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# Uncovering the Mystery of Response Spectrum Analysis - Part II

July 25, 2014 By: Nick Veikos

This is the second in a series of three posts about Response Spectrum Analysis. The first post discusses the overall approach and input curves. In this post, I will talk about obtaining the system response to this input.

Step II - Obtain the response of your system

A modal analysis of the system must first be performed. All of the modes that may participate significantly in the response of the system to the transient loading must be computed. This is vital for computing a proper solution.

The response is computed by treating each mode shape as a single-dof-system and scaling the response for each mode by the value of the response spectrum curve at the corresponding natural frequency. The computation for response, {U}i, for each mode, i, goes something like:

Where,

Si     is the response spectrum value at frequency i

is the participation factor for mode I based on the direction of loading

is the mass normalized mode shape for mode i

So, the response curve is just used along with the participation factor as a scale factor for each mass-normalized mode via a look-up table. Since the total response of the system to the transient load represented by the response spectrum will be a linear combination of all the modal responses, these individually-scaled modes must be combined in some way. Many options are available in the literature, but all are some variant of a “square root of the sum of the squares”, or SRSS, approach:

You could argue that this is somewhat arbitrary, and you’d be right. However, the response spectrum curve does not capture the relative phasing of each peak response, so there is no real alternative. The mode combination stage is where the approximation (and hopefully conservatism) comes into play. There are typically prescribed methods to be used ( Complete Quadratic Combination Method, Grouping Method, Double Sum Method, Square Root of the Sum of the Squares Method, etc.) depending on the regulatory agency involved. If no method is prescribed for you, I would suggest to use all the common ones available and pick the one which yields the peak response.

After the combination is complete, the resulting output will be the estimated peak response to the original transient loading at each location in the model. The response could be displacement, velocity, acceleration, stress, strain, etc. It is important to realize that the distribution of these quantities on the structure will not be physical – the peak values will generally not occur at the same time, so you are not looking at a snapshot in time. Instead, you are looking at a snapshot summary of all the anticipated peak values at each location in the structure during the entire duration of the transient event.

There is a lot more to properly performing a response spectrum analysis than what is outlined here, and there are many books and papers relating to this type of engineering analysis. Two I have found helpful are:

1. R.W. Clough, J. Penzien, Dynamics of Structures, 2nd Edition (Revised), ISBN 0-923907-50-5, Computers and Structures, Inc., Berkeley, California, 2004.

2. Combining Modal Responses and Spatial Components in Seismic Response Analysis, Regulatory Guide 1.92, Revision 2, Nuclear Regulatory Commission, Office of Nuclear Regulatory Research, July 2006.

In my next post, I will talk about some of the common errors and misconceptions related to response spectrum analysis.