# OS-V: 0300 Deep Simply-Supported Beam Harmonic Forced Vibration Response

Test 5H OptiStruct is used to investigate the Peak Displacement in y-direction and extreme fiber bending stress at undamped Natural Frequency (at the mid-span node).

## Benchmark Model

Timoshenko beam and Engineer’s beam elements are used to model the simply-supported beam which consists of 10 elements. The displacements in x, y, and z direction, as well as the rotation in x direction are fixed at the end A. In addition, the displacements in y and z direction are constrained at end B. A steady-state harmonic forced vibration F=F0 sin ωt is induced in the y-direction. (F0=106 N/m uniformly distributed, ω=2πf, f=0 to 4.16 Hz). For modal analysis solution, a damping ratio of 0.02 is applied in all 16 modes and for direct solution, Rayleigh damping factor α1=5.36 and α2=7.46×10-5 are given.

The material properties are:
Material Properties
Value
Young’s Modulus
200 × 109 N/m2
Poisson’s Ratio
0.3
Density
8000 kg/m3

## Frequency Response Summary

The frequency of each targeted mode is normalized with the closed form solution.
f*
Closed form solution
Peak Displacement (mm) Peak Stress (N/mm2) Frequency (Hz)
Reference Solution 13.45 241.9 42.65
PBEAML
Direct Solution 13.42 236.1 43.02
Normalized 1.002235469 1.024565862 0.991399349
Modal Solution 13.56 238.61 43.16
Normalized 0.991887906 1.01378819 0.988183503
PBEAM
Direct Solution 12.27 238.21 45.28
Normalized 1.096169519 1.015490534 0.941916961
Modal Solution 12.3 238.89 45.34
Normalized 1.093495935 1.012599941 0.94067049

## Model Files

The model files used in this problem include:
• Test5HPBEAMLD.fem
• Test5HPBEAMLM.fem
• Test5HPBEAMD.fem
• Test5HPBEAMM.fem

## Reference

NAFEMS R0016 - Selected Benchmarks for Forced Vibration, J Maguire, D J, Dawswell, L Gould 1989