Vibrations in structures are activated by dynamic periodic forces such as wind, people, traffic, and rotating machinery.
For normal floors with a span-to-depth ratio less than 25, vibrations are generally not an issue.
For lightweight structures with spans above 8 meters (24 feet), vibrations may occur.
As a rule of thumb, the natural frequency of a structure should be greater than 4.5 Hz to ensure it does not compromise strength, stability, or human sensitivity.
The numerical factor $a$ in the general formula for distributed mass is typically taken as 18 for practical accuracy, though it varies between 16 and 20 for similar systems.

Example: Natural Frequency Calculation for a Steel Beam

Consider a simply supported DIN 1025 I 200 steel beam with distributed mass:

Using the formula for natural frequency of a simply supported structure with distributed mass: $f = \frac{\pi}{2} \sqrt{\frac{E I}{q L^4}}$

Substitute the values: $f = \frac{\pi}{2} \sqrt{\frac{(200 \times 10^9) (2140 \times 10^{-8})}{(26.2) (12)^4}}$

First, compute $E I$ : $E I = (200 \times 10^9) \times (2140 \times 10^{-8}) = 4.28 \times 10^6 \, \text{N·m}^2$

Next, compute $q L^4$ : $L^4 = (12)^4 = 20736 \, \text{m}^4$ $q L^4 = 26.2 \times 20736 = 543619.2 \, \text{kg·m}^4$

Then, the ratio: $\frac{E I}{q L^4} = \frac{4.28 \times 10^6}{543619.2} \approx 7.87 \, \text{m}^{-2}$

Square root: $\sqrt{7.87} \approx 2.81 \, \text{m}^{-1}$

Finally, frequency: $f = \frac{\pi}{2} \times 2.81 \approx 1.57 \times 2.81 = 4.41 \, \text{Hz}$

Since $4.41 \, \text{Hz}$ is close to the 4.5 Hz threshold, the beam's vibration characteristics should be evaluated carefully for occupancy comfort.

References