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
Rensselaer Polytechnic Institute
Hartford, CT
November 2014
(For Graduation December 2014)
1
© Copyright 2014
by
Christopher Stubbs
All Rights Reserved
2
CONTENTS
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
LIST OF SYMBOLS ....................................................................................................... vii
ACKNOWLEDGMENT ................................................................................................ viii
ABSTRACT ..................................................................................................................... ix
1. Introduction.................................................................................................................. 1
1.1
Background ........................................................................................................ 1
1.2
History ................................................................................................................ 2
1.3
Goal .................................................................................................................... 9
2. Theory and Method.................................................................................................... 10
2.1
Theory .............................................................................................................. 10
2.2
Method ............................................................................................................. 10
2.2.1
Finite Element Model Development .................................................... 10
2.2.2
Parts and Geometry .............................................................................. 11
2.2.3
Meshed Components ............................................................................ 11
2.2.4
Material Properties ............................................................................... 14
2.2.5
Arrangement of Assembly ................................................................... 15
2.2.6
Analysis Steps and Loading ................................................................. 16
2.2.7
Boundary Conditions ........................................................................... 18
2.2.8
Contact Interactions ............................................................................. 20
2.2.9
Output................................................................................................... 21
3. Results and Discussion .............................................................................................. 22
3.1
Output Requests ............................................................................................... 22
3.2
Example Output ............................................................................................... 22
3.3
Regression Analysis ......................................................................................... 26
3.3.1
Lug: von Mises Stress .......................................................................... 26
3
3.4
3.3.2
Lug: Shear Tear-Out ............................................................................ 27
3.3.3
Lug: Bearing Stress .............................................................................. 28
3.3.4
Pin: von Mises Stress ........................................................................... 29
3.3.5
Pin: Shear Stress................................................................................... 29
3.3.6
Pin: Bearing Stress ............................................................................... 30
Verification Analysis ....................................................................................... 31
4. Conclusions................................................................................................................ 33
4.1
Future Work ..................................................................................................... 33
5. References.................................................................................................................. 34
6. Appendix 1................................................................................................................. 35
7. Appendix 2................................................................................................................. 50
4
LIST OF TABLES
Table 1 – Material properties of pin ................................................................................ 15
Table 2 - Parameters for verification analysis ................................................................. 31
Table 3 - Verification Analysis Error .............................................................................. 32
5
LIST OF FIGURES
Figure 1 – Example double shear clevis connection ......................................................... 1
Figure 2 – Madux’s parametric lug dimensions ................................................................ 3
Figure 3 – Wearing’s findings on hoop stress based on stepped analysis ......................... 4
Figure 4 – Stenman’s finite element model ....................................................................... 5
Figure 5 – To’s conforming bushing, section view ........................................................... 6
Figure 6 – Antoni’s pin-bushing-lug system ..................................................................... 6
Figure 7– Strozzi’s stress results from loaded clevis......................................................... 7
Figure 8– Antoni’s spring contact representation .............................................................. 8
Figure 9– Kwon’s contact stress distribution .................................................................... 8
Figure 10 – Overall view of the finite element model ..................................................... 11
Figure 11 – Meshed Pin ................................................................................................... 12
Figure 12 – Meshed Outer Lug ........................................................................................ 13
Figure 13 – Meshed inner lug .......................................................................................... 14
Figure 14 – Meshed assembly ......................................................................................... 16
Figure 15 – Load application ........................................................................................... 17
Figure 16 – Z symmetry plane ......................................................................................... 18
Figure 17 – X symmetry plane ........................................................................................ 19
Figure 18 – Fixed boundary condition............................................................................. 20
Figure 19 – von Mises stress of assembly ....................................................................... 21
Figure 20 – Shear tear-out plane ...................................................................................... 22
Figure 21 – Overall von Mises stress .............................................................................. 23
Figure 22 – Lug von Mises stress .................................................................................... 23
Figure 23 – Lug shear tear-out stress ............................................................................... 24
Figure 24 – Lug bearing stress......................................................................................... 24
Figure 25 – Pin von Mises stress ..................................................................................... 25
Figure 26 – Pin shear stress ............................................................................................. 25
Figure 27 – Pin bearing stress .......................................................................................... 25
6
LIST OF SYMBOLS
σ1
von Mises stress, lug
σ2
shear tear-out stress, lug
σ3
bearing stress, lug
σ4
von Mises stress, pin
σ5
shear stress, pin
σ6
bearing stress, pin
Epin
Young’s modulus of pin
Elug
Young’s modulus of lug
gap
lug gap
Kn1
concentration factor for σn, based on lug gap
Kn2
concentration factor for σn, based on ratio of pin to lug Young’s moduli
Kn3
concentration factor for σn, based on ratio of pin to lug Poisson’s ratios
Kn4
concentration factor for σn, based on lug width
Lw
lug width
Vpin
Poisson’s ratio of pin
Vlug
Poisson’s ratio of lug
7
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.
8
ABSTRACT
This thesis presents an approach for sizing frictionless double shear-plane clevis
connections to be under their material yield strengths for their given application. 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 closed-form
empirical formulae, using concentration 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.
9
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 shear-plane 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 loadsharing ability and symmetry, and the advantages of a single shear-plane clevis due to
their simplicity of fabrication, installation, and maintenance.
Figure 1 – Example double shear clevis connection
However, despite the commonality of many aspects of most clevis connections, no
single source addresses the design of such connections in the elastic domain. Some
sources examine varying aspects of the clevis connection, but only base the design on
1
data from tests that bring the connection to failure, and thus reflect a nonlinear response.
Other sources 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 the
Cozzone and Melcon, who presented a study in 1950 [2], 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. Using a
fairly simple approach, they were able to begin correlating empirical data, destructive
testing, and close-form formulae for predicting failure in the lug. Furthermore, because
of this simple and generalized approach, 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, it produced a series of empirical curves that 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. The
key results of this study was to present (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) a generalized method of developing a stress intensity
factor; (3) evaluate the stress components of both axial and transverse loading scenarios.
2
Figure 2 – Madux’s parametric lug dimensions
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 evaluation of the conceptual pin in a plate
layout. This allowed him to begin to evaluate the non-linearities of an initial clearance,
evaluating 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 concept 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 in 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 tensioncompression strains caused by the hoop stress and contact loading at the pin-lug
interface correlated with photo-elastic testing.
3
Figure 3 – Wearing’s findings on hoop stress based on stepped analysis
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 finite element software to perform simulated
empirical analyses to compare to Madux’s findings. 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. 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 using a uniform pressure
distribution that extended from the axial location of the peak pressure to the axial
location where the contact pressure was approximately 15% of the peak pressure.
4
Figure 4 – Stenman’s finite element model
Also in 2008, Q. D. To et al [8] began investigating more complex variations of
Roa’s pin-plate model. The investigation centered about a system in which the bushing
between the pin and lug was of non-trivial thickness, and was of a conforming nature.
Specifically, To evaluated a system in which a bolt is placed into a hole in a plate of
glass, in which the hole is much larger than the pin. The pin is subsequently glued to the
glass using a resin. This example is unique in that unlike the previous studies [1]
through [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 heavily influence of friction of the systematic
behavior of the pin-in-plate concept. It should be noted that upon review, although
friction does play an important role in the system, then general contour of stress at the
contact interface was markedly similar between To’s and Rao’s models.
5
Figure 5 – To’s conforming bushing, section view
Dissimilar to To, in 2010 Antoni [10] examined non-conforming intermediate
bushings. 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
6
In 2011, Strozzi [11] expanded on Wearing’s investigation into progressive contact,
contact surfaces that 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 furthering 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– Strozzi’s stress results from loaded clevis
Antoni [12] continued his previous investigations from 2010, and in 2013 added
non-linearities to his evaluations. 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 suggest
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 to the
stiffness.
7
Figure 8– Antoni’s spring contact representation
Finally, in 2013, Kwon [13] challenged Madux’s and Stenman’s postulations, and
evaluated an analytical solution 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.
Figure 9– Kwon’s contact stress distribution
8
1.3 Goal
The goal of this thesis is to present an approach for sizing frictionless double
shear-plane clevis connections to be under their material yield strengths for their given
application. Finite element analysis will be utilized to simulate testing 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 induced 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.
9
2. Theory and Method
2.1 Theory
The theory used in this thesis is a four step process of deriving closed-form
formulae for a standard pin-lug clevis:
Step 1:
Perform a convergence study on the model to ensure accuracy of analysis
Step 2:
Perform suite of analyses varying the parameters to be used in closedform solution – one parameter at a time
Step 3:
Perform regression analysis on results and derive equations using best fit
curves of the data
Step 4:
Verify solutions by comparing closed-form solutions with an additional
analysis
2.2 Method
A finite element model will be developed to analyze the stress in the pin and lugs.
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
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.
The following
files were used for the finite element analysis.
Pin-Parametric.cae
ABAQUS CAE database
Pin-Parametric.jnl
ABAQUS Journal File
10
The
Figure 10 – Overall view of the finite element model
2.2.2
Parts and Geometry
The finite element model consists of a one-quarter pin, one-half 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. These boundary conditions
are explained in further detail in Section 2.2.7.
2.2.3
Meshed Components
The three components are each meshed using an element seed size of 0.25”. All
elements are 3-dimensional 20-noded hexahedral reduced integration continuum
elements, denoted in ABAQUS as C3D20R. A figure of each component as meshed is
as follows.
Zero element errors and zero element warnings exist in the model.
11
Appendix 1 outputs the mesh verification data for each part. A mesh convergence study
was performed to verify the adequacy of the mesh, and is presented in Appendix 1.
Figure 11 – Meshed Pin
12
Figure 12 – Meshed Outer Lug
13
Figure 13 – Meshed inner lug
2.2.4
Material Properties
Each component is assigned a material for any given analysis. As a Newton-
Raphson solution ABAQUS/Standard is being used for a static analysis, density is not
considered. Only Young’s Modulus and Poisson’s ratio are varied. Table 1 depicts the
Young’s Modulus and Poisson’s ratio values used, and for which corresponding
analysis.
14
Young's Modulus (psi)
1.00E+07
2.00E+07
3.00E+07
4.00E+07
5.00E+07
6.00E+07
3.00E+07
3.00E+07
3.00E+07
3.00E+07
3.00E+07
3.00E+07
3.00E+07
3.00E+07
3.00E+07
3.00E+07
Poisson's Ratio
0.3
0.3
0.3
0.3
0.3
0.3
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.3
Analysis Name
YM10
YM20
YM30
YM40
YM50
YM60
PR05
PR10
PR15
PR20
PR25
PR30
PR35
PR40
PR45
All Other Analyses
Table 1 – Material properties of pin
2.2.5
Arrangement of Assembly
The arrangement of the assembly consists of one instance of each part. They are
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 coplanar with the mid-length of the inner lug, for reasons described in Section 2.2.7. 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.
15
Figure 14 – Meshed assembly
2.2.6
Analysis Steps and Loading
The analysis consists of two steps. These steps are Initial and Load.
Initial: In the Initial step, the appropriate boundary conditions for each run are created.
Further discussion and explanation of the boundary conditions are presented in Section
2.2.7.
16
Load: Load is a Static, General step. In this step, the loads for each run are applied to
the model. The load is ramped up from 0% to 100%, with static equilibrium solved for
at 10% increments. This will allow 10 data points for every analysis to be processed and
evaluated. The load is applied as a uniform anti-pressure at the top surface of the inner
lug; see Figure 15. It should be noted that due to the symmetry conditions described in
Section 2.2.7, only one-quarter of the load in the physical system is applied to the model
to achieve the same strains.
Figure 15 – Load application
17
2.2.7
Boundary Conditions
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. 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); see Figure 16.
Figure 16 – Z symmetry plane
The second symmetry boundary condition is imposed at the 0-180 azimuth plane of
the pin and both lugs. This allows for the X-axis extent of the model to be half of the
18
physical system. This symmetry boundary condition is imposed on continuum element
nodes using a zero-displacement condition in the x-axis degree of freedom (ABAQUS
degree of freedom U1); see Figure 17. 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. This boundary condition both reacts the applied
load and prevents the model from diverging based on free body modes; see Figure 18.
This fixed boundary condition is imposed on continuum element nodes using a zero19
displacement condition in the y- and z-axis degree of freedom (ABAQUS degrees of
freedom U2 and U3).
Figure 18 – Fixed boundary condition
2.2.8
Contact Interactions
One contact interaction constraint is imposed in the model, between the pin and the
lugs. The pin is designated the master surface, and the lugs comprise the slave surface.
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
20
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.
2.2.9
Output
An example output is presented as follows. However, it should be noted that for
each analysis, a full contour output was requested. 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
21
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, bearing stress for the lug and pin,
shear-tear out at the lug, bearing stresses 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 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.
22
Figure 21 – Overall von Mises stress
Figure 22 – Lug von Mises stress
23
Figure 23 – Lug shear tear-out stress
Figure 24 – Lug bearing stress
24
Figure 25 – Pin von Mises stress
Figure 26 – Pin shear stress
Figure 27 – Pin bearing stress
25
3.3 Regression Analysis
Through the use of regression analyses, concentration factors are determined for
each variable, and a final set of equations are developed. Equations are presented for
von Mises stress in the lug, shear tear-out in the lug, bearing stress in the lug, von Mises
stress in the pin, shear stress in the pin, and bearing stress in the pin.
For each stress mode, a general equation based upon the load applied is presented
and then stress concentration factors for each variable are presented.
K1 is the
concentration factor from the axial gap between lugs, K2 is the concentration factor from
the Young’s Moduli, K3 is the concentration factor from the Poisson’s ratios, and K4 is
the concentration 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 = Vpin/Vlug = 1), and dividing the equation by the result of the
baseline equation, i.e. normalizing each equation such that at the baseline condition, the
concentration factor is equal to 1.
3.3.1
Lug: von Mises Stress
The von Mises stress in the lug is determined to be:
Where:
26
3.3.2
Lug: Shear Tear-Out
The shear tear-out stress in the lug is determined to be:
Where:
27
3.3.3
Lug: Bearing Stress
The bearing stress in the lug is determined to be:
Where:
28
3.3.4
Pin: von Mises Stress
The von Mises stress in the pin is determined to be:
Where:
3.3.5
Pin: Shear Stress
The shear stress in the pin is determined to be:
Where:
29
3.3.6
Pin: Bearing Stress
The bearing stress in the pin is determined to be:
Where:
30
3.4 Verification Analysis
To confirm that the formulae presented are accurate, a verification analysis is
performed. One additional analysis is performed, with the following parameters:
Parameter
Load
Gap
Epin
Elug
v_pin
v_lug
Lug Width
Value
400 lbf
0.3"
6.00E+07
3.00E+07
0.15
0.3
1"
Table 2 - Parameters for verification analysis
The resulting finite element analysis showed a maximum von Mises stress in the lug
of 1612 psi. The computed concentration factors, and computed von Mises stress in the
lug is:
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:
31
Stress Component
Lug von Mises
Lug shear tear-out
Lug bearing
Pin von Mises
Pin shear
Pin bearing
FEA Value
(psi)
1612
800
1350
1501
377
1150
Calculated Value
(psi)
1586
794
1298
1409
372
1111
Error
(%)
1.6
0.8
3.9
6.1
1.3
3.4
Table 3 - Verification Analysis Error
The resulting error on the analysis is calculated to be 0.8% to a maximum of 6.1%,
within the acceptable limits of accuracy.
32
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 concentration 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.
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 vaiables 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.
33
5. References
[1]
N. S. Venkataraman, A Study into the Analysis of Interference Fits and Related
Problems. Ph.D. Thesis, Indian Institute of Science, Bangalore, India (1996)
[2]
F. P. Cozzone, Melcon, and Hoblit, Analysis of Lugs and Shear Pins Made of
Aluminum or Steel Alloys, Product Engineering. (1950)
[3]
M. A. Melcon and F. M. Hoblit, Developments in the Analysis of Lugs and Shear
Pins, Product Engineering. (1953)
[4]
Maddux et al. Stress Analysis Manual, Air Force Flight Dynamics Laboratory.
August 1969.
[5]
A. K. Rao, 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 Analsyis for Engineering Design. (1985)
[7]
C. A. Stenman, A Comparison of the Predicted Mechanical Behavior of Lug
Joints using Strength of Materials Models and Finite Element Analysis
[8]
Q.D. To et al, On the Cnforming Contact Problem in a Reinforced Pin-Loaded
Strucutre with a non-Zero Second Dundurs’ Constant
[9]
M. Ciavarella and P. Decuzzi, The State of Stress Induced by the Plane
frictionless Cylindrical Contact. I. The Case of Elastic Similarity. Int. J. Solids
Struct. (2001)
[10]
N. Antoni and F. Gaisne, Analytical Modeling for Static Stress Analysis of PinLoaded Lugs with Bush Fittings. Teuchos, Safran Group, Mechanics of
Structures Department, Montigny-Le-Bretonneux, France. (2010)
[11]
A. Strozzi et al, Maximum Equivalent Stress in a Pin-Loaded Lug in the Presence
of Initial Clearance. The Journal of Strain Analysis for Engineering Design.
(2011)
[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, Velizy-Villacoublay, France. (2013)
[13]
Kwon, A Critical Study of Pin Bending Behavior Using Finite Element Analysis.
Rensselaer Institute of Technology. (2013)
34
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.
analyses were performed using ABAQUS/Standard, Version 6.13-EF1.
The
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).
35
Figure A1-1 – Overall view of the finite element model
Parts and Geometry
The finite element model consists of a one-quarter pin, one-half 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. The pin has a radius of 0.5”
and a length of 5”. The lugs both have an inner radius of 0.5”, an outer radius of 1.0”, an
overall height of 2.4”, and a width of 2.0”.
Meshed Components
The three components are each meshed using varying seed size. All elements are
3-dimensional 20-noded hexahedral reduced integration continuum elements, denoted in
ABAQUS as C3D20R. A figure of each component as meshed is as follows. Zero
element errors and zero element warnings exist in the model.
36
Figure A1-2 – Meshed Pin (Pin Radius = 10 Elements)
37
Figure A1-3 – Meshed Outer Lug (Pin Radius = 10 Elements)
38
Figure A1-4 – Meshed Inner Lug (Pin Radius = 10 Elements)
Material Properties
As a Newton-Raphson solution ABAQUS/Standard is being used for a static
analysis, density is not considered. Only Young’s Modulus, valued at 30E6 psi, and
Poisson’s ratio, valued at 0.3, are defined.
39
Arrangement of Assembly
The arrangement of the assembly consists of one instance of each part. They are
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 axial gap between the inner lug
and outer lug is 0.5”, and the pin protrudes from the outer lug an arbitrary amount (0.5”),
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 A1-4 – Meshed assembly (Pin Radius = 10 Elements)
40
Analysis Steps and Loading
The analysis consists of two steps. These steps are Initial and Load.
Initial: In the Initial step, the appropriate boundary conditions for each run are created.
Load: Load is a Static, General step. In this step, the loads (100 lbf) for each run are
applied to the model. The load is applied as a uniform anti-pressure at the top surface of
the inner lug.
Figure A1-5 – Load application (Pin Radius = 10 Elements)
41
Boundary Conditions
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. 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 A1-6 – Z symmetry plane (Pin Radius = 10 Elements)
The second symmetry boundary condition is imposed at the 0-180 azimuth plane of
the pin and both lugs. This allows for the X-axis extent of the model to be half of the
42
physical system. This symmetry boundary condition is imposed on continuum element
nodes using a zero-displacement condition in 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 A1-7 – X symmetry plane (Pin Radius = 10 Elements)
The second type of boundary condition is a fixed boundary condition, which is
applied at the bottom of the outer lug. 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
43
condition in the y- and z-axis degree of freedom (ABAQUS degree of freedom U2 and
U3).
Figure A1-8 – Simply supported boundary condition (Pin Radius = 10 Elements)
Contact Interactions
One contact interaction constraint is imposed in the model, between the pin and the
lugs. The pin is designated the master surface, and the lugs comprise the slave surface.
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.
44
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.
Output
An example output (MR1) is shown in the following figures.
Figure A1-9 – Overall Stress
Figure A1-10 – Pin von Mises
45
Figure A1-11 – Lug von Mises
Figure A1-12 – Lug shear tear-out stress
46
Figure A1-15 – Pin shear stress
Figure A1-16 – Lug contact pressue
47
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.
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
48
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)
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.
49
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
50
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
51
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
52
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
53
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
54
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
55
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
56
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
57
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
58
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
3500
3000
2500
Lug Mises (S MISES)
2000
y = 580.98x2 - 2377.3x + 4078.9
R² = 0.9958
1500
Poly. (Lug Mises (S
MISES))
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
59
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
60
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
61
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
62
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
63
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
64
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
65