# 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:

f_{j/g} =
0.18 (g/Δ)^{1/2}
| 1 |

_{j/g}
| = | |

= | ||

= |

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

f_{n} =
0.18 [g/ (Δ_{j} + Δ_{g}) ]_{1/2}
| 2 |

_{n}
| = | |

= | ||

_{j}
| = | |

_{g}
| = |

Δ_{j}
and
Δ_{g}
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 |

_{j/g}
| = | |

_{j/g}
| = | |

_{g}
| = |

In addition to the terms
f_{j}
,
Δ_{j}
,
w_{j}
shown above , the following additional terms —D_{s}
,
D_{j}
,
B_{j}
,
and W_{j}
— which
are explained below, are also reported for the joist mode.

_{s}
| = | |

_{e}
| = | |

= | _{s} / 1.35
E_{c}
| |

_{c}
| = | |

_{j}
| = | =
I |

= | ||

_{j}
| = | =
C |

_{j}
| = | 2.0 for edge panels |

_{j}
| = | W |

For the girder mode, the terms reported include
f_{g}
,
Δ_{j}
,
w_{g}
which were explained earlier, and,
D_{g}
,
B_{g}
,
and W_{g}
which are
described below.

_{g}
| = |
=
I
=
I |

_{g}
| = |
= C
= 2/3 L |

_{g}
| = | = 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. |

_{g}
| = | =
w |

For the combined mode of vibration the parameters
reported are f_{n}
,
W,
β, Peak
Acceleration and Acceleration Limit.

f_{n}
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:

_{p}
| = | |

_{0}
| = | |

_{n}
| = |

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