# Strand7 spectral response solver

The spectral response solver calculates the response of a structure subjected to a random dynamic loading.

Two types of spectrum input (spectral curves) can be used: Response Spectrum and Power Spectral Density (PSD). In Strand7, a spectral curve can be defined as a function of either frequency or period.

In spectral response analysis, two types of random dynamic loads can be applied: earthquake (seismic) base excitation and general dynamic load.

The base excitation spectrum is applied as a translational excitation at the base, equally at all fixed degrees of freedom. The excitation may act in any arbitrary direction in the global X-Y-Z system and can be defined in terms of acceleration, velocity or displacement. Typical input spectra include those based on a particular earthquake or an averaged design spectrum given in the design codes.

The spectral response solver is based on the mode superposition technique and performs the following steps:

• If the loading is base excitation, the element mass matrix is assembled otherwise, the applied load vector is formed. Either a consistent or a lumped mass matrix can be used depending on the method used in the natural frequency analysis. Element equivalent load vectors may be either consistent or lumped according to the option setting.

• The modal excitation factors for each vibration mode are calculated. Vibration mode vectors from the natural frequency analysis are used. For seismic loading, mass participation factors are calculated.

• Determines the spectral values for all modes from the assigned spectral table by using the corresponding frequency value.

• Evaluates modal damping if Rayleigh damping is applied.

• Calculates the modal displacement magnitudes.

• Calculates maximum responses using both the CQC (Complete Quadratic Combination) and SRSS (Square Root of the Sum of the Squares) methods.

### Notes

1. Spectral response analysis is based on the mode superposition technique. The choice of the vibration modes used in the analysis has significant effect on the accuracy of the results.

2. The enforced constant terms for enforced displacements and shrink links are ignored in the spectral response solver.

3. The effect of material temperature dependency on stiffness is included indirectly via its inclusion in the natural frequency analysis.

4. The results of a spectral analysis are given as envelopes of maximum values of nodal displacements, element stresses, element strains, recovered reactions at constrained nodes and elastic forces at unconstrained nodes. The maximum response values are calculated by combining the maximum response of all modes included in the analysis. Contributions from individual modes are available as well as the combined maximum values.

5. It is important to recognize that all modal combination methods are approximate and generally, there can be no absolute assurance that the combined results are conservative. However, under normal conditions, the above two methods (CQC and SRSS) will produce results of acceptable accuracy.

6. Because of the way the maximum response is evaluated, the results have the following features:

Firstly, all computed terms are positive. To help with the visualisation of results, the Autosign option will apply the sign of the most significant mode, to the combined result. Usually this approach generates deformed displays that are plausible.

Secondly, the calculated response of each structural member may correspond to a different point in time. Thus member and nodal equilibrium cannot be checked; moments, shear forces, and deformations at points between the nodes in the model cannot be directly calculated. These points should be remembered when looking at the results.

For more information on spectral response analysis, see Strand7 Webnotes - Linear / Dynamics.