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000694

Dipl.-Ing. (FH) Gerhard Rehm

2021-03-29

Simplified Vibration Design for EC 5

With the RF-/TIMBER Pro add-on module, you can 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 the 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, the 6 mm limit value results in a minimum natural frequency of about 7.2 Hz.

f_{e} = \frac{π}{2 \cdot l^{2}} \cdot \sqrt{\frac{EI}{m}} \Rightarrow \frac{EI}{m \cdot l^{4}} = \frac{f_{e}^{2} \cdot 4}{π^{2}}

f_e	Natural frequency
l	Length of the beam
EI	Bending stiffness of the beam
m	Modal mass

Natural Frequency of Single-Span Beam with Constant Load

w = \frac{5 \cdot m \cdot g \cdot l^{4}}{384 \cdot EI} \Rightarrow \frac{EI}{m \cdot l^{4}} = \frac{5 \cdot g}{384 \cdot w}

w	Deflection
l	Length of the beam
m	Modal mass
g	Acceleration of gravity

Deflection of Single-Span Beam with Constant Load

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

\frac{f_{e}^{2} \cdot 4}{π^{2}} = \frac{5 \cdot g}{384 \cdot w}

f_{e, \min, DIN} = \frac{5}{\sqrt{0, 8 \cdot w}} = \frac{5}{\sqrt{0, 8 \cdot 0.6}} = 7.2 Hz

f_e,min,DIN	Minimum natural frequency according to DIN 1052
w	Deflection
g	Acceleration of gravity

Minimum Natural Frequency According to DIN 1052

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.

f_{e, \min, EC 5} = \frac{5}{\sqrt{0, 8 \cdot w}} = \frac{5}{\sqrt{0, 8 \cdot 0.5}} = 8 Hz

f_e,min,EC5	Minimum natural frequency according to EN 1995-1-1
w	Deflection

Minimum Natural Frequency According to EN 1995-1-1

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:

Three-Span Continuous Beam

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.

w_{\max, 8 Hz} \approx - k_{f}^{2} \cdot \frac{63 \cdot l^{2} \cdot (12 \cdot l_{k}^{3} - 10 \cdot l_{k} \cdot l^{2} - 3 \cdot l^{3})}{f_{e, 8 Hz}^{2} \cdot l_{k}^{4} \cdot (2 \cdot l_{k} + 3 \cdot l)} = - 1.00 mm

w_max,8Hz	Limit deflection of about 8 Hz to be reached
k_f	Correction factor (see PDF file)
l	Length of the middle span
l_k	Length of the outer panels
f_e,8Hz	Minimum natural frequency

Limit Deflection

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.

f_{e} \approx - k_{f} \cdot \sqrt{\frac{63 \cdot l^{2} \cdot (12 \cdot l_{k}^{3} - 10 \cdot l_{k} \cdot l^{2} - 3 \cdot l^{3})}{w_{0.0683 cm} \cdot l_{k}^{4} \cdot (2 \cdot l_{k} + 3 \cdot l)}} = 9.76 Hz

Natural Frequency

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.

Knowledge Base

KB 000694 | "Simplified" Vibration Design for EC 5

Author

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

Links

Structural Analysis & Design Software for Timber Structures

References

CEN. (2010). Eurocode 5: Design of Timber Structures – Part 1-1: General – Common Rules and Rules for Buildings; DIN EN 1995-1-1:2010-12. Berlin: Beuth Verlag GmbH.
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

Downloads

Limitation of Deflection (PDF)

7.04 MB

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