Homework 05
Reduced Order Models: Beam Plates and Shells
Holly Ibanez
MANE6960
Advanced Topics in Finite Elements
Summary
Reduced order models are simplified models of linear elastic solid mechanics systems. Beams and plates
are often simulated as 1D or 2D systems. Analytical solutions can be obtained in some cases and finite
element models can also be created.
This report analyzes two finite element models to estimate the displacement, stress, and strain
distributions:


Bending of a cantilever beam with a square cross section subjected to a concentrated load P
halfway across its length
Bending of a clamped square plate subjected to a uniform distributed pressure over its surface
Calculation and Discussion
The models created in COMSOL were assigned the following parameters:
Table 1: Model Parameters
Parameter
Length (m)
Height (m)
Thickness (m)
Elastic Modulus (Pa)
Poisson’s Ratio
Density (kg/m3)
1D Beam
1.0
0.1
0.1
2D Plate
1.0
1.0
0.02
2x1011
0.3
7850
Laplace’s equation is the governing equation for solid mechanics problems.
∇𝜎 = 0
Dirichlet boundary conditions are applied to the 1D beam and 2D plate, respectively:
𝑢(0) = 𝑢(𝐿) = 0
𝑢(𝑥, 0) = 𝑢(𝑥, 𝑏) = 𝑢(0, 𝑦) = 𝑢(𝑎, 𝑦) = 0
The variational formulation for both problems is as follows:
(𝑢′ , 𝑣 ′ ) = (𝑓, 𝑣)
The equations in Figure 1 show Roark’s formulas for stress and strain for a fixed cantilever beam with a
concentrated point load. These equations are used to verify the results obtained in COMSOL.
Figure 1: Roark's Formulas for Stress and Strain in a Cantilever Beam
The equations in Figure 2 show Roark’s formulas for stress and strain for a fixed plate with a uniform
pressure over the entire plate. These equations are used to verify the results obtained in COMSOL.
Figure 2: Roark's Formulas for Stress and Strain in a Fixed Plate
Results and Conclusions
Figures 3 and 4 show the resulting von Mises stress and displacement derived using COMSOL analysis. A
100 N point load is applied to the center of the 1D beam.
Figure 3: von Mises Stress in a 1D Cantilever Beam
Figure 4: Displacement in a 1D Cantilever Beam
Table 2 shows a comparison of the maximum displacement obtained using an extremely fine mesh in
COMSOL versus the maximum displacement calculated using Roark’s formulas. This comparison is used
to verify the COMSOL results.
Table 2: 1D Beam Results Verification
Maximum Displacement
COMSOL Results
-6.25 x 10-6 m
Roark’s Formula
-6.25 x 10-6 m
Figures 5 and 6 show the resulting von Mises stress and displacement derived using COMSOL analysis. A
100 Pa concentrated load is applied across the entire face of the 2D plate.
Figure 5: von Mises Stress in a 2D Fixed Plate
Figure 6: Displacement in a 2D Fixed Plate
Table 3 shows a comparison of the stress at the center of the plate and maximum displacement
obtained using an extremely fine mesh in COMSOL versus the values calculated using Roark’s formulas.
This comparison is used to verify the COMSOL results.
Table 3: 2D Plate Results Verification
Stress at the Plate Center
Maximum Displacement
COMSOL Results
3.50 x 104 N/m2
8.71 x 10-7 m
Roark’s Formula
3.47 x 104 N/m2
8.63 x 10-7 m
References
Young, W. C., Budynas, R. G., “Roark’s Formulas for Stress and Strain”. McGraw Hill. Seventh Edition.