# Inelastic Beam-Column Model

A hybrid element (see Fig.1) is employed to model planar inelastic beam-columns. The element consists of two potential plastic-hinges at the ends of an elastic beam-column. Each plastic-hinge is modeled by a nonlinear zero-length rotational spring (which is called a hinge-spring or spring hereafter). For the elastic beam-column, E, I, and A correspond to Young’s modulus, moment of

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M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 265-276.

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 inertia, and cross-sectional area of the element. All the plastification is assumed to be concentrated at end plastic-hinges. Fig. 1 Hybrid beam-column element and its end rotation

 Fig. 2 Yielding criteria under combined axial force and bending moment

The interaction between the axial force N and moment M is considered using a two-surface criterion, as shown in Fig. 2. For the first-yield surface, the equation is

M My + N Ny = 1 – Fr Fy (1)

where: My=Fy S is the first-yield moment of the section under a pure moment and S is the elastic modulus of the cross section; Ny=AFy is fully-plastic axial force capacity under a pure axial force; Fy is yielding stress of steel; and Fr is the peak residual stress in the flange of cross-section. In the meantime, the full-yield surface is expressed as

MMp +{n Ny У = 1 (2)

where Mp=FyZ is the plastic moment of the section under a pure moment and Z is the plastic modulus; and the exponent a depends on the shape of the section. For example, a=1.3 for a wide-flange section under strong axis bending (Duan and Chen 1990). Therefore, any section having a force point falling within the shaded area is partially yielded (see Fig. 2).