# RD-V: 0010 Beam Bending

A pressure load is applied to a beam to study simulating bending in a model.

## Input Files

Before you begin, copy the file(s) used in this problem to your working directory.

## Model Description

Units: mm, s, Mg

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

$\delta =\frac{Pw{L}^{4}}{8EI}+\frac{kPw{L}^{2}}{2GA}$

Where,
$P$
$W$
Width of beam
$L$
Length of beam
$E$
Modulus of elasticity
$I$
Area moment of inertia
$k$
Shear strain correction coefficient
$G$
Shear modulus
$A$
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