1. INTRODUCTION A symmetry limit theory has been presented recently by the present authors [1] for a new branching phenomenon in the steady states of a linear strain-hardening cantilever beam-column subjected to an idealized completely reversed tip deflection cycling. The purpose of this paper is to develop an extended theory of the symmetry limit for a steel beam-column with an arbitrary cross-section subjected to the same idealized deflection cycling program. The present theory is, however, not a straightforward extension of the previous theory bul deals with more general types of nonlinear hysteretic stress-strain relations. A general formulation on the basis of virtual work equations is presented so that it can be further extended to a more general form applicable to complex structural systems. 2. FUNDAMENTAL CONCEPTS FOR AN IDEALIZED PROBLEM A cantilever beam-column is subjected to an idealized tip deflection cycling program referred to as COIDA under a constant axial compressive dead load IV as shown in Fig. 1. COIDA is an idealized completely reversed tip deflection cycling program of stepwisely increasing amplitude Ψ consisting of infinitely many infinitesimal increments, such that every after an infinitesimal increment of Ψ, the deflection cycling at that increased amplitude is repeated as many times as is necessary for the beam-column to attain a steady state. The following hypotheses have been introduced in [1]. (H 1) A beam-column exhibits a sequence of symmetric steady states during some range of Ψ and then transition from a symmetric steady state to an asymmetric steady state at some value of Ψ. Once the transition occurs, the beam-column exhibits a sequence of asymmetric steady states. (H 2) All the state variables are continuous and piecewisely differentiable functions of Ψ. Sequences of steady states can be visualized as steady state paths in a "steady state space" as shown in Fig. 2. It has been pointed out in Ref. [1] that any two different steady state paths, say OSB and OSF in Fig. 2, can never be the possible responses of a beam-column to one and the same COIDA program and that the emanation of asymmetric branches at S is not a branching phenomenon on equilibrium paths, but may be interpreted as a segregation phenomenon in bundles of equilibrium paths and at the same time as a branching phenomenon in steady state paths. 3. BASIC EQUATIONS FOR STEADY STATES The strain-displacement relation, geometrical boundary conditions and equation of equilibrium are given by Eqs. (2), (3), and (4), respectively. The stress-strain relations for the virgin, unloading and loading paths shown in Fig. 4 can be written as Eqs. (8), [(9), (10)], and (11), respectively, in dimensionless forms. 4. FUNDAMENTAL HYPOTHESES The steady-state paths are computed here by means of an incremental procedure. The three fundamental hypotheses (H 3), (H 4) and (H 5) introduced, together with (Hi) and (H 2) stated above, in [1] are also introduced here in the forms suitable for the nonlinear hysteretic stress-strain relations. (H 3) The strain direction of a state point may be reversed only when the controlled tip deflection v (I) is reversed. For the strain responses during a convergent process to a steady state, however, this hypothesis does not exclude a possibility of strain reversal without reversal of υ (l). (H 4) At every symmetric steady state, the state point of an element of the beam-column travels along a constant closed locus that would reach the compressive virgin curve at the reversal of compressive strain. (H5) In the moment of transition from a symmetric steady state to an asymmetric steady state, no variation toward a steady-state locus whose compressive reversal point is away from the compressive virgin curve as shown in Fig. 7(b), will appear in any finite region in the beam-column. In other words, there occurs only such a variation toward a steady-state locus

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