## 200 100

Frequency, cps

FIGURE 3.10 Elastic and inelastic design spectra. (Reprinted from Ref. [1, p. 376] by courtesy of Marcel Dekker, Inc.)

Smooth design spectra for elastic and inelastic responses are available with consideration of various damping ratios, soil profiles, and earthquake records [1, pp. 366-379]. Sample elastic and inelastic design spectra are shown in Figure 3.10 for p — 0.05. The inelastic spectra given are only for ductility m — 4 and 8. Ductility factor is designed as the ratio of the maximum displacement to yielding displacement.

3.3 Response Analysis of Multiple d.o.f. Systems — Lumped-Mass Formulation

### 3.3.1 Nature of Spring-Mass Model

A structure that is assumed to be a discrete parameter (lumped mass) system must be conceived of as a model consisting of a finite number of masses connected by massless springs. The spring-mass model, depending on the characteristics of the structure, can be established in different ways. An example is shown in Figure 3.11, where M1 and M2 are masses lumped from girders and columns, and k1 and k2 represent column stiffnesses. When the girder is infinitely rigid, the structure has no joint rotations; this spring-mass model is shown in Figure 3.11b. When the girders are flexible and structural joint rotations exist, the spring-mass model differs as shown in Figure 3.11c. Note the reason for the difference: if x2 is displaced and the girders are rigid, no force is transmitted to the support. However, with flexible girders, the joints at the first floor rotate, the column below is distorted, and force is transmitted to the support.

3.3.2 Normal Modes, Modal Matrix, and Characteristics of Orthogonality

The motion equations associated with free undamped vibration of a spring-mass system can be expressed in matrix form as

where [M] and [K] are called the mass matrix and structural stiffness matrix, respectively. Let the displacement vector be {x} — (cospt){X}; then, Equation 3.29 may be expressed as (since cospt— 0)

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