Andrew Gandia
Mini-Project
MANE 4940
Fall 2011
Overview:
This analysis investigates the impact of applying a load on a beam fixed on both ends. Previously, a
beam that was fixed at one end and free at the other was analyzed to understand the deflection of the
beam if a load was applied at the free end. The study presented below will have a beam fixed at both
ends and the same load applied to the center of the beam.
Problem Statement:
A beam fixed on both ends is made of steel (E=1011 N/m2, ν =0.3) and has length L=1, height h = 0.1, and
breadth b = 0.1 (all in m). At its center, a downward force F=103 N is applied. The goal is to find the
deflection of the beam u(x) due to the applied force. Compare the COMSOL solution and the ANSYS
solution against the exact solution of a beam fixed at both ends and against the exact solution of a
cantilever beam.
Exact Solution:
For the exact solution of a beam fixed on both ends, the max deflection was calculated using the max
deflection at center equation in the figure below. Utilizing the problem statement parameters, the exact
solution was calculated in MAPLE, resulting with a max deflection of 6.25 x 10-6 m.
Figure 1: Exact Solution of Beam Fixed at Both Ends
COMSOL Approach:
The beam fixed at both ends was modeled in COMSOL utilizing the 10 x 1 x 1 P2 elements in the steady
state structural mechanics-linear elasticity module. The model was built up utilizing the inputs from the
problem statement. A point was created in the center of the beam and the downward force of 1000 N
was applied at this point. In the coordinates of the model, the point was (0.5, 0, 0)
Figure 2: Point and Beam prior to load being applied load
Figures 3 and 4 show the deflection of the beam when the force is applied:
Figure 3: Isometric View of Deformed Beam in COMSOL
Figure 4: Side View of Deformed Beam in COMSOL
As seen through the figures, the most deformation occurs at the center of the beam. Utilizing the point
evaluation tool, the deflection at this point was measured to be 6.905 x 10-6 m.
ANSYS Approach:
The beam fixed at both ends was also modeled in ANSYS. BEAM 189 was utilized and a rectangle was
built utilizing two lines (First line went from (0,0,0) to (0.5,0,0) of the model and the second line from
(0.5,0,0) to (1,0,0). A cross section based on the problem statement was created and a mesh of 40 x 2 x
2 was utilized. Like COMSOL, the beam was fixed at the endpoints and a force was applied at the center
of the beam (Coordinate (0.5, 0, 0)).
The following figures show the deflected beam.
Figure 5: Isometric View of Deformed Beam in ANSYS
Figure 6: Side View of Deformed Beam in ANSYS
From the results, like COMSOL, the max deflection occurs at the center of the beam. The max deflection
is 7.02 x 10-6 m.
Comparison of Results:
The following table shows the results for the maximum deflection of a beam.
Values
U max (meters)
Exact for cantilever beam
4 x 10-4
Exact for both ends fixed
6.25 x 10-6
COMSOL for both ends fixed
6.905 x 10-6
ANSYS for both ends fixed
7.02 x 10-6
Two major observations can be made. The first is that when compared to the values for the beam with
the fixed ends, the cantilever beam had a higher deflection than those values of the beam fixed at both
ends. This seems logical since the beam is only restrained at one end, versus a beam constrained at both
ends.
The second observation is that there is a slight difference between the ANSYS/COMSOL values versus
the exact. This could be due to the mesh size chosen and for more exact results, a higher element mesh
would be necessary.
Sources
stommel.tamu.edu/~esandt/Teach/Fall02/CVEN444/.../lecture35.ppt
HW 4 – MANE 4940 RPI