Dynamic analysis and response spectrum

A multi-storey building vibrates in several natural modes, each with its own period and deflected shape. The first mode is usually a simple sway with all floors moving the same way; higher modes have one or more reversals up the height. Modal analysis finds these periods and shapes from the mass and stiffness matrices, and the total response to an earthquake can be built from the responses of the individual modes.

The response spectrum method uses a design spectrum, a curve of spectral acceleration against period that envelopes the peak responses of single-degree systems to many possible ground motions, adjusted for soil and damping. For each mode, the spectral acceleration at its period, times its effective modal mass, gives that mode's base shear. Because the modal peaks do not occur at the same instant, they are combined statistically rather than added.

The square root of the sum of squares (SRSS) works when modal periods are well separated. When periods are close, as in torsionally coupled buildings, the complete quadratic combination (CQC) accounts for correlation between modes. Enough modes must be included to capture most of the building's mass; IS 1893 (Part 1) sets a minimum participating mass for this. Codes also compare the dynamic base shear with the equivalent static value and scale it up if it is lower.

For tall, irregular or important structures, response spectrum analysis is the standard approach. Nonlinear time-history analysis goes further, applying recorded or simulated ground motions and following yielding step by step, and is used for performance-based design and for base-isolated buildings.