# Response Spectrum Method Of Analysis – with simplified examples

## Introduction

Response spectrum method is favoured by earthquake engineering community because of:

•  It provides a technique for performing an equivalent static lateral load analysis.
•  It allows a clear understanding of the contributions of different modes of vibration.
•  It offers a simplified method for finding the design forces for structural members for earthquake.
•  It is also useful for approximate evaluation of   seismic reliability of structures.
• The concept of equivalent lateral forces for earthquake is a unique concept because it converts a  dynamic analysis partly to dynamic & partly to static  analysis for finding maximum stresses.
• For seismic design, these maximum stresses are of interest, not the time history of stress.
• Equivalent lateral force for an earthquake is defined as a set of lateral force which will produce the same peak response as that obtained by dynamic analysis of structures .
•   The equivalence is restricted to a single mode of vibration.

The response spectrum method of analysis is developed using the following steps.

•  A modal analysis of the structure is carried out to obtain mode shapes, frequencies & modal participation factors.
• Using the acceleration response spectrum, an equivalent static load is derived which will provide the  same maximum response as that obtained in each mode of vibration.
• Maximum modal responses are combined to find total   maximum response of the structure.
• The first step is the dynamic analysis while , the step is a static analysis.
• The first two steps do not have  approximations, while the third step has some approximations.
• As a result, response spectrum analysis is called an approximate analysis; but applications show that it  provides mostly a good estimate of peak responses.
• Method is developed for single point, single component  excitation for classically damped linear systems. However, with additional approximations it has been extended for multi point-multi component excitations & for non-classically damped systems.

## Equation of motion for MDOF system under single point excitation

• Since both response spectrum & mode shape    properties are required in obtaining , it is known  as modal response spectrum analysis.
• It is evident from above that both the dynamic &    static analyses are involved in the method of analysis as mentioned before.
• As the contributions of responses from different   modes constitute the total response, the total  maximum response is obtained by combining modal quantities.
• This combination is done in an approximate manner since actual dynamic analysis is now replaced by partly dynamic & partly static analysis.

Modal combination rules
Three different types of modal combination rules are  popular

• ABSSUM
• CQC

ABSSUM stands for absolute sum of maximum values  of responses;

The combination rule gives an upper bound to the computed values of the total response for two reasons:
It assumes that modal peak responses occur at the same time.
It ignores the algebraic sign of the response.
Actual time history analysis shows modal peaks occur at different times as shown in Fig. 5.1;further time history of the displacement has peak value at some other time.

Thus, the combination provides a conservative estimate of response.

SRSS combination rule denotes square root of sum of  squares of modal responses

• For structures with well separated frequencies, it provides a good estimate of total peak response.
• When frequencies are not well separated, some errors are introduced due to the degree of correlation of modal responses which is ignored.
• The CQC rule called complete quadratic combination rule takes care of this correlation.
• It is used for structures having closely spaced frequencies:
• Second term is valid for & includes the effect of degree of correlation.
• Due to the second term, the peak response may be estimated less than that of SRSS.
• Various expressions for are available; here only two are given :

• Both SRSS & CQC rules for combining peak modal responses are best derived by assuming earthquake as a stochastic process.
• If the ground motion is assumed as a stationary random  process, then generalized coordinate in  each mode is also a random pr

Application to 2D frames

Application to 3D tall frames

RSA for multi support excitation

Seismic code provisions