Page 82 - Dynamic Loading and Design of Structures

Page 60

Figure 2.27 Dynamic properties of the fixed end simply supported beam.

in the time evolution of the structural response; instead, we are interested in the maximum
values attained by the structure’s relative displacements, relative velocities and absolute
accelerations since those values control the maximum stresses that ultimately develop.
A response spectrum is defined as the maximum response (be it displacement, velocity or
acceleration) of all possible SDOF oscillators, which can be described by their natural
frequency and damping coefficient, to a given ground motion. Note that a response spectrum
is not the same as the DLF for a SDOF oscillator; both, however, can be used in the analysis
of SDOF, MDOF or continuous systems. In Figures 2.28 and 2.29, we respectively present
spectra resulting from artificially generated ground accelerations and the true, triple-scale
response spectrum for the main shock produced by the Kalamata, Greece 1986 earthquake
(Anagnostopoulos et al., 1986).
Response spectra can be classified as either elastic spectra, inelastic spectra, site specific
spectra, code prescribed spectra or as design spectra. Here we focus on the first type, as being
the most relevant to a first exposure in structural dynamics, and because they form the basis
from which the remaining ones can be derived. Specifically, and in order to complete the
presentation, we list the equations of motion of the SDOF oscillator subjected to ground
displacements y (t) and to ground accelerations ÿ (t), respectively, as
s
s

(2.79)

and

(2.80)

where

(2.81)

is the relative displacement between ground and structure.