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Response Spectrum Analysis

Response Spectrum

The response of a single degree of freedom system mode (frequency ff, spectral acceleration aspectrala_{spectral}) is

qi=aspectral(2πf)2q_{i} = \frac{a_{spectral}}{(2\pi f)^{2}}

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 ii in direction jj is

xji=Γijaspectral(2πfi)2ϕi\mathbf{x}_{ji} = \Gamma_{ij}\frac{a_{spectral}}{(2\pi f_{i})^{2}}\phi_{i}

Where Γij\Gamma_{ij} is the participation factor to account for the direction of excitation. The term

Γjiaspectral(2πfi)2\Gamma_{ji}\frac{a_{spectral}}{(2\pi f_{i})^{2}}

is the modal multiplier.

For global x and y we use Γx\Gamma_{x} & Γy\Gamma_{y}. So for excitation at an angle α we want to use Γα\Gamma_{\alpha}. Going back to the definition of the participation factor in x and y directions:

Γx=ϕTMxm^\Gamma_{x} = \frac{\phi^{T}\mathbf{Mx}}{\widehat{m}}
Γy=ϕTMym^\Gamma_{y} = \frac{\phi^{T}\mathbf{My}}{\widehat{m}}

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

rα=xcosα+ysinα\mathbf{r}_{\alpha} = \mathbf{x}\cos\alpha + \mathbf{y}\sin\alpha

And the orthogonal direction α\alpha' would have a rigid body vector

rα=xsinα+ycosα\mathbf{r}_{\alpha'} = - \mathbf{x}\sin\alpha + \mathbf{y}\cos\alpha

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.

Γα=ϕTMxm^cosα+ϕTMym^sinα\Gamma_{\alpha} = \frac{\phi^{T}\mathbf{Mx}}{\widehat{m}}\cos\alpha + \frac{\phi^{T}\mathbf{My}}{\widehat{m}}\sin\alpha
Γα=ϕTMxm^sinα+ϕTMym^cosα\Gamma_{\alpha'} = - \frac{\phi^{T}\mathbf{Mx}}{\widehat{m}}\sin\alpha + \frac{\phi^{T}\mathbf{My}}{\widehat{m}}\cos\alpha


Γα=Γxcosα+Γysinα\Gamma_{\alpha} = \Gamma_{x}\cos\alpha + \Gamma_{y}\sin\alpha
Γα=Γxsinα+Γycosα\Gamma_{\alpha'} = - \Gamma_{x}\sin\alpha + \Gamma_{y}\cos\alpha

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.


The main combination methods are:


x=ixix = \sum_{i}^{}\left| x_{i} \right|


x=ixi2x = \sqrt{\sum_{i}^{}{x_{i}}^{2}}


x=ijxiρijxjx = \sqrt{\sum_{i}^{}{\sum_{j}^{}{x_{i} \cdot \rho_{ij} \cdot x_{j}}}}


ρij=8ζiζj(ζi+βijζj)βij32(1βij2)2+4ζiζjβij(1+βij2)+4(ζi2+ζj2)βij2\rho_{ij} = \frac{8\sqrt{\zeta_{i}\zeta_{j}}\left( \zeta_{i} + \beta_{ij}\zeta_{j} \right){\beta_{ij}}^{\frac{3}{2}}}{\left( 1 - {\beta_{ij}}^{2} \right)^{2} + 4\zeta_{i}\zeta_{j}\beta_{ij}\left( 1 + {\beta_{ij}}^{2} \right) + 4\left( {\zeta_{i}}^{2} + {\zeta_{j}}^{2} \right){\beta_{ij}}^{2}}

where ζi\zeta_{i} and ζj\zeta_{j} are the damping associated with frequencies fif_{i} and fjf_{j} respectively.

βij=fifjfjfi\beta_{ij} = \frac{f_{i}}{f_{j}}\quad f_{j} \geq f_{i}

If the damping is constant this simplifies to

ρij=8ζ(1+βij)βij32(1βij2)2+4ζβij(1+βij)2\rho_{ij} = \frac{8\zeta\left( 1 + \beta_{ij} \right){\beta_{ij}}^{\frac{3}{2}}}{\left( 1 - {\beta_{ij}}^{2} \right)^{2} + 4\zeta\beta_{ij}\left( 1 + \beta_{ij} \right)^{2}}


x=ijxiρijxjx = \sqrt{\sum_{i}^{}{\sum_{j}^{}{x_{i} \cdot \rho_{ij} \cdot x_{j}}}}


ρij=2ζiζj(ζi+ζj)1+[(fifj)(ζifi+ζjfj)]2\rho_{ij} = \frac{\frac{2\sqrt{\zeta_{i}\zeta_{j}}}{\left( \zeta_{i} + \zeta_{j} \right)}}{1 + \left\lbrack \frac{\left( f_{i} - f_{j} \right)}{\left( \zeta_{i}f_{i} + \zeta_{j}f_{j} \right)} \right\rbrack^{2}}


In SRSS method, the spectra sx,sys_{x},s_{y} are applied to 100% on the principal directions. The responses obtained from SRSS combination has equal contributions from all the directions. However, in practice the same ground motion will not occurs in both the direction. Therefore, SRSS yields conservative results.

Menun and Der Kiureghian4^4 (1998) presented the CQC3 combination method for combination of the orthogonal spectrum. Let assume sx,sys_{x},s_{y} are the major and minor spectra applied at an arbitrary angle θ\theta from the structural axis. To simplify the analysis further assume the sys_{y} spectra is some fraction of sxs_{x} spectra.

sy=a×sxs_{y} = a \times s_{x}


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

Q=[Qx2+a2Qy2(1a2)(Qx2Qy2)sin2θ+2(1a2)Qxysinθcosθ+Qz2]12Q = \left\lbrack {Q_{x}}^{2} + a_{2}{Q_{y}}^{2} - \left( 1 - a^{2} \right)\left( {Q_{x}}^{2} - {Q_{y}}^{2} \right)\sin^{2}\theta + 2\left( 1 - a^{2} \right)Q_{xy}\sin\theta\cos\theta + {Q_{z}}^{2} \right\rbrack^{\frac{1}{2}}


Qx2=ijqxiρijqxj{Q_{x}}^{2} = \sum_{i}^{}{\sum_{j}^{}{q_{xi}\rho_{ij}q_{xj}}}
Qy2=ijqyiρijqyj{Q_{y}}^{2} = \sum_{i}^{}{\sum_{j}^{}{q_{yi}\rho_{ij}q_{yj}}}
Qxy=ijqxiρijqyjQ_{xy} = \sum_{i}^{}{\sum_{j}^{}{q_{xi}\rho_{ij}q_{yj}}}
Qz2=ijqziρijqzj{Q_{z}}^{2} = \sum_{i}^{}{\sum_{j}^{}{q_{zi}\rho_{ij}q_{zj}}}

and qxi,qyiq_{xi},q_{yi} are the modal quantities produced by spectrums applied at x and y directions, qziq_{zi} is the modal value produced by the vertical spectrum and θ\theta is the arbitrary angle at which the lateral spectra is applied.

Normally, the value of θ\theta is not known. The critical angle that produces maximum response can be calculated using

θcr=12tan1(2QxyQx2+Qy2)\theta_{cr} = \frac{1}{2}\tan^{- 1}\left( \frac{2Q_{xy}}{{Q_{x}}^{2} + {Q_{y}}^{2}} \right)

And the critical response becomes

Q=[Qx2+a2Qy2(1a2)(Qx2Qy2)sin2θcr+2(1a2)Qxysinθcrcosθcr+Qz2]12Q = \left\lbrack {Q_{x}}^{2} + a_{2}{Q_{y}}^{2} - \left( 1 - a^{2} \right)\left( {Q_{x}}^{2} - {Q_{y}}^{2} \right)\sin^{2}\theta_{cr} + 2\left( 1 - a^{2} \right)Q_{xy}\sin\theta_{cr}\cos\theta_{cr} + {Q_{z}}^{2} \right\rbrack^{\frac{1}{2}}

If the value of a=1a = 1, CQC3 combination reduces to SRSS combination. The peak response value is not dependent on the θ\theta and the peak response can be estimated using.


There is no specific guidelines available to choose the value of aa. Menun and Der Kiureghian presented an example for CQC3 combination with a value ranging from 0.50 to 0.85.

Storey Inertia Forces

The storey inertia forces can be calculated from the storey mass, m, and inertia, Izz, response spectrum and the modal results. The storey modal translations (ux,uy)and rotations (θz) are calculated (see below)

The force and moment for excitation in the ith direction are then determined from

fx=m.scodeCQC(Γiaspecux)fy=m.scodeCQC(Γiaspecuy)Mz=Izz.scodeCQC(Γiaspecθz)\begin{aligned}f_{x} &= m.s_{code}\sum_{CQC}^{}\left( \Gamma_{i}a_{spec}u_{x} \right)\\ f_{y} &= m.s_{code}\sum_{CQC}^{}\left( \Gamma_{i}a_{spec}u_{y} \right)\\ M_{z} &= I_{zz}.s_{code}\sum_{CQC}^{}\left( \Gamma_{i}a_{spec}\theta_{z} \right)\end{aligned}

Where scodes_{code} is the code scaling factor, aspeca_{spec}is the spectral acceleration and Γi\Gamma_{i} the participation factor.

2^2 Wilson, der Kiureghian & Bayo, 'Earthquake Engineering and Structural Dynamics', Vol 9, pp 187-194 (1981)

3^3 ASCE 4-09 Seismic analysis of safety related nuclear structures and commentary', Chapter 4.0 Analysis of Structures (2009)

4^4 Menun, C., and A. Der Kiureghian. 1998. “A Replacement for the 30 % Rule for Multicomponent Excitation,” Earthquake Spectra. Vol. 13, Number 1. February.