A CONTINUUM MECHANICS APPROACH OF
DERIVING STRESS TENSOR COMPONENTS
OF DOUBLE SHEAR-PLANE REVOLUTE
JOINTS IN THE ELASTIC DOMAIN
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
INTRODUCTION
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
• Time consuming
• Computationally intensive.
This thesis presents an approach for sizing frictionless double shearplane clevis connections to be under their material yield strengths for
their given application.
•
•
•
•
Simple empirical formulae
No specialized personnel required to develop finite element models,
Shorter analysis time
Shorter iteration time to understand what dimensions and variables are
most critical for a given clevis system.
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.
Widespread applications in mechanical systems, ranging from submarine
applications to house-hold door hinges.
• Although both singleand multi-shear-plane
clevis joints exist,
double shear-plane
joints are by far the
most common in
practical applications
(balance of
robustness/complexity).
– ability and symmetry
– simplicity of
fabrication, installation,
and maintenance.
HISTORY
Leveraging off FEA
Cozzone & Melcon (1950): first to be able to correlate empirical data,
destructive testing, and close-form formulae for predicting failure in the lug
Maddux (1953): evaluated both single shear and multi shear lug designs, for
use in designing components to be under their ultimate strength
Rao (1978): evaluated the non-linearities of an initial clearance, and
estimated the stresses in the pin, plate, and contact interface
Wearing (1985): used finite element analysis to transition from a pin within
an infinite plate, to a pin within a clevis lug joint
Stenman (2008): re-evaluated Maddux’s conclusions for contact pressure
To (2008): evaluated conforming bushing designs
Antoni (2010): evaluated non-conforming bushing desings
Strozzi (2011): further investigated initial clearance
Kwon (2013): further investigated Maddux’s and Stenman’s findings on
contact pressure
PROCEDURE
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 an
additional analysis
STEP 1: CONVERGENCE
STUDY
A plot is produced of mesh density vs. stress, and then is normalized to the most dense mesh result.
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
Value (Normalized)
1
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
MR #
7
8
9
10
PROCEDURE
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 an
additional analysis
STEP 2: MODEL
DEVELOPMENT
The finite element model
consists of:
• One-quarter pin
• One-half outer lug
• One-quarter middle lug
All elements are 3dimensional 20-noded
hexahedral reduced
integration continuum
elements, denoted in
ABAQUS as C3D20R.
BOUNDARY
CONDITIONS
Symmetry boundary conditions are
applied at
• mid-length of the inner lug and the
mid-length of the pin
• 0-180 azimuth plane of the pin and
both lugs
Fixed at bottom of outer lug
LOAD AND
INTERACTIONS
Load is as a anti-pressure at the
top of the inner lug
One contact interaction constraint
is imposed in the model, between
the pin and the lugs
• 3D surface smoothing
• Instances are places in initial fullclosure
• Normal contact: classical
Lagrange multiplier method
• standard pressure-overclosure
relationship.
• Tangential contact: zero-penalty
(frictionless / full-slip condition)
PARAMETRIC
ANALYSES
Minimu
Maximu
Increment
Analyses
m Value
m Value
Value
Performed
Parametric
Units
Load
lbf
400
4000
400
10
Lug Width
in
0.5
2
0.25
7
Lug Gap
in
0
0.9
0.1
10
6.00E+07
1.00E+07
6
0.55
0.05
9
Young's Modulus
of Pin
Poisson's Ratio
of Pin
psi
-
1.00E+0
7
0.05
EXAMPLE OUTPUT
PROCEDURE
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 an
additional analysis
REGRESSION
ANALYSES
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 tear-out 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
• 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 = 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 multiplication factor is equal to 1.
REGRESSION
ANALYSES: EXAMPLE
The various analyses are post-process, and best-fit curves are
created
The curves are created using a polynomial function of the power
needed to create a good correlation (R2 > 0.98)
• The curve for load applied was set about a y-intercept of 0, as no
stress exists with zero load applied
The equations taken from each curve, except for the load
applied curve, are then evaluated for the baseline configuration
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.
REGRESSION
ANALYSES: EXAMPLE
Lug von Mises vs.
Lug Width (in)
3000
Lug von Mises vs.
Lug Gap (in)
y = 636.44x3 - 2914.5x2 + 4151.7x +
541.86
R² = 0.9944
Lug Mises (S MISES)
2500
2000
3000
2000
1500
1500
Poly. (Lug Mises (S
MISES))
1000
500
y = 1279.7x + 1625
R² = 0.9996
Lug Mises (S
MISES)
2500
Linear (Lug Mises
(S MISES))
1000
500
0
0
0.5
1
1.5
2
0
2.5
0
Lug von Mises vs.
Young’s Modulus Ratio (-)
4000
y = 580.98x2 - 2377.3x + 4078.9
R² = 0.9958
3500
0.5
1
Lug von Mises vs.
Poisson’s Ratio Ratio (-)
y = -130.5x + 2392.1
R² = 0.9909
2400
2350
3000
2500
2000
1500
1000
Lug Mises (S
MISES)
2300
Lug Mises (S
MISES)
Poly. (Lug Mises (S
MISES))
2250
Linear (Lug Mises (S
MISES))
2200
500
0
2150
0
0.5
1
1.5
2
2.5
0
0.5
1
1.5
2
REGRESSION
ANALYSES: EXAMPLE
REGRESSION
ANALYSES: EXAMPLE
REGRESSION
ANALYSES: EXAMPLE
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:
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
2000
4000
6000
PROCEDURE
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 an
additional analysis
VERIFICATION
ANALYSES
Value
Parameter
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)
VERIFICATION
ANALYSES: EXAMPLE
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 is 1590 psi (1.36% Error)
VERIFICATION
ANALYSES
It is found that the average error percentage
among all stress components is less than 6.8%,
within the acceptable limits of accuracy
Stress Component
Average Error
(%)
Lug von Mises
4.2
Lug shear tear-out
3.7
Lug contact pressure
5.7
Pin von Mises
6.8
Pin shear
1.4
Pin contact pressure
5.7
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.
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.
• 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.