The Use of the Finite Element Method to Obtain a Simplified Formula
for the Estimation of Stress Concentrations in Misaligned Compression
Members
by
Jeffrey Doyon
A 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
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
August 2011
CONTENTS
LIST OF TABLES ............................................................................................................ iii
LIST OF FIGURES .......................................................................................................... iv
LIST OF SYMBOLS ........................................................................................................ iv
ACKNOWLEDGMENT ................................................................................................... v
ABSTRACT ..................................................................................................................... vi
1. Introduction.................................................................................................................. 1
2. Methodology ................................................................................................................ 3
2.1
Finite Element Point Studies of Misalignment Configurations ......................... 3
2.2
Finite Element Model Pre Processing ................................................................ 4
2.3
Effect of Taper Size ........................................................................................... 6
2.4
Effect of Compression Member Thickness (Mt) ............................................... 9
2.5
Effect of Compression Member Length (L)..................................................... 12
3. Results and Discussion .............................................................................................. 15
3.1
Empirical Equation Development .................................................................... 15
3.2
Equation Coefficient Development .................................................................. 15
3.3
Results Comparison ......................................................................................... 19
3.4
Equation Testing and Validation...................................................................... 21
4. Conclusions................................................................................................................ 25
5. References.................................................................................................................. 26
ii
LIST OF TABLES
Table 1: Constant Values for Evaluating the Effect of the Taper Size .............................. 7
Table 2: Data Points for Determining C Coefficient ......................................................... 9
Table 3: Constant Values for Evaluating the Effect of the Member Thickness .............. 11
Table 4: Constant Values for Evaluating the Effect of the Member Length ................... 12
Table 5: Data Points for Determining the A Coefficient ................................................. 16
Table 6: Data Points for Determining the B Coefficient ................................................. 16
Table 7: 25% Offset Results in Terms of Taper to Length Ratios .................................. 19
Table 8: 50% Offset Results in Terms of Taper to Length Ratios .................................. 19
Table 9: 75% Offset Results in Terms of Taper to Length Ratios .................................. 20
Table 10: Equation Test Cases ........................................................................................ 22
Table 11: Equation Cases Results.................................................................................... 23
iii
LIST OF FIGURES
Figure 1: Misalignment Configuration .............................................................................. 2
Figure 2: Typical FEM Configuration ............................................................................... 6
Figure 3: Typical Deflected Shape Contour Plot ............................................................... 6
Figure 4: Effect of Taper on Stress .................................................................................... 7
Figure 5: Initial Concerns with Varying Mt .................................................................... 10
Figure 6: Effect of Member Thickness (Mt) on Stress .................................................... 11
Figure 7: Effect of Overall Length on Stress ................................................................... 13
Figure 8: Effect of Overall Length on Stress ................................................................... 14
Figure 9: Compression Member Stress Versus RL .......................................................... 18
LIST OF SYMBOLS
Rt……………………………………………...……………….Taper Ratio (unit less)
Mt………………………………………...……Compression Member Thickness (in)
Md…………………………………………………Compression Member Depth (in)
L..…………………………………………………Compression Member Length (in)
Lt…………………………………………………………………...Taper Length (in)
α.………..………...……………..Percent Offset of Compression Member (unit less)
Δ…………….……………………………..Compression Member Misalignment (in)
RL……….Ratio of the Compression Member Length to the Taper Length (unit less)
σnom……………………......………Typical Stress away from the Discontinuity (psi)
σ*….………….……. Stress in way of the Discontinuity at the End of the Taper (psi)
iv
ACKNOWLEDGMENT
To Professor Hufner, thank you for all of your help.
To my Wife, thank you for everything.
To Bernard Nasser Jr., my English professor.
v
ABSTRACT
This project develops and tests a simplified formula to estimate the effect of
misalignment repair geometry on the stress concentration at the end of a taper (taper
toe). The ability for a builder to assess and correct out-of-tolerance conditions is very
important. Holding up a ship construction to perform a detail analysis can cost
significant time and money. Finite element models (FEM) using ABAQUS CPE4R
reduced integration plane strain elements were used in this project to compute the stress
concentrations in selected misalignment configurations. Point study FEM were evaluated
by determining the effect of a changing taper size (Rt), member thickness (Mt), and
member length (L), on the stress at the end of a taper due to a misalignment. The results
of these point studies were compiled and used to develop an equation to calculate the
stress at the end of a taper. The final equation of this project is based of a nominal
compression stress, the ratio between the taper length (Lt) and the member length (L),
and the offset percent (α). Additional FEM were developed and the results were
compared to the final equation to verify the accuracy of the equation. Comparisons
between the equation of this project, equivalent FEM results, and a simplified closed
form equation (From Reference [2]) demonstrates that the equation developed is
accurate to within 5% or +/- 2 ksi for most configurations with a 2:1 taper (Rt).
Comparison to the simplified closed form equation shows the equation derived in this
project is a better prediction of stress in a misaligned compression member. Therefore
the equation of this study is considered to be sufficient for predicting stress in a
misaligned compression member.
vi
1. Introduction
When building large structures that rely on compression members (such as external
pressure vessels), there are construction tolerances involved. The constructions tolerance
alignment of two adjacent compression members is focused upon herein. Reference [1]
specifies the tolerances for commercial pressure vessels. Sometimes these tolerances are
exceeded as a part of the building process creating an out-of-tolerance (OOT) condition.
When this occurs, the builder must either fix the OOT condition, or perform a detailed
analysis to demonstrate the adequacy of the OOT condition. Both solutions require time
and money that can effect the profit of the builder. A more time efficient and cost
effective method of correcting an OOT condition is to create a tapered transition
between the misaligned members. Figure 1 shows an illustration of the general
misalignment configuration this project will evaluate.
The ability for the builder to evaluate an OOT condition without the time or cost of
repair (breaking a weld and re aligning the compression member) or detailed analysis
would help support construction schedules, and result in a more profitable product. The
empirical method developed in this project allows a builder to ensure that the tapers used
to correct an OOT condition will provide an acceptable compressive stress state.
Finite element models were developed to study the effects of a changing taper size (Rt),
member thickness (Mt), and member length (L), on the stress at the end of a taper due to
misalignment.
This project develops and presents a simplified formula for the evaluation of the stresses
in construction misalignments. The formula was tested by arbitrarily selecting ten
misalignment configurations and comparing the calculated results against those obtained
by FEA, as well as against equations given in Reference [2].
1
Compression Member
Misalignment
Tapered Correction Shown in Blue
Compression Stress
(Both Sides)
Figure 1: Misalignment Configuration
This configuration can be simplified into a beam under a simultaneous moment load
from the OOT, and a compressive load. The moment is induced from the misalignment
of the compression member and the typical compressive stress. As such it does not
appear in Figure 1 as it is a result of the geometry and loading of the compression
member. Reference [2] provides both theoretical and numerical solutions to a simplified
problem for a beam under simultaneous compression loading and moment loading.
Equations from Reference [2] are compared to the solutions from the empirical equation
of this project. Other works that have investigated similar problems include Reference
[3]-[5]. Each of these references looks at variable cross sections different from that of
this project. Specifically, Reference [3] investigates thin walled I beams under
simultaneous axial and transverse loading. Reference [4] investigates the buckling of
Tapered I –Beams. Reference [5] investigates Linear and non-linear buckling of various
cross sections under various loading scenarios.
2
2. Methodology
In this project FEM’s (using ABAQUS pre and post processing software) are used to
analyze various configurations of OOT compression members. Results from each of the
FEM studies are compared and used to develop an empirical formula. The following
sections describe the pre and post processing of all the models used as well as the
thought process used in developing the empirical formula. Specifically the effects of the
taper ratio (Rt), the compression member thickness (Mt), and the effects of the
compression member length (L), are initially investigated. Each variable is investigated
for several different member offsets (α). Other variables such as the length ratio (RL) are
developed as a result of the investigation to the effects of each of the initial variables
(RT, Mt, and L).
2.1 Finite Element Point Studies of Misalignment Configurations
This study initially looks at the effects of four key variables on the stress at the end of
the taper; overall length (L), the taper ratio (Rt), member thickness (Mt), and the offset
percentage (α) of the member thickness. Over the course of this study it was found that
these assumptions could be further reduced into two variables; the ratio of the taper
length to the overall length (RL), and the ratio of the offset (Δ) to the member thickness
(Mt) described as the percent offset (α).
Figure 2 below describes the typical nomenclature that will be used in this document.
Taper Ratio (Rt)
Member Thickness (Mt)
Length (Lt)
Length (L)
Compression Force: Derived from The typical
stress and the compression member cross
sectional area (Mt*Md)
Offset Shown as a
percent of Mt (Δ)
Member Depth (Md). Thickness
in/out of the Page
3
𝑅𝑡 =
Taper Ratio (Rt)
𝐿𝑡
∆
Taper Length (Lt)
𝐿𝑡 = 𝑅𝑡 ∗ ∆
Present Offset (α)
∝= 𝑀𝑡
Ratio of Lengths (RL)
∆
𝑅𝐿 =
𝐿𝑡
𝐿
[1]
[2]
[3]
[4]
The taper ratio (Rt) shown above describes the transition from the discontinuity back to
the typical cross section of the compression member. The taper ratio (Rt) is defined by
equation [1].
The taper length defines the length of the taper in the direction of compression, the x
direction in this paper. The taper length (Lt) is directly related to the misalignment of the
compression member (Δ) and the taper ratio (Rt). This is shown in equation [2] above.
The percent offset (α) is a ratio of the compression member thickness (Mt) to the
compression member misalignment (Δ). This variable is used in the final equation of this
paper to predict stresses at the end of a taper in a compression member. This variable is
defined in equation [3] above.
The ratio of lengths is a ratio of the taper length (Lt) over the compression member
length (L). This variable is used in the final equation of this calculation in conjunction
with the percent offset to predict the stress at the end of a taper in a compression
member. This variable is defined in equation [4] above.
2.2 Finite Element Model Pre Processing
For each of the finite element analysis (FEA) configurations shown herein, the nominal
stress was assumed to be 10 ksi. This stress was induced by applying appropriate joint
loads at one end of the model. When applicable the mesh density remained as consistent
as possible. The element type remained consistent for all models analyzed. Element size
was approximately 0.25” by 0.25” for configurations with an Mt value of 1.0” or greater.
Smaller elements sizes were used for smaller values of Mt. For all values of Mt 1.0” and
4
less there were four elements through the thickness. The shape of the elements for all
configurations remained as close to a square as possible. This mesh and the loads applied
can be seen below in Figure 2.
The elements used in this study were ABAQUS CPE4R reduced integration elements.
The elements were assumed to be 5” thick. It is noted that one of the assumptions of this
study is a plane strain assumption through the depths of the members. So no strain, or
the effect thereof, is considered in this study.
Material for all cases considered herein is steel with and elastic modulus of 30e6 psi and
a Poisson’s ratio of 0.3.
The boundary conditions applied to this model assume that the compression member is
fixed on the right hand side. The left end of the compression member is assumed to be
fixed in rotation and fixed in the y direction. The x direction is constrained to be constant
across the left end (uniform compression). These conditions represent a stiffener in
compression that is constrained at some point far away from the discontinuity. These
boundary conditions can be seen below in Figure 2.
Due to the discontinuity, and the bending that it causes, an additional boundary condition
is required on the left end to restrict the compression member to deflect uniformly in the
x direction at the left end. This boundary condition applies a constraint to the nodes on
the left end to all have the same x deflection. With this constraint the left end of the
compression member will act more as a pinned condition, as the left end will rotate in
about the z axis (in and out of the page). This constraint is shown below in Figure 2.
5
Y
CPE4R ABAQUS elements sized
approximately 0.25” by 0.25”
X
Dy= Rz=0, Dx=Constant
Nodal Loads Applied to create appropriate stress
All Degrees of Freedom (DOF) Fixed
Figure 2: Typical FEM Configuration
Area From which High Stress is reported
Figure 3: Typical Deflected Shape Contour Plot
Sxx Shown
2.3 Effect of Taper Size
Initially five different taper sizes with three different offsets were used to evaluate the
effect of the taper on the increase in stress at the end of the taper. Tapers (Rt) of 1, 3, 5,
7, 9, were considered sufficient enough for initially determining the effect of the taper.
Offsets (α) of 25%, 50% and 75%, for each of these tapers were evaluated. Offsets (α) of
0% were not investigated as this would mean a beam in compression with no
misalignment. Therefore there would be no increase in stress and σ*= σnom. Table 1
below describes the parameters held constant in these elvatuations.
6
Table 1: Constant Values for Evaluating the Effect of the Taper Size
Constant Values (in inches unless otherwise noted)
Member Thickness Compression Force (F) Member
(Mt)
(Lbf)
Depth (Md)
1
50,000
5
Length from
Taper to End (L)
10
Nominal Stress
(psi)
10000
Figure 4 below shows the results for the configurations used to analyze the effect of the
taper on stress. A second order polynomial of the form 𝑌 = 𝐴𝑥 2 + 𝐵𝑥 + 𝐶 has been
fitted to each of the offset trends, where A, B, and C are coefficients, y is the final stress
and x is the taper ratio (Rt).
Effect of Taper (Rt) on Stress
L=10" Mt=1"
-30000
y = -65.511x 2 + 2211.3x - 27182
R² = 0.9998
Stress at the end of the Taper (psi)
-25000
y = -34.234x2 + 1163.3x - 21879
R² = 0.9998
y = -5.1286x 2 + 293.87x - 15937
R² = 1
-20000
Offset 25%
Offset 50%
Offset 75%
-15000
Poly. (Offset 25%)
Poly. (Offset 50%)
Poly. (Offset 75%)
-10000
No Misalignment, Typical Stress
-5000
0
1
2
3
4
5
6
7
8
9
10
0
Taper Ratio (Rt)
Figure 4: Effect of Taper on Stress
Based on the results seen in Figure 4, when the compression member length (L) is held
constant, larger taper ratio’s (Rt) will reduce the stress. As well as small offset ratio’s
(α). This project will later show that the effect the taper ratio (Rt) on stress is directly
related to the length ratio (RL).
7
It should be noted that an offset of 75% and a taper ratio (Rt) of 7 and 9 shown in Figure
4 do not seem to correlate with the rest of the data. This can be attributed to the length
ratio (RL) defined as the size of the taper length (Lt) compared to the size of the
compression member (L). This is better described in Section 2.4.
Initial plots that showed the value for a 1:1 taper for a 25% offset (not shown in Figure
4) didn’t seem to correlate with the rest of the data provided as it was higher than
expected in comparison the other data points. Reviewing the FEM mesh it was
determined that this inconsistency could be attributed to the mesh density, and a
refinement of a singularity, as opposed to the geometrical configuration of the
compression member. Therefore, this one data point was re-meshed to find a point that
correlated with the rest of the information. This is considered an acceptable method as
the stress difference was attributed to a singularity at the end of the taper. Therefore, a
coarser mesh was applied to obtain the general equation that describes the compression
members. A coarser mesh will minimize the effect of the mild singularity that occurs at
hard corners (1:1 taper ratio) in FEM’s. Additionally a 1:1 taper is not considered a
realistic configuration. Per Reference [1] a 3:1 taper is the minimum that should be used.
The trends in Figure 4 readily show how increasing the offset percent increases the stress
at the end of the taper. Comparing the stress state at the end of the taper (σ*) for Rt=1 in
Figure 4 to the typical stress of the compression member (σnom), seemed to show a
relationship. This new stress (σ*) can be further described by looking at the y
(Coefficient C described above) intercept of the best fit equations provided. The y
intercept provides a good prediction of the trends of the new stress for a condition of no
taper; this project does not investigate a condition with no taper applied due to the stress
concentration (it is noted the Y intercept for the equation provided would be describing
x=0 or Rt=0.). If we assume that there will never be a configuration with an offset that
doesn’t have a taper, then it becomes a reasonable assumption to assume that the C
coefficient for describing the new stress (σ*) can be appropriately described with the
following data points for a 1:1 taper (Table 2):
8
Table 2: Data Points for Determining C Coefficient
Percent Offset (α)
Stress for Rt=1
0.25
-15000
0.5
-20000
0.75
-25000
These points can be described by the equation [5] below. The full equation is multiplied
by -1.094 to obtain an approximation of the C coefficient seen in the best fit equations of
Figure 4. This factor is considered to be adequate for obtaining the C coefficient seen in
the best fit function from the initial points provided in Table 2. It is noted that this factor
is later changed as the empirical equation develops. The variable σnom seen in equation
[5] is the typical stress away from the discontinuity. σnom is -10 ksi for the initial point
studies of this project.
𝐶 = (𝜎𝑛𝑜𝑚 + (2 ∗ 𝛼 ∗ 𝜎𝑛𝑜𝑚 )) ∗ −1.094
[5]
Based on the information in Figure 4, the equation below describes how the coefficient
A changed with respect to the percent offset for the compression member. This equation
is an initial equation and changes for the final form as the empirical equation develops.
𝐴 = −140 ∗∝2
[6]
The B coefficient did not show any pattern that could be described in simpler terms.
Therefore a quadratic curve of best fit is used to describe how the B coefficient changes
with respect to α. This initial chosen curve of best fit is described below. As with the A
and C Coefficient, this curve of best fit will change for the final equation as empirical
equation develops.
𝐵 = 3548.6 ∗ 𝛼 2 + 816.3 ∗ 𝛼 − 131.99
[7]
2.4 Effect of Compression Member Thickness (Mt)
To study the effect of the member thickness, six different Mt values with three different
offset percents (α) were used to evaluate the effect of the Mt on the increase in stress at
the end of the taper. Mt values of 0.25”, 0.50”, 0.75”, 1.0”, 1.5” and 2.0” were
considered sufficient for determining the effect of the member thickness. Offset values
were adjusted accordingly so that offset percents (α) of 25%, 50%, and 75% were
9
maintained for each point study. Additionally, the ratio between the length of the taper
(Lt) and the total length of the compression member (L) was maintained at 10 (RL=10).
Initial point studies that varied the Mt values without maintaining RL did not yield
consistent results. Evaluating the data showed that as the taper length (Lt) increased, the
compression member did not experience as much bending and therefore the stress
decreased. This lack of bending was attributed to relative moment of inertia of the
compression member being increased. This increase in the moment of inertia was a
result of a larger percent of the compression member being composed of the taper which
has a larger cross sectional area. Figure 5 below describes this configuration.
Member thickness in way of taper is larger. This causes a greater
cross sectional area and thereby a larger moment of inertia
Taper Length (Lt)
Length (L)
Increased moment of inertia causes a decrease in
bending stress and therefore a lower overall stress.
This decrease in stress is not a result of the member
thickness (Mt).
Figure 5: Description of the Affect of Large values of RL on Mt
Maintaining RL ensures that the taper has the same overall effect on the model for
different configurations. Table 3 below describes the constant variables in the
evaluations of changing the value of Mt on the increased stress.
10
Table 3: Constant Values for Evaluating the Effect of the Member Thickness
Constant Values (in inches unless otherwise noted)
Nominal Stress
Member
Depth
(Md)
(psi)
Length Ratio (RL)
Taper Ratio (Rt)
10
3
5
-10000
Figure 6 below shows the stress values at the end of the taper as a function of the
compression member thickness (Mt). The trend seen in Figure 6 shows that the overall
stress patterns remain constant as the compression member thickness increases. Minor
local stress increases are attributed to relative mesh refinement in lieu of an actual
increase in stress.
Effect of Member Thickness (MT) on Stress
Rt=3 L=10
-30000
Stress at the End of the Taper (psi)
-25000
-20000
Offset 25%
-15000
Offset 50%
Offset 75%
-10000
-5000
0
0
0.5
1
1.5
2
2.5
Member Thickness (Mt) (in)
Figure 6: Effect of Member Thickness (Mt) on Stress
Mesh refinement here is considered to be relative to the overall size of the compression
member singularity. For smaller values of Mt (≤1.0”) there are 4 elements through the
thickness. For larger values of Mt the elements size remains about the same (0.25”) but
there are more elements through the thickness. This increase in elements is a relative
11
mesh refinement when compared to the overall size of the modeled compression
member. This refinement causes the stress at the end of the taper to artificially increase
as seen in Figure 6 as the refinement is in way of the mesh singularity. Therefore, this
study considers that the thickness of the compression member does not have an effect on
the stress at the end of the taper. Only the relative size of the offset compared to the
member thickness has an effect.
2.5 Effect of Compression Member Length (L)
To study the effect of the member Length five different length values (L) with three
different offsets (α) were used to evaluate the effect of the length on the increase in
stress at the end of the taper. Length (L) values of 6.0”, 8.0”, 10.0”, 12.0”, and 14.0”
were considered sufficient for initially determining the effect of the member thickness.
Delta values (Δ) were adjusted accordingly so that offset percents (α) of 25%, 50%, and
75% were maintained for each point study. Table 4 below describes the parameter
values held constant in these evaluations.
Table 4: Constant Values for Evaluating the Effect of the Member Length
Constant Values (in inches unless otherwise noted)
Member Thickness
Nominal Stress
(Mt)
Member Depth (Md)
(psi)
Taper Ratio (Rt)
3
1
5
-10000
Figure 7 below shows the effect of the member length on the stress at the end of the
taper base on the initial 15 configurations described above.
The data seen in Figure 7 seems to show that the stress decreases when the length
becomes smaller. This decrease could be from the smaller length of the compression
member, or from a high value of RL. Therefore as with Section 2.4, this section looks at
the data in terms of a ratio of the taper length (Lt) to the overall length (L)(the RL value)
to ensure that any decrease in stress is a result of a smaller length in lieu of larger
relative taper length. Figure 8 below shows the same data as Figure 7 in terms of a ratio
of the taper length to the compression member length. Several more data points have
12
been added for each offset. These points were added to ensure each offset had data
points for approximately the same range of taper length to overall length ratios.
Effect of Overall Length (L) on Stress
Rt=3 Mt=1"
-25000
Stress At the end of the Taper (psi)
-20000
-15000
Offset 25%
Offset 50%
y = 64.273x 2 - 1984.2x - 8305
R² = 0.9986
-10000
Offset 75%
Poly. (Offset 25%)
Poly. (Offset 50%)
y = 33.718x2 - 1022.4x - 12276
R² = 0.9982
Poly. (Offset 75%)
-5000
y = 11.171x2 - 330.03x - 13249
R² = 0.9974
0
0
2
4
6
8
10
12
14
16
Overall Length (L)
Figure 7: Effect of Overall Length on Stress
Stress Vs. Length (L)
Figure 8 shows that as the overall length increases, the stress at the end of the taper
seem to be approaching a limit. This limit seems to be consistent with the “C”
coefficient described in Section 2.3. Additionally the data shown in Figure 8 suggests
that RL-1 values of 10 and higher have little change in stress. This provides justification
for the ratio maintained in Section 2.4. Figure 8 also describes the same concern
described in Section 2.4. Specifically Values of RL-1 less than 10 seem to be artificially
low. This is considered to be the effect of the taper occupying a larger percent of the
compression member.
13
Effect of Overall Length (L)on Stress
Rt=3 Mt=1"
Stress At the end of the Taper (psi)
-30000
-20000
Offset 25%
Offset 50%
-10000
Offset 75%
0
0.00
5.00
10.00
15.00
20.00
25.00
Length Ratio Inverse RL-1
Figure 8: Effect of Overall Length on Stress
Stress Vs. RL-1
14
3. Results and Discussion
3.1 Empirical Equation Development
Section 2 provides the data points evaluated for various different configurations of
compressions members. The data correlates better when the stress at the end of the taper
is described in terms of a ratio of the taper length to the overall compression member
length. Therefore this section combines the information of Section 2 and shows it in
terms of ratios.
Figure 9 below shows the stress results displayed in terms of the compression member
Length to taper length ratio (RL). Also shown are quadratic equations of best fit for each
set of data. These equations are used to develop the empirical equation for this project.
3.2 Equation Coefficient Development
The general form of the data shown in Figure 9 seems to be well represented by a
quadratic formula. Therefore as in Section 2.3, this project assumes that the stress in the
compression member can be described by an equation of the form 𝑌 = 𝐴𝑥 2 + 𝐵𝑥 + 𝐶
where Y is the final stress value (σ*) at the end of the taper and x is the ratio between the
taper length and the total compression member length (RL). The coefficients are assumed
to vary as the offset percentage (α) changes.
The A coefficient of each of the best fit equations can be considered to follow a pattern
similar to Section 2.3. As with Section 2.3, if the “A” coefficient is simplified to the data
points seen below in Table 5, then the “A” coefficient can be described with the
following equation:
𝐴 = −20000 + (10000 ∗ 𝛼)
[8]
It is noted that this equation is based off of a compression member with a nominal
compression stress of -10 ksi. Therefore, the equation will require adjustment for
compression members with typical stress states that are different from 10 ksi. This
adjustment will be addressed in Section 3.3 of this project.
15
Table 5: Data Points for Determining the A Coefficient
Percent Offset
0.25
0.5
0.75
A Coefficient
-17500
-15000
-12500
The “B” coefficient does not present any readily discernible pattern that can be
expressed in more simplified terms other than the best fit linear regression curve.
Therefore similar to Section 2, the “B” coefficient will be based off of empirical data
only. A quadratic and a linear best fit curve were fit to the points seen in Table 6. The
linear best fit was found to provide a better correlation to the final results from
ABAQUS. Therefore, the linear curve was used in the final equation of this project. The
equation for the “B” coefficient can be seen below.
𝐵 = 33794 ∗ 𝛼 + 6590.7
[9]
Table 6: Data Points for Determining the B Coefficient
Percent Offset
0.25
0.5
0.75
B Coefficient
31359
24642
14462
The “C” coefficient remains unchanged with the exception that the empirical
multiplications of 1.094 from Section 2 has been changed to 1.11. This change has better
correlations with the ABAQUS FEM stress results. Therefore, this change is used in the
final equation of this project. The equation for “C”, as seen in Section 2, is re-written
below. The σnom stress from Section 2 has been changed below to reflect the -10ksi in
which this project was based upon. This change is for clarity in Section 3.3 where the
typical stress will no long be -10 ksi. As with the “B” coefficient, the “C” coefficient is
based off of a typical stress of -10 ksi. This difference will be addressed in Section 3.3.
𝐶 = (−10000 + (2 ∗ 𝛼 ∗ −10000)) ∗ −1.11
[10]
The final equation with A, B, C substituted into the 𝒀 = 𝑨𝒙𝟐 + 𝑩𝒙 + 𝑪 equation can be
seen below (Equation [11]). This is the equation that is used in Sections 3.3 for the
comparison to ABAQUS FEM stresses. This is also the equation that produced the
16
results in Figure 9 under “equation stress”. The stress at the end of the taper, (σ*) is
described with the variable σ* in equation [11] below.
𝜎 ∗ = (−20000 + (10000 ∗ 𝛼)) ∗ 𝑅𝐿 2 + (33794 ∗ 𝛼 + 6590.7) ∗ 𝑅𝐿 + ((−10000 +
(2 ∗ −10000 ∗ 𝛼)) ∗ 1.11)
[11]
As was stated before, all typical portions (σnom from previous equations) have been
substituted with -10ksi. This is done for clarity in later sections when the typical stress in
not -10ksi.
Using Equation [11] above, a user is able to provide values of RL, σnom, and α, and obtain
a predicted stress at the end of the taper. This value will be close (within 5% or 2 ksi) of
the equivalent ABAQUS stress when assuming a plane strain condition. The following
Sections test this equation by randomly picking values of RL, σnom, and α, and comparing
the predicted stress to the equivalent ABAQUS stress.
17
S11 Stress Versus R L
-30000
y = -12595x2 + 31359x - 27937
R² = 0.995
y = -15311x2 + 24642x - 22294
R² = 0.9938
-25000
S11 Stress (psi)
y = -17627x2 + 14462x - 16259
R² = 0.957
Offset 25%
Offset 50%
-20000
Offset 75%
Equation [11]
Stress
Poly. (Offset 25%)
-15000
Poly. (Offset 50%)
Poly. (Offset 75%)
-10000
-5000
0.0000
0.1000
0.2000
0.3000
0.4000
0.5000
0.6000
0.7000
0.8000
0
RL
Figure 9: Compression Member Stress Versus RL
18
3.3 Results Comparison
Tables 7-9 below show the configurations of Section 2 in terms of the overall
compression member length (L), the taper length (Lt) and the ratio between the overall
length and the taper length (RL). Also shown, is the stress at the end of the taper as
evaluated with ABAQUS, the stress at the end of the taper as evaluated with the final
equation of this project, and the percent different between the two stress values. The
member thickness (Mt) for all the configurations shown below is 1.0”.
Table 7: 25% Offset Results in Terms of Taper to Length Ratios
Compression Member with 25% Offset
All Values in inches unless otherwise specified
Overall Length (L) Taper Ratio (Rt) Taper Length (Lt)
10
1
0.25
14
3
0.75
12
3
0.75
10
3
0.75
10
3
0.75
8
3
0.75
10
5
1.25
6
3
0.75
10
7
1.75
10
9
2.25
3
3
0.75
1.5
3
0.75
Length Ratio (RL)
0.0250
0.0536
0.0625
0.0750
0.0750
0.0938
0.1250
0.1250
0.1750
0.2250
0.2500
0.5000
Stress FEM (psi)
-15649.5
-15690.2
-15581.2
-15102.4
-15428.4
-15198.8
-14590.8
-14815.4
-14138.8
-13705.5
-13930.3
-13437.6
Equation Stress (psi) Percent Difference
-16285.0
-4.06
-15894.6
-1.30
-15778.4
-1.27
-15620.5
-3.43
-15620.5
-1.25
-15393.9
-1.28
-15043.5
-3.10
-15043.5
-1.54
-14554.1
-2.94
-14152.1
-3.26
-13984.0
-0.39
-13505.4
-0.50
Table 8: 50% Offset Results in Terms of Taper to Length Ratios
Compression Member with 50% Offset
All Values in inches unless otherwise specified
Overall Length (L) Taper Ratio (Rt) Taper Length (Lt)
10
1
0.5
30
3
1.5
14
3
1.5
12
3
1.5
10
3
1.5
10
3
1.5
8
3
1.5
10
5
2.5
6
3
1.5
10
7
3.5
10
9
4.5
3
3
1.5
Length Ratio (RL)
0.0500
0.0500
0.1071
0.1250
0.1500
0.1500
0.1875
0.2500
0.2500
0.3500
0.4500
0.5000
Stress FEM (psi) Equation Stress (psi) Percent Difference
-20746.1
-21063.1
-1.53
-21201.7
-21063.1
0.65
-20008.9
-19855.7
0.77
-19634.8
-19498.4
0.69
-18690.6
-19014.3
-1.73
-19118.6
-19014.3
0.55
-18363.2
-18323.4
0.22
-16959.3
-17265.6
-1.81
-17164.6
-17265.6
-0.59
-15364.9
-15816.8
-2.94
-14199.5
-14668.0
-3.30
-13956.9
-14206.2
-1.79
19
Table 9: 75% Offset Results in Terms of Taper to Length Ratios
Compression Member with 75% Offset
All Values in inches unless otherwise specified
Overall Length (L) Taper Ratio (Rt) Taper Length (Lt)
10
1
0.75
22.5
3
2.25
18
3
2.25
14
3
2.25
12
3
2.25
10
3
2.25
10
3
2.25
8
3
2.25
10
5
3.75
6
3
2.25
10
7
5.25
10
9
6.75
Length Ratio (RL)
0.0750
0.1000
0.1250
0.1607
0.1875
0.2250
0.2250
0.2813
0.3750
0.3750
0.5250
0.6750
Stress FEM (psi) Equation Stress (psi) Percent Difference
-25087.3
-25425.1
-1.35
-25192.1
-24681.4
2.03
-24366.7
-23953.3
1.70
-23536.6
-22940.3
2.53
-22762.7
-22201.4
2.47
-21014.8
-21197.2
-0.87
-21704.7
-21197.2
2.34
-20181.2
-19756.7
2.10
-17822.3
-17531.7
1.63
-17840.4
-17531.7
1.73
-14956.4
-14428.8
3.53
-12554.9
-11888.4
5.31
In general, all stress predictions from the equation are within 5% of the predicted
ABAQUS stresses. It is noted that the difference in stress is less than 1 ksi for all cases
evaluated above. Considering the accuracy of FEM in general, and considering the
accuracy of structural tolerances for building pressure vessels, 5% and 1 ksi is
considered sufficiently accurate for this problem.
It is noted that the highest percentage difference between the equation and the FEM
results are found when the taper ratio is small (1:1 Taper) or when the taper length ratio
(RL) ratio is high (~> 0.5). The reason for both of these differences is described in the
previous section. Specifically a 1:1 taper has a small singularity due to a hard corner, and
a small value of RL results in an artificially stiff compression member. Additionally
these differences are considered acceptable as a 1:1 taper is not considered to be a
realistic condition for ship building practice. As described in Section 2, the change in
stress for RL-1 ratios greater then 10 (.1 RL ) changes minimally. Additionally, the stress
increases as the RL ratio decreases. Therefore, it is suggested that the RL ratio be less
than or equal to 0.1. This restriction will mitigate the accuracy concern for high RL
ratios, and ensure the compression member is not overly stiff due to the presence of the
tapers.
20
3.4 Equation Testing and Validation
The equation developed in Section 3.3 for an offset compression member was based
only on a nominal Mt value of 1” and a nominal typical stress of -10 ksi. This provides a
very limited data pool to choose from when evaluating real world conditions that could
be of any configuration or nominal stress. Therefore, the equation in Sections 3.3
(equation [11]) was modified slightly to accommodate different typical stress values.
This modified equation can be seen below. The value of σnom is a compressive stress and
thereby denoted as a negative stress.
𝜎 ∗ = (−20000 + (10000 ∗ 𝛼)) ∗ 𝑅𝐿 2 + (33794 ∗ 𝛼 + 6590.7) ∗ 𝑅𝐿 + ((−10000 +
(2 ∗ −10000 ∗ 𝛼)) ∗ 1.11) ∗
𝜎𝑛𝑜𝑚
[12]
−10000
The full equation is multiplied by the ratio of the nominal stress (σnom) over -10 ksi. As
the equation was developed based on a -10 ksi nominal stress, this equation assumes
linear behavior to scale the predicted stress accordingly. Therefore, the user must ensure
that the results provided by this equation do not exceed the yield strength of the
particular material being considered.
To test the equation, ten different configurations were evaluated. Stress was constrained
between -1 ksi and -60 ksi. This was considered to be representative of most
compression pressure vessel designs that would use High strength steel (HSS) or High
Strength steel alloys (HLSA) as their yield strength is generally on the order of 50-60
ksi.
The taper ratio (Rt) was held to a whole number to be consistent with shipbuilding
practices. The thickness was required to be less than or equal to 30”. This range will test
the equation for compression members that look like a frame web or a frame flange. A
very large flange in a pressure vessel might reach as high as 30”. Therefore, this limit
was considered sufficient. It should be noted that the equation of this project was based
upon a plane strain assumption. Therefore, a flange of 30” may not be considered plane
strain depending on it’s thickness. Depending on the level of accuracy required,
21
scenarios outside of the plane strain assumption might not be accurately predicted by the
equation of this project.
The offset ratio (α) was held to less than 0.95. A 0.95 offset is representative of a
compression member that is almost fully offset (no direct load transfer). For most
shipbuilders, a misalignment of this magnitude would effect more than just a
compression member. Most likely stability and surrounding structure would also have to
be considered. Therefore, this constraint was used.
The length ratio (RL) was constrained to be less than 0.9. At a ratio of 0.9, the
compression member is comprised mainly of the taper. As discussed in previous
sections, the compression member is considered to be overly stiff when this ratio is
above 0.1. Therefore, this constraint was considered to allow for all reasonable cases that
could be considered.
Table 10 below shows the final test configurations used based on the constraints
described above.
Table 10: Equation Test Cases
CASE #
1
2
3
4
5
6
7
8
9
10
Typical Stress σtyp
Taper (Rt)
Overall
Thickness (Mt)
Length (L) Offset (Δ) (Between -1,000 and -60,000) (Must be Whole Number) (Between 1 and 30)
10
42.6
63.76
21
98.22
57.825
33
4
17.2
55
0.25
0.758
2
0.672
7.3
0.125
0.4965
0.99
0.1
0.462
-8423
-9273
-1001
-59999
-48395
-4000.3
-9999.9
-34609
-2056
-10000.5
10
7
3
12
9
14
6
1
2
8
6
22
14
3
29
15
1.5
2
0.467
0.789
Table 11 below shows the results for each case. Results are shown as determined by the
equation of this study, as determined from ABAQUS FEA, and from the Reference [2]
Table 8.8 case 3.D equation. No tapers are considered in the Reference [2] equation.
This is considered a valid assumption as Reference [2] states that for a transition that is
22
gradual the modifications for a tapered cross section are not needed. Since we are
holding RL to be less than 0.1. When the size of the taper is compared to the overall
length, the effect of the taper can be considered gradual. Additionally the effect of the
taper will only reduce the applied moment used in the Reference [2] equation. Looking
at the results provided in Table 11, this would yield results that are less accurate than the
results provided. The length is two times the length used in the equation of this project.
Stress is calculated by using the adjusted moment from Reference [2] for the bending
stress and adding it to the typical compressive stress. The location of the moment is
assumed to be half the length input into the Reference [2] equation.
Table 11: Equation Cases Results
Equation Stress
FEA Stress
Roark's
Stress
-9,475.3
-10,389.4
-1,337.5
-79,522.5
-69,746.4
-4,438.4
-16,979.3
-59,664.3
-3,227.0
-22,391.6
-9549.8
-11897.2
-1739.86
-79296.5
-66759.2
-4340.24
-17888.2
-70900.9
-3145.17
-22617.4
-8949.7
-9752.6
-1215.6
-81421.7
-66873.6
-4050.4
-15499.5
-60377.5
-2755.5
-269218.3
Percent Difference Percent Difference
(Equation Vs FEA)
(Roarks Vs FEA)
0.8
12.7
23.1
-0.3
-4.5
-2.3
5.1
15.8
-2.6
1.0
6.28
18.03
30.13
-2.68
-0.17
6.68
13.35
14.84
12.39
-1090.31
Differnce in Stress
(FEA to Equation)
Differnce in Stress
(FEA to Roark's)
-74.5
-1507.8
-402.4
226.0
2987.2
98.2
-908.9
-11236.6
81.8
-225.8
-600.1
-2144.6
-524.3
2125.2
114.4
-289.9
-2388.7
-10523.4
-389.7
246600.9
In general the equation of this study correlates to the FEA results within 5%. Case #3
exceeds this 5 % as it is off by 23.1%. However, it is noted that the stress was very small
in this case and the equation is still within 0.5 ksi. Therefore, the equation is still
considered accurate for this case. Compression members with a member thickness that is
large (Case #2 and #5) have a difference from the FEM results that is larger than other
cases. Both Cases #2 and #5 are within 3ksi of the FEM stress. However, as Case #2
only has a typical stress of -9.3 ksi, this results in a 23% percent difference. As discussed
in previous sections, larger values of Mt results in a relative refinement of the mesh in
the compression member. This refinement would result in an increase in the predicted
stress from the FEM. When the mesh for these compression members was re-meshed to
be coarser, the FEM stress values correlate with the equation prediction better.
23
Case #8 evaluates a 1:1 taper. This level of taper is not considered to be a realistic
configuration. Reference [1] requires that any offset be faired (tapered) with a 3:1 taper.
Additionally, the 1:1 taper has more of a stress concentration then the other
configurations. For these reasons, it is determined that Case #8 is outside the usable
range of the equation.
When the equation is compared to Reference [2], the equation correlates closer to the
FEM. However, it is noted that Reference [2] does not take into account a taper.
Reference [2] does however have empirical information to account for varying cross
section beams. These ratios scale down the effective moment based on the ratios
between the initial and final cross section. This scaling was not considered in this study.
As seen in Table 11, the stress, as determined from Reference [2], both over and under
predicts the stress when compared to the FEM. Scaling down the moment would reduce
these numbers. Therefore, the decision was made to assume the taper is gradual enough
that these factors are not required.
24
4. Conclusions
This paper provides a quicker and easier method for the estimation of stress
concentrations in misaligned compression members. The given expression can be used
instead of a detailed finite element analysis (FEA) without adverse loss or accuracy.
Misaligned compression members can be described by suitably selected relative sizes.
Specifically the ratios of the taper length to the overall length (RL), the percent offset (α),
and the nominal stress (σnom), are used to describe the stress at the end of the taper (taper
toe). The specific formula obtained in this study (equation [12]) uses these variables to
give the stress (σ*) at the end of a taper.
𝜎 ∗ = (−20000 + (10000 ∗ 𝛼)) ∗ 𝑅𝐿 2 + (33794 ∗ 𝛼 + 6590.7) ∗ 𝑅𝐿
+ ((−10000 + (2 ∗ −10000 ∗ 𝛼)) ∗ 1.11) ∗
𝜎𝑛𝑜𝑚
−10000
Ten arbitrary configurations were analyzed and compared to test the equation. All cases
with the exceptions of Case #2 and #8 were within 5% or 1 ksi of the predicted stress as
determined by ABAQUS FEA. Case# 2 differs from the FEA stress by 1.5 ksi. Case #8
is not considered to be a realistic configuration as it models a 1:1 taper. Therefore, the
equation of this project is considered to be accurate to within 5% for all configurations
that have a taper of 2:1 or higher. For compression members that have a large value of
Mt (>15”), the accuracy may be +/- 2 ksi which may be more than 5% depending on the
initial typical stress. Additionally, special care should be taken to ensure that the plane
strain assumption on which this project was based is not violated.
25
5. References
1. Rules for Building and Classing Underwater Vehicles, Systems and Hyperbaric
Facilities, American Bureau of Ship Building (ABS), Copyright 2002, Houston
Tx.
2. Roark’s Formulas for Stress and Strain, Seventh Edition, Warren C. Young and
Richard G. Budynas, Copyright 2002, McGraw-Hill.
3. Lateral Buckling of Thin-Walled beam-Column Elements Under Simultaneous
Axial and Bending Loads, Foudil Mohri, Cherif Bouzerira, Michel Potier-Ferry,
Thin Walled Structures, Vol 46 2008 , 290-302.
4. Lateral Buckling of Web-Tapered I-Beams: A New Theory,Zhang Lei, Geng
Shu, Journal of Constructional Steel Research, Vol 64, 2008, 1379-1393.
5. Linear and Non-Linear Stability Analyses of Thin-Walled Beams With
Monosymmetric I Sections, Foudil Mohri, Noureddine Damil, Michel PotierFerry, Thin Walled Structures, Vol 48, 2010, 299-315.
26