# Missing Mass & Residual Rigid Response

A modal analysis takes account of how the mass is mobilised in a dynamic analysis. However only a relatively small number of modes are calculated, so not all of the mass if mobilised. Provided the modes up to a high enough frequency are calculated the remaining response can be considered as an essentially static response. There are procedures for establish the missing mass and taking this into account through a static analysis. One is given by the U.S Nuclear Regulatory Commission.

For a given excitation direction the mass associated with the modes can be calculated for each degree of freedom from

The missing mass is then

Given the ground zero period acceleration (ZPA) The missing response can be treated as a static load case

A set of static loads corresponding to the different directions can then be established.

# Gupta Method

The Gupta method is a way of including the residual rigid response along with the response spectrum analysis. This defines a rigid response coefficient, so that the periodic response is

The coefficient α is defined by Gupta as


U.S Nuclear Regulatory Commission, Regulatory Guide 1.92 Combining Modal responses and Spatial Components in Seismic Response Analysis, Revision 2, July 2006