Page 152 - Dynamic Loading and Design of Structures

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4.3.2 Elastic spectra

Response spectra
A response spectrum (i.e. a plot of the peak response (to an input motion) as a function of the
natural period) can be derived by analysing a series of Single Degree-Of-Freedom (SDOF)
systems, as explained in Chapter 2 (Section 2.5) and, in more detail, in the literature
(Newmark and Hall, 1982; Gupta, 1990; Clough and Penzien, 1993). The quantities typically
plotted are the spectral pseudo-acceleration Spa, pseudo-velocity Spv, and displacement Sd,
which are interrelated through the familiar expressions

(4.9)

Due to (4.9) the three spectral quantities can be plotted together on a log—log paper (see
Figure 4.10). It should be recalled that Spa and Spv are not the actual response acceleration and
velocity, respectively (see also Section 2.5). Nevertheless, Sp is practically the same as the
a
actual maximum acceleration for reasonable (i.e. not too high) values of damping, while S ,
pv
is nearly the same as the actual velocity except in the very short and the long period range
(Newmark and Hall, 1982). For design purposes, Spa is more useful than the actual response
acceleration, since the former can be used to calculate directly the maximum forces on the
structure, as discussed in Section 4.3.6.
An example of response spectra, referring to the longitudinal (horizontal) component of the
input motion of Figure 4.3, is given in Figure 4.8; for each spectrum five curves are plotted,
corresponding to damping ratios from 0 to 20 per cent. It is first noted that for lower values of
damping the variation of the spectral values with the natural period can be quite abrupt,
whereas for high damping values the spectra become much smoother. An important piece of
information provided by a spectrum is the range of periods for which the response of a
structure is peaking. The Spa curves in Figure 4.8(a) are typical in the sense that the peaks
occur in the short period range, mainly from 0.2 to 0.5 sec; this is a common feature of
motions recorded on rock sites. A second period range around 1 sec also shows some increase
in the amplification, but significantly lower than that in the short period range. However, if
the pseudo-velocity is used as the basis for identifying critical periods, it is seen in Figure
4.8(b) that the most critical range is that between 0.9 and 1.8 sec; the range of periods
between 0.3 and 0.5 sec is also characterized by local peaks, but is less critical than the
previous one. This illustrates an important problem in seismic design (i.e. the selection of the
parameter which best characterizes the damageability of a particular ground motion). Many
designers rely more on S which is a direct measure of the seismic energy input, since for
pv
negligible damping the energy stored in an oscillator with mass m is equal to . On the
other hand, recently developed displacement based design and assessment procedures are
based on the displacement spectrum. Despite the aforementioned trends, all current codes
base their design forces on Spa spectra