WRC Bulletin 537
3rd Edition, March 2022
ISSN 2372-1057
WRC PVRC MPC
Welding Research Council, Inc.
Bulletin
Precision Equations and
Enhanced Diagrams for
Local Stresses in Spherical
and Cylindrical Shells Due
to External Loadings for
Implementation of
WRC Bulletin 107
K.R. Wichman
A.G. Hopper
J.L. Mershon
Implementation Team
D.A. Osage, ASME Fellow, P.E.
M.E. Buchheim
D.E. Amos
T.C. Shaughnessy
D.A. Samodell
M. Straub
2
E G | The Equity Engineering Group, Inc.
WRC BULLETIN 537
PRECISION EQUATIONS AND
ENHANCED DIAGRAMS FOR LOCAL
STRESSES IN SPHERICAL AND
CYLINDRICAL SHELLS DUE TO
EXTERNAL LOADINGS FOR
IMPLEMENTATION OF
WRC BULLETIN 107
3RD EDITION, MARCH 2022
K.R. Wichman
A.G. Hopper
J.L. Mershon
Implementation Team
D.A. Osage, ASME Fellow, P.E.
M.E. Buchheim
D.E. Amos
T.C. Shaughnessy
D.A. Samodell
M. Straub
E2G | The Equity Engineering Group, Inc.
WRC Bulletin 537 – 3rd Edition
Precision Equations and Enhanced Diagrams for Local Stresses in Spherical and Cylindrical Shells Due to External Loadings for
Implementation of WRC Bulletin 107
WRC – Welding Research Council, Inc. (WRC) brings together science and engineering specialists in
developing the solutions to problems in welding and pressure vessel technology. They exchange
knowledge, share perspectives, and execute R and D activities. As needed, the Council organizes
and manages cooperative programs.
MPC – A Council of the WRC, the Materials Properties Council is dedicated to providing industry
with the best technology and the best data that can be obtained on the properties of materials to
help meet today’s most advanced concepts in design and service, life assessment, Fitness-ForService, and reliability and safety.
PVRC – A Council of the WRC, the goal of the Pressure Vessel Research Council is to encourage,
promote and conduct research in the field of pressure vessels and related pressure equipment
technologies, including evaluation of materials, design, fabrication, inspection and testing.
For more information, see www.forengineers.org
WRC Bulletins contain final reports from projects sponsored by the Welding Research Council, Inc.
important papers presented before engineering societies and other reports of current interest and lasting
value to engineers and scientists.
No warranty of any kind expressed or implied, in respect to data, analyses, graphs or any other information
provided in this publication is made by the Welding Research Council, Inc., and the use of any such
information is at the user’s sole risk.
All rights are reserved, and no part of this publication may be reproduced, downloaded, disseminated, or
otherwise transferred in any form or by any means, including photocopying, without the express written
consent of WRC.
Copyright © 2022 by Welding Research Council, Inc.
All rights, including translations, are reserved by WRC.
ISSN 2372-1057
Library of Congress Catalog Number: 85-647116
Welding Research Council, Inc.
20600 Chagrin Blvd.
Suite 1200
Shaker Heights, OH 44122
www.forengineers.org
ii
WRC Bulletin 537 – 3rd Edition
Precision Equations and Enhanced Diagrams for Local Stresses in Spherical and Cylindrical Shells Due to External Loadings for
Implementation of WRC Bulletin 107
FOREWORD
To WRC Bulletin 537, March 2022 Edition
WRC Bulletin 537 is intended to facilitate implementation of the widely required and used relations found in
the March 1979 revision of WRC Bulletin 107 for local stresses in spherical and cylindrical shells due to
external loadings. The original analytical and experimental work of WRC’s Pressure Vessel Research
Council has become an essential tool for pressure vessel design for 45 years and in its present form for over
30 years. WRC Bulletin 537 has been prepared in response to numerous requests over the years for the
precise equations depicted in the figures given in WRC Bulletin 107 (1979 Edition). The objective was to
eliminate potential errors in implementation, facilitate proper interpolation and extrapolation and permit
efficient computation with modern computers.
David A. Osage organized the extensive effort to precisely capture the details of each curve in every single
figure and develop the complex mathematical relations which render the new document useful for modern
engineering practice.
Involved in developing the equations and checking the results on Dave Osage’s team
were Mary Buchheim, David Amos, Tiffany Shaughnessy, and Debbie Samodell. Mike Straub offered WRC
his work at digitizing the numerous curves found in WRC Bulletin 107 to facilitate this effort.
Welding Research Council, Inc. will no longer deliver WRC Bulletin 107 when requested for purchase. WRC
Bulletin 537 provides the same content in a more useful and clearer format that also facilitates computer
implementation. Those responsible for codes, standards, and specifications that recommend or require use
of WRC Bulletin 107 should amend those documents to directly reference WRC Bulletin 537. WRC Bulletin
537 is the equivalent to WRC Bulletin 107 and provides the same acceptable basis for design.
WRC Bulletin 537 3rd Edition, published in March 2022, replaces 2nd Edition, published in October 2020.
The changes in the 3rd Edition are described below:
•
Added table numbers to all tables.
•
Added new paragraph 5.0 to provide rules on the use of the equations.
•
Added a cross reference between figures and table containing the curve fit data in Table 1S for
spherical shells and Table 1 C for cylindrical shells.
•
Updated paragraph 5.3.1 to indicate conditions where the stress in the nozzle may be greater than the
stress in the shell. Stresses in both the shell and nozzle may be determined using WRC Bulletin 297
Local Stresses in Cylindrical Shells Due to External Loadings on Nozzles-Supplement to WRC Bulletin
No. 107 (Revision I).
•
In Table 5, added a link to the Table 1C.
•
Modified Table 1C entitled List of Nondimensional Curves for Cylindrical Shells to clarify the correct
figures to use for stress calculations in Table 5. In addition, added a note to Table 1C on when to
Figures 1B-1 and 2B-1.
Dr. Martin Prager
Executive Director, WRC
iii
WRC Bulletin 537 – 3rd Edition
Precision Equations and Enhanced Diagrams for Local Stresses in Spherical and Cylindrical Shells Due to External Loadings for
Implementation of WRC Bulletin 107
FOREWORD
To WRC Bulletin 537, 2020 Edition
WRC Bulletin 537 is intended to facilitate implementation of the widely required and used relations found in
the March 1979 revision of WRC Bulletin 107 for local stresses in spherical and cylindrical shells due to
external loadings.
The original analytical and experimental work of WRC’s Pressure Vessel Research
Council has become an essential tool for pressure vessel design for 45 years and in its present form for over
30 years. WRC Bulletin 537 has been prepared in response to numerous requests over the years for the
precise equations depicted in the figures given in WRC Bulletin 107 (1979 Edition). The objective was to
eliminate potential errors in implementation, facilitate proper interpolation and extrapolation and permit
efficient computation with modern computers.
Mike Straub offered WRC his work at digitizing the numerous curves found in WRC Bulletin 107. David A.
Osage organized the extensive effort to precisely capture the details of each curve in every single figure and
develop the complex mathematical relations which render the new document useful for modern engineering
practice. The effort and assuring the accuracy of the results required a great deal of time and attention to
details. Involved in developing the equations and checking the results on Dave Osage’s team were Mary
Buchheim, David Amos, Tiffany Shaughnessy, and Debbie Samodell.
Welding Research Council, Inc. will no longer deliver WRC Bulletin 107 when requested for purchase. WRC
Bulletin 537 provides the same content in a more useful and clearer format and is not an update or revision
of WRC Bulletin 107. Instead, it is the 2010 printing of WRC Bulletin 107 and has been meticulously checked.
Those responsible for codes, standards and specifications that require use of WRC Bulletin 107 should amend
those documents to reflect the fact that WRC Bulletin 537 is the equivalent to WRC Bulletin 107 and provides
the same acceptable basis for design.
Since the 2010 Edition of WRC Bulletin 537 three Errata’s have been issued. The third Errata issued on
August 19, 2011 was accumulative and comprised of the first and second Errata’s issued on, respectively,
April 2011 and June 27, 2011.
WRC Bulletin 537 2nd Edition, published in October 2020, replaces previous versions of WRC Bulletin 537
as well as the first, second and third Errata. WRC Bulletin 537 2nd Edition is all-inclusive; changes have
been made to incorporate each item identified in the first, second and third Errata; they are:
•
Correct range for beta from 0.375 to 0.5 in the August 6, 1965 Foreword.
•
Correct page number to 43 for nondimensional curves for spherical shells.
•
Correct page number to 91 for nondimensional curves for cylindrical shells should.
•
Computation of Nø/(P/Rm) currently states 3C or 4C, should only State 3C in Table 5. (Note: Figure
3C is properly labeled.)
•
Computation of Nx/(P/Rm) currently states 3C or 4C, should only State 4C in Table 5. (Note: Figure
4C is properly labeled.)
iv
WRC Bulletin 537 – 3rd Edition
Precision Equations and Enhanced Diagrams for Local Stresses in Spherical and Cylindrical Shells Due to External Loadings for
Implementation of WRC Bulletin 107
•
Incorrect column name in last column, currently shows My should show Ny in the Curve Fit Table for
SPI-1.
•
Curve fit for data was adjusted to prevent negative values for Y within valid range for X in Figure SM-3
and Curve Fit Table.
•
Adjusted the placement of the labels for Nx and My for easier reading in Figure SM-9.
•
Equation shown is not the equation used to calculate the curve fit in the Curve Fit Table for SM-10.
•
Missing figure and table for Figure 4A and Curve Fit Table.
•
Figure 3B Title is incorrect and should be longitudinal moment not circumferential.
•
Figure 4B Title is incorrect should be longitudinal moment not circumferential.
•
Figure 1C-1 Original and Curve Fit Table display incorrect range for Y.
•
Figure 1C-1 Extrapolated and Curve Fit Table display incorrect range for Y.
•
Figure B-2 has incorrect values for x axis.
Additional changes made to WRC Bulletin 537 2nd Edition are as follows:
•
Changed WRC Bulletin number from 537 to 537 2nd Edition.
•
Corrected header throughout Bulletin.
•
Changed the cross reference from a specific page number to Section 8 for nondimensional curves for
spherical shells in Section 3.4.
•
Changed the cross reference from a specific page number to “clicking here” for nondimensional curves
for cylindrical shells in Section 4.4.1.
•
Changed in-line MathType text to Microsoft’s LaTeX Math Equation syntax.
Dr. Martin Prager
Executive Director, WRC
v
WRC Bulletin 537 – 3rd Edition
Precision Equations and Enhanced Diagrams for Local Stresses in Spherical and Cylindrical Shells Due to External Loadings for
Implementation of WRC Bulletin 107
FOREWORD
To WRC Bulletin 537, 2011 Edition
Bulletin 537 is intended to facilitate implementation of the widely required and used relations found in the
March 1979 Revision of WRC 107 for local stresses in spherical and cylindrical shells due to external loadings.
The original analytical and experimental work of WRC’s Pressure Vessel Research Council has become an
essential tool for pressure vessel design for 45 years and in its present form for over 30 years. In response
to numerous requests over the years for the precise equations depicted in the figures in the 1979 version of
WRC 107, WRC 537 has been prepared. The objective was to eliminate potential errors in implementation,
facilitate proper interpolation and extrapolation and permit efficient computation with modern computers.
Mike Straub offered WRC his work at digitizing the numerous curves found in WRC 107. Mr. David A. Osage
organized an extensive effort to precisely capture the details of each curve in each figure and develop the
complex mathematical relations which render the new document useful for modern engineering practice.
The effort and assuring the accuracy of the results required a great deal of time and attention to details.
Involved in developing the equations and checking the results on Dave Osage’s team were Mary Buchheim,
David Amos, Tiffany Shaughnessy, and Debbie Samodell.
WRC will no longer deliver WRC 107 when requested for purchase. WRC 537 provides the same content in
a more useful and clearer format. It is not an update or a revision of 107. It is the 2010 printing of WRC
107. It has been meticulously checked. Those responsible for codes, standards and specifications that
require use of WRC 107 should amend those documents to reflect the fact that WRC 537 is the equivalent to
WRC 107 and provides the same acceptable basis for design.
Since the 2010 edition of WRC Bulletin 537 three Errata’s have been issued. The first Errata was issued
in April 2011, the second Errata was issued on June 27, 2011 and the third Errata issued on August 19,
2011. Third Errata issued in August of 2011 was accumulative.
Dr. Martin Prager
Executive Director, WRC
vii
WRC Bulletin 537 – 3rd Edition
Precision Equations and Enhanced Diagrams for Local Stresses in Spherical and Cylindrical Shells Due to External Loadings for
Implementation of WRC Bulletin 107
FOREWORD
To WRC Bulletin 537, 2010 Edition
Bulletin 537 is intended to facilitate implementation of the widely required and used relations found in the
March 1979 Revision of WRC 107 for local stresses in spherical and cylindrical shells due to external loadings.
The original analytical and experimental work of WRC’s Pressure Vessel Research Council has become an
essential tool for pressure vessel design for 45 years and in its present form for over 30 years. In response
to numerous requests over the years for the precise equations depicted in the figures in the 1979 version of
WRC 107, WRC 537 has been prepared. The objective was to eliminate potential errors in implementation,
facilitate proper interpolation and extrapolation and permit efficient computation with modern computers.
Mike Straub offered WRC his work at digitizing the numerous curves found in WRC 107. Mr. David A. Osage
organized an extensive effort to precisely capture the details of each curve in each figure and develop the
complex mathematical relations which render the new document useful for modern engineering practice.
The effort and assuring the accuracy of the results required a great deal of time and attention to details.
Involved in developing the equations and checking the results on Dave Osage’s team were Mary Buchheim,
David Amos, Tiffany Chiasson and Debbie Samodell.
WRC will no longer deliver WRC 107 when requested for purchase. WRC 537 provides the same content in
a more useful and clearer format. It is not an update or a revision of 107. It is the 2010 printing of WRC
107. It has been meticulously checked. Those responsible for codes, standards and specifications that
require use of WRC 107 should amend those documents to reflect the fact that WRC 537 is equivalent WRC
107 and provides the same acceptable basis for design.
Dr. Martin Prager
Executive Director, WRC
viii
WRC Bulletin 537 – 3rd Edition
Precision Equations and Enhanced Diagrams for Local Stresses in Spherical and Cylindrical Shells Due to External Loadings for
Implementation of WRC Bulletin 107
FOREWORD
To WRC Bulletin 107, October 2002 Update of March 1979 Revision
The October 2002 Update to the March 1979 Revision of WRC Bulletin 107 includes minor editorial changes for
improvement and readability of several equations, curves, and some text. There are NO technical changes.
•
The calculation forms (Tables 2, 3 and 5) are improved, particularly to show the "+" and "-" quantities
more definitively.
•
The equation for stress in paragraph 3.6.3 is revised to be on one line.
•
The parameter definitions on several of the curves (beginning with Figure SR-1) are improved and
clarified.
•
Appendix B, exponents in Equations 1, 2, 3 and 4 are enlarged for readability.
PVRC thanks Mr. James R. Farr, Honorary Emeritus Member of the Pressure Vessel Research Council, for
his assistance in preparing this update.
NOTE: WRC Bulletins 107 and 297 should be considered (and purchased) as an integral set. In addition,
PVRC Technical Committees are working on a project that is envisioned to culminate in a new publication
to add to the WRC Bulletin 107 and 297 set. The new publication will provide significant new technical
information on local shell stresses from nozzles and attachments.
Greg L. Hollinger
The Pressure Vessel Research Council
ix
WRC Bulletin 537 – 3rd Edition
Precision Equations and Enhanced Diagrams for Local Stresses in Spherical and Cylindrical Shells Due to External Loadings for
Implementation of WRC Bulletin 107
FOREWORD
To WRC Bulletin 107, March 1979 Update of August 1965 Original Version
Welding Research Council, Inc. Bulletin No. 107 has been one of the most widely used bulletins ever
published by WRC. The original bulletin was published in August 1965. Since that time, a revised printing
was issued in December 1968; a second revised printing was issued in July 1970; a third revised printing
was released in April 1972; and a June 1977 reprint of the third revised printing was issued. As sometimes
happens with publications of this type, some errors were detected and then corrected in subsequent revised
printings.
In this March 1979 Revision of Bulletin 107, there are some additional revisions and clarifications. The
formulations for calculation of the combined stress intensity, 𝑆, in Tables 2, 3, and 5 have been clarified.
Changes in labels in Figures 1C-1, 2C-1, 3C, and 4C have been made and the calculated stresses for
Model "R" in Table A-3 and Model "C-l" in Table A-4 have been revised accordingly. The background for
the change in labels is given in a footnote on page 66.
Present plans call for a review and possible extension of curves to parameters which will cover most
openings in nuclear containment vessels and large storage tanks. Plans are to extend 𝑅⁄𝑇 from 300 to
600 and to extend 𝑑 ⁄𝐷 range from 0.003 to 0.10 for the new 𝑅⁄𝑇 range, review available test data to
establish limits of applicability, and develop some guidance for pad reinforcements.
Long range plans are to review shell theory in general, and Bijlaard's method in particular. The goal is to
extend the 𝑅⁄𝑇 up to 1200 for a 𝑑 ⁄𝐷 up to 0.1. This will include large deflection theory and other
nonlinear effects. In addition, available computer programs will be studied in hope of developing one which
will be an appropriate supplement to Bijlaard's method. Finally, a review will be made of limit loads related
to large 𝑅⁄𝑇 and small 𝑑 ⁄𝐷 .
J.R. Farr, Chairman
PVRC Design Division
x
WRC Bulletin 537 – 3rd Edition
Precision Equations and Enhanced Diagrams for Local Stresses in Spherical and Cylindrical Shells Due to External Loadings for
Implementation of WRC Bulletin 107
FOREWORD
To WRC Bulletin 107, August 1965 Original Version
Several years ago, the Pressure Vessel Research Committee sponsored an analytical and experimental
research program aimed at providing methods of determining the stresses in pressure vessel nozzle
connections subjected to various forms of external loading.
The analytical portion of this work was
accomplished by Prof. P.P. Bijlaard of Cornell University, and was reported in references [1] to [8] inclusive.
Development of the theoretical solutions involved several simplifying assumptions, including the use of
shallow shell theory for spherical vessels and flexible loading surfaces for cylindrical vessels. These
circumstances limited the potential usefulness of the results to 𝑑𝑖 ⁄𝐷𝑖 , ratios of perhaps 0.33 in the case
of spherical shells and 0.25 in the case of cylindrical shells. Since no data were available for the larger
diameter ratios, Prof. Bijlaard later supplied data, at the urging of the design engineers, for the values of
𝛽 = 0.375 and 0.50 ( 𝑑𝑖 ⁄𝐷𝑖 , ratios approaching 0.60 ) for cylindrical shells, as listed on page 12 of
reference [10]. In so doing, Prof. Bijlaard included a specific warning concerning the possible limitations
of these data, as follows: "The values for these large loading surfaces were computed on request of several
companies. It should be remembered, however, that they apply to flexible loading surfaces and, for radial
load, to the center of the loading surface. It should be understood that using these values for the edge of
the attachment, as was recommended for small loading surfaces, may be unconservative.''
Following completion of the theoretical work, experimental work was undertaken to verify the theory, the
results of which were published in references [17] and [18]. Whereas this work seemingly provided
reasonable verification of the theory, it was limited to relatively small 𝑑𝑖 ⁄𝐷𝑖 ratios-0.10 in the case of
spherical shells and 0.126 in the case of cylindrical shells. Since virtually no data, either analytical or
experimental, were available covering the larger diameter ratios, the Bureau of Ships sponsored a limited
investigation of this problem in spheres, aimed at a particular design problem, and the Pressure Vessel
Research Committee undertook a somewhat similar investigation in cylinders. Results of this work have
recently become available emphasizing the limitations in Bijlaard's data on cylindrical shells, particularly as
it applies to thin shells over the "extended range" (page 12 of reference [10]).
Incident to the use of Bijlaard's data for design purposes, it has become apparent that design engineers
sometimes have difficulty in interpreting or properly applying this work. As a result of such experience,
PVRC has felt it desirable that all Bijlaard's work be summarized in convenient, "cook-book" form to facilitate
its use by design engineers.
However, before this document could be issued, the above mentioned
limitations became apparent, presenting an unfortunate dilemma, viz., the test data indicate that the
calculated data are partially inadequate, but the exact nature and magnitude of the error is not known, nor
is any better analytical treatment of the problem available (for cylinders).
Under these circumstances, it was decided that the best course was to proceed with issuing the "cookbook," extending Bijlaard's curves as best we can base on available test data. This decision was based
on the premise that all of the proposed changes would be toward the conservative (or "safe") side and that
design engineers would continue to use Bijlaard 's extended range data unless some alternative were
offered. The following paper is therefore presented in the hope that it will facilitate the use of Bijlaard's
xi
WRC Bulletin 537 – 3rd Edition
Precision Equations and Enhanced Diagrams for Local Stresses in Spherical and Cylindrical Shells Due to External Loadings for
Implementation of WRC Bulletin 107
work by design engineers. Every effort has been made to point out any known limitations in the work and
to explain the exact nature of the changes which have been made to Bijlaard's original curves and data;
however, users are warned that the resulting work is not necessarily adequate for all cases. It is the hope
of the Subcommittee that additional theoretical work can be undertaken to provide more adequate data on
various phases of this problem.
F.S.G. Williams, Chairman
PVRC Subcommittee on Reinforced
Openings and External Loadings
xii
WRC Bulletin 537 – 3rd Edition
Precision Equations and Enhanced Diagrams for Local Stresses in Spherical and Cylindrical Shells Due to External Loadings for
Implementation of WRC Bulletin 107
TABLE OF CONTENTS
1
NOMENCLATURE ...................................................................................................................................... 1
1.1
1.2
1.3
GENERAL NOMENCLATURE ...........................................................................................................................1
NOMENCLATURE APPLICABLE TO SPHERICAL SHELLS .................................................................................1
NOMENCLATURE APPLICABLE TO CYLINDRICAL SHELLS ..............................................................................2
2
GENERAL EQUATION ............................................................................................................................... 3
3
SPHERICAL SHELLS ................................................................................................................................. 4
3.1
SIGN CONVENTION ........................................................................................................................................4
3.2
PARAMETERS .................................................................................................................................................6
3.2.1
Shell Parameter, (𝑼) ..........................................................................................................................6
3.2.2
Attachment Parameters .....................................................................................................................6
3.3
CALCULATION OF STRESSES .........................................................................................................................7
3.3.1
Stresses Resulting from Radial Load, 𝑷 .........................................................................................7
3.3.2
Stresses Resulting from Overturning Moment, 𝑴 .........................................................................8
3.3.3
Stresses Resulting from Torsional Moment, 𝑴𝑻 ...........................................................................9
3.3.4
Stresses Resulting from Shear Load, 𝑽 ..........................................................................................9
3.3.5
Stresses Resulting from Arbitrary Loading ....................................................................................10
3.4
LIST OF NONDIMENSIONAL CURVES FOR SPHERICAL SHELLS ...................................................................10
3.5
LIMITATIONS ON APPLICATION.....................................................................................................................10
3.5.1
Nozzle Stress .....................................................................................................................................10
3.5.2
Shell Stresses ....................................................................................................................................10
3.5.3
Ellipsoidal Shells ...............................................................................................................................11
3.6
ABRIDGED CALCULATION FOR MAXIMUM STRESSES DUE TO RADIAL AND MOMENT LOADING ONLY AT A
RIGID ATTACHMENT.................................................................................................................................................11
3.6.1
Maximum Stress Resulting from Radial Load, (𝑷) ......................................................................11
3.6.2
Maximum Stress Resulting from Overturning Moments, 𝑴𝟏 and 𝑴𝟐......................................11
3.6.3
Maximum Stress Resulting from Combined Load (𝑷) & Overturning Moment 𝑴 .................12
4
CYLINDRICAL SHELLS ........................................................................................................................... 12
4.1
SIGN CONVENTION ......................................................................................................................................12
4.2
PARAMETERS ...............................................................................................................................................14
4.2.1
Shell Parameter, 𝜸 ...........................................................................................................................14
4.2.2
Attachment Parameter, 𝜷 ................................................................................................................14
4.3
CALCULATION OF STRESSES .......................................................................................................................15
4.3.1
Stresses Resulting from Radial Load, 𝑷 .......................................................................................15
4.3.2
Stresses Resulting from Circumferential Moment, 𝑴𝒄 ...............................................................16
4.3.3
Stresses Resulting from Longitudinal Moment, 𝑴𝑳 ....................................................................17
4.3.4
Stresses Resulting from Torsional Moment, 𝑴𝑻 .........................................................................18
4.3.5
Stresses Resulting from Shear Loads, 𝑽𝒄 𝒂𝒏𝒅 𝑽𝑳 ......................................................................18
4.3.6
Stresses Resulting from Arbitrary Loading ....................................................................................19
4.4
NONDIMENSIONAL CURVES FOR CYLINDRICAL SHELLS .............................................................................19
4.4.1
List of Nondimensional Curves for Cylindrical Shells ..................................................................19
4.5
LIMITS ON APPLICATION ..............................................................................................................................19
4.5.1
External Radial Load ........................................................................................................................20
4.5.2
External Moment ...............................................................................................................................20
4.5.3
Attachment Stresses.........................................................................................................................20
5
NOTES ON USING THE EQUATIONS .................................................................................................... 20
6
ACKNOWLEDGMENT .............................................................................................................................. 21
xiii
WRC Bulletin 537 – 3rd Edition
Precision Equations and Enhanced Diagrams for Local Stresses in Spherical and Cylindrical Shells Due to External Loadings for
Implementation of WRC Bulletin 107
7
REFERENCES ........................................................................................................................................... 21
8
TABLES...................................................................................................................................................... 24
9
FIGURES .................................................................................................................................................... 36
APPENDIX A – BASIS FOR "CORRECTIONS" TO BIJLAARD'S CURVES ........................................... 141
A.1
INTRODUCTION...........................................................................................................................................141
A.2
SPHERICAL SHELLS ...................................................................................................................................141
A.3
CYLINDRICAL SHELLS ................................................................................................................................143
A.3.1
"Thick-Walled" Model Data ............................................................................................................143
A.3.2
"Thin-Walled" Model Data ..............................................................................................................144
A.3.3
Modification of Curves ....................................................................................................................146
A.4
TABLES ......................................................................................................................................................151
A.5
FIGURES ....................................................................................................................................................156
APPENDIX B – STRESS CONCENTRATION FACTORS FOR STRESSES DUE TO EXTERNAL
LOADS ............................................................................................................................................................. 175
B.1
INTRODUCTION AND TERMINOLOGY ..........................................................................................................175
B.2
STRESS CONCENTRATION FACTORS ........................................................................................................175
B.3
APPLICATION TO EXTERNAL LOADS ON A NOZZLE ...................................................................................177
B.3.1
Stresses at Fillet-Shell Juncture ...................................................................................................177
B.3.2
Stresses at Nozzle-Pipe Juncture ................................................................................................178
B.3.3
Special Case of a Tapered Nozzle ...............................................................................................178
B.4
APPLICATION TO BARS AND STRUCTURAL ATTACHMENTS .......................................................................178
B.5
ACKNOWLEDGMENT...................................................................................................................................178
B.6
FIGURES ....................................................................................................................................................179
xiv
WRC Bulletin 537 – 3rd Edition
Precision Equations and Enhanced Diagrams for Local Stresses in Spherical and Cylindrical Shells Due to External Loadings for
Implementation of WRC Bulletin 107
1
NOMENCLATURE
Symbols used in the formulas and equations in the text are listed below:
1.1
General Nomenclature
i
normal stress in the i-direction on the surface of the shell.
 ij
shear stress on the i-face of the j-direction.
S
stress intensity = twice maximum shear stress.
Ni
membrane force per unit length in the i-direction.
Mi
bending moment per unit length in the i-direction.
Kn
membrane stress concentration factor (pure tension or compression).
Kb
bending stress concentration factor.
i
denotes direction. In the case of spherical shells, this will refer to the tangential and radial
directions with respect to an axis normal to the shell through the center of the attachment as
shown in Figure 1. In the case of cylindrical shells, this will refer to longitudinal and
circumferential directions with respect to the axis of the cylinder as shown in Figure 2.
+
denotes tensile stress (when associated with 𝜎𝑖 ).
−
denotes compressive stress (when associated with 𝜎𝑖 ).

angle around attachment, degrees (see Figures 1 and 2).
E
modulus of elasticity.
P
concentrated radial load or total distributed radial load.
V
concentrated external shear load.
M
concentrated external overturning moment.
M
concentrated external torsional moment.
di
inside diameter and mean diameter, respectively, of the nozzle.
dm
inside diameter and mean diameter, respectively, of the nozzle.
Di
inside diameter and mean diameter, respectively, of the shell.
Dm
inside diameter and mean diameter, respectively, of the shell.
1.2
Nomenclature Applicable to Spherical Shells
V1
concentrated external shear load in 2-2 direction.
V2
concentrated external shear load in 1-1 direction.
1