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EML5523 Finite Element Analysis (Spring, 2015) April 19, 2015
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Modelling, Simulation and Optimization of a Torque-Arm
Akash Marakani (University of Florida)
Abstract
Finite element analysis for a Torque-arm is performed using a FEA software package called ABAQUS which is further
used for its simulation and optimization. The main objective of the optimization is to come up with the lightest structure
while having stress constraints. The torque arm is used for transmitting load from a shaft to a wheel. A preliminary
analysis is done to estimate the location of the maximum stress and the magnitude of the displacement under the
influence of load. Then the convergence study is carried out to determine the reasonable mesh size. The accurate
vertical displacement at the load application point is found out using the Richardson’s extrapolation. Finally, a Parameter
study is done to optimize the structure to minimize its mass and the optimal design for the torque arm is found out.
1. Introduction
Part 1: Preliminary analysis
A hand calculated analysis is done for the torque arm at
the initial design ( 𝑑1 = 12𝑐𝑚, 𝑑2 = 1𝑐𝑚, 𝑑3 = 27𝑐𝑚 ).
This analysis is basically done by approximating the
torque arm to a cantilever beam under the influence of a
point load at its tip to estimate its vertical displacement at
the tip and find out the location of the maximum stress.
Further these hand calculated values are then compared
with those obtained from the Abaqus analysis and then
the difference between them have been explained.
Part 2: Effect of Elements
The torque arm is modelled on Abaqus by sketching and
making the torque arm geometry fully constrained (as
shown in Figure 1). Various constraints such as
tangency, coincidence symmetry, fixed and proper
geometry is used to constrain sketch. When the geometry
becomes fully constrained the sketch turns green. The
necessary boundary conditions and loads are applied at
the left and the right hole of the shaft using the MPC
constraint. Then the mesh is generated using appropriate
partition lines for various element types such as Constant
Strain Triangle (CST), Linear Strain Triangle (LST), and
Quadrilateral with 4-node and 8-node. The vertical
displacement at the center of the right shaft and the
maximum Von-Mises stress is computed from the FE
analysis and further compared with those obtained in the
preliminary analysis.
Part 3: Convergence Study
A convergence study is carried out at the initial design
using a Quadrilateral 4-node element (Q4) on the vertical
displacement at the center of the right shaft with the
change in the number of elements in the structure. A
reasonable mesh size is found out. Further, we use
Richardson’s extrapolation to estimate the accurate
displacement at the load application point.
Part 4: Parameter Study
Using the mesh size obtained in the convergence study,
we try to minimize the design mass such that the
maximum stress is 240 MPa (Mega Pascal). We use a
one variable function as shown below,
The Relationship between‘d’ and design variable
(𝑑1, 𝑑2, 𝑑3) are
𝑑1 = 10𝑑 + 8
𝑑2 = −2𝑑 + 3
𝑑3 = −9𝑑 + 36
Where d∈ [0, 1]
Further a graph is plotted between the 𝑓(𝑑) and 𝑑 values
and the point where the function value is minimum, the
corresponding d value is used to find out the optimal
design.
Figure 1: Torque-Arm with dimensions in cm

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2. Approach
2.1 Part 1: Preliminary Analysis
The vertical displacement at the center of the right shaft
and the maximum Von-Mises stress are found out using
the techniques from Mechanics of Materials. As we know
that the cross sectional area of the torque arm is varying,
we approximate it to that of a cantilever beam under the
influence of a point load at one end and the other end is
fixed. The fixed end is considered as the center of the left
shaft and the right hole is considered as the other end.
The given material properties are as follows:
Young’s Modulus (E) = 206.8 𝐺𝑃𝑎
Poisson’s ratio (μ) = 0.29
Thickness = 1.0 cm
Density = 7850 𝑘𝑔/𝑚3
The approximated beam equivalent to the torque arm is
as shown in the below figure.
Figure 2: Cantilever beam used for preliminary analysis
The Displacement at the free end of the cantilever beam
is,
δ=𝑃𝐿3
/3𝐸𝐼
Where,
𝑃= Load (𝑖𝑛 𝑁𝑒𝑤𝑡𝑜𝑛𝑠)
𝐿= Length of the beam (42𝑐𝑚)
𝐼= Moment of Inertia (𝑏ℎ3
/12) in cm^4
b= 1 cm (thickness)
h= average height of 9.2 cm
The location of the maximum stress is computed by
calculating the stress at various points and then finding
out the maximum value for them. Due to the loading
conditions two kinds of stress exists,
Axial Stress (σx
1) = F/A
Where, F= Force applied i.e. -2789 N
A= Cross sectional area (cm2)
Bending Stress (σx
2) =
𝑀𝑦
𝐼
Where, M=Moment at the point (N-cm)
y =Distance from the neutral axis (cm)
I = Moment of inertia (cm4)
Moment of Inertia =
𝑏ℎ3
12
=
1×9.23
12
= 64.89 𝑐𝑚4
Therefore, using the axial and the bending stress the
stresses at every point are computed and hence the
maximum von-mises stress is found out.
2.2 Part 2: Effect of element types
Using the initial design constraints, a fully constrained
sketch is obtained by modelling it on the FE software
Abaqus as shown in the figure 3.
Figure 3: Fully Constrained sketch of the torque-arm
We use a plane stress element type, after the partitioning
has been done; a structured mesh is generated and
further the element type is varied to run the analysis to
find out maximum von-mises stress and vertical
displacement which is calculated in each case and
tabulated.
The number of nodes in each case is kept approximately
the same for all the element types (CST, LST, Q4, and
Q8) by varying the global size in each case.
2.3 Part 3: Convergence Study
In the convergence study, FE analysis is done using a Q4
element type. The mesh is made finer by drawing
partitioning lines. The mesh size is varied and the
analysis is done such that the number of nodes is
restricted to 1000 nodes as we are using a student
edition. A graph between the vertical displacement and
the number of element is plotted and we notice a

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converging curve to infinity and the reasonable mesh size
is estimated. However, the displacement value is not the
accurate value. To calculate the accurate value for
displacement we use Richardson’s extrapolation
technique to compute the value at infinity using the below
formula.
We obtain a similar graph.
Figure 4: Convergence study graph-Expected
Richardson’s Formula,
The values for the above equation are obtained from the
graph plotted between the mesh size and the
displacement values.
Figure 5: Fine Mesh obtained in Q4 analysis
Figure 6: Deformed contour after analysis for initial
design
2.4 Part 4: Parameter Study
We have to use the reasonable mesh size obtained from
the convergence study. The main objective of this study
is to minimize the mass and to keep the σmax ≤ 240 MPa.
We use a Q4 element with the reasonable mesh size to
create the mesh for the torque arm. But for this study
however, the values of d1, d2, d3 are varied to find out
the optimal design.
𝑑1 = 10𝑑 + 8
𝑑2 = −2𝑑 + 3
𝑑3 = −9𝑑 + 36
Where d∈ [0, 1]
Where d is incremented by 0.1 at each case and the
sketch is regenerated and an FE analysis is performed at
each and every step and the values for mass and the
maximum von-mises step are computed which are further
used to calculate the function value from the give
equation below.
Where,
Mass (d) = the mass obtained for a value of d
σmax (d) = Stress value obtained at the maximum value for
the corresponding value of d.
Further, a graph between 𝑓(𝑑) vs 𝑑 is plotted to find out
the optimal design of the structure and the final stress and
minimum mass is computed.
3. Results
Part 1: Preliminary analysis
For the initial design i.e. d1=12cm, d2=1 cm and
d3=27cm the analysis is done using the principles from
mechanics of materials. The torque arm is approximated
to a cantilever beam which tip load at one end. To find
the displacement at the center of the right hole, the
corresponding loads are acting as
𝐹𝑥 = −2789 𝑁
𝐹𝑦 = 5066 𝑁
The Displacement at the free end of the cantilever beam
is,
δ=𝑃𝐿3
/3𝐸𝐼
Where the vertical displacement is only caused because
of the vertical load in the y-direction.

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𝑃= 5066 N
𝐿= 42cm
𝐼= Moment of inertia
Displacement =
5066 ×423×12
3× 206.8×105×1×9.23 = 0.0932 𝑐𝑚
Displacement at the center of the right hole is calculated
to be 0.0932𝒄𝒎.
The maximum Von-Mises stress is calculated to be stress
due to horizontal force which leads to an axial stress and
the vertical force which leads to a bending stress in the x-
direction. The assumptions made are that the beam is
uniform and length of the beam is 42cm.
Stress due to horizontal force (σx1) =
𝐹
𝐴
=
−2789
1×9.2
Where,
F = Force in x direction (in newton)
A = Cross sectional are of the structure (𝑐𝑚^2)
Axial Stress = −𝟑𝟎𝟑. 𝟏𝟓 𝐍/𝐜𝐦 𝟐
Bending stress due to vertical load =
𝑀𝑦
𝐼
Where, 𝑀 = Moment at a location of 42 cm
= 5066 × 42 𝑁 − 𝑐𝑚
𝐼 = Moment of inertia= 64.89 𝑐𝑚4
𝑦 =
9.2
2
= 4.6𝑐𝑚
Bending Stress =
𝟓𝟎𝟔𝟔×𝟒𝟐×𝟒.𝟔
𝟔𝟒.𝟖𝟗
= 𝟏𝟓𝟎𝟖𝟑. 𝟐𝟑
𝑵
𝒄𝒎 𝟐
Maximum Von-Mises Stress = Bending Stress + Axial
stress
Max. Von-Mises Stress = (15083.23 − 303.15)𝑁/𝑐𝑚^2
= 14780.08
𝑁
𝑐𝑚2
Maximum Von mises stress = 147.8 𝑀𝑃𝑎
The maximum von mises stress value differs from the FE
analysis as we have approximated the structure to that of
a cantilever beam to minimize the hand calculation. The
maximum von mises stress location is at the top and
bottom of the left hole for the inner hole of the slot.
Part 2: Effect of element types
After a fully constrained sketch is generated on Abaqus,
a structured mesh is obtained by drawing partition lines.
The FE program results are obtained for each of the
different types of element and the result for its vertical
displacement at the left hole and the maximum von-mises
stress is calculated and the results are further tabulated
and compared as shown in the table below.
Table 1: Effect of element types
No. of
Nodes
Element
Type
Displacement
( in cm )
Max. Von-Mises
stress (in Mega-
Pascals)
416 CST
(CPS3)
0.113 155.6
428 LST
(CPS6)
0.122 188
416 Q4
(CPS4)
0.1204 173.4
416 Q8
(CPS8)
0.1229 193.8
Part 3: Convergence Study
At the initial design we consider a Q4 element to perform
a convergence study on the vertical displacement at the
center of the right shaft by plotting a graph between the
vertical displacement and the number of elements. As the
value converges we find out the reasonable mesh size of
the structure. The values have been tabulated as follows.
Table 2: Displacement, elements and mesh size for Q4
Using the above values obtained from the FE program,
we plot a graph as shown in the graph below.
Figure 7: Convergence study graph
Mesh Size Number of elements Vertical displacement (in cm)
2.5 97 0.114549
2 113 0.11738
1.5 148 0.1186
1 320 0.1204
0.8 457 0.1211
0.7 546 0.12141
0.65 746 0.12204
0.62 824 0.12216

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The reasonable mesh size is obtained to be computed to
be
Mesh Size = 0.62.
As we now are limited to 1000 nodes and need to
calculate the accurate displacement at infinity we use
Richardson’s extrapolation technique to compute the
value.
Richardson’s formula =
Where,
Φ = Displacement (𝑖𝑛 𝑐𝑚𝑠)
H = Mesh Size
Q = 2 (displacement)
Using the values obtained in the convergence study the
following graph is plotted.
Figure 8: Richardson Extrapolation
Using the above graph we calculate,
ϕ =
(0.12216×0.4225)−(0.1220×0.3844)
0.4225−0.3844
𝑐𝑚
Therefore,
Displacement = 0.1237 𝒄𝒎
Part 3: Parameter Study
A parameter study is conducted to find out the optimal
design to minimize the stress by constraining the stress
to ≤ 240 MPa. We use an objective function such as
f(d).
Initial mass = 2.31 𝑘𝑔𝑠
σmax (d) is in 𝑀𝑃𝑎
The relation between d and the design variable are as
follows
𝑑1 = 10𝑑 + 8
𝑑2 = −2𝑑 + 3
𝑑3 = −9𝑑 + 36
Where d∈ [0, 1]
As the value of d changes there is change in the design
variable and hence leads to a change in the FE geometry.
The study is performed by changing the value of d
between 0 and 1 with 10 equal increments.
Further a graph is plotted using the values obtained from
the function. 𝑓(𝑑) reaches a minimum value for a
particular value of d, which is considered as the optimal
d and for that value of d the design variables are
computed. Moreover, using those optimal values we
computed the stress and minimum mass of the structure.
Table 3: Parameter study
D D1 D2 D3 Mass (kg) Stress (N/cm^2) F(d)
0 8 3 36 1.566 4.64E+04 94.13626
0.1 9 2.8 35.1 1.687 4.15E+04 73.77197
0.2 10 2.6 34.2 1.79 4.04E+04 69.06656
0.3 11 2.4 33.3 1.8917 3.66E+04 53.40225
0.4 12 2.2 32.4 1.98557 3.07E+04 28.73455
0.5 13 2 31.5 2.072 2.81E+04 17.93864
0.6 14 1.8 30.6 2.15335 2.47E+04 3.807186
0.7 15 1.6 29.7 2.2268 2.15E+04 0.963983
0.8 16 1.4 28.8 2.293 1.90E+04 0.992641
0.9 17 1.2 27.9 2.35214 1.75E+04 1.018242
1 18 1 27 2.40514 1.60E+04 1.041186

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Using the tabulated values of 𝑓(𝑑) and d the graph is
plotted as shown in figure
Figure 9: Graph between f(d) vs d
Therefore we obtain a minimum value for the function at
d=0.7. the corresponding decision variables are as
follows.
Table 4: Optimal design values
Figure 10: The final constrained sketch - Optimal design
Figure 11: The Final analysis for optimal design
4. Discussions
The preliminary analysis is done using the methods from
mechanics of materials and the vertical displacement at
the center of the right shaft is calculated as well as the
location of the maximum stress is computed to be right
above the left circle of the inner slot. These results are
further compared to that of the FE analysis and the
results are approximately same with a slight difference
between them. This difference is mainly because while
doing the hand calculation to make them easier we
considered the geometry of the torque arm to be
approximate to that of a cantilever beam under tip
loading.
Further, various elements structured using the mesh and
the results in each case are tabulated. The convergence
study helps us to determine the reasonable mesh size for
the structure using a Quadrilateral with 4-node element
type. As the results are converging the accurate
displacement is found out using the Richardson
extrapolation and found out to be 0.1237 𝒄𝒎 .The
parameter stud helps us in determining the optimal
design by varying the design variables to minimize the
weight while a maximum stress constraint of ≤ 240 𝑀𝑃𝑎.
By minimizing the weight we are reducing the cost to
manufacture such a component which in turn helps in
making the design more optimized and economical. The
final mass of the structure reduces to 𝟐. 𝟐𝟐𝟔𝟖 𝒌𝒈𝒔.
5. Appendix
The input file for a quadrilateral with 4 nodes has been
briefly explained.
First, a part has been created corresponding the given
dimension.
** PARTS
**
*Part, name=Part-1
*End Part
**
**
An assembly instance is created for a quadrilateral with 4
node element type.
** ASSEMBLY
**
*Assembly, name=Assembly
d 0.7
15 cm
1.6 cm
29.7 cm
Mass 2.2268 kgs
Stress 215 Mpa
𝑑1
𝑑2
𝑑3
0
20
40
60
80
100
0 0.5 1 1.5
F(d)
d
Parameter Study
Optimal
Design
values

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**
*Instance, name=Part-1-1, part=Part-1
*Node
1, 2.76761985, -2.88795424
Under the mesh module the CPS4 element type is done.
*Element, type=CPS4 (Q4 element type)
1, 1, 43, 403, 67
*Node
983, 42., 1.5308085e-16, 0.
*Nset, nset=Part-1-1-RefPt_, internal
983, (Number of nodes are 983)
*Nset, nset=Set-1, generate
1, 982, 1
*Elset, elset=Set-1, generate
1, 824, 1 (Number of elements are 824)
** Section: Section-1
*Solid Section, elset=Set-1, material=Material-1
1.,
*End Instance
**
The constraints are further applied at the center of the left
and right shaft using multi point constraint (MPC).
** Constraint: Constraint-1
*MPC
BEAM, s_Set-3, m_Set-3
** Constraint: Constraint-2
*MPC
BEAM, s_Set-5, m_Set-5
*End Assembly
**
Further, the material properties are assigned,
** MATERIALS
**
*Material, name=Material-1
*Density
0.00785,
*Elastic
2.068e+07, 0.29 (Young’s modulus, Poisson’s ratio)
** ----------------------------------------------------------------
A step is created with linear perturbation.
**
** STEP: Step-1
**
*Step, name=Step-1, nlgeom=NO, perturbation
*Static
**
The boundary conditions are as follows,
** BOUNDARY CONDITIONS
**
** Name: BC-1 Type: Symmetry/Antisymmetry/Encastre
*Boundary
Set-10, ENCASTRE
**
The loads are as follows for the center of the right shaft,
** LOADS
**
** Name: Load-1 Type: Concentrated force
*Cload
Set-9, 1, -2789.
Set-9, 2, 5066.
**
** OUTPUT REQUESTS
**
**
** FIELD OUTPUT: F-Output-1
**
*Output, field, variable=PRESELECT
**
** HISTORY OUTPUT: H-Output-1
**
*Output, history, variable=PRESELECT
*End Step
Similarly the mesh is controlled by varying the element
type for LST, CST and Q8 element type. To perform the
parameter study the dimensions of the sketch are further
varied.

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FEA Project 2- Akash Marakani

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FEA Project 2- Akash Marakani