In the case of design of structures, complete time history analysis is not necessary; it is enough to valuate the peak values of forces and deformations.

**Spectrum analysis by modal response**

In the case of design of
structures, complete time history analysis is not necessary; it is enough to
valuate the peak values of forces and deformations. A good estimate, although
not accurate, of the peak response can be determined directly from response
spectrum for the ground motion. These peak modal responses are combined using
any one of the rules discussed below to get the value of the total response.
This method is known as RSA.

For any response value such as
force or deformation, the peak value *R _{n}*

where *A _{n}*(

1 Modal contribution rules

Knowing the peak value *R _{n}*

2 Absolute sum rule

If we assume that all peak values
of modal responses occur at the same time, and ignore the algebraic sign for
the peak value of the total response, we get

The above estimate is highly
conservative and not popular in structural design applications.

3 SRSS rule

The SRSS rule provides an
estimate of peak value according to the equation

The algebraic sign of *R _{n}*

4 CQC rule

The rule is applicable to wide
range of structures in which natural frequencies of closely spaced such as
those in unsymmetric buildings. According to the CQC rule (first developed by
Rosenblueth and Elorduy)

It is assumed that the damping ratio
*ς* is the same for all modes.
Figure 18.27 shows the variation of correlation coefficient with respect to *β** _{ij}*. It is
to be observed that the correlation coefficient

In *ρ* matrix the leading diagonal
terms are equal to 1. Hence Eq. 18.78 is rewritten as

The first summation of the
right-hand side is identical to SRSS combination rule, whereas the next term
may be positive or negative, depending the algebraic signs of *R _{n}*

5 Factors influencing the earthquake response

There are two parameters which
influence the earthquake response: (1) fundamental natural vibration period *T*_{1}
and (2) beam to column stiffness ratios based on the properties of the beam and
column in the storey closest to the mid-height of the frame.

where *EI _{b}* and

As the fundamental period
increases with the velocity and displacement-sensitive regions of the
structure, the higher mode response generally becomes an increasing percentage
of the total response. For design accuracy more modes should be included in the
analysis of buildings with longer periods than the buildings with shorter
period. More modes should be included in the analysis of buildings with smaller
*γ*. In some codal provisions, we
have to include the number of modes such that 90% of the total mass is the
participating mass.

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Civil : Structural dynamics of earthquake engineering : Spectrum analysis by modal response |

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