Mitesh Patel
MANE 6960 (Advanced Topics in Finite Elements)
Homework #4 (Plane Stress)
Introduction
The plane stress condition arises when one of the dimensions of the component is very small compared to
the rest of the dimensions. The plane stress condition is normally used and encountered while performing
structural analysis, and it is one of the important concepts used to simplify the problems in solid mechanics.
In this lesson, we will perform a problem with plane stress condition using the finite element software
COMSOL Multiphysics.
Problem Statement
In this problem, a beam was modeled as a rectangle with 1m width and 0.1m height. Because this problem
will be analyzed as a plane strain, the thickness of the beam was set to 0.001 (1e-3)m. A 100N point load
was applied at the free end of the beam, which is the right side, and the other end of the beam, which is the
left side, was fixed. The beam model is represented in Figure 1. The horizontal axis is considered as an xaxis and the vertical axis is considered as a y-axis. The 100N point load is applied in the negative y direction.
This is a standard cantilever beam problem that is frequently encountered during performance of structural
analysis. The beam was constraint on the left side in such a way that the beam can deflect in x, y, and z
directions.
Figure 1
Finite Element Method Model
In this problem, a beam was modeled as a rectangle with 1m width and 0.1m height. Because this problem
will be analyzed as a plane strain, the thickness of the beam was set to 0.001 (1e-3)m. A 100N point load
was applied at the free end of the beam, which is the right side, and the other end of the beam, which is the
left side, was fixed. The beam model is represented in Figure 1. The horizontal axis is considered as an xaxis and the vertical axis is considered as a y-axis. The 100N point load is applied in the negative y direction.
This is a standard cantilever beam problem that is frequently encountered during performance of structural
analysis. The beam was constraint on the left side in such a way that the beam can deflect in x, y, and z
directions. This model was analyzed five time with the extremely coarse, coarse, normal, finer, and
extremely fine, which are shown in Figures 2, 3, 4, 5, and 6, respectively.
Figure 2: Extremely Coarser Mesh
Figure 3: Coarser Mesh
Figure 4: Normal Mesh
Figure 5: Finer Mesh
Figure 6: Extremely Finer Mesh
It was assumed that the material of the beam is steel. The following material properties were used as inputs
to build the model in the COMSOL Multiphysics.
Boundary Conditions
If all the sides of the beams are constraint, then the beam is not allowed to deflect when the external load
is applied. If this the case, then the beam may break where it is constraint because it will experience higher
stresses at the constraint. This is because the beam cannot deform where it is fixed, and so the beam may
not absorb more loads. Therefore, there is one boundary condition applied to the beam on the left side. The
left side of the beam is constraint in such a way that the beam is not permitted to deflect and not permitted
to rotate about x, y, and z axes. The upper and the lower surfaces and the right side of the beam are free to
deflect and rotate in all the directions.
Analysis Results
The beam model was analyzed using the aforementioned five meshes. The plots and results obtained with
the five different meshes are shown in Figures7, 8, 9, 10, and 11. It can be seen from Figures 7 through 11
that the stress is maximum at the top and bottom surfaces near the left side because the beam wants to
deflect down because of the external force; however, it is constraint from moving down. Also, the maximum
deflection of the beam due to the applied external force is at the free end because the beam it not constraint.
The results of the deflection of the beam is in accordance with the visualized standard strength of material
fundamentals. The maximum deflections on the beam during extremely coarse, coarser, normal, finer, and
extremely finer meshes are 0.00199449m, 0.00200638m, 0.00201660m, 0.00201838m, and 0.00202084m,
respectively. There is no deflection at the surface of the beam where it is constraint.
Figure 7: Results with Extremely Coarse Mesh
Figure 8: Results with Coarser Mesh
Figure 9 Results with Normal Mesh
Figure 10: Results with Finer Mesh
Figure 11: Results with Extremely Fine
Validation
The plots from Figures 7 through 11 indicate that the deformation values get closer as the less meshes are
done. The deformation values decreases from the extremely fine mesh, which has the maximum number of
elements, to extremely coarse mesh, which has the minimum number of elements.
Conclusion
The maximum deflection occurs at the free end and there is no deflection where the beam is constraint in
all meshes. These results indicate that the COMSOL Multiphysics outputs are in accordance with the
standard visualized strength of material fundamentals.