Keywords: shells, plasticity, optimization, reliability-based design.

A unified formalism allows the reliability analysis of stretching plates, plates in bending and shallow curved plates discretized using triangular finite elements to be treated in a similar fashion. The formulation describing the fundamental relations of the problem reflects the finite element connectivity across the interelement boundaries and is thereby called a kinematic description. The material is assumed stable in Drucker's sense and the convex hypersurface is replaced by a set of convex hyperplanes. The formulation is unaltered just by re-interpretation of the symbols for thin flexural plates. The material is considered to satisfy a yield criterion formulated by Nielsen for reinforced concrete plates. In order to obtain linearized yield conditions, a safe linearization suggested by Wolfensberger which considers an octahedron is adopted.

The structural material is assumed to exhibit perfectly-plastic behaviour so that plastic collapse is the only possible failure mode. It consists of solving alternatively a reliability assessment (concave quadratic programming) and an optimal sizing problem (convex minimization) until the best solution is found. The reliability index is obtained by using the first order second moment approximation. The reliability assessment problem is formulated as the minimization of a concave quadratic function over a linear domain. This type of problem cannot be solved using convex programming techniques because of the possibility of nonglobal local minima. An enumerative strategy is employed to find the optimum solution giving the plastic deformations for the stochastic most important mechanism. The general nonconvex domain is transformed in the branch and bound (B&B) strategy into a sequence of intersecting convex domains by the use of convex underestimating functions. The two main ingredients are a combinatorial tree where the nodes are associated with linear programs and some upper and lower bounds to the final solution related to each node of the tree. In the context of reliability-based plastic optimization the objective is to minimize average resistance when the loading, the reliability level against plastic collapse and the shell thickness are prescribed. By fixing the design variables the inner problem gives the yield rotations and nodal displacements associated with the stochastic most important mechanism and other relevant modes. Since the approximation of the multimode constraint is convex, the optimum design problem can be solved using any convex programming technique.

Two numerical examples are solved, namely the reliability-based analysis of a concrete pier (plate stretching) and of a reinforced concrete floor (thin flexural plate).

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