1. 1
Investigation of the von Mises stress in an
Aluminum Membrane
Yixuan Jia
PID: A53105334
December 1st
, 2015
Contents
1. Executive summary 1
2. Introduction 2
3. Material properties 2
4. Geometry 2
5. Boundary conditions 3
6. Detailed calculations 4
7. Conclusions 8
1 Executive summary
The objective was to calculate accurately the von Mises stress, as a measure of stress intensity,
in an Aluminum membrane under thermal stress loads. The membrane was modeled using the
plane stress idealization. The membrane is constrained to the zero displacement in the normal
direction to the circular arc edges and the temperature is increased uniformly by 20°C. The region
of the highest von Mises stress shows up at the center of the arc edge, thus, we choose this point
as our point of interest. The sequence of computed solutions could be used to extrapolate to obtain
estimates of the true values at the point of interest.
The results of the study allow us to conclude that:
• The imcompatible mode 6 node triangular mesh from Abaqus performs well under the
studied conditions;
• The Richardson extrapolation is a valuable technique to estimate the true answers from the
finite element computations.

2. 2
2 Introduction
The goal of the present study is to illustrate calculation of accurate stress results under nontrivial
conditions. In the present
The procedure is to compute several solutions with uniform meshes of different resolutions (i.e.
built up of elements of different sizes), and then to attempt extrapolation to the limit of the element
size vanishing (h → 0).
The Abaqus/Standard, version 6.14.1, finite element program was adopted in this study.
3 Material properties
The material was assumed to be aluminum, with Young’s modulus of 70GPa, Poisson ratio of
0.33, and the thermal expansion coefficient 23×10-6
.
4 Geometry
The geometry of the membrane is shown in Figure 2. Note that the membrane is axis symmetric.
The shape of the membrane is rectangular, with two half-circular arc edges on both of the shorter
edges of the rectangle. The length of the short edge is 30mm and the length of the long edge is
70mm. The center of the circular arc edge is located at the center of the short edge. The radius of
the circular arc edge is 6mm.
The membrane is of uniform thickness of 1mm, which is a sufficiently small thickness to allow
for the plane-stress modeling to be used.

3. 3
Figure 1: Membrane shape with the study point of interest
Figure 2: Sketch of the membrane. The dimensions are in millimeters. The thickness is uniform,
equal to 1mm.
5 Boundary conditions

4. 4
The plane stress model is adopted. As shown in Figure 3, the membrane part s constrained to
zero displacement in the normal direction to the circular arc edges and the temperature is increased
uniformly by 20°C. As in the present case the thermal stress analysis is applied, the applied thermal
loads and boundary conditions are sufficient to remove all rigid body modes.
6 Detailed calculations
The meshes were generated of uniform element size, with element size taken as h = 3.75, 3, 2.4
[mm]. The advancing-front mesh generator produced meshes of reasonable quality, meaning that
no error messages were generated. Table 1 shows the numbers of nodes and elements for the
meshes used.
The element type adopted for all models was the uniform mesh of 6-node quadratic plane stress
triangles (CPS6).
Figure 3 Boundary conditions
To allow the application of finer meshes, apply the symmetric boundary condition, which is at
the center line of the rectangle, the x-axis displacement is zero. Thus only half of the membrane
area is needed to be calculated.
The results are illustrated in Figure 4 for the coarsest mesh employed. The mesh size is 3.73mm.
It shows the distribution of the von Mises stress. The highest Mises stress shows up at the center
of the arc edge. The value of the highest Mises stress is 83.22Mpa.

5. 5
Figure 5 shows the distribution of the von Mises stress for a less coarse mesh. The mesh size is
3mm. The highest Mises stress also shows up at the center of the arc edge. The value of the highest
Mises stress is 82.44Mpa.
Figure 6 shows the distribution of the von Mises stress for a finer mesh. The mesh size is 2.4mm.
The highest Mises stress always shows up at the center of the arc edge. The value of the highest
Mises stress is 83.07Mpa.
The computed values of the von Mises stress are summarized in Table 1. The computed data is
also summarized in Figure 5, 6 and 7. For the center point of the arc edge, the von Mises stress
values are plotted on a linear-linear scale as a function of the element size (Figure 8).
The true value of the von Mises stress was estimate with Richardson extrapolation. The results
for the three meshes ware used to estimate the true solution, and also as a byproduct the
convergence rate. Table 2 resents the estimates of the ture von Mises stress and the convergence
rate at the point of interest. Note that the convergence rate is relatively small at the center pont of
the arc edge.
The slope of the approximate error (difference of successive solutions) needs to be matched to
the slope of the estimated true error on a log-log plot. Figure 9 presents the approximate error
(dashed line) computed from the von Mises stresses for the 3 different meshes, and the estimated
true error (solid line) of theses three solutions. The convergence rate of these two linds of error
curve are in visual agreement for all points of interest. This is a confirmation of the suitability of
the computed data for extrapolation.

6. 6
Figure 4: von Mises stress distribution for the coarsest mesh
Figure 5: von Mises stress distribution for the second coarsest mesh
Figure 6: von Mises stress distribution for the finer mesh

7. 7
Mesh size (mm) h = 3.75 h = 3 h = 2.4
Von Mises stress
(MPa)
83.22 82.44 83.07
Table 1: Computed values of the von Mises stress for 3 uniform meshes of element sizes
Point of interest Estimated Mises stress (MPa) Convergence rate β
1 82.7885 0.957
Table 2: True values of the von Mises stress estimated with extrapolation, with the convergence
rate indicated
Figure 8: Computed values of the von Mises stress for 3 mesh size h = 3.75, 3, 2,4[mm], for which
the solution was obtained
Mesh size
2.4 2.6 2.8 3 3.2 3.4 3.6 3.8
ComputedMisessolution
82.4
82.5
82.6
82.7
82.8
82.9
83
83.1
83.2
83.3

8. 8
Figure 9: the approximate error (dashed line) computed from the von Mises stresses for the 3
different meshes, and the estimated true error (solid line) of theses three solutions.
7 Conclusion
The region of the highest von Mises stress shows up at the center of the arc edge, thus, we
choose this point as our point of interest. The sequence of computed solutions could be used to
extrapolate to obtain estimates of the true values at the point of interest shown in Figure 1. Table
1 shows the computed values of the von Mises stress for 3 uniform meshes of the elemet sizes.
Table 2 shows true values of the von Mises stress estimated with extrapolation and the convergence
rate.
References
[1] P. Krysl, Finite Element modeling with Abaqus and Matlab for Thermal and Stress Analysis,
San Diego, Pressure Cooker Press, 2015.
Refinement factor
2.4 2.6 2.8 3 3.2 3.4 3.6
Normalizederror
#10
-3
4
5
6
7
8
9
Estimated true errors
Normalized errors