# Simplified Vibration Design for EC 5

### Technical Article

With the RF‑/TIMBER Pro add-on module, it is possible to perform the vibration design known from DIN 1052 for the design according to EN 1995‑1‑1. In this design, the deflection under permanent and quasi-permanent action at the ideal one‑span beam may not exceed a limit value (6 mm according to DIN 1052). If you consider the relation between the natural frequency and the deflection for a hinged single-span beam subjected to a constant distributed load, a minimum natural frequency of about 7.2 Hz results from the 6 mm.

${\mathrm{f}}_{\mathrm{e}}=\frac{\mathrm{\pi}}{2\xb7{\mathrm{l}}^{2}}\xb7\sqrt{\frac{\mathrm{EI}}{\mathrm{m}}}\Rightarrow \frac{\mathrm{EI}}{\mathrm{m}\xb7{\mathrm{l}}^{4}}=\frac{{\mathrm{f}}_{\mathrm{e}}^{2}\xb74}{{\mathrm{\pi}}^{2}}$

f_{e} |
Natural frequency |

l |
Length of the beam |

EI |
Bending stiffness of the beam |

m |
Modal mass |

$\mathrm{w}=\frac{5\xb7\mathrm{m}\xb7\mathrm{g}\xb7{\mathrm{l}}^{4}}{384\xb7\mathrm{EI}}\Rightarrow \frac{\mathrm{EI}}{\mathrm{m}\xb7{\mathrm{l}}^{4}}=\frac{5\xb7\mathrm{g}}{384\xb7\mathrm{w}}$

w |
Deflection |

l |
Length of the beam |

m |
Modal mass |

g |
Acceleration of gravity |

If both equations are equated, the natural frequency of 7.2 Hz results in a deformation of 6 mm.

$\frac{{\mathrm{f}}_{\mathrm{e}}^{2}\xb74}{{\mathrm{\pi}}^{2}}=\frac{5\xb7\mathrm{g}}{384\xb7\mathrm{w}}\phantom{\rule{0ex}{0ex}}{\mathrm{f}}_{\mathrm{e},\mathrm{min},\mathrm{DIN}}=\frac{5}{\sqrt{0.8\xb7\mathrm{w}}}=\frac{5}{\sqrt{0.8\xb70,6}}=7,2\mathrm{Hz}$

f_{e, min, DIN} |
Minimum natural frequency according to DIN 1052 |

w |
Deflection |

g |
Acceleration of gravity |

If we take into account the fact that in most National Annexes of EC 5, a minimum natural frequency of 8.00 Hz is to be considered, we obtain a maximum deflection of about 5 mm.

${\mathrm{f}}_{\mathrm{e},\mathrm{min},\mathrm{EC}5}=\frac{5}{\sqrt{0.8\xb7w}}=\frac{5}{\sqrt{0.8\xb70,5}}=8\mathrm{Hz}$

f_{e, min, EC5} |
Minimum natural frequency according to EN 1995-1-1 |

w |
Deflection |

If the structural system deviates from a hinged single-span beam (for example, continuous beams, cantilevers, restraints), this must be taken into account accordingly in the deflection limitation.

#### Example:

We analyze a three‑span beam in a dwelling house. To avoid discomfort caused by persons, the system shall have a minimum fundamental frequency of 8 Hz. In order to consider this in RF‑/TIMBER Pro, you can use the formula for three-span beams (see the PDF document below) to determine the maximum allowed deflection of the central span.

${\mathrm{w}}_{\mathrm{max},8\mathrm{Hz}}\approx -{\mathrm{k}}_{\mathrm{f}}^{2}\xb7\frac{63\xb7{\mathrm{l}}^{2}\xb7(12\xb7{\mathrm{l}}_{\mathrm{k}}^{3}-10\xb7{\mathrm{l}}_{\mathrm{k}}\xb7{\mathrm{l}}^{2}-3\xb7{\mathrm{l}}^{3})}{{\mathrm{f}}_{\mathrm{e},8\mathrm{Hz}}^{2}\xb7{\mathrm{l}}_{\mathrm{k}}^{4}\xb7(2\xb7{\mathrm{l}}_{\mathrm{k}}+3\xb7\mathrm{l})}=-1,00\mathrm{mm}$

w_{max, 8Hz} |
Deflection limit to reach 8 Hz |

k_{f} |
Correction factor (see PDF file) |

l |
Length of the middle field |

l_{k} |
Length of the outer panels |

f_{e, 8Hz} |
Minimum natural frequency |

In this case, the middle span may deform by -1 mm to comply with the frequency criterion. The actual deflection under permanent load (2.1 kN/m) is obtained as -0.683 mm. Thus, the design of the natural frequency is complied with, and the natural frequency of the beam is greater than 8 Hz. A check calculation results in a natural frequency of 9.76 Hz.

${\mathrm{f}}_{\mathrm{e}}\approx -{\mathrm{k}}_{\mathrm{f}}\xb7\sqrt{\frac{63\xb7{\mathrm{l}}^{2}\xb7(12\xb7{\mathrm{l}}_{\mathrm{k}}^{3}-10\xb7{\mathrm{l}}_{\mathrm{k}}\xb7{\mathrm{l}}^{2}-3\xb7{\mathrm{l}}^{3})}{{\mathrm{w}}_{0,0683\mathrm{cm}}\xb7{\mathrm{l}}_{\mathrm{k}}^{4}\xb7(2\xb7{\mathrm{l}}_{\mathrm{k}}+3\xb7\mathrm{l})}}=9,76\mathrm{Hz}$

A more precise calculation with the RF‑/DYNAM Pro — Natural Vibrations add-on module results in a natural frequency of 9.86 Hz. The video shows the procedure.

Furthermore, it should be noted that further designs (stiffness criterion, vibration velocity, vibration acceleration) must be performed for the vibration design of apartment ceilings. Notes can be found in [1] or [2], for example.

#### Author

#### Dipl.-Ing. (FH) Gerhard Rehm

Product Engineering & Customer Support

Mr. Rehm is responsible for the development of products for timber structures, and provides technical support for customers.

#### Keywords

Vibration design Stiffness criterion Vibration velocity Vibration acceleration Natural frequency

#### Reference

#### Downloads

#### Links

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