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The two structures can then be designed by applying the seismic loads at the top and determining the associated forces, moments, and deflection. The lateral load, obtained Figure 2.17a. Examples of single-degree-of-freedom systems: (a) elevated water tank (b) restrau-rant atop tall concrete core. Note from Fig. 2.17, the acceleration = 26.25 ft/s2 for T = 0.5 s and p = 0.05 (water tank) and the acceleration = 11.25 ft/s2 for T = 1.00 and p = 0.10 (restaurant).

by multiplying the response spectrum acceleration by the effective mass of the system, is referred to as base shear, and its evaluation forms one of the major tasks in earthquake analysis.

In the examples, SDOF structures were chosen to illustrate the concept of spectrum analysis. A multistory building, however, cannot be modeled as an SDOF system because it will have as many modes of vibration as its degrees-of-freedom which are infinite for a real system. However, for practical purposes, the distributed mass of a building may be Time T, sees

Figure 2.18. Recorded ground acceleration: El Centro earthquake.

Time T, sees

Figure 2.18. Recorded ground acceleration: El Centro earthquake.

lumped at discrete levels to reduce the degrees-of-freedom to a manageable number. In multistory buildings, the masses are typically lumped at each floor level.

Thus, in the 2-D analysis of a building, the number of modes of vibration corresponds to the number of levels, with each mode having its own characteristic frequency. The actual motion of a building is a linear combination of its natural modes of vibration. During vibration, the masses vibrate in phase with the displacements as measured from their initial positions, always having the same relationship to each other. Therefore, all masses participating in a given mode pass the equilibrium position at the same time and reach their extreme positions at the same instant.

Using certain simplifying assumptions, it can be shown that each mode of vibration behaves as an independent SDOF system with a characteristic frequency. This method, called the modal superposition method, consists of evaluating the total response of a building by statistically combining the response of a finite number of modes of vibration.

A building, in general, vibrates with as many mode shapes and corresponding periods as its degrees-of-freedom. Each mode contributes to the base shear, and for elastic analysis, this contribution can be determined by multiplying a percentage of the total mass, called effective mass, by an acceleration corresponding to that modal period. The acceleration is typically read from the response spectrum modified for a damping associated with the structural system and the assumed return period of the design earthquake. Therefore, the procedure for determining the contribution of the base shear for each mode of an MDOF structure is the same as that for determining the base shear for an SDOF structure, except that an effective mass is used instead of the total mass. The effective mass is a function of the lumped mass and deflection at each floor with the largest value for the fundamental mode, becoming progressively less for higher modes. The mode shape must therefore be known in order to compute the effective mass.

Because the actual deflected shape of a building consists of a linear combination of its modal shapes, higher modes of vibration also contribute, although to a lesser degree, to the structural response. These can be taken into account through use of the concept of a participation factor. Further mathematical explanation of this concept is deferred to a later section, but suffice it to note that the base shear for each mode is determined as the summation of products of effective mass and spectral acceleration at each level. The force at each level for each mode is then obtained by distributing the base shear in proportion to the product of the floor weight and displacement. The design values are then computed using modal combination methods, such as CQC or SRSS.

Types of Response Spectrum. Three types of response spectra are used in practice.

1. Response spectra from actual earthquake records.

2. Smoothed design response spectra.

### 3. Site-specific response spectra.

Response Spectra from Actual Earthquake Records. To develop these response spectra, a series of damped SDOF mass-spring systems is subjected to an actual earthquake ground excitation, and by numerical integration of the maximum values for a range of periods of vibration is determined. However, the resulting spectral curves are quite jagged, being characterized by sharp peaks and troughs. Because the magnitude of these troughs and peaks varies significantly for different earthquakes, several possible earthquake spectra are used in the evaluation of the structural response.

Smooth Response Spectrum. As an alternative to the use of several earthquake spectra, a smooth spectrum representing an upper-bound response to several ground motions may be generated. The sharp peaks in earthquake records may indicate the resonant behavior of the system when the natural period of the system approaches a period of the forcing function, especially for systems with little or no damping. However, even a moderate amount of damping has a tendency to smooth out the peaks and reduce the spectral response.

Because buildings have some degree of damping, the peaks in the response spectra are of limited significance and therefore are smoothed out, as shown in Fig. 2.19. The other two response spectra for the velocity and displacement, shown in Figs. 2.20 and 2.21, are obtained from the acceleration spectrum, since they are related to one another. The three spectra can be represented in one graph, as shown in Fig. 2.22, in which the horizontal axis denotes the natural period and the ordinate the spectrum velocity, both on a logarithmic scale. The acceleration and displacement are represented on diagonal axes inclined at 45° to the horizontal. The plot, which presents all three spectral parameters, is called a tripartite response spectrum.

Consider, for example, that we wish to calculate the first-mode displacement of a building having a fundamental period T = 2 sec, subjected to a given base shear evaluated from the response spectrum given in Fig 2.22. One method is, of course, to perform a stiffness analysis of the building by defining the geometry, material and stiffness properties, and then subjecting it to the lateral loads evaluated from the response spectrum. However, an easier method, without having to go through an analysis, is to simply read off the lateral deflection from the tripartite diagram (Fig. 2.22), as will be shown presently. Figure 2.19. Smoothed acceleration spectra for the El Centro earthquake. Figure 2.20. Smoothed velocity spectra for the El Centro earthquake.

The displacement curve is plotted on the tripartite diagram using the relation 