j J


FIGURE 3.33 Simplified stress-strain models: (a) elasto-plastic; (b) bilinear; (c) curvilinear; and (d) Ramberg-Osgood.

FIGURE 3.33 Simplified stress-strain models: (a) elasto-plastic; (b) bilinear; (c) curvilinear; and (d) Ramberg-Osgood.

initial yield points in tension A and in compression A' are the same, but the values change for subsequent points B and C or D and E. A stress magnitude of 2sy is observed for AA', BC, and, DE which is known as Bauschinger effect. For practical purposes, some well-known simplified models are often used. Most typical are these shown in Figure 3.33 as elasto-plastic, bilinear, curvilinear, and Ramberg-Osgood. For dynamic nonlinear analysis, the main task is the derivation of incremental stiffness coefficients of a typical member. The motion equation is still expressed as Equation 3.34 or 3.93 but in incremental form. The direct integration methods presented in Section 3.6.3 are then employed for response analysis. The incremental stiffness coefficients of elasto-plastic and bilinear models are given in Sections 3.7.2 and 3.7.3, respectively. The derivations of incremental stiffness coefficients of curvilinear and Ramberg-Osgood models are lengthy and are not included here [1, pp. 555-578].

3.7.2 Elasto-Plastic Stiffness Formulation

The elasto-plastic model in Figure 3.34 shows that when the moment reaches the ultimate moment capacity of a member, the plastic moment cannot increase but the rotation of the plastic hinge at the cross-section can increase. A plastic hinge develops at the member's end where the magnitude of the moment is greater than at other locations as in the case of dynamic response. The member end behaves like a real center hinge with a constant ultimate moment, Mp. When the member end rotates in reverse, the moment decreases elastically and the plastic hinge disappears. Elastic behavior remains unchanged until the moment reaches ultimate moment capacity. Consequently, a plastic hinge forms again. Plastic hinge formation in a member has three possibilities: a hinge at the ¿-end, the j-end, or both ends (see i and j in Figure 3.35). Force-deformation relationships associated with elastic state and the three states of yield condition are given as follows [1, pp. 529-534].

fi I ''' Pei 2My EI ' ei

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