# 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).

## Model Files

## 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 =
F_{0} sin ωt is induced in the y-direction. (F_{0} = 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
α_{1} = 5.36 and α_{2} = 7.46×10^{-5} are given.

**Material Properties****Value**- Young’s Modulus
- 200 × 10
^{9}N/m^{2} - Poisson’s Ratio
- 0.3
- Density
- 8000 kg/m
^{3}

## Frequency Response Summary

`f*`

- Closed form solution

Peak Displacement (mm) | Peak Stress (N/mm^{2}) |
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 |

## Reference

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