# P. Floor Vibrations Engineering Theory

The fundamental natural frequency of the joist mode and the girder mode can be determined from equation 3.3 on page 11 of the design guide:

 fj/g = 0.18 (g/Δ)1/2 1

 fj/g = fundamental natural frequency of the joist or the girder mode g = acceleration due to gravity Δ = midspan deflection of the member due to the weight supported.

For the combined mode, the fundamental natural frequency can be determined from equation 3.4 on page 11 of the design guide:

 fn = 0.18 [g/ (Δj + Δg) ]1/2 2

 fn = fundamental natural frequency for the combined mode g = acceleration due to gravity Δj = joist deflection due to the weight supported Δg = girder deflection due to the weight supported

Δj and Δg are the local deflection of the joist and the girder determined from a secondary operation after the analysis. The stiffness analysis will yield the global deflection values for the girder beams. A line joining the start and the end nodes of the girder beam in its deflected position is created as a base line. Relative to this base line, the deflection values are zero for the start and end nodes. The local deflection values of the intermediate points of the girder beam are evaluated from the global deflection values relative to this base line.

It is this local deflection that is used in calculating the fundamental natural frequency as shown in the earlier equations. Further, the local deflection is also used in calculating the equivalent uniform loading on the joist and the girder, wj and wg, as shown in the equation on page 21 of the AISC Design Guide:

 3

 Ij/g = moment of inertia of the effective joist or girder section Lj/g = joist or girder span Eg = modulus of elasticity of steel

In addition to the terms fj , Δj , wj shown above , the following additional terms —Ds , Dj , Bj , and Wj — which are explained below, are also reported for the joist mode.

 Ds = the transformed slab moment of inertia per unit width and is determined from the equation at the bottom right corner of page 17 of the AISC Design Guide: de = effective depth of the concrete slab, usually taken as the depth of the concrete above the form deck plus one-half the depth of the form deck N = dynamic modular ratio = Es / 1.35 Ec Ec = modulus of elasticity of concrete Dj = the joist or the or beam transformed moment of inertia per unit width, and is determined from the equation shown at the top left of page 18 of the AISC Design Guide: = Ij/S S = joist or beam spacing Bj = the effective width for the beam or joist panel mode and is determined from equation 4.3a on page 17 of the AISC Design Guide: = Cj (Ds / Dj )1/4 Lj ≤ 2/3× floor width Cj = 1.0 for interior panels 2.0 for edge panels Wj = the weight of the beam panel and is calculated from equation 4.2 on page 17 of the AISC Design Guide and page 21 left side: Wj = wj Bj Lj ( x 1.5 if continuous )

For the girder mode, the terms reported include fg , Δj , wg which were explained earlier, and, Dg , Bg , and Wg which are described below.

 Dg = the girder transformed moment of inertia per unit width described on page 18 of the AISC Design Guide: = Ig/Lj for all but edge girders = Ig/2Lj for edge girders Bg = the effective width for the girder panel mode and is determined by equation 4.3b on page 18: = Cg (Dg / Dj )1/4 Lg ≤ 2/3× floor width for interior panels = 2/3 Lg for edge panels Cg = defined on page 18 as: = 1.6 for girders supporting joists connected to girder flange (e.g., joist seats ) = 1.8 for girders supporting beams connected to the girder web. Wg = the weight of the girder panel and is calculated by equation 4.2 on page 17 and described on page 21: = wg Bg Lg ( x 1.5 if continuous )

For the combined mode of vibration the parameters reported are fn , W, β, Peak Acceleration and Acceleration Limit.

fn is calculated from equation 2 shown above.

W is the equivalent panel weight in the combined mode and is calculated from the equation shown on page 21 of the AISC Design Guide:

 W

β is the value of the damping ratio as per Table 4.1 on page 18 of AISC Design Guide.

The peak acceleration due to walking excitation is then determined from the equation 4.1 on page 17 and on page 21 of the AISC Design Guide:

 ap = peak acceleration value due to walking excitation P0 = a constant force representing excitation and is determined as per Table 4.1 of the Design Guide. fn = fundamental natural frequency in combined mode

The acceleration limit is determined from Table 4.1 on page 18 of the Design Guide.