Advances in Seismic Design Methodologies - Earthquake Engineering for Structural Design - page 405

Advances in Seismic Design Methodologies 405

form the basis of seismic design forces in the majority of seismic codes. Response
spectrum represents the maximum response of a single degree of freedom (SDOF)
system to a given input motion, as a function of natural frequency and damping.
Determining the structural response requires that the ground motions be translated
into forces acting on the building. The forces have their origin in the fast change of
actions during a short time. These actions present dynamic characteristics, because the
variations are sufficiently rapid to involve inertial forces. Therefore, the dynamic
analysis must evaluate the structural response by taking into account these forces. The
general method for determining the seismic forces to apply to a structure is based on a
simple equation F = ma , where m is the mass of the building and a is the acceleration
of the structure. An elastic response spectrum is a simple plot of the peak response in
terms of acceleration, velocity or displacement of a series of frequency which are
forced into motion by the same base vibration. The resulting plot can then be used to
pick off the response of any linear system, its natural frequency of oscillation being
given. If the input used in calculating a response spectrum is steady-state periodic, then
the steady-state result is recorded.
After digitizing an accelerogram of a particular earthquake (Fig. 9.12a) and
assuming a numerical value for damping, the response of a SDOF elastic system can
be calculated. The dynamic motion is applied at the base of a cantilever having
different natural periods, which models the case of a structure restrained at the ground
level (Fig. 9.12b). The complete history of response for this elastic system can be
computed. The maximum values of accelerations, velocities and displacements are
determined from the obtained system response. By repeating the above process for a
great number of cantilever systems, for a given value of damping, a plot of the
response spectrum can be obtained (Fig. 9.12c) in function of system periods. One can
see that, due to the resonance effects, the spectra have the tendency to amplify the
response in the range of low periods.
Examining the resulting plot of response, one can see some amplification due to the
resonance effects, when the frequency of input excitation corresponds to the frequency
of the system. The main scope in constructing the spectra is to emphasize these
amplifications.
Separate plots for accelerations, velocities and displacements can be obtained by
using the same procedure but the most common formats for spectra are (Fig. 9.13):
- Spectral acceleration vs. period (frequency).

- Spectral displacement vs. period (frequency).