Finite Element Analysis
of a Beam and a Plate
Subjected to Mechanical
Loads
MANE 6960 – ADVANCED TOPICS IN FINITE ELEMENT
ANALYSIS
THOMAS PROVENCHER
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Introduction:
This model analyzes the stress and displacement of a steel beam and a thin steel plate subjected
to concentrated loads. One end of the beam was held firmly in a cantilevered manner while a
concentrated load was placed at the center. The edges of the plate were clamped firmly and not
allowed to move while a circular line load was applied around the center of the plate. The Finite
Element Analysis of both the beam and plate will be performed using two different methods.
The cantilevered beam will be analyzed separately in 1D and 3D environments and the plate will
be analyzed in both 2D and a 3D environments. The maximum deflection predicted by these
analyses will be compared a theoretical solutions.
Formulation and Solution:
The models created for this analysis will simulate the stress and displacement of a steel beam and
a plate subjected to a concentrated load. A 1.0 x 0.1 x 0.1 meter long steel beam and a 1.0 x 1.0
x 0.05 meter steel plate were modeled in COMSOL Multiphysics. The beam was modeled in
both the 1D and 3D environments and the plate was modeled in both the 2D and 3D
environments. In the case of the 1D beam and the 2D plate, the dimensions of the beam and the
thickness of the plate were entered into the simulation, respectively. One side of the beam and
the four edges of the plate were securely clamped with zero displacement boundary conditions.
For the 1D beam, a 1000 Newton downward force was placed at the beam’s center and for the
3D beam, a line was drawn across the width of the beam and a 10000N/m line load was applied
to it. For the 2D and 3D plate, a 1000N downward force was placed within the bounds of a 0.05
meter diameter circle drawn around the center of the plate. Figures 1 and 2 show the 1D and 3D
models used to analyze the beam the 2D and 3D models used to analyze the plate, respectively.
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Figure 1: Steel beam model and dimensions
Figure 2: Steel plate model and dimensions
An extra fine mesh was chosen to analyze these models as previous results have proven this
mesh density is sufficient for simple mechanical simulations. The mesh chosen was derived
from COMSOL’s physics based mesh creator and was constructed of Lagrange quadratic
elements. The extra fine mesh of the 3D beam and plate can be seen in Figures 3 and 4.
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Figure 3: Extra fine mesh density of the beam
Figure 4: Extra fine mesh density of the plate
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The variational formulation is shown below:
(𝑢′ , 𝜐′) = (𝑓, 𝜐)
The four beam and plate COMSOL models were run and analyzed. Their stress and
displacement values were compared to each other and their displacements were further compared
to theoretically determined displacements for their load cases. Figures 5 and 6 show the Roark’s
equations from Reference (a) used to determine the maximum theoretical displacements and
Figures 7-10 show the stress and vertical displacement maps of the four models. Table 1 shows
the maximum stress and displacement values provided by the COMSOL analysis as well as the
theoretical displacement.
Figure 5: Roark’s Formulas used to determine the beam’s theoretical displacement
Figure 6: Roark’s Formulas used to determine the plate’s theoretical displacement
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Figure 7: Stress and displacement plots of the centrally loaded 1D beam
Figure 8: Stress and displacement plots of the centrally loaded 3D beam
Figure 9: Stress and displacement plots of the centrally loaded 2D plate
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Figure 10: Stress and displacement plots of the centrally loaded 3D plate
Table 1: Stress and displacement maximum values
1D Beam
3D Beam
2D Plate
3D Plate
Maximum
Stress (Pa)
3.01E+06
2.95E+06
1.49E+05
1.48E+05
Maximum
Deflection (m)
-6.25E-05
-6.25E-05
-5.15E-07
-5.11E-07
Percent
Difference
Theoretical
Deflection
4.96E-02
-6.250E-05
0.76
-4.80E-07
Percent Error
2.168E-14
4.960E-02
7.24
6.44
The data provided in Table 1 shows that quadratic Lagrange elements are able to provide
accurate results for the all of the beam and plate models. The theoretical and analytical
displacement values agreed well within the accepted tolerance stipulated in Reference (a). The
results provided by the beam and plate models show that the 1D and 2D simplifications were
able to very accurately replicate the results from their respective 3D models.
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Conclusions:
This model was able to approximate the stress and displacement imparted on a thin cantilevered
beam and a thin square plate subjected to centrally located downward loads. One end of the
beam and the four sides of the plate were given zero displacement boundary condition to
simulate the beam being cantilevered and the plate being fully clamped, respectively. The
analysis results show that, as expected, the beam bends downward and that the center of the plate
bows out, away from the downward loads. The results also indicate that the 1D beam, 2D plate,
and both the 3D models were able to accurately replicate the displacement values provided by
theoretical equations.
References:
a) Young, Warren C., and Richard G. Budynas. "Chapter 11, Flat Plates." Roark's Formulas for
Stress and Strain: Warren C. Young; Richard G. Budynas. 7.th ed. New York: McGraw-Hill,
2002. 505. Print.
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