A Continuum Mechanics Approach of Deriving Stress Tensor
Components of Double Shear-Plane Revolute Joints in the Elastic
Domain
by
Christopher Stubbs
A Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Professor Ernesto Gutierrez-Miravete, Thesis Adviser
Additional Committee Members:
_________________________________________
Professor Norberto Lemcoff
_________________________________________
Professor Sudhangshu Bose
Rensselaer Polytechnic Institute
Hartford, CT
November 2014
(For Graduation December 2014)
i
© Copyright 2014
by
Christopher Stubbs
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
LIST OF SYMBOLS ....................................................................................................... vii
ACKNOWLEDGMENT ................................................................................................ viii
ABSTRACT ..................................................................................................................... ix
1. Introduction.................................................................................................................. 1
1.1
Background ........................................................................................................ 1
1.2
History ................................................................................................................ 2
1.3
Objective ............................................................................................................ 9
2. Theory and Method.................................................................................................... 10
2.1
Procedure.......................................................................................................... 10
2.2
Method ............................................................................................................. 10
2.2.1
Finite Element Model Development .................................................... 10
2.2.2
Output................................................................................................... 19
3. Results and Discussion .............................................................................................. 20
3.1
Output Requests ............................................................................................... 20
3.2
Example Output ............................................................................................... 20
3.3
Regression Analysis ......................................................................................... 24
3.4
Verification Analyses ....................................................................................... 31
4. Conclusions................................................................................................................ 34
4.1
Future Work ..................................................................................................... 34
5. References.................................................................................................................. 35
6. Appendix 1................................................................................................................. 37
7. Appendix 2................................................................................................................. 43
iii
LIST OF TABLES
Table 1 – Parametric Analyses Performed ...................................................................... 19
Table 2 - Parameters for verification analysis ................................................................. 32
Table 3 - Verification Analyses, Error ............................................................................ 33
Table 4 - Verification Analyses, Average Error .............................................................. 33
iv
LIST OF FIGURES
Figure 1 – Example double shear clevis connection ......................................................... 1
Figure 2 – Madux’s parametric lug dimensions [4]........................................................... 3
Figure 3 – Wearing’s findings on hoop stress based on stepped analysis [6] ................... 4
Figure 4 – Stenman’s finite element model [7] ................................................................. 5
Figure 5 – To’s conforming bushing, section view [8] ..................................................... 6
Figure 6 – Antoni’s pin-bushing-lug system [10].............................................................. 6
Figure 7– Strozzi’s stress results from loaded clevis [11] ................................................. 7
Figure 8– Antoni’s spring contact representation [12] ...................................................... 8
Figure 9– Kwon’s contact stress distribution [13] ............................................................. 8
Figure 10 – Overall view of the finite element model ..................................................... 11
Figure 11 – Meshed Pin ................................................................................................... 12
Figure 12 – Meshed Inner and Outer Lug ....................................................................... 12
Figure 13 – Convergence study results, normalized ........................................................ 13
Figure 14 – Meshed assembly ......................................................................................... 14
Figure 15 – Load application ........................................................................................... 16
Figure 16 – Z symmetry plane ......................................................................................... 16
Figure 17 – X symmetry plane ........................................................................................ 17
Figure 18 – Fixed boundary condition............................................................................. 18
Figure 19 – von Mises stress of assembly ....................................................................... 19
Figure 20 – Shear tear-out plane ...................................................................................... 20
Figure 21 – Overall von Mises stress .............................................................................. 21
Figure 22 – Lug von Mises stress .................................................................................... 21
Figure 23 – Lug shear tear-out stress ............................................................................... 22
Figure 24 – Lug contact pressure..................................................................................... 22
Figure 25 – Pin von Mises stress ..................................................................................... 23
Figure 26 – Pin shear stress ............................................................................................. 23
Figure 27 – Pin contact pressure ...................................................................................... 24
Figure 28 – Lug von Mises, load applied ........................................................................ 25
Figure 29 – Lug von Mises, lug width ............................................................................. 26
Figure 30 – Lug von Mises, lug gap ................................................................................ 26
v
Figure 31 – Lug von Mises, Young’s modulus ratio ....................................................... 26
Figure 32 – Lug von Mises, Poisson’s ratio ratio ............................................................ 27
vi
LIST OF SYMBOLS
Symbol
Definition
Units
σ1
von Mises stress, lug
psi
σ2
shear tear-out stress, lug
psi
σ3
contact pressure, lug
psi
σ4
von Mises stress, pin
psi
σ5
shear stress, pin
psi
σ6
Epin
Elug
gap
contact pressure, pin
Young’s modulus of pin
Young’s modulus of lug
lug gap
psi
Kn1
stress multiplication factor for σn, based on lug gap
-
Kn2
stress multiplication factor for σn, based on ratio of pin to lug
Young’s moduli
-
Kn3
stress multiplication factor for σn, based on ratio of pin to lug
Poisson’s ratios
-
Kn4
stress multiplication factor for σn, based on lug width
-
Lw
υ_pin
υ_lug
lug width
Poisson’s ratio of pin
Poisson’s ratio of lug
in
-
psi
psi
in
vii
ACKNOWLEDGMENT
I would like to thanks Professor Ernesto Gutierrez-Miravete for his advisement and
guidance – he saw this work through from inception to completion. My friends and
coworker at General Dynamics Electric Boat have all provided motivation and advice in
one way or another. Jeffrey Pierce has drawn on his vast knowledge of solid mechanics
to point me in the right direction or to provide me with the means to complete this work.
Lastly, I would like to thank my family and Mirrie Choi for their continued support; this
would not have been possible without them.
viii
ABSTRACT
The current engineering community uses finite element analyses in the development
and design of double shear-plane clevis connections. Although accurate, finite element
analyses of this nature are non-linear in nature, and are often both time consuming and
computationally intensive. This thesis presents an approach for sizing frictionless double
shear-plane clevis connections to be under their material yield strengths for their given
application. This will lessen the need for the engineering community to rely on finite
element analysis in designing these clevis connections.
Instead, simple empirical
formulae may be used. This has the advantage of (1) not needing specialized personnel to
develop finite element models, (2) much shorter analysis time, and (3) a much shorter
iteration time to understand what dimensions and variables are most critical for a given
clevis system.
Finite element analysis is utilized to simulate testing for purposes of developing
empirical formulae based on the load through the connection, lug widths, lug gaps,
Young’s moduli, and Poisson ratios. Regression analysis is then used to derive closedform empirical formulae, using multiplication factors based upon each parametric
evaluated. A verification analysis is performed to evaluate the error of the empirical
formulae, and acceptable levels of accuracy are verified.
ix
1. Introduction
1.1 Background
A clevis connection is defined herein as a revolute joint, comprising of two of more
lugs and a single pin. A double shear-plane clevis connection is defined as a clevis
connection with three lugs, and therefore two shear planes. These joints have widespread
applications in mechanical systems, ranging from submarine applications to house-hold
door hinges. Although both single- and multi-shear-plane clevis joints exist, double shearplane joints are by far the most common in practical applications. This is because they
balance the advantages of a multi-shear-plane connection due to their load-sharing ability
and symmetry, and the advantages of a single shear-plane clevis due to their simplicity of
fabrication, installation, and maintenance.
Figure 1 depicts an example clevis connection.
In clevis connections, two
components rotate relative to another about a single axis, with the other degrees of freedom
remaining constrained to one another. Clevis connections are used in a widespread variety
of application, including door hinges, automotive drive shaft joints, and any other hinged
connection in which two shear planes are required for the robustness of the system.
Figure 1 – Example double shear clevis connection
1
However, despite the commonality of many aspects of most clevis connections, most
documentation examines specific and unique joint designs [8]. Some sources examine
varying aspects of the clevis connection, but only base the design on data from tests that
bring the connection to failure, and thus reflect a nonlinear, elastic-plastic response [4].
Other sources [13] evaluate only one stress component of a single component of the clevis
system, such as bending in the pin, while ignoring pin shear and the entire lug tensor.
Additionally, each source is tailored towards maximum accuracy at certain radius-tolength aspect ratios, and thus various sources yield different results for a given clevis
system.
1.2 History
A thorough literature review of interference pin connections prior to 1966 was
performed by Venkatarman [1]. Included in the work done prior to 1966 was that of
Cozzone and Melcon [2], who presented a study in 1950, which was subsequently updated
by Melcon in 1953 [3], attempting to give generalized empirical data for pin and lug joints.
Particularly, they examined the cross sections in which (1) bearing and shear were highest,
and (2) in which the net tensile failure of the lug would occur. They were the first to be
able to correlate empirical data, destructive testing, and close-form formulae for predicting
failure in the lug. Furthermore, they were able to extrapolate these formulae for multi-lug
connections.
In 1969, Madux et al [4], in cooperation with the United States Air Force, attempted
to put forth a fully comprehensive document for use in the design and evaluation of pin
joints. Namely, they produced a series of empirical curves to estimate produce stress
intensity factors for the various failure modes. Madux evaluated both single shear and
multi shear lug designs, for use in designing components to be under their ultimate
strength. They parameterized a number of geometric aspects of the clevis connection, as
shown in Figure 2, to derive these generalized formulae based on the physical
characteristics of the system. Key conclusions of this study were (1) that the governing
failure mode changes depending on the pin and lug geometry, transitioning from shear to
compression in the lug at given geometry ratios; (2) that a generalized method of
2
developing a stress intensity factors was possible; (3) the method can be used to evaluate
the stress components of both axial and transverse loading scenarios.
Figure 2 – Madux’s parametric lug dimensions [4]
However, the work done prior to 1978 focused primarily on full contact along the
width of the pin, as well as presuming a state of either full- or no-slip between the pin and
clevis; it was therefore followed up with a continued literature review up to 1978 by Rao
[5]. Rao focused on the two-dimensional analysis of the problem of a pin in a plate layout.
He evaluated the non-linearities of an initial clearance, and estimated the stresses in the
pin, plate, and contact interface. He also investigated the effects of slip in the pin-to-plate
model.
In 1985, Wearing et al [6] investigated the effect of an initial clearance on a twodimensional framework. However, the advent of finite element analysis, in conjunction
with the use of analytical step function representations, allowed Wearing to transition from
a pin within an infinite plate, to a pin within a clevis lug joint. The linear stepping theory
used in Wearing’s investigation evaluated a finite number of distinct points on the contact
surface of the lug, and utilized the stiffness matrix of the system to define if a given point
was in contact or not. Slowly stepping through a load application, he was able to gradually
load the pin as to engage each point in series. The result of this work was first look at a
potential fully-deformable physical solution, in which the tension-compression strains
caused by the hoop stress and contact loading at the pin-lug interface correlated with
photo-elastic testing.
An example of Wearing’s tension-compression strain plot is
3
depicted in Figure 3. This plot shows how the tension and compression changes along the
contact surface of the lug.
Figure 3 – Wearing’s findings on hoop stress based on stepped analysis [6]
In 2008, with the vast improvement of computational capacity, and the popularization
of finite element software, Stenman [7] sought to re-evaluate Madux and the United States
Air Force’s findings from 1969. Specifically, she examined the third dimension of the
revolute joint; where Venkataraman, Rao, and Wearing all examined the radial and
azimuthal dimensions, Stenamn sought to verify Madux’s claims with respect to the axial
dimension. She used commercial finite element software to develop models to compare
with Madux’s findings. Figure 4 depicts the model developed by Stenman’s analyses, in
which symmetry constraints were used to increase the computation efficiency of the study.
In double shear clevis connections, the contact pressure as a function of axial distance
from the edge of a lug is non-linear, with the peak pressure existing at the shear plane, and
the pressure decreasing at some rate along the axial dimension of the pin. For purposes of
analytical solutions, Madux assumed the contact pressure distribution to be uniform over
the contact area. However, Stenman found that a more appropriate solution resulted from
4
using a uniform pressure distribution that did not extend the full length of the lug, but
instead extended to a representative distance from the edge of the lug as found by the finite
element model.
Figure 4 – Stenman’s finite element model [7]
In 2008, Q. D. To et al [8] began investigating more complex versions of Roa’s pinplate model. The investigation centered on a system in which the bushing between the pin
and lug was of non-trivial thickness, and was of a conforming nature.
Specifically, they
evaluated a system in which a bolt was placed into a hole in a glass plate, in which the
hole was much larger than the pin. The pin was subsequently glued to the glass using a
resin. A cross-sectional view of the system is depicted in Figure 5. This example is unique
in that unlike previous studies ([1]-[7]), this problem is driven in large part by a
conforming intermediate layer.
This study was an advancement from previous
investigation performed by Ciavarella and Decuzzi [9], as it took into account the
influence of friction on the systematic behavior of the pin-in-plate concept. However,
although friction does play an important role in the system, the general contours of stress
at the contact interface were markedly similar in To’s and Rao’s models.
5
Figure 5 – To’s conforming bushing, section view [8]
In contrast with To, in 2010 Antoni [10] examined non-conforming intermediate
bushings. As depicted in Figure 6, the system was comprised of a pin, a thin nonconforming bushing, and a lug. This work focused on shrink-fit, cold expansion, thermal,
or axial pin-loading conditions on bushings. Analytical solutions were derived, and
compared with highly refined two-dimensional finite element models. The finite element
analyses featured an analytic pin, with a highly refined mesh of a deformable bushing and
lug. This high level of mesh refinement was done to accurately capture the stress
phenomena at the interfacing surface between the bushing and the lug.
Figure 6 – Antoni’s pin-bushing-lug system [10]
6
In 2011, Strozzi [11] expanded Wearing’s investigation into progressive contact,
where contact surfaces increase as more load is applied. He investigated pin-lug joints in
which an initial clearance yielded different system behaviors.
The result of his
investigation was a further development of Madux’s stress concentration factor, K, to
include initial clearances as well as varying angles tapered thicknesses of the lug. The
design charts presented in Strozzi’s investigation are validated and presented for use in
the designing of clevis connections, where peak stress concentrations are of concern,
namely for fatigue applications. Figure 7 depicts the lug system studied, including the
offset load applied (offset at an arbitrary angle α).
Figure 7– Strozzi’s stress results from loaded clevis [11]
Antoni [12] continued his previous investigations from 2010, and in 2013 added nonlinearities to his evaluations. As depicted in Figure 8, Antoni used a matrix of springs to
represent the deformation of the lug as load is applied to the pin. The three non-linearities
focused on in his work were initial clearances – where he added to Strozzi’s study –
conforming contact – where he added to To’s work – and an evaluation of regressive and
progressive contact. Contrary to progressive contact, the study into regressive contact
postulated a system in which the area of the contact interface decreased as more load was
applied. His findings suggested that the regressive phenomenon of contact separation
results in a decrease in the overall stiffness of the system, while an initial clearance results
in an overall increase in the stiffness.
7
Figure 8– Antoni’s spring contact representation [12]
Finally, in 2013, Kwon [13] further investigated Madux’s and Stenman’s results, and
described a finite element model in which the contact pressure between the lug and pin
was represented as a linear pressure distribution, not a uniform pressure distribution. Like
Stenman, Kwon investigated the problem in all three dimensions, but he focused primarily
on the bending stress of the pin along its axis. From that analysis, he put forward a critical
pin diameter, at which the transition occurs between the bending and the shear stress
governing the failure of the pin. As an example of Kwon’s results, Figure 9 shows the
contact pressure of the pin as it varies along its length, with the two peaks representing the
inner edges of the two outer lugs.
Figure 9– Kwon’s contact stress distribution [13]
8
1.3 Objective
The objective of this thesis is to present an approach for sizing frictionless double
shear-plane clevis connections that will function under their material yield strengths for
their given application. Finite element analysis will be utilized to simulate the connections
for purposes of developing empirical formulae based on the load through the connection,
pin clearance, lug widths, lug gaps, Young’s moduli, and Poisson ratios.
For frictionless double shear-plane clevis connections, the development of closedform formulae for three-dimensional problems to date has been limited; based on the high
complexity and nonlinearity of the contact stiffness matrix, most formulae have been
deduced from physical testing, which is both expensive and data-restrictive. Therefore,
the industry-typical approach has become to derive equations for a given clevis connection
on a case-by-case basis for each design.
However, with the development of finite element analysis, the ability to obtain data
from a mass array of virtual experimentation is now possible. Still, the use of finite
element analysis does not change the fact that each clevis system needs to be evaluated on
a case-by-case basis. Substituting the current method with simple empirical formulae is
that no specialized personnel to develop finite element models are required, the analysis
time is greatly reduced, and a much shorter iteration time exists, allowing the engineer to
quickly understand what dimensions and variables are most critical for a given clevis
system.
9
2. Theory and Method
2.1 Procedure
The procedure used in this thesis is a four step process for deriving closed-form
formulae for a standard pin-lug clevis:
Step 1:
Perform a convergence study on a finite element model to ensure accuracy
and validity of analysis
Step 2:
Perform a carefully designed suite of analyses varying the parameters to be
used in the closed-form solution – one parameter at a time
Step 3:
Perform regression analysis on the computed results and derive equations
using best fit curves to the computed data
Step 4:
Verify solutions by comparing closed-form solutions with three additional
analyses
2.2 Method
A finite element model was developed to analyze the stress field in the pin and lugs
of a generic clevis connection. Parametric analyses will be performed, with the analyses
varying load through the connection, lug widths, lug gaps, Young’s moduli, and Poisson
ratios. The data will then be extracted and processed to derive generalized empirical
formulae for double shear-plane clevis connection. The clevis connection used has a 0.5”
pin radius and lug inner radius, a 1.0” lug outer radius, and an inner lug that twice the
thickness of a given outer lug. These dimensions were used as a baseline, as they represent
a typical clevis connection used in the engineering community. As the empirical formulae
become less accurate the more the design deviates from the baseline configuration, it was
important to represent a clevis design that was centered within the common design
envelope.
2.2.1
Finite Element Model Development
All geometry was modeled using ABAQUS/CAE, Version 6.13-EF1.
analyses were performed using ABAQUS/Standard, Version 6.13-EF1.
files were used for the finite element analysis.
10
The
The following
Figure 10 – Overall view of the finite element model
As depicted in Figure 10, the finite element model consists of a one-quarter pin, onehalf of the outer lug (depicted in green above), and quarter of the middle lug (depicted in
beige above). Symmetry boundary conditions were established to reduce the model size
and allow for a more detailed mesh for the same computational resource.
All elements are 3-dimensional 20-noded hexahedral reduced integration continuum
elements, denoted in ABAQUS as C3D20R. The meshed components are shown in
Figures 11 and 12. Zero element errors and zero element warnings existed in the models
after meshing.
11
Figure 11 – Meshed Pin
Figure 12 – Meshed Inner and Outer Lug
12
A mesh convergence study was performed to verify the adequacy of the mesh, and is
presented in Appendix 1. Upon collecting data from each of the convergence study
models, a plot is produced of mesh density vs. stress, and then is normalized to the most
dense mesh result. Figure 13 depicts the mesh convergence study data. As described in
Appendix 1, the von Mises stresses of the pin are within 10% accuracy with a mesh-radius
ratio of 1, and within 5% accuracy with a mesh-radius ratio of 2. The von Mises stresses
of the lug are within 10% accuracy with a mesh-radius ratio of 6, and within 5% accuracy
with a mesh-radius ratio of 8. Therefore, a mesh-radius ratio of 8 is maintained throughout
the study.
1.1
1.05
1
Value (Normalized)
0.95
Pin Mises (S MISES)
0.9
Lug Mises (S MISES)
0.85
Shear Tear Out (S12)
0.8
Net Tensile (S22)
0.75
Pin Bending (S33)
0.7
Pin Shear (S12)
0.65
Lug Bearing (CPRESS)
0.6
Pin Bearing (CPRESS)
0.55
0.5
1
2
3
4
5
6
7
8
9
10
MR #
Figure 13 – Convergence study results, normalized
Each component is assigned a material for any given analysis. As a NewtonRaphson solution ABAQUS/Standard is being used for a static analysis, density is not
considered. Only Young’s Modulus and Poisson’s ratio are varied. For the baseline
analysis, the Young’s Modulus and Poisson’s ratio values are set to 3E7 psi and 0.3
respectively, typical for steel.
13
The system is arranged such that the pin is initially in contact with the bottom of the
inner lug contact surface, and the top of the outer lug contact surface. The mid-length of
the pin is co-planar with the mid-length of the inner lug. The gap between the inner lug
and outer lug are varied, and the pin protrudes from the outer lug an arbitrary amount, as
it has been shown by Kwon [13] that the stresses in the system are not a function of pin
length beyond the outer lug.
Figure 14 – Meshed assembly
14
Each ABAQUS analysis consists of two steps. These steps are Initial and Load.
Initial: In the Initial step, the appropriate constraints for each run are created. These
boundary conditions set both the edge conditions and the symmetry constraints necessary
to fully capture the kinematics of the system.
Load: Load is a Static, General step. In this step, the load for each run is applied to the
model. The load was applied as a uniform anti-pressure at the top surface of the inner lug;
see Figure 15.
15
Figure 15 – Load application
Two types of boundary conditions are imposed on the model. The first type are
symmetry boundary conditions. These boundary conditions are imposed at the two
symmetry planes in the model. The first is imposed at the mid-length of the inner lug and
the mid-length of the pin (red highlighted surface shown in Figure 16). This allows for
the axial extent of the model to be half of the physical system. This symmetry boundary
condition is imposed on continuum element nodes using a zero-displacement condition in
the z-axis degree of freedom (ABAQUS degree of freedom U3).
Figure 16 – Z symmetry plane
The second symmetry boundary condition is imposed at the 0-180 azimuth plane of
the pin and both lugs (red highlighted surface shown in Figure 17). This allows for the Xaxis extent of the model to be half of the physical system. This symmetry boundary
condition is imposed on continuum element nodes using a zero-displacement condition in
16
the x-axis degree of freedom (ABAQUS degree of freedom U1). The combination of the
two symmetry conditions allows for the model extent to be one quarter of the physical
system, allowing for a much more detailed mesh for the same computational resources.
Figure 17 – X symmetry plane
The second type of boundary condition is a fixed boundary condition, which is
applied at the bottom of the outer lug (red highlighted surface shown in Figure 18). This
boundary condition both reacts the applied load and prevents the model from diverging
based on free body modes. This fixed boundary condition is imposed on continuum
element nodes using a zero-displacement condition in the y- and z-axis degree of freedom
(ABAQUS degrees of freedom U2 and U3).
17
Figure 18 – Fixed boundary condition
One contact interaction constraint is imposed in the model, between the pin and the
lugs. As the curvature of contact is significant in this problem, surface smoothing at the
interaction level is enlisted. Furthermore, the instances are places in initial full-closure,
so no nodal adjustment is required. The interaction property normal to the contact surface
is characterized by a classical Lagrange multiplier method, with the constraint utilizing a
standard pressure-overclosure relationship. The interaction property tangential to the
contact surface is characterized by a zero-penalty constraint. This is used to create a fully
frictionless, full-slip condition.
A total of 42 analyses were performed, in which the load applied, lug width, lug gap,
Young’s modulus of the pin, and Poisson’s ratio of the pin were varied. Table 1 depicts
all the analyses performed.
18
Minimum Maximum
Value
Value
Increment
Value
Analyses
Performed
4000
2
0.9
400
0.25
0.1
10
7
10
1.00E+07
6.00E+07
1.00E+07
6
0.05
0.45
0.05
9
Parametric
Units
Load
Lug Width
Lug Gap
Young's Modulus of
Pin
Poisson's Ratio of Pin
lbf
in
in
400
0.5
0
psi
-
Table 1 – Parametric Analyses Performed
2.2.2
Output
For each analysis, the full set of results was examined. Here, an example of typical
results is presented. Figure 19 displays an overall view of the von Mises stresses of the
assembly when 100% of the load is applied.
Figure 19 – von Mises stress of assembly
As can be seen in Figure 19, the maximum stress in the system is at the contact surface
between the pin and the lug, at each lug edge. The stress dissipates along the axial length
of the pin, as was described by both Stenman [7] and Kwon [13]. In addition, the pin
shear stress, dominated by pure shear due to the short moment arm, but still with the
existence of transverse shear, can be seen in the unsupported section of the pin between
lugs.
19
3. Results and Discussion
3.1 Output Requests
Output is requested for specific key variables that can be evaluated and
independently verified with previous historical and empirical data. The variables of
interest are von Mises stresses for the lug and pin, contact pressure for the lug and pin,
shear-tear out at the lug, contact pressure at the lug and pin, and shear stress at the pin.
The following figure shows a diagram of the shear-tear out plane of interest for clarity.
Figure 20 – Shear tear-out plane
3.2 Example Output
The following figures show example outputs for a single loading condition. The
analysis shown is part of the analysis suite of varying loads. The particular analysis
depicted is of a load of a load of 4000 lbf applied to the system. The full results, best fit
curves, and R2 values can be found in Appendix 2. Figure 21 depicts the overall von Mises
stress.
20
Figure 21 – Overall von Mises stress
Figure 22 depicts the von Mises stress of the two lugs. The high stresses are at the edges
of each lug, based on the increased contribution of the contact pressure.
Figure 22 – Lug von Mises stress
Figure 23 depicts the shear stress of the lug. The shear tear-out plane has spread out from
a singular plane, which is what exists with a small load, to two distinct shear planes at
some angle away from the symmetry plane. The angle away from the symmetry plane is
dependent on the load applied, geometry, and material properties.
21
Figure 23 – Lug shear tear-out stress
Figure 24 depicts the lug contact pressure. As expected, the contact pressure dissipates
along the axial length of the pin, as was described by both Stenman and Kwon.
Figure 24 – Lug contact pressure
Figure 25 depicts the von Mises stress in the pin, with varying contributions from contact
pressure, shear stress, and bending stress.
22
Figure 25 – Pin von Mises stress
Figure 26 depicts the shear stress in the pin. The shear stress is a result of both transverse
and pure shear, mainly occurring at the unsupported section of the pin between the lugs.
Figure 26 – Pin shear stress
23
Figure 27 depicts the contact pressure on the pin. As expected, the contact pressure is
highest at the edge of the lugs, and rapidly decreases in stress down to almost zero along
the axial length of the pin.
Figure 27 – Pin contact pressure
3.3 Regression Analysis
Through the use of regression analyses, multiplication factors are determined for
each variable, and a final set of closed form expressions representing the finite element
results are developed. Equations are presented for von Mises stress in the lug, shear tearout in the lug, contact pressure in the lug, von Mises stress in the pin, shear stress in the
pin, and contact pressure in the pin.
For each stress mode, a general equation based upon the load applied is presented and
then stress multiplication factors for each variable are presented. K1 is the multiplication
factor from the axial gap between lugs, K2 is the multiplication factor from the Young’s
Moduli, K3 is the multiplication factor from the Poisson’s ratios, and K4 is the
multiplication factor from the lug width. The equations were derived by creating a best
fit curve for the data, evaluating them at the baseline condition (0.5” lug gap, 2” lug width,
and Epin/Elug = υ _pin/ υ_lug = 1), and dividing the equation by the result of the baseline
equation, i.e. normalizing each equation such that at the baseline condition, the
multiplication factor is equal to 1.
24
An example of the regression analysis is shown here, for shear tear-out in the lug.
However, the other five stress components evaluated are performed similarly. For the full
set of results, see Appendix 2. First, the various analyses were post-process, and best-fit
curves were created. The curves were created using a polynomial function of the power
needed to create a good correlation (R2 > 0.98). In addition, the curve for load applied
was set about a y-intercept of 0, as no stress exists with zero load applied. The following
figures show the post-processing for lug shear tear-out as a function of load, gap, Young’s
modulus (ratio of pin to lug), Poisson’s ratio (ratio of pin to lug), and lug width.
Lug von Mises vs
Load (lbf)
25000
y = 5.8691x
R² = 1
20000
15000
Lug Mises (S MISES)
10000
Linear (Lug Mises (S
MISES))
5000
0
0
1000
2000
3000
4000
5000
Figure 28 – Lug von Mises, load applied
Lug von Mises vs
Lug width (in)
3000
y = 636.44x3 - 2914.5x2 + 4151.7x + 541.86
R² = 0.9944
2500
2000
Lug Mises (S MISES)
1500
Poly. (Lug Mises (S
MISES))
1000
500
0
0
0.5
1
1.5
2
25
2.5
Figure 29 – Lug von Mises, lug width
Lug von Mises vs
Lug gap (in)
3000
y = 1279.7x + 1625
R² = 0.9996
2500
2000
Lug Mises (S MISES)
1500
Linear (Lug Mises (S
MISES))
1000
500
0
0
0.2
0.4
0.6
0.8
1
Figure 30 – Lug von Mises, lug gap
Lug von Mises vs
Young's modulus Ratio (-)
y = 580.98x2 - 2377.3x + 4078.9
R² = 0.9958
4000
3500
3000
2500
Lug Mises (S MISES)
2000
1500
Poly. (Lug Mises (S
MISES))
1000
500
0
0
0.5
1
1.5
2
2.5
Figure 31 – Lug von Mises, Young’s modulus ratio
26
Lug von Mises vs
Poisson's Ratio Ratio (-)
y = -130.5x + 2392.1
R² = 0.9909
2400
2350
2300
Lug Mises (S MISES)
2250
Linear (Lug Mises (S
MISES))
2200
2150
0
0.5
1
1.5
2
Figure 32 – Lug von Mises, Poisson’s ratio ratio
The equations taken from each curve, except for the load applied curve, are then
evaluated for the baseline configuration, which is a lug gap of 0.5”, a lug width of 2.0”,
and a Young’s modulus and Poisson’s ratio pin-to-lug ratio of 1. This is shown in the
equations below. The first equation is the best fit curve for the lug gap (“gap”). The
second is the best fit curve based for the Young’s modulus ratio (“Epin/Elug”). The third
is the best fit curve for the Poisson’s ratio ratio (“Vpin/Vlug”). The fourth is the best fit
curve for the lug width (“Lw”). The following equations show each equation set at the
baseline configuration.
27
The equations are then normalized by their value at the baseline configuration. This
is done to set each equation to 1.0 at the baseline configuration, such that the equations
can be used as multiplication factors, or factors that affect the stress as a function of the
system’s deviation from the baseline configuration. That means, for example, if all four
equations were set to a value of 2 (i.e. the pre-normalized equations resulting in double
the stress as the baseline configuration), the resulting overall multiplication factor would
be 16. The equations therefore become:
These equations are then combined with the curve for the applied load, such that each
multiplication factor is multiplied together, and then multiplied with the load applied
curve. This yields a final equation for von Mises stress in lug of:
(1)
As was done in the preceding example, the shear stress in the pin and lug, the contact
pressure in the pin and the lug, and the von Mises stress in the pin are also derived. The
shear tear-out stress in the lug is determined to be:
(2)
28
Where:
The contact pressure in the lug is determined to be:
(3)
Where:
29
The von Mises stress in the pin is determined to be:
(4)
Where:
The shear stress in the pin is determined to be:
(5)
Where:
30
The contact pressure in the pin is determined to be:
(6)
Where:
3.4 Verification Analyses
To confirm that the formulae presented are accurate, three verification analyses are
performed, with the parameters shown in Table 2.
31
Parameter
Value
Verification 1
Verification 2
Verification 3
Load (lbf)
400
800
80
Gap (in)
0.3
0.5
0.4
Epin (psi)
6.00E+07
4.00E+07
3.00E+07
Elug (psi)
3.00E+07
2.00E+07
3.00E+07
v_pin
0.15
0.3
0.2
v_lug
0.3
0.2
0.4
1
1.4
1.2
Lug Width (in)
Table 2 - Parameters for verification analysis
As an example, the resulting finite element analysis from Verification 1 showed a
maximum von Mises stress in the lug of 1612 psi. For that configuration, the computed
multiplication factors, and computed von Mises stress in the lug are:
The resulting error on the analysis is calculated to be 1.36%, well within the
acceptable limits of accuracy. Table 3 depicts the error percentages for all six stress
components evaluated, for all four verification analyses.
The resulting error of the analyses varies depending on the analysis and stress
component.
Among all four analyses, the average error percentage of each stress
component is shown in Table 4. It is found that the average error percentage among all
stress components is less than 6.8%, within the acceptable limits of accuracy.
32
Stress Component
Lug von Mises
FEA Value (psi)
1612
Calculated Value (psi)
1586
Error (%)
1.6
Verification 1
Lug shear tear-out
800
794
0.8
Lug bearing
1350
1298
3.9
Pin von Mises
1501
1409
6.1
Pin shear
377
372
1.3
Pin bearing
1150
1111
3.4
Lug von Mises
3658
3338
8.7
Lug shear tear-out
1826
1696
7.1
Lug bearing
2857
2563
10.3
Pin von Mises
2377
2087
12.2
Pin shear
759
764
0.7
Pin bearing
2450
2194
10.4
Lug von Mises
459
469
2.2
Lug shear tear-out
218
225
3.2
Lug bearing
435
422
3
Pin von Mises
201
205
2
Pin shear
87
85
2.3
Pin bearing
368
356
3.3
Verification 3
Verification 2
Analysis
Table 3 - Verification Analyses, Error
Stress Component
Lug von Mises
Error (%)
4.2
Lug shear tear-out
3.7
Lug bearing
5.7
Pin von Mises
6.8
Pin shear
1.4
Pin bearing
5.7
Table 4 - Verification Analyses, Average Error
33
4. Conclusions
This thesis presented an approach for sizing frictionless double shear-plane clevis
connections to be under their material yield strengths for their given application. Finite
element analysis was utilized to simulate testing for purposes of developing empirical
formulae based on the load through the connection, lug widths, lug gaps, Young’s moduli,
and Poisson ratios. Regression analysis was then used to derive closed-form empirical
formulae, using multiplication factors based upon each parametric evaluated.
A
verification analysis was performed to evaluate the error of the empirical formulae, and
acceptable levels of accuracy were verified.
The results presented in this thesis will lessen the need for the engineering community
to rely on finite element analysis in designing these clevis connections. Instead, the simple
empirical formulae presented may be used. This has the advantage of (1) not needing
specialized personnel to develop finite element models, (2) much shorter analysis time,
and (3) a much shorter iteration time to understand what dimensions and variables are
most critical for a given clevis system.
4.1 Future Work
This thesis laid the groundwork and approach that can be used in developing a
closed-form solution for the full range of clevis connections. In this thesis, a number of
stress components and a suite of variables were examined. However, this approach can
be extended to study the effects of other variations on clevis connections, such as the
existence of bearings, pin clearance, pin radius, and lug outer radius. In addition, stress
components such as pin bending and lug hoop stress can be evaluated further using this
same technique.
34
5. References
[1]
Venkataraman, N. S. A Study into the Analysis of Interference Fits and Related
Problems. Ph.D. Thesis. Indian Institute of Science, Bangalore, India, 1996.
[2]
Cozzone, F. P., M. A. Melcon, and F. M. Hoblit. "Analysis of Lugs and Shear Pins
Made of Aluminum or Steel Alloys." Product Engineering (1950): n. pag. Print
[3]
M. A. Melcon, and F. M. Hoblit. “Developments in the Analysis of Lugs and Shear
Pins, Product Engineering.” Product Engineering (1953): n. pag. Print
[4]
Maddux et al. Stress Analysis Manual, Air Force Flight Dynamics Laboratory.
(August 1969)
[5]
Rao, A. K. Elastic Analysis of Pin Joints. Department of Aeronautical
Engineering, Indian Institute of Science, Bangalore, India. 1978.
[6]
J. L. Wearing et al. “A Study of the Stress Distribution in a lug loaded by a free
fitting pin.” The Journal of Strain Analysis for Engineering Design (1985): n. pag.
Print
[7]
Stenman, C. A. A Comparison of the Predicted Mechanical Behavior of Lug Joints
using Strength of Materials Models and Finite Element Analysis. Master’s Thesis.
Rensselaer Polytechnic Institute, Hartford, CT, 2008.
[8]
To, Q. D., and Q. C. He. "On the Conforming Contact Problem in a Reinforced
Pin-Loaded Structure with a Non-Zero Second Dundurs’ Constant." International
Journal of Solids and Structures 45 (2008): 3935-950. Print.
[9]
Ciavarella, M., and P. Decuzzi. "The State of Stress Induced by the Plane
Frictionless Cylindrical Contact. I. The Case of Elastic Similarity." International
Journal of Solids and Structures 38 (2001): 4525-533. Print.
[10]
Antoni, N., and F. Graisne. "Analytical Modeling for Static Stress Analysis of PinLoaded Lugs with Bush Fittings." International Journal of Non-Linear Mechanics
58 (2014): 258-82. Print.
[11]
Strozzi, A., A. Baldini, M. Giacopini, E. Bertocchi, and L. Bertocchi. "Maximum
Equivalent Stress in a Pin-Loaded Lug in the Presence of Initial Clearance." The
Journal of Strain Analysis for Engineering Design 9 (2011): n. pag. Print.
35
[12]
N. Antoni, “A Study of Contact Non-Linearities in Pin-Loaded Lugs: Seperation,
Clearance and Frictional Slipping Effects.” Safran Group, Analysis Methods of
Structures (2013): n. pag. Print.
[13]
Kwon, E. A Critical Study of Pin Bending Behavior Using Finite Element Analysis.
Rensselaer Institute of Technology. Master’s Thesis. Rensselaer Polytechnic
Institute, Hartford, CT, 2013.
36
6. Appendix 1
Method
A finite element model will be developed to determine the appropriate mesh density
for the model used. A parametric analysis will be performed, with the analyses varying
mesh density. The data will then be extracted and processed to derive a relationship
between the solution convergence and the density of the mesh as compared to the pin
radius.
Finite Element Model Development
All geometry was modeled using ABAQUS/CAE, Version 6.13-EF1. The analyses
were performed using ABAQUS/Standard, Version 6.13-EF1.
The following files
were used for the finite element analysis.
MeshConvergenceModel.cae
ABAQUS CAE database
MeshConvergenceModel.jnl
ABAQUS Journal File
The input and output database files were formatted such that the ratio of element size
to pin radius is easily determined. Each output database was named “MR” followed by
the number of element lengths in the pin radius (e.g. “MR2.odb” indicates an output
database in which the seed size is half that of the pin radius).
The model used was the same as the baseline configuration model used for all of
the anlyses
Output
An example output (MR1) is shown in the following figures.
37
Figure A1-9 – Overall Stress
Figure A1-10 – Pin von Mises
38
Figure A1-11 – Lug von Mises
Figure A1-12 – Lug shear tear-out stress
Figure A1-15 – Pin shear stress
39
Figure A1-16 – Lug contact pressue
Figure A1-17 – Pin contact pressure
Upon collecting data from each of the convergence study models, a plot is produced
of mesh density vs. stress, and then is normalized to the most dense mesh results.
40
2500
2000
Pin Mises (S MISES)
Lug Mises (S MISES)
1500
psi
Shear Tear Out (S12)
Net Tensile (S22)
1000
Pin Bending (S33)
Pin Shear (S12)
500
Lug Bearing (CPRESS)
Pin Bearing (CPRESS)
0
1
2
3
4
5
6
7
8
9
10
MR #
Figure A1-18 – Stress vs. mesh density
1.1
1.05
1
Value (Normalized)
0.95
Pin Mises (S MISES)
0.9
Lug Mises (S MISES)
0.85
Shear Tear Out (S12)
0.8
Net Tensile (S22)
0.75
Pin Bending (S33)
0.7
Pin Shear (S12)
0.65
Lug Bearing (CPRESS)
0.6
Pin Bearing (CPRESS)
0.55
0.5
1
2
3
4
5
6
7
8
9
10
MR #
Figure A1-19 – Stress vs. mesh density (normalized)
41
Based on the normalized stress plots, it is shown that the von Mises stresses of the
pin are within 10% accuracy with a mesh-radius ratio of 1, and within 5% accuracy with
a mesh-radius ratio of 2. The von Mises stresses of the lug are within 10% accuracy with
a mesh-radius ratio of 6, and within 5% accuracy with a mesh-radius ratio of 8. Therefore,
a mesh-radius ratio of 8 is maintained throughout the study.
42
7. Appendix 2
Load
400
800
1200
1600
2000
2400
2800
3200
3600
4000
Lug
Lug
Shear Tear
Mises (S
Out (S12)
MISES)
2349
1148
4697
2296
7046
3377
9394
4592
11741
5740
14088
6888
16435
8035
18781
9183
21127
10330
23472
11477
Pin
Lug Bearing
(CPRESS)
Pin Mises (S
MISES)
2089
4176
6263
8349
10433
12517
14598
16679
18758
20835
Pin Shear
(S23)
Pin Bearing
(CPRESS)
428
856
1283
1710
2137
2563
2989
3415
3841
4266
1779
3566
5362
7166
8980
10802
12635
14479
16333
18196
791
1581
2371
3162
3952
4742
5532
6322
7112
7901
Shear Tear Out (S12)
14000
y = 2.8683x
R² = 1
12000
10000
Shear Tear Out (S12)
8000
6000
Linear (Shear Tear Out
(S12))
4000
2000
0
0
1000
2000
3000
4000
5000
43
Lug Bearing (CPRESS)
25000
y = 5.2122x
R² = 1
20000
15000
Lug Bearing (CPRESS)
10000
Linear (Lug Bearing
(CPRESS))
5000
0
0
1000
2000
3000
4000
5000
Pin Mises (S MISES)
9000
y = 1.9756x
R² = 1
8000
7000
6000
Pin Mises (S MISES)
5000
4000
Linear (Pin Mises (S
MISES))
3000
2000
1000
0
0
1000
2000
3000
4000
5000
44
Pin Shear (S23)
4500
y = 1.0673x
R² = 1
4000
3500
3000
2500
Pin Shear (S23)
2000
Linear (Pin Shear (S23))
1500
1000
500
0
0
1000
2000
3000
4000
5000
Pin Bearing (CPRESS)
20000
18000
16000
14000
12000
10000
8000
6000
4000
2000
0
y = 4.5234x
R² = 0.9999
Pin Bearing (CPRESS)
Linear (Pin Bearing
(CPRESS))
0
1000
2000
3000
4000
5000
45
Lug Mises (S MISES)
25000
y = 5.8691x
R² = 1
20000
15000
Lug Mises (S MISES)
10000
Linear (Lug Mises (S
MISES))
5000
0
0
1000
0.5
0.75
1
1.25
1.5
1.75
2
Lug
Lug
Shear Tear
Mises (S
Out (S12)
MISES)
1962
995
2300
1087
2414
1161
2407
1153
2357
1144
2309
1128
2272
1117
Lug Width
2000
3000
4000
5000
Pin
Lug Bearing
(CPRESS)
1889
2161
2205
2156
2085
2029
1990
Pin Mises (S
MISES)
Pin Shear
(S23)
Pin Bearing
(CPRESS)
378
414
430
430
423
416
410
1600
1832
1871
1832
1773
1726
1692
1766
1636
1285
1019
884
822
791
Lug Mises (S MISES)
3000
y = 636.44x3 - 2914.5x2 + 4151.7x + 541.86
R² = 0.9944
2500
2000
Lug Mises (S MISES)
1500
Poly. (Lug Mises (S
MISES))
1000
500
0
0
0.5
1
1.5
2
2.5
46
Shear Tear Out (S12)
1200
y = 174.22x3 - 837.52x2 + 1262.5x + 548.43
R² = 0.9831
1150
1100
Shear Tear Out (S12)
1050
Poly. (Shear Tear Out
(S12))
1000
950
0
0.5
1
1.5
2
2.5
Lug Bearing (CPRESS)
y = -452.85x4 + 2891.8x3 - 6727.7x2 +
6515.7x - 19.714
R² = 0.9999
2250
2200
2150
2100
2050
2000
1950
1900
1850
Lug Bearing (CPRESS)
Poly. (Lug Bearing
(CPRESS))
0
0.5
1
1.5
2
2.5
Pin Mises (S MISES)
y = -1214.1x4 + 6497x3 - 11836x2 + 7726.2x +
129
R² = 0.999
2000
1500
Pin Mises (S MISES)
1000
Poly. (Pin Mises (S
MISES))
500
0
0
0.5
1
1.5
2
2.5
47
Pin Shear (S23)
y = 65.778x3 - 311.24x2 + 454.27x + 220.57
R² = 0.9987
440
430
420
410
Pin Shear (S23)
400
Poly. (Pin Shear (S23))
390
380
370
0
0.5
1
1.5
2
2.5
Pin Bearing (CPRESS)
1900
y = -382.06x4 + 2436.5x3 - 5669.5x2 +
5502.2x - 14.071
R² = 0.9998
1850
1800
Pin Bearing (CPRESS)
1750
1700
Poly. (Pin Bearing
(CPRESS))
1650
1600
1550
0
0.5
1
1.5
2
2.5
Lug
Lug Gap
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Lug
Shear Tear
Mises (S
Out (S12)
MISES)
1611
825
1749
886
1885
946
2017
1004
2146
1061
2272
1117
2397
1173
2521
1229
2644
1285
2767
1340
Pin
Lug Bearing
(CPRESS)
1357
1489
1619
1746
1869
1990
2108
2225
2341
2457
48
Pin Mises (S
MISES)
647
676
699
717
734
791
865
942
1022
1102
Pin Shear
(S23)
Pin Bearing
(CPRESS)
301
328
353
374
393
410
427
451
502
477
705
1268
1379
1486
1590
1692
1792
1892
1990
2089
Lug Mises (S MISES)
3000
y = 1279.7x + 1625
R² = 0.9996
2500
2000
Lug Mises (S MISES)
1500
Linear (Lug Mises (S
MISES))
1000
500
0
0
0.2
0.4
0.6
0.8
1
Shear Tear Out (S12)
1600
y = 570.06x + 830.07
R² = 0.9998
1400
1200
1000
Shear Tear Out (S12)
800
Linear (Shear Tear Out
(S12))
600
400
200
0
0
0.2
0.4
0.6
0.8
1
49
Lug Bearing (CPRESS)
3000
y = 1218.2x + 1371.9
R² = 0.9994
2500
2000
Lug Bearing (CPRESS)
1500
Linear (Lug Bearing
(CPRESS))
1000
500
0
0
0.2
0.4
0.6
0.8
1
Pin Mises (S MISES)
y = -2294.6x4 + 4225.2x3 - 1914.3x2 + 477.49x
+ 646.43
R² = 0.9992
1200
1000
800
Pin Mises (S MISES)
600
Poly. (Pin Mises (S
MISES))
400
200
0
0
0.2
0.4
0.6
0.8
1
50
Pin Shear (S23)
600
y = 210.18x + 307.02
R² = 0.9678
500
400
Pin Shear (S23)
300
Linear (Pin Shear
(S23))
200
100
0
0
0.2
0.4
0.6
0.8
1
Pin Bearing (CPRESS)
2500
y = 34449x5 - 89283x4 + 85811x3 - 37462x2 +
8229.7x + 712.48
2000
R² = 0.9978
1500
Pin Bearing (CPRESS)
1000
Poly. (Pin Bearing
(CPRESS))
500
0
0
0.2
0.4
0.6
0.8
1
Lug
Young's
Modulus
(Epin/Elug)
0.333333333
0.666666667
1
1.333333333
1.666666667
2
Pin
Lug
Mises (S
MISES)
Shear Tear
Out (S12)
Lug Bearing
(CPRESS)
3391
2686
2272
1972
1768
1618
1582
1294
1117
988
898
833
3297
2465
1990
1642
1416
1256
51
Pin Mises (S
MISES)
1031
852
791
838
881
924
Pin Shear
(S23)
Pin Bearing
(CPRESS)
515
452
410
381
380
379
2794
2092
1692
1398
1207
1072
Lug Mises (S MISES)
4000
y = 580.98x2 - 2377.3x + 4078.9
R² = 0.9958
3500
3000
2500
Lug Mises (S MISES)
2000
Poly. (Lug Mises (S
MISES))
1500
1000
500
0
0
0.5
1
1.5
2
2.5
Shear Tear Out (S12)
1800
y = 235.13x2 - 982.51x + 1868.7
R² = 0.9968
1600
1400
1200
Shear Tear Out (S12)
1000
800
Poly. (Shear Tear Out
(S12))
600
400
200
0
0
0.5
1
1.5
2
2.5
52
Lug Bearing (CPRESS)
3500
y = 700.07x2 - 2807.8x + 4107
R² = 0.9954
3000
2500
Lug Bearing (CPRESS)
2000
1500
Poly. (Lug Bearing
(CPRESS))
1000
500
0
0
0.5
1
1.5
2
2.5
Pin Mises (S MISES)
1200
y = 441.45x5 - 2551.5x4 + 5271.8x3 - 4396.5x2
+ 1007.8x + 1018
R² = 1
1000
800
Pin Mises (S MISES)
600
Poly. (Pin Mises (S
MISES))
400
200
0
0
0.5
1
1.5
2
2.5
53
Pin Shear (S23)
600
76
500
400
Pin Shear (S23)
300
Poly. (Pin Shear (S23))
200
100
0
0
0.5
1
1.5
2
2.5
Pin Bearing (CPRESS)
3000
y = 589.98x2 - 2367.4x + 3476.9
R² = 0.9954
2500
2000
Pin Bearing (CPRESS)
1500
Poly. (Pin Bearing
(CPRESS))
1000
500
0
0
0.5
1
1.5
2
2.5
54
Poisson's Ratio
(Vpin/Vlug)
0.166666667
0.333333333
0.5
0.666666667
0.833333333
1
1.166666667
1.333333333
1.5
Lug
Lug
Shear Tear
Mises (S
Out (S12)
MISES)
2368
1155
2346
1146
2324
1138
2305
1130
2287
1123
2272
1117
2245
1107
2216
1095
2187
1084
Pin
Lug Bearing
(CPRESS)
Pin Mises (S
MISES)
2115
2086
2058
2032
2009
1990
1950
1908
1865
823
816
810
804
797
791
785
779
773
Lug Mises (S MISES)
2400
y = -130.5x + 2392.1
R² = 0.9909
2350
2300
Lug Mises (S MISES)
2250
Linear (Lug Mises (S
MISES))
2200
2150
0
0.5
1
1.5
2
55
Pin Shear
(S23)
Pin Bearing
(CPRESS)
437
432
426
421
416
410
406
401
395
1791
1768
1746
1725
1707
1692
1659
1625
1590
Shear Tear Out (S12)
1160
1150
y = -51.2x + 1164.3
R² = 0.9912
1140
1130
Shear Tear Out (S12)
1120
Linear (Shear Tear Out
(S12))
1110
1100
1090
1080
0
0.5
1
1.5
2
Lug Bearing (CPRESS)
2150
y = -85.818x3 + 167.01x2 - 250.64x + 2152.8
R² = 0.9986
2100
2050
Lug Bearing (CPRESS)
2000
Poly. (Lug Bearing
(CPRESS))
1950
1900
1850
0
0.5
1
1.5
2
56
Pin Mises (S MISES)
830
y = -37.4x + 828.72
R² = 0.9996
820
810
Pin Mises (S MISES)
800
Linear (Pin Mises (S
MISES))
790
780
770
0
0.5
1
1.5
2
Pin Shear (S23)
440
435
430
425
420
415
410
405
400
395
390
y = -31.2x + 442
R² = 0.999
Pin Shear (S23)
Linear (Pin Shear
(S23))
0
0.5
1
1.5
2
57
Pin Bearing (CPRESS)
1850
y = -70x3 + 134.09x2 - 198.71x + 1820.9
R² = 0.9984
1800
1750
Pin Bearing (CPRESS)
1700
Poly. (Pin Bearing
(CPRESS))
1650
1600
1550
0
0.5
1
1.5
2
58