The Correlation of Closed Form Solutions of a
Stiffener to Finite Element Analyses of Stiffeners
with Varying Geometry
by
Bernard S. Nasser, Jr
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Advisor
_________________________________________
Ken Brown, Project Advisor
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2011
(For Graduation May, 2012)
CONTENTS
CONTENTS ...................................................................................................................... ii
LIST OF TABLES ............................................................................................................ iii
LIST OF FIGURES .......................................................................................................... iv
NOMENCLATURE ......................................................................................................... vi
ACKNOWLEDGMENT ................................................................................................. vii
ABSTRACT ................................................................................................................... viii
1.0 INTRODUCTION/BACKGROUND .......................................................................... 1
2.0 THEORY/METHODOLOGY ..................................................................................... 6
2.1 Closed Form Solutions ....................................................................................... 6
2.1.1 Assumptions ........................................................................................... 6
2.1.2 Closed Form Analysis for Plate ............................................................. 6
2.1.3 Closed Form Analysis for Stiffener ....................................................... 7
2.2 Finite Element Modeling.................................................................................... 9
2.2.1 Assumptions ........................................................................................... 9
2.2.2 Baseline Models ................................................................................... 11
2.2.3 Plate / Stiffener Models ........................................................................ 13
2.2.4 Final Configuration Models ................................................................. 13
3.0 RESULTS/DISCUSSION ......................................................................................... 15
3.1 Closed Form Solution ...................................................................................... 15
3.1.1 Closed Form Solution for Rectangular Plate ....................................... 15
3.1.2 Close Form Solution for Stiffener ........................................................ 16
3.2 FEA Results ..................................................................................................... 23
3.2.1 Baseline Model Results ........................................................................ 23
3.2.2 Plate / Stiffener Model Results ............................................................ 30
3.2.3 Final Configurations ............................................................................ 36
4.0 CONCLUSIONS ....................................................................................................... 40
5.0 REFERENCES .......................................................................................................... 49
APPENDIX A: SUPPLEMENTAL CALCULATIONS ................................................ A1
APPENDIX B CONVERGENCE RESULTS ................................................................. B1
ii
LIST OF TABLES
Table 1:
Analysis Dimensional Properties…………………………………………... 5
Table 2:
Summary of Maximum Deflections and Stresses Using Closed
Form Solutions……………………………………………………………... 15
Table 3:
Supporting Values, Clamped Rectangular Plate…………………………… 16
Table 4:
Supporting Values, Simply Supported Rectangular Plate…………………. 16
Table 5:
Supporting Values, Clamped Stiffener…………………………………….. 18
Table 6:
Supporting Values, Simply Supported Stiffener…………………………… 19
Table 7:
Supporting Values, Clamped Plate/Stiffener………………………………. 21
Table 8:
Supporting Values, Simply Supported Plate/Stiffener……………………...23
Table 9:
Summary of Baseline Models vs. Closed Form Solutions………………… 24
Table 10: Summary of Plate/Stiffener FEA Models vs. Closed Form Solutions……...31
Table 11: Summary of Final Configuration Models vs. Closed Form Solutions…….. 37
Table 12: Summary of Percent Error between Closed Form Solutions and
Clamped Model Results……………………………………………………. 41
iii
LIST OF FIGURES
Figure 1: Continuous plate divided into individual panels ............................................................................1
Figure 2: Uniformly distributed load along flat rectangular plate ..................................................................2
Figure 3: Uniformly distributed load along stiffener .....................................................................................2
Figure 4: General arrangement used for final stiffener configurations ..........................................................3
Figure 5: Design configuration, dead-end stiffener .......................................................................................4
Figure 6: Design configuration, sniped stiffener ............................................................................................4
Figure 7: Design configuration, wrapped stiffener ........................................................................................5
Figure 8: Plate and stiffener cross section view .............................................................................................5
Figure 9: Baseline FEA model ..................................................................................................................... 10
Figure 10: FEA of typical stiffener with effective width of plate ................................................................ 10
Figure 11: FEA of final configuration ......................................................................................................... 11
Figure 12: Deflection of Clamped Stiffener under Uniformly Distributed Load ......................................... 17
Figure 13: Moment Force of Clamped Stiffener under Uniformly Distributed Load .................................. 17
Figure 14: Bending Stress of Clamped Stiffener under Uniformly Distributed Load .................................. 17
Figure 15: Deflection of Simply Supported Stiffener under Uniformly Distributed Load .......................... 18
Figure 16: Moment Force of Simply Supported Stiffener under Uniformly Distributed Load .................... 19
Figure 17: Bending Stress of Simply Supported Stiffener under Uniformly Distributed Load ................... 19
Figure 18: Deflection of Clamped Plate/Stiffener under Uniformly Distributed Load ................................ 20
Figure 19: Moment Force of Clamped Plate/Stiffener under Uniformly Distributed Load ......................... 20
Figure 20: Bending Stress of Clamped Plate/Stiffener under Uniformly Distributed Load ......................... 21
Figure 21: Deflection of Simply Supported Plate/Stiffener under Uniformly Distributed Load ................. 22
Figure 22: Moment of Simply Supported Plate/Stiffener under Uniformly Distributed Load ..................... 22
Figure 23: Bending Stress of Simply Supported Plate/Stiffener under Uniformly Distributed Load .......... 22
Figure 24: Clamped rectangular panel FEA results ..................................................................................... 24
Figure 25: Simply supported rectangular plate FEA results ........................................................................ 25
Figure 26: Clamped stiffener FEA results ................................................................................................... 26
Figure 27: Peak bending stress in clamped stiffener .................................................................................... 26
Figure 28: Peak deflection in clamped stiffener........................................................................................... 27
Figure 29: Simply supported stiffener FEA results ...................................................................................... 28
Figure 30: Peak bending stress in simply supported stiffener ...................................................................... 28
Figure 31: Peak deflection in simply supported stiffener............................................................................. 28
Figure 32: Sniped stiffener FEA results ....................................................................................................... 29
Figure 33: Peak bending stress in sniped stiffener ....................................................................................... 30
Figure 34: Peak deflection in sniped stiffener .............................................................................................. 30
Figure 35: Clamped plate/stiffener FEA model results ................................................................................ 32
Figure 36: Peak bending stress in clamped plate/stiffener ........................................................................... 32
Figure 37: Peak deflection in clamped plate/stiffener .................................................................................. 32
Figure 38: Simply supported plate/stiffener FEA model results ................................................................. 33
Figure 39: Peak bending stress in simply supported plate/stiffener ............................................................. 34
Figure 40: Peak deflection in simply supported plate/stiffener .................................................................... 34
Figure 41: Sniped plate/stiffener FEA model results ................................................................................... 35
Figure 42: Stress comparison of sniped plate/stiffener FEA results to closed form solutions for
clamped/simply supported stiffeners .................................................................................................. 36
Figure 43: Deflection comparison of sniped plate/stiffener FEA results to closed form solutions for
clamped/simply supported stiffeners .................................................................................................. 36
Figure 44: Stress comparison of final configuration FEA results to closed form results ............................. 37
Figure 45: Deflection comparison of final configuration FEA results to closed form solutions .................. 37
Figure 46: Butted final configuration FEA model results ............................................................................ 38
Figure 47: Wrapped final configuration FEA model results ........................................................................ 39
Figure 48: Sniped final configuration FEA model results ........................................................................... 40
Figure 49: Iteration comparison of clamped stiffener (in percent error) ...................................................... 42
Figure 50: Stress comparison of model iterations to closed form solutions ................................................. 43
Figure 51: Deflection comparison of model iterations to closed form solutions ......................................... 43
iv
LIST OF FIGURES (CONT.)
Figure 52: Iteration comparison of simple supported stiffener (in percent error) ........................................ 44
Figure 53: Stress comparison of model iterations to closed form solutions ................................................. 44
Figure 54: Deflection comparison of model iterations to closed form solutions ......................................... 45
Figure 55: Butted final configuration FEA model stress concentration ....................................................... 46
Figure 56: Wrapped final configuration FEA model stress concentration ................................................... 47
Figure 57: Sniped final configuration FEA model stress concentration ...................................................... 48
v
NOMENCLATURE
a…………..length of long side of panel (in)
α pc………………..
dimensionless constant for clamped plate
α pss………………..
dimensionless constant for simply supported plate
b…………….
length of short side of panel (in)
βpc………………
dimensionless constant for clamped plate
βpss……………dimensionless constant for simply supported plate
c ps……………….
distance from edge of plate/stiffener to neutral axis (in)
c s…………………..
distance from edge of stiffener to neutral axis (in)
δpc…………………
deformation of clamped plate (in)
δpsc………………..
deformation of clamped plate/stiffener (in)
δpss………………
deformation of simply supported plate (in)
δpsss………………
deformation of simply supported plate/stiffener (in)
δsc………………..
deformation of clamped stiffener (in)
δsss……………….
deformation of simply supported stiffener (in)
E………………
modulus of elasticity (lb/in2)
FEA………….
finite element analysis
FEM…………..
finite element model
in…………….
inch
area moment of inertia of plate/stiffener (in4)
Ips…………………
area moment of inertia of plate (in4)
Is…………………..
L……………..
length of stiffener (in)
Mpsc……………..
moment of clamped plate/stiffener (lb·in)
Mpsss…………….
moment of simply supported plate/stiffener (lb·in)
moment of clamped stiffener (in4)
Msc……………..
Msss……………….
moment of simply supported stiffener (lb·in)
NA……………
neutral axis
Pef f ……………….
effective width of plate (in)
uniformly distributed pressure load (lb/in2)
po………………….
SD………………..
stiffener depth (in)
Ss…………………
stiffener spacing (in)
stress in clamped plate (lb/in2)
σpc……………….
stress in clamped plate/stiffener (lb/in2)
σpsc……………….
stress in simply supported plate (lb/in2)
σpss……………….
stress in simply supported plate/stiffener (lb/in2)
σpsss……………
stress in clamped stiffener (lb/in2)
σsc…………………
stress in simply supported stiffener (lb/in2)
σsss……………..
t……………….
plate thickness (in)
t f …………………..
stiffener flange thickness (in)
t w……………………
stiffener web thickness (in)
wf ………………..
stiffener flange width (in)
wps……………….
line load for plate/stiffener (lb/in)
ws……………….
line load for stiffener only (lb/in)
x…………….
distance along stiffener (in)
vi
ACKNOWLEDGMENT
I would like to take this opportunity to thank the following people for their support: Mr.
Royle, a true friend, I thank you for your help and support throughout the masters
program at RPI. You helped me to get to this point, and I would never have gotten here
without you. Professor Doyon, thank you for your continued support throughout this
project. I truly appreciate all your input on helping me to get this project done. You’ve
inspired me to not only be a better engineer, but to also be a better person. I would also
like to thank my parents, for everything they’ve done for me over the years. Their
guidance, encouragement, and support have played such an important role in me getting
to this point. And finally, I would like to thank Diana, for always being there for me.
You’ve been very patient with me especially with all those long nights when I was
hiding from you and working on this project. You’ve never wavered in your support for
me. I thank you and I love you.
I would also like to thank my project advisors for their support and their patience with
me over the course of this project.
vii
ABSTRACT
This study focuses on the effect of engineering assumptions made when
designing a plate panel/stiffener system under a uniformly distributed load. The
plate/panel system can be used in numerous shipbuilding applications, such as in
designing/reinforcing a pressurized tank. Initial engineering assumptions design the
panel using Classical Deflection Theory (small deflection theory) for a flat rectangular
plate under a uniformly distributed load that is either fixed or clamped around the edges.
The stiffeners are initially designed as being under a uniformly distributed load and
either simply supported or fixed at the ends. The initial panel and stiffener sizes
generated are an approximation, as a finite element analysis is required to evaluate and
authorize the final configuration. Final configurations can be very different from the
simple initial closed form evaluations, as stress issues and fabrication constraints are
accounted for. The study herein performs a comparative analysis between the closed
form solutions (stress and deflection for a fixed or pinned stiffener under a uniformly
distributed load) and the values generated by a finite element analysis. The finite
element analyses focus on common “final configurations” used.
The study herein introduces the closed form equations necessary to determine an
initial stiffener configuration (see Section 2.1). A series of finite element models are
presented that compare the final stiffener configurations to the initial stiffener
configurations (see Section 2.2). Using the closed form equations, stress and deflection
values are calculated for the initial configurations (see Section 3.1 and Appendix A).
Section 3.2 provides the results for the finite element models, with mesh convergence
studies and boundary condition studies being performed in Appendix B. Conclusions are
drawn and comparisons are provided in Section 4.0 for the initial configurations to the
final FEA configurations. Sources of error are introduced and possible additional studies
are presented.
The analyses performed herein demonstrate the final stiffener configurations that
are commonly seen in shipbuilding design can be compared to the closed form solutions
used to generate the initial simplistic configurations. The results show that mesh
refinement and boundary conditions can dramatically alter the results of the finite
element models.
viii
1.0
INTRODUCTION/BACKGROUND
When considering a flat plate under a uniformly distributed load, it is common
engineering practice to reinforce it with a supporting stiffener. Support stiffeners, if
designed correctly, will absorb the majority of the applied load and prevent the
previously unsupported plate from prematurely yielding or deflecting to an unacceptable
magnitude. This is beneficial in the sense that the stronger the support stiffeners, the less
load the plate needs to handle, the smaller/thinner the plate can be. A proper balance
between the two is vital in creating a more buildable product that is weight optimized
with minimal weld volume and a low quantity of stiffeners.
A uniform plate that extends over a support and has more than one span along its
length or width is considered “continuous”. A continuous plate may be analyzed by
subdividing it into individual panels (Figure 1). The analysis is based on equilibrium
conditions of individual panels and the compatibility of displacements or force at the
adjoining edges, as invoked by the Classical Elastic Deflection Theory (small-deflection
theory). In other words, for a continuous plate with bi-directional stiffening, analysis of
any one panel can be simplified into that for a flat plate under a uniformly distributed
pressure, where the stiffeners represent the boundary conditions. This can be seen in
Figure 1. Depending on the design of the stiffeners, the individual panels can be
evaluated as simply supported or clamped flat rectangular plates, as shown in Figure 2.
Bi-directional stiffeners
Individual “panel” created as a
result of boundary conditions
(bi-directional stiffeners)
Continuous plate divided into
individual panels as a result of bidirectional stiffening
Figure 1: Continuous plate divided into individual panels
1
po
a
b
a)
po
a
b
b)
Figure 2: Uniformly distributed load along flat rectangular plate
a) simply supported b) clamped
The rigidity of the adjoining panel edges is dependent on the support provided. A
common form of support used is a T-frame. It is necessary for the T-frame to be sized to
support the load of the panel while remaining within allowable elastic stress limits. It is
noted that depending on the purpose of the stiffener that the design may allow for some
permanent set. However for the purposes of this paper the stiffeners and the panels are
always assumed to remain elastic. The T-frame can be evaluated as a stiffener under a
uniformly distributed load that is either simply supported or clamped at the ends, as
shown in Figure 3.
L
po
L
po
a)
po
b)
Figure 3: Uniformly distributed load across stiffener
a) simply supported b) clamped
The fundamental theories discussed above serve as the foundation for common
shipbuilding designs, such as for a pressurized tank. A pressurized tank, which roughly
resembles a box that is reinforced with support beams, needs to be strong enough to
2
withstand the forces created by the uniformly distributed pressure load. The initial
design phase of these tanks, as indicated [1], implements the fundamental theories of
beams and flat rectangular plates. Specifically, each tank surface (as a result of the
supporting beams) is broken into individual panels for the purpose of evaluating them
using the fundamental theories of thin shell deflection, as shown in Figure 1. As
identified [1], the supporting beams can be evaluated using the fundamental theories for
designing a beam. Beams are designed rigidly enough to enable the plate to be separated
into individual panels. Through these theories an initial tank design (panel/stiffener
configuration) can be created.
The initial tank design generated is an approximation, as a finite element analysis
is required to evaluate and authorize the final configuration. Final configurations can be
very different from the initial designs created (using the closed form theories), as
fabrication constraints and stress issues created by asymmetric loading/geometry are
accounted for. Design adjustments and alterations can be made during the finite element
analysis phase.
This study focuses on the discrepancy between the initial panel/stiffener
configurations (as determined via closed-form solutions) and the final panel/stiffener
configurations (as determined via finite element analysis). The arrangement shown in
Figure 4 is used to evaluate the final configurations throughout this analysis.
Individual “panel” created as a
result of boundary conditions
(stiffeners)
Stiffeners
See Figures 5, 6, and 7
for stiffener
configurations
Continuous plate divided into
individual panels as a result of
stiffeners
See Figure 8
Side panel
Figure 4: General arrangement used for final stiffener configurations
3
The final configuration analyzed is composed of a continuous plate reinforced
longitudinally by stiffeners and supported at the ends by vertical side panels. It is noted
side panels were used in lieu of the bi-directional stiffener configuration shown in Figure
1. Figures 5-7 detail three stiffener design configurations (final configurations)
commonly used in shipbuilding practices that account for the boundary breaks for
individual panels as well as for satisfying asymmetric loading/geometry and fabrication
issues. These design configurations include butting (or dead-ending) the stiffener into
the side panel, sniping the stiffener prior to contacting the side panel, and wrapping the
stiffener around the side panel.
Stiffened Panel
Stiffener dead-ended
into side panel
T-frame
(stiffener web and flange)
Side Panel
Figure 5: Design configuration, dead-end stiffener
Stiffened Panel
Stiffener sniped prior
to contacting side
panel
T-frame
(stiffener web and flange)
Side Panel
Figure 6: Design configuration, sniped stiffener
4
Stiffened Panel
Stiffener wrapped
into side panel
T-frame
(stiffener web and flange)
Side Panel
Figure 7: Design configuration, wrapped stiffener
The goal of the comparison (and subsequently this study) is to determine how
“approximate” the closed form beam solutions are to the common design configurations
that are actually being modeled / fabricated.
For the purposes of the analysis, an arbitrary pressure load of 100 psi is
uniformly distributed across the plate and/or stiffener (depending on the model
iteration). Stiffener length (L) is 48 inches, and the distance between stiffeners (Ss) is 24
inches. Table 1 and Figure 8 summarize the dimensions used throughout the analysis:
Table 1: Analysis Dimensional Properties (all dimensions in inches)
Stiffener Length (L )
Stiffener Spacing (S S )
48.00
24.00
Effective Plate Width (P eff )
Plate Thickness (t )
Stiffener Depth (S D )
24.00
0.75
5.00
Stiffener Flange Width (w f )
3.00
Stiffener Web Thickness (t w )
0.50
Stiffener Flange Thickness (t f )
1.00
Peff = 24.00”
t = 0.75”
Stiffener web
Plate (effective width)
tw = 0.50”
Fd = 5.00”
Stiffener flange
tf = 1.00”
wf = 3.00”
Figure 8: Plate and stiffener cross section view
(all dimensions in inches)
5
2.0
THEORY/METHODOLOGY
2.1 Closed Form Solutions
2.1.1
Assumptions
The calculations and analyses conducted herein pertain to thin plates with small
deflections. In accordance with [2] the criterion often applied to define a thin plate is the
ratio of the thickness to the smaller span length should be less than 1/20. Furthermore,
[3] identifies the maximum deflection shall not be more than one half of the thickness.
The formulas of this analysis are also based on the following assumptions [3]:



The plate is flat, of uniform thickness, and of homogeneous isotropic material.
All forces- loads and reactions- are normal to the plane of the plate.
The plate is nowhere stressed beyond the elastic limit.
The equations defined herein for stress and deflections of stiffeners are based off the
following assumptions [3]:








2.1.2
The stiffener is a homogeneous material with the same modulus of elasticity in
tension as well as compression
The stiffener is straight
The stiffener has a uniform cross section
The stiffener has at least one longitudinal plane of symmetry
All loads and reactions are perpendicular to the axis of the stiffener, and they lie
in the same plane
The stiffener is long compared to its depth (span/depth ratio >8 for metal
stiffeners of compact section)
The stiffener is not disproportionately wide
The stiffener is nowhere stressed beyond the elastic limit
Closed Form Analysis for Plate
Clamped Rectangular Plate under Uniformly Distributed Load
For a rectangular plate (length a, width b, thickness t) with all edges fixed that is
subjected to a uniform load over the entire plate, the maximum deflection of the plate is
given by [3]:
 pc 
 pc pob 4
(1)
Et 3
6
where αpc is a constant [3], that depends on the ratio between the short side of the panel
to the long side of the panel, and E is the Modulus of Elasticity of the material. The
deflection of the plate must satisfy the requirements for thin plate deflection, as stated in
Section 2.1.1. The maximum stress is located at the center of the long edge of the
plate [3]:
 pc 
 pc pob 2
(2)
t2
where βpc is a constant [3] that depends on the ratio between the short side of the panel to
the long side of the panel, po is the pressure, b is the short side of the panel, and t is the
thickness.
Simply Supported Rectangular Plate under Uniformly Distributed Load
For a rectangular plate with all edges simply supported subjected to a uniform load over
the entire plate, the maximum deflection of the plate is determined as follows:
 pss 
 pss pob 4
(3)
Et 3
where αpss is a constant [3] that depends on the ratio between the short side of the panel
to the long side of the panel. The deflection of the plate must satisfy the requirements for
thin plate deflection, as stated in Section 2.1.1. The maximum stress is located at the
center of the plate [4]:
 pss 
 pss pob 2
(4)
t2
where βpss is a constant [3] that depends on the ratio between the short side of the panel
to the long side of the panel.
2.1.3
Closed Form Analysis for Stiffener
Clamped Stiffener under Uniformly Distributed Load
For a stiffener with both ends fixed that is subjected to a uniform load over the surface,
the deflection at any point along its length can be determined as follows [4]:
wx 2 L  x 
 ( x)  
24 EI s
2
(5)
7
where w represents the uniformly distributed load (applied as a line load), x represents
any point along the length of the stiffener, L is the overall length of the stiffener, and Is is
the area moment of inertia of the stiffener. The maximum deflection of the stiffener is
determined as follows [4]
 sc  
wL4
384 EI s
(6)
The moment of the stiffener, Msc, at any point along its length, x, can be determined by:
M ( x) 

1
w 6 Lx  L2  6 x 2
12

(7)
The maximum moment of the stiffener is determined by [4]:
M sc 
wL2
12
(8)
The bending stress of the stiffener is determined from the following equation [5]:
 sc 
M sccs
Is
(9)
where cs represents the distance from the edge of the stiffener to it’s neutral axis (NA). It
is noted equations (5)-(9) are also applicable to the clamped plate/stiffener configuration
analyzed in this study.
Simply Supported Stiffener under Uniformly Distributed Load
For a stiffener with both ends simply supported that is subjected to a uniform load over
the surface, the deflection at any point x along its length can be determined as
follows [4]:

wx L3  2 x 2 L  x 3
 ( x)  
24 EI s

(10)
The maximum deflection of the stiffener is determined as follows [4]:
 sss  
5wL4
384 EI s
(11)
The moment of the stiffener at any point along its length can be determined as
follows [4]:
M ( x) 
1
wL  x x
2
(12)
8
The maximum moment of the stiffener, Msss, is determined by [4]:
M sss 
wL2
8
(13)
The bending stress of the stiffener, can be determined from the previously defined
Equation (9). It is noted equations (10)-(13) are also applicable to the simply supported
plate/stiffener configuration analyzed in this study.
2.2 Finite Element Modeling
2.2.1
Assumptions
Since final design configurations (using finite element analysis) can be very
different from the initial designs configurations (created using the closed form
equations), a series of sub-models, or model iterations, are necessary. These sub-models
will follow the simple stiffener/plate (as shown in Figures 2 and 3) through a series of
transformations (iterations) into the final stiffener configurations (as shown in Figures 5,
6, and 7). The intent of the model iterations is to address potential sources of error
encountered along the way (from baseline design to final configuration), as created by
mesh refinement, boundary conditions, etc. All models analyzed herein were created
using 20 noded quadratic solid elements with material properties of steel ( E = 30 x 106
psi and v = 0.3). The FEA program used was Electric Boat in-house software
COMMANDS.
The first series of models created are of the simple flat rectangular plate and
stiffener evaluated above (see Figure 9), and are referred to herein as the Baseline
Models (see Section 2.2.2). The baseline models are necessary to determine an optimal
mesh density and appropriate boundary conditions, which will be applied to subsequent
model iterations. The determination of the baseline models (including mesh convergence
studies and boundary condition evaluations) can be found in Appendix B.
9
a)
b)
Figure 9: Baseline FEA model
a) stiffener b) plate
The next model iteration takes the baseline models a step further, by modeling a
portion of the baseline plate as it is supported by the baseline stiffener (see Figure 10 and
Section 2.2.3). The extent of baseline plate modeled is the effective width of the plate.
This is the maximum width of plate considered to be supported by a single stiffener,
which is one half bay of plate on either side of the stiffener (it is assumed a distance
from the subject stiffener greater than a half bay away will receive more support from
the adjacent stiffener). Since a half bay spacing is 12 inches, the effective width of the
plate is considered to be 24 inches (12 inches on either side of the stiffener). This model
iteration incorporates the optimal mesh density and appropriate boundary conditions
derived from the baseline models, and applies the pressure load to the plate surface (no
longer directly to the top of the stiffener frame web).
Figure 10: FEA of typical stiffener with effective width of plate
10
The final iteration represents the final stiffener configurations, which takes the
second iteration a step further; vertical plates are added to the ends of the stiffeners, and
the model is extended to include a stiffener/shell combination on both sides of the
baseline condition (see Figure 11 and Section 2.2.4). The intent of this iteration is to
focus on the results of the middle stiffener, which has been isolated from the boundary
conditions. The thought here is the only difference between the final configuration
models will be the geometry of the stiffeners, and conclusions can be drawn as to how
the results of these different geometries compare to each other as well as to the closed
form solutions.
Note: Portion of
shell top and side,
removed for clarity
Figure 11: FEA of final configuration
2.2.2
Baseline Models
In order to accurately compare the results of the FEA test cases to the results of
the closed form solutions, a proper mesh density for the model iterations needs to be
determined. A proper mesh density is important as too few elements will generate a
coarser mesh that may not capture the peak stress and deflection regions. Too many
elements will generate an excessively refined mesh that will increase analysis time and
may possibly create artificial peak stresses. A proper mesh density is selected by
performing a mesh convergence study using the baseline models described below. By
11
choosing an initial mesh density and then increasing the number of elements, a density
can be selected once the FEA results are considered within an acceptable margin to the
closed form solutions. A mesh convergence study for the iterations is detailed in
Appendix B.
The baseline models are also used to determine an effective application of
boundary conditions. The method or position in which boundary conditions are applied
can dramatically alter the results of that given model, so it is important to iterate the
baseline model (where the geometries of the FEA and the closed form solutions are the
same, thus the results should be theoretically the same) in order to determine the most
effective application of boundary conditions for the subsequent test case models (where
the geometries of the FEA and the closed form solutions are not the same, creating
potential inconsistencies). For the purpose of this study, the baseline models will be
refined such that the FEA results correlate to within approximately 5% of the closed
form solutions.
Panel Baseline Models
The panel baseline models represent a single panel that is either clamped or
simply supported on all edges and has a uniformly distributed load applied to its surface
(see Figure 2 and Figure 9). The models are 0.75” thick and 48” x 24”, which represents
the size of the panel (largest area of unsupported plate, 48 inches long, 24 inches
between stiffeners). Boundary conditions are applied to the perimeter of the models to
replicate the respective constrained condition of the panel (see Section 3.2.1 and
Appendix B for more details).
Stiffener Baseline Model
The stiffener baseline models represent a stiffener that is in the shape of a Tframe that is either clamped or simply supported. A uniformly distributed load is applied
to the top surface of the stiffener webs and the models are constrained at both ends (see
Figure 3 and Figure 9) to replicate the simply supported or clamped condition. Cross
section dimensions are 5.0 x 3.0 x 0.5 x 1.0 (as shown in Figure 8), and the model
extends 48”. The sniped stiffener model has the same model length and typical cross
section dimensions as the clamped and simply supported baseline stiffener models.
However, the sniped stiffener is chamfered, or sniped, at approximately 45 degrees as it
12
approaches the ends of the stiffener. Boundary conditions are applied to the ends of the
stiffeners to replicate the respective constrained condition of the stiffener (see Section
3.2.1 and Appendix B for more details).
2.2.3
Plate / Stiffener Models
As discussed in the previous section, mesh convergence and boundary condition
studies for the baseline models are necessary in order to validate the final configuration
models. The plate/stiffener models combine the effective width of plate with the baseline
stiffener model (see Figure 10). For this iteration, a uniformly distributed load is applied
across the baseline plate, which is supported by the baseline stiffener. The stiffener is
centered length-wise under the plate. Separate models are created to replicate the
clamped model, the simply supported model, and the sniped model (including altered
geometry). The significance of these models is to document the additional errors
resulting from complicated boundary conditions and different loading conditions.
2.2.4
Final Configuration Models
The final configuration models are necessary to analyze stiffener results that are
isolated from the boundary conditions. The rigidity of the boundary conditions could
have an adverse effect on the results. Therefore the final configuration models attempt to
resolve this situation by creating a plate that is supported transversely by three uniform
stiffeners, equally spaced at 24” (see Figure 11). The ends of the models are supported
by vertical plates, fully fixed at the base and are short and thick. They are designed to
provide a more realistic configuration of a panel supported by a stiffener, while still
maintaining the approximate rigidity that would be given by boundary conditions at the
perimeter of the plate. The modeling assumptions of this iteration attempt to create an
internal stiffener (the center stiffener) that is completely isolated from any boundary
conditions. The plate/stiffener combination is allowed to deflect and bend based upon
the reactions of the adjacent stiffeners, and not based upon a perfectly stiff connection.
Focus can now be directed towards the alternate stiffener configurations, which are
described below.
13
Sniped Final Configuration Model
In the sniped final configuration model, the model configuration described above
(and shown in Figure 11) is supported by sniped stiffeners (as described in Section 2.2.2)
The stiffener does not connect directly to the vertical end plates (see Figure 6). This
configuration is a common configuration used in shipbuilding designs, as it is easy to
fabricate and provides accessibility for tradesmen during fit-up. It is assumed the results
of this configuration resemble a simply supported configuration.
Butted Final Configuration Model
In this model configuration, the shell plate is supported by the baseline stiffener
(as shown in Figure 11). Unlike the sniped frame model, the stiffener butts directly into
the shell wall (see Figure 5). This configuration can create difficulties during the
fabrication process, as well as stress concentrations in the wall plate (since the stiffener
is butting directly into the wall). This configuration is also a common configuration used
in shipbuilding designs, as long as chocks are placed on the outboard sides of the vertical
walls to mitigate stress concentrations. It is assumed the results of this configuration
resemble a clamped configuration.
Wrapped Final Configuration Model
In the wrapped final configuration, the geometry of the model is the same as that
used on the butted final configuration model. However, the stiffener wraps down the side
of the vertical plates (see Figures 7 and 11), as opposed to ending abruptly in the
aforementioned configurations. This configuration is a much more efficient fabrication
alternative to the other configurations, as few parts are involved and the outer wall is
more evenly and continuously supported. It is assumed the results of this configuration
resemble a clamped configuration.
14
3.0
RESULTS/DISCUSSION
3.1 Closed Form Solution
Table 2 summarizes the maximum deflections and stresses using the equations given in
Section 2.1 and the values in Tables 3-8.
Table 2: Summary of Maximum Deflections and Stresses Using Closed Form Solutions
Symbol
Comment
Value
δpc
deformation of clamped plate (in)
-0.0726
δpsc
deformation of clamped plate/stiffener (in)
-0.0158
δpss
deformation of simply supported plate (in)
-0.2910
δpsss
deformation of simply supported plate/stiffener (in)
-0.0790
δsc
deformation of clamped stiffener (in)
-0.0022
δsss
deformation of simply supported stiffener (in)
-0.0111
σpc
σpsc
σpss
σpsss
3.1.1
2
stress in clamped plate (lb/in )
50933
2
stress in clamped plate/stiffener (lb/in )
-29830
2
stress in simply supported plate (lb/in )
62484
2
stress in simply supported plate/stiffener (lb/in )
2
44745
σsc
stress in clamped stiffener (lb/in )
3226
σsss
stress in simply supported stiffener (lb/in2)
4838
Closed Form Solution for Rectangular Plate
Clamped Rectangular Plate Under Uniformly Distributed Load
For a rectangular plate with all edges fixed that is subjected to a uniform load over the
entire plate, the maximum deflection of the plate is determined to be -0.0726 in, as
shown in Table 2. Since the peak deflection is less than half of the plate thickness, and
since the plate thickness is less than 1/20th of the plate length, the plate size is acceptable
for thin plate theory. The maximum stress is located at the center of the long edge of the
plate is determined to be 50933 psi as shown in Table 2. Table 3 summarizes the
supporting values used to calculate the peak deflection and stress above.
15
Table 3: Supporting Values, Clamped Rectangular Plate
α pc = 0.0277 (a/b = 2.0) [2]
b = 24in
β pc = 0.4974 (a/b = 2.0) [2]
E = 30 x 106 psi
p o = 100 psi
t = 0.75in
Simply Supported Rectangular Plate Under Uniformly Distributed Load
For a rectangular plate with all edges simply supported subjected to a uniform load over
the entire plate, the maximum deflection of the plate is determined to be -0.2910 in, as
shown in Table 2. Since the peak deflection is less than half of the plate thickness, and
since the plate thickness is less than 1/20th of the plate length, the plate size is acceptable
for thin plate theory. The maximum stress is located at the center of the long edge of the
plate is determined to be 62484 psi as shown in Table 2. Table 4 summarizes the
supporting values used to calculate the peak deflection and stress above.
Table 4: Supporting Values, Simply Supported Rectangular Plate
α pss = 0.1110 (a/b = 2.0) [2]
b = 24in
β pss = 0.6102 (a/b = 2.0) [2]
E = 30 x 106 psi
p o = 100 psi
t = 0.75in
3.1.2
Close Form Solution for Stiffener
Clamped Stiffener Under Uniformly Distributed Load
For a stiffener with both ends fixed subjected to a uniform load over the surface, the
maximum deflection of the stiffener is determined to be -0.0022 in as shown in Table 2.
The peak bending stress is determined to be 3226 psi as shown in Table 2. The
deflection, moment, and bending stress at any point along the length of the stiffener is
shown in Figures 12, 13, and 14, respectively. Table 5 summarizes the supporting values
used to calculate the peak deflection and stress above.
16
Clamped Beam Deflection
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
Deflection (in)
-0.0005
-0.001
Calculated
-0.0015
-0.002
-0.0025
Beam Length (in)
Figure 12: Deflection of Clamped Stiffener under Uniformly Distributed Load
Clamped Beam Moment Distribution
6000
4000
Moment Force (lb*in)
2000
0
0.00
-2000
4.00
8.00
12.00 16.00 20.00 24.00 28.00 32.00 36.00 40.00 44.00 48.00
Calculated
-4000
-6000
-8000
-10000
-12000
Beam Length (in)
.
Figure 13: Moment Force of Clamped Stiffener under Uniformly Distributed Load
Clamped Beam Bending Stress
4000
3000
Stress (psi)
2000
1000
0
0.00
Calculated
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
-1000
-2000
Beam Length (in)
Figure 14: Bending Stress of Clamped Stiffener under Uniformly Distributed Load
17
Table 5: Supporting Values, Clamped Stiffener
c s = -3.5 in (see appendix A)
E = 30 x 106 psi
I s = 10.42 in4 (see Appendix A)
L = 48 in
M sc = -9600 lb·in
w s = 50 lb/in
Simply Supported Stiffener Under Uniformly Distributed Load
For a stiffener with both ends simply supported subjected to a uniform load over the
surface, the maximum deflection of the stiffener is determined to be -0.0111 in as shown
in Table 2. The peak bending stress is determined to be 4838 psi as shown in Table 2.
The deflection, moment, and bending stress at any point along the length of the stiffener
is shown in Figures 15, 16, and 17, respectively. Table 6 summarizes the supporting
values used to calculate the peak deflection and stress above.
Simply Supported Beam Deflection
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
Deflection (in)
-0.002
-0.004
-0.006
Calculated
-0.008
-0.01
-0.012
Beam Length (in)
Figure 15: Deflection of Simply Supported Stiffener under Uniformly Distributed Load
18
Simply Supported Beam Moment Distribution
16000
Moment Force (lb*in)
14000
12000
10000
8000
Calculated
6000
4000
2000
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
Beam Length (in)
Figure 16: Moment Force of Simply Supported Stiffener under Uniformly Distributed Load
Simply Suported Beam Bending Stress
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
-1000
Stress (psi)
-2000
-3000
Calculated
-4000
-5000
-6000
Beam Length (in)
Figure 17: Bending Stress of Simply Supported Stiffener under Uniformly Distributed Load
Table 6: Supporting Values, Simply Supported Stiffener
c s = -3.5 in (see appendix A)
6
E = 30 x 10 psi
4
I s = 10.42 in (see Appendix A)
L = 48 in
M sss = 14400 lb·in
w s = 50 lb/in
19
Clamped Plate/Stiffener Under Uniformly Distributed Load
For a stiffener with an effective length of shell that has both ends fixed subjected to a
uniform load over the surface, the maximum deflection of the stiffener is determined to
be -0.0158 in as shown in Table 2. The peak bending stress is determined to be -29830
psi as shown in Table 2. The deflection, moment, and bending stress at any point along
the length of the stiffener is shown in Figures 18, 19, and 20, respectively. Table 7
summarizes the supporting values used to calculate the peak deflection and stress above.
Clamped Plate/Stiffener Deflection
0
0.00
-0.002
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
-0.004
-0.008
Calculated
-0.01
-0.012
-0.014
-0.016
-0.018
Beam Length (in)
Figure 18: Deflection of Clamped Plate/Stiffener under Uniformly Distributed Load
Clamped Plate/Stiffener Moment Distribution
300000
200000
Moment Force (lb*in)
Deflection (in)
-0.006
100000
0
0.00
-100000
4.00
8.00
12.00 16.00
20.00 24.00
28.00 32.00
36.00 40.00
44.00 48.00
Calculated
-200000
-300000
-400000
-500000
Beam Length (in)
Figure 19: Moment Force of Clamped Plate/Stiffener under Uniformly Distributed Load
20
Clamped Plate/Stiffener Bending Stress
20000
15000
10000
Stress (psi)
5000
0
0.00
-5000
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
Calculated
-10000
-15000
-20000
-25000
-30000
-35000
Beam Length (in)
Figure 20: Bending Stress of Clamped Plate/Stiffener under Uniformly Distributed Load
Table 7: Supporting Values, Clamped Plate/Stiffener
c ps = 4.53 in (see appendix A)
6
E = 30 x 10 psi
4
I ps = 70.02 in (see Appendix A)
L = 48 in
M psc = -460,800 lb·in
w ps = 2400 lb/in
Simply Supported Plate/Stiffener Under Uniformly Distributed Load
For a stiffener with an effective length of shell that has both ends simply supported
subjected to a uniform load over the surface, the maximum deflection of the stiffener is
determined to be -0.0790 in as shown in Table 2. The peak bending stress is determined
to be 44745 psi as shown in Table 2. The deflection, moment, and bending stress at any
point along the length of the stiffener is shown in Figures 21, 22, and 23, respectively.
Table 8 summarizes the supporting values used to calculate the peak deflection and
stress above.
21
Simply Supported Plate/Stiffener Deflection
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
-0.01
-0.02
Deflection (in)
-0.03
-0.04
Calculated
-0.05
-0.06
-0.07
-0.08
-0.09
Beam Length (in)
Figure 21: Deflection of Simply Supported Plate/Stiffener under Uniformly Distributed Load
Simply Supported Plate/Stiffener Moment Distribution
800000
Moment Force (lb*in)
700000
600000
500000
400000
Calculated
300000
200000
100000
0
0.00
4.00
8.00 12.00 16.00 20.00 24.00 28.00 32.00 36.00 40.00 44.00 48.00
Beam Length (in)
Figure 22: Moment of Simply Supported Plate/Stiffener under Uniformly Distributed Load
Simply Supported Plate/Stiffener Bending Stress
50000
45000
40000
Stress (psi)
35000
30000
25000
Calculated
20000
15000
10000
5000
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
Beam Length (in)
Figure 23: Bending Stress of Simply Supported Plate/Stiffener under Uniformly Distributed Load
22
Table 8: Supporting Values, Simply Supported Plate/Stiffener
c ps = 4.53 in (see appendix A)
6
E = 30 x 10 psi
4
I ps = 70.017 in (see Appendix A)
L = 48 in
M psss = 691,200 lb·in
w ps = 2400 lb/in
3.2 FEA Results
It is noted for the graphs provided herein, the term “calculated” refers to the results
plotted via the closed form solutions provided in Section 2.1. FEA results are labeled
separately.
3.2.1
Baseline Model Results
Mesh and boundary condition convergence studies were conducted for the baseline
models described above. The final peak results of each baseline condition are presented
herein (see Table 9); the results of the convergence study iterations are provided in
Appendix B.
23
Table 9: Summary of Baseline Models vs. Closed Form Solutions
Simply Supported Panel
Clamped Panel
Simply Supported Stiffener
Clamped Stiffener
Sniped - Simple Stiffener
Sniped - Clamped Stiffener
Note:
Stress
Deflection
Calculated
FEA
% Error
Calculated
FEA
-62484
-63136
1.0%
-0.290980
-0.29394
-50934
-54698
7.4%
-0.072613
-0.07266
-4838
-4866
0.6%
-0.011059
-0.01158
3226
6251
0.8%
-0.002212
-0.00232
-4838
-1803
-150.0%
-0.011059
-0.00560
-1613
-1803
10.5%
-0.002212
-0.00560
+ % error = FEA result greater than Closed form Solution
- % error = FEA result less than Closed form Solution
% Error
1.0%
0.1%
0.2%
5.1%
-97.0%
60.0%
Clamped Panel Baseline Model Results
For the clamped panel baseline model, a finite element mesh 48 elements along the
length by 24 elements wide, and 2 elements thick was used (Figure 9b). All edge nodes
were fully constrained (all translations and rotations) to replicate the clamped condition.
A peak deflection of -0.072662in was reported, which is approximately 0.07% greater
than the closed form value (-0.072613in). A peak stress of -54698 psi was reported, as
shown in Figure 24. This is approximately 7.39% greater than the closed form value
(-50934 psi). Based on this result, a 48 x 24 x 2 element mesh is considered to be an
acceptable plate/shell mesh density for future model iterations.
Figure 24: Clamped rectangular panel FEA results
Baseline Model
24
Simply Supported Panel Baseline Model
For the simply supported panel baseline model, a finite element mesh 48 elements along
the length by 24 elements wide, and 2 elements thick was used (Figure 9b). All midplane edge nodes were fully constrained (translations and rotations) to replicate the
simply supported condition. A peak deflection of -0.29394 in was reported. This is
approximately 1.01% greater than the closed form value (-0.29098). A peak stress of
-63136 psi was reported, as shown in Figure 25. This is approximately 1.03% greater
than the closed form value (-62484 psi). Therefore, a 48 x 24 x 2 element mesh is
considered to be an acceptable plate/shell mesh density for future model iterations.
Figure 25: Simply supported rectangular plate FEA results
Baseline model
Clamped Stiffener Baseline Model Results
For the clamped stiffener baseline model, a finite element mesh consisting of 48
elements along the length, 2 elements deep and 2 elements thick was used to represent
the stiffener web (Figure 9a). A finite element mesh 48 elements long by one element
thick by two elements wide was used to represent the stiffener flange. End elements
were fully constrained (translations and rotations) to replicate the clamped condition.
Figure 26 depicts the computed stress. It is noted the overall response of the model is
taken into consideration when determining the optimal model, not just the correlation of
25
the peak FEA stress to the closed form solution. Figures 27 and 28 provide node-to-node
stress and deflection comparisons, respectively, of the clamped stiffener model to the
closed form solutions. Nodes along the top centerline of the stiffener web (in way of the
peak bending stress) are compared with the exception of the nodes at the ends, which are
influenced by the boundary conditions. A peak stress of 3250.60 psi was reported, as
shown in Figure 26. This is approximately 0.78% greater than the closed form value
(3225.60 psi). A peak deflection of -0.0023249in was reported, which is approximately
5.11% greater than the closed form value (-0.0022118in). Therefore, the stiffener
element mesh and model boundary conditions described herein are considered effective
as they provide FEA results comparable to the closed form solutions. Additional model
convergence studies are performed and discussed in more detail in Appendix B.
Node-to-node
comparison, see
Figures 27 and 28
Neutral Axis
(NA)
Figure 26: Clamped stiffener FEA results
Baseline Model
Stress Results: Clamped FEA vs Closed Form Solution
4000
Stress (psi)
3000
2000
Calculated
1000
0
0.00
-1000
stf_fix_edge_elems
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
-2000
Beam Length (in)
Figure 27: Peak bending stress in clamped stiffener
26
48.00
Deflection Results: Clamped FEA vs. Closed Form Solution
0
0.00
4.00
8.00
12.00 16.00
20.00 24.00 28.00
32.00 36.00 40.00
44.00 48.00
Deflection (in)
-0.0005
-0.001
Calculated
stf_fix_edge_elems
-0.0015
-0.002
-0.0025
Beam Length (in)
Figure 28: Peak deflection in clamped stiffener
Simply Supported Stiffener Baseline Model Results
The simply supported stiffener baseline model used the same mesh as the clamped
baseline model. Edge nodes along the stiffener’s neutral axis were fully constrained
(translations and rotations) to replicate the simply supported condition. Figure 29 depicts
the stress results of the FEM. It is noted the overall response of the model is taken into
consideration when determining the optimal model, not just the correlation of the peak
FEA stress to the closed form solution. Figures 30 and 31 provide node-to-node stress
and deflection comparisons, respectively, of the simply supported stiffener model to the
closed form solutions. Nodes along the top centerline of the stiffener web (in way of the
peak bending stress) are compared with the exception of the nodes at the ends of the
beam, which were considered to be influenced by the boundary conditions. A peak
bending stress of -4866.00 psi was reported, as shown in Figure 29. This is
approximately 0.57% greater than the closed form value (-4838.40 psi). A peak
deflection of -0.0115770 in was reported, which is approximately 0.16% greater than the
closed form value (-0.011059in). Based on the analyses described above, the stiffener
element mesh described herein is considered to provide FEA results comparable to the
closed form solution. Model convergence studies are performed and discussed in more
detail in Appendix B.
27
NA
Node-to-node comparison, see
Figures 30 and 31
Figure 29: Simply supported stiffener FEA results
Baseline model
Stress Results: Simply Supported FEA vs Closed Form Solution
1000
Stress (psi)
0
0.00
-1000
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
-2000
Calculated
-3000
stf_pin48elem
-4000
-5000
-6000
Beam Length (in)
Figure 30: Peak bending stress in simply supported stiffener
Deflection Results: Simply Supported FEA vs Closed Form Solution
Deflection (in)
0
0.00
-0.002
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
-0.004
-0.006
Calculated
-0.008
stf_pin48elem
-0.01
-0.012
-0.014
Beam Length (in)
Figure 31: Peak deflection in simply supported stiffener
28
Sniped Stiffener Baseline Model
The baseline model of the sniped stiffener was developed to understand how the unique
geometry of this stiffener correlates to that of the baseline clamped and simply supported
stiffeners. Using the same typical finite element mesh as the clamped and simply
supported baseline stiffeners, the sniped stiffener was chamfered at a 45 degree angle
from both ends. Nodes at the top edges of the stiffener were fully constrained (DOF 1-6).
Figure 32 depicts the stress results of the FEM. It is noted the overall response of the
model is taken into consideration when determining the optimal model, not just the
correlation of the peak FEA stress to the closed form solution. As shown in Figure 32,
the boundary conditions applied to this model have a significant influence on the nodes
closest to the ends of the stiffener, creating large stress concentrations that are dismissed
for the purpose of comparisons. Figures 33 and 34 provide node-to-node stress and
deflection comparisons, respectively, of the sniped stiffener model to the closed form
solutions. Nodes along the top centerline of the stiffener web (in way of the peak
bending stress) are compared. The peak compressive bending stress along the top
centerline of the stiffener was -1802.90 psi, as shown in Figures 32 and 33. This is
approximately 10.5% greater than the closed form value (-1612.80 psi) for a clamped
stiffener, and over 150% less than the closed form value (-4838.40 psi) for a simply
supported stiffener. The peak deflection along the top centerline of the stiffener was
-0.0056 in, as shown in Figure 34. This is approximately 60% larger than the closed
form value (-0.002212 in) for a clamped stiffener, and approximately 97% smaller than
the closed form value (-0.011059 in) for a simply supported stiffener.
Node-to-node comparison, see
Figures 33 and 34
-1802.90
Figure 32: Sniped stiffener FEA results
Baseline model
29
Stress Results: Sniped FEA vs. Closed Form Solutions
4000
3000
Stress (psi)
2000
1000
0
-10000.00
Calculated Simple
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
-2000
Calculated Clamped
stf_sniped
-3000
-4000
-5000
-6000
Beam Length (in)
Figure 33: Peak bending stress in sniped stiffener
Deflection Results: Sniped FEA vs. Closed Form Solutions
Deflection (in)
0
0.00
-0.002
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
-0.004
Calculated Simple
-0.006
Calculated Clamped
stf_sniped
-0.008
-0.01
-0.012
Beam Length (in)
Figure 34: Peak deflection in sniped stiffener
As shown in Figures 33 and 34, the constrained nodes at the ends cause the sniped
stiffener to act more rigidly than a simply supported stiffener, but the chamfered material
that has been removed causes it to act less rigidly than a clamped stiffener. Therefore,
the sniped stiffener is considered to act between a simply supported stiffener and a
clamped stiffener.
3.2.2
Plate / Stiffener Model Results
Table 10 summarizes the peak results of the plate/stiffener model results as compared to
the closed form solutions. The results of the boundary condition convergence study
iterations are provided in Appendix B.
30
Table 10: Summary of Plate/Stiffener FEA Models vs. Closed Form Solutions
Simply Supported Plate/Stiffener
Clamped Plate/Stiffener
Sniped - Simple Plate/Stiffener
Sniped - Clamped Plate/Stiffener
Note:
Calculated
44745
14915
44745
14915
Stress
FEA
43901
10541
36953
36953
% Error
1.9%
-30.0%
-17.0%
150.0%
Deflection
Calculated
FEA
-0.078975
-0.09329
-0.015795
-0.01867
-0.078975
-0.07818
-0.015795
-0.07818
% Error
18.0%
18.0%
-1.0%
400.0%
+ % error = FEA result greater than Closed Form Solution
- % error = FEA result less than Closed Form Solution
Clamped Plate/Stiffener Model Results
For the plate/stiffener model, the first five elements of each end (5 inches total, each
side) were fully constrained (translations and rotations) to replicate the clamped
condition (see Appendix B for discussion of selection of boundary conditions for this
model). Figure 35 depicts the stress results of the FEM. It is noted the overall response
of the model is taken into consideration when determining the optimal model, not just
the correlation of the peak FEA stress to the closed form solution. As shown in Figure
35, the boundary conditions applied to this model have a significant influence on the
nodes and elements closest to the ends of the stiffener, creating large stress
concentrations that are dismissed for the purpose of comparisons. Thus, peak tensile
bending stress regions (located at the center of the stiffener, far away from boundary
conditions) are compared. Figures 36 and 37 provide node-to-node stress and deflection
comparisons, respectively, of the clamped stiffener model to the closed form solutions.
Nodes along the inboard side of the stiffener flange (in way of the peak bending stress)
are compared. A peak deflection of -0.018667in was reported, which is approximately
18% greater than the closed form value (-0.0157950in). A peak stress of 10541 psi was
reported, as shown in Figures 35 and 36. This is approximately 30% less than the closed
form value (14915.09 psi).
31
Note: beam shown only, shell
removed for clarity
NA
Node-to-node comparison, see
Figures 36 and 37
10.5 ksi
Figure 35: Clamped plate/stiffener FEA model results
Stress Results: Clamped FEA vs. Closed Form Solution
20000.00
15000.00
10000.00
Stress (psi)
5000.00
0.00
-5000.00
p/s_fix_edge_5elem
0
4
8
12
16
20
24
28
32
36
40
44
48
Calculated
-10000.00
-15000.00
-20000.00
-25000.00
-30000.00
-35000.00
Beam Length (in)
Figure 36: Peak bending stress in clamped plate/stiffener
Deflection Results: Clamped FEA vs. Closed Form Solution
0.000
-0.002
0
4
8
12
16
20
24
28
32
36
40
44
48
-0.004
Deflection (in)
-0.006
-0.008
p/s_fix_edge_5elem
-0.010
Calculated
-0.012
-0.014
-0.016
-0.018
-0.020
Beam Length (in)
Figure 37: Peak deflection in clamped plate/stiffener
As shown in Figure 36, the model overall predicts a smaller bending stress across the
stiffener, when compared to the closed form solution. This can be attributed to the extent
of the boundary conditions, and the large stress seen at the first unconstrained node at
32
either end of the stiffener (approximately -26 ksi, which is over twice as large as the
calculated (using closed form equations) bending stress at this same extent). As shown in
Figure 37, the stiffener deflects more than the closed form solution at the center, but not
as much closer to the ends (approaching the boundary conditions). Based on the
convergence analyses described above, the stiffener element mesh described herein is
considered to be an acceptable representation of a stiffener with an effective length of
shell that is clamped. Additional studies are performed in Appendix B which further
validate this assumption.
Simply Supported Plate/Stiffener Model Results
For the simply supported plate/stiffener model, edge nodes along the stiffener’s neutral
axis were fully constrained (translations and rotations) to replicate the simply supported
condition. Figure 38 depicts the stress results of the FEM. It is noted the overall response
of the model is taken into consideration when determining the optimal model, not just
the correlation of the peak FEA stress to the closed form solution. Figures 39 and 40
provide node-to-node stress and deflection comparisons, respectively, of the clamped
stiffener model to the closed form solutions. Nodes along the inboard side of the
stiffener flange (in way of the peak bending stress) are compared, with the exception of
the nodes at the ends of the beam that are influenced by the boundary conditions. A peak
deflection of -0.093291in was reported, which is approximately 18% greater than the
closed form solution value (-0.078975in). A peak stress of 43901 psi was reported, as
shown in Figures 38 and 39. This is approximately 1.89% less than the closed form
solution value (44745.27 psi).
Note: beam shown only, shell
removed for clarity
NA
Node-to-node comparison, see
43.9 ksi
Figures 39 and 40 Figure 38: Simply supported plate/stiffener FEA model results
33
Stress Results: Simply Supported FEA vs. Closed Form Solution
50000
40000
Stress (psi)
30000
Calculated
p/s_pin_midtona
20000
10000
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
-10000
Beam Length (in)
Figure 39: Peak bending stress in simply supported plate/stiffener
Deflection Results: Clamped FEA vs. Closed Form Solution
0
-0.010.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
Deflection (in)
-0.02
-0.03
-0.04
Calculated
-0.05
p/s_pin_midtona
-0.06
-0.07
-0.08
-0.09
-0.1
Beam Length (in)
Figure 40: Peak deflection in simply supported plate/stiffener
As shown in Figure 39, the bending stress of the model is comparable to the closed form
solution (to within several percentage points) towards the center of the stiffener (away
from the boundary conditions). The model overall predicts a smaller bending stress than
the closed form solution. However, the model predicts a larger deflection across the
entire length of the stiffener, when compared to the closed form solution. Based on the
convergence analyses described above, the stiffener element mesh described herein is
considered to be an acceptable representation of a stiffener with an effective length of
shell that is simply supported. Additional studies are performed in Appendix B which
further validate this assumption.
34
Sniped Plate/Stiffener Model Results
For the sniped plate/stiffener model, edge nodes at the mid-plane of the plate were fully
constrained (translations and rotations). As shown in Figure 41, the boundary conditions
applied to this model have a significant influence on the nodes closest to the ends of the
stiffener, creating large stress concentrations that are dismissed for the purpose of
comparisons. Figures 42 and 43 provide node-to-node stress and deflection comparisons,
respectively, of the sniped stiffener model to the closed form solutions. Nodes along the
inboard face of the stiffener flange (in way of the peak bending stress) are compared.
The peak tensile bending stress along the inboard face of the stiffener was 36953 psi, as
shown in Figures 41 and 42. This is approximately 150% greater than the calculated
peak tensile value (14915.09 psi) for a clamped plate/stiffener, and approximately 17%
less than the closed form value (44745.27 psi) for a simply supported plate/stiffener. The
peak deflection along the inboard face of the stiffener flange was -0.078180 in, as shown
in Figure 43. This is approximately 400% larger than the calculated value (-0.015795 in)
for a clamped plate/stiffener, and approximately 1.00% smaller than the calculated value
(-0.078975 in) for a simply supported plate/stiffener.
Node-to-node comparison, see
Figures 42 and 43
37.0 ksi
Figure 41: Sniped plate/stiffener FEA model results
35
Stress Results: Sniped FEA vs. Closed Form Solution
50000
40000
Stress (psi)
30000
Calculated Simple
20000
Calculated Clamp
10000
p/s_sniped
0
0.00
-10000
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
-20000
-30000
-40000
Beam Length (in)
Figure 42: Stress comparison of sniped plate/stiffener FEA results to closed form solutions for
clamped/simply supported stiffeners
Deflection Results: Sniped FEA vs. Closed Form Solution
0
0.00
-0.01
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
Deflection (in)
-0.02
-0.03
Calculated Simple
-0.04
Calculated Clampe
-0.05
p/s_sniped
-0.06
-0.07
-0.08
-0.09
Beam Length (in)
Figure 43: Deflection comparison of sniped plate/stiffener FEA results to closed form solutions for
clamped/simply supported stiffeners
As shown in Figures 42 and 43, the constrained nodes at the ends cause the sniped
stiffener to be more rigid than a simply supported stiffener, but the chamfered material
that has been removed causes it to be less rigid than a clamped stiffener. Based on the
analyses described above, the sniped stiffener is considered to act somewhere between a
simply supported stiffener and a clamped stiffener.
3.2.3
Final Configurations
Table 11 summarizes the peak results of the final configurations (Figure 11) as
compared to the closed form solutions.
36
Table 11: Summary of Final Configuration Models vs. Closed Form Solutions
Stress
Deflection
Calculated
FEA
% Error
Calculated
FEA
% Error
Butted Final Configuration
14915
20573
38.0%
-0.015795
-0.05392 240.0%
Wrapped Final Configuration
14915
14154
-5.0%
-0.015795
-0.02969 88.0%
Sniped Final Configuration (simple)
44745
38939
-13.0%
-0.078975
-0.09876 25.0%
Sniped Final Configuration (clamped)
14915
38939
160.0%
-0.015795
-0.09876 525.0%
Note:
+ % error = FEA result greater than closed form solution
- % error = FEA result less than closed form solution
Figures 44 and 45 provide node-to-node stress and deflection comparisons, respectively,
of the final configuration models to the closed form solutions. Nodes along the inboard
side of the stiffener flanges (in way of the peak bending stresses) are compared.
Stress Results: Final FEA vs. Closed Form Solutions
60000
40000
Stress (psi)
20000
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
-20000
Calculated Simple
Final Butted
Calculated Clamped
Final Sniped
Final Wrapped
-40000
-60000
Beam Length (in)
Figure 44: Stress comparison of final configuration FEA results to closed form results
Deflection Results: Final FEA vs. Closed Form Solutions
Deflection (in)
0.02
0
0.00
-0.02
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
Calculated Simple
Final Butted
Final Clamped
Final Sniped
Final Wrapped
-0.04
-0.06
-0.08
-0.1
-0.12
Beam Length (in)
Figure 45: Deflection comparison of final configuration FEA results to closed form solutions
37
Butted Final Configuration Model Results
For the butted final configuration model, Figure 46 depicts the stress results of the FEM
(elevation cut looking at middle stiffener). As shown in Figure 46, the butting of the
stiffener into the side panel has a significant influence on the nodes closest to the ends of
the stiffener, creating large stress concentrations that are dismissed for the purpose of
comparisons. Thus, peak tensile bending stresses (located at the center of the stiffener,
far away from boundary conditions) are compared. Figures 44 and 45 provide node-tonode stress and deflection comparisons, respectively, of the butted final configuration
model to the closed form solutions. Nodes along the inboard side of the stiffener flange
(in way of the peak bending stress) are compared. A peak deflection of -0.053924in was
reported, which is approximately 240% greater than the calculated value (-0.0157950in).
A peak stress of 20573 psi was reported, as shown in Figure 46. This is approximately
38% more than the calculated value (14915.09 psi).
20.6 ksi
Node-to-node comparison, see
Figures 44 and 45
Figure 46: Butted final configuration FEA model results
Wrapped Final Configuration Model Results
For the wrapped final configuration model, Figure 47 depicts the stress results of the
FEM (elevation cut looking at middle stiffener). As shown in Figure 47, the wrapped
end configuration of the stiffener creates a large stress concentration in the radiused
portion of the flange, an area that is dismissed for the purpose of comparisons. Thus,
peak tensile bending stresses (located at the center of the stiffener, far away from
boundary conditions) are compared. Figures 44 and 45 provide node-to-node stress and
deflection comparisons, respectively, of the wrapped stiffener model to the closed form
38
solutions. Nodes along the inboard side of the stiffener flange (in way of the peak
bending stress) are compared. A peak deflection of -0.029692in was reported, which is
approximately 88% greater than the calculated value (-0.0157950in). A peak stress of
14154 psi was reported, as shown in Figure 47. This is approximately 5% less than the
calculated value (14915.09 psi).
14.2 ksi
Node-to-node comparison, see
Figures 44 and 45
Figure 47: Wrapped final configuration FEA model results
Sniped Final Configuration Model Results
For the sniped final configuration model, the boundary conditions applied to this model
have a significant influence on the nodes closest to the ends of the stiffener (as shown in
Figure 48, elevation cut looking at middle stiffener), creating large stress concentrations
that are dismissed for the purpose of comparisons. Figures 44 and 45 provide node-tonode stress and deflection comparisons, respectively, of the sniped stiffener model to the
closed form solutions. Nodes along the inboard face of the stiffener flange (in way of the
peak bending stress) are compared. A peak deflection of -0.098756 in was reported,
which is approximately 525% larger than the calculated value (-0.015795 in) for a
clamped plate/stiffener, and approximately 25% larger than the calculated value
(-0.078975 in) for a simply supported plate/stiffener. The peak tensile bending stress
along the inboard face of the stiffener was 38939 psi, as shown in Figure 48. This is
approximately 160% greater than the calculated peak tensile value (14915.09 psi) for a
clamped stiffener, and approximately 13% less than the calculated value (44745.27 psi)
for a simply supported plate/stiffener.
39
Node-to-node comparison, see
38.9 ksi
Figures 44 and 45
Figure 48: Sniped final configuration FEA model results
4.0
CONCLUSIONS
The results provided herein demonstrate a relationship exists between the final
model configurations analyzed and the closed form calculations for a simply supported
and/or clamped stiffener under a uniformly distributed load. Figure 44 provides a nodeto-node comparison of the peak stresses in the stiffeners of each final configuration, as
compared to the closed form solutions. As shown in this figure, the wrapped and butted
model configurations more closely resemble the clamped configuration. The sniped
configuration, on the other hand, more closely resembles the simply supported
configuration. Figure 45 provides a node-to-node comparison of the peak deflections of
each final configuration, as compared to the closed form solutions. All three
configurations predict an overall higher model deflection response than the closed form
solutions, which suggests the plate portion of the stiffener cross section is not providing
as much stiffness/support as the closed form solutions suggest it should (see Section 3.1
and Appendix A for closed form solutions).
Similar trends observed in Figure 44 can be concluded here. Specifically, the
wrapped configuration deflects the least out of the three configurations, most closely
resembling the closed form clamped solution. This can be attributed to the fact the
wrapped configuration has the smallest unsupported stiffener length between the three
models being compared. The sniped configuration deflects the most and most closely
resembles the simply supported closed form solution. It is noted the sniped configuration
40
overall deflects more across the model length than the closed form solution. This can be
attributed to the fact the sniped configuration is the smallest stiffener size and is
providing the least amount of stiffening support. Due to the sniped ends of the stiffener,
it is not intersecting with the side panels and is therefore not being supported by the side
panels, contrary to the other configurations. This further confirms the previously made
conclusion that the sniped stiffener is not as structurally strong as a typical simply
supported stiffener, and should not be designed to support the same load as that of a
simply supported stiffener. The deflection response for the butted final configuration is
also higher than initially expected, as this configuration deflects almost two and a half
times more than the closed form solution dictates.
For the clamped condition, Table 12 and Figure 49 summarize the relationship of
the model iterations to each other as well as to the closed form solution. The clamped
stiffener (1), the clamped plate/stiffener (2), the butted final configuration (3), and the
wrapped final configuration (4) iterations are compared. Each model is represented by a
single average percent error value, as shown in Table 12. The average percent error is
determined from a comparison between each node and each calculated value (using the
closed form solution) at that same distance along the beam. In other words, the average
percent error is determined from comparing the average nodal stress to the average
calculated value. Nodes from the inboard face of the flange are used (area in way of peak
bending stress).
Table 12: Summary of Percent Error between Closed Form Solutions and Clamped Model Results
% Error
Stress Deflection
Stiffener Clamped (stf_fix_edge_elem )
-6.23
5.42
Plate/Stiffener Clamped (p/s_fix_edge_5elem ) -33.93
16.26
Butted Final Configuration (final butted )
49.38
260.88
Wrapped Final Configuration (final wrapped ) -8.94
88.46
Note:
+ % error = FEA result greater than Closed form Solution
- % error = FEA result less than Closed form Solution
41
Iteration Comparison, Clamped Stiffener
300.00
250.00
% Error
200.00
150.00
Stress
Deflection
100.00
50.00
0.00
0
1
2
3
4
5
-50.00
Iteration
Figure 49: Iteration comparison of clamped stiffener (in percent error)
Figure 49 indicates a relatively close relationship with the closed form solution
compared to the clamped stiffener and clamped plate/stiffener models (iteration 2 and 3
in Figure 49, respectively). However, the percent error increases (particularly in
deflection) when comparing the butted and wrapped final configurations (iteration 3 and
4 in Figure 49, respectively). This reiterates the conclusion previously stated, which
suggests the plate portion of the stiffener cross section is not providing as much
stiffness/support as the closed form solutions assume it should (see Section 3.1 and
Appendix A for closed form solutions). Based on these comparisons, the closed form
solution for a clamped stiffener is considered to roughly compare to the results for the
butted and wrapped final configuration stiffeners. Figures 50 and 51 provide the node-tonode comparisons for the stress and deflections, respectively, that were used to
determine the average percent error shown in Figure 49 (nodes influenced by boundary
conditions were omitted).
42
Stress Results: Clam ped FEA vs. Exact Solution
Stress Results: Clamped FEA vs. Closed Form Solution
30000.00
20000.00
Stress (psi)
10000.00
0.00
-10000.00 0
4
8
12
16
20
24
28
32
36
40
44
48
-20000.00
Iter1 Clamp LE
Exact Clamp
Butted Final
Wrapped Final
-30000.00
-40000.00
-50000.00
-60000.00
Beam Length (in)
Figure 50: Stress comparison of model iterations to closed form solutions
DeflectionClamped
Results: Clam
vs. Exact
Solution
Deflection Results:
FEAped
vs.FEA
Closed
Form
Solution
0.010
Deflection (in)
0.000
-0.010
0
4
8
12
16
20
24
28
32
36
40
44
48
Iter1 Clamp LE
-0.020
Exact Clamp
-0.030
Butted Final
Wrapped Final
-0.040
-0.050
-0.060
Beam Length (in)
Figure 51: Deflection comparison of model iterations to closed form solutions
For the simply supported condition, Figure 52 summarizes the relationship of the
model iterations to each other as well as to the closed form solution. The simply
supported stiffener (1), the sniped stiffener (2), the simply supported plate/stiffener (3),
the sniped plate/stiffener (4), and the sniped final configuration (5) iterations are
compared. Each model is represented by a single average percent error value,
determined using the same method described previously for the clamped iteration.
43
Iteration Comparison, Simply Supported Beam
40
20
0
% Error
0
1
2
3
4
5
6
-20
Stress
Deflection
-40
-60
-80
-100
Iteration
Figure 52: Iteration comparison of simple supported stiffener (in percent error)
Figure 52 indicates a close relationship with the closed form solution compared to the
simply supported model (iteration 1) and the sniped final configuration model (iteration
5). Based on these comparisons, the closed form solution for a simply supported stiffener
is considered to roughly compare to the results for the sniped final configuration
stiffener. Figures 53 and 54 provide the node-to-node comparisons for the stress and
deflections, respectively, that were used to determine the average percent error shown in
Figure 52 (nodes influenced by boundary conditions were omitted).
Stress
Results:
Sim ply Supported
FEA
vs. Exact
Solution
Stress Results:
Simply
Supported
FEA vs.
Closed
Form
Solution
50000
40000
Exact Simple
Iter3 Simple LE
Sniped LE
Sniped Final
Stress (psi)
30000
20000
10000
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
-10000
Beam Length (in)
Figure 53: Stress comparison of model iterations to closed form solutions
44
Deflection Results: Simply Supported FEA vs. Exact Solution
Deflection Results: Simply Supported FEA vs. Closed Form Solution
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
Deflection (in)
-0.02
-0.04
Exact Simple
Iter3 Simple LE
-0.06
Sniped LE
Sniped Final
-0.08
-0.1
-0.12
Beam Length (in)
Figure 54: Deflection comparison of model iterations to closed form solutions
While the FEA results confirm the closed form solutions provide results that are
comparable to the final configurations, it is imperative that the final configurations are
evaluated via an finite element analysis. The simplistic two dimensional analysis of the
closed form solution does not account for the unique geometry and/or loading conditions
that are included in the finite element analyses. For instance, the initial assumption that
the butted final configuration could be evaluated initially as a clamped stiffener was
confirmed by the analyses herein. However, the FEA shows a large stress concentration
exists in way of the intersection between the stiffener flange and the wall, as shown in
Figure 55 below. This stress concentration can be attributed to the perpendicular
intersection between the stiffener and the wall. The addition of a radius will mitigate this
stress. This peak stress is a stress that is not accounted for in the closed form solutions
provided in Section 2.
45
Shell
Butted
Frame
Stress
concentration
Wall
Figure 55: Butted final configuration FEA model stress concentration
As another example, the initial assumption that the wrapped final configuration
could be considered a clamped stiffener was confirmed by the analyses herein. However,
the FEA shows a large stress concentration exists in way of the wrapped portion of the
stiffener, as shown in Figure 56 below.
46
Shell
Wall
Wrapped
Frame
Stress
concentration
Figure 56: Wrapped final configuration FEA model stress concentration
If this stress concentration were to exceed the stress requirements of a design, an
increase in the radius would be necessary. Similarly, this is not taken into consideration
with the two dimensional analysis.
Another instance is observed with the sniped configurations. While the initial
assumption that the sniped configuration could be compared to a simply supported
condition was confirmed by the analyses performed herein, the FEA shows large stress
concentrations exist in way of the sniped stiffener-to-shell intersections, as shown in
Figure 57 below.
47
Shell
Sniped
Frame
Stress
concentration
Wall
Figure 57: Sniped final configuration FEA model stress concentration
These stress concentrations indicate the sniped stiffener configuration should not be used
to support an area this length; rather, it should be used as a secondary stiffener to support
a smaller region of plate.
A source of error in this analysis could be attributed to the walls used in the final
configurations. The size of the wall chosen was arbitrary. It was assumed that the overly
thick wall that was relatively short in depth (twice as deep as frame) would be stout
enough to resemble a rigid condition. While this wall was considered to sufficiently
depict a rigid connection, it was not 100 percent rigid as the boundary conditions used
on the prior iterations were. The lack of 100 percent rigidity in the wall could account
for a slight (minimal) effect in the stiffener stress and deflection results.
48
5.0
REFERENCES
[1]
Non-Circular Pressure Vessels –Some Guidance Notes for Designers, M.
Starczewski, British Engine Technical Report 1981 Volume XIV, page 62.
[2]
Stresses in Plates and Shells, A. Ugural, second edition, McGraw-Hill , 1999.
[3]
Roark’s Formulas for Stress and Strain, W. Young, R. Budynas, seventh edition,
McGraw-Hill, 2002.
[4]
Beam Design Formulas with Shear and Moment Diagrams, The American Wood
Council, Design Aid No. 6, 1997.
[5]
Mechanics of Materials, F. Beer, R. Johnston, second edition, McGraw-Hill,
1992.
49
APPENDIX A:
SUPPLEMENTAL CALCULATIONS
A1
A1.0 PURPOSE
The purpose of this Appendix is to provide the results of supplemental calculations that
support the results provided in the main body of this paper.
A2
A2.0 ANALYSIS RESULTS
A2.1 Moment of Inertia for Stiffener Only
In order to determine the area moment of inertia of the stiffener cross section with
respect to its neutral axis (x’), the stiffener is broken down into rectangles, as shown in
Figure A.1.
x
1
0.50”
y1
= 2.00”
c
= 3.50”
5.00”
y2
=4.50”
c
x’
2
1.00”
3.00”
Figure A.1: Stiffener cross section
Cross sectional properties are summarized in Table A.1
Table A.1: Stiffener Cross Sectional Properties
1
2
Total
Area (in2)
(0.50)(4.00) = 2.00
(1.00)(3.00) = 3.00
ΣA = 5.00
y (in)
2.00
4.50
yA (in3)
(2.00)(2.00) = 4.00
(4.50)(3.00) = 13.50
ΣyA = 17.50
The distance from the x axis to the neutral axis is determined by the following equation:
Y A   y A
(A-1)
Y  3.5in
The moment of inertia of the cross section can then be determined using the Parallel
Axis Theorem [5]. This theorem calculates the moment of inertia of each individual
rectangle and then adds them together to determine the total moment of inertia of the
A3
cross sectional area with respect to its neutral axis. The total moment of inertia is
calculated as shown below:
1
I x '  ( I  Ad 2 )  ( bh 3  Ad 2 )
12
I x' 
(A-2)
1
1
(0.5)( 43 )  (2)(1.52 )  (3)(13 )  (3)(12 )
12
12
I  10.4167in 4
Based on these calculations, c, the distance from cross section neutral axis to the location
of the maximum compression/tension stress, is determined as shown in Figure A.2
Load
Max Compression
surface
ccompression
=-3.50”
NA
ctension
=+1.50”
Max Tension
surface
Figure A.2: Location of peak compression/tension surfaces
A2.2 Moment of Inertia for Plate/Stiffener
In order to determine the moment of inertia of the stiffener cross section with respect to
its neutral axis (x’), the stiffener is broken down into rectangles, as shown in Figure A.3.
A4
y1
24.00”
= 0.375”
x
0.75”
1
c
x’
c
= 1.217”
y2
= 2.75”
y3
=5.25”
5.00”
0.50”
2
3
1.00”
3.00”
Figure A.3: Plate/stiffener cross section
Cross sectional properties are summarized in Table B.2
Table A.2: Plate/Stiffener Cross Sectional Properties
1
2
3
Total
Area (in2)
(24.00)(0.75) = 18.00
(0.50)(4.00) = 2.00
(1.00)(3.00) = 3.00
ΣA = 23.00
y (in)
0.375
2.75
5.25
yA (in3)
(0.375)(18.00) = 6.75
(2.75)(2.00) = 5.50
(5.25)(3.00) = 15.75
ΣyA = 28.00
The distance from the x axis to the neutral axis is determined by recalling A-1:
Y A   y A
(A-1)
Y  1.217in
The moment of inertia of the cross section can then be determined using the Parallel
Axis Theorem [5]. Recalling equation (A-2):
1
I x '  ( I  Ad 2 )  ( bh 3  Ad 2 )
12
(A-2)
1
1
1
(24)(0.753 )  (18)(0.84 2 ) (0.5)( 43 )  (2)(1.532 )  (3)(13 )  (3)( 4.032 )
12
12
12
4
I  70.017in
I x' 
A5
Based on these calculations, c, the distance from cross section neutral axis to the location
of the maximum compression/tension stress, is determined as shown in Figure A.4.
Load
Max Compression
surface
ccompression
c
NA
=-1.217”
ctension
=+4.533”
Max Tension
surface
Figure A.4: Location of peak compression/tension surfaces
A6
APPENDIX B
CONVERGENCE RESULTS
B1
B1.0 PURPOSE
The purpose of this Appendix is to provide the results of the finite element analysis
convergence studies performed by this analysis. The finite element results presented
herein supported the conclusions made in the main body of this paper.
B2.0 Analysis Results
Convergence studies were performed for the following modeling iterations:

Simply supported rectangular panel under uniformly distributed load

Clamped rectangular panel under uniformly distributed load

Simply supported stiffener (T-frame shape) under uniformly distributed load

Clamped stiffener (T-frame shape) under uniformly distributed load

Simply supported plate/stiffener under uniformly distributed load

Clamped plate/stiffener under uniformly distributed load
B2.1 Simply Supported Rectangular Panel under Uniformly Distributed Load
The area of the rectangular panel evaluated was the equivalent of the unsupported
portion between frames, or 48” x 24”, 0.75” thick. All meshes were two elements thick
and four elements wide. Convergence studies evaluated the number of elements
lengthwise across the plate, beginning with mesh sizes of 6 elements (plt_pin8elem, 8”
long), 12 elements (plt_pin12elem, 4” long), 24 elements (plt_pin24elem, 2” long), and
48 elements (plt_pin48elem, 1” long). For a simply supported panel, translations are
constrained however rotations are unconstrained. In order to allow the edges of the panel
to rotate freely while restricting any form of translation, the mid-plane nodes along the
perimeter of the panel were constrained. This prevents the edges of the panel from
deflecting, yet allows the panel to rotate as necessary. This scenario was applied to all
simply supported rectangular panel convergence studies discussed in this section. Table
B.1 summarizes the peak stress and deflection results of the model iterations and
compares these results to the closed form solutions. Figure B.1 compares the percent
error of the peak stress and deflection of each model to the closed form solution. It is
noted a positive percent error means the model predicted a higher result than the closed
B2
form solution, and conversely, a negative percent error means the model predicted a
lower result than the closed form solution.
Table B.1: Summary of Peak Stress and Deflection for Simply Supported Panel
Calculated
plt_pin8elem
plt_pin12elem
plt_pin24elem
plt_pin48elem
Stress % Error Deflection % Error
62484
0.29098
61908 -0.92%
0.27668
-4.91%
62127 -0.57%
0.28920
-0.61%
63002 0.83%
0.29282
0.63%
63136 1.04%
0.29394
1.02%
Simply Supported Panel Results
Closed Form Solution vs FEA
6.00%
% Difference
4.00%
2.00%
Stress
0.00%
0
1
2
3
4
5
Deflection
-2.00%
-4.00%
-6.00%
Iteration
Figure B.1: Summary of Percent Error Between Closed Form Solutions and Model Results for a
Simply Supported Panel
The results presented herein for a simply supported panel demonstrate that all four
model iterations are within approximately five percent of the closed form solutions.
Based upon the results presented herein, the simply supported panel with 48 elements
along its length (plt_pin48elem) is best considered to represent the closed form solution.
The primary reason for selecting the 48 element model iteration (in lieu of the 24
element model iteration) is the refined mesh will be a larger factor in the more
complicated iterations.
B2.2 Clamped Rectangular Panel under Uniformly Distributed Load
Model iterations for a clamped rectangular panel focused on the mesh refinement
iterations, similar to those for a simply supported plate. The same plate size, as well as
mesh size, was evaluated. As discussed in the main body of this paper, a panel that is
clamped has all translations and rotations constrained around the perimeter. Two
B3
potential boundary condition scenarios were considered: fully constraining the perimeter
nodes of the panel, or fully constraining the perimeter elements of the panel. For the first
scenario, the perimeter nodes of the plate were fully constrained and the same mesh sizes
used in section B2.1 were evaluated (8 elements to 48 elements). Results provided in
Table B.2 and Figure B.2 show as the number of elements increase, the closer the model
responds to the derived calculations. These results yielded the same ideal mesh as
chosen in Section B2.1. The second boundary condition scenario was evaluated for the
48 element mesh only, as a check to see how constraining the edge elements restricts the
model response (plt_clmp48elem_bc). As shown in Figure B.2 and Table B.2,
constraining the edge elements over constrains the model, causing it to be too stiff,
which results in smaller deflections and lower stresses. The second scenario was
dismissed and the first scenario was chosen, with the same ideal mesh being chosen as
that of Section B2.1.
Table B.2: Summary of Peak Stress and Deflection for Clamped Plate
Calculated
plt_clmp8elem
plt_clmp12elem
plt_clmp24elem
plt_clmp48elem
plt_clmp48elem_bc
Stress
50934
22865
40833
52103
54698
37232
% Error
-55.11%
-19.83%
2.30%
7.39%
-26.90%
Deflection
0.0726139
0.0438980
0.0664140
0.0714930
0.0726620
0.0518900
% Error
-39.55%
-8.54%
-1.54%
0.07%
-28.54%
Clamped Panel Results
Closed Form Solution vs FEA
20.00%
10.00%
% Difference
0.00%
-10.00%
0
1
2
3
-20.00%
4
5
6
Stress
Deflection
-30.00%
-40.00%
-50.00%
-60.00%
Iteration
Figure B.2: Summary of Percent Error between Closed Form Solutions and Model Results for a
Clamped Plate
B4
B2.3 Simply Supported Stiffener under Uniformly Distributed Load
Similar to the approach discussed in sections B2.1 and B2.2, a mesh convergence study
was necessary to determine the optimal mesh density for a simply supported stiffener
under a uniformly distributed load. Mesh studies were conducted for stiffeners
composed of 6 elements, 12 elements, 24 elements, and 48 elements along the length.
Boundary conditions for this model were trivial, as the stiffener was fully constrained at
the neutral axis at both ends. This constraining method is consistent with the
assumptions of the closed form solution, which also constrains the stiffener along the
neutral axis.
Recalling from the main body, the calculated peak bending stress along the stiffener is
-4.84 ksi. This peak compressive stress is located along the top of the stiffener at the
center of the stiffener (see main body and Appendix A for calculations). Table B.3
compares the results of the closed form solution for stress to the results determined by
the model iterations. Figures B.3 through B.5 provide a node-to-node comparison of the
bending stress along the top centerline of the web between the closed form solution and
the model iterations.
Table B.3: Summary of Peak Compressive Bending Stress in Simply Supported Stiffener
Calculated
stf_pin6elem
stf_pin12elem
stf_pin24elem
stf_pin48elem
Stress
-4838.40
-4933.60
-4884.80
-4870.30
-4866.00
B5
% Error
1.97%
0.96%
0.66%
0.57%
Stress
1000
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
Stress (psi)
-1000
Calculated
-2000
stf_pin6elem
stf_pin12elem
stf_pin24elem
-3000
stf_pin48elem
-4000
-5000
-6000
Beam Length (in)
Figure B.3: Summary of Peak Compressive Bending Stress in Simply Supported Stiffener
Stress
0
0.00
1.00
2.00
3.00
4.00
5.00
6.00
Stress (psi)
-400
Calculated
-800
stf_pin6elem
stf_pin12elem
stf_pin24elem
-1200
stf_pin48elem
-1600
-2000
Beam Length (in)
Figure B.4: Summary of Peak Compressive Bending Stress in Simply Supported Stiffener
(close-up in way of Boundary Conditions)
B6
Stress
-4500
20.00
21.00
22.00
23.00
24.00
25.00
26.00
27.00
28.00
Stress (psi)
-4600
Calculated
-4700
stf_pin6elem
stf_pin12elem
stf_pin24elem
-4800
stf_pin48elem
-4900
-5000
Beam Length (in)
Figure B.5: Summary of Peak Compressive Bending Stress in Simply Supported Stiffener
(close-up in way of center of stiffener)
The calculated peak deflection along the stiffener is -0.01106 in and is located at the
center of the stiffener (see main body and Appendix A for calculations). Table B.4
compares the results of the closed form solution for deflection to the results determined
by the model iterations. Figures B.6 and B.7 provide a node-to-node comparison of the
deflection of the stiffener.
Table B.4: Summary of Peak Deflection in Simply Supported Stiffener
Calculated
stf_pin6elem
stf_pin12elem
stf_pin24elem
stf_pin48elem
Deflection
-0.0110592
-0.0114890
-0.0115340
-0.0115580
-0.0115770
B7
% Error
3.89%
0.39%
0.21%
0.16%
Deflection
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
-0.002
Deflection (in)
-0.004
Calculated
-0.006
stf_pin6elem
stf_pin12elem
-0.008
stf_pin24elem
stf_pin48elem
-0.01
-0.012
-0.014
Beam Length (in)
Figure B.6: Summary of Peak Deflection in Simply Supported Stiffener
Deflection
-0.01
20.00
-0.0102
21.00
22.00
23.00
24.00
25.00
26.00
27.00
28.00
Deflection (in)
-0.0104
-0.0106
Calculated
-0.0108
stf_pin6elem
-0.011
stf_pin12elem
stf_pin24elem
-0.0112
stf_pin48elem
-0.0114
-0.0116
-0.0118
-0.012
Beam Length (in)
Figure B.7: Summary of Peak Deflection in Simply Supported Stiffener
(close-up in way of center of stiffener)
Table B.5 and Figure B.8 summarize the percent error of the model iterations relative to
the closed form solutions for stress and deflection. The percent error of each model was
determined by comparing the average compressive stress or deflection from the model
iterations to the average compressive stress or deflection from the calculated results.
Nodes along the stiffener length in way of the peak bending stress and deflection were
used. This comparison provides a rough estimate of how the model behaves relative to
the closed form solution. Nodes influenced by boundary conditions were not used in this
comparison. In all iterations, the model predicted a higher average stress and deflection
B8
than the closed form calculations. It is noted a positive percent error means the model
predicted higher results than the closed form solution, and conversely, a negative percent
error means the model predicted lower results than the closed form solution.
Table B.5: Summary of Percent Error between Closed Form Solutions and Model Results for a
Simply Supported Stiffener
% Error
Stress Deflection
stf_pin6elem
0.46
4.17
stf_pin12elem
0.61
4.63
stf_pin24elem
0.62
4.91
stf_pin48elem
0.52
5.16
Simply Supported Beam %Error
6.00
5.00
% Error
4.00
Stress
3.00
Deflection
2.00
1.00
0.00
0
1
2
3
4
5
Iteration
Figure B.8: Summary of Percent Error between Closed Form Solutions and Model Results for a
Simply Supported Stiffener
The results presented herein for a simply supported stiffener demonstrate that all four
model iterations are within approximately five percent of the closed form solutions.
Based upon the results presented herein, the simply supported stiffener with 48 elements
along its length is best considered to represent the closed form solution. The primary
reason for selecting the 48 element model iteration is the refined mesh will be a larger
factor in the more complicated iterations.
B2.4 Clamped Stiffener under Uniformly Distributed Load
For a clamped stiffener under a uniformly distributed load, the optimal mesh determined
via Section B2.3 was used for all model iterations. Unlike the simply supported stiffener,
boundary conditions had a much larger influence on the overall response of the models,
and were the primary focus of the iterations shown herein. The following model
B9
iterations were performed to determine the optimal boundary condition for a clamped
stiffener under a uniformly distributed load:
a) iteration 1 (stf_fix_edge_nodes): edge nodes on both ends of model were completely constrained
b) iteration 2 (stf_fix_edge_elems): edge elements on both ends of model were completely constrained
c) iteration 3 (stf_fix_xtra_elems): the stiffener model was extended on both ends by one element, which
was completely constrained
d) iteration 4 (stf_fix_edge_constraint): edge nodes on one end of the model were completely
constrained, while edge nodes on opposite end were constrained in yz plane but allowed to move in x
direction.
Figure B.9 depicts the model boundary conditions of each iteration.
L
L
L
a)
L
Y
b)
c)
d)
X
Z
Figure B.9: Clamped Stiffener Model Boundary Conditions
a) iteration 1 b) iteration 2 c) iteration 3 d) iteration 4
Recalling from the main body, the calculated peak bending stress along the stiffener is
3.23 ksi. The peak tensile stress is located along the top of the stiffener at the edges (see
main body and Appendix A for calculations). Figures B.10 and B.11 provide a node-tonode comparison of the bending stress along the top centerline of the web. It is noted
peak bending stresses in the iterations did not provide comparable results, since the peak
stresses are located at the edges, which are highly influenced by the respective boundary
conditions. The basis of determining the optimal iteration was based off how well the
model converged towards the closed form solution away from the boundary conditions.
B10
Stress
5000
4000
Stress (psi)
3000
Calculated
2000
stf_fix_edge_nodes
stf_fix_edge_elems
stf_fix_xtra_elems
1000
stf_fix_edge_constraint
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
-1000
-2000
Beam Length (in)
Figure B.10: Summary of Peak Bending Stress in Clamped Stiffener
Stress
-1300
20.00
21.00
22.00
23.00
24.00
25.00
26.00
27.00
28.00
-1350
Stress (psi)
-1400
-1450
Calculated
-1500
stf_fix_edge_nodes
stf_fix_edge_elems
stf_fix_xtra_elems
-1550
stf_fix_edge_constraint
-1600
-1650
-1700
Beam Length (in)
Figure B.11: Summary of Peak Bending Stress in Clamped Stiffener
(close-up in way of center of stiffener)
The calculated peak deflection along the stiffener is -0.00221 in and is located at the
center of the stiffener (see main body and Appendix A for calculations). Table B.6
compares the results of the closed form solution for deflection to the results determined
by the model iterations. Figures B.12 and B.13 provide a node-to-node comparison of
the deflection of the stiffener.
B11
Table B.6: Summary of Peak Deflection in Clamped Stiffener
Deflection
-0.002212
-0.002711
-0.002325
-0.002711
-0.002718
Calculated
stf_fix_edge_nodes
stf_fix_edge_elems
stf_fix_xtra_elems
stf_fix_edge_constraint
% Error
22.6
5.1
22.6
22.9
Deflection
0
0.00
4.00
8.00
12.00 16.00 20.00 24.00 28.00 32.00 36.00 40.00 44.00 48.00
Deflection (in)
-0.0005
-0.001
Calculated
stf_fix_edge_nodes
stf_fix_edge_elems
-0.0015
stf_fix_xtra_elems
stf_fix_edge_constraint
-0.002
-0.0025
-0.003
Beam Length (in)
Figure B.12: Summary of Peak Deflection in Clamped Stiffener
Deflection
-0.002
20.00
-0.0021
21.00
22.00
23.00
24.00
25.00
26.00
27.00
28.00
Deflection (in)
-0.0022
-0.0023
Calculated
-0.0024
stf_fix_edge_nodes
stf_fix_edge_elems
-0.0025
stf_fix_xtra_elems
stf_fix_edge_constraint
-0.0026
-0.0027
-0.0028
Beam Length (in)
Figure B.13: Summary of Peak Deflection in Clamped Stiffener
(close-up in way of center of stiffener)
Table B.7 and Figure B.14 summarize the percent error of the model iterations relative
to the closed form solutions for stress and deflection. The percent error of each model
was determined by comparing the average compressive stress or deflection from the
B12
model iterations to the average compressive stress or deflection from the calculated
results. Nodes along the stiffener length in way of the peak bending stress and deflection
were used. This comparison provides a rough estimate of how the model behaves
relative to the closed form solution. Nodes influenced by boundary conditions were not
used in this comparison. It is noted a positive percent error means the model predicted
higher results than the closed form solution, and conversely, a negative percent error
means the model predicted lower results than the closed form solution.
Table B.7: Summary of Percent Error between Closed Form Solutions and Model Results for a
Clamped Stiffener
% Error
Stress Deflection
stf_fix_edge_nodes
3.9
24.5
stf_fix_edge_elems
-6.2
5.4
stf_fix_xtra_elems
2.9
23.2
stf_fix_edge_constraint
3.1
23.5
Clamped Beam % Error
30
25
20
% Error
15
Stress
10
Deflection
5
0
0
1
2
3
4
5
-5
-10
Iteration
Figure B.14: Summary of Percent Error between Closed Form Solutions and Model Results for a
Clamped Stiffener
The results presented herein for a clamped stiffener demonstrate that all four model
iterations are within approximately five percent of the closed form solutions for bending
stress and within approximately twenty five percent of the closed form solutions for
deflection. Based upon the results presented herein, model iteration 2 is best considered
to represent the closed form solution. While this model actually represents a smaller
unsupported stiffener length (since elements on either side of stiffener are fully
B13
constrained) the stress and deflection percent error is balanced more so than the other
model iterations. That is, the other three models predict a stress within five percent of the
actual stress, but the deflection is as high as twenty five percent off. Additionally, the
smaller unsupported length will better compare to the wrapped final configuration.
B2.4 Simply Supported Plate/Stiffener under Uniformly Distributed Load
For a simply supported stiffener with an effective length of shell under a uniformly
distributed load, the optimal mesh determined via the previous sections was used for all
model iterations. Boundary conditions had a large influence on the overall response of
the models, and were the primary focus of the iterations shown herein. The following
boundary condition model iterations were performed to determine the optimal model for
a simply supported plate/stiffener under a uniformly distributed load:
a) iteration 1 (p/s_pin_edge_mid): shell edge midnodes completely constrained
b) iteration 2 (p/s_pin_na): neutral axis of plate/stiffener completely constrained
c) iteration 3 (p/s_pin_midna): the shell edge midnodes and the neutral axis of the plate/stiffener were
completely constrained
d) iteration 4 (p/s_pin_midtona): edge nodes from the midnodes of the shell to the neutral axis of the
plate/stiffener were completely constrained
e) iteration 5 (p/s_pin_edge_inbd): inboard side of shell nodes completely constrained
f) iteration 6 (p/s_pin_rotations): neutral axis of plate/stiffener completely constrained, all other edge
nodes constrained in y, z directions and allowed to move in x direction
All model iterations were constrained along the sides of the effective length in the shell
in the xy plane. This method replicated the fact that the edge nodes of the shell are the
midbays of the frame, a point of inflection, where there is no rotation. Figure B.15
depicts the boundary conditions of the model iterations.
B14
L
L
b)
a)
L
c)
d)
L
L
e)
f)
Figure B.15: Simply Supported Plate/Stiffener Model Boundary Conditions for:
a) iteration 1 b) iteration 2 c) iteration 3 d) iteration 4 e) iteration 5 f) iteration 6
Recalling from the main body, the calculated peak bending stress along the stiffener is
44.75 ksi. This peak tensile stress is located along the inboard face of the stiffener flange
at the midpoint (see main body and Appendix A for calculations). Table B.8 compares
the results of the closed form solution for stress to the results determined by the model
iterations. Figures B.16 through B.18 provide a node-to-node comparison of the bending
stress along the inboard side of the flange
B15
Table B.8: Summary of Peak Tensile Bending Stress in Simply Supported Plate/Stiffener
Stress
Calculated
44745.27
p/s_pin_edge_mid 45459.00
p/s_pin_na
46794.00
p/s_pin_midna
44072.00
p/s_pin_midtona 43901.00
p/s_pin_edge_inbd 46268.00
p/s_pin_rotations 46871.00
% Error
1.6%
4.6%
-1.5%
-1.9%
3.4%
4.8%
Stress
50000
Calculated
p/s_pin_edge_mid
40000
p/s_pin_na
p/s_pin_midna
p/s_pin_midtona
Stress (psi)
30000
p/s_pin_edge_inbd
ps_pin_rotations
20000
10000
0
0.00
4.00
8.00
12.00 16.00 20.00 24.00 28.00 32.00 36.00 40.00 44.00 48.00
-10000
Beam Length (in)
Figure B.16: Summary of Peak Tensile Bending Stress in Simply Supported Plate/Stiffener
Stress
30000
25000
Calculated
20000
p/s_pin_edge_mid
Stress (psi)
p/s_pin_na
p/s_pin_midna
15000
p/s_pin_midtona
p/s_pin_edge_inbd
10000
p/s_pin_rotations
5000
0
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
-5000
Beam Length (in)
Figure B.17: Summary of Peak Tensile Bending Stress in Simply Supported Plate/Stiffener
(close-up in way of Boundary Conditions)
B16
Stress
48000
47000
Calculated
46000
p/s_pin_edge_mid
Stress (psi)
p/s_pin_na
p/s_pin_midna
45000
p/s_pin_midtona
p/s_pin_edge_inbd
44000
p/s_pin_rotations
43000
42000
41000
40000
20.00
21.00
22.00
23.00
24.00
25.00
26.00
27.00
28.00
Beam Length (in)
Figure B.18: Summary of Peak Tensile Bending Stress in Simply Supported Plate/Stiffener
(close-up in way of center of stiffener)
The calculated peak deflection along the stiffener is -0.07897 in and is located at the
center of the stiffener (see main body and Appendix A for calculations). Table B.9
compares the results of the closed form solution for deflection to the results determined
by the model iterations. Figures B.19 and B.20 provide a node-to-node comparison of
the deflection of the stiffener.
Table B.9: Summary of Peak Deflection in Simply Supported Plate/Stiffener
Calculated
p/s_pin_edge_mid
p/s_pin_na
p/s_pin_midna
p/s_pin_midtona
p/s_pin_edge_inbd
p/s_pin_rotations
Deflection % Error
-0.0789749
-0.0989480 25.3%
-0.1117800 41.5%
-0.0938030 18.8%
-0.0932910 18.1%
-0.1087500 37.7%
-0.1051200 33.1%
B17
Deflection
0
0.00
4.00
8.00 12.00 16.00 20.00 24.00 28.00 32.00 36.00 40.00 44.00 48.00
Deflection (in)
-0.02
Calculated
-0.04
p/s_pin_edge_mid
p/s_pin_na
-0.06
p/s_pin_midna
p/s_pin_midtona
-0.08
p/s_pin_edge_inbd
p/s_pin_rotations
-0.1
-0.12
Beam Length (in)
Figure B.19: Summary of Peak Deflection in Simply Supported Plate/Stiffener
Deflection
-0.06
20.00
21.00
22.00
23.00
24.00
25.00
26.00
27.00
28.00
-0.07
Deflection (in)
Calculated
p/s_pin_edge_mid
-0.08
p/s_pin_na
p/s_pin_midna
-0.09
p/s_pin_midtona
p/s_pin_edge_inbd
p/s_pin_rotations
-0.1
-0.11
-0.12
Beam Length (in)
Figure B.20: Summary of Peak Deflection in Simply Supported Plate/Stiffener
(close-up in way of center of stiffener)
Table B.10 and Figure B.21 summarize the percent error of the model iterations relative
to the closed form solutions for stress and deflection. The percent error of each model
was determined by comparing the average compressive stress or deflection from the
model iterations to the average compressive stress or deflection from the calculated
results. Nodes along the stiffener length in way of the peak bending stress and deflection
were used. This comparison provides a rough estimate of how the model behaves
relative to the closed form solution. Nodes influenced by boundary conditions were not
used in this comparison. It is noted a positive percent error means the model predicted
B18
higher results than the closed form solution, and conversely, a negative percent error
means the model predicted lower results than the closed form solution.
Table B.10: Summary of Percent Error between Closed Form Solutions and Model Results for a
Simply Supported Plate/Stiffener
% Error
Stress Deflection
p/s_pin_edge_mid
0.65
27.37
p/s_pin_na
5.84
45.32
p/s_pin_midna
-2.69
19.57
p/s_pin_midtona
-3.14
18.84
p/s_pin_edge_inbd
2.70
40.13
p/s_pin_rotations
5.25
33.97
Simply Supported Beam/LE % Error
50.00
40.00
% Error
30.00
Stress
20.00
Series2
10.00
0.00
0
1
2
3
4
5
6
7
-10.00
Iteration
Figure B.21: Summary of Percent Error between Closed Form Solutions and Model Results for a
Simply Supported Plate/Stiffener
The results presented herein for a simply supported plate/stiffener demonstrate that all
six model iterations are within approximately five percent of the closed form solutions
for bending stress. There is a larger percent error for the deflection results, where
predicted model deflections range from twenty to forty five percent higher. Model
iteration 4 is considered to best represent the closed form solution. While model iteration
3 presents similar results, the actual boundary conditions of iteration 3 are not
considered an accurate representation for a simply supported stiffener. That is, the ends
of the stiffener are constrained in two completely separate areas, creating an inaccurate
model response in the area in between these constraints. On the other hand, model
iteration 4 is constrained in a larger, but more condensed area.
B19
B2.4 Clamped Plate/Stiffener under Uniformly Distributed Load
For a clamped plate/stiffener under a uniformly distributed load, the optimal mesh
determined from the previous studies was used for all model iterations. Boundary
conditions had a large influence on the overall response of the models, and were the
primary focus of the iterations shown herein. The following boundary condition model
iterations were performed to determine the optimal model for a clamped plate/stiffener
under a uniformly distributed load:
a) iteration 1 (p/s_fix_edge_nodes): edge nodes of both ends completely constrained
b) iteration 2 (p/s_fix_edge_1elem): outside elements on both ends (one inch long each) fully
constrained
c) iteration 3 (p/s_fix_constraint): neutral axis of plate/stiffener completely constrained, all other edge
nodes constrained in yz plane and allowed to move in x direction
d) iteration 4 (p/s_fix_edge_5elem): outside five elements on both ends (one inch long each) fully
constrained
All model iterations were constrained along the sides of the effective length in the shell
in the x and the z directions. This method replicated the fact that the edge nodes of the
shell are the midbays of the frame, a point of inflection, where there is no rotation.
Figure B.22 depicts the model boundary conditions for these iterations.
L
L
L
b)
a)
L
L
c)
d)
Figure B.22: Clamped Plate/Stiffener Model Boundary Conditions
a) iteration 1 b) iteration 2 c) iteration 3 d) iteration 4
B20
Recalling from the main body, the calculated peak bending stress along the stiffener is
-29.83 ksi. This peak compressive stress is located along the inboard face of the flange
of the stiffener at the edges (see main body and Appendix A for calculations). Figures
B.23 through B.25 provide a node-to-node comparison of the bending stress along the
top centerline of the web. It is noted peak bending stresses in the iterations did not
provide comparable results, since the peak stresses are located at the edges, which are
highly influenced by the respective boundary conditions. The basis of determining the
optimal iteration was based off how well the model converged towards the closed form
solution away from the boundary conditions.
Stress
40000
Stress (psi)
20000
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
-20000
32.00
36.00
40.00
44.00
48.00
Calculated
p/s_fix_edge_nodes
p/s_fix_edge_1elem
-40000
p/s_fix_constraint
p/s_fix_edge_5elem
-60000
-80000
-100000
Beam Length (in)
Figure B.23: Summary of Peak Compressive Bending Stress in Clamped Plate/Stiffener
B21
Stress
40000
20000
Stress (psi)
0
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
Calculated
-20000
p/s_fix_edge_nodes
p/s_fix_edge_1elem
p/s_fix_constraint
-40000
p/s_fix_edge_5elem
-60000
-80000
-100000
Beam Length (in)
Figure B.24: Summary of Peak Bending Stress in Clamped Plate/Stiffener
(close-up in way of Boundary Conditions)
Stress
29000
Stress (psi)
25000
Calculated
21000
p/s_fix_edge_nodes
p/s_fix_edge_1elem
p/s_fix_constraint
17000
p/s_fix_edge_5elem
13000
9000
20.00
21.00
22.00
23.00
24.00
25.00
26.00
27.00
28.00
Beam Length (in)
Figure B.25: Summary of Peak Compressive Bending Stress in Clamped Plate/Stiffener
(close-up in way of center of stiffener)
The calculated peak deflection along the stiffener is -0.00221 in and is located along the
center of the stiffener (see main body and Appendix A for calculations). Table B.11
compares the results of the closed form solution for deflection to the results determined
by the model iterations. Figure B.26 provides a node-to-node comparison of the
deflection of the stiffener.
B22
Table B.11: Summary of Peak Deflection in Clamped Plate/Stiffener
Calculated
p/s_fix_edge_nodes
p/s_fix_edge_1elem
p/s_fix_constraint
p/s_fix_edge_5elem
Deflection
-0.0157950
-0.0378100
-0.0332400
-0.0596930
-0.0186670
% Error
139.4%
110.4%
279.2%
18.2%
Deflection
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
-0.01
Deflection (in)
-0.02
Calculated
-0.03
p/s_fix_edge_nodes
p/s_fix_edge_1elem
-0.04
p/s_fix_constraint
p/s_fix_edge_5elem
-0.05
-0.06
-0.07
Beam Length (in)
Figure B.26: Summary of Peak Deflection in Clamped Plate/Stiffener
Table B.12 and Figure B.27 summarize the percent error of the model iterations relative
to the closed form solutions for stress and deflection. The percent error of each model
was determined by comparing the average compressive stress or deflection from the
model iterations to the average compressive stress or deflection from the calculated
results. Nodes along the stiffener length in way of the peak bending stress and deflection
were used. This comparison provides a rough estimate of how the model behaves
relative to the closed form solution. Nodes influenced by boundary conditions were not
used in this comparison. It is noted a positive percent error means the model predicted
higher results than the closed form solution, and conversely, a negative percent error
means the model predicted lower results than the closed form solution.
B23
Table B.12: Summary of Percent Error between Closed Form Solutions and Model Results for a
Clamped Plate/Stiffener
% Error
Stress Deflection
p/s_fix_edge_nodes
10.07
142.18
p/s_fix_edge_1elem
-0.12
115.03
p/s_fix_constraint
97.32
295.25
p/s_fix_edge_5elem
-33.93
16.26
Clamped Beam/LE % Error
350
300
250
% Error
200
150
Stress
Deflection
100
50
0
0
1
2
3
4
5
-50
-100
Iteration
Figure B.27: Summary of Percent Error between Closed Form Solutions and Model Results for a
Clamped Plate/Stiffener
The results presented herein for a clamped stiffener demonstrate that all four model
iterations do not compare very well to the closed form solutions. In all but one case, the
models predicted higher stress and deflection results than the closed form solutions.
Based upon the results presented herein, model iteration 4 is best considered to represent
the closed form solution. While this model actually represents a smaller unsupported
stiffener length (since elements on either side of stiffener are fully constrained) the stress
and deflection percent error is balanced. The other three models predict an average
deflection that is greater than 100% higher than the closed form solutions, which is
unacceptable. Additionally, the smaller unsupported length will better compare to the
wrapped final configuration.
B24