Response Spectrum

The equation of motion for a single degree of freedom system subject to the \(- \bscalar{m}{\bscalar{\bscalar{\ddot{u}}}_{g}}\) force is as follows:

\[ \begin{equation} \bscalar{\ddot{u}} + 2\bscalar{\omega}\bscalar{\xi}\bscalar{\dot{u}} + \bscalar{\omega}^{2}\bscalar{u} = - {\bscalar{\bscalar{\ddot{u}}}_{g}} \end{equation} \]

Where \(\bscalar{\omega}\) is the circular frequency of the structure in \(\bunit{rad/s}\), and \(\bscalar{\xi}\) is the damping ratio. This equation is solved for many single degree of freedom structures having different periods, each subjected to the same ground acceleration, and each with the same damping ratio. The plot of maximum response \(\bscalar{S_{D}} = \max\left| \bscalar{u(t)}\right|\) against structural frequency is called the response spectrum (Gupta, 1990). Design response spectra are normally presented as a plot of the maximum acceleration against the period (the inverse of the natural frequency of the structure), written as \(\bscalar{S_{A}} = \bscalar{\omega}^{2}\bscalar{S_{D}}\).

Last updated 26-02-2024