Abstract
The problem of optimal design of the statically indeterminate arch girder which constitutes the primary structural system of the arch bridge is presented. The task is to determine the optimal shape of the axis of the arch girder, as well as the optimal distribution of the cross section height, ensuring the minimum arch volume as well as fulfillment of the standard requirements. This optimisation task, with numerous control functions and constraints, is formulated as a control theory problem with maintaining the formal structure of the minimum principle and then transformed to the multipoint boundary value problem and solved by means of numerical methods. The numerical results are obtained with optimal control methods, using the Dircol software. Since the changes in the shape and cross-section of the arch affect the distribution of the dead and moving loads transferred on the girder from the bridge deck, the optimisation procedure is combined with the finite element method analysis, which together with the complexity of the multidecision arch optimisation problem accounts for the novelty of the proposed approach. The numerical analysis reveals that the optimal girder shape is the frame-arched structure, with considerable lengths of straight sections and only short arch elements, in the areas of the application of concentrated forces and moments. The presented method can be successfully extended to optimisation of structures with different static schemes and load categories taken into account.
Highlights
Arches have been widely used as bridge structural elements for centuries
Arch girders are important, expensive, and often unique structural elements of many bridge structures, so the crucial issue is searching for optimal girder shapes and optimal geometry of their cross sections that will reduce construction costs while simultaneously meeting all the constraints arising both from standards and imposed by developers
Examples of the determination of the optimum shape of brick masonry arches under dynamic loads by cellular automata were presented by Kumarci et al [10]
Summary
Arches have been widely used as bridge structural elements for centuries. In contemporary engineering, arch girders are highly attractive to designers and architects due to their resistance, safety, and the possibility of shaping their various forms; as a result, they actively influence the landscape. The need to take into account all these constraints, as well as many load combinations and control variables significantly increases the dimensions of optimisation problems and complexity of mathematical modelling. This often results in the necessity of task simplification in order to fit into the model. The application of Pontryagin’s minimum principle in combination with FEM computations allow global solutions to design problems to be reached They provide all the information on shaping a girder of variable curvature and stiffness in a way that guarantees the minimisation of the accepted objective function while simultaneously meeting all the constraints arising from standards and the constraints imposed by developers
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