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2021-03-29

Simplified Vibration Design for EC 5

fe =π2 · l2 · EIm  EIm · l4 = fe2 · 4π2
fe Natural frequency
l Length of the beam
EI Bending stiffness of the beam
m Modal mass

w = 5 · m · g · l4384 · EI  EIm · l4 = 5 · g384 · 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.

fe2 · 4π2 = 5 ·g384 · wfe,min,DIN = 50,8 ·w = 50,8 ·0.6 = 7.2 Hz
fe,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.

fe,min,EC5 = 50,8 ·w = 50,8 ·0.5 = 8 Hz
fe,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.

wmax,8Hz  -kf2 · 63 · l2 ·( 12 · lk3 - 10 · lk · l2 - 3 · l3 )fe,8Hz2 · lk4 · ( 2 ·  lk + 3 · l ) = -1.00 mm
wmax,8Hz Limit deflection of about 8 Hz to be reached
kf Correction factor (see PDF file)
l Length of the middle span
lk Length of the outer panels
fe,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.

fe  -kf · 63 · l2 ·( 12 · lk3 - 10 · lk · l2 - 3 · l3 )w0.0683cm · lk4 · ( 2 ·  lk + 3 · l ) = 9.76 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

Mr. Rehm is responsible for developing products for timber structures, and he provides technical support for customers.

Links
References
  1. Eurocode 5: Bemessung und Konstruktion von Holzbauten - Teil 1-1: Allgemeines - Allgemeine Regeln und Regeln für den Hochbau; DIN EN 1995-1-1:2010-12
  2. Winter, S.; Hamm, P.; Richter, A.: Schwingungs- und Dämpfungsverhalten von Holz- und Holz-Beton-Verbunddecken. Schlussbericht Juli 2010. München: TU München, Lehrstuhl für Holzbau und Baukonstruktion, 2010
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