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3.6.5.3 Comments on Nonlinear Systems

The stochastic response of nonlinear structures cannot be obtained by the aforementioned method because of the requirement of superposition. One of the approaches is to generate an ensemble of ground motion accelerograms by the techniques based on filtered white noise and then determine deterministically the time-history response of the nonlinear structure to each input accelerogram, and finally examine the output response process using Monte Carlo methods. Usually, one is interested primarily in the mean and standard deviation values of the extreme response [8].

### 3.6.6 Comments on Wave Propagation Analysis

Theoretically, mode superposition is an effective method of obtaining the free or forced vibration response of a continuous system of finite extent. In practice, several difficulties may arise in application of the method. For instance, it may not be possible to determine the mode shapes and frequencies of the continuous system being analyzed. A system of infinite extent has a continuous band of frequencies, and the term ''mode shape'' loses its meaning. Obviously, modal analysis is not possible in such a case. However, an alternative method known as wave propagation analysis may prove to be quite effective in obtaining the response of the system [9].

3.7 Hysteresis Models and Nonlinear Response Analysis 3.7.1 Introduction

The previous sections focused on elastic structures with emphasis on mathematical models, analytical methodologies, and response characteristics. When a structure is subjected to dynamic force or ground motion, its constituent members may deform beyond their elastic limit, such as yielding stress of steel or crack stress of concrete. If we assume that the members continue to behave elastically, then their response behavior is based on linear or elastic analysis as presented previously. When the stress-strain relationship beyond the elastic stage is considered, the response then results from nonlinear or inelastic analysis. Naturally, nonlinear analysis always encompasses linear analysis because of elastic material behavior at the early loading stage. When inelastic material behavior is considered in formulating the force-deformation relationship of a structural member, the relationship is called hysteresis model. A typical stress-strain relationship of structural steel is shown in Figure 3.32. The linear relationship between O and A is defined as elastic behavior. After initial yielding, oy, the slope of the stress-strain curve is not constant and material behavior becomes inelastic. Unloading path B-C and reloading path D-E are elastic and form straight lines parallel to the initial elastic path O-A. Absolute values of the

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