Solvers Background: Spectral Response

Description

The Spectral Response solver calculates the response of a structure subjected to a random dynamic loading for which a frequency response function is available (Response Spectrum analysis). It also calculates the response of a structure subjected to a random dynamic loading described in the form of a PSD (Power Spectral Density) function.

Random Dynamic Loading

For Response Spectrum analysis, the Factor vs Frequency table describes the response of a single degree of freedom system, of given frequency (the horizontal axis in the table), subject to the random event.
For Power Spectral Density (PSD), the Factor vs Frequency table describes the input of the random event in terms of its frequency content (the horizontal axis in the table).

Types of Random Dynamic Loading

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 restrained degrees of freedom. The excitation may act in any direction in the global XYZ 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. Strand7 does not support rocking seismic excitation of the base, nor multiple support excitation.
The load spectrum simulates a random dynamic loading applied to the structure. Typical applications include the analysis of structures loaded with random wind loads, ocean wave loads and machinery vibration. All nodal, element and gravity loads are included in this type of load input, except for thermal loads and pre-loads, which are excluded. The effects of thermal loads and pre-loads are considered in terms of stress stiffening effects by the inclusion of initial conditions in the natural frequency analysis, but they are not considered as spectral loads. Any number of load cases may be used to define the loading condition.

Procedure

The Spectral Response solver is based on the mode superposition method and executes the following steps:

Calculates and assembles the element mass matrix if base excitation loading is applied, otherwise calculates and assembles the element load and nodal load vectors.
Calculates the modal excitation factors for each vibration mode. Vibration mode vectors from the natural frequency analysis are used. For seismic loading, mode participation factors and mass participation ratios are calculated.
Determines the spectral values for all modes from the assigned spectral table by using the corresponding frequency value.
Calculates the modal displacement magnitudes.
Calculates modal element stresses and strains, nodal reactions, etc.
Calculates maximum responses using the CQC (Complete Quadratic Combination) and/or the SRSS (Square Root of the Sum of the Squares) methods.

Notes

Spectral response analysis is based on the mode superposition method. The choice of vibration modes used in the analysis has a significant effect on the accuracy of the results.
The analysis relies on natural frequency results, therefore it is important to use the correct natural frequency solution file.
The same natural frequency result file can be used to run the Spectral Response solver multiple times with different settings.
The constant terms for enforced displacements, multi-point and shrink links are ignored by the Spectral Response solver. These are treated as fixed restraints.
The effect of material temperature dependence (see Special Topics: Temperature Dependence) on stiffness is included indirectly via its inclusion in the natural frequency analysis.
The combined results of a spectral response analysis (i.e., the CQC and SRSS results) are given as envelopes of maximum values of nodal displacements, element stresses, element strains, recovered reactions at constrained nodes and elastic or inertia forces at unconstrained nodes.
Under normal conditions, both the CQC and SRSS methods will produce results of acceptable accuracy. However, as all modal combination methods are approximate, there can be no absolute assurance that the combined results will be conservative.
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 and to provide a pseudo equilibrated set of results, the Autosign function applies the sign of the mode with the most significant contribution to the combined result.
- Secondly, the calculated maximum response of all the actions on each structural member or node may correspond to a different point in time (i.e., they are not all occurring at the same time ). Thus member and nodal equilibrium cannot be checked.