# Seismic Analysis and the Shock Response Spectrum Explained

*By Bill Kelly *

Prediction of the response of a structure to an earthquake or a shock requires some detailed study in this specialized field. These events are complex and transient in nature. The loading generally comes up through the foundation of the structure as a seismic excitation. For example, the time history of the ground motion during an earthquake is noisy and random with a range of frequency content.

With powerful computers, a full blown time domain transient analysis can be performed to simulate the entire event and its aftermath. However, the most common analysis method for these types of problems is a *Response*** Spectrum Analysis (RSA)**. In this case, the solution is done in the frequency domain which greatly reduces computational time, but requires an understanding of the modal behavior.

*5 Modes Contributing to Seismic Response*

I’d like to give a general description of the key aspects of such an analysis because it is not well understood, even by experienced structural analysts who may be more accustomed to static, modal, or single-frequency harmonic forced response stress analyses. We’ll discuss it in the context of an earthquake, although the same methods apply to general shock analysis.

In RSA, the key input is the ** Seismic or Shock Response Spectrum (SRS). ** A common misconception is to assume that this frequency-based curve represents the response of the ground surface to the earthquake, and that this excitation should be applied to the base of the structure. In reality, it’s the peak response of a single degree of freedom (SDOF) system to the earthquake. So if your structure is just a simple mass on a spring, then the SRS is basically the answer you’re looking for.

In reality, every structure is more complex than that, and that’s why you need a finite element analysis. The FEA first solves a modal analysis of the structure, which then reduces the complex structure into an equivalent set of SDOF systems or modes. Then, referring to the SRS, the code determines the peak response for each SDOF system to the earthquake. It then has to estimate what the total response is by somehow combining all those modes together. The most common method is to use a square root sum of squares (SRSS) combination. Just adding them together would be too conservative since it’s unlikely that the peaks of the sinusoidal responses of all the modes would line up. Of course there are other details such as damping, modal mass, and participation factors, but it all starts with a basic understanding of the RSA and SRS, which hopefully this blog provided.

The following animation is a time transient solution of an electronics system experiencing a standardized shock input. It could have also been analyzed with RSA although the RSA method does not lend itself to animations due to the SRSS step.