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may be estimated using the relationship shown in Table 2.1. The table can also be used to estimate modal participation factors at building roof levels and base shear participation factors corresponding to the first five modes.

For most buildings, inelastic response can be expected to occur during a major earthquake, implying that an inelastic analysis is more proper for design. However, in spite of the availability of nonlinear inelastic programs, they are not used in typical design practice because: 1) their proper use requires knowledge of their inner workings and theories; 2) results produced are difficult to interpret and apply to traditional design criteria; and 3) the necessary computations are expensive. Therefore, analyses in practice typically use linear elastic procedures based on the response spectrum method.

### 2.2.14.1. Response Spectrum Method

The word spectrum in seismic engineering conveys the idea that the response of buildings having a broad range of periods is summarized in a single graph. For a given earthquake motion and a percentage of critical damping, a typical response spectrum gives a plot of earthquake-related responses such as acceleration, velocity, and deflection for a complete range, or spectrum, of building periods.

Thus, a response spectrum (Figs. 2.15 and 2.16) may be visualized as a graphical representation of the dynamic response of a series of progressively longer cantilever

pendulums with increasing natural periods subjected to a common lateral seismic motion of the base. Imagine that the fixed base of the cantilevers is moved rapidly back and forth in the horizontal direction, its motion corresponding to that occurring in a given earthquake. A plot of maximum dynamic response, such as accelerations versus the periods of the pendulums, gives us an acceleration response spectrum as shown in Fig. 2.15 for the given earthquake motion. In this figure, the absolute value of the peak acceleration response occurring during the excitation for each pendulum is represented by a point on the acceleration spectrum curve. As an example, an acceleration response spectra for the 1940 El Centro earthquake is illustrated in Fig. 2.17. Using ground acceleration as an input, a family of response spectrum curves can be generated for various levels of damping, where higher values of damping result in lower spectral response.

To establish the concept of how a response spectrum is used to evaluate seismic lateral forces, consider two single-degree-of-freedom structures: 1) an elevated water tank supported on columns; and 2) a revolving restaurant supported at the top of a tall concrete core (see Fig. 2.17a). To simplify, we will neglect the mass of the columns supporting the tank, and consider only the mass mx of the tank in the dynamic analysis. Similarly, the mass m2 assigned to the restaurant is the only mass considered in the second structure. Given the simplified models, let us examine how we can calculate the lateral loads for both these structures resulting from an earthquake, for example, one that has the same ground motion characteristics as the 1940 El Centro earthquake shown in Fig. 2.18. To evaluate the seismic lateral loads, we shall use the recorded ground acceleration for the first 30 seconds. Observe that the maximum acceleration recorded is 0.33g. This occurred about 2 seconds after the start of the record.

T= Natural period, sec

Figure 2.17. Acceleration spectrum: El Centro earthquake.

T= Natural period, sec

Figure 2.17. Acceleration spectrum: El Centro earthquake.

As a first step, the base of the two structures is analytically subjected to the same acceleration as the El Centro recorded acceleration. The purpose is to calculate the maximum dynamic response experienced by the two masses during the first 30 seconds of the earthquake. The maximum response such as displacement, velocity, and acceleration response of an SDOF system such as the two examples considered here may be obtained by considering the earthquake effects as a series of impulsive loads, and then integrating the effect of individual impulses over the duration of the earthquake. This procedure, the Duhamel Integration Method, requires considerable analytical effort. However, in seismic design, it is generally not necessary to carry out the integration because the maximum response for many previously recorded and synthetic earthquakes are already established. The spectral acceleration response for the north-south component of the El Centro earthquake, shown in Fig. 2.17, is one such example.

To determine the seismic lateral loads, assume the tank and restaurant structures weigh 720 kips (3202 kN) and 2400 kips (10,675 kN), with corresponding periods of vibration of 0.5 sec and 1 sec, respectively. Since the response of a structure is strongly influenced by damping, it is necessary to estimate the damping factors for the two structures. Let us assume that the percentage of critical damping b for the tank and restaurant are 5 and 10% of the critical damping, respectively. From Fig. 2.17, the acceleration for the tank structure is 26.25 ft/s2, giving a horizontal force in kips equal to the mass of the tank, times the acceleration. Thus,

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