Plate with Circular Hole
MANE-4240: Introduction to Finite Elements
by
Mahamoud Coulibaly
Master of Science in Mechanical Engineering
Rensselaer Polytechnic Institute
Hartford, CT
May 11, 2015
pg. 1
Table of Contents
Section
Page
1. Abstract………………………………………………………
3
2. Introduction and Background………………………………..
4
3. Computing in COMSOL…………………………………….
6
4. COMSOL Results……………………………………………
9
5. Computing in Abaqus………………………………………..
12
6. Abaqus Results………………………………………………
7. Summary……………………………………………………
12
15
pg. 2
ABSTRACT:
This paper is to study the impact of having a discontinuity as a result of having a small hole in a
plate. An infinite large plate with a small was used to evaluate the stress the stress factor in a
vicinity of the hole in COMSOL and ABAQUS. The results obtained was compared to the exact
solution in Maple. Using the “Theory of Elasticity” by S. Timoshenko, which evaluated the
impact of having a small hole in a relatively large plate, a formula was derived that will calculate
the exact solution.
The stress in the part is calculated using a standard formula based on the surface area under
the load. The stress in the vicinity of the hole was calculated using the formula obtained in
“Theory of elasticity”. The ration of the stress in the vicinity of the hole to the stress of the part
away from the hole will give the stress concentration factor.
That stress concentration factor otherwise known as “Kt” is three. However, this formulation is
only be true if the size of the plate is wide enough and the hole is small. This study investigates
the effect of material discontinuity on the stress of the plate.
Increased stress gradient is seen at the vicinity of the discontinuity. The magnitude of the stress
factor “Kt” will change as the size of the plate varies.
Two variables will be investigated using two different software packages.
The first variable will look at different mesh sizes and types to see which one will get us close to
the exact solution derived in Maple. This experiment will be conducted in COMSOL.
The second experiment will change the size of the plate and mesh sizes to understand the
correlation. There is definitely a relationship between hole sizes and width. We will vary the
plate size and hole size while somehow maintaining the ratio between them to understand the
impact.
In each case we will compute the percent error
pg. 3
Introduction and Background:
In aerospace, materials are used in application that could exceed their capabilities. As a result,
several methods were developed to provide adequate cooling to the part. One of such method
is cooling the part by drilling several small impingement cooling.
The impact of having several small holes in the part without sacrificing its structural capability
will need to be understood and I will spend some time exploring that in this study.
Can the relationship between hole sizes and plate dimensions and hole to hole spacing be
exploited in the preliminary design phase?
The application of such design is endless. For example, there is a need to provide
instrumentation routing in the structure. In most case, holes will have to be drilled in the
structure to provide space for the cable to pass thru. Knowing ahead of time if it is possible to
accommodate that demand without compromising the design is key to make quick decision in
the field where the analysis package is not available.
Overview of Large Thin Plate with Hole:
This is a rectangular plate of the size of 2m x 1m X 0.025m with a hole of 0.025mm dia in the
middle. The plate is made of structural steel. The structural properties are listed in Table 1:
Parameter
Value
E
2E11 Pa
v
0.3
pressure
1E6 Pa
Table 1: Structural Steel Mechanical Properties
Theory and Methodology:
pg. 4
The formula given to calculate stress in the x and y direction in the plate are given by the below
formula. Using these formulas in Maple, the exact solution of the stress around the hole on the
plate was obtained.
>
>
>
Notice that the stress is small in the x direction and larger in the y direction which is the
direction of the load applied to the part.
pg. 5
S/Sy = 3
Therefore the stress concentration factor is calculated to be three. This will be the exact
solution to be used in comparison with our estimations using couple of the structural analysis
packages (COMSOL and ABAQUS).
Computing in Comsol Multiphysics:
Wikipedia has a following definition for COMSOL: “COMSOL Multiphysics is a finite element
analysis, solver and Simulation software / FEA Software package for various physics and
engineering applications, especially coupled phenomena, or multiphysics. The packages are
cross-platform (Windows, Mac, Linux). In addition to conventional physics-based user
interfaces, COMSOL Multiphysics also allows for entering coupled systems of partial differential
equations (PDEs). The PDEs can be entered directly or using the so-called weak form (see finite
element method for a description of weak formulation).”
The following sections shall provide detailed steps that was taken to compute approximations
in COMSOL.
In this analysis, 2D was used to look at the effect of the hole in a large plate. The plate size, the
hole size and the thickness were modeled accordingly in COMSOL. The material was then
specified to be steel and the properties were downloaded from the library imbedded in the
software.
After the material was assigned, the boundary conditions and the load was applied.
Loads and Boundary Condition:
Loads and boundary conditions were applied in the plate in SI units.
pg. 6
Loads -Red
Boundary Conditions – Blue Triangles
Figure 1: Load and Boundary Condition Locations
Hole
Symmetry was used to minimize the file size and to decrease to computing time. As it can be
seen in the picture above (fig. 1), the plate was adequately constrained before solving the
problem. The forward edge of the plate is constrained in the X- direction and the bottom of the
plate was constrained in the y direction. The load was then applied to the top of the plate in the
Y- direction.
Meshing:
Two different types of linear brick elements mesh were looked at: The normal and the extra
fine mesh.
One type of quadratic mesh was also looked at to predict the stress concentration of the plate
near the discontinuity.
pg. 7
Figure 2: Normal Linear Brick Mesh
Figure 3: Extra Fine Linear Brick Mesh
pg. 8
Figure 4: Quadratic Corse Mesh
Looking at these meshes, one can observe that no matter how fine the linear brick mesh is, it
does not do a great job following the contour of the hole. There are gaps between the arc and
the mesh. As the meshes are made finer and finer, the gaps will decrease but they will not
disappear. Therefore, there will always be an error in the approximation.
The quadratic mesh, even the coarse one does a better job hugging the contour of the circle.
Results:
Each of the models analyzed does a better job predicting the Kt around the hole. One can
deduct from the results obtained that the finer the mesh, the better the approximation. Using
the linear brick mesh, the approximation is very close to the exact solution with the extra fine
mesh.
Better accuracy can be obtained using the coarse quadratic element mesh compare to the fine
8-node linear brick mesh (See picture below). Table 2 provides the percent error with each of
the approximation that was used.
Figure 5: Normal Linear Brick Mesh
pg. 9
Figure 6: Extra Fine Linear Brick Mesh
Figure 7: Quadratic Corse
Below are the results obtained from performing the analysis in COMSOL. Results confirm than
Quadratic mesh provides better approximation in solving stress in plate with discontinuity as a
result of having a circular hole.
(experimental value) − (true value)
% error = ――――――――――――― × 100
true value
pg. 10
Stress in plate away Stress in plate
from the hole
close to the hole
Exact Solution in Maple
Linear Brick -Normal
Linear brick -Extra Fine
Quadratic
10^6
10^6
10^6
10^6
3.00*10^6
2.9792*10^6
2.982910^6
2.9961*10^6
Stress
Concentration
Percent
error
3
2.9792
2.9829
2.9961
exact
-0.007
-0.006
-0.001
Table 2: Percent Error between different mesh sizes and types
Looking at the stress plot, it is clearly visible that there is a stress die-out as the measure point
is further away from the hole. One can take advantage of that observation by putting several
holes in the plate without further impacting the stress concentration around the discontinuity.
For this illustration, a plate was modeled in COMSOL with several holes. The distance between
holes was set to be the equivalent of one and half times the diameter of the hole. As it can be
seen in figure 8, the stress concentration is not significantly increased. In fact, the result is very
close to the exact solution derived with Maple. This latest study with multiple holes provides a
Kt of 2.9164 compare to the exact solution of 3.
In conclusion, multiple small holes could be drilled in the plate as long as adequate distances
between the holes are maintained. Another study could look at optimum distance between
holes as a function of loads or plate thickness.
Figure 8: Multiple holes – Quadratic Mesh
pg. 11
Computing in Abaqus:
From Wikipedia, Abaqus FEA is a software suite for finite element analysis and computer-aided
engineering, originally released in 1978. The name and logo of this software are based on
the abacus calculation tool. Abaqus/CAE, or "Complete Abaqus Environment" (a backronymwith an
obvious root in Computer-Aided Engineering). It is a software application used for both the
modeling and analysis of mechanical components and assemblies (pre-processing) and visualizing
the finite element analysis result.
This software is very user friendly and can process 3D model very rapidly. For that reason,
symmetry was not used in this analysis. Instead the full 3D model was used to provide the
approximation.
Aplate with the following size was used: 254mm x 127mm X 12.7mm with a hole of 12.7 mm
The same constrains were used as in COMSOL with the same load applied in the y direction.
The material was selected to be steel with the same properties as what was used in the analysis
with COMSOL.
Meshing:
For meshing in Abaqus, three different meshes were used to perform the approximation. The
12,6,3 “8-nodal” brick element were used. When performing the meshing operation, just as in
COMSOL , it was apparent that the brick element was not the optimum way to mesh a plate
with a hole. Around the hole, it can be seen that the brick element will not completely hug the
edge of the hole.(fig 9, 10 and 11)
Quadratic element however will conform to the edge and seems to be best fit for this
application. (fig. 12)
Results:
Figure 9 (12 size 8- Nodal brick element), the equivalent of coarse size mesh; the result
obtained was in line with what was expected. The stress field in the axis perpendicular to load
direction will see larger stress results when compared with the x direction. Keep in mind that
this time around the tension load is applied in the x direction. When looking at the stress with
the load applied, the stress in the plate without the hole is 0.685 compared to the stress around
the hole of 1.558. The equivalent stress factor in this particular case is 2.274.
pg. 12
Table 3 provide the stress concentration factor for the rest of the different case studies and the
percent error.
Quadratic element does a better job even with a coarse mesh in approximating the result of the
stress around the hole in the plate.
Stress (actual) = F/A
Stress (actual)/Stress (hole)=3 (exact solution)
Stress in plate away Stress in plate
from the hole
close to the hole
Exact Solution
12 size -8 nodal brick
6 size -8 nodal brick
3 size -8 nodal brick
12 size -Quadratic
0.685
0.685
0.685
0.685
0.685
2.055
1.558
1.75
1.848
2.054
Stress
Concentration
Percent
error
3
2.274
2.555
2.698
2.999
exact
-0.242
-0.148
-0.101
0.000
Table 3: Percent Error between different mesh sizes and types (Abaqus)
Figure: 9 – 12 size “8- Nodal brick element”
pg. 13
Figure: 10 – 6 size “8- Nodal brick element”
Figure: 11 – 3 size “8- Nodal brick element”
Figure: 12 – 12 size “Quadratic element”
Figure: 13 – Summary
pg. 14
Summary:
Comsol and Abaqus both do a great job approximating stresses in the plate with discontinuity.
The finer the mesh, the lower the percent error when the results are compared to the exact
solution.
Quadratic meshes are preferred since they do a better job approximating the results with
coarse mesh.
Mesh size determine the speed in which to get the result and the accuracy of the
approximation. Therefore it is important to know how to apply them.
Around discontinuities and sharp corners, it is advisable to use fine mesh to accurately capture
the geometry. In large flat area a larger mesh is acceptable.
When the results have enough safety margin, a refinement may not be needed to determine
the capability of the part.
However, when the results are borderline or have negative margins, a more refined model may
be needed to further investigate the issue.
Speed is an important factor as well. Structural approximation results are often needed in a
short to take action to meet deliveries or support deadlines. Even though a finer mesh is always
preferred, it may not be best option depending on how fast one has to deliver results.
Symmetry and coarser mesh may be use to increase the computation time. However, one has
to understand the accuracy of the results with the choice and also understand the safety
margin before opting to do a faster evaluation.
Having analyzed several concepts, one can deduct that large plate with a hole have a
stress die out and that knowledge allows us to drill several holes in the part without
compromising the structural integrity of the part. This is extremely important in gaging the
viability of the design in the preliminary study period.
pg. 15
References
a) Timoshenko, Stephen P., Krieger, S. Woinowsky, Theory of Plates and Shells, 2nd Edition,
McGraw-Hill International Editions, 1959
pg. 16