Human perceptibility to transient floor vibrations depends on three factors: damping, frequency, and initial amplitude. Scales relating these parameters to human reaction have been developed.
No attempt will be made here to present a comprehensive discussion of Murray's vibration analysis methodology. Rather, the user is referred to the references listed above. There are a number of items, however, which need mentioning:
Equation 11‑1
The first natural frequency of a simply supported beam is calculated by
| fb = 1.57 [gEIt /WL3] 1/2 | |
where
| g | = | 386 in/sec2 |
| L | = | span, inches |
| E | = | modulus of elasticity, ksi |
| It | = | transformed moment of inertia, in4 |
| W | = | total weight supported by the beam, kips. |
Equation 11 ‑ 2
In the alternate method, if a beam is supported by flexural members rather than rigid supports, the flexibility of these members is taken into account in calculating the frequency of the floor system. The system frequency can be approximated by:
where
| fs | = | frequency of the system, cps |
| fb | = | frequency of the beam, cps |
| fg | = | frequency of the girder, cps. |
Equation 11 ‑ 3
The frequency of the supporting girder is calculated by:
| fg = 1.57 [gEIg/WgLg 3] 1/2 | |
where
| Wg | = | total supported weight, kips |
| Ig | = | girder moment of inertia, in4, calculated as for normal composite construction design |
| Lg | = | girder span, in |
| E | = | modulus of elasticity, ksi. |
Equation 11 ‑ 4
For the methodology involved in calculating the amplitude of vibration of an individual member the user is referred to the references given above. However, the following should be noted. For slabs supported by at least five parallel, equally spaced beams, the effective number of beams resisting a heel‑drop impact is given by:
| Neff = 2.97 - S/(17.3de) + L4/(1350EIt) | |
where
| S | = | beam spacing, in |
| de | = | effective slab depth in inches, including concrete in the rib of the decking. |
This equation is not appropriate for beams spaced less than 2.5 ft.
Equation 11 ‑ 5
The effective amplitude of vibration of a beam is given by
where
| A0t | = | is the amplitude of an individual beam (see Reference 1). |
Equation 11 ‑ 6
The amplitude of the floor system (when the influence of supporting girders is included) is given by:
where
| A0g | = | is the amplitude of the supporting girder. |
