Response Spectrum

The response spectrum provides the maximum response (in terms of relative displacement, relative velocity or absolute acceleration) of an harmonic single-degree-of-freedom (dof) oscillator, subject to an arbitrary accelerogram, as a function of the structural period T_n and of thedamping ratio ξ (usually a standard value equal to 5% of the critical damping is used, generally applicable to structures). The maximum amplitude of the response is obtained by integrating the equation of motion of the harmonic oscillator:

where: y(t) is the relative displacement of the oscillator with respect to the ground, ä(t)is the absolute acceleration of the oscillator, and ωn is the oscillator natural circular frequency

The following definitions are introduced:

displacement spectrum (relative)

velocity spectrum (relative)

acceleration spectrum (absolute)

The pseudo-acceleration and pseudo-velocity spectra are also widely used in practice. They are defined as a function of the displacement spectrum as follows:

pseudo-acceleration spectrum:

pseudo-velocity spectrum:

The meaning of the acceleration response spectrum and the procedure for computing the spectral ordinates for selected periods on one of the ITACA records is illustrated in the following figure:

Example of construction of the acceleration response spectrum for the Gemona accelerogram (EW component, bottom) recorded during the Friuli earthquake, 15/09/1976, 03:15.

where: y(t) is the relative displacement of the oscillator with respect to the ground, ä(t)is the absolute acceleration of the oscillator, and __n is the oscillator natural circular frequency

The following definitions are introduced:

displacement spectrum (relative)

velocity spectrum (relative)

acceleration spectrum (absolute)

The pseudo-acceleration and pseudo-velocity spectra are also widely used in practice. They are defined as a function of the displacement spectrum as follows:

pseudo-acceleration spectrum:

pseudo-velocity spectrum:

The meaning of the acceleration response spectrum and the procedure for computing the spectral ordinates for selected periods on one of the ITACA records is illustrated in the following figure:

Example of construction of the acceleration response spectrum for the Gemona accelerogram (EW component, bottom) recorded during the Friuli earthquake, 15/09/1976, 03:15.