Response Spectrum Analysis

The equilibrium equation for a structure performing free vibration appears as the eigenvalue problem:(1) $\left(K-\lambda M\right)A=0$

Where,

$K$

Stiffness matrix of the structure

$M$

Mass matrix. Damping is neglected.

It is assumed each individual mode behaves like a single degree-of-freedom system. The transient response at a degree of freedom is:(2) $$u_k={\displaystyle \sum _i{A}_{ik}{\psi }_i\chi }$$

Where,

$A$

Eigenvector

$\psi$

Modal participation factor

$\chi$

Response spectrum

For loading due to base acceleration, the modal participation factor can be expressed as:(3) ${\psi }_i=A_i^{T}MT$

Where,

$A$

Eigenvector

$M$

Mass matrix

$T$

Rigid body motion due to excitation

In ABS modal combination, the peak response is estimated by:(4) $$u_k={\displaystyle \sum _i\left|A_{ik}\right|\left|{\psi }_i\chi \right|}$$

In CQC modal combination, the peak response is estimated by:(5) $$u_k=\sqrt{{\displaystyle \sum _m{\displaystyle \sum _n{v}_m{\rho }_{mn}{v}_n}}}$$

Where,

$$v_m$$

Modal response associated with mode $m$

$${\rho }_{mn}$$

Cross-modal coefficient

The cross modal coefficient $${\rho }_{mn}$$ between modes $m$ and $n$ is calculated as:(6) $${\rho }_{mn}=\frac{8\sqrt{{\xi }_m{\xi }_n}\left({\xi }_m+r_{nm}{\xi }_n\right)r_{nm}^{1.5}}{{\left(1-r_{nm}^2\right)}^2+4{\xi }_m{\xi }_n{r}_{nm}\left(1+r_{nm}^2\right)+4\left({\xi }_m^2+{\xi }_n^2\right)r_{nm}^2}$$

Where,

$r_{nm}=\frac{{\lambda }_n}{{\lambda }_m}$

Ratio of eigenvalues of the modes

$${\xi }_m\text{ and }{\xi }_n$$

Modal damping values of the two modes.

In SRSS modal combination, the peak response is estimated by:(7) $$u_k=\sqrt{{\displaystyle \sum _i{\left(A_{ik}{\psi }_i\chi \right)}^2}}$$

The NRL method combines ABS and SRSS methods. It adds the maximum modal response by ABS method and the rest of the modes by SRSS method. The peak response is estimated by:(8) $$u_k=\left|A_{ik}\right|\left|{\psi }_i\chi \right|+\sqrt{{\displaystyle \sum _{j\ne i}{\left(A_{jk}{\psi }_j\chi \right)}^2}}$$

The following data selectors are recognized for an RSA subcase definition.

METHOD

References an eigenvalue extraction Bulk Data Entry definition (EIGRL). Only METHOD(STRUCTURE) is supported. This reference is required.

RSPEC

References an RSPEC Bulk Data Entry where the combination rules, excitation DOF, and the input spectra are identified. This reference is required.

SDAMPING

References damping table Bulk Data Entries (TABDMP1 or TABDMP2) to specify modal damping. This reference is required.

SPC

References single point constraint Bulk Data Entries (SPCADD, SPC and SPC1). For RSA analysis, these entries define the base degrees of freedom where excitation is applied.

MPC

References multi-point constraint Bulk Data Entries (MPCADD or MPC).

Bulk Data Entries which have particular significance for RSA include:

RSPEC

Specifies combination rules, excitation DOF, and references the input spectra.

DTI,SPECSEL

Defines response spectra.

EIGRL

Defines parameters for eigenvalue extraction.

PARAM, LFREQ and PARAM, HFREQ

Defines the range of modes used in modal combinations.

TABDMP1

Specifies modal damping as a function of frequency.

TABDMP2

Specifies modal damping as a function of a range of mode indices.

SPC, SPC1, and SPCADD

Specifies base where excitation is applied and other constraints.

SUBCASE 100
	RSPEC = 2
	SPC = 5
	SDAMPING = 12
	METHOD = 24
$
BEGIN BULK
$
PARAM, LFREQ, 0.1
PARAM, HFREQ, 1000.
EIGRL, 24, 0.0, 1000.
RSPEC, 2, ABS, CQC, 0.1
, 99, 2.0, 1.0, 0.0, 0.0
DTI, SPECSEL, 99, , A, 2, 0., 3, 0.02,
, 4, 0.04, ENDREC
TABDMP1, 12, …
TABLED1, 2
+,…
TABLED1, 3
+,…
TABLED1, 4
+,…
ENDDATA
$