cad project 05 report final
TRANSCRIPT
PROJECT 05
FINITE ELEMENT ANALYSIS
WITH SOLIDWORKS
UB
SIDHANT SHARMA
#36819679
MAE -577
11/3/2010
2
Contents
1 Introduction…………………………………..………………3
2 Problem statement…………………..……….….........…....…3
3 Result…………………………….….……….……..………..6
3.1 2-D Truss...………………..………….………………………….…..6
3.2 Beams………………………………………………………….……14
3.2.1 Simply supported beam…………………………...………….…..……14
3.2.2 Cantilever beam………………………………………………..………19
3.3 C-link analysis……………...………………...............................….23
3.3.1 Standard mesh…………………………………..………………..……23
3.3.2 Curvature based mesh…………………………..……………..………28
3.3.3 Rounded inside edges………………………….……………..…….….34
3.3.4 Design change consideration………………...……………..……….…44
4 Discussion………………………………….....….……..….50
5 Conclusion…………………………….…...…………..…..51
6 References………………………………...……...….…….52
3
1 Introduction
This project is intended towards learning the use of Solidworks 2010 for carrying out finite
element analysis of structures and mechanical devices such as a truss or a cantilever beam. The
CAE package is very handy for doing such a type of finite element study as it is fairly easy to
define the various conditions or the constraints under which the device is being subjected to, and
it gives sufficiently accurate results required for the further analysis of the system.
The problems considered in this project such as the 2-D truss or the loaded beam problem can be
solved analytically using the basic equilibrium equations of force and moment, and the results
can be visually analyzed using the powerful MATLAB tool. The solutions obtained in
Solidworks are compared to the solutions in MATLAB with the help of various plots to explore
the accuracy of the CAE package.
FEA is being tremendously used in the industry today for design and analysis purposes.
SolidWorks provides the ability to carry out this technique in a very user friendly manner, it is
fast and the design changes can be explored by making changes in the CAD models and design
constraints. This ability of Solidworks has been depicted in this project by considering a detailed
study of a C-bracket.
In this project, we have not only learnt the basics of implementing FEA in this software through
lessons provided in the book “Introduction to Finite Element Analysis using SolidWorks
Simulation 2010” but also used explored the capability of the package for analysis using
different mesh types, mesh sizes, and observing their corresponding effects on computational
time and convergence towards the analytical solution.
2 Problem statement
We are required to study and complete the lessons 4, 7, 11 provided in the book “Introduction to
Finite Element Analysis using SolidWorks Simulation 2010”. The results obtained are to be
compared with analytical solutions plotted in MATLAB in order to explore the accuracy of the
solutions in Solidworks.
The first part of the problem deals with the analysis of a 2-D Truss element (lesson 4). This
problem has to be solved analytically using the direct stiffness method of truss elements in 2-D
space by formulating the stiffness matrix. The truss structure consists of two elements and three
joints. The two base joints are fixed and only allow rotation as a degree of freedom. The middle
joint allows rotation as well as translation and it is subjected to a force of 50 lb in the horizontal
direction as shown in Figure 1. The material is steel rod of diameter 0.25 inches.
4
Figure 1 Truss element configuration with loading
We are required to solve this problem to compute the stress in the elements and the displacement
of the middle joint. These results are to be plotted in MATLAB.
Similarly, the same problem has to be solved in Solidworks using the steps provided in Chapter 4
and thus generate the stress, displacements plots for comparison.
The second part of the problem is to solve simple beam problems analytically using the force-
moment equilibrium equations. The shear force and bending moment equations are to be
formulated and the plots are to be shown using MATLAB. The loaded beams are shown below;
Figure 2 (a) which shows a supported beam and Figure 2 (b) which shows a cantilever beam. The
material for the beams is alloy steel.
By following the steps given in Chapter 7, the same problems have to be modeled, and FEA
analysis to be carried out in SolidWorks. Finally, the solutions have to be compared with the
MATLAB plots.
5
(a) (b) Figure 2 (a) supported beam problem; and (b) cantilever beam problem
The third part of the problem deals with the Von Mises stress analysis of a C-link shown in
Figure 3. The FEA has to be carried out using standard mesh sizes of 0.15 inch, 0. 10 inch, 0.10
inch with selective refinement and automatic transition. Plots of Von Mises stress, maximum
Von Mises stress and computational time v/s mesh size are to be generated in MATLAB.
Figure 3 c-link dimensions and loading (all dimensions in inches)
6
Then the effectiveness of a curvature based mesh is to be explored. The number of elements in a
circle (parameter) is to be varied from 4 to 36 in steps of 8. We have to record the number of
elements and the solving time in each case and plot the same graphs as in first case for
comparison. Based on this, we have to decide if a curvature based mesh is necessary in this
problem.
We have to model the C-link with inner edges rounded to a certain radius and carry out the
convergence analysis as in the first case with a standard mesh. Based on this analysis, we have to
decide how this added feature affects the solution, and whether using a curvature based mesh
instead, would improve the solution. Is this feature necessary to include in the FEM analysis?
A design change has to be suggested for the same C-link to reduce the maximum stress. The
design has to be modeled and the convergence analysis has to be carried out for the same in order
to show its performance.
Another requirement is to read and discuss the following topics
- SolidWorks Simulation Help;
- Meshing Advice in SolidWorks Simulation.
3 Result
The lessons 4, 7 and 11 given in the book were completed using the step by step instructions
provided. Once the parts have been modeled, the next step is to add the Solidworks Simulation
option from Tools menu and Add Ins option. Then we start a new static study. We begin by
defining the material type, the no. of joints, degree of freedom of these joints, the forces on the
elements as per the problem. The 2-D truss and beam problems are also solved analytically in
MATLAB using valid equations. Results from both the softwares are presented in the following
sections.
3.1 2-D Truss
The 2-D truss problem is solved using the direct stiffness method in MATLAB. The global
system of equations is given by Eq. (A) where {F} is the nodal force vector, {X} is the global
displacement vector and [K] is the global stiffness matrix
{F} = [K] {X} --- Equation (A)
The transformation from global to local co-ordinates can be achieved by the Eq. (B) where Ɵ is
the angle of truss element.
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Eq. (A) can be expanded in the form given by Eq. (C)
A – cross sectional area of element = 0.049 sq. inches
E – Youngs modulus of elasticity = 30 x psi
L – length of element
u – global displacement at node
f – global force at node
c – cos(Ɵ)
s – sin(Ɵ)
Ɵ– angle of element
Formulation of global stiffness matrix for the two elements:
Element 1:
Ɵ = (6/8) = 36.87 degree
L =
Therefore, EA/L = 147262 lb/in
--- Equation (B)
--- Equation (C)
[K] = --- Equation (D)
8
Substituting these values in Eq. (D) , the global stiffness for element 1 is obtained as below
(from MATLAB):
[K1] =
[94248 70686 -94248 -70686
70686 53014 -70686 -53014
-94248 -70686 94248 70686
-70686 -53014 70686 53014]
Element 2:
Ɵ = (6/4) = -56.31 degree
L =
Therefore, EA/L = 147262 lb/in
Substituting these values in Eq. (D) , the global stiffness for element 2 is obtained as below
[K2] =
[ 62836 -94253 -62836 94253
-94253 1.4138e+005 94253 -1.4138e+005
-62836 94253 62836 -94253
94253 -1.4138e+005 -94253 1.4138e+005]
The combined global stiffness matrix can be obtained by writing the following matrices:
[K11] =
[ 94248 70686 -94248 -70686 0 0
70686 53014 -70686 -53014 0 0
-94248 -70686 94248 70686 0 0
-70686 -53014 70686 53014 0 0
0 0 0 0 0 0
0 0 0 0 0 0]
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[K22] =
[0 0 0 0 0 0
0 0 0 0 0 0
0 0 62836 -94253 -62836 94253
0 0 -94253 1.4138e+005 94253 -1.4138e+005
0 0 -62836 94253 62836 -94253
0 0 94253 -1.4138e+005 -94253 1.4138e+005]
Therefore the combined global stiffness matrix [K] can be formulated as shown:
[K] = [K11] + [K22]
= [ 94248 70686 -94248 -70686 0 0
70686 53014 -70686 -53014 0 0
-94248 -70686 1.5708e+005 -23568 -62836 94253
-70686 -53014 -23568 1.9439e+005 94253 -1.4138e+005
0 0 -62836 94253 62836 -94253
0 0 94253 -1.4138e+005 -94253 1.4138e+005]
10
Figure 4 Truss structure plotted in MATLAB
Figure 4 shows a MATLAB plot of the truss structure. There are three joints or nodes shown by
red dots and the two truss elements are shown by blue lines.
Figure 5 magnified displacement of node 2 under load
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Figure 5 shows the plot for displacement of joint 2 under the effect of a load of 50 lb. The
displacement has been magnified by a factor of as it is very small in magnitude. The X and
Y displacements of node 2 can be observed closely in Figure 6.
Figure 6 zoomed displacement of node 2
Comparison of results and discussion:
Figure 7 shows the X-displacement plot for the truss obtained from Solidworks. It can be
observed that node 2 is displaced by 3.193 x inches towards the right as compared to the
analytical value of 3.24 x obtained from MATLAB which can be seen in Figure 6.
Figure 8 shows the Y-displacement plot for the truss obtained from Solidworks. It can be
observed that node 2 is displaced by 3.871 x inches towards the right as compared to the
analytical value of 3.93 x obtained from MATLAB which can be seen in Figure 6. The
error computation will be discussed later.
12
Figure 7 X-displacement plot from Solidworks
Figure 8 Y-displacement plot from Solidworks
13
The MATLAB results for the stresses (in psi) in the two elements are given below:
stress_elem1 =
850.34
stress_elem2 =
-613.19
>>
Figure 9 shows the axial stress plot for the truss obtained from Solidworks. It can be observed
that stress developed in element 1 is 848.8 psi and that in element 2 is -612.1 psi as compared to
the analytical values of 850.34 psi and -613.19 psi obtained from MATLAB.
Figure 9 axial stress plot from Solidworks
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3.2 Beams
Two beam problems have been solved both in MATLAB and Solidworks and the results have
been compared in this section.
3.2.1 Simply supported beam
The first beam problem is solved in the following manner. The force and moment equations are
given by Eq. (1) and Eq. (2). The points A, B, C, D refer to the points on the beam which are
shown in Figure 2(a)
Calculation of reaction forces:
∑ - Equation (1)
∑ - Equation (2)
From Eq. (2),
= 850 lb
Therefore from Eq. (1),
= 550 lb
Formulation of shear force (V) and bending moment (M) equations:
Between A and C:
is the distance from point A to point C,
V = = 550lb - Equation (3)
M = = 550 - Equation (4)
15
Between A and D:
is the distance from point A to point D,
V = 50lb - Equation (5)
M = = 50 - Equation (6)
Between A and B:
is the distance from point A to point B,
V =
- Equation (7)
M =
= - Equation (8)
16
Comparison of results and discussion:
Figure 10 shows the shear force and bending moment diagrams for the beam problem plotted in
MATLAB. Shear force changes from 550 lbf to 50 lbf at x = 24 inches, and then reduces to -850
lbf at the extreme right end. It becomes zero at x=50 inches. The highest bending moment is
14450 lbf-in at x = 50 inches. Bending moment is zero at the two ends.
Figure 10 shear force and bending moment diagrams for beam from MATLAB
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The same problem is solved in Solidworks using the steps given in Chapter 7.
Figure 11 shows the highest axial and bending stress plot for the beam. The maximum stress of
21674.1 psi is observed in the middle portion shown in red, and the least stress of 455.2 psi is
observed in the end portions of the beam which is shown in blue.
Figure 11 highest axial and bending stress plot from Solidworks
Figure 12 shows the shear force diagram for the beam. The maximum shear force is 550
lb uptill x= 24 inches, then it reduces to 50 lb uptill x = 48 inches, after that it linearly
decreases to a minimum is -850 lb at the right end. The solution is the same as obtained
from MATLAB as seen in Figure 10.
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Figure 12 shear force plot from Solidworks
Figure 13 shows the bending moment diagram for the beam. The diagram is similar in
trend to the one obtained using MATLAB as seen in Figure 10. The maximum bending
moment is 14450 lbf-in near x= 50 inches, which is the same as the one obtained from
MATLAB solution. Also the bending moment is zero at the two ends.
Therefore the solution using Solidworks is very close to the analytical solution and the
values for both the shear force and the bending moment are very agreeable.
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Figure 13 shear force plot from Solidworks
3.2.2 Cantilever beam
The cantilever beam problem is solved in the following manner. The points A, B, C, D refer to
the points on the beam which are shown in Figure 2(b)
Calculation of reaction forces:
∑ - Equation (9)
∑ - Equation (10)
Formulation of shear force (V) and bending moment (M) equations:
Between A and C:
is the distance from point A to point C,
V = = 170lb - Equation (11)
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M = = 170 - 910 - Equation (12)
Between A and D:
is the distance from point A to point D,
V = 80lb - Equation (13)
M = = 80 - Equation (14)
Between A and B:
is the distance from point A to point B,
V =
- Equation (15)
M =
= - Equation (16)
Comparison of results and discussion:
Figure 14 shows the shear force and bending moment diagrams for the beam problem plotted in
MATLAB. Shear force changes from 170 lbf to 80 lbf at x= 36 inches and then linearly reduces
to 0 at the extreme right end which is the free end of the cantilever beam. Figure 14 also shows
the bending moment diagram. The BM is -10920 at the leftmost end and increase to zero at the
right most end. Two linear sections are followed by a quadratic section in the graph.
21
Figure 14 shear force and bending moment diagrams for cantilever beam from MATLAB
The same problem is solved in Solidworks using the steps given in Chapter 7.
Figure 15 shows the shear force diagram for the beam. The maximum shear force is 170 lb upto
x= 36 inches and it becomes constant 80lb from x= 36 to 72 inches, after that reduces linearly to
0 lb, which is the same trend as obtained from the MATLAB solution seen in Figure 14.
Figure 16 shows the bending moment diagram for the beam. The bending moment is -10920 lbf-
in at the leftmost end which is the same as the one obtained from MATLAB solution as seen in
Figure 14. It is zero at the right end. The graph is very agreeable to the one in Figure 14 from the
MATLAB solution.
Therefore the solution using Solidworks is very close to the analytical solution and the
values for both the shear force and the bending moment are accurate.
22
Figure 15 shear force plot from Solidworks
Figure 16 bending moment plot from Solidworks
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3.3 C-link analysis
This section deals with results of stress analysis of an AL6061 c-link. Different mesh sizes and
mesh types are used for the FEA and their corresponding effects on the solution are observed.
3.3.1 Standard mesh
Four cases are considered for this type of mesh. The convergence analysis can be studied using
the Von Mises stress plots. Computation time of each mesh is noted for further analysis.
(a) (b)
Figure 17 (a) standard mesh with global element size 0.15 inch ; (b) Von Mises stress plot from Solidworks
Figure 17 (a) shows the generated standard mesh using global element size of 0.15 inch. The
maximum Von Mises stress can be observed at the inner radius as 20858.3 psi from Figure 17
(b).
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Figure 18 (a) shows the generated standard mesh using global element size of 0.10 inch. It has
more elements as compared to 0.15 inch mesh. The maximum Von Mises stress can be observed
at the inner radius as 21155.6 psi from Figure 18 (b). The stress is converging towards the
analytical solution with a finer mesh.
(a) (b)
Figure 18 (a) standard mesh with global element size 0.10 inch ; (b) Von Mises stress plot from Solidworks
Figure 19 (a) shows the generated standard mesh using global element size of 0.10 inch and
selective refinement along the lower inner surface where the mesh is finer than the rest of the
surfaces. It has more elements as compared to previous mesh without selective refinement.
The maximum Von Mises stress can be observed at the inner radius as 21125 psi from Figure 19
(b). The stress has become lesser than the previous mesh without selective refinement and
therefore the convergence is being observed towards the analytical solution of 20946 psi.
25
(a) (b)
Figure 19 (a) standard mesh of 0.10 inch with selective refinement ; (b) Von Mises stress plot from Solidworks
Figure 20 (a) shows the generated standard mesh using global element size of 0.10 inch with
selective refinement and automatic transition. Finer mesh can be observed at the lower inner
surface and on the adjoining side surface. It has more elements as compared to previous mesh.
The maximum Von Mises stress can be observed at the inner radius as 21152.4 psi from Figure
20 (b). The stress is more than previous mesh, therefore which implies that the mesh seen in
figure 19 (a) is sufficient for our problem.
26
(a) (b)
Figure 20 (a) standard mesh of 0.10 inch with selective refinement and auto transition ; (b) Von Mises stress plot
The computational times and Max Von-Mises stress are noted for each of the above four meshes
considered. This is shown in Table 1.
mesh element size (inches)
tolerance (inches)
computation time (sec)
max von mises stress(psi)
0.15 0.015 1.00 20858.30
0.10 0.015 4.00 21155.60
0.10 + selective refinement
0.005 5.00 21125.00
0.10 + selective refinement +
automatic transition
0.005 11.00 21152.40
Table 1: computational time and max stress record
It can be observed from Figure 21, max Von Mises stress v/s mesh size plot, how the solutions
for different meshes converge towards the analytical solution of 20946 psi. The analytical value
of the max. Von-Mises stress is shown by the thick red line on the top plot. The thin blue line
shows the convergence characteristics as the mesh size is changed and finer meshes are
27
considered. The convergence is very good and the variation above and below the analytical value
is very small as can be observed in the graph. The value is less for 0.15 inch mesh and then it
becomes almost a straight line for the other three meshes.
From Figure 21, computational time v/s mesh size bar chart, it can be observed that as the mesh
becomes finer, the time of computation increases from 1 second in case of 0.15 inch mesh to 11
seconds for a 0.10 inch mesh with selective refinement and automatic transition.
Figure 21 max von-Mises stress and computational time v/s mesh size (standard mesh type) in MATLAB
28
3.3.2 Curvature based mesh
Here, a curvature based mesh has been explored for the stress analysis of the c-link. The global
element size has been fixed as 0.10 inches and the parameter that is varied is the minimum no. of
elements in a circle (N). It is varied from 4 to 36 in steps of 8. As this parameter is increased, it
also increases the total no of elements in the mesh.
(a) (b)
Figure 22 (a) curvature based mesh with N=4 ; (b) Von Mises stress plot from Solidworks
Figure 22 (a) shows the generated curvature mesh using N=4. The maximum Von Mises stress
can be observed at the inner radius as 21107.4 psi from Figure 22 (b).
Figure 23 (a) shows the generated standard mesh using N=12. It has more elements as compared
to the mesh with N=4. The maximum Von Mises stress can be observed at the inner radius as
21080.2 psi from Figure 23 (b). The stress is converging towards the analytical solution with a
finer mesh and has decreased in value than in the first case.
29
(a) (b)
Figure 23 (a) curvature based mesh with N=12 ; (b) Von Mises stress plot from Solidworks
Figure 24 (a) shows the generated standard mesh using N=20. It has more elements as compared
to the previous mesh, and the regions with curvature like the hole surface and radial profiles have
relatively finer meshes compared to the rest of the surfaces on the c-link. This property is
observed as the value of N increases.
The maximum Von Mises stress can be observed at the inner radius as 21073 psi from Figure 24
(b). The stress has become lesser than the previous mesh with N=12. The solution has become
closer to the analytical value of 20946 psi. Therefore convergence is being observed with
increasing value of N.
30
(a) (b)
Figure 24 (a) curvature based mesh with N=20 ; (b) Von Mises stress plot from Solidworks
Figure 25 (a) shows the generated standard mesh using N=28. The mesh has become very fine in
the hole region and the radial edges of the c-link. It has more elements as compared to the
previous meshes.
The maximum Von Mises stress can be observed at the inner radius as 21090.4 psi from Figure
25 (b). The value has increased slightly from the previous mesh with N=20 but it is still in very
close range to the analytical solution. Thus the convergence towards the analytical solution using
a curvature based mesh is good. It also follows from this that the value of N=20 is sufficient for
our purpose as it gives the closest solution.
31
(a) (b)
Figure 25 (a) curvature based mesh with N=28; (b) Von Mises stress plot from Solidworks
Figure 26 (a) shows the generated standard mesh using N=36. The mesh has become more fine
in the hole region and the radial edges of the c-link. It has more elements as compared to the
previous meshes.
The maximum Von Mises stress can be observed at the inner radius as 21072.4 psi from Figure
26 (b). The value decreases from the previous mesh with N=20 and it is still in very close range
to the analytical solution. Thus the convergence towards the analytical solution using a curvature
based mesh is good. Thus using N=36 gives the solution closest to the analytical solution. The
difference between the max stress value for N=20 and N=36 is almost zero depicting good
convergence characteristics.
32
(a) (b)
Figure 26 (a) curvature based mesh with N=36; (b) Von Mises stress plot from Solidworks
It has been observed before that as the no. of elements in a circle increases from N= 4 to 36, the
total elements making up the mesh on the c-link surface also increases. Table 2 below shows the
total no. of elements for each value of N.
curvature mesh (no of elements in
circle ‘N’) no of elements
computation time (sec)
max von mises stress(psi)
4 13711 4 21107.40
12 14707 4 21080.20
20 16795 5 21073.00
28 20550 6 21090.40
36 25720 9 21072.4
Table 2: no of elements (N), time and max stress record
The computational times and Max Von-Mises stress are noted for each of the above mesh sizes
considered. It can be observed from Figure 27, max Von Mises stress v/s mesh size plot, how the
solutions for different meshes converge towards the analytical solution of 20946 psi. The
analytical value of the max. Von-Mises stress is shown by the thick red line on the top plot.
The thin blue line shows the convergence characteristics as the mesh size is changed and finer
meshes are considered.
From Figure 27, computational time v/s mesh size bar chart, it can be observed that as the mesh
becomes finer, the time of computation increases.
33
Figure 27 max von-Mises stress and computational time v/s mesh size (curvature mesh type) in MATLAB
Comparison of results and discussion:
time comparison max stress comparison diff between max and min
value
std mesh computation
time (sec)
curvature mesh computation
time (sec)
std mesh max von
mises stress(psi)
curvature mesh max von
mises stress(psi)
std mesh stress(psi)
curvature mesh stress(psi)
1.00 4.00 20858.30 21107.40
297.30 35.00
4.00 4.00 21155.60 21080.20
5.00 5.00 21125.00 21073.00
11.00 6.00 21152.40 21090.40
9.00 21072.4 Table 3 : comparison of standard and curvature mesh results
34
Table 3 shows the comparison of results of standard and curvature based mesh. The
computational time for the finest mesh (0.10 + refinement + auto transition) is higher for
standard mesh than a curvature mesh (N=36) being 11 sec. and 9 sec. respectively. Observing the
max stress comparison, we see that the variation in stress is higher for a standard mesh being 298
psi, and it is very low for curvature mesh being only 35 psi.
Thus we can conclude that the curvature based mesh gives faster and more accurate results than
a standard mesh and has better convergence properties towards the analytical solution which is
20946 psi. Therefore in this specific example it is more advantageous to use a curvature base
mesh.
3.3.3 Rounded inside edges
The inside edges of the c-link are rounded to a radius of 0.05 inches. The model with the rounded
edges is shown below in Figure 28. The analysis is done using a standard mesh with the same
four cases (mesh sizes) that were considered in Section 3.3.1.
Figure 28 c-link model with rounded inner edges
Figure 29 (a) shows the generated standard mesh using global element size of 0.15 inch. The
maximum Von Mises stress can be observed at the inner radius as 21254.3 psi from Figure 29
(b). The effect on solution is that the stress value has increased as compared to the stress
value without rounded edges (reduced area) as observed in Figure 17 (b) which is 20858.3 psi.
Figure 30 (a) shows the generated standard mesh using global element size of 0.10 inch. It has
more elements as compared to 0.15 inch mesh. The maximum Von Mises stress can be observed
at the inner radius as 21459.7 psi from Figure 30 (b).
35
(a) (b)
Figure 29 (a) standard mesh with global element size 0.15 inch ; (b) Von Mises stress plot from Solidworks
(a) (b)
Figure 30 (a) standard mesh with global element size 0.10 inch ; (b) Von Mises stress plot from Solidworks
36
Figure 31 (a) shows the generated standard mesh using global element size of 0.10 inch and
selective refinement along the lower inner surface where the mesh is finer than the rest of the
surfaces. It has more elements as compared to previous mesh without selective refinement.
The maximum Von Mises stress can be observed at the inner radius as 21467.7 psi from Figure
31 (b). The max Von-Mises stress has become greater than the previous mesh without selective
refinement.
(a) (b)
Figure 31 (a) standard mesh of 0.10 inch with selective refinement ; (b) Von Mises stress plot from Solidworks
Figure 32 (a) shows the generated standard mesh using global element size of 0.10 inch with
selective refinement and automatic transition. Finer mesh can be observed at the lower inner
surface and on the adjoining side surface. It has more elements as compared to previous mesh.
37
The maximum Von Mises stress can be observed at the inner radius as 21518.6 psi from Figure
32 (b). The max. stress here is greater than all the four cases
(a) (b)
Figure 32 (a) standard mesh of 0.10 inch with selective refinement and auto transition ; (b) Von Mises stress plot
The computational times and Max Von-Mises stress are noted for each of the above mesh sizes
considered as shown in Table 4.
mesh element size (inches)
tolerance (inches)
computation time (sec)
max von mises
stress(psi)
0.15 0.015 1.00 21254.30
0.10 0.015 4.00 21459.70
0.10 + selective refinement
0.005 5.00 21467.70
0.10 + selective refinement +
automatic transition
0.005 7.00 21518.60
Table 4: computational time and max stress record
38
It can be observed from Figure 33, max Von Mises stress v/s mesh size plot, how the solution is
affected by the rounding feature. The analytical value of the max. Von-Mises stress is shown by
the thick red line on the top plot. The thin blue line shows the max stress variation as the mesh
size changes. It can be seen that the max stress becomes higher as the mesh size becomes finer
and deviates from the analytical solution. From Figure 27, computational time v/s mesh size bar
chart, it can be observed that as the mesh becomes finer, the time of computation increases.
Figure 33 max von-Mises stress and computational time v/s mesh size (standard mesh type) in MATLAB
39
Comparison and discussion of results: The comparison between the solutions is shown in figure
34. It can be observed that the max Von-Mises stress is greater for a c-link with rounded inner
edges (thick blue line) than with sharp edges (thin red line). The black line shows the analytical
solution. It can be seen that the red line converges towards the solution, while the blue line
deviates away from the analytical value. Thus using rounded inner edges in the analysis does not
show good convergence as compared to the c-link without rounded edges.
Also the time of computation is the same for the first three cases; however for the automatic
transition case, the time of computation is lower for a c-link with rounded inner edges.
Figure 34 comparison of plots for max von-Mises stress and computational time v/s mesh size in MATLAB for c-link with sharp and rounded inner edges.
The max. stress is increased only by about 2% when the rounded feature is added to the c-link
design, while the time of computation is also not very different from first case. Therefore the
analysis gives sufficiently accurate results without the rounded inner edges and the
convergence characteristics are better. Therefore including the rounded feature in the analysis
is not necessary in our case.
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Using a curvature based mesh for the c-link with rounded inside edges gives the results that are
described below. The cases considered are with N=4, 12, 36. The convergence characteristics
can be observed from the following plots.
(a) (b)
Figure 35 (a) curvature based mesh with N=4 ; (b) Von Mises stress plot from Solidworks
Figure 35 (a) shows the generated curvature mesh using N=4. The maximum Von Mises stress
can be observed at the inner radius as 21483.9 psi from Figure 35 (b).
Figure 36 (a) shows the generated standard mesh using N=12. It has more elements as compared
to the mesh with N=4. The maximum Von Mises stress can be observed at the inner radius as
21450.7 psi from Figure 36 (b). The max stress is lesser than in the previous case.
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(a) (b)
Figure 36 (a) curvature based mesh with N=12 ; (b) Von Mises stress plot from Solidworks
Figure 37 (a) shows the generated standard mesh using N=20. It has more elements as compared
to the previous mesh, and the regions with curvature like the hole surface and radial profiles have
relatively finer meshes compared to the rest of the surfaces on the c-link. This property is
observed as the value of N increases.
The maximum Von Mises stress can be observed at the inner radius as 21442.2 psi from Figure
37 (b). The stress has become lesser than the previous mesh with N=12.
Figure 38 (a) shows the generated standard mesh using N=36. It has more elements as compared
to the previous mesh, and the regions with curvature like the hole surface and radial profiles have
relatively finer meshes compared to the rest of the surfaces on the c-link.
The maximum Von Mises stress can be observed at the inner radius as 21483 psi from Figure 38
(b). The stress has become more than the previous mesh with N=20.
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(a) (b)
Figure 37 (a) curvature based mesh with N=20 ; (b) Von Mises stress plot from Solidworks
(a) (b)
Figure 38 (a) curvature based mesh with N=36 ; (b) Von Mises stress plot from Solidworks
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The computational times and Max Von-Mises stress are noted for each of the above mesh sizes
considered. It can be observed from Figure 39, max Von Mises stress v/s mesh size plot. The
analytical value of the max. Von-Mises stress is shown by the thick red line on the top plot. The
thin blue line shows the convergence characteristics as the mesh size is changed and finer
meshes are considered. The convergence is not very good with a curvature based mesh.
From Figure 39, computational time v/s mesh size bar chart, it can be observed that as the mesh
becomes finer, the time of computation increases. The times are much higher than the
computational times from standard meshes.
Figure 39 max von-Mises stress and computational time v/s mesh size (curvature based mesh type) in MATLAB
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3.3.4 Design change consideration
The c-link used for the stress analysis previously has an analytical maximum Von-Mises stress
value of 20946 psi. In this section, our aim is to modify the design of the c-link in order to reduce
the max Von-Mises stress the c-link is being subjected to under the loading of 200 lbs. The stress
depends on the cross sectional area and the distance from the center of curvature to the neutral
axis. By changing any of these parameters, we can change the max. Von-Mises stress produced
in the link. The new design of the c-link is shown in Figure 40 (a) and 40 (b)
It can be observed in figure 40 (b) that the cross sectional area is not uniform throughout the c-
link. The c-link is now thicker in the areas where maximum stress was being observed from the
results of the previous analysis, that is along the bends of the inner surface. Increasing the cross
section in those areas will reduce the max. Von-Mises stress. The results of the stress analysis of
the redesigned c-link have been presented in the following sections.
(a) (b)
Figure 40 (a) CAD model of redesigned c-link (fillet radius of 0.05 inches); (b) 2-D view (all dimensions in inches)
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Figure 41 (a) shows the generated standard mesh using global element size of 0.15 inch. The
maximum Von Mises stress can be observed at the inner middle section as 13425.3 psi from
Figure 41 (b). Therefore the stress value has decreased as compared to the stress value in the
older design which was in the range of 20,000 – 22000 psi
(a) (b)
Figure 41 (a) standard mesh with global element size 0.15 inch ; (b) Von Mises stress plot from Solidworks
Figure 42 (a) shows the generated standard mesh using global element size of 0.10 inch. It has
more elements as compared to 0.15 inch mesh. The maximum Von Mises stress can be observed
at the inner mid portion as 13420 psi from Figure 42 (b) which is lower than the previous mesh.
Figure 43 (a) shows the generated standard mesh using global element size of 0.10 inch and
selective refinement along the middle inner radial section where the mesh is finer than the rest of
the surfaces. It has more elements as compared to previous mesh without selective refinement.
The maximum Von Mises stress can be observed at the inner radius as 13308.5 psi from Figure
43 (b). The max Von-Mises stress has become lesser than the previous mesh without selective
refinement. Thus convergence is being observed as the mesh is becoming finer.
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(a) (b)
Figure 42 (a) standard mesh with global element size 0.10 inch ; (b) Von Mises stress plot from Solidworks
(a) (b)
Figure 43 (a) standard mesh of 0.10 inch with selective refinement ; (b) Von Mises stress plot from Solidworks
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Figure 44 (a) shows the generated standard mesh using global element size of 0.10 inch with
selective refinement and automatic transition. Finer mesh can be observed at the inner mid
surface and in some spots on the adjoining surfaces. It has more elements as compared to the
previous mesh. The maximum Von Mises stress can be observed at the inner middle section as
13307.1 psi from Figure 44 (b) which is lowest in all the four cases. Therefore the convergence
is very good.
(a) (b)
Figure 44 (a) standard mesh of 0.10 inch with selective refinement and auto transition ; (b) Von Mises stress plot
The computational times and Max Von-Mises stress are noted for each of the above mesh sizes
considered as shown in Table 5.
mesh element size (inches)
tolerance (inches)
computation time (sec)
max von mises
stress(psi)
0.15 0.015 2.00 13425.3
0.10 0.015 3.00 13420.4
0.10 + selective refinement
0.005 4.00 13308.5
0.10 + selective refinement +
automatic transition
0.005 6.00 13307.5
Table 5: computational time and max stress record
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It can be observed from Figure 45, max Von Mises stress v/s mesh size plot, how the solution
converges. The thin blue line shows the max Von-Mises stress variation as the mesh size
changes. It can be seen that the max stress becomes lesser as the mesh size becomes finer. From
Figure 45, computational time v/s mesh size bar chart, it can be observed that as the mesh
becomes finer, the time of computation increases.
Figure 45 max von-Mises stress and computational time v/s mesh size (standard mesh type) in MATLAB
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Comparison and discussion of results: The comparison between the solutions is shown in figure
46. It can be observed that the max Von-Mises stress is lesser for the new design of a c-link lying
in the range of 13300 -13500 psi(thick blue line) than the old design where it lies in the range of
20800 - 21200 psi (thin red line). Thus there is a high reduction the value of maximum stress.
Also the time of computation is almost same for the first three cases; however for the automatic
transition case, the time of computation is lower for the redesigned c-link as seen in the bar chart
in Figure 46, the difference being 5 seconds.
Figure 46 comparison of plots for max von-Mises stress and computational time v/s mesh size in MATLAB for old and redesigned c-link
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4 Discussion
As a part of this project, we have also read through the Solidworks Simulation documentation for
a better understanding of the system, and acquired some information on meshing. A brief
discussion about them follows.
Solidworks Simulation Help: Solidworks simulation help is a feature available that is very
useful for learning and getting familiar with the techniques used for implementing simulation.
The web-based help provides options for topic search, navigation, and upto date documentation.
It has user friendly conventions, bold indicates user interface items, and italic green is for
jumping to specific topics. The analysis background section gives explanation and steps to
perform each type of analysis for example static, dynamic, thermal, frequency, fatigue, etc.
Solidworks simulation is the system which we have used in this project; it has fast solvers and
one screen resolution for results. The documentation provides help with steps, meshing, material
and specifying conditions for analysis.
The Solidworks simulation system has an interactive simulation tree and a good graphical
interface. The simulation studies section provides detailed information about the studies possible,
mesh types, types of solvers, steps to create and run studies along with exporting data.
Solidworks simulation has the property of working with composite materials which are used in
many real-life applications. Different types of loading according to the problem can be specified;
also movement and boundary constraint help is available to fully define a problem. To carry out
FEA, a mesh has to be generated. Two types of meshes, standard and curvature based mesh are
available. Help provides documentation related to changing mesh parameters, analyzing changes,
control and mesh plot probing. Through the help option, we can learn about applying materials,
creating libraries, specifying properties, etc. Optimizations in design can be carried out using
Design Study option using parameters. It is possible to create analysis libraries for frequently
used analysis features. There is a lot of help available for plotting graphs and report making in
order to support analysis. Checking stress results is possible; factor of safety definitions help
realize failure criteria.
Meshing advice in Solidworks Simulation: Reading through this article by Brian Zias from
Alignex, we not only realize the importance of meshing in an FEA analysis software but also the
problems related to it. Meshing heavily depends on the amount of RAM and the processor
speed. A slow processor leads to high solution times and a less amount of RAM leads to crashes
as it does have the capacity for large no. of elements in the mesh. Therefore with these
restrictions, we must strive to optimize the meshing.
One of the important techniques is the mixed mesh creation. Mixed-mesh refers to the
simultaneous use of solid, shell, and beam elements in the same study. Thus it helps in reducing
the no. of elements in the mesh.
In Solidworks 2008, we had to specify a „mixed mesh' in the study. Once the study was created,
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every component was solid type by default unless specified. The great thing about Solidworks
2009 is that there is no longer an option to create a „mixed mesh' study. All studies are enabled to
be mixed mesh always. Also, there are some default options embedded for example any surface
bodies will be treated as shell elements, any structural members (weldment parts) will be treated
as beam elements, and everything else will be treated as solid elements.
For carrying out assembly analyses, one of the important considerations is the contact sets or
connectors. The three important types of contacts are bonded, free and no penetration. For large
models, it is best to start with everything bonded and then start adding other interactions in a step
by step manner.
Mesh size and no. of nodes are restricted by the amount of RAM. In order to maximize the
available RAM we can try fresh reboots, simplify meshes and resolve issues, use mixed mesh or
curvature based meshes, improvement in modeling, or upgrading as the last option.
5 Conclusion
This project has made us aware of the capabilities of Solidworks in carrying out finite element
analysis of devices such as beams, trusses and c-links. The lessons provided in the book are very
helpful in understanding the method of carrying out such analyses as it provides step by step
instructions. The solutions obtained through these simulations may not always be accurate and
therefore a thorough understanding of the equations behind them are to be studied carefully to
understand these inaccuracies. We have explored different mesh types, sizes, loading and
constraints in this project, and have developed a good understanding of the importance and
problems related to meshing and loading in FEA simulations.
We have also learnt how to use MATLAB for solving trusses by the direct stiffness method and
formulation of shear force and bending moment equations in beam problems. These are very
easy to formulate and give accurate results.
Thus the project has given us an insight into carrying out FEA through the use of packages such
as MATLAB and Solidworks.
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6 References
BOOKS
[1] Introduction to Finite Element Analysis using SolidWorks Simulation 2010–by RH Shih
WEBSITES
[1] www.wikipedia.com
[2] www.youtube.com
[3] http://help.solidworks.com
[4] http://blog.alignex.com/mechanical-technical-blog