The Effect of T-Stiffener Web and Flange Tilt on Frame Stress
Evaluated using Finite Element Analysis
by
Dean Pasquerella
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
Major Subject: MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2014
CONTENTS
CONTENTS……………………………………………………………………………...ii
LIST OF TABLES………………………………………………………………………iii
LIST OF FIGURES……………………………………………………………………...iv
NOMENCLATURE……………………………………………………………………...v
LIST OF KEY WORDS…………………………………………………………….…...vi
ACKNOWLEDGEMENT………………………………………………………………vii
ABSTRACT……………..……………………………………………………………..viii
1.0 INTRODUCTION/BACKGROUND………………………………………………...1
2.0 THEORY/METHODOLOGY………………………………………………………..6
2.1 Specification Requirements…………………………………………………...…6
2.2 Assumptions……………………………………………………………………..6
2.3 Closed Form Analysis………………………………………………………...…8
2.4 Finite Element Analysis………………………………………………………....9
3.0 RESULTS/DISCUSSION…………………………………………………………..15
3.1 Closed Form Solution………………………………………………………..…15
3.2 Finite Element Analysis Results……………………………………………….15
3.2.1 Baseline Model Results………………………………………………….15
3.2.1 Model Iteration Results…………………………………………………15
4.0 CONCLUSIONS……………………………………………………………………18
5.0 REFERENCES……………………………………………………………………...20
APPENDIX A: SUPPLEMENTAL CALCULATIONS……………………………….A1
APPENDIX B: CONVERGENCE RESULTS…………………………………………B1
ii
LIST OF TABLES
Table 1: Web and Flange Tilt Model Iteration Results……………………………...………..15
Table A.1: Input Values for Appendix A Calculations……………………………………....A4
Table A.2: Output Values for Appendix A Calculations……………………………………A4
Table B.1: Summary of Peak Stress for Boundary Condition Evaluation……………………B3
Table B.2: Number of Elements and Element Size Convergence Analysis………………….B4
Table B.3: Summary of Peak Stress for Convergence Analysis……………………………...B5
iii
LIST OF FIGURES
Figure 1: Schematic Showing the Portions of a T-Stiffener………………………………...….1
Figure 2: Examples of Flat Plate Stiffener Arrangements…………………………………..….1
Figure 3: Examples of Cylindrical Plate Stiffener Arrangement……………………………….2
Figure 4: Elevation View of a Submarine Pressure Hull with Internal Stiffeners……………...2
Figure 5: Cross Section View of Different Web and Flange Tilt Combinations……………….3
Figure 6: Example of a Typical Shell and Frame Assembly Modeled using FEM…………….4
Figure 7: Sketch Showing Geometry Values for the Stiffener System………………………...8
Figure 8: Engineering Sketch of a Submarine Pressure Hull and Sub-Model Location……...10
Figure 9: Depiction of the Location of End Load and Displacement Boundary Conditions in
Relation to Frame Locations………..…………………………………………………………10
Figure 10: Depiction of the Location of End Load and Displacement Boundary Conditions in
Relation to Frame Locations …………………………………………………….……………12
Figure 11: Example FEM Showing End Load and Displacement Boundary Conditions….…12
Figure 12: Section View at Cut Plane Showing Location of Displacement Boundary
Conditions……………………………………………………………………………………..13
Figure 13: FEM Results for 2.0 Degree Web and Flange Tilt Condition……………………16
Figure 14: FEM Results for 2.0 Degree Web and Flange Tilt Condition Showing the Peak
Compressive Stress in the Flange……………………………………………………………..17
Figure 15: Un-deflected and Deflected Shape of FEM at 2.0 Degree Web and Flange Tilt
Condition……………………………………………………………………………………...17
Figure B.1: Section View of Pressure Hull Showing Frame Number Locations………….…B2
Figure B.2: Elevation View Showing Typical Frame Bay Deflections………………………B4
Figure B.3: Description Key for Table B.2 Column Titles…………………………………...B5
iv
NOMENCLATURE
Ac:
Aeff:
Af:
Afl:
Ap:
At:
Aw:
b:
d:
Do:
E:
FEA:
FEM:
g:
h:
hf:
L:
Lf:
p:
R:
Rcg:
Rf:
Ro:
tf:
wf:
yf:
yp:
yt:
yw:
:
:
:
θ:
:
:
f:
σc:
σmax :
σy:
Circumferential Area (in2)
Effective Cross-Sectional Area of the Frame (in2)
Area of the Frame (in2)
Area of Flange (in2)
Area of Plating (in2)
Total Section Area (in2)
Area of Web (in2)
Web Thickness (in)
Stiffener Depth (in)
Diameter to External Surface of Shell (in)
Modulus of Elasticity (lb/in2)
Finite Element Analysis
Finite Element Model
Acceleration of Gravity (in/s2)
Shell Plating Thickness (in)
Height of Fluid (in)
Unsupported Length of Shell (in)
Center-to-center Distance between Adjacent Frames (in)
Hydrostatic Pressure (psi)
Radius to Mean Surface of Shell (in)
Radius to Neutral Axis of the Frame (in)
Radius to Face of Flange (in)
Outside Radius of Shell (in)
Flange Thickness (in)
Flange Width (in)
Centroid of Flange (in)
Centroid of Plating (in)
Total Section Centroid (in)
Centroid of Web (in)
Ratio of the Effective Frame Area to Shell Area (dimensionless)
Ratio of the Flange Width to Frame Spacing (dimensionless)
Shell Stress Parameter (dimensionless)
Shell Flexibility Parameter (dimensionless)
Poisson Ratio (dimensionless)
Material Density (lb/in3)
Fluid Density (lb/in3)
Compressive Stress (psi)
Maximum Stress Allowable (psi)
Material Yield Strength (psi)
v
LIST OF KEY WORDS
Finite Element Analysis, Finite Element Model, Flange Tilt, Frame, Midbay, Pressure
Hull, Shell, Submarine, T-Stiffener, Tolerance, Web Tilt
vi
ACKNOWLEDGMENT
I would like to take this opportunity to thank my mother for her help and support throughout
the master’s program at RPI. I would never have reached this point in my career without her
guidance.
I would like to thank the entire RPI staff for their continued support and help in order to
complete my master’s degree.
Finally, I would like to thank my project advisor, Ernesto Gutierrez-Miravete, for his support
over the course of this project. It was a struggle at times and difficult to reach this point, but
your persistence and determination has been very helpful throughout this process.
vii
ABSTRACT
The goal of this project is to use finite element analysis to determine at what angular
offset a T-stiffener will exceed yield when subjected to uniform loading from submergence
pressure in a submarine application. It is unavoidable during the fabrication and construction
process of a plate/stiffener assembly to create geometrical imperfections. These imperfections
result in an increase in stress and deflection of the stiffener due to a modified load path.
However, during typical initial design, these deficiencies are not taken into account, as the
assembly is analyzed as having perfect geometry. Therefore, the effect on specification
requirements for stress due to tilt in the web and flange of a T-stiffener is unknown.
The evaluation will be performed on a cylinder supported by T-stiffeners, which is
externally pressurized to simulate a submarine under sea pressure. Initial engineering design
will be performed on a T-stiffener in which the web and flange are perpendicular to one
another and the web and cylinder are perpendicular to one another. This will serve as the
baseline model. The vertical web and horizontal flange pieces of the T-stiffener will then be
offset to certain angular imperfections resulting in increased stress and deflection in the Tstiffener. Varying offsets of the web and flange will be evaluated to determine what
construction tolerances are required to assure that the design does not result in stresses
exceeding specification tolerances, which in turn could lead to failure in the design.
viii
1.0 INTRODUCTION/BACKGROUND
This study focuses on the effect of construction tolerances inherent in the manufacturing
and assembly of T-stiffeners used to support a cylinder under a uniformly distributed load. Tstiffeners are used in numerous shipbuilding applications, such as in the design of tanks,
bulkheads, holding trunks, and especially in the cylindrical pressure hull of a submarine. A Tstiffener consists of a horizontal flange connected to a plate with a vertical web. The Tstiffener and plate assembly creates an I-beam cross-section. The web serves to resist the shear
force in the stiffener, while the flange serves to resist the bending moment experienced by the
stiffener. A schematic showing the web, flange and plate portions of a T-stiffener is shown in
Figure 1. Where “stiffener” is used herein, it refers to the T-stiffener design.
Figure 1: Schematic Showing the Portions of a T-Stiffener
Some examples of how stiffeners can be arranged in tanks, bulkheads or holding trunks
(i.e. flat plate designs) and also in pressure hull and non-pressure hull applications (i.e.
cylindrical designs) can be seen in Figures 2 and 3, respectively. Also, a schematic showing
the stiffener design used in a submarine pressure hull application is shown in Figure 4.
Figure 2: Examples of Flat Plate Stiffener Arrangements
1
Figure 3: Examples of Cylindrical Plate Stiffener Arrangement
Figure 4: Elevation View of a Submarine Pressure Hull with Internal Stiffeners
The engineering design of the stiffeners starts by creating in an initial design for each
specific application that assumes ideal geometry, meaning 90 degrees between the vertical
web and the base (i.e. pressure hull shell, tank structure, holding bulkhead) and 90 degrees
between the vertical web and horizontal flange. However, this ideal design does not take into
account the imperfections that result from the manufacturing and construction process of these
stiffeners. Stiffeners are used all over a submersible, in many different applications. Due to the
high volume of use and large scale and size of these stiffeners, imperfections are inherent in
the construction process and are unavoidable. Additionally, welding at high temperatures
results in perpendicularity of the plate/web and web/flange to be nearly impossible.
2
Tolerances are set forth to allow for a certain amount of inaccuracy during this process,
however, it is necessary to understand if these tolerances are adequate or not. Several different
scenarios using different combinations of flange and web tilt can exist, as shown in the
engineering sketches in Figure 5.
Figure 5: Cross Section View of Different Web and Flange Tilt Combinations
Angular offsets, or tilt, of the web and flange pieces leads to increased stresses in the
stiffeners. This could result in stresses exceeding the allowable limits set forth by the
specification documents, ending in failure of the design. Based on each specific application,
stress limits are set forth by specification documents, which are usually based on a percentage
of the materials yield strength. Fabrication specifications provide allowances for the amount
of tilt in degrees the web can have in relation to the plate, and the flange can have in relation
to the web. To assure these tolerances are satisfied, web and flange tilt measurements are
performed after construction of the assembly has been completed. If a web or flange tilt
tolerance is exceeded, the construction tradesman either fixes the out of tolerance condition, or
provides the condition to engineering to evaluate the specific application to determine its
acceptability via analysis.
Another important aspect of understanding the effects of imperfect stiffener conditions is
based on the rework that needs to be performed due to the stiffener exceeding construction
tolerances. Fixing the condition can be completed in several different ways. Some of these
solutions are for the web or flange pieces to be jacked back into place with jacking bars, the
3
welds of the web to plate, or of the flange to web to be cut loose and re-welded, to apply
additional weld beads over the existing welds (the heat from the welding can draw back or
pull the web/flange back into tolerance), or a combination of these methods. Whichever
solution is chosen, additional work is required which leads to increased cost, schedule
impacts, quality impacts, and the possibility to result in other construction errors and rework.
Therefore, by having a better understanding of how stiffener tilt affects the strength of the
stiffener, more conditions can be determined to be acceptable as-is using analytical
techniques, which results in less rework.
Finite Element Analysis (FEA) was used to determine at what web and flange tilts the
specification requirements are met and when they are exceeded. The first step will be to
perform an initial engineering evaluation using closed form solutions to determine a baseline
model. The initial engineering evaluation will analyze a cylindrical plate and a stiffener under
a uniformly distributed external load on the cylinder, which represents sea pressure on a
submarine. This initial engineering evaluation using a closed form solution will then be
compared with the results of a Finite Element Model (FEM) created with the same geometry.
An example of a typical shell and frame assembly modeled using FEM is shown in Figure 6.
In figure 6, elements used to model the shell, web and flange are shown in green, yellow and
red, respectively.
Figure 6: Example of a Typical Shell and Frame Assembly Modeled using FEM
4
This is typical in submarine design, as final configurations determined from FEA can be
very different from the initial design, due to modifications in the structure. Structural
modifications can be due to fabrication constraints, material availability or stress
concentrations from asymmetric structure and foundations that alter certain areas of the
design. Typically, an FEA based of an initial design will be modified several times until an
adequate solution is reached. This step of the design process will not be addressed in this
paper, as it is not within the scope of this project. However, it is important to note that
numerous iterations are performed before a final solution is reached.
Once a baseline FEM is developed, including a mesh convergence study and boundary
condition evaluation, the baseline FEM will then be modified by varying tilts in the flange and
web to evaluate at what angle the peak stresses in the stiffeners exceed tolerances.
5
2.0 THEORY/METHODOLOGY
2.1 Specification Requirements
Specification requirements are set based on the application being evaluated. Stiffeners
are used in many different applications on a submarine and are thus evaluated for many
different specification requirements based on the purpose of the stiffener. In some cases,
stiffeners are allowed to enter into the inelastic or plastic region, while in other cases,
stiffeners are required to remain within the elastic region. For the purpose of this paper, we
will assume that the stiffeners evaluated will be designed to remain within the materials elastic
region. This is an effective method to evaluate an application that experiences fatigue loading
to assure no permanent deformation exists throughout the life of the stiffener. The evaluated
loading condition experienced by a specific application is also based on the function of the
assembly. The application examined in this paper is a pressure hull submersible under sea
pressure, and hence, the loading is considered to be a uniformly distributed pressure load,
rather than a point load, partially distributed load or non-uniformly distributed load.
2.2 Assumptions
For the purpose of this analysis, several geometric properties, material properties,
loading and specification requirements will be chosen to resemble a cylinder/stiffener
assembly used in a typical submarine application.
Loading
A uniformly distributed pressure load (p) of 1100 psi will be applied on the external side
of the cylinder, which represents a submarine in the ocean at approximately 2475 feet deep. It
is assumed that all forces (loads and reactions) are normal to the plane of the plates. Ocean
pressure can be calculated using the following equation:
p = f * g * hf
(1)
In this equation, f is the density of water, g is the acceleration of gravity and hf is the height
of the fluid. The density of salt water depends on several factors, with the main factors being
the water salinity (the amount of salt content in the body of water) and the temperature of the
water. As for temperature, the temperature of the ocean varies greatly depending on location
6
on earth, the depth under water and the salinity of the water. For the purpose of this
evaluation, the density of water is assumed to be 62.4 lb/ft3.
Material Properties
To represent the typical properties used on a submarine pressure hull, the cylinder and
stiffer will be made of the same material, which is high yield steel having a yield strength (σy)
of 100 ksi, i.e. HY-100 steel. The steel has a density () of 0.283 lb/in3, a Poisson Ratio (ν) of
0.3, and a modulus of elasticity (E) of 30E6 psi. Also, material assumptions are made such
that the steel is of uniform thickness and homogenous isotropic material. The steel has the
same modulus of elasticity in tension and compression.
Stress Allowable
For the purpose of this evaluation, we will assume that to achieve a maximum stress
equal to the materials yield strength, as specified in Section 2.1, the maximum allowable stress
in our baseline analysis is less than 95% of the material yield strength. This 5% allowance will
be placed into the design to account for the manufacturing uncertainty analyzed with the
models. Thus, the maximum stress (σmax) in the baseline model will be held to 0.95*(100 ksi)
= 95 ksi.
Geometry
Many factors contribute to the geometry of the cylinder and stiffener assembly, with an
overall goal of creating an assembly that meets the design intent, but results in the lowest cost
to build and weighs the least. Additionally, before the stiffeners of a submarine can be
analyzed, the arrangement of all internal components is evaluated. This is important as
limiting free space is essential in cutting down on weight and size of the submarine, which
results in increased submarine capabilities. This process of the design can take several months
or even years, with a multitude of iterations. Therefore, for the purpose of this analysis, we
will assume an arrangement team review and a cost/weight analysis has been performed which
results in the following geometry and thickness values:
7
Pressure Hull Shell Outside Diameter = d = 30’ = 360”
Pressure Hull Shell Thickness = h = 1.5”
Center to Center Distance Between Adjacent Frames = Lf = 30”
Stiffener Depth = d = 10.0”
Flange Width = wf = 6.0”
Web Thickness = b = 0.5”
Flange Thickness = tf = 0.7”
A sketch showing the stiffener system is shown in Figure 7:
Figure 7: Sketch Showing Geometry Values for the Stiffener System
Additionally, it is assumed that the plates have no imperfections, i.e. radii are constant and
exact, and there are no variations in length/width/depth of the stiffeners, so the cross section is
uniform.
2.3 Closed Form Analysis
The ability of a circular cylindrical shell reinforced by uniformly spaced ring frames to
support a pressure load is documented in Reference [1]. The paper investigated the
axisymmetric elastic deformations and stresses in a ring stiffened circular cylindrical shell
8
under hydrostatic pressure and properly considered the nonlinear “beam-column” effect
produced by the combination of longitudinal compression with longitudinal bending. In the
case of a ring-stiffened cylinder under hydrostatic pressure, a portion of the shell will act
effectively with the ring frame to resist direct stress and bending moment caused by the
interaction between the shell and frames. The equation for the effective width of the shell
plating is provided in Appendix A. Appendix A also provides all the equations used to
calculate maximum stress on the internal face of the flange in the stiffener/plating assembly
discussed herein.
2.4 Finite Element Analysis
For the FEMs evaluated herein, 2-dimensional, or 2-D, elements were utilized.
Furthermore, they were four noded, quadrilateral general shell elements. The FEM geometry
was constructed in Patran, Reference [2], while the finite element computation and postprocessing was performed with Electric Boats in-house software COMMANDS, Reference
[3]. The FEMs could have been constructed using either 2-D or 3-D elements, as both would
have yielded accurate results with the geometry of this study. 2-D elements can be used in lieu
of 3-D elements when applications have some form of symmetry and minimal complexity in
the geometry to be model, which is the case for the assemblies evaluated herein. 2-D elements
allow for quicker modeling time, shorter runtime and reduced cost. Additionally, 2-D
elements are frequently utilized through submarine design, and were therefore used herein to
conform to typical industry standards.
There are two types of 2-D elements that are commonly used, which are triangular
elements and quadrilateral elements. For the models created herein, only quadrilateral
elements were utilized. Quadrilateral elements were used to create a structured grid, which
was easy to construct and mirror on planes of symmetry. Triangular elements were not needed
since the geometry and mesh were relatively simple to construct. Triangular elements are
much more efficient when modeling asymmetric structure and areas where multiple parts
intersect.
Since many engineering applications evaluated using FEA, such as a submarine pressure
hull, can be of a large scale, it is important to create sub-models to evaluate only the portion of
the problem that is being analyzed. Creating a sub-model to reflect the condition of an entire
9
submarine can be done using end loading and boundary conditions. Figure 8 presents an
engineering sketch of an entire submarine pressure hull and depicts the smaller area of the
submarine pressure hull which will be modeled.
Figure 8: Engineering Sketch of a Submarine Pressure Hull and Sub-Model Location
To represent the portion of the pressure hull which is not modeled, nodal end load boundary
conditions and displacement boundary conditions are used, as shown in Figure 9. Nodal end
loads were arbitrarily chosen to be on the aft end of the model, while displacments were
chosen to be on the forward end of the model. These locations could have been reversed and
the reuslts would not have been affected. The purpose of the nodal end load and displacement
boundary conditions are to represnet the portions of the submarine that have been removed
from the model. Also, external pressure loads are applied radially around the entire pressure
hull shell.
Figure 9: Depiction of the Location of End Load and Displacment Boundary Conditions
10
The nodal end load boundary conditions are applied axially to the aft end of the model
and represent the total pressure load exerted over the circumferential area of the submarine
that has been removed. Therefore, the nodal end load that is applied can be calculated using
the equation below, where p is the hydrostatic pressure, Ac is the circumferential area
removed, R is the radius at the shell mid-surface, and n is the total number of nodes for which
the end load is being applied:
End Load = p*Ac/n = p*π*R2/n
(2)
The displacement boundary conditions in the evaluation are applied on the forward end
of the model using nodal displacements restricting certain directional movements. These are
applied as constrained, or fixed, boundary conditions. This is discussed in more detail in the
upcoming paragraphs.
The coorindate system for the sketch in Figure 9, and all FEMs used herein, is as
follows:
x-axis: fore/aft direction with zero being at the models forward perpendicular
y-axis: up/down with zero being at the models main axis of revolution
z-axis: left/right or athwartship (port/starboard) with zero being at the models main axis of
revolution
The longitudinal location at which the boundary conditions will be applied are such that
they are at a midbay (the location half way between two consecutive webs). This is because
the deflected shape of the shell is sinusodal, with the midbay being the plane of symmetry. A
depiction of this can be seen in Figure 10.
11
Figure 10: Depiction of the Location of End Load and Displacement Boundary
Conditions in Relation to Frame Locations
Figure 11 presents an isometric view showing an FEM example of the end load and
displacement boundary conditions. End loads are shown in yellow and applied to the aft end
of the model, while displacement are shown in blue and are applied to the forward end of the
model.
Figure 11: Example FEM Showing End Load and Displacement Boundary Conditions
12
Based on the mesh density of the shell, the number of nodes (n) at which the end load is
applied varies. Therefore, while the total end load stays constant for FEMs with different mesh
densities, the nodal end load applied to each node can vary.
The displacement boundary conditions restricting directional movements are applied
along the same y-z plane to represent the area of the submarine that has been removed from
the model. The displacements for the nodes on the y-z plane of the model where it has been
cut are as follows:
 For all nodes in the y-z plane, the displacement in the x axis, and the rotations in the
x, y and z axes are fixed (constrained to zero movement). These boundary conditions
are used due to the plane of symmetry for the deflected shape, where the displacement
is expected to be entirely radial. This is the point of maximum deflection, so no
rotations are expected.
 The points at top centerline and bottom centerline will have a fixed boundary
condition in the z axis, as the deflection at these locations is completely vertical, with
no z displacement expected.
 To the same fact, the points at ships centerline on the port and starboard sides will
have a fixed boundary condition in the y axis, as the deflection at these locations is
completely side to side, with no y displacement expected.
Figure 12 shows an engineering sketch depicting these displacement boundary conditions.
Figure 12: Section View at Cut Plane Showing Location of Displacement Boundary
Conditions
13
The two remaining finite element assumptions required before results can be obtained
are to determine the size of the model (how many frames and frame bays are required to be
modeled) and to perform a mesh convergence study. These evaluations are documented in
Appendix B.
14
3.0 RESULTS/DISCUSSION
3.1 Closed Form Solution
Appendix A presents the input and output values, as well as the equations, used in the
closed form analysis. As shown in Table A.2 of Appendix A, the closed form evaluation
shows the peak compressive stress on the internal face of the flange is 94.79 ksi, which will be
used in Section 3.2 to confirm the FEA solution for the baseline model. The peak compressive
stress of 94.79 ksi is less than the 95 ksi stress allowable presented in Section 2.2. Therefore,
since the peak stress is less than the maximum allowable stress, the initial design is
acceptable.
3.2 Finite Element Analysis Results
3.2.1 Baseline Model Results
Mesh and boundary condition convergence studies were conducted for the baseline
model, as presented in Appendix B. The baseline FEM results in a peak compressive stress on
the internal face of the flange of -94.69 ksi. The peak compressive stress of -94.69 ksi
determined by the FEA is within 0.11% of the peak compressive stress determined by the
closed form analytical solution presented in Section 3.1. This percent difference is considered
adequate for the level of accuracy required for pressure hull structure design.
3.2.2 Model Iteration Results
The web and flange tilt were varied by 0.5 degree intervals until the resulting peak
compressive stress in the frame exceeded the 100 ksi allowable. The results of this evaluation
are presented in Table 1.
Table 1: Web and Flange Tilt Model Iteration Results
Web and Flange
Tilt (Degrees)
0.0
0.5
1.0
1.5
2.0
Peak Compressive
Stress in Frame (ksi)
-94.69
-96.36
-97.87
-99.38
-101.42
15
As shown in Table 1, the peak compressive stress in the frame exceeds the allowable
requirement of 100 ksi at a web and flange tilt of 2.0 degrees, which resulted in a peak
compressive stress of -101.42 ksi. Stress plots showing the condition for the 2.0 degree web
and flange tilt are provided in Figures 13 and 14. Figure 13 shows an isometric view of the
web and flange of the FEM, so only the shell elements have been removed. The fringe plot
scale can be seen on the right side of the plot. Figure 14 shows a zoomed in view from the
Figure 13 plot and shows the location of the peak compressive stress in the flange.
Figure 13: FEM Results for 2.0 Degree Web and Flange Tilt Condition
16
Figure 14: FEM Results for 2.0 Degree Web and Flange Tilt Condition Showing the
Peak Compressive Stress in the Flange
For informational purposes, Figure 15 shows the un-deflected and deflected shape of the FEM
for the 2.0 degree web and flange tilt condition. Figure 15 is an elevation view cut down the
center of the model, with the deflected shape at 20 times magnification
Un-Deflected Shape
Deflected Shape
Figure 15: Un-deflected and Deflected Shape of FEM at 2.0 Degree Web and Flange Tilt
Condition
17
4.0 CONCLUSIONS
The results presented herein demonstrate the increase in stress as a result of imperfect
geometry in a plate/stiffener assembly, which could exist in real world applications such as
frames in a submarine pressure hull. Imperfect frame geometry, mainly the tilt of the web in
relation to the pressure hull and the flange in relation to the web, is unavoidable in the
construction/fabrication process and results in an increase in stress in the frame due to a
modified load path. During typical initial design, these deficiencies are not taken into account,
as the assembly is analyzed as having perfect geometry. However, in this evaluation, an initial
design was performed on a pressure hull shell application using a closed form solution to
create an assembly that contains a 5% allowance in the peak compressive stress in the frame.
Specifically, design requirements were chosen such that the material does not enter into the
inelastic region, or in this case, above 100 ksi. The initial closed form solution resulted in a
peak compressive stress in the frame of -94.79 ksi, which results in an acceptable initial
design.
Using 2-dimension quadrilateral elements, the initial design was constructed using FEA
to validate the closed form solution and allow for modifications of the stiffener geometry. The
baseline FEM resulted in a peak compressive stress in the frame of -94.69 ksi, which has a
percent difference from the closed form solution of 0.11%. A difference of 0.11% is
considered adequate for the level of accuracy required for pressure hull structure design.
The baseline FEM was then modified by varying the angular offset of the web to the
pressure hull and the flange to the web until a peak compressive stress exceeded the 100 ksi
allowable limit. Using this approach, a web and flange tilt of 2.0 degrees resulted in a peak
compressive stress of -101.42 ksi, which exceeded the allowable by 1.42 ksi, while a web and
flange tilt of 1.5 degrees resulted in a peak compressive stress of -99.38 ksi, which is within
the allowable limits. Therefore, for the pressure hull design evaluated herein, the specification
documentation shall require a web and flange tilt of less than or equal to 1.5 degrees to assure
yield of the material is not exceeded, which could result in failure of the design.
The 1.5 degree requirement for web and flange tilt is within the capabilities of welding
and assembly of typical pressure hull submarine structure. Knowing the allowable web and
flange tilt tolerance permits engineering, design and trades personnel to make informed
18
decisions about construction of the pressure hull. During measurement of these geometrical
properties, any value below 1.5 degrees for web or flange tilt can be accepted without
requiring rework of the structure. Additionally, steps can be taken in the fabrication and
assembly processes to utilize this tolerance to construct the submarine while taking into
account quality, time and cost.
19
5.0 REFERENCES
[1]
John G. Pulos and Vito L. Salerno, Axisymmetric Elastic Deformations and Stresses in a
Ring-Stiffened, Perfectly Circular Cylindrical Shell Under External Hydrostatic Pressure,
Department of the Navy David Taylor Model Basin, Report 1497, dated September 1961.
[2]
MSC Software Corporation, Patran, Version 2010.2.3.
[3]
J. F. Waters and M. L. Woratzeck, CMT-2.1.1 Input Formats/Data Entry, General
Dynamics Electric Boat Division, dated March 1984.
[4]
F. Beer, F. Johnston and J. DeWolf, Mechanics of Materials, Forth Edition, McGraw-
Hill, dated 2006.
20
APPENDIX A:
SUPPLEMENTAL CALCULATIONS
A1
A1.0 PURPOSE
The purpose of this Appendix is to provide the supplemental calculation for the
evaluation discussed in Section 2.3 and the results documented in Section 3.1. This appendix
will show the calculations performed to determine the peak compressive stress on the internal
face of a nominal stiffener flange.
A2.0 RESULTS
The calculation for the total section area, total section centroid (Reference [1]), nondimensional parameters (Reference [4]) and peak compressive stress on the internal face of a
nominal stiffener flange (Reference [4]) are given as follows. Symbols for inputs used in this
section can be seen in the Nomenclature section of this paper.
A2.1 Total Section Area
Area of Plating = Ap = Lf*h
(3)
Area of Web = Aw = b*(d-tf)
(4)
Area of Flange = Afl = wf*tf
(5)
Area of Frame = Af = Aw + Afl
(6)
Total Section Area = At = Ap + Af
(7)
Centroid of Plating = yp = d + 0.5*h
(8)
Centroid of Web = yw = tf + 0.5*(d - tf)
(9)
A2.2 Total Section Centroid
Centroid of Flange = yfl = 0.5*tf
(10)
Total Section Centroid = yt = ((Ap*yp)+(Aw*yw)+(Afl*yfl))/At
(11)
Radius at Frame Center of Gravity = Rcg = R – h – d + yt
(12)
A2.3 Non-Dimensional Parameters
Aeff is the effective cross-sectional area of the frame, which is given by:
(13)
A2
α is the ratio of the effective frame area to shell area, which is given by:
(14)
β is the ratio of the flange width to frame spacing, which is given by:
(15)
 is the shell stress parameter, which is given by:
(16)
The following non-dimensional parameters are introduced to calculate the stress function of
the load and gemoetry of the shell:
(17)
(18)
ϴ is the shell flexibility parameter, which is given by:
(19)
Where L is the unsupported length of shell, which is given by:
L = Lf - b
(20)
F1 is the stress function of the load and geometry of the shell, which is given by:
(21)
A2.4 Direct Compressive Stress
Finally, σ is the direct compressive stress on the internal face of the stiffener flange, which is
given by:
(22)
A3
Where Rf is the radius to the inside face of the flange, which is given by the following:
Rf = Ro – h – d
(23)
Table A.1 shows the input values used in this appendix. Table A.2 shows the results of the
equations documented in this appendix.
Table A.1: Input Values for Appendix A Calculations
Symbol
Ro
Value
180
Units
in
R
179.25
in
Rf
168.5
in
h
1.5
in
Lf
30
in
L
b
d
29.5
0.5
10
in
in
in
tf
0.7
in
wf
6
in
p
1100
psi
E
v
3E+07
0.3
psi
-
Table A.2: Output Values for Appendix A Calculations
Symbol
Value
Area
Units
Ap
45.0
in2
Aw
4.7
in2
Af l
4.2
in2
Af
8.9
in2
At
in2
yp
53.9
Centroid
10.8
yw
5.4
in
yf l
0.4
in
yt
9.5
in
in
Rcg
178.0
in
Non-Dimentaionl Parameters
in2
Aef f
8.914



η1
0.198
0.017
0.433
0.377
-
η2
ϴ
0.598
2.313
-
F1
0.776
-
σ
Stress
94.79
ksi
A4
APPENDIX B
CONVERGENCE ANALYSES
B1
B1.0 PURPOSE
The purpose of this appendix is to provide the results from the two initial evaluations
performed to determine the most accurate baseline model to use in this evaluation. These two
evaluations are a boundary condition evaluation and a mesh convergence study. These
evaluations support the FEA results presented in the main body of this paper.
B2.0 RESULTS/DISCUSSION
B2.1 Boundary Condition Evaluation
Based on the location of boundary conditions, they can have a local effect on the FEM
and impact the results. Therefore, this evaluation determined how many frame bays were
required to produce results that are far enough away from the boundary conditions to not have
an effect. This was accomplished by comparing the peak stress in the flange to the closed form
solution.
The FEM used in this evaluation used all of the same modeling assumptions discussed in
the main body of this report, which includes materials, thickness, geometry, loading and
boundary conditions. The FEM used for this evaluation (titled AppendixB.BC.Eval.cdb)
modeled six stiffeners, which was determined to be a large enough to mitigate any boundary
condition effects. Figure B.1 shows the location of each frame in relation to the boundary
conditions and end loads.
Figure B.1: Section View of Pressure Hull Showing Frame Number Location
B2
The results for this evaluation are shown in Table B.1, which presents the peak stress in
the flange at each frame location, as well as a percent change in stress from the adjacent
frame.
Table B.1: Summary of Peak Stress and Deflection for Boundary Condition
Evaluation
Frame # Frame Stress (ksi) % Change
1
-92.56
2
-94.21
1.78
3
-94.35
0.15
4
-94.32
0.03
5
-94.32
0.00
6
-94.32
0.00
B2.2 Mesh Convergence Study
The mesh used for modeling FEM elements can affect the FEM and therefore impact the
results. If a mesh is too fine or too coarse, the results can be uncertain. Additionally, the finer
the mesh, the longer the model run time, so a mesh that is too fine can be unnecessary and lead
to extensive analysis run time. Thus, this convergence analysis was performed to assure the
element size and mesh density of the FEM is such that it provides the most accurate results
compared to the analytical closed form solution. The mesh convergence study was only
performed on the baseline FEM and not the subsequent web and flange tilt modified FEMs, as
all of the models have the same geometrical dimensions, excluding web and flange angular tilt.
It can be assumed that the results of the mesh convergence study performed for the baseline
FEM provides the same level of accuracy as with the modified FEMs. As with the model
discussed in Section B2.1, the FEMs evaluated in this study used all of the same modeling
assumptions discussed in the main body of this report, which includes materials, thicknesses,
geometry, loading and boundary conditions.
There are several geometric properties that result in initial constraints to this
convergence study. The constraints used for determining the number of elements in the mesh
and thus the size of the elements are as follows:

For the elements used to model the pressure hull shell, the longitudinal size of the
elements as they relate to the distance between the webs was chosen such that the
B3
number of elements was a multiple of four. This is because based on the expected
deflection of the shell being sinusoidal, the maximum deflection is located half way
between the webs, and the points of inflection for the deflected shape of the shell are
located at one quarter and there quarters between the webs. Therefore, the mesh was
chosen so that nodes were located at these locations to most accurately represent the
expected deflection. An engineering sketch showing this can be seen in Figure B.2.
Figure B.2: Elevation View Showing Typical Frame Bay Deflection

The number of elements chosen for the circumferential extent of the shell was the
same for the flanges and webs.

The flange uses an even number of elements so that nodes could be evenly spaced
and will line up at the intersection of the web and flange.
Based on these constraints, four convergence models were evaluated with the number of
elements and corresponding element sizes as shown in Table B.2. The sketch provided in
Figure B.3 depicts the column titles of Table B.2.
B4
Table B.2: Number of Elements and Element Size for Convergence Analysis
Model Name
Converge1.cdb
Converge2.cdb
Converge3.cdb
Converge4.cdb
Shell
# of Elements
Circumferentially Longitudinally Per Bay
120 = 3 degrees
4
180 - 2 degrees
4
270 - 1.5 degrees
8
360 = 1 degrees
8
Web Height
Flange Width
Size (in.) # of Elements Length (in.) # of Elements Length (in.)
9.4 x
6.3 x
4.2 x
3.1 x
7.5
7.5
3.8
3.8
2
3
3
4
5.2
3.5
3.5
2.6
2
2
4
4
3.5
3.5
1.8
1.8
Figure B.3: Description Key for Table B.2 Column Titles
The results for this evaluation can be seen in Table B.3, which presents the peak stress in
the flange for each of the four convergence models, as well as a percent change from the
closed form solution of -94.79 ksi presneted in Appendix A.
Table B.3: Summary of Peak Stress and Deflection for Convergence Study
B3.0 CONCLUSION
Based on the results presented in Table B.1, the peak flange stress in each frame is
within 2% of one another. However, between the fourth and fifth frame, there is no change in
percent difference (when rounding values to two decimal points), meaning that the accuracy of
the results at the fifth frame is no better than that shown at the fourth frame. Therefore, for the
B5
evaluation documented in the main body of this report, all of the FEMs will be modeled with
four frames. Although the models could be performed with more frames and lead to similar
results, this is not necessary and would lead to increased modeling time and analysis run time.
The peak stress results will be evaluated at the fourth frame away from the boundary
conditions as this is determined to be within the accuracy required for this analysis.
Based on the results presented in Table B.3, the peak flange stress in each convergence
model is less than or equal to 0.50% different from the closed form solution pressneted in
Appendix A. However, the model with the smallest percent difference is Converge4.cdb, with
a percent difference of 0.11%. This model is constructed with the tightest mesh density, so it
contains the samllest quadratic elements and the largeest number of total elements out of the
four convergence models. Due to the relatively small number of nodes and elements in the
convergence models, there is no significant cost or time increase due to the tighter mesh
desnity in the convergnece models. Therefore, for the evaluation documetned in the main body
of this report, all of the FEMs will contain a mesh density equivalent to that of the
Convere4.cdb model, which is shown in Table B.2.
B6