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, Thesis Advisor
_________________________________________
Ken Brown, Thesis Advisor
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2011
(For Graduation May, 2012)

LIST OF TABLES
LIST OF FIGURES
NOMENCLATURE
ACKNOWLEDGEMENT
ABSTRACT
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Type the text of your acknowledgment here. vi

This study focuses on the effect of engineering assumptions made when designing a plate panel/stiffener system under a uniformly distributed load. 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 beams under a uniformly distributed load that are 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 will perform a comparative analysis between the closed form solutions (stress and deflection for a fixed or pinned rectangular plate with a uniformly distributed load) and the values generated by a finite element analysis. The finite element analyses will focus on common “final configurations” used. The goal is to determine how “approximate” the closed form solutions are to what is actually being modeled / fabricated.
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When considering a flat plate under a uniformly distributed load, it is common engineering practice to reinforce it with a supporting beam. Support beams, if designed correctly, will absorb the majority of the applied load and prevent the previously unsupported plate from prematurely yielding. This is beneficial in the sense that the stronger the support beams, 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 weight optimized design. This thought process can be applied to general shipbuilding practice when designing a pressurized tank.
A uniform plate that extends over a support and has more than one span along its length or width is considered “continuous”. The continuous plate may be analyzed by subdividing the plate into individual panels. 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 Deflection Theory (small-deflection theory). The individual panels can be evaluated as flat rectangular plates under a uniformly distributed load and are either simply supported or clamped around the outer edge, as shown in Figure 1. a p o a) b
1

p o p o a b b)
Figure 1: Uniformly distributed load across 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 cold-rolled T-frame. It is necessary for the T-frame to be sized to support the load of the panel while remaining within allowable stress limits.
The T-frame can be evaluated as a beam under a uniformly distributed load that is either simply supported or clamped at the ends, as shown in Figure 2.
L p o
L p o p o a) b)
Figure 2: Uniformly distributed load across beam 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 withstand the forces created by the uniformly distributed pressure load. The initial design phase of these tanks involves implementing the fundamental theories of beams and flat rectangular plates. In this application, each tank surface can be evaluated separately. The surface (as a result of the supporting beams) is broken into individual panels for the purpose of evaluating it using the fundamental theories of thin shell deflection. The supporting beams can be evaluated using the fundamental theories for designing a beam. The design of the beams is created rigidly enough to enable the plate
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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. Figures 3-5 detail three design configurations commonly used in shipbuilding practices that account for these design adjustments and alterations. These design configurations include dead-ending the beam into the side panel, sniping the beam prior to contacting the side panel, and wrapping the beam around the side panel.
Figure 3: Design configuration, dead-end beam
Figure 4: Design configuration, sniped beam
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Figure 5: Design configuration, wrap beam
This study focuses on the discrepancy between the initial panel/stiffener configurations
(as determined via exact analysis) and the final panel/stiffener configurations (as determined via finite element analysis). The goal is to determine how “approximate” the closed form solutions are to the common design configuration that are actually being modeled / fabricated.
For the purposes of the analysis, the following dimensions will be used, as shown in
Table 1 and Figure 6:
Table 1: Frame Dimensional Properties (all dimensions in inches)
Frame Length (L)
Frame Spacing (F s
)
Effective Length (F eff
Shell Thickness (t)
)
Frame Depth (F d
)
Flange Width (w)
Web Thickness (t w
)
Flange Thickness (t f
)
48.00
24.00
24.00
0.75
5.00
3.00
0.50
1.00
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24.00”
5.00”
0.50”
Shell
Frame
(Effective Length)
Web
Frame
Flange
1.00”
3.00”
Figure 6: Frame cross section view (all dimensions in inches)
0.75”
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2.1.1
Assumptions
The calculations and analyses conducted herein pertain to thin plates with small deflections. In accordance with Reference (a), 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, Reference (b) 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, as identified by Reference (xx):

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 beams are based off the following assumptions, as stated in Reference (a):

The beam is a homogeneous material with the same modulus of elasticity in tension as well as compression

The beam is straight

The beam has a uniform cross section

The beam has at least one longitudinal plane of symmetry

All loads and reactions are perpendicular to the axis of the beam, and they lie in the same plane

The beam is long compared to its depth (span/depth ratio >8 for metal beams of compact section)

The beam is not disproportionately wide

The beam is nowhere stressed beyond the elastic limit
2.1.2
Exact Analysis for 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 as follows in accordance with Reference (b):
 
 p o b
4
Et
3
(1)
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where α is a constant, as provided by Reference (b), 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. As defined by Reference (b):
 max


1 p o b
2 t
2
(2) where
β
1
is a constant , as provided by Reference (b), that depends on the ratio between the short side of the panel to the long side of the panel, p o
is the pressure, b is the short side of the panel, and t is the thickness. The stress at the center of the plate is defined by
Reference (b) as:
 

2 p o b
2 t
2
(3) where
β
2
is a constant, as provided by Reference (b), that depends on the ratio between the short side of the panel to the long side of the panel.
Simply Supported Rectangular Plate under Uniformly Distributed Load
For a rectangular plate with all edges simply supported that is subjected to a uniform load over the entire plate, the maximum deflection of the plate is determined as follows:
 
 p o b
4
Et
3
(4) where α is a constant, as provided by Reference (b), 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 plate. As defined by Reference (b):
 max

 p o b
2 t
2
(5) where
β
is a constant, as provided by Reference (b), that depends on the ratio between the short side of the panel to the long side of the panel.
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2.1.3
Exact Analysis for Beam
Clamped Beam under Uniformly Distributed Load
For a beam 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 in accordance with
Reference (xx):

( x )
  wx
2

L
 x

2
24 EI
(xx) where w represents the uniformly distributed load (applied as a line load), L is the overall length of the beam, E is the Modulus of Elasticity for the steel, and I is the moment of inertia of the beam. The maximum deflection of the beam is determined as follows in accordance with Reference (xx):
  wL
4

384 EI
(xx)
The moment of the beam, M , at any point along its length, can be determined by:
M ( x )
 
1
12 w

6 Lx

L
2 
6 x
2

(xx)
The maximum moment of the beam, M , is determined by:
M
 wL 2
12
(xx) as provided by Reference (xx). The bending stress of the beam is determined from the following equation, as defined by Reference (xx):
 
Mc
I
(xx) where c represents the distance from the edge of the beam to it’s neutral axis, and I represents the moment of inertia of the beam.
Simply Supported Beam under Uniformly Distributed Load
For a beam with both ends simply supported that is subjected to a uniform load over the surface, the deflection at any point along its length can be determined as follows in accordance with Reference (xx):
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
( x )
  wx

L
3 
2 x
2
L
 x
3

24 EI
(xx)
The maximum deflection of the beam is determined as follows in accordance with
Reference (xx):
 
5 wL
4

384 EI
(xx) where w represents the uniformly distributed load (applied as a line load), L is the overall length of the beam, E is the Modulus of Elasticity for the steel, and I is the moment of inertia of the beam.
The moment of the beam at any point along its length can be determined as follows, as provided by Reference (xx):
M ( x )
 
1
2 w

L
 x
 x
The maximum moment of the beam, M , is determined by:
(xx)
M
 wL
2
8
(xx) as provided by Reference (xx). The bending stress of the beam is determined from the following equation, as defined by Reference (xx):
 
Mc
More detail on c? distance farthest away from neutral
(xx)
I axis = max compression, closest = max tension where c represents the distance from the edge of the beam to it’s neutral axis, and I represents the moment of inertia of the beam.
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 theories), a series of sub-models, or model iterations, are necessary. These sub-models will follow the simple beam/plate (as shown in Figures 1 and 2) through a series of transformations
(iterations) into the final beam configurations (as shown in Figure 3, 4, and 5). These iterations will address potential sources of error encountered along the way, as created
9

by mesh refinement, boundary conditions, etc. All models analyzed herein were created using Hex20 solid elements.
The first series of models created are exact replicas of the simple flat rectangular plate and beam evaluated above (see Figure 7), 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 future model iterations. a) b)
Figure 7: Baseline FEA model for a) beam b) plate
The next model iteration takes the baseline models a step further, by modeling the baseline extent of shell (referred to as the effective length) as supported by the baseline beam (see Figure 8 and Section 2.2.3). This 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 beam frame web).
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Figure 8: FEA of typical beam with effective length of plate
The final iteration represents the final beam configurations, which takes the second iteration a step further; vertical plates are added to the ends of the beams, and the model is extended to include a beam/shell combination on both sides of the baseline condition
(see Figure 9 and Section 2.2.4). The intent of this iteration is to focus on the results of the middle beam, which has been isolated from the boundary conditions. The thought here is the only difference between the models will be the geometry of the beams, and conclusions can be drawn as to how these different geometries affect the results of the models.
Figure 9: 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 exact analysis, a proper mesh density for the test cases 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 too refined of a mesh that will increase analysis time and may possibly create artificial peak
11

stresses. A proper mesh density is selected by performing a mesh convergence study using the baseline models described below. By choosing an initial mesh density and iterating the number of elements, a density can be selected once the FEA results are considered within an acceptable margin to the exact solutions.
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 exact 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 future test case models (where the geometries of the FEA and the exact 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 exact solutions.
Clamped Rectangular Plate Baseline Model
The first baseline model created represents a single panel that is clamped on all edges and has a uniformly distributed load applied to its surface (see Figure 7b). The model is
48” x 24”, which represents the largest area of unsupported plate (48 inches long, 24 inches between frames). The model is 0.750” thick with a material property of steel (E =
30 x 10
6
, v = 0.3). Boundary conditions are applied to the perimeter of the model to replicate the clamped condition of the panel. The significance of this model is to not only validate the exact solutions previously derived for a clamped plate but also to serve as the basis of comparison for the test case models.
Simply Supported Rectangular Plate Baseline Model
The second baseline model created represents a single panel that is simply supported on all edges and has a uniformly distributed load applied to its surface (see Figure 7b). The model is 48” x 24”, which represents the largest area of unsupported plate (48 inches long, 24 inches between frames). The model is 0.750” thick with a material property of steel (E = 30 x 10 6 , v = 0.3). Boundary conditions are applied to the perimeter of the model to replicate the simply supported condition of the panel. The significance of this model is to not only validate the numerical solutions previously derived for a simply supported plate but also to serve as the basis of comparison for the test case models.
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Clamped Beam Baseline Model
The next baseline model created represents a beam that is in the shape of a T-frame and is clamped at both ends (see Figure 7a). Cross section dimensions are 5.0 x 3.0 x 0.5 x
1.0 (as shown in Figure 6), and the model extents 48”. A uniformly distributed load is applied to the top surface of the web only. The model has a material property of steel (E
= 30 x 10
6
, v = 0.3). Boundary conditions are applied to the ends of the beam to replicate the clamped condition of the beam. The significance of this model is to not only validate the exact solutions previously derived for a clamped beam but also to serve as the basis of comparison for the test case models.
Simply Supported Beam Baseline Model
The next baseline model created represents a beam that is in the shape of a T-frame and is simply supported at both ends (see Figure 7a). Cross section dimensions are 5.0 x 3.0 x 0.5 x 1.0 (as shown in Figure 6), and the model extents 48”. A uniformly distributed load is applied to the top surface of the web only. The model has a material property of steel (E = 30 x 10
6
, v = 0.3). Boundary conditions are applied to the ends of the beam to replicate the simply supported condition of the beam. The significance of this model is to not only validate the exact solutions previously derived for a simply supported beam but also to serve as the basis of comparison for the test case models.
Sniped Beam Baseline Model
In this baseline model, a beam is created in the shape of a T-frame and is clamped at both ends. Unlike the previously two baseline models described, the beam flange is chamfered, or sniped, into the beam web, which is then sniped at approximately 45 degrees as it approaches the end of the beam. Typical cross section dimensions are 5.0 x
3.0 x 0.5 x 1.0, and the model extends 48”. A uniformly distributed load is applied to the top surface of the web only. The model has a material property of steel (E = 30 x 10 6 , v
= 0.3). Boundary conditions are applied to the ends of the beam. The significance of this model is to understand how the unique geometry of this beam correlates to that of the baseline clamped and simply supported beams.
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2.2.3
Frame & Effective Length of Shell Models
Once an acceptable mesh and ideal boundary conditions have been selected, the next step is to determine the effect of more accurately applying the pressure load. This iteration takes the base models one step further by combining the baseline plate model with the baseline beam model (see Figure 8). For this iteration, a uniformly distributed load is distributed across the baseline plate, which is supported by the baseline beam.
The beam is centered length-wise under the plate, creating an I-beam. The length of shell used is referred to as the effective length of the shell, which is equidistant to a half frame bay of shell on either side of the beam. Separate models are created using separate boundary conditions for 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 presented as a result of complicated boundary conditions and different loading conditions.
2.2.4
Full Configuration
With the completion of the previous iterations, the next step is to analyze results that are as isolated as possible from the boundary conditions. The rigidity of the boundary conditions could have an adverse condition on the results, so the full configuration attempts to resolve this situation by creating a length of plate that is supported transversely by three uniform beams, equally spaced at 24” (see Figure 9). 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 concept of a tank design, 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 create an internal frame (the center frame) that is completely isolated from any boundary conditions. The plate/beam combination is allowed to deflect and bend based upon the reactions of the adjacent frames, and not based upon a perfectly stiff connection. Focus can now be directed towards the alternate beam configurations, which are described below.
Sniped Frame Model
In this model configuration, the shell plate is supported by the previously described baseline beam. However, in this instance, the beam is sniped at approximately 45
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degrees as it approaches the tank walls. The beam does not connect directly to the tank wall. This configuration is a common configuration used in shipbuilding designs, as it is easy to fabricate and provides accessibility for tradesmen during fit-up. Engineering assumptions assume this configuration to resemble a simply supported configuration.
Block Frame Model
In this model configuration, the shell plate is supported by the baseline beam. Unlike the sniped frame model, the beam butts directly into the shell wall. This configuration can create difficulties during the fabrication process, as well as stress concentrations in the wall plate (since the beam 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 tank walls to mitigate stress concentrations. Engineering assumptions assume this configuration to resemble a clamped configuration.
Wrapped Frame Model
In this model configuration, the shell plate is supported by the baseline beam. The beam wraps down the side of the tank walls, 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 of the tank is more evenly and continuously supported. Engineering assumptions assume this configuration to resemble a clamped configuration.
3.1.1
Exact 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 can be determined using Equation (1):
 
 p o b
4
Et
3
(1) where
α = 0.0277 (see Reference (b), for a/b = 1) p o
= 100 psi
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b = 24in
E = 30 x 10
6
psi t = 0.75in the resulting maximum deflection is
δ = 0.0726in since the peak deflection is less than half of the plate thickness, and since the plate thickness is less than 1/20 th of the plate length, the plate is acceptable for thin plate theory.
The maximum stress is located at the center of the long edge of the plate. Recalling
Equation (2):
 max


1 p o b
2 t
2 where
β
1
= 0.4974 (see Reference (b), for a/b = 2.0)
(2) p o
= 100 psi b = 24in t = 0.75in the resulting maximum stress is
σ max
= 50933 psi
The stress at the center of the plate is defined by Equation (3):
 

2 p o b
2 t
2 where
β
2
= 0.1386 (see Reference (b), for a/b = 1) p o
= 100 psi b = 30in t = 0.75in the resulting stress is
σ
=22176 psi
(3)
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Simply Supported Rectangular Plate Under Uniformly Distributed Load
For a rectangular plate with all edges simply supported that is subjected to a uniform load over the entire plate, the maximum deflection of the plate can be determined using
Equation (4):
 
 p o b
4
Et
3 where
α
= 0.1110 (see Reference (b), for a/b = 2.0)
(4) p o
= 100 psi b = 24in
E = 30 x 10 6 psi t = 0.75in the resulting maximum deflection is
δ
= 0.29097in since the peak deflection is less than half of the plate thickness, and since the plate thickness is less than 1/20 th
of the plate length, the plate is acceptable for thin plate theory.
The maximum stress is located at the center of the plate. As defined by Reference (b):
  max
 p o b
2 t
2
(5) where
β
= 0.6102 (see Reference (b), for a/b = 2.0) p o
= 100 psi b = 24in t = 0.75in the resulting maximum stress is
σ max
= 62484 psi
3.1.2
Exact Solution for Beam
Clamped Beam Under Uniformly Distributed Load
17

For a beam with both ends fixed that is subjected to a uniform load over the surface, the maximum deflection of the beam can be determined using Equation (xx):
  wL
4

384 EI
(xx) where w = 50 lb/in
L = 48 in
E = 30 x 10
6
psi
I = 10.4167 in
4
(see Appendix A for calculation) the resulting maximum deflection is
δ = -0.0022118 in
Using Equation (xx), the deflection pattern across the length of the beam can be seen in
Figure xx.
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
-0.0005
-0.001
-0.0015
-0.002
Exact Clamped
-0.0025
Beam Length (in)
Figure xx: Deflection of Clamped Beam under Uniformly Distributed Load
The maximum moment of the beam can be determined by Equation (xx):
M
 wL
2
12
(xx) where: w = 50 lb/in
L = 48 in
The resulting maximum moment is
18

M = 9600 lb·in
Recalling Equation (xx), the moment of the beam along the length of the beam can be calculated, as shown in Figure xx below.
Clamped Beam M oment Distribution
-4000
-6000
-8000
-10000
-12000
6000
4000
2000
0
-2000
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)
Exact Clamped
Figure xx: Moment Force of Clamped Beam under Uniformly Distributed Load
The peak stress of the beam can be determined from Equation (xx):
 
Mc
I
(xx) where
M = 9600 lb·in c = 3.5 (see appendix A for calculation)
I = 10.4167 in 4 (see Appendix A for calculation)
The resulting peak stress is
σ
= 322.56 psi
The bending stress distribution across the length of the beam can be seen in Figure xx, as calculated using Equation (xx).
19

Clamped Beam Bending Stress
4000
3000
2000
1000 Exact Clamped
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 xx: Bending Stress of Clamped Beam under Uniformly Distributed Load
Simply Supported Beam Under Uniformly Distributed Load
For a beam with both ends simply supported that is subjected to a uniform load over the surface, the maximum deflection of the beam can be determined from Equation (xx):
 
5 wL
4

384 EI
(xx) where w = 50 lb/in
L = 48 in
E = 30 x 10
6
psi
I = 10.417 in
4
(see Appendix A for calculation) the resulting maximum deflection is
δ = -0.011059 in
Recalling Equation (xx), the deflection of the beam anywhere along its surface can be determined, as shown in Figure xx.
20

Simply Supported Beam Deflection
0
0.00
4.00
-0.002
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
-0.008
Exact Simple
-0.01
-0.012
Beam Length (in)
Figure xx: Deflection of Simply Supported Beam under Uniformly Distributed Load
The maximum moment of the beam can be determined by Equation (xx):
M
 wL
2
8
(xx) where: w = 50 lb/in
L = 48 in
The resulting maximum moment is
M = 14400 lb·in
The moment distribution across the length of the beam can be seen in Figure xx, as calculated using Equation (xx).
21

Simply Supported Beam M oment Distribution
16000
14000
12000
10000
8000
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)
Exact Simple
Figure xx: Moment Force of Simply Supported Beam under Uniformly Distributed Load
The peak stress of the beam can be determined from Equation (xx):
 
Mc
I
(xx) where
M = 14400 lb·in c = 3.5 (see appendix A for calculation)
I = 10.4167 in
4
(see Appendix A for calculation)
The resulting peak stress is
σ
= 483.84 psi
Recalling Equation (xx), the bending stress across the length of the beam can be calculated, as shown in Figure xx.
22

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
-2000
-3000
-4000
Exact Simple
-5000
-6000
Beam Length (in)
Figure xx: Bending Stress of Simply Supported Beam under Uniformly Distributed Load
Clamped Beam/LE Under Uniformly Distributed Load
For a beam with an effective length of shell that has both ends fixed and is subjected to a uniform load over the surface, the maximum deflection of the beam can be determined in the same method as for a simple beam. Recalling Equation (xx):
  wL
4

384 EI
(xx) where w = 2400 lb/in
L = 48 in
E = 30 x 10
6
psi
I = 70.017 in
4
(see Appendix A for calculation) the resulting maximum deflection is
δ
= -0.0157950 in
Recalling Equation (xx), the deflection pattern along the length of the beam can be calculated, as shown in Figure xx.
23

Clamped Beam/LE Deflection
0
0.00
-0.002
-0.004
-0.006
-0.008
-0.01
-0.012
-0.014
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
Exact Clamp
-0.016
-0.018
Beam Length (in)
Figure xx: Deflection of Clamped Beam/LE under Uniformly Distributed Load
The maximum moment of the beam can be determined by Equation (xx):
M
 wL
2
12
(xx) where: w = 2400 lb/in
L = 48 in
The resulting maximum moment is
M = 460,800 lb·in
Recalling Equation (xx), the moment distribution across the length of the beam can be calculated, as shown in Figure xx.
Clamped Beam/LE Moment Distribution
300000
200000
100000
0
-100000
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
-200000
-300000
-400000
-500000
Beam Length (in)
Exact Clamped
24

Figure xx: Moment Force of Clamped Beam/Le under Uniformly Distributed Load
The peak stress of the beam can be determined from Equation (xx):
 
Mc
I
(xx) where
M = 460,800 lb·in c = 4.5 (see appendix A for calculation)
I = 10.4167 in
4
(see Appendix A for calculation)
The resulting peak stress is
σ
= 29830.18 psi
Recalling Equation (xx), the bending stress along the length of the beam can be calculated, as shown in Figure xx.
Clamped Beam/LE Bending Stress
20000
15000
10000
5000
0
-5000
0.00
-10000
-15000
-20000
-25000
-30000
-35000
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
Exact Clamp
Beam Length (in)
Figure xx: Bending Stress of Clamped Beam/LE under Uniformly Distributed Load
Simply Supported Beam/LE Under Uniformly Distributed Load
For a beam with an effective length of shell that has both ends simply supported and is subjected to a uniform load over the surface, the maximum deflection of the beam can be determined in the same method as for a simple beam. Recalling Equation (xx):
 
5 wL
4

384 EI where w = 2400 lb/in
L = 48 in
(xx)
25

E = 30 x 10
6
psi
I = 70.017 in
4
(see Appendix A for calculation) the resulting maximum deflection is
δ = -0.078975 in
Recalling Equation (xx), the deflection pattern along the length of the beam can be calculated, as shown in Figure xx.
Simply Supported Beam/LE Deflection
0
0.00
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
4.00
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
44.00
48.00
Exact Simple
-0.07
-0.08
-0.09
Beam Length (in)
Figure xx: Deflection of Simply Supported Beam/LE under Uniformly Distributed Load
The maximum moment of the beam can be determined by Equation (xx):
M
 wL
2
8
(xx) where: w = 2400 lb/in
L = 48 in
The resulting maximum moment is
M = 14400 lb·in
Recalling Equation (xx), the moment distribution across the length of the beam can be calculated, as shown in Figure xx.
26

Simply Supported Beam/LE Moment Distribution
800000
700000
600000
500000
400000
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)
Exact Simple
Figure xx: Moment of Simply Supported Beam/LE under Uniformly Distributed Load
The peak stress of the beam can be determined from Equation (xx):
 
Mc
I where
M = 460,800 lb·in c = 4.5 (see appendix A for calculation)
I = 70.017 in
4
(see Appendix A for calculation)
The resulting peak stress is
σ
= 44745.27 psi
(xx)
Simply Supported Beam/LE Bending Stress
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
0.00
4.00
8.00
12.00
16.00
20.00
24.00
28.00
Beam Length (in)
32.00
36.00
40.00
44.00
48.00
Exact Simple
27

Figure xx: Bending Stress of Simply Supported Beam under Uniformly Distributed Load
3.2.1
Baseline Models
Mesh and boundary condition convergence studies were conducted for the baseline models described above. The final results of each baseline condition are presented herein
(see Table xx); the results of the convergence study iterations are provided in Appendix
B.
Table xx: Summary of Numerical vs FEA for panel
Mesh Density
6 x 6
8 x 8
12 x 12
16 x 16
Deflection
(Numerical w max
= 0.088)
Stress
(Numerical σ max
= 49248 psi, σ center
= 22176 psi)
FEA % Difference FEA (max) % Difference FEA (@center) % Difference
0.085777
0.086898
0.087762
0.088064
2.59
1.27
0.27
-0.07
44111
45992
47640
48306
11.65
7.08
3.38
1.95
21260
21514
21766
21862
4.31
3.08
1.88
1.44
Clamped Rectangular Plate Baseline Model
The baseline model of the clamped rectangular plate was created to verify the exact solutions as well as to serve as a baseline for comparison to the test cases. A finite element mesh of 2304 elements (48 x 24 x 2) was used. All edge nodes were fully constrained (DOF 1-6) to replicate the clamped condition. A peak deflection of -
0.072662in was reported, which is approximately 0.01% greater than the exact value (-
0.072614in). A peak stress (compression/tension) of -54698 psi was reported, as shown in Figure xx. This is approximately 6.9% greater than the exact value (-50934 psi). The stress at the center of the plate was reported to be 21862 psi, as shown in Figure xx. This is approximately 1.43% from the numerical value (22176 psi). Based on the convergence analyses performed herein, a 48 x 24 x 2 element mesh is considered to be an acceptable plate/shell mesh density for future test case applications.
28

Figure xx: Clamped Rectangular Plate
Baseline Model
Peak Tension Stresses on outboard and inboard sides of plate
Simply Supported Rectangular Plate Baseline Model
The baseline model of the simply supported rectangular plate was created to verify the exact solutions as well as to serve as a baseline for comparison to the test cases. A finite element mesh of 2304 elements (48 x 24 x 2) was used. All midplane edge nodes were fully constrained (DOF 1-6) to replicate the simply supported condition. A peak deflection of 0.29394 was reported. This is approximately 1.01% greater than the exact value (0.29098). A peak stress (compression/tension) of 63136 psi was reported, as shown in Figure xx. This is approximately 1.03% greater than the exact value (62484 psi). Based on the convergence analyses performed herein, a 48 x 24 x 2 element mesh is considered to be an acceptable plate/shell mesh density for future test case applications.
29

Figure xx: Simply Supported Rectangular Plate
Baseline model
Peak tension stresses on outboard and inboard sides of plate
Clamped Beam Baseline Model
The baseline model of the clamped beam was created to verify the exact solutions as well as to serve as a baseline for comparison to the test cases. A finite element mesh 48 elements long, by 2 elements deep, by 2 elements thick, was used to represent the beam web. End elements were fully constrained (DOF 1-6) to replicate the clamped condition.
A peak deflection of -0.0023466in was reported, which is approximately 5.74% greater than the exact value (-0.0022118in). A peak stress (compression/tension) of 325.06 psi was reported, as shown in Figure xx. This is approximately 0.77% less than the exact value (-322.56 psi). Based on the convergence analyses described above, the beam element mesh described herein is considered to be an acceptable beam mesh density for future test case applications.
30

Figure xx: Clamped Rectangular Plate
Baseline Model
Peak Tension Stresses on outboard and inboard sides of plate
Simply Supported Beam Baseline Model
The baseline model of the simply supported beam was created to verify the exact solutions as well as to serve as a baseline for comparison to the test cases. A finite element mesh 48 elements long, by 2 elements deep, by 2 elements thick, was used to represent the beam web. Edge nodes along the beam’s neutral axis were fully constrained (DOF 1-6) to replicate the simply supported condition. A peak deflection of
-0.011667in was reported, which is approximately 5.21% greater than the exact value (-
0.0110592in). A peak stress (compression/tension) of 486.60 psi was reported, as shown in Figure xx. This is approximately 0.57% greater than the exact value (483.84 psi).
Based on the analyses described above, the beam element mesh described herein is considered to be an acceptable beam mesh density for future test case applications.
31

3.2.2
Frame & Effective Length of Shell
3.2.3
Full Configuration, Frames Chamfered
3.2.4
Full Configuration, Frames Squared
3.2.5
Full Configuration, Frames Wrapped
32

This study is expected to provide finite element model results for three different plate/stiffener assemblies that are comparable to numerical results (stress and deflection values for a fixed or pinned rectangular plate).
33

Roark’s Formulas for Stress and Strain, W. Young, R. Budynas, seventh edition,
McGraw-Hill, 2002.
Stresses in Plates and Shells, A. Ugural, second edition, McGraw-Hill , 1999.
Mechanics of Materials, F. Beer, R. Johnston, second edition, McGraw-Hill, 1992.
Theory of Plates and Shells, Timoshenko, S. and S. Woinowskky-Krieger, McGraw-
Hill.
Specification for Structural Steel Buildings, ANSI/AISC 360-10, approved by the AISC
Committee on Specifications, June 22, 2010.
An Exact Solution for the Deflection of a Clamped Rectangular Plate under Uniform
Load, C.E. Imurak and I. Gerdemeli, Applied Mathematical Sciences, Vol 1, 2007.
Bending of Rectangular Plates with Finite Deflections, Bauer, F., Bauer, L., W. Becker,
E. Reiss, ASME J. Appl Mech., vol 30, no.2, 1963.
Large and Small Defletions of a Cantilever Beam, T. Belendez, C. Nelpp, A Belendez,
European Journal of Physics, May 8, 2002.
34

A1

A2

A3

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.
A4

A2.1 Moment of Inertia for Beam Only
In order to determine the moment of inertia of the beam cross section with respect to its neutral axis ( x
’), the beam is broken down into rectangles, as shown in Figure B.1. x
1
0.50
” y
1
= 2.00
”
5.00
” c
= 3.50
” y
2
=4.50
” c x
’
1.00
”
2
3.00
”
Cross sectional properties are summarized in Table B.1
1
2
Total
Area (in
2
)
(0.50)(4.00) = 2.00
(1.00)(3.00) = 3.00
ΣA = 5.00
y (in)
2.00
4.50
yA (in
3
)
(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 (xx)
Y

3 .
5 in
The moment of inertia of the cross section can then be determined using the Parallel
Axis Theorem, as defined by Reference (xx). 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 cross sectional area with respect to its neutral axis. The total moment of inertia is calculated as shown below:
A5

I x '
 
( I

Ad
2
)
 
1
(
12 bh
3 
Ad
2
)
I x '

1
12
( 0 .
5 )( 4
3
)

( 2 )( 1 .
5
2
)

1
12
( 3 )( 1
3
)

( 3 )( 1
2
)
I

10 .
4167 in
4
(xx) c = distance from cross section neutral axis to location of maximum compression/tension stress = 3.5”
A2.2 Moment of Inertia for Beam/Effective Length of Shell
In order to determine the moment of inertia of the beam cross section with respect to its neutral axis ( x ’), the beam is broken down into rectangles, as shown in Figure B.2.
24.00
” y
1
= 0.375
” x
0.75
”
1 x ’ c c
= 1.217
” y
2
= 2.75
”
5.00
” y
3
=5.25
”
0.50
”
2
3
1.00
”
3.00
”
Cross sectional properties are summarized in Table B.2
1
2
3
Total
Area (in
2
)
(24.00)(0.75) = 18.00
(0.50)(4.00) = 2.00
(1.00)(3.00) = 3.00
ΣA = 23.00
y (in) yA (in
3
)
0.375
(0.375)(18.00) = 6.75
2.75
(2.75)(2.00) = 5.50
5.25
(5.25)(3.00) = 15.75
ΣyA = 28.00
The distance from the x axis to the neutral axis is determined by recalling equation (xx):
Y

A
  y A (xx)
A6

Y

1 .
217 in
The moment of inertia of the cross section can then be determined using the Parallel
Axis Theorem, as defined by Reference (xx). Recalling equation (xx):
I x '
 
( I

Ad
2
)
 
1
(
12 bh
3 
Ad
2
) (xx)
I x '

1
12
( 24 )( 0 .
75
3
)

( 18 )( 0 .
84
2
)
1
12
( 0 .
5 )( 4
3
)

( 2 )( 1 .
53
2
)

1
12
( 3 )( 1
3
)

( 3 )( 4 .
03
2
)
I

70 .
017 in
4
A7

Appendix B
Convergence Results
B1

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.
Convergence studies were performed for the following modeling iterations:

Simply supported rectangular plate under uniformly distributed load

Clamped rectangular plate under uniformly distributed load

Simply supported beam (T-frame shape) under uniformly distributed load

Clamped beam (T-frame shape) under uniformly distributed load

Simply supported beam/LE (I-beam shape) under uniformly distributed load

Clamped beam/LE (I-beam shape) under uniformly distributed load
B2.1 Simply Supported Rectangular Plate under Uniformly Distributed Load
The area of the rectangular plate 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 ( iter1 , 8” long), 12 elements ( iter2 , 4” long), 24 elements ( iter3 , 2” long), and 48 elements ( iter4 , 1” long). For a simply supported plate, translations are constrained however rotations are unconstrained. In order to allow the edges of the plate to rotate freely while restricting any form of translation, the mid-plane nodes along the perimeter of the plate were constrained. This prevents the edges of the plate from deflecting, yet allows the plate to rotate as necessary. This scenario was applied to all simply supported rectangular plate 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 exact solutions. Figure B.1 compares the percent error of the peak stress and deflection of each model to the exact solution. It is noted a positive percent error means the model
B2

predicted a higher result than the exact solution, and conversely, a negative percent error means the model predicted a lower result than the exact solution.
Table B.1: Summary of Peak Stress and Deflection for Simply Supported Plate
Stress % Error Deflection % Error
Exact Value 62484
Iter1 61908
Iter2
Iter3
Iter4
62127
63002
63136
-
-0.92%
-0.57%
0.83%
1.04%
0.29098
0.27668
0.28920
0.29282
0.29394
-
-4.91%
-0.61%
0.63%
1.02%
Simply Supported Plate Results
Exact vs FEA
6.00%
4.00%
2.00%
0.00%
0
-2.00%
-4.00%
1 2 3 4 5
Stress
Deflection
-6.00%
Iteration
Figure B.1: Summary of Percent Error Between Exact Calculations and Model Results for a Simply
Supported Plate
The results presented herein for a simply supported plate demonstrate that all four model iterations are within approximately five percent of the exact calculations. Based upon the results presented herein, the simply supported plate with 48 elements along its length
( iter4 ) is best considered to represent the exact 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.
B2.2 Clamped Rectangular Plate under Uniformly Distributed Load
B3

Model iterations for a clamped rectangular plate 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 plate that is clamped has all translations and rotations constrained around the perimeter. Two potential boundary condition scenarios were considered: fully constraining the perimeter nodes of the plate, or fully constraining the perimeter elements of the plate. 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 ( iter1-iter4 ). 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
( iter5 ). 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.
Table B.2: Summary of Peak Stress and Deflection for Clamped Plate
Stress % Error Deflection % Error
Exact Value 50934
Iter1
Iter2
22865
40833
Iter3
Iter4
Iter5
52103
54698
37232
-
2.30%
7.39%
0.0726139
-55.11% 0.0438980
-39.55%
-19.83% 0.0664140
0.0714930
0.0726620
-
-8.54%
-1.54%
0.07%
-26.90% 0.0518900
-28.54%
Clamped Plate Results
Exact vs FEA
20.00%
10.00%
0.00%
-10.00%
0
-20.00%
-30.00%
-40.00%
-50.00%
-60.00%
1 2 3 4 5 6
Stress
Deflection
Iteration
B4

Figure B.2: Summary of Percent Error between Exact Calculations and Model Results for a
Clamped Plate
B2.3 Simply Supported Beam 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 beam under a uniformly distributed load. Mesh studies were conducted for beams composed of 6 elements, 12 elements, 24 elements, and 48 elements along the length. Boundary conditions for this model were trivial, as the beam was fully constrained at the neutral axis at both ends. This constraining method is consistent with the assumptions of the exact solution, which also constrains the beam along the neutral axis.
Recalling from the main body, the calculated peak bending stress along the beam is -
4.84 ksi. This peak compressive stress is located along the top of the beam at the center of the beam (see main body and Appendix A for calculations). Table B.3 compares the results of the exact 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 exact solution and the model iterations.
Table B.3: Summary of Peak Compressive Bending Stress in Simply Supported Beam
Stress
Exact Value -4838.40
Iter1
Iter2
-4933.60
-4884.80
Iter3
Iter4
-4870.30
-4866.00
% Error
-
1.97%
0.96%
0.66%
0.57%
B5

-2000
-3000
-4000
-5000
-6000
Stress
1000
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
Beam Length (in)
Figure B.3: Summary of Peak Compressive Bending Stress in Simply Supported Beam
Stress
0
0.00
1.00
2.00
3.00
4.00
5.00
6.00
Exact Simple
6elem
12elem
24elem
48elem
-400
-800
-1200
Exact Sim ple
6elem
12elem
24elem
48elem
-1600
-2000
Beam Length (in)
Figure B.4: Summary of Peak Compressive Bending Stress in Simply Supported Beam
(in way of Boundary Conditions)
B6

Stress
-4500
20.00
-4600
21.00
22.00
23.00
24.00
25.00
26.00
27.00
28.00
-4700
-4800
-4900
Exact Simple
6elem
12elem
24elem
48elem
-5000
Beam Length (in)
Figure B.5: Summary of Peak Compressive Bending Stress in Simply Supported Beam
(in way of center of beam)
The calculated peak deflection along the beam is -0.01106 in and is located at the center of the beam (see main body and Appendix A for calculations). Table B.4 compares the results of the exact 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 beam.
Table B.4: Summary of Peak Deflection in Simply Supported Beam
Exact Value
Iter1
Iter2
Iter3
Iter4
Deflection
-0.0110592
-0.0114890
-0.0115340
-0.0115580
-0.0115770
% Error
-
3.89%
0.39%
0.21%
0.16%
B7

-0.008
-0.01
-0.012
-0.014
Deflection
0
-0.002
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
-0.004
-0.006
Beam Length (in)
Exact Simple
6elem
12elem
24elem
48elem
Figure B.6: Summary of Peak Deflection in Simply Supported Beam
Deflection
-0.01
20.00
-0.0102
-0.0104
-0.0106
-0.0108
-0.011
-0.0112
-0.0114
-0.0116
-0.0118
-0.012
21.00
22.00
23.00
24.00
25.00
26.00
27.00
28.00
Exact Simple
6elem
12elem
24elem
48elem
Beam Length (in)
Figure B.7: Summary of Peak Deflection in Simply Supported Beam
(in way of center of beam)
Table B.5 and Figure B.8 summarize the percent error of the model iterations relative to the exact 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 beam 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 exact 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

4.00
3.00
2.00
1.00
than the exact calculations. It is noted a positive percent error means the model predicted higher results than the exact solution, and conversely, a negative percent error means the model predicted lower results than the exact solution.
Table B.5: Summary of Percent Error between Exact Calculations and Model Results for a Simply
Supported Beam
6elem
12elem
24elem
48elem
% Error
Stress Deflection
0.46
0.61
0.62
0.52
4.17
4.63
4.91
5.16
Simply Supported Beam %Error
6.00
5.00
Stress
Deflection
0.00
0 1 2 3 4 5
Iteration
Figure B.8: Summary of Percent Error between Exact Calculations and Model Results for a Simply
Supported Beam
The results presented herein for a simply supported beam demonstrate that all four model iterations are within approximately five percent of the exact calculations. Based upon the results presented herein, the simply supported beam with 48 elements along its length is best considered to represent the exact 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 Beam under Uniformly Distributed Load
For a clamped beam under a uniformly distributed load, the optimal mesh determined via Section B2.3 was used for all model iterations. Unlike the simply supported beam, boundary conditions had a much larger influence on the overall response of the models,
B9

and were the primary focus of the iterations shown herein. The following model iterations were performed to determine the optimal boundary condition for a clamped beam under a uniformly distributed load: a) iteration 1 ( iter1 ): edge nodes on both ends of model were completely constrained b) iteration 2 ( iter2 ): edge elements on both ends of model were completely constrained c) iteration 3 ( iter3 ): the beam model was extended on both ends by one element, which was completely constrained d) iteration 4 ( iter4 ): 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
L a) b) Y c) d)
X
Z
Figure B.9: Clamped Beam Model Boundary Conditions for: a) iter1 b) iter2 c) iter3 d) iter4
Recalling from the main body, the calculated peak bending stress along the beam is 3.23 ksi. The peak tensile stress is located along the top of the beam at the edges (see main body and Appendix A for calculations). Figures B.10 and B.11 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 exact solution away from the boundary conditions.
B10

Stress
5000
4000
3000
2000
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
-1000
Exact Clamped
Iter1 Edge Nodes
Iter2 Edge Elems
Iter3 Extra Elems
Iter4 Edge Constraint
-2000
Beam Length (in)
Figure B.10: Summary of Peak Bending Stress in Clamped Beam
Stress
-1300
20.00
-1350
21.00
22.00
23.00
24.00
25.00
26.00
27.00
28.00
-1400
-1450
-1500
-1550
Exact Clamped
Iter1 Edge Nodes
Iter2 Edge Elems
Iter3 Extra Elems
Iter4 Edge Constraint
-1600
-1650
-1700
Beam Length (in)
Figure B.11: Summary of Peak Bending Stress in Clamped Beam
(in way of center of beam)
The calculated peak deflection along the beam is -0.00221 in and is located at the center of the beam (see main body and Appendix A for calculations). Table B.6 compares the results of the exact 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 beam.
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Table B.6: Summary of Peak Deflection in Clamped Beam
Exact Value
Iter1
Iter2
Iter3
Iter4
Deflection
-0.002212
-0.002711
-0.002325
-0.002711
-0.002718
% 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
-0.0005
-0.001
-0.0015
-0.002
-0.0025
Exact Clamped
Iter1 Edge Nodes
Iter2 Edge Elems
Iter3 Extra Elems
Iter4 Edge Constraint
-0.003
Beam Length (in)
Figure B.12: Summary of Peak Deflection in Clamped Beam
Deflection
-0.002
20.00
-0.0021
-0.0022
-0.0023
-0.0024
21.00
22.00
23.00
24.00
25.00
26.00
27.00
28.00
Exact Clamped
Iter1 Edge Nodes
Iter2 Edge Elems
Iter3 Extra Elems
Iter4 Edge Constraint -0.0025
-0.0026
-0.0027
-0.0028
Be am Le ngth (in)
Figure B.13: Summary of Peak Deflection in Clamped Beam (in way of center of beam)
Table B.7 and Figure B.14 summarize the percent error of the model iterations relative to the exact 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.
B12

Nodes along the beam 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 exact 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 exact solution, and conversely, a negative percent error means the model predicted lower results than the exact solution.
Table B.7: Summary of Percent Error between Exact Calculations and Model Results for a
Clamped Beam
Iter1
Iter2
Iter3
Iter4
% Error
Stress Deflection
3.9
-6.2
2.9
3.1
24.5
5.4
23.2
23.5
Clamped Beam % Error
30
25
20
15
10
5
0
0
-5
1 2 3 4 5
Stress
Deflection
-10
Iteration
Figure B.14: Summary of Percent Error between Exact Calculations and Model Results for a
Clamped Beam
The results presented herein for a clamped beam demonstrate that all four model iterations are within approximately five percent of the exact calculations for bending stress and within approximately twenty five percent of the exact calculations for deflection. Based upon the results presented herein, model iteration 2 is best considered to represent the exact solution. While this model actually represents a smaller unsupported beam length (since elements on either side of beam are fully constrained) the stress and deflection percent error is balanced more so than the other model
B13

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 Beam/LE under Uniformly Distributed Load
For a simply supported beam with an effective length (LE) 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 beam/LE under a uniformly distributed load: a) iteration 1 ( iter1 ): shell edge midnodes completely constrained b) iteration 2 ( iter2 ): neutral axis of beam/LE completely constrained c) iteration 3 ( iter3 ): the shell edge midnodes and the neutral axis of the beam/LE were completely constrained d) iteration 4 ( iter4 ): edge nodes from the midnodes of the shell to the neutral axis of the beam/LE were completely constrained e) iteration 5 ( iter5 ): inboard side of shell nodes completely constrained f) iteration 6 ( iter6 ): neutral axis of beam/LE 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 a)
L b)
L d) c)
L
L e) f)
Figure B.15: Simply Supported Beam/LE Model Boundary Conditions for: a) iter1 b) iter2 c) iter3 d) iter4 e) iter5 f) iter6
Recalling from the main body, the calculated peak bending stress along the beam is
44.75 ksi. This peak tensile stress is located along the inboard face of the beam flange at the midpoint (see main body and Appendix A for calculations). Table B.8 compares the results of the exact 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

50000
Table B.8: Summary of Peak Tensile Bending Stress in Simply Supported Beam/LE
Stress % Error
Exact Value 44745.27
Iter1 45459.00
-
1.6%
Iter2
Iter3
Iter4
Iter5
Iter6
46794.00
4.6%
44072.00
-1.5%
43901.00
-1.9%
46268.00
3.4%
46871.00
4.8%
Stress
40000
30000
20000
10000
Exact Simple
Iter1 Shell Midnodes
Iter2 NA Only
Iter3 Shell & NA
Iter4 Shell to NA
Iter5 Inbd Shell
Iter6 Rotations
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
Beam Length (in)
Figure B.16: Summary of Peak Tensile Bending Stress in Simply Supported Beam/LE
Stress
30000
25000
20000
15000
10000
5000
Exact Simple
Iter1 Shell Midnodes
Iter2 NA Only
Iter3 Shell & NA
Iter4 Shell to NA
Iter5 Inbd Shell
Iter6 Rotations
0
0.00
-5000
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
Beam Length (in)
Figure B.17: Summary of Peak Tensile Bending Stress in Simply Supported Beam/LE
(in way of Boundary Conditions)
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Stress
48000
47000
46000
45000
44000
43000
Exact Simple
Iter1 Shell Midnodes
Iter2 NA Only
Iter3 Shell & NA
Iter4 Shell to NA
Iter5 Inbd Shell
Iter6 Rotations
42000
41000
40000
20.00
21.00
22.00
23.00
24.00
Beam Length (in)
25.00
26.00
27.00
28.00
Figure B.18: Summary of Peak Tensile Bending Stress in Simply Supported Beam/LE
(in way of center of beam)
The calculated peak deflection along the beam is -0.07897 in and is located at the center of the beam (see main body and Appendix A for calculations). Table B.9 compares the results of the exact 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 beam.
Table B.9: Summary of Peak Deflection in Simply Supported Beam/LE
Exact Value
Iter1
Iter2
Iter3
Iter4
Iter5
Iter6
Deflection
-0.0789749
-0.0989480
-0.1117800
-0.0938030
-0.0932910
-0.1087500
-0.1051200
% Error
-
25.3%
41.5%
18.8%
18.1%
37.7%
33.1%
B17

-0.04
-0.06
-0.08
Deflection
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
-0.1
-0.12
Beam Length (in)
Exact Simple
Iter1 Shell Midnode
Iter2 NA Only
Iter3 Shell & NA
Iter4 Shell to NA
Iter5 Inbd Shell
Iter6 Rotations
Figure B.19: Summary of Peak Deflection in Simply Supported Beam/LE
Deflection
-0.06
20.00
-0.07
21.00
22.00
23.00
24.00
25.00
26.00
27.00
28.00
-0.08
-0.09
-0.1
-0.11
Exact Simple
Iter1 Shell Midnode
Iter2 NA Only
Iter3 Shell & NA
Iter4 Shell to NA
Iter5 Inbd Shell
Iter6 Rotations
-0.12
Beam Length (in)
Figure B.20: Summary of Peak Deflection in Simply Supported Beam/LE
Table B.10 and Figure B.21 summarize the percent error of the model iterations relative to the exact 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 beam 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 exact 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 exact solution, and conversely, a negative percent error means the model predicted lower results than the exact solution.
B18

Table B.10: Summary of Percent Error between Exact Calculations and Model Results for a Simply
Supported Beam/LE
% Error
Stress Deflection
Iter1
Iter2
Iter3
Iter4
Iter5
Iter6
0.65
5.84
-2.69
-3.14
2.70
5.25
27.37
45.32
19.57
18.84
40.13
33.97
Simply Supported Beam/LE % Error
50.00
40.00
30.00
Stress
Series2
20.00
10.00
0.00
0 1 2 3 4 5 6 7
-10.00
Iteration
Figure B.21: Summary of Percent Error between Exact Calculations and Model Results for a
Simply Supported Beam/LE
The results presented herein for a simply supported beam/LE demonstrate that all six model iterations are within approximately five percent of the exact calculations 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 exact 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 beam. That is, the ends of the beam 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.
B2.4 Clamped Beam/LE under Uniformly Distributed Load
For a clamped beam/LE under a uniformly distributed load, the optimal mesh determined from the previous studies was used for all model iterations. Boundary
B19

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 beam/LE under a uniformly distributed load: a) iteration 1 ( iter1 ): edge nodes of both ends completely constrained b) iteration 2 ( iter2 ): outside elements on both ends (one inch long each) fully constrained c) iteration 3 ( iter3 ): neutral axis of beam/LE completely constrained, all other edge nodes constrained in yz plane and allowed to move in x direction d) iteration 4 ( iter4 ): 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 a)
L b)
L
L c) d)
Figure B.22: Clamped Beam/LE Model Boundary Conditions for: a) iter1 b) iter2 c) iter3 d) iter4
Recalling from the main body, the calculated peak bending stress along the beam is
-29.83 ksi. This peak compressive stress is located along the inboard face of the flange of the beam 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
B20

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 exact solution away from the boundary conditions.
-40000
-60000
-80000
-100000
Stress
40000
20000
0
0.00
-20000
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)
Exact Clamped
Iter1 Edge Nodes
Iter2 Edge Elems 1in
Iter3 Constraint
Iter4 Edge Elems 5in
Figure B.23: Summary of Peak Compressive Bending Stress in Clamped Beam/LE
Stress
40000
20000
0
0.00
-20000
-40000
-60000
-80000
-100000
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
Beam Length (in)
Figure B.24: Summary of Peak Bending Stress in Clamped Beam/LE
(in way of Boundary Conditions)
Exact Clamped
Iter1 Edge Nodes
Iter2 Edge Elems 1in
Iter3 Constraint
Iter4 Edge Elems 5in
B21

Stress
29000
25000
21000
17000
13000
Exact Clamped
Iter1 Edge Nodes
Iter2 Edge Elems 1in
Iter3 Constraint
Iter4 Edge Elems 5in
9000
20.00
21.00
22.00
23.00
24.00
Beam Length (in)
25.00
26.00
27.00
28.00
Figure B.25: Summary of Peak Compressive Bending Stress in Clamped Beam/LE
(in way of center of beam)
The calculated peak deflection along the beam is -0.00221 in and is located along the center of the beam (see main body and Appendix A for calculations). Table B.11 compares the results of the exact 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 beam.
Table B.11: Summary of Peak Deflection in Clamped Beam/LE
Deflection % Error
Exact Value -0.0157950
Iter1 -0.0378100
-
139.4%
Iter2
Iter3
Iter4
-0.0332400
-0.0598930
-0.0186670
110.4%
279.2%
18.2%
B22

Deflection
0
-0.01
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
-0.02
-0.03
-0.04
-0.05
-0.06
Exact Clamped
Iter1 Edge Nodes
Iter2 Edge Elems 1in
Iter3 Constraint
Iter4 Edge Elems 5in
-0.07
Beam Length (in)
Figure B.26: Summary of Peak Deflection in Clamped Beam/LE
Table B.12 and Figure B.27 summarize the percent error of the model iterations relative to the exact 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 beam 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 exact 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 exact solution, and conversely, a negative percent error means the model predicted lower results than the exact solution.
Table B.12: Summary of Percent Error between Exact Calculations and Model Results for a
Clamped Beam/LE
Iter1
Iter2
Iter3
Iter4
% Error
Stress Deflection
10.07
-0.12
97.32
-33.93
142.18
115.03
295.25
16.26
B23

Clamped Beam/LE % Error
350
300
250
200
150
100
50
Stress
Deflection
0
0
-50
1 2 3 4 5
-100
Iteration
Figure B.27: Summary of Percent Error between Exact Calculations and Model Results for a
Clamped Beam/LE
The results presented herein for a clamped beam demonstrate that all four model iterations do not compare very well to the exact results. In all but one case, the models predicted higher stress and deflection results than the exact solutions. Based upon the results presented herein, model iteration 4 is best considered to represent the exact solution. While this model actually represents a smaller unsupported beam length (since elements on either side of beam 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 exact solutions, which is unacceptable. Additionally, the smaller unsupported length will better compare to the wrapped final configuration.
B24