Response spectrum analysis can be used to estimate the peak response (displacement, stress, etc.) of a structure to a particular base motion. The method is only approximate, but it is often a useful, inexpensive method for preliminary design studies.

The response spectrum procedure is based on using a subset of the modes of the system, which must first be extracted by using the eigenfrequency extraction procedure. The modes will include eigenmodes and, if activated in the eigenfrequency extraction step, residual modes. The number of modes extracted must be sufficient to model the dynamic response of the system adequately, which is a matter of judgment on your part.

While the response in the response spectrum procedure is for linear vibrations, the prior response may be nonlinear. Initial stress effects (stress stiffening) will be included in the response spectrum analysis if nonlinear geometric effects (“General and linear perturbation procedures,” Section 6.1.2) were included in a general analysis step prior to the eigenfrequency extraction step.

The problem to be solved can be stated as follows: given a set of base motions, (), specified in orthogonal directions defined by direction cosines (), estimate the peak value over all time of the response of any variable in a finite element model that is simultaneously subjected to these multiple base motions. The peak response is first computed independently for each direction of excitation for each natural mode of the system as a function of frequency and damping. These independent responses are then combined to create an estimate of the actual peak response of any variable chosen for output, as a function of frequency and damping.

The acceleration history (base motion) is not given directly in a response spectrum analysis; it must first be converted into a spectrum.

The response spectrum method is based on first finding the exact value of the peak response to each base motion of a one degree of freedom system that has a natural frequency equal to the frequency of interest. The single degree of freedom system is characterized by its undamped natural frequency, , and the fraction of critical damping present in the system, , at each mode . The equations of motion of the system are integrated through time to find peak values of relative displacement, relative velocity, and absolute acceleration for the linear, one degree of freedom system. This process is repeated for all frequency and damping values in the range of interest. Plots of these responses are known as displacement, velocity, and acceleration spectra: , , and . “Analysis of a cantilever subject to earthquake motion,” Section 1.4.13 of the ABAQUS Benchmarks Manual, contains a FORTRAN program that can be used to build a spectrum from an acceleration record in this way. The response spectrum can also be obtained directly from measured data.

You define a response spectrum by giving a table of values of as a function of frequency, , and damping, . For each damping value the magnitude of the response spectrum must be given over the entire range of frequencies needed, in ascending value of frequency. ABAQUS/Standard interpolates linearly between the values given on a log-log scale. Outside the extremes of the frequency range given, the magnitude is assumed to be constant, corresponding to the end value given. (See “Material data definition,” Section 9.1.2, for an explanation of data interpolation.)

Any number of spectra can be defined, and each spectrum must be named. The response spectrum procedure allows up to three spectra to be applied to the model in orthogonal physical directions defined by their direction cosines.

*SPECTRUM, NAME=name

*SPECTRUM, NAME=name, TYPE=DISPLACEMENT 
*SPECTRUM, NAME=name, TYPE=VELOCITY
*SPECTRUM, NAME=name, TYPE=ACCELERATION
*SPECTRUM, NAME=name, TYPE=G, G=g

*SPECTRUM, NAME=name, INPUT=file name

Since the response spectrum procedure uses modal methods to define a model's response, the value of any physical variable is defined from the amplitudes of the modal responses (the “generalized coordinates”), . The first stage in the response spectrum procedure is to estimate the peak values of these modal responses. For mode and spectrum this is

Similar expressions for and can be obtained by substituting velocity or acceleration spectra in the above equation.

The individual peak responses to the excitations in different directions will occur at different times and, therefore, must be combined into an overall peak response. Two combinations must be performed, and both introduce approximations into the results:

Directional summation methods

You choose the method for combining the multidirectional excitations depending on the nature of the excitations.

Directional response for excitation in approximately a single direction in space

If the input spectra in the different directions are components of a base excitation that is approximately in a single direction in space, then for each mode the peak responses in the different spatial directions are summed algebraically by

After this summation is performed, the modal responses are summed. (Choosing the method used for modal summation is described below in “Modal summation methods.”) Since the directional components are summed first, the subscript is not relevant and can be ignored in the modal summation equations that follow.

Input File Usage:	Use the following option to choose the algebraic summation approach:
	*RESPONSE SPECTRUM, COMP=ALGEBRAIC, SUM=sum

ABAQUS/CAE Usage:

Step module: Create Step: Linear perturbation: Response spectrum: Excitations: Single direction

Directional response for multiple independent excitations

If the spectra in different directions represent independent excitations, the modal summation is performed first, as explained below in “Modal summation methods.” Then the responses in different excitation directions are combined by

Input File Usage:	Use the following option to choose the square root of the sum of the squares summation approach:
	*RESPONSE SPECTRUM, COMP=SRSS, SUM=sum

ABAQUS/CAE Usage:

Step module: Create Step: Linear perturbation: Response spectrum: Excitations: Multiple direction square root of the sum of squares

Modal summation methods

The peak response of some physical variable (a component i of displacement, stress, section force, reaction force, etc.) caused by the motion in the th natural mode excited by the given response spectra in direction at frequency with damping is given by

where is the ith component of mode , and there is no sum on . (In the case of algebraic summation the subscript is not relevant and can be ignored in this equation and in those that follow.)

There are several methods for combining these peak physical responses in the individual modes, , into estimates of the total peak response, . These methods are each described below.

The absolute value method

The absolute value method is the most conservative method for combining the modal responses. It is obtained by summing the absolute values resulting from each mode:

This method implies that all of the responses peak simultaneously. It will overpredict the peak response of most systems; therefore, it may be too conservative to help in design.

Input File Usage:

*RESPONSE SPECTRUM, COMP=comp, SUM=ABS

ABAQUS/CAE Usage:

Step module: Create Step: Linear perturbation: Response spectrum: Summations: Absolute values

The square root of the sum of the squares method

The square root of the sum of the squares method is less conservative than the absolute value method. It is also usually more accurate if the natural frequencies of the system are well separated. It uses the square root of the sum of the squares to combine the modal responses:

Input File Usage:

*RESPONSE SPECTRUM, COMP=comp, SUM=SRSS

ABAQUS/CAE Usage:

Step module: Create Step: Linear perturbation: Response spectrum: Summations: Square root of the sum of squares

The Naval Research Laboratory method

The absolute value and square root of the sum of the squares methods can be combined to yield the Naval Research Laboratory method. It distinguishes the mode, , in which the physical variable has its maximum response and adds the square root of the sum of squares of the peak responses in all other modes to the absolute value of the peak response of that mode. This method gives the estimate:

Input File Usage:

*RESPONSE SPECTRUM, COMP=comp, SUM=NRL

ABAQUS/CAE Usage:

Step module: Create Step: Linear perturbation: Response spectrum: Summations: Naval Research Laboratory

The ten-percent method

The ten-percent method, recommended by Regulatory Guide 1.92 (1976) of the U.S. Nuclear Regulatory Commission, improves the estimation for structures with closely spaced frequencies. The ten-percent method modifies the square root of the sum of the squares method by adding a contribution from all pairs of modes and whose frequencies are within 10% of each other, giving the estimate:

The frequencies of modes and are considered to be within 10% of each other whenever

The ten-percent method reduces to the square root of the sum of the squares method if the modes are well separated with no coupling between them.

Input File Usage:

*RESPONSE SPECTRUM, COMP=comp, SUM=TENP

ABAQUS/CAE Usage:

Step module: Create Step: Linear perturbation: Response spectrum: Summations: Ten percent

The complete quadratic combination method

Like the ten-percent method, the complete quadratic combination method improves the estimation for structures with closely spaced eigenvalues. The complete quadratic combination method combines the modal response with the formula

where are cross-correlation coefficients between modes and , which depend on the ratio of frequencies and modal damping between the two modes:

where .

If the modes are well spaced, their cross-correlation coefficient will be small () and the method will give the same results as the square root of the sum of the squares method.

The complete quadratic combination method can be used only with algebraic summation over different spectra applied in different directions because the signs of the modal peak responses play a significant role in this approach. This method is usually recommended for asymmetrical building systems since, in such cases, other methods can underestimate the response in the direction of motion and overestimate the response in the transverse direction.

Input File Usage:

*RESPONSE SPECTRUM, COMP=ALGEBRAIC, SUM=CQC

ABAQUS/CAE Usage:

Step module: Create Step: Linear perturbation: Response spectrum: Summations: Complete quadratic combination

You can select the modes to be used in modal superposition and specify damping values for all selected modes.

*SELECT EIGENMODES, DEFINITION=MODE NUMBERS
*SELECT EIGENMODES, GENERATE, DEFINITION=MODE NUMBERS

*SELECT EIGENMODES, DEFINITION=FREQUENCY RANGE

*MODAL DAMPING, DEFINITION=MODE NUMBERS

*MODAL DAMPING, DEFINITION=FREQUENCY RANGE

*MODAL DAMPING, DEFINITION=FREQUENCY RANGE