Novel Computational Implementations for
Ultimate Limit State Analysis and Design

A thesis submitted in partial fulfilment for the
degree of Doctor of Philosophy

Wael Darwich

in the
Engineering Faculty
Department of Civil and Structural Engineering
March 2010

Computational formulations for plastic analysis and design, originally developed in the 1960s and 1970s, have been largely neglected in the last few decades due to the prohibitive computational expense of solving the associated mathematical optimization problems for real-world structures. Instead elastic analysis formulations (e.g. finite element analysis) have became predominant in computational analysis and design practice. However, the computational power and efficiency of mainstream optimization solvers have increased dramatically in recent years and a recently developed adaptive refinement technique proposed by Gilbert and Tyas for plastic layout optimization problems has helped further increase the size of design problem that can be tackled using available computational resources. This allows the whole field of computational limit analysis and design to be revisited. Given that mainstream civil and structural engineers are typically concerned with analysis and design of the collapse limit state, such tools - which can directly model the collapse state - have the potential to be of significant practical use in industry, potentially replacing some of the cumbersome manual techniques still in use today. In this thesis, the adaptive refinement technique proposed by Gilbert and Tyas for truss1 optimization problems, is extended to enable inter-node connections to be 'removed' as well as 'added'. The technique is illustrated in a simple MATLAB script to encourage others to make use of the procedure, and is successfully applied to a series of benchmark problems (e.g. the Hemp cantilever and MBB beam problems). The refinement scheme also lies at the heart of a new object-oriented software framework, which encapsulates the similarities between limit analysis and limit design formulations, and provides an effective means of rapidly developing both limit analysis and design software applications. Essential details of the framework are described and simple examples are provided. Furthermore, as the original goal of the thesis was to develop practical tools for engineers, details of a number of design and analysis software applications, which have been built on the software framework developed in this work, are described. Two of these applications, LimitState:RING and LimitState:GEO, are already in use in industry, demonstrating beyond doubt the practical usefulness of the technology developed.
In the field of truss layout optimization, the ability to treat transmissible loads has also been used for the first time, providing a novel form finding tool based on the 'new, advanced numerical techniques' developed. Using a material that has equal resistance against tension and compression, it was always thought that the parabola was the optimum structural form to carry a uniformly distributed load between two pinned supports. However, using the new numerical tool, a significant scientific finding of this thesis is that the parabola is demonstrably not the most optimum form. Instead a more optimal form comprises a central parabolic rib in the mid-span region, with lobes which appear to be of the form of Hencky-nets near each support. The computed volume is approx. 0.35% lower than that of the single layer parabolic arch. Although the reduction in volume is relatively small, the change in structural layout is radical. This finding overturns a belief widely held by scientists and engineers since the time of Christian Huygens in the 17th century.

Wael Darwich,
18 Mar 2010, 08:32