Bi-directional hysteresis models are useful for dynamic response analyses of structures under two dimensional seismic excitation. The theory of plasticity for work hardening materials has been utilized for the bi-directional extension of uni-directional hysteresis models. The work softening (or strength degrading) hysteresis models are, however, difficult to be extended to bi-directional models by means of the theory of plasticity alone, because the theory utilizes Drucker's postulate of 'stable inelastic material'. This paper firstly presents a mathematical formulation to extend the uni-directional hysteretic model with work softening to bidirectional model. Secondly, this paper presents a series of earthquake response analyses to study the effects of work softening and bi-directional exitation on the seismic response. Following conclusions were obtained. (1) Hysteresis models with work softening can be classified into 2 types : (a) monotonic degrading type, in which the load-deformation relation by monotonic loading coincides with the envelope curve of the cyclic load-deformation relation as shown in Fig. 2, and (b) cumulative degrading type, in which the envelope curve of the cyclic load-deformation relation lies beneath the monotonic curve as shown in Fig. 1. (2) Restoring force-deflection relations of reinforced concrete members will be monotonic degrading type, if the strain of the centroids of the members does not exceed the strain at the compressive strength. Otherwise, the restoring force-deflection relations will be cumuratiye degrading type. (3) The fracture potentional defined on the deflection plane enables to extend the uni-directional hysteretic model with work softening to bi-directional model. (4) Assuming the expansion rule of the fracture potential as shown in Fig. 18 or 19 enables to express the cumurative degrading type or monotonic degrading type, respectively. (5) The virtual restoring force is defined as the actual restoring force minus the fracture portion of the restoring force. The plastic potential defined on the virtual restoring force plane enables the plastic potential to move and contract during the fracturing. (6) The proposed model agrees with the plane-section-remain-plane analysis of reinforced concrete sections. (7) The response of work softening system under either uni-directional or bi-directional earthquake exitation tends to go to one particular direction much more than that of work hardening system. (8) In the cases of work hardening system, the ratio of the uni-directional ductility response to the hysteretic energy absorbed is larger than that of the bi-directional one. In other words, the ductility response of the work hardening system by an uni-directional earthquake excitation is larger than that by a bi-directional earthquake excitation with the same amount of energy input. (9) In the cases of work softening system of the ratio of cumulative degrading type, the uni-directional ductility response to the hysteretic energy absorbed is larger than that of the bi-directional one. In other words, the ductility response of, the work hardening system by an uni-directional earthquake excitation is larger than that by a bi-directional earthquake excitation with the same amount of energy input. (10) The amount of energy input by earthquake excitation to work softening systems is about the same as that to work hardening systems. (11) The 'effective repetition number : N, is defined by N=E/Q_OD_y(μ-1) where E is the hysteretic energy absorbed by the system, μ is the yield strength of the system, D_y is the yield defletion of the system, and ju is the maximum ductility reponse of the system during the earthquake. The number, N, can be approximated as the function of the ratio of K_2K_1 or K_3/K_1 as shown by the solid line or the broken line in Fig. 27. This approximation as well as the assumption of same amount of

(View PDF for the rest of the abstract.)