## Q

y p y p applied loads and the change in structural geometry. Although most of the spread-of-plasticity analysis methods have been developed for planar analysis (White 1985; Vogel 1985), three-dimensional spread-of-plasticity techniques are also available involving various degrees of refinements (Chen and Atsuta 1977; Clark 1994; Wang 1988; White 1988; Jiang et al. 2002).

The simplest second-order inelastic analysis is the elastic-plastic hinge approach. The analysis assumes that the element remains elastic except at its ends where zero-length plastic hinges are allowed to form. Plastic hinge analysis of planar frames can be found in Orbison 1982; Ziemian et al. 1992a,b; Liew et al. 1993; White et al. 1993; Chen and Toma 1994; Chen and Sohal 1994; Chen et al. 1996, among others. Advanced analyses of three-dimensional frames are reported in Chen et al. (2000) and Liew et al. (2000). Second-order plastic hinge analysis allows efficient analysis of large-scale building frames. This is particularly true for structures in which the axial forces in the component members are small and the behavior is predominated by bending actions. Although elastic-plastic hinge approaches can provide essentially the same load-displacement predictions as second-order plastic-zone methods for many frame problems, they cannot be classified as advanced analysis for use in frame design. Some modifications to the elastic-plastic hinge are required to qualify the methods as advanced analysis, and they are discussed in Section 2.12.7.

Figure 2.111 shows the load-displacement curve (a smooth curve with a descending branch) obtained from the second-order inelastic analysis. The computed limit load should be close to that obtained from the plastic-zone analysis.

2.12.2 Column Stability 2.12.2.1 Stability Equations

The stability equation of a column can be obtained by considering an infinitesimal deformed segment of the column as shown in Figure 2.113. Considering the moment equilibrium about point b, we obtain or upon simplification

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