How to Perform a Modal Response Spectrum Analysis

The complete workflow for modal response spectrum analysis — from model preparation and modal extraction through spectral acceleration lookup, modal combination and directional combination to recovered structural response.

Seismic15 min read

seismicmodal analysisresponse spectrumSRSSCQCmodal combination

When to use this guide

Use this guide when you need to perform a seismic response spectrum analysis on a structure or equipment model from start to finish. This is the workhorse method of seismic structural assessment: it captures the dominant dynamic behaviour through modal decomposition and uses a response spectrum to represent the ground or floor motion. The method is appropriate when the structure remains essentially linear elastic under the seismic event, which is the assumption underlying most equipment qualification and many building code procedures. If significant nonlinearity is expected — yielding, contact opening, gap closure — a nonlinear time-history analysis is required instead.

Required inputs

Structural or equipment finite element model with mass correctly distributed (lumped or consistent) and stiffness representing the in-service condition
Boundary conditions representing the support: base fixity for a building model, floor or frame attachment for equipment
Response spectrum input: ground response spectrum (GRS) for a primary structure, or floor response spectrum (FRS) for equipment, at the appropriate damping ratio
Damping ratio for the analysis — structural damping for the building, equipment damping for the equipment model
Direction information: which global axes correspond to the horizontal and vertical spectral inputs
Governing code or specification defining the modal combination method (SRSS or CQC), directional combination rule (100-40-40 or SRSS), and mass participation threshold

Step-by-step method

Prepare the model: confirm mass representation includes all significant mass — structural self-weight, equipment, contents, non-structural mass. Verify the mass source and that mass has been assigned to the correct degrees of freedom
Apply boundary conditions: fix the base or attach to the support structure. Ensure the model has no unconstrained rigid-body modes that would contaminate the modal extraction
Run a real eigenvalue (modal) analysis. Extract sufficient modes to meet the mass participation criterion — typically 90 percent cumulative effective modal mass in each of the three translational directions. For models with local modes, this may require many modes; assess whether local modes are physical or numerical
Review the modal results: tabulate natural frequency, effective modal mass per direction, and mode shape description. Identify the dominant modes in each direction and any closely spaced mode pairs
For each mode and each direction of excitation, compute the spectral acceleration Sa,i by entering the response spectrum at the modal frequency. Interpolate in log-log space if between tabulated points
Compute the modal response for each quantity of interest. For displacement at node j in mode i: ui,j = phi_i,j * Gamma_i * Sa_i / omega_i^2, where phi is the mode shape, Gamma the participation factor, and omega the circular frequency. For forces and stresses, the analogous modal contribution is computed element by element
Combine modal responses per direction: apply SRSS if all modal frequency pairs are separated by more than 10 percent, otherwise use CQC. The combination is performed on each scalar response quantity separately — displacement, force component, stress component
Combine directional responses: for three-directional excitation, apply the 100-40-40 rule (100 percent of one direction plus 40 percent of each orthogonal direction, with the governing direction cycled) or SRSS of directions per the governing specification. Three combinations are produced, one with each direction governing
Apply the missing-mass correction if the cumulative effective modal mass is below the threshold. The residual rigid response is computed from the zero-period acceleration and the missing mass fraction
Envelope the three directional combinations to produce the final seismic response for each quantity of interest
Recover element forces, stresses and reactions from the combined and enveloped results

Choosing the modal combination method

The choice between SRSS and CQC depends on the modal frequency spacing. SRSS assumes modal responses are statistically independent, which is valid when modal frequencies are well separated. When two modes have frequencies within approximately 10 percent of each other, their responses are correlated and SRSS underestimates the combined response. CQC accounts for this correlation through a coupling coefficient derived from the modal frequencies and damping. As a practical rule, use CQC as the default — it reduces to SRSS for well-separated modes and is always at least as accurate. Most commercial solvers now offer CQC as standard.

CQC is the safer default. The computational cost is negligible and it eliminates the risk of underestimating response for closely spaced modes that were not identified during the frequency spacing check.

Checks

Cumulative effective modal mass ≥ 90 percent in each translational direction (or apply missing-mass correction)
No rigid-body modes present in the extracted modal set — these indicate a boundary condition problem
Modal frequencies fall within the frequency range of the response spectrum. Modes above the ZPA cut-off contribute only rigid response
Closely spaced modes identified and CQC used if any pair is within 10 percent
Directional combination applied after modal combination, not before
For stress recovery, each stress component is combined separately before computing equivalent (von Mises) stress. Do not combine von Mises values from individual modes
Reaction forces from the combined result are physically plausible — check against a simplified static equivalent

Interpretation

The output of a modal response spectrum analysis is a set of peak probable response quantities: peak displacement at each node, peak force in each element, peak stress at each location. These peaks are statistical estimates — the SRSS or CQC combination gives the expected value of the peak response under the assumption that modal peaks do not occur simultaneously. The recovered values do not represent a simultaneous structural state. For margin assessment, compare the recovered peak stresses against allowables under the governing load combination. For deflection-sensitive equipment, compare peak displacement against the functional clearance. The inherent conservatism of the method — typically 5–15 percent above a time-history mean — is usually acceptable for qualification but should be acknowledged when comparing with test results.

Common mistakes

Insufficient modes extracted — cumulative mass participation below 90 percent with no missing-mass correction applied
Using SRSS when closely spaced modes are present. This underestimates response and is a common cause of failed qualification that passes on re-analysis with CQC
Combining von Mises stresses across modes instead of combining stress components and then computing von Mises. This overestimates equivalent stress
Applying directional combination before modal combination — the statistical basis is different and the result is incorrect
Using a response spectrum at the wrong damping ratio without scaling
Treating the recovered displacements as a simultaneous deflected shape for clearance checking. Each displacement is a peak; the peaks do not occur at the same time

The most consequential error in modal response spectrum analysis is insufficient modal extraction. Always check the cumulative mass participation before accepting any result.