# # Response Spectrum Analysis

## # Response Spectrum

The response of a single degree of freedom system mode (frequency

Modal analysis reduces a complex structure to an equivalent system of
single degree of freedom oscillators so this can be applied to the
structure as a whole for any selected mode. The response in a given mode

Where

is the modal multiplier.

For global x and y we use

Where x and y corresponds to a rigid body displacement in the respective
direction. So the rigid body vector at

And the orthogonal direction

This means that for a rotated excitation direction we just need to rotate the participation factors and we don’t need to transform the displacements, etc.

or

That leaves the only transformation we need being the transformation of global displacements to local for nodes in constraint axes. For these we want to transform modal results from global to local, do the combination and transform combined value from local to global.

The modal responses are then combined using one of several combination methods.

## # Combinations

The main combination methods are:

**ABSSUM**

**SRSS**

**CQC**

where

where

If the damping is constant this simplifies to

**Rosenbluth**

where

**CQC3**

In SRSS method, the spectra

Menun and Der Kiureghian

The peak response value can be estimated using the fundamental CQC3 equation

where

and

Normally, the value of

And the critical response becomes

If the value of

There is no specific guidelines available to choose the value of

## # Storey Inertia Forces

The storey inertia forces can be calculated from the storey mass, m, and
inertia, I_{zz}, response spectrum and the modal results. The
storey modal translations (u_{x},u_{y})and rotations
(θz) are calculated (see below)

The force and moment for excitation in the i^{th} direction are
then determined from

Where

*Wilson, der Kiureghian & Bayo, 'Earthquake Engineering and Structural Dynamics', Vol 9, pp 187-194 (1981)*

*ASCE 4-09 Seismic analysis of safety related nuclear structures and commentary', Chapter 4.0 Analysis of Structures (2009)*

*Menun, C., and A. Der Kiureghian. 1998. “A Replacement for the 30 % Rule for Multicomponent Excitation,” Earthquake Spectra. Vol. 13, Number 1. February.*