the ratio X /X roughly equals 1/(2£). The peak displacement at resonance is thus very sensitive to damping, and is infinite for the theoretical case of zero damping. For a more realistic damping ratio of 0.05, the displacement of the structure is around ten times the ground displacement.

This illustrates the key principles of dynamic response, but it is worth noting here that the dynamic amplifications observed under real earthquake loading are rather lower than those discussed above, both because an earthquake time-history is not a simple sinusoid, and because it has a finite (usually quite short) duration.

3.3 Response spectra and their application to linear structural systems e no go on to consider the linear response of structures to realistic earthquake time-histories. An earthquake can be measured and represented as the variation of ground acceleration ith tie in three orthogonal directions ^i-S, E-W and vertical). An example, recorded during the 1940 El Centro earthquake in California, is shown in Figure 3.6. Obviously, the exact nature of an earthquake time-history is unknown in advance, will be different for every earthquake, and indeed ill vary over the affected region due to factors such as local ground conditions, epicentral distance etc.

The time-domain response to an earthquake ground motion can be determined by a variety of techniques, all of which are quite mathematically complex. For example, in the Duhamel's integral approach, the earthquake

Figure 3.6 Accelerogram for 1940 El Centro earthquake (N-S component).

Figure 3.6 Accelerogram for 1940 El Centro earthquake (N-S component).

record is treated as a sequence of short impulses, and the time-varying responses to each impulse are summed to give the total response.

Although the method of evaluation is rather complex, the behaviour under a general dynamic load can be quite easily understood by comparison with the single-frequency, sinusoidal load case discussed in Section 3.2.4. In that case, we saw that large dynamic amplifications occur if the loading period is close to the natural period of the structure. Irregular dynamic loading can be thought of as having many different components at different periods. Often the structure's natural period will lie within the band of periods contained in the loading. The structure will tend to pick up and amplify those components close to its own natural period just as it would with a simple sinusoid. The response will therefore be dominated by vibration at or close to the natural period of the structure. However, because the loading does not have constant amplitude and is likely to have only finite duration, the amplifications achieved are likely to be much smaller than for the sinusoidal case. An example is shown in Figure 3.7, where a 0.5 s period structure is subjected to the El entro earthquake record plotted in Figure 3.6. The earthquake contains a wide band of frequency components, but it can be seen that the 0.5 s component undergoes a large amplification and dominates the response.

3.3.2 Response spectrum

The response of a wide range of structures to a particular earthquake can be summarised using a response spectrum. The time domain response of

Figure 3.7 Acceleration of 0.5 s period SDOF structure subject to the El Centro (N-S) earthquake record

Figure 3.7 Acceleration of 0.5 s period SDOF structure subject to the El Centro (N-S) earthquake record numerous SDOF systems having different natural periods is computed, and the maximum absolute displacement (or acceleration, or velocity) achieved is plotted as a function of the SDOF system period. If desired, a range of curves can be plotted for SDOF systems having different damping ratios.

So the response spectrum shows the peak response of an SDOF structure to a particular earthquake, as a function of the natural period and damping ratio of the structure. For example, Figure 3.8 shows the response spectrum for the El Centro (N-S) accelerogram in Figure 3.6, for SDOF structures with 5 per cent damping.

A key advantage of the response spectrum approach is that earthquakes that look quite different when represented in the time domain may actually contain similar frequency contents, and so result in broadly similar response spectra. his akes the response spectru a useful design tool for dealing with a future earthquake whose precise nature is unknown. To create a design spectrum, it is normal to compute spectra for several different earthquakes, then envelope and sooth the, resulting in a single curve that encapsulates the dynamic characteristics of a large number of possible earthquake accelerograms.

Figure 3.9 shows the elastic response spectra defined by EC 8 (2004). EC8 specifies ttvo categories of spectra: type 1 for areas of high seismicity (defined as M > 5.5), and type 2 for areas of moderate seismicity (M < 5.5). Within each category, spectra are given for five different soil types: A - rock; B - very dense sand or gravel, or very stiff clay; C - dense sand or gravel, or stiff clay; D - loose-to-medium cohesionless soil, or soft-to-firm cohesive soil; E - soil profiles with a surface layer of alluvium of thickness 5-20 m. he vertical axis is the peak, or spectral acceleration of the elastic structure, o

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