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3.7.3 Bilinear Stiffness Formulation

The moment-rotation relationship of a member can be idealized as a bilinear model, as shown in Figure 3.36. Initial elastic slope and subsequent inelastic slope of the bilinear moment-curvature curve are EI and EI1, respectively; I1 can be in terms of I. The Bauschinger effect on the moment magnitude of 2Mp also exists between two subsequent plastic hinges as signified by points B and C. The momentrotation relationship of a member is composed of two imaginary components as linear component and elasto-plastic component sketched in Figure 3.36b,c respectively. Initial stiffness of the hysteresis loop and of the elastic, elasto-plastic components is a, a1, a2, respectively, where a = a1 + a2, a1 = pa, a2 = qa,

Actual length = 0

Actual length = 0

FIGURE 3.37 Bilinear member: (a) nonlinear beam; (b) linear component; and (c) elasto-plastic component.

and p + q = 1. p is the fraction of stiffness apportioned to the linear component and q is the fraction of stiffness apportioned to the elasto-plastic component. The second slope a1 of the hysteresis loop is the same as the initial slope of the linear component. The incremental stiffness coefficients are given as follows [1, pp. 534-538].

1. Both ends linear

a b —c —c

0 0

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