Keywords: concrete beams, plastic analysis, optimisation, complementarity.

When concrete continuous beams are loaded to failure the bending moment diagram at collapse is generally different from the linearly elastic diagram. This is due to redistribution of moments resulting from nonlinear deformations at critical sections. In the case of unlimited ductility the collapse load may be found by applying the usual theorems of limit analysis. But as reinforced concrete sections have only limited ductility the formation of the collapse mechanism may be preceded by expulsion of V-shaped damaged concrete compression region or fracture of the reinforcing bars. Therefore rotation at plastic hinges must be checked not to exceed allowable values. The design codes have provisions to compute rotation capacity at a plastic hinge as a function of the ratio of the depth of the neutral axis to the effective depth of the section and distance between points of zero bending moments containing the hinge section.

A procedure for analysis of continuous reinforced concrete beams is proposed that computes collapse load by maximizing the loading factor subject to constraints that enforce equilibrium, section strength and limit plastic hinge rotations to allowable values. Two implementation details deserve special attention. Firstly, a general flexibility approach is developed to compute plastic hinge rotations. Secondly, plastic rotations can only occur at plastic hinges, therefore rotations can only exist at sections subject to fully plastic bending moments. The resulting mathematical programming problem contains complementarity constraints (MPCC), and may be written as [1]:

where and is the complementarity operator, which requires that either a component or the corresponding component . This problem can be replaced with the following equivalent nonlinear programming problem (NLP):

where .

This NLP problem violates the Mangasarian-Fromovitz constraint qualification. Until recently it has been assumed that due to degeneracy of Lagrange multipliers the use of standard and powerful nonlinear programming algorithms was numerically unsafe. It has been shown that some SQP methods exhibit local convergence near strongly stationary points [2]. Hundreds of MPCCs were successfully solved using SQP with filter techniques [3]; therefore no special purpose methods with penalty parameters are needed any more.

The example presented demonstrates that MPCC is a powerful tool to perform plastic analysis of concrete structures with limited ductility.

1

S. Leyffer, "Complementarity Constraints as Nonlinear Equations: Theory and Numerical Experience", Preprint ANL/MCS-P1054-0603, MCS Division, Argonne National Laboratory, Argonne, 2003.

2

R. Fletcher, S. Leyffer, D. Ralph, S. Scholtes, "Local Convergence of SQP Methods for Mathematical Programs with Equilibrium Constraints", Numerical Analysis Report NA/209, Dep. of Mathematics, Univ. of Dundee, Dundee, 2002.

3

R. Fletcher, S. Leyffer, "Numerical Experience with Solving MPECs as NLPs", Numerical Analysis Report NA/210, Dep. of Mathematics, Univ. of Dundee, Dundee, 2002.

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