical development of solid mechanics and in the
distinguished personalities who contributed to this development are referred to Todhunter and
Pearson, A. E. H. Love, S. A. Timoshenko, and H. M. Westergaard. [1]
Table 1-1
Basic assumptions
Assumption
[1]
Transverse characteristic dimension is small
in comparison
to lateral
characteristic
dimension
[2]
Displacements are
small in comparis on to
transverse characteristic
dimension
[3]
Transverse
shearing strains
which act on
planes parallel
to middle (section, plane,
surface) are
neglected
Theory
Consequence
Justification
Shallow
beam
Beam is shallow in comparison to length, h < 1
Medium-thin
plate
Plate is thin in comparison to lateral dimension, h < 1
Shell is thin in comparison to minimum radius
of curvature, h « 1
0.01 ::::; h/l ::::; 0.5
deep
cable
beam
0.001 ::::; h/l ::::; 0.4
membrane thick
plate
0.001 ::::; h/l ::::; 0.05
thick
curved
membrane shell
Thin shell
All
Equilibrium may be formula ted with respect to
the initial undeformed
geometry. Products of
deformation parameters may be neglected.
The system may be described by a system of
geometrically linear
equations
Validity may be
established by
calculation in
the course of
the solution
Shallow
beam
Plane sections before deformation remain plane
after deformation
Straight fibers which are
perpendicular to the
middle plane before
deformation remain
perpendicular to the
middle plane after
deformation
Straight fibers which are
perpendicular to the
middle surface before
deformati.on remain
perpendicular to the
middle surface after
deformation
h<l
Medium-thin
plate
Thin shell
[4]
Normal stresses
acting on planes
parallel to middIe (section,
plane, surface)
are neglected
4
Shallow
beam
Medium-thin
plate
Thin shell
Beam depth does not
change during deformation
Plate thickness does not
change during deformation
Shell thickness does not
change during deformation
h<l
h«l
h<l
h< 1
h« 1
5
1.3 Load Resistance Mechanisms
(c)
(d)
Fig. 1-2
All forces shown in
Fig. (c) also are acting
Means of Load Resistance
for resisting applied loading among these structural forms may be quite different. Idealized free-body diagrams of the three cases, along with an additional
form, the arch, are shown in figure 1-2. For simplicity, only a vertical loading
is shown in each case, but the following observations are generally valid for any
distributed loading.
The beam, being straight, depends on the shear V to resist transverse loading.
In turn, V, acting together with the loading and reaction R, requires a moment
M for equilibrium. The beam may be classified as a one-dimensional flexural
member.
The arch, because it is curved, can develop a thrust N to resist the applied
loading, in addition to the shear V. Although V and M are still present in the
general case, the efficiency ofthe arch form lies primarily in resisting the loading
with N and minimizing V and M. The arch may be called a one-dimensional
extensional member.
The plate, being flat, relies on the transverse shear V to resist transverse
6
1 Introduction
Table 1-2
Classification of structural forms
Configuration
2- Dimensional
Primary Resistance Mechanism
1- Dimensional
Flexural
Extensional
Beam
Arch
Plate
Shell
loading in the same manner as the beam. The two-dimensional configuration
results in bending moments M and, additionally, twisting moments T on each
internal face. Because the loading is generally carried in both directions and
because the twisting rigidity in isotropic plates is quite significant, a plate is
considerably stiffer than a beam of comparable span and thickness. The plate
may be considered a two-dimensional flexural member.
The shell, being curved, can develop thrusts N to form the primary resistance
mechanism in addition to those forces and moments present in the plate. Also,
the two-dimensional curved configuration mobilizes an in-plane shear S in each
direction. Although V, M, and T are still present in the general case, the
efficiency of the shell form rests with the reliance on Nand S as the primary
means of resistance, with V, M, and T minimized. The shell may be termed a
two-dimensional extensional member. Furthermore, whereas an arch of a specific shape can resist only one loading pattern in a purely extensional state, a
shell can remain virtually momentless for a variety of loadings. The classification of these forms is summarized in table 1-2.
Of course, there is some overlap within the classification in table 1-2. One
common situation is the presence of axial loading in a beam and, correspondingly, in-plane loading in a plate. Within the limitations of assumption
[2J, these effects, called rod and diaphragm action, respectively, are uncoupled
from the primary action and may be combined with the flexural behavior by
superposition. The study of instability, however, involves a coupling between
extensional and flexural effects.
Another possibility with respect to the beam or plate is the presence of axial
or in-plane forces which develop a transverse component through curvature
induced by the flexural action. This case may be treated only with a relaxation
of assumption [2J.
A third combination, which is of considerable practical importance, is a form
curved in only one direction, such as a cylinder or cone. Although these
so-called developable surfaces are generally regarded as shells , the resistance
mechanism in the uncurved direction is basically flexural, whereas that in the
curved direction may be primarily extensional.
As a further introductory comment, it should be pointed out that structural
materials are generally far more efficient in an extensional rather than in a
flexural mode, making the arch and shell preferable over the beam and plate.
The extensional mode, within the scope of small deformations, can be developed
7
1.3 Load Resistance Mechanisms
Fig. 1-3 (a)
Pantheon, Rome, Italy, Dome Span
= 43.4 m; Dome Rise = 21.6 m
only through initial curvature; this limits the application of arches and shells
both from a fabrication and from a utilization standpoint. The structural form
is subject to many constraints apart from the most efficient use of the material
and will be considered as specified a priori in most of this text.
It is of present as well as historic!!l interest to recognize that significant
structures were erected utilizing the efficient doubly curved form of the dome
long before the development of modern engineering analysis. 3 Several of these
structures survive today. The Pantheon of ancient Rome (figure 1-3), attributed
to Marcus Agrippa and Emperor Hadrian and constructed of a cementitious
material, has stood for about two thousand years; the Hagia Sophia in Istanbul
(figure 1-4), originally completed by Isidorus, Jr., has epitomized Byzantine
architecture for fifteen centuries. Beautifully tile-covered mosques from the
Persian empire (figure 1-5) survive in Iran. Increased geometrical refinement is
exhibited in the Renaissance cathedrals of Santa Maria del Fiore in Florence
(figure 1-6), constructed without shoring by Brunelleschi, and St. Peter's in
Rome (figure 1-7), designed by Michelangelo. Also, Sir Christopher Wren's St.
Paul's Cathedral (figure 1-8) remains to grace the London skyline.
Of these ancient domes, the Pantheon and Santa Maria are regarded as
among the greatest construction achievements of all time. 4,5 In addition to the
obviously spectacular clear span, the roof of the Pantheon is formed with
intersecting ribs (figure 1-3b) which provide greater stiffness and stability than
an equivalent amount of material of uniform thickness. Further weight reduc-
8
1 Introduction
Fig. 1-3(b) The Interior of the Pantheon; National Gallery of Art, Washington; Samuel H. Kress
Collection
tion is achieved by incorporating three grades of lightweight aggregate in the
concrete. 4 Brunelleschi's dome, documented in considerable detail by Parsens, 5
marks a deliberate departure from the then traditional hemisphere to a more
efficient pointed (circular) arch profile. The cross section is composed of a
9
1.3 Load Resistance Mechanisms
Fig. 1-4 Hagia Sophia, Istanbul, Turkey. Dome Span
Dr. I. Mungan)
Fig. 1-5
= 31.9 m; Dome Rise = 13.8 m (Courtesy
Dome of Shah's Mosque, Iran
10
Fig.1-6(a)
1 Introduction
S. Maria Del Fiore, Florence, Italy. Dome Span = 42.4 m; Dome Rise = 36.6 m
separated double wall (figure 1-6b), tied together by the meridional arches and
a secondary system of horizontal arches. Also incorporated was a timber chain
circling the dome to resist the somewhat vaguely understood outward thrust
resulting from the arched meridional profile. It is remarkable that the two
techniques used for increasing the structural efficiency of the walls of these
ancient structures have modern counterparts, i.e., stiffened and multilayered
shells as described in chapter 6.
Although the ancient domes were not thin or engineered in the modern sense,
they exhibit the.unique capability of the curved surface to bridge considerable
space without intermediate supports utilizing construction materials capable
of resisting only compressive forces. Modern techniques of structural analysis,
1.3 Load Resistance Mechanisms
11
11 Fw.# /rN4. , I.I·"""~.I .,..,
II J.....,~I.H',.JttI..,...- ;"·I.·,, ' ...
('" ,......... ...!!..1. ,... J".... ...
"~/.
/) ,J,., .. I, f;·.r,J.·,,,
J f, .lI..itlr" ,.IJ.J~·'iJ.
,.. 1," ,., , .., ... f. '"
r: .,.",,', 'r- .... J..
~
I~I J..,.,.
.I
I
~
"
I
,J
....
00
J: J'
~
"f, "
L ..... .
1.. .,. .... ~
••
L ......
'-_,."
Fig. 1-6(b) Cross-Section of Dome of Santa Maria del Fiore from Sgrilli's Descrizione e studi
dell' insigne fabbrica di S. Maria del Fiore (Florence, 1733). From W. B. Parsons "Engineers and
Engineering in the Renaissance", the Williams and Wilkins Co., Baltimore, © 1939
12
1 Introduction
Fig. 1-7
St. Peter's, Rome, Italy. Dome Span = 41.6 m; Dome Rise = 35.1 m
most recently computer-based finite element modeling, have revealed the intuitive understanding of structural mechanics exhibited by the Romans and by
the men of the Renaissance in designing and building their spectacular domes.
Perhaps one reason that several shells remain from antiquity is the ability of
surface structures to survive extreme loading. It was reported that a cooling
tower shell was among the few surviving structures in the Tangshun, China,
earthquake of 1976. 6 The hyperbolic paraboloid shown in figure 1-9 resisted
the 1985 Mexico City earthquake without apparent structural damage amid
totally destroyed conventional structures. 7 Surely, the ancient domes from the
Persian empire, such as that in figure 1-5, have weathered many earthquakes.
These examples inspire confidence in the toughness of well-designed and constructed surface structures, and the remainder of this book is an exposition of
13
1.3 Load Resistance Mechanisms
Fig.1-8
St. Paul's, London, England, Dome Span
=
30.8 m; Dome Rise
=
33.5 m
the basic principles by which such structures withstand the forces of nature and
man.
The connection between the modern developments in thin shell technology
and corresponding significant scientific events in the post-industrial revolution
period are documented in chapter 1 of Fung and Sechler. 4 The remaining
chapters of that modern collection of papers (which is devoted to the unification
of theory, experiment, and design of thin shell structures) is probably best
appreciated after the fundamental material presented in this book has been
mastered.
14
Fig. 1-9
1 Introduction
Hyperbolic Paraboloid, Mexico City, After 1985 Earthquake (Courtesy M. Celebi)
1.4 References
1. I. Todhunter and K. Pearson, A History of the Theory of Elasticity and of the Strength
of Materials. 2 vols. (Cambridge: Cambridge University Press, 1886 and 1893).
Also A. E. H. Love, A Treastise on the Mathematical Theory of Elasticity, 4th ed.
(New York: Dover Publications, 1944), Introduction; S. A. Timoshenko, History
of Strength of Materials (New York: McGraw-Hill, 1953); and H. M. Westergaard, Theory of Elasticity and Plasticity (New York: Dover Publications, 1964),
chap.lI.
2. J. T. Oden, Finite Elements of Nonlinear Continua (New York: McGraw-Hill, 1972),
pp. 144-145.
3. C. T. Grimm, "Brick Masonry Shells," Journal of the Structural Division, ASCE 101,
no. STI (January 1975): 79-95; discussion by K. Anadol and E. Arioylu, no. STll
(November 1975): 2451-2455.
4. J. Bobrowski, "Design Philosophy for Long Spans in Building and Bridges," The
Structural Engineer, 64A, no. 1 (January 1986): 5-12.
5. W. B. Parsens, "Engineers and Engineering in the Renaissance," (Baltimore, MD:
Williams and Wilkins, 1939): pp. 587-607.
6. J. K. Wu, private communication.
7. M. Celebi, private communication.
8. Y. C. Fung and E. E. Sechler, eds. Thin Shell Structures (Englewood Cliffs, NJ:
Prentice-Hall, 1971).
15
1.5 Exercises
1.5 Exercises
The numerical problems in this book are given without specific units to allow English,
metric, or SI unit dimensions to be selected.
1 It has been claimed that structural materials are generally more efficient in an
extensional rather than a bending mode. To illustrate this, consider the two cases
shown in figure 1-10. Both structures span the same distance 2L, carry the same load
P, and contain the same amount of material. For a span of L = 40, compare the
maximum fiber stress in each case.
Ip
1
5
L
L
Fig. 1-10
CHAPTER
2
Geometry
2.1 Curvilinear Coordinates
Consider a portion of the middle surface of a shell as shown in figure 2-1.
The surface is defined with respect to the X - Y -Z global Cartesian coordinates by
(2.1)
Z =f(X, Y)
Then, a set of coordinates a and 13 are selected which are related to the Cartesian
system by
X
= f1(a,f3)
Y
= f2(a, 13)
Z
= f3(a,f3)
(a)}
(b)
(2.2)
(c)
where f1' f2' and f3 are continuous, single-valued functions. Since each ordered
pair (a, 13) corresponds to only one point on the surface, the surface is uniquely
described in terms of a and 13, which are called curvilinear coordinates. If one
of the coordinates, e.g., 13, is incremented 13 = 131, 13 = 132, ... ,13 = 13m then we
define a series of parametric curves on the surface, along which only a varies.
These curves are termed the SIX coordinate lines. Similarly, if a takes on the values
a = a l ' a = a 2, ...• a = am. we get the sp coordinate lines. The coordinate lines
are shown in figure 2-1.
If the coordinate lines SIX and sp are mutually perpendicular at all points on
the surface, the curvilinear coordinates are said to be orthogonal. Orthogonal
curvilinear coordinates are used exclusively in this book.
2.2 Middle Surface Geometry
Equation (2.1), which describes the middle surface, may be written in terms of
a position vector emanating from the origin as shown in figure 2-2:
16
17
2.2 Middle Surface Geometry
z
x
Fig. 2-1
r
= Xu + Yv + Zw
Middle Surface of a Shell
(2.3)
in which u, v, and ware unit vectors along the X, Y, and Z axes, respectively.
Substituting equation (2.2) into (2.3), the position vector is defined in terms
of the curvilinear coordinates by
(2.4)
The derivatives ofr with respect to the curvilinear coordinates are considered
next:
or
-=r
oa.''''
(2.5a)
and
or
op
= r,p
(2.5b)
are vectors that are tangent to the s'" and sp coordinate lines, respectively. Since
18
2 Geometry
z
r
~--------------------------------y
x
Fig.2-2
Position Vector to a Point on the Middle Surface
the coordinate lines are orthogonal, these tangent vectors are orthogonal as
well, and their scalar product r,1X r,(J = O.
The vector joining the two points on the middle surface (oc, f3) and (oc + doc,
f3 + df3) shown in figure 2-2 is
0
ds = r.lXdoc
+ r,
lX
(2.6)
df3
If we form the scalar product of ds with itself, we have
ds ds = ds 2 = (r,IX r ,IX ) doc 2 + (r,(J r ,(J ) df32
0
0
0
(2.7)
Defining
A2
= r,1X or,1X
(2.8a)
19
2.2 Middle Surface Geometry
and
B2 = r.{J or.{J
ds 2 = A 2 dIX 2 + B2 dfJ2
(2.8b)
(2.9)
which is known as the first quadratic form of the theory of surfaces.
The quantities A and B are called Lame parameters or measure numbers and
are fundamental to the understanding of curvilinear coordinates. To interpret
their meaning physically, consider two cases in which each of the coordinates
IX and fJ is varied individually and independently. For these cases, equation (2.9)
becomes
dsa.
= AdlX
(2. lOa)
dS{J
=
BdfJ
(2. lOb)
and
Thus, dsa. is the change in arc length along coordinate line sa. when IX is incremented by dlX, and dS{J is the change in arc length along coordinate line s{J when
fJ is incremented by dfJ. We see that the Lame parameters are quantities which
relate the change in arc length on the surface to the corresponding change in
the curvilinear coordinate; hence the alternate name, measure number.
As a simple example, consider a circular arc of radius R in the Y - Z plane
shown in figure 2-3. If the curvilinear coordinate is chosen as the polar angle fJ,
z
dZ~
dY
------------~------~------~~----------y
Fig. 2-3 Lame Parameter for a Circular Arc
20
2 Geometry
ds
= RdP
(2.11)
and the Lame parameter is R. Alternately, the curvilinear coordinate could be
chosen as the Z coordinate, in which case we have (from figure 2-3)
(2.12)
From the equation of the circle y2
+ Z2 = R2,
2YdY= -2ZdZ
or
so that equation (2.12) gives
ds
Z2)1/2
= ( 1 + y2
dZ
(2.13)
and the Lame parameter is
Z2)1/2
( 1+y2
A third possibility is to choose the arc length itself as the curvilinear coordinate.
Then
ds
= 1 (ds)
(2.14)
and the Lame parameter is the constant 1. It is apparent from this elementary
example that the Lame parameter may be a constant or a rather involved
function. For a particular problem, the choice of curvilinear coordinates which
correspond to the simplest possible expressions for the Lame parameter can
serve to expedite the mathematics of the solution greatly.
The preceding example illustrates that the Lame parameters may sometimes
be found by the geometric representation of equations (2.9) and (2.10). In more
complicated cases, we may compute the Lame parameters from equations (2.8)
and (2.3) as
+ (y,a)2 + (Z,a)2
= (X ,p )2 + (Y,p )2 + (Z,p )2
A2 = (X,a)2
(2.15a)
B2
(2.15b)
Finally, with respect to the first quadratic form, note that it generally pertains
to the measurement of distances on the surface, but does not involve the specific
shape of the surface.
21
2.3 Unit Tangent Vectors and Principal Directions
Fig.2-4
Unit Tangent Vectors
2.3 Unit Tangent Vectors and Principal Directions
2.3.1 Definition of Unit Tangent Vectors and Normal Section: It is convenient
to refer all vector point functions to a triad of unit vectors composed of tangent
vectors to the coordinate lines and a normal vector which define a right hand
system, as shown in figure 2-4.
We have already defined tangent vectors to the coordinate lines in equation
(2.5). Hence, in view of equation (2.8),
r
r
t =-'-"'-=~
'"
t
±lr,,,,1
A
_~_r,p
p - ±Ir,pl - B
(2.16a)
(2.16b)
and the normal vector is found by forming the vector product of t", and tp:
1
tn = t", x tp = AB(r,<z x r,p)
(2.16c)
A normal section of the surface is defined as a plane curve obtained by cutting
22
2 Geometry
the surface with a plane containing the normal to the surface, tn. In particular,
normal sections containing the unit tangent vectors t", and tp are of interest.
2.3.2 Principal Directions: Now, consider the general point (tXi' /3j) on the middle surface as shown in figure 2-1. At this point, each coordinate line, Sp(tXi) and
Sat(/3j), may be regarded as a normal section with a corresponding radius of
curvature Rp(tXi' /3j) and Rat(tXi' /3j), respectively, which is directed from the center
of curvature to the point (tXi' /3j) along tn' Obviously, there are an infinite number
of possible Rat and Rp at any point, since an infinite number of orientations for
the curvilinear coordinates exist.
From the theory of surfaces, it may be shown that there is a system of
orthogonal curvilinear coordinates (tX*, /3*) oriented such that one radius of
curvature
= IR:I is the maximum of all possible IRati, whereas the other
radius of curvature R1 = IR1 I is the minimum of all possible IRp I. 1 We call this
system the principal orthogonal curvilinear coordinates corresponding to the
principal directions, tX* and /3*. The associated coordinate lines are known as
the lines of principal curvature, and R: and R1 as the principal radii of curvature.
In the subsequent derivation of unit tangent vector derivatives, principal directions will be used exclusively, so that tX and /3 imply tX* and /3*.
R:
2.3.3 Derivatives of Unit Tangent Vectors: To establish the relationships between the Lame parameters and the principal radii of curvature for a surface,
it is necessary to derive a set of relationships for the derivatives of the tangent
vectors, tat' t p• tn, with respect to tX and /3. 2 The derivatives of the unit tangent
vectors are expressed in terms of the unit tangent vectors themselves:
ta,a
ta,p
tp,a
tp,p
tn,,,,
tn,p
-A
0
-A,p
B
0
B,a
A
0
A,p
B
0
0
-B,a
A
A
R",
0
0
0
B
Rp
Rat
-B
Rp
{::}
(2.17)
0
0
We now consider the verification of equation (2.17). It is convenient to write
the equation once again with the elements grouped as shown:
23
2.3 Unit Tangent Vectors and Principal Directions
0
-A,p
B
(I)
(II)
0
B,a.
A
-A
Ra.
0
(IV)
ta.,a.
ta.,p
tp,a.
tp,p
tn,a.
tn,p
A,p
B
0
(II)
(I)
-B,a.
A
A
Ra.
0
0
(III)
0
(III)
B
Rp
0
-B
H}
Rp
The grouping refers to the arguments in the following sections.
2.3.3.1 (I) Component in Direction of Differentiated Vector. The derivative of
any unit tangent vector is normal to the vector itself so that there are no
components in the direction of the vector being differentiated; e.g., there is no
ta. component for ta.,a. or ta.,p'
This assertion is well known from elementary vector calculus. As a proof,
consider the scalar product of two unit tangent vectors in the curvilinear
coordinate system:
t; • t j
= 1 i = j' 15 .. = 0 i '# j
'
"J
'
I,j = IJ(, [3, n
b ..
= bij { . IJ.
Differentiating the product with respect to k(k
(t;· tj),k = 0
from which
Then set j
= i
to get
2(t;· t;,k) = 0
=
IJ(
or [3)
24
2 Geometry
so that ti,k has no component along t i . Taking i = Q(, P and nand k =
in tum, verifies the indicated zero terms in this group.
Q(
and
p,
2.3.3.2 (II) Components of Derivatives of t .. and tp in Q( and P Directions. We
multiply equations (2.16a) and (2.16b) by A and B to get r, .. and r,p, respectively,
and form the mixed second partial derivatives,
(2.18a)
r, ..p = r,p ..
With equation (2.16) substituted into equation (2.18a), we obtain
=
(At .. ),p
t .. A,p
+ At.. ,p = (Btp), .. = tpB, .. + Btp,..
(2.18b)
from which
(2.18c)
Now consider the component oft.. ,.. in the tp direction, given by tp' t .. ,... We take
the derivative of the product t ... tp with respect to Q(
(t.. • t p), .. = tp' t .. ,..
+ t .. · t p,..
Since t ... tp
=
0,
tp·t .. , ..
=
- t.. ·tp, ..
(2.18d)
(2.18e)
We replace t p, .. by equation (2.18c) to get
1
tp' t .. ,.. = - B t .. · [ - tpH, ..
t.t
13
a,a
+ t .. A,p + Ata,pJ
-A ,_13
= __
B
(2.18f)
as given in the first row of equation (2.17). The other components of the
derivatives of ta and tp in the Q( and P directions may be verified in a similar
manner.
2.3.3.3 (III) Derivatives of tn. Consider the normal section at the point Pl on
the Sa coordinate line, as shown in figure 2-5. The vector tn is shown at the point
Pl and also at point Pz, a small distance As", away. The vector construction at
PI shows that the change in tn> At n, is approximately parallel to the tangent to
the curve at Pl and the chord PlP2' Therefore,
Atn
=
IAtnlt",
By similar triangles, as Aoc diminishes,
(2. 19a)
25
2.3 Unit Tangent Vectors and Principal Directions
~n
t n2 \
\ p.
tm
I
Fig. 2-5
Idtnl
Itn11
Normal Section
P1PZ
(2.19b)
Ra
Considering figure 2-5 and equation (2. lOa),
P1PZ ~ dS a
= Adoc
Recognizing Itn11
(2.19b), we have
dtn
doc
A
Ra.
-=-t
(2.19c)
= 1, and substituting equations (2.19a) and (2.19c) into
(2. 19d)
a
Taking the limit of both sides as doc
--+
0,
(2.1ge)
as given in the fifth row of equation (2.17). tn,p is evaluated in a similar manner.
Note that the argument employed in (III) is more general than that used in (II),
since the entire derivative is computed instead of just one component.
26
2 Geometry
2.3.3.4 (IV) Components of Derivatives of ta and tp in Normal Direction. Consider the normal component of t a •a given by t n" t a • a . Proceeding as before,
(2.20a)
Since tn "til = 0,
(2.20b)
But, we have already evaluated t n •a in equation (2. 1ge); hence
-A
tn"ta.a =R
(2.20c)
a
as given in the first row, third column, of equation (2.17). The remaining normal
components of the derivatives of ta and tp may be verified in a similar manner.
2.4 Second Quadratic Form of the Theory of Surfaces
Recall that in section 2.2, we derived the first quadratic form of the theory of
surfaces, which pertains to the measurement of distances on the surface but not
specifically to the shape of the surface. In this section, we seek information with
respect to the latter property, the shape.
Consider a normal section which traces a plane curve with arc length coordinate Si' An example is the normal section along Sa shown in figure 2-5;
however, Si is not necessarily restricted to only principal direction coordinate
lines in the ensuing development. The curvature of such a section is known as
the normal curvature Ki and is defined as a function of the position vector r,
shown in figure 2-2, by the Frenet-Serret formula 3 as
(2.21)
where the negative sign corresponds to the selection oftn as the outward normal,
i.e., directed toward the convexity of normal sections with positive curvature. 3
We want to express the normal curvature in terms of the curvilinear coordinates oc and {3. Starting with
r.s,s, = (r,s,),s, = (r,aoc,s,
we first evaluate
and
+ r,p{3,s,),s,
27
2.4 Second Quadratic Form of the Theory of Surfaces
(r.pP"i)"i = P.•.r.P'i
+ r.pp.SiSi
p.Si(r.p«a. Si + r.flPp.•i) + r.pp.SiSi
=
from which
r" iSi = r.««(a. Si )2 + 2r.«pa, Si P" i + r. pp(p.•i)2 + r.«a" i' i + r.pP" i' i
(2.22)
Next, we scalar multiply each side of equation (2.22) by t". The last two terms
on the right-hand side (r.h.s.) vanish, since t" is normal to r.« and r.p by equation
(2.16). In order to simplify the first three terms of equation (2.22), we first
consider
t
t
"
or.« = 0
(2.23a)
"
or.p = 0
(2.23b)
and differentiate both equations with respect to a and then p, which gives
tIl r.«« = -r.« t".«
(2.23c)
tIl r.«p = -r.« t".fI
(2.23d)
t" r. p« = -r.p t".«
(2.23e)
tIl r.pp = - r.p t".P
(2.23f)
0
0
0
0
0
0
0
0
We now continue the scalar product oft" with the first three terms of equation
(2.22). In view of equations (2.21) and (2.23c-f), we have
"i =
-
= -
r.« t".«(a .•.)2 - 2r.« t".pa,SiP" i - r.p t".P(P.•.)2
0
0
0
At« t".«(a .•.)2 - 2At« t".pa"iP" i - Btp t".P(P.sf
0
0
(2.24)
0
We may evaluate the scalar products in equation (2.24) from equation (2.17).
Therefore,
_A2
B2
(2.25)
= ~(a .•f + O(a .•ip.•,) - R(P.•f
"i
p
«
It is convenient to multiply equation (2.25) by
dsf
ds?-•
whereupon
"i =
L(a , I
)2jds?+ N(P,51.)2 ds?-l
'
dsf
(2.26a)
where
(2.26b)
28
2 Geometry
Using equation (2.9), we may consolidate equation (2.26a) to get
L da 2 + N dfJ2
Ki
= A2 da 2 + B2 dfJ2
(2.27)
In equation (2.27), da and dfJ represent (oalos i ) dS i and (ofJlosi) ds i, respectively,
for a particular direction i on the surface.
Finally, we write equation (2.26) as
II
I
(2.28a)
K·=-
,
where
II = Lda 2 + N dfJ2
(2.28b)
and
1= A2 da 2 + B 2dfJ2
= First quadratic form (equation 2.9).
(2.28c)
The second quadratic form II thus relates to the shape of the curve through the
presence of radii of curvature R" and Rp.
2.5 Principal Radii of Curvature
Thus far, we have assumed that the principal radii of curvature of the shell,
R" and R p , are known or easily found. We now examine the details of this
calculation.
Consider the case where the normal section corresponding to a is a curve
Z = f(X) in the X-Z plane.
R
=
"
-[1
+ (Z,x)2J3 /2
Z,xx
(2.29)
When the surface is specified parametrically in terms of the curvilinear
coordinates as in equation (2.4), we may compute the radius of curvature using
equation (2.21). If we take the Si direction as one of the coordinate lines, e.g., S,,'
then
(2.30)
Equation (2.30) was evaluated from equation (2.22), recognizing that
Since tn· r,a = 0, equation (2.30) reduces to
R
-1
..
= t,,· r, ....(a,s) 2
P, •• = O.
29
2.6 Gauss-Codazzi Relations
Noting that o(,s. = IjA from equation (2.10a), and
equation (2.16c),
tn
=
(ljAB)(r,<x x r,p) from
-A 3 B
R =----<X
(r,<x x r,p)' r,<x<x
(2.31)
Similarly,
-B 3 A
Rp=-----(r,<x x r,p)' r,pp
(2.32)
Equations (2.31) and (2.32) may be evaluated from equations (2.4), (2.15), and
(2.2) for a given geometry.
Another technique for computing the radii of curvature that is useful for shells
of revolution is illustrated in section 4.3.2.3.
2.6 Gauss-Codazzi Relations
No connective relationships between the Lame parameters, A and B, and the
principal radii of curvature, R<x and Rp, have been set forth. To explore this
further, consider the equality of the second mixed partials
(2.33)
or, from equation (2.17),
(2.34a)
(2.34b)
Again, using the differential relationships in equation (2.17), we find
(2.34c)
With t<x and tp being mutually orthogonal, equation (2.34c) may only be satisfied
if
(2.35)
and
(2.36)
30
2 Geometry
If we now consider one of the other second mixed partial derivative identities
or
and manipulate these equations using equation (2.17), a third differential
relationship
AB
(2.37)
results. Equations (2.35), (2.36), and (2.37) are known as the Gauss-Codazzi
relations and define the connectivity among A, B, R"" and R p , such that these
parameters define a surface. Equation (2.37) is particularly useful in the derivation of the equations of equilibrium and is discussed later.
Although we will not pursue the details, note that a parallel set of equations may be derived for the deformed middle surface by starting with the
normal vector to the deformed surface in place of tn in equation (2.33).3 The
resulting equations, properly termed Gauss-Codazzi relations for the deformed
middle surface, are the conditions for the continuity of the middle surface displacements and the compatibility between the strains and displacements.
They serve the same role as the St. Venant equations in the theory of elasticity. We refer to such equations in connection with the bending of shells in
chapter 9.
2.7 Gaussian Curvature
On the right side of equation (2.37), note the fraction 1/R",R p , which is the
product of the principal curvatures. This is known as the Gaussian curvature
and plays an important role in the characterization of shells. Although the
Gaussian curvature may be readily computed by using the equations of section
2.5, such a calculation is seldom required for purposes of classification; it is
often sufficient to know only the algebraic sign.
If we consider the normal sections corresponding to the principal directions,
the Gaussian curvature is positive if both centers of curvature lie on the same
side of the surface and is negative if the centers lie on opposite sides. If one of
the radii of curvatures is equal to infinity, the Gaussian curvature is zero.
Representative cases are shown in figure 2-6. A plate is the degenerate case of
a shell with zero Gaussian curvature, since both radii are infinite.
Technically, the Gaussian curvature is a scalar point function, and a particular shell may have regions with positive, negative, and/or zero values. Nevertheless, a single sign predominates for most practical cases. Calladine4 has
31
2.7 Gaussian Curvature
Rp
ZERO
NEGATIVE
POSITIVE
Fig. 2-6 Gaussian Curvature
Positive
Negative
Zero
Surface
Doubly Curved
Synclastic
Doubly Curved
Antic1astic
Singly Curved
Developability
Nondevelopable
Nondevelopable
Developable
Type of Equation
Discriminant)
Elliptic (Positive)
Hyperbolic
(Negative)
Parabolic (Zero)
Straight or Ruled Lines
on Surface
None
Two Sets
(A and B Below)
One Set (C Below)
Sphere
r
Hyperboloid
of Revolution
Cylinder
Paraboloid
of Revolution
Hyperbolic
Paraboloid
Cone
Examples
0
~c
/I
1
Flat Plate
(Degenerate
Case)
Fig. 2-7
Classification of Shells by Gaussian Curvature
32
2 Geometry
Fig. 2-8(a) Open Hyperbolic Paraboloid Roof, Ponce Coliseum, Puerto Rico (Courtesy Professor
A. C. Scordelis)
recently reexamined the relationship between Gaussian curvature and shell
theory, referring to Gauss's original ideas and extending the concept to faceted
surfaces as well as traditionally curved forms.
A classification of shells by Gaussian curvature is given in figure 2-7. With
respect to this classification, note that only the zero curvature shells are developable and hence may be formed from flat material. This property of zero
curvature shells contributes to their wide usage. Also, note that the negative
curvature shells have two sets of straight or ruled lines on the surface which
correspond to the two sets of real characteristics associated with hyperbolic
surfaces, s so that the formwork for such a shell can be fabricated from straight
materials. This feature of negative curvature shells is largely responsible for the
economical usage of reinforced concrete hyperbolic paraboloid (HP) shells in
a variety of applications. 6
The mathematical classification refers to the type of partial differential equation associated with quadratic surfaces having the indicated Gaussian curvature. If a shell is or can be approximated logically as a quadratic surface
Z = f(X, Y), then the algebraic sign of the discriminant
(2.38)
determines the type of partial differential equation and, further, coincides with
the sign of the Gaussian curvature, 5 as indicated in figure 2-7.
A variety of thin shell and plate structures are shown in figure 2-8 and will
be referred to frequently throughout the book.
33
2.7 Gaussian Curvature
Fig. 2-8 (b)
Fig.2-8(c)
Hyperbolic Paraboloid Roof, Theater, Northbrook, IL
Vertical Hyperbolic Paraboloids, Church, San Francisco, CA
34
2 Geometry
Fig. 2-8(d)
Fig.2-8(e)
Hyperbolic Paraboloids, Church, Mexico City, Mexico
Hyperbolic Paraboloid, Bethesda, MD (Courtesy National Institutes of Heath)
35
2.7 Gaussian Curvature
Fig.2-8(f)
Fig. 2-8(g)
Parabolic Vaults, Church, St. Louis County, MO
Hyperboloid of Revolution, Planetarium, St. Louis, MO
36
2 Geometry
Fig. 2-8(h)
Fig. 2-8(i)
Open Cylindrical Roof, Airport, Barcelona, Spain
Spherical Roof, Auditorium, Cambridge, MA
37
2.7 Gaussian Curvature
Fig.2-8(j)
Kingdome, Seattle, WA (Courtesy Dudley, Hardin & Yang, Inc.)
Fig.2-8(k)
Intersecting Barrel Shells, Airport, St. Louis, MO
Fig. 2-8(1)
Shallow Spherical Roof with Cutouts, Restaurant, Miami Beach, FL
DIPLOMAl
,;)
Q
~
~
N
00
w
39
2.7 Gaussian Curvature
Fig. 2-8(m)
Fig.2-8(n)
Reticulated Spherical Roof, Astrodome, Houston, TX
Saddledome, Calgary, Alberta, Canada (Courtesy Dr. Jan Bobrowski, F. Eng.)
40
2 Geometry
Fig. 2-8(0)
Cooling Towers, Schmeehausen, West Germany
Fig. 2-8(p) Supporting Columns for Hyperbolic Cooling Tower (Courtesy Professor W.
Schnobrich)
41
2.7 Gaussian Curvature
•••• I •
I
Fig. 2-8(q) Human Aortic Heart Valve. Source: P. L. Gould et aI., "Stress Analysis of the Human
Aortic Valve," Journal of Computers and Structures 3 (1973): 379. (Courtesy Dr. R. Clarko reprinted
with permission of Pergamon Press)
Fig.2-8(r)
Stiffened Cylindrical Shell (Courtesy Chicago Bridge & Iron Co.)
Fig. 2-8(s)
Fig. 2-8(t)
42
Spheroidal Water Tower (Courtesy Chicago Bridge & Iron Co.)
Column-Supported Spherical Tanks (Courtesy Chicago Bridge & Iron Co.)
Fig. 2-8(u)
Fig. 2-8(v)
Inc.)
Column-Supported, Stiffened Water Tower (Courtesy Chicago Bridge & Iron Co.)
Steel Hyperbolic Paraboloid Roof, Aircraft Hangar (Courte~y Lev Zetlin Associates,
43
2 Geometry
44
Fig.2-8(w) WaIDe Slab, Library, St. Louis, MO
45
2.7 Gaussian Curvature
Fig. 2-8(x)
Folded Plate Roof, Law School, St. Louis, MO
Fig.2-8(y)
Torospherical Head
46
2 Geometry
2.8 Specialization of Shell Geometry
Because of the wide variety of plate and shell structures encountered in engineering practice, several geometrical classes are of particular interest.
2.S.1 Shallow Shells: The theory of shallow shells has wide application for roof
shells that have a relatively small rise as compared to their spans. Considering
~O~__________~~-+__~__________~y
Fig. 2-9
Shallow Shell Geometry
47
2.8 Specialization of Shell Geometry
the surface specified in Cartesian coordinates as in equation (2.1), the shell is
said to be shallow if, in the subsequent mathematical analysis, (Z,X)2 and (Z, y)2
may be neglected by virtue of smallness in comparison to unity. The effect of
this simplification is seen by considering figure 2-9.
Consider a differential element of the middle surface bound by the intersections with two planes parallel to the Y-Z plane and separated by a distance
dX, and two planes parallel to the X-Z plane separated by a distance dY. As
illustrated by the inset in figure 2-9,
ds x ~ dX[1
ds y
~
dY[1
+ (Z,x)2J1/2
+ (Z,y)2]1/2
(2.39a)
(2.39b)
which, when the geometric simplification is applicable, become
ds x
~
dX
(2.40a)
ds y
~
dY
(2.40b)
The practical interpretation of equations (2.40) is that the curvilinear coordinates 0( and f3 may be selected as the Cartesian coordinates X and Y with the
Lame parameters A = B = 1. Additional approximations are introduced into
shallow shell theory with respect to the equilibrium equations in chapter 9.
The practical range of the theory of shallow shells is restricted to shells with
a central rise of one-fifth or less than the span. 7 However, this theory has been
applied to shells that may not fit the criteria in a global sense, by treating pieces
axis of rotation
generator
......
meridian-
",
parallel circle
Fig. 2-10
Shell of Revolution
48
2 Geometry
or finite elements of the shell which can be considered shallow, and then
assembling these elements to satisfy the global geometry. B
2.8.2 Shells of Revolution: A surface of revolution is described by rotating a
plane curve generator around an axis of rotation to form a closed surface, as
illustrated in figure 2-10. The lines of principal curvature are called the meridians (normal sections formed by planes containing the axis of rotation) and
parallel circles (normal sections traced by planes perpendicular to the axis of
rotation).
In figure 2-11, we show the meridian of a shell of revolution illustrating both
positive and negative Gaussian curvature. The equation ofthe meridian is given
by
Z=Z(R)
(2.41a)
where
·R
----R
center of
curvature r.>-===----1\---t-~---I
MERIDIAN
MERIDIAN
Positive Gaussian
Curvature
Negative Gaussian
Curvature
PARALLEL
Fig. 2-11
CIRCLE
Geometry for Shells of Revolution
2.8 Specialization of Shell Geometry
49
(2.41 b)
Consider a reference point on the surface. The angle formed by the extended
normal to the surface at this point and the axis of rotation is defined as the
meridional angle ,p; and, the angle between the radius of the parallel circle at
the point and the X axis is designated as the circumferential angle e. Correspondingly, the meridians are taken as the Sa. coordinate lines, and Ra. = R,p,
the meridional radius of curvature. The parallel circles are the sfJ coordinate
lines, with RfJ = R o, the circumferential radius of curvature. The radius of the
parallel circle, which is equal to R as defined in equation (2.41b), is termed the
horizontal radius and is denoted by Ro. Note that Ro is not a principal radius
of curvature, since it is not normal to the surface. Rather, it is the projection of
Ro on the horizontal plane, i.e.,
(2.42)
A closed shell of revolution is frequently called a dome, and the peak of such
a shell is termed the pole. A pole introduces certain mathematical complications
because, at this point, Ro -. O.
In most applications, the curvilinear coordinate in the f3 or e direction is
chosen as the circumferential angle e. Therefore, from equation (2.10b),
dSfJ
= Bdf3 = ds o = Rode
(2.43a)
and thus
B = Ro = Rosin,p
(2.43b)
In the a. or ,p direction, there are at least three useful choices for the curvilinear
coordinate: (a) the meridional angle ,p; (b) the axial coordinate Z; and (c) the
arc length s,p. The respective Lame parameter A for each case is:
(a) Meridional angle, ,p:
ds,p
= R,pd,p
(2.44a)
and from equation (2.10)
(2.44b)
A=R,p
(b) Axial coordinate, Z:
Considering an element of arc length, similar to that shown on the inset of
figure 2-3, but with dY = dR o
ds~
=
ds~
= dZ 2 + dR6
= dZ 2 + (R O,Z)2 dZ 2
ds z = dZ[1 + (R o,z)2r/2
(2.45a)
(2.45b)
and therefore
A = [1
+ (R o,z)2r/2
(2.45c)
50
2 Geometry
(c) Arc length s",:
=
A
(2.45d)
1
but since the meridian is defined by Z = Z(R o ), the arc length coordinate
must be computed by integrating equation (2.45b). Therefore,
Sz
=
SZ [1
+ (RO,Z)2] 1/2 dZ
(2.46)
Zo
The limits of the integral indicate the coordinate Z at the origin for Sz (perhaps
the base or top of the shell), and at the section where the coordinate is being
evaluated, respectively.
The choice of an appropriate curvilinear coordinate for the meridional
direction is problem-dependent and involves several considerations. In towertype shells, which are essentially vertical structures, the axial coordinate Z has
the greatest significance with respect to the physical construction of the shell.
For relatively flat domes, however, the axial coordinate approaches zero even
for points relatively far from the pole. The meridional angle rjJ may behave in
a similar manner, so the arc length will be the most stable coordinate for such
cases. Also, if the meridian has an inflection point, the coordinate rjJ might cause
difficulties because it may not provide a one-to-one correspondence with all
points on the shell surface.
The most popular, but by no means universal, preference of theoreticians in
the field of shells of revolution has been to choose the meridional angle rjJ as the
basis for the development of the government equations. This may be due
somewhat to historical precedent, since the early work on shells of revolution
focused on spherical shells 9 for which A = R", = constant, thereby greatly
simplifying the ensuing treatment.
We noted in section 2.6 that the parameters A, B, R,%, and Rp must satisfy the
three Gauss-Codazzi conditions to define a surface. These conditions, as given
by equations (2.35), (2.36), and (2.37), can be checked for the rjJ-(J curvilinear
coordinate system:
Equation (2.35):
Satisfied identically, since none of the parameters are functions of P((J).
Equation (2.36):
(
= (R9sinrjJ),,,,
R9sinrjJ)
R9
,'"
cosrjJ
R",
= (R9 sinrjJ) ' '"
R",
or
(R 9 sinrjJ),,,, = Ro,,,,
Equation (2.37):
= R",cosrjJ
(2.47)
2.8 Specialization of Shell Geometry
Fig.2-12
51
Geometrical Interpretation of Gauss-Codazzi Condition
(2.48)
Substituting equation (2.47) for the numerator on the left-hand side (l.h.s)
equation (2.48) gives an identity. Thus, the Gauss-Codazzi conditions for a
shell of revolution described by the coordinates rp and () are satisfied, provided
equation (2.47) is valid. This equation is very useful in what follows, and it is
instructive to derive it from a purely geometric argument. Consider the meridian shown in figure 2-12:
Referring to points Band D
At B,
Ro(rp)
At D,
Ro(rp
+ ilrp) =
ilRo
=
=
AB
CD
CD - AB
52
2 Geometry
= BD sin
(I - ~)
= BDcos~
BD =
Since
R¢A~
= R¢ cos ~A~
ARo = A(R6sin~) = R¢cos~A~
ARo
Or,
A(R6 sin ~) _
A.
A~
- R¢cos,!,
lim A~ --+ 0 gives equation (2.47).
One may desire to calculate the value of Z corresponding to a particular value of rP. With R6 = R6(~)' Ro(~) is found from equation (2.42) and substituted into equation (2.41a) to obtain the corresponding value of Z(~). If the
arc length also is required, equation (2.46) should be evaluated. Also, a direct
transformation between s¢ and ~ can be derived by integrating equation
(2.44a).
It is expedient to be able to transform derivatives with respect to ~ to
derivatives with respect to Z, i.e., ( ),z = ( ),¢ (d~/dZ). If we consider equation
(2.45c) with Ro,z given by R o,¢ (d~/dZ) and R o,¢ given by equation (2.47), we
have
A = [1
drP)2J1/2
+ R~ cos 2 ~ ( dZ
for the axial coordinate Z.
Next, we write ds¢ for both the
~
(2.49)
and Z coordinates
drP)2J1/2
ds¢ = R¢d~ = [ 1 + R~COS2~ ( dZ
dZ = ds z
or
Squaring both sides and solving for (#/dZ)
d~ =
dZ
+
1
- R¢(1 - cos 2 rP)1/2
1
=+---
R¢sin~
(2.50)
53
2.8 Specialization of Shell Geometry
x
Fig. 2-13 Alternate Cylindrical Coordinates
Substituting equation (2.50) into (2.49) gives
A = ±cscl/J
(2.51)
for the Z coordinate. Equation (2.50) is quite useful, since it is often preferable
to differentiate with respect to l/J first in order to utilize equation (2.47).
The ± signs in equations (2.50) and (2.51) indicate that an increment dl/J may
produce a change in arc length equal in magnitude but opposite in sense to that
produced by dZ. For example, in figure 2-11 where Z is directed upward, this
would be the case. Hit is desired to use Z as the primary meridional coordinate,
it may be preferable to direct Z as positive downward. 10
An alternate set of coordinates for which Z increases in direct proportion to l/J is shown in figure 2-13. The right-hand convention and the outward
directed normal, i.e., t z x to = tn, are preserved, but e is now measured from
the Yaxis.
2.9 References
1. H. Kraus, Thin Elastic Shells (New York: Wiley, 1967), pp. 12-14.
2. V. V. Novozhilov, Thin Shell Theory [translated from 2nd Russian ed. by P. G.
Lowe (Groningen, The Netherlands: NoordhofT, 1964), pp. 9-30].
54
2 Geometr:
3. A. L. Gol'denveizer, Theory of Elastic Thin Shells [translated from Russian by G.
Herrmann (New York: Pergamon Press, 1961), p. 13].
4. C. R. Calladine, Gaussian Curvature and Shell Structures, The Mathematics of
Surfaces [J. A. Gregory Ed.] (Oxford: Clarendon Press, 1986), pp. 179-196.
5. A. M. Haas, Design of Thin Concrete Shells vol. 2 (New York: Wiley, 1967), pp. 5-11.
6. C. Faber, Candela: The Shell Builder (New York: Reinhold, 1963).
7. Novozhilov, Thin Shell Theory, pp. 94-99.
8. G. R. Cowper, G. M. Lindberg, M. D. Olson, "A Shallow Shell Finite Element of
Triangular Shape," International Journal of Solids and Structures 6, no. 8 (1970):
1133-1156.
9. H. Reissner, Spannungen in Kugelschalen (Leipzig: Muller-Breslau-Festschrift,
1912), pp. 181-193.
10. V. Z. Vlasov, General Theory of Shells and Its Application in Engineering, NASA
Technical Translation TTF -99 (Washington, DC.: National Aeronautics and Space
Administration, 1964), pp. 5-13.
2.10 Exercises
2.1
Verify the second, third, fourth, and sixth equations of equation (2.17).
2.2
Derive equation (2.37).
2.3
Verify that the Gauss-Codazzi relations are satisfied for a shell of revolution with
A given by equation (2.45c).
2.4
The parametric equation for a shell of revolution is given in Kraus, Thin Elastic
Shells, as
r
=
Rocos8u
+ Ro sin 8v + Zw
where Ro = Ro(Z) follows the notation of equation (2.4) and figure 2-11.
(a) Choosing Z and 8 as the curvilinear coordinates, compute expressions for the
first and second quadratic forms. Note that if the alternate Cartesian axes
shown in Figure 2-13 are used, the parametric equation must be modified.
(b) Repeat (a) using the meridional angle I/J instead of Z.
2.5
Compute the first and second quadratic forms for the curvilinear coordinates
specified in exercise 2.4, considering the following geometries:
(a) Right circular cylinder.
(b) Ellipsoid of revolution.
(c) Hyperboloid of one sheet.
2.6
Investigate the relative merits of using r,a and r,(I' nonunit tangent vectors, as the
base vectors instead oft,a and t,(I' See Gol'denveizer, Theory of Elastic Thin Shells,
for an example of such a formulation.
CHAPTER
3
Equilibrium
3.1 Stress Resultants and Couples
Consider the element of the shell shown in figure 3-1(a), bounded by the normal
sections oc, oc + doc, /3, and /3 + d/3. The geometry of the middle surface of such
an element was considered in chapter 2 (see figures 2-2 and 2-4). Here, we show
the entire thickness h with the coordinate, defined in the direction of tn and
depict a differential volume element doc d/3 d, with thickness d" parallel to and
displaced from the middle surface a distance ,.
In figure 3-1(b), the stresses acting on the volume element are indicated. The
sign convention is that of the theory of elasticity, with the first subscript
indicating the surface on which the stress acts, as identified by the normal to
that surface, and the second subscript indicating the direction in which the stress
acts. The solution for the various stresses, shown in figure 3-1(b) at a particular
point of the continuum (oc, /3, 0, is the fundamental problem of the theory of
elasticity. Plate and shell theories deal with a simplification of the stress at a
point problem. Instead of the stresses, it is considered sufficient to solve for the
total forces and moments per length of middle surface, which are known as stress
resultants and stress couples, respectively.! The stress at any point presumably
can then be evaluated by back-substitution, although we will find that this can
sometimes be done only approximately.
We have thus alluded to the major simplification present in the plate and
shell problem as compared to the theory of elasticity problem: the plate or shell
problem is formulated and solved for the forces and moments per unit length
ofthe middle surface, rather than for the stresses at each point of the continuum.
Toward this end, we now consider the definition of the stress resultants and
couples in terms of the stresses.
In figure 3-2, the stress resultants and couples are shown on an element of
the middle surface. The extensional forces N.. and N p , transverse shear forces
Q", and Qp, and bending moments M", and Mp are represented by single
subscripted variables indicating the surface on which the force or moment acts,
whereas the in-plane shear forces N",p and N p"" and the twisting moments M",p
and M p"" carry a double subscript. The first subscript corresponds to the surface
on which the force or moment acts; the second indicates the direction in which
55
56
3 Equilibrium
middle surface
a+da
dad,8d' =VoIume Element
(0)
(b)
Fig. 3-1
Shell Volume Element and Stresses
the force or moment acts. Though this notation is not completely consistent
with the theory of elasticity definitions shown in figure 3-1(b), it has been used
almost exclusively in classical texts and references, and a departure here is felt
to be inappropriate.
We now focus on an arc ofthe middle surface
(3.1)
along the
Sa
coordinate line and the corresponding arc of the volume element
57
3.1 Stress Resultants and Couples
"bp+Nap,ada ~ dsp
Na+Na,a da
/f
Qa +Qa,ada
dS a ~Npa+NPa,pdP
1""'Np+Np ,pdP
Op+Qp,pdP
(a) Stress Resultants
(b) Stress Couples
Fig. 3-2 Stress Resultants and Stress Couples
lying a distance, along the normal tn from the middle surface as shown in figure
3-3, which represents a view of figure 3-1(a) normal to tp. The length of the
volume elemep.t is
(3.2)
3 Equilibrium
S8
d
ta
Fig. 3-3 N
or m al se ct
a n d th e p ro
je c te d a re a
da/l(O = ds",
(O d~
= dS",(l
A si m il a r se
dS/I(O
+
l E le m en t
is
(3.3)
~~)d~
ction n o rm a
= dS/I
io n on a Shel
l to t", gives
(1 + ~~)
da",(O = dS/I
(l
+
(3.4)
~~)d~
h edS
m/Ia g=n it u d e ,
w hTe re
B dP· o f th e ,t re " r"
,u
(3.5)
lt a n " acting
o n th e ,",ctio
n n o rm a l to
t. are
(3.6)
(3 .7 )
59
3.1 Stress Resultants and Couples
The magnitudes of the stress couples are
{ M '" }
M",p
1
dsp
= -
-
and
{MM}
p",
P
(da (0
'"
fh/2 { (J",,,, }
-h/2 (J",p
S
h/2 { (J",,,, }
-h/2 (J",p
(
(
1+
= _ 1 fh/2 { (Jpp }
ds", -h/2 (Jp",
-
S
h/2 { (Jpp }
-h/2 (Jp",
(
(
(3.8)
Y) d(
_'"
Rp
(dap(O
1+
_(
(3.9)
)
R",
d(
At this point, we reiterate that once the plate or shell problem has been solved
for the stress resultants and couples, the recovery of the elasticity problem
would necessitate the inversion of equations (3.6)-(3.9), which might prove
difficult in a mathematically exact sense. In a way, this is the price paid to realize
a simplified theory-the sacrifice of mathematical precision to achieve a solution that is satisfactory from an engineering standpoint.
In practice, the stresses at a particular point in the continuum are usually
recovered from the stress resultants and couples by (a) neglecting the terms
(1 + (IR",) and (1 + (IRp) in equations (3.6)-(3.9); and, (b) assuming that all
stresses, except for the transverse shear stresses (J",n and (Jpn, vary linearly across
the cross section. The preceding arguments imply that
(J;;( 0
=
(J;;(O)
(Jij( () = (Jij(O)
+ ((Jii (~) -
h;2
(J;;(O»)
+ ((Jij (~) -
(Jij(O»)
h;2
(3. lOa)
(i,j = ex, f3)
It is consistent with elementary theory to assume that the transverse shear
stresses are distributed through the thickness as a second-degree parabola with
a maximum at the middle surface. Thus,
(3.10b)
When these relationships are introduced into equations (3.6)-(3.9), the maximum stresses on the cross section are
(i
(J.. (
lJ
h) = --.!!.
N· + 6M··
+_
__lJ
-
2
h -
h2
= ex, f3)
(i. =
)
ex,f3)
= f3, ex
(3.10c)
(3.10d)
60
3 Equilibrium
(i
= IX, {3)
(3.10e)
Then, the stresses at any other level within the cross section are found from
equations (3.10a) and (3. lOb).
3.2 Equilibrium of the Shell Element
Now that all ofthe stresses acting on the shell are referred to quantities defined
on the middle surface, we have reduced the problem to two dimensions and
seek a set of relationships between the stress resultants and couples that reflects
the equilibrium of the middle surface. In this derivation, we basically follow the
approach of Novozhilov,2 which is based on vector algebra. More mathematical approaches based principally on tensor calculus, 3 as well as more physical approaches relying mainly on the free-body diagram and trigonometry,4.S
are available; in selecting the vector derivation, we hope to achieve an accommodation between mathematical elegance and sometimes cumbersome physical reasoning. The physical interpretation of the resulting equations will be
explored later for selected geometrical forms.
The stress resultants and couples on each face of the middle surface element,
shown in figure 3-2, are combined into stress and stress-couple vectors in figure
3-4. Also shown is the load vector q. In terms of the stress resultants and couples,
the resulting vectors are
+ N"ptp + QlZtn)dsp
= (Np"t" + Nptp + Qpt n ) ds"
F" = (N"t"
(3.lla)
Fp
(3. 11 b)
~+~,ada
/1
Ca + Ca,ada
Fig.3-4
J~+Fp,pdP
Cp +Cp,pdP
Stress and Stress-Couple Vectors
61
3.2 Equilibrium of the Shell Element
c'" = (-M",pt", + M",tp)dsp
C p = (-Mpt", + Mp",tp)ds",
(3.11c)
(3.11d)
and the load vector is
(3.11e)
where q"" qp, and qn are the respective load intensities per unit area of the middle
surface. Note with respect to equations (3.11c) and (3.lld) that the negative
signs on the first terms arise because the stress couple vectors were chosen to
correspond to the positive sense ofthe stresses (see figures 3-1 and 3-2).
We now apply the equations of static equilibrium
(3.12a)
(3.12b)
Referring to figure 3-4, the force equilibrium equation, (3.12a), becomes
(F",
+ F",.",doc -
F",)
+ (Fp + Fp.pdP -
Fp)
+ qds",dsp =
0
or
(3.13)
Substituting equations (3.lla), (3.11b), and (3.lle) into equation (3.13), and
dividing through by doc dP, we have
[(lV",t",
+ lV",ptp + (2",tn )l1],,,, + [(lVp",t", + lVptp + (2pt n)ff],p
+ (q",t", + qptp + qntn)ffl1 = 0
(3.14)
The differentiations indicated in equation (3.14) are carried out in a straightforward manner using equation (2.17) to evaluate the derivatives of the unit
tangent vectors. The resulting vector equations may be factored into the form
F",t", + Fptp + Fntn = 0
(3.15)
Since the unit tangent vectors are independent, equation (3.15) can only be
satisfied if
F", = 0;
Fp = 0;
Fn = 0
(3.16)
which gives the three scalar equations of force equilibrium
F", = [(l1lV",),,,,
+ (fflVp",),p + ff,plV",p
ffl1
+ (2",- + q",ffl1 =
R",
Fp
=
[(l1lV",p),,,,
ffl1
(3.17a)
0
+ (fflVp),p + l1,,,,lVp,,, -
+ (2p- + qpffl1 = 0
Rp
- l1,,,,lVp]
ff,plV",]
(3.17b)
62
3 Equilibrium
(3.17c)
The individual terms in equations (3.17a-c) will be interpreted in section 3.3
for the shell of revolution geometry. However, note that there is considerable
coupling of the equations; e.g., Nil. appears in Fp and F" as well as in Frz as
expected. This coupling is due to two general sources: (a) the curvature of the
surface; and, (b) the possibility that the Lame parameters A and B vary with
both O!: and fJ.
If, for example, A = A(/X) and B = B(fJ) only, several of the coupling terms
drop out. This emphasizes the importance of choosing the curvilinear coordinates so as to achieve the simplest mathematical formulation consistent with
the problem.
The moment equilibrium equation, (3.12b), is evaluated about axes through
point 0 in figure 3-4.
1. First, we have the contributions of the stress couple vectors
Crz + Crz,rzdO!: - Crz + C p + Cp,pdfJ - C p
(3.18a)
or from equations (3.11c) and (3.11d),
[( -Mrzptrz + Mrztp)BJ,rz dO!: dfJ + [( -Mptrz + Mprztp)AJ,pd/XdfJ
(3.18b)
2. Next, we investigate the contributions of the stress resultants. Observe in
equation (3.18b) that the stress couple terms are of second differential order,
dO!: dfJ, so that terms of third or higher differential order will drop out in the
limit. From figure 3-2, we see that the extensional stress resultants Nil. and
Np contribute moments about t" of
- Nil., II. dO!: dsp d~p t" --+ third order
and
Np,p dfJ dsrz d~rz t"
--+
third order
and the in-plane shear stress resultants Nrzp and N prz give
Nrzp dsp dsrz t"
--+
second order
and
- N prz ds rz dsp t"
--+
second order
The transverse shear stress resultant Qrz contributes
- Qrz dsp dsrz tp
--+
second order
about tp whereas Qp gives
Qp dsrz dsp til. --+ second order
63
3.3 Equilibrium Equations for Shells of Revolution
about t". Collecting the second order contributions of the stress vector to
moment equilibrium, we have
(3.19)
3. Then, we note from figure 3-4 that the contribution of q is of third differential
order and therefore negligible.
4. Finally, we expand equation (3.18b) in accordance with equation (2.17),
combine the resulting terms with equation (3.19), factor in terms of the unit
vectors, and divide by dcx dP, giving
G"t"
+ Gptp + Gntn = 0
(3.20)
which is satisfied if and only if
G"
= 0; Gp = 0; Gn = 0
(3.21)
Thus, we have the three scalar equations of moment equilibrium
=
Gp =
G"
-(BM"p),,, - (AMp),p - B,,,Mp,,
(BM"),,,
+ (AMp,,),p + A,pM"p -
+ A,pM" + QpAB = 0
B,,,Mp - Q"AB
= O.
(3.22a)
(3.22b)
(3.22c)
The set of six equations of static equilibrium, equations (3.17a-c) and
(3.22a-c), may immediately be reduced to five, since it is easily shown by the
substitution of equations (3.6)-(3.9) into equation (3.22c) that the later equation,
commonly known as the sixth equation of equilibrium, is satisfied identically if
the symmetry of the stress tensor (lp" = (l"p is invoked. 2 Thus, equation (3.22c)
may be disregarded, although we may mention it from time to time. There are
several modified shell theories which attempt to redefine the stress resultants
and couples so that the sixth equation can be satisfied in the form of equation
(3.22c).6 None of these alters the fact that the sixth equation is identically
satisfied using a proposition from a higher theory, the theory of elasticity, and
consequently cannot be germane in any further development. Therefore, five
equilibrium equations in ten unknowns remain. Such a problem is classified as
statically indeterminate, indicating that the solution awaits the introduction of
additional relationships between the stress resultants and couples, and the
deformations of the shell.
3.3 Equilibrium Equations for Shells of Revolution
3.3.1 Specialization of General Equations: The shell of revolution is a widely
used geometry and affords a sufficiently general, yet easily visualized model for
the physical interpretation of the equilibrium equations.
3 Equilibrium
64
For the shell of revolution shown in figure 2-11, we set ex = rjJ and f3 = O. From
equations (2.44b) and (2.43b), A = R,p(rjJ) and B = Ro(rjJ) = Re(rjJ) sin rjJ. Also,
recall the Gauss-Codazzi condition, equation (2.47), which gives
B,a = Ro,,p = (Re sin rjJ),,p = R,p cos rjJ
(3.23)
Writing the equilibrium equations including these relations, we have from
equations (3.l7a-c), and (3.22a-c),
+ R,pNe,p,e - R,pcosrjJNe + RoQ,p + q,pR,pRo = 0
(RoN,pe),,p + R,pNe,e + R,pcosrjJNe,p + R,pQesinrjJ + qeR,pRo =
(RoQ,p),,p + R,pQe,e - RoN,p - R,psinrjJNe + qnR,pRo = 0
-(RoM,pe),,p - R,pMe,e - R,pcosrjJMe,p + R,pRoQe = 0
(RoM,p),,p + R,pMe,p,e - R,pcos rjJ Me - R,pRoQ,p = 0
(3.24a)
(RoN,p),,p
N,pe - Ne,p
M,pe
R,p
0
M 8,p
= 0
R8
+ -- - -
(3.24b)
(3.24c)
(3.24d)
(3.24e)
(3.24f)
The middle surface element for a shell of revolution geometry with positive
Gaussian curvature is shown in figure 3-5. Stress resultants and couples are
shown on an enlarged view of this element normal to tn in figure 3-6. Also shown
in figure 3-6 are horizontal and meridional sections, HVl, MVl, and MV2,
dX
horizontal
plane
~~;::-;:;--.~D
0
dX ·OP =RcPB
dX=.BsL dB
\
OP
=cosrpdB
Fig. 3-5
Middle Surface Element and Geometrical Relationships for a Shell of Revolution
65
3.3 Equilibrium Equations for Shells of Revolution
Force Element
M;R o d8
Me ~ d¢+(M8 R¢ d¢)'8 d8
Q¢Rod8+(Q¢RJ18),4'
~~R~"'I'"
N¢ Rod8+ (N¢Rod8 ),¢d¢
Moment
Element
Fig. 3-6 Forces and Moments Acting on a Shell of Revolution Element
with selected resultants and couples indicated. In comparing the terms of
equation (3.24) with the forces and moments on figure 3-6, recall that we divided
through by da. dP, or, in this case, dtP dO, in the course of the derivation, and that
we have retained terms of second differential order only. Also, for the differential
element, sindtPI2 = dtPl2, sindOl2 = d012, and cosdtPI2 = cosdOl2 = 1.
3.3.2 Physical Interpretation of Equilibrium Equations
3.3.2.1 Force Equilibrium in the tP Direction: Equation (3.24a). The first term
represents the increment in the force N;Ro dO as tP changes, whereas the second
66
3 Equilibrium
gives the increment in the force N8fJ RfJ dr/J with respect to e, as seen on the force
element in figure 3-6. Since R~ is not a function of e, it is treated as a constant
in the second term. The third term indicates a contribution of the force N 8RfJ dr/J
to the r/J direction equilibrium and is explained by first referring to view HV1,
where the radial component 2N8R~dr/Jde/2, which is in the horizontal plane, is
developed. This radial force is transferred to view MV1 and is further resolved
in the r/J and n directions, with the r/J component being the third term in equation
(3.24a). The negative sign indicates that the component acts in the negative r/J
direction as defined on figure 2-11. The fourth term is the contribution of
QfJR o de and is substantiated by considering view MV2. The last term is, of
course, the applied loading.
3.3.2.2 Force Equilibrium in the e Direction: Equation (3.24b). The first and
second terms are analogous to the corresponding terms of equation (~.24a) and
are easily verified on the force element. The third term is the circumferential contribution of force N8~RfJ dr/J and arises because forces N8fJR~ dr/J and N 8fJ RfJ dr/J +
(N8~R~dr/J).8de are not parallel to one another, but rather are directed along
tangents to the meridians at e and e + de, respectively. As shown in figure 3-5,
the extension of these tangents defines an intersection on the axis of rotation
at an angle dX, which is expressed as de cos r/J. Therefore, from the force element,
we ha ve 2N8fJR~ dr/J dX/2 = N8~RfJ cos r/J dr/J de in the t8 direction. The fourth term
is the contribution ofQ8R~dr/J and is shown in view HV1, whereas the fifth term
is the applied loading.
3.3.2.3 Force Equilibrium in the n Direction: Equation (3.24c). The first two
terms are the increments of the transverse shear resultants QfJ and Q8 in the r/J
and e directions, respectively, as shown on the force element. The third term
represents the contribution of force NfJR o de and is obtained by projection in
view MV2. The fourth term, representing a similar contribution of N8R~ dr/J, is
the normal projection of the radial force N 8RfJ dr/J de in view M V 1. This force
was introduced previously in connection with the third term in equation (3.24a).
Both of these normal components are directed in the negative tn direction. The
fifth term again is the applied loading.
3.3.2.4 Moment Equilibrium about the t fJ Axis: Equation (3.24d). The first two
terms of this equation are analogous to the corresponding terms of equation
(3.24a) and may be verified accordingly on the moment element, noting that
the increments are directed in the negative t fJ direction. The third term is found
analogously to the projection of N8RO dr/J on views HV1 and MV1. The fourth
term is the contribution of the force Q8RfJ dr/J, multiplied by Ro de, as seen on
the force element.
3.3.2.5 Moment Equilibrium about the t8 Axis: Equation (3.24e). Again, the first
three terms are analogous to the force equilibrium equation, (3.24b), with the
67
3.5 Nature of the Applied Loading
third term being opposite in sense to N(Jq,Rq, d</J, as seen from the moment
element. The last term is the couple Qq,Ro dO Rq, d</J, which is directed along the
negative t(J direction.
3.3.2.6 Moment Equilibrium about the tn Axis: Equation (3.24f). The first two
terms represent the moments of the in-plane shear resultants Nq,(J and N(Jq, about
the normal tn' The third term is opposite in sense but analogous to the normal
projection of Nq,Ro d</J on view MV2; the fourth term follows the normal
projection of N(JRq,d</J on views HVI and MVl.
3.4 Equilibrium Equations for Plates
Equations (3.17a-c) and (3.22a-c) may easily be reduced to sufficiently general
equilibrium equations for medium-thin plates by allowing R" and Rp ..... 00.
After these simplifications, we have
+ (ANp,,),p + A,pN"p - B,,,N,J + qaAB =
[(BNap),a + (ANp),p + B,aNpa - A,pNaJ + qpAB =
[(BQa),a + (AQp),pJ + qnAB = 0
[(BN").,,
-(BMap),a - (AMp).p - B,aMPa
(BMa),a
+ (AMpa),p +
0
(3.25a)
0
(3.25b)
+ A,pMa + QpAB = 0
A,pM"p - B,,,Mp - QaAB
=0
(3.25c)
(3.25d)
(3.25e)
The last equation becomes an identity.
Further simplifications will be made for regular geometries and for bending
in the absence ofin-plane forces; for now, we note in equations (3.25a-c) that
the in-plane stress resultants (Na, N p, NaP, N pa ) are uncoupled from the stress
couples (Ma, Mp, MaP' M prz ) and the transverse shear stress resultants (Q", Qp).
This suggests that the plate resists in-plane loading (q", qp) exclusively through
extensional action and transverse loading (qn) only by flexure. This is an
important contrast to shell action as demonstrated by equation (3.17c), where
the in-plane stress resultants provide resistance against transverse loading by
virtue of the curvature of the shell.
3.5 Nature of the Applied Loading
Thus far, we have assumed that the applied loading is expressible as a distributed
force per unit area of middle surface, with components in the directions of the
unit vectors. This leaves open the cases of concentrated forces, and distributed
and concentrated moments.
68
3 Equilibrium
In general, shells are most efficient when the loading is distributed over the
surface, because-provided certain geometrical and support conditions are
met-such loading can be resisted primarily by the extensional and in-plane
shear stress resultants, rather than by the transverse shear stress resultants and
the bending and twisting stress couples. In addition to the distributed surface
loading included in the formulation by q, distributed moments about the t¢
and t9 axes can easily be accommodated by including appropriate terms in
equations (3.22a) and (3.22b). Moments about the tn axis, however, are not
admissible, since the corresponding equilibrium equation, (3.22c), was suppressed. A plate or shell generally offers very great stiffness to twisting about
the normal, provided rigid body motion is restrained.
Concentrated forces and moments require special attention in plate and shell
theory. Often, these loadings produce singular points, in which case admissible
solutions may be found only away from the point of application of the load. In
any case, these forces are usually resisted primarily by the transverse shears that
are directly related to the bending and twisting stress couples. We may view
concentrated forces and moments as limits of the corresponding distributed
effect. Consider the area L1a subjected to a uniformly distributed loading of
intensity p in figure 3-7. We define a concentrated force Pc as
Pc
=
lim
(pAa)
(3.26)
4A--+O
p4a remains constant
Now consider two concentrated coplanar forces as defined in equation (3.26)
and shown in figure 3-7(b). The forces Pc are equal in magnitude and opposite
in direction a distance L1, apart. We define a concentrated moment Mc about
an axis normal to the plane of the forces as
p
II 1III
(b)
(0)
Fig.3-7
Concentrated Force and Moment
3.7 Exercises
69
M=
c
lim
(3.27)
a~-O
P cAe: remains constant
Equations (3.26) and (3.27) are somewhat abstract in this form, but are useful
in obtaining many solutions for plates and shells.
3.6 References
1. H. Kraus, Thin Elastic Shells (New York: Wiley, 1967), p. 33.
2. V. V. Novozhilov, Thin Shell Theory [translated from 2nd Russian ed. by P. G. Lowe
(Groningen, The Netherlands: Noordhoff, 1964), pp. 34-39].
3. A. E. Green and W. Zerna, Theoretical Elasticity (Oxford; Oxford University Press,
1968).
4. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (New
York: McGraw-Hill, 1959).
5. W. Fliigge, Stresses in Shells, 2nd ed. (Berlin: Springer-Verlag, 1973).
6. Kraus, Thin Elastic Shells, pp. 58-65.
3.7 Exercises
3.1
Derive Fa' Fp, and F., as given in equation (3.17), from equation (3.14).
3.2
Derive G a , Gp , and G., as given in equation (3.22), from equations (3.18b) and (3.19).
3.3
Verify that equation (3.22c) is identically satisfied if (Jap = (Jpa.1t is often incorrectly
concluded that the identical satisfaction of equation (3.22c) implies that NaP = NPa
and MaP = Mpa. For what geometries do these relations hold?
3.4
Redraw figures 3-5 and 3-6 for a shell of revolution with negative Gaussian
curvature and verify the physical interpretation shown in the views HV1, MV1,
and MV2 for the negative curvature element.
3.5
Derive the equilibrium equations for a shell of revolution where the coordinate
is taken as the axial coordinate Z.
IX
CHAPTER
4
Membrane Theory
4.1 Simplification of the Equilibrium Equations
We examine the individual terms of the force equilibrium equations, (3. 17a-c),
and the moment equilibrium equations, (3.22a-c). We see that the equations
are coupled only through the transverse shear stress resultants, QI% and Qp. If
we suppose that for a certain class of shells, the stress couples are an order of
magnitude smaller than the extensional and in-place shear stress resultants,
we may deduce from equations (3.22a-c) that the transverse shear stress resultants are similarly small and thus may be neglected in the force equilibrium
equations, (3.17). This implies that the shell may achieve force equilibrium
through the action of in-plane forces alone. From a physical viewpoint, this
possibility is evident for the first two equilibrium equations which reflect
in-plane resistance to in-plane loading, a natural and obvious mechanism.
On the other hand, the third equilibrium equation refers to the normal direction, and the possibility of resisting transverse loading with in-plane forces
alone is not as apparent. It is evident from equation (3.17c) that this mode of
resistance is possible only if at least one radius of curvature is finite; i.e., RI%
and/or Rp #- 00. Thus, flat plates are excluded from resisting transverse loading
in this manner, within the limitations of small deformation theory (assumption
[2], table 1-1.
The applications and limitations of this idealized behavior, termed the membrane theory of thin shells, is examined in this chapter for a variety of shell forms.
At present, it is constructive to consider some ofthe consequences of membrane
behavior. First, it is quite desirable from a material efficiency standpoint. Recall
from chapter 1 that structural materials are generally far more efficient in
extension rather than in flexure. Second, if we introduce an additional simplification with respect to the shear stress resultants,
N"p ~ N p"
=S
(4.1)
the force equilibrium equations become
(4.2a)
70
4.2 Applicability of Membrane Theory
71
(4.2b)
(4.2c)
In these three equations, there are but three unknowns, N a , N p , and S. Such a
system is said to be statically determinate and thus independent of compatibility or constitutive considerations. Of course, this simplifies the subsequent
mathematics considerably. The convenient assumption stated in equation
(4.1) may be easily justified by considering the definition of Nap and Npa as
given in equations (3.6) and (3.7). Since (Tap = (Tpa because the stress tensor is
symmetric, NaP and NPa may differ only in the disparity of the terms 1 + 'IRp
and 1 + 'IRa' respectively. Since 'IRp and (IRa are themselves necessarily small
in comparison to 1, the difference in Nap and Npa is obviously negligible. Thus,
the simplification introduced in equation (4.1) is justified apart from other
considerations.
Equations (4.2a-c), together with the requisite boundary conditions, constitute the essential components of the membrane theory. Strictly speaking, the
entire solution includes the evaluation of the displacements as well. This requires the constitutive and compatibility relationships that have not yet been
developed here. However, just as in the analysis of statically determinate beams
and frames, the membrane theory displacements may be computed following
the establishment of the force field with only a few exceptions. 1 Consequently,
consideration of the membrane theory displacements is deferred until chapter 6.
4.2 Applicability of Membrane Theory
The shells for which the membrane theory is applicable fall into two general
classes: (a) absolutely flexible shells or true membranes, which by virtue of their
thinness have a negligible bending stiffness; and, (b) shells with finite bending
stiffnesses which still develop relatively small bending stresses. 2
Concerning the absolutely flexible shells, there is little question of the dominance of membrane action, provided that the principal stresses remained tensile.
The presence of compressive stresses in this type of shell would likely cause
buckling of the membrane. For flexible shells, the requirement of small displacements as stated in assumption [2] of table 1-1 is often not attainable, and the
equations as developed herein are not applicable without modification. This
complication notwithstanding, membrane surfaces are widely used for longspan roofs, as either tentlike or air-supported structures. An unusual example
of a thin, flexible, curved object to which shell theory has been applied for the
72
4 Membrane Theory
purpose of estimating the stresses present is the human aortic heart valve [figure
2-8 (q)]. 3
A wide variety of shells with finite bending rigidities can be designed and
constructed to resist external loading almost exclusively through membrane
action. We should recognize that the establishment of the precise bounds of the
membrane theory requires the consideration of the governing equations of the
general theory of shells. Rigorously, a valid membrane theory solution must be
a close approximation to the response that would be computed using the full
set of shell equations. However, we may make some observations regarding the
geometry, boundary conditions, and loading generally consistent with membrane
behavior.
With respect to the geometry, a continuous curved surface is conducive to
membrane action. Also recall that the second quadratic form of the theory of
surfaces, equation (2.28b), contains the principal radii of curvature, and a
smooth variation of these parameters is essential if equilibrium is to be achieved
without transverse shears and bending and twisting moments.
The boundary conditions must be specified with respect to the membrane
stress resultants, N", N p , and S, or the corresponding displacements, and must
indicate either a constraint that will fully develop the force on the boundary; a
release that will enable the shell to displace freely in the corresponding direction;
or, possibly, a linear combination of both, as in the case of an elastic support.
Moreover, the boundary should not provide constraints that would develop
moments and transverse shears. This implies that the ideal boundary must, in
some cases, permit rotations and transverse displacements while providing
complete restraint in the plane of the shell.
A loading that can be resisted by membrane action must be distributed
over the surface without severe variations or concentrations. In the membrane equilibrium equations, (4.2a-c), concentrated loadings are not admissible. From a physical standpoint, it is apparent that a load normal to and
concentrated over a small area of the shell surface would be resisted principally
by transverse shears which, in turn, produce bending, as demonstrated by
equations (3.22a-c).
From this brief discussion of the conditions which correspond to ideal
membrane behavior, it is obvious that to design and construct an actual shell
that satisfies all of these conditions may be quite difficult. However, even shells
which do not meet all of the requirements entirely can be built so that membrane action predominates throughout most of the continuum, except for
localized regions in which the required conditions are violated. In short, many
shells exhibit basically membrane behavior augmented by locally prominent
bending action. This is not to say that all shells may be made to act primarily as membranes or that all bending effects are localized, but such behavior is a desirable and often attainable result of good design and careful
construction.
73
4.3 Shells of Revolution
4.3 Shells of Revolution
4.3.1 Specialization of Equilibrium Equations: We now specialize the membrane theory equilibrium equations, (4.2), for the shell of revolution geometry,
where the curvilinear coordinates are taken as the meridional angle, ifJ, and the
circumferential angle e.
With (X = ifJ and f3 = e, A = Rt/J(ifJ) and B = Ro(ifJ) = RIJ sin ifJ as previously
established in equations (2.44b) and (2.43b). If we make these substitutions into
equations (4.2a-c) and carry out the differentiations using the Gauss-Codazzi
condition, equation (2.47), we find
(4.3a)
(4.3b)
Nt/J
Rt/J
Nil
+ RIJ =
qn
(4.3c)
We immediately note that equations (4.3a-c) are identical to equations (3.24a-c)
divided through by Rt/JR o, with the bending terms suppressed and the in-plane
shear stress resultants Nt/JIJ and Nllt/J set equal to S. As such, the physical interpretation of these equations is contained within section 3.3.2. We also observe that
the third equation, (4.3c), is an algebraic equation. Hence, either Nil or Nt/J may
be directly eliminated from equations (4.3a) and (4.3b), leaving a second order
system of partial differential equations to be solved.
To simplify the solution of the equilibrium equations, it is convenient to
introduce certain transformations at this stage. Timoshenko and WoinowskyKrieger use integrating functions,4 whereas Novozhilov suggests the auxiliary
variables 5
(4.4a)
and
(4.4b)
We first solve equation (4.3c) for
Nil
= RII ( qn -
:;)
(4.5)
and then introduce equations (4.4) and (4.5) into equations (4.3a) and (4.3b) to
get
74
4 Membrane Theory
R~Rsin ifJ ,I,
'I',q, +
q,
J=
<",/1
• A.) 3 . 2 A.
= ( qncos'l'A. - qq,sm'l'
R/I sm 'I'
(4.6a)
(4.6b)
Equations (4.6a) and (4.6b) are the transformed membrane theory equilibrium
equations. Once t/I and'; are determined, Nq, and S follow from equations (4.4a)
and (4.4b), and N/I may then be calculated from equation (4.5). The subsequent
combination and the eventual solution of the transformed equilibrium equations are dependent on the circumferential distribution of the applied surface
loading. If the loading is distributed uniformly around the circumference, it is
termed axisymmetric, whereas if the loading is a variable function of e, it is
obviously nonsymmetric. Each possibility is considered further in the subsequent sections.
4.3.2 Axisymmetrical Loading
4.3.2.1 Integration of Equilibrium Equations. The case of axisymmetrically
loaded shells of revolution is perhaps the most widely studied class of problems
in the shell literature. From a standpoint of applications, this case is often the
most important, since it corresponds to gravity loading and internal pressurization, as well as to many other practical loading situations.
We may drop the e-dependent terms in equations (4.6a) and (4.6b) to derive
the uncoupled set of ordinary differential equations
t/I,q, = (qn cos ifJ
- qq, sin ifJ)Rq,R/I sin ifJ
';,q, = -q/lRq,R~sin2ifJ
(4.7a)
(4.7b)
Considering equation (4.7a) and integrating, we get
t/I(ifJ) =
f (qn cos ifJ -
qq, sin ifJ)Rq,R/I sin ifJ difJ
(4.8)
In many cases, an alternate form of equation (4.8) is convenient. We recognize
that evaluating the integral for a particular shell geometry will produce a single
integration constant, and that this constant must be determined from a boundary condition on t/I, or on Nq,. We designate the boundary where the condition
is specified as ifJ = ifJ' and the corresponding boundary condition as t/I(ifJ'). Then,
equation (4.8) may be written as
t/I(ifJ) = t/I(ifJ')
+
fq, (qn cos ifJ
q,'
- qq, sin ifJ)Rq,R/I sin ifJ difJ
(4.9)
Of course, at ifJ = ifJ', t/I = IjJ(ifJ') = Nq,(ifJ')R/I(ifJ') sin 2 ifJ'. The solution for Nq, follows from equations (4.4a) and (4.9) as
75
4.3 Shells of Revolution
(4.10)
+
1.
RlJsIn
2,J.
'f'
f,p (qn COS ¢ - q,p sin ¢)R,pRIJ sin ¢ d¢
,p'
Then, NIJ is computed from equation (4.5).
The in-plane shear stress resultant S is completely uncoupled from N,p and
N IJ , as is apparent from equations (4.7a) and (4.7b). For common axisymmetric
loading cases, qlJ = 0 and the shear stress is also 0; however, this is not necessarily
so, since equation (4.7b) may be integrated in an identical fashion to equation
(4.7a), yielding
(4.11)
or
S(¢) =
S(¢")R~(¢") sin 2
R2 . 2¢
<//'
IJ SIn
(4.12)
In equations (4.11) and (4.12), ¢(¢") and S(¢") are the corresponding values of
the functions specified at the boundary ¢ = ¢". This solution represents a state
of pure shear produced by a torsional loading.
In summary, we observe that the membrane theory equilibrium equations
for axisymmetrically loaded shells of revolution reduce to two uncoupled first
order ordinary differential equations. This system admits the prescription of
two boundary conditions, one on the meridional stress resultant N,p' and the
other on the in-plane shear stress resultant S.
One special case is worthy of consideration before we turn to the integration
of equations (4.10) and (4.12) for specific shell geometries. If we consider a closed
shell as first discussed in section 2.8.2, we have the two possibilities illustrated
in figure 4-1. As mentioned in chapter 1, the form of domed roofs evolved from
the smooth to the pointed top in the Renaissance. In case (a), which we will call
a dome [figures 2-8(i) and (1)], the meridian remains continuous as ¢ and Ro -+ 0;
whereas in case (b), which may be termed a pointed or ogival shell, ¢ = ¢t at
Ro = O. If we take the pole angle ¢ = ¢p as the boundary ¢' in equation (4.9),
where ¢p = 0 for case (a) and ¢p = ¢t for case (b), and evaluate ljJ(¢p), we find
ljJ(¢p)
= N,p(¢p)RIJ(¢p) sin 2 ¢p
= N,p(¢p)Ro(¢p) sin ¢p
=0
(4.13)
76
4 Membrane Theory
(b)
(0)
Fig. 4-1
Closed Shells
since Ro(rPp) = O. Similarly, considering equation (4.11) for
the solutions for closed shells simplify to
Ntfo
=
1
. 2
Re SIll
S= .
2
Re
A.
'I'
1. 2
SIll
ltfo (qn cos rP - qtfo sin rP)RtfoR e sin rP drP
tfo p
rP
l'"'"p
. 2 'I'
A. dA.
qeR",Re2 SIll
'I'
~, ~(rPp)
= O. Thus,
(4. 14a)
(4. 14b)
For the common case of a dome [figure 4-1(a)], rPp = O.
We may observe several interesting points about the preceding solutions for
closed shells:
1. The pole condition does not necessarily imply that N",(rPp) or S(rPp) = o.
2. For the case of a dome, rPp = 0, the integrated expressions may produce an
indeterminate form which must be evaluated by some limiting operation.
3. Since the available boundary conditions on N", and S are implied by the pole,
there is no opportunity to specify any further boundary conditions on the
stress resultants within the membrane theory. This means that to maintain
equilibrium, the remaining boundary of the shell must develop whatever
values of the membrane stress resultants that are computed from equations
(4.14). We recall that such an ideal boundary may not impose constraints
which will develop bending forces or moments.
4. For a dome, all meridians meet at the pole, and any direction is parallel to
one meridian and at right angles to another; that is, the curvilinear coordinates rP and () are interchangeable and indistinguishable. Therefore, R",(O) =
Re(O) = Rp and, corresponding, Ntfo(O) = Ne(O). Then, from equation (4.3c),
N",(O)
= Ne(O) =
Rp
qn(O)T
(4.15)
77
4.3 Shells of Revolution
z
cos¢K
,,
/cp
q
qn=-q
"
q =q sin,l,
't'
./
axis of rotation
~
Fig.4-2
Spherical Shell Under Self-Weight Load
4.3.2.2 Spherical Shells. As the first example, consider the spherical shell illustrated in figure 4-2. The shell has radius a, constant thickness h, mass density
p, and is bound by angles ifJt and ifJb' Initially, the weight of the shell alone, known
as the self-weight, is taken as the loading condition.
The principal radii of curvature are the definitive geometric parameters. The
radius of curvature of the meridian is, of course, the radius of the generating
circle a. This is easily verified by considering the equation of the meridian
Z2 + R'5 = a 2 and substituting into equation (2.29) with ex; = ifJ and X = Ro.
With Z,Ro = -Ro/Z and Z,RoRo = -(1/Z)(1 + R'5/Z 2), we find
R,p
=
- [1 + (R'5/z 2 )]3/2
-(1/Z)[1 + (R'5/Z 2 )]
=
.../(Z2
+ R'5)
(4. 16a)
=a
The other principal radius may be computed from equation (2.42). We note
from figure 4-2 that the horizontal radius Ro = a sin ifJ, so that
78
4 Membrane Theory
RIJ
a sin tP
sin tP
= -- =
(4.16b)
a
Equations (4.16a and b) obviously satisfy the Gauss-Codazzi condition, equation 2.47.
Next, consider the applied loading. The self-weight of the shell per unit area
of the middle surface is given by
q
=
pgh
(4.17)
in which 9 = the acceleration of gravity. The load q acts vertically, of course.
The governing equilibrium equations require the loading to be resolved in the
tP, e, and n directions. Since the vertical load has no component in the circumferential direction, qlJ = O. On the inset in figure 4-2, we have the resolution of
q into q</l and qn. The signs are established by comparison with the unit vectors
shown in figure 2-11.
At present, the boundaries are assumed to satisfy ideal membrane theory
restrictions. The upper boundary is presumed to be free of stresses and the lower
boundary is taken to be unyielding in the tP or meridional direction, free to
displace in the n or normal direction, and unrestrained against rotation about
an axis along the e or circumferential direction. The base conditions imply that
the necessary N</I to maintain equilibrium will be developed, but no Q</I or M</I
can occur. The roller symbol used on the figure corresponds to a similar
condition in plane structural analysis; however, here the boundary is a space
curve, and for shells which have circumferential loading, a condition in the e
direction also must be specified. In the latter instance, the ideal membrane
boundary must be extended beyond the simple conceptual model of a roller.
We now substitute the geometry, loading, and boundary conditions into
equation (4.10). These calculated quantities are R</I = RIJ = a; q</l = qsintP and
qn = -qcostP· We take tP' = tPt so that N</I(tPt) = O. Then, we have
N</I(tP)
=
.1 2 tP f</l (- q cos 2 tP - q sin 2 tP)a 2 sin tP dtP
asm
</I,
qa
= ~(cos tP
Sin
'f'
(4.18)
- cos tPt)
From equation (4.5), the circumferential or hoop stress resultant is given by
NIJ
=
a( -qcostP -
:</1)
(4.19)
We now examine a spherical dome by letting tPt = O. Using an elementary
trigonometric identity, equation (4.18) becomes
qa
N</I
= (1 + cos tP)(1
-qa
1 + cos tP
_ cos tP) (cos tP - 1)
(4.20)
79
4.3 Shells of Revolution
-8
-6
-4
N4>lqa
Fig. 4-3
-2
o
o
2
4
6
Nglqa
Self-Weight Stress Resultants for Spherical Domes of Constant Thickness
and equation (4.19) follows as
No
=
a ( - q cos <P
= qa (
+ 1 + ~os <P )
(4.21)
1 <p - cos <p)
1 + cos
At <p = 0, we have
-qa
N",= No =-2-
(4.22)
which is in accordance with equation (4.15). It is also notable that the indeterminate form mentioned as point (2) in section 4.3.2.1 was resolved by evaluating
the integral first, and then going to the limit as <p -. O.
It is quite instructive to consider equations (4.20) and (4.21) for increasing
values of <p as shown in figure 4-3, where nondimensional plots for N", and No
are provided. These graphs actually represent the self-weight stresses for all
constant thickness spherical domes. For any particular dome for which the
lower boundary is located by a given value of <Pb' the relevant parts are the
regions <p ::5; <Pb' A shallow dome (rjJ" « 90°), a hemispherical dome (<Pb = 90°),
and a deep dome (<Pb > 90°) are illustrated in the insets of the graphs. An
interesting feature of the graphs is that N", is always negative or compressive,
whereas No changes from negative to positive. This exact transition point may
be computed from equation (4.21) as the value of <P satisfying
1
----,-- cos<p = 0
1 + cos<p
which has the solution <p = 51 °49'. The transItIon angle has some general
practical ramifications, since a shell with <Pb < 51 ° will be entirely in com pres-
4 Membrane Theory
80
sion under gravity loading, which is especially desirable for shells constructed
of concrete.
'
With reference to the ancient masonry domes described in chapter 1, the
necessity for maintaining the entire dome in compression is obvious and is
apparently reflected in the change from the spherical to the more parabolic
profile used in the Renaissance.
The apparent desirability of providing an entirely compressive state of stress
is countered by another practical consideration. Recall that the support for the
spherical shell must develop the calculated value of Nt/J at ¢ = ¢b' An idealized
typical support is shown in figure 4-4(a), where a circumferential ring beam is
employed to resist the thrust. The shell is assumed to extend to the centroid of
the ring beam to eliminate the introduction of eccentricity.6 A section through
the ring beam is shown in the inset. The vertical component of the thrust,
V = N"'(¢b) sin ¢b' is transmitted to the foundation, whereas the horizontal
component must be developed by the beam. A half-plan of the ring is shown
in figure 4-4(b), with a unit length segment in the inset; there, the horizontal
reaction H = - N",(¢b) cos ¢b is countered by the radial component of the hoop
tension T.
We may evaluate T by summing forces in the X direction on figure 4-4(b).
(4.23a)
or the magnitude
(4.23b)
F or the spherical shell
b = a sin ¢b
(4.24)
so that
(4.25)
For the self-weight case, we find by substituting equation (4.20) into equation
(4.25)
TDL
2
cos ¢b sin ¢b
= qa 1 + COS'l'b
A.
(4.26)
It is clear from comparing equation (4.26) and the graph for NIJ in figure 4-3
that there will be a strain incompatibility at the junction between the shell and
ring beam. For ¢b < 51 °49', the hoop stress NIJ is compressive; the ring force
TDL is tensile for all ¢b < 90°. Thus, the dome, which has been shown to be in
a state of compression through membrane theory analysis, must somehow
accommodate the circumferential expansion ofthe base ring accompanying the
tensile force TDL . Clearly, this cannot be accomplished with membrane action
81
4.3 Shells of Revolution
b
(0)
----·- - T
-f-- t - -- - - - , - - --.A-
- -_a X
---+--T
(b)
Fig. 4-4 Ring Beam for a Shell of Revolution
82
4 Membrane Theory
alone, and bending forces must be considered to satisfy deformational continuity. In the terminology of section 4.2, we have violated the ideal boundary
conditions. In some instances, such as for the dome shown in figure 2-8(j),
prestressing of the base ring is effective in reducing the circumferential strain
incompatibility. In general, the bending effects introduced by this type of
support are usually confined to a relatively small portion of the shell adjacent
to the base, and membrane action will still predominate throughout most of
the shell. It should also be noted that equation (4.23) is not restricted to spherical
shells, but is valid for all shells of revolution supported in this fashion.
It is apparent from figure 4-3 that the stresses increase rapidly for deep
spherical shells with rPb > 120°. For such shells, it is logical to support the shell
somewhere above the base. Such a situation is depicted in figure 2-8 (t), although
the description of discrete column supports requires an extension of the ideal
membrane case, as we will see later. If we have, for example, a complete sphere
supported by an ideal membrane boundary at an angle rP = rPr as shown in figure
4-5, we can easily obtain the stress pattern from figure 4-3. For rP s rPr we
compute the stresses as before. For rP > rP" the lower portion ofthe shell behaves
just like a spherical cap with a boundary angle rPb = 11: - rP" and with a loading
which is just opposite in sense to that shown on figure 4-2, as shown in the inset
Ideal Membrane
Boundary
Fig. 4-5
Complete Spherical Shell with an Intermediate Continuous Support
83
4.3 Shells of Revolution
of figure 4-5. Thus, if we enter (f = 11: - ¢> for ¢> on figure 4-3, the dead load stress
resultants for the lower portion of the shell are identical in magnitude but
opposite in sign to those read from the graph.
Another common loading condition for spherical domes is that of uniform
normal pressure. Consider a positive internal pressure p on the spherical dome.
We substitute R.p = RIJ = a; q.p = 0; qn = P and ¢>p = 0 into equation (4.14a),
whereupon
.1 2 ¢> f.p P cos ¢> a2 sin ¢> d¢>
a SIn
0
N.p(¢» =
pa 2
=--
[1 .
-sm 2 ¢>
a sin ¢> 2
J.p
0
(4.27)
pa
2
From equation (4.5)
NIJ(¢»
=
a(p - :.p)
(4.28)
pa
2
so that we have a uniform state of stress, N.p = NIJ = paj2, throughout the shell.
A variation on the spherical water tank is the spheroidal shell, figure 2-8(s),
which is an ellipsoidal shell of revolution. 7
4.3.2.3 Hyperboloidal Shells. As a second illustration of a membrane theory
solution for an axisymmetrically loaded shell of revolution, we consider the
form illustrated in figures 4-6 and 2-8(g). This surface is generated by the rotation of a hyperbola of one sheet and is a shell with negative Gaussian curvature, since the centers of curvature corresponding to R.p and RIJ lie on opposite
sides of the meridian (see section 2.7 and figures 2-6 and 2-11). This form is of
considerable practical importance, because of the wide use of such shells for
reinforced concrete hyperbolic cooling towers [figure 2-8(0)]. These massive
structures may approach a height of 600 ft (183 m), span 300 ft (92 m) in
diameter, and yet have an average thickness ofless than 9 in (23 cm), a striking
testimony to the efficiency of thin shell structures.
The equation of the generating curve is
R~
Z2
---=
a2
b2
1
(4.29)
in which b is a characteristic dimension of the shell that may be evaluated by
substituting the base coordinates (s, S) or the top coordinates (t, T) into equation (4.29) as
84
4 Membrane Theory
rotation
a
t-
%
o
Throat
U)
Bose
s
Fig. 4-6
as
Hyperboloidal Shell Geometry
(4.30)
The ratio alb is the slope of the asymptote to the generating hyperbola shown
on figure 4-6, and the parameter
(4.31)
may be viewed as an indicator of the deviation of the profile from the degenerate
85
4.3 Shells of Revolution
case of the cylinder, k = 1, with a larger k corresponding to a more pronounced
curvature of the meridian. As may be seen from equation (4.30), b, and hence
the geometric profile, can be set independently using specified top or base
dimensions. For cooling tower applications, it is sometimes desirable to use
different values of b for the top and bottom, and thus create a compound shell
with a smooth transition at Z = O.
With the parameter k defined, we may rewrite equation (4.29) as
R5 - (k 2 - l)Z2 = a 2
(4.32)
The next step is to derive expressions for the principal radii of curvature in
terms of the curvilinear coordinate ,po Direct substitution of the equation of the
meridian into equation (2.29), as previously done for the spherical shell, leads
to cumbersome expressions since Z and Ro are not given as explicit functions
of ,po Instead we consider equation (4.32) along with the expression for the
differential arc length, equation (2.45a), which becomes for this case,
ds~
= dZ 2 + dR5
or
R2r/> dA.2
= Z .r/>2 dA.2
'I'
'I'
+ R2O,/f dA.2
'I'
(4.33)
We then solve equation (4.32) for Z
Z
= (R5 - a 2 )1/2
(4.34)
k2 _ 1
and compute
Z
_
.r/> -
RoRo,r/>
Z(k2 - 1)
Since
Ro,r/> = Rr/> cos,p
(4.35a)
.
RoRrpcos,p
L,r/> = [(R5 _ a2)(k 2 _ 1)]1/2
(4.35b)
After substituting equations (4.35a) and (4.35b) into equation (4.33) and clearing
fractions, we may cancel R~ d,p2 in each term, so that
(R5 - a 2)(k 2 - 1)
=
R5COS2,p
+ (R5
- a 2)(k 2
-
1)cos 2 ,p
(4.36)
from which we find, after some manipulation,
Ro
sin,p
RIJ = - - =
a.J(k 2 - 1)
[k 2 sin 2 ,p - 1J 1/2
-::c:-;;-:''-;;--,----::-;-;;;-
Finally, from equations (4.37) and (4.35a),
(4.37)
86
4 Membrane Theory
(4.38)
We now consider the elementary case of the self-weight. The principal radii
of curvature are given by equations (4.37) and (4.38), and the loading components are q" = q sin iP and qn = - q cos iP as derived in figure 4-2. Also, we'take
iP' = iP" in equation (4.10), which becomes
N,,(iP)
=
sin 2 iPt [k 2 sin 2 iP - 1J 1/2
N,,(iPt) sin2iP [k2 sin2iPt - 1]1/2
f"
[k 2 sin 2 iP - 1]1/2
a sin2iPJ(k2 - 1)
(4.39)
. 2 a2(k 2 - 1) sin iP diP
.J",(-qCOS ,p-qsm iP) [k2sin2iP-1J 2
2
Assuming a stress-free top edge, N,,(iPt) = 0, and q = constant (uniform thickness), equation (4.39) integrates t0 8
N,,(iP)
qa[k 2 sin 2 iP - 1]1/2
(4.40a)
= sin2iPJ(k2 _ 1) ['1 (iP) - '1 (iPt)J
in which the integrated function
'1
-cosiP
1
In(J(k2 - 1) - kCOS iP )
_
(iP) - 2[k 2 sin 2 iP - 1J + 4kJ(k 2 - 1)
J(k 2 - 1) + k cos iP
(4.40b)
The circumferential stress resultant N9 is computed from equation (4.5) as
aJ(P - 1) [
N9 (iP) = [k2sin2 iP _ 1]1/2 -qcosiP
+
N,,(iP) [k 2 sin 2 iP - 1J3/2
aJ(k2 - 1)
J
(4.41)
Some further comments are in order regarding the preceding solution for the
self-weight stress resultants. If the shell thickness changes with the height of the
shell, the solution as derived must be generalized slightly. Assuming that the
shell thickness is or may be approximated as piecewise constant within a defined
subregion of the shell, the basic solution, equation (4.39), may be applied in
a stepwise fashion. For any subregion r bounded by iP: and iP: as shown in
figure 4-7,
r = 1, ... ,n
(4.42a)
and, from equation (4.17),
qr
= pghr
(4.42b)
where h r is the local constant thickness. We start at the top of the shell, r = 1,
for which the solution is given by equation (4.40a). Then, we evaluate equation
(4.40a) at iP = iP; = iP: to compute N,,(iP;) = N,,(iPn which is substituted into
the first term of equation (4.39). Thus, the solution for region r = 2 is given by
87
4.3 Shells of Revolution
Fig. 4-7
Hyperboloidal Shell Divided into Constant Thickness Regions
equation (4.40a), plus the first term of equation (4.39), with
the general region r, we have
,
, sin2,p; [k 2 sin 2 ,p' _ 1]1/2
N~(,p ) = N~(,pt ) sin2 ,p' [k2 sin2,p; _ 1J1/2
+
, [k 2 sin 2 ,p' - 1J1/2
,
,
q a sin2,p',J(k2 _ 1) [~1(,p) - ~l(,pt)]
(A taken as (p,2. For
(4.43)
No(,p') is calculated as before from equation (4.41).
Although the coordinate ,p has proved to be convenient for integrating the
88
4 Membrane Theory
membrane theory equations, it is somewhat awkward for physically locating a
particular position on the shell. It is obvious that the axial coordinate Z would
be more meaningful from the standpoint of practical construction. Corresponding to any value of <p, Ro may be computed from equation (4.37), and then
Z can be found from equation (4.34). Conversely, for a specified Z, Ro is
computed from equation (4.32), and <p is conveniently obtained by solving
equation (4.37) for sin <p:
<p
=
.
SID
-1 {
Ro
}
(a 2 + k2(R~ _ a2)]1/2
(4.44)
Values of <p in the first and second quadrant correspond to the lower and upper
portions of the shell, respectively.
To compute the limits of integration, <Pt and <P.. when the corresponding value
of R o, t, and s, are known, equation (4.44) can be used directly.
For spherical shells, we were able to deduce some general characteristics of
the response to self-load by making a nondimensional plot of the membrane
theory stress resultants. It is also useful to attempt such a study for the hyperboloidal shell. If we define a nondimensional meridional coordinate
(4.45)
and assume the thickness to be constant, we may plot nondimensional membrane stress resultants for various values of k 2 , als, and alt. Such a plot is shown
in figure 4-8 for a shell of typical proportions, alt = 0.90 and als = 0.55. To
give some idea of the actual size of a corresponding cooling tower with k 2 =
1.18 and a = 70 ft (23 m), the dimensions would be s = 127 ft (39 m), t = 78 ft
(24 m), S = 258 ft (79 m), and T = 80 ft (25 m) so that the total height is 338 ft
(104 m).
From figure 4-8, we see that under self weight load the entire shell is in a
state of compression, except for a portion above the throat, which has a
relatively small tensile hoop stress. From the standpoint of reinforced concrete,
this stress pattern is quite favorable. With respect to the base region, <D = 1.0,
there are several points to be noted. First, the state of circumferential compression does not match the anticipated tension in the ring support, as previously illustrated in the discussion of spherical shells. Second, hyperboloidal
shells which are used in cooling tower applications must have openings at the
base to allow air to enter. This is accomplished by supporting the shell on an
annular ring of closely spaced columns, as shown in figure 2-8(p), producing a
so-called mixed boundary, where the shell may be presumed to have zero
meridional displacement within the column width and zero meridional stress
between the columns. The combination of the discrete support and the ring
beam clearly indicates a boundary far more complex than the idealized membrane theory case. This is considered further in subsequent sections. (Complete
89
4.3 Shells of Revolution
nA,=
'f"
~
qa
-1.5 . . . - - - - - - - - - - - - - - .
-1.0
NS
nB= qo
-0.5
o
0.5
L...-_"---_...L..-_-+-_.....I.-_....J
o
0.2
04
0.6
¢
0.8
1.0
Fig. 4-8 Nondimensional Stress Resultants for a Hyperboloidal Shell with aft = 0.90 and a/s =
0.55. Source: P. L. Gould and S. L. Lee, "Hyperbolic Cooling Towers under Seismic Load," Journal
of the Structural Division, ASCE, 93, no. ST3 (June 1967): 95.
tabulations of membrane theory stress resultants for hyperboloidal shells of
typical dimensions are provided in several available articles. S )
4.3.2.4 Toroidal Shells. The toroidal shell is quite useful and efficient for pressure vessel applications. Also, segments of toroidal shells are frequently used as
transitions between cylindrical tanks and shallow spherical or flat caps [figure
2-8(y)]. This type of compound shell will be treated in some depth later.
Examining the toroidal geometry, figure 4-9, we see that the surface is
generated by the rotation of a closed curve, usually a circle, about an axis lying
inside the curve.
The definitive geometry is conveniently established from figure 4-9. The
meridian is the circle ABeD, with radius a. If we examine a normal to the surface
defined by a meridional angle <p, we observe that it pierces the surface at two
points; that is, there is not a one-to-one correspondence between curvilinear
coordinate <p and a unique point on the surface. The consequence of this
90
4 Membra ne Theory
axis of rotation
-l
r
A
A
b
Fig.4-9
Toroidal Shell Geometry
anomaly is that the segment of the shell within the semicircle BAD must be
considered separately from that within BCD.
Considering the exterior segment BAD denoted by the superscript e,
RZ = bcscf/J e + a
(4.46)
where b = the constant distance from the center of the generating circle to the
axis of rotation.
The center of curvature of the meridian BAD is on the same side of the
meridian as the center of curvature of RZ signifying a shell of positive Gaussian
curvature, so that
R~=
+a
(4.47)
For the interior segment BCD denoted by the superscript i,
R~ =
b csc f/J i_a
(4.48)
Observe that, for this segment, the center of curvature of the meridian lies on
the opposite side of the meridian from the center of curvature of R~ , indicating
that this segment has negative Gaussian curvature. Thus,
R~ =-a
(4.49)
4.3 Shells of Revolution
91
At this point, it is appropriate to comment further on the choice of the correct
algebraic sign of the principal radii of curvature when the magnitudes are
obtained from a geometric rather than from a mathematical argument, as we
have done here. Specifically, the condition of negative Gaussian curvature
implies only that the product I/R,pRo is negative and does not indicate which
ofthe radii has the negative sign. Also, a shell with positive Gaussian curvature
may seemingly have both principal radii with negative signs. From the fundamental geometrical definition of the shell ofrevolution, figure 2-11, the horizontal radius Ro = Ro sin ¢J is always positive. With sin ¢J remaining positive for
o ~ ¢J ~ n, Ro and hence Ro will always be positive, and thus the signs on
equation (4.47) and (4.49) are necessary to give the correct sign for the Gaussian
curvature. It is easily verified that equations (4.46) and (4.48) satisfy the GaussCodazzi condition. In the toroid, we have encountered a shell which has positive
Gaussian curvature in one region and negative curvature in the other, with
transition points at Band D. At Band D, Ro - 00, indicating a zero Gaussian
curvature condition.
The most common loading case for toroids is a uniform internal pressure p.
Referring to figure 2-11, this loading corresponds to the positive sense oftn for
the positive curvature segment and the negative sense of tn for the negative
curvature segment. Therefore, for segment BAD, qn = q: = + p, and for segment BCD, qn = q! = - p. Further, because of symmetry, we may restrict
our consideration to the quarter-circles AB for e(n/2 ~ ¢Je ~ 0) and BC for
i(n ~ ¢Ji ~ n/2). We select the indefinite integral form, equation (4.8), for 1/1,
since this shell does not have an external boundary where 1/1 or N,p is known
or can be specified a priori. For e, qn = q: = p, Rq, = R~ = + a and Ro = R~ =
b csc ¢Je + a, so that
1/11
=
=
f
pcos¢Je(a)(bcsc¢Je + a) sin ¢Je d¢Je
pa ( a sin ¢Je +
~ sin 2 ¢Je + C1 )
and
= (b
=
,/,
CSC'f'e
C)
pa
(b ·,/,e a. 2,/,e
. 2,/,e
sm'f' + -2 sm 'f' + 1
+ a) stn
'f'
b+ pa. ""e (b + -2a sin¢Je + .C~e)
asm'f'
sm'f'
For a finite meridional stress at ¢Je = 0 or n (points B and D), C 1 = 0, so that
Ne =
,p
pa(2b + a sin ¢Je)
2(b + a sin ¢Je)
=-::--::-------,,------:-':::--'--
(4.50)
92
4 Membrane Theory
To evaluate the circumferential stress resultant
,/.e
N ,pe = (b csc If'
+ a) [ p -
N(l>
we use equation (4.5)
pa(2b + a sin ,pe)]
2a(b + a sin ,pe)
=-=--'c-::-----,-----'---'-
Multiplying the numerator and denominator of the r.h.s. by sin,pe and simplifying, we find
e
Ne =
p
~
Slnlf'e
(
b
. e
+ a sm
(J -
b-
a sin ,pe)
2
pa
2
which, of course, does not vary with ,p.
Using a parallel argument for segment BCD with qn
-a and Re = R~ = bCSC(Ji - a,
N i = pa(2b - a sin ,pi)
,p
2(b - a sin ,pi)
(4.51)
= q! = - p, R,p = R~ =
(4.52)
and
NJ =
pa
2
At the common points Band D, ,pe
(4.53)
= 0 and n, and ,pi = nand 0, respectively, and
(4.54)
Interesting, we see that the circumferential stress resultant Ne throughout the
shell, as well as the meridional stress resultant N,p at the top and bottom circles,
are independent of the relative plan size of the torous, as represented by b, and
are only dependent on the radius of the generating circle a. Also, as the mean
plan radius b becomes large as compared to a, N3 and N~ --+ pa throughout the
shell. These observations have rather simple physical interpretations which give
some insight into the general load resisting characteristics of shells.
First, to investigate the circumferential stress resultant N e, we consider the
half-plan free body as shown in figure 4-10. We have a resultant force of
magnitude p x [Projected area of outer circle - Projected area of inner circle],
acting in the Y direction. From figure 4-9, the difference in the projected
areas is 4ab + 2(tna 2 ) - [4ab - 2(tna 2 )J = 2na 2 , or simply twice the crosssectional area, so that the resultant force is 2npa 2 • This force must be balanced
by the force in the shell wall arising from N e. Since the pressure p is constant throughout the cross section, it is reasonable to regard Ne as constant.
Therefore,
2Ne(2na) = 2pna 2
or
93
4.3 Shells of Revolution
y
inner
circle
a
b
SECTION AA (FIG. 4-9)
Fig.4-10
pa
2
No = -
Toroidal Shell under Uniform Pressure Load
(4.55)
which verifies equations (4.51) and (4.53). In the next section, we further illustrate that equations of overall (as opposed to differential) equilibrium on
strategically chosen sections of shells can frequently lead to simple solutions
for the membrane theory stress resultants.
Next, we seek to interpret the values of Nt/> at the top and bottom circles.
From equations (4.46) and (4.48), the circumferential radius Ro -+ 00 at Band
D. Considering the so-called third equation of membrane equilibrium, equation
(4.3c),
Nt/>
No
R:+R:=qn
0
(4.3c)
t/>
we have
or
(4.56)
which verifies equation (4.54). Once No has been derived from the consideration
of overall equilibrium [Le., figure 4-10 and equation (4.55)J, Nt/> may be computed at any point on the meridian using equation (4.3c).
94
4 Membrane Theory
In general, we make frequent recourse to equation (4.3c) to interpret the
results of mathematical analyses from a physical standpoint. This equation,
which describes the resistance of the shell to transverse loading, is perhaps the
most incisive single equation in the theory of shells. Moreover, it clearly
illustrates the basic difference between plate and shell action in terms of the
influence of the initial curvature of the surface.
4.3.3 Alternate Formulation for Axisymmetric Loading
4.3.3.1 Overall Equilibrium. In the previous section, the circumferential stress
computation for a symmetrically loaded toroidal shell was verified using an
equation expressing the overall equilibrium across an internal section. This
procedure is quite efficient in a variety of cases. Consider the general axisymmetrically loaded shell of revolution shown in figure 4-11 and pass a reference
section normal to the axis of rotation. The trace of this section on the shell is
the horizontal circle defined by the meridional angle t/J.
To maintain equilibrium in the Z direction, the axial component of the
meridional stress resultant N", multiplied by the circumference of the section
must balance the resultant axial load above the section, Q(t/J), taken positive in
the negative Z direction to ensure that a compressive force produces a negative
meridional stress.
The axial component of N",(t/J) is given by N",(t/J) sin t/J so that
(4.57)
axis of rotation
Fig. 4-11 Symmetrically Loaded Shell of Revolution
95
4.3 Shells of Revolution
In terms of the magnitude Q(rP), we find
(4.58)
Then, No(rP) is found from equation (4.3c).
To compute the resultant vertical load Q(rP) when the shell is subject to a
distributed surface loading, we examine a differential ring element above the
section. This ring element is defined by the auxiliary meridional angle 1] and
has a meridional length equal to Rt/J(1])d1]. The distributed surface loading,
represented by qn and qt/J' contributes axial components + qn(1]) cos 1] and
- qt/J(1]) sin 1] per unit area of middle surface, respectively, as shown in figure
4-11. Then
Q(rP) = ft/J [qn(1]) cos 1] - qt/J(1]) sin 1]J [2nR o(1])J [Rt/J(1]) d1]J
t/J,
(4.59)
which may be directly substituted into equation (4.58) to evaluate Nt/J(rP).
As an elementary example, we will again solve the spherical dome under
self-weight load, previously considered in section 4.3.2.2 and illustrated in figure
4-2. From figure 4-2, we have qn = -qcos1], qt/J = qsin1], Ro = a sin 1], and
Rt/J = a, so that
= ft/J (- q COS 2 1] - q sin21])(2na sin 1])a d1]
Q(rP)
t/J,
= 2nqa 2 (cos rP - cos rPt)
(4.60a)
From equation (4.58),
N =
Q(rP)
t/J
2nasin 2 rP
qa
= ~(cos rP
sm
'P
(4.60b)
- cos rPt)
which is identical to equation (4.18).
We may also consider a closed rotational shell of arbitrary shape under
uniform normal pressure p. Generalizing equation (4.60b) gives
pn[R o(rP)J 2
1
Nt/J(rP) = 2nR o(rP) sin rP = 2. pRo
whereupon
No(rP)
=
pRo
[1 -
Ro ]
2Rt/J
(4.61a)
(4.61b)
from equation (4.3c). These simple equations hold for any shell of revolution.
96
4 Membrane Theory
(0 )
I' I
(b)
Fig.4-12 Symmetrically Loaded Cylindrical Shell
4.3.3.2 Circular Cylindrical Shells. A similar approach may be used for a circular cylindrical shell subject to an internal pressure p(Z). Consider a unit length
axial slice of such a cylinder with radius a as shown in figure 4-12(a). In this
case, the magnitude of the total resultant force along the e = 0 (vertical) axis is
Q(e)
=2
f:
p(Z)cos1J(l)(ad1J)
= 2p(Z)a sin e
(4.62a)
The corresponding component of the circumferential force, which must balance
Q(O), is
(4.62b)
Equating forces, we have
2N9 (O) sin 0 = 2p(Z)a sin 0
or
N9 = p(Z)a
(4.62c)
97
4.3 Shells of Revolution
For a uniform pressure distribution, p(Z)
= p
and
(4.62d)
N(J=pa
which is, of course, well known from elementary strength of materials. Obviously, equations (4.62c and d) follow directly from equation (4.3c), since R¢ =
00. This solution also holds for incomplete or open cylindrical shells, bounded
at a maximum 8 = ± 8". This type of shell, illustrated in figure 2-8(h), is
examined in detail in chapter 7.
We now turn to the computation of the meridional stress resultant. Here, rP
is not suitable for the meridional coordinate, since rP = nl2 all along the axis.
Instead, the axial coordinate Z is appropriate. We return to equations (4.2) with
0( = Z, f3 = 8, A = 1, and B = a. For axisymmetricalloading, equation (4.2b)
vanishes and equation (4.2c) confirms the result given in equation (4.62c). This
leaves equation (4.2a), which becomes
1
-(aNz
).z
a
+ qz =
°
or
N z .z
= -qz
so that
Nz
=
-
f
(4.63a)
qzdZ
We may also write the integral in the alternate form introduced in section
4.3.2.1.
Nz(Z)
= Nz(O) -
LZ qz dZ
(4.63b)
Equations (4.63a) and (4.63b) indicate that for a symmetrically loaded cylindrical shell, the meridional stress resultant is uncoupled from both the normal
loading and the circumferential stress resultant and is a function only of the
axial loading and the boundary conditions. For example, if the shell shown in
figure 4-12(a) is subjected to a uniform axial edge load N per unit length of
circumference, as shown in figure 4-12(b), and the pressure qz = 0, then equation
(4.63b) gives
Nz(Z)
= N = constant
(4.64)
so that the effect of the edge load penetrates along the axis of the cylinder and
must be developed on the opposite boundary. This unattenuated propagation
of edge loads is quite plausible if one recalls that the cylindrical shells have
meridians composed of straight lines parallel to such axial loads. A similar
situation arises in the membrane behavior of shells of negative Gaussian curva-
4 Mem brane Theory
98
I~
)\
Vi
lt
/
/
\
OX;'
of
rotation
\
\
(
.
~
-
te
restrain~
no
In
this direction
cone
s
(a)
=rqtl2
Reference
Section
( b)
Fig. 4- 13
Conical Shell
99
4.3 Shells of Revolution
ture, where tangential edge loads tend to propagate along the straight lines
on the surface. 9 This is illustrated in section 4.4.3.5.
It is interesting to compare the solutions for the toroidal shell and the
cylindrical shell under uniform normal pressure loading. For this purpose, the
coordinates <p and () on the toroid should be regarded as equivalent to () and
Z, respectively, in the cylinder. Now, if b » a, we find [from equation (4.50) or
(4.52) for the toroid and equation (4.62d) for the cylinder] that (Nqj)toroid =
(NO)Cylinder = pa. Furthermore, if the cylindrical shell is closed by a flat circular
plate, N z = P x (Area/Circumference) = pa/2, which would match No as computed for the toroid in equations (4.51) and (4.53). Thus, the membrane resistance mechanism of the two geometries is remarkably similar, with the torus
representing, in effect, a self-closing cylinder.
4.3.3.3 Conical Shells. We now investigate conical shells and initially examine
the frustum, as shown in figure 4-13(a). The shell is conveniently described in
terms of the axial coordinate Z, since the meridional angle <p = n/2 - rx =
constant. At any section,
Ro =
t
+ Ztanrx
(4.65)
and
Ro
t
+ Ztanrx
(4.66)
R -------o - sin <p cos 0(
Since the meridian is straight, R qj =
obtain
t + Ztanrx
No=qn---cosrx
00,
and from equation (4.3c) we immediately
(4.67)
Then, we concentrate on the meridional stress resultant N z .
Assume that the shell is subjected to an internal suction q, and that the top
is closed by a flat plate which is free to move radially and to rotate freely at the
junction with the conical shell. The idealized junction detail is shown in
the inset. Under these conditions, the plate transmits a total force of only
qnt 2 in the + Z direction to the shell. Therefore, the resultant axial force at the
top is of magnitude
Ql
=
-qnt 2
(4.68)
This force is distributed over the circumference of the top circle, 2nt, providing
an edge load of intensity qt/2 on the shell, as shown in figure 4-13(b).
We now want to derive the resultant force Q2(Z) due to the uniform suction.
We define an auxiliary axial coordinate 11 and resolve the load accordingly.
Then, we have
100
4 Membrane Theory
Q2(Z)
=
rZ [-qsin1X][2n(t + lltan1X)]~
COS 1X
J
0
= -2nq tan 1X
=
f:
-2nqZ tan 1X(t
(t
+ lltan1X)dll
(4.69)
+ ~tan1X)
Since
.
Nz(Z) sm rP
Q(Z)
= 2nRo(Z)
(4.70a)
and
(4.70b)
then,
Nz(Z)
=
Ql + Q2(Z)
2ncos1X(t + Ztan1X)
_q{t 2
+ 2Ztan1X[t + (Z/2) tan 1X]}
2cosa(t + Ztan1X)
(4.71a)
From equation (4.67), with qn = -q,
t+Ztana
() = - q
NeZ
---COS1X
(4.71 b)
We note in passing that if a is set equal to 0, we have a cylinder with radius t.
Equations (4.71) (a) and (b) would reduce to
Nz =
qt
-2
(4.72a)
and
(4.72b)
Since the N z comes entirely from the top plate reaction in the axial direction,
these equations agree with the solution for cylindrical shells found in the
previous section.
We may now investigate a complete cone by letting t - 0, whereby
q tan 1X
Nz = - - Z - 2 cos 1X
and
(4.73a)
101
4.3 Shells of Revolution
tan ex
cos ex
(4.73b)
NIJ= - q Z - -
The complete cone is an illustration of a pointed closed shell, shown in figure
4-1 (b). In contrast to the dome, which was examined in detail in sections 4.3.2.1
and 4.3.2.2, N z and NIJ ~ 0 at the pole. Also, recall that for a closed spherical
shell under uniform pressure, we found N,p = NIJ throughout; whereas, for the
closed conical shell, the circumferential stress is double the meridional stress.
4.3.4 Compound Axisymmetrically Loaded Shells
4.3.4.1 Transition Segments. We now examine shells which have a meridian
defined by more than one geometric curve; such combinations may be called
compound shells. A common usage of this form is for pressure vessels, in which
a basic cylindrical tube may be capped at each end by a so-called head and
bottom, which are often shallow spherical shells. Examples of such shells are
shown in figures 2-8(r, s, and u), and an idealized case is depicted in figure 4-14.
One's first inclination might be to make the ends hemispherical in order to
provide a smooth transition; however, such a shell is nondevelopable and is not
readily formed by bending flat sheet. Rather, it is considerably easier to produce
a realtively shallow spherical cap by rolling and dishing operations guided by
suitable templates.
Turning to the shell shown in figure 4-14 under a positive internal pressure
p, we find from equations (4.27) and (4.28) that for the spherical segments
(4.74a)
For the cylindrical shell, from equation (4.62c),
-+--....1.....------0_ Z
~%
cylinder
~pa';~inf' ~_
(po 12)cos-..J
'VfI
Fig.4-14
pa,/2
V--Z-sphericol head
~o, 12)cos¢,
r
Cylindrical Shell with a Spherical Head
102
4 Membrane Theory
(4.74b)
As far as the meridional stress resultant in the cylindrical shell is concerned, we
have shown in equation (4.64) that Nz = a constant dependent on the boundary
value and the axial loading, if present. As shown in the inset of figure 4-14,
the meridional stress in the sphere imparts an edge force per unit length of
(pad2) sin <Pl to the cylinder so that
Nz(Z)
=
(P;l) sin <Pl
(4.75)
We attempt to ascertain the admissibility of the membrane theory solution
for the compound shell. First, note that the sudden transition between the
cylindrical and spherical shell is a violation of the geometrical guideline for the
existence of a membrane state of stress, as discussed in section 4.2. It is rather
obvious that the radial component of N",(<Pl)' (pad2) cos ,pl' cannot be balanced
by an extensional force in the cylinder, but will introduce transverse shears and,
subsequently, bending which will affect both shells. Another result apparent
from equations (4.74) is that the circumferential or hoop stress resultant N9 will
likely be different for both shells at the junction. This leads to discontinuities
in the circumferential strain and the radial deformation, which we cannot
quantify at this stage of the development but which serve as further evidence
of the inadequacy of the membrane theory in the junction region. Note that
even if a complete hemisphere is used for a cap, with a l = a2 = a,
N() (hemisphere)
N() (cylinder)
=
P;
(4.76a)
=
pa
(4.76b)
so that the circumferential strain discrepancy will still remain, even though the
unbalanced radial force (paj2) COS,pl vanishes.
An attractive method of balancing the radial component of the thrust in the
cap is to provide a ring stiffener or collar at the junction. An example is provided
by figure 2-8(r), and an idealization is shown in figure 4-15. The ring tension T
T
w
-
-
-
-----l~-
Fig.4-15
Ring Stiffener at a Shell Junction
T
103
4.3 Shells of Revolution
_ ':::J
02
1
--------------:r-I-;n-d-er--I~~ ~r~
(b)
Fig.4-16
~ t~rojd
Compound Shells
in the stiffener can balance the radial thrust in the same manner as a cylindrical
shell resists normal pressure, so that 2T = (pa 1 /2) cos ¢(2a2), with the assumption that w « a 2 • The ring, however, will restrain the radial expansion of the
cylinder, and thus violate the ideal membrane boundary requirements.
We consider two additional possibilities of providing a smooth transition.
In figure 4-16(a), we show an ellipsoidal cap; in figure 4-16(b), we insert a
segment of a toroidal shell between the cylinder and the spherical cap. The latter
form is sometimes called a torospherical head.
Considering the ellipsoidal head first, we note that a smooth transition is
provided between the cylinder and the cap along the meridian, so that transverse shear in the cylinder is not required for radial equilibrium. However, the
hoop stresses computed for an ellipsoid again do not match those for a cylinder
as calculated from equation (4.76b)10; thus, an incompatibility in the circumferential strain and subsequent radial deformation occurs, which, as before,
must be corrected by transverse shears on the cylinder and on the ellipsoid.
Also, the problem of forming a deep curved cap remains.
The torospherical head shown in figure 4-16(b) circumvents the deep cap
104
4 Membrane Theory
problem but it turns out that the hoop stresses and the corresponding radial
deformations, as computed from membrane theory, result in incompatibilities
at both the cylindrical-toroidal and toroidal-spherical junctions. The torospherical shell will be treated in detail in the following section.
Meanwhile, we may illustrate an interesting property of compound pressure
vessels by considering the normal equilibrium equation
N,p
No
(4.3c)
R+R=qn
,p
0
which was earlier termed the single most incisive equation in thin shell theory.
We first apply the overall equilibrium approach developed in section 4.3.3.1 to
the shells shown in figure 4-16, and particularly consider equation (4.61a). At
any section defined by the coordinate rjJ, Nq,(rjJ) cannot change abruptly as long
as Ro does not change. With this in mind, we rewrite equation (4.3c) with qn = P
as
No = Ro(P -
~;)
With the foregoing arguments in mind, at a smooth transition
No
=[ -R:,p + pJ
(4.77)
Thus, even if the transition is smooth along the meridian and the loading is
also smooth, a sudden change in Rq, can radically change the magnitude and
even the algebraic sign of No. Recall that the cylindrical shell in figure 4-16
resists the pressure p by No/Ro, since R,p is 00. This occurs regardless of the
magnitude of N,p. Suddenly, at the transition, we encounter a finite value of Rq"
R,pe for the ellipsoid and a3 for the toroid. Since N,p is positive, equation (4.77)
reveals that No decreases in proportion to the now finite value of Rq,' even
perhaps becoming negative. Both of these cases may produce a large negative
value of the hoop stress resultant N O, l l and such shells may fail because of
circumferential buckling if proper strengthening and stiffening is not provided.
An actual case where this occurred is described in Fino and Schneider. 12 The
shell segment shown in figure 2-8(y) was fabricated to quantify this failure
mechanism further through an experimental study.
4.3.4.2 Torospherical Head. In the preceding section, the possibility of inserting a toroidal segment between a spherical cap and a cylindrical shell was
discussed. The geometry of this shell is shown in more detail in figure 4-17.
In terms of the radii of curvature of the sphere, a 1, the toroid, a3' and the
cylinder, a 2 , the meridional angle at the sphere-toroid junction is given
by
(4.78)
105
4.3 Shells of Revolution
sin 1/>1
Cyllndrlco.l
CROSS SECT,
Fig.4-17
Torospherical Shell Geometry
In practice, the radius of the toroid is selected so as to provide smooth
transitions at both ends.
To derive the membrane theory stress resultants, we again use the overall
equilibrium method. At a general section within the toroid, rPI < rP < n12, the
resultant axial load consists of two parts: QI = the load on the spherical cap,
and Q2(tP) = the load between rPI and rP on the toroid. For a uniform positive
pressure p, the magnitude of the first force is the pressure multiplied by the
projected base area of the cap,
QI = pn(a l sin tPd 2
(4.79a)
For the second force, we first compute the horizontal radius of the toroid
(4.79b)
whereupon the magnitude equals the pressure multiplied by the area of the
projected annulus,
Q2(tP)
= pn[R6 - (a l sinrP)2]
= pn[(a l - a3)2 sin 2rPI + a~ sin 2rP
+ 2(al - a3)a3sinrPIsinrP - arsin2 tPI]
106
4 Membrane Theory
which reduces to
Q2(</J) = pna 3 (sin</J - sin</JI) [a 3 (sin</J - sin</JI)
+ 2a I
sin</JIJ
(4.79c)
Summing equations (4.79a) and (4.79c), we have
Q(</J)
= pnQ(</J)
(4.80a)
= (a l sin </JI)2 + a 3 (sin</J - sin</JI)
. [a 3 (sin</J - sin</JI) + 2a I sin</JIJ
(4.80b)
where
Q(</J)
From equation (4.58),
1
Nt/>
= 2nRo sin </J Q(</J)
(4.81a)
pQ(</J)
To compute the circumferential stress resultant, we use equation (4.3c), which
gives
(4.81b)
From equation (4.79b) since Ro = Rosin</J,
(4.82a)
From figure 4-17,
(4.82b)
Substituting equations (4.81a) and (4.82a and b) into equation (4.81b) gives
_ [(a l
No-p
.{1-
-
a 3 ) sin ifJI
. A.
sm",
+ a3 sin ifJ]
2a 3 sin </J[(a l
(4.83)
Q(r/J)
-
a 3 ) sin </JI
+ a3 sin r/JJ
}
Equations (4.81a) and (4.83) constitute the membrane theory solution for the
so-called toroidal knuckle portion of the torospherical head. As implied in the
previous section, the form of equations (4.81b) and (4.83) suggests that No
107
4.3 Shells of Revolution
PLAN VIEW
HEAD NO.1 h=O.20·
HEAD NO.2 h .. O.25Dollar
It.
(2 pes.)
-
.....-h
2ilAround
SECT.
Fig.4-18(a)
Torospherical Test Specimen
will probably be negative, whereas the circumferential stress in both adjacent
segments will be tensile.
The dimensions of the test specimen shown in figure 2-8(y) are given in figure
4-18(a). This shell has been subjected to extensive analytical and experimental
108
4 Membrane Theory
~
Spherical
~
Toroicial
\
\
32.74'
\
~
~ Cy""dCle.'
\
.___ ....___.. ____ .___ ._ ..... _____ ....J
\\
CROSS SECT.
\
Fig.4-18(b) Test Specimen Middle Surface Geometry
investigation and will be used here to compute the linear stress pattern due to
internal pressure.
From figure 4-18(a), allowing for the wall thickness h = 0.2 in., the geometrical properties are
a 1 = 172.9 in.
a 2 = 96.1 in.
a 3 = 32.74 in.
as shown in figure 4-18(b), so that from equation (4.78)
sin rPl
96.1 - 32.74
= 179.9 _ 32.74
= 0.45205
and
rPl
=
26.88°
The equation of the meridian was established with the origin set at the pole,
and the resulting equations are as follows:
Spherical cap:
Z2
+ R2
- 345.8Z
=0
(4.84a)
109
4.3 Shells of Revolution
Toroidal knuckle:
Z2
+ R2
- 95.7574Z - 126.72R
+ 5234.9519 = 0
(4.84b)
Cylindrical segment:
R - 96.1 = 0
(4.84c)
Pertinent material properties are Young's modulus E = 30,000 psi and Poisson's ratio J.l = 0.3. Equations (4.84) may be used to verify the continuity of
slopes at the junctions.
In figure 4-19, the circumferential stress resultant N(J is shown as a function
of the arc length siP' The change from tension in the spherical cap to compression
in the torus and again to tension in the cylinder is clearly illustrated. Along
with N(J, a scale for the extensional component of the circumferential stress,
G(J(J = N(J/h, is also given to facilitate comparisons with bending theory solutions,
which are somewhat thickness-dependent. The abrupt jumps are mitigated by
bending effects, which will be discussed in chapter 9.
4.3.5 Nonsymmetric Loading
4.3.5.1 Transformation of Equilibrium Equations. Thus far, we have considered
the membrane theory analysis of shells of revolution subject to loading that is
nonvarying with respect to the circumferential coordinate e, so-called symmetric loading. The solutions obtained in the previous section are based principally on equation (4.7a), which is written in terms of the auxiliary variable t/I.
This equation was obtained by suppressing the e dependence. We now return
to equations (4.6) and eliminate by taking alat/J of equation (4.6a); alae of
equation (4.6b); then taking the difference of the two equations; and finally by
dividing both sides by the leading term RiPR(J sin t/J, which gives!3
e
1
RiPR(J sin t/J
(R~ sin t/J t/I )
RiP
.iP ,iP
1
+ RiP sin 2 t/J
t/I
,(J(J
1 . t/J [(qn cos t/J - qiP sin t/J)Ri sin 2 t/J] iP
RiPR(Jsm
'
(4.85)
+ R(J(qn,(J(J + q(J,(J sin t/J)
Equation (4.85) is a second order partial differential equation in t/I. Once t/I is
determined, may be found by solving equation (4.6a) or (4.6b).
e
4.3.5.2 Fourier Series Representation. Since we have a partial differential equa-
tion to solve, we try the standard method of separation of variables. Specifically,
we apply the Fourier series technique, whereby all loadings and dependent
variables are taken in the form of Fourier series:
110
4 Membrane Theory
200
~
150 30
Membrane
Theory
100 20
50 10
Arc Length from 'Pole
1/1
Q.
50
80
70
60
90
-50 -10
/
-100 -20
-150 -30
~OO
/
V
100
110
120
130
1~
~
V
-40
SPHERE
Fig.4-19
~I~
TORUS
-I ..
CYLINDER--
Membrane Theory Circumferential Stress in Torospherical Shell
{q~COSj(}}
{q;}
qs =.fa q~ sin j .(}
00
{q} =
J
Ns
= ~o
{N;}
S
(4.86a)
q~ COSJ(}
qn
00
J-
{N~COSj(}}
NJcosj(}
SJ sinj(}
(4.86b)
111
4.3 Shells of Revolution
{~} = j~O {~~~:f:}
(4.86c)
Substituting equations (4.86) into equations (4.6a), (4.6b), and (4.85), we get
~
L...
j=O
{[Ri
sin ¢ .
---'----t/l/,p
R,p
'J
+ j~J cosj()
(4.87a)
= -
[(q~ sin ¢ - jqDR,pR~ sin ¢J Sinj ()}
and
~
L...
j=O
{[
(4.87b)
'J .()
1
(R~
sin tP . ) j2
./,
---t/lJ,p
'I'J COS]
R,pRo sin ¢
R,p
'.,p
R,p sin 2 ¢
= [
-
1.
A.
R,pRoSlD'I'
[(q~ cos tP - q~ sin ¢)RJ sin 2 ¢l,p
jRo(jq~ - q~ sin ¢)
(4.87c)
J
COSj ()}
We may treat each equation of the series separately, and then obtain the total
solution by summing to a suitable truncation limit onj. We recognize that the
individual equations for each harmonic are ordinary, rather than partial, differential equations, since the ()-dependent terms may be cancelled. This is, of
course, a formidable mathematical simplification. Also, we see that equations
(4.87a) and (4. 87b), withj = 0, reduce to equations (4.7a) and (4.7b), respectively,
which have previously been investigated with respect to axisymmetric loading.
Thus, the previous solution for axisymmetric loading, as given by equation (4.8)
and (4.9), also serves as the solution of the j = 0 harmonic for the non symmetric
loading situation, with the loading taken as q~ and q~, the j = 0 components
of the surface loading. These solutions do not include the case of a j = 0
component of qo which, as mentioned in section 4.3.2.1, is quite rare. This case
can be accommodated, however, by simply interchanging the sinj() and cosj()
terms in equations (4.86a-c), obtaining an analogous set of equations to equations (4.87a-c), and combining the results of the two solutions.
Since the j = 0 case has already been solved, we now seek a solution to
equation (4.87c) for a general harmonicj > O. After cancelling the cosj() terms,
we have for harmonic j
112
4 Membrane Theor:
1
RI/>RIJ sin,p
(R~ sin,p IjIi) _
RI/>
,I/> ,I/>
j2
RI/> sin 2 ,p
1
IjIi
.
.
3
. A.[{q~cos,p-q~sin,p)RlJsin2,pJ,1/>
RI/>RIJSIll'f'
(4.88)
- jRIJ{jq~ - qJ sin ,p)
We postpone consideration of the general solution of equation (4.88) until
further specializations are introduced. For now, note that once IjIi has been
determined, we may solve for ~i from either equation (4.87a) or (4.87b). Since
~i occurs as an undifferented term in equation (4.87a), this equation is commonly selected. For each harmonic, the stress resultants at any circumferential
angle may then be computed by evaluating N~ and Si, using equations (4.4),
and Nj, using equation (4.3c), and then substituting the appropriate value of e
into equation (4.86b). In passing, note that the "price paid" for the mathematical
simplification of reducing the partial differential equation to an uncoupled
series of ordinary differential equations is reflected in the total number of
harmonic solutions that must be summed to adequately represent the effect of
the particular surface loading in a given case. Each harmonic solution entails
a complete analysis of the shell, whereas the partial differential equation, (4.85),
need be solved only once. In spite of this, the Fourier series approach has been
widely used to solve shells of revolution subject to unsymmetric loading,
because it is fundamentally simpler from a mathematical viewpoint.
The eventual solution of equation (4.88) is strongly dependent on the harmonic number j. The case j = 1 is called antisymmetric loading and is somewhat
more tractable than the case j > 1, which will be called asymmetric loading.
4.3.6 Antisymmetrical Loading
4.3.6.1 Integration of Equilibrium Equations and Boundary Requirements. Consider equation (4.88) withj = 1 and introduce a further auxiliary variable 13
(4.89a)
which gives
[
1.,p 1,I/>J
RI/> sm
,I/>
=
1.,p
RIJ SIll
[(q~ cos,p -
- RIJRI/>{q~ -
qJ sin ,p)Ri sin 2 ,pJ,1/>
qJ sin,p)
(4.89b)
The transformation between equations (4.88) and (4.89b) is most easily verified
by substituting equation (4.89a) into equation (4.89b) and performing the
indicated differentiation with the aid of the Gauss-Codazzi relation, equation
(2.47).
Equation (4.89b) is in a form which permits 1 to be evaluated by successive
113
4.3 Shells of Revolution
integrations with respect to r/J. Then, 1/1 1 is found from equation (4.89a), and the
remaining auxiliary variables and stress resultants are determined from equations (4.87a), (4.86a), (4.4a), (4.4b), and (4.5). This procedure is straightforward
and, since it is discussed in detail in Novozhilov,14 it is not repeated here.
Rather, we present the results which may be easily verified as a solution by
direct substitution into the membrane theory equilibrium equations, (4.3a-c).
The stress resultants corresponding to j = 1 are
NtP = NJ cos ();
N(J = Nl cos ();
S = Sl sin ()
(4.90a, b, c)
where the Fourier coefficients are given by
NJ =
N.(J1
=
2 1. rP {C 2
RosIn
R(J
+ C1 f
tP
RtP sin r/J dr/J
tP'
+f
tP
<P(rP)R tP sin rP drP}
tP'
1
q1R
NtP
n
(J - -Rq,
sl = NJ cos rP -
C1
Ro
(4.90d)
(4.90e)
+ X(rP)
(4.90f)
and the functions <P(rP) and X(rP) are
<P(rP) = (q~ cos rP -
qJ sin r/J)RoRe
- ftP (q; sin rP
+ q~ cos rP - qJ )Rq,Ro drP
X(rP) = - 1 ftP (q~ sin rP
Ro tP'
+ qJ cos rP - qJ )RtPRo drP
q,'
(4.91a)
and
(4.91b)
Here, we assume that both NtP and S are specified on the same boundary,
rP = rP'· C 1 and C2 are integration constants evaluated from equations (4.90d)
and (4.90f) as
C2
= NJ (rP')R'5(rP') sin rP'
(4.92a)
and
(4.92b)
so that a stress-free top edge gives C1 = C2 = O. Also, for a dome we take rP' = O.
Then, Ro = 0 and C1 = C2 = O. The more general case, where NtP and S are
specified at different boundaries, may be treated in an analogous manner.
To evaluate the stress resultants at the pole of a dome for j = 1, we first
consider equation (4.90d), with rP' = 0 and C1 = C2 = 0, and recognize that we
have a % indeterminate form. We apply L'Hospital's rule by differentiating
the numerator and denominator of equation (4.90d). Noting that
114
4 Membrane Theory
(R~ sin ,p),q, = (sin ,p)2R oRq, cos,p
= Ro cos ,p(2Rq, sin,p
+ R~ cos,p
(4.93)
+ Ro)
we get
· N1q,Y'=1m
("')
l'
11m
q,~o
q,~o
<DRq, sin,p
.
Ro cos ,p(2Rq, sm,p
(4.94)
+ Ro)
which remains an indeterminate form. Therefore, we must apply L'Hospital's
rule once more to equation (4.94). Considering the numerator, after cancelling
sin ,p with Ro = Ro sin ,p,
(<DRq,),q,
= Rq,<D,q, + <DRq"q,
Since <D(O) = 0, the second term vanishes. Evaluating <D,q,(O), we find that
this term also vanishes provided q~ (0) = 0, which is a reasonable smoothness
requirement as will be apparent later. Thus,
lim (<DRq, sin ,p),q, = 0
q,~o
+0
(4.95)
= 0
Differentiating the denominator gives
2(Rosin,pRq,cos,p),q,
+ (RoRocos,p),q,
(4.96)
At,p = 0, the second term in equation (4.96) becomes Ro(O)Rq,(O). Considering
the first term, we have
2(Rosin,pRq,cos,p),q,
= 2[Rq,cos,p(Rq,cos,p) +
Rosin,p(Rq,cos,p),q,]
(4.97)
As ,p -+ 0, the second term of equation (4.97) also goes to 0, and the first term
becomes 2R~(0). Thus, we finally conclude that
lim NJ(,p) =
0
= 0
q,-o
Rq,(0)[2Rq,(0) + Ro(O)]
(4.98a)
NJ(O) = 0
(4.98b)
or
We may then establish from equation (4.90f) that
Sl(O) = 0
(4.99)
since C 1 = 0 and X(O) = 0 by L'Hospital's rule applied to equation (4.91b).
Examining equation (4.90e) for Ni as,p -+ 0, the second term goes to 0, and
the first term remains. Thus, we find
(4.100)
In section 4.3.2.1, we proved that Nq,(O) = N(J(O) for a dome. Equation (4.98b)
indicates that NJ(O) = 0 which, in view of equation (4.100), implies that q~(O)
4.3 Shells of Revolution
115
must be equal to 0 for a properly defined anti symmetric loading. An example
is presented in the following section.
Finally, with respect to the general analysis of anti symmetrically loaded shells
of revolution, note that we have evaluated the constants of integration through
the consideration of (a) specified values of NJ and Sl at given boundaries for
open shells; or, (b) pole conditions for domes. It is required that the remaining
boundary develops the calculated values of Nq, and S to ensure membrane
action. As illustrated in figure 4-20(a), the displacements corresponding to these
stress resultants, in the meridional direction for Nq, and in the tangential
direction for S, must be O.
Such a set of boundary conditions (meridional and tangential displacements
equal to 0 on one boundary) are apparently sufficient to ensure membrane
behavior, provided that the corresponding geometrical and loading restrictions
are observed. The question arises as to the possibility of specifying an alternate
set of conditions for an open shell, e.g., meridional displacement equal to 0 on
one boundary, and circumferential displacement equal to 0 on the other, as
shown in figure 4-20(b). The study of this possibility requires the consideration
of pure bending deformations, which are deformations which produce neither
extension nor in-plane shearing of the middle surface.
Although a formal study of pure bending deformations is beyond the scope
of this text, one can show that shells for which this behavior is possible are
inherently unstable. The requisite circumstances are not primarily a function
of the applied loading, but are related to the geometrical properties and the
boundary conditions. Mathematically, one can show that such deformations
constitute a nontrivial solution to the equations describing the middle surface
displacement of the shell when the loading terms qq,' qo, and qn are equal to
0, e.g., a nontrivial homogeneous solution to the equations for the displacements of the middle surface. A closely related configuration, inextensional
deformation, originally conceived by Lord Rayleigh in his study of the vibrations of cylindrical shells,15 has been used to facilitate instability analysis of
shells.
Obviously, the possibility of pure bending deformations in actual structures
is to be avoided. One can show that a support for which both the meridional
and circumferential displacements are constrained on one boundary, figure
4-20(a), is sufficient to prevent pure bending deformations for all shells of
revolution and is also necessary to remove this possibility for shells with
negative Gaussian curvature. 16 In other words, there are some geometries for
which a shell with negative Gaussian curvature might exhibit pure bending
deformations if both the meridional and tangential displacements are not
prevented at one boundary, as in figure 4-20(b).
An interesting parable based on an awareness of the possibility of these
deformations' drastically reducing the stable configuration of lattice structures
is found in the treatise of V. Z. Vlasov.17 He wrote
116
4 Membrane Theory
no constraints
~------~------~
Developed by
Tangential Constraint
meridional and
tangential /
constraints
Developed by
' \ Meridional Constraint
-, S
(a)
~__
--- ---
s
tangential
constraint on
__
m_e_"_a_ion
__
a_'__l~__~__r-__~~,,
constraint only/
~
\N '\
Developed by
Tangential Constraint
Developed by
Meridional Constraint
¢
(b)
Fig. 4-20
Boundary Requirements for Shells of Negative Gaussian Curvature
117
4.3 Shells of Revolution
Fig. 4-21
Lattice Structure Attributed to Engineer Shukov
The lattice designs of Engineer Shukov which are well known in construction
practice, outlined by the surface of a hyperboloid of one sheet and consisting of
rods arranged in straight lines, forming and tied together in horizontal rings, may
constitute instantaneously varying unstable systems under certain combinations of
geometric si2es and types of asymmetrical loads (wind loads, for example). Unless
such structures are properly reinforced by additional structural elements (for example, tension members located in the planes of the rings), they possess very little
load-bearing capacity."
A diagram of a type of structure designed by Shukov, although not with
negative curvature in this case, is shown in figure 4_21. 18
4.3.6.2 Spherical Shell under Antisymmetrical Wind Loading. It is common
practice to represent dynamic loadings such as wind and earthquake by statically
equivalent or pseudostatic loading. Although this procedure is not universally
a
Reprinted with permission of National Aeronautics and Space Administration.
118
4 Membrane Theory
applicable and can often lead to erroneous conclusions, it is expedient. If due
care is exercised, it can provide an approximation that is adequate for design,
particularly when the pseudo static loading is developed from consideration of
the dynamic forces and the response characteristics of the structure.
With this preface, we show a static wind loading on a spherical dome in figure
4_22.19 The wind pressure is considered to be normal to the surface, and the
x
x
...
wind
f--_ Y
p
Fig. 4-22
Static Design Wind Load on a Spherical Shell
119
4.3 Shells of Revolution
components are
=0
qIJ = 0
qn = - p sin <6 cos e
(4.101a)
qt/>
(4.101b)
(4.101c)
where p = the static wind pressure intensity.
Therefore, from equation (4.86a),
q~
= -psin<6
(4.102)
At this point, a word of warning is in order. It is fairly obvious that the
pseudo static wind loading given on figure 4-22 and in equation (4.101c) has
been conceived as a reasonable facsimile of a possible pressure distribution to
fit the antisymmetricalload form, j = 1, and is not necessarily based on measured dynamic wind pressures. Extreme caution should be exercised in using
this loading for actual design. In section 4.3.7.5, a spherical shell is analyzed for
an experimentally determined pressure distribution and quite different results
are obtained. Nevertheless, the antisymmetrical pressure case is interesting.
Proceeding with the results for the spherical dome, we have Rt/> = RIJ = a;
C 1 = C 2 = 0; <6' = 0; and q~ = qJ = 0, q" = - p sin <6 substituted into equations
(4.90) and (4.91). After integration, the solution iS 19
NJ = Nl =
Sl
cos <6
3 <6 (2
sm
pa 3 .
-(pa sin <6
+ cos <6)(1
- cos <6)2
+ NJ)
= _ pa (2 + cos <6)(1 - cos <6)2
3 sin 3 <6
(4.103a)
(4.103b)
(4.103c)
This solution is readily verified by direct substitution in the equilibrium
equations, (4.3a-c).
Equations (4.102) and (4.103) are graphed in non dimensional form on figure
4-23 for a hemisphere. As discussed earlier with respect to figure 4-3, the graph
holds as well for spherical domes with <6b < n/2 by considering the portion
above that ordinate. Two points of special interest on this graph are (a) Sl is
maximum at <6 = n/2; and, (b) NJ = 0 at <6 = n/2. To provide a physical explanation of these results, we again find the overall equilibrium approach to be
helpful. The appropriate equilibrium equations, written with respect to the
X - Y coodinates shown in figure 4-24, are !:Mx = 0 and l:Fy = O. Here, the X
axis is drawn through the center of curvature and Y is parallel to the assumed direction of the wind. We show the appropriate free-body diagrams
in figure 4-24, where the positive signs of Nt/> and S are established from
figure 3-6.
First, considering !:Mx = 0, we have
,.po
N'¢,.po
-0.5 -/.0 0 -.5 -I.
o
NO fpo
-0.5
-/.0 0
S' /po
-0.5 -1.0
Pressure and Stresses on Windward Quadrant
q~
a
Fig. 4-23 Stress Resultants Due to Antisymmetrical Wind Load on a Spherical Dome
-/.0
'<
....
~
'"
~
8
""~
~
-
121
4.3 Shells of Revolution
Q
sine/>
S(f,B)
Fig. 4-24
==-i-==~--Y
Equilibrium of a Spherical Shell Under Antisymmetrical Wind Load
I" (n)
-" NtfJ I,8 (a d8)(a cos 8)
("/2
+ Jo
I"-"
qn(t/J,8)(ad8)(adt/J)(0)
+ f~" S(~,8)<ad8)(0) = 0
(4.104)
The last two integrals vanish because qn(t/J, 8) and S(n/2, 8) pass through the X
axis, so that we have remaining
122
4 Membrane Theory
f:x
f:x
(i)
N¢(i,o)a 2 cosOdO = NJ(i)a 2
= na 2 NJ
cos 2 0dO
=0
We thus confirm that
(4.105)
as shown on figure 4-23.
Next, we evaluate ~Fy
i
x
JII=-X
SX/2
¢=o
=
0
-qn(¢,O)sin¢cosO(ad¢)(asin¢dO)
+
f:x
f:x
= pa 2
s(i,o)sinO(adO)
1"/2
+ as! (i)
sin 3 ¢ cos 2 0 d¢ dO
f:x
sin 2 OdO
(4.106)
=0
so that
2
3
(4.107)
as shown in figure 4-23.
The verification of the stress resultants using the overall equilibrium equations suggests that these two equations, written with reference to a general
coordinate ¢, should provide an independent derivation for NJ and S1, whereupon Nl would be computed, as usual, from equation (4.3c). Note that, in the
general case, ¢b "# n12, so that N¢ and S would participate in ~Mx = 0, and N¢
in ~Fy = O. The procedure of summing forces and moments across a section to
compute bending and shear stresses is strikingly similar to beam theory. In fact,
4.3 Shells of R~olution
123
the j = 1 case is indeed equivalent to elementary flexure except, of course, that
theory alone does not admit the computation of the circumferential stress
resultant N(). The use of beam theory to compute the stresses in antisymmetrically loaded shells ofrevolution was developed in detail by Popov. 20
4.3.6.3 Rotational Shells under Seismic Loading. Another dynamic force frequently simulated by a static representation is an earthquake or seismic load.
Records of actual or artificial earthquakes are generally available in the form
of two horizontal components (N -S) and (E- W), and a vertical component.
The vertical component is an axisymmetric loading which fits the solution
developed in section 4.3.2.1. Regarding the horizontal components, it is frequently assumed that they are not in phase; i.e., the peak effects from the two
directions do not occur at the same instant on the structure. Since shells of
revolution are axisymmetric in construction, it is here regarded as sufficient to
consider only the stronger of the two components. The X axis in figure 4-25 is
presumed to be oriented to this direction.
From the results of a linear dynamic analysis or from some approximations
thereto, a total design base shear V may be computed corresponding to the
strong horizontal motion earthquake component. The base shear is frequently
written as a percentage C of the dead weight of the structure (e.g., 15% g).
Details of the computation of V are beyond the scope of this book since the
dynamic characteristics of the structure must enter into any rational treatment.
An excellent comprehensive discussion of this subject is given in Clough,21
and an application to shells of revolution is contained in Gould, Sen, and
Suryoutomo. 22 The remainder of this section is concerned with the determination of the distribution of surface loading that produces a specified value of the
base shear V and the evaulation of the corresponding membrane theory stress
resultants.
As depicted in figure 4-25, the magnitude of the base shear, V, is equal to the
total horizontal resultant of N", and S, or
V
=
I"" [S(<pb,B)sinB - N",(<Pb' B)cos <Pb cos B]RO(<Pb) dB
(4.108a)
We assume that the base moves uniformly without circumferential distortion,
so that only the j = 1 harmonic participates. Then, we have
V
= RO(<Pb>[ Sl (<Pb) I~" sin 2 B dB - NJ(<Pb) cos 1>" I~" cos 2 B dB]
= nRO(<Pb) [Sl(<Pb) - NJ(tA,)COS<Pb]
(4.108b)
The subsequent analysis for the membrane theory stress resultants is dependent on the surface loading which produces V. We know that the equivalent
surface loading should have a resultant in the direction of V; i.e.,
124
4 Membrane Theory
v
~
x
II
Fig. 4-25
Base Shear Due to Horizontal Seismic Loading
qds"dsp=V=Vtx
(4.109)
Surface
where q = qtx is defined in equation (3.11e). Since V is assumed to act in the
horizontal plane, at any level q acts in the horizontal plane and is parallel to
the assumed strong direction of the earthquake, indicated here by the X axis.
Referring to figure 4-26(b), we see that q(tP) has a circumferential component of
125
4.3 Shells of Revolution
Direction of
Horizonfol
GroUl~
Mofion
(b)
Fig. 4-26
Static Horizontal Seismic Loading on a Shell of Revolution
q(¢) sin 0, and a radial component of q(¢) cos O. Considering the meridional
view, figure 4-26(a), the radial component, q(¢) cos 0, is projected from the lower
figure and resolved into a meridional component, qt/J cos 0 cos ¢, and a normal
component, q(¢) cos 0 sin ¢. Thus. referring to figure 3-6 for the correct signs,
we have
qt/J = -q(¢)cos¢cosO
(4. 110a)
ql) = q(¢) sin 0
(4.110b)
qn
= -q(¢)sin¢cosO
(4. 110c)
We compare equations (4. 110a-c) with equation (4.86a) and conclude that
q~
=
qJ =
q~
-q(¢)cos¢
(4. 110d)
q(¢)
(4.110e)
= -q(¢)sin¢
(4. 11 Of)
126
4 Membrane Theory
It remains to determine the meridional variation of q, q(t/J), so that equation
(4.109) is satisfied.
Consider a cooling tower structure in the form of a hyperboloid of revolution
as illustrated in figure 4-27. An actual cooling tower is shown in figure 2-8(0).
First, we relate the total base shear magnitude V to the total shell weight W.
Following the overall equilibrium method,
(4.111a)
where N¢(,p,,) is due to self-weight load. Substituting equation (4.40a) and (4.37)
into equation (4.111a),
W
= 2nq DL a2 ['1(,p,,)
- 'l(t/Jt)]
(4.111b)
where '1 is defined in equation (4.40b). Then,
V
W
C=-
(4. 112a)
V=CW
(4. 112b)
or
Either C or V is assumed to be specified by consideration of the dynamic
characteristics of the system, as described, for example, in Clough.21 Note also
that equation (4.111) may be useful for calculating the total weight of a tower
of given dimensions.
Now, we refer to some characteristics of the linear dynamic response of the
shell in harmonic j = 1. We can show that this response can be represented as
a linear combination of the longitudinal modes offree vibration,23 and that the
first mode is the most important for the seismic analysis. The first mode normal
displacement for a tower of typical dimensions is shown in figure 4-27. Keeping
in mind that the total base shear V is presumed to be known, the actual
magnitude of the normal displacement is not needed for this discussion;
what is important here is the shape. The inertial forces produced by a harmonic response are, in fact, proportional to the mode shape. This follows from
D'Alembert's principle. Thus, it is reasonable to try to arrive at a distribution
of the base shear V which resembles the dynamic deflected shape. Of course,
the actual longitudinal response is a combination of the participating modes
and is an ever-changing function of time; the best we can attempt with a static
approach is a one-term approximation at a given time, and this is reasonably
selected as the first mode shape. Thus, we shall try to arrive at a distribution
q(t/J), which has a shape similar to the first mode of the normal displacement.
The most common distribution used for shell design has been to assume that
the seismic load is everywhere proportional to the gravity load, or
q(t/J) = CqDL
(4.113)
This is referred to as the Mass (M) distribution on figure 4-27 and the compo-
U)
:1: 1
I-
~ /
+
N
1
J
"lJ
01
4>t
~
z
Normal
Displacement
Fundamental
Mode
r
Mass
.,.;:,
q=CqOL E,13
Mass Height
r
Fig. 4-27 Seismic Loading Distributions for Hyperboloidal Shells
,ST
\
- .-.~-~---~--~--~
-. - ._-._----_.-.
Mass Height 2
r q=CqoL~132
., -
tv
-...)
-
=
g.2"
<
0
('I>
:;,g
0
-,
~
!!.
t:T
til
w
.;..
128
4 Membrane Theory
nents are computed by substituting equation (4.113) into (4.110). It is obvious
from the figure that this distribution does not realistically represent the first
mode shape.
Another possibility, frequently adopted in the analysis of tall building structures, is to assume that the load is distributed in proportion to the weight and
the height of each element of the structure above the base. We refer to this as
the mass-height (MH) distribution on the figure. This distribution has been
adapted to cooling towers by the author. 24 To derive this shape, we refer to a
differential element dE, which is located at a distance E = S + Z above the base,
as shown on figure 4-27. E is thus an axial coordinate measured from the base.
The surface loading acting on the differential element dE, which represents the
appropriate portion ofthe base shear V computed by equation (4.109), is given
by
(4.114)
where W is given by equation (4.111). Then, F(E) is divided by the differential
area
A(E) = 2nRo(E) dE
(4.115)
to get q(E) = F(E)/A(E).The resulting expression is easily transformed into a
function of rP
(4.116)
through equation (4.34). (1 is a constant defined in Appendix 4A. From the
graph of the MH distribution, it is obvious that this appears to be far more
realistic than the M distribution when compared to the mode shape.
A third possibility may be included, whereby the shear is distributed according to the weight and the square of the height of each element above the base.
This is termed the mass-height2 (MH2) distribution. Following the same procedure, we geeS
(4.117)
where (2 is a constant defined in the Appendix 4A. From the graph, we see that
perhaps the MH2 distribution is even a better approximation than the MH.
The components of surface load for each of the assumed distributions may
be substituted into equations (4.90), (4.91), and (4.92) to evaluate the stress
resultants, as has been done in various publications. Since the results are not
readily available in collected form, the pertinent formulae are summarized in
Appendix 4A. All quantities are nondimensionalized and are tersely presented
in a form suitable for computer input preparation. Tabulated results for a wide
range of tower dimensions are also available for the M distribution. 26
129
4.3 Shells of Revolution
2 ..0
SHELL DATA
als = 0.65
alt =.0.9.0
1.0
k2
= 1.12
a
Zia
-1 ..0
-2.0
-3.0
a
1.0
2 ..0
3.0
4.0
5.0
6.0
Fig. 4-28 Effect of Base Shear Distribution on the Meridional Stress Resultant for a Hyperboloidal Shell
Although the detailed derivation of the various solutions are probably of
interest only to the specialist, it is instructive to look at a comparative study of
the stress resultants. In figure 4-28, we show the nondimensional meridional
stress resultant n~ = NJ /(qDLa) for a shell of typical proportions using the three
proposed distributions. Also shown is the negative of the dead-load stress
resultant, equation (4.40a), suitably nondimensionalized. When the seismic load
graph falls to the right of the dead load graph, this indicates a net tension at
() = 0 due to the combined loading. It is for this net tension that a concrete shell
should be reinforced.
Also, it is appropriate to note here that the designer should be very careful
when considering combined lateral and gravity loading which produces a net
tension situation. Each loading should be factored by an appropriate amount
prior to combining effects with opposite signs. For example, consider figure 4-28
at Z/a = -1.0, where N~DL) = - 2.5 and NJ(MH) = 3.3. Using load factors from
a standard code,27
130
4 Membrane Theory
N~NET)
= 1.3(3.3) - 0.9(2.5) = 2.04
An alternative (not recommended) would be to apply a factor of safety to the
net stress. For example, using a nominal factor of safety of 1.5,
N~NET)
= 1.5(3.3 - 2.5) = 1.20
It is apparent that the latter procedure can lead to unconservative results when
the lateral and dead load stresses are close in magnitude but opposite in sign.
4.3.7 Asymmetrical Loading
4.3.7.1 Integration of Equations. The analysis of shells of revolution for harmonicsj> 1 is quite different from thej = 0 andj = 1 cases. Whereas there are
a relative abundance of quadrature solutions available for the symmetric and
antisymmetric loading cases, this is not the case for asymmetric loading; the
governing equation, equation (4.88), is not solvable in general terms but, rather,
whatever solutions that exist are dependent on some special form of the variable
coefficients.
At this point, recall that the j = 0 and j = 1 solutions were directly related
to and derivable from static equations of overall equilibrium. The j = solution represents force equilibrium parallel to the axis of rotation, whereas the
j = 1 case reflects equilibrium of forces normal to the axis and equilibrium of
moments about the axis of overturning. In contrast,j > 1 does not correspond
to any equation of overall equilibrium. All loading cases for j > 1 are said to
be self-equilibrated with respect to the overall equilibrium of the shell. Thus,
the differential equations offer the most promising approach. One should note
at the outset (and it is demonstrated in subsequent sections) that the membrane
theory stress resultants in shells of revolution may be very sensitive to the
influence of the higher harmonic components of the surface loading, and that
shells which are subject to such loading must be investigated very thoroughly.
°
4.3.7.2 Spherical Shell with Edge Loading. One specific geometry that can be
solved for j > 1 is the spherical shell. We limit our investigation to the case of
no surface loading, which produces a homogeneous set of equations. This form
is sufficient to describe an important practical situation, the edge-loaded shell
of revolution. Once a homogeneous solution is available, a particular solution
for a given surface loading is easily obtained using the standard method of
variation of parameters.
It is expedient to return to the original untransformed membrane equations,
equation (4.3). We introduce the separated form of the stress resultants, equation (4.86b), for harmonic j into these equations. Noting that equation (4.3c)
gives N(J = - N", since qn = 0, we have
.
N$.",
.
j.
+ 2NJcotrjJ + sinrjJ SJ = 0
(4.118a)
131
4.3 Shells of Revolution
(4. l1Sb)
These equations may be separated by introducing the auxiliary variables 28
NJ + sj
Q~ = NJ - sj
Q{
(4.119a)
=
(4. 119b)
giving
Q{,~ + ( 2 cot <,6 + Si~ <,6) Q{ = 0
(4. 120a)
Q~ ,~ + (2cot<,6 - -!-)Q~
=0
sm<,6
(4. 120b)
which have solutions 28
(4.121a)
C j tan j (<,6/2)
22 sin 2 <,6
Qj _
(4.121b)
From equation (4.119),
.(¢l)
"2 + q .tan).(<,6)J
"2
(4. 122a)
.(¢l)
"2 - q .tan}.(<,6)J
"2
(4. 122b)
1 <,6 [q. cotl
N~. = 2sin2
1 <,6 [q. cotl
S). = 2 sin2
The complete homogeneous solution is given in the Fourier series form of
equation (4.S6b) by
N~(<,6,
e) = - N o(<,6, e)
=
(¢l) + q .tan).(<,6)J
cosjO
2
~ [C{
. cotl. 2 .1 2 d..f...,
sm 'I' }=o
2
1 <,6 /~'o
~ [C{. cotl.(<,6)
S(<,6, e) = 2 sin2
"2
- q . tan}. (<,6)J
"2 sinje
(4.123a)
(4.l23b)
The constants for each harmonic, C{ and C~, must be evaluated from a set
of nonzero force boundary conditions which are also expanded in Fourier
series. One possibility is
L NJ(<,6') cosjO
(4.1 24a)
L sj(<,6") sinje
(4. 124b)
00
N~(<,6', e) =
j=O
00
S(<,6", e) =
j=O
132
4 Membrane Theory
In these equations, ¢/ and ¢/' denote the respective boundaries of the spherical
shell where N¢ and S are specified. Also, the form of equation (4.123) indicates
that N¢ and S may both be designated at the same boundary-e.g., ¢/ = ¢l" in
equation (4.124)-or that N¢ or S, alone, may be chosen at both boundaries.
In any case, only two boundary conditions are available, and the computed
values of N¢ and/or S at the other boundaries must be developed by the support
in order for the solution to remain valid.
For a spherical dome, cot(¢l/2) ~ 00 at the pole, so that Ci = 0 and only one
condition, N¢ or S, may be specified on the lower boundary.
4.3.7.3 Shell of Revolution Supported on Columns. It is frequently necessary to
provide interruptions at the base of a shell above the ground for functional
reasons. One obvious possibility is to support the shell on a series of concentrically placed columns, figures 2-8(p, t, u). This obviously introduces an
interruption in the continuous, nondeforming boundary required to maintain
membrane action and suggests that the stress pattern, particularly in the vicinity
of the support, may be severely altered.
A fairly common approach is to analyze the shell by a superposition technique as shown in figure 4-29. In what follows, the discussion is restricted to
axisymmetric loading and membrane theory to introduce the concept expeditiously, but this type of model has been generalized for nonsymmetric loading
and bending effects as well. 29 Also, here the columns are assumed to be equispaced around the circumference, and the axis of the column is taken to be
tangent to the meridian at the boundary ¢l = ¢lb' For columns oriented vertically,
the same theory can be applied with some minor modifications.
We now consider the superposition in figure 4-29 in detail. We assume that
the axisymmetrical surface loading q is carried equally by the nc equispaced
columns. As shown in figure 4-29(a), the column reaction Rc is taken to be
uniformly distributed over the column width We> so that the intensity of the
reaction per unit length of circumference Rei is Rc/wc. Loading case (a) is
represented by the superposition of cases (b) and (c). Case (b) is the familiar
axisymmetric loading case previously studied in great detail in sections 4.3.2
and 4.3.3. The continuous boundary meridional stress resultant at ¢l = ¢lb is
denoted by N¢(b) (fjJ,,). Case (c) has an edge loading applied along ¢l = ¢lb' It
consists of (i) the negative of the continuous boundary reaction from (b) in the
region between the columns; and, (ii) the difference between the column reaction
and the continuous boundary reaction within the column width. Since case (b)
is readily solvable, the remaining problem is to address case (c).
First, we evaluate the column reaction in terms of the known meridional
stress resultant from the continuous boundary analysis, case (b), as
Rc
=
2nRo(fjJ,,)
N¢(b)(fjJ,,)--=--:.:...::..:...
nc
The intensity is given by
(4.125)
133
4.3 Shells of Revolution
axis of rotation
-L./ q
+
N.p(b)( 4>~-l-~'r-I-.1..rl-"+-:L,L.-hl,....L-.J,..rl.,..L.:~
~
(d)
~
(c )
Fig. 4-29
Superposition Representation of a Column-Supported Shell of Revolution
Referring to figure 4-29(d), we write
measured in the horizontal plane
We
in terms of the subtended angle
213
(4.127)
Then, substituting equations (4.127) and (4.125) into equation (4.126), we have
R
_ rcN</J(b)(tPb)
cl -
f3 nc
(4.128)
Rcl has units of force/length.
Now, we wish to express the edge loading shown in figure 4-29(c) as a function of the circumferential coordinate 8. Referring to figure 4-29(d), we have,
134
4 Membrane Theory
formally,
N,p(ekP", 0) = [ -N,p(b)(~)
+ bRc/]
(4.129)
where b = 1 if m(2nlnJ - f3 ::; 0 ::; m(2nlnJ
(m = - nel2, ... , - 1,0, 1, ... , nel2)
b = for all other O.
°
+ f3
The limits on b are set presuming an even number of columns and can be
adjusted accordingly for an odd number, and N,p(e) is taken as positive in the
positive sense of N,p(b).
We expand N,p(e) in a Fourier series similar to equation (4. 124a),
N,p(e)(ffJ", 0)
=
L
00
(4.130)
N~(e)(~) cosjO
j=O
First, the Fourier coefficient N3(e) is given by
(4. 131a)
From equations (4.129) and (4.128),
o
1
N,p(e)(rPb) = 2n
f1t [
-1t
"") + un~ N,p(b)«(P,,)]
dO
f3 ne
N,p(b)('Pb
-
= N,p(b) (rPb) [-2n + 2f3 nen ]
2n
(4.131b)
f3ne
=0
This confirms that the net meridional resultant in figure 4-29(c) is 0, since the
total of the column reactions is equal and opposite to the uniform boundary
reaction. This edge loading is then said to be self-equilibrated.
Next, for j > 0, we have
(4.132)
Referring to figure 4-29(d), where the circumferentil'!-l spacing of the columns is
given as 2nlne, we have
N~(e)(rPb) = ~{f:1t -N,p(b)(ffJ,,)cosjOdO + Rc/[f: cosjOdO
+
f
(21tinc)+p
f
cosjO dO
(21tin c )-p
... +
(nc- 1)(2ftinc)+p
(nc-
1)(2ftin c )-p
+
f2(2ftinc)+p
cosjO dO
+ ...
2(2ftinc )-p
cosjO dO
+
f2ft
21t-P
cosjO dO
]}
(4. 133a)
135
4.3 Shells of Revolution
. "") Nq,(b)(t/J,,) { 2 . .
NJ(c)('I'b =
--;-slnJn
n
]
1
+ "c~ [
m-1
n (. . )
+ ]--:---p
SlnJP
nc
2n
sinj (m+ p)
nc
+ sinj2n -
sinj(2n -
-
2n
sinj (m- )
p]
nc
(4. 133b)
P») }
For an integer value ofj, the first and next to last terms drop out. Furthermore,
it can be shown that for j #- inc, where i is an integer, the remaining terms sum
to 0, so that N~(c)(t/J,,) = 0 ifj #- inc. Whenj = inc, the complete symmetry results
in the total integral being equal to nc times the integral over one column, or
oi ("")
NtJ>(c)
'l'b
=
nc
Nq,(b)(t/J,,) 2n sinjp
.p
n
] nc
2
= jp Nq,(b)(t/J,,) sinjp (j
(4.134)
=
inC>
The preceding assertions are easily verified for a specified numerical value of nco
We thus conclude that only those harmonicsj which are integer multiples of
the total number of columns nc participate in the solution, and we may rewrite
equation (4.130) as
Nq,(c)(tPb,{J)
=
00
L
(4.135)
N~(c)((A)cosjO
j=nc, 2nc···
This is an example of cyclic symmetry for rotational shells.
We now turn to the spherical shell treated in section 4.3.7.2 as an example
of a column supported shell. An actual case is shown in figure 2-8(t). If the shell
is not a dome, there are two constants of integration to be evaluated. We have
already specified Nq,(tPb), and since we cannot provide an ideal condition because
of the columns, the lower boundary must develop the corresponding value of
S(tPb). This leaves us the option of specifying Nq,(tPt) or S(tPt), and Nq,(tPt) = 0 is
probably the more realistic choice. Thus, we must provide suitable resistance
to S at both boundaries in the case of the open shell, or at the base in the case
of a dome. This can be accomplished with a circumferentially stiff edge member.
For our example, we chose a hemispherical dome supported on four columns
as shown in figure 4-30, subject to dead load q. From equation (4.20), the
solution for case (b) of the superposition is
-qa
Nq,(b)
At tP
=
(4.136)
1 + cos tP
= tPb = n12, Nq,(b) = Nq,(b)(t/J,,) = -qa, and equation (4.134) becomes
.
2qa . .
NJ(c)(tPb) = - jp SlnJP
(413
. 7a)
136
4 Membrane Theory
o
-4
-8
o
..
8
o as
lot
S(3()O)lqa
Fig. 4-30
Stress Resultants for a Column-Supported Hemispherical Dome under Dead Load
From equation (4.135) with nc = 4 and ¢Jb = n12,
n)
N¢>(c) ( -, e
2
2qa..
L
00
=
j=4,8,12
.
(4, 137b)
---:--fJ SIDjfJcOSje
j
Recalling the Fourier series expansion for the stresses in the spherical shell,
equations (4.123a) and (4.l23b); taking q = 0 because we are considering a
dome; and evaluating equation (4.123a) at ¢Jb = n12,
(2'n ) = 2"1 jL;;o~ q . tan}.(n)4
N¢>
e
(4.138)
cosje
Equating the coefficients of cosje for each term in the series in equations
(4. 137b) and (4.138), we find
q
.
-4qa
= ----;t3sin jfJ
(j
= 4,8,12)
(4.139)
since tan j (nI4) = 1, and q = 0 for j -:f=. 4,8, 12. Now, we may write the solution
to case (c) of the superposition by substituting C~ into equations (4.123)
N¢>(c)(¢J, e)
Ne(c)(¢J, 8)
= -
- 2qa
fJ . 2 ¢J
SID
2qa
Sc(¢J, e) = fJ . 2 ¢J
SID
sinjfJ
L -.j 00
j=4,8,12
. (¢J)
.
(4. 140a)
tan} -2 COSje
sinjfJ
. (¢J) . .
L -.
-tan} -2 SIDje
j
00
j=4, 8, 12
(4. 140b)
The complete solution consists of case (b) plus case (c), which gives
N¢>(¢J, e)
=
1
-qa
¢J
+ cos
Ne(¢J, e) = qa (
+ N¢>(c)(¢J, e)
1 ¢J - cos ¢J) - N¢>(c)(¢J, e)
1 + cos
(4.141a)
(4.141b)
4.3 Shells of Revolution
137
(4.141c)
in view of equations (4.20) and (4.21). Again, we stress that the value of S = Sc
computed from equation (4.141c) must be developed at the lower boundary.
To study the characteristics ofthis solution, we show the nondimensionalized
stress resultants for the hemisphere in figure 4-30. The half angle f3 is taken as
5°; N,p and No are graphed at () = 0°, the center line of the column; and S is
shown at () = 30°. The series converges rather slowly with about 14 terms
(n = 56) required to compute, within 1%, the known value of N,p at the base,
[360° /(4 x 2 x 5°)] N,pb(¢Jb)' or 9 times the continuous boundary reaction. The
continuous boundary stress resultants, which correspond to figure 4-29(b) and
are available from figure 4-3, are superimposed and shaded to indicate the
penetration of the discontinuous boundary reaction amplification into the
shell.
Although we have obtained a convergent membrane theory solution for the
spherical dome supported on columns, all shells with concentrated edge effects
are not able to sustain such loading in this way. Only a limited number of
specific geometries have been investigated, and no general proof is offered;
however, it appears that the attenuation of concentrated meridional edge effects
through principally membrane action occurs only in positive curvature shells.
Physically, it can be surmised that the straight characteristic lines present on
zero- and negative-curvature shells offer a path for the propagation of the edge
effect to the opposite boundaries, unless bending distortions occur.
While the quantification of bending effects is beyond the scope of this chapter,
we may expect that the shell may have to be strengthened to provide adequate
resistance. In concrete shells, the bending stresses are conveniently resisted by
local thickening. The wall thicknesses of the shells shown in figure 2-8(g) and
(p) are approximately equal to the column depth near the base and are gradually
reduced away from this region. For metal shells, thickening alone may not be
sufficient to provide an economical solution, since buckling in the region of
high compressive stresses must be considered. Auxiliary stiffeners, such as the
ring beam shown in figure 2-8(u), are one possibility. Another configuration
used for bin structures is to provide full-height columns, along with eave and
transition ring beams, as shown in figure 4_31. 30
Rotter has proposed a membrane theory solution for the case ofaxisymmetrical loaded bins, based on the cyclic symmetry of the structure. 30 As shown
in figure 4-32, the shell is loaded locally by a normal pressure qn(Z) = Pn(Z) and
a wall friction loading qz(Z) = JLPn(Z), where the normal pressure Pn(Z) and the
friction coefficient JL are determined from the properties of the material. Containments for solids differ markedly from liquid storage tanks with the additional need to provide resistance to the wall friction loading from the solids.
These forces may be amplified during drawdown and should be considered
carefully be the designer. 31
Also acting are the self-load of the cylindrical shell qz = q = Wd(2naL),
138
4 Membrane Theory
Eaves
Ringbeam
~
~
Fig.431
Transition
Ringbeam
Column-Supported Bin with Columns Extending to the Eaves Ring
where Wi is the total weight of the cylindrical portion of the bin with length L;
and a circumferential line load qz(O) = W2 /(2na), where W 2 = the superimposed
vertical load at the top of the cylinder, Z =.0. Note that Nz(O) = - qz(O).
The vertical load in the shell wall is transferred into the adjacent column by
the in-plane shear stress resultant, S. By symmetry, S = 0 along the meridian
equidistant between two columns. Assuming S to be uniform along the column
and to vary linearly in the circumferential direction, we have
S(O)
=
~
2neL
(1 _
ne O)
n
(4. 142a)
139
4.3 Shells of Revolution
Z
!q
LPn(ZI
L
~ Pn(z)
Fig.4-32
Half-Panel of Column-Supported Bin
where
W
= total load carried by the nc columns
=W1 +W2 +W3
and
W3
= total weight of the stored material in the bin
140
4 Membrane Theory
This implies that the total load is uniformly transferred into the columns along
the length L.
The circumferential stress resultant is easily found from equation (4.3c) as
(4. 142b)
For the meridional stress resultant, Nz(Z), we follow the overall equilibrium
approach, section 4.3.3.1. We have the resultant force from the weight of
the shell above the section Z, W 1 (Z/L); the resultant force from the superimposed vertical load, W z; the resultant force from the frictional loading,
2naJ-l J~ Pn(Z) dZ; and the total force in the columns, W(Z/L). Substituting into
equation (4.58) and recalling that a resultant axial load in the negative Z
direction is taken as positive, we have
(4. 142c)
At the top of the panel, Z = 0, N z is due to only the superimposed load Wz ; at
the bottom of the panel, Z = L, the stress is proportional to the difference of
the weight of the contents and the frictional load on the wall.
Just as in the case of a spherical shell supported on columns, the ring beams
must resist the shear force S(8) applied around the circumference, probably
eccentric to the axis. This is examined in greater detail by Rotter. 3o The shell
thickness must be adequate to resist buckling due to the maximum principal
compressive stresses produced by S(O), N 9 (L), and Nz(L), which probably occurs
near the intersection of the shell, the column, and the transition ring beam.
4.3.7.4 Cylindrical Shell Under Circuniferentially Varying Normal Pressure.
Consider a circumferentially varying normal pressure p(O) on a cylindrical shell.
This could simulate a pseudostatic wind loading on a cylindrical tower, as
shown in figure 4-33. The circumferential variation of the pressure would
generally be established through field measurements or wind tunnel tests. For
this example, p is taken as constant with respect to Z, but a variation with Z
may be treated by a simple extension of the following development.
We follow a procedure suggested in Rish and Steel. 32 It is convenient to
return to the membrane theory equations in terms of curvilinear coordinates IX
and p, equations (4.2). We take IX = Z, P = 0; A = 1, B = a, Ra = 00, Rp = a;
qa = qp = 0, and qn = - p(O). Then, we have
N z .z
S. z
1
°
=°
+ -S()
a' =
1
+ -N9
a ' ()
(4. 143a)
(4. 143b)
141
4.3 Shells of Revolution
:::
::n
Z
I
H
I
////(////J '///
I
....
Wind
Fig. 4-33
-Ne =
a
a
'I
I
~~,--A----+.m-----Y
Wind Loading on a Cylindrical Shell
(4. 143c)
-p(O)
Because these equations are relatively simple, we are able to treat the partial
differential equation form directly rather than to introduce separated variables.
However, we later find it convenient to represent the integrated solution in
terms of the separated variables.
Proceeding from equation (4.143c), we have
Ne(Z,O) = - p(O)a
(4.144)
Next, we take V/VO of equation (4.144), substitute into equation (4.143b), and
integrate to get
S(Z, 0)
=
p(O),eZ
+ 11 (0)
142
With a stress-free top edge, S(O, 0)
°
4 Membrane Theory
= 0; therefore, 11 (0) = and
S(Z,O) = P(O),6Z
(4.145)
Finally, we take iJliJO of equation (4.145), substitute into equation (4. 143a), and
integrate once more to obtain
Z2
Nz(Z,O) = - p(O) ,6 6
2a- + 12(0)
Again invoking the stress-free condition at Z = 0, Nz(O, 0) = 0,12(0) = 0, and
Nz(Z,O)
Z2
p(O) ,6 6
2a-
= -
(4.146)
We now represent p(O) by the Fourier series
p(8) =
L
00
i=O
pi cosjO
(4.147)
and write the stress resultants in the separated form, similar to equation (4.86b),
with Z replacing </J:
NZ}
{ N6
S
=.~
1
0
{Nl(Z)COSjO}
N~(Z) cosjO
Sl(Z) sinjO
(4.148)
where
.
2 .Z2
Nl(Z) =j pl_
2a
= -pia
Si(Z) = _jpiZ
N/(Z)
(4. 149a)
(4. 149b)
(4. 149c)
These very simple expressions for the stress resultants for a general harmonic
j offer a rare insight into the interesting properties of the higher harmonic
components. Consider, for example, the solution expressed by equations (4.148)
and (4.149) contrasted with a solution for the resultant lateral force and overturning moment found by elementary beam theory:
1. The circumferential stress resultant N6 is not computable by beam theory.
2. The shear stress resultant is linearly proportional to the axial coordinate Z,
just as in beam theory. However, the shell solution shows that S is also
proportional to the harmonic number j. This discrepancy, of course, increases
in proportion to the contribution of the higher harmonics.
3. The meridional stress resultant is proportional to the square of the axial
coordinate by either theory; however, the shell solution reveals that Nz is
also proportional to j2, indicating that the meridional stress resultant is
highly sensitive to the higher harmonic components.
143
4.3 Shells of Revolution
To help put these observations in perspective, note two characteristics of the
circumferential normal pressure distribution p(O):
1. High harmonic number components of p(O) results from distributions of the
pressure that vary rapidly around the circumference. To a point, the more
rapid the fluctuations, the greater relative participation of the higher harmonics. However, the case of extremely rapid fluctuations associated with
large values of j would be tempered by the physical limitations of accompanying significant amplitudes. That is, the pi values would usually tend to
diminish as j becomes very large.
2. Harmonic components j = 0 and j > 1 do not contribute to the resultant
lateral force or overturning moment on the shell. Only j = 1 affects these
resultants. If the meridian were curved, the j = 0 component would influence
the axial force equilibrium, but thej > 1 components are always self-equilibrated with respect to the overall equilibrium of the shell.
As an illustration, consider the cylinder shown in figure 4-33 subject to the
pressure distribution shown in figure 4-34. This loading was derived from
, Wind
B =00
8= 72.5 0
8= 1000 - - \.
0 .21
P,
j
~
0
1
2
3
4
5
6
7
8
9
·10
11
.229
.277
.598
.472
.063
-.124
-.027
.046
.000
-.023
.009
.013
12
-.009
Pr
Fig. 4-34 Design Wind Pressure for Circular Towers (Reprinted with permission of American
Society of Civil Engineers)8
144
4 Membrane Theory
measurements taken from a wind tunnel experiment 32 and is representative of
the circumferential pressure distribution on large cylindrical and hyperboloidal
towers with a roughened surface. The Fourier coefficients for this distribution
are also shown on the figure. The loading is given in terms of pressure coefficients multiplied by a reference pressure Pro which is generally a function of
geographical location, terrain, exposure, and height; it is usually taken as the
stagnation pressure of the wind (1/2)p V 2 , where p is the mass density and V is
the velocity of the wind. The 0.524 increase shown on figure 4-34 is due to
internal suction, generally fairly constant around the circumference. The maximum external suction, however, has been found to be a function of the angle
where flow separation occurs, which, in turn, is dependent on the surface
roughness. The value here at () = 72.5°,0.69 + 0.52 or about 1.2 Pr without the
internal suction, is characteristic of rough towers, whereas a peak approaching
3.0 Pr is theoretically possible for smooth towers. Roughness in the form of
vertical ribs is often built into cooling tower shells to reduce the suction 33 [see
figure 2-8(0)].
For simplicity, Pr is taken as unity here, and also as constant with height. A
stepwise variation in Pr is easily handled by applying the solution, equations
(4.143) through (4.149), in a piecewise fashion as detailed in section 4.3.2.3.
We focus the computation on the meridional stress N z at () = 0 for the base,
Z = H. With Pr = 1.0, equations (4.148) and (4.149) give
H2 00
Nz(H,O) = 2a i~O j2 pi
(4.150)
for the Fourier coefficients shown on figure 4-34. Summing for harmonics 0 to
12, we get
H2
(4.151)
Nz(H, 0) = 5.422a
Now, for comparison, we evaluate Nz(H, 0) from beam theory. Since only the
first harmonic contributes to the resultant force per unit height R y , we have
Ry
=
f~" pl cos ()(a d()) cos ()
= plan
and the bending moment is
H2 p l anH2
Mx=R yT =
2
Since we are considering the stress resultant that acts on the circular middle
surface line, we compute Nz using the familiar linear stress formula N z = Mza/
Ix, where Ix is the moment of inertia of a circle of radius a about a diameter,
na 3. Thus, we have
145
4.3 Shells of Revolution
Fig.4-35
Slice-Beam Model
(4.152)
For the tabulated value of pi found in figure 4-34, equation (4.152) becomes
H2
Ni(H, O) = 0.277 2i
(4.153)
Comparing equations (4.153) and (4.151) clearly illustrates the sensitivity of N z
to the higher harmonic components. This emphasizes the practical importance
of obtaining realistic circumferential pressure distributions on tower-type
structures.
The perhaps unexpected sensitivity of Nl and Si to the harmonic number j
and the resulting large departure from the values that might be anticipated from
elementary beam theory are clearly demonstrated in the preceding solution.
From a physical standpoint, it is instructive to consider a "slice-beam" model
proposed by Brissoulis and Pecknold. 34 On figure 4-35 a longitudinal slice
bounded by the meridians (J = ±(n/2j) and loaded by a normal pressure is
shown. At the section Z = L, the loading p(Z, 8) produces a bending moment
about an axis parallel to the chord of the circular sector cross-section of the
slice beam. The applied moment is resisted by a couple consisting of (a) the
resultant of the meridional stresses Nl(L) cosj8 acting within the depth of the
beam cross section, a[1 - cos(n/2j)]; and, (b) the accumulated shearing force,
It Si(Z) dZ, acting along both edges of the slice. As the harmonic number j
increases, the depth of the slice beam and, hence, the available internal lever
arm decreases. Likewise, the resultant transverse component of p(Z, 8) from
Z = 0 to Z = L must be resisted by a similar component of the resultant of
Si(L,8) acting on the sector cross section, which is shortening as j increases.
Therefore, asj increases, both Nl and Si must increase proportionally to balance
both the applied bending moment and transverse force. This model has been
quantified by Gould to verify equations (4.148) and (4. 149)?5
Although some additional resistance may be provided by meridional bending
146
4 Membrane Theory
and transverse shear, the cantilevered "slice-beam" appears to reveal the essentials of the resisting mechanism of cylindrical-type shells subject to normal
pressures with substantial (i.e., significantj > 1 content) circumferential fluctuation. In addition, Brissoulis and Pecknold suggested that the basic beam could
be generalized into a "propped-cantilever," with the reaction at the free end
representing the effect of a ring stiffener or a roof. 34
4.3.7.5 Spherical Shell Under Measured Pseudostatic Wind Pressure. The separated flow pattern for wind acting on a cylindrical shell, figure 4-34, can also
be expected to occur on spherical shells, but is more complex since the flow
moves over as well as around the body. A set of pressure coefficients from a
wind tunnel test of a truncated sphere is shown on figure 4-36(a). It is apparent
that separation occurs both meridionally and circumferentially, with external
suction acting over most of the surface.
Each curve in figure 4-36(a) would require a set of Fourier coefficients to be
properly represented. This would complicate the harmonic-by-harmonic solution of equation (4.88), requiring the shell to be divided into meridional segments within which the Fourier coefficients for the loading could be taken as
constant. Ultimately, this produces a step-by-step solution to an initial value
problem starting at the free edge, which is similar to the procedure for variable
thickness shells discussed in section 4.3.2.3 and illustrated on figure 4-7.
In such cases, much of the efficiency of the harmonic superposition method
is lost, and it may become attractive to consider the partial differential equations
(4.3) in the variables rP and O. This approach requires only one solution over
the surface, but must usually be accomplished by a numerical method such as
finite differences. The curves for N", and No shown in figure 4-36(b) were obtained
by Bellworthy and Croll 36 in that way.
It is of interest to compare these results with those presented on figure 4-23
for the anti symmetric loading. The latter values are scaled by 0.9 to correspond
to the pressure coefficient at rP = 90°, 0 = 0° on figure 4-36(a), and are plotted
on figure 4-36(b). Any similarity between the curves is probably coincidental.
However, it might be fortuitous for thin metal shells, which are susceptible to
buckling and may have been designed for an antisymmetric pressure distribution, that the more realistic loading produces conservative values for the maximum negative hoop stress.
As in the previous section, the importance of establishing a realistic distribution of wind pressure on the shell surface is apparent.
4.3.7.6 Hyperboloidal Shells Under Wind Loading. As mentioned previously,
analytical solutions to the membrane theory equations for j > 1 are only
available for isolated cases. As a means of demonstrating a possible approach
to the general solution of equation (4.88), and also of describing a problem with
considerable practical utility, consider the hyperboloidal shell of revolution,
figure 4-6, subject to the normal wind pressure shown in figure 4-34.
-1·6
- 14
-1·2
1
15(.
at
~.o-
100·
Angl. to wind '
eo(J
120·
~~~
eo, .. 0-
e~.
160"
180·
~WIND DIRECTION
Fig.4-36(a) External Wind Pressure Distribution for Truncated Spheres (From Bicknell, 1. and Davis, P., "Wind Tunnel
Studies of Spherical Tower Mounted Radomes", MIT, Lincoln Laboratories, Group Report 76-7(1958).) Reprinted with
permission of Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts.
ou
v
-•
c:
..•
£
.j:>.
.j:>..
-.)
-
o
::s
§.
::,g
~
o
'"-,
o
::r
@:
IZl
<,.,
•
-~
CD
o
o
~
W
\0
"0'
,R
o
~
W
I _____ 3
$
i 3d'
I
-
J
--
I
--
Circumferential stress resultants
CD,
/
Lo
•
/
/
i
/
o·
{6
Test Results
Ne/pq
Simplified Distribution
Fig. 4-36(b) Membrane Stress Resultant for Spherical Shell Under External Wind Pressure
Meridional stress resultants N¢/pq
0
S6
l
_0 -
"'"
....
~
~
~
po
~
8
a::
.j>.
00
4.3 Shells of Revolution
149
In this case, the loading takes the form
=0
(4.154a)
qe = 0
(4. 154b)
qtf>
00
L
qn =
i=O
(4. 154c)
pi cosj8
where the Fourier coefficients are tabulated in figure 4-34.
The solutions for harmonicsj = 0 andj = 1 are obtained in identical fashion
to the dead-load case, section 4.3.2.3, and the seismic load case, section 4.3.6.3,
respectively. They are available in Lee and Gould. 37
For the casej > 1, a different approach is required. We begin with equation
(4.88) and differentiate the first term on each side using the product rule. The
resulting equation is
r/I,tf>tf>
+ Xl(,p)r/I,tf> + X2(,p)I/I = X3(,p)
(4.155)
after dividing through by the leading coefficient. In equation (4.155),
Xl(,p)
Rtf»
= ( 2- - 1
Re
1
(4.156a)
cot,p - -Rt/>,t/>
Rtf>
j2Rtf>
(4.156b)
Ro sin,p
X3(,p)
=
~[(q~cos,p - q~sin,p)R5Reltf> - jR~(jq~ RoRe
qjsin,p)
(4.156c)
The expansion in equations (4.155) and (4.156) has been retained in a general
form to point out that we are confronted with a second order linear differential
equation with variable coefficients. It is obvious from equations (4.156a) and
(4.156b) that the variable coefficients may be very complicated expressions for
many surfaces.
A strategy for dealing with such an equation is suggested in Novozhilov. 38
The technique involves a straightforward elimination of the first order term r/I,tf>
through the introduction of an additional auxiliary variable, and also another
less obvious transformation on the independent variable,p. The resulting equation has the form
(4.157)
in which If! and 11 are the transformed variables, and X4(,p) is the appropriately
modified r.h.s. The detaiis of the transformation are left to the cited references;
here, we examine the remaining variable coefficient p in more detail. The
coefficient is a very involved function of the geometry and harmonic number j
and is given in Lee and Gould 37 in terms of a new function, Xs, as
(4.158)
150
4 Membrane Theory
The remaining solution procedure is outlined as follows:
1. Expand [p(,B)]2 in a Fourier cosine series in
[p(,B)]2 =
co
I
k=l
rx k cos 2k,B
,B
(4.159)
and substitute into equation (4.157).
2. The equation is now in the form of Hill's equation, which is solved using
techniques described in Hill, and also in Whittaker and Watson. 39 The
solution is in the form of a rapidly converging infinite series. From a historical viewpoint, it is interesting to note that this equation was first studied by
the noted astronomer, G. W. Hill, with reference to the motion of the lunar
perigee, which is the point of the orbit of the moon that is nearest to the earth.
3. Using the solution derived in (2), the original stress resultant variables are
recovered through the various transformations.
4. Through some numerical studies using the rather involved series solution
obtained from Hill's technique, a very close approximation was found by
considering only one term k = 0, of the Fourier series expansion for [p(,B)2],
i.e., rxo. 40 In effect, this reduces equation (4.157) to a second order linear
differential equation with a constant coefficient. With the coefficient [p(,B)]2
replaced by the constant rxO in equation (4.157), the equation is easily solvable
in terms of elementary trigonometric functions.
A comprehensive set of tables for hyperboloidal shells with the general
properties of hyperbolic cooling towers that are subject to the wind loading
shown in figure 4-34 is given in Lee and Gould. 37
The solution for a hyperbolic cooling tower exhibits similar characteristics
to those previously observed for the cylindrical shell under wind loading. In
particular, two points should be reemphasized:
1. The solution is very sensitive to the harmonic number j.
2. The wind load stresses are opposite in sign to the gravity load stresses, and
the net stresses, critical for design, should be calculated using proper load
factors. The sensitivity of the cooling tower to the net tension is similar to
the combined seismic-gravity loadings discussed in section 4.3.6.3 and illustrated on figure 4-28.
4.3.7.7 Commentary. The preceding example indicates that the solution to the
j > 1 case may be quite complicated for other than cylindrical, spherical, and,
perhaps, conical geometries. The transformation to the Hill's equation form has
apparently only been investigated for the hyperboloidal shell and may be useful
for other geometries as well. Also, the hyperboloidal shell under wind load has
been solved in the partial differential equation form, similar to that used for the
cylindrical shell in section 4.3.7.4, as opposed to the harmonic decomposition
approach stressed here. 41
The investigation of a doubly curved shell for asymmetric loading probably
approaches the limit to which analytical solutions are efficiently employed.
4.4 Shells of Translation
151
Computer-based numerical techniques have been used extensively for rotational shell analyses,42 but the independent check of selected cases, with such
analytical solutions that are available, serves to establish confidence in the
numerical procedures. Also, note that a comprehensive collection of analytical
solutions to rotational shell problems has been compiled in a semi tabular form
by Baker et a1. 43 and provide a valuable resource for the design engineer.
In addition to illuminating some of the difficulties encountered in solving an
unsymmetrically loaded shell of revolution, equation (4.155) also provides a
good basis for discussing a widely used and apparently logical approximate
solution technique for such shells. That is, to replace the meridian with a
combination of cylindrical, conical, and spherical segments, all of which can be
solved for the general harmonic. Although this approach seems reasonable
when comparing corresponding coordinates along the meridian, it should be
noted that the definitive geometric parameters present in equations (4.156),
most notably RIP' are not necessarily approximated closely by these simpler
curves. With mathematically based, computer-implemented solution techniques now available for such problems, it is recommended that ad hoc substitute
curve procedures not be relied upon.
4.4 Shells of Translation
A surface of translation may be generated by passing one plane curve over
another. The translational shell is often treated separately from the rotational
shell, especially in a theoretical context; however, there may be considerable
overlap. For example, zero-curvature shells of revolution, such as cylinders and
cones, may be formed by the translation of a straight line over a circle, or, in
the case of the cylinder, by the translation of the circle along the straight line.
Even the negative curvature hyperboloid of revolution may be produced by the
trace of a skewed line around a circle. In the context of shell analysis, it is
generally simpler to regard shells as rotational rather than translational surfaces when .there is a choice. Therefore, in our subsequent treatment, we are
only concerned with shells that do not posses rotational symmetry.
A popular use of the translational shell is to provide a roof without interior
supports over what can be a quite large plan area. Some notable examples are
shown in figures 2-8. As can be seen from figures 2-8(a) and (b), such shells are
often supported at only a few points to provide an uninterrupted open space.
Because of the wide variety of boundary conditions encountered, we may find
that the membrane theory solution, by itself, is a poor approximation to the
stress distribution in such a shell, and that bending effects will play an important
or even dominant role in the design. The bending analysis of translational shells
is discussed in chapter 9.
Further, we find it convenient to separate the membrane analysis of transla-
152
4 Membrane Theory
tional shells into two parts: (a) circular cylindrical shells; and, (b) shells of double
curvature.
4.4.1 Circular Cylindrical Shells: Single or multiple circular cylindrical shells
have been widely employed in barrel shell roofs, such as that illustrated in figure
2-8(h). Extensive discussions of this application are available. 44 Here, we concentrate on the basic features of such shells.
The geometry is defined in figure 4-37. We have attempted to remain consistent with section 4.3.3.2 and figure 4-12. An exception is the adoption of X
rather than Z as the axial coordinate to conform to the prevalent practice in
the literature. Thus, we have at: = X, f3 = (); A = 1, B = a, and Rrz = 00, Rp = a.
The shell is bounded by the two arcs at X = 0 and X = L, and the two straight
edges at () = ± (}k'
We start from the membrane theory equilibrium equations, (4.2), with the
appropriate geometric parameters inserted:
N x ,x
Sx
,
1
+ -S
(J + qx(X, 8) = 0
a'
(4. 160a)
1
+ -a
N(J (J + q(J(X, ()) = 0
'
(4. 160b)
(4. 160c)
If we compare equations (4.160a-c) with equations (4.143), we find that they
are identical except for the loading terms.
As such, with reference to section 4.3.7.4, the equations are readily solved to
give
N(J(X, (})
S(X,(})
Nx(X,(})
=
= =
(4.161a)
qna
f ~N(J,(JdX - f q(JdX + fl((})
-~
f
S,(JdX -
f
qx dX
+ f2((})
(4.161b)
(4.161c)
As an elementary example, we consider a circumferentially uniform but
axially varying gravity load qd(X), as shown in figure 4-37:
qx(X,(})
=0
(4. 162a)
q(J(X, (})
= qd(X) sin ()
(4. 162b)
qn(X, (})
= - qAX) cos ()
(4. 162c)
Proceeding, we find
153
4.4 Shells of Translation
z
qd
,IIIl
q=qE~
I l/q,,=-qcf
~
d
~
~..
I
OSS
l/qu
I
I
qs=q cos8sinS
ddY =~cosS
u
I ~ I
Sy
~~jq =-q cos 2S
ds y
I
n
dY
Uniform Load
Gravity Load
Fig. 4-37
Open Cylindrical Shell
u
154
4 Membrane Theory
Ne
= -aqAX) cos ()
(4.163a)
Ne,e = aqAX) sin ()
S
= -2sin8 f qAX)dX + fl«())
(4.163b)
S,e= -2cos() fqAX)dX+fl«()),e
Nx =
-~{ -2cos() f[f qAX)dX JdX + fl«()),e f
dX}
(4.163c)
+ f2(8)
We must now inquire further into the longitudinal distribution of the dead
load qAX). There are two cases of practical interest: (a) uniform distribution
qAX) = q = constant
and (b) periodic distribution
qAX)
=
L
00
j=O
'.
jnX
q~SlDL
For the latter case, the function qd(X) may be considered as an odd function
on the interval - L ::; X ::; L, as shown in figure 4-38, whereupon
qAX)
=
jnX
L qa S l DL j=l,3,odd
00
••
(4. 164a)
where the Fourier coefficients are
.
q~
2
=-
L
fL qAX)SlD-dx
. jnX
L
0
(j
= 1,3, ... odd)
(4.164b)
As an example we may approximate the uniform distribution qd by the first
term (j = 1) of the Fourier series, for which
...
-L "
---
-
Fig. 4-38
Odd Function
155
4.4 Shells of Translation
qJ =2qd
L
i
L
0
. nX
slD-dX
L
(4. 165a)
4
=-qd
n
and
qAX)
~
4
. nX
-qd SlD n
(4. 165b)
L
Finally, we evaluate the stress resultants, equations (4.161a-c), for both a
uniform dead load qd = constant, and for the first term approximation of qd as
given by equation (4. 165b). We proceed with the integration from the boundary
X = 0, and assume that the shell is simply supported; e.g., Nx(O) = Nx(L) = O.
For qd(X) = qd = constant, we have from equations (4. 163b) and (4. 163c)
S = - 2qdX sin 0 + 11 (0)
1
(4. 166a)
2
N x = - - [ -qdX cos 0 + XII (0) 9] + 12(0)
a
'
(4. 166b)
Applying the boundary conditions to equation (4.166b),
12(0) = 0
11 (0),9
= qdLcos 0
from which
/1(0) = qdLsinO
and
S(X,O) = qdL ( 1 Nx(X,O) =
2:)
sin 0
-qd~X( 1 - ~)COSO
(4. 167a)
(4. 167b)
Also, equation (4.163a) becomes
N 9(X,0) = -qdacosO
(4. 167c)
For the single term Fourier series approximation, we substitute equation
(4. 165b) into equations (4. 163b) and (4.163c) and carry out an analogous integration, which gives
nX.
S = -Sqd(L)
- cos-smO
+ /1(0)
n n
L
1 [Sqd
(L)2
. nX
] +12«(})
Nx = -sm-cos(}+X/l«(})9
ann
L
'
(4. 16Sa)
(4. 16Sb)
156
4 Membrane Theory
Table 4-1
Comparison of maximum stress resultants
Coefficient
Maximum Stress Resultant
Common Factor
Exact
Approximate
qdLsin8
-qd(L/a) cos 8
-qdacos8
1
L/4 = 0.2SL
1
S/n 2 = 0.S1
SL/n 3 = 0.25SL
4/n = 1.27
S(O)
N x (L/2)
NiJ(L/2)
We first apply the boundary conditiol1 Nx(O) = 0, from which f2(0) = O. Also,
we note that the loading qiX) is symmetrical about X = L/2, so that S(L/2) =
O. Then, from equation (4. 168a),fl (0) = 0, and the stress resultants Sand N x are
S(X, 0)
8qdL-
nX.
= ~cosLsmO
8qdL2 . nX
Nx(X,O) = --3-sm-cosO
na
L
(4. 169a)
(4.169b)
Also, equation (4.163a) becomes
4qda . nX
N(J(X,O) = ---sm-cosO
n
L
(4. 169c)
It is interesting to compare the stress resultants computed from the actual
loading qd with those evaluated from the single term Fourier approximation.
Note the maximum stress resultants S(O), N x (L/2), and N(J(L/2), as shown in
table 4-1. From this comparison we see that except for S (0), the maximum stress
resultants are conservatively computed from the one term solution. If further
refinement is desired, additional terms of the series may be included. 45
An interesting aspect of this solution was the enforcement of the symmetry
boundary condition, S(L/2) = O. In general, the appropriate boundary conditions, in addition to Nx(O) = Nx(L) = 0, would be constraints which can
develop the shear stress resultant S at the supports.
For this problem, the end point values of S are easily found from either
equation (4. 167a) or equation (4. 169a) as
S(O,O)
=
S(O,O)
= -
-S(L,O)
= qdLsinO
(4. 170a)
or
S(L, 0)
8qd L .
= - 2 sm 0
n
(4. 170b)
respec~ively.
If the symmetry situation was not recognized or enforced, the appropriate
kinematic conditions would have been
(4. 170c)
4.4 Shells of Translation
157
The direct application of the later constraints creates a statically indeterminate membrane theory problem, since the stress resultants cannot be evaluated without consideration of the displacements. This is discussed further in
section 6.3.4.1.
Another case of practical interest is that of a load acting in the Z direction
and uniformly distributed in the Y direction, quo The resolution of this load is
shown in figure 4-37.
qx(X,e)
=0
(4.171a)
q/J(X, e)
= qu cos e sin e
(4. 171b)
qn(X, e)
= - qu cos 2 ()
(4.171c)
A parallel set of solutions for the stress resultants, using both the actual loading
and the Fourier series approximation, can be derived in a straightforward
manner.
When the load qu is uniformly distributed in the X direction as well, it is then
constant over the X - Y plane. This form is commonly used to represent snow
loading on a shell roof.
We now inquire into the accuracy of the membrane theory stress analysis for
a barrel shell. Obviously, the cases we have discussed thus far satisfy the
requirements of smooth loading and regular geometry. When it comes to the
adequacy of the physical boundaries to meet the requirements ofthe membrane
theory, however, we need a closer examination.
The transverse boundaries at X = 0 and X = L have been assumed to be
simply supported, implying that N x should vanish. Moreover, our analysis
indicates that the maximum value of S must be developed there, as shown by
equations (4.170). These conditions suggest the use of a diaphragm that is
perfectly flexible in the X, or longitudinal direction, but infinitely rigid in the
e, or circumferential direction. A typical practical solution is shown in figure
2-8(h), where a slight overhang beyond the diaphragm is also provided. In
general, the membrane theory condition is considered to be fairly well approximated by such a support.
Now we examine the longitudinal boundaries at e = ± ()k' The solution
requires N/J(ek ) and S(ek ), as evaluated from equations (4.167) or (4.169), to be
developed on the boundary. Furthermore, the boundary must be free to displace in the normal direction so that no transverse shear will develop. In
practice, several situations are found. 46 Two are illustrated in figure 2-8(h),
where the outer edge is free and the interior edges are continuous with the
adjacent barrel, but unrestrained in the vertical direction. Also, supplementary
stiffening beams have been employed in some structures. However, unless a
continuous support, such as a bearing wall or a stiff frame, is provided beneath
all longitudinal edges, there will be a serious violation of the membrane theory
conditions and a resulting drastic alteration of the load-resisting mechanism.
This situation is easily understood by reference to figure 4-39, where we can
158
4 Membrane Theory
Supported Longitudinal
Boundary
Fig. 4-39
Unsupported Longitudinal
Boundary
Longitudinal Boundary Conditions for Cylindrical Shells
see that the membrane theory solution implies that the shell resists transverse
loading primarily as a series of circular arches. In contrast, where supports are
not provided along the edge, the shell must resist transverse loading primarily
by flexural action in the longitudinal direction. Because the stress resultant N x
must develop the required longitudinal resisting couple at any section, the value
computed from membrane theory will be considerably altered by bending, as
we will see in chapter 9. Since the contribution of N(J( ± lU in resisting the
vertical load will be lost, the vertical equilibrium must be provided by S(O) and
S(L). From these heuristic observations, we may anticipate a significant readjustment of all in-plane stress resultants, as well as the possible addition of
significant bending moments and transverse shear forces from the solution of
the general equations for open cylindrical shells.
4.4.2 Shells of Double Curvature: If we refer to the shell element shown in the
center of figure 4-40, the membrane theory equilibrium equations may be
directly written from equations (4.2a-c) by selecting the arc lengths as the
curvilinear coordinates. Therefore, we have IX = Sx and f3 = sy- However, a
somewhat different approach is widely used because it simplifies the subsequent
mathematics. 47 ,48 First, we choose the Cartesian coordinate system X - Y-Z as
shown on the figure. Then, the differential area element ds x dsy located at
Z(X, Y) is mapped onto the lower element dXdY in the X-Y plane. Also, a
shorthand notation for the incremental change of stress resultants is introduced
on the figure: N x+ = N x + Nx,xdx, etc.
Projected stress resultants N x , Ny, and SXY are defined, such that the X - Y
components of the resulting forces on the edges ds x or dsy are equal to the total
159
4.4 Shells of Translation
f-y
I
Mapped Element
(NyZ.ydXt
Z Components
(NXZ'xdYt
X
Shell Element
Z(X,Y)
Mapped Element
X and Y Components
Fig. 4-40
Shell Element and Mapping into Cartesian Coordinates
forces on the corresponding mapped edges dX or dY. On the shell element, we
have noted the local tangents
tan yx(X, Y)
= Z,x
tan yy(X, Y)
=
Z, y
which are useful in what follows. Proceeding, we set
NxdY = Nxdsycosyx
}
(4.172)
160
4 Membrane Theory
SXY dY
= S dsy cos yy
Ny dX
=
SXY dX
Ny ds x cos '}'y
= S ds x cos Yx
Since
dY
dsy
=
dX
-d
Sx
= cOSYx
-
}
cosYy
(4.173)
then
(4. 174a)
N
y
S
= Ny cosYx
(4.174b)
=
(4. 174c)
cos Yy
SXY
Now, we derive the equilibrium equations for the mapped element. After
solving for the projected stress resultants, the actual membrane resultants are
recovered from equations (4. 174a-c). Since SXy = S from equation (4. 174c), we
drop the subscript on this term.
Summing forces in the X and Y directions, we have
(Nx
+ Nx.xdX)dY -
NxdY
+ (S + S,ydY)dX
- SdX
+ qxdX dY =
0
or
Nx,x
+ S,y + qx =
0
(4.175a)
Ny,y
+ S,x + qy =
0
(4.175b)
and
To form the third equilibrium equation, we project onto the plane the Z
components of the forces corresponding to the actual stress resultants, as shown
on the top element in figure 4-40.
We have from N x
.
N x d sysmyx
=
cosYy dY .
Nx----smyx
cos Yx cosYy
= Nxtanyx dY
and from Ny
(4.176a)
161
4.4 Shells of Translation
.
cosYx dX .
Nydsxsmyy = Ny----smyy
cosyy cosYx
= Ny tan yy dX
(4. 176b)
= NyZ,ydX
From the shear forces, we find
· yy = S --sm
dY . yy
S d Sy sm
cosyy
= Stan yydY
(4.176c)
= SZ,ydY
·
S dX .
S dSx SIn Yx = --sm Yx
cOSYx
(4.176d)
= Stanyx dX
= SZ.x dX
Summing the projected forces in the Z direction,
(NxZ,x),x
+ (NyZ,y),y + (SZ,y),x + (SZ,xb + qz = 0
(4.177)
It is convenient to expand and rearrange equation (4.177) as
Z,xxNx
+ 2Z, xy S + Z,yyNy + Z,x(Nx,x + S,y)
+ Z,y(Ny,y + S,x) + qz =
0
(4.178)
We then replace the terms in parentheses in equation (4.178) by qx and qy from
equations (4. 17Sa) and (4. 175b), leaving
(4.179)
as the third equilibrium equation.
We now introduce the Pucher stress function $l',49 which is defined such that
/F,yy = Nx +
/F,xx = Ny
-/F,XY
=S
+
f
qx dX
(4. 180a)
f qydY
(4. 180b)
(4.18Oc)
It is easily verified by substitution that if $l' is a continuous function with
continuous partial derivatives, equations (4,175a) and (4.175b) are identically
satisfied. Therefore, the membrane theory stress analysis for shells of double
curvature reduces to the single equation
162
4 Membrane Theory
Z,xx!#',yy - 2Z,xy~xy
+ Z,yy!#',xx
= -qz + Z,xqx + Z,yqy + Z,xx
f qx dX + Z,yy f qydY
(4.181)
The mapping onto the X - Y plane, and the subsequent introduction of the
Pucher stress function, has produced a single equation which encompasses
a large class of shells. Once the stress function Ji' is evaluated, the projected
stress resultants are computed from equation (4.180), and the actual resultants
from equation (4.174). The mathematical complications in solving equation
(4.181) for a given geometry, and the resulting properties of the solution, are
dependent on the type of the surface equation Z(X, Y)-e.g., elliptic, hyperbolic, or parabolic-and the complexity of the applied loading, qx, qy, and qz,
as specified in the Cartesian coordinate system.
For the purposes of illustrating the Pucher stress function technique, we
consider three representative surfaces encountered quite often in shell roof
structures: (a) hyperbolic paraboloid, (b) elliptic paraboloid, and (c) conoid.
4.4.3 Hyperbolic Paraboloids
4.4.3.1 Surface Properties. The hyperbolic paraboloid (HP) is perhaps the most
fascinating of the translational shells. The basic form is generated by passing a
convex parabola over a concave parabola, producing an anticlastic surface, as
shown in figure 4-41. For obvious reasons, such a hyperbolic paraboloid is
commonly referred to as a "saddle" shell.
trace of vertical plane
Z
/
OIl
I
Fig. 4-41
Hyperbolic Paraboloid Saddle Shell
XY plane
163
4.4 Shells of Translation
The equation of the surface shown in figure 4-41 is
(4.182)
We observe that the hyperbolic paraboloid is a surface of negative Gaussian
curvature. This is easily verified from equation (2.38) with Z.xx = 2(cx/a2),
Z.yy = -2(c y/b 2) and Z.Xy = 0, from which the discriminant 0 = Z.xxZ,Yf(Z.xy)2 < O.
The stress analysis of the basic form of the hyperbolic paraboloid shown in
figure 4-41 is postponed so that we can first consider a more common configuration. We observe from figure 4-41 that the surface has negative Gaussian
curvature. Referring to the classification in figure 2-7, we expect that there
should be two sets of straight lines on the surface. Mathematically, these families
of straight lines are called the real characteristics. In order to arrive at the
equations for the characteristics, we use a construction suggested by Billington. 50
First, we pass an arbitrary vertical plane through the shell. This plane
intersects the X - Y plane along the straight line
X
=
a1
-
(::)
Y
(4.183)
shown as a dashed line in figure 4-41. In equation (4.183), a 1 and blare the
intercepts of the line on the X and Yaxes, respectively, and -adb 1 is the slope
of the line. Next, we locate the intersection of the vertical plane with the shell
surface. Since the X - Y projection of the intersection is the straight line given
by equation (4.183), we substitute the latter equation into equation (4.182) to
get the Z coordinate of the intersection of the plane and the surface as a function
of Y:
(4.184)
Equation (4.184) describes a straight line if the coefficient of y2 vanishes, which
occurs when
a1 = + [(c y/b 2)]1/2
b1
-
(cx/ a2 )
(4.185)
Recalling that (-adbd is the slope of the projection of the intersection onto
the X - Y plane, the families of straight lines on the surface are defined by those
vertical planes intersecting the X - Y plane which produce traces with slopes
given by the r.h.s. of equation (4.185). Finally, we substitute the computed
(adb 1 ), equation (4.185), into the equation of the intersection with the surface,
equation (4.184), to get the equations of the families of straight lines, or characteristics
164
4 Membrane Theory
Z
+2..j(cX c y ) y
ab a l
=
+ Cx
2
a2 a l
(4.186)
Through any point on the surface pass two characteristics. To identify the
straight lines passing through a given point (X, Y), we substitute, X, Y and
equation (4.185) into equation (4.183) and find
_ _ - _ [(C y /b 2)]1/2_
al
X
-
+
y
(c x /a 2 )
(4.187)
Then, each of the computed values of al is substituted into equation (4.186) to
obtain the equations of the intersecting straight lines.
Now that we have identified two families of intersecting straight lines on the
surface, a new possibility arises. Instead of visualizing the shell as produced by
the translation of one parabola over another, why not generate the shell by the
translation of one or the other of the sets of straight lines? Correspondingly, we
could choose our reference coordinates to coincide with a pair of the straight
lines. This is especially attractive if the boundaries of the shell coincide with
these straight lines.
To explore this possibility in terms of our previous equations, we restrict our
discussion to a hyperbolic paraboloid with a square plan-form, for which b = a
and C x = Cy = c. Then, the slopes ofthe straight lines on the surface in the X - Y
plane are computed from equation (4.185) as
(:J
=
(4.188)
±1
Therefore, the families of straight lines are orthogonal. and we can define a new
coordinate system X - Y at 45° from the X - Y system, as shown in figure 4-42.
The more general transformation is treated in Billington. 51
The relationship between the original X - Y coordinates and a new pair
rotated 45° is easily derived from figure 4-42:
X
y
= X cos 45 - Y sin 45
=
X sin 45 + Y cos 45
(4.189)
which, upon substitution into equation (4.182), gives
Z
=
c2 [X 2 cos 2 45
a
+
ji' 2 sin 2 45 - 2XYcos45 sin 45
(4.190)
Now, we choose new boundaries for the surface that are parallel to the straight
lines, instead of to the parabolas. These are indicated by dashed lines on figure
4.4 Shells of Translation
165
x
Fig. 4-42
Coordinate Rotation for Hyperbolic Paraboloid
4-42 as EF, FG, GH, and HE. The plan dimension referred to in the rotated
coordinate system is d = (.J2/ 2)a, so that we may rewrite equation (4.190) as
-2c
Z
_
= (2d/ .J2)2XY
-c __
=
d 2 XY
(4.191)
=kXY
where k = -c/d 2 •
We show a quadrant of the hyperbolic paraboloid defined by equation (4.191)
in figure 4-43. W may visualize this surface as being formed by a straight line
parallel to the X axis, called a generator, translated between the two straight
lines OJ and KG, known as directrices, where point G has been displaced
vertically a distance - c from K in the Z direction. This construction produces
a warped hyperbolic paraboloidal surface with straight boundaries.
Hyperbolic paraboloids of this general form have become a symbol of grace
and elegance in architectural applications of thin shell roofs. Popularized by F.
Candela in Mexico,52 several examples of actual shells are shown in figures
2-8(a- e). A noteworthy construction feature ofthis surface is that the formwork
for reinforced concrete shells may be fabricated entirely from straight pieces.
166
4 Membrane Theory
z
directrix
G
Fig.4-43
Hyperbolic Paraboloid Formed by Straight-Line Generators
Various combinations of four quadrants, similar to those shown in figure 4-43,
are depicted in figure 4-44. These basic forms are referred to freely in the ensuing
discussion. In turn, these units may be combined into larger roofs, such as the
multiple saddles in figures 2-8(a) and (e).
4.4.3.2 Stresses in HP Shell with Straight Boundaries. Now, we proceed to the
stress analysis. We rewrite the governing equation for the Pucher stress function, equation (4.181), to match the X-Y coordinate system:
Z,iiil',Yf - 2Z,ifil',if
+ Z:ffil',ii
= -qz + Z.-fqf + Z,Hi + Z.ii
f
qi dX
+ Z,ff
f
(4.192)
qf dY
with equations (4.180) correspondingly redefined. Then, from equation (4.191),
Z.i
= kY
Z,f
= kX
Z,ii = 0
167
4.4 Shells of Translation
(b) Hipped
(0) Umbrella
0 K
J
ro~
View 00
(c) Cant iI ever
(d) Saddle
~
G
View cc
e....J
A
G'
G'
G~
tie
~
G K
View ff
View
dd
~JG
View ee
Fig.4-44 Hyperbolic Paraboloids with Straight Boundaries (a) Umbrella, (b) Gable (Hipped), (c)
Cantilever, (d) Saddle
Z,yy = 0
Z,Xy
=
k
whereupon equation (4.192) becomes
-jF'x-y-
,
1
_.;-;
= -(
-qz + kXqy + kIqx)
2k
(4.193)
168
4 Membrane Theory
Now, referring to equation (4. 180c), we see that
1
__
-Ji',Xy = S = 2k( -qz + kXqy + kYqx)
(4.194)
Note that there are no terms in S that can be affected by the boundary
conditions. Rather, the boundaries of the shell are required to develop the
computed values at X = d or Y = d.
We now proceed with the solution of equation (4.193) for $'. First, we take
a/ax and integrate with respect to f, giving
Ji',xx
If [-qz,x + k(Xqy
- + Yqx),x]
- + 11 (X)= - 2k
dY
= Ny +
(4.195)
f qydY
by equation (4. 180a). Similarly,
Ji',yy
= - 21k
= Nx +
f
f
[-qz,y
+ k(Xqy + Yqx),y] dX + 12(Y)
(4.196)
qx dX
From equations (4.194), (4.195), and (4.196), we may write explicit expressions
for the projected stress resultants, and then evaluate the actual stress resultants
from equation (4.174), with X and Y replaced by X and f. The functions of
integration, 11 (X) and 12(Y)' can then be determined from the specified boundary conditions along X = ± d and Y = ± d.
Frequently, when the hyperbolic paraboloid is used as a roof and subjected
principally to a distributed vertical loading, further assumptions are justified.
A first step is to consider only the Z component of loading, which greatly simplifies the integrals in equations (4.195) and (4.196). A further step is to assume
that the remaining load qz is constant and uniformly distributed over the projected surface. If this is the case, equations (4.194), (4.195), and (4.196) reduce to
S = -qz
2k
(4.197)
and
Ny = 11 (X)
(4.198a)
N x = 12(Y)
(4. 198b)
Referring to figure 4-43, if the boundary X = 0 or X = d is assumed to be
stress-free in the X direction [Nx(O, Y) = 0 or Nx(d, Y) = 0], then as Y varies
along that boundary, N x = 12(Y) = O. Similarly, a free boundary at Y= 0 or
Y = d gives 11 (X) = 0, so that the expressions for the stresses on the hyperbolic
4.4 Shells of Translation
169
paraboloid simplify to
(4.199)
and
Nx
=
Ny
=0
(4.200)
where x and yare curvilinear coordinates on the shell surface corresponding
to X and Y on the projected surface.
A loading of the form qz = constant is frequently termed live load and is often
taken to represent snow loading, as discussed for cylindrical shells in section
4.4.1. For shells that are relatively shallow, a uniform distribution for qz may
also be used to approximate the dead load closely. If a uniform load approximation is used for the dead load, an equivalent value qz should be computed
by dividing the total shell weight by the plan area. This will ensure that overall
statics will not be violated.
We now reflect on the remarkably simple stress distribution found for a
uniform load on a hyperbolic paraboloid. Equations (4.199) and (4.200) indicate
a state of pure shear in the shell. To compute S, it is necessary only to evaluate
k = -cjd 2 and divide qz by 2k. In designers' jargon, this is termed a "back of
the envelope" calculation. But, one should not be deceived by the apparent
simplicity of the mathematics. We must inquire further into the physical
boundary conditions which have to be provided to sustain the shell in this
simple-to-calculate state of stress.
As an example, consider a uniform load qz = - Pu, constant per unit area of
projected surface, acting on the umbrella shell shown in figure 4-44(a). The shell
is shown in detail in figure 4-45, where the four segments are numbered. From
equation (4.199) and figure 4-45, k = -c/d 2 and
S = -(-Pu) = -Pu d2
-2(c/d 2 )
2c
(4.201)
The sense of S corresponding to the negative value in equation (4.201) is
obtained by referring to figure 4-40, and is shown in figure 4-45(a). An enlarged
view of quadrant 1 is shown in figure 4-45(b), with the shell proper separated
from the edges OJ, JG, GK, and KO. Clearly, the uniform shear S must be
developed by axially loaded edge members along each of these lines. Edge OJ
is subject to a tensile force T(Y), which increases from 0 at point 0 to a
maximum value of T(d) = Sd at J, where it is balanced by an equal tension
coming from the edge member of adjacent quadrant 2. An identical tensile force
is present along edge OK. The situation along the interior edge members JG
and KG is somewhat different. Considering JG, the compression C(X) contributed from quadrant 1 builds from 0 at J to a maximum of S.j(d 2 + c 2 ) at G.
Adjacent quadrant 2 contributes an identical force, so the maximum com-
170
4 Membrane Theory
(0 )
K
J
~
c
~
2S ~ G~2S
(c)
View 00
(b)
C(d)
Fig. 4-45
C(d)
Umbrella Shell
pression C(d) = 2S.J(d2 + c 2). An identical situation occurs along KG and the
other two interior edge members. Although the horizontal component of the
maximum compression at G for each interior edge member, C(d)[d/.J(d 2 + c2)],
is balanced by the similar component from the opposite interior edge member,
171
4.4 Shells of Translation
the vertical components of these four forces, 4C(d) [e/.j(d 2 + e 2)], is transmitted through the column to the foundation. Of course, the total vertical force
4[2S.j(d 2 + e 2)] [e/.j(d 2 + e 2)] = 8Se should balance the total applied load
4Pud2. From equation (4.201),
8Se =
8Pu d2e
= 4pu d 2
2e
(4.202)
We observe from figure 4-45(c) that the compressive forces flow naturally to
the support. Also note that the internal self-balancing of the maximum tensile
forces and the horizontal components of the maximum compressive forces
occurs only if the dimensions and loading are identical for the four segments. Any unbalanced loading will produce a resultant shear and overturning
moment, which would have to be resisted by the single column. For this reason,
the single free-standing umbrella is usually restricted to rather modest dimensions. The umbrella has also been used in multiple repetitions to cover large
plan areas where occasional columns are acceptable. 53 In such applications,
greater lateral stability can be provided through frame action of the interconnected shells.
The stress pattern for any of the other cases shown in figure 4-44 is easily
obtained by locating the corresponding quadrant OJGK on each figure. Then,
the forces in the edge members are computed by starting from a point of
assumed zero axial force, usually the junction of the edge member with an
external boundary.
Next, consider a dead load Pd' constant per unit area of middle surface.
Referring to figure 4-40,
-pddsxdsy = qzdXdY
From equation (4.173), dX/ds x = cos /'x and dY/ds y = cos /'y, so that
qz =
-Pd
cosYx cos /,y
(4.203)
--~'-----
In terms of the
X - 1" coordinates
tanyx = Z,i = k1"
tanyy = Z,y = kX
and
1
1
cos Yx = .j(1 + tan 2 Yx) = .j(1 + k21"2)
1
cos yy = .j(1 + tan 2 yy) = .j(1
so that
1
+ k 2X2)
)
(4.204)
172
4 Membrane Theory
qz
=
=
-Pd[(1
-Pd[1
+ k21"2)(1 + k2X2)]1/2
+ k2(X2 + 1"2) + k4X21"2]1/2
(4.205a)
For most practical cases, k4 « k 2, and the last term in equation (4.205a) can be
neglected. We may also approximate the remaining square root term with the
binomial expansion
qz
~ p{ 1 + ~k2(X2 + 1"2) + ...
J
(4.205b)
Substituting equation (4.205b) into equation (4.199) gives
S=
-qz
2k
= Pd
2k
[1 + ~k2(X2 +
2
1"2)J
(4.206)
Noting that !#',iy = -S, we integrate equation (4.206) to get
ff =
~:d[ X1"+ ~2 (X 3 1" + 1" 3 X) + f1(1") + f2(X)J
(4.207)
Then, we compute the extensional stress resultants from equations (4. 180a) and
(4. 180b) as
Ni
=
!#',yy
=
-Pd[k 2 X1"+f(1")--]
2k
1
,ff
(4.208)
and
Ny
= !#',ii
(4.209)
Whereas S is independent of any boundary conditions, just as in the case of the
uniform loading, the extensional stress resultants are dependent on the boundary conditions. Also, since we may specify Ni amd Ny on only one of the two
parallel boundaries, the opposite boundaries must develop the computed stress
resultants. If, for example, we consider the umbrella shell shown in figure 4-45,
it is reasonable to presume that these stress resultants vanish on the exterior
boundaries, so that
Ni(O, 1") = 0
(4.21Oa)
=0
(4.210b)
Ny(X,O)
giving
f1(1"),Yf = f2(X),ii
=
0
(4.211)
4.4 Shells of Translation
173
The expressions for the dead-load stress resultants in this shell are evaluated
from equations (4.206), (4.211), (4.208), (4.209), (4.174), and (4.204).
(4.212a)
(4.212b)
(4.212c)
The forces in the edge members are computed in a similar manner as for the
uniform load case. However, since S = SeX, Y), we must proceed more formally.
Referring to figure 4-45,
IY
T(Y)
=
C(X)
=2
(4.213a)
S(O, Y)dY
I
i
o
SeX, d) dX
(4.213b)
cos Yx
Note that the compressive force C(X) is evaluated in equation (4.213b) by
integrating along the inclined edge member. The integrands for equations
(4.213a) and (4.213b) are computed by substituting X = 0 and Y = d, respectively, in equation (4.212a); then, the maximum values of T and C are found by
inserting Y = d and X = d as the upper limits of integration.
Earlier, we alluded to the possibility of replacing the dead load which is
constant over the shell surface, with a uniform load which is constant over the
projected area. A comparison of the solutions for the loadings Pu and Pd' as
given by equations (4.201) and (4.212), provides an obvious motivation in terms
of algebraic simplicity if the errors introduced are not significant. As mentioned
before, if we replace Pd by an equivalent Pu, we should use an adjusted load qz,
which is the total weight of the shell divided by the plan area, to satisfy overall
statics. We find the total weight of the shell using equation (4.205b) as
Q=
S: S: p{
1
+ ~k2(X2 +
Y 2)JdX dY
(4.214)
and then compute the equivalent uniform load
(4.215)
for use in equation (4.199). The effect of this simplification on the accuracy of
the ensuing solution is not great for many practical geometries.
An interesting alternative to the free edge boundary conditions stated in
equation (4.210) is implied in a design example based on an actual shell having
174
4 Membrane Theory
the basic gable form shown in figure 4-44(b).s4 If the exterior edges KG and GJ
were taken as stress-free, the integration functions could be readily evaluated
by substituting
Nx(d, Y)
=0
and
Ny(X,d) = 0
into equations (4.208) and (4.209), respectively. However, the subsequent
expressions for N x and Ny, analogous to equations (4.212b) and (4.212c), would
give nonzero (tensile) values along the ridge lines OJ and OK; that is,
Nx(O, Y) > 0
and
Ny(X,O) > 0
The horizontal components of these stress resultants would be balanced by
identical forces from the adjacent segments, but the vertical (Z) components
would add to produced an unbalanced force that would have to be sustained
by the edge member. This is illustrated in the inset of view bb in figure 4-44(b).
To avoid this situation, the designer apparently decided it was preferable to
assume that N x and Ny vanish along the ridge lines, and then to develop the
calculated values of the stress resultants on the perimeter. For this case, Nx and
Ny become compressive as well. We surmise that occasionally there is some
latitude in specifying boundary conditions in a membrane theory solution, and
that the subsequent problem of providing physical constraints consistent with
the assumed conditions should be kept in mind. This shell is discussed further
in sections 4.4.3.3 and 4.4.3.4.
4.4.3.3 Arch Action. The state of pure shear found for uniformly loaded hyperbolic paraboloid shells can be interpreted from another standpoint. We study
the same quadrant OJGK shown in figure 4-45, but, for variety, we consider
the saddle shell illustrated in figure 4-44(d). The membrane theory stress analysis is the same, and the uniform negative shear S is shown in figure 4-46. From
elementary strength of materials, it is easily shown that a state of pure shear
with respect to the X - Y axes corresponds to principal tensile and compressive
stresses of the same magnitude acting in the X and Y directions as shown. If
we then take sections through the element parallel to the X and Y axes, we
reveal segments of the parabolic generators of the basic hyperbolic paraboloid
shown in figure 4-41. The convex parabola MN parallel to the Yaxis acts as a
compression arch subject to a uniformly distributed load Pu, producing a constant axial compression of S per unit width. This is the elementary problem of
the second degree parabola subject to a uniform load per unit projected length.
Now, examining the parabola parallel to the X axis PQ, we find a similar
175
4.4 Shells of Translation
Compression Arch
s
Tension Arch
s
s
..Fig. 4-46
-
s
s
s
Arch Action for a Uniformly Loaded Hyperbolic Paraboloid
situation, except that we have a concave parabola forming a tension arch. Each
parabola imparts a constant reaction S per unit width of shell on the edge
members, e.g., at points Nand P on JG. Of course, the other ends of the
parabolas, indicated by dashed extensions, will intersect edge members in
adjacent quadrants.
To complete the arch representation, we investigate the edge forces. Considering a typical point on the boundary (e.g., N), observe that the unit width
compression arch imparts a force of S in the + Y direction, whereas the tension
arch acting on the same segment of the edge beam also produces a force of S
directed along the + X axis. The intersecting arches are coplanar at N, and we
may resolve the intersecting forces into components along and normal to the
176
4 Membrane Theory
edge member. The normal components cancel, while the axial components
produce a resultant force of ..)2S. This force acts over a length of edge beam
= ")2, so that we find a constant force S per unit length of edge beam as before.
Thus, the arch representation of the hyperbolic paraboloid is certainly equivalent to the previous shell solution, and often provides valuable insightespecially if irregular shapes or boundaries are encountered.
The gable or hipped HP, figure 4-44(b), which was discussed in section 4.4.3.2,
should carry load primarily by compression arches parallel to OG and tension
arches parallel to KJ. Subsequent studies by Shaabon and Ketchum 55 have
shown that the tension arches are ineffective, and that practically the entire load
is transmitted to the corners by the intersecting compression arches passing
through the crown. Of course, the stress level is essentially doubled. This
behavior is attributed to the inability of the edge members to resist the tensile
forces in the shell; rather, they displace inward, forcing the load to flow to the
corners. 54,55
4.4.3.4 Edge Members. We have seen that the membrane theory stress analysis
for hyperbolic paraboloids with straight-line boundaries is fairly straightforward. We have also established the necessity of developing the in-plane
shear forces on the boundary. We now look at the requirements for ideal edge
members:
1. In order to develop the shear fully, the edge members should be inextensible.
2. Since the entire weight of the shell is ultimately carried to the foundation
through only four to eight compression members for the cases illustrated in
figure 4-44, the edge members should be capable of sustaining sizable axial
forces. These forces will produce corresponding strains that will conflict with
the first requirement.
3. The edge members should be self-supporting and should not be carried by
the shell. Since these members may be rather sizable, they would impart a
sizable additional load on the shell that might be unacceptable.
From this list, it is obvious that in practice it may be difficult to attain any of
these ideal conditions. As a specific illustration, consider the cantilever shell
shown in figure 4-44(c). For this shell, we have four different edge members, as
seen in table 4-2. Typical sections for such members in rs:inforced concrete shells
Table 4-2
Maximum forces in edge members
Member
Location
OK
KG
Exterior
Exterior
Ridge
OJ
JG
Valley
Maximum Force
(+ = tension)
-Sd@O
-S-J(c 2 + d 2 ) @ G
+2Sd @O
-2S-J(c 2 + d 2 ) @ G
177
4.4 Shells of Translation
Downturned
Upturned
Exterior
Ridge
Volley
Eccentricity
Fig.4-47
Edge Members for Hyperbolic Paraboloids
are shown in figure 4-47. The dashed lines indicate possible smooth transitions
between the shell and the edge member that may be desirable for architectural
and/or constructional purposes. The edge members may be proportioned as
struts under axial tension or compression.
Next, consider the self-weight of the edge members. Referring to figure
4-44(c), the ridge member OJ could be considered as a beam spanning between
o and J. The valley member GJ forms a triangular frame GJG' with the
corresponding member on the other half of the shell and resists the beam
reactions at 0 from the two ridge members, OJ and 0' J, in addition to their
self-weight and the shear forces imparted by the shell. Exterior member KK' is
178
4 Membrane Theory
regarded as a beam spanning between K and K' and carrying the beam reaction
at 0 from OJ, their self-weight and the shell shear forces. This brings us to the
exterior members KG and K' G', which are required to take the beam reactions
from KK' along with their self-weight and the shear forces, seemingly as
cantilevers.
Other alternatives are possible. In some cases, it is feasible to utilize exterior
walls or window wall framing to support the perimeter edge members. The shell
shown in figure 2-8(b) admits this possibility, whereas the shell shown in figure
2-8(e) obviously does not. Also, some designers might convert the self-weight
of some of the edge members~particularly those which cannot sustain themselves~to equivalent distributed loads, and include these loads in the uniform
load qz. This essentially requires the shell to support the edge beams.
Perhaps the most challenging cases from the design standpoint are cantilevered compression edge members, such as KG, or the even longer members,
such as G' G, that occur in saddle shells as shown in figure 4-44(d). A spectacular
example is shown in figure 2-8(a).
Referring again to figures 4-44(c) and 4-45, if the neutral axis of the edge beam
for member KG does not coincide with the middle surface of the shell, the
shear force acts through an eccentricity that produces a uniformly distributed
moment. For the down turned beam (figure 4-47), the moment is opposite to
that produced by the cantilevered self-weight and thus counters the self-weight
somewhat; whereas for the upturned beam, the moment would add to that
arising from the self-weight. Depending on whether tension or compression is
introduced into the edge member by the shell and the particular exterior
support condition, a countering prestressing moment can generally be produced by appropriate upturning or downturning of the member to give the
required eccentricity.
Another possibility~particularly effective for tension edge members~is to
produce mechanical prestressing with rods or cables. Also, as shown on the
examples of figure 4-44, it is desirable to use ties between the supported ends
of the inclined compression members to counter the horizontal component of
the thrust. Insofar as the reduction of the dominance of the membrane type
action and the subsequent introduction of bending into the shell and the edge
member are concerned, a potentially detrimental effect is a relative horizontal
displacement of the abutments. 56 A comprehensive discussion of the forces in
the edge members of a hipped hyperbolic paraboloid and some pertinent design
recommendations for such shells are presented in Schnobrich and in Shaaban
and Ketchum. 57
Again, referring to the hipped shell discussed in the preceding two sections,
it has been suggested that if the edge members are not effective in transmitting
the in-plane shear stresses to the supports, then they may be reduced in size to
avoid loading the shell excessively. 57 Additionally, for a continuous group of
hipped hypars, the effect of prescribing a relative displacement of the abutments consistent with the strain in a steel tie was investigated. 58 The state of
179
4.4 Shells of Translation
stress in the shell closely resembled the membrane theory pattern, but the edge
members received a nonsymmetrical loading. Limited field observations supported this characterization, as opposed to the unrestrained or undeforming
idealizations.
Finally, with respect to hyperbolic paraboloids with straight boundaries,
note that although the presentation has been pointed somewhat at reinforced
concrete shells, there are many interesting applications of this very efficient
structural form in other materials: for example, in steel 59 and wood. 60 A
cantilevered, cable-supported hyperbolic paraboloid forms the roof for the
jumbo jet hangar shown in figure 2-8(v), and metal hyperbolic paraboloid units
were used between long-span arched trusses to cover a large arena in Mexico
City constructed for the 1968 Olympic games.
4.4.3.5 Hyperbolic Paraboloid with Curved Boundaries. We now resume our
investigation of the basic hyperbolic paraboloid described by equation (4.182)
and shown in figure 4-41. We restrict ourselves to a qz loading, for which
equation (4.181) becomes
Z,xx.f/'.yy - 2Z,xy.f/'.xy
+ Z,yy.f/'.xx =
-qz
(4.216)
From equation (4.182),
2cx
Zxx=,
a2
2c y
-bl
Z,yy
=
Z,XY
=0
and
so that equation (4.216) reduces to
= 72cx ~,yy
2cy
bl.f/'.xx
= -qz(X, Y)
(4.217)
As a specific case, we take the uniform load qz = - Pu' There are obviously
multiple particular solutions, since the derivatives on fF are uncoupled. Two
possibilities are
(4.218)
and
(4.219)
180
4 Membrane Theory
Also, we may have any linear combination of ffl and ff2
ff(X, Y)
= 2 1 ff1 (X) + 2 2 ff2 (Y)
where
(4.220)
This presents us with a multiplicity of particular solutions for the projected
stress resultants, which can be written using equation (4.180) as
a2
22
Nx(X, Y)
= -2 Pu-
Ny
(X, Y)
= - -2 P
u Cy
(4.221b)
=0
(4.221c)
seX, Y)
Cx
21
b2
(4.221a)
The actual stress resultants, N x and Ny, may then be found from equation
(4.174).
For a given shell, the solution which will prevail is a function of the relative
stiffnesses of the boundaries. Referring to figure 4-41, if the boundaries at
X = ± a are fully developed in the x direction, and the shell is unrestrained at
Y = ± b, we would choose 21 = 0 and 22 = 1. This, of course, corresponds to
a tension arch in the x direction, such as we encountered in a previous section.
Similarly, 21 = 1 and 22 = 0 indicate full support in the y direction at Y = ± b,
and no restraint at X = ± a, or a compression arch in the y direction. Obviously,
with finite in-plane resistance along all boundaries, we would have nonzero
values of both 21 and 2 2, but the necessary requirement is that at least one pair
of the boundaries must develop the ensuing forces.
The possibility of "directing" the extensional stresses by adjusting the support
conditions makes this version of the hyperbolic paraboloid quite versatile.
Although a designer may prefer the state of uniform compression, 21 = 1, for
a reinforced concrete shell, a tensile state, 22 = 1, might be more desirable for
a steel structure. Numerous structures in this shape have been built using
high-strength steel cables as the principal members. In such applications, sagging cables resist the principal tension and hogging cables span in the normal
direction. A roofing system consisting of precast panels or a fabric membrane
may be applied to such a structure.
Another variation on the hyperbolic paraboloid with curved boundaries is
to leave the edges free of extensional stresses and to resist the surface loading
by shear forces along only two parallel boundaries. In turn, the resulting
reactions can be provided by a pair of axially loaded boundary arches. As an
illustration, consider the hyperbolic paraboloid shown in figure 4-41 with
boundary conditions
(4.222a)
181
4.4 Shells of Translation
Ny(X, ±b) = 0
(4.222b)
S(X, ±b) = 0
(4.222c)
Moreover, the boundaries AD and BC can develop the in-plane shear stresses,
i.e.,
(4. 222d)
S(±a, Y) -# 0
A plan view of this shell is shown as figure 4-48(a), with b assumed to be
equal to 2a. Assume Cy = 4c x , so that cx/a2 = cy/b 2 = c/a 2. Equation (4.182)
then becomes
(4.223)
These geometrical simplifications facilitate the following development. Also, the
applied loading is taken as qz = - Pu, and the sense of S( ± a, Y) is shown on
the figure so as to provide an upward ( + Z) resultant.
This problem is handily treated by first selecting a particular solution, and
then enforcing the prescribed boundary conditions through a homogeneous
solution.
Initially, the loading is assumed to be resisted by extensional stresses along
one pair of boundaries (e.g., AB and CD), as shown in figure 4-48 (b). This
corresponds to A.1 = 1 and A.2 = 0 in equations (4.220) and (4.221), for which
1 b2 =
Ny(X, Y) = --Pu2 Cy
Nx(X, Y) = S(X, Y) = 0
(a
P -~
2
2
C
)
= -N*
(4.224)
as shown on the figure. The actual reactions, which are equivalent to Ny(X, ± 2a),
act in the middle surface and may be found from equations (4.223), (4.172), and
(4.174b); however, it is convenient to proceed in terms of the projected resultants. The solution given in equation (4.224) balances the applied loading but
violates the boundary conditions stipulated in equations (4.222). Hence, it
represents only a particular solution for this system.
The homogeneous solution may be efficiently obtained through a physical
rather than a purely mathematical approach. We first identify the straight lines
or real characteristics on the surface. The slopes of these lines in the X - Y plane
are found from equation (4.185) specialized for the current geometry:
a1
b l = ±1
(4.225)
Some typical X - Y traces are shown in figure 4-48(c). We concentrate on those
which pierce the boundaries AB and CD at the reference points (X, =+= 2a),
respectively. Taking a segment of length ,J2 along each boundary (see inset,
figure 4-46), the edge forces ,J2N*, are resolved and translated along the
182
4 Membrane Theory
S(-oY)
8 ----... ~ ----.. ~ -
.-...-.. ..-.- --- C
a
Y
a
A
-
--- - ---
S(a,Y)
~
X
a
a
II
~
a
~_D
a
(a)
r---------------.---------------~---Y
X
(b)
~--------~~--+----Y
(c)
Fig. 4-48
x
Hyperbolic Paraboloid with Curved Boundaries
183
4.4 Shells of Translation
_c
B_
(d)
X
r--+----*-------~-----y
CD
(e)
(f)
X
X
Fig. 4-48 (continued)
characteristics to the X = ± a boundaries. The translated forces are equal to
N* and act over boundary segments of length ../2 along BC and CD.
Next, as shown on figure 4-48(d), additional characteristics that pass through
the points on BC and AD which were pierced by the translated forces from AB
4 Membrane Theory
184
and CD are added. To produce resultant boundary forces that are parallel to
BC and AD, self-equilibrated edge forces N* acting along the newly denoted
characteristics are supplied. These forces give a resultant force in the Y direction
equal to 2N*(,J2/2). Dividing by the length of the boundary segment, J2, we
obtain
JS(±a, Y)J
= N* = JNy(X, ±2a)J
(4.226)
as the projected boundary reactions.
It is obvious that the foregoing argument can be repeated for each point
along AB and CD, so that the additional solution represented by figure 4-48(d)
combined with the solution given by equation (4.224) fulfills the boundary
conditions stated in equations (4.222a-c).
It is easily shown that the additional solution is indeed a homogeneous
solution to equation (4.217). Allowing for symmetry, the shell is divided into
four regions in figure 4-48 (e), and the stress state for each is indicated. In regions
1 and 4, N x = Ny = ±N* and S = O. Referring to equations (4.180a) and
(4.180b), ~yy = ~xx. Since c x /a 2 = cy/b 2 , equation (4.217) is satisfied. For
regions 2 and 3, the state of pure shear-i.e., N x = Ny = 0; S = ± N*-is
obviously a solution to equation (4.217). Thus, the combination of figures
4-48(b) and 4-48(e) gives the total stresses in each region, figure 4-48(f).
Note that the discontinuities between the various regions make the membrane theory state of stress only an idealized possibility for this case. Careful
detailing and construction are required to approach this stress distribution in
an actual structure.
The Saddled orne, figure 2-8(n), is an elegant anticlastic shell with the boundary defined by a space curve. The form is developed from a sphere of about 136
m in diameter, as shown in figure 4-49(a) The floor is circular, formed by the
intersection of a horizontal plane and the sphere (b), and the roof perimeter is
the intersection of an HP with the sphere (c) This produces the definitive form
of the appropriately named Saddled orne (d) bounded by an undulating ring
beam.
The roof is a stressed cable network with precast lightweight concrete panels
placed on the cables and the gaps filled with concrete. The result is a solid,
ribbed thin shell.
The ring beam is subjected to a complex system of loading which is basically
counterbalanced by prestressing. A member of this sophistication could only
be analyzed accurately by a numerical procedure, the finite element method,
which is well within the state of the art of modern technology.
The loading on the roof is a function of the stiffness of the ring and the
restraining A-frames, located at the two low points at the ends of the center
hogging cables and visible on figure 2-8(n). A very flexible ring would minimize
the forces in the net but allow large horizontal deflections; the restraint produced by a rigid A-frame would minimize the deflections at the expense of a
larger horizontal reaction transferred from the ring. The optimized final design
produced an elastic support which controlled the deflections while resisting the
185
4.4 Shells of Translation
la)
(d)
I
/
"
..... - - - ,
,\
\
~'
~
, ...
,,'--"
/
Fig. 4-49 Calga ry's Olympic Saddledome Design Concept
Groin Arches get axial
and transverse loading
little stress in this direction
loads resisted by these arches
Fig.4-50
Groined Vault
reactions due to the prestressing, dead and live loads, creep and thermal
efTects.61
Comparing this structure with the straight boundary HPs, figure 4-44, the
evolution of this form, facilitated by high-strength cables, prestressing, and
computer technology, is striking.
An interesting extension of the basic hyperbolic paraboloidal geometry can
be achieved by intersecting two shells, as shown in figure 4-50. Each quadrant
is known as a groined vault. The basic resistance is essentially in the directions
parallel to the exterior boundaries. Along the edges, the loading is transferred
186
4 Membrane Theory
by the shell, which acts as an assemblage of compression arches, to the groin
arches, which pick up reactions from two adjacent vaults and transmit the forces
to the foundation. Again, it is desirable to provide horizontal restraint at the
base of the groin arches through a system of ties. Many variations on the
groined vault concept are found. An elegant example with a parabolic cross
section is shown in figure 2-8(f). This shell form is treated in detail by Billington,
who provides a simplified model to compute the stresses in the groin arches as
well as a detailed analysis procedure for the shell. 62
4.4.4 Elliptic Paraboloid: The elliptic paraboloid is generated by the translation of one convex paraboloid over another, as shown in figure 4-51(a). The
equation of the shell is given by
(4.227)
It is easily observed and may be demonstrated numerically using equation (2.38)
that this shell has positive Gaussian curvature, and, as such, we do not find
straight lines on the surface.
The name elliptic paraboloid comes from the fact that if a horizontal plane
Z = Z = constant (Z < 0) is passed through the shell, the equation of the
intersection
(4.228)
is an ellipse. Also, if cx/a2
X2
+
y2
)z
=
cy/b 2 = c/d 2, we have a paraboloid of revolution
= _(dC2
(4.229)
with the intersection at Z = 2(2 < 0) given by
X2
+
y2
= _(dC2
)z
(4.230)
which is, of course, a circle, since Z is negative.
Evaluating the required partial derivatives from equation (4.227) and substituting into equation (4.216), we obtain
(2Ca2
X)
:!F,yy
+ (2CY)~_
b2 .:#',xx - qz(X, Y)
(4.231)
which is very similar to equation (4.217), and suggests that this shell would resist
a uniform load qz = - Pu as an intersecting system of compression arches
parallel to the X and Y axis. It is easily shown by comparison with equations
(4.217), (4.218), and (4.221) that particular solutions of the form
187
4.4 Shells of Translation
z
Fig. 4-51
y
Elliptic Paraboloid Shell
(4.232)
and
a;;
.'1'"4
=
1 a2 2
--Pu- y
4 ex
(4.233)
will produce projected stress resultants N x = 0 and Ny = 0, respectively. However, these simple unidirectional patterns are often not compatible with the
188
4 Membrane Theory
physical situation, because it may be somewhat difficult to provide the reactions
required to develop Ny along Y = ± band/or N x along X = ± a. Rather, we
wish to explore the possibility of this system transmitting the loading directly
to the four corners, without the necessity of providing in-plane restraint along
the exterior boundaries.
The specific requirement is to find a solution to equation (4.231), such that
N x ( ± a, Y) = Ny(X, ± b) = 0 except at the corners, (± a, ± b). Obviously, it is
only possible to satisfy one or the other condition using equation (4.232) or
(4.233), so that, at best, !?3 or !?4 will serve as a particular solution. We may
add a homogeneous solution in the form of an infinite series that identically
satisfies the same boundary conditions as the selected particular solution, !?3
or !?4' Then, the amplitudes of the terms of the infinite series are adjusted so
that the total solution, homogeneous plus particular, will satisfy the remaining
boundary conditions.
As an illustration, we select
(4.234)
where
ff's
=
m
=
L
00
j=1,3,S ...
AjcoshmX cos nY
jnJc x
2a
Cy
jn
n = 2b
ffs is easily shown to be the homogeneous solution of equation (4.231), and we
have seen that !?4 produces Ny = 0 throughout. From equation (4.180b), we
observe that because of :?s, Ny is proportional to cos(jn/2b) Y, which vanishes
at Y = ± b. Thus, the boundary conditions on Ny(X, ± a) are identically satisfied.
To proceed with the solution for the stress resultants, .?4 must be expanded
in a Fourier series compatible with .?s. N x ( ± a, Y) is then written from equation
(4.180a), and the Aj coefficients are chosen to satisfy the boundary condition
N x ( ± a, Y) = 0, Then, the stress function and stress resultants are found in series
form from equations (4.234), (4.180), and (4.174). The details are rather involved
algebraically and are not given here. Complete solutions, including parametric
tabulations, are available. 63
It is apparent that the resulting membrane theory solution requires that
the boundary arches develop the in-plane shear and thereby transmit a portion of the applied load to the corners, as shown in figure 4-51(b). However,
the enforced boundary conditions prohibit direct stresses along the exterior
boundaries. The shear S builds up to a theoretically infinite value at the corners,
189
4.4 Shells of Translation
z
skewed generators
straight
generator
directrix
L
Fig. 4-52
Conoidal Shell
where a concentrated reaction acts, and the remaining load is carried directly
to the supports by arch strips AOe and BOD in a fashion similar to that of the
hipped HP, as discussed in section 4.4.3.3. It is apparent that the actual stress
state cannot be fully described by membrane theory, and, therefore, bending
effects should be considered as well. Also, the details of construction might well
include some stiffening and rounding of the corner to spread the high shear
forces dictated by membrane theory equilibrium.
4.4.5 Conoids: A surface produced by the translation of a straight line generator between two directrices- one of which is itself a straight line and the other
a plane curve- is called a conoid (figure 4-52). If the generator is perpendicular to the directrices, we have a square conoid; if not, the conoid is said to be
skewed.
Shells in the form of conoids are used for roof applications when it is desired
to maintain a relatively flat profile and still provide openings to admit natural
sunlight.
The coordinate system shown in figure 4-52 has the Y-Z plane normal to the
symmetry line of the curved directrix, whereas the X axis is taken along the
straight directrix. With respect to this coordinate system, the equation of the
curved directrix is of the form Z = f(X) and the square conoid is described by
Z(X, Y) = Y f(X)
L
(4.235)
190
4 Membrane Theory
For a skewed conoid, the surface equation is more complicated. In either case,
the conoid is not a second order surface, and the discriminant test for Gaussian
curvature, equation (2.38), is inapplicable. However, it is readily observed from
figure 4-52 that the skewed conoid has negative Gaussian curvature, because
there are two sets of straight line generators, whereas the square conoid has
zero Gaussian curvature, since the two sets of generators become coincident.
Consider a square parabolic conoid,
h(1 - ~:)
(4.236a)
=!: Y(1 _X2)
2
(4.236b)
f(X) =
and
z
L
a
We now write equation (4.181) for a uniform load qz = - Pu:
-C~~ Y) ~yy
+ C;~ X )
~Xy = Pu
(4.237)
Although, as we have observed with respect to the hyperbolic and elliptic
paraboloids, multiple solutions for stress functions can frequently be generated,
a solution of practical utility must be consistent with the physical boundary
conditions. Because of symmetry, we should have S vanish along X = 0, so that
-~Xy(O, Y) = O. Also, ifthe curved directrix cannot sustain forces normal to
the plane of the curve, Nx(X,L) = 0, implying that ~yy(X,L) = O.
Again, comprehensive results are readily available in the literature,64 so that
we will not dwell on the details of the various solutions for conoidal shells.
However, it is of general interest to consider a simplified model of the conoid
that is suggested by our earlier study of hyperbolic paraboloids. From figure
4-52, it is apparent that the conoid could possibly transmit the uniform load as
a series of arches in the X -Z plane, with the attendant thrusts developed along
the boundaries X = ± a. The provision ofthe appropriate boundary conditions
is plausible, since the horizontal components of the thrust would balance along
the interior boundaries, and the vertical components could be resisted by walls
or beams in the Y direction. Further, the shell would seemingly be entirely in
compression.
With respect to the equation (4.237), we may obviously select a solution of
the form
(4.238)
which would yield the one-way arch action with Ny and S = O. Then, if we return
to equation (4. 174a) for the actual stress resultant N", and compute
4.5 References
191
2h
tan Yx = Z.x = ---XY
a2 L
h
tanyy = Z .Y = __
a 2 L X2
from equation (4.236b), we find
__ a 2L{1 + [(-2h/a 2L)Xy]2}1 /2
x
N 2hY
1 + [(h/a 2L)X 2J2
Pu
(4.239)
from equations (4.204) and (4.238). For small values of Y, N x grows very large,
with a singularity at Y = O. Essentially, this reflects that the shell becomes very
shallow and the arches correspondingly flatter, until they degenerate into a
straight line along the direction Y = O. Obviously, this region of the conoid
cannot be sustained in equilibrium by extensional forces alone, and bending
must be considered. The use of a plate approximation for this region would
seem appropriate.
4.5 References
1. W. Fliigge, Stresses in Shells, 2nd ed. (Berlin: Springer-Verlag, 1973), pp. 100-102.
2. V. V. Novozhilov, Thin Shell Theory [translated from 2nd Russian ed. by P. G. Lowe
(Groningen, The Netherlands: NoordhotT, 1964), pp. 105-107].
3. P. L. Gould, A. Cataloglu, G. Dhatt, A Chattopadhyay, and R. E. Clark, "Stress
Analysis of the Human Heart Valve," Journal of Computers and Structures, 3,1973,
pp. 377-384.
4. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed.
(New York: McGraw-Hill, 1959), pp. 449-450.
5. Novozhilov, Thin Shell Theory, pp. 117-119.
6. D. P. Billington, Thin Shell Concrete Structures (New York: McGraw-Hill, 2nd ed.
1982), p. 114.
7. E. H. Baker, L. Kovalevsky, and F. L. Rish, Structural Analysis of Shells (New York:
McGraw-Hill, 1972), p. 256.
8. P. L. Gould and S. L. Lee, "Hyperbolic Cooling Towers under Seismic Design
Load," Journal of the Structural Division, ASCE 93, no. ST3, (June 1967): 87-109.
Closure, Journal of the Structural Division, ASCE 94, no. ST10 (October 1968):
2487-2493; S. L. Lee and P. L. Gould, "Hyperbolic Cooling Towers under Wind
Load," Journal of the Structural Division, ASCE 93, no. ST5 (October 1967): 487-514.
9. Fliigge, Stresses in Shells, pp. 171-179.
10. Novozhilov, Thin Shell Theory, pp. 130-138.
11. Ibid.
12. A. Fino and R. W. Schneider, "Wrinkling of a Large Thin Code Head under Internal
Pressure," Welding Research Council Bulletin no. 69 (New York: Welding Research
Council, June 1961), pp. 11-13.
192
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
4 Membrane Theory
Novozhilov, Thin Shell Theory, pp. 117-119.
Ibid., pp. 147-151.
H. Kraus, Thin Elastic Shells (New York: Wiley, 1967), pp. 307-314.
Novozhilov, Thin Shell Theory, pp. 151-163.
V. Z. Vlasov, General Theory of Shells and Its Application in Engineering, NASA
Technical Translation TTF -99 (Washington, D.C.: National Aeronautics and Space
Administration, 1964), pp. 200-201.
H. K. CHMTKO, CTPOI1TEJIhHA5I MEXAHMKA (Construction Mechanics)
(MOCKBA: "BhLCIIlAH IIlKOJIA", 1980).
Timoshenko and Woinowsky-Krieger, Theory of Plates and Shells, pp. 449-451.
E. P. Popov, "Earthquake Stresses in Spherical Domes and in Cones," Journal of
the Structure Division, ASCE 82, no. ST3 (May 1956): 974-1-974-14.
R. W. Clough, "Earthquake Response of Structures," in R. L. Wiegel, ed., Earthquake
Engineering (Englewood Cliffs, N.J.: Prentice-Hall, 1970), pp. 307-334.
P. L. Gould, S. K. Sen, and H. Suryoutomo, "Dynamic Analysis of ColumnSupported Hyperboloidal Shells," Earthquake Engineering and Structural Dynamics
2 (1974): 269-280.
Ibid.
P. L. Gould, "Hyperbolic Cooling Towers under Seismic Design Loading," Proc. of
the Fourth Symposium. on Earthquake Engineering I., University of Roorkee,
Roorkee, India, November 1970.
P. L. Gould, "Quasistatic Seismic Loading Distributions for Hyperbolic Cooling
Towers," Bulletin of the Indian Society of Earthquake Technology 8, no. 4 (December
1971): 163-168.
Gould and Lee, Closure to "Hyperbolic Cooling Towers under Seismic Design
Load."
Building Code Requirements for Reinforced Concrete Structures (ACI 318-71)
(Detroit: American Concrete Institute, 1971), pp. 26-27.
Timoshenko and Woinowsky-Krieger, Theory of Plates and Shells, pp. 453456.
P. L. Gould and S. L. Lee, "Column-Supported Hyperboloids under Wind Load,"
Publications of the International Association for Bridge and Structural Engineering,
Zurich, Switzerland 31-11, 1971, pp. 47-64; S. K. Sen and P. L. Gould, "Hyperboloidal Shells on Discrete Supports," technical note, Journal of the Structural
Division, ASCE 99, no. ST3 (March 1973): 595-603.
J. M. Rotter, "Membrane Theory of Shells for Bins and Silos," Dept. of Civil and
Mining Engineering, Univ. of Sydney, NSW, Australia.
M. E. Killion, "Design Pressures in Circular Bins," Journal of the Structural Division,
ASCE 111, no. 8 (August 1985): 1760-1774.
R. F. Rish and T. F. Steel, "Design and Selection of Hyperbolic Cooling Towers,"
Journal of the Power Division, ASCE 85, no. P05, (October 1959): 89-117.
Reinforced Concrete Cooling Tower Shells-Practice and Commentary (ACI
334.2R-84) (Detroit: American Concrete Institute, 1984): pp. 6-7.
D. Brissoulis and D. A. Pecknold, "Behavior of Empty Steel Grain Silos under Wind
Loading: Part I: The Stiffened Cylindrical Shell," Engineering Structures (October
1986): 260-274.
4.5 References
193
35. P. L. Gould, "The Cylindrical Shell Slice Beam," Journal of Engineering Mechanics,
ASCE 114, no. 7 (July, 1988).
36. A. J. Bellworthy and J. G. A. Croll, "Dielectric Space Frame Domes," Space Structures, 1, no. 1, (1985): 41-50.
37. Lee and Gould, "Hyperbolic Cooling Towers under Wind Load."
38. Novozhilov, Thin Shell Theory, pp. 315-319.
39. G. W. Hill, Collected Mathematical Works, vol. 1 (Washington, D.C.: Carnegie
Institute of Washington, 1905-1907), pp. 243-270; E. T. Whittaker and G. N.
Watson, A Course of Modern Analysis, 4th ed. (Cambridge: Cambridge University
Press, 1935), pp. 412-417.
40. P. L. Gould, "Unsymmetrically Loaded Hyperboloids of Revolution," Journal of
the Engineering Mechanics Division, ASCE 94, no. EM5 (October 1968): 1029-1043.
41. E. Ingerslev, "Design of Cooling Tower Shells," Proceedings of the June 1966
Bratislava, Czechoslovakia, Symposium on Tower Shaped Steel and Reinforced
Concrete structures, lASS. Madrid, Spain, 1968.
42. P. L. Gould, "Finite Element Analysis of Shells of Revolution" (London: Pitman,
1985).
43. Baker, Kovalevsky, and Rish, Structural Analysis of Shells.
44. "Design of Cylindrical Concrete Roofs," ASCE Manual of Engineering Practice no.
31 (New York: American Society of Civil Engineers, 1952); Billington, Thin Shell
Concrete Structures, chaps. 5 and 6.
45. Billington, Thin Shell Concrete Structures, pp. 186-191.
46. Billington, Thin Shell Concrete Structures, pp. 210-211.
47. Billington, Thin Shell Concrete Structures, pp. 258-262.
48. A. L. Parme, "Shells of Double Curvature," Trans. ASCE, vol. 123, 1958, pp.
990-1025.
49. Billington, Thin Shell Concrete Structures, pp. 261-262.
50. Billington, Thin Shell Concrete Structures, pp. 272-274.
51. Ibid.
52. C. Faber, Candela: The Shell Builder (New York: Reinhold, 1963).
53. "The New Newark Airport," Civil Engineering 44, no. 9 (September 1974): 74-76.
54. D. P. Billington, "Thin Shell Concrete Structures" (New York: McGraw Hill, 1965):
pp. 250-254; 2nd Ed., 1982, pp. 277-283.
55. A. Shaaban and M. Ketchum, "Design of Hipped Hypar Shells," Journal of the
Structural Division, ASCE, 102, no. ST11, (November 1976): pp. 2151-2161.
56. W. C. Schnobrich, "Analysis of Hyperbolic Paraboloid Shells," Symposium on
Concrete Thin Shells, ACI Special Publication SP-28, paper SP-28-13 (Detroit:
American Concrete Institute, 1971), pp. 275-311.
57. W. C. Schnobrich, "Analysis of Hipped Roof Hyperbolic Paraboloid Structures,"
Journal of the Structural Division, ASCE 98, no. ST7 (July 1972): 1575-1583;
discussion by M.S. Ketchum, vol. 99, no. ST4 (April 1973): 796-797; closure, vol.
100, no. ST2 (February 1974): 467-469; A. Shaaban and M. S. Ketchum "Design of
Hipped Hypar Shells," Journal of the Structural Division, ASCE 102, no. ST11
(November 1976): 2151-2161.
58. S. H. Simmonds, "Continuous Hypar Roofs for Water Treatment Plant," ACI Fall
Convention, Baltimore, November 1986.
194
4 Membrane Theory
59. S. S. Tezcan, K. M. Agrawal, and G. Kostro, "Finite Element Analysis of Hyperbolic
Paraboloid Shells," Journal of the Structural Division, ASCE 97, no. ST1 (January
1971): 407-424; P. V. Banavalkar and P. Gergely, "Thin-Steel Hyperbolic Paraboloid Shells," Journal of the Structural Division, ASCE 98, no. STU (November
1973): 2605-2621.
60. R. C. Liu and N. C. Teter, "Hyperbolic Paraboloid Shells Built with Wood Products," Bulletin of lASS 30 (June 1967): 3-8.
61. J. Bobrowski, "Calgary's Olympic Saddledome," Space Structures, V. 1, no. 1, 1985,
pp. 13-26; and "The Saddledome': The Olympic Ice Stadium in Calgary," Proc.
Canadian Society for Civil Engineering Annual Coriference, Saskatoon, SK, May
27-311985.
62. Billington, ThirJ Shell Concrete Structures (1982), pp. 29-32, 265-272.
63. Billington, Thin Shell Concrete Structures, pp. 283-290.
64. A. M. Haas, Thin Concrete Shells, vol. 2 (New York: Wiley, 1967), p. 103-133.
ApPENDIX4A
Summary of Surface Loading and Stress Resultants
for Quasistatic Seismic Loading on Hyperboloidal
Shells of Revolution
See figure 4-27 for Notation.
Nondimensional Geometry
a = throat radius
s
s=a
R9
r: - - 9 -
r,p
-J(k 2
1)
-
a - [k 2 sin 2 tfJ -1J 1/2
=-:;;~--;;-:--","=-;;;;-
R,p
--J(k 2
-
1)
= -a = =[k'""2-S1'""·n-:<-2-:-tfJ-----'1J=3/::;-2
ro = r9 sin tfJ
Surface Loading
q(tfJ)
=
q(tfJ)
qDL
qJ =
q(tfJ) cos tfJ
qt
=
q(tfJ)
q;
=
-q(tfJ)sintfJ
Stress Resultants
195
196
Appendix 4A Summary of Surface Loading and Stress Resultants
Functions
M Distribution
q(¢J) = CqDL
<l>1(¢J) = C{2(1(¢J,) [(3(<6) - (3(¢J,)] - [(4(¢J) - (4(¢J,)]}
<l>2(¢J) = 2C[(1(¢J) - (l(¢J,)]
MH Distribution
MH2 Distribution
q(¢J)
<1>1
<1>2
=
C'2~2
2C(2 [( 1(8
=
+ (9 + ~(5(¢J')]rPtrP
= 2C'2[~5(¢J) - ~5(¢Jt)]
General
1
[J(k 2 - 1) .
2
]
(1(¢J) = 4(k2 _ 1)
k i n K - 2r8 cos¢J
K = ~J,.:.,,(k-..-2_----,1-'--)_---::-k_co_s-'--¢J
J(k 2 - 1) + kcos¢J
4.6 Exercises
197
1 [1
(4(<P) = 2k2(3 ,J(k2 _ l)lnK
(5(<P) = (1(6
2
r,p ]
+ kcos<p + 3k2
+ ~(2 + (7
1
(6(<P) = 4k2(k2 _ 1) - (3 S
(s(<p) =
4k2~k; ~ 1) + ({2k2(k~ _
r,p
(9(<P) = 3k2
[(3(~
+ s)
4
(1
=
(l(<Pb) - (l(<Pt)
(2(<Pb) - (2(<Pt)
(2
=
(l(<Pb) - (l(<Pt)
(5(<Pb) - (5(<Pt)
+s
2
1) - S2]
1
- 2k 2(k 2 - 1)
+
r;]
lOk 2(k 2 - 1)
4.6 Exercises
4.1
Consider the spherical shell shown in figure 4-2, with a snow load of p, constant
per unit area of plan projection.
(a) Complete a membrane theory analysis for this loading, including a graphical
study similar to figure 4-3.
(b) At what angle t/> does the circumferential stress change from compression to
tension?
4.2
For the spherical shell under snow load described in the preceding exercise and
bound by a base angle ¢>h, compute the tensile force that would be developed in
the ring beam at t/> = t/>b.
4.3
Consider the spherical lantern shell as shown in figure 4-53.
(a) Compute the membrane theory stress resultants due to
(1) A uniform downward vertical line load of 500 (force/unit length of circumference) on the upper ring.
(2) A live load of 30 (force/unit area of horizontal projection).
(3) A dead load of 75 (force/unit area of middle surface).
Compute each case separately, and then combine the results.
198
Appendix 4A Summary of Surface Loading and Stress Resultants
Fig. 4-53
(b) Compute the forces that must be developed by horizontal ring beams at the
upper and lower edges, so that the shell will behave approximately as a
membrane.
4.4
In section 4.3.2.2, we have seen that the self-weight meridional stress resultant in
a uniform thickness spherical dome increases as t/J increases. The corresponding stress (I",,,, = N",/h will, of course, increase in the same fashion. To try
to equalize the self-weight meridional stress in a hemispherical dome, it is proposed to increase the thickness from a basic value h = ho at t/J = 0 to h = hb at
t/J = 90° by using the transition h(t/J) = ho + (hb - h o ) sin t/J.
(a) Compute the N", and Ns for the resulting self-weight load, where the shell
material has mass density p.
(b) Show the variation of the meridional stress (I,p,p from t/J = 0 to t/J = 90°.
(c) Select a value for hb as a function of h o , so that the condition at the base will
govern.
4.5
Consider the ellipsoid of revolution shown in figure 4-54.
(a) Compute the expressions for the principal radii of curvature.
(b) Derive expressions for the membrane theory stress resultants due to a uniform
internal suction q.
(c) Is is possible to have tensile stresses in this shell under the suction q? For what
dimensional combination will this occur?
4.6
An open ogival shell is shown in figure 4-55.
(a) Determine the membrane theory stress resultants due to
(1) Dead load, q (force/unit area of middle surface).
(2) Live load, p (force/unit area of plan projection).
(3) Ring load, w (force/unit length of circumference).
Evaluate the effects of each loading separately. For cases (1) and (2), derive
the expressions by using the general formulation for axisymmetrically loaded
shells of revolution. Check cases (1) and (2) as well as (3) by using the alternative formulation based on overall vertical equilibrium.
(b) Investigate the singularity as t/Jl ~ O.
199
4.6 Exercises
axis of rotation
--t---------t
I
------I--b
---t------
a
Fig. 4-54
LL=p
I I I I
~
I
Fig. 4-55
4.7
Consider a hyperboloid of revolution as shown in figure 4-6.
(a) Derive R~, equation (4.38), directly from equation (2.29).
(b) Show that a transformation between tP and Ro equivalent to equation (4.44)
can be derived from equation (4.36).
(c) In terms of the parameter k and the meridional angle tP or the axial coordinate
Z, determine the membrane theory stress resultants due to a hydrostatic
loading qh = 1'(T - Z), where l' is the unit weight of the fluid.
4.8
For a constant thickness vertical cylindrical tank of radius a and height H,
compute the membrane theory stress resultant distribution due to hydrostatic
pressure, using a fluid with unit weight 1', for the following cases:
200
Appendix 4A Summary of Surface Loading and Stress Resultants
0/2
0/2
0/2
o
Fig. 4-56
(a) The tank is full.
(b) The tank is half-full.
4.9
Consider the conical shell shown in figure 4-56. The shell has a line load pa applied
to the collar at midheight, and a uniform live load p applied only on the area
outside the collar.
(a) Compute the membrane theory stress resultants in the shell.
(b) Compute the force in the ring beam at the base.
4.10
The double conical pressure vessel shown in figure 4-57 is supported at mid-height
by a circular ring beam in the horizontal plane.
(a) Determine the membrane theory stress resultants due to a uniform internal
pressure p = 30 (force/unit area) and a self-weight of q = 3 (force/unit area).
(b) For each loading case, show the variations of N,p and No with Z and
compute the reactions on the ring beam. In particular, note the maxima for
N,pand No.
4.11
Analyze the paraboloidal shell of revolution with dimensions as shown in figure
4-58 for
(a) Self-weight, q.
(b) Uniform internal pressure, p.
4.12
Re-solve the compound shell shown in figure 4-14 for a hydrostatic loading yZ,
where Z is measured from the pole ofthe spherical cap at the left end. Determine
the incompatability in the circumferential stress resultant at the junctions of the
two shells.
4.13
A cylindrical tank with an ellipsoidal bottom is shown in figure 4-59. The shell is
subject to a hydrostatic load from a fluid with unit weight y.
(a) Compute the membrane theory stress resultants in the shell.
(b) Plot the stress resultants along the meridian.
(c) For what range of C and D will the circumferential stress in the ellipsoidal
bottom become compressive?
4.6 Exercises
201
z
50
50
Fig. 4-57
a
Fig. 4-58
4.14
Consider a hyperboloidal shell of revolution as shown in figure 4-6. The shell is
subject to a vertical static seismic loading q = fqDV where f is a known fraction.
Determine the membrane theory stress resultants.
4.15
(a) Verify the first harmonic shell of revolution solution presented in equations
(4.90) and (4.91).
(b). Verify that for a dome, 8 1 (0) = 0, by considering equations (4.90f) and (4.91b).
202
Appendix 4A Summary of Surface Loading and Stress Resultants
z
H
D
c
Fig. 4-59
4.16
For the spherical shell shown in figure 4-2
(a) Determine the membrane theory stress resultants due to a horizontal quasistatic seismic loading.
(b) For a hemisphere, determine the total base shear and overturning moment.
(c) Verify the values of N~ and S at ifJ = 90° computed by (a), using the results
of (b).
4.17
To optimize the efficiency of the shell material, the meridional and circumferential
stress resultants Nt/J and No should be equal at all points on the shell. This can
be accomplished by specifying that Nt/J(ifJ,8) = No(ifJ,8) = N* at every point on
a dome of constant thickness for a given axisymmetric loading condition and then
finding the requisite shell of revolution geometry, i.e., R.p(ifJ), Ro(ifJ), Ro(ifJ).
Investigate this possibility for the following loading cases:
(a) Uniform normal pressure, q = constant.
(b) Hydrostatic pressure q = yZ, where Z is measured from the pole.
Part (b) is analogous to the problem of determining the shape of a drop of liquid
resting on a plane surface, where N* is the surface tension. (See A. Pfluger,
Elementary Statics of Shells, 2nd ed., trans. E. Galantay (New York: F. W. Dodge,
1961), pp. 42-44.)
4.18
Consider the column-supported spherical shell shown on figure 4-29.
(a) Reanalyze this shell for a uniform live load p.
(b) For eight equispaced columns and a value of f1 = 2°, compute the membrane
theory stress resultant distribution and show the results in a graph similar to
figure 4-30.
4.19
Consider the wind-loaded cylindrical shell shown in figure 4-33. Assume that the
4.6 Exercises
4.20
4.21
203
vertical pressure distribution is qn(Z) = -[(Z - H)2j(H2)]p(8) and derive the
corresponding expressions to equations (4.149).
For an open cylindrical shell as shown in figure 4-37, develop membrane theory
solutions for the case where the load qu is uniformly distributed in the Y direction
and
(a) The load is uniformly distributed in X direction.
(b) The load is harmonically distributed in X direction.
Consider the open cylindrical shell shown in figure 4-37. In some respects, it may
be visualized as a beam of span L. With this in mind, it is proposed to increase
the thickness progressively from the ends to the center according to the relation
heX)
= h(O) + [ h (~) - h(O) ] sin ~
Compute the membrane theory stress resultants due to the self-weight of this shell.
4.22
For the hyperbolic paraboloid defined by equation (4.182), verify that the shell
has negative Gaussian curvature.
4.23
For the hyperbolic paraboloid shell shown in figure 4-60, compute
(a) N x and Ny due to a dead load q and a live load p.
(b) The axial force in the edge member AC at points A and B.
(c) Evaluate (a) and (b) for the following numerical values:
(1) Dead load of 40 (force per unit area of middle surface).
(2) Snow load of 20 (force per unit area of horizontal projection).
4.24
Consider the hyperbolic paraboloid shown in figure 4-61. Compute the membrane
theory stress resultants due to the fill load shown. Take the boundary conditions
as Nx(O, Y) = Ny(X,O) = o.
30
H
30
I
I
J
o
E
8
A
F
C
G
V~]5
60
Y
E,H
G,J
ELEVATION
12I~c
12~~
H
---tferOd------ G
CORNER
Fig. 4-60
ELEVATION
204
Appendix 4A Summary of Surface Loading and Stress Resultants
z
Fig. 4-6 1
z
(-
A
~ L,a,O)
c~
1
~----I&..4y
./3L
(0,0, H)
.~-I------l 1- L ,L,O)
L
Fig. 4-62
4.25
Verify the positive Gaussian curvatures of the elliptic paraboloid defined by
equation (4.227).
4.26
Consider the shell shown in figure 4-62. The shell is a paraboloid of revolution
on an equilateral triangular plan. It was shown in section 4.4.4 that this geometry
4.6 Exercises
205
is a specialized case of the elliptic paraboloid. Assume that extensional stress
resultants cannot be developed normal to the boundaries.
(a) Solve for the membrane theory stress resultants in the shell due to a dead load
q.
(b) Determine the stress distribution along the boundaries and the stresses at the
corners.
4.27 Consider the hyperbolic paraboloid shown in plan view in figure 4-48(a) with the
projected length along the Y axis extended from 4a to 5a.
(a) Are the boundary conditions stated in equation (4.222) still applicable for this
shell?
(b) If not, what revisions are necessary?
4.28
4.29
Consider a conical shell as shown in figure 4-13 and derive expressions for the
membrane theory stress resultants for circumferentially varying normal pressure,
similar to the cylindrical shell analysis in section 4.3.7.4.
Consider the "slice-beam" shown in figure 4-35 with p(Z, lJ) = - pi cosjlJ, and
verify equations (4.149a) and (4. 149c) using equations of overall equilibrium.
CHAPTER
5
Deformations
5.1 General
In the earlier chapters, we developed a geometric description of the middle
surface of a shell which proved to be adequate to derive the equations of
equilibrium. In turn, a simplified subset of the equilibrium equations formed
the basis of the membrane theory of shells, for which many important practical
applications were illustrated. Although membrane action in a shell is desirable
from the dual standpoints of mathematical simplification and material efficiency,
the requisite conditions for corresponding behavior cannot always be simulated
in an actual structure. Consequently, to expand our base of understanding of
shell behavior, we must develop relationships between the forces and the
deformations of the shell. The first step is a description of the displacements,
where we follow the vector approach suggested by Novozhilov. 1
5.2 Displacement
5.2.1 Displacement Vector: In figure 5-1(a), a differential element ofthe middle
surface is shown before and after deformation. The reference point 0 located
on the middle surface moves to the position 0', and this displacement is denoted
by A. Considering a second point 0('), located a distance' from 0 along the
original unit normal tn' the corresponding position after deformation is 0;(0
(i = 1 or 2), and the displacement between 0(0 and oa,) is A;(n
We have shown two possible positions of oao, o~ (0 and 0~(0. Point o~ (0
lies on the unit normal to the deformed middle surface, t~, at the same distance
,from the middle surface. The point o~ (0 is located on t~ because of assumption
[3], table 1-1, and remains, from the middle surface because of assumption [4].
On the other hand, point 0~(0 is also located, from the deformed middle
surface, but no longer necessarily lies on the normal t~; that is, the location of
0~(0 only requires the enforcement of assumption [4]. Since assumption [3]
refers to the suppression of transverse shearing strains, we may surmise that
the difference in 0 ~ (0 and 0 ~ (0 is the effect of the transverse shearing strain.
This is clearly shown in figures 5-1(b) and 5-1 (c), where views normal to tp and
206
207
5.2 Displacement
(b)
(e)
Fig. 5-1
Deformation of a Middle Surface Element
are shown with the transverse shearing strains defined as ,},,, and '}'p. The
positive sense shown is chosen arbitrarily to highlight '}'~ and Yp.
It was stated in the introduction that most classical work in shell and plate
theories has been based on the suppression of transverse shearing deformations
to achieve mathematical simplification. Currently, however, the analysis of
complex plates and shells is frequently carried out by powerful numerical
techniques that are not necessarily dependent on this simplification. Also, some
authors have expressed an opinion that the inclusion of transverse shearing
strains extends the bounds of the theory to include somewhat thicker plates
and shells. This is somewhat difficult to quantify, since a true thick shell theory
should account for transverse normal stresses as well, but it appears to be
correct in a heuristic sense. In the interest of generality, therefore, we will retain
t~
208
5 Deformations
the transverse shearing strains and treat o~(C) as the deformed point away from
the middle surface, and A2 ({) as the corresponding displacement. Subsequently,
we will show that a theory which neglects transverse shearing strains is easily
obtained from the more general theory.
From figure 5-1,
(5.1)
or
(5.2)
Equation (5.2) expresses the deformation of a point off the middle surface in
terms of the deformation of the corresponding middle surface point and the
unchanging distance between the middle surface and the point in question.
This establishes the pattern for subsequent developments: to relate the behavior
of a point away from the middle surface to the behavior of the corresponding
point on the middle surface.
5.2.2 Displacements in Terms of Middle Surface Parameters: Now, we express
the displacement vectors in terms of the unit tangent vectors of the undeformed
middle surface:
A = Data + Dptp + D.t.
(5.3a)
A2 (0 = Da(Ota + Dp({)tp + D.(Ot.
(5.3b)
Here, DOl' D p, and D. are the components of the middle surface displacement
in the respective directions, and Da(O, Dp({), and D.(O are the corresponding
components away from the middle surface. Also, the position vector to the
deformed middle surface is expressed in terms of the displacement vector by
r'=r+A
(5.4)
In equation (5.2), the unit vectors of the deformed middle surface, t~, t p, and
are present. In line with equation (5.4) for the vector r', we seek to write t~,
t p, and t~ in terms of the unit vectors of the undeformed middle surface. Since
t~,
t~=t~xtp
(5.5)
it is sufficient to develop suitable expressions for t~ and tp.
By analogy with the definitions in equations (2.16) and (2.8), we have
t'
r'
=~
(5.6a)
r'
t' -~
(5.6b)
a
p-
A'
B'
209
5.2 Displacement
where
r: . )1/2
(S.7a)
B' = -+(r'.p or'.p )1/2
(S.7b)
A' =
± (r: .
0
Thus, we have defined A' and B' as the Lame parameters for the deformed
middle surface. Now, taking %a of equation (S.4)
r: ..
=
r ... + (D.. t .. )... + (Dpt p)... + (D"tn)...
(S.8)
The first term in equation (S.8) is replaced using equation (2.16a), and the
remaining terms are differentiated by the product rule, with the derivatives of
the unit tangent vectors obtained from equation (2.17). After carrying out the
indicated operations, we have
r: . =
A[(1 + B.. )t.. + B..ptp -
I/I.. t n ]
(S.9)
where the coefficients of the unit vectors are given by
111
B.. = AD..... + ABA.pDp+R"Dn
..
(S.lOa)
1
1
B.. p = A Dp.ac - AB A.pDa.
(S.lOb)
-(~ Dn.a. - ~a. Da.)
(S.lOc)
1/1..
=
Similarly,
r:p
= B[Bp .. t ..
+ (1 + Bp)tp -
I/Ipt n]
(S.l1)
in which
(S.12a)
(S.12b)
I/Ip =
-(~D".p - ~p Dp)
(S.12c)
The coefficients defined in equations (S.10) and (S.12) are extremely important
in what follows and are interpreted from a physical standpoint at a later stage
of this chapter.
We are ready to substitute equations (S.9)-(S.12) into equations (S.6a) and
(S.6b). As we carry out this operation, recall assumption [2] oftable 1-1, which
restricted the development to the domain of comparatively small deformations.
This limitation enables products of the coefficients defined in equations (S.lO)
210
5 Deformations
and (5.12) to be neglected, and ensures that the subsequent relationships will
be linear. We wish to emphasize at this point that although assumption [2J
greatly simplifies the problem from the mathematical standpoint, the justification
is based on verification of the magnitudes of the neglected terms for a particular
problem or class of problems. In most practical applications, the linear theory
has been verified to be adequate, but, if not, a variety of nonlinear theories are
available. 2
Performing the indicated substitutions, we have from equations (5.7a) and
(5.9)
+ B~)2 + B;p + 1/1;]1/2
+ B~)
A' = A[(l
A(l
~
(5.13)
and from equations (5.7b) and (5.11),
B' = B[B~~
(5.14)
+ Bp)
B(l
~
+ (1 + Bp)2 + I/IffJl/2
after the product terms are dropped. Then, from equation (5.6a),
t'
=
A [(1
~
~ t~
+ B~)t~ + B~fJtp A(l + B~)
+ B~ptfJ -
l/I~tnJ
(5.15)
I/I~tn
and, from equation (5.6b),
t' _ _
B-=[,---,Bp':.:::~---.::t~~+-----.:(_l_+-----.!Bp:-=-)t..!:.p_------.:...I/I~p.....::tn:.::J
p-
B(l
~ Bp~t~
+ tp -
+ Bp)
(5.16)
I/Ipt n
where
an d
1jI;
./,
~~~'I';
1
+ B;
(i = O(,P)
p,O(
j =
Finally, we compute the vector product t~ x t p, which, after simplification, is
(5.17)
Equation (5.17) is the desired expression of the unit normal to the deformed
middle surface, in terms of the tangent vectors to the undeformed middle surface.
Substituting equations (5.15), (5.16), and (5.17) into equation (5.2) gives
A 2 (C)
=
A
-
+ C[(I/I~ + y~ + YpBp~)t~ + (I/Ip + Yp + Y~B"p)tp
(y"l/I" + ypljlp)tnJ
(5.18a)
Again, products of the deformation parameters, such as ypBp", are neglected, so
that equation (5.18a) simplifies to
211
5.2 Displacement
(S.18b)
In view of equation (S.3a), equation (S.18b) may be rewritten as
A2 «()
=
[D"
+ (1/1" + y")]t,, + [Dp + (I/Ip + Yp)]t p + Dntn
(S.19)
so that the components of A2 (O are
= D" + (1/1" + y,,)
Dp«() = Dp + «I/Ip + Yp)
Dnm = Dn
(S.20a)
D,,«()
(S.20b)
(S.20c)
Here, we have expressed the components of the displacement of an arbitrary
point on the shell in terms of the displacement of the corresponding point on the
middle surface and the unchanging separation of the points along the unit normal
vector. Equation (S.20c) is, of course, a direct restatement of assumption [4].
At this point, we observe that the popular shell theories which incorporate
assumption [3], as well, may be obtained by simply setting y" = Yp = 0 in the
previous equations.
5.2.3 Rotations: Recalling that the coefficients D" and Dp represent the respective
displacements of the middle surface along the s" and sp coordinate lines, the
terms 1/1" and I/Ip may be viewed as rotations of the normal t~ about the tp
and t~ axes, respectively. To show this, we construct figures S-2(a) and S-2(b),
which are, in turn, identical to the views shown in figures S-1(b) and S-l(c).
On figure S-2, the projections of the unit vectors to the undeformed surface t"
and tp may be considered to remain unit vectors within the scope of this theory;
i.e., t,,· t~ ~ 1 from equation (S.lS). If we scalar multiply equation (S.17) by t",
and then t p , we get
----~-------4~~~f
f3
t'ME----~----~r---
(a)
(b)
Fig.5-2
a
Transverse Shearing Strains
212
5 Deformations
(S.21a)
and
I/Ip =
tp·t~
(S.21b)
Equation (S.21a) is interpreted in figure S-2(a). Continuing,
t" . t~
= cos 1fI"
(S.21c)
For small rotations
(S.21d)
or, 1/1" is the rotation of the normal to the deformed middle surface about the
tp axis. Similarly, from figure S-2(b), I/Ip is the rotation of the normal about
the t~ axis. Within the scope of the small deformation theory, 1/1" and I/Ip may
be visualized as rotations about tp and t" as well.
We now consider the transverse shearing strains, initially defined in figures
S-l(b) and S-I(c), superimposed on the rotations 1/1" and I/Ip in figures S-2(a) and
S-2(b). We see that the terms 1/1" + y" and I/Ip + yp, which initially appeared in
equation (S.19), may be interpreted as rotations. If we consider a line connecting
a point away from the deformed middle surface, O 2, and the corresponding
middle surface point, and a second line along the normal to the undeformed
middle surface tn' the angles formed by the intersection of these lines are,
respectively,
D"p = I/Ill
in the
(X-n
Dp"
=
+ Yll
(S.22a)
plane and
I/Ip + Yp
(S.22b)
in the fJ-n plane. We also may observe from figure S-2 that if y" and Yp = 0,
DIlP and Dp" are, in turn, the angles between t" and t~, and tp and tp. DIlP
and Dpll are termed middle surface rotations and, together with D", Dp, and
Dn , constitute the set of generalized displacements for the shell theory under
study.
The complete description of the displacement of any point on or within the
shell surface is given in terms of the five generalized displacements. Seemingly,
rotation about the normal has no utility, provided that the structure is constrained against rigid body movement. However, there are occasionally cases
where the shell has a slope discontinuity or is approximated by an assemblage
of flat (facet) surfaces. 3 It may be then convenient to introduce the normal
rotation, which is known as the drilling degree of freedom.
213
5.3 Strain
p
p'
after deformation
deformation
Fig. 5-3
Deformation of a Coordinate Line
5.3 Strain
5.3.1 Strain on Middle Surface: Referring to figure S-3, consider an arc op on
the Sa coordinate line that has an initial length of
dS a = Ada.
(S.23)
After deformation, the points
becomes
ds~ =
=
0
and p move to
A'da.
A(l
0'
and p', and the arc length
(S.24)
+ Ba)da.
from equation (S.13). The extensional strain is defined as the relative increase
in length of a curvilinear line element, or the final length minus the initial length,
divided by the initial length. So, in the a. direction, we have
ds~
- dSa
dS a
A(l
+ Ba) da. -
Ada.
Ada
(S.2S)
Thus Ra, which appeared in the coefficient of ta in equation (S.9) and is defined
by equation (S.lOa), is the strain in the a direction. Correspondingly in the p
direction, the strain is given by Bp, which is defined in equation (S.12a). We have
thus found that the extensional strains occur quite naturally in the expressions
for the derivatives of the position vector to the deformed middle surface.
The in-plane shearing strain is defined as the change in angle between ta and
tp during deformation. Referring to figure S-4, the shearing strain is
(J)
= eap + epa = sin(Bap + epa)
=
cos(i - eap- epa)
(S.26)
214
5 Deformations
t'p
."./2- £.D-~
aIJ /3a
Fig.5-4
In-Plane Shearing Strain
We form the indicated scalar product using equations (S.lS) and (S.16), which
gives
w
=
t~·
tp = epa + eap + l/Jal/Jp
(S.27)
Since the last term is negligible, we have
(S.28)
as the shearing strain. Again, eap and epa are found among the coefficients of
the equations defining the derivatives of the position vector of the deformed
middle surface as equations (S.lOb) and (S.12b). The transverse shearing strains
Ya and Yp have previously been defined.
5.3.2 Strain on Parallel Surface: The differential arc length along coordinate
line sa of the parallel surface is
dsa(O
=
A(OdO(
(S.29)
Referring to the normal section, figure S-S,
dsa(O
dS a
R a(,)
Ra
(S.30a)
Since
(S.30b)
and
dS a = AdO(
(S.30c)
then
(S.31)
215
5.3 Strain
surface
P, (C)
middle surface
Normal
Fig. 5-5
Section
Displacement of a Parallel Surface
Comparing equations (5.29) and (5.31), the Lame parameter for the parallel
surface is
A(O = A(1 + ~J
For a parallel surface on a normal section along the
Rp(O
=
Rp
+(
ds «() = B(1 + ~Jdf3
p
(5.32)
sp
coordinate line
(5.33)
(5.34)
and
B«()
=
B(l
+ ~J
(5.35)
We recognize that the parallel surface is described by a system of curvilinear
coordinates that is identical to those which describe the middle surface. Therefore, the relationships between the strains and displacements defined for the
middle surface in equations (5.10a) and (5.lOb) and (5.12a) and (5.12b) may be
written for the parallel surface by replacing R"" R p, A, and B by R",(n Rp(n
A«(), and B«), respectively; and, by replacing D"" D p, and Dn by D",(O, Dp(O,
and Dn(n as defined in equations (5.20).
For the extensional strains, we get
8",(0
= 8", + (K",
(5.36)
8p(0
=
8p + (Kp
(5.37)
216
5 Deformations
after dropping terms of order (h/R) compared to 1; the latter order of magnitude
comparison is widely used in shell theory and is concisely stated as O(h/R): 1.
The coefficients of ( are
1
1
Ka = A DaP,a
+ AB A,pDpa
(5.38)
and
1
Kp = B Dpa,p
1
+ AB B,aDaP
(5.39)
where DaP and Dpa are defined in equation (5.22). The terms Ka and Kp correspond
to the familiar curvature terms in linear beam theory. For plates, they are indeed
curvatures, but, for shells, they are more properly called changes in curvature,
since shells by definition are initially curved.
For the in-plane shearing strain w(o, we first combine eap and epa as indicated
in equation (5.28). Adding equations (5.10b) and (5. 12b), we have
1
1
1
1
w=-D
A Da +-D
B Dp
A p,a - AB'P
B a,p - AB,a
(5.40)
Now, referring to the parallel surface, we perform the indicated substitutions
for the radii of curvature, Lame parameters, and displacements, and also drop
terms of o (h/R): 1. The resulting expression for the in-plane shearing strain on
the parallel surface is
w(()
= eap + (Ta + epa + (Tp
(5.41)
where
1
T a =-D
A pa,a
1
-A ,p Dap
AB
(5.42)
and
(5.43)
It is convenient to introduce
T
= t(Ta + Tp)
(5.44)
Then, equation (5.41) is written as
w(O
= w + 2(T
(5.45)
where T is called the twist or the torsion of the middle surface.
We have now completely defined each strain on the parallel surface, except
for the transverse shearing strain, as a linear combination of two terms: (a) the
corresponding strain on the middle surface, which is expressed in terms of the
middle surface displacements by one of equations (5.10) or (5.12); and (b) a change
5.3 Strain
217
in curvature or a twist, which is, in turn, given in terms of the middle surface
displacements by equations (5.38), (5.39), or (5.42)-(5.44), respectively. The transverse shearing strains are also defined in terms of middle surface displacements
by equations (5.22), (5.lOc), and (5.12c), but remain constant over the shell depth.
The differences in the variation of the strains through the thickness when
transverse shearing strains are included (linear vs. constant) give rise to a
curious phenomenon known as shear locking in finite element models when
numerical integrations through the thickness are required. 4 If suitable precautions are not taken, this can lead to serious errors; this is one reason that
solutions incorporating these strains have been slow to gain popularity.
These relationships among strains, changes in curvature, and twist on the
one hand, and the displacements of the middle surface on the other, are of
primary importance in the theory of shells and are called generalized straindisplacement or kinematic equations. We have thus established the second basic
component of the elasticity problem as outlined in section 1.1, the conditions
of compatibility of the strains and displacements, suitably specialized for the
shell theory under consideration:
(5.46a)
1
1
1
ep = BDp,P + ABB,,,D,, + RDn
(5.46b)
p
1
OJ
1
1
1
= A Dp,,, - AB~,pD" + JjD",p - ABB,,,Dp
(5.46c)
(5.46d)
(5.46e)
(5.46f)
y" =
(~ Dn,,, - ~" D,,) + D"p
(5.46g)
yP =
(~Dn,p - ~p Dp) + Dp"
(5.46h)
If we choose to neglect transverse shearing strains, we set y" = YP = 0 in
equation (5.46), so that D"p = tjJ" and Dp" = tjJp. Then, replacing D"p and Dp" in
equations (5.46d)-(5.46f) by tjJ" and tjJp, as given by equations (5.lOc) and (5.12c),
we get the changes in curvatures directly in terms ofthe middle surface displacements:
218
5 Deformations
1(1
1(1
1) 1 (1 . 1)
1) 1 (1 1)
Ka -- - Da ,a - A ,p -D
Dp
A -D
A n,a - -R~
AB
B n,p - -Rp
(5.47a)
Kp -- - Dp ,p - B,a -D
Da
B -D
B n,p - -Rp
AB
A n,a - -Ra
(5.47b)
T
!{-~(~D
- ~(~D
A B n,p - ~Dp)
Rp
B A n,a - ~D)
R a a ,p
,a
= 2
+ AlB [ A,p
(~ D.,a - ~a Da) + B,a (~ Dn,p - ~p Dp) ]}
(5.47c)
Note with respect to equations (5.46) and (5.47) that no force-deformation
relationships are involved and, hence, linear material behavior is not a prerequisite for the application of these equations. Also, as previously demonstrated
for the equilibrium equations, a considerable simplification of these equations
often occurs for specific geometries through the elimination of some of the
coupling terms.
5.4 Strain-Displacement Relations for Shells of Revolution
2-11,
5.4.1 Specialization of Equations: The specialization of the curvilinear coordinate system to the shell of revolution geometry, shown in figure
is carried
out in section 3.3.1. The corresponding strain-displacement equations for r:J. = ¢I
and p = e are
1
R(Dt/>,t/>
t/>
+ Dn)
Bt/>
=
Bo
= R(Do,o + cos ¢lDt/> + sm ¢lDn)
W
= R Do,t/> + ji- (D~,IJ - cos ¢lDo)
(5.48a)
1
.
o
1
~
1
(5.48b)
(5.48c)
0
1
(5.48d)
Kt/>=RDt/>o,t/>
if>
Ko
=
1
-R DOt/>,o
T
=
~[~t/> DOt/>,t/> + ~O (Dt/>o,o -
Yt/>
=
o
cos f/J
+ --p:-Dt/>o
(5.48e)
0
1
+R(Dn,t/> - Dt/»
t/>
+ Dt/>o
COSf/JDot/»]
(5.48f)
(5.48g)
5.4 Strain-Displacement Relations for Shells of Revolution
Yo
+ ~O (Dn,o
=
- sin fjJDo)
+ Do~
219
(S,48h)
If the transverse shearing strains are neglected, we have the alternate form
K~ = -~[~(Dn,~
- D~)]
R~ R~
,~
Ko
1
,
cosfjJ
R5 (Dn,o - sm fjJDo),o - R~Ro (Dn,~ - D~)
= -
= -21 {_~[-l-(Dn,o
r
(S,49a)
R~
+
Ro
- sinfjJDo)]
cosfjJ
.
}
R5 (Dn,o - sml/JDo)
,~
-
(S,49b)
-l-(Dn,~ - D~),o
R~Ro
(S.49c)
We may also write the strains and displacements for the shell of revolution in
the Fourier series form that proved expedient in the treatment of non symmetric
loading on shells of revolution. Following the procedure of section 4,3.S.2, we
have
G~
G~cosj()
Go
G~cosj()
W
w j sinj()
00
K~
I
Ko
j=O
K~COSj()
K~COSj()
(S.50)
r j sinj()
Y~ cosj()
yl sinj()
r
Y~
Yo
and
D~ cosj()
D~
Do
Dn
00
I
j=O
D~o
D~ sinj()
D~ cosj()
(S.Sl)
~o cosj()
D~~ sinj()
Do~
Equations (S.SO) and (S,Sl) are substituted into equations (S,48) and (S,49) to
derive the kinematic equations for general harmonic j:
.1.
G~ = R(D~,~
~
.
G~
1.
.
+ D~)
(S,S2a)
'.'
= -(jD~ + cos fjJD~ + sm l/JD~)
Ro
(S.52b)
220
5 Deformations
(S.S2c)
(S.S2d)
(S.S2e)
(S.S2f)
(S.S2g)
(S.S2h)
or with transverse shearing strains neglected
K~
o
o
K~
1 [ -(D~.tfi
1
= -j
= R2
o=
T;J
D~)
0
0
Rtfi Rtfi
0
o
(jD~
+ sin rjJD~) 0
1[1 [1
- -
-(jD~
0
2 Rtfi Ro
-
cos rjJ
- 2 - (jD~
Ro
0
J
(S.53a)
.tfi
cos rjJ
R R (D~.tfi - D~)
tfi 0
0
(S.S3b)
0
j
+ sinrjJD~)oJ + --(D~,tfi
0
+ sm rjJD~)
•
0
J
.tfi
RtfiRo
-
D~)
0
(S.S3c)
5.4.2 Physical Interpretation: The first two equations of(S.48a-h) for the extensional strains may be physically interpreted by considering figure S-6, which
shows a segment of the meridian of a shell of revolution. The first term of
equation (S.48a) follows from the basic definition of strain as the change in
length of the differential segment op, Dtfi .tfi drjJ, divided by the initial length Rtfi drjJ.
The second term of the equation represents the strain from the normal displacement Dn; this is easily visualized as a change in the arc length op due to
a change in R tfi , (Rtfi + Dn) drjJ - Rtfi drjJ, again divided by the initial length, Rtfi drjJ.
In equation (S.48b), the first term is again straightforward. The remaining two
terms occur because of the change in the radius of the parallel circle due to Dtfi
and D n, which is shown as Aoo' on figure S-6. The horizontal projection of Dtfi
is Dtfi cos rjJ, and that of Dn is Dn sin rjJ. Together, we have the change in the length
of the circular arc, (Dtfi cos rjJ + Dn sin rjJ) dO, divided by the original length, Ro dO.
Next, we consider equations (S.48d) and (S.48e), the changes of the curvature.
These are defined as the change in the surface rotations over the segments drjJ
5.4 Strain-Displacement Relations for Shells of Revolution
Fig. 5-6
221
Displacement of the Meridian of a Shell of Revolution
and dO, respectively, divided by the length of the segments. Equation (5.48d) is
the change in the rotation in the f/J direction, D"IJ, " df/J, divided by the arc length,
R" df/J, or (l/R,,)D"IJ,'" We may also interpret this result directly in terms of
the displacements by using figure 5-6. The rotation at the initial point in the
segment, point 0, as it moves to o~ is (l/R")D,,. From o~ to 0;, we have the
transverse shearing strain contribution of YIP' where the positive sense was
established by figure 5-2. There is also a rotation at o~ due to Dn. This is
illustrated in figure 5-7, where the same differential segment is shown with
an exaggerated Dn displacement. The rotation at o~ due to the change in Dn is
seen to be (l/R")Dn,,,-in a sense opposite that ofthe rotation due to D". Hence
the total change in rotation over the segment
+ YIP - ~Dn,") df/J
(~D"
R"
R"
,"
is divided by the segment length, R" df/J, to derive the meridional change in
curvature. This expression is verified by substituting equation (5.48g) into
(5.48d).
Equation (5.48e) has two components. The first term is the change in the
rotation DIJ", DIJ",IJ dO, divided by the arc length, Ro dO, or (l/Ro)DIJ",IJ' Again,
if a segment of the shell along a parallel circle is examined, this term may be
222
5 Deformations
,
0,
normal at 0
Fig.5-7 Normal Displacement of the Meridian
interpreted directly in terms of the displacements. Note that there is a second
term present in the expression for Ko. This term is (cos ,p/Ro)D",o, which acts
about t",. The contribution of rotation D",o may be followed by referring to
figure 3-6 and replacing the term NoR", d,p by D",o on the force element and on
view HVI. Then, moving to view MV1, we see the resolution of this vector into
the meridional direction and the normal direction. The drilling component,
corresponding to rotation about the normal, is not admissible in the present
theory. The meridional component, acting in the negative ,p direction, is the
rotation D",o cos,p and represents a change in slope along the parallel circle
over the segment. The sign of this component is positive since, as shown in
figure 5-2(b) with IX = ,p and P = e, the positive sense of the rotation D",o points
in the negative ,p direction. Mter division by the arc length, Ro de, we have
the second term of equation (5.4Se).
The preceding use of an argument from the equilibrium equations to explain
a compatibility relationship is not an isolated coincidence. This is a representative part of a static-geometric analogy that exists between the equations of
equilibrium and the equations of compatibility. This analogy is sometimes
useful in constructing dual solutions. A complete development of the staticgeometric analogy may be found in Gol'denveizer. 5
We now turn our attention to the in-plane shearing strain w, which is given
by equation (5.4Sc) and is shown on figure 5-S as the sum of
and Wo0 The
w'"
5.4 Strain-Displacement Relations for Shells of Revolution
Fig. 5-8
223
Shearing Displacements
term w¢> is given by D¢>,8 de/(R o de) = (1/R o)D¢>,8' To evaluate w 8, note that point
+ D8,¢> dfjJ in the e direction. When the e displacement
of point 0, D8 , is projected onto the corresponding displacement of p to p', it
becomes
p has moved a distance D8
D8(RO
+ ~o,¢>#) =
DB( 1 + R¢>;:SfjJ dfjJ)
because of the change in the horizontal radius Ro. The difference,
[D8,¢> - D8(R¢>/Ro) cos fjJ] dfjJ
R¢>#
= W8
and the sum of w¢> and W8 is w, as given by equation (5.4Sc).
Finally, we examine the twist ", which is given by equations (5.4Sf). Since"
is obtained from "¢> and "B as defined in equation (5.44), we concentrate on
the basic terms "¢> and "8' Referring to figures 5-S and 5-9(a), the twist in the fjJ
direction, "¢>' is given by the change in the rotation in the e direction at 0' as
fjJ varies, D8¢>,¢> dfjJ, divided by the arc length, R¢> dfjJ, which is the first term in
equation (5.4Sf). The twist in the e direction, "8' consists partially of the
corresponding change in the rotation in the fjJ direction at 0' as e varies, D¢>8,8 de,
divided by the arc length, Ro de, as shown in figure 5-9(b). Another contribution
to the twist in the e direction is due to the change in the horizontal radius
Ro as fjJ changes. From equation (5.4Sh), D8¢> is approximately inversely proportional to Ro so that DB¢> is reduced to
224
5 Deformations
~o'
//
r'
D8¢~
~
___-____q'
p'
Fig. 5-9
(a)
Twisting Displacements
Ro
-----DIJt/>
Ro + Ro,t/>d</J
over the segment, or we have a change of
Ro,t/>d</J
-'-'------DIJt/>
Ro
=
Rt/>
--cos</JDIJt/>d</J
Ro
After division by the arc length over which the change in horizontal radius
occurs, Rt/> d</J, we have the third term of equation (5.48f).
We have attempted to provide an explanation of the individual terms of
the compatibility equations by using physical arguments for the specialized
shell of revolution geometry. In a general shell, the physical illustrations would
become untenable; yet, the initial derivation by vector algebra was straightforward. In the past, considerable effort has been expended in deriving compatibility relationships for specialized geometries using strictly physical reasoning.
Here, in both the equilibrium and compatibility treatments, we have sought to
vary the approach somewhat by initially deemphasizing the physical in favor
of the mathematical approach, but then closely supporting the results with
specific physical illustrations.
5.5 Strain-Displacement Relations for Plates
As initially set forth in section 3.4, the shell equations are transformed into a
sufficiently general form for medium-thin plates by letting the radii of curvature
5.5 Strain-Displacement Relations for Plates
225
approach infinity. From equations (5.46),
s'"
1
1
= AD""", + ABA,pDp
(S.S4a)
(S.S4b)
(S.S4c)
(S.S4d)
(S.S4e)
1[1
1 (A,pDa,p + B,a,Dp",)]
=2
A Dpa,a + B1 Da,p,p - AB
(S.S4f)
Ya
= (~Dn,a, + Da,p)
(S.S4g)
Yp
= (~Dn,p + Dpa)
(S.S4h)
T
When the transverse shearing strains are neglected, we have
(S.SSa)
(S.SSb)
- AlB [Art Dn,a,
+ ~'" DR,p]}
(S.SSc)
It has been noted several times that the adoption of assumption [3J and
the subsequent suppression of transverse shearing strains generally leads to
significant mathematical simplifications. An example of this may be seen for
the plate case by comparing equations (S.S4d)-(S.S4f) with equations (S.SSa)(S.SSc). In the latter equations, the curvatures and hence the entire flexural
behavior of the shell are dependent on only the normal displacement DR and
are essentially uncoupled from the extensional behavior. The form of the
curvature-displacement relations is identical to that encountered in elementary
beam theory. On the other hand, the retention of transverse shearing strains
226
5 Deformations
leads to a considerably more complicated mathematical formulation. 6 This is
discussed further in chapter 8.
5.6 References
1. V. V. Novozhilov, Thin Shell Theory [translated from 2nd Russian ed. by P. G. Lowe
(Groningen, The Netherlands: NoordhotT, 1964), pp. 14-27].
2. H. Kraus, Thin Elastic Shells (New York: Wiley, 1967), chap. 1.
3. P. Bergan and M. K. Nygard, "Nonlinear Shell Elements with Six Freedoms per
Node," Proc. First World Congress on Computational Mechanics, University of
Texas at Austin, Austin, September 1985.
4. C. K. Choi, "Reduced Integrated Nonconforming Plate Element," Journal of Engineering Mechanics, ASCE 112, No.4 (April 1986): 370-385.
5. A. L. Gol'denveizer, Theory of Elastic Thin Shells [translated from Russian ed.
(New York: Pergamon Press, 1961), pp. 92-96].
6. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed.
(New York: McGraw-Hill, 1959), pp. 165-173.
5.7 Exercises
5.1
Perform the ditTerentiations indicated in equation (5.8) to verify equation (5.9); also,
verify equation (5.11).
5.2
Verify the expressions given for "", and "fJ in equations (5.38) and (5.39); also verify
the expressions for 't"", and 't"fJ' as given in equations (5.42) and (5.43).
On an appropriate sketch, show the physical meaning of the first term in equation
(5.48e) in terms of the middle surface displacements.
5.3
5.4
Derive the strain-displacement relations for a shell of revolution where the Lame
parameter A is taken to correspond to the axial coordinate Z. Give a physical
explanation of each term using appropriate sketches.
5.5
Show that the four terms £"'fJ'
£fJ'"
£",fJ
't"",+-='t"fJ+-
R",
ify",
= YfJ = O.
RfJ
£fJ"" 't"""
and 't"fJ satisfy the identity
CHAPTER
6
Constitutive Laws, Boundary Conditions,
and Displacements
6.1 Constitutive Laws
The constitutive law or stress-strain relationship is the third basic component of
the elasticity problem. This subject is quite broad, but the detailed examination
of various alternatives that are dependent on the characteristics of particular
engineering materials is not properly within the scope of this book. Rather,
it remains within the purview of the theory of elasticity, since we may accommodate a variety of material laws within our formulation of the shell or plate
problem. Initially, we use the basic Hooke's law for isotropic materials, and
then we illustrate how some extended material laws can be accommodated.
6.1.1 Isotropic Material Law
6.1.1.1 Strain-Stress Relationship Including Temperature Effects. The strains, as
defined in section 5.1, are related to the stresses, introduced in section 3.1, by
8",(OC,
p, 0 =
8p(OC, p, 0
w(oc,
=
p, 0 =
1
+ aT(oc, p, 0
(6.1a)
1
+ aT(oc, p, 0
(6.1b)
E (0"",,,, - J1.O"pp)
E (O"pp - J1.0"",,,,)
1
G O"",p
=
1
G (Jp",
(6.1c)
(6.1d)
I'p(oc, P)
1
= G O"pn
(6.1e)
n
In equation (6.1), E = Young's modulus, G and Gn = in-plane and normal
shearing moduli, and J1. = Poisson's ratio. Also, a represents the coefficient of
thermal expansion, which will be taken as piecewise constant, and T(oc, p, 0 =
the temperature, measured from a specified reference value. Thermal effects are
included snce they are very important in some applications. 1
227
228
6 Constitutive Laws, Boundary Conditions, and Displacements
Matrix 6-1
II
0"•• (0(, p, ()
0
0
0
6.+(".
0
0
0
6(1+("/1
0
1
2(1 +11)
0
0
(1)+2(,
0
0
0
1
2(1 +11)
0
Y.
0
0
0
0
0
Y/I
0
1-112
I_p,2
II
1-112
1-112
0
p)
0"/1.(0(, p)
0"(1(1(0(, p, ()
0"./1(0(, p, 0
0"•• (0(,
=E
1
2(1 + II)
I-II
1
I-II
- EaT(O(, p, ()
0
We find it convenient to invert equation (6.1) and substitute the expressions
8,,(a, fl, 0, 8p(a, fl, 0, and w(a, fl, 0, as given by equations (5.36), (5.37), and (5.45),
respectively. After performing the indicated substitutions and assuming for the
present that Gn = G = E/[2(1 + Jl)], we obtain the stress-strain law
{a(a, fl, ()} = [CJ{ £(0(, fl, ()) - {aT(a, fl· ()}
(6.2)
which is given in det.ail in matrix 6-1.
We use the matrix representation to illustrate that any other consistent
stress-strain relationship may be accommodated by altering the elements of
[C] and {aT}, which are essentially independent from the remainder of the
development.
6.1.1.2 Stress Resultant-Strain and Stress Couple-Curvature Relations. In the
surface structure problem, the constitutive equations are required to relate the
stress resultants and stress couples, instead of merely stresses, to the corresponding
strains and curvatures. These relationships are obtained in a straightforward
fashion. We substitute the stress-strain law, matrix 6-1, into equations (3.6)(3.9). For example, consider the expression for N" in equation (3.6):
N" =
f,
"/2
(1""
"12
=E f
"/2
-"/2
(
()
1+-
(8"
Rp
1 - J1.2
d(
(""
Jl8p
J1.("p
"fiT )
+
--+
--+
- - -11 - Jl2
1 - Jl2
1 - J1.2
- Jl
(6.3)
.(1 + ~Jd(
Since (/Rp « 1, it may be neglected in this integration. We split the integral into
N" = 1
E
- Jl
2
f"/2
-"/2
(8"
E"fi f"/2
+ ("" + Jl8p + Jl("p)d( - -1-
- J1. -"/2
T(Od(
(6.4)
229
6.1 Constitutive Laws
In this development, the cross section is assumed to be symmetric about the
middle plane or surface, so that
f ,d,
h l2
=0
-h12
and the second and fourth terms of the first integral vanish, leaving
Eh
Na. = -1-2 (B",
-Jl
+ JlBp) -
(6.5)
N"'T
where
Eri
N",T = - 1 Jl
fh l2 T(Od,
-h12
which may be called the thermal load, is left in general form to accommodate
the possibility that T = T(O. The remainder of the stress resultant-strain
equations are obtained in the same fashion.
Now, consider the expression for M", in equation (3.8)
(6.6)
(6.7)
For a symmetric cross section, the first and third terms of the first integral
vanish, leaving
Eh 3
M", = 12(1 _ Jl2) (/(",
+ /1/(p)
-
M",T
(6.8)
where
is known as the thermal moment. If Tis constant through the thickness, there are
no thermal moments. The remainder of the stress couple-curvature equations
are obtained in an identical fashion. Collecting the equations, we can develop
matrix 6-2, where
Na.T
=
NPT
=
NT
Eri
= -1 - Jl
fh l2 T(Od,
-h12
(6.9a)
0
0
0
0
0
0
0
0
0
Q.
Q/I
0
0
0
0
0
0
h3
12(1 - p.2)
p.h 3
12(1 - p.2)
0
12(1 - p.2)
12(1 - p.2)
0
p.h 3
h3
M.pl
0
0
0
0
h
2(1 - p.)
0
M.I
0
0
0
0
0
0
0
MfJl
0
I
N. fJ
=E
0
I
h
2
1 - /1
1 - J1.2
I
p.h
h
p.h
1 - /1 2 1 - p.2
r
Np
NJ
Matrix 6-2
+ p.)
0
0
12(1
h3
0
0
0
0
0
2(1
0
+ p.)
p.h
0
0
0
0
0
0
2(1
+ p.)
J..h
0
0
0
0
0
0
0
YfJ
Y.
t
KfJ
K.
ill
BfJ
I s. I
0
0
0
MpT
M.T
0
NpT
I N.T
w
tv
...
~
~
!II
l
~
Q.
I\>
e:g.=
=
!I'
=
0
"<
(j
Q.
I\>
=
=
0
!I'
0:1
I\>
~
l""'
<i
::t.
'"
=
a::t.
(j
0
0\
o
231
6.1 Constitutive Laws
MaT
=
MpT
=
MT
ECi.
1 - Jl
= --
fh l2 T(Oe de
(6.9b)
-h12
and where the shear moduli are now assumed to be related by Gn = AG. A,
sometimes called the shear correction factor or the transverse warping shape
factor, is relatively complex to define explicitly, but is commonly taken as 5/6
for isotropic shells. 2
Matrix 6-2 in the form
(6.10)
which serves to illustrate again that more complex material properties, expressed
through a modification of the stress-strain law [CJ, can be represented through
a corresponding generalization of [DJ without affecting the remaining development. Similarly, the {NT} matrix can be suitably altered if {aT} is changed. We
consider some of these possibilities in the next section. Furthermore, we must
emphasize that two terms of [DJ, D33 and D66 , often appear in slightly different
forms, modified by a factor of2. This arises because of the somewhat ambiguous
definitions of wand 't" as given in equations (5.28) and (5.44); there is little
consensus on whether w should be defined as in equation (5.28), or as one-half
of that quantity. Similarly, the factor of 1/2 introduced into the definition of
't" could be omitted. The only requirement is to maintain consistency in the
constitutive matrix, and [DJ, as given in Matrix 6-2, is correct in this regard.
Also, note that for plates with no in-plane loading, the first three rows and
columns of matrix 6-2 may be deleted. If transverse shearing strains are not
included in the formulation-either for shells or for plates-the last two rows
and columns are not relevant; rather, the transverse shear resultants are obtained from the fourth and fifth equilibrium equations, (3.22a) and (3.22b), once
the stress couples are computed.
It is also of interest to note that although the transverse shear resultants are
readily computed from the last two equations in matrix 6-2, the corresponding
stresses in matrix 6-1 violate the stress-free conditions on the faces ± h/2, since
Yo< and YP are constant through the thickness. This has been a major consideration in the formulation of a consistent theory of plates, as discussed later
in section 8.6.4. However, within the context of our treatment, this violation is
only at the point (elasticity) level and may be circumvented by computing the
shear stresses directly from the stress resultants, as described in equations
(3.10).
6.1.2 Specifically Orthotropic Materials: A generalization of the isotropic condition which is of interest to us is the specifically orthotropic material. The strict
definition of this configuration is that elastic properties are specified along two
mutually orthogonal material axes, whereas the shell still retains the piecewise
constant thickness and cross-sectional symmetry presumed so far in this text.
Further, it is assumed in our development that the orthogonal material axes
232
6 Constitutive Laws, Boundary Conditions, and Displacements
Matrix 6-3
[Dor] =
ShC11 d(
ShC12 d(
0
0
0
0
0
0
Sh C 12 d(
Sh C22 d(
0
0
0
0
0
0
0
0
Sh C44 d(
0
0
0
0
0
0
0
0
Sh C 11 (2 d(
ShC12(2 d(
0
0
0
0
0
0
Sh C12(2d(
S.C 22 (2d(
0
0
0
0
0
0
0
0
S. C44 (2 d(
0
0
0
0
0
0
0
0
A,ShC55 d(
0
0
0
0
0
0
0
0
Ap S. C66 d(
coincide with the coordinate lines of the middle surface of the shell. The development of composite material technology has made it feasible to produce very
efficient shells and plates having such orthotropic properties.
With this as a preface, we introduce a modified matrix [C] into equation (6.2)
[Cor] =
C ll
C l2
Cl2
C 22
0
0
0
0
0
0
0
0
C44
0
0
0
0
0
C55
0
0
0
0
0
C66
(6.11)
The elements of matrix [Cor] are generally determined experimentally.
[Cor] may be used to derive a corresponding stress resultant-strain matrix
[Dor] and a corresponding thermal load vector {N Tor } by following the same
procedure described in section 6.1.1.2. We also redefine the thermal coefficients
and transverse warping shape factors to permit different values in each direction. The resulting generalization of equation (6.10) is
(6.12)
where [Dor] is given in general form as shown in matrix 6-3, and the elements
of {N Tor} are
1
1
=1
1
+ C 12 (iP)T(Od(
(6.13a)
a" + C 22 (iP)T(Od(
(6.13b)
(clla"
+ C 12 (iP)T(Od(
(6.13c)
(c l2 a"
+ C 22 (iP)T(Od(
(6.13d)
N"Tor
=
(clla"
N pTor
=
(C 12
M"Tor
M pTor =
6.1 Constitutive Laws
233
For homogeneous shells, the integrals in matrix 6-3 and equations (6.13a-d)
are easily evaluated in explicit form. For multilayered sections, the integrals
may be computed by summing over the thickness. Multilayered shells and
plates are widely used in applications of high-performance materials. For such
layered shells, the shape factors A./X and A.{J are determined for specific cases
by Dong and Tso, 3 through correlation with selected reference solutions based
on the theory of elasticity. A complete bibliography on the calculation of shape
factors for layered plates and shells is also given there.
6.1.3 Stiffened Shells and Plates and Reticulated Shells: There are two additional structural systems which are neither shells nor plates by the strict definition, but may be conveniently treated as such. These are (a) shells or plates to
which discrete stiffeners, surface undulations, or folds have been added to
supplement the strength and/or rigidity of the basic constant thickness section;
and, (b) frames composed of intersecting grids of closely spaced members that
follow the topography of a curved surface. Such frameworks are termed shelllike structures or reticulated shells and combine the favorable structural characteristics of the basic geometric form with the use of efficient prefabricated
components, requiring little or no falsework for erection. 4 The ancient Pantheon
(figure 1-3) is a stiffened shell; the contemporary Astrodome in Houston, Texas
(figure 2-8 [m]), is a reticulated shell.
Although it is theoretically possible to model these relatively complex systems as ensembles of the various constituent components, it is often feasible,
considerably simpler, and more economical to use surface structure equations.
Naturally, the question arises as to the appropriate value for the "thickness"
since, by definition, a shell or plate has a piecewise constant or smoothly varying
thickness. It appears to be proper to consider this question at the level of the
stress resultant-strain relationships, matrix [D], rather than at the stress-strain
level, matrix [C], since the latter matrix is based on the stress-at-a-point focus
of the theory of elasticity-which obviously has little meaning when we must
consider equivalent thicknesses.
A fairly common and quite illustrative example is that of a shell or plate with
equispaced stiffening ribs running in one direction, as shown in figure 6-1, or
in both directions. A common method of treating this problem is to combine
the properties of the basic shell or plate and the stiffener over a repeating
interval of the cross section, such as d{J' and then to compute an equivalent
thickness in the direction of the stiffener. This process is often referred to by
the colloquial title of verschmieren, after the German verb "to smear over."
For a shell which behaves essentially in accordance with membrane theory,
the equivalent thickness is logically computed with respect to the extensional
rigidity, whereas for a plate subjected to bending, it would be based on the
flexural rigidity.
To illustrate such a computation, consider the situation shown in figure 6-1.
For a shell, we compute the area of the repeating cross section, hd{J + lib, and
234
6 Constitutive Laws, Boundary Conditions, and Displacements
f3
Fig. 6-1
Shell or Plate with Stiffening Ribs
divide by the spacing, d p , to get
hex
bh
= h +T
p
(6.14)
In the other direction, we use hp = h. If the stiffened structure is a plate acting
primarily in bending, we would compute the moment of inertia of the repeating
T section about its centroidal axis, 1Mi . Then, the equivalent thickness her is found
from setting
(6.15a)
or
h =
"
(121hii)1/3
dp
(6.15b)
The thickness in the {J direction is again taken as hp = h.
The equivalent thicknesses are then used to compute the corresponding terms
of matrix [D]. Referring to the elements of matrix 6-2, we would use her to
compute the terms in the first and fourth rows, and hp for the terms in the second
and fifth rows. It has been suggested 5 that the terms corresponding to the
shearing rigidity (rows three, seven, and eight) be evaluated by using the basic
6.1 Constitutive Laws
235
thickness h, whereas the twisting rigidity (row six) may be computed using
the basic thickness or, in some cases, be neglected altogether. This technique
may be easily extended to cases in which the stiffeners run in both directions
by computing appropriate hp thicknesses from formulas analogous to equations
(6.14) and (6.15). A form of a plate supplemented by beams running in two
directions known as a waffle slab (figure 2-8[w]) is quite widely used in reinforced
concrete building construction.
This simple illustration of a verschmieren procedure raises several questions
which we consider at this point. If only one equivalent thickness is defined in
each direction, should it be the extensional or the flexural thickness for a shell
that may have significant bending? We have opted for the extensional, but will
this equivalent thickness adequately represent the flexural characteristics of
the stiffened shell? Also, what about the shearing and twisting thicknesses? To
include only the basic shell or plate, as mentioned in the preceding paragraph,
seems to be a rather crude approximation. Finally, the use of different equivalent thicknesses in the at: and f3 directions for the extensional and bending
rigidities would cause matrix [D] to be unsymmetric because of the off-diagonal
terms in rows one, two, four, and five. This leads to contradictions and computational difficulties that should be avoided.
From the preceding questions, it is obvious that a procedure which embodies
more of the characteristics of the actual structure is desirable. This is especially
true as the stiffeners become more prominent with respect to the basic cross
section, up to the case of reticulated structures, where they may constitute
practically the entire means of resisting the primary forces and moments.
A promising approach for this problem is the split rigidity concept introduced
by Buchert. 4 The method is a generalization of a development by E. Reissner,
who has been one of the most prolific contributors to the theory of shells in the
middle and late twentieth century.6 Essentially, different equivalent thicknesses
for the extensional, shearing, bending, and twisting terms may be specified. To
avoid difficulties with nonsymmetry in [D], Poisson's ratio is neglected. This
leads to a diagonal matrix [Deq], which replaces [D] in equation (6.10):
[D
eq
J=
r
E h h
hap(m) h;(b) h~(b) h;P(b) Ah,.(m) )ohp(m)J
arm) p(m) 2
12 12 12
2
2
I
(6.16)
The additional subscripts (m) and (b) indicate membrane or extensional and
bending or flexural, respectively.
To illustrate the split rigidity concept, we choose a reticulated shell composed
of a gridwork of structural members, as shown in figure 6-2. The members are
assumed to have locally constant spacing, d,. and d p ; areas A,. and Ap; moments
of inertia I,. and Ip; and St. Venant torsion constants K,. and K p, in each
direction. Any infilled material between the main members is assumed to be
either restricted to local loading distribution, and therefore negligible in this
analysis, or included in the section properties of the main members.
With these idealizations, we may immediately compute
236
6 Constitutive Laws, Boundary Conditions, and Displacements
Aj3, 113 ,Kj3
Fig. 6-2
A"
h,,(m)=T
Reticulated Shell
(6.17)
p
and
Ap
=d
hp(m)
(6.18)
"
The equivalent in-plane shear thickness is a bit harder to visualize. A first
approximation, considering only the primary members of the frame, would be
to take the average of the equivalent extensional thicknesses, t[h,,(rn) + hp(rn)],
so that
h"p(rn)
Ap)
="21 (A"
dp + d;
(6.19)
We note that there is the possibility of increasing the in-plane shear resistance
substantially through the integration of the infilling material with the primary
frame structure.
Now, we turn to the equivalent flexural thicknesses. Considering the Q(
direction first, we equate the moments of inertia per unit width of the frame
member and the equivalent shell
f" _ ~h3
d - 12
p
,,(b)
from which
h,,(b)
= ( 12 ~:) 1/3
(6.20)
6.2 Boundary Conditions
237
Similarly, in the fJ direction
hp(b)
= (12 ~) 1/3
(6.21)
Last, we turn to the equivalent twisting thickness. It is in this mode of resistance that a shell-like structure often differs most from a true shell. Whereas
a true shell or plate has relatively great twisting rigidity as compared with
the flexural rigidity, the reticulated structure is likely to have comparatively
little twisting rigidity, especially if it comprises thin-walled members with open
cross section.
In his study of orthotropic plates, Huffington suggested a procedure based
on the St. Venant torsional rigidity of the members. 7 This is, by no means,
a general solution; however, it serves to illustrate a rational approach applicable
to a specific class of problems. It is likely that the twisting rigidity will be of
greater significance in plates, which behave in a primarily flexural mode and
may have little influence on shell-like forms. Since we are following a general
course applicable to shells as well, we take only the average extensional thickness for computing the twisting rigidity. Therefore,
(6.22)
(6.23)
The use of an equivalent stress resultant-strain matrix, such as the one
defined here, enables a large variety of curved frames and gridworks to be
efficiently analyzed by using solutions based on shell and plate theories, i.e., by
using a continuum as opposed to a discrete approach. We must remember,
however, that the stress resultants and couples computed from such solutions
are intensities per unit length of middle surface and must subsequently be converted to forces and moments in the individual members. A logical procedure
for accomplishing this conversion is to take the average value of the stress
resultant or couple over a repeating interval, d" or d p , multiplied by the interval.
6.2 Boundary Conditions
6.2.1 Role of Boundary Conditions in the Formulation of Shell and Plate Theories:
In the preceding section, we have presented the constitutive laws that serve to
connect the equilibrium equations derived in chapter 3 and the compatibility
relations developed in chapter 5. The classical formulation of the solid mechanics problem embodies these three components, which may be collected as
a set of partial differential equations or as an energy principle. We explore both
238
6 Constitutive Laws, Boundary Conditions, and Displacements
ofthese formats in depth in the succeeding chapters; however, there is one more
ingredient necessary to complete the theory in either case: a suitable set of
boundary conditions. The possibilities may be classified as follows: (a) static
(a stress resultant or stress couple is specified; (b) kinematic (a displacement
or rotation is specified); and (c) elastic (a linear combination of a static and
a kinematic quantity is specified). This may be represented by a spring-type
support.
In the following discussion, we place two restrictions on the boundary conditions to simplify the formulation somewhat: (a) all boundaries coincide with
either an Sa or sp coordinate line and, (b) all kinematic boundary conditions are
homogeneous. Exceptions to the first restriction include such problems as skew
intersections and cutouts and are best treated using discrete approximation
techniques such as the finite element method. Nonhomogeneous boundary
conditions are not necessarily more difficult to deal with than the homogeneous
variety, but since they occur comparatively infrequently, it is usually expedient
to exclude this possibility from the general formulation and to treat such
situations that may arise as special cases.
6.2.2 Static and Kinematic Boundary Conditions: We consider two representative
boundaries of a surface in figure 6-3. The boundary coinciding with the Sa coordinate line is located by f3 = 11, and the boundary along the sp coordinate line
is designated by 0( = a. On the boundary 0( = a, there are five static quantities
acting: N a, Nap, Qa, M a, and Map. Also, there are five kinematic components
present: Da, Dp, Dm DaP ' and Dpa. Note that each static quantity performs work
through one and only one kinematic quantity within the small deformation
theory. We group the associated quantities as static-kinematic correspondents
as listed on figure 6-3. For the boundary f3 = 11, the correspondents are also
given on the figure. This pairing is quite important, for at any point on either
boundary, only one of each pair of static-kinematic correspondents may be
specified-except for an elastic boundary represented by a linear combination
of the two. In no case, however, may both or neither of the correspondents
be designated on a boundary. Further, when one correspondent is specified,
the other must be accommodated by the boundary; e.g., if a displacement, such
as Da, is set equal to 0, the boundary must resist whatever value of Na is
calculated from the subsequent solution; or if the stress resultant Na = 0, the
boundary must be free to displace through the computed Da. In the case of
static boundary conditions, nonzero values are often prescribed as a known
loading term, i.e., in figure 4-12, Nz(O) = - N.
Before examining some specialized forms of the boundary conditions, we
should mention one further requirement, the elimination of rigid body displacements. To accomplish this, it is sufficient that Da, D p, Dm Dap , and Dpa each be
restrained to vanish at one point on the surface. Seemingly, we would also need
to restrict the drilling rotation (about the normal), even though this degree of
freedom is not germane to the theory. There are also certain cases when the
239
6.2 Boundary Conditions
n
a
B.C. of a=a ()Q =0)
B.C. of P=/J (yp=O)
N,s or Dp
Na or Da
Nap or
Qa
Dp
or Dn
a
Dp)
(Oap or
q,)
(NaP
o.e or Do
a
a
(~a or Dp)
((Sa/3 or Dn)
Mp or Dpa
Ma or Dap
MaP
or D
~
M,8a or
D,8a
Fig. 6-3
DaP
Boundary Conditions
loading is self-equilibrated, such as those corresponding to thej > 1 harmonics
in section 4.3.7, for which the rigid body displacements do not have to be
eliminated explicitly.
As specific examples of easily visualized boundary conditions which are
frequently encountered, we consider the boundary (l( = ~:
Condition Desired
Specified Quantities
Free
Fixed
Hinged
Roller (1.. to middle surface)
Simply supported (II to middle surface)
Sliding (1.. to middle surface)
N« = N«fJ = Q« = M« = M«fJ = 0
D« = DfJ = D. = D«{J = D{J« = 0
M« = 0; D« = D{J = D. = Dp« = 0
Q« = M« = 0; D« = D{J = D{J« = 0
N« = M« = 0; DfJ = D. = D{J« = 0
Q« = 0; D« = D{J = D«{J = D{J« = 0
240
6 Constitutive Laws, Boundary Conditions, and Displacements
Recognize that the terms used to describe the boundaries-e.g., hinged and
roller-are essentially two-dimensional in implication and do not fully describe
the constraint in the f3 direction. Consequently, the conditions on D p , and/or
Dpa may be interchanged with NaP and/or Map, respectively, if the physical
situation seems appropriate. Also, the term simply supported is often imprecisely
used to also describe the hinged or roller cases. Technically, this condition
should be reserved for beam-like situations for which a meridional constraint
is not necessary. A similar set of conditions can be derived for the boundary
f3
=
11.
6.2.3 Kirchhoff Boundary Conditions: We have mentioned several times that
much of the classical theory of shells and plates is based on the suppression
of the transverse shearing strains. From the statement of the admissible boundary conditions presented in the preceding section, we can anticipate that the
most general system of governing differential equations is of tenth order in
each direction, permitting prescription of five conditions on each of the four
boundaries. When transverse shearing strains are not included, the governing
equations are reduced to eighth order in each direction, and a contracted set
of boundary conditions is required.
The derivation of the static conditions consistent with the suppression of
transverse shearing strains is one of the more interesting theoretical problems
in the theories of shells and plates. These are commonly known as the Kirchhoff
boundary conditions after the noted mathematician G. Kirchhoff, who first
Fig. 6-4
Forces and Moments on a Shell Bo undary
241
6.2 Boundary Conditions
<- middle surface
Fig. 6-5
Kelvin-Tait Argument
estabiished the correct relationships for a plate using an energy formulation. 8
Although Kirchhoff's derivation served to clarify the situation somewhat, it
was mathematical in basis, and it remained for two distinguished nineteenthcentury scientists, Lord Kelvin and P. Tait, 9 to provide a physical interpretation
of the Kirchhoff conditions.
To illustrate the Kelvin-Tait argument and to provide an extension into
the shell formulation, refer to the ex = a boundary as shown on figure 6-4. We
show four adjacent differential half-segments ab, bo, oc and cd, each of length
!J.sp/2, and indicate the resultant forces acting on segment boc due to N",p and
Q",. These resultants act at point o. Also, we show the resultant couples on
two adjacent segments, abo and oed, due to M",p. These act at points band
c, respectively. The directions shown correspond to the sign convention of
figure 3-2. Now, in figure 6-5, we examine a normal section showing the middle
surface of the differential segment and replace the moments due to the stress
couples by equivalent forces at a, 0, and d. The replacement of a moment on
the boundary by a pair of equal and opposite forces separated by a differential
distance is justified by the St. Venant principle, 1 0 one of the many enduring contributions of this outstanding nineteenth-century scientist. From M",p, we have
the forces (M",p!J.sp)/!J.s p = M",p acting at a and 0 normal to the chord ao. From
M",p + M",p,p!J.{3, we derive a similar pair of forces acting normal t~ chord od.
Also shown by dashed vectors are the effective in-plane shear N",p and the
effective transverse shear Q. which we wish to evaluate at point o.
242
6 Constitutive Laws, Boundary Conditions, and Displacements
Summing forces in the
Pdirection, we have
NlZpdsp = NlZpds p + M IZP sin (dS p )
2Rp
+ (MIZP
+ MlZp,pdP)Sin(:~:)
Since MIZP,pdP is of higher order than the remaining terms and sin(dsp/2Rp)
dS p/2R p, the effective in-plane shear is
NlZp = N IZP
M lZp
Rp
+--
(6.24)
Summing forces in the n direction, we have
QlZdsp = QlZdsp -
MlZpCOS(:~:) + (MIZP + MIZP,pdP) cos
Noting that cos(dsp/2Rp)
shear is
Q
17.
=
Q
17.
~
(:::J
1, and that ds p = BdP, the effective transverse
+ M IZP . p
(6.25)
B
Along the boundary
~
P=
p, we can develop a similar set of effective shears
MplZ
NplZ=NplZ+T
(6.26)
17.
and
(6.27)
Now, referring to figure 6-3, we have the five static boundary conditions
contracted to four along each boundary.
For the corresponding kinematic conditions, we first examine equations (5.46g)
and (5.46h) with YIZ = Yp = O. Then DIZP and DplZ , corresponding to the bending
couples, are equal to "'17. and
respectively. For the effective shear stresses,
which incorporate the twisting couples, we have Dp and Dn on the IX = (X
boundary, and DIZ and Dn on the P = P boundary. These are also listed on
figure 6-3.
"'p,
6.2.4 Boundary Conditions for Plates with No Transverse Shearing Strains:
Although the general boundary conditions described in section 6.2.2 are applicable for plates as well as shells, the Kirchhoff conditions are somewhat
simplified for plates. An examination of equations (6.24) and (6.26) reveals Rp
and RIZ in the denominators of the twisting stress couple terms, M IZP and M plZ .
6.2 Boundary Conditions
243
Since these radii ~ 00, the effective in-plane shear is superfluous, and we are
left with only the effective transverse shears, as given by equations (6.25) and
(6.27).
6.2.5 Boundary Conditions for Shells of Revolution: The shell of revolution
geometry was specialized in section 2.8.2, with the meridians corresponding to
the Sa. coordinate lines and the parallel circles to the sp coordinate lines. For
this shell form, we only have boundary conditions along the parallel circles. If
the meridional coordinate is taken as </J, and the circumferential coordinate as
e, a parallel circle boundary corresponds to </J = (j. The Kirchhoff conditions
follow from equations (6.24) and (6.25) as
M,pl)
~
= N,pl) + - -
N,pl)
RI)
(6.28)
and
(6.29)
6.2.6 Symmetry: In addition to external boundary conditions such as those
we have discussed in the preceding paragraphs, there may be equations of
condition that arise because of symmetry properties of both the geometry
and the loading. The practical manifestation of the recognition of symmetry
conditions is that only a part of the entire continuum must be analyzed.
A fairly general modern treatment of the topic of symmetry was presented
by Glockner,ll and we focus here on the applications to shells and plates.
Of course, we have already used the concept of symmetry extensively in our
treatment of shells of revolution as a distinct class of surface structures. Such
shells are distinguished geometrically by their rotational symmetry. Further,
in the analysis by the Fourier series technique in chapter 4, symmetric and
anti symmetric components of the loading were identified. Formally, we may
state that a loading function q(x) is symmetric about x = 0 if q( -x) = q(x), and
antisymmetric if q( -x) = -q(x). Furthermore, higher harmonic components,
which are self-equilibrated, are said to possess cyclic symmetry. This was
illustrated by the column-supported shell solution in section 4.3.7.3.
As a first illustration, consider a complete rotational shell, such as a sphere
or an ellipsoid subject to a uniform normal pressure qn = q. Using the notation
of figures 4-2 and 6-3, the following symmetry conditions may be enforced at
</J =
n12:
D,p
= 0;
Q,p
=0
These conditions require that only half of the shell be represented in the ensuing
analysis. This may not be important in a membrane theory analysis which is
statically determinate~e.g., equations (4.27) and (4.28)~but may result in
244
6 Constitutive Laws, Boundary Conditions, and Displacements
considerable savings when an analysis using the general theory is performed.
In some numerical solutions, it also may be necessary to enforce such symmetry
conditions to eliminate rigid body displacements.
Another case of interest is a shell of translation, such as the elliptic paraboloid
shown in figure 4-51(a) subject to a uniform vertical loading qz = q. We may
consider only the quadrant OEDF and, using the notation of figure 4-40, take
0( = x and f3 = yon figure 6-3. Then, we have
along OF and
Dy
= 0; Nyx = 0; Qy = 0; Dyx = 0; Myx = 0
along OE. It is possible to affect further savings by defining symmetry along
the diagonal OD, but since this involves displacements along non-principal lines
of curvature, it lies outside our present scope. The conditions for the shell of
translation are also applicable to uniformly loaded, symmetrically supported
plates.
.
As a final example, we choose the same elliptic paraboloid illustrated in
figure 4-51(a) subjected to a hydrostatic loading acting normal to the X-Y
plane. The distribution in the Y direction is given by
qAX, Y) = q
(b - Y)
2
and the loading is assumed to be uniform across the X direction. This loading
is readily decomposed into a constant component (j = 0)
and a linear component (j
=
1)
Y
q}(X, Y) = -q2
The constant component q~ is symmetrical with respect to both the X and
Yaxes, and is treated identically to the previous example. The linear component
q} remains symmetric with respect to the Yaxis, so the conditions along OF
are unaltered. However, this component is anti symmetric with respect to the
X axis, since qi(X, - Y) = - qi(X, Y). The resulting conditions along OE are
Ny
= 0; Dx = 0; D. = 0; My = 0; Dxy = 0
which are the correspondents of the conditions on a line of loading symmetry
in each case. Again, the anti symmetry conditions are applicable to plates as
well.
6.3 Membrane Theory Displacements
245
6.3 Membrane Theory Displacements
6.3.1 General Aspects: In the preceding sections of this chapter, we have completed the derivation of the individual components required to formulate the
general theories of shells and plates. In chapter 4, we investigated a very
important class of problems, the membrane theory of shells, which requires only
the solution of a reduced set of equilibrium equations to determine the in-plane
stress resultants. The membrane theory is generally statically determinate as
opposed to the general theory, which is statically indeterminate and therefore
requires simultaneous consideration of equilibrium, compatibility, and constitutive relations. Nevertheless, we cannot say that the membrane solution is
complete until the displacements are determined. As is the usual procedure for
statically determinate problems, we may compute these displacements from the
equilibrium solution, assuming that the stress field is known.
At the outset, it is instructive to consider the relevance of the membrane
theory displacements in shell analysis and design. First, since the design of
many shells is based on the membrane theory, the magnitudes of the associated
displacements would be useful to verify the applicability of small displacement
theory (assumption [2], table 1-1). Also, knowledge of the expected displacements can be important in providing a supporting system to accommodate
the required boundary deformations, as mentioned in the previous section.
Further, it is possible to obtain values for the bending stress resultants and
couples by substituting the membrane theory displacements into the general
strain-displacement and constitutive equations, equations (5.46) or (5.47), and
equation (6.10), respectively. At best, the values obtained in this manner may
be useful for order-of-magnitude estimation in the regions of the shell dominated by membrane action, but do not account for boundary effects.
Next, even if a solution is obtained using a bending theory, the values of
the in-plane stress resultants are often not changed significantly from those
computed from a membrane theory analysis except, perhaps, in localized regions. Since certain of the stress resultants, such as Ntfo in symmetrically and
anti symmetrically loading shells of revolution, are dependent on the overall
equilibrium of the shell, they cannot be altered very much by bending action.
This means that the middle surface displacements, which are primarily functions
of the in-plane stress resultants, may be estimated quite closely from the
membrane theory solution.
Additionally, we may anticipate that a logical procedure to analyze shells
which do not fully meet the requirements of the membrane theory because of
inappropriate boundary conditions, would be to use the flexibility method.
An example would be a spherical shell, such as that illustrated in figure 4-2,
with the lower boundary pinned or fixed, as defined in section 6.2. Briefly,
the flexibility method consists of introducing a sufficient number of kinematic
releases into the system to make it statically determinate. Then, the conditions
of deformation compatibility consistent with the original system constraints
246
6 Constitutive Laws, Boundary Conditions, and Displacements
are enforced on superposed primary and secondary systems to determine the
static correspondents of the kinematic releases, known as the redundants. This
procedure is familiar to students of structural mechanics and is described in
most standard texts on the analysis of indeterminate structures, so we need
not elaborate here. Briefly, we note that the compatibility equations take the
form
+ [F] {X}
=
{X}
{A}
=
the vector of the displacements corresponding to each of the releases
on the original structure. Commonly, {A} is null.
{X}
=
the vector of the displacements of each of the releases due to the
applied loading. These displacements are computed on the released
(now statically determinate) structure.
{X}
=
the vector of the statical correspondents of the kinematic releases,
the redundants, initially unknown. For plates and shells, the elements of {X} are usually transverse shear forces and moments on
the boundary.
[F]
=
the matrix of flexibility influence coefficients, each element of which
is a displacement of a release due to a unit value of one of the
elements of {X}. The influence coefficients are computed on the
released, statically determinate, structure.
{A}
(6.30)
where
Once {X} and [F] are computed and {A} is specified, we solve for
{X} = [F-l][{A} - {A}]
(6.31)
For shells, {X} can usually be evaluated with sufficient accuracy by using the
membrane theory, as discussed in the previous paragraph. The elements of [F],
being displacements due to boundary transverse shear forces and bending
moments, must still be computed by using the bending theory; however, since
no surface loads are involved, only the homogeneous bending equations need
be considered, which frequently simplifies the solution. A specific illustration
of this method is presented in chapter 9 for the bending analysis of cylindrical
shells.
Thus, we have suggested a variety of reasons for studying the middle surface
displacements of shells in equilibrium under a known internal stress field
computed by the membrane theory.
6.3.2 Governing Equations: To connect the middle surface displacements Da.,
Dp, and Dn with the stress resultants Na., Np, and Na.p, compare equations (5.46a)(5.46c) with the first three rows of matrix 6-2. Since matrix 6-2 gives the stress
resultants as functions of strain, we must invert the relations
(6.32a)
247
6.3 Membrane Theory Displacements
Np = /lD 1 6"
S
+ D 1 6p -
(6.32b)
NPT
= D2 w
(6.32c)
recalling that N"p is replaced by S in the membrane theory. From matrix 6-2,
Dl = Ehj(1 - /l2) and D2 = Ehj[2(1 + /l)]. Solving for the strains in view of
equations (6. 1a), (6.1b) and (6.9a), we have
1
/lNp)
+ (1
- /l)NT]
(6.33a)
Eh [(Np - /IN,,)
+ (1
- /l)NT]
(6.33b)
6"
= Eh [(N" -
6p
=
W
= 2(1 + /l) S
1
(6.33c)
Eh
By equating the r.h.s. of equations (S.46a-c) and equations (6.33a-c), we arrive
at the complete set of equations relating the displacements and the stress
resultants. With these stress resultants having been determined by using the
methods of chapter 4, we have three partial differential equations in the three
unknown middle surface displacements, D", Dp, and Dn.
We may also obtain expressions for the rotations D"p and Dp" in terms of
the middle surface displacements from equations (S.46g) and (S.46h), if we
neglect the transverse shearing strains. This is consistent with the membrane
theory where Q" and Qp are assumed to be zero.
Many forms used for shell structures facilitate simplifications in the functional dependence of the geometrical parameters. Therefore, it is expedient to
specialize the equations for specific geometrical types.
6.3.3 Shells of Revolution
6.3.3.1 Transformation of Governing Equations. Since we have already specialized the strain-displacement relations for shells of revolution with ex = t/J and
f3 = () in section S.4, we have from equations (S.48a-c) and (6.33a-c)
R",6",
R
= DtP,¢ + Dn = E~ [(N", - /lNo) + (1 - /l)NT]
R o6o = Do,o
A.
•
+ cos 'I' D", + sm ifJ Dn
Ro
Row = RDo,,,,
+ D""o -
Ro
= Eh [(No - /IN,,,)
cosifJ Do =
2(1
+ /l)Ro
Eh
S
(6.34a)
+ (1 - /l)NT] (6.34b)
(6.34c)
'" resultants and thermal loads are both presumed to be known
Since the stress
at this stage, we drop the thermal terms, which can simply be absorbed into
N", and No if present.
Now, we return to the technique of section 4.3.1, whereby a judiciously
chosen set of auxiliary variables was used to reduce the set of equations. Again,
248
6 Constitutive Laws, Boundary Conditions, and Displacements
we follow Novozhilov 12 by introducing
IjI = DIJ
(6.35a)
¢ = Dt/>
(6.35b)
Ro
sin¢
Next, we substitute equations (6.35a) and (6.35b) into equations (6.34a-c) and
eliminate Dn between the first two equations, giving
RIJ ~,t/> - sin¢
1
R~ sin ¢ ",Rt/>
,t/>
2(1 + /1)
Eh R 0 S
"',0 = Ehsin¢ [(Rt/> + {lRIJ)Nt/> + -~,IJ --
(RIJ
+ /1Rt/»NIJ]
(6.36a)
(6.36b)
These equations may be compared with the transformed equilibrium equations,
(4.6b) and (4.6a), respectively. We see that the l.h.s. of both would be identical
if we were to interchange", and 1jI, and ~ and ¢; whereas, the r.h.s., in both cases,
consist entirely of known functions. Therefore, whatever solution strategies
were deduced for the membrane theory equilibrium equations are applicable for
the equations governing the corresponding displacements. Also, we recognize
that once ¢ and IjI are determined, Dt/> and DIJ follow from equations (6.35a),
and (6.35b) and Dn may be calculated from equation (6.34a) or (6.34b).
As a practical matter, when we attempt to use the various solutions already
derived in chapter 4 for the determination of the corresponding displacements,
we soon realize that the r.h.s. of equations (6.34a-c) are likely to be more
complicated algebraically than the corresponding functions in the equilibrium
equations. This is because the expressions for the stress resultants are already
quite involved for some of the cases that we solved in chapter 4. After being
multiplied by the radii of curvature expressions on the r.h.s., integrals that can
only be evaluated numerically are often produced. This slight complication
notwithstanding, the similarity of the system of governing equations for the
membrane theory stress resultants in terms of the known applied loading,
on the one hand, and the in-plane displacements and the computed stress
resultants on the other, is quite striking. This serves as another example of
the static-geometric analogy mentioned in section 5.4.2.
Once we have evaluated the in-plane displacements, we may compute the
rotations from equations (5.48g) and (5.48h) with Yt/> = Yo = 0:
Dt/>IJ
=
1
-R:(Dn,t/> - Dt/»
(6.37a)
t/>
DIJt/>
= -
~o (Dn.o -
sin ifJ Do)
(6.37b)
We now find it convenient to again consider the axisymmetric (j = 0) and
nonsymmetric (j > 0) cases separately.
249
6.3 Membrane Theory Displacements
6.3.3.2 Axisymmetric Displacements. We drop the O-dependent terms from
equations (6.36) and (6.37) and integrate the first two of the set. Following
section 4.3.2.1 and using equation (6.34b), we find
-
~
Dt/J
=
sin¢
=
~E f-~-[(Rq)
+ j.tRo)Nq) hsm¢
If! =
Do
Ro
=
+ j.t)
(6.38a)
(Ro
+ j.tRq»)NoJd¢
f Rq) S d¢
hRo
(6.38b)
Ro
Dn = Eh (No - j.tNq») - cot¢ Dq)
(6.38c)
1
2 (1
E
Dq)o = -R(Dn.q) - Dq»)
q)
Do
Ro
DOq)=~
(6.38d)
(6.38e)
The indefinite integrals in equations (6.38a) and (6.38b) may be written in
the alternate form
(6.39a)
(6.39b)
where ¢", and ¢"" are the boundaries at which Dq) and Do are specified. We have
used ¢", and ¢"" to emphasize that these boundaries are not the same boundaries where the corresponding stress resultants are specified, ¢' and ¢", as
identified in equations (4.10) and (4.11). Recalling the arguments of section 6.2.2,
we cannot specify Nq) and Dq), nor S and Do on the same boundary. Moreover,
since we have presumably specified the boundary values Nq)(¢') and S(¢") in
the equilibrium solution, Dq) and Do must be imposed at the other boundary in
each case. Thus, we have little latitude in the choice for ¢", and ¢"".
When Ro is not finite, as for ¢ = 0 on a toroidal shell, equation (6.38c) is not
applicable, but an alternate equation for Dn can be found from equation (6.34a).
For a dome, we again encounter indeterminate forms for Dq) and Do at the
pole. It is easily shown using L'Hospital's rule that Dq)(O) = Do(O) = O. Then,
equations (6.38a-c) gives
(6.40)
250
6 Constitutive Laws, Boundary Conditions, and Displacements
As an example, we investigate a spherical dome under dead load, using the
solution for stresses derived in section 4.3.2.2 and referring to figure 4-2 with
(A = O. We wish to compute the displacement normal to the middle surface, Dn ,
at the lower boundary rP = rPb'
Examining equation (6.38c) we observe that since D",(rPb) = 0 as dictated by
the requirement to develop N",(rh,) fully, we need only the values of N", and No
at rP = rPb' From equations (4.20) and (4.21), we have
qa
(6.41)
and
NO(rPb)
= qa (1
1 rP
+ cos
b
(6.42)
- cos rPb)
We substitute these values into equation (6.38c) with Ro = a, to get
qa 2 (
Dn(rh,) = -h
E
1 + J-L t/J - cos rPb )
1 + cos b
(6.43)
At the pole, the normal displacement is found using equations (4.22) and
(6.40):
_qa 2
Dn(O) = 2Eh (1 - J-L)
(6.44)
As a further example, we consider the displacements of a hyperboloidal
shell under self-weight load, previously analyzed in section 4.3.2.3. Referring to
equations (6.38a) and (6.38b), since S = 0, IfJ is O. Then, ~ is evaluated from
equation (6.39a), with equations (4.40) and (4.41) substituted for N", and No
and rP'" taken as rPb' The resulting integral is quite complicated, but it is easily
evaluated using a numerical algorithm, such as the trapezoidal method. Finally,
Dn is found from equation (6.38c).
A study of the displacements for a parametric range of hyperboloidal shell
dimensions, similar to that shown on figure 4-8 for the stress resultants, is given
in figure 6-6. The nondimensionalizing parameters are indicated on the figure.
Also a complete tabulation is available in Gould. 13
6.3.3.3 Nonsymmetric Loading. To complete the analogous treatment of the
equations of equilibrium and compatibility, we apply to equations (6.36b) and
(6.36a), respectively, the procedure initially used to obtain equation (4.85) from
equations (4.6a) and (4.6b) in section 4.3.5.1. The result is
1
R",Ro sin t/J
[R~ sin rP - ]
R",
1
t/I,,,, ,'" + R", sin 2 t/J
IfJ
_
,00 -
x {2(1 + J-L)(RoS),,,, + (Si~rP [(R", + J-LRo)N",
which is in the same form as equation (4.85).
1
R",Ro sin rPEh
- (Ro
+ J-LR",)NoJ}J
(6.45)
251
6.3 Membrane Theory Displacements
o oL--~::"':"""-~--='-=:----=':=---7.
0.4
I/J
1- 4 r - - - . - - - r - - - - . - - - - r - - - - ,
k2: 1.05
d
n
=Dn
1.10
Eh
Q02
o r---~~~~--+---_+--~
0.2
0.4
0.6
0.8
1.0
I/J
Fig. 6-6 Nondimensional Displacements for a Hyperboloidal Shell with alt = 0.90 and als =
als = 0.55. Source: P. L. Gould and S. L. Lee, "Hyperbolic Cooling Towers under Seismic Design
Load," Journal of the Structural Division, ASCE 93, no. ST3 (June 1967): 95.
All strains and displacements have previously been expressed in harmonic
form by equations (S.SO) and (S.SI), and the corresponding strain-displacement
relations have been written as equations (S.S2). With the stress resultants having
been expanded in Fourier series in equation (4.86b), equations (6.34a-c) are
easily written in separated form. Further, we take the variables IfI and ~ as
{:} =
j~O {~~::;:}
(6.46)
and the separated form of equation (6.4S), analogous to equation (4.88), is
1
R",R(J sin <,b
x
{2(1 +
[R~ sin <,b IfIj ]
R",
P
.,p.,p - R", sin 2 <,b
J.l)(R(JSj).", - j
IfIj _
1
- R",R(J sin <,bEh
Ci~ <,b [(R", + J.lR(J)NJ -
(R(J
+ J.lR",)NJ] )
(6.47)
}
The final major equation for the displacements corresponds to the antisymmetrical case, j = 1, earlier discussed in section 4.3.6. Following the same
252
6 Constitutive Laws, Boundary Conditions, and Displacements
steps used in section 4.3.6.1, we introduce the variable
Y(¢)
=
(6.48)
ljI1(¢)R(Jsin¢
into equation (6.47) withj taken as 1. The equation reduces to
(
x
1
-)
R",sin¢ ~</> ,</>
1
= R(J sin ¢Eh
{2(1 + J-l)(R(JS1
),</> -
Ci~ ¢ [(R", + J-lR(J)Nt -
(6.49)
(R(J
+ J-lR",)Ni] )
}
which may be solved as indicated in section 4.3.6.
We now turn to the asymmetric case,j > 1, which is described by equation
(6.47). For spherical and cylindrical shells, it is possible to arrive at solutions
for the asymmetric displacements that are analogous to the stress solutions
given in section 4.3.7.2 and section 4.3.7.4, respectively. 14 For more complicated
geometries, the transformation applied in section 4.3.7.6 may be of some value;
however, little work has apparently been done in this area in terms of analytical
solutions. Rather, for the most part, numerical techniques are used to eva~uate
displacements for asymmetrically loaded shells of revolution. 15
Now, we have completed, in some detail, the displacement analysis for shells
of revolution. Once equation (6.47) is solved for ljIi to determine Di, ~i is
computed from equation (6.36b) to get D~ and then D~ is found from equation
(6.34b).
It does not seem particularly useful to evaluate the membrane displacements
for specific shell geometries in the same detail that is devoted to stress analysis,
since the remaining treatment is self-evident on the basis of the analogy established. However, it is of general interest to summarize some results of various
studies that can be consulted by the interested reader.
First, for closed shells as shown in figure 4-1, certain relationships must exist
among the displacements at the pole to avoid singularities in the solution. These
have been derived in Brombolich and Gould 16 and are summarized in table 6-1.
The derivation of these relationships is discussed further in section 9.3.4.4.
Table 6-1
Displacement constraints for closed shells
Discontinuous Meridian
j
Continuous Dome
@¢ = 0
(Figure4-1[a])
o
D", = Do = D",o = Do", = 0
Do = D",o = Do", = 0;
Do = - tan ¢,D.
D. = 0
D",e = Do", = 0; D", = -cos¢,De;
Dn = - sin ¢,Do
Harmonic Number
1
D",
>1
+ Do = 0; D",o + Do", = 0
@¢=¢,
(Figure 4-1 [b])
253
6.3 Membrane Theory Displacements
Next, since hyperboloidal shells under seismic loading were considered in
much detail, note that the nondimensionalized displacements for the M distribution are tabulated in Gould and Lee. 1 7
Last, we should discuss the general influences of the various harmonic components of displacementj = O,j = 1, andj > 1 on a rotational shell of the tower
type. It is convenient to do this with respect to the R-Z Cartesian coordinates
system shown on figure 2-11.
For thej = 0 case, the resultant displacement will be principally an elongation
or shortening ofthe shell in the Z (axial) direction, with any R (radial) deformation being a uniform expansion or contraction of the parallel circles, labeled
as DO on figure 6-7(a).
For the j = 1 case, the resultant displacement will consist of a uniform lateral
movement in the R direction, illustrated in figure 6-7(b), along with flexural-like
elongation or shortening across the cross section in the Z direction. The
maximum lateral displacement Di must be checked to verify the applicability
of the small deformation theory (assumption [2J, table 1-1).
As discussed in chapter 4, both the j = 0 and j = 1 cases may be solved from
the elementary theories of axial deformation and beam bending. Consequently,
there is no distortion of the cross section. However, for the j > 1 case, the
displacements will vary around the circumference in proportion to sinj8 and
cosj8, and the cross section will distort as shown in figure 6-7(c). In problems
with considerable higher harmonic participation, the limits of the small deformation theory are frequently taxed by the circumferential distortions, where
the relative displacements represented by the difference in the peak amplitudes
should be checked. This difference is labeled as 2Di on the figure. Also, significant distortions in the higher harmonics would necessarily be accompanied
by circumferential bending, which would weaken the basis of a membrane
theory solution.
For an actual loading that may consist of contributions from more than one
harmonic, the maximum relative circumferential displacement, after summing
all participating harmonics, should be used for the check.
6.3.4 Shells of Translation
6.3.4.1 Cylindrical Shells. We consider the general strain-displacement relationship, equations (S.46a-c); and the strain-stress resultant relationships,
equations (6.33a-c), with the curvilinear coordinates adopted for the cylindrical
geometry in section 4.4.1. Thus, we take (X = X, fJ = 8; A = 1, B = a; and
R(7. = 00, RfJ = a to get
ex
=
D x .x
=
1
Eh [(Nx - I-lNo)
+ (1
- I-l)NT J
(6.S0a)
(6.S0b)
254
6 Constitutive Laws, Boundary Conditions, and Displacements
0'
(b)
Fig. 6-7
Harmonic Components of the Displacement of a Shell of Revolution
+ Dx .e =
aw = aDe.x
2(1
+ Jl}a
Eh
S
(6. SOc}
As explained in section 6.3.3.1, the thermal terms are conveniently absorbed
into the stress resultants and are not carried forth explicitly.
We first integrate equation (6.50a) to get
Dx = ;h
f
(Nx - JlNe}dx
+ f3(()}
From equation (6.50c), we write
(6.51)
6.3 Membrane Theory Displacements
De
=
1
-~
f Dx,e dX + 2(1Eh+ /l) f S dX + f4(O)
255
(6.52)
and then from equation (6.50b), we find
(6.53)
The membrane theory rotations are found from equations (5.46g) and (5.46h),
with I'll = I'p = 0:
(6.54)
and
(6.55)
A ready example is the simply supported cylindrical shell subject to the first
harmonic of the Fourier series expansion for the dead load. The stress analysis
for this loading was carried out in section 4.4.1, and the stress resultants are
given by equations (4.169). The boundary conditions must be stated with respect
to the displacements. The longitudinal symmetry dictates that Dx(L/2) = O.
Since the boundary corresponds to the simply supported condition discussed
in section 6.2.2, we also have De(O) = O. These boundary conditions yield
f3(O) = f4(O) = O.
Proceeding, we find by substituting equations (4.169) into equations (6.51)(6.55), and carrying out the integrations and substitutions,
Dx
4qd(L)[2L2
= - -- -
nEh n
n2a
nX
/la ] cos-cosO
L
(6.56a)
(6.56b)
(6.56c)
(6.56d)
4qd [
2/lL2]. nX .
Dex = ---h a - - 2 - sm-smO
nE
n a
L
(6.56e)
The preceding expressions are useful in the bending analysis of open cylindrical
shells, when the membrane theory solution serves as the particular solution. The
boundary displacements, as computed from equations (6.56a-c), are corrected
by edge forces and moments to satisfy prescribed compatibility conditions on
the longitudinal boundaries. 18 The procedure is quite analogous to the classical
256
6 Constitutive Laws, Boundary Conditions, and Displacements
flexibility method of structural analysis, discussed in section 6.3.1, and is
explored in more detail in chapter 9.
At this point, we consider an example that represents an exception to the
general association of membrane theory analysis with statically determinate
systems. Occasionally, boundary conditions are encountered that do not grossly
violate the requirements discussed in section 4.2, and yet do not permit the
a priori determination of the stress resultants before considering the displacements. We might call this a statically indeterminate membrane theory problem,
which was mentioned in section 4.4.1.
As an illustration, we reconsider the previous example in section 4.4.1 with
an axial constraint condition
(6.57a)
replacing the condition
(6.57b)
The preceding static condition enabled!1 (e) and!2(e) in equations (4.168a) and
(4.168b) to be set equal to zero and, subsequently, the explicit expressions for
Sand N x to be written as equations (4.169a) and (4.169b). We may retain the
other static boundary condition S(L/2) = 0, since the symmetry is not altered;
thus, !1 (e) is still zero. Proceeding, the expressions for the stress resultants in
the modified problem are written from equations (4.168) as
8qdL
nX
S = --2-cos-sine
n
L
(6.58a)
(6.58b)
No
4qda . nX
= ---sm-cose
n
L
(6.58c)
We note from equation (6.58b) that the function !2(e) remains to be determined, obviously from consideration of the displacements. If we substitute
equations (6.58a) and (6.58b) into equations (6.51) and (6.52), we have
Dx
1
= Dxs + Eh !2(e)X + !3(e)
(6.59a)
(6.59b)
where Dxs and Dos represent the corresponding displacements from the simply
supported cases, equations (6.56a) and (6.56b), respectively. We now have three
functions of integration to be evaluated from the boundary conditions
(6.60a)
257
6.3 Membrane Theory Displacements
together with the condition stated in equation (4. 170c)
Do(O) = Do(L) =
°
(6.60b)
and an additional symmetry condition
(6.60c)
These equations form only three independent conditions because of the symmetry
of the problem. Since Dos and Dxs automatically satisfy equations (6.60b) and
(6.60c), we find by evaluating Dx(O), Do(O), and Dx(L/2) that
Dxs(O, 0)
+ 13(0) = 0
(6.61a)
=0
(6.61b)
14(0)
(6.61c)
from which
12(0)
=
2Eh
(6.62a)
LDxs(O,o)
= -Dxs(O,O)
14(0) = 0
(6.62b)
13(0)
(6.62c)
It is easily verified that the conditions on Dx(L) and Do(L) are similarly satisfied
by equations (6.62a-c).
To complete the analysis, we substitute equations (6.62a-c) into equations
(6.58) and (6.59) and take Dxs and Dos from equations (6.56a) and (6.56b). The
modified expression for the stress resultants N x is
Nx
[L2
. nX
= -8qd
-SlO- 2
n
na
L
(2L2
--
n2 a
-
J.la )] cosO
(6.63)
with S and No given by equations (6.58a) and (6.58c), respectively. The corresponding displacements are
Dx
4qd
=- (L)[2L2
- 2 - J.la ][cos -nX + -2X - 1] cos 0
nEh n
n a
L
L
4qd (L)[L
nXDO=--h
- (2U
22+4+3J.l) S. l O
nE n
n n a
L
+
(6.64a)
(6.64b)
2L2- J.la ) (X2
X)] sinO
--(n2 a
aL
a
The remaining displacements Dn , Dxo , and Dox are easily evaluated from
equations (6.53)-(6.55).
258
6 Constitutive Laws, Boundary Conditions, and Displacements
We should remember that the applications of this statically indeterminate
membrane analysis are somewhat restricted, since the admissible boundary
conditions may not grossly violate the membrane theory requirements as
discussed previously in section 4.4.1.
6.3.4.2 Shells with Double Curvature. For doubly curved shells of the form considered in section 4.4.2, the computation of displacements due to the membrane
theory stresses is not widely treated in the literature. This is probably due to
two main reasons: (a) One ofthe principal uses for the membrane displacements
is to incorporate them into a flexibility type general solution, as described in
the previous section. Although this approach is quite applicable for shells of
revolution and for cylindrical shells, it is not particularly suited for doubly
curved translational shells due to the lack of homogeneous bending solutions.
(b) The integration of the stress resultant-displacement relations generally must
be carried out numerically, even for the shell of revolution geometry, although
the stress resultants may have been evaluated analytically. Rather than deal
with partially analytical, partially numerical solutions, it is often expedient to
employ a strictly numerical approach. Such a technique for translational shells
is described in Hedgren and Billington. 19 Also, the general techniques of finite
differences 20 and finite elements 21 have been applied to this class of problem.
6.4 References
1. R. D. Lowrey and P. L. Gould, "Thermal Analysis of Orthotropic Layered Shells
2.
3.
4.
5.
6.
7.
8.
of Revolution by the Finite Element Method," Proc. of the lASS Symposium on
Shell Structures and Climatic Influences, University of Calgary, Alberta, Canada,
July 1972, pp. 315-325.
G. R. Heppler and J. S. Hansen, "A Mindlin Element for Thick and Deep Shells,"
Computer Methods in Applied Mechanics and Engineering 54 (1986): 21-47.
S. B. Dong and F. K. W. Tso, "On a Laminated Orthotropic Shell Theory Including
Transverse Shear Deformations," Journal of Applied Mechanics, Trans. ASME 39,
series E, no. 4 (December 1972): 1091-1097.
K. P. Buchert, Buckling of Shell and Shell-Like Structures (Columbia, Mo.: K. P.
Buchert and Associates, 1973), pp. 5-10, 31-34.
S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed.
(New York: McGraw-Hill, 1959), pp. 368-369.
E. Reissner, "Some Aspects of the Theory of Thin Elastic Shells," Journal, Boston
Society of Engineers, Boston, Mass., 42, no. 2, (April 1956): 100-133.
J. J. Huffington, Jr., "Theoretical Determination of Rigidity Properties of Orthogonally Stiffened Plates," Journal of Applied Mathematics, ASM E, paper no. 55-A-12
(March 1956): 15-20.
G. Kirchhoff, "Vorlesungen tiber Mathematische Physik," vol. 1. Mechanik, 1877,
p.450.
6.5 Exercises
259
9. Lord Kelvin and P. G. Tait, Treatise on Natural Philosophy, vol. 1, pt. 2, 1883,
p.188.
10. S. Timoshenko and J. N. Goodier, Theory of Elasticity, 2nd ed. (New York:
McGraw-Hill, 1951), p. 33.
11. P. G. Glockner, "Symmetry in Structural Mechanics," Journal of the Structural
Division, ASCE 99, no. STl (January 1973): 71-89.
12. V. V. Novozhilov, Thin Shell Theory [translated from 2nd Russian ed. by P. G. Lowe
(Groningen, The Netherlands: Noordhoff, 1964), p. 118].
13. P. L. Gould and S. L. Lee, "Hyperbolic Cooling Towers under Seismic Design
Load," Journal of the Structural Division, ASCE 93, no. ST3, (June 1967): 87-109.
14. W. Fliigge, Stresses in Shells, 2nd ed. (Berlin: Springer-Verlag, 1973), pp. 85-87,
121-124.
15. D. W. Martin, J. S. Maddock, and W. E. Scriven, "Membrane Displacements in
Hyperbolic Cooling Towers Due to Wind- and Dead-Loading," Proc. Institution of
Civil Engineers 28, 1964, pp. 327-337.
16. L. J. Brombolich and P. L. Gould, "Finite Element Analysis of Shells of Revolution
by Minimization of the Potential Energy Functional," Proc. Conference on Applications of the Finite Element Method in Civil Engineering, Vanderbilt University,
Nashville, Tenn., November 1969, pp. 279-307; L. J. Brombolich and P. L. Gould,
"A High-Precision Curved Shell Finite Element," Synoptic, AIAA Journal 10, no. 6
(June 1972): 727-728.
17. Gould and Lee, "Hyperbolic Cooling Towers."
18. "Design of Cylindrical Concrete Roofs," ASCE Manual of Engineering Practice 31
(New York: American Society of Civil Engineers, 1952).
19. A. W. Hedgren and D. P. Billington, "Numerical Analysis of Translational Shell
Roofs," Journal of the Structural Division, ASCE 92, no. STI (February 1966):
223-244.
20. M. Soare, Application of Finite Difference Equations to Shell Analysis (Oxford:
Pergamon Press, 1967).
21. R. W. Clough and C. P. Johnson, "Finite Element Analysis of Arbitrary Thin Shells,"
Concrete Thin Shells, ACI publication SP-28 (Detroit: American Concrete Institute,
1971), pp. 333-363.
6.5 Exercises
6.1
With reference to section 6.2.2, list the boundary conditions on the boundary
fJ = If for the free, fixed, hinged, roller, and sliding idealizations.
6.2
Generalize the hinged boundary condition on ex. = Cl to represent elastic displacement and rotation constraints.
Consider the fJ = If boundary and derive equations (6.26) and (6.27).
6.3
6.4
Consider a shell of revolution geometry and directly derive equations (6.28) and
(6.29).
6.S
Determine the normal displacements at points A and B on the toroidal shell shown
in figure 4-9 due to an internal pressure p.
260
6.6
6.7
6.8
6 Constitutive Laws, Boundary Conditions, and Displacements
Determine the decrease in diameter of a complete sphere of radius a and thickness
h under an internal suction q.
For the cylindrical shell under the harmonically dependent loading considered in
section 4.3.7.4, derive expressions for the displacements for the casesj = O,j = 1,
and j > 1. Sketch the deflections on an elevation and on a cross section for each
harmonic case.
Write equations (6.34a-c) in harmonic form and consider separately the cases
j = O,j = 1, andj > 1. Verify the pole conditions for D" DB' and D•.
6.9
Determine the membrane theory displacements for an open cylindrical shell, as
shown in figure 4-37, subject to a uniform loading q., as given by equations (4.171),
for the following cases:
(a) Load uniformly distributed in X direction.
(b) Load harmonically distributed in X direction.
6.10 Derive the governing equations for the membrane theory displacements of a
hyperbolic paraboloid, as shown in figure 4-41, for a uniform live load qz = -po
If the shell is supported by two vertical arches spanning between the corners of
the shell above A and D, and Band C, respectively, state the corresponding
boundary conditions.
CHAPTER
7
Energy and Approximate Methods
7.1 General
In solid mechanics, an energy formulation is often viewed as an alternative to
the differential equation statement. There exists a direct connection between
the energy and the differential equation approaches through the principle of
virtual displacements and through various extremum principles. However, we
will not pursue this connection, since it is beyond our immediate scope.
Our interest in introducing energy considerations here is twofold. First,
energy principles are the basis for many powerful numerical methods for solving
shell and plate problems. Second, energy-based solutions are an important
resource even for the classical formulations that we stress in this book. We
illustrate the latter in some detail in chapter 9, and the materials in this chapter
may be deferred until the appropriate sections are encountered without loss in
continuity.
In this chapter, we highlight some of the more important aspects of energy
methods. We stress primarily those items essential to support the ensuing applications. Also, we introduce some approximations that lead to the development
of numerically based solution techniques.
7.2 Strain Energy
The strain energy for an elastic body Ue may be written in terms of the strain
energy density dUe as
Ue =
Iv
dUe
(7.1)
where V represents the volume of the continuum.
In terms of the orthogonal curvilinear coordinates, as defined in chapter 2,
and the differential thickness d(, as shown on figure 3-1(a), the strain energy
density follows from the linear theory of elasticity:
261
262
7 Energy and Approximate Methods
= [O"««(OC, p, ()6«(OC, p, 0 + O"pp(OC, p, ()6p(OC, p, ()
+ O"n,,(OC, p, 06n (OC, p, () + O"«p(OC, p, Ow(oc, p, 0
+ O"«n(OC, Ph'«(oc, P) + O"pn(OC, P)Yp(OC, P)] dV
dUe
(7.2)
The stress terms are introduced in section 3.1; the strains are defined in section
5.3.
For a shell or plate, the differential volume dV may be expressed in terms of
the differential area dS and thickness d( as
dV
= dSd(
= A(Odoc B(OdPdC
=A
(1 + ~«) (1 + ~p)
doc B
(7.3a)
dP d(
or, after dropping terms of o (hjR) : 1,
dV
= ABdocdPd(
(7.3b)
It is generally preferable to write the strain energy in terms of a single field
variable, i.e., stress, strain, or displacement. To retain generality, we choose the
strains, which are connected to each of the others by a single set of equations,
(6.1) and (5.46), respectively.
We proceed by noting that ann = 0 (assumption [4J, table 1-1), and by
rewriting equation (7.1) in the matrix notation introduced in equation (6.2):
U·=~LLO"J{E}dV
(7.4)
Now, substituting equation (6.2) into (7.4), we get
U.
=
~L
[[CJ{E} - {O"T}Y{E}dV
(7.5)
Keeping in mind that the elements of {O"T } may be regarded as known, equation
(7.5) is an expression ofthe strain energy in terms of strain-type quantities alone.
We may now multiply out the terms in equation (7.5). Since the arithmetic is
quite lengthy and straightforward, it is omitted here. After integration through
the thickness and some rearrangement, the resulting expression is
263
7.3 Potential Energy of the Applied Loads
- ex(1
- ex(1
+ Jl)(8" + 811 )
J
h/2
T(e) de
-h/2
+ Jl)(K" + K(J) J
h/2
}
T(Oe de AB drx df3
-h/2
where the volume integral has been replaced by a surface integral over S.
Modified expressions for U. which correspond to other stress-strain laws may
be written in the same manner using the appropriate [C], as discussed in
chapter 6.
Iffor the moment we neglect the temperature dependent terms, we may group
equation (7.6) as
U.
= 2(1 _E
r{
Jl2) Js h[l]
2
3
h
h
}
+ 4"[11]
+ 12
[III]
(7.7)
ABdrxdf3
It is convenient to interpret equation (7.7) with respect to the three component
expressions [I, II, III], which are, respectively, linear, quadratic, and cubic in
the thickness h. First, the linear component Eh/[2(1 - Jl2)] [I] represents
extensional and shearing energy, which is predominant in shells that behave
primarily in accordance with the membrane theory. The quadratic component
Eh2/[8(1 - Jl2)] [II] is a coupling between extensional and shearing, and twisting
and bending terms, and enters into some geometrically nonlinear formulations.
Finally, the cubic component Eh 3 /[24(1 - Jl2)] [III] contains both bending
and twisting energy which are the primary resistance mechanisms in the flexure
of thin plates. In many cases, only one or two of the three components are
required.
7.3 Potential Energy of the Applied Loads
In equation (3.11e), the surface loading vector q is defined in terms of loading
intensities per unit area of middle surface q,,(rx, f3), q(J(rx, f3) and qn(rx, f3). Here,
we may also admit loads which act along a coordinate line. These line loads are
defined in the same fashion as the surface loading vectors
(7.8a)
for a load acting along the s" coordinate line corresponding to f3 =
11, and
(7.8b)
for a load acting along the s(J coordinate line corresponding to rx = ex. Boundary
reactions are frequently represented by such line loads. Furthermore, we may
accommodate concentrated loads defined by
(7.9)
for a load acting at (ex,11). No body forces distributed through the thickness are
264
7 Energy and Approximate Methods
explicitly included in this development. Rather, for a thin continuum, it is
generally sufficient to refer such forces to the middle surface, as we did in
chapter 3 for gravity loading.
Corresponding to each of the loading terms are the components of the middle
surface displacement vector A(tX,f3). These components have been defined in
equation (5.3a) as D,AtX, 13), Dp(tX, 13), and Dn(tX, 13).
It is convenient to define the change in potential energy of the applied loading
Uq as the product of each loading component and the corresponding displacement component. Further, when a positive displacement occurs, Uq decreases,
so that the product of the correspondents is given a negative sign. Thus, we have
Uq
= -
{Is
q(tX, 13)' A(tX, f3)A(tX, f3)B(tX, 13) dtX df3
+L
[r
+
4(a, 13) . A(a, f3)B(a, 13) df3 ]
JS(l
L
s
4(tX, jJ). A(tX, jJ) A (tX, jJ) dtX
(7. lOa)
+ ~ q(a, jJ) . A(a, jJ)}
or, in terms of the components,
Uq
= -
{Is
+~
+
q;(tX, f3)D;(a, f3)A(tX, f3)B(tX, 13) dtX df3
[1.
L
1];(tX, jJ)D;(tX, jJ)A(tX, jJ) dtX
(7. lOb)
/];(a, f3)D;(a, f3)B(a, 13) df3 ]
+ ~ i'i;(a, jJ)D;(a, jJ)}
(i = tX, 13, n)
In equations (7.10a) and (7. lOb), the second term signifies that the load potentials
for each line load acting on the shell surface are summed, and the third term
indicates that load potentials for each concentrated load are similarly summed.
Note that equations (7.8) and (7.9) can be generalized to include distributed,
line, and concentrated couples corresponding to the rotations Drzp and Dprz .
7.4 Energy Principles and Rayleigh-Ritz Method
7.4.1 Principle of Virtual Displacements: Consider an elastic body subject to
a set of surface tractions composed of the distributed load vectors q(tX,f3), as
defined in equation (3. 11 e), along with line load vectors 4(tX, jJ) and 4(a, 13) and
concentrated load vectors q(a, jJ). If we assume that there are no thermal or
265
7.4 Energy Principles and Rayleigh-Ritz Method
inertial effects, then the law of conservation of energy requires that the work
done by the surface tractions be equal to the strain energy stored in the material.
Now, a change (5A is imposed on the displacement field A, which moves to
a new position A + (5A. (5( ) is the variational operator and, for our purposes,
(5 A may be regarded as a linear increment of the vector A. Also, (5 A is assumed
to be compatible with the constraints of the system, although this restriction is
not necessary but only convenient. 1 The source of (5 A is not specified; i.e., it
does not necessarily result from any particular loading system. Hence, (5A is
called a virtual displacement.
Next, we write the energy balance corresponding to the virtual displacement.
The virtual work performed by the surface tractions is given by
(5(}q =
L
q(r:x, [3). (5A(r:x, [3)A(r:x, [3)B(r:x, [3) dr:x d[3
[1
+I
1
s
+
(7.11 )
4(r:x, TJ)· (5A(r:x, TJ)A(r:x, TJ) dr:x
s.
4(Ci, [3). (5 A(a, [3)B(a, [3) d[3]
~
+I
s
q(Ci, TJ)· (5 A(a, TJ)
and the strain energy is changed by (5UE' which is written from equation (7.1) as
(5U,
= (5
Iv
dUE
(7.12)
The r.h.s. of equation (7.12)
(5
Iv
dUE
may be evaluated from equation (7.5); however, since equation (7.5) is not yet
written as an explicit function of the displacement A, and since variations
or increments of A are being considered, we choose to leave the r.h.s. of
equation (7.12) in the general form.
Then, the energy balance is written by equating equations (7.11) and (7.12):
(5(}q = (5
Iv
dUE
(7.13)
which is a statement of the principle of virtual displacements. This principle
is a special form of the principle of virtual work, which is regarded by many
contemporary mechanicians as the cornerstone of solid mechanics. (For a complete discussion of the interrelation of the various energy principles of solid
mechanics, the interested reader is referred to Washizu. 2 )
The principle of virtual displacements, as stated in this section, can be taken
as an alternate statement of the condition of static equilibrium, as derived in
chapter 3. This interpretation is used freely in the ensuing sections.
266
7 Energy and Approximate Methods
7.4.2 Principle of Minimum Total Potential Energy: It is convenient to introduce some further assumptions at this point. First, we specify that the volume V
does not change during the virtual displacement and that the surface tractions
also do not vary. Next, the variational operator b( ) is restricted to variations
or increments in displacement only. We now recognize that bUq , as defined in
equation (7.11), is simply -bUq , where the load potential Uq was defined in
equation (7.1Oa). Hence, we may rewrite equation (7.13) as
(7.14)
where Ut is known as the total potential energy. Bear in mind that U t itself is
an integral function of various algebraic functions of the dependent variables,
as demonstrated by equations (7.6) and (7.10). Such a function is called a
functional in the terminology of the calculus of variations. Equation (7.14)
states that the total potential energy of an elastic system in equilibrium must be
stationary with respect to all displacements satisfying the boundary conditions.
Technically, the term stationary corresponds to the first variation of the functional U" b U" vanishing, and may indicate an absolute or relative maximum
or minimum; however, for our applications here, the stationary condition may
be regarded as an absolute or relative minimum.
From the standpoint of solution strategy, a dual interpretation of the principle
is helpful: we seek a displacement field A(Il(, (3) which satisfies the kinematic
boundary conditions and makes U a minimum; then, the stress resultants and
stress couples computed from A via the strain-displacement and constitutive
laws will satisfy the equilibrium considerations.
The classical solution for the problem of finding the minimum of a functional
is the fundamental problem of the calculus of variations and is not within our
scope here. The interested reader is referred to F orray 3 for an introductory
treatment of this subject. For our purposes, we wish to pursue the application
of one of the direct methods of the calculus of variations for the solution of
equation (7.14). Of the several possibilities available, the Rayleigh-Ritz method
is perhaps the best known and simplest to understand, and thus we consider
this technique first.
However, before proceeding, we might reflect on the first sentence of this
chapter, where we noted that an energy formulation may be regarded as an
alternative to a differential equation statement. Historically, the latter was
called the method of effective causes, whereas the former was known as the
method of final causes. The renowned mathematician Euler, who was the
inventor of the calculus of variations, observed in an argument which concisely
blends science and theology: "Since the fabric of the universe is most perfect,
and is the work of a most wise Creator, nothing whatsoever takes place in the
universe in which some relation of maximum and minimum does not appear.
Wherefore there is absolutely no doubt that every effect in the universe can be
explained as satisfactorily from final causes, by the aid of the method of maxima
and minima, as it can from the effective causes themselves .... "4 The resounding
t
7.4 Energy Principles and Rayleigh-Ritz Method
267
versatility and success ofthe contemporary energy-based finite element method
appears to leave even Euler's lofty claim understated.
7.4.3 Rayleigh-Ritz Method: Consider the displacement vector A(ex, /3), which
was defined originally in equation (5.3a) to include D", Dp, and Dn. Because we
may have additional generalized displacements, such as D"p and Dp" as defined
in equations (5.22a) and (5.22b), and because in some cases not all of the
displacement components are included, we write A in the general form
(7.15)
where the index m can be set according to the problem at hand.
The next step is to assume that each I1k(k = I,m) in equation (7.15) is of the
form
(7.16)
In equation (7.16), each (A represents a linearly independent coordinate function,
and each Ckl is an undetermined constant coefficient. It is not necessary that
each coordinate function satisfies all of the kinematic boundary conditions, but
only that the sequence 11k meets this requirement. The truncation index for
the sequence n may be set at a different value for 11 k , subject to some guidelines mentioned further on, and it is anticipated that improved results will be
obtained by increasing n, which means including more terms in the sequence.
Commonly, polynomials or elementary trigonometric functions are selected for
the coordinate functions ¢kl, but a wide variety of possibilities are available
according to the problem.
Now, the expressions for 11k are substituted into the strain and the change in
potential energy terms in equation (7.14). In the strain energy U" as given by
equation (7.6), we first must evaluate the corresponding strains by using the
appropriate strain-displacement relationships. After the integrations indicated
in equation (7.6) and (7.10) are carried out, U, becomes an algebraic rather
than an integral function. Therefore, the extremum problem of the calculus of
variations is transformed into the maximum~minimum problem of differential
calculus. The later problem, in turn, is solved by satisfying
(k
=
I,m)
1= 1,n
(7.17)
which produces a set of simultaneous algebraic equations for the coefficients
Ckl • Following the solution of these equations, the displacement functions 11k
can readily be used to find strains and curvatures, from which stress resultants
and couples may be computed.
The convergence ofthis method in a rigorous mathematical sense is considered
268
7 Energy and Approximate Methods
to be a difficult theoretical question. 5 In practice, improved results are obtained
in two ways: first, by selecting coordinate functions which closely resemble the
actual displacement field-the value of physical intuition in this regard is
obvious; second, by using an increasing number of terms in each sequence.
Of course, this increases the number of simultaneous equations and the computational effort required.
The careful reader may have noted that no specific reference was made as to
the chosen coordinate functions' a priori satisfying the internal compatibility
or strain-displacement conditions. This illustrates one of the advantages of
a displacement formulation. For reasonably chosen coordinate functions, the
ability to compute the strains from the strain-displacement conditions in the
course of evaluating U. represents, in fact, the satisfaction of the compatibility
requirements, so that little difficulty is encountered in this regard.
Another point of general interest is to note the limitations of the method.
Obviously, the requirement of the displacement function sequences satisfying
the boundary conditions can be quite restrictive in the case of irregularly shaped
continua. The application of the Rayleigh-Ritz method in a piecewise fashion
to subdivisions or finite elements of the entire medium, with the undetermined
coefficients chosen to enforce continuity requirements along interelement boundaries as well as to satisfy the external boundary conditions, has become one of
the most powerful techniques in applied mechanics: the finite element method.
A wide variety of applications of this technique are presented by Zienkiewicz, 6
who has contributed greatly to its popularity.
Even before the emergence of the finite element method in the form presently familiar to engineers, Timoshenko suggested that the Rayleigh-Ritz
technique-originated by Nobel laureate Lord Rayleigh in his epic treatise
"Theory of Sound" and elaborated by W. Ritz, who considered applications
to the analysis of thin plates-has spurred more research in the strength of
materials and the theory of elasticity than any other single mathematical tool. 7
The vast literature accompanying the finite element method resoundingly reinforces this claim.
In regard to the use of the finite element method to model shells, the explicit
inclusion of rigid body modes has proved to be troublesome as a result of the
curvilinear coordinates. It has been suggested by Heppler and Hansen 8 that
the more convenient implicit representation is best implemented with approximations of equal degree for the translational degrees of freedom contained in
{A}, equation (7.15). This implies a single value of n for all such variables.
Also, we should mention that the Rayleigh-Ritz method may be generalized
by taking each of the Ckl coefficients in equation (7.16) as an unknown function
of one of the independent variables, rather than as simply a constant. In this
case, equation (7.16) is generalized to
~k = (Ao(rJ., {3)
+
n
L Ck/(rJ. or {3)rPk/(rJ., {3)
1=1
(7.18)
269
7.5 Galerkin Method
It is usually convenient to choose rAo to satisfy all of the boundary conditions;
then, each term of the sequence Ck1tPkl(1 = 1, n) must vanish on the boundaries.
This extension ofthe basic Rayleigh-Ritz technique is known as the Kantorovich method. If applied skillfully, it can reduce the numerical work considerably
for some problems. 9
7.5 Galerkin Method
The Galerkin method is quite similar in execution to the Rayleigh-Ritz method
in that the solution is postulated to be represented by a sequence of coordinate
functions, such as equation (7.16). The most apparent operational difference
is that in the Galerkin technique, these coordinate functions are tested in the
governing differential equations rather than in an energy expression.
As an illustration, assume that the system has been reduced to a single
equation of the form
(7.19)
where Ak is one element of the general displacement vector {A}, equation (7.15),
and B is a linear differential operator. For example, the Laplacian operator in
Cartesian coordinates, obtained by setting
B( ) = ( ).xx
+(
(7.20)
).yy
is frequently encountered in plate problems. The r.h.s. of equation (7.19) represents known quantities, e.g., the applied surface loading. When equation (7.16)
is substituted into equation (7.19), we have
B
C~ Ck1tPk) -
fk(rx, /3, n)
=
Rk(rx, /3, n)
(7.21)
where Rk is a residual error function. Recall that Ak has been postulated to
satisfy the boundary conditions of the problem. The essence of the Galerkin
method is to force Ak to also satisfy the governing equation as closely as
possible. This means that the residual error Rk should be minimized.
The condition for minimizing Rk is to require that Rk be orthogonal to each
<Pkl over the domain of the problem. 9 That is,
Iv
RktPkl dV = 0
(7.22a)
or
(7.22b)
270
7 Energy and Approximate Methods
This produces a system of n linear simultaneous algebraic equations in the n
unknown coefficients Ckl.
Obviously, this procedure becomes increasingly involved when there are
multiple equations and unknowns in the system, instead ofjust a single equation
in d k as we have considered here. But, for a large class of second order ordinary
differential equations, the Galerkin method leads to an identical set of coefficients as produced by the Rayleigh-Ritz method 10 and may be preferable if
it is more convenient to work with the governing equations rather than with
the energy functional. Moreover, there are problems for which no satisfactory
variational principle has been formulated, but for which a set of governing
differential equations is available. This suggests that the Galerkin method may
be even broader in applicability than the Rayleigh-Ritz method.
7.6 References
1. F. L. DiMaggio, "Principle of Virtual Work in Structural Analysis," Journal of
the Structural Division, ASCE 86, no. STU (November 1960): 65-77.
2. K. Washizu, Variational Methods in Elasticity and Plasticity (New York: Pergamon
Press, 1968).
3. M. J. Forray, Variational Calculus in Science and Engineering (New York: McGrawHill, 1968), chaps. 1 and 2.
4. S. Timoshenko, History of Strength of Materials (New York: McGraw-Hill, 1953),
p. 31. Copyright, 1953 by McGraw-Hill, Inc., and used with permission of McGrawHill Book Co.
5. R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 1 (New York:
Interscience Publishers, 1966), pp. 175-176.
6. O. C. Zienkiewicz, The Finite Element Method in Engineering Science (London:
McGraw-Hill, 1971).
7. Timoshenko, History of Strength of Materials, pp. 338-339,399-401.
8. G. R. Heppler and J. S. Hansen, "A Mindlin Element for Thick and Deep Shells,"
Computer Methods in Applied Mechanics and Engineering 54 (1986): 21-47.
9. Forray, "Variational Calculus," pp. 199-201.
10. Forray, "Variational Calculus," pp. 189-199.
7.7 Exercises
7.1
Generalize equations (7.8) to include the possibility of surface loading consisting
of distributed, line, and concentrated couples.
7.2
Consider a simple supported beam oflength L, with a uniformly distributed load w.
(a) Compute the deflected shape D(x), by using a sequence of coordinate functions
271
7.7 Exercises
D(x) = C 1
. 2nx
. nx
+ C2SlOT + C3SlOT + ...
and observe the convergence. Use both the Rayleigh-Ritz and Galerkin
methods.
(b) Repeat the procedure, using a sequence of polynomials of the form
D(x) =
7.3
C1
+ c 2x(L -
x)
+ C3X2(L -
x)
+ c4 x(L -
X)2
+ ...
For the polynomials, recall that the sequence must satisfy the boundary
conditions
Consider equations (7.10) and generalize the load potential to include distributed,
line, and concentrated applied moments.
CHAPTER
S
Bending of Plates
8.1 Governing Equations
8.1.1 General Formulation: The equilibrium equations for initially flat plates
are stated as equations (3.25a-e); the strain-displacement relations are given by
equations (5.54) or, with transverse shearing strains suppressed, as equations
(5.55). Taken together with the stress resultant-strain relationships in the form
of equation (6.10), the requisite boundary conditions discussed in section 6.2,
and specifically the Kirchhoff conditions equations (6.25) and (6.27), the elements of a quite general plate theory are available and substantiated.
It is convenient to consider the development of the theory of plates in several
stages, initially neglecting transverse shearing strains and in-plane stress resultants and also assuming isotropic material properties. Then, the theory will be
generalized as necessary.
When the in-plane forces are neglected, equations (3.25a) and (3.25b) are
removed, leaving
+ (AQp),p] + qnAB = 0
(BM"p)." + (AMp),p + B,,,Mp,, - A,pM" (BM"),,, + (AMp,,),p + A,pM"p - B,,,Mp -
(S.la)
[(BQ"),,,
QpAB = 0
(S.lb)
Q"AB = 0
(S.lc)
Along with the elimination of the in-plane forces, we also disregard D", Dp ,
and the extensional and in-plane shearing strains, e", ep, and OJ. With transverse
shearing strains also suppressed, the complete set of kinematic equations is
given by equations (5.55). Of course, there remain extensional and in-plane
shearing strains away from the middle plane as computed from equations (5,36),
(5.37), and (5.45):
= (K"
(S.2a)
ep(O = (Kp
(S.2b)
w(O = 2(.
(S.2c)
e,,(O
where K" and Kp are the curvatures and. is the twist of the middle plane.
On substituting equations (S.2) into equations (5.55a-c), we find
272
273
8.1 Governing Equations
Ka(O
[1(1 )
A,p
]
= - A A Dn,a ,a + AB2
Dn,p
(S.3a)
(S.3b)
1
- AB
D
[AA,PDn,a + Ii
B]}
n,p
(S.3c)
,a
These terms are seen to be functions of the normal displacement Dn only.
The stress couples are easily expressed in terms of Dn by substituting equations (S.3a-c) into the appropriate constitutive laws. For isotropic materials,
we have, from matrix 6-2 and equations (S.3a-c),
(S.4a)
(8.4b)
1
- AB
[AA
,p
D
B]}
n,a + Ii n,p
D
(8.4c)
,a
in which D is the flexural rigidity of the plate, given by
Eh 3
D=-----=-
12(1 - J1.)2
(S.4d)
We will assume that Map = M pa , with the justification following the same
argument offered in taking Nap = Npa in section 4.1. Note that when the transverse shearing strains are excluded, the transverse shear stress resultants can
no longer be evaluated from the corresponding constitutive relationship, given
by the last two rows of matrix 6-2. Rather, equations (S.lb) and (S.lc) are solved
for
(S.5a)
274
8 Bending of Plates
and
(8.5b)
so that Q" and Qp can be computed once the stress couples have been determined. This is analogous to the elementary theory of beams, in which the shear
force is evaluated from equilibrium considerations rather than from a constitutive law. Only transverse shearing strains can be suppressed; the corresponding
forces are required for equilibrium.
The formulation of the plate bending problem is completed as follows:
1. Introduce equations (8.4) into equations (8.5).
2. Differentiate equations (8.5a) and (8.5b) by f3 and
3. Substitute into equation (8.1a).
IX,
respectively.
The resulting equation expresses equilibrium in the normal direction in terms
of the single displacement Dn and constitutes the governing equation of the
system. This is a classical displacement formulation. Once Dn is evaluated, the
stress couples may be computed by differentiation from equations (8.5a) and
(8.5b).
We also have the effective transverse shears, Q" and Qp,
Q
"
=
Q
"
+ M"p,p
-
B
Mp
Qp=Qp+~
A
(8.6a)
(8.6b)
which we obtain from equations (6.25) and (6.27). Also, the rotations Dap and
Dpa are found from equations (5.54g) and (5.54h) with y" = YP = 0:
DaP =
l/I" = -
Dp" =
l/IfJ =
1
A Dn,,,
(8.7a)
1
-liDn,p
(8.7b)
It is evident from the complexity of the foregoing expressions that the algebra
involved in carrying out the steps outlined in the previous paragraphs becomes
quite involved. For our purposes, it is sufficient to specialize the further development for two predominant cases: Cartesian coordinates and polar coordinates,
8.1.2 Cartesian Coordinates: For Cartesian coordinates, shown on figure 8-1,
IX = X, f3 = Y, n = Z, and A = B = 1. Then equations (8.4a-c) become
+ JlDz,yy)
My = -D(Dz,yy + JlDz,xx)
Mx = -D(Dz,xx
(8.8a)
(8.8b)
275
8.1 Governing Equations
z
r-------------------~---------y
x
~+Xy
Fig. 8-1
Mxy
=
Myx
=
Forces and Moments in Cartesian Coordinates
-D(l - fl)Dz,xy
(8.8c)
Likewise, equations (8.5a) and (8.5b) reduce to
Qy
= [Mxy,x + My,y]
= -D[Dz,yyy + jlDz,xxy + (1 - fl)Dz,xxy]
=
-D[Dz,yyy
+ Dz,xxy]
(8.9a)
276
8 Bending of Plates
= [Mx . x + M yx•y ]
(8.9b)
= -D[Dz . xxx + IlDz.xyy + (1 -1l)Dz .xyy ]
= -D[Dz . xxx + Dz.xyy]
Next, we take a/ay of equation (8.9a) and a/ax of equation (8.9b) and substitute
Qx
into equation (8.1a) to get
Qx.x + Qy.y + qz
=0
(8.10)
or
-D[Dz.xxxx
+ 2Dz .xxyy + Dz.yyyy] + qz =
0
(8.11)
Equation (8.11) may be written concisely as
V 2 (V 2 Dz )
= V 4 Dz = qz
(8.12)
+(
(8.13a)
D
where
V2( ) = ( ),xx
).yy
and
(8.13b)
V2( ) and V4( ) are commonly called the Laplacian and biharmonic operators,
respectively,l and (8.12) is commonly known as the plate equation.
The homogeneous part of the governing equation of the engineering theory
of medium thin plates is the biharmonic equation
(8.14)
The recognition of this form is important for at least two reasons. First, there
is a considerable reservoir of mathematical knowledge dealing with biharmonic
equations which can be transferred to our plate theory.2 Second, because V4( )
is invariant, we may obtain the corresponding equation for any other system
of orthogonal curvilinear coordinates by simply transforming the biharmonic
operator. We will verify this for polar coordinates in the next section and utilize
the invariant property of the V2 and V4 operators repeatedly.
Of particular interest with respect to the preceding formulation is the reduction of the system to an equation with a single dependent variable. Stepping
back through the derivation, we find that the key step was the expression ofthe
extensional and shearing strains in terms of the normal displacement Dz only,
equations (8.2) and (8.3). In order to interpret this form a physical standpoint,
we examine a segment of the middle plane along a normal section traced by
the X -Z plane as shown in figure 8-2.
We show the segment deformed in the positive Z direction, where both the
displacement Dz and the derivative Dz.x are positive. Also, we recall the basic
277
8.1 Governing Equations
z
Plate
:::"."::::'~x
•
:
'
:.
:.
Plate
Fig. 8-2
Displacements for a Plate
assumptions listed in table 1-1. We focus on an arbitrary point CD which
initially lies at a distance , from the undeformed middle plane. Since all
geometry is referred to the middle plane, we locate the corresponding point on
the undeformed middle plane, point Q). During deformation, point Q) displaces a distance Dz parallel to the original Z direction (assumption III) to point
0). Since we are neglecting transverse shearing contributions, the deformed
position of point CD must lie on the normal to the deformed middle plane, Dz,x,
erected at point 0) (assumption rn). Likewise, the thickness remains constant
during deformation (assumption @]), so that the point must remain a distance
(from the deformed middle plane. The final position is denoted by @.
Note that point CD also has a displacement in the X direction denoted by
Dx. Since Dx is negative corresponding to the positive sense of Dz,x, as shown
on figure 8-2,
Dx = -(Dz,x
(8.15)
278
8 Bending of Plates
Because the Lame parameters are constant, the strain in the X -direction is
found from equation (S.S4a), as simply
ilx(O
= Dx,x
(S.16)
= -(Dz,xx
which corresponds to equation (S.2a), after lat, in equation (S.3a) is specialized
for Cartesian coordinates.
An identical argument referred to a normal section traced by the YZ plane
gives
(S.17)
Dy = -(Dz,y
and
(S.lS)
corresponding to equation (S.2b), with equation (S.3b) suitably specialized.
Equations (S.lS) and (S.17) clearly illustrate the coupling of the in-plane displacements to the normal displacement, which is the key to the relatively simple
form of the plate equation.
The in-plane shearing strain, adapted from equation (S.S4c), is
OJ
= Dx,y + Dy,x
= -2(Dz ,xy
(S.19)
which matches equation (S.2c) with T specialized for Cartesian coordinates.
To complete the formulation, we also list the effective transverse shears Qx
and Qy. From equations (S.6), (S.Sc), and (S.9).
Qx
+ Mxy,y
=
Qx
=
-D[Dz,xxx
+ (2 -
J.l)Dz,xyy]
+ (2 -
J.l)Dz,xxy]
(S.20a)
and
Qy = Qy
=
+ Mxy,x
-D[Dz,yyy
(S.20b)
The rotations DXY and Dyx are found from equation (S.7) as
= -Dz,x
(S.21a)
Dyx = r/ly = -Dz,y
(S.21b)
Dxy = r/lx
It is notable that equation (S.l1) is the two-dimensional counterpart of the
familiar Bernoulli-Euler equation, which is the cornerstone of beam theory.
Taking Dz = Dz(X) or Dz(Y) only in equation (S.ll) and neglecting Poisson's
ratio in the elastic constant D, we have the governing differential equation for
a beam of unit width and depth h.
8.1 Governing Equations
279
Equation (8.11) also serves to articulate the differences between a plate and
a two-dimensional gridwork of beams. Whereas the first and third terms, Dz,xxxx
and Dz , yyyy, clearly describe bending resistance in the respective directions, the
second term, 2Dz ,xxyy, represents the contribution of the twisting rigidity to
the load resistance. This is easily verified by examining equations (8.8), in which
the directional derivatives Dz,xx and Dz,yy form the bending stress couples and
the mixed derivative Dz,xy produces the twisting stress couple.
The presence of the twisting couples discerns between two-dimensional flexural
behavior and plate behavior. In this regard, it is of interest to note that in 1789,
J. Bernoulli proposed approximating a plate with a system of perpendicular
intersecting beams, based on an earlier model of a flexible membrane by Euler.
S. Germain, aided by Lagrange, incorporated the twisting term to find the
homogeneous part of equation (8.11), and the correct strain energy expression
was derived by Poisson. A comprehensive plate theory, including loading terms
and boundary conditions, was finally advanced by Navier in 18203 , although
the final resolution of the free edge boundary conditions in terms of effective forces appeared somewhat later, as discussed in section 6.2.3. We shall
thoroughly explore Navier's approach later in this chapter.
8.1.3 Polar Coordinates: For polar coordinates, shown on figure 8-3, oc = R,
fJ = 0, and n = Z. Obviously, A = 1, but considering the S/J coordinate line,
(8.22)
so that B = R. Since B = B(oc) in the general formulation, the B,er term is R,R = 1
and A,p = o. Also shown in figure 8-3 are the stress resultants and stress couples
corresponding to the polar coordinates, as adapted from figure 3-2. Equations
(8.4a-c) become
MR=
-D[Dz'RR+*(~Dz'/J/J+Dz'R)J
(8.23a)
M/J =
-D[~2 Dz,/J/J + ~DZ'R + jl.DZ,RRJ
(8.23b)
MR/J
M/JR
=
=
-D(1 -
jl.>[~ (Dz'R/J - ~ DZ'/J)]
(8.23c)
Likewise, equations (8.5) specialize to
Q/J
=
1
R [(RMR/J),R
=
R [2MR/J + RMRO,R + M/J,/J]
1
+ Mo,/J + MR/J]
(8.24a)
280
8 Bending of Plates
z
A---------------------y
x
o~
,0;
Fig. 8-3
Forces and Moments in Polar Coordinates
281
8.1 Governing Equations
and
QR
=
1
R [(RMRh
1
=R
=
[MR
+ M R6 ,6 -
M 6]
+ RMR,R + M R6,6 -
-D [ DZ,RRR
1
+R
(8.24b)
M 6]
1
DZ,RR - R2 DZ,R
1
+ R2 DZ,R66 -
2
]
R3 DZ,66
To express equation (8.1a) in polar coordinates,
(RQRh
+ Q6,6 + Rqz = 0
(8.25)
in terms of D z , we take the appropriate derivatives of equations (8.24) to get
DZ,RRRR
2
+ R DZ,RRR -
1
R2 (DZ,RR - 2Dz,RR66)
1
+ R3 (DZ,R
- 2Dz,R66)
1
qz
+ R4 (4Dz ,66 + DZ,6666) = Ii
(8.26)
It may be verified that equation (8.26) can be written as the biharmonic equation
V 4 Dz
=
V 2 [V 2 Dz ]
= qz
(8.27)
D
where, in polar coordinates,
2
V ( )=
(
),RR
1
+R(
),R
1
+ R2 (
),611
(8.28a)
or
(8.28b)
Comparing equations (8.12) and (8.27) demonstrates the invariant property
of the biharmonic operator. That is, equation (8.12) may be directly transformed into any other coordinate system by transforming V4( ) = V2[V2( )]
accordingly.
It is sometimes convenient to write
(8.29a)
and
(8.29b)
which are easily verified.
282
8 Bending of Plates
We also have the effective transverse shears
(8.30a)
Q(j
= Q(j + MR(j,R
(8.30b)
and the rotations
DR(j = tPR = -DZ,R
(8.30c)
1
D(jR = tP(j = - R Dz,(j
(8.30d)
A preponderance of circular plate problems encountered in engineering are
axisymmetrically loaded, in the same manner as the axisymmetrically loaded
shell of revolution treated in section 4.3.2. For such cases, all functions of () drop
out of the preceding equations, so that
(8.31a)
MR = -D(Dz'RR +-iDZ'R)
M(j = -D
(~ DZ,R + P.Dz,RR)
(8.31b)
(8.31c)
MR(j=O
Also,
(8.32a)
Q(j=O
QR
= D (Dz'RRR + ~ DZ,RR -
~2 DZ'R)
(8.32b)
and
QR= QR
(8.32c)
Finally, the governing equation simplifies to
2
1
1
qz
DZ,RRRR + R DZ,RRR - R2 DZ,RR + R3 DZ,R = Ii
(8.33a)
or
(8.33b)
where V:( ) signifies the axisymmetric biharmonic operator. This equation is
conveniently solved in operator form, as we will demonstrate in a later section.
8.1.4 Force and Moment Transformations: Once we have selected the 0( and P
coordinates, we may evaluate the transverse shear resultants and stress couples
283
8.1 Governing Equations
from the preceding equations only on planes parallel to the Sa. and sp coordinate
lines. Frequently, we are interested in values on other planes, since the maximum transverse shear and/or bending moment may act in directions other than
those coinciding with the coordinate lines.
The approach is similar to that used to evaluate principal stresses in linear
elasticity. In figure 8-4(a), a second set of orthogonal axes (e, 11), which represent
arbitrary directions for which we wish to evaluate the forces and moments, are
z
MaptJ.s( siny
I Ma~'(
,jnY
~.~ : :: ~
M~;~:-~-:-o:-;-t .:r"'M~(:~
VI
<J
M17~
(b)
Fig.8-4
(C )
Forces and Moments in Rotated Coordinates
284
8 Bending of Plates
defined. These axes are oriented by the angle y measured clockwise in the 1X-f3
plane. On the differential rectangular element in the inset, bound by Sa and sp
coordinate lines, we superimpose the rotated coordinates and consider shaded
triangle CD, which is enlarged in figure 8-4(b). The positive senses of the stress
resultants acting on the element are obtained from figure 3-2.
Summing forces in the Z direction, we have
-QaAs~sin y
- QpAs~cosy
Q" = Qasin y
+ Qpcosy
+ Q"As~ = 0
or
From the shaded triangle
~,
(8.34)
we may find
(8.35)
Qa cos y - Qp sin y
Q~ =
Next, we treat the stress couples acting on the triangular element CD, as
shown in figure 8-4(c). Again, the positive senses of the couples are set from
figure 3-2. From moment equilibrium about the ~ axis, we have
- M"As~
With Mpa
+ Ma sin 2 yAs~ + Mp cos 2 yAs~ + Mpa cos y sin yAs~
+ Map sin y cos yAs~ = 0
= MaP'
M" = Masin2y
+ Mpcos 2 y + 2Mapsinycosy
(8.36a)
This equation may be written in terms of the double angle 2y as
M" = -t(Ma + Mp) - t(Ma - Mp) cos 2y + Map sin 2y
(8.36b)
Moment equilibrium about the rJ axis yields
M,,~As~
- Ma sin y cos yAs~
+ Mp cos y sin yAs~ + Map sin 2 yAs~
- Mpacos2yAs~ = 0
from which
M,,~
= (Ma - Mp)sinycosy - Map(sin 2 y - cos 2 y)
(8.37a)
or, in terms of the double angle,
M,,~
= -t(Ma - Mp)sin2y + Mapcos2y
A similar calculation for triangle
M~
~
(8.37b)
gives
= Ma cos 2 Y + M{J sin 2 y - 2Map sin y cos y
(8.38a)
or
(8.38b)
and
285
8.1 Governing Equations
=
M~"
M,,~
(8.39)
Equations (8.34)-(8.39) enable the transverse shear resultants and the stress
couples to be calculated on any specified section of the plate. They play the
same role in the theory of plates as the familiar Mohr's circle transformations
occupy in the analysis of stresses at a point in the theory of elasticity.
8.1.5 Alternate Formulation: In the preceding sections, we have almost routinely referred to the biharmonic equation as the plate equation and, indeed,
this is common terminology. Nevertheless there are alternative formulations,
one of which may be anticipated when we reflect on our previously discussed
analogies to beam theory. Just as in beam theory, we can separate the fourth
order governing equation, deflection = f(load), into two coupled second order
equations, deflection = f(bending moment) and bending moment = f(load). The
latter form is directly applicable to statically determinate beams and is the basis
for some methods of analyzing statically indeterminate beams as well.
For the plate problem, we add equations (8.36b) and (8.38b) to get
M~+ M"
= Mil. + Mp
(8.40)
Equation (8.40) states that the sum of the flexural stress couples acting about
two mutually orthogonal axes is invariant with respect to coordinate transformation. Thus, we may proceed using the specialized Cartesian coordinate
expressions for Mx + My while retaining complete generality. Summing equations (8.8a) and (8.8b),
Mx
+ My =
=
-D(l
-D(l
+ /l)(Dz . xx + Dz.yy)
+ /l)V 2 Dz
(8.41)
Equations (8.40) and (8.41) constitute a formal proof of the invariant property
ofV2.
We now set
M
=
Mx+My
1+/l
= -DV 2 Dz
(8.42)
so that
V2 Dz =
-~M
(8.43)
Then, from equations (8.12) and (8.43),
V4 Dz
= V2(V2Dz) = -~V2M = qz
D
D
(8.44)
or
(8.45)
286
8 Bending of Plates
Equations (8.43) and (8.45) comprise the alternate formulation and allow the
plate problem to be solved in an analogous manner as a uniformly stretched,
laterally loaded membrane. The membrane problem is regarded as somewhat
simpler to treat numerically.4
8.1.6 Boundary Conditions: The boundary conditions corresponding to the
theory of plates without transverse shearing strains are easily synthesized from
section 6.2.2 and figure 6-3. Since there are no extensional forces considered,
three basic conditions apply: fixed or clamped, simply supported, and free.
These are listed for boundaries IX = ex and f3 = li in table 8-1. They are stated
in homogeneous form but may be generalized, as described at the conclusion
of this section.
With respect to the fixed condition, often called a clamped boundary, note
from equations (8.7) that DlI.p = 0 and Dpll. = 0 imply Dz,lI. = 0 and Dz,p = 0,
respectively.
The simply supported boundary may be written in an alternate form for
Cartesian coordinates when the boundaries are coincident with the coordinate
lines. We consider first a boundary defined by X = X, as shown for example
in figure 8-5. The condition of simple support states that Dz(X, Y) = O. Further,
it is implied that the boundary is straight, and therefore, Dz , yy(X, Y) = O. A
similar argument with respect to a boundary Y = Y leads to Dz,xx(X, Y) = O.
Equations (8.8a) and (8.8b) for simply supported boundaries become
Mx(X, Y)
= -DDz,xx(X, Y) = 0
(8.46a)
My(X, Y)
= -DDz,yy(X, Y) = 0
(8.46b)
and
Table 8-1
Boundary conditions for plates
Condition
and
Symbol
Fixed or Clamped
Simply
Supported
Free
Boundary
d
o
o
Dz = DlI.p = 0
or
Dz = Dz,lI. = 0
Dz=Dpll.=O
or
Dz = Dz,p = 0
Dz = 0; Mil. = 0
Dz
QII.=MII.=O
Qp=Mp=O
or
Dz = Dz,lI.lI. = 0
O;Mp =0
or
Dz = Dz,pp = 0
=
287
8.1 Governing Equations
t..-""l..- Mx/c)-MXy(b)
I
X=X
I
I
I
tMXY (d)+ Myx(e) = 2MXY (X,Y)
Fig. 8-5
Forces at Right-angle Comer
which, in view of the preceding discussion, generalize to
Mx(.X, Y)
= -DV 2 Dz (X, Y) = 0
(8.47a)
My(X, Y)
= -DV 2 Dz (X, Y) = 0
(8.47b)
and
Thus, on each boundary,
If Mx
= 0 -+ Dz,xx = 0 -+ V2 Dz = 0
and
If My
= 0 -+ Dz,yy = 0 -+ V2 Dz = 0
The statement of the simply supported boundary condition as V2 Dz = 0 is
predicated on the edge's being straight, as well as undeflecting. In polar coordinates, the second term of equation (8.28a) would not vanish on a simply
supported boundary, since it is proportional to the rotation DRS' equation
(8.3Oc), which is unconstrained. Therefore, MR = 0 does not imply V2 Dz = O.
Now consider the free edge condition. In section 6.2.3, we presented the
contraction of the general boundary conditions into the Kirchhoff conditions
as a result of the suppression of transverse shearing strains. For the plate
problem, equations (8.6) indicate that the transverse shear and the twisting
stress couple are combined into an effective transverse shear force which is
prescribed to vanish on a free edge.
In addition to the free edge situation, the effective transverse shear forces
288
8 Bending of Plates
have some interesting implications for edges which are supported. In figure 8-5,
we show two boundaries of a simply supported rectangular plate, where the
Kelvin-Tait approach developed on figures 6-4 and 6-5 is used to replace the
twisting stress couples by closely spaced forces. The positive sense of these
couples is established from figure 8-1.
We first study two typical differential segments, say band c, on the X = X
boundary. At the junction, we have a net contribution to the effective shear
equal to
(8.48)
Again, the positive sense is established from the direction of Qx in figure R-l (cd.
If Mxy is a continuous function with continuous derivatives, equation (8.48)
becomes
(8.49)
Mxy,yL\Y
which combines with QxL\Y to give the effective shear Qx, as written in equation
(8.20a) after the L\ Y terms have been cancelled. On the other hLmd, consider the
case in which Mxy may be continuous, but not the first derivative in the Y
direction. This corresponds to a jump discontinuity in M Xy along the boundarv,
which produces a concentrated force at the point of discontinuity, with the
magnitude given by equation (8.48). Conversely, if a concentrated force of a
given magnitude is applied along the boundary, the twisting stress couple must
undergo a jump equal to the magnitude of the applied force.
We carry the argument one step further by considering the intersection of
the two boundaries X = X and Y = Y at the corner. Here, there is definitely a
jump discontinuity producing a concentrated force of magnitude
(8.50al
Since Myx
= M xy ,
Reorner = 2Mxy
we have the corner force
(8.50b)
The concept of the corner force is not limited to the intersection of two free
boundaries, where it obviously must vanish. In general, any right angle corner
where at least one ofthe intersecting boundaries can develop Mn or MIX will
have a corner force. Physically, a plate having twisting stress couples acting as
shown in figure 8-5 would require that the force 2Mxy (.X, Y) be developed at
the corner to prevent the corner from uplifting. The case of non-right angle
intersections and the detailed distribution of the effective transverse shear along
the entire boundary will be examined for some specific examples in the following
sections.
It should also be mentioned that the homogeneous boundary conditions
discussed in this section may be generalized to include prescri bed edge displacements and forces. For example, within the definition of fixed and hinged
conditions, a nonzero value of the transverse displacement })z may be accom-
289
8.2 Rectangular Plates
modated. This may be of interest in support settlement problems. Also, a
specified edge moment may be inserted in place of MIZ or Mp = 0 for the hinged
and free conditions. Depending on the solution routine, known edge moments
and edge forces may be grouped with the applied loading terms, but should
properly be regarded as static boundary conditions. Several illustrations of this
generalization are found in the ensuing sections.
Furthermore, a linear combination of a static-kinematic correspondent pair
may be encountered in the description of an elastic support. For example,
M,lfi, {3) = kDIZP(fi, {3) would represent a linear rotational spring with spring
constant k along the boundary ex = "fi.
8.2 Rectangular Plates
8.2.1 Bending Under Edge Loading
8.2.1.1 General Solution. Consider a rectangular plate subject to the uniformly
distributed edge moments Mx = Ml and My = M 2 , as shown in figure 8-6. The
sign convention for the moments is consistent with figure 3-2(b). Also, the
deflected middle plane is shown on the figure.
For this problem, it is expedient to shortcut the formal procedure of solving
the governing differential equation, (8.12). Rather, we surmise that the state of
bending is constant throughout the plate. That is,
z
Middle Plane
-F-----y
x
Fig. 8-6
Rectangular Plate Loaded by Edge Moments
290
8 Bending of Plates
=
Ml
(S.51a)
My(X, Y) = M2
(S.51b)
Mx(X, Y)
Then, we may write from equations (S.Sa) and (S.Sb)
Mx
= Ml =
My = M2 =
+ IlDz,yy)
-D(Dz,yy + IlDz,xx)
(S.52a)
-D(Dz,xx
(S.52b)
We eliminate Dz,yy to get
(Ml -IlM2)
D(1 _ 1l2)
Dz,xx = -
(S.53)
which may be integrated to give
Dz,x=
(Ml -IlM2)
D(1 _ 1l2) X
+ fl(Y) +
C1
(S.54)
and
(S.55)
Dz(X, Y) =
We may proceed further by noting from equations (S.51)-(S.53) that DZ,yy
must also be constant. Therefore,
Dz,yy
=
Xfl(Y),yy
+ f2(Y),yy = constant
(S.56)
Since there is no X in the second term of equation (S.56) to cancel the variable
X in the first term, fl(Y),yy = 0 and
fl(Y) = C 3 Y
+ C4
f2(Y) = C s y2
+ C6 Y + C7
(S.57a)
(S.57b)
and
Dz,yy = 2Cs
(S.57c)
We then substitute equation (S.55) into equation (S.52b), taking into account
equations (S.53) and (S.57c), to find
Il (Ml - IlM2)]
M2 = -D [ 2Cs - D (1 -1l2)
(S.5Sa)
from which
= _ (M2
C
5
- IlMd
2D(1 - 1l2 )
Also, from equations (S.Sc), (S.54), and (S.57a),
(S.5Sb)
291
8.2 Rectangular Plates
Mxy
=
(8.59)
-D(1 - /l)C 3
Upon substituting equation (8.57a) and (8.57b) into equations (8.55), we may
combine the term XC 4 , which arises from Xfl(Y), with the term C1X to produce
a term CsX, and also absorb the constant C 7 into C2 to make C 9 . Then, from
equations (8.55)-(8.59), we have the general solution
Dz
-1
= 2D(1 _ /l2/(M 1
+
C 3 XY
-
/lM2)X
2
+ (M2
2
- /lMd Y ]
(8.60)
+ CsX + C6 Y + C9
At this point, we inquire further into the nature of the constants in equation
(8.60). Constant C 3 multiplies the term XY and thus will contribute to Mxy via
equation (8.8c). The remaining constants C s , C 6 , and C 9 are coefficients oflinear
or constant terms; therefore, they cannot contribute to the stress couples or to
the transverse shear stress resultants, which are dependent on quadratic and
cubic terms, respectively, as is evident from equations (8.8) and (8.9). These are
rigid-body terms which depend only on the location of the origin and the
orientation of the coordinate system, but not on the applied loading.
With the origin of the coordinate system placed in the center of the plate,
as shown in figure 8-6, double symmetry is created. Therefore, Dz(a, Y) =
D z ( - a, Y), from which
C3
=
Cs
=0
Also, Dz(X, b)
=
(8.61a)
Dz(X, - b) so that
C6 = 0
Since C 3
Mxy
=
(8.61 b)
0, equation (8.59) gives
=0
(8.62)
This leaves only C9 still to be determined.
Since we have no readily apparent kinematic boundary conditions to constrain the displacement, we may arbitrarily select a point on the deflected plate
as the reference for measuring Dz . It is convenient to choose the origin X = 0,
Y = 0, as a point of zero displacement, i.e.,
Dz(O,O)
=0
(8.63)
meaning that Z is measured from the middle surface of the deflected plate. Once
again, we stress that this arbitrary choice does not affect the stress resultants
or couples, since C9 is a rigid-body term. Imposing equation (8.63) on equation
(8.60) gives C 9 = 0 so that, in view of equations (8.61),
Dz(X, Y)
-1
2
_ /l2) [(Ml - /lM2)X
= 2D(1
+ (M2
2
- /lMdY ]
(8.64)
292
°
°
8 Bending of Plates
Finally, since Dz,xx and Dz , yy are constant, Qx = Qy = from equation (8,9).
Likewise, with Mxy = 0, Qx = and Qy = 0 from equation (8.20), and there are
no corner forces.
We have treated the rectangular plate subject to edge moments in detail. It
is noteworthy that the computational effort was substantially reduced by
starting with a physically plausible stllte of constant bending in each direction
and then proceeding to derive a consistent solution for the problem at hand.
Although this approach may appear to be ad hoc and is obviously not completely general, it is often productive. It is in the spirit ofthe semi-inverse method,
attributed to St. Venant, 5 which is so widely applied in the theory of elasticity,
It is also important to reflect on the role of the preceding solution in the
engineering application of the theory of plates. First, we will uSe some special
cases to demonstrate that certain loading combinations are valuable for experimental verification of the theory. Second, we will find that the preponderance
of readily obtainable analytic solutions are for plates with simply supported
boundaries. The well-known flexibility method of structural analysis, outlined
in section 6.3.1, suggests that solutions for plates with fixed boundaries can be
obtained by combining the corresponding solution for a simply supported
boundary condition with a solution due to edge moments of appropriate
magnitude, so as to satisfy the no-rotation condition at the boundaries.
8.2.1.2 Parabolic Bending. Referring to figure 8-6, we take Ml = M2 = M,
which represents edge moments of an equal intensity applied uniformly around
the boundary of the plate. The solution, equation (8.64), reduces to
Dz(X, Y)
M
2
2
= - 2D(1 + p.) (X + Y )
(8.65)
which is the equation of a paraboloid of revolution. We previously encountered
a surface initially in this form in our study of translational shells, equation
(4.229).
8.2.1.3 Cylindrical Bending. Another interesting case is to determine what com-
bination of edge moments would be required to produce a cylindrical deflected
shape. Since a cylindrical surface is only singly curved, we want Dz = Dz(X) or
Dz = Dz(Y) only. From equation (8.64), it appears that Ml = jlM2 or M2 =
jlMl would be required to achieve these conditions. We illustrate Dz = Dz(Y)
on figure 8-7 and choose M2 = M and Ml = jlM. The deflected shape is given
by
Dz
= - -MY 2
2D
(8.66)
It is instructive to interpret this solution from a physical standpoint. At first
thought, one might surmise that a plate could be bent into a cylindrical shape
293
8.2 Rectangular Plates
z
2b
Loading o n Plate
Deflected Surface
due fo M2 only
z
20
Cross - Section at Y
=
0
Fig. 8-7 Cylindrical Bending of Rectangular Plate
by simply applying equal couples to two parallel edges and leaving the other
edges unloaded. Closer scrutiny reveals that the Poisson effect under this
loading would produce a contraction of the top fibers and an extension of the
bottom fibers, with a corresponding curvature as shown in the cross section on
figure 8-7. Thus, the moment J.lM is required on the other edges to produce
294
8 Bending of Plates
compensating extension in the top and contraction in the bottom so as to
restore the rectangular cross section.
Parenthetically, this simple problem motivates a possible experimental means
of determining Poisson's ratio, f.l. We consider a plate initially deflected by a
known couple M 2 acting on the edges parallel to the X axis, and then loaded
incrementally by a couple Ml along the edges parallel to the Yaxis. Observing
the deflection in the X direction, the ratio of M IiM 2 for which a line on the
surface parallel to the X axis becomes straight would give the effective value of
f.l. As mentioned in chapter 6, the determination of the appropriate elastic
constants to insert into a solution is often a difficult proposition. The utilization
of experiments based on known analytical solutions in which the constants are
indirectly determined is often productive in this regard.
8.2.1.4 Anticlastic Bending. The loading case Ml = -M2 = M is shown in
figure 8-8(a). Substituting into equation (8.64), we have
Dz(X, Y)
=
-M
(
2D 1- f.l
)(X
2
2
- Y )
(8.67)
which we recognize from equation (4.182) as the equation of the hyperbolic
paraboloid, describing an anticlastic surface.
We are intrigued by this result because we know that the two sets of real
characteristics should occur as straight lines on the deflected surface. In our
study of shells of this shape, we found the families of straight lines quite
important in the physical understanding of the load resistance mechanism.
If we restrict our inquiry to square plates, we find from section 4.4.3.1 that
the straight lines are oriented at 45° to the X - Y coordinates, as shown in figure
8-8(b). We wish to examine the state of stress along these straight lines, which
correspond to a rotation of y = + 45° as defined in figure 8-4. We take ~ and ,.,
as X and f, respectively, as shown in figure 8-8(b), and evaluate Mgg, Mg y, and
Myy from equations (8.36b), (8.37b), and (8.38b). With M" = Mx = M, Mp =
My = -M, and M"p = Mxy = 0, we find
Mgg
= Myy = 0
(8.68)
and
Mgy=M
(8.69)
Constructing the Kelvin-Tate argument, figure 8-8(c), we see that the state of
stress on the portion of the plate bound by the 45° straight lines which connect
the midpoints of each side is equivalent to that obtained by taking a square
plate with sides of length .j2a, loaded only with opposing corner forces of
magnitude 2M. Naturally, the moments within the perimeter of the straight
lines on the original plate are identical to those on the corner loaded square
plate.
295
8.2 Rectangular Plates
z
f---I---- y
I
2M
,
I
I
2M
2M
(c)
2M
Fig. 8-8 Antic1astic Bending of Rectangular Plate
The bending of a plate into an anticlastic surface by two pairs of opposed
corner forces was first investigated by Lamb, 6 who is renown for his work in
hydrodynamics. It is obvious that the situation depicted in figure 8-8(c) is
relatively simple to simulate physically, and classical experiments based on this
configuration were performed by Nitdai to verify the flexural theory of plates. 7
296
8 Bending of Plates
Z
/
b
I
/
-7
l'tl",
q (X;f)
z
I
7
y
I
/
/
/
/
x
Fig. 8-9
Arbitrary Transverse Loading on Rectangular Plate
8.2.2 Navier Solution for Simply Supported Rectangular Plates
8.2.2.1 General Solution
v4 Dz(X, Y) = qz(X,
Y)
D
(8.70)
Among his many accomplishments, Navier is credited with developing the
first satisfactory theory of plates, in the form of equation (8.70). 3 A widely used
solution technique, which bears his name, is to expand both Dz and qz into
double Fourier sine series, to solve the resulting equations for a general harmonic, and to obtain the complete solution by superimposing a sufficient
number of harmonic components to attain an acceptable precision. We may
recall that Fourier series were introduced for shells of revolution in section
4.3.5.2. Here, sine series are chosen because the simply supported boundary
conditions are satisfied identically, as we will show later.
Proceeding, we assume
Dz(X, Y)
=
L L
00
00.
}=I k=1
X
Y
Dzksinjn-sinknb
a
(8.71)
where Dfk is the Fourier coefficient for the displacement in the general harmonics j and k. Then,
297
8.2 Rectangular Plates
L'" L'"
V4 Dz -_
j=l k=l
[(jn)4
- +2 (jn)2(kn)2
+ (kn)4]
a
a
X.
.k.
b
b
Y
b
·Di smjn-smkna
L
(8.72)
X
Y
-a + (k)2]2.
Diksinjn-sinkn-b
b
a
'" '" [(j)2
= n4 ~
}=1 k=l
The r.h.s. of equation (8.70) is expanded as
1 ~ ~ .k. . X.
Y
L.. L.. qi smjn-smknb
D j=l k=l
a
qz(X, Y)
--- = D
(8.73)
where qik is the Fourier coefficient for the loading.
We proceed to evaluate qik by the standard method. To keep the notation
consistent,j and k should be regarded as specific, though general, indices in the
following operations; I and m are introduced as the corresponding variable
indices:
1. Multiply both sides by sinmn(Y/b)dY and integrate from 0 to b.
2. Note that
f
b
o
k Y .
Y
sm nbsmmn b d Y =
•
o
{b
2"
(m
=1=
k)
(m
= k)
(8.74a)
3. Multiply both sides by sinln(X/a)dX and integrate from 0 to a.
4. Note that
f
x. I Xd
smjn-sm n- X
a.
o
a
a
=
o
{a
2"
(l
=1=
(I
= j)
j)
(8.74b)
Steps (2) and (4) demonstrate the orthogonality property that greatly facilitates
the evaluation of the Fourier coefficients. Therefore,
qik = a:
s: S:
qz(X,
Y)Sinjn~ SinknfdYdX
(8.75)
We now substitute equations (8.72) and (8.73) into equation (8.70) and write
the equation for a specific j and k:
n 4 [(-j)2
a
+ (k)2]2.
Dik =qik
b
D
(8.76)
where qik is given by equation (8.75). We then solve equation (8.76) for Dik and
substitute into equation (8.71) to write the complete solution as
298
8 Bending of Plates
(8.77)
It remains to be shown that equation (8.77) satisfies the simply supported
boundary conditions. Referring to section 8.1.6, we require Dz(O, Y) = Dz(a, Y) =
Dz(X,O) = Dz(X, b) = 0, which are obviously satisfied by equation (8.77); and
Mx(O, Y) = Mx(a, Y) = My(X, 0) = My(X, b) = O. The latter conditions can be
written as V2 Dz(O, Y) = V2 Dz(a, Y) = V2 Dz(X, 0) = V2 Dz(X, b) = 0 from equation (8.47). Obviously, [( ),xx + ( ),yy] operating on equation (8.77) contains
the product sinjn(X/a) sinkn(Y/b) which vanishes along the boundaries. Thus,
the identical satisfaction of the simply supported boundary conditions by the
Navier double sine series solution is verified. Unfortunately, it is not so easy to
find appropriate functions to satisfy other boundary conditions identically.
Although we claim to have found the general solution as equation (8.77), the
analysis is not complete. We must proceed through the tedious differentiations
required to compute the stress couples and transverse shear resultants. However, we can regard the operations as being routine, since the displacement
function is obviously continuous and repeatedly differentiable. This ensures
that we do not violate the requirements of internal compatibility, which is
obviously a positive feature of the displacement formulation selected. Also, it is
noteworthy that here we were able to derive a solution sufficiently general to
apply for all harmonics; whereas, for example, in the analysis of doubly curved
shells of revolution, we found it necessary to treat the symmetrical and antisymmetrical harmonics separately from the general case.
Since we have obtained a quite general solution, we will summarize the
various rotations, stress resultants, and couples. These are obtained from equations (8.8), (8.9), (8.20), and (8.21), and all have the form
F(X, Y)
=
F(Dz )
(8.78)
where the various functions are collected in table 8-2.
8.2.2.2 Uniformly Distributed Loading. For a uniformly distributed load,
qz(X, Y)
=
qo
(8.79)
equation (8.75) is
.
4q
qfk = abO
16
fa fb sinjrc--;;sinkn[;dYdX
X
Y
0
= n 2jk qo
0
(j and k odd)
Table
8-2
w), :'G)'t (in ~)F. (k·f)
Functions for double sine series solution
F(D,) ~ F, j~ .~, qj'
F(Dz )
F2
F3
F4
sin
sin
cos
sin
sin
cos
-sin
sin
(~)G)
cos
cos
(~Y
sin
-sin
-cos
sin
(~YG)
-sin
cos
G)GY
cos
-sin
(~Y
sin
-cos
sin
sin
sin
sin
cos
cos
cos
sin
sin
cos
cos
sin
sin
cos
Fl
1
Dz
1
n 4D
Dz.x
=
-Dxy
1
j
n 3D
a
Dz .y
=
-Dyx
1
k
n 3D
b
1
GY
Dz .xx
Dz.xy
Dz . yy
Dz .xxx
Dz .xxy
Dz .xyy
Dz . yyy
Mx
My
Mxy
Qx
Qy
Qx
Qy
n 2D
1
n2D
1
n2D
GY
1
nD
1
nD
1
nD
1
nD
n2
n2
1
n2
1
n
n
1
n
1
n
GY +~(~Y
~GY +GY
(1-
~)G)G)
G)[(~Y +GYJ
(~)[(~Y +(~YJ
(~)[GY +(2-~)GYJ
G)[(2-~)GY +GYJ
299
300
8 Bending of Plates
or
=0
(j and k even)
(8.80)
from which we find
Dz(X, Y)
=
16qo.f.f
11:6 D
._ L.-
')2 + (k)2J2
b
. . X . k Y
1
_ L.-
}-1,3 ... k-1,3 ...
jk [ (~
smJ1I: -a sm
11:
(8.81)
b
from equation (8.77).
From a computational standpoint, it is instructive to examine the convergence of this series. For simplicity, we take a square plate, a = b, and consider
the deflection in the center, X = Y = a/2.
Dz(~ ~) _16~qo,,=-a_4 f
2' 2
=
1I: 6 D
f [sinj~Sink~]
j=1,3, ... k=1,3,...
jk(j2
+
k 2 )2
(8.82)
The term in the square brackets takes on the following values:
j
k
1
1
3
1
1
3
1
5
]
[
+0.2500
-0.0033
-0.0033
+0.0003
This indicates quite rapid convergence, so that even the first term of the series
should be quite close to the actual displacement. However, in this regard, we
recall that the bending and twisting stress couples are computed from the
second, and the transverse shear resultants from the third derivatives of Dz .
From table 8-2, it is obvious that each differentiation adds a j or k to the
numerator, thereby reducing the convergence rate. Therefore, although the
displacements may converge quite quickly, many more terms of the series may
be required to evaluate moments and shears accurately.
8.2.2.3 Approximate Analysis for Uniformly Distributed Loading. Since the j =
1, k = 1 term ofthe series gives a very close approximation for the displacement
due to the uniform load qo, we may study the essential characteristics of the
transverse shear resultants and the stress couples using a single harmonic
approximation of a uniformly distributed loading, as shown in figure 8-10.
From equation (8.80), we compute
301
8.2 Rectangular Plates
2M)(Y
Fig.8-10 Single Harmonic Distributed Loading on Rectangular Plate
(S.S3)
Substituting equation (S.S3) into equation (8.77), we have
(S.S4)
from which we find the effective transverse shears Qx and Qy from table S-2:
_
Qx(X,Y)=
(16qo)[~ + (2 - Jl)~]
7
(1
a a2
1)2
+ b2
x.
Y
COS1taStn1t b
(S.S5a)
and
(S.S5b)
On figure 8-10 we show the effective transverse shears on the boundaries,
302
8 Bending of Plates
Qx(O, Y), Qx(a, Y), Qy(X,O), and Qy(X, b). The positive sense of these forces is
found by referring to figure 8-l.
We now consider equilibrium in the Z direction and define the total applied
load
QZl
=
f: f:
(8.86a)
qzdY dX
With the load intensity given by
qz(X, Y)
= (16/n 2 )qo sin n(X/a) sin n(Y/b)
(8.86b)
as specified by equations (8.73) and (8.83),
(8.86c)
which acts in the positive Z direction.
Considering now the effective transverse shears on the boundaries, we have
the resultant QZ2 acting in the positive Z direction given by
QZ2
f:
-f:
= -
[I Qx(O, Y)I + IQx(a, Y)IJ dY
(8.87)
[IQy(X,O)1
+ IQy(X,b)IJdX
After considerable algebra we compute
QZ2
=
64
-4qO 2
n
(a
ab
4
b2 )2 [a
+
+ 2(2 -
22
J1.)a b
4
+b J
MUltiplying the numerator and denominator of equation (8.86c) by (a 2
and adding it to equation (8.88), we have
(8.88)
+ b2 )2
(8.89)
as an apparently unbalanced force. But, we must also include the comer forces.
From equation (8.50b) and figure 8-5, we see that there are comer forces of
magnitude 2Mxy as shown on figure 8-10. Referring to table 8-2 and equation
(8.83), we find
Mxy(X, Y)
(8.90)
303
8.2 Rectangular Plates
In evaluating Rxy at each comer, we note from figure 8-5 that a positive Mxy
produces a comer force in the negative Z direction. At each comer, (X = 0, a;
Y = 0, b) Mxy is negative, so that the comer forces are directed in the positive
Z direction. Each force is given by
(8.91)
producing a total force in the positive Z direction of
QZ3 = -4Rxy
(1 - p.)
a 3 b3
= 128 n4 qo (a2 + b2)2
(8.92)
Now, adding equation (8.92) to equation (8.89) we get
QZl
+ QZ2 + QZ3
(8.93)
= 0
Thus, we demonstrate the contribution of the comer forces to the overall
equilibrium of a rectangular plate. As an order of magnitude estimate, each
comer force will be about 10% of the total load for a square plate.
It may be shown that for non-right angle comers of simply supported
polygonal plates, there is no concentrated comer force, but, as the included
angle approaches 90°, the intensities of the effective transverse shears on the
boundaries, Q~ and Q", increase rapidly near the comer. This may be demonstrated for a triangular plate which is discussed in section 8.4.2.
We now consider the stress couples, which are easily computed using table
8-2 and equation (8.83):
(8.94a)
and
My(X, Y)
=
( 16
n 2Qo
)( 1)
n2
p.GY + GY . X. Y
[GY + (~YJsmnasmnb
(8.94b)
It is obvious that both functions will follow the load distribution shown in figure
8-10 with maxima at the center, X = a12, Y = b12.
In order to assess the relative magnitude ofthe bending moments at the center
of a square plate where b = a, we evaluate
8 Bending of Plates
304
Mx
aa) = My (a2'2a) = 4(1 + Jl) qoa
(2'2
1C 4
2
~ (0.04 - 0.05)qoa
2
(8.95)
as the maximum moment.
For comparison, consider a beam of unit width carrying the same sinusoidal
loading over the span a. With a loading
q(X)
=
16
. X
zqoStn1C
a
(8.96a)
the differential equation for a prismatic beam
(8.96b)
E1Dz ,xxxx = q
has a solution
qo
1 (16
Dz(X)=--2- )
E1 1C
(a)4.
-
1C
X
Sln1C-
(8.96c)
a
from which
M(X) = E1Dz ,xx
(8.96d)
16qo 2 .
X
= __
a SIn 1C4
a
1C
with a maximum value of
(8.96e)
By comparing equations (8.95) and (8.96e), it is apparent that the plate is a far
more efficient flexural member than the equivalent beam due to the two-way
flexural action and the twisting rigidity. Equation (8.95), which gives the maximum bending moment for a uniformly loaded square plate, also confirms the
observations of the seventeenth-century French physicist Mariotte, who surmised that the total load Qo corresponding to the
moment, in this
case Qo = qoa 2 , should remain constant and independent of the size ofthe plate
if the thickness is not changed. 8
It is of interest to compute the moments in the vicinity of the corners by
considering rotated axes ~, Yf at 45° to the X - Y coordinates, as shown on
figure 8-9. Referring to figure 8-4(c) and taking 0( = X, fJ = Y, and y = 45° in
equations (8.36b), (8.38b), and (8.37b), respectively, give
maximum
Mq = M XY
M~=
R
="2
R
-MXY = -
2
305
8.2 Rectangular Plates
M,,~
=
M~"
=0
Thus, in the vicinity of the corner, we find a state of anticlastic bending, as
described in section 8.2.1.4. The moments ±Mxy or ±R/2 are the same order
of magnitude as the moments in the center of the plate, e.g., O.68Mmax for a
square plate. 9 Mq is essentially a clamping moment, similar to that developed
at a fixed boundary.
Another manifestation of the corner force phenomenon may be observed by
visualizing a rectangular plate under self-weight set on a continuous knife edge.
Seemingly, this boundary would correspond to an ideal simple support condition. However, for an applied loading in the downward (negative Z) direction,
the preceding solution would indicate corner forces acting downward on the
plate. If the plate simply rests on the knife edge with no tied owns, the corners
will tend to rise away from the support. It may also be remarked that the lack
of restraint to develop the corner forces will remove the clamping effect of Mq
and thereby cause the maximum moments at the center to increase,9 perhaps
by 35% for a square plate.
8.2.2.4 Rectangular Patch Loading. In figure 8-11, we show a loading uniformly
distributed over a rectangular patch on the plate. The total magnitude of the
z
x
Fig. 8-11
Patch Loading on Rectangular Plate
306
8 Bending of Plates
(8.97)
= 0
~
otherwise
and '1 denote the X and Y coordinates, respectively, ofthe center ofthe patch.
Substituting equation (8.97) into equation (8.75), we have
.
q~k =
(Q )
( -b
4 ) -do
a
c
16 Q
11;+C/2 S"+d/2
l;-c/2
,,-d/2
X
Y
sinjn-sinknb dYdX
a
. . ~ . k '1 . . c . k d
(8.98)
= n2cdjk osm]n-a sm n hsm]n 2a sm n 2b
As a check, we select the uniformly distributed load considered in section 8.2.2.2.
For that case, we have Qo = qoab; c = a, d = b; and ~ = a12, '1 = b12. Substituting these values into equation (8.98), we get the identical result to that
obtained in equations (8.80). The solution for the deflection function and the
subsequent stress resultants and couples follow from table 8-2.
8.2.2.5 Concentrated Loading. The solution just obtained for the patch loading
may be used to describe a concentrated load at any point on the plate. To
illustrate, we consider Qo acting at X = ~ and Y = '1, as shown on figure 8-12.
We take equation (8.98) with c and d approaching zero and get
6
~
qik=~~gn~jkQosinjn-asinkn~
(. . c) (.
k d)
sm]n"2
sm n 2b
cad
(8.99)
The limit is evaluated by L'Hospital's rule, which gives, for the last two terms,
.. c . k d
sm]n 2a sm n 2b
lim - - - ---::-c
d
c-+o
d-+O
jn kn
2a 2b
reducing equation (8.99) to
·k
q~ =
4Qo . . ~ . k '1
ab sm]n-a sm n h
From equation (8.77), we write the complete solution as
(8.100)
307
8.2 Rectangular Plates
z
x
Fig.8-12
Dz(X,
Y;~,rJ)
=
Concentrated Load on Rectangular Plate
. . ~ . k rJ
sInJn-sIn
n-b
QO-4- L L [(-.)2 + (k)2J2
sinjn-sinkn4
n abD
0000
j=1 k=1
a
]
a
X
a
b
Y
b
(8.101)
We introduce the notation Dz(X, Y;~, rJ) to emphasize that Dz is dependent on
two sets of coordinates: (a) the point of observation of the deflection, (X, Y), and
(b) the point of application of the applied load Qo, (~, rJ). The coefficient of Qo
in equation (8.101) is defined as the Green's function, G(X, Y;~, rJ), and represents the displacement at the point of observation (X, Y) due to a unit load at
the point of application (~, 1]). Graphically, the Green's function for the displacement of the plate may be visualized as an irifluence surface representing the
deflected shape of the entire plate due to a unit load at a particular point.
Once we establish the Green's function for a system, we may readily determine the deflection at any point of observation due to a prescribed surface
loading qz = qz(~, rJ)· Then, by superposition, we have
Dz(X, Y)
=
II qz(~,rJ)G(X, Y;~,rJ)d~drJ
Area
The Green's function approach is widely used in mechanics. 10
(8.102)
308
8 Bending of Plates
If we consider the solution given by equation (8.101) in more detail, we expect
the series for Dz to converge rapidly because of the rand
terms in the
denominator. However, the series expressing the stress couples and transverse
shear resultants are found to be divergent right at the point of application of
the concentrated load, e.g., X = ~, Y = f/. Of course, the series for the load itself
will not converge. Thus, this solution, although simple in form, is of limited
value for numerical computations and the approach described in the following
section is more useful. However, it is of interest to compute Dz for a square
plate, b = a, with a concentrated load Qo at the center. The first four terms of
the series for this case give
e
D ( -a -a)
z 2'2
Qo
D
~O.Ol-a
2
(8.103)
To compare equation (8.103) with a beam solution, we must take the entire
plate as the beam width a. Then (EI)beam ~ aD and
D
= Qoa 2 ~ 0.02 Qo a 2
48D
D
(8.104)
again indicating the relative rigidity of the plate as compared to a flexural
member of equal span.
8.2.3 Levy-Nadai Solution for Rectangular Plates
8.2.3.1 General Technique. Although the double sine series solution developed
in the previous section is straightforward, the resulting expressions often require
that many terms be included in the summation to attain acceptable precision.
Of course, this objection is largely anachronistic in the era of the computer;
nevertheless, it is of interest from a fundamental standpoint to explore the
possibility of generating more efficient solutions. Also, the Navier solution is
well suited for the simply supported boundary only, and the desire to treat other
boundary conditions gave rise to the work of Levy, a pupil of St. Venant, and
Estanave. 6 The ensuing single series approach was exploited by NtLdai as well,
and carries the name of the Levy-NtLdai solution.
We introduce this technique by taking the displacement function Dz as the
sum of (a) DZ1 ' a beam solution which satisfies equilibrium and the boundary
conditions in one direction; and, (b) DZ2 ' a homogeneous solution which can
be used to enforce the boundary conditions in the other direction, while not
violating the boundary conditions in the first direction.
We refer to figure 8-13 where the X axis is located along the center line of
the plate, for reasons which will become obvious. We select DZl = DZl (X) as
the beam bending solution for a loading qz which satisfies the boundary
conditions at X = 0 and X = a. For example, a simply supported beam under
a uniform load qz = qo has a deflection function
309
8.2 Rectangular Plates
y
Q
I.
X
b/2
Fig.8-13
b/2
Coordinates for U:vy-Nadai Solution
(8.105)
A number of similar formulae for various loadings and boundary conditions
are collected in the AISC Manual of Steel Construction. l l
If we make the adjustment for the Poisson effect and change the term El to
D, equation (8.105) will satisfy the plate equation, equation (8.12), as well as the
simply supported boundary conditions at X = 0 and X = a. However, the
boundary conditions at Y ± b/2 are obviously not addressed since DZI =
DZI (X) only. To correct this, we need D Z2 , which satisfies the boundary conditions in both directions. Since DZI already will balance the loading, DZ2 should
be a homogeneous solution of equation (8.12).
8.2.3.2 Simply Supported Plate Under Uniform Load. We consider the case of
a uniformly loaded, simply supported plate, previously treated in section 8.2.2.2.
Using equation (8.105) for D Z1 ' with El replaced by D, we require the functions
to satisfy
V4 D z
= V 4 DZI + V 4 DZ2 = ~ + 0 = ~
For DZ2 ' we choose a single series solution
(8.106)
310
8 Bending of Plates
DZ2
=
X
L I/IJ(Y) sinj1t-a
i=l
00.
(8.107)
where I/Ii is a function of Yalone. This expression for DZ2 automatically satisfies
the boundary conditions in the X direction. Substituting equation (8.107) into
equation (8.12), we have
.
4
V DZ2
)4I/IJ =?: -J1t.
a
00
[(.
J=l
(. )2
J1t.
2 a
.
+ I/I!yyyy
I/I.Jyy
J
'.
X
smJ1ta
(8.108)
=0
Equation (8.108) is satisfied for all j if the terms within [ ] = 0, leading to the
linear ordinary differential equation for I/Ii(y):
.
J1t
.
(J. 1)2tI/I,Jyy
+ (.~)4 I/IJ
.
I/I!yyyy - 2 ~
=
(8.109)
0
which may be solved by classical means as
.
I/IJ(Y) =
Y.
Y.
Y
'.
Y
q . coshj1t+ Qj1t-smhj1t- + q smhj1t-
a
.
a
Y
Y
a
a
a
a
(8.110)
+ Cij1t-coshj1tThe selection of the origin at the middle of the side parallel to the Y axis now
becomes useful. Since the loading is uniform, we recognize that the plate
deflection is symmetrical about the X axis. This being the case, only even
functions of Y[f( - Y) = f(Y)]' can participate in the solution and the coefficients of the odd terms in equation (8.110), [f( - Y) = - f(Y)], drop out.
Therefore, we have q and ci = 0 and the complete solution to equation (8.106)
reduces to
Dz(X, Y) = DZI
+ DZ2
qoX 3 - 2aX 2 + X 3 ) + L...
~ [ q . coshj1tY
= -2-(a
W
~l
a
(8.111)
. Y' h j1tYJ'
X
+ Qj1t-sm
smj1t-
a
a
a
We have the integration constants C{ and q available to satisfy the simply
supported boundary conditions at Y = ± bj2. Since we have already invoked
the requirement of symmetry to suppress the odd terms in equation (8.110), the
conditions at Y = + bj2,
Dz(
X,~) = DZ,yy( X'~) = 0
(8.112)
311
8.2 Rectangular Plates
are sufficient. It is necessary to expand the expression for DZ1 in a Fourier series
so that the constants can be chosen in harmonic form. This series is found
routinely9 as
DZ1
4qoa4
1t D
= -5-
1 .. X
L3.odd --;sSlDJ1tJ
a
00
(8.113)
j=l.
and, after substitution of equation (8.113), equation (8.111) can be written in
the form
Dz(X, Y)
4
qa
°D
=
L
00
j=1.3.odd
(8.114)
[4
.
Y
Y
YJ
X
s:s + q coshj1t- + C~j1t-sinhj1t- sinj1t1t J
a
a
a
a
where the original constants q and q have been modified by the common
factor. Now, the constants are evaluated by applying equation (8.112):
+ 2 cosh A.i)
1t SjS cosh 2 A.l
Ci = - 2(A.i sinh ,V
5
(8.11 Sa)
and
(8.115b)
where
,V = j1tb
2a
(8.115c)
The rotations, stress couples, and transverse shear stress resultants can be
obtained directly from equations (8.20), (8.8), and (8.9).
In retrospect, it is apparent that the "beam" solution could have been taken
in the form of a series initially. This would lead directly to equation (8.114)
without encountering equation (8.111).
A detailed study of this solution, including the convergence properties, is
contained in Timoshenko and Woinowsky-Krieger. 9 We have already examined this problem in detail in section 8.2.2.2, and the results found there can
be confirmed with the single series solution. Also available are parametric tables
for the various functions for uniform and hydrostatic loading. 9 Additionally,
single series solutions are developed for some additional cases including the
bending moments under a concentrated load,12 a somewhat difficult computational exercise as noted in section 8.2.2.5.
Some results of the comprehensive study in Reference 12 are of general
interest. The single series Levy-Nadai solution is usually more rapidly converging than the double series Navier solution. Heuristically, this indicates that
starting with the known beam function as a first approximation leads efficiency
8 Bending of Plates
312
I
y
I
I~
It
I
I
I
I
I
I
I
I
k,
a
I~
I
I
I----- ___I ~
I
I
b/2
bl2
x
Fig.8-14
Rectangular Plate Loaded by Edge Moments
in the computational algorithm. Particularly, since smaller computers are
becoming increasingly popular in engineering, the savings achieved by using a
solution based on a well-founded initial trial may be significant.
An interesting observation is that when the aspect ratio b/a ~ 3, a plate
behaves essentially as a one-way flexural member spanning in the short direction, except near the edges. This may be appreciated by noting the denominators of equations (S.115a) and (8.115b), which get large with increasing b/a. With
the influence of DZ2 diminishing, the deflection function is dominated by the
beam solution DZ1 '
S.2.3.3 Other Single-Series Solutions. Another classical application of the
U:vy-Nadai approach is shown in figure 8-14, where a rectangular plate is
loaded by constant edge moments. If Mn(X) =F M Y2 (X), the solution is facilitated by the superposition of a symmetrical and an antisymmetrical case. The
details are left as an exercise.
Also, we observe that rectangular plates which have only two opposite sides
simply supported may be treated by inserting a beam solution with the appropriate boundary conditions for DZl in equation (S.111) and proceeding accordingly. This will be adequate for plates which have two sides simply supported, two sides clamped; three sides simply supported, one side free; and two
sides simply supported, one side clamped, one side free.
More complicated boundary conditions are treated with Levy-Nadai type
solutions in Timoshenko and Woinowsky-Krieger. 13 • 14 Also, some additional
8.3 Circular Plates
313
solution procedures are discussed in Kiattikomol, Keer and Dundurs, 15 and extensive tabulated results are provided by Szilard. 16 These analytical approaches
become rather involved, and one must soon turn to numerical techniques. The
classical solutions remain valuable, however, as a basis for incisively evaluating
the results of numerical solutions through quantitative comparisons and orderof-magnitude bounds.
8.3 Circular Plates
8.3.1 Axisymmetric Bending
8.3.1.1 Governing Equation. Axisymmetric bending of circular plates is described by equations (8.33). Using equation (8.27) and (8.28b) with the () terms
omitted, we may rewrite equation (8.33a) in the form
~ [ R (~ [R(DzhlR }J,R
(8.116)
which is quite convenient for solution.
8.3.1.2 Homogeneous Solution. We first consider the homogeneous part of
equation (8.116) and integrate consecutively as follows:
[
[
J.R = 0
]
(
(
=
C1
C1
)
R
,R
)
= C1 lnR + C2
[
lR
=
[
] =C
C 1 RlnR
1(~2
+ C2 R
InR _
:2) + 2~2 + 3
C
= C1 R 2 lnR + C2 R 2 + C3
where C 1 and C2 have been redefined. Continuing
C
314
8 Bending of Plates
and, finally,
D
z
=
DZh
=
C 1 R21nR
+ C 2R 2 + C3 1nR + C4
(8.117)
where C 1 and C2 have again been redefined. The terms composing equation
(8.117) are known as biharmonic functions, because they individually and collectively satisfy the biharmonic equation in polar coordinates.
It is informative to compute the rotation, stress couples, and transverse shear
corresponding to DZh ' These are found from equations (8.30c), (8.31), and (8.32b)
and are listed in table 8-3. We note the following characteristics of the tabulated
functions:
1. R 2 1n R is singular at R = 0 and R = 00, and is the only term which contributes to QR'
2. R 2 is singular at R = 00; for this function MR = Mo (homogeneou.s bending).
3. In R is singular at R = O.
4. 1 is regular throughout.
These characteristics are useful for expediting solutions to various problems,
as we will see later.
8.3.1.3 Solid Plates. Consider the case of a constant edge moment M applied
around the circumference of a simply supported circular plate with radius a, as
shown in figure 8-15.
The boundary conditions are
=0
(8. 118a)
MR(a) = M
(8.118b)
Dz(a)
Of course, we may write equation (8.118b) as a function of Dz , but, as we shall
see, this is unnecessary. There are two additional boundary conditions required,
since we have a fourth order system. These conditions follow from the requirement that the solution remains finite at R = O. This is a familiar situation, which
we first encountered in the treatment of domes in chapter 4. In our preceding
characterization of the various biharmonic terms, we noted the singularity of
the In Rand R 2 1n R terms at R = 0, so that we must set C 1 = C3 = O. Then,
we find, by substituting the remaining terms in equation (8.117) into equations
(8.118) with the help of table 8-3, that
C2 a 2
-C2 2(1
+ C4 = 0
+ Ji)
=
M
Ii
(8.119a)
(8.119b)
X
C4
x
x
C3
Particular
qz(R) = qo
qo/64D
x
x
C1
C2
Homogeneous
DZh(R)
I R4
1
lnR
R 2 1nR
R2
I D,(R)
-4R3
-R(21nR
-2R
1
R
0
D1M(R)
= -DZ,R
+ 1)
DZ,RR
+
-4(3
0
+ p,)R2
(1 - p,)/R 2
+ 3 + P,J
~ DZ,R -
-4(1
0
+ 3p,)R2
-(1 - p,)/R 2
-[2(1 + p,)lnR
-2(1 + p,)
=-
M'(Re
Functions of Dz
iDz,RJ
-[2(1 + p,)lnR
-2(1 + p,)
=-
M,(R(D
Solution components for a circular plate
Solution
Component
Table 8-3
+ 1 + 3p,]
P,Dz,RRJ
-32R
0
0
-4/R
0
1
- -R2DZ,R
QR(R)
D
= DZ,RRR
1
+ JiDZ,RR
64
0
0
0
0
qz(R)
D
= V:Dz
Vl
w
-
'"
ct
iii
"C
....
E..
I>l
(')
2
w
00
316
8 Bending of Plates
ff.
I
L
t~--Q--~~~-R-Q--~J
Mr.
Fig.8-15
:,\M
Edge Moments on Circular Plate
from which
-M
C2 = 2D(1
(S.120a)
+ Jl)
Ma 2
C = 2D(1
4
(S.120b)
+ Jl)
and
Dz(R) = 2(1
M
+
2
2
Jl)D (a - R )
(S.121)
Also, we find
(S.122)
which indicates that the bending moment is constant throughout the plate in
all directions. A state of homogeneous bending was previously encountered for
a rectangular plate in section S.2.1.2.
The first logical case of applied surface loading is a uniform load qz = qQ, as
shown on figure S-16. It is easily verified by direct substitution into equation
(8.116) that the particular solution is
Dzp(R)
= 6~~ R4
(S.123)
For convenience, the various functions corresponding to this particular solution are recorded in table S-3.
Now, combining equation (S.117) with equation (S.123) after setting C1 =
C 3 = 0 as previously discussed, we have
Fig.8-16
Uniform Load on Circular Plate
317
8.3 Circular Plates
(8.124)
with the boundary conditions corresponding to the simple support,
Dz(a)
MR(a)
=
=0
(8.125)
With the aid of table 8-3, we write
(8.126a)
and
+ Il)C2
-2(1
6~~ [4(3 + ll)a 2 ] = 0
-
(8. 126b)
from which
C =
2
C =
4
=
_~[2(3 + Il)J 2
64D (1 + Il) a
~[2(3 + Il) 64D (1 + Il)
~[5 +
64D 1
(8.127a)
IJa4
(8.127b)
IlJa 4
+ Il
and then
(8.128)
We may also find the moments from table 8-3 as
MR(R)
= ::[ -(~(~
= qo (3 +
16
M,iR)
a2 [ -2(1
+ Il)]
+ Il)R 2]
(8. 129a)
:t)
a 2[ - 2(1
+ Il)]
~~ [(3 + ll)a 2 - (1 + 31l)R2]
At the boundary, R
- 4(3
ll)(a 2 _ R2)
= :: [ -(~(~
=
:t)
= a,
- 4(1
+ 31l)R 2]
(8.129b)
318
8 Bending of Plates
Fig.8-17 Clamped Circular Plate
(8.130)
while at the center, R = 0,
(8.131)
so that the maximum moment occurs at the center of the plate. At the pole,
MR(O) = M 6 (0), which again illustrates the isotropy condition first encountered
in the study ofaxisymmetrica1ly loaded shells of revolution, section 4.3.2.1.
We now investigate a clamped plate under uniform load, as shown in figure
8-17. There are two evident procedures: (a) We may superimpose the solutions
for the cases shown in figures 8-16 and 8-15 and enforce the compatibility
condition DR6 (a) = 0 to compute MR(a); or, (b) we may take the solution as
equation (8.124), with the constants determined from the boundary conditions
(8.132)
We select (a), the superposition approach, and enter table 8-3 with constants
from equations (8.120) and (8.127) to find
2
qo [-2(3 + p,)
DR6(a) = 64D
(1 + p,) a (- 2a) - 4a
3J -
2(1
M
+ p,)D ( - 2a)
(8.133)
=0
from which
qoa 2
M max = MR(a) = - 8
(8.134)
The deflection function for the clamped plate is found by superimposing equation (8.128) and equation (8.121) with M = MR(a) as given by equation (8.134),
319
8.3 Circular Plates
M
,
.!
M
(0
't\j'""'1
0
f
_
~
-
Base Line
Clamped Plate
M
CII
0
r
---''--_~_--I._ _~_ _ _ _ _ _ _ _~~ ___L--~B~as~
Line - Simply
Supported Plate
R
R=O
R=a
Fig. 8-18
Moments in Circular Plates
and may be written in the perfect square form
Dz(R)
= 6~~ (a 2 -
(8.135)
R2)2
We may consisely plot the stress couples for both the simply supported and the
clamped boundary conditions on a common graph, figure 8-18, since the edge
moment MR(a) produces a constant value of MR and M8 throughout.
8.3.1.4 Plates with Annular Openings. Another case of interest is a circular plate
with an annular opening, as illustrated in figure 8-19. We first consider the plate
under an exterior edge moment Ml and an interior edge moment M 2 , both
uniformly distributed around the circumference. Only the exterior boundary is
constrained against displacement in the Z direction. We have the homogeneous
solution, equation (8.117), subject to the boundary conditions
a
a
Fig.8-19 Circular Plate with Annular Opening
320
8 Bending of Plates
Q
Fig. 8-20
Dz(a)
=0
Q
Annular Plate with Peripheral Edge Moment
(a)
MR(a)
= M1 (b)
QR(b)
=0
MR(b)
= M2 (d)
(8.136)
(c)
A check of table 8-3 reveals that only the C 1 term contributes to QR(R). From
equation (8.136c), QR(b) = 0 so that C 1 = O. The remaining conditions, equations (8.136a), (8.136b), and (8.136d), lead to the complete solution 17
Dz(R)
1
= D(a2 _
b2)
[a 2M1 - b 2M2
2(1
+ /1)
2
2
(a - R )
(8.137)
A special case is M2 = 0 and M1 = M, as shown in figure 8-20. Equation
(8.137) then reduces to
Dz(R)
=
Ma 2
[(a 2 - R2)
b2
D(a 2 _ b 2 ) 2(1 + /1) - (1 - /1) In
aRJ
(8.138)
We calculate the circumferential bending moment M9 from table 8-3, where it
is indicated that only the terms R2 and In R contribute:
(8.139)
The maximum value of Mo is at R = b,
321
8.3 Circular Plates
M
Fig. 8-21
Arbitrary Plate with Circular Hole
(8.140)
Now, if we let the hole shrink, e.g., b -+ 0, then Mo(b) -+ 2M, indicating that there
will be a stress concentration around the hole approaching twice the value of
the moment applied on the outer edge.
This deduction may be generalized a bit more. If we consider a plate of
arbitrary shape which has a state of homogeneous bending M and a circular
hole of radius d, as shown in figure 8-21, the solution in polar coordinates should
contain the same biharmonic terms as equation (8.138),
(8.141)
The influence of the In R term on the moments diminishes rapidly away from
the hole, whereas the R2 term produces homogeneous bending. If we use the
boundary conditions
MR(d)
MR(R -+ (0)
=0
(8. 142a)
=
(8.142b)
M
we find from table 8-3 that
(8. 143a)
and
(8.143b)
from which
322
8 Bending of Plates
(8.144)
and
(8.145)
so that for R = d, Me -+ 2M.
Now consider a simply supported open circular plate with a total load P
uniformly distributed around the inner circumference, so that
(8.146)
as shown in figure 8-22. The algebraic sign of QR is established in accordance
with figure 8-3. We note from table 8-3 that only the term C1 R 2 ln R of the
displacement function can contribute to QR. Hence,
(8. 147a)
or
(8.147b)
The remaining constants are determined from the boundary conditions
r
I
I
l,~
b
a
Fig. 8-22
qJL
b
a
Uniform Transverse Shear on Annular Plate
8.3 Circular Plates
323
=0
=0
(8.148b)
MR(a) = 0
(8.148c)
MR(b)
Dz(a)
(8.148a)
and the complete solution finally takes the form 1 7
(8.149)
where
c - ~ [1 5 -
J.l _
1 + J.l
4nD
__ ~[(1 +
J.l) a 2 b2
4nD (1 - J.l) (a 2 - b 2 )
C6 -
C
7
2b 2 ln~]
a2 - b2 a
= Pa 2 [1 + (1 - /1) _
8nD
2(1 + /1)
a2
(8.150a)
ln~]
b2
-
(8.150b)
a
b2
ln~]
(8.150c)
a
If we now let b ~ 0, noting that
limlnb=O
1
(8.151)
b~O
b2
by L'Hospital's rule, we find
Dz(R) =
~[R2In R + i~ + /1)
8nD
a
1 + /1)
(a 2 - R2)]
(8.152)
which is the solution for a solid plate under a concentrated load P at the center.
It is of interest to review the order of the limiting operations which were
employed in the preceding paragraphs. In going from equation (8.149) to
equation (8.152), the effect of the hole was eliminated because we treated the
general function Dz(R) rather than a specific value. On the other hand, the
limiting operation based on equation (8.140) retained the effect ofthe hole, since
it was performed on a specific value Me(b) rather than the general function,
Me (R).
There are a number of additional cases that can be handled with the solutions
developed here. For example, a plate with a rigid annular insert or plug is shown
in figure 8-23. If the plug is subjected to an arbitrary axisymmetric loading,
qz(R), the total load on the plug is
P = 2n
f:
qz(R)R dR
We may then represent the region of the plate R
(8.153)
~
b by the superposition of
324
8 Bending of Plates
~+ QR(b)~
a~)
M2
+
l
I
Fig. 8-23
~
6
M2 =MR (b)
Annular Plate with Rigid Plug
(a) a solution corresponding to figure 8-22 with QR = - P /(2nb) and (D) a
solution based on figure 8-19 with MI = 0 and M2 = MR(b), as shown in
figure 8-23. Then, MR(b) is evaluated from the compatibility condition,
DRO(b) = O. There are numerous variations of the rigid insert problem which
are important in machine design. A compilation ofthese solutions may be found
in Timoshenko and Woinowsky-Krieger l7 and in Szilard. 18
In figure 8-24, we show a plate with an annular opening which is axisymmetrically loaded outside the opening. It is convenient to first solve the case of
the same loading applied to a solid plate, figure 8-24(b). In this first solution,
we may take qz(R) in the region 0 ::::;; R ::::;; b as any function of R. For convenience, we show qz(R) = qz(b) for 0::::;; R ::::;; b on figure 8-24(b). For the solid
plate, we evaluate QR(b) and MR(b). For example, with a uniform load qz(R) =
qo, we would have
QR
= -
nb 2
qo 2nb
=
b
(8.154)
-Q0"2
and also calculate MR(b) from equation (8.129a) as
MR(b) =
(~~)<3 + ll)(a
2 -
b2 )
(8.155)
For more complex loadings, the corresponding values of QR(b) and MR(b)
325
8.3 Circular Plates
(0)
( b)
(c)
( d)
Fig.8-24
Radially Varying Load on Annular Plate
would be determined by combining the homogeneous solution, equation (8.117),
with an appropriate particular solution. Once QR(b) and MR(b) are evaluated
for the solid plate, we correspondingly apply a force and a moment of equal
magnitude and opposite sense to the open plate, as shown in figures 8-24(c)
and (d). These latter solutions correspond to the cases shown in figures 8-22
and 8-20, respectively, and the complete solution for b ::; R ::; a is found by
superposition.
It should be noted that the preceding solutions can also represent a clamped
plate, with the addition of a solution for MR(a) = M 1 , as shown in figure
8-19. The value of Ml would be determined from the compatibility condition,
DRIJ(a) = O.
8.3.2 General Bending
8.3.2.1 Solution of Governing Equation. We return to the general biharmonic
equation, equation (8.26), and seek a set of biharmonic functions which satisfy
the 0 dependence as well. The solution to the homogeneous equation may be
written in the separated from attributed to the noted mechanician Clebsch 19
DZh
= FO(R)
+
L Fi(R) cosjO
i=l
00
(8.156)
326
8 Bending of Plates
This approach is identical to that employed for the shell of revolution in section
4.3.5.2. From a physical standpoint, this is plausible since the circular plate is
the degenerate case of the shell of revolution as Rtf> -+ 00.
The function FO(R) is the solution of the axisymmetric case as given by
equation (8.117). For j ~ 1, we substitute equation (8.156) into equation (8.26)
to get
Just as in the case of shells of revolution, the solution for j = 1 is somewhat
simpler and is given by
Fl(R)
= c}RlnR + qR 3 + qR + cl ~
(8.158)
Forj> 1,
Fi(R)
=
C1Ri
+ C~R-i + qRi+ 2 + CiR-(j-2)
(8.159)
These homogeneous solutions may be combined with appropriate particular
solutions to solve specific loading cases, with the integration constants being
determined from the boundary conditions. In each case, the loading function
and hence the particular solution should be described in a Fourier cosine series
to conform to DZh ' as given by equation (8.156).
8.3.2.2 Linearly Varying Load. We investigate a circular plate under a linearly
varying load, as shown in Figure 8-25. This can represent hydrostatic pressure
and may be resolved into symmetric and antisymmetric components, as shown
on the figure. We write the symmetric component as
° ql + q2
qz=--2
(8.160)
the antisymmetric component as
(8.161)
and the displacement function as
Dz
= D~ + Di
(8.162)
For the symmetric component, q~, we have already obtained solutions for
simply supported and clamped boundaries in section 8..3.1.3. To apply the
327
8.3 Circular Plates
y
w;;...-~--+--x
I
Q,;Q2
IIIII
Fig. 8-25
Hydrostatically Loaded Circular Plate
previously obtained solutions to this problem, q~ is inserted for qo in equation
(8.128) for the simply supported boundary, or in equation (8.135) for the
clamped boundary.
For the antisymmetric component, q}, we write
IX
qz(R, lJ) = qza
(8.163)
With X = R cos lJ,
IR
qz(R, lJ) = qz-cos lJ
a
(8.164)
328
8 Bending of Plates
Since the governing equation, equation (8.26), requires four derivatives with
respect to R, we select a fifth order function for the particular solution
Dip
= CJR s cos e
(8.165)
Substituting equation (8.165) into equation (8.26), we find
CH120
+ 2(60) -
[20
+ 2(20)] + [5 + 2(5)] + (-4 +
l)}Rcose
qi R
= - -cos e (8. 166a)
D a
or
192CJR cos e
=
ql R
~ -cos e
D a
(8.166b)
from which
1
Cl_~
5 -
(8.167)
192Da
and
(8.168)
Combining equation (8.168) with the homogeneous solution Dih = Fl cos e,
where Fl is given by equation (8.158), we have
Di(R,e)
= [CfRlnR + CiR 3 + CjR + Cl ~ + l:;D :SJcose
(8.169)
So that the solution remains finite at R = 0, we take ct = Cl = O. The
remaining constants are evaluated from the appropriate boundary conditions.
For a clamped outer boundary,
Di(a, e) = Di.R(a, e) = 0
(8.170)
which leads to the constants
(8.171)
and the deflection function
Di(R, e)
ql
R
= 19;D ~(a2 - R2)2 cos e
(8.172)
For a simply supported outer boundary,
Di(a, e)
= Mj(a, e) = 0
which gives the constants
(8.173)
329
8.4 Plates of Other Shapes
q} 2(5 + JL)
192D (3 + JL) a;
c1 _
2 -
-
C1
_
3 -
q} (7
192D (3
+ JL)
+ JL) a
3
(8.174)
and the deflection function
Di(R,8)
=
q
192(3 : JL)D
[7 + JL - (3 + JL)(~rJ
. Ra(a 2 - R2)cos8
(8.175)
A detailed solution for the antisymmetric loading condition with a simply
supported boundary is presented in Timoshenko and W oinowsky-Krieger, 19
where the stress couples, the transverse shear forces, and the locations and
magnitudes of the maximum moments are established. Also, the problem of a
clamped circular plate under an eccentric concentrated load and some rather
extensive generalizations thereof are examined in Timoshenko and W oinowskyKrieger19 and in Michell. 20 Among other things, the latter solutions provide an
illustration of the Maxwell-Betti reciprocal theorem for a plate. 21
8.4 Plates of Other Shapes
8.4.1 General Approach: There are a number of solutions for plates with other
than rectangular or circular planform which can be expressed in Cartesian or
polar coordinates. In Cartesian coordinates, solutions for triangular and elliptical plates are found by starting with a deflection function Dz that is proportional
to the equation of the boundary of the plate, which ensures that Dz = 0 on the
boundary. The function may be augmented in order to satisfy either a simply
supported or a clamped condition. In polar coordinates, a sector shape can
readily be treated using the general solution described in section 8.3.2.1. Some
representative illustrations are presented in the following sections. Also, a variety of tabulated results for various shaped plates are contained in Szilard. 16. 18
8.4.2 Triangular Plates: Considering the equilateral triangular plate shown in
figure 8-26, the coordinate system is located with the origin at the centroid of
the triangle. With respect to this coordinate system, the equation of each side
is shown on the figure and the product of the three terms on the l.h.s.,
DZ1
=
_![X 3
3
-
3Xy2 - a(X2
+ y2) + ~a3J
27
(8.176)
gives us a start toward a deflection function, since it will satisfy the condition
Dz = 0 on all boundaries. The function DZl must be modified to satisfy the
equilibrium condition V 4 D z = 0 and the remaining boundary conditions.
For an edge moment M uniformly distributed around the boundary, the
complete solution is
8 Bending of Plates
330
y
~.~y+-'-x-~o=o
v'3 3v'3
o
+----·!--------~----x
2-0=O
y- - 'X+_
./3
3V'3
0/3
o
Fig. 8-26
3M
Dz = ---DZl
4aD
M- [
= -
4aD
Triangular Plate
4 a3 ]
X 3 - 3Xy 2 - a(X2 + y2) + 27
(8.177)
To verify this solution we must show that (a) V 4 D z = 0 on the entire plate; (b)
Dz = 0 on the boundaries; (c) Mx( -a/3, Y) = M·(the other boundaries may
also be checked but this will not be necessary because of the symmetry).
Proceeding, we consider
1. V 4 Dz = 0 throughout: From equations (8.13) we note that any nonzero term
remaining after the application of V2( ) must be at least quadratic in X and
y. No such terms are present in equation (8.177).
2. D z = 0 on boundary: This is obviously satisfied from the derivation of DZl •
3. Mx( -a/3, Y)
=
M: From equation (8.8a)
Mx = -D(Dz,xx + J1.Dz ,yy)
=
M
-D 4aD [6X - 2a
+ J1.( -6X -
M
2a)]
Mx( -a/3, Y) = - 4a[ -2a - 2a
and the solution is verified.
+ J1.(2a -
2a)] = M
331
8.4 Plates of Other Shapes
For a uniformly distributed load qo on a simply supported triangular plate,
the deflection function DZI is modified t0 22
D =
Z
~[X3
64aD
.(~a2 _
3Xy2 - a(X2 + y2) +
~a3J
27
(8.178)
X2 _ Y2)
This solution may be verified in a similar manner to that for equation (8.177).
It is instructive to perform a detailed stress analysis on this plate, following the
general computational format presented in section 8.2.2.3. Among other things,
this will confirm the absence of concentrated corner forces at a non-right angle
corner. This is left to the exercises.
A solution for a simply supported isosceles right triangular plate is also given
by Timoshenko and Woinowsky-Krieger. 22
8.4.3 Elliptical Plates: A clamped elliptical plate is shown in figure 8-27 and
the equation of the boundary is used to write
DZI
X2
y2
= - -2 - - 2+ 1
a
b
(8.179)
For a uniform load qo, this solution may be modified t0 23
DZ2 = C(DZlf
(8.180a)
y
----+--.,-.-x
b
a
Fig. 8-27
Elliptical Plate
332
8 Bending of Plates
where
c=
8D
(3 3+ 2)
a4
+b 4
a2
(8.180b)
b2
A complete stress analysis may be performed in the format of section 8.2.2.3
and is available in Timoshenko and Woinowsky-Krieger.23
8.4.4 Circular Sector Plates: A plate which is a slice of a complete circle may
be solved using a procedure similar to that described in section 8.3.2.1, except
that the displacement function for the harmonics j ;;::: 1 is taken as a Fourier
sine series. The arithmetic is rather involved, and the interested reader is referred
to Szilard. 24
The relatively meager array of available solutions for irregularly shaped
plates points to the tremendous breakthrough made possible when the finite
element technique became available. This method enables almost any form to
be realistically modeled and solved.
8.5 Energy Method Solutions
8.5.1 Strain Energy for Plates in Flexure: Refer to the general expression for
U. as given by equation (7.6). We are presently considering no extensional
strains, and, in accordance with our previous development, we choose to neglect
transverse shearing strains. Therefore, we have only the component III in
equation (7.7) remaining along with the bending thermal term.
U.
=
~ Is {[(K" + Kp)2 -
~(t + Il)(K" +
2(1 - Il)(K"K p -
,2)J
(8.181)
l
Kp) fh 2 T(O( d(} AB dex d{3
-h12
Since we are interested in a displacement formulation, we may substitute the
strain-displacement relationships, as given by equations (5.55), into equation
(8.181). This will be specialized for the two coordinate systems that we have
considered, Cartesian and polar.
For Cartesian coordinates, we take ex = X, {3 = Y, n = Z, and A = B = 1 in
equation (5.55) to get
= -Dz .xx
Ky = -Dz,YY
KX
,=
-Dz,xy
(8.182a)
(8.182b)
(8. 182c)
333
8.5 Energy Method Solutions
Then, substituting equations (8.l82a-c) into equation (8.181), we have
V.
r
D Js {
2
="2
(Dz,xx +
Dz,yy) -
2
2(1 - f.l)[Dz,xxDz,yy - (Dz,xy) ]
+ a(l + f.l)(Dz,xx + Dz,yy)
f
h'2
-h/2
(8.183)
}
T(OC d, dY dX
We may rewrite equation (8.183) as
V. = V. l
+ V. 2 + V,
r {(V
D Js
="2
2
D z ) 2 - 2(1 - f.l)[Dz,xxDz,yy - (Dz,xy) 2 ]
(8.184)
+ a(l + f.l)V 2D z fh '2 T(O' d,}dY dX
-h12
Since the strain energy density dV" which is the integrand of equation (8.184),
is obviously independent of the choice of coordinate axes and since V2 Dz has
been shown to be invariant, the term
(8.185)
also must be invariant. This observation can be useful for transforming V. to
other coordinate systems.
For polar coordinates, we may repeat the preceding calculations with IX = R,
P = 0, n = Z, A = 1, and B = R. However, for variety, we will use the invariant
property of dV. as discussed in the preceding paragraph, since V 2 Dz is already
available in polar coordinates and is given by equation (8.28a) or (8.28b). The
thermal term will be omitted since it is proportional to V 2 Dz . Therefore, we
need only transform the term stated in equation (8.185).
Referring to figure 8-3, we have
R2
= X2+
(8.186a)
y2
X
R ,x
= -R = cos()
(8.186b)
R ,y
Y
. ()
= -R
= SIn
(8.186c)
Also, taking
1
a/ax of both sides of X = R cos (), we find
= cos () R,x -
(R sin ())O,x
from which
o
,x
=
cosOR,x -
Rsin()
1
sin ()
R
using equation (8.l86b). Similarly,
(8.187)
334
8 Bending of Plates
() y = cos(}
,
(8.188)
R
Equations (8.186a-c), (8.187), and (8.188) constitute the basic relations required
to transform equation (8.185) into polar coordinates.
We now evaluate the required derivatives using the chain rule:
Dz,xx
= cos (}(DZ,RRR,x + DZ,R8(},X) + DZ,R( -sin (}) ((), x)
sin ()
+ DZ,88(},X)
- T(Dz,BRR,x
_ D
Z,8
=
+
sin 2 ()
~ DZ,R
R2
2 sin () cos ()
2
cos (}DZ,RR -
R
+
(8.189)
sin (}(R,x»)
(cos 9«(},x)
R
sin 2 ()
+ ~DZ,88
DZ,R8
2 sin () cos ()
R2
DZ,9
Similarly, we compute
• 2
+2
Dz,yy = sm (}DZ,RR
cos 2 ()
+ ~DZ,R -
sin () cos ()
R
DZ,R8
2 sin () cos () D
R2
cos 2 ()
+ ~DZ,88
(8.190)
Z,8
The sum of equations (8.189) and (8.190) checks with equation (8.28a) for V 2 Dz .
Finally, we evaluate
Dz,xy
== cos (}(DZ,RRR,y + DZ,R8(},Y) + DZ,R[ -sin9(O,y)]
sin ()
-T(Dz ,8R R ,y
-D
Z,8
(COS9((},y) _ Sin9(R,y»)
.
-
+
(cos 2 ()
sin () cos () D
R
(cos 2 0 - sin 2 9) D
R2
-
R
sin 2 ())
sin 0 cos 0 D
Z,88 -
R2
(8.191)
R2
R
= sm(}cosODZ,RR
-
+ DZ,88(},Y)
Z,O
DZ,RfJ
Z,R
335
8.5 Energy Method Solutions
We may then substitute equations (8.189)-(8.191) into equation (8.184) to get
the strain energy in polar coordinates. This is rather involved in the general
case, but we can write the axisymmetric expression fairly concisely. We first
evaluate equation (8.185), dropping the O-dependent term. This reduces to
Dz,xxDz,rY - (DZ,Xy)2 = (l/R)Dz,RDz,RR' which, along with the first two terms
in equations (8.28a), gives
Us
=~
Is f[
(Dz'RR
+ kDz,R
r-2(1;-
Jl) DZ,RDz,RR JRdRdO
(8.192)
for equation (8.184) with the thermal term omitted. For the common case of a
solid circular plate with radius a, equation (8.192) becomes
Us = nD
S
a [(
0
DZ,RR
1)2
+ /iDZ,R
-
J
2(1 - Jl)
R
DZ,RDz,RR RdR
(8.193)
8.S.2 Simply Supported Plate Under Concentrated Load: We reconsider the
problem illustrated in figure 8-12 and attempt to confirm the solution using the
virtual work approach derived in section 7.4.1. Our first step is to take the
solution in the form of a Navier-type double series, as given by equation (8.71),
and to evaluate U•. With
L L
00
Dz =
00
j=1 k=l
X
Y
D~ksinjn-sinkn-b
(8.194)
a
we have
Dz,xx
= -
Dz,rY = Dz,xy
=
X.
Y
L L Dfk (jn)2.
smjn-smknb
a
a
(8.195a)
X
Y
L Lco IYzk (kn)2
-b sinjn-sinknb
a
(8. 195b)
00
•
j=1k=1
00
j=1 k=1
L L
00
00
00
j=1 k=1
(kn)
X
Y
~k• (jn)
-b cosjn-cos
knb
a
a
(8.195c)
When we substitute equations (8.195a-c) into equation (8.184), terms of the
L L L L are generated.
00
form
00
00
00
j=1 k=1 j=1 k=1
However, because of the orthogonality relationships, only those terms corresponding to the same j and the same k will remain after integration, and the
double summations over j and k reduce to single summations. Therefore,
omitting the terms which subsequently drop out, we have
(8.196)
336
8 Bending of Plates
Noting that
fo sin
b
2
Y
b
kn-dY = - and
b
2
reduces to
_ n 4 abD
Uel 8
j~ k~l
00
00
jk
fax
sin
2 jn~dX
a
0
a
=-, equation (8.196)
2
2[(1)2 + (~)2J2
(Dz )
a
b
(8.197)
We now consider
Ue2
=
D(1 - /1)
fo fb .L I
a
00
00
0 J=l k=l
•
[(jn)2(kn)2
X
(Dfkf
sin 2 jn~
a
b
a
(8.198)
Since
and
x
foa
sin
a
2 jn~dX
=
fa cos
0
X
a
a
2
2 jn~dX = -
the entire expression for Ue2 vanishes and Ve = Vel, with no thermal effects.
We are now prepared to impart a virtual displacement to the system. Recalling the derivation in section 7.4.1, the virtual displacement is required to
conform to the constraints of the system, which means in this case the boundary
conditions. If the virtual displacement is twice differentiable, the straindisplacement conditions, equations (5.55a-c), will automatically satisfy compatibility. Therefore, it is logical to take the virtual displacement in the form of
equation (8.194).
We proceed for a single general term of the series and choose for the virtual
displacement
(8.199)
where bD~k represents the amplitude of the virtual displacement. Also, note that
the source of the virtual displacement need not be specified.
Referring to figure 8-12, the virtual work done by the external load Qo acting
at X = ~ and Y = 11 is
MJq =
QobDz(~,l1) = QobD~kSinjn~Sinkn~
(8.200)
The corresponding change in strain energy is
bUe = V e, DzbDz
(8.201)
337
8.5 Energy Method Solutions
For U. we take Ulf, the single general term of the series given by equation
(8.197), and evaluate
'k
'k
U.,D z = UI1' Dj1
ab [( ]')2
'k
= -n4D
8~ + (k)2J2
b (2Di)
(8.202)
Substituting equation (8.202) into equation (8.201),
bU.
= n4~bD [(~r + (~rJ ~kbDik
(8.203)
Now, equating equations (8.200) and (8.203) and cancelling the terms b~k, we
find
(8.204)
and the total deflection is given by
Dz
=
~~
'k'
X·kY
~ ~ Di smjn-sm nb
j=l k=l
a
(8.205)
which checks with equation (8.101).
This example serves to illustrate that the principle of virtual work can serve
as an alternate statement of equilibrium.
8.5.3 Clamped Plate Under Uniformly Distributed Loading
8.5.3.1 Approximate Solutions. The solution of a fully clamped plate cannot be
readily accomplished with the Navier-type double series solution and the
solution of the Levy-Nadai procedure is fairly involved. 25 This seems to be an
ideal case for the application of the Rayleigh-Ritz or Galerkin method. Since
the first steps in either method are common, we will proceed with a unified
solution as far as possible. We consider the problem shown in figure 8-28, with
a uniformly distributed qo acting in the positive Z direction. Referring to section
7.4.3, we have here only one generalized displacement D z so that in equations
(7.15) and (7.16), m = 1 and Lll = Dz . We select a single coordinate function
rPz = rPl = (X2 - a2)2(y2 - b 2)2
(8.206)
and the one-term approximation for Dz is
(8.207)
It is obvious that the coordinate function rPl satisfies the stated boundary
conditions: Dz(±a, Y) = Dz,x(±a, Y) = Dz(X, ±b) = Dz,y(X, ±b) = O. However, it is sometimes difticult to select appropriate coordinate functions when
338
8 Bending of Plates
y
b
b
a
a
Fig. 8-28
Clamped Rectangular Plate
more complicated boundary conditions are encountered. The coordinate function selected here is obviously twice differentiable, so that the compatibility
relationships will also be satisfied.
For both the Rayleigh-Ritz and the Galerkin solutions, the Laplacian, V 2 Dz ,
is required. We compute
Dz,xx
= 4c(3X 2 - a 2)(y2 - b 2)2
(8.208a)
Dz,yy
= 4C(X2 - a 2)2(3y2 - b 2)
(8.208b)
Dz,xy
= 16cXY(X 2 - a 2)(y2 - b 2)
(8.208c)
and write
V 2 Dz
= Dz,xx + Dz,yy
= 4c[(3X2 - a 2)(y2 _ b 2)2 + (X2 _ a2)2(3y2 _ b 2 )]
(8.209)
8.5.3.2 Rayleigh-Ritz Solution. We proceed first with the Rayleigh-Ritz solution. Regarding the second term in equation (8.184) as given by equation (8.185),
it has been found to be negligible for plates in which the plan-form is polygonal
and the edges remain straight 26 and, in fact, was shown to vanish entirely for
339
8.5 Energy Method Solutions
the problem considered in the previous section. It may be omitted here as well
and
D fa fb
u. ="2
-a -b (V2 DZ )2 dY dX
(8.210)
where V2 Dz is given by equation (8.209).
The potential energy ofthe applied loads Uq , as defined in equation (7.10), is
given by
Uq = -
f~a f~b qoDz dY dX
(8.211)
where Dz is given by equation (8.207).
We then write the total potential energy as
(8.212)
Now invoking equation (7.17), we set
U,.c
=0
(8.213)
We may simplify the arithmetic by commuting the operations in equation
(8.213) and equation (8.212):
U,.c
=
:e f~a f~b [~
(V 2Dz )2 - qoDz JdY dX
(8.214)
Evaluating d/de of the integrand using equations (8.209) and (8.207),
d
-[ ] =
de
De{4[(3X2 - a 2)(y2 _ b 2)2 + (X2 _ a 2)2(3y2 _ b 2)]y
(8.215)
_ qo(X 2 - a 2)2(y2 _ b 2)2
Then, substituting this expression into equation (8.214) and solving for e, we
have
The integrations are somewhat tedious and are not given here in detail, but the
340
8 Bending of Plates
result is 26
c
~
;;.
{S.20 [1 + ~):J + 2.97 (~)' }
(8.217)
Then Dz(X, Y) is found by substituting c into equation (8.207).
In comparison to more accurate solutions, the results for the central deflection are found to vary from an error of about 5% for a square plate, alb = 1,
to over 30% for alb = 0, representing a long, narrow plate. 26 With such discrepancies on the displacements, the stress couples can be expected to be further
in error and intolerable, since these quantities are obtained from twice differentiating the deflection function. In general, when appraising numerical
solutions, one should compare the most sensitive meaningful quantities, which
are generally those computed by differentiation.
Increased accuracy may be obtained for this problem by adding coordinate
functions. A three-term approximation is
(8.218)
Only even terms are included because of the double symmetry of the problem.
8.5.3.3 Galerkin Solution. The one-term Galerkin solution follows from equations (7.19)-(7.22). We first compute the residual error term R k , defined by
equation (7.21), by substituting equation (8.206) into equation (8.12). Continuing from equations (8.208a-c), we find
Dz,xxxx
= 24c(y2 - b2)2
Dz,yyyy
= 24c(X2 -
(8.219a)
a 2f
(8.219b)
Dz,xxyy = 16c(3X2 - a 2)(3 y2 - b 2 )
(8.219c)
and
Rk
= V4 Dz _ qo
D
=
8c[3(y2 - b 2 )2
+ 4(3X2
- a 2)(3y2 - b 2) + 3(X2 _ a 2)]
(8.220)
Then, the orthogonality condition, equation (7.22a), produces
f~a f~b {8C[3(Y 2 -
b 2)2
+ 4(3X2 -
a 2)(3y2 - b 2) + 3(X2 - a 2 )]
- ~} {(X2 - a 2)2(y2 - b 2)2} dY dX
(8.221)
=0
341
8.5 Energy Method Solutions
from which
qo
I
c = ----------D
III
(8.222)
where I is identical to the numerator of equation (8.216) and
III =
f~a f~b 8[3(y2 + 3(X2
b 2) + 4(3X2 - a 2 )(3y2 - b 2)
(8.223)
- a 2)] (X2 - a 2)2(y2 - b 2)2 dY dX
In order to show that the one-term Rayleigh-Ritz and Galerkin solutions are
identical, we must prove II = III, where II is defined in equation (8.216). We
rewrite both II and III in terms of the coordinate function rPz. In view of
equations (8.208a) and (8.208b),
II =
f~a f~b (rPz,xx + rPz, yy)2 dY dX
= f~a
f~b [(rPZ,XX)2 + 2rPz,xxrPz,yy + (rPZ,yy)2] dX dY
(8.224)
Also, by comparison with equations (8.219),
III =
f~a f~b (rPz,xxxx + 2rPz,xxyy + rPz,yyyy)rPz dY dX
(8.225)
We now integrate equation (8.224) by parts. For the first term, we have
f fb
a
-a
-b
rPz,xxrPz,xx dY dX
=
fb
-b
rPz,xxrPz,x
- fa fb
LdY
a
(8.226)
rPz,xrPz,xxx dY dX
Integrating by parts again, we have
- f~a f~b rPz,xrPz,xxx dY dX
=
f~b rPz,xxxrPz LdY
+
fa f~b
(8.227)
rPzrPz,xxxx dY dX
If the boundary terms at X = ± a drop out, we have reduced the first term of
equation (8.224) to that of (8.225). Considering the physical boundary conditions discussed in section 8.1.6, we see that
a
rPz,xxrPz,x I = 0
-a
342
8 Bending of Plates
implies that either the moment or rotation is zero on the boundary. Similarly,
for
a
rPZ,xxxrPz
I =0
-a
either the transverse shear or the deflection must vanish. For the clamped plate,
the rotation and deflection are zero on X = ± a so that the boundary terms
drop out and the first terms of II and III are identical. Similar integrations by
parts reduce the second and third terms of II to the corresponding terms of III,
so that the one-term Rayleigh-Ritz and Galerkin solutions are identical.
8.6 Extensions of the Theory of Plates
8.6.1 Variable Flexural Rigidity: Consider a simple extension of plate theory,
where the flexural rigidity D, as defined by equation (S.4d), is generalized to
D(rx, f3). Then, the plate equation may be reformulated by taking D = D(rx, f3) in
equations (S.4a-c) and following the subsequent steps outlined in section S.1.1.
To illustrate for Cartesian coordinates, equations (S.Sa-c) are written as
+ jJ.Dz,yy)
My = -D(X, Y)(Dz,yy + jJ.Dz,xx)
Mx
=
Mxy
(S.22Sa)
-D(X, Y)(Dz,xx
(8.22Sb)
= Myx = -D(X, Y)(1 - jJ.)Dz,XY
(S.22Sc)
and equations (S.9a) and (S.9b) generalize to
Qy
=
-[D(X, Y)(Dz,yy
+ (1
Qx
=
(S.229a)
- jJ.) [D(X, Y)Dz,xy Ix
-[D(X, Y)(Dz,xx
+ (1
+ jJ.Dz,xx)]'y
+ jJ.Dz,yy)],x
(8.229b)
- jJ.)[D(X, Y)Dz,xrJ,y
whereupon the equilibrium equation (S.l1) becomes
[D(X, Y)(Dz,xx
+ jJ.Dz,yy)],xx + 2(1 - jJ.)[D(X, Y)Dz,xyIxy
+ [D(X, Y)(Dz,yy + jJ.Dz,xx)]'yy =
qz
(S.230)
Equation (S.230) is used to analyze rectangular plates of variable thickness
using the Levy approach in Timoshenko and Woinowsky-Krieger 27 and in
Conway.28
Axisymmetrical circular plates with an axisymmetric thickness variation are
of importance in machine part design. The appropriate governing equation is
derived in an analogous manner as for the rectangular plate,29,3o and possible
solution schemes and examples are provided in those references.
In practical applications, solutions for plates of variable thickness are often
conducted using finite difference or finite element procedures.
343
8.6 Extensions of the Theory of Plates
8.6.2 Specifically Orthotropic Plates: Another direct extension of the theory of
plates is the accommodation of specifically orthotropic properties, as defined in
section 6.1.2. In this case, the material properties are given by [Cor]' equation
(6.11), and specifically by the fourth, fifth, and sixth rows and columns of [Dor],
matrix 6-3. In Cartesian coordinates, the generalized moment-curvature expressions are found from equation (6.10) by replacing rows and columns 4 through
6 in matrix 6-2 with the corresponding elements of [D or ]; by taking X, Y, Z for
0(, {3, and n; and then by using equations (8.3) for the curvatures K x , K y, and -r:
=
+ Dor45DZ,YY)
My = -(Dor55Dz,yy + Dor54Dz,xx)
Mx
(8.231a)
-(Dor44Dz,xx
(8.231b)
(8.231c)
Mxy = Myx = -Dor66DZ,XY
The elements of [D or ]' DOrij ' are presumed to be known, as discussed in section
6.1.2, and Dor45 = Dor54 '
If we take each Dorij as constant, we may write the equilibrium equation by
first substituting equation (8.231) into equation (8.9a) and (8.9b) to get
Qy
Qx
+ Dor45DZ,Xxy + Dor66DZ,xxy)
= -(Dor44Dz,xxx + Dor45DZ,XYy + Dor66DZ,XYY)
=
(8.232a)
-(Dor55DZ,yyy
(8.232b)
and then introducing equations (8.232a) and (8.232b) into equation (8.10), which
becomes
+ qz = 0
(8.233)
To verify the consistency of this derivation, we may check the isotropic case,
From matrix 6-2, Dor44 = Dor55 = D, Dor45 = JlD, and Dor66 = D(1 - Jl). Therefore, Dor45 + Dor66 = D and equation (8.233) reduces to equation (8.11).
Equation (8.233) is of interest in the study of stiffened plates. In this application, the term Dor45 may be neglected, as discussed in section 6.1.3, and the terms
Dor44 , Dor55 , and Dor66 are taken as the corresponding elements of [Deq],
equation (6.23). Other methods of finding the material constants for specific
configurations are discussed in Timoshenko and Woinowsky-Krieger. 31
For a rectangular plate simply supported on all sides, equation (8.233) may
be solved using the Navier approach, following section 8.2.2. Starting with
equation (8.71) for Dz , the l.h.s. of equation (8.233) becomes
(i1t)2(k1t)2
j~ k~l [(i1t)4
Dor44 -;; + 2(Dor45 + Dor66 ) -;;
b
00
00
(8.234)
( k1t)4]
+ Dor55 b
X.
Y
D~ SlDi 1t-;;SlDk1t b
'k'
With the r.h.s expanded in a Fourier series as given in equation (8.73) and qik
344
8 Bending of Plates
evaluated from the specified load distribution qz(X, Y) by equation (8.75), we
get
whereupon Dz is found by entering equation (8.235) into equation (8.71):
D z ( X, Y ) =
~~
jft
kf-l
'k'
X·kY
Di sm j 7t-;;-sm 7tb
(8.236)
Thus, available solutions for simply supported isotropic plates are easily extended to include orthotropic plates, once the material properties are defined.
8.6.3 Multilayered Plates: Another form of anisotropy which is of considerable
practical importance is the multilayered plate, which may be composed of two
or more bonded layers of isotropic or anisotropic materials. The simplest
method of dealing with plates composed of isotropic layers is to use a modified
form of the basic plate equation, equation (8.12):
V 4 Dz = qz
Dr
(8.237)
'where Dr is a transformed flexural rigidity which is computed from the basic
properties, E and fl, of the individuallayers. 32 • 33
One type of layered plate is called a sandwich plate and is composed of at
least three plies. The outer layers, or skin, are usually relatively thin, but of high
strength, and resist the flexural and twisting moments by dev~loping couples of
opposing in-plane forces; the inner core transmits the shear stresses between
the outer layers. This behavior is similar to that of an H-shaped beam, where
the two outer layers would represent the flanges and the inner fayer, the web.
The analysis of this type of plate may be based on a large deflection theory
developed by E. Reissner. 34
8.6.4 Inclusion of Transverse Shearing Deformations: Preceding the development ofthe Kirchhoff boundary conditions, section 6.2.3, we saw that the order
of governing equations derived by neglecting transverse shearing deformations
permits only two of the three obvious force conditions to be enforced at a free
edge. Also, the presence of concentrated corner forces in rectangular plates is
attributable to the suppression of the transverse shearing deformations. For
thin plates, these shortcomings appear to be largely academic and the elementary plate theory is adequate; however, as a plate becomes relatively thicker,
transverse shearing effects may be more important. Moreover, eliminating the
transverse shearing strain permits the in-plane displacements to be written
8.6 Extensions of the Theory of Plates
345
directly in terms of the normal displacement [see equations (8.15) and (8.17)]
and leads to much simplified governing equations.
Although the contradictions incorporated in elementary plate theory have
been evident since the time of Kirchhoff, a satisfactory alternative which includes transverse shearing deformations appeared relatively recently (1944) and
is attributed to E. Reissner, 35,36 with some significant embellishments by A.
Green. 37 More recently, refined plate theories have been classified into first
and higher order shearing deformation theories,38 which carry the respective
names of Hencky-Mindlin and Kromm-Reddy. The distinction is drawn
because the first order theory (commonly known as Mindlin plate theory) does
not satisfy the shear stress-free conditions on the surfaces ± hj2. This is easily
seen by referring to the last two equations of matrix 6-1, where the shear stresses
are proportional to the shearing strains which are constant through the thickness and, hence, do not necessarily vanish at the surfaces. The higher order
theories attempt to correct this shortcoming by including thickness-dependent
factors, at the expense of adding unknowns into the equations. As discussed in
section 6.1.1.2, the deficiency is minor in the context of elementary plate and
shell theory. It has also been observed that the effect of shearing deformations
is more pronounced in orthotropic than in isotropic plates. 38 The interested
reader is referred to Timoshenko and Woinowski-Kreiger 39 and to Reddy38
for some examples of solutions including transverse shearing deformations.
Although the inclusion of transverse shearing deformations complicates the
problem considerably for a differential equation formulation, the incorporation
of these effects is relatively easy in an energy-based approach. If we consider
equations (7.6) and (7.7), we see that the linear component Ehj[2(1 - Jl2)] [I]
contains the transverse shearing strains. Thus, we may add
Eh
4(1
+ Jl)
r
2
Js (Ya
2
+ Yp)ABdad/3
(8.238)
to equation (8.181) to get the expanded version of UB :
(8.239)
When we proceed with the displacement formulation, we substitute the
general strain-displacement relationship equations (5.54a-h) into equation
(8.239). For Cartesian coordinates, we have
KX
= Dxy,x
(8.240a)
Ky
=
Dyx,y
(8.240b)
r
=
t(Dyx,x
+ Dxy,y)
(8.240c)
346
8 Bending of Plates
+ Dxy
Dz,y + Dyx
Yx = Dz,x
(8.240d)
yy =
(8.240e)
and equation (8.239) becomes
U. =
Is {~
[(Dxy,x
+ DYX ,y)2
- 2(1 - J.L)[Dxy,xDyx,y - i(Dyx,x
Eh
+ DXy,y)2]
2
+ 4(1 + J.L)[(Dz,x + Dxy) + (Dz,y + Dyx)
- ~(1
+ J.L)(Dxy,x + Dyx,y)
f
h' 2
2
(8.241)
]
}
T(oe de dX dY
-h12
A similar specialization is easily accomplished for polar coordinates. This is
left as an exercise.
In anticipation of the use of equation (8.241) in a Rayleigh-Ritz type of
solution, it is instructive to compare this generalized form with the version
where transverse shearing strains are neglected, equation (8.183). Referring to
the procedure described in section 7.4.3, we see that the index m in equations
(7.15) and (7.16) will be equal to 3 instead of 1, corresponding to the generalized
displacements
(8.242)
This means that there will be three times as many coordinate functions and
considerably more numerical calculations, but this is not foreboding in the
computer age.
There is another, more subtle, difference between the energy expressions
which can be important from a computational standpoint. In equation (8.241),
the generalized displacements are present only up to the first derivative, whereas
in the earlier form, equation (8.183), there are second derivatives of the dependent variables. The order of the highest derivative appearing in the strain energy
functional dictates the minimum continuity required at junctions of finite
elements. 4o The lower order continuity necessary for a functional based on
equation (8.241) rather than equation (8.183) somewhat compensates for the
increased number of generalized displacements and makes the formulation
including transverse shearing deformations attractive. The relative ease and'
efficiency of including these deformations in an energy-based formulation for
shells of revolution are demonstrated in Brombolich and Gould. 41 The necessity for employing precautions to avoid overstiffening by "shear locking" was
noted in section 5.3.2.
8.6.5 Folded Plates: Plates are basically shallow flexural members and are
somewhat inefficient in flexural action. An appealing procedure to increase the
347
8.6 Extensions of the Theory of Plates
Fig. 8-29
Folded Plate
flexural rigidity of a given plate is to introduce undulations or folds in one, or
possibly more than one, direction. This serves to increase the section modulus
in these directions markedly. Of course, because there is no longer a flat surface,
this procedure is impractical for many situations.
A widely used application of this concept is the folded plate, which is primarily used for roofs over large, column-free areas. One such structure, partially
designed by the author, is shown in figure 2-8(x).
A typical configuration for a folded plate roof is shown in figure 8-29.
Basically, this structure is a relatively wide beam with a saw-tooth cross section
of depth H spanning the distance L from support to support. The idealized
beam behavior is violated by distortions of the cross section, which make
elementary beam theory, alone, inapplicable.
The load-carrying mechanism for folded plates may be conveniently visualized in two parts. The surface loading is resolved into in-plane and transverse
components. Then, the transverse surface loads are resisted by one-way plate
bending over a span D. The reactions produced by the transverse loads on the
plates are applied at the ridge lines, which act as supports for the plates. Near
the ends, some of the loading is directly transferred to the end supports, but
since H « L, the plates are basically one-way and most of the load is carried
to the ridge lines. This is called plate action.
The ridge lines are subjected to the plate reactions, which are resolved in
oblique coordinates to act in the planes of the intersecting plates at each joint.
Along with the in-plane component of the surface load on each plate, these
348
8 Bending of Plates
forces are resisted by the flexural action of the plate acting as a beam of width
h, and depth D and span L. This is termed diaphragm action.
The end blocks are generally solid infills or stiff frames, rigid in the vertical
plane but flexible in the longitudinal direction, and a supporting structure, such
as a wall or a line of columns. Away from the ends, the ridge lines deflect in
accordance with the diaphragm action of the plates acting as beams with span
L, width h, and depth D. Thus, an interaction occurs between the plate and the
diaphragm behavior along the ridge lines, since the supports for the plate action
are not unyielding but elastic. This conceptual model is the basis for many of
the folded plate theories used in engineering design. Because these design
methods are generally developed in term~ of planar structural analysis theory
rather than in terms of the theory of plates, they are not treated here. The
interested reader is referred to Yitzhaki 42 and to Simpson. 43
The governing equations for folded plates may be written in the context of
the theory of plates by combining the equations describing the in-plane forces
and corresponding displacements with the basic plate equation, equation (8.12).
To do this, we take local Cartesian coordinates such that X and Yare in the
plane of each plate and write equations (3.25a) and (3.25b) as
Nx,x
+ Nxy,y + qx =
0
(8.243a)
+ Ny,y + qy
0
(8.243b)
Nxy,x
=
Next, we replace the stress resultants by the strains using matrix 6-2, without
thermal effects.
Eh
[
1 _ J-l2 (ex
- J-l w ]
+ J-ley).x + -1 2
- ,y + qx
[1 -
=
0
(8.244)
Eh
J-l
1 _ J-l2 - 2 - w ,x
]
+ (ey + J-lex),y + qy =
0
Finally, we express the strains in terms of the displacements using equations
(5.54a-c)
(8.245a)
Eh
[1 -
J-l
1 _ J-l2 -2-(D y ,x
+ Dx,y),x + (Dy,y + J-lDx,x),y ] + qy = 0
(8.245b)
Equations (8.245a) and (S.245b) together with equation (8.12),
(8.246)
form the governing equations for the so-called exact theory of folded plates. 44
Solutions using this theory are presented by several authors,45-48 and a critical
evaluation of various solution procedures is available. 49
349
8.7 Instability and Finite Deformation
8.7 Instability and Finite Deformation
8.7.1 Modification of Equilibrium Equations: We now direct our interest to the
analysis of plates which are subjected to forces acting in the middle plane, along
with the transverse loading. If these middle plane forces are compressive in at
least one direction, plate instability is a possibility. Further, if there are no
transverse forces but only in-plane compressive forces acting, we have the
two-dimensional analogue of the classical Euler column buckling problem. In
order to show this, we must relax our basic assumption 11], which enabled the
equilibrium equations to be formulated with respect to the undeformed middle
surface, and now include the effect of the in-plane forces on the equilibrium in
the normal direction.
Refer to figure 3-2(a) and consider a section of the middle surface along an
Sa coordinate line with length ds a. We show two such sections on figure 8-30,
both before and after deformation. Figure 8-30(a) is for Na and Figure 8-30(b)
pertains to N pa . The surface rotation Dap is denoted in accordance with figure
5-2 and may be expressed in terms of Dn• a by equation (8.7a) if transverse
shearing strains are neglected. Note that figure 8-30 corresponds to the positive
sense of Dap and the negative sense of Dn.a, since the ( + )ta direction is opposite
to that shown on figure 5-2(a).
We now reexamine the stress resultant vectors Fa and F p, as defined in
equations (3.11a) and (3.11b). First, from figure 8-30(a), we see that the stress
resultant Na has a normal component - Na sin Dap ~ - NaDap which adds a
force - NaDapt n ds p to Fa in equation (3.11a). The negative sign is in accordance
with the sign convention defined on figure 3-2(a). Next, from figure 8-30(b), we
have the stress resultant Npa with a normal component - NpaDaP which contributes a force - NpaDaptn dS a to Fp in equation (3.11b). Similar sections along an
sp coordinate line show the forces - Nap Dpatn ds p adding to Fa in equation (3.11a)
and - NpDpatn dS a going to Fp in equation (3.11b). Since these additional normal
forces associate only with Qa and Qp, respectively, as coefficients oftn in the two
equations, they are easily traced through to equations (3.25a-c), the scalar force
equilibrium equations for a plate. The in-plane equations are unaffected, and
the generalization of equation (3.25c) is
[B(Qa - NaDap - NapDp.)J.a
+ [A(Qp
- NpaDap - NpDpa)J.p
+ qnAB =
0
(8.247)
It is convenient first to expand equation (8.247) and then eliminate certain
terms by introducing the in-plane equilibrium equations, (3.25a) and (3.25b).
We will proceed for the two coordinate systems which are of interest in the
theory of plates, Cartesian and polar.
For Cartesian coordinates, 0( = X, P = Y, n = Z, and A = B = 1. Then,
equations (3.25a) and (3.25b) become
(8.248a)
350
8 Bending of Plates
Original Plate
--~~------------- Sa
~--~--~--~~~~--~
Fig.8-30
Nxy,x
In-Plane Forces in a Plate
+ Ny,y + qy = 0
(8.248b)
and equation (8.247) is written as
Qx,x
+ Qy,y -
Dxy(Nx,x
+ Nyx,y) -
Dyx(Nxy,x
+ Ny,y)
- NxDxy,x - NyDyx,y - NyxDxy,y - NxyDyx,x
+ qz = 0
(8.249)
Substituting equations (8.248a) and (8.248b) into equation (8.249) and setting
351
8.7 Instability and Finite Deformation
N xy
=
N yX' we have
Qx,x
+ Qy,y -
+ Dxy,y)
+ Dxyqx + Dyxqy + qz = 0
NxDxy,x - NyDyx,y - Nxy(Dyx,x
(8.250)
A more familar form of equation (8.250), applicable for transverse loading
only, is found by letting qx = qy = 0 and also by suppressing transverse shearing
strains, allowing
(8.251)
to be introduced from equations (8.7). Then, substituting equations (8.10)-(8.12)
into equation (8.250), we have
(8.252)
which becomes the governing equation for the deflection of a plate in the
presence of lateral forces.
In polar coordinates, oc = R, f3 = 8, and n = Z along with A = 1 and B = R.
Equations (3.25a) and (3.25b) become
+ NOR,o - No + qRR = 0
(RNRO),R + No,o + NOR + qoR = 0
(8.253a)
(RNR),R
(8.253b)
and equation (8.247) expands to
QR
+ RQR,R + Qo,o -
DRI!(NR + RNR,R
+ NOR,I!)
D9R(NRI! + RNRI!,R + NI!,o) -
(8.254)
NRRDRI!,R
- NI!DI!R,I! - NRI!RDI!R,R - NORDRI!,I!
+ qzR = 0
Substituting equations (8.253a) and (8.253b) into equation (8.254) and taking
N RIJ = N IJR , we have
+
NRIJ(DI!R - RDIJR,R - DRO,I!)
Now, we let qR
+ qRDRIJR + qlJDoRR + qzR =
0
(8.255)
= qlJ = 0 and suppress the transverse shearing strains. From
equations (8.30), this gives DRIJ
= -
DZ,R and DIJR
= -
1
R Dz,I!' Introducing
equations (8.24)-(8.27) into equation (8.255), we get
NIJ( DZ,R
DV 4 Dz - NRDz,RR - R
+ R1 Dz,lJo)
2NRI!
(1
)
+~ RDz,I!-Dz,RIJ
for plates with transverse loading only.
(8.256)
=qz
352
8 Bending of Plates
For the axisymmetric case, equation (8.256) reduces to
4
DVaDZ
-
NIJ
NRDz,RR - RDz,R
=
(8.257)
qz
where V:( ) = the axisymmetric biharmonic operator defined in equations
(8.33) or, more conveniently, in equation (8.116). Thus, we establish equations
(8.252), (8.256), and (8.257) as the expanded forms of equations (8.12), (8.27), and
(8.33b), respectively, when basic assumption
is relaxed.
An example using a modified form of equation (8.257) is solved by Timoshenko
and Woinowsky-Krieger. 5o Some classical problems in Cartesian coordinates
are treated in the subsequent sections. Also, more complex cases including shear
loading, orthotropic plates, sandwich plates, and clamped boundaries are investigated by Brush and Almroth. 51
rn
8.7.2 Modification of Strain Energy: The expression for strain energy, as given
by equations (7.6) and (7.7), must be supplemented when coupling of the normal displacements and the in-plane strains is included, since nonlinear strain
terms are required. We refer to the basic description of deformation as discussed in sections 5.2 and 5.3. However, we will not attempt to develop a
complete nonlinear theory, but only to retain those higher order terms containing Dn. This corresponds to a modified finite deformation theory, in which
the in-plane displacements remain small but the rotations are regarded as
moderate.
First consider equations (5.13) and (5.14) for A' and B'. Noting from equations
(5.10c) and (5.12c), respectively, that "'" and
contain Dn terms, we have
A' = A [ (1
/
"'p
J1 2
1
+ e,,)2 + A2
(Dn ,,,)2
(8.258)
and
(8.259)
Using the binomial theorem, we may simplify these expressions to
A' = A[1
and
B'
=
B
+ e" + 2~2(Dn,,,)2J
[1 +
ep
+
2~2 (Dn.p)2 J
(8.260)
(8.261)
We now return to the definition of the middle surface strains in section 5.3.1.
We denote the modified strains as e", ep, and ro, respectively, and retain the basic
definitions of these components of strain. Then, equation (5.25) becomes
353
8.7 Instability and Finite Deformation
(8.262)
in view of equation (8.260). Correspondingly, equation (8.261) generalizes to
_
6p
=
6p
+"21 (D
13)2
n• p
(8.263)
For the shearing strain, we note the product 1jI"ljIp in equation (5.27), which
previously had been neglected. Referring to equations (5.lOc) and (5.12c) and
retaining only the Dn terms,
Therefore, from equation (5.27), we find
(8.264)
In order to obtain a modified expression for the strain energy, refer to the
basic expression, equation (7.4). Assume that the strain energy due to bending
is not changed by the axial forces and remains the cubic component of equation
(7.7), (D/2) [III]. Also assume that the in-plane stress resultants are due entirely
to applied edge loading in the plane of the plate, in which case they are
unchanged during bending. This implies that the external and internal work
done by these constant stress resultants acting through the corresponding
external and internal in-plane displacements will cancel in the energy balance
when a virtual transverse displacement is introduced. This may be formally
substantiated by rather involved arguments,50 which are not repeated here.
Therefore, the additional strain energy V., is due entirely to the straining of the
middle surface as a result of the bending. With these assumptions, we can write
V.
=
Is
[N"(B,, - 6,,)
+ Np(Bp -
6
p) + N"p(ro - w)]ABdexdf3
(8.265)
Note that there is no 1/2 coefficient in equation (8.265), since the in-plane stress
resultants are already acting when the additional middle surface strains occur.
Substituting equations (8.262)-(8.264) into equation (8.265) gives
r
V. = ~ Is [N,,(D~"r + Np(Di
+ 2N"p (D~,,) (DiP)] AB dex df3
p
(8.266)
and the total strain energy for plate bending in the presence of constant in-plane
forces is
354
8 Bending of Plates
U.
+
U.
=
U.+S
D
= 2[III] +
U.
(8.267)
where III is defined in equations (7.6) and (7.7).
For Cartesian coordinates, equation (8.266) becomes
U. =
r
2"1 Js [Nx(Dz,x) 2 + Ny(Dz,y) 2 + 2Nxy Dz ,x Dz,y] dX dY
(8.268)
and U. is given by equation (8.184); for polar coordinates,
1
U. = 2"
Jsr [NR(Dz,R) + Ne (Dz,e)2
R
+ 2NReDz ,R (Dz,
R e) RdRdfJ]
2
(8.269)
For axisymmetric loading on a solid plate of radius a, equation (8.269) simplifies
to
U. =
1t
f:
NR(Dz ,R)2RdR
(8.270)
and U. is given by equation (8.193).
8.7.3 Simply Supported Rectangular Plate Under Transverse and Unidirectional
In-Plane Loadings: Consider the rectangular plate shown in figure 8-9 with the
addition of constant in-plane loads Nx(O, Y) and Nx(a, Y) along the two boundaries parallel to the Y axis, as shown on the inset in figure 8-31.
We choose the differential equation approach from section 8.7.1, whereby
equation (8.252) becomes
DV 4 Dz - NxDz,xx - qz
=0
(8.271)
Following the procedure used in section 8.2.2.1 with Dz given by equation (8.71)
and qz by equation (8.73), we arrive at the equation for specific harmonics j
andk
(8.272)
from which the Fourier coefficient is
(8.273)
As before, the Fourier coefficient for the applied surface loading, qik, is evaluated
from equation (8.75). The total plate deflection is then computed from equation
(8.71).
If the in-plane load is tensile, Dik, and hence the deflection, is reduced from
the case with no in-plane forces; however, if Nx is compressive, the deflection
355
8.7 Instability and Finite Deformation
.01.0
........,
8
-II·.... 6
+
....
.010
~
'----'
4
2
o
~
______ __ ___ __
~
~
~
~
- L_ _~_ _ _ _ L
3&
__~____~~_
4./20
alb
Fig. 8-31
Lower Bound for Critical Load in Rectangular Plate
is increased. In fact, there are critical values of N x , one for each of the specific
harmonicsj = Jand k = k, for which the denominator of { } in equation (8.273)
vanishes and the deflection becomes infinite:
NYcr
=
-n 2
D(f)T(fY + (~Yr
(8.274)
The N XCT are the elastic buckling loads for a uniaxially compressed, simple
supported rectangular plate and are unaffected by the transverse load qz.
It is evident from equation (8.274) that Nxcr is quite similar in form to the
Euler buckling formula for columns. It includes the term n 2 , the term D =
Eh 3 /[12(1 - J.l2)], which is equivalent to the EI term in the column formula,
and the length terms a and b. However, the length terms are not only in the
denominator; rather, the dimension in the direction of loading, a, appears also
in the numerator. 1\lso note that the J term, corresponding to the harmonic
numbers of the buckling modes in the direction of loading, occurs in both the
numerator and denominator. The value of Jwhich yields the lowest critical load
is not obvious from equation (8.274). It is quite clear, however, that k = 1 will
356
8 Bending of Plates
Grr
give the lowest value of NXer> so that we may write
NXer!
=
-n2D(7Y[(~r +
(8.275)
which may be conveniently rearranged as
-
N Xer1 =
-n 2
D[-ba+ }t;
1 aJ2
-,:;z-
(8.276)
j
where alb is the aspect ratio.
Now, the problem is to determine which value of] will produce the lowest
Ncr
for a given aspect ratio. This may be done by plotting the term
[]~+T ~
r
against the aspect ratio alb for various integer values of J, as shown in figure
8-31. It may be observed from this figure that the minimum value
[ j-b
a
aJ2
1
+ =j b
~
(8.277)
4
is a close lower bound for most cases, so that
N Xcr1 ~
-4n 2 D
b2
(8.278)
Of course, the number of waves in the buckled shape will still depend on alb.
The deflected shape for] = 2, corresponding to .j2 ::s; alb ::s;
is shown on
the inset.
In contrast to column buckling, note that the lowest critical load is practically
independent of the length ofthe member in the direction of the loading. Rather,
only the length of the loaded edge, which is a quantity not present in the column
problem, is significant.
It should be noted that the single series Levy-Nadai approach, introduced
in section 8.2.3, is effective for plate instability investigations as well. The latter
is the logical alternative to the Navier solution when boundary conditions
other than all sides simply supported are encountered. The single series solution
is applied in Brush and Almroth,51 and an application is suggested in the
exercises.
J6,
8.7.4 Simply Supported Rectangular Plate Under Transverse and Bidirectional
In-Plane Loadings: Now consider the rectangular plate with a concentrated
load Qo(~, 11), as shown in figure 8-12, with the addition of constant in-plane
loadings Ny(X,O) and Ny(X, b) along the boundaries parallel to the X axis, as
well as the Nx loads introduced in the preceding section. Nxy is taken as zero.
We use the virtual work method developed in section 8.5.2, with Dz given by
equation (8.194) and V. by equation (8.197). The first step is to evaluate V., as
357
8.7 Instability and Finite Deformation
given by equation (8.196). Following the procedure detailed in section 8.5.2, the
additional strain energy due to the in-plane forces is
(8.279)
Vii
2
= -8- jf; k~l (D~kf
n ab
00
00
[_
(j)2 + Ny- (k)2]
b
Nx ~
(8.280)
Adding equation (8.280) to equation (8.197) gives the total strain energy U'+ii.
Next, we take for U. H the single general term of the series, vl!ii> and find
JV.+ ii . From equation (8.280), we have
(8.281)
and adding equation (8.281) to equation (8.203), we get
(8.282)
The principle of virtual work is applied by equating equations (8.200) and
(8.282), from which we find
Then, the total deflection may be evaluated by equation (8.205).
We see that if Nx and Ny are both tensile, the deflection is reduced from the
case where no in-plane forces are acting. If Nx and/or Ny are compressive, we
may have instability indicated by the denominator -+ 0, whereupon
(8.284)
Equation (8.284) can lead to several classes of buckling problems, such as (a)
are proportional, i.e., Nx = INcr and Ny = gN"" where I and g are
specified constants which permit Ncr to be evaluated; and, (b) Nx (or Ny) is a
fixed value. Correspondingly, Nycr (or Nxcr ) can be computed. As an example,
Nx and Ny
358
8 Bending of Plates
when Ny = 0 we have an identical situation to that discussed in the previous
section and covered by equation (8.274). Note also that buckling may occur
even if one of the in-plane forces is tensile, although it retards the instability.
As a simple example of buckling under bidirectional in-plane loading, consider a square plate with a = band Nx = Ny. Equation (8.284) reduces to
_
n2D _
Ncr = -V(j2
_
+ k 2)
(8.285)
Obviously the lowest critical load is found for J = k = 1 as
_
2n 2 D
Ncr = -~
(8.286)
This indicates aO single wave in each direction.
Another simple illustration for the square plate is provided by taking
gNx . Then, we have
_
N Xcr
n2D (p + p)2
= -V (j2 + gk 2 )
Ny =
(8.287)
NXcr1 will again correspond to J = k = 1, so that
N
-4 n 2 D
---(1 + g) b 2
Xcr! -
(8.288)
The effect of unidirectional tension on the retardation of buckling is demonstrated by considering a negative value of 9 in equation (8.287). In many
practical cases of unidirectional compression, the boundaries in the other
direction, parallel to the direction ofloading, are restrained in the middle plane.
As the plate then deflects, these boundaries develop tensile forces in the middle
plane which oppose the onset of instability.
8.7.5 Finite Deformation of Plates: If we consider the modified expressions for
the middle surface strains, equations (8.262)-(8.264), and restrict ourselves for
the moment to Cartesian coordinates, we have
(8.289)
in view of equations (5.54a-c). Equations (8.289) may be combined into a single
compatibility equation in terms of Dz and the strains.
We form certain second partials, assuming sufficient continuity so that the
order of differentiaton can be altered:
359
8.7 Instability and Finite Deformation
=
By,xx =
ro,XY =
BX,yy
+ (DZ ,xy)2 + Dz,xDz,XYy
(a)
Dy,xxy + (Dz ,xy)2 + Dz,yDz,xxy
(b)
+ Dx,xyy + (Dz ,xy)2 + Dz,xxDz,yy
+ Dz,xDz,XYY + Dz,yDz,xxy
(c)
Dx,xyy
Dy,xxy
(8.290)
Adding equations (8.290a) and (8.290b) and subtracting (8.290c), we obtain
(8.291)
We next eliminate the strains in favor of the stress resultants. From matrix
6-2, in the absence of thermal terms,
NX =1
Ny
=
1
Eh
-J-L
Eh
-J-L
2(BX + J-LBy)
(a)
+ J-LBx)
(b)
2
(By
Eh
Nxy = 2(1
+ J-L) OJ
(8.292)
(c)
which may be inverted to
1
Bx
= Eh (Nx
By
= Eh (Ny
OJ
=
1
- J-LNy )
(a)
- J-LNx )
(b)
2(1 + J-L)
Eh Nxy
(8.293)
(c)
Now, replacing Bx, By, and OJ by Bx , By and ro in equation (8.293a-c) and
introducing this equation into equation (8.291), we have
1
Eh [Nx,yy
+ Ny,xx
- J-L(Nx,xx
+ Ny,yy)
- 2(1
+ J-L)Nxy,xyJ
= (Dz ,xy)2 - Dz,xxDz,yy
(8.294)
Equation (8.294) and equation (8.252) constitute the compatibility and equilibrium equations, respectively.
There are still four unknowns remaining in the two equations, so that further
refinement is necessary.
If a stress function ~ is defined such that
N x = /F,yy
Ny
=
/F,xx
(8.295a)
(8.295b)
360
8 Bending of Plates
(8.295c)
and equations (8.295a-c) are introduced into equations (8.294) and (8.252), we
obtain
(8.296)
and
(8.297)
which are known as von Karman equations for the large deflection of plates,
after the famous contemporary mechanician.
The von Karman equations are coupled and nonlinear. The nonlinearity
arises from the relaxation of assumption [lJ, and the enforcement of this
assumption immediately reduces equation (8.297) to the equation of the linear
theory, equation (8.12). Also, the equations are written in invariant form and
thus may be readily transformed to other coordinate systems. Y. C. Fung 52
observed that the r.h.s. of equation (8.296) is related to the Gaussian curvature
of the deformed surface. From equation (2.38), we confirm that (D z ,Xy)2 Dz,xxDz, yy = b, the discriminant of the deformed surface, so that if the plate is
bent into a developable surface (zero Gaussian curvature) such as a cylinder,
the r.h.s. of equation (8.296) vanishes.
We may investigate this in more detail by referring to the cylindrically bent
plate shown on figure 8-7, which was treated in section 8.2.1.3. Because of the
single curvature, Dz,xx = Dz , yX = 0 and the r.h.s. of equation (8.297) reduces to
1
Dz,yyyy = D (qz
+ NyDz,yy)
(8.298)
D z , yy is left in general form rather than being evaluated from equation (8.66),
since the presence of Ny will modify D z somewhat. Equation (8.298) is a fourth
order ordinary linear differential equation with a constant coefficient, which is
readily solved by classical methods. The homogeneous solution is written as
(8.299a)
where
(8.299b)
with the particular solution depending on the form of qz.
For example, if qz = qo = constant,
(8.300)
361
8.7 Instability and Finite Deformation
With Ny initially specified, this problem is analogous to an axially loaded
beam and may be investigated further for various boundary conditions along
the Y axis. However, when the boundaries parallel to the X axis are restrained
in the middle plane, Ny is initially unknown. Referring to the cylindrical cross
section in figure 8-7, this force may be determined from the compatibility
condition whereby the extension of the plate in the Y direction produced by Ny
must be equal to the difference between the final arc length, Sy, and the initial
length in the Y direction, taken as 2b.
To compute the extension of the plate in the Y-direction, we neglect the
contribution of Dz to By in equation (8.289b) so that By :::,; By. We then substitute
By = Dy,y into equation (8.293b) to find
Dy,y
1
= Eh (Ny - IlNx )
(8.301)
Further, we assume that ex = 0 in equation (8.293a) and use the resulting
= IlNy in equation (8.301) to get
Nx
N y (1 - 1l 2 )
Eh
Dy,y =
(8.302)
from which
D
=
y
N y(1 - IlZ) Y
Eh
(8.303)
since Dy(O) = O.
To express Dy as the difference between the final arc length and the initial
length in the Y direction, refer to equation (2.46) with Sz replaced by Sy, dZ
by dY, and dR o by dD z . Since the deformation is symmetric, we need consider
only half the plate (0 < Y < b). Then, from equation (2.46),
Sy
=
f:
+ (DZ ,y)Zr/2 dY
[1
(8.304)
We use the binomial expansions 1 to simplify equation (8.304) as
Sy:::'; b
+~
f:
so that
Dy
=
Sy -
b
=
(8.305)
(Dz,yfdY
~
f:
(8.306)
(Dz,y)ZdY
Equation (8.303), evaluated at Y = b, and equation (8.306) give
D
y= N
y (1
-1l 2)b
Eh
= ~ (b (D
2
Jo
Z,y
)2dY
(8.307)
8 Bending of Plates
362
Equation (8.307) along with equations (8.299a) and (8.299b) and a particular
solution such as equation (8.300) constitute the solution to the problem of
cylindrical bending when the in-plane displacements of the middle surface are
restrained. An obvious iterative solution algorithm is to begin with an assumed
value of Ny; compute the integration constants in equation (<8.299a) using the
appropriate boundary conditions on Dz ; and finally, check the assumed Ny
using equation (8.307). Solutions of the cylindrical bending problem are available in Timoshenko and Woinowsky-Krieger. 53
For equations (8.296) and (8.297), a limited number of analytical solutions
using the Navier approach are available,54 but the calculations are quite
involved and numerical solutions are attractive.
An important point which might be investigated with solutions based on the
finite deflection theory of plates is the limit of the small deflection theory.
Specifically, we are interested in two points: (a) how large the normal displacement D z must be in order to obtain stress couples significantly different from
those calculated using elementary plate theory; and, (b) how significant are
the in-plane stress resultants accompanying the finite transverse displacement?
These questions are difficult to answer in general but may be studied for circular
plates, as we will show in the next section.
8.7.6 Finite Deformation ofAxisymmetrically Loaded Circular Plates: Consider an axisymmetrically loaded circular plate. Instead of the general expression, equation (8.2), it is convenient to start with the equilibrium equations
specialized for axisymmetric loading. Equation (8.2S3a) gives
(8.308)
Another relationship is found from generalizing equation (8.32b), which followed from equation (3.2Sc) through equations (8.1)-(8.S). The first term in the
axisymmetric form of equation (8.247) is [R(QR + NRDz,R)J.R when a = R,
B = R, and transverse shearing strains are suppressed. This leads to a modified
transverse shear, QR + NRDz,R' so that equation (8.32b) becomes
QR
+
NRDz,R
= D (DZ'RRR + ~ DZ,RR -
~2 DZ'R)
(8.309)
From section 8.3.1.4, we recall that for an axisymmetric loading qR(R), QR(R)
can be directly expressed in terms of qR' so that QR may be regarded as a known
quantity in what follows.
Now, we use the strain-displacement relations for polar coordinates, which
are obtained from equations (8.262) and (8.263) together with (S.S4a) and (S.S4b),
to write
(8.310)
and
363
8.7 Instability and Finite Deformation
(8.311)
Substituting 8 R and 80 in the stress-strain laws, equation (6.10), we find
NR
Eh
= 1 _ J12
[
DR,R
1
2
DRJ
,R) + J11{
+ 2(Dz
(8.312)
2J
(8.313)
and
NIJ = 1 _EhJ12 [DR
I { + J1DR,R
J1
+2
(DZ,R)
Finally, we substitute equations (8.312) and (8.313) into equations (8.308) and
(8.309) to get
(8.314)
and
(8.315)
which are again coupled and nonlinear.
These equations are studied for several cases of applied loading and for a
variety of boundary conditions in Timoshenko and Woinowsky-Krieger. 55 Of
particular interest for our purposes is the case of a simply supported solid plate
with edge moment M as shown in figure 8-15. For this case, QR = qz = O.
Timoshenko and Woinowsky-Krieger present a numerical solution for a plate
of thickness h, radius a ~ 23h, and M = MR(a) = 2.93 x 1O- 3 (D/h). With respect
to the comparison with elementary plate theory as discussed in the previous
section, there are two items of major interest: (a) the magnitude of the displacements and moments calculated from the nonlinear theory, as compared to those
computed from the elementary theory; and, (b) the significance of the in-plane
stress resultants NR and No, which are not computable from the elementary
theory.
Considering first the displacements, the maximum transverse deflection D z ~
0.55h. From equation (8.121) Dz(O) = 0.62h for J1 = 0.25, so that the refined
theory gives about a 10% smaller deflection. The elementary theory has stress
couples MR = Mo = M = constant throughout the plate, whereas the refined
theory yields values for MR and Mo at the center which are about 12% less.
364
8 Bending of Plates
Thus, the elementary theory appears to be conservative and fairly accurate for
deflections up to about one-half the plate thickness.
To assess the significance of the in-plane stress resultants, we must refer to
actual stresses, rather than the stress resultants. From elementary strength of
materials and the definitions in chapter 3,
O'u
(
h)
N;
6M;
±"2 =h±V
(8.316)
The stress resultants NR and N9 are both tensile and relatively small in the
central portion of the plate. In the outer portion, NR -+ 0 and N9/h becomes
compressive and reaches a magnitude of almost 20% of 6M/h 2 , where M is the
constant moment of the elementary theory. From this, we conclude that with
deflections approaching the half-thickness of the plate, the seemingly simple
case of uniform moment may be susceptible to circumferential instability because of the relatively high compressive values of N 9 •
This study has provided us with an opportunity to assess the validity and
limitations of one of the important assumptions in the elementary theory of
plates, assumption [lJ.
8.8 References
1. R. Szilard, Theory and Analysis of Plates, Prentice-Hall, Inc., Englewood Cliffs, NJ,
1974, p. 36.
2. M. Filonenko-Borodich, Theory of Elasticity, [trans. from Russian, (New York:
Dover Publications), pp. 260-268].
3. S. Timoshenko, History of the Strength of Materials (New York: McGraw-Hill,
1953), pp. 119-122.
4. S. Timoshenko, and S. Woinowsky-Krieger, Theory of Plates and Sheils, 2nd ed.,
(New York: McGraw-Hill, 1959), p. 92-97.
5. S. Timoshenko and J. N. Goodier, Theory of Elasticity, 2nd ed., (New York:
McGraw-Hill, 1951), pp. 316-317.
6. S. Timoshenko, History of the Strength of Materials, pp. 333-340.
7. A Nadai, Die Elastichen Platten, (Berlin: Springer-Verlag, 1925), p. 42.
8. S. Timoshenko, History of the Strength of Materials, pp. 110-113.
9. S. Timoshenko and W. Woinowsky-Krieger, Theory of Plates and Shells, pp. 113135.
to. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, (New York:
Interscience Publishers, 1966), pp. 351-396.
11. Manual of Steel Construction, 8th ed., (New York: American Institute of Steel
Construction, 1980), pp. 2.114-2.125.
12. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Sheils, pp. 136162.
13. Ibid., chap. 6.
8.8 References
365
14. Ibid., chap. 7.
15. K. Kiattikomol, L. M. Keer, and J. Dundurs, "Application of Dual Series to
Rectangular Plates," Journal of the Engineering Mechanics Division, ASCE, vol. 100,
no. EM 2, Proc. Paper 10486, April 1974, pp. 433-444.
16. R. Szilard, Theory and Analysis of Plates, pp. 650-673.
17. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, pp. 58-62.
18. R. Szilard, Theory and Analysis of Plates, pp. 613-649.
19. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, pp. 282293.
20. J. H. Michell, "On the Flexure of Circular Plates", Proc. London Math. Soc., 34,
1902, pp. 223-228.
21. N. J. Hoff, The Analysis of Structures (New York: Wiley, 1956), pp. 373-377.
22. S. Timoshenko and S. Woinoiwsky-Krieger, Theory of Plates and Shells, pp. 313319.
23. Ibid., pp. 310-313.
24. R. Szilard, Theory and Analysis of Plates, pp. 126-129.
25. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, pp. 197205.
26. M. J. Forray, Variational Calculus in Science and Engineering (New York: McGrawHill, 1968), pp. 163-164.
27. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, pp. 173179.
28. H. D. Conway, "A Levy-type Solution for a Rectangular Plate of Variable Thickness," J. of Applied Mechanics, ASME, Vol. 2J, June 1958, pp. 297-308.
29. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, pp. 298-308.
30. R. Szilard, Theory and Analysis of Plates, pp. 130-136.
31. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, pp. 366-369.
32. Ibid., pp. 390-394.
33. K. S. Pister and S. B. Dong, "Elastic Bending of Layered Plates," Journal of the
Engineering Mechanics Division, ASCE, Vol. 84, No. 10, October 1959, pp. 1-10.
34. E. Reissner. "Finite Deflections of Sandwich Plates." Pmc. nf the First U.s National
Congress of Applied Mechanics, ASME, New York, 1952.
35. E. Reissner, "Effect of Shear Deformation on Bending of Elastic Plates", Journal of
Applied Mechanics, ASME, vol. 12, no. 55, 1947, pp. A69-A77.
36. E. Reissner, "On Bending of Elastic Plates," Quarterly of Applied Mechanics, vol. 5,
1947, pp. 55-68.
37. A. E. Green, "On Reissner's Theory of Bending of Elastic Plates," Quarterly of
Applied Mathematics, vol. 7,1949, pp. 223-228.
38. J. N. Reddy, Energy and Variational Methods in Applied Mechanics (New York:
Wiley, 1984), pp. 354-388.
39. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, pp. 165173.
40. C. S. Desai and J. F. Abel, Introduction to the Finite Element Method (New York:
Van Nostrand-Reinhold, 1972), pp. 80-82; 178-181.
366
8 Bending of Plates
41. L. J. Brombolich and P. L. Gould, "A High-Precision Curved Shell Finite Element,"
AIAA Journal, vol. 10, no. 6, June 1972, pp. 727-728.
42. D. Yitzhaki, Prismatic and Cylindrical Shell Roofs (Haifa, Israel: Haifa Science
Publishing, 1958).
43. H. Simpson, "Design of Folded Plate Roofs," Journal of the Structural Division,
ASCE, vol. 84, no. 1, January 1958, pp. 1-21.
44. D. P. Billington. Thin Shell Concrete Structures (New York: McGraw-Hill, 1965),
chaps. 8 and 9.
45. M. Pultar, "Foundations of Folded Plate Theories," lASS International Colloquium
on the Progress of Shell Structures in the Last 10 Years and Its Future Development,
Session VI, Madrid, Spain, September-October 1969, pp. 1-16.
46. J. E. Goldberg and J. L. Leve, ''Theory of Prismatic Folded Plates," Memoirs,
IABSE, vol. 17, 1957, p. 59.
47. V. A. Pulmano and S. L. Lee, "Prismatic Shells with Intermediate Columns," Journal
of the Structural Division, ASCE, vol. 91, no. ST6, December 1965, pp. 215-237.
48. M. Pultar, D. B. Billington, and J. Riera, "Folded Plates Continuous over Flexible
Supports," Journal of the Structural Division, ASCE, vol. 93, no. ST5, October 1967,
pp.253-277.
49. Phase I Report of the Task Committee on Folded Plate Construction, Journal of
the Structural Division, ASCE, vol. 89, no. ST6, December 1963, pp. 365-406.
50. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Sheils, pp. 380387.
51. D. O. Brush and B. O. Almroth, Buckling of Bars, Plates and Shells (New York:
McGraw-Hill, 1975), pp. 94-112.
52. Y. C. Fung, Foundations of Solid Mechanics (Englewood Cliffs, NJ: Prentice-Hall,
1965), pp. 463-470.
53. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, chap. 1.
54. Ibid., pp. 421-428.
55. Ibid., pp. 396-415.
8.9 Exercises
The numerical problems are given without specific units to allow English, metric, or SI
unit dimensions to be selected.
8.1
Using an appropriate free-body diagram and the definition
verify that for a plate described in Cartesian coordinates
Dr = -C;Dz.r
and
367
8.9 Exercises
rrF
------,
r -Y- - - -
X
I
I
I
I
+
I
I
I
I
I
L
_______ _ _ _ _ _ _ ....JI
b/2
0/2
=
0/2
Fluid, unit wt. = y
bI2
Fig. 8-32
8.2
A rectangular plate rests on an elastic foundation with modulus K. The units of
K are force/surface area/distance in the transverse direction. Derive the governing
equations for the bending of this plate.
8.3
Using an appropriate free-body diagram and the Kelvin-Tait argument, verify the
effective shear expression in polar coordinates as given by equations (8.30).
8.4
Consider triangle
8.5
The simply supported plate shown in figure 8-32 is half-submerged in a fluid of
unit weight y.
(a) Obtain a solution for the displacement function using both the Navier and
Levy-Nadai methods.
(b) Using one of the preceding solutions:
(1) Determine the midspan deflections.
(2) Determine the maximum value of Mx and the location on the plate.
(3) Plot the variations of (a) Mx and My along the center lines of the plate
and (b) Qx and Qy along each boundary.
(4) Compute the corner forces.
8.6
Consider the plate shown in figure 8-9. The loading qz(X, Y) = qo = constant.
All edges are simply supported, except the edge along the Yaxis at X = 0, which
is clamped. Using the L6vy-Nadai approach, derive the deflection function and
compare the maximum moments in the X and Y directions to those found when
all sides are simply supported. To facilitate the numerical work, a square plate
a = b may be used as the basis of comparison.
8.7
Consider the axisymmetrically loaded, simply supported circular plate as shown
in figure 8-33.
(a) Obtain the solutions for the displacement, the radial moment, and the circumferential moment.
(b) Evaluate the maximum radial and circumferential moments.
(c) Verify the solution for a concentrated load at the center of the plate of
magnitude Po = qo1tb 2 , given by equation (8.152).
a> in figure 8-4 and derive equations (8.38) and (8.39).
368
8 Bending of Plates
t t
/5.
QO
f f t
D-
b
0
Fig. 8-33
qo
t t t
~
~
b
0
Fig. 8-34
Q
1
II
1 I
I 1J
(0)
~(bJ
Fig. 8-35
8.9 Exercises
369
Rigid Plug
1
I III I
I
II~ I
Fig. 8-36
8.8
Consider the clamped circular plate shown in figure 8-34 and re-solve exercise 8.7.
8.9
Consider the circular plate as shown in figure 8-35. The plate is subjected to (a) a
uniformly distributed load qo; and (b) a hydrostatic loading with maximum intensity ql, and is supported on a knife-edge circular support which has a
radius b.
Derive the general solution for the deflection function for this plate. The
solution should consist of a diagram of the superposition representation (if
required) plus a careful statement of the appropriate conditions required to
evaluate the integration constants. It is suggested that each loading case be treated
separately.
8.10
Consider the circular plate with a rigid insert as shown in figure 8-36. The plate is
subjected to a uniformly distributed load of intensity qo on the insert and ql on the
annulus.
(a) Derive the general solution for the deflection function for this plate. The
solution should consist of a diagram of the superposition representation (if
required) plus a careful statement of the appropriate conditions required to
evaluate the integration constants.
(b) Compute the expressions for the radial and circumferential moments and
determine the maximum values and locations.
8.11
Consider the triangular plate shown in figure 8-26 subject to a uniformly distributed load qo.
370
8 Bending of Plates
y
b
+-----f'---X
b
t--~o
-~
-==r:rrrrnn
---+----=-0
qo
Fig. 8-37
(a) Verify the solution as given by equation (8.178).
(b) Plot the moments and the transverse shear on a typical edge.
(c) Verify that the total edge shear balances the applied loading and hence, that no
corner forces are required or present.
8.12
Re-solve the case of the clamped plate, shown in figure 8-28, with a uniformly
distributed loading qo, using the virtual work method.
8.13
Consider the clamped rectangular plate subject to a hydrostatic loading as shown
in figure 8-37. Obtain a one-term and a two-term numerical solution for Dz using
the Rayleigh-Ritz method.
8.14
Consider a simply supported rectangular plate under a uniform tension T and
compression C, loaded uniformly in one quadrant by a force q, as shown in figure
8-38.
(a) Using the Green's function approach, derive the expression for the deflected
surface.
(b) Determine the critical value of C assuming that T = Cf2.
(c) Determine the lowest critical value of C and the corresponding buckling mode
as a function of the aspect ratio.
8.15
8.16
Consider the plate shown in figure 8-38 subject to the loading q over two quadrants,
~ a/2, 0 ~ y ~ b, and re-solve exercise 8.14 using the virtual work method
with T = Cf4.
o~ x
Show that the buckling load for a rectangular plate with dimensions as shown on
figure 8-31 cannot be less than N xcr = n 2 D/a 2 if the plate is simply supported on
8.9 Exercises
371
c
_1~~!---,---,--,'---'---'--1
mtmf:~:m~~:1:f:f:f:::: - - - - - - -:
-
T
):::::::::::::::::::::::\::::::::::::.,::::::::
q
I
---.---.-y
T
0/2
::r:::::::::::::::::::::::::::::::::::::::::::::::::+ - -- - - -....+-----tI
lL _____________ -1I
b/2
0/2
b/2
x
Fig. 8-38
the loaded edges (0, Y) and (a, Y) and the boundary conditions on the other edges
are unspecified.
8.17
Consider the rectangular plate as shown on figure 8-31.
(a) Re-solve the all sides simply supported case treated in section 8.7.3 using a
U:vy- Nlidai solution in the form of equations (8.106) and (8.107).
(b) Determine the critical load for the case when the plate is clamped along the
unloaded edges (X , 0) and (X, b).
8.18
Consider the circular plate shown in figure 8-33 with an added uniform radial
compression force N R • Determine the critical value of N R .
8.19
In section 8.2.2.3, the maximum bending moment in a square simply supported
plate under a uniform load was shown to be proportional to the total load on the
plate, q oa 2 .
(a) Investigate the variation of the maximum normal displacement as the radius a
increases while q oa 2 remains constant.
(b) Investigate this proportionality for a case where two opposite sides are fixed,
whereas the other opposite sides are simply supported.
CHAPTER
9
Shell Bending and Instability
9.1 General
In earlier chapters, we derived the equilibrium, strain-displacement, and constitutive equations and stated the required boundary conditions for the bending
theory of shells, referred to a system of orthogonal curvilinear coordinates.
Also, we developed strain energy and potential energy expressions that can be
incorporated into an energy formulation of the shell theory. In this chapter,
these equations are specialized for various classes of shells, as we have done for
the membrane theory equations in chapter 4.
Before proceeding, again note that many shells may achieve equilibrium
through membrane action alone, provided the requisite conditions are closely
approached by the actual shell. For such shells, bending is a secondary phenomenon often confined to narrow regions near boundaries, geometric discontinuities, and concentrated loads. On the other hand, there are shells for
which the membrane theory idealization is grossly violated by the physical
situation. The bending behavior may alter the stress pattern from that computed by the membrane theory in two ways: (a) significant transverse shearing
forces and bending and twisting moments can develop; and, (b) the pattern of
the in-plane stress resultants may be altered markedly by the bending deformations. Although a shell may seriously violate the membrane theory requirements, there still remains the possibility of resisting transverse loading primarily
with in-plane forces, which is the basic initial attraction of this structural form.
It is this latter possibility, whereby the transverse loading may be resisted by a
combination of in-plane forces and transverse shearing forces, which distinguishes bending of shells from the elementary behavior of plates.
In the study of shell bending, cylindrical and conical shells are frequently considered apart from rotational and translational shells, although they technically
may fall into one or both of these classes. The bending behavior of shells with
zero Gaussian curvature is quite distinct, however, since in the direction where
the radius of curvature is infinite, these shells cannot develop any membrane
forces to resist the transverse loading. As we have seen in chapter 4, the
membrane theory solutions for such shells generally show the entire transverse
loading being sustained in the curved direction, a one-way resistance pattern.
When the membrane boundary conditions in the curved direction are violated
372
9.2 Circular Cylindrical Shells
373
and rendered incapable of developing the required reactions, drastic alteration
of the stress pattern is often the result because the zero curvature direction is
of no help except in a bending mode.
Another important reason for studying cylindrical shells, especially, as a
distinct class is that fairly simple analytical solutions may be found, whereas
for more complex geometries, such solutions are relatively scarce. Moreover,
many of the techniques for solving the governing equations for other geometries
involve various approximations and simplifications that cast the equations into
a form similar to that of the cylindrical shell. Thus, the cylindrical shell solutions
assume added importance from the standpoint of generality.
Finally, we note that in practice, cylindrical shells are probably the most
frequently encountered form and certainly merit careful attention.
9.2 Circular Cylindrical Shells
9.2.1 Specialization of Equations: In the discussion of the membrane theory of
circular cylindrical shells (section 4.3.3.2), we chose cx = Z, f3 = e, A = 1, and
B = a, as shown in figure 4-12(a). For this geometry, R" = 00 and Rp = a.
Then, in section 4.4.1, the axial coordinate was taken as X in deference to its
widespread usage in current literature. We should feel comfortable with either
choice. For vertical vessels in which the axis of rotation coincides with the global
Z axis, rx = Z is the logical choice, whereas for horizontal shells, cx = X seems
equally suitable. We use cx= X in this chapter.
The force equilibrium equations, (3.l7a-c), and moment equilibrium equations, (3.22a) and (3.22b), become
+ Nox.o + qx Q = 0
aNxo . x + No,o + Qo + qoa = 0
aQx,x + Qo,o - No + qn Q = 0
-aMxo,x - Mo,o + QoQ = 0
aMx,x + Mox,o - Qxa = 0
aNx . x
(9.1a)
(9.1b)
(9.1c)
(9.1 d)
(9.1e)
The strain-displacement relations, equations (5.46a-h), are
(9.2a)
ex = Dx,x
(9.2b)
OJ
KX
=
Do • x
1
+ -Q Dx • 0
= Dxo.x
(9.2c)
(9.2d)
374
9 Shell Bending and Instability
(9.2e)
T
=
~[Dex,x + ~Dxe,eJ
(9.2f)
+ Dxe)
(9.2g)
Yx = (Dn,x
Ye
1
= -(Dn
ea '
De)
+ Dex
(9.2h)
If transverse shearing strains are neglected, equations (5.47a-c) replace
equations (9.2d-h):
KX
=
(9.3a)
-Dn,xx
(9.3b)
T =
~GDe,x -
Dn,X9)
(9.3c)
Also, we have the effective shear forces evaluated from equations (6.24)-(6.27):
(9.3d)
(9.3e)
N ex = N ex
Qe
=
Qe
+ Mex,x
(9.3f)
(9.3g)
The constitutive relations are given by matrix 6-2, with IX and f3 taken as X and
e, respectively. These are restated here only in the matrix form of equation (6.10)
(9.4)
with the subscripts serving to remind us of our choice of coordinates.
It is easily verified that the membrane theory equations are recoverable from
the preceding expressions. With the bending terms neglected and N ex = N xe =
S, equations (9.1a-c) reduce to equations (4.160) and equations (9.2a-c), and
the corresponding parts of equation (9.4) are identical to equations (6.50).
9.2.2 Axisymmetrical Loading
9.2.2.1 Displacement Formulation and Solution. Axisymmetrically loaded circular cylindrical shells are employed as pressure vessels and tanks in many
industrial applications. Some examples are shown in figures 2-8(r) and (u). For
375
9.2 Circular Cylindrical Shens
axisymmetricalloading, the terms qe, Nex, N xe , Mex, M xe , Qe, co, "e, 't, Ye, De,
Du , and all terms differentiated with respect to () drop out. This leaves the three
equilibrium equations
+ qx = 0
N x .x
(9.5a)
Ne
Qxx--+
qn
.
a
=0
(9.5b)
Mx.x - Qx
=0
(9.5c)
and the strain-displacement relationships
6X
=
(9.6a)
Dx.x
Dn
(9.6b)
6e=-
a
"x =
Yx
=
(9.6c)
Dxe.x
Dn.x
+ Dxe
(9.6d)
or, in the absence of transverse shearing strains,
(9.7)
We also have the pertinent constitutive relationships from equation (9.4). These
are written explicitly from matrix 6-2 as
Nx =
Eh
I-JJ
- - - 2 (6 x
+ JJ6e ) -
NXT
(9.Sa)
(9.Sb)
Mx =D"x -MXT
(9.Sc)
Me = JJD"x - MeT
(9.Sd)
(9.Se)
where D has been defined explicitly in equation (S.4d). When transverse shearing
strains are suppressed, equation (9.Se) is of no use, so that Qx must be evaluated
from equation (9.5c).
We now express the equilibrium equations in terms of the displacements in
the classical fashion of a displacement formulation.
First, we substitute equations (9.6a-d) into equations (9.Sa-e) to get the
stress resultants and couples in terms of the displacements. Also, in view of
equations (6.9), we set NXT = NeT = NT and MXT = MeT = M T . Then, we
have
376
9 Shell Bending and Instability
Nx =
Eh
--2 (
1-j1
Dn) - NT
DX X + j1.
a
N(J = 1 _Ehj12 ( j1Dx .x
Dn) + ---;;
(9.9a)
(9.9b)
NT
Mx = DDx(J.x - MT
(9.9c)
M(J = j1DDx (J.x - MT
(9.9d)
AEh
Qx = 2(1 + j1) (Dn.x
(9.ge)
+ DX(J)
In the absence of transverse shearing strains, we use equation (9.7) for
find
Kx
and
Mx = -DDn. xx - MT
(9. lOa)
M(J = -j1DDn. xx - MT
(9. lOb)
and, from equations (9.5c) and (9. lOa),
Qx = -DDn. xxx - M T. X
(9.lOc)
Also, in equation (9.3e), with MX(J.(J = 0, Qx = Qx.
Now, substituting equations (9.9a-e) into (9.5a-c), we get
(9. 11 a)
(9. 11 b)
(9. 11 c)
The loading and thermal terms are presumed as known and transposed to
the r.h.s. The resulting set of three equations in the three unknowns Dx , Dn ,
and DX(J constitutes the displacement formulation. We will not reduce these
equations further, but concentrate on the theory in which transverse shearing
strains are neglected.
With Yx = 0, DX(J is expressed in terms of Dn by equation (9.6d). We need
retain only the first of equations (9.11), along with a new equation found by
substituting equations (9.9b) and (9.lOc) into equation (9.5b):
(9.12a)
Eh 2) ( j1Dxx+Dn)
-DDnxXxX(1
.
a -j1
.
a
=
NT
-qn--+MTXX
a'
(9.12b)
377
9.2 Circular Cylindrical Shells
This set may be reduced to a single equation. We first integrate both sides of
equation (9. 12a), which gives
1 :hJ-l2 (Dx,x
+ ~Dn)
= -
f
qx dX
+ NT + C
(9.13a)
where C is an integration constant.
Comparing equation (9.13a) with equation (9.9a), we find
Nx
f
qx dX
= -
+C
(9.13b)
We may rewrite the integral in equation (9.13b) in the alternate form introduced
in chapter 4 by choosing C = Nx(O). Then, we have
Nx(X)
= -
LX qx dX + Nx(O)
(9.13c)
Next, we solve equation (9.13a) for
Dx,x
1 - J-l2 [ =~
IX qx dX + NT + Nx(O)] 0
J-l
-;;Dn
(9.13d)
which we substitute into equation (9. 12b) to obtain
(9.14)
which consolidates to
DDn xxxx
+ -Eh2 Dn =
'a
qn
(1 -
+ -a-J-l) NT -
+ ~[LX qxdX -
MT xx
'
(9.15)
Nx(O)]
Equation (9.15) is often written in the form
Dn,xxxx
+ 4k4 Dn =
(9. 16a)
where
k4
= 3(1 - J-l2) = ~
a 2h 2
4Da 2
(9. 16b)
Equation (9.16a) may be recognized as the governing equation for a well-
378
9 Shell Bending and Instability
documented problem in mechanics, the bending of a prismatic beam on an elastic foundation.! Just as an elastic foundation takes a portion of any transverse
loading applied to a beam, the shell parameter Eh/a 2 , which appears explicitly
in equation (9.15), supplements the flexural rigidity D of the cylindrical shell
in resisting the transverse loading qn. Once the governing equation is solved
for D n , the extensional stress resultants are computed from equations (9.9a)
and (9.9b), and the stress couples and transverse shear resultant from equations
(9.10a-c).
It is instructive to consider a somewhat specialized case, in which the thermal
terms are dropped and the axial load qx = O. Then, from equation (9.13c),
N x = constant. If one boundary-e.g., X = O-is unconstrained against axial
deformation, Nx(O) = 0 and, therefore, N x = 0 throughout the shell. With these
simplifications, equation (9.9a) gives
(9.17)
and equation (9.9b) becomes
Eh
Ne =-Dn
(9.18)
a
which is a remarkably simple algebraic relationship between the major stress
resultant and the primary displacement, considering the complexity of the
system with which we began.
The displacement formulation of the axisymmetrically loaded cylindrical
shell may be generalized in a straightforward matter to accommodate shells
in which the thickness varies as h = h(X).
Continuing with equation (9. 16a), the homogeneous solution is written from
the auxiliary equation
(9.19a)
with roots
m = ±k(1
+ i),
±k(1 - i)
(9.19b)
(9.20)
where C 1 -C4 are integration constants.
The particular solution is, of course, dependent on the loading, and the constants C 1 -C4 are found from applying the appropriate boundary conditions,
as we see in several examples later.
9.2.2.2 Semi-infinite Cylindrical Shells. The complete solution of equation
(9. 16a) involves four integration constants, as indicated in equation (9.20). This
constitutes a two-point boundary value problem. Within the class ofaxisym-
379
9.2 Circular Cylindrical Shells
n
-x
a
Fig. 9-1
Semi-infinite Edge-Loaded Cylindrical Shell
metrically loaded cylindrical shells, however, there are many cases for which
the forces at one boundary do not materially affect those at the other boundary.
Such shells are termed semi-infinite.
We may demonstrate this behavior by referring to equation (9.20). If X is
measured from one boundary, the factor e kX will grow very large as X increases
unless C3 = C4 = 0, whereas the factor e- kX will cause the other terms to
attenuate. Thus, the solution simplifies to
(9.21)
with C 1 and C2 evaluated from the boundary conditions at X = O.
The semi-infinite approach simplifies the ensuing arithmetic considerably.
The range of shell parameters for which this assumption is valid may be
determined by considering a shell with an arbitrary edge loading and observing
the behavior of the solution as the distance from the loaded boundary increases.
As shown in figure 9-1, a transverse shear force Qo and bending moment Mo
are applied uniformly around the circumference at X = O. The positive signs
are chosen in accordance with figure 3-2. We have, from equations (9. lOa) and
(9.10c),
Mo
=
Mx(O)
=
Qo
=
Qx(O)
= -
-DDn,xx(O)
DDn,xxx(O)
(9.22a)
(9.22b)
from which
(9.23a)
and
(9.23b)
380
9 Shell Bending and Instability
Before we carry the solution further, it is convenient to define the functions
= e-kX(coskX + sinkX)
F2(kX) = e-kX(cos kX - sin kX)
F1(kX)
(9.24a)
(9.24b)
F3(kX) = e- kX cos kX = t(Fl
+ F2)
(9.24c)
F4(kX) = e- kX sin kX = t(F1
-
F2)
(9.24d)
Now, we may express Dn and the various derivatives required to obtain other
components of the solution by matrix 9-l.
Since Fl (0) = F2 (0) = 1 and decrease as X increases, and since F3 and F4 are
linear combinations of Fl and F2, the behavior of Fl (X) and F2(X) is indicative
of the propagation of the effects of the boundary loads. These functions are
plotted on figure 9-2 and are tabulated in Timoshenko and Woinowsky-Krieger
and Tsui. 2 It is apparent from the figure that the edge loads will produce
insignificant effects at about kX = 3.
Matrix 9-1
D_
D_.x
D_.xx
O
1
D
Mo
-2k 2
Qo
-2k3
Mo
0
F1(kX)
(a)
0
F2 (kX)
(b)
Qo
2k2
0
-Mo
0
0
Qo
k
F3(kX)
(c)
0
-Qo
0
2kMo
F4 (kX)
(d)
D_.xxx
k
1.0
0.8
0.6
0.4
,
F.
0.2
4.0
0
-0.2
kX
-0.4
Fig. 9-2
Solution Functions for Semi-infinite Shells
5.0
381
9.2 Circular Cylindrical Shells
In the literature, it is common to find L ~ n/k designated as the length for
which a given shell is considered to be a long shell and, therefore, treatable by
the semi-infinite approach. L is called the half-wave length of bending and may
be written as
L=
n
~3(1 - f1.)2
fo
(9.25)
using equation (9. 16b).
Beyond this particular case, L is often used as a measure of the penetration
of the effect of various singularities, such as concentrated loads, holes, and
discontinuities, into the interior of a shell. Also, Calladine has established
a relationship between the extensional and flexural resisting modes of shells
based on the change in Gaussian curvature during distortion. In his scheme,
the characteristic
term plays a central role. 3
Note that the edge effects that have been discussed here, Qo and M o, are
self-equilibrating with respect to the overall equilibrium of the shell. The corresponding dissipation of these edge disturbances may be regarded as a demonstration of St. Venant's principle, which we have mentioned in connection with
the derivation ofthe Kirchhoff boundary conditions in section 6.2.3. Recall that
there may be another type of edge effect which is not self-equilibrated and
readily penetrates to the opposite boundary. An example is the axial line load
N shown in figure 4-12(b) and discussed in section 4.3.3.2. This is clearly
shown by equation (9.13c) with Nx(O) = N. We should be careful to reserve
the semi-infinite simplification for cases with self-equilibrating edge loads, such
as Qx and Mo·
Finally, with respect to the problem illustrated on figure 9-1, we may obtain
explicit expressions for the stress resultants from equations (9. lOa-c), (9.18), and
matrix 9-1:
fo
Mx(X)
=
-DDn,xx
=
MoFl(kX)
+ ~o F4(kX)
(9.26b)
Mo(X) = f1.Mx
Qx(X)
=
-DDn,xxx = QOF2(kX) - 2kMoF4(kX)
Eh
(9.26a)
-Eh [
No(X) = ~Dn(X) = 2ak 2D
MoF2(kX)
Qo
]
+ TF3(kX)
(9.26c)
(9.26d)
This solution is utilized frequently in the ensuing examples.
9.2.2.3 Circuniferential Line Loading. Consider a shell subjected to a circumferentialline load P (force/length), as shown in figure 9-3(a), and assume that
the shell extends a distance of at least n/k in each direction from the point of
application of P. A free-body diagram of the narrow ring under the load,
382
9 Shell Bending and Instability
p
,-----------~------------~------x
o
p
----'~ 1 t
(0)
J,--(b)
Fig. 9-3
Symmetric Line Loading on a Cylindrical Shell
figure 9-3(b), reveals that the problem reduces to the case of an edge-loaded
semi-infinite shell, similar to that treated in the previous section, with boundary
conditions
p
(9.27a)
Qo = Qx(O) = -2"
Dxo(O)
=
-D".x(O)
=0
(9.27b)
From matrix 9-1(b) and equations (9.27a) and (9.27b), we have
Mo
Qo
P
= -2k = 4k
(9.27c)
The displacement function and stress resultants are written by substituting
equations (9.27a) and (9.27c) into matrix 9-1 and equations (9.26a-d), and then
simplifying the ensuing expressions using equations (9.24c) and (9.24d):
(9.28a)
(9.28b)
383
9.2 Circular Cylindrical Shells
M8(X)
=
P
-"2 [F2 (kX) + F4 (kX)] =
Qx(X) =
N8(X)
(9.28c)
JlMx(X)
=
Eh D,,(X)
=
(9.28d)
(9.28e)
a
Under the load at X
D,,(O)
P
-"2F3(kX)
=
0, we find the maximum values
P
8k 3 D
(9.29a)
P
Mx(O) = 4k
(9.29b)
P
Qx(O)
= -"2
N 8 (0)
=
PEh
8ak 3 D
(9.29c)
Pak
(9.29d)
=2
These effects die out with increasing X, as previously indicated.
This solution is of interest in the case of a cylindrical shell with a circular ring
stiffener, which we consider later, and also in generating solutions for loading
distributed in the X direction by using the Green's function approach.
9.2.2.4 Axially Distributed Loading. We now consider a radial load uniformly
distributed around the circumference but arbitrarily distributed along the X
axis, as shown in figure 9-4. To treat this problem from a general standpoint,
we choose the Green's function technique. In this approach, the response of
the shell to a single concentrated load acting at a point of application C, which
is arbitrarily located within the loaded region, is studied first. The influence of
this load is evaluated at a specified point of observation. Then, the effect of the
entire distributed load at the point of observation is computed by integrating
over the loaded region. In this general example, we must treat two separate
cases: (a) point of observation outside the loaded region of the shell, such as
point A in figure 9-4; and, (b) point of observation within the loaded region of
the shell, such as point B in figure 9-4.
We first select the point of observation as A, outside the loaded region at
a distance X from a convenient origin o. We then define the line load produced
by p(,,) acting over the differential length d" at the point of application C as
p(,,) d". The resulting displacement at A is given by equation (9.28a), with the
argument of Fl (kX) taken as the distance between A and C, " - X:
D,,(A,C)
=
D"(X,,,)
=
p(,,) d"
8k 3D F1(k[,,-X])
(9.30)
384
9 Shell Bending and Instability
x
o
Fig. 9-4
p(X)
A
Distributed Symmetrical Loading on a Cylindrical Shell
Then, the displacement due to the entire distributed loading is found as
Dn(A) = Dn(X) =
In=~~2 Dn(X,1'/)d1'/
(9.31)
which is easily evaluated once p(X) [or p(1'/)] is specified.
Second, locate the point of observation at B, within the loaded region. Point B
is a distance X = d 1 + b1 from the origin 0, and is a distance ~ from the point
of application C. The displacement at B due to the line load p(~) d~ is
(9.32)
and the displacement of B due to the entire load is found to be
Dn(B)
= Dn(X) =
I~=:l Dn(X,~) d~ + I~=:2 Dn(X,~) d~
(9.33)
where b2 = d 2 - d 1 - b1 . For both integrals in the preceding equation, the
coordinate ~ is positive as measured from point B.
For a point of observation at the edge of the loaded region, A and Bare
coincident and equations (9.31) and (9.33) should be identical. With b1 = 0,
b2 = d 2 - d 1 in equation (9.33), and the identity is confirmed.
The stress resultants and couples for specific loadings p(X) may be evaluated
by similiar integrations or by subsequent differentiations of the computed
Dn(X), as indicated in equations (9.26).
385
9.2 Circular Cylindrical Shells
h
L
(0 )
12
~~)=
hl2
72
T
(1j-12 )12
(T,+T2 )12
T,
=
8+ 1
IS
+
To
(b)
Fig.9-5
Built-In Cylindrical Shell under Pressure and Thermal Loadings
9.2.2.5 Built-In Shell under Internal Pressure and Temperature Gradient. Next,
we examine the cylindrical shell shown in figure 9-5(a), which is subject to
a uniform internal pressure p and a linear temperature gradient- Tl on the
outside and T2 on the inside. Both the pressure and the thermal gradient are
taken as constant along the length of the shell, and the ends are fixed against
translation and rotation. We again assume that L is sufficiently large so that
semi-infinite analysis is valid. A similar problem is considered by Kraus using
the more general short-shell solution,4 but for only a uniform temperature
change through the thickness.
Proceeding, we have the homogeneous solution from equation (9.21)
Dnh
= e-kX(C 1 cos kX
+ C2 sin kX)
= C 1 F 3 (kX)
+ C 2 F4 (kX)
(9.34)
We now examine equation (9.16a) with respect to the construction of a particular
solution and consider each term on ther.h.s. individually.
First, for the pressure term qn = p, we select
p
Dnp1
= 4k4D
pa 2
Eh
(9.35)
9 Shell Bending and Instability
386
To find the contribution of the temperature gradient, we first evaluate NT and
MT from equations (6.9a) and (6.9b). It is helpful to resolve the linear gradient
into symmetric and antisymmetric components, as shown in figure 9-5(b):
T(O
T.
=
+ (1'"
(9.36a)
where
(9.36b)
T
= _T,__1_-_T_2
(9.36c)
2
a
and ( = an auxiliary normal coordinate measured from the middle surface.
Then, from equation (6.9a) and (6.9b),
NT
=
E?i.
-1- J.l
E?i.
= 1 _ J.l
MT
f
h/2
-h/2
fh/2
-h/2
T(Od(
E?i.h
(9.37a)
-1-T.
- J.l
=
E?i.h 2
(9.37b)
T(O( d( = 6(1 _ J.l) T"
Now, in equation (9. 16a), we have
4k 4 D np2
=
[1 a-;J.lJ[l~:
(9.38)
T. ]
from which
(9.39)
Since MT,xx = qx = 0 in this case, no particular solutions are required for these
terms.
Finally, we have the axial stress resultant term Nx(O), which gives
(9.40a)
or
D np3
J.l
= -k 4 Nx(O) =
4 aD
J.la
(9.40b)
--h Nx(O)
E
Collecting all of the particular solution contributions, we have
Dnp
=
D np1
+ D np2 + Dnp3 =
pa 2
Eh
_
+ rxaT. -
J.la
Eh Nx(O)
(9.41)
We may make an interesting observation concerning the particular solution
by referring to our treatment of membrane theory displacements for cylindrical
9.2 Circular Cylindrical Shells
387
shells in section 6.3.4.1. Specifically, we consider the relevant part of equation
(6.53) for the membrane theory normal displacement
(9.42a)
Now, we substitute the membrane theory stress resultant N(l> as given by equation
(4.62b),
(9.42b)
N()=pa
along with
(9.42c)
N x =0
into equation (9.42a), which gives
Dn
=
pa 2
(9.42d)
= Eh
Dnml
which is the same as equation (9.35). For the thermal term, referring to
equations (6.9a) and (6.9b), we introduce
Nx
=
N()
=
E~h
NT =--T.
1-f.1
(9.43a)
into equation (9.42a). Then,
a Mh
Dn = Dnm2 = -h - - T.(1 - f.1) = ~aT.
E 1 - f.1
(9.43b)
which is identical to equation (9.39). Similarly, with N x = Nx(O) in equation
(9.42a),
(9.43c)
which matches equation (9.40b). We thus have quantified the earlier assertion
that the membrane theory solution frequently serves as a particular solution to
the bending theory equations.
We now proceed with the general solution, which is the sum of equations
(9.34) and (9.41), by enforcing the boundary conditions
(9.44a)
and
Dx()(O) = -Dn,x(O) = 0
(9.44b)
The first condition gives
C1
+
pa 2
Eh
_
+ cxaT. -
f.1a
Eh Nx(O) = 0
(9.45a)
388
9 Shell Bending and Instability
while the second yields
k( - C1
+ C2 ) =
(9.45b)
0
whereupon
2
C 1 = - [ -pa
Eh
+ -rxaT.S -
Jl.a
]
-Nx(O)
(9.46a)
Eh
and
(9.46b)
To evaluate the term Nx(O) explicitly, we integrate equation (9.13d), giving
Dx =
LX Dx,xdX + Dx(O)
(9.47a)
with
Dx,x(O) =
1 _Jl.2
---m;- [NT + Nx(O)]
in view of equation (9.44a). With the end of the shell restrained at X
(9.47b)
= 0,
(9.47c)
and
Nx(O)
=
-NT
&.h
= ---1'.
l-JI.
(9.47d)
Then, substituting equation (9.47d) into equations (9.46a) and (9.46b),
pa 2
C 1 = - [ Eh
+ ( 1 _1 JI.)_rxa1'. ]
(9.48a)
and
C2
=
(9.48b)
C1
Correspondingly, equation (9.41) can be consolidated into
Dnp
pa 2
(
1 )_
= Eh + 1 _ JI. rxa1'.
(9.49)
and together with equation (9.34),
D n" = C 1 F 3 (kX)
+ C 2 F4 (kX)
(9.50)
constitutes the general solution. We also need Dx,x, which is found from
equation (9.13d), as
(9.51)
389
9.2 Circular Cylindrical Shells
Explicit expressions for the stress resultants and couples may be written by
substituting the preceding solution into equations (9.9) and (9.10). This is
routine and is omitted for brevity. However, we may surmise from our previous
studies of the semi-infinite cylindrical shell that the homogeneous part of the
solution Dnh will be most influential near the ends and will diminish as X
increases, whereas the particular part Dnp , which is also the membrane theory
solution, will predominate away from the ends. Also, note from equation (9.10)
that there will be bending in the shell~even after the effects of Dnh diminish~
because of the constant thermal moment M T •
9.2.2.6 Short Cylindrical Shells. When the distance between boundary points,
L, is such that L < n/k, the semi-infinite assumption is no longer valid, and the
general solution, (9.20), should be used. We consider the extension of the
edge-loaded cylinder problem, shown in figure 9-1, to the case where line
moments and transverse shear forces are applied to both ends (figure 9-6).
We first rewrite equation (9.20) as
(9.52a)
where
Fs(kX)
= e- kX cos kX
F6(kX)
= e- kX sin kX
F7 (kX)
= e kX cos kX
(9.52b)
F8(kX) = e kX sin kX
The boundary conditions for the loading in figure 9-6 are
r-----x
~~--------------~:
~
Fig.9-6
Edge-Loaded Cylindrical Shell
B
390
9 Shell Bending and Instability
Mx(O)
Qx(O)
Mx(L)
Qx(L)
=
=
=
=
= MA
-DDn.xxx(O) = QA
-DDn.xx(kL) = MB
-DDn.xxx(kL) = QB
-DDn.xx(O)
(9.53)
We now turn to the tedious calculation of the derivatives of Dnh • We first
evaluate
and then continue with the differentiation in operator notation, e.g. D n =
d n ( )/dXn
Fs(kX)
= Fs
D1FS = -kFs - kF6
D2FS = Dl(D1FS) = -kD1Fs - kD1F6
F6(kX)
= F6
D1F6 = -kF6
+ kFs
+ kD1FS
-kD2F6 + kD2Fs
D2F6 = Dl(D1F6) = -kD1F6
D3F6 = Dl(D2F6) =
F7(kX)
=
F7
D1F7
=
kF7 - kFs
D2F7
=
Dl(D1F7)
=
kD1F7 - kDIFg
D3F7 = Dl(D2F7)
=
kD2F7 - kD2Fg
=
kDIFg
Fg(kX) = Fg
D1Fs = kFg + kF7
D2Fg = Dl(DlFg)
D3Fg
= Dl(D2Fg) = kD2Fg
+ kD1F7
+ kD2F7
By repeated substitution, the various derivatives can be written in terms of
Fs-Fg:
D1Fs
= -k(Fs + F6)
D2FS
=
2k2F6
D3 Fs = 2k 3(Fs - F6 )
391
9.2 Circular Cylindrical Shells
Matrix 9-2
0
k
F6(kL)
k[Fs(kL)
-F6(kL)]
-1
k
-Fs(kL)
k[Fs(kL)
+F6(kL)]
0
-k
-Fs(kL)
-k[F7(kL)
+ Fs(kL)]
1
k
F7(kL)
k[F7(kL)
-Fs(kL)]
r} c}
C2
~:
-1
= 2k 2 D
QA
~:
Dl F6 = k(Fs - F6 )
D2F6 = -2PFs
D3 F6 = 2P(Fs
+ F6 )
Dl F7 = k(F7 - Fs)
D2F7 = - 2k2FS
D3F7 = -2k 3(F7
Dl Fs = k(F7
+ Fs)
+ Fs)
D2Fs = 2k2F7
D2 Fs = 2P(F7 - Fs)
Now, we insert the appropriate derivatives into equation (9.53), noting that
= F7(0) = 1 and F6 (0) = Fs(O) = 0, to get matrix 9-2, or
Fs(O)
1
[FJ{C} = - 2k 2 D {B}
(9.55a)
from which
(9.55b)
Although {C} may be written explicitly in algebraic form, the resulting expressions are cumbersome; also, in the computer era, it is routine to evaluate {C}
by specifying numerical values for the shell parameters and performing the
inversion and multiplication. We shall regard the problem as essentially solved
at this point and consider an application.
9.2.2.7 Ring-Stiffened Cylindrical Shells. Ring-stiffened cylindrical tubes are
commonly used for pressure vessels, submersible vehicles, and rockets. An
example is shown in figure 2-8(r), and an idealization is depicted in figure 9-7.
The rings are often relatively close together, so that the semi-infinite simplification may not be valid.
392
9 Shell Bending and Instability
p.
L
~~~
b
PI2 P/2
Stiffener Removed
Fig. 9-7
Ring-Stiffened Cylindrical Shell
As the cylinder deforms in the radial direction, the ring will retard the expansion or contraction. The resulting contact force between the cylinder and
the stiffener is shown as P in figure 9-7. This problem is somewhat similar in
concept to that shown in figure 4-4, where the ring beam of a dome is analyzed.
In that case, the strain incompatibility at the interface of the shell and the beam
could not be resolved because of the limitations of the membrane theory. Here,
however, we are able to enforce the deformation compatibility between the ring
stiffener and the shell.
We assume that the stiffener thickness hs « L, so that the reaction can be
considered to act on the shell as a radial line load. We take the inside radius of
the stiffener as a = a + hj 2, where a is the radius at the shell middle surface and
h is the shell thickness. Also, b is the width of the stiffener. In practice, the
stiffeners may be located on the inside of the shell, requiring a slight modification
of the solution.
The radial deformation ofthe ring stiffener due to the line load P (forcejlength
of circumference) is easily computed by considering the stiffener to be a short
cylindrical shell with radius = a + bj2, thickness = b, and length = hs • Then we
apply equation (6.53) which, for the loading and geometry under consideration,
reduces to
Dn
= Ds =
[a
+ (bj2)]
Eb
No
(9.56a)
Assuming the line load to be uniformly distributed over the length h., and using
equation (4.62c),
9.2 Circular Cylindrical Shells
No
=
P
hs
(a+~)
2
393
(9.56b)
so that the stiffener deformation in terms of the unknown contact force P is
D =
pea + (b/2)Y
s
Ebh s
(9.56c)
Now, we examine the forces acting on the shell. We assume the loading
is a uniform internal pressure p, so that the particular solution is given by
equation (9.35):
(9.57)
Also, we have the force P applied to the shell by the ring stiffener. Assuming P
is directed outward on the ring, as depicted on the figure, an equal and opposite
force reacts on the shell, as shown in the inset. This is the same situation depicted
in figure 9-3, but with the sense reversed. Thus, we may find end conditions
from equations (9.27), referring to figure 9-6 for the correct algebraic signs for
Qx(O) = QA and Qx(L) = QB:
Qx(O)
P
="2
(9.58a)
(9.58b)
and
Qx(L)
P
= -"2
Dxo(L) = - Dn.x(L) = 0
(9.58c)
(9.58d)
Substituting equations (9.58a) and (9.58c) into the second and fourth rows of
matrix 9-2. and equations (9.58b) and (9.58d) into equation (9.54) yields four
equations in the four unknown integration constants C 1 -C4 and the contact
force P. We obtain an additional relationship by equating the radial displacements of the shell and the stiffener:
(9.59)
where Dnh is given by equation (9.52a), Dnp by equation (9.57), and Ds by
equation (9.56c).
As mentioned previously, the short-shell solution is best carried out by
inserting numerical values pertaining to the case at hand. We may, however,
easily proceed for the case where L > n/k and the semi-infinite solution is valid.
We need only the end conditions given by equations (9.58a) and (9.58b), and
may immediately write
394
9 Shell Bending and Instability
(9.60)
from equation (9.29a) with the appropriate sign change. Correspondingly,
equation (9.59) becomes
P
-8k 3 D
+
pa 2
Eh =
pea + (bj2)]2
bhs
(9.61)
from which P is found. Then, the remaining stress resultants and couples are
routinely calculated.
A slight variation in this problem is noted in Timoshenko and WoinowskyKrieger. 2 If the shell is closed at both ends, an axial stress resultant
pna 2
pa
Nx = - - = 2na
2
is present. This contributes to the particular solution given by equation (9.40b).
Thus,
(9.62)
so that in equation (9.57) and the ensuing equations,
D
np
2
= pa
Eh
(1 -~)
2
(9.63)
For loading cases which are nonaxisymmetrical, circumferential shear forces
N xo are developed between the shell and the stiffeners. An additional compati-
bility relationship, equating the circumferential displacements of the shell and
the stiffener, is needed to evaluate these contact forces, which may be eccentric
to the centroid of the stiffener and also produce circumferential bending moments Mo. In reinforced concrete barrel shells, discussed in sections 9.2.3.1 and
9.2.3.3, the stiffeners are generally integral with the shell and can be treated
using an "effective width" concept. 5
9.2.3 General Loading
9.2.3.1 Displacement Formulation and Solution. The analysis of cylindrical shells
for nonaxisymmetrical surface loading commands a prominent place in the
literature, and a comprehensive presentation is beyond our objectives in this
text. We should mention, however, that the formulation and application of
this theory have attracted the contributions of some of the most prominent
twentieth-century mechanicians and engineers to the extensive literature on
the subject, including L. H. Donnell, J. Kempner, H. Schorer (Swiss), N. J. Hoff
and A. Parme in the United States; A. Aas Jakobsen in Norway; H. Reissner,
395
9.2 Circular Cylindrical Shells
W. Fliigge, U. Finsterwalder, F. Dischinger, and W. Zerna in Germany; J. E.
Gibson and R. S. Jenkins in Great Britain; E. Torroja in Spain; and V. V.
Novozhilov, A. I. Lur'e, and V. Z. Vlasov in the USSR. Comprehensive historical
reviews of these works are found in Fliigge 6 from the Western standpoint, and
in Novozhilov 7 from the Soviet standpoint. Apparently, the Soviet work in this
field did not have much impact in the West until the publication of the English
language translation of Vlasov's monograph. 8 This formulation is applicable
to open, as well as closed, cylindrical shells. As such, one of tbe first prominent
applications was to the design of so-called barrel shell roofs, figure 2-8(h). An
interesting account of the development of this form for shell roofs in Germany
by Finsterwalder and his associates, and the subsequent technology transfer to
the United States by A. Tedesko, is provided by Billington. 9
We begin the displacement formulation, with transverse shearing strains
neglected, by eliminating the transverse shear stress resultants from equations
(9.1b) and (9.1c), using equations (9.1d) and (9.1e). We also take N ox = N xo
and Mox = M xo , leaving the three equilibrium equations
(9.64a)
aNxo.x
1
+ No,o + M xo.x + ~Mo,o + aqo = 0
aMx , xx
1
+ -Mo
a ' 00 + 2Mxo , xo -
No
+ qna = 0
(9.64b)
(9.64c)
Next, we substitute equations (9.2a-c) and (9.3a-c) into (9.4) to get the stress
resultant-displacement equations
(9.65a)
N xo
(9.65b)
l
1
= 2(1nh
_ Jl) Do,x + ~Dx,o
J
(9.65c)
Mx= -D[Dnxx+
Jl2(Dnoo-Doo)J-MxT
,
a
'
,
Mo = -D [JlDn , xx
Mxo =
+ ~2
(Dn' 80 a
DC: Jl)[~Do,x
Do , o)J - MOT
- Dn,xoJ
(9.65d)
(9.65e)
(9.65f)
Then, these expressions for the resultants and couples are inserted into equations (9.64a-c) to obtain equilibrium equations in terms of the displacements.
396
9 Shell Bending and Instability
Following equations (6.9), we set NXT
Dx,xx
+
(1-
2a 211) Dx,oo
+
(1 +
=
NOT
11) Do,xo
~
=
NT
and
MXT
+
11) Do, xx
1~:2
MOT
=
M T:
+ (11)
~ Dn,x
(9.66a)
1 - 112)
= ( ~ (-qx
C;
=
+
(:2)
[C ;
Do, 00
11 ) Do, xx
+
+
C~11)Dx.xo
(:2 )
Do, 00
-
1 2(
+
(:2)
(:2 )
Dn,o
Dn,xxo -
-_ ---11- -qo
Eh
+ NT,x)
Dn,oooJ
(9.66b)
M)
+N T,O
- + - -T,O
a
a2
(9.66c)
We may simplify equations (9.66b) and (9.66c) by examining the terms
multiplied by h 2/a 2. We will drop all such terms as being ofO(h2/a 2): 1. At this
point, this step involves some presumption. Whereas the first two terms of
the (h 2 /12a 2 ) [ ] expression in equation (9.66b) have corresponding terms in
the same equation that are of 0(1), the remaining terms and all terms contained
in (h 2 /12a 2 )[ ] in equations (9.66c) involve derivatives not found elsewhere
in the equation. Nevertheless, this assumption, if not fully justified, has been
found to give good results for a wide class of problems and greatly simplifies
the remaining derivation and solution. A discussion of the possible errors
introduced by this step is contained in Kraus. 10
Insofar as equation (9.66b) is concerned, we may regard the suppression
of the 0(h 2 /a 2 ) terms as being equivalent to neglecting the influence of the
stress couples on the in-plane equilibrium equation. Also, ignoring the
0(h2 /a 2 ) terms in equation (9.66c) is equivalent to replacing the displacementcurvature relationships, equations (9.3), by the corresponding equations for
a plate. The later equations are given by equations (5.55), with IX = X and f3 =
e, as
KX =
-Dn,xx
(9.67a)
(9.67b)
397
9.2 Circular Cylindrical Shells
T =
1
--Dn X(J
a '
(9,67c)
These interpretations give a reasonable physical basis to the elimination of
the O(h2 /12a 2 ) terms.
We now collect the simplified equations as
Dx,xx
C;
C~fl)D(J,x(J + (~)Dn,x
+ (:2
+C
~fl)Dx,x(J + (:2 )
+ (\~/)Dx,(J(J +
fl)D(J,xx
h2
-V4Dn
12
1
+2
a
=
)D(J,(J(J
(flaDx,x
+ D(J,(J + Dn) =
(9.68a)
Px
Dn,(J
=
p(J
(9.68b)
(9.68c)
Pn
where
(9.69a)
(9.69b)
(9.69c)
and
2
V ( ) = ( ),xx
1
+ 2( ),(J(J
(9.69d)
a
is the harmonic operator in the X -() cylindrical coordinate system. Kraus l1
has derived a similar set of equations, applicable for noncircular cylindrical
shells as well, by neglecting the stress couples in equation (9.64b) and using
equations (9.67) for the changes in curvatures at the outset. The only perceptible
difference appears to be that the thermal moment gradient term in P(J, M T ,(J/a 2 ,
is not present in his equations.
It is easily verified that for axisymmetric loading, equations (9.68a) and
(9.68c) are identical to equations (9.12a), and (9.12b), which reduce ultimately
to equations (9.16a) and (9.16b).
The formulation continues by forming a judicious set of mixed partial derivatives as suggested by Kraus: l1 (a) iJ2/oX 2 (equation [9.68a]); (b) 02/0()2
(equation [9.68a]); (c) 02/0 XO() (equation [9.68b]); and, (d) solution of (a) and
(b) for the mixed partials of De:
D(J,xxxe
=
1 2a
+ fl [ Px,xx - Dx,xxxx -
(1 -
2a2 fl) Dx,xx(J(J
(9.70a)
398
9 Shell Bending and Instability
(9.70b)
and
c;~)
Do.xxxo
+
(:2 )
Do.xooo
+
C~~)DX.XXOO
+
(9.70c)
(:2 )
Dn.xoo - Po.xo
=0
We now eliminate the first two terms in equation (9.70c) by using equations
(9.70a) and (9.70b), and consolidate the remaining terms over the common
denominator a(1 + ~) to get
(9.71)
Equation (9.71) becomes the governing equation for D x , once Dn has been
determined.
We may derive a similar equation for Do by interchanging equations (9.68a)
and (9.68b) in the aforementioned mixed partial operations (a), (b), and (c) and
solving (a) and (b) for the mixed partials of Dx. This produces
Dx.xxxo = 1 2a
+ ~ [ Po.xx -
(1 - ~)
-2- Do.xxxx -
(1)
-(:2 )
(:2
a2
Do.xxoo
(9.72a)
Dn.xxo ]
Dx.xlilio = 1
and
Dx.xxx/l
~ ~[PO.09 -
(1-
C; ~ )
~) DX.X999 +
+ 2a2
DO.XX9/1 -
(1 + ~)
~ D/I.XX06
)D/I./I/1/1/1
(9.72b)
+ (~)
~ Dn.xX/I
- Px.X/I
=0
(9.72c)
Eliminating the mixed partials of Dx from equation (9.72c) and clearing, we
have
399
9.2 Circular Cylindrical Shells
(9.73)
which becomes the governing equation for De, once Dn has been determined.
Finally, we may isolate the normal displacement Dn. The procedure is: (a)
a/ox (equation [9.71]); (b) a/ae (equation [9.73]); and, (c) V4 (equation [9.68c]).
These operations give
(V 4Dx) , x
= V4(Dx , x) =
-(~)Dn
a ' XXXX + (~3)Dn
a ' xxee + Px , xxx
(9.74a)
(V 4De),e
-c;
= V 4(De,e)
=
fJ)Dn,xxe9 -
(:4 )
(9.74b)
Dn,9999
- d;1 ~fJ~)PX'X99 + C: fJ)P9,XX9 + (:2 )P9,999
and
(9.74c)
After substituting equations (9.74a) and (9.74b) into equations (9.74c), we have
the desired equations for Dn:
h2 8
12 V Dn
+
(1 -
fJ2)
~ Dn,xxxx
1
= V Pn - ~ fJPx,xxx - a2 PX,X99
4
1[
+
(2
+a fJ) P9,XX9 + ( a13)
(9.75)
P9,999
]
Equations (9.71), (9.73), and (9.75) are among the most famous equations in
the theory of thin shells and are commonly referred to as, alternately, Donnell's
equations, Jenkins's equations, or Vlasov's equations. The eighth order system
is uncoupled for the displacements, so that once Dn is determined from
equation (9.75), Dx and De may be found by solving equations (9.71) and (9.73),
respectively. These equations are also used, with slight elaboration, to study
the dynamic response of circular cylindrical shells. 12
The appropriate boundary conditions for the foregoing equations are established from figure 6-3, with a = X and P = e. The relationships in which there
400
9 Shell Bending and Instability
are no transverse shearing strains included are applicable here. Note that the
equations derived pertain to open, as well as to closed, circular cylindrical shells.
For closed shells, the boundary conditions in the () direction are replaced by
continuity conditions of the form f(O) = f(2n), where f(() is a function which
is continuous at () = O.
Before considering some specific applications, it is instructive to make some
general comments on the analytical solution of the governing equations for
this theory, equations (9.71), (9.73) and (9.75), or, alternately, the coupled
equations (9.68a-c). Complete solutions are fairly complicated algebraically,
but are extensively documented in specialized works, such as Fliigge. 13
Briefly, the solutions of the homogeneous equations for edge loadings at
X = it = constant are of interest for both open and closed· shells and are
obtained by taking the Fourier expansions
DXh}
{ DlJh
Dnh
=
i~O
{DJch(X) COS j ()}
D~h(X) sinjO
(9.76)
D~h(X)COSjO
Note that for closed shells, Dih(O) = Dih (2n)(i = X, (), n), which serve as the continuity or periodicity conditions. Then, for a general harmonic j, the substitution
of equations (9.76) into the governing equations gives an eighth order algebraic
system. The solution ultimately takes the general form
Ri(X,()
= CG{C{[e-glx(f!Cosg3x - fising3x)]
+ CHe-g X(flcosg 3x + f/s ing 3x)]
+ Q[e- g2x (fi cosg 4x - fjsing 4 x)]
+ Ci[e- g2x (f1 cos g4X + fi sin g4X)]
+ Q[eg x(f!cosg3 x + fis ing 3x)]
+ CHeg X(-flcosg 3x + f/s ing3 x )]
+ CHeg 2X(ficosg 4x + fjsinY4 x )J
1
(9.77)
1
1
+
q[eY2X( -flcosg 4x
+ fis ing3 x )J}
(C~~())
SInJ()
in which Ri(X, () = a typical displacement, stress resultant, or stress couple for
harmonic j; x = X/a; 91 - g4 are quantities dependent on j, the dimensions
of the shell, and the material properties; f!-fi are quantities dependent onj,
the dimensions of shell, and the material properties for each individual Ri; CG
is a constant dependent on the shell geometry and material properties; and
C{ -q are constants of integration.
In the general case, all eight constants are needed, and hence eight boundary
conditions in the X direction are required, whereas for a semi-infinite shell,
401
9.2 Circular Cylindrical Shells
those terms associated with q-q are suppressed, and therefore only four
boundary conditions are needed.
For an edge loading at X = X, such as we studied for axisymmetric loading,
the integration constants are determined by first substituting the respective
displacements, Ri(X, 0), into the stress resultant-displacement relations, equations (9.65); and then setting those expressions corresponding to the specified
edge loadings equal to their boundary values, Nx(X), NX9 (X), Qx(X), and/or
Mx(X). Additional equations may be obtained by the specification of kinematic
boundary conditions Dx(X), D9(X), Dn(X), and/or D9X(X), For a semi-infinite
shell, X corresponds to X = 0, whereas for a complete shell, X stands for X = L
as well.
Next, we consider the homogeneous solutions for a shell with an edge loading
applied at a boundary 0 = (j = constant. Obviously, this case is pertinent only
for open cylindrical shells, such as those discussed in section 4.4.1. Here, series
solutions of the form
k
00
D9h
Dnh
X
Dxh(O) cos kn L
DXh
L
k=O
.
X
D;h(O)smkn L
(9.78)
X
D!h( 0) cos kn L
are widely used. An eighth order algebraic system must be evaluated for each
harmonic component k, from which the solution may be expressed in the same
form as equation (9.77), with j replaced by k, and the coordinates X and 0
interchanged:
Rk
=
CG{CHe-91//(Rcosg30 - ftsing3 0 )
+ Q[e- 9'//(ftcosg 30 + Rsing 3 0)]
+ Q[e- 929 (Rcosg 4 0 - f.t sin g4 0 )
+ C![e-929(f.tcosg40 + Rsing 4 0)]
+ Q[e 919 (Rcosg 30 + Rsing 3 0)]
+ CHe 9,9( -ftcosg 30 + R sin g3 0 )]
(9.79)
X
coskn L
+ CHe 929 ( -f.tcosg40 + ff sin g3 0 )]}
or
X
' kn L
sm
402
9 Shell Bending and Instability
Again, the effects at opposite boundaries may be uncoupled by using the
semi-infinite approach. Because the coordinate lines normal to the loaded
boundaries are curved in this case, the effects of the boundary forces may be
expected to dissipate more rapidly than those for the previously studied case, in
which in-plane forces Nx (8, X) applied at X = X can propagate along straight
lines. After the integration constants are obtained from a specified combination
of static and kinematic edge conditions, the homogeneous open shell solution
is completely described. The possible static edge conditions involve
whereas the kinematic conditions refer to
Such solutions are used extensively in the analysis of open cylindrical shell roofs,
and extensive tabulations are found in Design of Cylindrical Concrete Roofs. 14
We now consider particular solutions for various applied loadings and
thermal effects. A general approach is to take
.k
X
Dkp cos kn L cosj8
00
00
II
(9.80)
j=O k=O
.k
D~p cos kn
X
L cosj8
Such solutions may be combined with the edge load solutions generated from
equations (9.76) and (9.78) to solve a wide range of cylindrical shell problems.
Recall from our earlier calculation of the membrane theory displacements for
cylindrical shells (section 6.3.4.1) that the solution for the example considered
was precisely in the format of equations (9.80), with j = k = 1. This strongly
suggests that the membrane theory solution will serve as a particular solution
for the bending theory equations in many instances.
Finally, in our general discussion of solution procedures, we outline the technique of Vlasov,15 which is very attractive for certain problems. Starting with
the three coupled equations, (9.68a-c), and using a nondimensional axial coordinate X/L, he introduces a stress function
(9.81)
from which the three displacements Dx , De, and Dm as well as the stress
resultants and couples, can be found by differentiation. The particular solutions
correspond to the individual surface loading components Px, Pe, and Pn' respectively, and the resulting equation is in the same form as equation (9.75) with cD
replacing Dn. He specializes the equation for a normal loading Pn(X,8) and
403
9.2 Circular Cylindrical Shells
further stipulates that the shell is simply supported on all four sides. Referring
to section 6.2.2, this corresponds to boundaries such that where X = X =
constant, N x = Mx = D8 = Dn = 0; and where () = = constant, N8 = M8 =
Dx = Dn = O. Next, he uses the Navier approach, which is familiar from plate
solutions, to write
e
'"
'V
~ L...
~
= L...
(). k X
slnj7t=sln
7t-=
",;k'
'V"
j=l k=l
(9.82)
X
()
This stress function automatically satisfies the simply supported boundary
conditions. ~k is then determined so as to satisfy the governing equation for
each harmonic jk.
The solution by Vlasov's technique is fairly simple to implement, although
we do not pursue the details here. An interesting and useful result is the solution
for a cylindrical shell under a concentrated load, which isolates the Green's
function for such structures. 16
9.2.3.2 Column-Supported Cylindrical Shells. We now consider a cylindrical
tank resting on a concentric ring of equispaced columns, as shown in figure 9-8.
The shell carries an axisymmetric loading q(X), and the transition between the
shell and the columns is assumed to be facilitated by a ring beam. To avoid
~rz-. q(X)
L
ing Beam
B
a
Fig. 9-8
Column-Supported Cylindrical Shell
404
9 Shell Bending and Instability
ambiguity in the subsequent development, it is convenient to set the origin for
X at the top of the ring beam and to presume that the ring beam depth B « L.
Thus, the tops of the supporting columns are also located at X = O.
This situation is similar to that shown in figure 4-29, except that, of course,
here the shell is cylindrical. Whereas we were able to treat the spherical geometry
by using the membrane theory, a membrane description is inadequate for the
column-supported cylindrical shell. This is illustrated in section 4.3.2 and by
figure 4-12(b), which shows that a force N applied to one end simply propagates
along the straight meridian in the absence of bending.
Now, we turn to the problem at hand. The representation of the discrete
supports is identical to that shown in figure 4-29, with th = nl2 and RO(rPb) = a.
Here, we have X in place of rP for the meridional coordinate. Case (4-29 [bJ) of
the superposition is described by a particular solution, which can be taken as
the membrane theory solution, and a homogeneous solution, which satisfies
the boundary conditions. The in-plane stress resultants for the particular
solution are calculated from equations (4.58) and (4.62a), with the appropriate
coordinate modifications:
NXbp(X)
Q(X)
= -2na
(9.83a)
(9.83b)
The homogeneous solution for case (4-29[bJ) is obtained from the treatment
of axisymmetrically loaded semi-infinite cylindrical shells in section 9.2.2.2 as
equation (9.21).
If the tank is supported by a circumferentially rigid ring beam, the appropriate
boundary conditions would be
(9.84)
and either
(9.85a)
or
(9.85b)
depending on whether the ring beam is assumed to prevent or to permit
rotations about the circumference. The condition for an actual shell would
probably fall somewhere in between the two idealized situations. We will
proceed with the former for purposes of illustration.
To find C1 and C2 , the expressions for Dn and DXIJ evaluated at X = 0 are
substituted into equations (9.84) and (9.85a). In tum, Dn(O) and DX(J(O) are
composed of (a) the particular solution contributions obtained by substituting
equations (9.83a) and (9.83b), evaluated at X = 0, into equations (6.53) and
(6.54); and, (b) the homogeneous solution contributions, found from equation
405
9.2 Circular Cylindrical Shells
(9.21). Alternately, if boundary condition (9.85b) is selected, Mx(O) comes from
equations (9.22a), whereupon C 1 and C2 may be found.
To complete case (4-29 [b]) of the superposition, MXb(X), M 8b (X), and QXb(X)
are computed from equations (9.26a-c) and N8Xb(X) is the sum of equations
(9.83b) and (9.26d). NXb(X) remains as the membrane theory value N Xbp' as given
by equation (9.83a).
Case (4-29 [c]) of the superposition is a homogenous loading condition that
is solved following equation (9.76), with the stress resultants, couples, and
displacements having the form of equations (9.77). The boundary conditions at
X = 0 are given by equations (9.84) and (9.85a) or (9.85b), plus two additional
equations. First, the ring beam may be regarded as circumferentially rigid, so
that
(9.86)
The final boundary condition at the base is found from the representation of
the meridional stress resultants Nx(O) by (a) the negative of the continuous
boundary reaction and (b) the intensity of the column reactions, as shown on
figure 4-29(c). This condition has been expressed earlier by equation (4.129),
which is rewritten for the case at hand as
Nxc(O,O) = [ - NXb(O)
where b
(m
1
=
= -
+ bRcl]
(9.87)
ifm(~:) - fJ ~ 0 ~ m(~:) + fJ
ne /2, . .. , - 1,0,1, ... , ne /2)
b = 0 for all other 0
ne
=
the total number of columns
As before, equation (9.87) is expanded in a Fourier series in 0
Nxe(O,O) =
00
L:
j=l
Nl-AO) cosjO
(9.88)
which gives the Fourier coefficients [see equation (4.134)]
.
2
.
NJcc(O) = jfJ NXb(O) smjfJ (j = inJ
(9.89a)
Nl-AO) = 0
(9.89b)
and
(j -:/= inJ
Thus, we have expressed the fourth boundary condition at X = 0 [equation
(9.87)] in the harmonic form
406
9 Shell Bending and Instability
NXc(O, 9)
=
00
L
j=nc. 2nc . . ,-
Nkc(0)cosj9
(9.90)
We presume that the semi-infinite assumption is adequate for our purposes
in stress analysis, so that we need not be concerned with the boundary conditions at X = L. However, from a practical standpoint, note that open cylindrical
tanks are generally stiffened at the top to prevent ovaling due to the propagation of the concentrated column reactions Nxc(O, 9).
For some specific tank structures, the idealized boundary conditions stated
in equations (9.84) and in (9.85a) or (9.85b) might warrant refinement. One
possibility is to incorporate the ring stiffener model introduced in section 9.2.2.7
to generalize equation (9.84). A second possibility concerns the common use of
such a tank to contain a liquid or solid. In such vessels, a circular plate or
perhaps a shallow cone or other rotational shell bottom would be attached
near the ring beam level. Depending on the connection detail, additional radial
and, perhaps, rotational restraints would be introduced. The possibilities are
many, depending on the actual case in question, and the realistic incorporation
of additional constraining elements into the mathematical model may alter
the computed stress pattern significantly. 1 7 The bottom plate or shell, itself,
would behave essentially as an axisymmetric member supported by the ring
beam. Such structures have been discussed extensively in the preceding chapters.
If the weight of a portion of the contents of the tank is carried directly to
the ring beam by a bottom plate or shell, instead of being transferred through
the shell wall, then the resultant force Q(X) in equation (9.83a) would not
include this portion of the loading for the calculation of NXb(X) for X > 0;
however, for case [4-29(c)] of the superposition, where NXb(O) is encountered,
the entire load from the tank must be included in Q(O) for the calculation of
NXb(O) and the column reaction Rei. For example, if a circular plate bottom
were provided at X = 0 in the tank shown in figure 9-8, all of the weight of the
internal contents would be carried directly to the ring beam if no wall friction
was assumed. This would leave only the dead weight of the tank and roof in
Q(X), X > 0, but the weight of the contents would be added for Q(O) in case
[4-29(c)J.
With the boundary conditions for the idealized problem established as equations (9.84), (9.85a) or (9.85b), (9.86), and (9.90), we set the specific expressions
for Ri = D~, D~6 or M~, D~, and Nk [which are in the form of equation (9.77)J
equal to the boundary values at X = 0, which are 0, 0, 0, and Nkc(O), respectively.
Then the integration constants for harmonic j, C{ -Ci, are found from the
simultaneous solution of the four equations. The stress resultants, couples,
and displacements for harmonic j are then calculated from the corresponding
specific form of equation (9.77).
Convergence is established by the progressive diminution of the results from
consecutive participating harmonics (j = nco 2n" .. . ). For the usual cases encountered, four or five harmonics should be adequate. The complete results
407
9.2 Circular Cylindrical Shells
are obtained by summing the stress resultants and couples and displacements
obtained from the specific forms of equation (9.77) over the participating
harmonics at various circumferential locations e, and then adding these values
to the case [4-29(b)] results.
Fliigge 18 has considered a case with a/h = 150, nc = 8, and fJ = 6°. This gives
an amplification of 360°/(8 x 2fJ) = 360/96 = 3.75 for N x at the base; that is,
Rcl = 3.75NXb(0). The effect of this amplification and the corresponding change
in No dies out at about X = a, with the stress resultants above this level
practically equal to the membrane theory values.
In Gould 19 the column-supported shell model is generalized to represent
a longitudinally distributed, as opposed to a line, attachment between the column
and the shell wall. This study is particularly applicable to elevated storage tanks,
such as that shown in figure 2-8(u).
9.2.3.3 Multiple Barrel Shells. A multiple barrel shell composed of adjacent
segments of circular cylindrical shells is depicted in figure 9-9(a), with the
positive sense of the coordinates taken from figure 4-37. An actual shell roof of
this type is shown in figure 2-8(h). The membrane theory analysis of a single
shell of this form was conducted in sections 4.4.1 and 6.3.4.1.
First, consider the boundary conditions in the X direction. The great majority of shell roofs of this configuration have been designed assuming simply
supported boundaries:
at X = 0
and
X = L
(9.91)
If the support is a wall or frame in the Y-Z plane, it must be sufficiently stiff to
prevent displacements in the Y or Z directions; yet, it must be sufficiently flexible
to permit displacements along the X axis and rotations about the Y axis.
Obviously, these requirements are difficult to satisfy exactly in a physical structure, but they are probably the best representation which can be incorporated
into an elementary analytical solution.
Now, consider the boundary conditions in the e direction. Referring to figure
9-9(a), three situations are identified: exterior, interior-general, and interiorsymmetry line. In some cases, the resistance is augmented by stiffening beams
spanning the length L; here, we will not consider this situation but only the
basic shell. First, the exterior edge may be regarded as free, so that
(9.92)
The interior edges are somewhat more complicated to characterize, since they
are continuous with the adjacent shell. Clearly, the displacements Dx, Do, and
the rotation Dox must be compatible between the intersecting shells. Strict
enforcement of this condition could result in four simultaneous equations per
interior valley line, which greatly complicates the calculations. A simplification,
widely used in design, is to treat every interior edge as if it were located
408
9 Shell Bending and Instability
Top
~
Base ~
Base
Compression 0
Tension
Nx
(b)
Fig. 9-9
Open Cylindrical Shell
409
9.2 Circular Cylindrical Shells
on a line of symmetry where the lateral displacement, D y , and the rotation
about the longitudinal axis, DlJx , vanish. For the coordinate system defined on
figure 9-9(a), these conditions become
(9.93a)
for the interior edge of the outside barrel shell, and
(9.93b)
for all of the interior barrels, where from figure 9-9(a),
(9.93c)
If YIJ
= 0, the rotation DlJx is given in terms of Dn and DIJ from equations (9.2h):
(9.93d)
which is identical to equation (6.55) obtained for the membrane theory rotation.
In addition, we may obtain two static conditions from the symmetry assumption. First, the transverse shear stress resultants NlJx must be equal and opposite
on any two coincident edges. When such edges lie along a symmetry line,
-
NIJX( - Ok) = 0
(exterior)
b
I;
arre
-
NlJx( ± Ok) = 0
(interiOr)
b
I
arre s
(9.94a)
is the only possibility. Similarly, any resultant vertical force Fz on one edge
must have an equal and opposite counterpart on the other edge. Thus, for edges
along symmetry lines,
Fz ( - 0 ) = 0 (exterior);
barrel
k
Fz ( ± 0 ) = 0 (interior)
k
barrels
(9.94b)
where, from figure 9-9(a),
Fz
= QlJcosO - NlJsinO
(9.94c)
Equations (9.91)-(9.94) prescribe the boundary conditions for a simplified
bending theory analysis of cylindrical shells, in which only two cases are
considered-an exterior and a typical interior barrel. Further, the two cases
are uncoupled, thus expeding the calculations.
We first compare the expressions for the membrane theory stress resultants
(derived in section 4.4.1) and the corresponding displacements (found in section
6.3.4.1) against these boundary conditions. In particular, consider the case of a
uniform dead load qd (force/area) represented by one term of a Fourier series,
as given by equation (4.165b):
qAX) =
~qd sin 1C X
1C
L
(9.95)
410
9 Shell Bending and Instability
For clarity, all quantities computed from the membrane theory equations will
be designated by the subscript m. We check the expressions for the stress
resultants, equations (4. 169a-c), and find that the static boundary conditions in
the X direction, N x = Mx = 0 at X = 0 and L, are satisfied. An examination of
the membrane theory displacements for this case, given by equations (6.56a-c),
reveals that the kinematic boundary conditions in the X direction, Do = Dn = 0
at X = 0 and L, are similarly satisfied. Thus, the only possible violation of
the membrane theory conditions may occur on the () = ± ()k boundaries.
Observe from figure 4-39 that the idealized membrane boundary must develop
NOm and Sm at () = ± ()k' This is in clear conflict with the exterior edge condition,
equation (9.92), where No and Nox, which corresponds to Sm, are required to
vanish. An interior edge can develop only the Y component of NOm' FYm =
Nom cos (), as shown on figure 9-9(a), leaving the Z component, Fzm = - NOm sin (),
unbalanced. With respect to the N ox or Sm force, however, a careful examination
of figure 4-39 reveals that the sense of Sm along two adjacent boundaries as
computed from the membrane theory solution would be in the same direction,
either in the ( +) or ( - ) X direction. On the other hand, two adjacent shells
would be expected to develop equal and opposite shear resultants along their
common boundary, since no resultant force along the X axis can exist. Moreover, if the interior valley is a symmetry line as assumed in developing equations
(9.94a), then Nox must be equal to O. It is apparent, then, that the longitudinal
boundary conditions implied for the development of the state of stress predicted
by the membrane theory are grossly violated.
The basic procedure to rectify these violations consists of starting with the
computed membrane theory stress resultants, which are in equilibrium with the
applied loading, and the corresponding displacements. Then, corrective edge
loads are applied to satisfy the boundary conditions at () = ± ()k' Mathematically,
this is equivalent to adopting the membrane theory solution as the particular
solution to the governing equations. This has been shown quantitatively to be
a sufficiently close approximation,20 which is hardly surprising in view of our
previous discussions.
With the membrane theory solution taken as the particular solution, we need
consider only the homogeneous portion of Donnell's equation for the corrective
edge loadings. Specifically, refer to the solutions given by equations (9.78), which
were proposed for loading along edges e = constant. Notice that these expressions are in the same Fourier series form as our membrane theory solution.
This is, of course, no coincidence, but the very reason for taking the membrane
theory solutions as Fourier series in the first place. As we have stated earlier, the
displacements, stress resultants, and couples corresponding to equations (9.78)
take the general form of equation (9.79). Now, we further assume that the semiinfinite simplification is applicable, so that the effects at opposite boundaries
() = ± ()k are uncoupled. This leaves only the first half of equation (9.79), with
undetermined constants C~-C!.
We first consider the exterior edge. The violation of the boundary conditions
411
9.2 Circular Cylindrical Shells
stated in equation (9.92) are precisely equal to the membrane theory stress
resultants evaluated at () = Ok. From equations (4. 169a-c),
Sm(X, ()k)
8qdL
X .
= --z-cos n-sm ()k
n
L
(9.96)
and
(9.97)
Next, from the semi-infinite form of equations (9.79), with k = 1, we select
expressions for Nix, Ni, MJ, and QJ in terms of the four integration constants
ct-Cl. Then, we enforce the free edge boundary conditions, equation (9.92),
by setting
No\
(~ , ()k) + Sm (~ , Ok) =
0
(9.98a)
0
(9.98b)
MJ(~ ,Ok) = 0
(9.98c)
Ni (~ , ()k) + NOm (~ , Ok)
=
QJ (~ , ()k) =
0
(9.98d)
after which ct-CJ are determined. Note that the functions of (X/L) [sin n(X/L)]
or [cos n(X/L)] are the same for both the membrane theory and bending theory
solutions, so that once the free edge condition is enforced at one point in the
interval 0 :::;; X :::;; L, it is satisfied at every point. This is another advantage of
the Fourier series representation.
Once the constants of integration are determined for the exterior edge solution,
the stress resultants and couples from the bending theory solution can be
evaluated at a sufficient number of coordinates [(X/L), ()] to obtain the stress
distribution. Of course, the membrane theory stress resultants are added to Nix,
Ni, and Ni to get the complete expression for these functions. Finally, the total
displacements may be computed as the sum of the membrane theory values,
equations (6.56a-e), and those found from the bending solutions with the
constants inserted. This solution is valid in the exterior portion of the shell
«()k ;:::: () ;::::
0).
We have not provided sufficient detail here with respct to the bending
solutions to perform calculations on actual shells, but detailed procedures
and examples are available (see Design of Cylindrical Concrete Roofszl and
BillingtonZZ).1t is often convenient to construct the bending theory corrections
to the membrane theory solutions as linear combinations of unit load solutions
412
9 Shell Bending and Instability
of the type
(9.99a)
and
Ni =
1; N(Jx
=
MJ
= QJ = 0
(9.99b)
Stich solutions are tabulated for a wide range of shell parameters in Design of
Cylindrical Concrete Roo/s. 23
We now take up the idealized interior edge, referring to the boundary
conditions given by equations (9.93) and (9.94). We use an analogous procedure,
superimposing the membrane theory solution with bending solutions for edge
loadings. The constants of integration are evaluated from the conditions
(~ ,Ok) + DYm(~ ,Ok) = 0
(9.100a)
DJx(~ ,Ok) + D(Jxm(~ ,Ok) = 0
(9.100b)
D;
Nix(~ ,Ok) + Sm(~ ,Ok) = 0
(9.100c)
Fi (~ ,Ok) + Fzm(~ ,Ok) =
(9.100d)
0
where the bending components are taken from expressions of the form of
equation (9.79), and the membrane theory parts are given by equations (6.56a-e)
and (4. 169a-c). Also, D y , and D(Jx are expressed in terms of D(J and Dn by
equations (9.93c) and (9.93d), respectively, and Fz is stated in terms of Q(J and
N(J by equation (9.94c). Again, these computations may be expedited by the use
of the tables in Design of Cylindrical Concrete Roofs. 24
Finally, with respect to simply supported circular cylindrical shells without
intermediate supports, we should assess the relevance of the rather complex
analysis we have outlined. We made a start in this direction at the conclusion
of section 4.4.1. Now we can amplify some of these remarks, since we have, in
principle, satisfied the boundary conditions on all edges.
Basically, at every cross section, the shell must resist the statical bending
moment due to the applied loading, regardless of the theory used. For a uniform
load w (force/length) the statical moment at the center line X = L/2 is the
well-known
_
L2
Mx=w-
8
(9.101)
9.3 Shells of Revolution
413
It is of interest to compare the magnitude and distribution ofthe stress resultant
N x over the cross section calculated by using the bending theory of shells to
the like quantity found from elementary beam theory
Mx Z
NX=-I-
(9.102)
where I = the moment of inertia of the cross section with unit thickness about
the centroidal axis. This may be regarded as a gross measure of the relevance
of the shell theory, as opposed to beam theory, calculations. In figure 9-9(b),
we show the distribution of N x over the cross section for a single barrel shell
analyzed in Design of Cylindrical Concrete Roofs,25 along with the straight line
distribution given by equation (9.102). Note that both the maximum tensile
and compressive stress resultants are more than double the values computed
from the linear strain, beam theory formula. It has been verified that both
distributions of N x yield practically the same statical moment Mx about the
centroid of the cross section. 26 Obviously, the analysis of an open cylindrical
shell by elementary beam theory is grossly inaccurate, insofar as the elastic
response is concerned.
9.3 Shells of Revolution
9.3.1 General: The bending of shells of revolution is a relatively complex
subject from a mathematical standpoint. Also, the availability of very efficient
and powerful numerically based computer programs has made many of the
classical means of solving such problems somewhat archaic. Moreover, the
classical solutions are developed in considerable detail in a number of popular
monographs (Novozhilov,27 Fliigge,28 and Kraus 29 ), and little would be accomplished here by restating these lengthy derivations. Instead, we explore
only some of the important classical findings, and then present an energy
formulation suitable for a finite element solution.
9.3.2 Axisymmetrical Loading-Analytical Solutions
9.3.2.1 Governing Equations. An early classical formulation of the shell bending
problem was presented by H. Reissner (the father of E. Reissner), who studied
the spherical shell. 30 This "mixed" formulation, generalized by E. Meissner,31
represents the fourth order system by a coupled set of two second order
equations in terms of the meridional rotation D,,6 and the transverse shear
resultant Q,,; here, the derivation is generalized to include specifically orthotropic
materials.
We begin by specializing the equilibrium equations, equations (3.24a, c, e),
for the case of axisymmetric bending:
414
9 Shell Bending and Instability
(RoN",).", - R",cosr/lNe + RoQ",
+ q",R",R o = 0
R",sinr/lNe + qnR",Ro = 0
(9.103a)
(RoQ",).", - RoN", -
(9.103b)
(RoM",).", - R", cos r/lMe - R",RoQ", = 0
(9.103c)
We next combine the strain-displacement relationship for axisymmetric shells,
equations (5.48a, c, d, g), and the constitutive law, equation (6.12), into a set of
stress resultant and stress couple-displacement equations:
N",
= Dorll [~'" (DM + Dn)]
+ Dorl2[~o (cosr/lD", + sinr/lD
n )] -
Ne = Dorl2
(9.104a)
N"'Tor
[~'" (DM + Dn)]
+ Dor22[~o (cosr/lD", + sinr/lDn) ] -
(9.104b)
NeTor
(9.104c)
Me =
Dor45[~", D",e.",] + Dorss[c;/ D",e] -
M"'Tor
(9.104d)
Symmetry of the constitutive matrix is presumed, so that Dor12 = Dor21 and
Dor45 = Dor54 ' Also, as before, the thermal terms are presumed to be known at
the outset.
The boundary conditions corresponding to the axisymmetric formulation
are specialized from 1igure 6-3 as N", = fi", or D", = D",; Q", = Q", or Dn = Dn; and
M", = M", or D",e = D",e'
We now substitute equations (9.104c) and (9.104d) into equation (9.103c).
To simplify the derivation here somewhat, the elements of the constitutive
matrix [Dor] are taken as constant, but they can be generalized for shells with
continuously varying material properties and/or thickness.
Continuing, we have
D",e.#
+ ~'" (~o)
o
D",e.",
"'.'"
(9.105)
which is the first of two governing equations in D",e and Q",.
415
9.3 Shells of Revolution
To obtain the second equation, we first consider equations (9.104a) and
(9.104b). We form the combinations
~[(9.104a) Dor11
Dor12 (9.104b)]
Dor22
and
~[(9.104b) Dor22
Dor12 (9.104a)]
Dor11
and rewrite the expression (1/Ro)(cost/JD",
to get
+ sin t/J Dn)
as (1/R e)(cott/JD",
+ Dn)
(9.106a)
and
where
111
Dor12
Dor11
= --
an d
(9.106c)
Next, we eliminate Dn between equations (9.106a) and (9.106b) to find
D",.", - cot t/J D",
=
(1
1
+ /11/12
R", + ~
111Re) (N", + N"'Tor)
) [(Vor 11
or22
(9.107a)
and differentiate equation (9.106b) with respect to t/J, which gives
2
cott/JD",.",-csc t/JD",+Dn.",=
Dor22
X
(1
1
)
- /11/12
{Re[ -/11(N",
+ (Ne
+ N"'Tor)
(9.107b)
+ NeTor )]},'"
We then eliminate D",.", from equations (9.107a) and (9.107b) by taking
[(9.107b) - cot t/J (9.107a)]. The l.h.s. becomes
(9.108)
It is now convenient to suppress the transverse shearing strains. From
equation (S.48g), note that for Y", = 0,
416
9 Shell Bending and Instability
D",o
=
1
-R(Dn.", - D",)
(9.109)
so that -(1/R",)'" [l.h.s. (9.108)] = D",o, which is one of our dependent variables.
To complete the derivation, the stress resultants N", and No on the r.h.s. of
equation (9.107b) must be expressed in terms of the other variable Q",. This is
most easily accomplished by using the overall equilibrium concept introduced
in section 4.3.3.1. With Q", now included, equation (4.57) generalizes to
(9.110)
where Q(¢) is the resultant vertical load in the negative Z direction, expressed
as a function of the surface loading by equation (4.59). Therefore,
N",
Q(¢)
= 2 nRo SIll
. ¢ + Q", cot ¢
(9.111a)
Now, considering equation (9.103b) in view of equation (9.111a),
1
No = R", sin ¢ [(RoQ",),,,, - RoN",
+ qnR",Ro]
= R 1. ¢ [(RoQ",),,,, - Ro cot ¢ Q", - 2 Q(~) ¢ + qnR",Ro]
(9.111b)
n SIll
'" SIll
Equations (9.111a) and (9.111b)may be used to remove N", and No from the r.h.s.
of the combined equations (9.107a) and (9.107b), i,e., [(9.107b) - cot ¢ (9.107a)],
which then becomes a second equation in D",o and Q",. This is tedious to carry
through in the general form; consequently, at this point, the specialization for
a commonly encountered case seems prudent.
9.3.2.2 Homogeneous Solution for Spherical Shells. Consider an isotropic shell
and no surface loading or thermal effects. Then, qn = Q(¢) = 0; J11 = J12 = J1;
Dorll = Dor22 = Eh/(1 - J12); Dor44 = Dor55 = D; and Dor45 = J1D, where D has
been previously defined as Eh 3 /[12(1 - J12)]. Also, for the spherical geometry,
R", = Ro = a and Ro = a sin fjJ. We may anticipate that the forthcoming solution
will represent the influence of edge forces and moments which, in combination
with a membrane theory solution for the surface loading, constitutes a complete
solution to the spherical shell problem.
Now, we proceed with the aforementioned ([9.107b] - cot ¢ [9.107a]) combination, which, in view of equation (9.109), gives
1
D",o = Eh [(1
+ J1) cot fjJ(N",
- No) - No.",
+ J1N",.",]
(9.112)
Equations (9.111a) and (9.111b) simplify to
N", = Q",cotfjJ
(9.113a)
No=Q""",
(9. 113b)
417
9.3 Shells of Revolution
where we have employed the Gauss-Codazzi relationship Ro.; = R; cos ,p via
equation (2.47).
Substituting equations (9. 113a) and (9.113b) into (9.112), we have
1
D;8 = - Eh [Q;.#
+ cot,p Q;.; -
2
(cot ,p - Jl)Q;J
(9. 114a)
The specialized form ofthe first of our governing equations, equations (9.105), is
a2
])Q; = D;8.;;
+ cot ,pD;B.; -
(cot 2 ,p
+ Jl)D;8
(9. 114b)
Noting the similarity between the preceding two equations, we may define the
operator32
L( ) = ( ).#
+ cot,p(
).; - cot 2 ,p( )
(9.115)
whereupon the equations become
L(Q;) + JlQ;
L(D;8) - JlD;8
=
-EhD;B
a2
= ])Qt/>
(9. 116a)
(9.116b)
We substitute equation (9. 116b) into (9.116a), and (9. 116a) into (9. 116b), to get,
respectively,
LL(D;8) - Jl 2D;8
=
a 2 Eh
---VD;8
a 2 Eh
LL(Q;) - Jl2Q; = ---V Q;
(9. 117a)
(9. 117b)
so that the equations are uncoupled. When one variable is determined, the other
may be immediately found from equations (9. 116a) and (9.116b), and all other
terms follow from the previous formulae.
It is of interest to rewrite equation (9. 117b) in the form
LL(Q;)
+ 4k4 Q; =
0
(9. 118a)
where
(9. 118b)
The resulting equation (9.118a) is quite similar to equation (9. 16a), which
describes cylindrical shells. Here, of course, the operator L( ) causes more
complication in the eventual solution.
It is also convenient to express equation (9.118a) in complex form as 33
418
9 Shell Bending and Instability
+ 2ik 2QI,bJ
- 2ik 2 [L(QI,b)
+ 2ik 2QI,bJ =
0
(9. 119a)
L[L(QI,b) - 2ieQI,bJ
+ 2ik 2 [L(QI,b)
- 2ik 2QI,bJ = 0
(9. 119b)
L[L(QI,b)
which can be easily verified to be equivalent to the original equation.
Thus, the solution of the two second order equations:
L(QI,b)
+ 2ik 2QI,b = 0
L(QI,b) - 2ik 2QI,b
=
(9. 120a)
(9. 120b)
0
will constitute the solution to the fourth order equation (9.118a).
The formulation of the governing equations for other geometries follows
similar, but sometimes more complicated, steps. The interested reader is referred
to Novozhilov,34 Fliigge,35 and Kraus. 36
An exact solution to the governing equations for the spherical geometry
exists. If one of the equations, e.g., equation (9. 120a), is fully expanded, we have
QI,b.#
+ cot rp QI,b.1,b -
cot 2 rp QI,b
+ 2ik 2QI,b
=
0
(9.121)
which is a second order linear differential equation with variable coefficients.
The further transformations
(9. 122a)
and
QI,b
=
QI,b sin rp
(9. 122b)
put the equation into the form
QI,b,1/t1/t
1 - 51/1 1 - 2il/l2 - 1/1) QI,b,,,, - 41/1(1 _ 1/1) QI,b
+ 21/1(1
=
0
(9.123)
which is known as the hypergeometric equation. Such equations may be solved
with some rather slowly converging power series expressions. 37
9.3.2.3 Asymptotic Solutions for Spherical Shells. One of the most powerful
methods for solving the complicated differential equations that arise in the
bending theory of shells is that of asymptotic integration. This method is
developed in Kraus 38 and Novozhilov 39 and applied to both symmetrically
and asymmetrically loaded shells of revolution in the latter reference. A detailed
development is beyond our objectives in this book. It is sufficient to mention
here that the asymptotic integration technique, as applied to second order
equations such as equation (9.121), transforms the equation into a form in which
the first derivative of the dependent variable is eliminated.
The attractiveness of solving the bending theory equations for rotational
shells in a relatively simple way has given rise to some useful approximate
methods, in which it is postulated that the solutions to the homogeneous
equations have the form
419
9.3 Shells of Revolution
e±k¢ sin k¢ }
e±k¢cos k¢
(9.124)
Although expressions of this type can be rigorously derived using asymptotic
integration, such solutions may be obtained directly by simply neglecting certain
terms in the original differential equation. This is applicable for shells in which
certain geometric parameters remain essentially constant over a significant
portion of the shell. For example, note in equation (9.121) that the second
and third terms are multiplied by cot ¢ and cot 2 ¢, which are relatively small
when ¢ z n/4, whereas the fourth term has a coefficient of k 2 , which, from
equation (9.118b), is proportional to a/h. Thus, it is reasonable to use the
simplified equation
(9.125)
to find approximate solutions for the bending of spherical shells under edge
loads. If we are interested in solutions in the vicinity of the lower boundary of
spherical shells due to edge loading, as shown in figure 9-10, it is only necessary
that ¢b not be too small. This reasoning is also applicable for other shells with
smoothly varying geometry, such as conical shells, and is sometimes referred
to as the Geckeler approximation. 40 Despite the lack of mathematical rigor,
such "quick and dirty" solutions are often the only relatively simple analytical
means available to check complex, computer-based analyses.
We may write the solution of equation (9.125) from the auxiliary equations
(9.126)
Fig.9-1O
Edge-Loaded Spherical Shell
420
9 Shell Bending and Instability
which give
m
=
(9.127)
±(1 ± i)k
so that the complete solution is
Q</>
= B 1e(1+i)k</> + B 2e(1-i)k</> + B 3e-(1+i)k</> + B4e-(1-i)k</>
(9.128)
where Bl - B4 are integration constants.
We now wish to redefine the integration constants so that Q</> is expressed
in terms of real functions. First, we use the identities
e~k</>
e-· k</>
=
=
cos k¢J + i sin k¢J}
cos k¢J - i sin k</J
(9.129)
to rewrite equation (9.128) as
Q</>
= (Bl +
B 2)e k</> cos k</J
+ i(Bl - B 2)e k</> sin k¢J
It is convenient to express this solution as a function ofthe auxiliary coordinate
(9. 130b)
which we show on figure 9-10 and introduce into the first term of equation
(9. 130a):
(Bl
+ B 2)e k</> cos k¢J = (Bl + B 2)e k(</>b- ih cos k(¢Jb - ¢)
= (Bl + B 2)ek</>be-k¢(cos k¢Jb cos k¢)
+ sin k¢Jb sin k¢)
= [(Bl + B 2 )ek</>bcos k¢JbJ e- k¢ cos k¢)
+ [(Bl + B2)e k</>b sin k,p"J e- k¢ sin k¢)
(9.l30c)
Similarly, the second term becomes
i(Bl - B 2)e k</> sin k¢J = [i(Bl - B2)e k</>b sin k¢JbJ e- k¢ cos k¢Jb
- [i(Bl - B2)e k</>b cos k¢JbJe- k¢ sin k¢)
(9. 130d)
The terms in [ J in the preceding equations may be collected as coefficients
of the variable terms e- k¢ cos k¢) and e- k¢ sin k¢), and can be redefined as new
arbitrary constants C 1 and C 2 • We retain the last two expressions in equation
(9. 130a) in terms of ¢J, with (B3 + B4) and -i(B3 - B4) redefined as C3 and C4,
respectively, so that the revised form of the solution is
Q</>
= e-k¢(C1COsk~ + C2 sink¢) + e- k</>(C3 cos k¢J + C4 sin k¢J)
(9.131)
Equation (9.131) is identical in form to equation (9.20), if¢) is replaced by </Jb - </J
and the constants are redefined. This comparison gives rise to the physical
interpretation of the Geckeler approximation as the replacement of the actual
shell by a cylindrical shell of suitable dimensions.
421
9.3 Shells of Revolution
The foregoing transformation is an alternate means of introducing the semiinfinite concept that we utilized extensively for cylindrical shells. The solution
is uncoupled into parts that decay from the boundaries ¢J = ¢Jb and ¢J = ¢Jt,
respectively. If the influence of the loading at one boundary does not penetrate
into the region of the shell affected by the loading at the other boundary, we
have a semi-infinite shell. Considering edge loading originating at ¢J = ¢Jb' we
retain
(9.132)
We now evaluate the corresponding stress resultants, couples, and displacements. In performing the required differentiations, take care to note that
( ).91 = -( ),.~
(9.133)
From equation (9.111a), with Q(¢J)
=
0,
N9I(¢J) = Q9I cot ¢J
(9. 134a)
= e- k¢ cot ¢J(C1 cos k(j) + C2 sin k(j))
(9. 134b)
N 9I ((j)) = e- k¢ cot(¢Jb - (j))(C1 cos k(j) + C2 sin k(j))
(9. 134c)
Expanding equation (9.103b) with qn
=
0, we have
a(sin ¢J Q9I).9I - a sin ¢J Nq, - a sin ¢J No =
°
(9. 135a)
Solving for No, after replacing Nq, with equation (9. 134a), we find
No((j))
=
QM
(9.135b)
= - Q9I.¢
= (C 1
-
C 2 )ke-k¢cosk(j) + (C 1
+ C2 )ke- k ¢sink(j)
Next, we turn to the rotation Dq,o, which is expressed by equation (9.116a). It is
consistent with our previous approximations to take L( ) ~ ( ).q,q, and also to
neglect the IlQ9I term, whereby
-
1
D9Io(¢J) = - Eh QMq,
=
-
1
Eh Q9I.¢¢
(9.136)
The actual displacements are sometimes of interest. Dq, and Dn may be individually
found from the simultaneous solution of equations (6.34a) and (6.34b). Of more
use for our purposes is the radial displacement DR, which is defined in the inset
of figure 9-10 as
(9.137a)
422
9 Shell Bending and Instability
Referring to equation (6.34b) with Ro(¢)
DR(</J)
=
iiBo
=
=
a sin ¢) = ii,
ii
Eh (No - /1N",)
(9. 137b)
which is easily evaluated from equations (9.134c) and (9.135b).
We now turn to the stress couples, which can be obtained from equations
(9.104c) and (9.104d). Again, it is consistent with our previous arguments to
neglect the second terms in these equations in comparison to the first, whereby
M",
=
D
(9. 138a)
--;;D",o,,,,
which is conveniently written, using equations (9.136) and (9.118b) with /1 2
neglected, as
a
a
M", = - 4k4 Qt/J,t/J",,,, = 4e Q"',iii
(9. 138b)
which gives
M",(¢)
= ;ke-ki[(Cl + C2 )cosk¢) - (C 1
Also, from equation (9.104d) with Dor45
Mo(¢)
=
= /1,
-
C 2 ) sin k¢)J
(9. 138c)
and (9. 138a)
(9.139)
/1M",(f/J)
We have thus derived approximate, but explicit, expressions for the forces,
moments, and displacements in an edge-loaded spherical shell. From these
equations, it is obvious that the assumption leading to equation (9.125)-the
neglect of Q", and Q""", as compared to Q""",,,,-is reasonable since each subsequent differentiation gives a factor k that is of o (a/h). Moreover, the presence
of the cot </J and coe </J coefficients of Q""", and Q"" respectively, in equation
(9.121) further reduces the influence of the latter terms when </J > rc/4. The limits
of application of the semi-infinite solution can be established using figure 9-2,
with kX replaced by k</J where k is given by equation (9.118b).
For practical application, it is helpful to represent the previous solution in
the same format as we used for the edge-loaded cylindrical shell. Consider the
loading shown in figure 9-10, which is analogous to figure 9-1 for the cylindrical
shell. We take
Mo = M",(</Jb)
(9. 140a)
Qo = Q",(t/J,,)
(9. 140b)
whereupon, from equations (9.132) and (9.138c) with
Qo
=
C1
f/J = 0,
(9.141a)
(9.141b)
9.3 Shells of Revolution
423
from which
Cl
= Qo
2k
C2 =-Mo - Qo
a
(9. 142a)
(9. 142b)
A matrix of derivatives of Q¢, similar to that written for Dn in matrix 9-1, is
useful for further computations.
9.3.2.4 Compound Shells. We are now able to resume our consideration of the
cylindrical shell with a spherical head, as shown in figure 4-14 and discussed
in section 4.3.4. Previously, we found that the membrane theory, by itself, was
inadequate, since (a) radial deformations between the two shells were incompatible; and, (b) except for the complete hemisphere, the radial component of
the meridional force in the spherical shell was unbalanced. We may now fully
analyze the shell by the flexibility method described in section 6.3.1, using the
notation introduced in equation (6.30).
On figure 9-11(a), we show the pressurized shell from figure 4-14, with the
axial coordinate taken as X, and the origin placed at the end of the cylinder.
The respective thicknesses of the spherical and cylindrical segments are defined
as hI and h2' and the Young's moduli and Poisson's ratios as El and E2 and
fl.l and fl.2· Note that sintfrl = a2la l ·
Loading case [9.11(b)] requires the membrane theory solutions for the cylindrical and the spherical shells, given in equations (4.74)-(4.76) and equations
(6.38a-e), as well as a bending solution for the cylindrical shell, given by
equations (9.21)-(9.29) and matrix 9-1, with Qo = p(ad2) cos tfrl and Mo = O.
As we have noted previously, the membrane theory solution for the internal
pressure loading gives incompatibilities for both the relative radial displacement [figure 9-11(d)] and the relative meridional rotation [figure 9-11(e}] of
the spherical and cylindrical shells at their junction. We denote these quantities
in the original shell, case [9-11(a)], as Lll and Ll2' and in case [9-11(b)] as ~1
and ~2' There is no contribution to ~2 from the membrane theory solutions
for either segment, since, under the uniform internal pressure loading, each shell
deforms without any rotation of the meridian. This is easily visualized. However, there still remains a contribution of ~2 from the line load p(ad2) cos tfrl'
To restore the compatibility, we provide loading case [9-11(c)], consisting
of the static correspondents of Lll and Ll 2 -namely, a radial transverse shear
force Xl and a meridional moment X2' equal and opposite on each segment.
Solutions for these loading systems are given for the cylindrical shell by equations (9.22)-(9.29) and matrix 9-1 with Qo = Xl and Mo = X2' where the signs
are established in accordance with figure 9-1; and for the spherical shell by
equations (9.132)-(9.142) with Qo = Xl sin tfr, and Mo = X2' where the signs are
established in accordance with figure 3-6. For the flexibility formulation that
we are following, it is expedient to solve the following loading cases: (i) Qo = 1,
424
9 Shell Bending and Instability
X---lI.---=
(0)
---0--0
(b)
7
+
X,C05t/J,
(c)
Fig. 9-11
Compound Shell
Mo = 0 on the cylindrical shell, and Qo = sin tPl' Mo = 0 on the spherical shell;
and, (ii) Qo = 0, Mo = 1 on the cylindrical shell, and Qo = 0, Mo = 1 on the
spherical shell. For case (i), the relative radial displacement and rotation are
designated as Fll and F 21 ; whereas, for case (ii), the corresponding quantities
are labeled F12 and F22 •
We now consider the details of the evaluation of the individual terms in
425
9.3 Shells of Revolution
equation (6.30) for this problem:
{.::\l}
.::\2
=
{~l} + [Fll F12]{Xl}
.::\2
F21
F22
(9.143)
X2
First, note that we are dealing with relative displacements and rotations,
which are computed from the absolute displacements of the two segments of
the shell. Therefore, we must carefully define the sign convention for these
quantities. Corresponding to the senses assumed for the redundants Xl and X2
in figure 9-11(c), it is consistent to take the positive sense of the relative radial
displacement as increasing, i.e., inward on the cylinder (del) and outward on
the sphere (d.d. Similarly, a positive relative meridional rotation will open
the internal angle between the cylinder (de2 ) and the sphere (d. 2 ), originally
n/2 + rPl' These are illustrated on figure 9-1l(d) and (e). For example, if we
compute the radial displacement in the cylinder Dn due to one of the loading
cases, and it is positive (outward) by the cylindrical shell convention, it takes
a negative sign in the relative displacement expression, since it is opposite to
in sense to del' On the other hand, a positive computed value for the radial
displacement of the spherical shell DR is consistent with dsl , so that it has a
positive sign in the relative displacement expression. Similarly, a positive meridional rotation DX/J in the cylindrical shell may be seen from figure 5-2(a) (with
0( = X and fJ = 0) to be opposite in sense to de2 , so that it gets a negative sign.
The rotation of the spherical shell is established from figure 5-2(a) (with 0( = rP
and fJ = 0), where a positive meridional rotation DI/J/J(rP), corresponds to a
positive d. 2 •
Next, note that the compatibility condition requires that the relative radial
displacement and relative meridional rotation between the two segments vanishes, or
(9.144)
For the evaluation of the elements of {X} and [F], we refer to the chart of
basic solutions given in table 9-1. Note that the cylindrical and spherical
segments may each have different constant thicknesses and material properties.
In the last column, the most probable algebraic sign of the computed displacement or rotation on the shell segment is assumed, and the corresponding
algebraic sign of the contribution of this quantity to the relative displacement
is given.
First we calculate the displacements with the continuity constraints relaxed,
Xl and X2 •
For the cylindrical shell, we compute the normal displacement du~ to the
internal pressure from equation (9.42a) with N x == p(ad2) sin rPl and N/J = pa 2 as
Dn(O)
=
p
E:~2 (a 2- Ial sin rPl) = p E:t (1 -
which is a negative contribution to
Xl'
I)
(9.145a)
X=O
k4 _ 3(1 - jJ.n
2 2
(a2h2)
,
E = E2 D = D2 =
a = a2
h = h2, jJ. = 112
E2h~
12(1 - jJ.~)
Qo =0
Mo = 1
Qo = 1
Mo=O
Mo=O
matrix 9-1, row (a)
D.
F12
F22
matrix 9-1, row (a)
(9.6d) with Yx = 0;
then
matrix 9-1, row (b)
D.
DX8
F21
(9.6d) with Yx = 0;
then
matrix 9-1, row (b)
DX8
F11
[\2
(9.6d) with Yx = 0;
then
matrix 9-1, row (b)
DX8
~1
Contributes
to
~1
Qo = p(ad2) cos 1P1
N8 = pa2
and
N" = p(ad2)sinIP1
(9.42a) with
Equations
matrix 9-1, row (a)
D.
p
Compute
D.
(DX9 = 0)
Loading
Basic solutions for a compound shell
Segment
Cylindrical
Table 9-1
-DX8 -+ +dc2
-D. -+ +dc1
-DX8 -+ +dc2
-D. -+ +dc1
-DX8 = +dc2
-D. -+ +de1
+D. -+ -del
Sign
Convention
tv
.".
.:;-
~
I»
QQ
"e:=
=
...=Po.
=
~
t::tI
~
c.-
til
\Q
01
= <PI
<Pb
=
k1 = 3(1 -
¢=O
<P
Iln(::Y
= DI
I,
=
l
h=h
,Il=1l1
E E D
Spherical
R¢>=Ro=a
a = al
Table 9-1 Continued
Qo = 0
Mo = 1
or
CI = 0
Cz = 2k l la l
Qo = sin <PI = azla l
Mo=O
or
C I = sin <PI = azla l
Cz = -sin<pI = -azlal
P
F12
Fzz
(9. 137b) with
N¢ and No from
(9.134c) and (9. 135b)
(9.136)
D¢>8
FZI
Fl1
(9.137b) with
N¢> and No from
(9.134c) and (9.135b)
(9.136)
~I
(9.137b) with
N¢> = No = p(ad2)
DR
D¢>o
DR
DR
(D¢>o = 0)
+dsz
+D¢o-> +dsz
-DR-> -dsl
+ D¢>o ->
+DR -> +dsl
+DR -> +dsl
tv
-...)
oj::.
I:)
o·
[
o
~
8.,
'"
g,
Vl
::r"
'"w
428
9 Shell Bending and Instability
Also, we have the normal displacement due to the unbalanced transverse
shear imparted by the spherical cap to the cylinder, Qo = p(ad2) cos ¢JI' which
is found from matrix 9-1, row (a), as
D (0) = _ a l COS¢JI
n
p 4D2k~
(9. 145b)
providing a positive contribution to d l . The accompanying rotation is found
from matrix 9-1, row (b), as
Dxo(O)
=
-Dn.x
=
a l cos ¢JI
-p 4D2k~
(9. 145c)
which is a positive contribution to A2 .
For the spherical shell, we have the radial displacement due to the internal
pressure from equation (9.137b) with NtjJ = No = p(ad2) and a = a l sin¢JI'
D (0)
R
= ai(l -
Jl) sin ¢JI =
a l a 2 (1 _ )
2Elhl
P 2Elhl
Jl
p
(9.145d)
giving a positive contribution to AI' This loading does not produce a contribution
to A2 .
Next, we compute the contribution of the unit actions on the cylindrical shell
to the flexibility influence coefficients.
Solving for Dn with Qo = 1 and Mo = 0 inserted into matrix 9-1, row (a),
we find
1
(9. 146a)
which is a positive contribution to F II .
The same loading in matrix 9-1, row (b), with the use of equation (9.6d), gives
Dxo(O)
1
= -Dn.x(O) = - 2k~D2
(9. 146b)
which is a positive contribution to F21 .
Proceeding, we repeat the computations with Qo = 0 and Mo = 1, which
gIVes
(9. 146c)
providing a positive contribution to F12 and
Dxo(O)
1
= -Dn.x(O) = - - k 2D2
which is a positive contribution to F 22 •
(9. 146d)
429
9.3 Shells of Revolution
Finally, we have the contributions to the flexibility influence coefficients from
the unit actions on the spherical shell.
Solving for DR with Qo = sin £P1 and Mo = 0, we find C1 = sin £P1 and C2 =
-sin£P1 from equations (9.142). Then, from equations (9. 134c) and (9. 135b), we
evaluate NtP(O) = cos £P1 and Ne(O) = 2k1 sin £P1' which are inserted into equation
(9.137b) to produce
DR(O) =
a 1 sin £P1
.
h (2k1 sm £P1 - 11 cos £Pd
E1 1
2 (2k1
= -a - -a-2 EIhl
a1
(9. 147a)
A. )
{lCOS'!'1
giving a positive contribution to F 11 .
The same loading used in equation (9.136) yields
2
DtPe(O) = - El hI ki sin £PI =
2a2ki
E 1h 1a 1
(9. 147b)
which is a negative contribution to F21 .
Proceeding, we repeat the computations with Qo = 0 and Mo = 1. This gives
C I = 0 and C2 = 2(kdad; and NtP(O) = 0 and Ne(O) = -2ki!a 1. Then, we find
(9.147c)
which is a negative contribution to F12 and
4k~
(9.147d)
which is a negative contribution to F22 .
Collecting the results, we have from equations (9. 145a), (9.145b), and (9. 145d),
-
~
_ p [ - -a~E2h2
1 -
(1
- -112)
2
a1a2(1- 11 1)]
+ aIcos£P1 + --=--~:--::---'-::':"
4D2k~
2E1h1
(9.148a)
and from equation (9.145c),
-
a l cos £PI
~2 = P 4D2k~
(9.148b)
Also, from equations (9. 146a) and (9. 147a),
(9.148c)
(9.1 48d)
430
9 Shell Bending and Instability
from equations (9.146c) and (9.147c),
2a2 k2
F 12 -_ _1_ 1
2k~D2
E1 h1 a 1
(9. 148e)
and, from (9. 146d) and (9.147d),
_
1
4
3
F22 - - - k1
k2D2
E 1 h 1a 1
(9. 148f)
Note that F21 = F12 , a check of symmetry.
The equations can be put in better order for evaluation if we write the shell
parameter k2 for the cylindrical shell in the same nondimensional form as k1
for the spherical shell, i.e.,
ki = 3(1 -
Iln (::
Y
(9.149a)
= kiai
Then, we may state equations (9. 148a-e) as
K 1 -
P
[_ a~
E h
(1_1l2)
22
2
1 2
A..
+ Ea 2ah2 k 2 cos 'f'1
+a
1a 2(12E h
lld
J
(9. 149b)
11
(9. 149c)
(9.149d)
(9. 14ge)
F22 =
4
E 2h 2 a 2
P
2 -
4
E 1h 1a 1
k3
1
(9. 149f)
If the materials and/or the thicknesses for the two shells are the same, the
equations can be further simplified.
Once all of the terms are evaluated,
(9.150)
in view of equation (9.144). After determining Xl and X2' the stress resultants,
couples, and displacements in the cylindrical and spherical segments are found
as linear combinations of the solutions for the loading cases shown in figure
9-11(b) and (c).
We shall not pursue further details of the calculations here, but refer to
a numerical study in Fliigge. 41 There, two cases are considered, a hemispherical
head and a shallower cap with ¢J1 = 45°. For both cases, h1 = h2 = h, h/a = 0.01,
431
9.3 Shells of Revolution
Fig.9-12(a-b)
Stress Resultants and Moments in Compound Shells
and J1 = 0.3. On figures 9-12(a) and (b), comparative values of the meridional
stress couple, Mx or Mt/>, and the circumferential or hoop stress resultant, No,
are shown for the two examples.
The meridional moments increase greatly for the discontinuous case. Similarly, for the circumferential stress resultant, the hemispherical head provides
a smooth transition for the hoop stresses in the two segments; the shallow
cap results in an increased magnitude of No as well as a change in sense to
compression, which gives rise to the possibility of circumferential buckling or
wrinkling. An independent calculation 42 gave a maximum No = -7.8pa for
the shallow cap.
Although the hemispherical head might seem more attractive on the basis of
this comparison, the fabrication advantages of shallow caps have been pointed
out previously, and the practical solution is often to include a circumferential
ring stiffener, as we see in figure 2-8(r). Details regarding the design of such
stiffeners and other practical considerations are found in Pirok and Wozniak. 43
When an elastic stiffener is used, it is possible to generalize the model to include
the stiffener, following the same reasoning we employed in section 9.2.2.7.
A comprehensive study of ring-stiffened cylindrical-conical shells under hydrostatic loading was performed by the author and his co-workers, and the results
are reported in Gould et al. 44 and Wang and Gould. 45 In general, an adequate
circumferential stiffener greatly moderates the extreme amplifications caused
by a geometric discontinuity.44
Another possibility for reducing the stress amplification due to a geometric
discontinuity is the use of a transition segment, such as a torospherical head,
discussed in section 4.3.4.2 and illustrated in figures 4-16(b) and 4-17. Ranjan
432
9 Shell Bending and Instability
08 mi ddle
surface
I
I
I
I
---.-.- 0.2 in thickness 150 psi Uni form
Pressure
200
150
-
Membrane
Theory
100
."",---- ......
I
\
50
\
\
\
60
70
....
--
I
I
\
50
/
Arc Length from Pole
90
~O
-50
\
\
t'
100
V
110
..........
~-
j
I
I
120
130
140
;,,-
\ /,
-100
-150
I
-200
~
Fig.9-12(c)
SPHERE
~I~
TORUS
~14
CYLINDER - -
Circumferential Stress on Middle Surface of Torospherical Head
and Steele 46 have examined the range of applicability of approximate analytical
solutions derived from asymptotic expansions of the dependent variables to
the solution of torospherical shells. From their study, one can draw some
conclusions as to the optimum radius of the toroidal knuckle as a function of
the cylindrical and spherical radii and the shell thickness. Also, their study
suggests a lower bound approximation for the critical internal pressure corresponding to circumferential wrinkling. Although they obtain fairly accurate
9.3 Shells of Revolution
433
solutions with their analytical formulae, based on comparisons with numerical
and experimental results, it would appear that the best present technology for
an actual design case would include a finite element analysis using doubly
curved rotational shell elements.
The finite element approach is introduced in section 9.3.4 and is widely
documented in the technical literature. Results from such an analysis ofthe shell
treated in section 4.3.4.2 are shown in figure 9-12(c), where the smoothing of
the peak membrane theory stresses is apparent. Further results for this shell,
such as displacements and stress couples, are also available. 47
9.3.3 Asymmetrical Loading-Analytical Solutions: When the loading on a
shell of revolution is not axisymmetric, the Fourier series technique that we
employed extensively in chapter 4 is again useful to separate the independent
variables. The problem then reduces to the solution of a set of ordinary differential
equations for each harmonic j.
General solutions for spherical and conical shells are given by Fliigge. 48
These solutions are based on the classical displacement method and are exact
in the sense that only quantities of O(h2/R2): 1 are neglected. These solutions
are of value in studying such cases as static wind loading and discrete column
supports, which we discussed in sections 4.3.6.2 and 4.3.7.3., respectively.
For geometries other than spherical and conical and, of course, cylindrical
(which we have treated separately), exact solutions are scarce.
As a general approach, the formulation of Novozhilov 49 is very attractive.
He follows the classical force method rather than the displacement procedure
that we have employed almost exclusively in our study of plate bending in
chapter 8 and of cylindrical shells in this chapter. We did use a force formulation
for the membrane theory in chapter 4, but this involved only the equilibrium
equations, since that problem is statically determinate. In the bending theory,
however, the situation is more complicated, and it is necessary to express the
strain-displacement or compatibility equations, (5.52) and (5.53), in terms of
the stress resultants and couples. Once this is accomplished via the constitutive
law, the resulting expressions are combined with the equilibrium relations to
provide a consistent set of equations in which the stress resultants and couples
are the unknowns. In this context, it is of interest to note that the H. ReissnerMeissner approach presented in section 9.3.2.1 is a combination of a force and
a displacement formulation, since the dependent variables are Qt/J and Dt/J8' This
is often termed a mixed formulation.
We may also point out an interesting feature of the Novozhilov formulation
of the shell bending problem, whereby the dependent variables are grouped
into complex variables. This results in a halving of the order of the eventual set
of equations to be solved and thereby simplifies the algebra. The solution of
the ensuing equations is accomplished by the asymptotic integration technique,
to which we have alluded previously. Novozhilov's solution is applicable for
surface loading which is described by the first two harmonics, j = 0 and j = 1.
434
9 Shell Bending and Instability
Moreover, shells with a special form of the meridian, i.e., (l/Rq, - 1/Ro)csc 2rP =
constant, can be solved for the general harmonic loadingj > 1, which suggests
a substitute curve approximation for other meridional profiles. Beyond this,
it is generally necessary to neglect terms of O[P(h/R)] to achieve a solution;
thus, for higher harmonics, the accuracy diminishes. However, a large number
of practical problems can be described rather closely in terms of the lower
harmonic load components-such as the wind loading shown on figure 4-34so that the Novozhilov method has wide applicability. The author has extended
this approach to include cases in which terms of o [P(h/R)] may be significant,
such as in the column-supported shell problem discussed in section 4.3.7.3. 50
It is felt, however, that currently the most feasible approach to the general
solution of shells of revolution under asymmetrical as well as symmetrical
loading is to use an energy-based numerical solution. We develop the required
energy expressions for shells of revolution in the next section.
9.3.4 Energy Formulation
9.3.4.1 General Considerations. The total potential energy functional U defined
in section 7.4.2 is specialized here for a shell of revolution. It is convenient to
consider an arbitrary segment of a shell bound by two parallel circles defined
by the meridional angles rPi and rPi+l' as shown in figure 9-13. Here, the Z axis
is oriented so that Z = 0 corresponds to the upper boundary of the shell. This
segment may be a portion of a shell or even an entire shell. Also, rP; may be 0°,
which indicates a closed shell or dome. It is evident that the total potential
energy for a shell composed of the assemblage of a number of such segments
is the sum of the energies of all the segments. Therefore, the development, for
the most part, may focus on a single general segment, such as that shown in
figure 9-13. Of course, the segments adjoining external boundaries require some
specialization to accommodate prescribed boundary conditions. We also should
remember that the stationary condition, stated previously as equation (7.14),
t
bUt = 0
(9.151)
implies a global extremum and should be applied to the entire assembled shell.
Nevertheless, the numerical techniques generally applied to this problem permit
almost all of the calculations to be performed at the segment or local, as
opposed to the global level.
9.3.4.2 Geometry. Recall from section 2.8.2 that three possibilities were presented
for the meridional coordinate: the meridional angle rP, the axial coordinate Z,
and the meridional arc length sq,. In fact, the various computations that are
required in an actual problem sometimes involve functions of two, or even all
three of these possibilities. For this topic, it is convenient to use the R-Z
Cartesian coordinates to locate stations on the shell, but to take the arc length
as the primary dependent variable. It is convenient to define for each segment
435
9.3 Shells of Revolution
Axis of Revolution
R·I
(R i ,Z;)
R------~--~-~~---------------4-------,
R.1+ I
Z
Fig.9-13 Shell of Revolution Finite Element Geometry (Reprinted with permission of American
Institute of Aeronautics and Astronautics. Source: L. J. Brombolich and P. L. Gould, "A HighPrecision Curved Shell Finite Element," AIAA Journal, 10, no. 6 [June 1972]: 727)
a nondimensional arc length coordinate
s
=!<L
Li
(0
s
s S 1)
(9. 152a)
where s¢ is measured from (R i , Z;), and where, from equation (2.46),
Li =
S:i+l [1 + (R
o . z fJ1 /2 dZ
(9.152b)
We also find it convenient to express the following geometric relations that
are required in the formulation as functions of Z:51
436
9 Shell Bending and Instability
Ro(Z)
=
R,p(Z)
=
(9.153a)
R
[1
+ (R
O.Z
)2]3/2
Ro.zz
+ (R o.z )2r 1/2
cos ¢(Z) = Ro.z[1 + R o. z )2r 1/2
sin ¢(Z) = [1
Ro
R/I(Z) = - . -
(9.153b)
(9.153c)
(9.153d)
(9.153e)
sm¢
9.3.4.3 Fourier Series Representation. As is customary for shells of revolution,
all external loading, stress resultants and couples, strains and changes in curvature, and displacements and rotations are represented by Fourier series in the
circumferential variable (). These expressions are given as equations (4.86a) and
(4.86b) for the loading and in-plane stress resultants; by equations (5.50) for the
strains and changes in curvature; and by equations (5.51) for the displacements
and rotations. To make our discussion complete, we write the corresponding
expressions for the bending stress resultants and couples:
Q~cosj()
Q,p
Q/I
M,p
00
L
i=O
M/I
Q~sinj()
M*cosj()
(9.154)
M~cosj()
M*esinj()
M,p/l
We may then further restrict our development to typical harmonic j of
segment i, for which the stress resultants and couples and the strains and
changes in curvature are represented by vectors {Ni} and {.:i}, respectively, as
defined in matrix 6-2 and equation (6.10). Typical elements of these vectors are
NJ cosj(), wi sinj(), etc. Also, we have the displacement vector {Ai}, defined in
accordance with equation (5.51), as
{AI} = {D* cosj() D~ sinj() D~ cosj() D*o cosj() D~,p sinj()}
(9.155)
9.3.4.4 Strain-Displacement Relationships. If we anticipate a displacement formulation, we will have need for the strains and changes in curvature expressed
as functions of the displacements and rotations. We derived such compatibility
relationships for harmonicj as equations (5.52) and (5.53). We may transform
these equations to the nondimensional arc length variable s by noting, from
figure 9-13, that
R,pd¢
=
ds,p
=
Lids
from which we may express
(9.156a)
437
9.3 Shells of Revolution
Matrix 9-3
1
L. ( ).s
0
cos ,p
Ro
j
,
Ro
R,p
sin <p
Ro
Ro
1
cos,p
L i ( ).s-R;
0
0
0
0
0
0
0
-j
[Bf] =
0
R,p
0
0
0
0
1
- L. ( ).s
,
sin,p
j
Ro
Ro
0
0
0
0
0
0
1
0
L., ( ).s
cos ,p
Ro
-j
2Ro
j
Ro
cos,p
2Li ( ).s - 2Ro
1
-1
0
0
-1
(9.156b)
as
(9.157)
The resulting relations are conveniently written in the matrix form
(9.158)
where [B{] is given by matrix 9-3.
Note that [B{J is an operator matrix and must always premultiply {AI}.
Modified expressions for K~, K~, and oj, with transverse shearing strains
suppressed, are easily written, but are not particularly advantageous in the
formulation.
We also require a modification of the strain-displacement relationships if
a closed shell is encountered. We have shown two cases in figure 4-1 and have
recorded the pole condition for both possibilities in table 6-1. Here, the origin
for the coordinate system is generally selected so that s = 0 when Ro = O.
To establish the pole conditions and to modify the strain-displacement
relationships suitably, we simply postulate that all strains and changes in
438
9 Shell Bending and Instability
curvature remain finite as s --+ 0. Consider, for example, the expression for
which we write from equation (9.158) and matrix 9-3, row (2), as
.
f,J
fJ
cos <p.
j
.
sin <P
.
= --m
+ -DJ
R
R fJ + --DJ
R
n
'I'
o
0
f,~,
(9.159)
0
For both shells shown in figure 4-1, Ro(O) = 0, so that for the first term to remain
finite as s --+ 0, (cos <P D~)s=o = 0, which creates an indeterminate form. The
second term will remain finite only if (jD~)s=o = 0, and the third term requires
that (sin <p D~)s=o = 0. Hence, we have the general pole condition
+ jD~ + sinq)DDs=o =
(cos<pD~
°
(9.160)
°
for which we may investigate the individual pole conditions. We have six
separate possibilities to study, consisting of two types of closed shells, (<p)s=o =
and (<p)s=o = <Pt, for j = 0, j = 1, and j > 1. It is easiest to treat the harmonics
individually.
j = 0: For this case, equation (9.160) reduces to
(cos<pD3
+ sinq)D~)s=o
=
°
(9.161a)
If (<p)s=o = 0, the second term drops out, so that no restriction is needed on D~;
whereas if (<p)s=o = <Pt>
D3(0) =
°
(9.161b)
as recorded in the second line, second column, of table 6-1. Also, for the latter
case, equation (9.161a) gives
D3(0) = - tan <ptD~(O)
(9.161c)
as listed in the second line, third column, of table 6-1.
j = 1: Here, equation (9.160) becomes
(cos<pDJ
If (<p)s=o
+ DJ +
sinq)D;)s=o
=
0
(9.162a)
= 0,
DJ(O)
+ DJ(O) =
0
(9. 162b)
as recorded in the third line, second column, oft able 6-1. Note that D;(O) = 0
is also listed in the table. This arises out of consideration of YfJ and is not required
here-but neither is it contradicted. Now, if (<p)s=o = <Pt, we have
(9.163a)
Examining the corresponding entry in row 3, column 3, of table 6-1, we find
(9.1 63b)
which obviously cannot be completely obtained from equation (9.163a). However, equation (9.163b) does satisfy equation (9.163a), so that no contradiction
is encountered.
439
9.3 Shells of Revolution
j > 1: In this case,
(9.164)
D~(O) = D~(O) = D~(O) = 0
as listed in line 4 of table 6-1 for both types of closed shells, obviously satisfies
equation (9.160), but, as before, consideration of some of the other individual
relationships is required to certify the correctness of these conditions.
Proceeding with similar arguments for the other strains that have singularities
at Z = O-i.e., wi, K~, ri, and yJ-the remaining entries in table 6-1 may be
verified.
Having isolated the necessary constraints on the displacements to avoid singularities, we now turn to the actual modified strain-displacement relationships.
We again use e~ as given by equation (9.159) to illustrate the approach. In view
of equation (9.160), equation (9.159) evaluated at s = 0 becomes
( i)
_ [cos r/J D~
ee s=o -
o
o
+ jD~ + sin r/J D~J
R
o
s=O
(9.165)
which is an indeterminate form. Using L'Hospital's rule, we may investigate the
convenient to differentiate with respect to r/J, and then to convert
to s using equation (9.157). Proceeding, we have
lims~o e{ It is
lim e~ = [-sinr/JD~(LdR¢J + cosr/JD~,s
s~o
Rr/Jcosr/J(LdRr/J)
+ jD~,sJ
s~o
+ [COSr/JD~(LdRr/J) + sinr/JD~,sJ
Rr/J cos r/J (Ld Rr/J)
=
1.
[ -Vi'Y'S
Li
For the dome, r/J
.
1.
-+
+L
j
i cos
.
A.
'I'
DJe,s
s~o
1.
+ -DJ
R nr/J
(9.166)
tanr/J . tanr/J . ]
-R- V i'Y + --DJ
L
n,s
r/J
i
s~o
0 as s -+ 0 and the last two terms drop out, giving
j.
1.
eMO) = -D~,s(O) + -D~,s(O) + -(O)D~(O)
Li
Li
Rr/J
(9.167)
In practice, when a dome is modeled by multiple segments, equation (9.167) is
used for e~(s) in the uppermost segment where i = O.
To make this discussion complete, we present the entire set of modified
strain-displacement equations for a dome: 51
(9.168)
where rows 2, 3, 5, 6, and 8 (corresponding to eo, w, Kr/J' r, and Ye) of [BiJ are
replaced by the modified equations, such as equation (9.167), as depicted in
matrix 9-4.
Additionally, there are circumstances in which the pole cannot conveniently
be located at s = 0, but coincides with s = L i . For example, a shell which is
closed at both ends will present such a situation at one end unless complete
440
9 Shell Bending and Instability
Matrix 9-4
1
1
Lo ( ),s
0
1
Lo ( ),s
Lo ( ),s
R",
Lo ( ),s
0
0
0
0
0
1
Lo ( ),s
0
0
0
1
Lo ( ),s
0
0
0
-j
2Lo ( ),s
0
0
-1
Lo ( ),s
-1
0
1
j
Lo ( ),s
0
-1
-j
j
R",
1
[Bb] =
R",
0
R",
0
0
0
0
0
0
0
j
Lo ( ),s
loading and geometric symmetry are present, whereupon only half the shell
must be considered. Similar modified strain-displacement equations may be
developed by enforcing the finite strain conditions at s = L; instead of s = O.
An analogous set of equations can be developed for the case shown in
figure 4-1(b) where rP = rPt·
Thus, we have the strain-displacement relationships given as a function of
the nondimensional arc length variable s by matrix 9-3 and equation (9.158),
augmented by special equations for pole segments, matrix 9-4 and equation
(9.168.)
9.3.4.5 Total Potential Energy Functional. U<, as defined by equation (7.14), is
the sum of U., the strain energy, and Uq , the potential energy of the applied
loads.
First, considering the strain energy U., as given by equation (7.4), we may
integrate through the thickness to obtain the strain energy for segment i in
harmonicj:
Ui;
=~
2
i1 fX
0
LNiJ {Ei} dA;
(9.169a)
-x
where the differential surface area
(9.169b)
441
9.3 Shells of Revolution
We proceed with the displacement formulation, generalizing for specifically
orthotropic shells, by replacing LNi J using equation (6.12):
Vii
=
~ Sal f~" [[DorJ{Ei} -
= ~ Sal
{NTor}Y{Ei}dAi
f~" ({Ei)T[DorJ{Ei} -
LNTorJ {Ei})dAi
(9.170a)
(9. 170b)
noting that [Dor] is symmetric. Finally, we replace the strains by the displacements using matrix 9-3 and equation (9.158), and get
Vii.
="21
f f"
1
0
-"
. { '}
. {
([[Rf] Ai ] T [Dor] [[Rf]
Ai'} ]
- LNTorJ [[Rf] {A{} ])dAi
(9.171)
Now, we turn to the potential energy of the applied loads V q , as defined in
equations (7.10). Of the three terms contained in the general expression, equation (7.10a), which accommodate distributed, line, and concentrated loadings,
only the first term and the second integral of the second term are pertinent here.
Using Fourier series, line loads along the Sa or st/J coordinate line, as represented
by the first integral in the second term, may be included in the first term, and
any concentrated loads, as represented by the third term, may be included in
the second integral of the second term. Therefore, in terms of the present
notation,
Vji
= -
Sal f~" LA{J {q{} dAi - f~" LA{(Z;)J {t}{(Z;)} R(Zi) d8
-f~"
(9.172)
LA{(Zi+1)J {t}{(Zi+1)}R(Zi+d d8
where {q{} {Ai(Z;)}, and {t}{(Zi)} are defined corresponding to equation (7.10a).
Note that in a numerical calculation, the second and the third terms of
equation (9.172) would both not ordinarily appear. The second term might be
included with segment i - 1, or the third with segment i + 1, depending on the
computational order.
For the entire shell,
V/
=
p
L (Vi; + vj;)
i=l
(9.173)
where the shell is assumed to be composed of p segments. The principle of minimum total potential energy [Equation (7.14)] gives the equilibrium condition
{)V/ = 0
(9.174)
442
9 Shell Bending and Instability
For asymmetric loading, the complete solution is composed of the superposition of the solutions to equation (9.174) for the relevant participating
harmonics.
9.3.4.6 Rayleigh-Ritz Solution. The Rayleigh-Ritz method was presented in
section 7.4.3 as a promising numerical approach to the solution of equations
such as equation (9.174). In the context of the present development for a single
general harmonic j, the vector {A}, as introduced in equation (7.15), is the
assemblage of the vectors {An for all of the segments comprising the shell.
Recall that {An was defined in equation (9.155) as the vector of displacements
for segment i. The index m introduced in equation (7.15) is equal to 5 in this
case, since there are five displacement components in {Ai}, except for thej = 0
case when De and Detfo = 0 and m is reduced to 3. But, for the general case, there
will be five comparison or trial functions of the form of equation (7.16) for each
of the p segments of the shell. Polynomials of the form
1,5)
(~1== 1,p
(9.175)
where A{l = D~, A{2 = Dj, A{3 = D~, A{4 = D~e, and Ai5 = Djtfo are quite popular
for this application. It is important to note that n, the order of the approximation for each A{k' may vary for each displacement k, segment i, and harmonic j.
Upon substitution of the set of 5p comparison functions of the form of
equation (9.175) for {An into equations (9.171) and (9.172), and the operation
by [B{] on the polynomials, the extremum problem represented by equation
(9.174), is transformed into the maximum-minimum problem of the calculus
I
(k
=
=
o,n)
1,5
(9.176)
i = 1,p
The maximum-minimum conditions, equation (9.176), are necessary, but
not sufficient, to determine the n coefficients (elk') that are present for each of
the five comparison functions A{k in each segment i = 1, p. In addition to
equation (9.176), there are intersegment continuity conditions, which recognize
that the displacements are continuous along the common boundary of adjacent
segments. Also, kinematic boundary conditions which fix the values of the corresponding displacements may be prescribed on the external boundaries. Thus,
in summary, the maximum-minimum conditions, the intersegment continuity
conditions, and the kinematic boundary conditions comprise the necessary
and sufficient conditions to satisfy the principle of minimum total potential
energy approximately. The further development ofthis approach is beyond our
objectives here and is left to the province of the finite element method. 52
One further point should be noted, however. Static boundary conditions are
also encountered. These enter as the natural boundary conditions of the varia-
9.4 Shells of Translation
443
tional problem; this means, essentially, that if the kinematic correspondent of
a certain force or moment is not prescribed on the boundary, that force or
moment will take the value of the corresponding external loading component
along the boundary. As an example, consider a typical stress resultant Nifo on
a given external boundary l/J = l/J'. If the displacement Difo(l/J') is not prescribed
a priori, along that boundary, then Nifo(l/J') is equal to the corresponding element
of the external loading vector tlifo(l/J'). If tlifo(l/J') = 0, then Nifo(l/J') = O.
We have outlined a quite general, energy-based approach that may be
developed and extended to solve many complex problems in the field of shells
of revolution on a fairly routine, automated basis. This technique is developed
quite completely in a companion volume,53 and an example of the capability
for analyzing a torospherical shell was shown in figure 9-12(c).
9.4 Shells of Translation
9.4.1 General: The general theory of shells of translation, apart from cylindrical
shells, is lightly treated in contemporary English-language textbooks. Although
there are many papers in the literature that describe analysis methods for such
shells, most are based on idealized boundary conditions that do not necessarily
match the physical situation very closely. For example, the inclusion of flexible
edge members, which are often present in reality, greatly complicates the analysis.
From a practical standpoint, however, numerous translational shells having
spectacularly large spans have been constructed. As an example, we may take
the shell shown in figure 2-8(a). Such major structures certainly warrant a complete investigation, beyond the scope of the membrane theory. The means for
carrying out such investigations are often computer-based numerical solutions,
such as finite element or finite difference techniques, or carefully executed
experimental studies using physical models. The current availability and capabilities of numerical analysis procedures for translational shells are described
in Schnobrich. 54 Here, we introduce two theories upon which many of the
currently popular numerical algorithms are founded.
9.4.2 Mushtari-Donnell-Vlasov (MDV) Theory: This theory is a simplification
of the general theory of shells in orthogonal curvilinear coordinates and is based
on the assumptions that (a) the effect of the transverse shear forces Q", and Qp
in the in-plane equilibrium equations is negligible; and, (b) the influence of the
normal displacement Dn will predominate over the influences of the in-plane
displacements Da. and Dp in the bending response of the shell. We have discussed
the first assumption at length in connection with other derivations, and it
should be acceptable provided the membrane theory boundary conditions
are not grossly violated. The second assumption is most feasible as the shell
becomes relatively flatter, approaching a plate. Such a shell is said to be shallow.
444
9 Shell Bending and Instability
Shallow shells were introduced in section 2.8.1, and we build on this geometric
concept in the following paragraphs. Some quantitative criteria for a shell to
be classified as shallow are also stated later.
Proceeding, we first consider the force equilibrium equations in general form,
equations (3.17). We omit the transverse shear stress terms as stated in assumption (a) and take N lZp = NplZ = S:
+ (AS),p + A,pS (BS),IZ + (ANp),p + B,IZS (BNIZ),IZ
(BQIZ),IZ
+
+ qlZAB = 0
A,pNIZ + qpAB = 0
B,lZNp
AB
AB
(AQp),p - NIZR - NpR
IZ
P
+
qn AB
=0
(9. 177a)
(9. 177b)
(9. 177c)
We first wish to express the transverse shear forces in terms of the normal
displacement Dn. With transverse shearing strains suppressed, QIZ and Qp are
given in terms of the stress couples through equations (3.22a) and (3.22b):
Qp
=
1
AB [(BMIZP),IZ
+ (AMp),p + B,IZMplZ -
A,pMIZJ
(9.178a)
(9.178b)
Next, we apply the constitutive law matrix 6-2, omitting the thermal terms, and
take M IZP = MplZ to write the stress couples as functions of KIZ , Kp, and r:
MIZ
Mp
M IZP
= D(KIZ + JlKp)
= D(JlKIZ + Kp)
(9.179b)
=
(9.179c)
D(1 - Jl)r
(9. 179a)
Before substituting equations (9. 179a-c) into (9.178a) and (9.178b), we express
the changes in curvature as functions of the displacements. These relationships
are given, in general, by equations (5.47a-c), but we invoke assumption (b) and
eliminate the contributions of DIZ and Dp. The simplified compatibility equations
are
K
=
1(1 ) 1
1(1 ) 1
-A ,p D n,p
A -D
A n,lZ ,IZ - AB2
(9.180a)
- B,IZ Dn,lZ
p- -B -D
B n,p ,p - A2B
(9.180b)
1- ( D
1
1
)
r=AB
n,lZp --A
A ,p Dn,lZ --B
B ,IZ Dn,p
(9. 180c)
K IZ
The next step is to substitute equations (9.180a-c) into equations (9.179a-c)
9.4 Shells of Translation
445
and then into equations (9.178a) and (9.178b). The algebra is considerable, and
we defer to N ovozhilov 5 5 to write
(9.181a)
(9.181b)
where the Laplacian, in general terms, is given by
V2( )
=
AlB
{[~ ( ),aJ.a + [~( ),pJ.J
(9.181c)
Finally, substituting equations (9.181a) and (9.181b) into equation (9. 177c) and
dividing through by AB, we obtain
-D
{[B
2
]
AB
A (V Dn),a
,a
+ [A"B(V 2 Dn),p
J}
,p
Na - Rp
Np
- Ra
+ qn = 0
(9.182)
or, in view of equation (9.181c),
4
DV D
n
Na
Np
+ -Ra
+ -Rp
-
q = 0
n
(9.183)
which we recognize as being very similar to the equilibrium equation for plates.
Thus, we have equations (9. 177a) and (9. 177b), together with equations (9.183)
and the requisite boundary conditions, constituting the shell theory which
was apparently derived independently by Mushtari in the Soviet Union and
Donnell in the United States. 55
The governing equations at this stage of the development are somewhat
similar to those encountered in section 8.7.5 following the derivation of equation
(8.294). Equations (8.252) and (8.294) formed a coupled set of equations for the
finite displacement theory of plates. The dependent variables in those equations
were the normal displacement and the in-plane stress resultants; that is essentially
the same situation we now have with equations (9.177a) and (9.177b) and
equation (9.183). Therefore, the introduction of a stress function !II', defined
such that
N
)
1B -!II'
a= -B1 (1-!II'
B ,p ,p + AB
,a A1 ,a
(9.184a)
1A -!II'
1
+AB
,p B ,p
(9. 184b)
1(1
)
N p -- -A
-!II'
A ,a ,a
S
=
1! I( I ' - -A
1
1
)
-AB
,ap A ,fJ !II',a - -B
B ,a !II',p
is quite logical. 56
(9.184c)
446
9 Shell Bending and Instability
The steps in the back substitution are tedious and, basically, are similar to
our previous exercises in the development of the von Karman equations in
section 8.7.5. We omit the details and follow the procedure suggested by
Vlasov. 57 Also, we consider the special case in which the in-plane components
of the surface loading q/Z and qp are equal to O. The resulting equations are
1- [(-B
A2B
A1 ,/Z ) ,/Z
+ (1-A
B ,p ) ,p ] ff',/Z --
-1- [(-B
AB2
A1 ,/Z ) ,/Z
+ (1-A
B ,p ) ,p ]
DV 4 Dn +
- qn = 0
ff',p --
- -1- -1§ -
R/ZRp A
0
(9.185a)
,p - 0
(9.185b)
,/Z -
- -1- -1§
R/ZRp B
-
and
V2 ff'
(9.186a)
where
V2( ) = AlB
{[~p ~( ),/Z1/Z + [~/Z ; ( ),p1J
(9.186b)
The equalities in equations (9.185a) and (9.185b) have been obtained by invoking
equation (2.37), the third Gauss-Codazzi relationship; the second term in
equation (9.186a) is expressed in terms of the operator V2( ) by using the first
two Gauss-Codazzi relations, equations (2.35) and (2.36).
It may now be argued that for certain classes of problems, equations (9.185a)
and (9.185b) may be considered to be satisfied. First, for shells of zero Gaussian
curvature, such as cylindrical or conical shells, equations (9.185a) and (9.185b)
are identically satisfied. Also, if we think of shells that are relatively flat or
shallow, the Gaussian curvature will be small and the coefficients of ff',/Z and
ff',p in equations (9.185a) and (9.185b) will be an order of magnitude smaller
than the coefficients of the same terms in equations (9.186), so the former equations can be regarded to be approximately satisfied. A third class of problems
for which the first two equations may be ignored are those cases in which the
stress resultants are rapidly varying functions of ex and p. Then, the higher
derivatives of § predominate, and the contribution of the terms associated
with ff',/Z and ff',p will be relatively small as compared to those multiplying ff',,.,.
and ff',pp. Similar reasoning was used to motivate the Geckeler approximation
in section 9.3.2.3.
We now seek a second relationship between Dn and ff'. This accomplishment
is attributed to Vlasov by Novozhilov 58 and requires two general steps.
The first operation is the derivation of the compatibility equations for the
deformation of the middle surface in terms of strains and curvatures only. These
equations are analogous to the S1. Venant equations of the theory of elasticity
and may be established by following the same steps employed in the derivation
of the Gauss-Codazzi relations in section 2.6, if the tangent vectors to the deformed middle surface are used in place of the tangent vectors to the undeformed
447
9.4 Shells of Translation
middle surface. Specifically, t n , .. and tn,p in equation (2.33) are replaced by ( ..
and p , as given by equations (5.9) and (5.11), respectively. Then the identical
steps are carried out on these tangent vectors to derive the compatibility
equations, which may be viewed as Gauss-Codazzi relations for the deformed
middle surface. 57
The second operation required is the substitution of equations (9. 180a-c) for
the changes in curvature; the replacement of the in-plane strains by the corresponding stress resultants using matrix 6-2; and the subsequent introduction
of the stress function ff via equations (9. 184a-c).
The final result, after eliminating terms similar to those previously dropped
in equations (9.185a) and (9.185b), is
r:
1
Eh
-
- V4 ff - V 2 D
n
=0
(9.187)
Thus, the solution of the coupled equations (9.186a) and (9.187) is sufficient to
describe the complete pattern of stress and deformation within the limitations
of the approximations introduced in the course of the derivation. Novozhilov
observed that these equations form the basis for the investigation of many
practical problems. 59 Also, explicit solutions are provided by Vlasov 60 and are
based on some further approximations: (a) the shell is described by Cartesian
coordinates, with
(9.188)
A=B::::::1
and, (b) R .. and Rp are taken as constant (average values) and are given by
R .. (X, Y)
= Rx
(9.189)
Rp(X, Y)
=
(9.190)
fiy
These assumptions are specifically tied to the shallow shell notion and, according
to Vlasov, can be utilized for shells that cover rectangular plan areas if the rise
is no more than 1/5 of the smaller side of the rectangle.
The introduction of the simplifications given in equations (9.188)-(9.190) into
the previous equations reduces the operators V2( ) and V2( ) to
V2( )
=(
-2
=
),xx
+ ( ), yy
(9.191)
and
V ( )
1
fiy (
),xx
1
+ fix (
),yy
(9.192)
where fi y and Rx are average values of the radii of curvature along the respective
coordinate lines. The solution then proceeds with the introduction of a potential
function G, such that
(9.193a)
448
9 Shell Bending and Instability
and
(9.193b)
ff = EhV 2 G
When introduced into equation (9. 186a), equations (9.193) give 61
VBG
+ 12(1
- Jl2) V4 G _ qn = 0
h2
D
(9.194)
There is a marked similarity between equation (9.194) and equation (9.75),
which governs the general bending of cylindrical shells, and the solution strategies are similar. Once G is found, ff is determined from equation (9.193b).
The interested reader is referred to Vlasov 62 for the complete development
of this solution. As an example, the bending of an elliptic paraboloid shell of
a similar configuration to that shown on figure 4-51 can be investigated using
equation (9.194). Also, these equations can be readily generalized to the study
of the instability of certain shells, as we see later.
9.4.3 Theory of Shallow Shells: The theory of shallow shells is roughly equivalent
to the simplification of the MDV theory that is facilitated by equations (9.189)
and (9.190) and has been widely used for shell analysis. The approach is to
employ some simplifying assumptions at the outset to derive the governing
equations in a form which is particularly suited for shell roofs covering rectangular plan areas. This theory has also been used to analyze shells that
become locally shollow when the original shell is divided into finite segments
or elements-a procedure to which we have often alluded in our previous
discussions.
Refer to the basic geometric relationships introduced in section 2.8.1. Recalling
the assumption that (Z.x)2 and (Z'y)2 may be neglected in comparison to unity,
we may evaluate the principal radii of curvature from equation (2.29)
1
Rx= - - Z,xx
(9.195a)
Similarly,
R y
1
= ---
(9.195b)
Z,yy
The expressions for the extensional strains
from equation (5.46a) and (5.46b) as
Bx
and
By
are obtained directly
(9.196a)
and
(9. 196c)
The equation for the in-plane shearing strain could similarly be written from
449
9.4 Shells of Translation
equation (S.46c). However, it is more correct to use a somewhat different form
that arises be,cause the Cartesian coordinates are not orthogonal when projected
on the surface. The expanded form of ill is 63
(9.197)
The term Z,XY = -(l/Rxy) may be called the twisting curvature, and was
encountered previously in section 4.4.2 when planar X - Y coordinates were
also used. Additionally, we have the expressions for ,,«, "p, and 'r, which, with
A = B = 1, reduce to
-D,.,xx
(9. 198a)
Icy
=
=
-D,.,yy
(9.198b)
'r
=
-Dn,xy
(9.198c)
"x
Equations (9.196)-(9.198) are the strain-displacement or compatibility equations for this theory.
We now proceed as in the previous section. The analogue to the normal
equilibrium equation, (9.186a) and (9.186b), is 64
(9.199)
where the stress function is related to the stress resultants by the simplified form
of equations (9.184a-c),
(9.200a)
N x = ff,yy
=
S=
Ny
ff,xx
(9.200b)
-ff,XY
(9.200c)
Similarly, an analogous equation to equation (9.187) is developed. This
equation is written as 65
;h V 4 ff
+ (Z,yyD,.,xx - 2Z, xy D,.,xy + Z,xxD,..yy) = EhP
(9.201a)
where P is derived from the applied surface loading as
P
= (1 + fl)(Px,yy + Py,xx) -
Px(X, Y)
=
Py(X, Y)
=
f
f
flV2(Px
+ Py)
(9.201b)
qx dX
(9.201c)
qydY
(9.201d)
+(
(9.201e)
and
V2( )
=(
),xx
),yy
450
9 Shell Bending and Instability
Px and P y in equations (9.201c) and (9.201d) are cumulative in-plane forces and
are set to 0 on boundaries that are unrestrained in the X and Y directions,
respectively. Note that this theory is slightly complicated by starting with
non-orthogonal coordinates, but retains the generality of including the in-plane
components of the surface loading qx and qy. For most shells encountered in
practice, the MDV theory and the theory of shallow shells are fairly equivalent
and both are sometimes referred to as shallow shell theories. From an applications standpoint, the most significant difference is probably in the nature of the
external boundaries. If the boundaries are coincident with the coordinate lines
of the middle surface, the general form of the M ushtari - Donnell-Vlasov theory
is appropriate; if the boundaries fall along the Cartesian coordinate lines, the
theory of shallow shells, or the even simpler specialization of the MDV theory,
seems to be more attractive.
9.4.4 Strain Energy for Shallow Shells: The energy expressions and principles
derived in chapter 7 are directly adaptable to shells of translation, but we have
not carried out the details of the specializations based on the theories presented
in the last two sectons. However, it is of general interest to examine the expression for strain energy, equation (7.6), using the simplified strain-displacement
relationship, equations (9.196)-(9.198). With the transverse shearing strains
neglected, term [I] of equation (7.7) becomes
[I] = ex2
1 - J1 2
+ ey2 + 2J1ex ey + -2-w
= (Dx,x - _1 Dn)2 + (Dy,y _ _
1 Dn)2
Z,XX
+ 2J1 (Dx,x
- -I-Dn) (Dy, y
Z,xx
Z,yy
-
(9.202)
_1_Dn)
Z,yy
Term [II] is dropped in the linear theory, and
[III]
= (Kx + Ky)2 - 2(1 - J1)(KX Ky -
1"2)
= (Dn,xx + Dn,yy)2 - 2(1 - J1)(Dn,xxDn,yy - D;,xy)
(9.203)
The resulting expression,
(9.204)
is frequently encountered in the literature. The thermal terms, as defined in
equation (7.6), can easily be added if required.
451
9.5 Instability and Finite Deformations
• Bifurcation Poi nt
/
/
-~ -~-~~- -- - _~1]9P--Jfl.!..o..I49b_ - - - -7/
\
limit point
.
,,',~
/
"..........
'/1./ -
unloadmg/
-
/ \'\
./
--
reloading
L--------------------------------------Dn
Fig.9-14
Paths of Shell Instability
9.S Instability and Finite Deformations
9.5.1 General: Local instabilities, characterized by displacements of comparatively small wave length similar to those occurring in plate structures, may be
significant in thin-walled vessels where compressive stresses are present. This
type of instability is delineated from general instability, which could correspond to the buckling of an entire stack or tower as a beam-column. Since the
amplitudes encountered in local instabilities are comparatively small, this phenomenon can be analyzed in many cases by using the simplifications of the
shallow shell theories presented in the previous sections.
There are several possible approaches to the elastic instability analysis of
shells. These are illustrated schematically on figure 9-14, where we have plotted
the normal displacement, D n , against a loading parameter, N, which represents
some combination of the in-plane loading.
First, if we can arrive at a general solution for the normal displacement in
the presence of transverse and in-plane loading, and then we observe that
certain discrete values of N cause the displacements to become excessively large,
we term these values of N the critical loads for a perfect shell, Ncr. This is
illustrated by the curves labeled 1 in figure 9-14, with the various branches
corresponding to different magnitudes of the known transverse loading. This
452
9 Shell Bending and Instability
approach was used for the study of plate instability in section 8.7.3. Note that
the same Ncr is obtained regardless of the transverse loading, which we call qn
for shells.
The second approach follows from the realization that ~r is independent of
qn' This suggests that the transverse loading can be ignored in the stability
analysis, which means that for N < Ncr> Dn = O. This is indicated by curve 2,
which is the special case of 1 for qn = O. Then, at N = ~r> the displacements
become unbounded as before, following path 3. This approach is attractive
because the complete solution for Dn as a function of qn and N is not required.
Rather, the states of equilibrium just prior to and just after instability can be
examined without concern for the remainder of the load-deflection curve. At
the transition point (~r> 0), a bifurcation of equilibrium is said to occur, since
there are at least two possible states of equilibrium, Dn = 0 and Dn --+ 00. This
approach is, of course, familiar from the classical solution of the Euler column
buckling problem. In regard to the unbounded displacements following bifurcation, curve 3, a more sophisticated analysis may yield an unstable (descending)
post-buckling path, curve 4.
Although it is implied by curve 2 that the equilibrium path prior to buckling
is linear, buckling of shells may also be preceded by a nonlinear pre buckling
response, such as a curve 1. 66 The bifurcation would then occur at the intersection
of curves 1 and 4. 67
A third approach which should be mentioned is that of using geometrically
nonlinear strain-displacement relationships to determine the complete loaddeflection curve-unloading as well as loading. As illustrated by curve 5,
a critical load Ncr(nl)' which occurs at a limit point, may be found somewhat
below Ncr. A classical illustration is an axially compressed conical shell, such
as that shown in figure 4-13, where the axial shortening and accompanying
rotational deformation would cause an immediate degradation of the stiffness. 66
Then, the curve would show unloading, and perhaps reloading, as other elements of resistance are mobilized by the finite deformations. If this latter
resistance is adequate, the load may eventually surpass NCr(nl) and even Ncr>
provided the material capacity is not exceeded. An actual shell cannot unload
in most cases, so there may be a sudden transition from the loading to the
reloading branches ofthe curve at N = ~r(nl)' with a corresponding jump in Dn.
This is termed snap-through buckling and is characteristic of shallow arches
and shells.
Shells with initial imperfections follow a path such as 5. As the magnitude
of the imperfections decreases, the descending branch of curve 5 approaches
curve 4.
Note that all three of the general approaches discussed have been predicated
on elastic buckling. If material nonlinearities are present for N :5; ~r> then
additional complications arise. Sometimes these can be handled conveniently
by the tangent modulus theory.
9.5 Instability and Finite Deformations
453
9.5.2 Equilibrium Equation Method: As we mentioned in our discussion of the
bifurcation of equilibrium approach to elastic instability in the previous section,
we need only consider the immediate neighborhood of the transition point
(Ncr> 0). Up to this point, the shell will be regarded as being subject to a membrane
state of stress N xo , N yo , and 80 . 68 At the transition point, the greatly increased
normal displacements result in the development of normal components of these
in-plane stress resultants. Such normal components may be treated as surface
loading in the tn direction, since the starting point for the analysis is the state
of stress N xo , N yo , and So. With this in mind, the normal components of the
membrane stress resultants are used for qn in equations (9.183) and (9.186a) if
the MDV theory is used; in equation (9.194) if the simplified version ofthe MDV
theory is used; and in equation (9.199) if we choose the theory of shallow shells.
It now remains to determine the normal components of the membrane stress
resultants.
Refer to figure 8-30, where the normal components of the in-plane stress
resultants for a plate are apparent. (The analysis corresponding to this figure
is contained in section 8.7.1.) Since the net normal force due to any single
stress resultant is the difference of the normal projections at the two ends of
the differential segment, that analysis is valid for shells as well. Following
equation (8.250), with the transverse shear forces and in-plane loads omitted,
we have for Cartesian coordinates
(9.205)
With transverse shearing strains suppressed and N xyo = So, and with Dx and
Dy neglected in comparison with Dn in the Cartesian coordinate version of
equations (5.46g) and (5.46h) in the spirit of the shallow shell assumptions,
we have
Dxy = -Dn,x}
Dyx = -Dn,y
(9.206)
so that
qnl = NxoDn,xx
+ NYODn,n + 2S0 Dn,xy
(9.207a)
which is sometimes written as
qnl = - [NxoKx
+ NYOKy + 2So.]
(9.207b)
in view of equations (9.198). For orthogonal curvilinear coordinates, the same
analysis is valid, and
qnl = - [NaOKa + NpoKp + 2So.]
(9.207c)
We proceed with the simplest of the three theories, the MDV theory in
Cartesian coordinates. The governing equation is written in terms of the potential function G from equation (9.194) and qn is taken as qnl' as given by
454
9 Shell Bending and Instability
equation (9.207a):69
8
V G
+
12(1 - 112) - 4
1
h2
V G = Jj(NxoDn.xx
+ NYODn,yy + 2S0 Dn,xy)
(9.208a)
where again
V2( ) = ( ),xx
+(
1
-2
V ( ) = Ry ( ),xx
(9.208b)
),yy
1
+ Rx (
(9.208c)
),yy
and G is related to Dn and ff by
(9.208d)
Dn = V 4 G
and
(9.208e)
If we write the r.h.s. of equation (9.208a) in terms of G, we have
V8G
+
12(1 -11 2 )V 4 G
h2
=
1
4
D [Nxo(V G),xx
(9.209)
+ Nyo(V
4
G),yy
+ 2So(V
4
G),XY]
We shall apply this equation to the stability analysis of spherical shells in
a later section, but first we develop the energy method.
9.5.3 Energy Method for Cylindrical Shells: From an energy standpoint, the
transition between the pre buckled and post buckled states may be represented
by the following stability criterion: 70 (a) there is no bending prior to the onset
of buckling, so that the total strain energy is due to the in-plane stress resultants;
(b) at the onset of instability, there are additional contributions to the strain
energy due to middle surface straining and bending; and, (c) the increase in
strain energy as buckling occurs must be equal to the work done by the external
loading and by the components of the in-plane forces that act through normal
displacements. The latter source of "external" work is analogous to the load
component qnl introduced in the previous section and is not present during
infinitesimal deformations. We illustrate this for a cylindrical shell under uniform axial loading, as shown in figure 9-15.
We first consider the strains and charges in curvature prior to and following
buckling.
In the pre buckled state, which we designated by the subscript 0, we assume
that there is no bending. There is no normal loading acting, so that the third
equilibrium equation, (9.5b), with Qx = qnO = 0, gives NfJO = O. We now refer to
the constitutive law, equations (6.1) with IX = X, f3 = (), and no thermal terms.
455
9.5 Instability and Finite Deformations
a
Fig.9-15
L
Axially Loaded Cylindrical Shell
The axial strain just before buckling is found from equation (6.1a) as
Ncr
Eh
(9.21Oa)
exo = - -
The corresponding circumferential strain follows from equation (6.1b) as
(9.210b)
As a result of buckling, the normal displacement becomes Dn , producing
a circumferential strain e 91 . This strain is proportional to Dn through equation
(9.6b), so that
(9.211)
Therefore, the total circumferential strain after the shell has buckled is
e9
=
e90
Dn
+ e91
=- a
(9.212)
Jlexo
To find the corresponding meridional strain, we now refer to the first equation
of matrix 6-2 with Na. = N x = -Ncr:
(9.213a)
456
9 Shell Bending and Instability
from which
(9.213b)
Substituting for N"rlEh from equation (9.2lOa) and for 80 from equation (9.212),
we find
8x
=
(1 - 11 2 )8 XO - 11 (Dn
-;; - 118 x o)
(9.214)
Also, we have the change in curvature after buckling
KX = -Dn • XX
(9.215)
Of course, Kx = 0 prior to buckling.
Now, we may evaluate the change in strain energy during buckling. We may
specialize equation (7.6) for the case of cylindrical shells with axisymmetric
loading to
Ean
U. = 1 _ 112
fL {h(8X2+ 82+ 2118x 8
0
0
0)
+ h12 Kx2} dX
2
(9.216)
The change in strain energy is given by
I1U.
=
U.(8 x , 80 , Kx) - U. o(8 xo , 800 , 0)
(9.217)
We substitute equations (9.214), (9.212), and (9.215), respectively, for 8 x , 80' and
Kx into equation (9.216) to compute U.; also, using equations (9.21Oa) and
(9.21Ob), we take 8xo and -118xo for 8 x and 80 , respectively, to evaluate U. o .
Equation (9.217) then becomes
I1U. = an
SaL [Eh(~ny -
2I1Eh8XO(~n) + D(Dn.xx )2]dX
(9.218)
Next, we consider the work done by the external loading and the normal
components of the in-plane forces during buckling. For the external loading,
N = -N"r and
I1UqN = 2na
SaL
-N"r(8 x - 8xo )dX
(9.219)
= 2l1nN"r SaL Dn dX
in view of equation (9.214). Note that there is no 1/2 coefficient in this expression, since the full N"r is already present prior to the onset of buckling. For the
normal components of the in-plane forces, refer to the loading qnl given by
457
9.5 Instability and Finite Deformations
equation (9.207a). The work done over the circumference by qnl acting through
Dn is
= ~ SoL (qnlDn)(2nadX)
AUqn
In equation (9.207a) for qnl' we take N xo
equation (9.220a) becomes
AUqn
= -
(9.220a)
= -
~r and N yO
SOL TCaNcrDn.xxDndX
= So = 0, whereupon
(9.220b)
Adding equations (9.219) and (9.220b), we have
AUq = AUq[il + AUqn
= 2TCNcr {1l SoL [Dn -
~Dn.xxDnJdX}
(9.221)
Finally, equating equations (9.218) and (9.221) completes the application of the
stability criterion:
(9.222)
These equations can be treated most easily with a Rayleigh-Ritz-type approximation for D•. We provide an example in a later section.
9.5.4 Buckling of Pressurized Spherical Shells: Vlasov has investigated the
stability of pressurized spherical shells by using equations (9.208a-e) and
(9.209).71 For a sphere of radius a,
V2( )
= !V 2(
a
(9.223)
)
The membrane theory of axisymmetrically loaded spherical shells has been
presented in section 4.3.2.2. For a uniform internal suction or external pressure
p, q. = - p, and equations (4.27) and (4.28) give
pa
(9.224)
N,p = No = - 2
from which the prebuckling stresses
pa
N xo = N yO = - -
2
(9.225a)
and
So
=0
Then the r.h.s. of equation (9.209) becomes
(9.225b)
458
9 Shell Bending and Instability
(9.226)
Now, in view of equations (9.224)-(9.226), equation (9.209) reduces to
V8G
+
12(1 - 1l2)V 4 G + pa V6G
h2 a 2
2D
=
0
(9.227)
We seek solutions of the form
(9.228)
where 2 is a characteristic or eigenvalue related to P
22 [22
+ Pcra 2 + 12(1
- 1l 2 )] = 0
h2 a 2
2D
The trivial solution 2
equation
22
+ Pcr a 2 +
2D
=
= Pcr>
and obtain
(9.229)
0 is discarded, and we have remaining the quadratic
12( 1 - 1l 2 ) = 0
h2 a 2
(9.230)
which we write in terms of Pcr as
(9.231)
It is initially unclear as to which value of 2 corresponds to the minimum
buckling load.
We may proceed by invoking the minimization condition to find the lowest
value of Pcr> Pcrl. This condition is
Pcr.A
=0
(9.232a)
from which
(9.232b)
or
2
=
+ 2~[3(1
- 1l 2 )]
ha
(9.232c)
The negative root of 2 is meaningful, since it results in a positive Pcr> which has
already been defined as an internal suction or external pressure.
With
2 = -2~[3(1 - 1l 2 )]
ha
(9.233)
9.5 Instability and Finite Deformations
Perl =
2E
(h)2
J[3(t - 112)] ~
459
(9.234)
is the lowest buckling pressure for a spherical shell. This equation is applicable
for complete spheres, as well as for spherical shells that have close to ideal
membrane boundary conditions. A direct treatment of spherical shell buckling
in polar coordinates is developed in Brush and Almroth. 72
Note that we have arrived at the lowest critical load, without first finding
a general solution for the normal displacement Dn. Subsequently we may
investigate the normal displacement Dn , which is the mode shape of the buckle.
If we return to equation (9.208d),
Dn = V 4 G
= V2(AG)
(9.235)
= AV 2 G
The potential function G, in turn, is found from equation (9.228)
V 2 G - AG
=
0
(9.236)
where A is given by equation (9.233). The solutions of equation (9.236) are the
eigenfunctions of the problem. Recall, however, that we have used the simplest
form of the MDV theory based on Cartesian coordinates. Consequently,
solutions of equations (9.235) and (9.236) that satisfy the kinematic boundary
conditions are only feasible if the shell covers a rectangular plan area. For
boundaries coincident with the coordinate lines, the original form of the theory,
equations (9.186a), (9.t86b), and (9.187), should be used in place of the subsequently simplified equation (9.194). This is carried out in some detail in
Vlasov,73 where it is shown that the lowest critical pressure only differs from
Perl' as computed from equation (9.234), by a term of 0 [11(h/a)]: 1. Nevertheless,
the more rigorous solution provides a basis for determining the eigenfunctions
and subsequent displacement or mode shapes. In many applications, only Perl
is of interest, and the solution obtained from equation (9.234) is sufficient to
estimate the buckling pressure. Note that, in this regard, quite large factors
of safety against elastic buckling-perhaps three to five or even more-are
commonly specified for the design of thin shells. Since the bifurcation analysis
accounts for neither initial imperfections nor nonlinear behavior, the critical
pressure obtained theoretically is very difficult to approach in reality, even
in carefully controlled experiments,74 and a very ample factor of safety is
prudent unless a more sophisticated analysis can be accomplished. A means for
introducing additional refinements into the analysis is described in section 9.5.6.
However, many design procedures are based on the elastic critical load, reduced
substantially by a "knockdown factor."
9.5.5 Buckling of Axially Loaded Cylindrical Shells: We now investigate the
buckling ofthe shell shown in figure 9-15. We use the energy criterion, equation
460
9 Shell Bending and Instability
(9.222), and assume a one-term Rayleigh-Ritz-type solution for Dn
D
C Slnmn
.
X
n =
L
(9.237)
Substituting equation (9.237) into I1U. and I1Uq as given by equations (9.218)
and (9.221), respectively, gives
EhC 2L
2p,Ehe
C
fL sinmn-dX
X
m4n4C 2 }
+D 2
I1U.
=
an {
I1Uq
=
2n~r {P,C LL sin mn ~ dX + ~ m~n2 C 2}
and
2
a
2
-
a
xo
0
L
L
3
(9.238)
(9.239)
After substituting equations (9.238) and (9.239) into equation (9.222), and noting
from equation (9.210a) that exo = ( - Ncr/Eh), we find the coefficients of the C
terms cancelling and the coefficients of C 2 producing
Ncr
=D
[m 2n 2
Eh L2 ]
-u + a2D
m2n2
The minimum
Ncr .m=
(9.240)
Ncr is found from
0
(9.241a)
giving
(9.241b)
and
(9.242)
a.J[3(1 - p,2)]
The length of the buckled waves is given, in terms of the characteristic
term noted in section 9.2.2.2, by
fo
(9.243)
which may be useful for establishing the proper meridional spacing of ring
stiffeners.
The identical result may be obtained by solving the equilibrium equation,
equation (9.209).
Various other specific solutions based on the two approaches developed in
this chapter are available in the literature for spherical and cylindrical shells.
461
9.5 Instability and Finite Deformations
Using either the differential equation or the energy approach, the solutions
are quite simple and may be used for other geometries with some judicious
geometric approximations (see Vlasov 75 and Fliigge 76). Also, an extensive
array of more complex situations, including body forces, fluid pressure, torsion, orthotropic shells, and initial imperfections, is treated in Brush and
Almroth. 77
9.5.6 Finite Deformation of Shallow Shells: Recall from section 8.7.5 that
the introduction of the geometrically nonlinear strain-displacement relations, (8.289a-c), lead to the von Karman equations, describing finite deformations for plates. Vlasov has derived a similar set of equations for shallow
shells. 78
In this case, the nonlinear strain-displacement relations are the same as for
plates, except that ex and By each have an additional term proportional to
the radii of curvature, as we may verify from equations (5.46a) and (5.46b).
Therefore, the appropriate equations in Cartesian coordinates are
_
ex = Dx .x
Dn
1
+ Rx + :2(Dn,x)
2
(9.244a)
(9.244b)
(9.244c)
If we follow the identical steps leading to equation (8.296), we obtain 79
4= _ [2
1 Dn,yy - Ry
1 Dn,xx]
V .'#' - Eh (Dn,xy) - Dn,XXDn,yy - Rx
(9.245)
as the generalized form of that compatibility equation. As before,.JF is the stress
function defined by equations (8.295).
Now, considering the second equation which was derived for plates as
equation (8.297), note that this equation followed from equation (8.252), which
is the finite displacement version of the basic biharmonic equation for plate
bending, equation (8.12). Following the same reasoning, the generalization of
equation (8.12) to shallow shell theory is equation (9.183), where the term
NIX
Np
RIX
Rp
-+is added to the l.h.s. This term enters into the shallow shell equivalent of
equation (8.252) as
Nx
Ny
Rx
Ry
-+and subsequently into the shallow shell form of equation (8.297) as
462
9 Shell Bending and Instability
-(;x
~yy + ;y ~xx)
on the r.h.s. Finally, we have the generalization of equilibrium equation (8.297):
V4Dn
=
~ [qn + ~yyDn,XX + ~XXDn,yy - 2~xyDn,Xy
(9.246)
- _1_.1i'
Rx
,yy
- _1.1i'
Ry
,xx
]
Equations (9.245) and (9.246) constitute the generalized von Karman equations,
which form a finite deformation theory for shallow shells. In these equations,
the in-plane stress resultants Ny and Ny are assumed as positive. The corresponding plate equations (8.296) and (8.297), of course, follow for Rx = Ry = 00;
also, for a cylindrical shell of radius a, Rx = 00 and Ry = a. Similarly, for
a spherical shell with radius a, Rx = Ry = a.
We have noted in the previous section that the elastic buckling load based
on infinitesimal deformation theory, as presented in sections 9.5.1 through 9.5.5,
may only be a rough indication of the load at which an actual shell may buckle.
We have also suggested that a more elaborate analysis may provide theoretical
results which are in reasonable agreement with physical experiments. Such
an analysis has been provided by two of the most distinguished mechanicians
and engineers of the twentieth century, T. von Karman and H. S. Tsein, in
their study of the postbuckling behavior of cylindrical shells based on the finite
deformation equations presented in this section. so They used a Rayleigh-Ritz
approach and assumed an approximate expression for D n , selecting the unspecified parameters based on minimization of the potential energy functional
and on some test results. This analysis provides a strain energy expression
similar to equation (9.238), modified by a factor proportional to the buckled
normal deflection to thickness ratio Dn/h. For Dn/h = 0, the critical load is given
by equation (9.242); as Dn/h increases, the critical load decreases rapidly to only
about 1/3 of this value. S !
A frcq uently used equation for the buckling of axially loaded cylindrical shells
based on their analysis is
Eh2
Ncr!
= 3ay'[3(1 _
Eh2
J.l2)]
(9.247)
~0.2-
a
implying a "knockdown factor" of 3 when compared to equation (9.242).
By way of confirmation of this theory, an experimental study of open aluminum cylindrical shells with simply supported, uniformly compressed ends and
free longitudinal edges by Yang and Guralnick gave measured buckling loads
9.5 Instability and Finite Deformations
463
that correlated quite well with numerical predictions based on equations (9.241a),
(9.241 b), and (9.242). 82 It was observed that the correlation between analytically
predicted and experimentally measured buckling loads was considerably better
than similar comparisons for closed cylindrical shells. It was also noted that
open shells have a theoretical critical stress of only 10% to 15% of the corresponding values for comparable closed shells in the range of parameters
investigated. A relatively complete compilation of the rather extensive literature
in this area is contained in the study.
9.5.7 Postbuckling and Imperfection Sensitivity: Inasmuch as the theoretically
determined buckling load calculated by a small deflection theory is rarely
attained or even approached in experiments, attention has been focused on
the sources of discrepancy such as (a) the influence of boundary conditions;
(b) the presence of prebuckling deformations caused by edge constraints; and,
(c) the effect of initial imperfections. 83 It is generally assumed that the initial
imperfections are the most influential contributor.
The effect of initial imperfections is closely related to the load-displacement
response following the onset of buckling, the postbuckling path. Although the
calculation of the secondary equilibrium path is beyond our present scope, we
may observe some possibilities from figure 9-14. Curve 4 has a negative slope
and is said to be unstable, whereas a system which follows curve 5 and snaps
through to the positive slope branch is said to be stable in the post buckling
region. The possibilities for slightly imperfect shells are indicated more clearly
in figure 9_16,83 where three cases are shown. Between the cases of stable
symmetric post buckling (I) and unstable symmetric post buckling (III) is the
case of un symmetric post buckling (II), which may follow a stable or unstable
course.
Structures which behave as case I are said to be imperfection-insensitive. For
this case, the classical buckling load is a useful measure, whereas for the other
cases, a more refined calculation is usually necessary. Imperfection-sensitive
structures may be prone to a more violent buckling failure and sudden collapse,
so that a careful analysis is warranted. The seminal contribution in the field of
postbuckling analysis and imperfection sensitivity is the theory of Koiter. 83 . 84
Rotational shells subject to axial compression are generally highly imperfection sensitive; in the case of plates or shallow roof shells which can develop
a secondary mode of resistance such as cantenary or tension field action,
buckling is relatively insensitive to initial imperfections.
Shells may also buckle under external normal pressure or internal suction.
Spherical shells under such loading are generally imperfection-sensitive. More
generally, Wittek 85 considered free-ended rotational shells of varying Gaussian
curvature, i.e., an ellipsoid (+), a cylinder (0), and a hyperboloid ( -), under
an axisymmetrical normal pressure. In figure 9-17, results are shown for the
critical pressure A. as a function of RdR2' where Rl = Ro (,p = n12) which is proportional to R e, and R2 = R~. Three different computations were performed.
464
9 Shell Bending and Instability
P
A=PCL
P
A=-PCL
1.0
Fig. 9-16
Case I
Case II
Case III
(a)
(b)
(c)
Postbuckling Paths for Slightly Imperfect Structures (a) Case I: Stable Postbuckling,
(b) Case II: Unsymmetric Postbuckling, (c) Case III: Unstable Postbuckling
The upper curve, Aero is the small deformation (linear) critical load, whereas
the lower curve, Ab' is a reduced critical load based only on the bending portion
of the strain energy, roughly equivalent to neglecting the group I terms in
equation (7.7). The latter calculation is claimed to yield a lower bound solution
to the linear buckling value. The third computation, a nonlinear analysis
indicated by the plotted points, will be discussed subsequently.
Buckling may be idealized as a shift in the load resistance mechanism from
a stiff membrane mode, associated with the group I terms in equation (7.7),
to a more flexible bending mode. Accordingly, the initial imperfections would
be primarily acted upon by the in-plane stress resultants. It follows that this
sensitivity would be measured by the difference of the curves, Acr - Ab = Am.
This difference ()bviously reduces with decreasing Gaussian curvature and is
quantified by the factor (1.g. In the negative curvature region, the points where
the curves intersect correspond to geometrical combinations for which pure
bending or inextensional deformations, discussed in section 4.3.6.1, may occur.
Between the curves for Acr and Ab' the plotted points indicate values calculated
by a nonlinear analysis. The difference between these values and the lower curve
is regarded as confirmation of Ab as the lower bound, not only for the linear but
also for the nonlinear critical load.
Shells must also be investigated for buckling due to asymmetric normal
pressure, such as the wind load depicted on figure 4-36(a) for a spherical shell
and on figure 4-34 for a cylinder or hyperboloid. We expect buckling to initiate
in regions of high compression under those loading conditions.
j~6
I
Ellipsoid
I----l
I
-0,04
-0.0304
-0,02
Circular Cylinder
+0,02
wave number
i
Hyperboloid
1-'-C-I~R1
-0,06
-0,08 R2
o
VI
0'1
"'"
::I
'"
k=2
I>l
a.
9
0'
~
0'
[
Po
§
Q
5i
5'
'"
Iii
}=4
ag=Acr-Ab = Am
Acr
Acr
a.'.
Rl/h=190
A _ qR12
,'/Rl =6,00
- Eh2
q
p. =0.2
k =1(2)
glob
imperfection j =circumferential
wave number
sensitIvity factor
k = meridional
{f?'s::--zf7fh21
r
jRl
T
I
0
2
Fig. 9-17 Global Imperfection Sensitivity of a Rotational Shell
+0,04
I
I
0,2
-+------
I
I
0,5
-I-
--+------j--
+0,06
Nonlinear Au
/=5
I
~-20
I«({{{(f-{(~
R1~,
R2 +0,08
~
I A
l,
+ A~
!---l- 5,0
q
Acr
Rl
1!t
rE! 1
2'
'"v.
466
9 Shell Bending and Instability
For the spherical shell, the resultants plotted in figure 4-36(b) indicate that
there are regions along the windward meridian where both N,p and NIJ are
compressive. This is true for the simplified as well as for the realistic pressure
distribution. The biaxial state of compression tends to increase the imperfection
sensitivity.
In contrast, for the cylindrical shell considered in section 4.3.7.4, the circumferential stress resultant No is proportional to the load, equation (4.149b), and
thus would be compressive along the windward meridian, () = 0°. However,
the meridional stress, equation (4.149a), would be tensile in this region as a
result of overturning and would tend to counter a circumferentially initiated
buckle. Similarly, in the separation zone, () ~ 70°, the tendency for a meridionally initiated buckle due to a compressive N,p is countered by a tensile
value of NIJ. The same behavior is true for hyperboloidal shells. Such shells,
where the compression is only uniaxial and perhaps opposed in the perpendicular direction, may be regarded as imperfection-insensitive for that particular
loading case. For an axial or self-load, a state of biaxial compression may be
present in the same shell, e.g., figure 4-8, and the shell would be more sensitive
to imperfections.
9.6 References
1. M. Hetenyi, Beams on Elastic Foundations (Ann Arbor: University of Michigan
Press, 1946).
2. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (New York:
McGraw-Hill, 1959), pp. 472-481; E. Y. W. Tsui, Stresses in Shells of Revolution
(Menlo Park, CA: Pacific Coast Publishers, 1968), p. 63.
3. C. R. Calladine, "Theory of Shell Structures," (Cambridge: Cambridge University
Press, 1983); Gaussian Curvature and Shell Structures, The Mathematics of Shell
Structures [J. A. Gregory, Ed.] (Oxford: Clarendon Press, 1986) pp. 179-196.
4. H. Kraus, Thin Elastic Shells (New York: Wiley, 1967), pp. 136-139.
5. "Design of Cylindrical Concrete Roofs," ASCE Manual of Engineering Practice
no. 31 (New York: American Society of Civil Engineers, 1952).
6. W. Fliigge, Stresses in Shells, 2nd ed. (Berlin: Springer-Verlag, 1973), chap. 5 and
pp. 513-515.
7. V. V. Novozhilov, Thin Shell Theory [translated from 2nd Russian ed. by P. G. Lowe
(Groningen, The Netherlands: Noordhoff, 1964), pp. 215-218].
8. V. Z. Vlasov, General Theory of Shells and Its Application in Engineering, NASA
TTF-99 (Washington, D.C.: National Aeronautics and Space Administration, April
1964).
9. D. P. Billington, Thin Shell Concrete Structures, 2nd ed. (New York: McGraw-Hill,
1982) pp. 10-21.
10. Kraus, Thin Elastic Shells, pp. 221-229.
11. Ibid., pp. 200-204.
9.6 References
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
467
Ibid., pp. 297-314.
Fliigge, Stresses in Shells, pp. 217-250.
Design of Cylindrical Concrete Roofs.
Vlasov, General Theory of Shells, pp. 359-394.
Ibid., pp. 376-394.
P. L. Gould et ai., "Column Supported Cylindrical-Conical Tanks," Journal of the
Structural Division, ASCE 102, no. ST2 (February 1976): 429-447.
Fliigge, Stresses in Shells, pp. 231-235.
Gould et ai., "Column Supported Cylindrical-Conical Tanks," pp. 429-447.
D. P. Billington, Thin Shell Concrete Structures (New York: McGraw-Hill, 1965),
pp.171-172.
Design of Cylindrical Concrete Roofs.
Billington, Thin Shell Concrete Design, chap. 6.
Design of Cylindrical Concrete Roofs.
Ibid.
Ibid., pp. 48-57.
Ibid., pp. 48-57.
Novozhilov, Thin Shell Theory, chap. IV.
Fliigge, Stresses in Shells, chap. 6.
Kraus, Thin Elastic Shells, chaps. 5, 6, and 7.
H. Reissner, Spannungen in Kugelschalen (Leipzig: Muller-Breslau-Fetschrift, 1912),
pp.181-193.
E. Meissner, "Das Elastizitatsproblem fUr diinne Schalen von Ringflachen-, Kugel-,
and Kegelform," Physik Zeit. 14 (1913): 343-349.
Fliigge, Stresses in Shells, pp. 326-344.
Ibid.
Novozhilov, Thin Shell Theory, chaps. III-IV.
Fliigge, Stresses in Shells, chap. 6.
Kraus, Thin Elastic Shells, chaps. 5, 6, and 7.
Fliigge, Stresses in Shells, pp. 326-334.
Kraus. Thin Elastic Shells. pp. 271-283.
Novozhilov, Thin Shell Theory, pp. 315-319.
Kraus, Thin Elastic Shells, pp. 261-271.
Fliigge, Stresses in Shells, pp. 341-351.
H. Harintho, private communication.
J. N. Pirok and R. S. Wozniak, "Steel Tanks," in Structural Engineering Handbook,
E. H. Gaylord, Jr., and C. N. Gaylord, eds. (New York: McGraw-Hill, 1968),
chap. 23.
Gould et ai., "Column Supported Cylindrical-Conical Tanks."
R. S. C. Wang and P. L. Gould, "Continuously Supported Cylindrical-Conical
Tanks," Journal of the Structural Division, ASCE 100, no. STlO (October 1974):
2037-2052.
468
9 Shell Bending and Instability
46. G. V. Ranjan and C. R. Steele, "Analysis ofTorospherical Pressure Vessels," Journal
of the Engineering Mechanics Division, ASCE 102, no. EM4 (August 1976): 643-657.
47. P. L. Gould, J. S. Lin and J. M. Rotter, "Linear Stress Analysis of Torospherical
Head," Journal of Engineering Mechanics, ASCE III, no. 10 (October 1985): 12961300.
48. Fliigge, Stresses in Shells, pp. 386-413.
49. Novozhilov, Thin Shell Theory, chap. 4.
50. P. L. Gould and S. L. Lee, "Hyperboloids of Revolution Supported on Columns,"
Journal of the Engineering Mechanics Division, ASCE 95, no. EM5 (October 1969):
1083-1100; P. L. Gould and S. L. Lee, "Column-Supported Hyperboloids under
Wind Load" (Zurich, Switzerland: Publications, International Association for Bridge
and Structural Engineering, 1971), vol. 31-11, pp. 47-64.
51. L. J. Brombolich and P. L. Gould, "Finite Element Analysis of Shells of Revolution
by Minimization of the Potential Energy Functional," Proceedings ofthe Symposium
on Applications of Finite Element Methods in Civil Engineering (Nashville, Tenn.:
Vanderbilt University, 1969, pp. 279-307.
52. Ibid. and L. J. Brombolich and P. L. Gould, "A High-Precision Curved Shell Finite
Element," Synoptic, AIAA Journal 10, no. 6 (June 1972): 727-728.
53. P. L. Gould, Finite Element Analysis of Shells of Revolution, (London: Pitman, 1985).
54. W. C. Schnobrich, "Analysis of Hyperbolic Paraboloid Shells," Concrete Thin Shells,
publication SP-28 (Detroit: American Concrete Institute, 1971), pp. 275-311; "Different Methods of Numerical Analysis of Structures," Proc. of Symposium on
Practical Aspects in the Computation of Shell and Spatial Structures, (K. U. Leuven,
Belgium, July 1986).
55. Novozhilov, Thin Shell Theory, pp. 88-94.
56. Ibid.
57. Vlasov, General Theory of Shells, p. 345-346.
58. Novozhilov, Thin Shell Theory, p. 92.
59. Ibid.
60. Vlasov, General Theory of Shells, pp. 495-514.
61. Ibid.
62. Ibid.
63. Novozhilov, Thin Shell Theory, pp. 94-99.
64. Ibid.
65. Ibid.
66. B. O. Almroth and J. H. Starnes, "The Computer in Shell Stability Analysis," Journal
of the Engineering Mechanics Division, ASCE 101, no. EM6 (December 1975):
873-888.
67. D. Bushnell, "Static Collapse: A Survey of Methods and Modes of Modes of
Behavior," Finite Elements in Analysis and Design 1, (1985), pp. 165-205,
68. Vlasov, General Theory of Shells, pp. 521-525.
69. Ibid.
70. S. Timoshenko and J. Gere, Theory of Elastic Stability (New York: McGraw-Hill,
1961), pp. 457-519.
71. Vlasov, General Theory of Shells, pp. 525-529.
469
9.7 Exercises
72. D. O. Brush and B. O. Almroth, Buckling of Bars, Plates, and Shells (New York:
McGraw-Hill, 1975), pp. 142-258.
73. Vlasov, General Theory of Shells, pp. 525-529.
74. Timoshenko and Gere, Theory of Elastic Stability, pp. 468-473.
75. Vlasov, General Theory of Shells, pp. 529-538.
76. Flugge, Stresses in Shells, chap. 7.
77. Brush and Almroth, Buckling of Bars, Plates, and Shells.
78. Vlasov, General Theory of Shells, pp. 538-543.
79. Ibid.
80. T. von Karman and H. S. Tsien, "The Buckling of Thin Cylindrical Shells under
Elastic Compression," Journal of Aeronautical Sciences 8 (1941): 303.
81. Timoshenko and Gere, Theory of Elastic Stability, pp. 471-472.
82. T. H. Yang and S. A. Guralnick, "Buckling of Axially Loaded Open Shells," Journal
of the Engineering Mechanics Division, ASCE 102, no. EM2 (April 1976): 199-211.
83. H. Kolisnik and C. Tahiani, "A Survey of Methods of Analysis of Stiffened Shell
Structures," Civil Engineering Research Report No. 85-3, Royal Military College of
Canada (August 1985).
84. W. T. Koiter, "Over de Stabiliteit van het elastisch evenwicht," Dissertation Delft,
1945, English Translation NASA TT F-lO, 833 (1967).
85. U. Wittek, "Beitrag zum Tragverhalten der Strukturen bei endlichen Verformungen
unter besonderer Beachtung des Nachbeulmechanismus dunner Flachentragwerke,"
Mitteilung Nr. 80-1, Technical Reports, Institut fur Konstruktiven Ingenierbau,
Ruhr-Universitat Bochum (May 1980).
9.7 Exercises
9.1
(a) Consider exercise 4.8. Assume that the lower boundary is fixed and determine
the bending stress resultants and couples in terms of the radius a, height H,
thickness h, Young's modulus E, Poisson's ratio J.t, and the unit weight y.
(b) Re-solve part (a), except now assume that the lower boundary is hinged.
9.2
Consider the solution for exercises 9.1(a) or (b) with properties typical of two
common engineering materials, and graph the variations of No and M x for the
tank full and half-full. Locate the points of maximum stress.
Material properties
Property
Steel
E
J.t
30 x 10 6
0.25
0.25
250
250
0.20
h
H
a
y
Concrete
3
X 106
0.10
10
250
250
0.08
Units
force/area
length
length
length
force/volume
470
9 Shell Bending and Instability
For the English system, length units
inches 3 ; force units = pounds.
9.3
Consider the shell shown in figure 9-5 with the following numerical values.
Additionally, E, Jl, and h are as listed under Steel in exercise 9.2.
a = 36
p = 100
Tl = 50°;
9.4
9.5
= inches; area unit = inches 2 ; volume units =
(length);
(force/area)
T2 = 80°
L = 360
(length)
Compute Dn, No, and Mx due to the pressure and temperature effects separately
and in combination. Locate the points of maximum stress by graphing the results.
Consider the ring-stiffened cylindrical shell shown in figure 9-7, but with the rings
located on the inside of the shell. Derive the equations to evaluate the contact
force P.
Consider the tapered cylindrical shells as shown in figure 9-18. The shell is
subjected to hydrostatic pressure with the unit weight of the fluid = y.
(a) Derive the general solution for the displacement function, and also expressions
for Mx and No.
(b) If the analytical solution becomes too involved, set up a numerical solution
for (a).
(c) Consider an alternative to the gradual increase in thickness, whereby the shell
is constructed in two constant thickness segments
x
H
a
Fig. 9-18
471
9.7 Exercises
h3
=
3h 1 :
h2 (0 < X
~~)
and
h4
= h1
: 3h 2
(~ < X ~ H).
For this configuration, repeat the calculations in (a) and compare the results to
those found for the tapered wall.
9.6
Consider exercise 4.1, which refers to figure 4-2. Now assume that the base is
hinged rather than free to displace in the normal direction and perform a complete
analysis using an approximate bending solution. If the algebra becomes unwieldy,
assume reasonable values for the material and geometric parameters.
9.7
Continue the solution of the compound shell shown in figure 9-11 from equation
(9.150) and verify the key points on the graphs shown in figure 9-12(a). Use
h1 = h2 = h, h/a = 0.01, and Jl = 0.3. Assume reasonable values for any additional
properties required.
9.8
Verify the pole conditions for a shell of revolution listed in table 6-1.
9.9
Derive [Bb], matrix 9-4, from [B{], matrix 9-3.
9.10
Derive a matrix analogous to [Bb] for the closed shell shown in figure 4-1(b).
9.11
Specialize the MDV and the shallow shell theories for cylindrical shells.
9.12
Investigate the bending theory solution under gravity loading for the elliptic
paraboloid shown in figure 4-51. Assume that the shell is shallow.
9.13
Compute the buckling pressure for the spherical shell treated in section 9.5.4 using
the energy method.
9.14
Compute the buckling load for the cylindrical shell treated in section 9.5.5 using
the equilibrium equation method.
9.15
Consider a closed cylindrical shell oflength L and radius a and derive an expression
for the critical uniform internal suction.
9.16
Compute the value of the critical density of material y (force/volume) for which a
hemispherical shell of thickness h and radius a would buckle under self-weight.
9.17
Consider the cylindrical shell shown in figure 9-11, but with a flat circular plate
end. Determine the meridional moment and radial force at the junction. Take the
plate thickness as twice the shell thickness h and assume reasonable values for
any other properties required.
9.18
Try to obtain the solution to exercise 9.17 as the degenerate case of the spherical
head considered in exercise 9.7.
CHAPTER
10
Conclusion
10.1 General
The preceding chapters have treated the analysis of thin elastic plates and shells
from a unified viewpoint, insofar as possible. Specializations to specific formse.g., membrane shells, flat plates-were introduced in a logical sequence after
the general aspects of the theory were set forth. The emphasis was on explaining
the mechanics of surface structures, with ample examples to illustrate the most
important aspects. In general, relatively simple and idealized cases, which could
readily be solved analytically, were used to illustrate the salient points. Fairly
complete compilations of known analytical solutions for both plates 1 and thin
shells 2 are available elsewhere to supplement the examples given in the text.
The solution of more complex structures is relegated to the domain of numerical
analysis, and several citations to the literature in this area were provided.
Although the calculations for those situations may be more involved, such
structures generally reflect the same basic behavioral characteristics as the
relatively simple illustrations contained in this book.
10.2 Proportioning
One aspect of the shell design process that has not been extensively addressed
in this text is proportioning. Rather, the emphasis here has been on analysis.
In a practical sense, proportioning is vital, and it is our hope that interest in
this topic has been aroused. It is clear that the specific form of a surface
structure is mainly determined by the anticipated function: flat plates for
carrying pedestrian and vehicular traffic, shells of revolution for containment,
and shallow shells for long, clear-span roofs. Also, the construction materialsgenerally metals and/or concrete-playa very important role in the proportioning process.
Although the proportioning of surface structures is obviously very much
related to the anticipated function and the material of construction, it is possible
to offer a few general comments on this topic. We consider separately (a) regions
472
10.2 Proportioning
473
of the structure where the stresses are primarily in-plane or membrane; and,
(b) regions where there is significant bending action.
In case (a), direct tensile stresses should be resisted entirely by reinforcing
steel in concrete shells, or by providing ample wall thicknesses for metal shells.
Regions with direct compressive stresses are generally controlled by stability
requirements, particularly in metal plate structures.
For case (b), the bending moments or stress couples may be resisted by considering a concrete section with reinforcement near the surfaces to act as a wide
flexural member. However, the relatively small effective depth available may
complicate the provision of ample reinforcing steel. For thin metal structures,
it is often impractical to resist high local bending stresses with the basic shell
thickness, so that stiffening members are attached.
Beyond these brief generalizations, the proportioning ofthin surface structures
is a somewhat subjective and always challenging branch of structural design
which has become somewhat specialized for various structural forms and
materials.
For thin concrete shells, the Building Code Requirements for Reinforced
Concrete (ACI 318) give some general guidelines. 3 The shell thickness and the
reinforcement area and spacing should satisfy the conditions of strength and
serviceability which prevail for all concrete structures. Additionally, thin shells
must be investigated for general and local instability. Often, the shell thickness
is determined by the later consideration.
Reinforcement is required to resist tensile stresses, as well as to control
shrinkage and thermal cracking. The bars which resist the in-plane stress
resultants should be placed in two or more directions and should ideally be
oriented in the general directions of the principal tensile stresses, especially in
regions of high tension. Often, however, this is impractical, so that the required
resistance in every direction must be provided by an orthogonal mesh. A
representative situation is illustrated in figure 10-1, where the various types
of reinforcement required in an umbrella shell are shown. The orthogonal grid
for the shell proper, the additional shell steel in the vicinity of the columns, and
the edge beam reinforcement are difficult to accommodate in a relatively thin
section, and careful planning and supervision are necessary to ensure a sound,
crack-free structure.
Reinforcement to resist stress couples should be placed near both faces even
though moment reversal is not anticipated, since the bending may vary rapidly
along the surface. The two layers may also include the membrane reinforcement,
or there may be three or more layers. The provision of adequate clearance and
cover may necessitate increasing the shell thickness in such cases.
Of course, the edge members must be proportioned to resist the forces
imparted by the shell. Often, an adjacent portion of the shell can be assumed
to act with the member, or the edge member may be formed by a smooth
transition from the shell cross section. Frequently in concrete shells, posttensioning is employed to improve the efficiency of these members.
474
10 Conclusion
Fig. 10-1
Reinforcement for an Umbrella Shell (Courtesy Dr. A. Monsey)
For steel tanks, J. N. Pirok and R. S. Wozniak 4 have treated several aspects
of proportioning. They discuss the calculation of the plate thickness in tensile
regions with regard to efficiency of the welded joints. In regions of compression,
the proper consideration of stability effects on the allowable stress is elaborated.
In junction regions (rings and knuckles), guidelines are provided for determining the portion of the shell wall that might be effective as a ring beam,
acting alone or with an additional structural member. Stiffeners and opening
reinforcement are also discussed briefly. The tank shown in figure 2-8(r) serves
as a representative example of these concepts.
Beyond the rather general treatments elaborated in the preceding paragraphs,
there are a number of authoritative works for concrete cooling towers, 5 steel
tanks, 6 pressure vessels, 7 and bins. 8, 9 Also, a number of unusual concrete tanks
are analyzed and partially detailed in a recent volume on shell structures. 10
10.3 Future Applications of Thin Shells
At the conclusion of a book which focuses on a rather specialized class of
structures, it seems appropriate to comment on future applications of such
constructions. In particular, roof shells of reinforced concrete have been built
far less frequently in the last quarter of the twentieth century than in the
10.4 References
475
immediately preceding decades. This has been attributed to the increased
availability of cheaper prefabricated systems. 11
One of the world's most distinguished engineers, J. Schlaich, has recently
considered this issue. 11 Schlaich eloquently states:
1. Shells are the most honest structures.
2. He, who cares about the shape of his structures, needs the shells.
3. He, who cares about the genuine use of materials, does not want to miss the
concrete shells.
This being the case, he concludes that "he, who cares about beautiful and genuine
structures, cannot accept the fact that concrete shells are slowly disappearing."
He conjectures that with the imaginative application of modern falsework
techniques, such as pneumatic forms, and new materials, such as fiber concrete,
light and versatile concrete shells may again be competitive, especially if some
premium is allowed for quality.
Newspaper editor Horace Greeley, a renowned observer of the American
scene, wrote in 1860: "I think concrete walls, rightly made, wi11last a thousand
years." 12 The same is surely true of concrete shells, and it is a challenge to the
present and future generations of structural engineers to see that such structures
are indeed "rightly made."
10.4 References
1. R. Szilard, Theory and Analysis of Plates (Englewood Cliffs, NJ: Prentice-Hall, 1974).
2. E. H. Baker, L. Kovalevsky, and F. L. Rish, Structural Analysis of Shells (New York:
McGraw-Hill, 1972).
3. "Building Code Requirements for Reinforced Concrete" (ACI 318-83) with Commentary (ACI 318R-83) (Detroit: American Concrete Institute, 1983), chap. 19.
4. J. N. Pirok and R. Wozniak, "Steel Tanks," in Structural Engineering Handbook,
E. H. Gaylord, Jr., and C. N. Gaylord, eds. (New York: McGraw-Hill, 1968), sec.
23.
5. "Reinforced Concrete Cooling Tower Shells: Practice and Commentary," Journal
of the ACI, Title 81-52, 81, no. 6 (November-December 1984): 623-631.
6. "Recommended Rules for Design and Construction of Large, Welded, Low Pressure
Storage Tanks," API Standard 620 American Petroleum Institute, 1966, "Welded
Steel Tanks for Oil Storage," API Standard 650, American Petroleum Institute,
1973; "AWWA Standard for Steel Tanks-Standpipes, Reserving and Elevated
Tanks-for Water Storage," AWWA D100-67, American Water Works Association,
or AWS D5.2-67, American Welding Society.
7. "Rules for Construction of Pressure Vessels," div. 1, 1971 ed., ASME Boiler and
Pressure Vessel Code, sec. VIII, American Society of Mechanical Engineers, 1971;
"Alternative Rules for Pressure Vessels," div. 2, 1971 ed., ASME Boiler and Pressure
Vessel Code, sec. VIII, American Society of Mechanical Engineers, 1971.
476
10 Conclusion
8. N. S. Trahair, A. Abel, P. Ansourian, H. M. Irvine, J. M. Rotter, Structural Design
of Steel Bins for Bulk Solids, (Sydney: Australian Institute of Steel Construction,
1983).
9. "Design and Construction of Welded Steel Bins," AWRA Technical Note 14, Australian Welding Research Association (Milsons Point, Australia, December 1974).
to. V. S. Kelkar and R. T. Sewell, Fundamentals of the Analysis and Design of Shell
Structures (Englewood Cliffs, NJ: Prentice-Hall, 1987): chap. 6.
11. J. Schlaich, "Do Concrete Shells Have a Future?" Bulletin of the lASS, no. 89
(December 1985): 38-46.
12. St. Louis Construction News & Review, vol. 18, no. 4 (April 15, 1987): 36.
Index
Aas-Jacobsen, A., 394
Abel, A., 474, 476
Abel, J., 346, 364
A-frame, 184
Agrawal, K., 179, 194
Air-supported structures, 71
Almroth, B., 352, 356, 366, 452, 459,
461,468,469
Alternate formulation for shells of revolution, 94-101
Alternate rules for pressure vessels, 474475
AISC Manual of Steel Construction, 309,
364
Amplification of stress, 407
Anadol, K., 14
Annular insert, See Plug, rigid; opening,
319,321,323; plate, See Plate, annular
Anisotropic material, 344
Ansourian, P., 474, 476
Anticlastic bending, 294-295, 305; shell,
162, 184; surface, 31, 294,295
Antisymmetrical loading, 112-130, 146,
148
Aortic heart valve, 41, 72
Approximate methods, v, See also specific methods
Are, 56; length, 20, 49, 50, 53, 85, 109,
213, 214, 220-224, 361, 434, 435;
length coordinate, 48, 49, 435, 436,
440
Arch, 5-8, 10; action, 5, 158, 174-176,
190; boundary, 180, 190; compression, 174-176, 180, 186; groin,
186; strip, 189; tension, 175, 180
Area, differential, 262; of a member,
235; plan, 173; projected, 173; surface, 262, 440
Arioylu, E., 14
Aspect ratio, 312, 356, 370
Assumptions, basic, 4; for boundaries,
176; membrane theory, 70; plates,
67
Astrodome, 39, 233
Asymmetrical load, See Loading, asymmetrical
Asymptotic integration, 418-420, 433,
solution, 418-423, 432
Attentuation of edge effects, See Semiinfinite shells
Auxiliary variable, 73, 109, 112, 113,
131, 149, 247
AWWA Standard for Steel Tanks, 474,
475
Axial compression, 176-178; coordinate,
49, 50, 52, 88, 95, 434; load, 94,
95,98-100, 105,455; strain, 176,
455; tension, 80, 81,176-178
Axially distributed loading, See Loading,
axially distributed
Axis of rotation, 48, 77, 84,90, 94,99,
373; orthogonal, 283, 285
Axisymmetrical shells, See Shells of revolution
Axisymmetric(al) bending, 313-325,
413-433; displacements, See Displacement, axisymmetric; loading,
74-112, 132, 140, 198, 318, 362,
374-394, 397
Back-of-the-envelope calculation,
169
Baker, E., 151, 191, 193,472,475
Banavalkar, P., 179, 194
Barrel shells, 37, 152, 394, 395, 407413
478
Base shear, 123, 124, 126-129, 202; distributions, 126-129, 195-197
Beam, 5, 270, 271, 279, 285, 304, 344;
action, 158, 378; deep, 4; downturned, 177-178; edge, See Edge
member; gridwork, 279; ring, See
Ring beam; simply supported, 308;
slice, See Slice-ibeam; solution, 308,
311; stiffening, 407; theory of, I, 3,
4, 122, 142, 144, 145, 225, 274,
278, 408, 413; upturned, 177-178
Bellworthy, A., 146, 193
Bending, energy, 263; general, 137, 158,
325-329; homogeneous, 246, 258,
321; moment, 55, 144, 145,316,
379, 394, 412, See also Overturning
moment; of plates, See Plates, bending of; of shells, v, 132, 151, 178,
189, 372-450; resistance, 279; stiffness, See Flexural rigidity; stress
couples, 68, 242, 279; stresses, 137,
473; theory, 109,471; See also
Flexural
Bergen, P., vii, 212, 226
Bernoulli, J., 279
Bernoulli-Euler equation, 278; theory, 3
Bifurcation, 451, 452
Biharmonic equation, 276, 279, 285,
314,325,326,461; function, 314,
321,325,326; operator, 276, 281,
282, 352
Billington, D., 80, 156-158,161, 163,
164, 174, 176, 186, 188, 191, 193,
194,258,259,348,366,395,410,
411,466,467
Bin, 137-140, 474
Binomial expansion, 172, 361; theorem,
352
Bobrowski, J., vii, 14, 185,194
Boiler end, See Bottom and Hemispherical head
Bottom, 406
Boundary, 155, 157, 164, 175, 180, 190,
249,286,287,317,372,402,407;
arch, 180, 188; assumptions, See
Assumptions for boundaries;
clamped, See fixed; column-supported, See Column-supported shell;
constraints, See Constraints; continuous, 132, 137, 157; exterior, 410;
Index
external, 171, 188,434,450; fixed,
239,259,286,292,305,318,328,
329,331,352,367,370,371; force,
184, 344, 401, 402; free, 78, 113,
142, 146, 157, 168, 173, 239, 259,
286, 344,407,462; hinged, 239,
240, 259; idealizations of, 238-240;
interior, 407-412; knife edge, 305,
369; loaded, 419; longitudinal, 157,
255; points, 389; reaction, 263;
requirements, 135; roller, 78, 239,
240, 259; shell, 240; simply-supported, 239, 240, 286, 308, 317,
328,329,363,367,402,407; sliding, 239, 259; straight, 166, 167,
176, 179; symmetry line, 409; transverse, 157; uncoupled, 410; See also
Boundary conditions and specific
types of shells
Boundary conditions, 71, 72, 74-76,
115, 151, 155, 156, 168, 169, 172,
181, 184, 188, 190,237-244,259,
268,269,272,308-310,312,314,
318,319,321,322,326,328,329,
336-338,341,356,361-363,371,
372,382,387,389,395,400,401,
404-407, 409, 410, 412, 444, 463;
axial, 256, 317; clamped, See fixed;
displacement, See kinematic; elastic,
238; fixed, 239, 288, 326, 327; free,
239,279,287,411; force, 131, 132,
344; hinged, 239, 288; homogeneous, 238, 288; ideal, 72, 82, 103,
443; kinematic, 156, 238-240, 255,
291,401,402,410,442,459; Kirchhoff, See Kirchhoff boundary conditions; membrane, 72, 372, 410, 443,
459; mixed, 88; natural, 442-443;
nonhomogeneous, 238; no-rotation,
292; plates, 242, 243, 286, 312;
roller, 239; shells of revolution, 243;
simply supported, 239, 287, 292,
296, 298, 309, 310, 326, 327, 329;
sliding, 239; static, 238-240, 402,
410, 442; symmetry, 156; See also
Boundaries and specific types of
shells
Brissoulis, D., 145, 146, 192
Brombolich, L., 252, 259, 346, 366,
435,439,442,468
Index
Brunelleschi, 7, 8
Brush, D., 352, 356, 366, 459, 461, 469
Buchert, K., 233, 235, 258
Buckling, See Instability
Building Code Requirements for Reinforced Concrete, 129, 192,473, 475
Built-in shell, 385-389
Bushnell, D., vii, 452, 468
Cables, 4, 184; hogging, 180; sagging,
180
Calculus of variations, 266, 267
Calladine, C., vii, 31,54,381,486
Candela, F., 165
Cantilever(ed), 178; edge member, 176,
178; HP shell, 167
Cataloglu, A., 72, 191
Celebi, M., 14
Ceftter of curvature, See Curvature
Change in curvature, 216, 22~223, 397,
436, 437, 444, 454, 456
Characteristic, 32, 183, 294; dimension,
v, 1,2,4; real, 163, 164,181; value, See Eigenvalue
Chattopadhyay, A., 72, 191
Chntko, H., II7, 192
Choi, C.K., 217, 226
Circle, 186
Circular cylindrical shells, See Cylindrical shells
Circular, plates, See Plates, circular; sector plates, 329, 331
Circular ring, See Ring beam
Circumference, 132
Circumferential angle, 48, 49, 73, 112;
buckling, 104, See also Wrinkling;
coordinate, 49, 109, 133, 243, 407;
deformation, 223, 252, 253; distribution of pressure, 1~145; force,
See Hoop tension and Ring tension;
loading, 124, 125, 137; radius of
curvature, See Principal radius of
curvature; stiffener, 431; strain, 102,
455; stress (resultant), 78, 86, 92,
95, 102, 106, 107, 109, 1l0, 140,
142, 197, 198,200,202,431,432,
466; See also Hoop stress
Clamped plate, 318, 319, 325,337-342
Clark, R., 72, 191
479
Classical solutions, v
Clearance, 473
Clebsch, 325
Closed shell, 75-76, 252, 400, 437-440,
471; See also Dome
Clough, R., 123, 192,258
Codazzi, See Gauss-Codazzi relations
Codes and standards, See specific titles
Collar, See Ring-stiffener
Column, reaction, 132-134; -supported
shells, 38, 40, 42, 43, 88, 132-140,
243,403-407,433; width, 192
Combined shells, See Compound shells
Comparison function, 442
Compatibility, vi .. condition( s), I, 71,
217,255,268,318,324,325,336,
361,394,425,433,444; equation(s), 30, 222, 224, 237, 245,
250,358,359,447,449,461; See
also Strain-displacement relations
Complex variables, 433
Composite material, 232
Compound shells, 85, 89, 101-109, 200,
423-433, 471
Compression, See Axial compression
Computer, v, vi, 312, 419, 443
Concentrated force, 67, 68; load, 3, 72,
263,264,306-308, 3II, 322, 323,
329,356,367,372,381,383,441;
moment, 67, 68
Concrete, fiber, 475
Cone, 6, 151,406
Conical segment, 151; shells, 98-10 I ,
200, 205, 372, 419, 433, 446,
452
Conoid, 162, 189-191
Conservation of energy, 265
Constitutive law, v, 71,227-237,245,
253, 258, 266, 273, 363, 372, 374,
375, 414, 433, 444, 454
Constraints, vi, 115, 116
Contact force, 392, 393
Continuity, 400; conditions, intersegment,442
Continua, v
Continuum approach, 237
Conway, H., 342, 365
Cooling tower, 40, 126, 474; See also
Hyperbolic cooling tower; Hyperboloidal shell
480
Coordinate line, 16-19, 24, 28, 49, 56,
211,213-215,238,243,263,279,
283,284,349,402,441,447
Coordinates, 16, 166, 171, 189,282,
291,407; axial, 142, 152,226,373,
402; Cartesian, 16, 47, 54, 158,
162, 164, 274-279, 285, 329, 332,
342, 343, 348, 349, 352, 354, 358,
366,434,447,449,450,453,459,
461; cylindrical, 397, 407-409;
function, 267, 268, 337,338,340,
341, 346; (orthogonal) curvilinear,
vi, 16, 17, 19-21,62,73,76,85,
88, 89, 104, 140, 158, 169, 215,
218,253,261, 268, 269,372,443,
453; polar, 279-283, 287, 329,
332-335,346,349,351,362,367;
transformation of, 164, 165, 285;
See also specific geometrical forms
Corner force, 188,205,287,288,294,
302-305,331, 367, 370
Couple, applied, 264, 270, 271
Courant, R., 268, 270, 307, 364
Cowper, G., 48, 54
Cracking, 473
Critical load, 355-358
Croll, J., vii, 146, 193
Cross-section, 229, 231, 233, 234
Curvature, 6, 62, 216, 225, 228, 267,
272, 293; center of, 90, 119; line of,
244; normal, 26; twisting, 449; See
also Gaussian curvature and Change
in curvature
Curve, shape of, 28
Cyclic symmetry, See Symmetry, cyclic
Cylinder, 6, 85, 100, 137, 140-146, 151
Cylindrical bending, 292-294, 360, 362;
conical shell, 431; coordinates, 53;
segment, 151; surface, 31; tank,
199, 200, 403-407;
Cylindrical shells, 54, 89, 96-98, 100109, 140-145, 150, 169, 205, 372,
373,420,421,423-426,428-433,
446, 463; bending theory of, 246,
373-413,448,471; buckling, See
instability of; column-supported,
403-407; concentrated load on, 403;
displacement, 252-258, 260; equilibrium equations for, See Equilibrium equations; instability of, 454-
Index
457,459-461,464-466,471; intersecting, 407-413; membrane theory
of, 152-158, 373, 386, 387,402,
409; open, 36, 153, 158, 203, 402,
407-413, 462, 463; pressure loaded,
See Pressure loading; ring-stiffened,
41,391-394,470; segment, 423,
430; stiffened, 41; tapered, 470;
towers, 143, 144; vibrations of, 115;
wind-loaded, 140-145, 202-203;
See also Barrel shells
Dead loading, See Self-weight
Decay length, See Wave length of bending
Deflected shape, 307
Deflection, See Deformation and Displacement; function, 308, 309, 318,
328, 329, 340, 357; large, 344
Deformation, 3, 184, 206-226; axial,
378; finite, 452, linear, v; radial,
423; See also Displacement
Desai, c., 346, 365
Design of Cylindrical Concrete Roofs,
152, 193,255,259,394,402,411413, 466, 467
Developable surface, 6
Dhatt, G., 72, 191
Diaphragm, 157; action, 6, 348
Differential equation approach, 261, 269,
270,285,289,345,354,360,413
DiMaggio, F., 265, 270
Direction, principal, See Principal direction
Directrix, curved, 189, 190; straight,
165, 189
Dischinger, F., 395
Discontinuities, 372, 381, 431
Discrete approach, 237; supporting systems, See Column-supported shells
Discriminant, 32, 163, 360
Displacement, 1I5, 206-212, 215-217,
238,262,267,337,348, 360-364,
375, 378, 384, 394, 399, 400-402,
405,407,410,411,421,430,433,
436, 442; antisymmetric, 251-253;
asymmetric, 252; axisymmetric,
248-250, 253; -curvature relations,
396; field, 265, 266, 268; finite,
362; formulation, 268-269, 274,
Index
298, 332, 376-378, 395-400, 436,
441; function, 267, 298, 326, 382,
470; generalized, 346; harmonic
components of, 253, 254; in-plane,
344, 353; membrane theory, See
Membrane theory displacements,
meridional, 115; method, 433, See
also formulation; middle surface,
246-258; nonsymmetric, 248, 250253; normal, 126, 127, 225, 250,
259,345,371,428,444,451,452,
454, 455, 459; of plates, See Plates,
deflected; radial, 421, 423-425; relative, 253, 254, 423-425; rigid
body, 238, 239; of shells with double curvature, 258; of shells of revolution, 247-253; of shells of translation, 253-258; small, 4, 245;
tangential, 115; vector, 206-208,
264,267,269,442; virtual, See
Virtual work, principle of
Distances, measurement of, 20, 26
Distributed load, 264, 383, 384, 441;
moment, 67, 68
Dome, 49, 50, 75, 76, 101, 113, 114,
135, 136, 201, 202, 314, 392, 434,
439; ancient, 10; masonry, 80; See
also Spherical dome
Dong, S., 233, 258, 344, 365
Donnell, L., 394, 399, 444, 445; equation, 399; See also MushtariDonnell-Vlasov theory
Doubly-curved shells, 150, 158-162
Drawdown, 137
Drilling degree-of-freedom, 212, 222,
238
Dundurs, J., 312, 365
Dynamic force, 118, 123, 126
Earthquake, See Seismic
Eccentricity effects, 178
Edge, See Boundary; force (load), 97,
98, 102, 132, 133, 137, 173-176,
184,255,289-295,378-381,389,
400,401,410,416,419,421,422;
member, 135, 169, 170-179, 443,
473; moment, 416
Effective causes, method of, 266; depth,
473; forces and couples, 241-243,
481
367, 374; width, 394; See also
Kirchhoff, G.
Eigenfunction, 459
Eigenvalue, 458
Elastic boundary, See Boundary conditions, elastic; constants, 294; foundation, 367, 378; law, See Constitutive law; modulus, See Young's
modulus
Elasticity, Theory of, vi. 1,30,55,56,
63,217,231,233,261,268,279,
285, 292, 446
Elkin, R., vii
Ellipse, 186
Elliptic equation, 31, 162; paraboloid,
162, 186-190,204,205,244,448,
471
Elliptical plate, 329, 331; transition, See
Ellipsoidal cap
Ellipsoid of revolution, 54, 198,
463
Ellipsoidal cap (head), 103, 200
End conditions, See Boundary conditions
Energy, criterion, See Stability criterion;
formulation for shells, 261, 372,
413,434-443; methods, 261-271,
332-342, 346; minimum total potential, See Minimum total potential energy, principle of; potential, See Potential energy; principle, v. 237,
264-269; solutions for plates, 322,
342; solutions for shells, 322-342;
strain, See Strain energy; See also
specific types of
Engineering theories, 1-3
Equilibrium, vi. 4, 55-70, 245, 265,
266, 274, 337, 375, 441; of an element, 284; equations, I, 61-67, 7376, 104, 152, 160, 206, 222, 237,
245,248,250,272,276,281,342,
343, 349, 359, 362, 372, 373, 395,
396, 413, 433, 449, 453, 454, 462;
overall, 92-95, 104-106, 119, 121,
122, 126, 130, 140, 143, 198,205,
245,302,381,416; See also specific geometric types
Error function, 269, 340
Estanave, 308
Euler, L., 266, 267, 279, 452
Expansion, See Thermal
482
Extensional energy, 263; force, 55; member, 5; mode, 6, 381; rigidity, 233;
shearing, 263; strain, 213, 215, 220,
332, 448; stress resultant, 62-68,
70, 378; thickness, 235-237
Extremum problem, 267, 442
Faber, C, 32, 54, 165, 193
Facet surface, 212
Factor of safety, 130, See also Knockdown factor
Falsework, See Formwork
Field measurements, 140
Fill load, 203
Filonenko-Borodich, M., 276, 364
Final causes, Method of, 266
Finite deformation of plates, 352, 358364, 444; deformation of shells,
461-463; difference method, 258,
342,443; element method, 48, 184,
217,258,268,342,346,433,442,
443
Fino, A., 104, 191
Finsterwalter, U., 395
First quadratic form, See Quadratic form,
first
Fliichentragwerke, v
Flexible shell, 71
Flexibility influence coefficient, 246,
424-430; matrix, 246, 430; method,
245,246,256,258,292,423-425
Flexural action, 6, 279, 304; member, 5,
6; mode, 6, 381; rigidity, 233,273,
342,344,347,378; thickness, 235,
236
FlUgge, W., 60, 69, 71, 89, 98,191,
252,259,395,400,407,413,417,
418, 430, 433, 461, 466-469
Folded plate, 45, 233, 346-348
Force, axial, 171; body, 263, 461;
boundary conditions, See Boundary
conditions; concentrated, 288; compressive, 169-171, 173, 371; -deformation relationship, 218; equilibrium, See Equilibrium equations;
extensional, 191; in-plane, See Inplane force; method, 433; tensile,
169-171; vertical, 171,325; See
also specific types of
Index
Form work , 165, 475
Forray, M., 266, 269, 270, 339, 365
Fourier coefficient, 113, 134, 144, 146"
296,297,354; series, 109-112,
131, 134, 136, 137, 142, 150, 154156, 188,219,243,251, 255, 296298,311,326,343,404,409-411,
433, 436, 441
Free boundary, See Boundary and
Boundary conditions; vibration, 126
Frenet-Serret formula, 26
Friction coefficient, 137; load, See Wall
friction loading
Frustum of a cone, 98
Function, 16; integration, 174; odd, 154;
stress, See Stress function
Functional, 266, 434, 440-442
Fung, Y.C., 13, 14,360, 366
Future applications, 474, 475
Gable HP shell, 167, 174, 176, 178
Galerkin method, 269-271, 337, 340342
Gauss, K., 31
Gauss-Codazzi conditions, 29, 30, 50,
51,54,64,73,78,91,417,466
Gaussian curvature, 30-32, 48, 64, 69,
83,89,90,97, 115, 116, 137, 151,
163, 186, 190,203,204,360,372,
373,381, 446, 463, 464
Geckeler approximation, 419, 420, 446
Generator, parabolic, 174; plane curve,
48; skewed, 189; straight, 165, 166,
189
Geometry, v, vi, 16-54, 72, 115,400,
434-436; smooth, 419; of middle
surface, 16-20; regular, 157; See
also specific types of shells
Gere, 1., 454, 459, 462, 468, 469
Gergely, P., 179, 194
Germain, S., 279
Gibson, 1., 395; equation, 399
Glockner, P., 243, 259
Goldberg, 1., 348, 366
Gol'denveizer, A., 26, 30, 54, 222,
226
Goodier, 1., 241, 259, 292, 364
Gould, P., 72, 86, 89, 123, 126, 128,
132, 149, 151, 191-193,227,250,
Index
252, 253, 258, 259, 346, 366, 406,
407, 431, 433-435, 439, 442
Gravity loading, See Self-weight load
Greeley, H., 475, 476
Green, A., 60, 69,345,365
Green's function, 307, 370, 383, 403
Gregory, J., 466
Gridwork system, 233
Grimm, C., 14
Groined vault, 185
Guralnick, S., 462, 469
Haas, A., 32, 54, 190, 194
Hadrian, Emperor, 7
Hagia Sophia, 7, 9
Han, K., vii
Hansen, J., 237, 258, 270
Harintho, H., vii, 431, 467
Harmonic analysis, See Fourier series;
component, 401,406,433; loading,
260, 300-305,434; number, 355,
436, 438, 440, 442; solution, 112,
405; See also Fourier series
Heart valve, 41, 72
Hedgren, A., 258, 259
Hemisphere, 8
Hemispherical head, 430-432, shell, 79,
101-102, 119, 135-137, 198,202;
See also Spherical shells
Hencky-Mindlin theory, 345
Heppler, C., 231, 258, 268, 270
Hemnann, G., 54
Hetenyi, M., 378, 466
Hilbert, D., 268, 270, 307, 364
Hill, G., 150, 193; equation, 150
Hipped HP shell, See Gable HP shell
Hoff, N., 329, 365, 394
Holes, 381, See also Annular plates
Homogeneous bending, 314; equation,
130; solution, 130-131, 405
Hooke's law, 227
Hoop stress (resultant), 78, 80, 102, 104;
tension, 80
Horizontal circle, 94; radius, 49, 77, 91,
105, 223, 224
Huffington, J., 237, 258
Hydrostatic loading, 199, 200, 202, 244,
311,326,327,367,369,370,431,
470
483
Hyperbolic cooling tower, 39, 40, 83,
126, 150; equation, 31, 162; surface, 31, 32; See also Hyperboloidal
shell of revolution
Hyperbolic paraboloid shell (HP), 14,
32-35,39,43, 162-186, 190,203205; arch action in, 174-176, 190;
displacements, 260; edge members,
169-179, 203; steel, 43, 179; with
curved boundary, 179-186; with
straight boundary, 166--174; wood,
179
Hyperboloid of one sheet, 54, 83, 117,
199; of revolution, 126, 151,463466
Hyperboloidal shell of revolution, vi, 8389, 146, 149-151, 195-197,201,
250; displacements of, 250, 251,
253; geometry of, 83-86; seismic
loading on, 126--130, 195-197; selfweight load on, 86--89; wind loading
on, 146, 149-151; See also Hyperbolic cooling tower
Hypergeometric equation, 418
Imperfection, 452, 461
Imperfection sensitivity, 463-466
Incompatible defonnation, 80
Indeterminate fonn, 76, 79, i 13, 114,
249, 438, 439
Inertial effects, 264
Inextensional deformation, 464, See also
Pure bending deformation
Infilled material, 235, 236
Influence surface, 307
Ingerslev, E., 150, 193
In-plane displacement, 278, 352, 353,
362,443; force, 70, 176,272, 348358,402,450,456; loading, 6, 67,
354-358,450,451,453; shear, 6,
176, 188,241,242; shearing strain,
213, 214, 216, 278, 353, 448;
strain, 447; stress resultant, 62-68,
70, 75, 142, 444, 453, 462, 464,
473
Instability, energy method, 454-457,
459-461; equilibrium method, 453,
454,457-459; general, 6, 171,451,
453; local, 451; of plates, 349-358,
484
Instability (cont.)
364, 370, 371; of shells, 115, 140,
431,448,451-461,474
Integrating functions, 73
Intersegment continuity conditions, 268
Invariant property, 276, 279, 285, 333,
360
Irvine, H., 474, 476
Isidorous, Jr., 7
Isotropic material, 227-231, 272, 273,
343-345
Isotropy condition, 318
Jenkins, R., 395; equation, 399
Johnson, c., 258, 259
Kantorovich method, 269
Kato, S., vii
Kanmin, T. von, 462, 469; equations for
finite deformations, 360,446,461
Keer, L., 312, 365
Kelkar, Y., 474, 476
Kelvin, Lord, 241, 259; Kelvin-Tait argument, 241, 242, 288, 294, 367
Kempner, J., 394
Ketchum, M., vii, 174, 176, 178, 193
Kiattikomol, K., 312, 365
Killion, M., 137, 192
Kinematic boundary conditions, See
Boundary conditions; equations, See
Strain-displacement relations
Kinematic releases, 245, 246
Kingdome, 37
Kirchhoff, G., 3, 240, 241, 258, 345;
boundary conditions, 240-243, 272,
287, 344, 381; effective stress resultants and couples, 242-244; theory,3
Knockdown factor, 459, 462
Knuckle, See Toroidal knuckle
Koiter, W., 463, 467
Kolisnik, H., 463, 469
Kostro, G., 179, 194
Kovalesky, L., 151, 191, 193,472,475
Kriitzig, W., vii
Kraus, H., 22, 53-55, 63, 69, 115, 192,
210,226,385, 396, 397,399,413,
418,419,466,467
Kromm-Reddy theory, 345
Index
Lamb, H., 295
Lagrange, J., 279
Lame parameter, 19,20,22,29,47,49,
62, 209, 215, 216, 226, 278
Laplacian operator, 269, 276, 338,444
Large deflections, See Finite defonna tions
Lateral loading, See Transverse loading
Lattice structures, 115
Layered plate, See Sandwich plate
Lee, B. J., vii
Lee, S.L., 86, 89, 128, 132, 149, 190-193, 250, 253, 259, 348, 366,434,
468
Length of shell meridian, See Arc length
Leve, J., 348, 366
Levy, M., 308
Levy-Nadai solution, 308-313, 337, 356,
367,371
L'Hospitals rule, 113, 114, 249, 306,
323,439
Limitations on applicability of theory, 4,
363, 364
Limit point, 452
Linear increment, 265
Lin, J.S., vii, 433, 468
Lindberg, G., 48, 54
Line load, 137,200,263,264,381,383,
392,423,441; moment, 389; of
principal curvature, See Coordinate
line
Liquid, Drop of, 202
Liu, R., 179, 194
Live load, 169, 197, 198,200,202,203
Load buckling, 458; -deflection curve,
452; distribution, 153, 154, 162;
factor, 129, 130; resistance, 5; total,
304, 371; vector, 60, 61
Loading, v, 72, 74, 104, 115,291,309;
asymmetric, 130--151,433,434,
442; axial, 6, 95; axially distributed,
383, 384; axisymmetric, 403; components, 77, 78, 86; edge, See Edge
force; external, 443; gravity, 78, 80,
471; in-plane, See In-plane loading;
intensity, 61; nature of, 67; smooth,
157; vertical, 244; See also specific
types of
Long cylindrical shells, See Semi-infinite
shells
Love, A., 3, 14; first approximation to
the theory of plates, 3
485
Index
Lowe, P., 53, 69, 191, 226, 259, 446
Lowrey, R., 227, 258
Lur'e, A., 395
Maddock, J., 252, 259
Mapping, 158-162
Marcus Agrippa, 7
Mariotte, 304
Martin, D., 252, 259
Mass density, 144, 198
Mass, Mass-Height distributions, 126129, 196
Material, axis, 231; law, See Constitutive
law; nonlinearity, 452; properties, v,
400, 414, 425, 430; structural, 6,
15, 472
Maximum-minimum conditions, 442
Maxwell-Betti reciprocal theorem, 329
Measurement of distance on a surface, 20
Measure number, 19
Medium-thin plates, v, See also Plates
Meissner, E., 413, 433, 467
Members, classification of, 5, 6
Membrane, 4, 279, 286
Membrane theory, 70-205, 233, 243,
245,263,372,374,389,392,402,
404,407, 411, 423, 433, 443, 472;
applicability, 71, 72; displacements,
71, 245-258, 386, 387, 402,410;
for shells of revolution, 73-151,
247-253; for shells of translation,
151-191,253-258; state of stress,
453; statically indeterminate, 256258; stress resultants, See Stress resultant; solution in bending theory,
404, 405, 410-412, 416, 423
Mexico City earthquake, 12, 14
Meridian, 48, 49, 50, 76, 89, 90, 98,
104, 108, 132, 200, 220, 404, 434,
466
Meridional angle, 48-50, 54, 73, 94, 95,
199,434; bending, 145; coordinate,
50-53, 88, 95, 243, 404, 434; direction, 50; force, 243; length, 95;
load, 125; moment, 423, 425, 431;
radius of curvature, See Principal radius of curvature; segment, 146;
strain, 455; stress (resultant), 75, 92,
94,97,98, 102, 129, 140, 142,
144, 198, 202, 466
Methods, See specific titles
Michelangelo, 7
Michell, J., 329, 365
Middle axis, 2, 4; plane, 2, 4, 229, 276,
277,289,358; surface, 2,4, 16-18,
30,55,60,64, 1I5, 144, 171,206218, 229, 232, 237, 241, 250, 264,
352,447
Mindlin, R., plate theory, 345
Minimum total potential energy, principle
of, 266, 267, 441, 442
Mixed formulation, 413, 433
Mode shape, 126-128, 459
Model, mathematical, v
Modulus of elasticity, See Young's modulus
Mohr's circle, 285
Moment, bending, 5, 158,317-319,329,
344; clamping, 305; concentrated,
See Concentrated moment; -curvature relations, 343; edge, 289, 292,
320, 325, 329, 342; equilibrium, See
Equilibrium equations; of inertia,
144, 234-236,413; overturning,
143, 171,202; twisting, 6, 344; See
also Stress couple
Momentless state of stress, See Membrane theory of shells
Monse, A .. 474
Multilayered shells, 10, 233
Multiple barrel shells, See Barrel shells
Mushtari, K., 444, 445; -Donnell-Vlasov
(MDV) Theory, 443--448, 450, 453,
459,471
Nadai, A., 295, 308; See also Levy-Nadai solution
Navier, 3, 279, 296, 308; hypothesis, 3;
solution, 29fr.308, 311, 335, 337,
343,356,362,367,402
Negative curvature shells, See Gaussian
curvature
New Newark Airport, The, 171,
193
Newton's laws, 1
Nonlinear equations, 452, 461-463; formulation, 263; plate theory, 3, 352354, 358-364; prebuckling state,
451,452; shell theory, 3,461-465;
See also Finite deformation of . . .
486
Nonsymmetric loading, 74, 109-112,
132, 151,219
Normal curvature, See Curvature, normal;
direction, 26; displacement, 273,
276-278, 352, 353, 362, 399, 443;
loading, 125, 140-146, 205, 402,
463; rotation of, 2, 212; to middle
plane, 4, 211, 212, 277, 278; to
middle surface, 4; section, 21, 24,
25,28,55,58,214,215,241,276,
278; vector, See Unit normal vector
Novozhilov, V., 22, 47, 53, 54, 60, 63,
69,71,73, 103, 104, 109, 112,
113, 115, 149, 191-193,206,226,
247,259,395,413,418,433,445447,449,466--468
Numerical solutions, 261, 313, See also
Finite difference method and Finite
element method
Nygard, M., 212, 226
Oden, J., 14
Ogival shell, 75, 198
Olson, M., 48, 54
Open shell, 36, 115, 400, 401
Opening reinforcement, 474
Operator, 266, 417, 418, 437, See also
Biharmonic operator
Orthogonal curvilinear coordinates. See
Coordinates
Orthogonality, 297, 335, 340
Orthotropic, materials, 231-233; plate,
237,344,345,352; shell, 461; specifically, 231-233, 343,344,413,
441
Overall equilibrium, See Equilibrium
overall
Pannuni,8
Pantheon, 7, 8, 233
Parabola, 174, 175
Parabolic bending, 292; equation, 31,
162; surface, 31; vault, 186
Paraboloid, convex, 186; of revolution,
186, 200, 204, 205, 292
Parallel circle, 48, 49, 220-224, 243,
434; surface, 214-218
Parme, A., 158, 193,394
Index
Parsens, W., 8, 14
Partial differential equation, set of, 237,
240,248
Particular solution, 316, 385-387
Patch loading, 305, 306
Pearson, K., 3, 14
Pecknold, D., vii, 145, 146, 192
Periodic distribution, 154; function, See
Fourier series
Periodicity condition, See Continuity
Persia, 7, 12
Pfluger, A., 202
Phase I Report of the Task Committee on
Folded Plate Construction, 348, 366
Physical approach, vi, 60; interpretation
of equations, 65-67, 220-224
Pirok, J., 431, 467, 474, 475
Pister, K., 344, 365
Plate, action, 191, 279, 347; annular,
319-325; bending of, v, 231, 234,
263,272-371,433,461; buckling
of, See instability of; circular, 292,
313-329, 335, 342, 354, 362-364,
367,369,371,471; deflected, 291,
310, 351, 354, 406; equilibrium
equations for, 67, 276, 285, 309,
342, 344, 396, 444; finite deformations of, 360, 461-463; flat, vi. 70,
98; flexure, See Bending of; folded,
See Folded plate; in-plane stress resultants, 453; instability of, 349358; irregularly shaped, 332; long,
narrow, 340; metal, 473; multilayered, 233, 344; rectangular, 288313,342,343,354-358,367,370,
371; solid, 314-319, 363; square,
294, 300, 304, 305, 308, 358, 367;
stiffened, 343; strain-displacement
relations for, 224-226; strain energy
for, 263, 352-354; symmetry conditions, 244; theory of, 1, 3, 4, 231,
240,245,268,286,287,292,295,
348, 362, 363, 403, 472; thick, 4;
thickness, 474; See also specific
types of plates
Plug, rigid, 323, 324, 369
Pneumatic forms, 475
Pochanart, S., vii
Point of application, 307, 383; of observation, 307, 383
Point load, See Concentrated load
Index
Pointed shell, 75, 101
Poisson, S., 279, effect, 293, 309
Poisson's ratio, 109, 227, 235, 278, 294,
423
Pole angle, 75, 76; condition, 76, 115,
252,260,318,437-439,471; of a
shell, 49, 108, 113, 132
Polygonal plate, 303
Polynomial solution, 442
Popov, E., 123, 192
Position vector, 16, 17, 213
Positive curvature shells, See Gaussian
curvature
Postbuckling, 452, 463-466
Potential energy, 263-264, 266, 267,
339,434,440-442; function, 447,
453, 459, 462; See also Minimum
total potential energy
Prebuckling, 454, 463
Pressure, coefficient, 143, 144, 146, 147;
critical, 459, 463; fluid, 461; loading, 74, 104, 105, 137, 145,205,
243, 423; external, 457, 458, 463;
internal, 91, 92, 96, 200, 259, 385,
393,428; uniform, 91, 93, 95, 97,
105, 202, 393; wind, 140-146, 464
Pressure vessels, 89, WI, 104,374,391,
474
Prestressing, 178, 185
Principal curvature, 22; direction, 22; orthogonal curvilinear coordinates, 22;
radius of curvature, 22, 28, 29, 49,
77,85,86,91; stress, 140,174; See
also specific types of shells
Principles, See spec({ic titles
Propagation of edge loads, 97, 137
Proportioning, 472-474
Pucher stress function, 161, 162, 166
Pulmano, V., 348, 366
Pultar, M., 348, 366
Pure bending deformations, 155,464
Quadratic form, first, 19; second, 26-28,
54,72
Quadrature, 130
Radial deformation, 102-104, 392, 393;
displacement, See Displacement, radial; equilibrium, 103; expansion,
487
103; force, 66, 102,471; loading,
125, 325, 383
Radius of curvature, 4, 28, 70, 72, 93,
104, 198, 199,216,224,248,372,
447,448,461; computation of, See
spec(fic shell geometries; principal,
See Principal radius of curvature
Ranjan, G .• 431,468
Rayleigh. Lord, 115, 268; -Ritz method,
264-271,337-342,346, 370,442,
443,457,460,462
Reactions, See specific types of
Recommended Rules for Design and
Construction of Large. Welded, Low
Pressure Storage Tanks, 474, 475
Reddy, J., 345, 365
Redundant, 245, 246
Reference position, 2, 291
Reinforced Concrete Cooling Tower
Shells: Practice and Commentary,
144, 192,474,475
Reinforcement, 473
Reissner, E., vii, 235. 258, 413
Reissner, H., 50, 54, 344, 345, 365,
394, 433, 467
Release, 245, 246
Renaissance, 7
Resistance mechanism, 6
Resultant force, 143, 406, 409, 410,
416
Reticulated shell, 39, 233-238
Ribbed plate, See Waffle slab; shell. 7,
144, 184; structure, 235
Ridge beam, See Member; line, 174,
347; member, 177, 178
Riera, J., 348, 366
Rigid body displacement, 238, 239, 244;
insert, See Plug, rigid; mode, 268;
motion, 68, 212, 291
Rigidity, See specific types of
Ring beam, 80, 81, 88, 137, 140, 184,
197, 198,200,392,403-405,460,
474; element, 95; load, 198, See
also Line load; stiffener, 102, 391394,406,431; tension, 102-103
Rish, F., 151, 191, 193
Rish, R., 140, 143, 192
Ritz, W., 268, See also Rayleigh-Ritz
method
Rockets, 391
Rod action, 6
488
Roller, See Boundary; Boundary conditions
Roof, See Shells of translation
Rotation, 211, 212, 220-222,247,255,
274,278,279,287,298,299,311,
314,342,349,352,407,409,413,
421, 423-425, 428, 436; of coordinate axis, See Coordinate transformation; relative, 423-425
Rotational shells, See Shells of revolution
Rotter, J., vii, 137, 140, 192,433,468,
474,476
Rules for Construction of Pressure Vessels, 474, 475
Saddledome, 39, 184-185
Saddle HP shell, 162, 166, 174, 178
Saint Venant, 241, 292, 308; equations,
30; principle, 241,381; torsion constants, 235; torsional rigidity, 237
St. Louis Construction News & Review,
475,476
St. Maria del Fiore dome, 7, 10, 11
St. Paul's dome, 7, 13
St. Peter's dome, 7, 12
Sandwich plate, 344, 352; shell, See also
Multilayered plate, shell
Schallert, K., vii
Schlaich, J., 475, 476
Schneider, R., 104, 191
Schnobrich, W., vii, 176, 178, 193,443,
468
Schorer, H., 394
Scriven, W., 252, 259
Sechler, E., 13, 14
Second order terms, 62, 63; quadratic
form, See Quadratic form, second
Section modulus, 347
Seismic load(ing), 117, 123-130, 149,
194-197,201-202
Self-equilibrated edge effects, 134, 381;
loading, 130, 143, 239
Self-weight, 74, 80, 86, 95, 126, 137,
149, 150, 152-155, 171, 177, 178,
203, 255, 264, 409, 466; stress resultant, 78, 79, 86, 129
Semi-infinite shells, 378-389, 421
Semi-inverse method, 292
Sen, S., 123, 126, 132, 192
Separation of variables, 141, 142
Index
Sewell, R., 474, 476
Sgrille, 11
Shaaban, A., 174, 176, 178, 193
Shah's Mosque, 7, 9
Shallow arch, 452; shells, 46-48, 169,
191,443-450,461-463,471,472
Shape factor, 233, See also Transverse
warping shape factor
Shape functions of surface, See Coordinate functions
Shape of the surface, 26, 28
Shear, 5, 169, 171; correction factor,
231; force, 176, 177; loading, 352;
lOCking, 217, 346; pure, 75, 169,
174, 184; uniform, 169
Shearing deformation, See Deformation;
force, 274, 394; modulus, 227, 231;
resultant, 410; rigidity, 234; thickness, 235, 236; See also In-plane
shear and Transverse shear
Shell-like structure, See Reticulated shell
Shell, 7; assembled, 434; bending of, See
Bending of shells; deformation of,
See Deformation and Displacement
of shells . . . ; displacement of, See
Displacement of shells . . . ; of doublecurvature, 258; element, 56, 58,
158, 159; equilibrium of, See Equilibrium equations; finite deformations of, See Finite deformation of
shells; geometry of, See Geometry;
specific types; homogeneous, 233;
instability of, See Instability of
shells; multilayered, See Multilayered shells; of revolution, vi, 29,
47-53,62-67,69,73-151, 195203, 218-224, 226, 243, 247-253,
258, 259, 282, 298, 318, 326, 372,
413-443,463,471,472; of translation, 151-191,244,292,372,474,
475; segment, 434, 440-442; theory
of, 1,3,4,240, 245; thick, 4; See
also Membrane theory of shells and
specific types of shells
Short shell, 385, 389-391
Shukov, Engineer, 11 7
Sign conventions, 55, 56
Simmonds, S., vii, 178, 193
Simple support, See Boundary and
Boundary conditions
Index
Simply supported plate, 296-319, 335337,367
Simpson, H., 348, 366
Singularity, 439
Sinusoidal loading, 296-298, 300-305
Sixth equation of equilibrium, 63, See
also Drilling degree-of-freedom
Slice-beam, 145:-146, 205
Sliding boundary, See Boundary and
Boundary conditions
Snap-through buckling, 452
Snow load, 169, 203
Soare, M., 258, 259
Solution components, 315
Specifically orthotropic condition, See
Orthotropic, specifically
Sphere, 184, 260
Spherical shell, 36, 38, 50, 77-83, 85,
88, 197, 198,202; axisymmetrically
loaded, 77-83; bending of, 416423, 433; buckling of, See instability of; cap, 82, 89, 101-109,423,
428,430,431; See also head;
closed, 10 1, See also dome; columnsupported, 38,42, 135-137,202;
complete, 82; deep, 79, 82; displacement, 249, 250, 252; dome,
36, 37, 78-83, 95, 118-120, 132,
135-137; edge-loaded, 130-132,
140,421,422; geometry of, 77,
404,416; head, 101,423; instability
of, 454, 457-460, 464-466, 471;
membrane theory of, 77-83, 197,
198, 457; pressure loaded, 83; reticulated. 39; segment. 151. 423.430;
self-weight loaded, 77-83, 249;
snow-loaded, 197; wind-loaded,
117-123., 146-148
Spheroidal tower, 42, 83
Split rigidity, 235
Spring, 289
Stability, See Instability; criterion, 454,
457,459
Starnes, J., 452, 468
Statically determinate system, 71, 245,
246, 256-258; indeterminate shell,
63, 157
Static, -geometric analogy, 222, 248;
-kinematic correspondents, 238, 239,
289, 423, 443; relationship, vi
489
Stationary condition, 266, 434
Steel, T., 140, 143, 192
Steele, C., 432, 468
Stiffened plate, 233-237; shell, 10, 41,
233-237
Stiffener, 137,233, 383, 392, 393, 474
Stiffening beam, 157; rib, 234; ring, See
Ring beam
Strain, 213-228, 246, 262, 267, 268,
272, 278, 332, 348, 353, 358, 359,
436, 437, 439, 441, 454; -displacement relations, 209, 215-226, 247,
251, 253, 266, 268, 272, 332, 336,
348, 362, 372, 373, 375,414, 433,
436-440,450,461; energy, 261263,265,267, 332-337, 346, 352354,357,440,441,450,452,454457, 464; incompatibility, 392; See
also specij1c types
Stress, 55-60, 166, 168,227,228, 262;
at-a-point, See Elasticity, Theory of;
bending, 71; compressive, 174; concentration, 321; couple, 55-68, 229,
237,238,241,266,267,274,279,
283-285,291,298-300,303, 306,
311,314,315-320,329,340,362,
363, 367, 369, 370, 372, 375, 378,
384, 389, 394, 395, 397, 400, 402,
405-407,410,421,422,430,431,
433,436,444, 449, 473; extensional, 180, 181; function, 161, 188,
359,402,403,444,447,461; normal, 4; prebuckling, 457; principal,
283; resultant, 55-68, 92, 112, 113,
115, 130, 155-162, 168, 172-174,
180, 181, 188, 190, 195-205,237,
238, 245-248, 254-256, 258, 266,
267, 279, 284, 288, 291, 298, 299,
306, 348, 349, 353, 358, 363, 364,
375,378,381, 382, 384, 386, 387,
389,394,395,400,402,404-407,
409-411,421,430,431,433,436,
443,447,449,464; resultant-displacement relations, 395, 401, 414;
-strain, 1, 227-237, 263, 272, See
also Constitutive law; tensile, 174;
tensor, 63, 71; vector, 60, 349; uniform, 83; See also specij1c types of
shells
Structures, Theory of, 348
490
Submersible vehicles, 391
Substitute curve, 434
Suction load, 98, 144, 260, 457, 458,
463,471
Superposition, 132, 133, 135, 136,318,
369
Support elastic, 289; line, 347; settlement,289
Surface(s), 55; developable, 6, 360; load,
vi, 123, 130, 195,263,269,270,
347, 354, 416, 444, 449, 453; middle, See Middle surface; of translation, 151, See also Shell of translation; projected, 158-162, 169;
roughness, 144; shape of, See Shape
of the surface; structure, v, 472; tension, 202; undulations, 233; See also
specific types of plates and shells
Suryoutomo, H., 123, 126, 192
Symmetry, 135, 151,243,244,256,
291,310,340,361,409,430,440;
cyclic, 135,243; cyclic, 135,243;
longitudinal, 255; rotational, 243
Synclastic surface, 31
Szilard, R., 276, 313, 324, 329, 332,
342, 364, 365, 472, 475
Tahiani, c., 463, 469
Tait, P., 241, 259; See also Kelvin-Tait
argument
Tangent local, 159; vector, 18, 54; See
also Unit tangent vector
Tangent modulus theory, 452
Tangshun, China earthquake, 12
Tank, column-supported, See Columnsupported shells; cylindrical, 374,
403; elevated, 407; spherical, 42,
135-137; steel, 474; stiffened, 41,
43; water, 43
Tedesko, A., 395
Temperature, 227, 263, See also Thermal
Tension field, 463
Tensor, approach, 60; stress, See Stress
tensor
Teter, N., 179, 194
Tezcan, S., 179, 194
Theorems, See specific titles
Thermal coefficient, 232; energy, 263,
450; expansion, coefficient of, 227;
Index
gradient, 385, 386; load, 229, 232,
247,254,264, 359, 376-378, 387,
402, 414, 454; moment, 229, 332,
333, 336, 376-378, 389, 397, 402,
414
Thick plates, 4, 207; shells, 4, 207
Thickness, 55, 86, 137, 140,229,231,
233,261-263,277,304,363,392,
425,430,432,440,470,471,473;
equivalent, 233-235; variations in,
4,86,87, 146,231,342,378,414
Thin shells, v, See also Shells
Third equation of equilibrium, 93, 94; order terms, 62, 63
Thrust, 5, 6
Tie, 178, 186
Timoshenko, S .. 14,241,259,266,268,
270, 279, 292, 295, 296, 304, 308,
364, 454, 459, 468, 469; and S.
Woinowsky-Kreiger. 3, 60, 69, 73,
118, 131, 191, 192,226,234,258,
286,305,311,312,320,324,325,
329,331,332,337,342-345,352,
353, 362-366, 380, 394
Todhunter, I., 3, 14
Toroidal knuckle, 106, 109, 432, 474;
shell, 89-94, 105-109, 249, 259
Torospherical head, 45, 103-\09,431.
432,443
Torroja, E., 395
Torsional loading, 75
Torsion of surface, See Twist of surface
Total load, 171, 173, 176
Tower, 50, 143-145, See also Hyperbolic cooling tower
Trahair, N., 474, 476
Transition segments, 10 1-104
Translational degree of freedom, 268
Transverse loading, 67, 70, 142, 351,
355, 372, 378, 452
Transverse shear(ing) deformation, 344346; force, 55, 72, 102, 103, 145,
157, 158,246,273,287,288,322,
329, 370, 372, 379, 389, 423,428,
443, 444, 453; strain, 3, 4, 206208,212,214,216,217,219-221,
225, 231, 240-242, 247, 272-274,
286,287,349,351,362,374-376,
395, 400, 415, 437; stress, 59, 60;
stress resultant, 62-68, 70, 231,
491
Index
241-243, 278, 282-284, 291, 298,
300-302,308,311,314,370,378,
395, 413, 450
Transverse warping shape factor,
231
Trapezoidal method, 250
Trial function, 442
Triangular plate, 329-331, 369
Tsein, H., 462, 469
Tso, F., 233, 258
Tsui, E., 380,466
Twist of surface, 216, 223, 272
Twisting energy, 263; moment, 6, 55;
rigidity, 6, 235, 237, 279, 304;
stress couple, 68, 242, 279, 287,
288; stress resultant, See Stress resultant; thickness, 235, 237
Umbrella HP shell, 167, 169-172,473,
474
Uniformly distributed load, 153, 154,
168, 169, 173, 178, 179,203,298,
300-306,309-312,316,318,331,
337, 369, 370, 409
Unit load, 307; normal vector, 21-26,
57,206-212; tangent vector, 21-26,
61, 208-212, 446, 447; vector, 17,
21,78,211; weight, 199,200
Unsymmetrical loading, See Nonsymmetrical loading
Valley, interior, 410; member, 177
Variable coefficient, 149
Variational operator, 265; principle, 265,
270
Variation of parameters, 130
Vault, 35, 185, 186
Vector algebra, 60, 224; calculus, vi; See
also specific types
Verschmieren, 233, 235
Virtual displacement, 265, 266, 353;
principle of, 261, 264-266
Virtual work, 265, 335-337, 356--358,
370; principle of, 265, 357
Vlasov, V., 53, 54, 115, 192,395,399,
402,403,447,448,453,454,457,
466--469; equation, 399; See also
Mushtari-Donnell-Vlasov theory
Volume, 266, differential, 262; element,
56,57
Von Kannan, See Karman, T. von
Waffle slab, 44, 235
Wall friction loading, 137, 140,406
Wang, R., 431, 467
Water tower, See Tower, water
Washizu, K., 265, 270
Watson, G., 150, 193
Wave length of bending, 381; buckling,
460
Weight of a shell, 126, 140, 173, 176,
See also Self-weight
Welded joints, 474
Welded Steel Tanks for Oil Storage, 474,
475
Westergaard, H., 3, 14
Whittaker, E., 150, 193
Williams, M, vii
Wind, load(ing), 117, 118, 202,433,
434; pressure, 118, 119, 146-151;
tunnel tests, 140, 143, 146-148
Wittek, U., vii. 463,469
Woinowsky-Kreiger, S., See Timoshenko
and Woinowsky-Kreiger
Wozniak, R., 431, 467, 474, 475
Wren, Sir c., 7
Wrinkling, 431,432
Wu, 1.K., vii. 14
Yang, T., 462, 469
Yitzhaki, D., 348, 366
Young's modulus, 109,227,423
Zema, W., 60, 69, 395
Zero curvature shells, See Gaussian curvature
Zienkiewicz, 0., 268, 270