Analysis Procedure - Response Spectrum Analysis

The normal modes are calculated first to decouple the equations of motion with the use of generalized modal coordinates. The maximum modal responses are determined from the base excitation response spectrum. With the use of modal combination techniques, the maximum structural response is calculated by summing the contributions from each mode.

Maximum Modal Response

Let Sd(ωI, ξI), Sv(ωI, ξI), and Sa(ωI, ξI) be the input response spectra values for the displacement, pseudo-velocity, and pseudo-acceleration, respectively.

As each mode can be idealized as a SDOF system, the maximum responses over excitation duration in terms of modal coordinates are obtained from the input response spectra values:

are the modal relative displacements, the modal relative velocities, and the modal absolute accelerations.

Γi is the modal participation factor for each mode i, which is equal to .

is the transpose of eigenvector i. , which is an influence vector that relates the base motion to the rigid body structure displacements.

For each mode, the structure's maximum responses in relation to the modal coordinates are:

Note that the maximum modal responses do not occur at the same time. For different modes, the maxima are reached at different time instances during the excitation. To calculate the structure's total maximum response, several mode combination techniques are available.