A beam with dimensions length = 508 mm, height = 50.8 mm, and width = 25.4 mm is
meshed using solid elements that are length and width of 12.7 mm. Steel material
properties are used with a Poisson’s ratio of zero to eliminate artificial
stiffening. A uniform pressure of 0.3447 MPa is applied across the top of the beam.
The pressure is applied using a smooth function to minimize the vibration of the
beam during loading. The pressure is applied from 0.0 – 0.15 s and then held until
0.25 s. To measure the stress at the top of the beam, two shell elements are placed
120.65 mm from the fixed end of the beam.
Results
The analytical solution to the end displacement can be calculated using Castigliano’s
theorem for a cantilever beam which includes the terms for bending, as well as
shear. 1(1)
Where,
Pressure load
Width of beam
Length of beam
Modulus of elasticity
Area moment of inertia
Shear strain correction coefficient
Shear modulus
Cross sectional area of beam
The calculated displacement at the center of the end of the beam was 1.3713 mm
compared to the analytical result of 1.3752 mm. This results in a numerical error of
0.28%.
The calculated stress in the shell elements on the top of the beam was 59.9716 MPa
compared to the analytical result of 58.5562 MPa. 1 This results in a numerical error of 2.42%.
Conclusion
The numerical error in displacement and stress is small
and could be reduced further by refining the mesh.
1 Buechler, Miles,
Amanda McCarty, Derek Reding, and R. D. Maupin. "Explicit finite element code
verification problems" In 22nd SEM International Modal Analysis
Conference, Dearborn, MI. 2004