An incremental procedure for deformation analysis of elastic-plastic frames.

In the present paper an incremental procedure is formulated for first-order elastoplastic analysis of plane frame structures discretized into a finite number of beam elements and described by piecewise linear elastic-perfectly plastic constitutive laws, under the assumption of both reversible and irreversible behaviour of material and using piecewise linear yield conditions at any desired degree of discretization in the space of the active stress resultants (axial force, shear force and bending moment). The proposed method, by using the independent elastic-plastic kinematical compatibility equations, restrains the problem sizes within not more than twice the number of the redundant unknowns in the complete elastic frame, regardless of the degree of discretization of the piecewise linear yield conditions, still maintaining the advantage exhibited by the Mathematical Programming methods of requiring only one factorization of the matrix governing the problem when no local unloading occurs. Furthermore, the outlined algorithm allows the additional computational effort to be restrained in the case of local unloading, inasmuch as it requires a new factorization to be performed of a part of the matrix governing the problem, whose size is small with respect to the total size of the matrix

Displacement analysis in elastic-plastic frames at plastic collapse

A linear programming method is proposed for displacement analysis at the collapse-load level for certain frame structures. These structures are discretized into finite elements and described by piecewise-linear elastic-perfectly plastic constitutive laws. The formulation is for monotonically increasing loads. Neither the spread of plastic zones from hinges nor the effects of change of geometry on the conditions of equilibrium are included. An extremum energy principle and the kinematic formulation of the linear programming method are employed to solve the problem in the case of regular collapse mechanism. By contrast, prior iterative procedure involving the solution of a set of linear equations is required when a partial collapse mechanism occurs, in order to establish the actual stress resultant distribution over the whole structure. The computational effort required in addition to the previously performed rigid-plastic limit analysis is independent of the total number of yield modes in the structure. It is expended upon the diagonalization of a set of linear equations, the number of which equals that of redundancies in the complete elastic frame, in the case of regular collapse mechanism and, in addition, upon the solution of a set of linear equations, equal in number to that of residual redundant unknowns less one. Two sample frames are analyzed in order to investigate the influence of combined stress resultants on deformation state at plastic collapse