Keywords: structural optimization, buckling, welded joints, costs, reliability-based design.

A belt-conveyor bridge is built inside a ring-stiffened cylindrical shell. The unknown variables are the span length, shell thickness as well as the thickness and the number of flat rings. The design constraints relate to the local shell buckling strength, to the panel ring buckling and to the deflection of the bridge. In this paper the design rules of Det Norske Veritas are used for ring-stiffened cylindrical shells. The shape of the rings is a simple flat plate, which is welded to the shell by double fillet welds. The cost function includes material and fabrication costs. In the calculation of the fabrication cost, the cost of forming the shell elements into the cylindrical shape and the cutting of the flat ring-stiffeners is also taken into account. A level II reliability method is used to find the probability of failure. The overall structural reliability is obtained by using the Ditlevsen method of conditional bounding. The costs of the plate designed to ensure a stipulated probability of failure are compared with the solutions obtained for a code based method, which employs partial safety factors. A branch and bound strategy coupled with an entropy-based algorithm is used to solve the reliability-based optimization. The entropy-based procedure is employed to find optimum continuous design variables giving lower bounds on the decision tree and the discrete solutions are found by implicit enumeration.

The optimum solutions as a function of the length by using level I procedures (safety loading and resistance factors) are as follows: The local shell buckling is in general the most important constraint for the lower span optimum solution (span length 29m). For intermediate spans the panel ring buckling is often decisive. The deflection constraint is relevant for longer spans. Both to increase the number of rings (and reduce the thickness of the flat rings) and the opposite are associated with more expensive solutions. Material cost is about half of total cost. The forming cost of the shell elements is significant. The most cost effective spans (lower cost/span ratios) range from 56 to 69 m.

The partial safety factors were calibrated to introduce randomness. A high mode correlation was observed considering both failure modes (local shell buckling and maximum deflection). The reliability-based solutions are the same as the deterministic if a probability of failure 10-4 is required. However if a probability of failure 10-5 is imposed the optimum solutions found for the span lengths 29, 43, 56, 57, 69 and 72 m are no longer valid. The higher reliability requirement would lead to a thicker shell. The remaining optimum solutions are indifferent to the specified probability of failure because the most important constraint is the panel ring buckling and this requirement is deterministic.

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