Long-span bridges have traditionally employed suspension or cable-stayed forms, comprising vertical pylons and networks of cables supporting a bridge deck. However, the optimality of such forms over very long spans appears never to have been rigorously assessed, and the theoretically optimal form for a given span carrying gravity loading has remained unknown. To address this we here describe a new numerical layout optimization procedure capable of intrinsically modelling the self-weight of the constituent structural elements, and use this to identify the form requiring the minimum volume of material for a given span. The bridge forms identified are complex and differ markedly to traditional suspension and cable-stayed bridge forms. Simplified variants incorporating split pylons are also presented. Although these would still be challenging to construct in practice, a benefit is that they are capable of spanning much greater distances for a given volume of material than traditional suspension and cable-stayed forms employing vertical pylons, particularly when very long spans (e.g. over 2 km) are involved.
The well-known ‘ground structure’-based truss layout optimization method has recently been extended to allow accurate modelling of distributed self-weight. By incorporating equally stressed catenaries in the ground structure, non-conservative errors caused by neglecting bending effects within members carrying their own weight are eliminated. However, in cases where the self-weight of a structure has a favourable role in supporting the applied loads, solutions that include convoluted arrangements of overlapping elements may often be generated. To address this, an enhanced layout optimization formulation is proposed that explicitly allows inclusion of favourable unstressed masses, such as counterweights. Frictional supports are also modelled and the cost of abutments and anchorages taken account of in the formulation. The efficacy of the proposed methodology is demonstrated through application to benchmark examples and to the conceptual design of a simplified long-span bridge structure, considering both ground anchored and self-anchored alternatives.
Traditional truss layout optimization employing the ground structure method will often generate layouts that are too complex to fabricate in practice. To address this, mixed integer linear programming can be used to enforce buildability constraints, leading to simplified truss forms. Limits on the number of joints in the structure and/or the minimum angle between connected members can be imposed, with the joints arising from crossover of pairs of members accounted for. However, in layout optimization, the number of constraints arising from 'crossover joints' increases rapidly with problem size, along with computational expense. To address this, crossover constraints are here dynamically generated and added at runtime only as required (so-called lazy constraints); speedups of more than 20 times are observed whilst ensuring that there is no loss of solution quality. Also, results from the layout optimization step are shown to provide a suitable starting point for a non-linear geometry optimization step, enabling results to be obtained that are in agreement with literature solutions. It is also shown that symmetric problems may not have symmetric optimal solutions, and that multiple distinct and equally optimal solutions may be found.
A new interactive truss layout optimization web-app has been developed for educational use. This has been designed to be used on a range of devices, from mobile phones to desktop PCs. Truss designs are first generated via numerical layout optimization and then rationalized via geometry optimization. It is then shown that these designs can be simplified using a computationally inexpensive process that allows the user to control the trade-off between complexity and structural volume. The process involves the use of smooth Heaviside representations of member existence variables, with nodal slack forces employed that allow unstable intermediate truss structures. Full details of the web-app are provided in this contribution, from underlying formulation to cloud computing implementation. A range of numerical examples are used to demonstrate the efficacy of the web-app, and to show how it can potentially be used in educational and practical engineering settings.
<p>This paper explores the potential for structural optimization to be used at the conceptual design stage for long span bridge structures. A discrete numerical layout optimization procedure is used to automatically identify the most structurally efficient forms for prescribed design problems. The study builds on previous work which identified designs ranging from cable stayed to tied arch bridge forms, depending on the specified limiting tensile/compressive stress ratio. Here this work is extended to encompass a wider range of support types and the impact of self-weight is now considered to allow very long spans to be treated, beyond what is realisable today. To permit comparison with traditional bridge forms an idealized 3-span bridge design problem is considered. It is found that optimized forms that consume considerably less material than traditional bridge forms are identified using the optimization procedure described. This theoretically extends the distance that can be spanned when using a given material, notwithstanding that these forms are likely to be challenging to construct using current practices.</p>
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