Analysis of Geometrically Non-Linear Elastic-Plastic Framed Structures/Tampriai plastinų geometriškai netiesinių strypinių konstrukcijų analizė

A stress-strain field (SSF) evaluation of elastic-plastic structures under the action of a completely specified external loading that doesn't exceed its limiting value, but produces plastic deformation, is under consideration. An assumption of the discrete structure, possesing the small bar strains and large displacements assumptions, is applied. It is known that for disipative structures SSF depends on the loading history. When solving the analysis extremum problems on the basis of extreme energetical principles [1]-[8] for global external loads (final load magnitudes) one cannot fix directly the unloading phenomenon in cross-sections. Therefore, the possibilities to evaluate the SSF change of the structure, as well as the possibility to take into account various plastic deformation stages, finishing prior to the stage when the final magnitudes of loads are achieved, are lost. Therefore, the residual displacement magnitudes, obtained by solving the problem for global loads only, can be determined only very approximately. The proposed in the manuscript complementary load method to investigate the SSF for geometrically non-linear structure enables to avoid the above-mentioned negative aspects. Applying the method, design process of a structure is iterative step-by-step procedure. A load step ΔF is introduced. Monotonically increasing the load by step ΔF, increments of SSF are obtained. The analysis problem in static and kinematic formulations is stated for this purpose. Formulations are based on the dual extreme energetic principles formulated for residual strains and residual displacements increments for any structure deformation stage. The actual field of residual forces proceeding the plastic failure is obtained by using the extreme principle of the minimum of complementary energy increment—the ststic formulation (8). Applying the Lagrange function of this formulation, the dual problem is formulated to determine the displacements of the structure. Thus the mathematical model (11) corresponds to the extremum principle of a maximum of complementary work increment. Changing the sign of the objective function the problem (11) corresponds to the extremum principle of a minimum of total complementary energy increment. Finally, we conclude that increments of residual forces (15) and displacements (14) are the linear functions only one of plastic multipliers Δλv. By substituting that functions into problem (11) the elastic-plastic geometrically non-linear framed structures SSF analysis problem can be transformed to modified problem (16). Then applying formulae (14) and (15) the SSF increments for the V—th step are determined. The actual SSF is obtained applying the relationships (1) and (2). The SSF analysis problem for a two-storey frame made of reinforcement concrete at the stage prior to plastic collapse is presented for the numerical example. Numerical experiments show that proposed mathematical model (16) allows to determine the actual forces and displacements for real structures and to achieve sufficient results for practical design needs.

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