Homework Project 4
Plane Stress, Plane Strain, Thermoeleasticity
Advanced Topics in Finite Elements
Summer 2015
John Connor
System Description
Plate systems are used to simplify a complex system such that a particular failure mode or effect
can be evaluated using previously derived calculations. The plate system usually represents a
“sample” of the much larger complex part, most often in 2-D. The plate can then be
appropriately loaded to observe hoe the system may respond.
In this project a 1m X 1m X 0.01m plate will be analyzed for thermal loading, pressure loading,
and combined thermal pressure loading. We will assume a thermal gradient of 10K between the
top and bottom surface of the plate, as well as a 1000Pa pressure applied to the top surface of the
plate. The plate material chosen for this project is low carbon steel, inputs for the steel are
provided in the input section.
Governing Equations, Boundary Conditions and Input Data
Flat plate equations are defined in Roark’s Formulas for Stress and Strain, the equations
presented in Roark’s provide stress and displacement equations for several types of boundary
conditions and loading scenarios. The flat plate equations for this project can be found on page
508 and is case number 8a of Table 11.4. The formula for stress defined in Roark’s is:
𝛽2 ∗ 𝑞 ∗ 𝑏 2
𝑀𝑎𝑥 @ 𝑐𝑒𝑛𝑡𝑒𝑟: 𝜎 =
𝑡2
Where σ is stress, q is force per unit area, b is the short side length, t is the thickness, and β2 is a
constant based off the ratio of short to long side. The equation for displacement is then defined
as:
𝛼 ∗ 𝑞 ∗ 𝑏4
𝑀𝑎𝑥 @ 𝑐𝑒𝑛𝑡𝑒𝑟: 𝑦 =
𝐸 ∗ 𝑡3
Where E is the Modulus of Elasticity and α is a constant based off the ratio of short to long side.
These equations form the basis of half of the analysis. The other half of the analysis is based off
the thermal expansion equation subject to external constraints defined on page 759 for a uniform
flat plate to be:
𝛥𝑇 ∗ 𝛾 ∗ 𝐸
𝜎=
(1 − 𝑣)
Where γ is coefficient of thermal expansion, ν is the Poisson’s Ratio, and ΔT is the change in
temperature over the plate.
Input data from MatWeb, Roark’s, and the homework assignment sheet are shown below in
Table 1.
Table 1- Input Data
Name
Value [Unit]
Length
1 [m]
Width
1 [m]
Thickness
0.01 [m]
Pressure
1000 [Pa]
Elastic Modulus
200 [GPa]
Poisson’s Ratio
0.3
Density
7750 [kg/m3]
Coefficient of Thermal 13.5 [µm/m-K]
Expansion
Thermal Conductivity 93 [W/m-K]
Constant, β2
0.1386
Constant, α
0.0138
Source
Assignment Sheet
Assignment Sheet
Assignment Sheet
Assignment Sheet
MatWeb
MatWeb
MatWeb
MatWeb
MatWeb
Roark’s
Roark’s
The Galerkin finite element variation formulation used for the finite element analysis is:
∫ 𝑢̇ 𝑣𝑑𝑥 + ∫ 𝑢′ 𝑣 ′ 𝑑𝑥 = 0
Description of the Finite Element Model
The model was developed in Abaqus CAE; the plate was modeled as a solid, shown in Figure 1.
Material properties defined in Table 1 were then assigned to the plate model. The part was then
added to the assembly section and the loading step (Step-1) was created.
Figure 1 – Model of the Plate
Mesh was applied then to the part with a global seed size of 0.05 and a local seed size to create 4
elements throughout the thickness of the plate. C3D20T (A 20-node thermally coupled brick,
triquadratic displacement, trilinear temperature) elements were assigned to the mesh. There are
no element errors or warnings from the mesh. The mesh is shown in Figure 2.
Figure 2 – Meshed Plate
To evaluate the given loading conditions a boundary condition was placed at the outside edges of
the plate to lock the plate in all directions. To apply the temperature difference, two temperature
boundaries were placed on top (T_low=0K) and bottom (T_high=10K) surfaces of the plate these
boundary conditions are suppressed when the analysis does not evaluate temperature. For the
pressure evaluation a uniform pressure loading is added to the top surface of the plate and is
suppressed when the analysis only evaluates for the temperature difference.
Three input files are created to run the different loading scenarios:
 HW4_T – Temperature difference evaluation
 HW4_P – Pressure evaluation
 HW4_T – Combined Pressure and Temperature difference evaluation
Results, Discussion and Conclusion
The results of the analyses are shown in Figures 3-5. Each Figure shows the stress and
displacement for a given loading condition. The results show that the temperature difference has
a greater impact on the stress in the plate then the pressure. However, the pressure has a greater
impact on the displacement of the plate. The combined loading shows that the displacement can
be additive or decreasing depending on the direction for which the load is applied while stress at
least at this pressure is neligable.
Figure 3 – Temperature Only Evaluation
Figure 4 – Pressure Only
Figure 5 – Combined Loading
To verify that the model was valid the results were compared to the solutions produced using the
Roark’s equations, See Table 2. This shows that for the stress evaluations the FEA and the
Roark’s equations do not agree 100%, however there are in the same magnitude range. Roark’s
does offer an answer to this stating, “For rectangular plates with simply supported edges the
maximum
stresses are shown to be near the center of the plate. There are, however, stresses of similar
magnitude near the corners if the corners are held down as has been assumed for all cases
presented. Reference 21 discusses the increase in stress at the center of the plate when the
corners are permitted to rise. For a uniformly loaded square plate this increase in stress is
approximately 35%.” However the displacement solution is very accurately predicted for the
pressure only case.
Table 2 – FEA Validation
Analysis
Solution Type [Unit] FEA
Roark’s Equation
HW4_T
Stress [MPa]
42.3
38.6
HW4_P
Stress [MPa]
1.02
1.38
HW4_P
Displacement [µm]
68.2
69.0
References
1. Low Carbon Steel; MatWeb;
<http://www.matweb.com/search/DataSheet.aspx?MatGUID=034970339dd14349a8297d
2c83134649&ckck=1>
2. Young, Warren; Bundynas, Richard; Roark’s Formulas for Stress and Strain Ed. 7;
McGraw-Hill; Dated 2002
3. Zienkiewicz, O. (2013). Finite Element Method: Its Basis and Fundamentals.
Butterworth-Heinemann Ltd.