Finite Element Analysis of Thermal
Stresses in a Circular Plate
Holly Ibanez
MANE 4240 Introduction to Finite Element Analysis
May 15, 2015
Abstract
The objective of this project is to determine the thermal stresses and deflection in a flat, circular plate
that is clamped along the outer radius. The bottom face of the plate shall be subjected to heating, while
the top face of the plate is cooled. The thermal stress and displacement are calculated in a steady state
using COMSOL. The process and analysis completed in COMSOL shall also be discussed.
1.0
Introduction
Finite Element Analysis (FEA) is a numerical method that offers a means to find an approximate solution
to boundary value problems for partial differential equations. It is applied in engineering as a
computational tool to divide a complex problem into smaller elements. FEA is commonly used for a
variety of multiphysics problems including:
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Heat conduction
Solid state diffusion
Potential flows
Lubrication flows
Galvanic corrosion
Electromagnetics
Solid mechanics
Fluid mechanics
Acoustics
Financial systems
Many of these complex problems do not yield an exact solution; however, by utilizing FEA, an
approximate solution can be obtained by either theoretical analysis or software modeling. COMSOL was
used to analyze a circular plate subjected to a temperature gradient, as discussed throughout this paper.
2.0
Formulation
The system under review is a flat circular plate with a radius of 50 cm and thickness of 2 cm. The chosen
material of the plate is Aluminum 6063-T83, and a temperature of 373.15 K is applied to the bottom
surface of the plate, while a temperature of 273.15 is applied to the top surface. The outer edges of the
plate are clamped, or in a fixed position, and thermally insulated.
Meters
The first step in COMSOL pre-processing is to specify the type of analysis to be used. In this case, an
analytic function is selected in the software. The system is defined as two dimensional in a steady state.
The next step is to define the geometry, which is a circular plate. To define this geometry, a rectangle is
created and revolved about an axis, as shown in Figure 1.
Meters
Figure 1: Circular Plate Geometry Creation
A material must be specified after the geometry is created. Aluminum 6063-T83 is the material selected
for the circular plate analysis. The properties of this material are shown in Table 1.
Table 1: Aluminum 6063-T83 Properties
The next step is to apply the boundary conditions to the Aluminum plate. The bottom face is subjected
to a thermal stress of 373.15 K, while the top face is simultaneously subjected to a thermal stress of
273.15 K. The outer edges of the plate are clamped, or in a fixed position. See Figure 2 for an
illustration of the boundary conditions.
Meters
Top Face
T = 273.15 K
Fixed Outer Edge
Bottom Face
T = 373.15 K
Meters
Figure 2: Circular Plate Boundary Conditions
Finally, before computing a solution to the problem, a mesh size must be specified. The following
COMSOL-defined meshes were chosen for analysis and comparison:
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Coarser mesh
Normal mesh
Finer mesh
Extra fine mesh
Extremely fine mesh
3.0
Solution
With all of the pre-processing information completed, it is possible to compute the final results for each
mesh size. Once the results are computed, the user may select which results to view. The focus of this
study is thermal stress and deflection that result from the temperature gradient applied to the plate in a
steady state. The stress and deflection can be viewed graphically by creating surface plots for each
desired result. The plots may be created in both 2- and 3- dimensions.
4.0
Results
Table 2 contains a comparison of maximum tensile stress, maximum compressive stress, and maximum
deflection for each of the previously defined meshes. It must be noted that an exact solution is
impossible to obtain for the circular plate system subjected to thermal stresses; therefore, mesh
extensions were performed to validate the COMSOL results. It can be seen in Table 2 that the stress and
deflection converges as the mesh is analyzed from course to extremely fine.
Mesh Size
Maximum
Tensile Stress
(N/m2)
Coarser
Normal
Finer
Extra Fine
Extremely Fine
1.79E+07
1.82E+07
1.80E+07
3.04E+07
3.44E+07
Maximum
Maximum Axial
Compressive Stress
Deflection
2
(cm)
(N/m )
2.58E+08
2.59E+08
2.51E+08
2.91E+08
3.08E+08
0.011
0.010
0.012
0.007
0.005
Table 2: COMSOL Tabular Results
5.0
Discussion
The circular plate system can be visualized as a real world problem in which a pan is placed on a stovetop. The bottom face of the plate is subjected to 373.15 K, which is the boiling point of water. The
bottom face of the plate, or pan, is 273.13 K, which is the freezing point of water. The temperature
applied to the bottom face of the plate is chosen because 273.15 K is the reference temperature for the
system defined in COMSOL.
As seen in the following figures, a tensile stress is developed on the top face of the plate and a
compressive stress is present on the bottom face due to thermal expansion, which occurs when
materials are subjected to high temperatures. The axial displacement reaches a maximum at the center
of the plate. These results are consistent for each variation of meshing. The various meshes are
required to compare the convergence of results, which validates the accuracy of the stresses and
deflections. An analytical method is typically used to determine an exact solution the given problem;
however, it is impossible to calculate an exact solution for this system. As a result, an approximate
solution is computed in COMSOL using the different meshes.
The Figures 3-12 show a visual representation of the tabular results. Both maximum and tensile
stresses, as well as maximum deflections are shown.
Figures 3 and 4: Coarser Mesh Analysis
Figure 5 and 6: Normal Mesh Analysis
Figure 7 and 8: Finer Mesh Analysis
Figure 9 and 10: Extra Fine Mesh Analysis
Figure 11 and 12: Extremely Fine Mesh Analysis
A comparison of coarser and finer meshes can be seen in Figures 13 and 14 for information. It is evident
that the finer mesh considers more nodes, and therefore elements. This results in a more accurate
analysis of the system as a whole.
Figure 13: Coarser Mesh
Figure 14: Finer Mesh
6.0
Conclusion
Based on the COMSOL analysis and mesh extension, it can be concluded that the resulting stresses and
deflection are accurate. Special emphasis was placed on pre- and post- processing in COMSOL,
specifically mesh extensions, to validate the results obtained. The values all converged when analyzed at
finer meshes. It can also be concluded from thermoelastic theory that a material will expand when
thermal energy is added. A real world example to support this theory is a pan placed on a hot stove.
It is recommended that for future studies, the stress and deflection should be calculated for different
materials and geometries for comparison to the results of this analysis. Altering the boundary
conditions, such a free edge or greater temperature gradient, would also yield different results.
7.0
References
[1] Bower, Allan F., “Approximate theories for solids with special shapes: rods, beams, membranes,
plates and shells.” Chaper 10, Solid Mechanics. 2012.
[2] Kulkarni, V.S., “Quasi-static thermal stresses in a thick circular plate”. Science Direct, 12 April
2006.