Keywords: limit analysis, holonomic constraints, complementarity, stress resultant interaction, mathematical programming.

In this paper limit elastoplastic analysis of frame structures with hardening behaviour and axial-shear force-bending moment interaction is examined in the framework of mathematical programming. The maximum load carrying capacity of the structure is determined by solving an optimization problem with linear equilibrium, compatibility and yield constraints together with a complementarity constraint that is treated using the penalty function formulation. Incorporation of the shear force effect at yield determines the nonlinear three-dimensional yield surface that is linearized with appropriate polyhedra. The proposed method pioneered by Maier et al. [1,2] is based on a combination of limit and holonomic elastoplastic analysis. The problem is formulated in terms of equilibrium, compatibility, piecewise linear constitutive laws, associated flow rules and a complementarity condition that excludes the simultaneous activation of plastic deformation and unloading. As a result of the presence of the latter condition, the problem constitutes a mathematical programming problem with equilibrium constraints (MPEC) that is nonsmooth, nonconvex and often numerically unstable. This problem has been solved as a quadratic programming problem or a restricted basis linear programming problem or a parametric linear complementarity problem [3]. Subsequent development of mathematical algorithms for MPEC problems has led to the proper treatment of complementarity condition [4] and consequently to an extensive use of this approach for elastoplastic analysis of frames [5,6]. In this paper, the maximum load carrying capacity of the structure is determined by solving a nonlinear programming problem that provides simultaneously the maximum load together with the corresponding stresses, displacements and strains of the structure. Herein the generalized Gendy-Saleeb yield criterion, appropriately linearized, accounts for the axial-shear force-bending moment interaction in critical sections. Numerical results of steel frame analyses for rigid-perfectly plastic and isotropic hardening behaviour are compared to those of axial force-bending moment interaction demonstrating the role of shear force. The convex hull of the behaviour of the cross sections is introduced, the evolution of which becomes quite informative in describing the different stages of the structure reaching an ultimate state.

1

G. Maier, "A matrix structural theory of piecewise linear elastoplasticity with interacting yield planes", Meccanica, 5, 54-66, 1970. doi:10.1007/BF02133524

2

O.D. Donato, G. Maier, "Mathematical programming methods for the inelastic analysis of reinforced concrete frames allowing for limited rotation capacity", International Journal for Numerical Methods in Engineering, 4, 307-29,1972. doi:10.1007/BF02128846

3

G. Maier, D.E. Grierson, M.J. Best, "Mathematical programming methods for deformation analysis at plastic collapse", Computers and Structures, 7, 599-612, 1977. doi:10.1016/0045-7949(77)90002-5

4

Z.Q. Luo, J.S. Pang, D. Ralph, "Mathematical programs with equilibrium constraint", Cambridge University Press, 1996. doi:10.1017/CBO9780511983658

5

G. Cocchetti, G. Maier, "Elastic-plastic and limit-state analyses of frames with softening plastic-hinge models by mathematical programming", International Journal of Solids and Structures, 40, 7219-44, 2003. doi:10.1016/S0020-7683(03)00363-9

6

S. Tangaramvong, F. Tin-Loi, "Simultaneous ultimate load and deformation analysis of strain softening frames under combined stresses", Engineering Structures, 30, 664-74, 2008. doi:10.1016/j.engstruct.2007.05.014

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