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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 Commission5^5.

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

m^i=∑n=1NΓnjφni{\widehat{m}}_{i} = \sum_{n = 1}^{N}{\Gamma_{nj}\varphi_{ni}}

The missing mass is then

mi−m^im_{i} - {\widehat{m}}_{i}

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

fi=(mi−m^i)aZPAf_{i} = \left( m_{i} - {\widehat{m}}_{i} \right)a_{ZPA}

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, αi\alpha_i so that the periodic response is

rpi=[1−αi2]12rir_{pi} = \left\lbrack 1 - {\alpha_{i}}^{2} \right\rbrack^{\frac{1}{2}}r_{i}

The coefficient α is defined by Gupta as

αi={0fi≤f1ln⁡(fif1)ln⁡(f2f1)f1≤fi≤f21fi≥f2\alpha_{i} = \left\{ \begin{matrix} 0\quad&f_{i} \leq f_{1} \\ \frac{\ln\left( \frac{f_{i}}{f_{1}} \right)}{\ln\left( \frac{f_{2}}{f_{1}} \right)}\quad&f_{1} \leq f_{i} \leq f_{2} \\ 1\quad&f_{i} \geq f_{2} \\ \end{matrix} \right.

where

f1=max⁡(aspectral)2πmax⁡(vspectral)f2=frigid\begin{aligned}f_{1} &= \frac{\max ({a_{spectral}})}{2\pi \max({v_{spectral}})}\\ f_{2} &= f_{rigid}\end{aligned}

5^5 U.S Nuclear Regulatory Commission, Regulatory Guide 1.92 Combining Modal responses and Spatial Components in Seismic Response Analysis, Revision 2, July 2006