Matthew Gilbert

Application of discontinuity layout optimization to plane plasticity problems

A new and potentially widely applicable numerical analysis procedure for continuum mechanics problems is described. The procedure is used here to determine the critical layout of discontinuities and associated upper-bound limit load for plane plasticity problems. Potential discontinuities, which interlink nodes laid out over the body under consideration, are permitted to crossover one another giving a much wider search space than when such discontinuities are located only at the edges of finite elements of fixed topology. Highly efficient linear programming solvers can be employed when certain popular failure criteria are specified (e.g. Tresca or Mohr–Coulomb in plane strain). Stress/velocity singularities are automatically identified and visual interpretation of the output is straightforward. The procedure, coined ‘discontinuity layout optimization’ (DLO), is related to that used to identify the optimum layout of bars in trusses, with discontinuities (e.g. slip-lines) in a translational failure mechanism corresponding to bars in an optimum truss. Hence, a recently developed adaptive nodal connection strategy developed for truss layout optimization problems can advantageously be applied here. The procedure is used to identify critical translational failure mechanisms for selected metal forming and soil mechanics problems. Close agreement with the exact analytical solutions is obtained.

Layout optimization of large‐scale pin‐jointed frames

Computerized layout (or “topology”) optimization was pioneered almost four decades back. However, despite dramatic increases in available computer power and the application of increasingly efficient optimization algorithms, even now only relatively modest sized problems can be tackled using the traditional “ground structure” approach. This is because of the need, in general, for the latter to contain every conceivable member connecting together the nodes in a problem. A simple, but effective solution method capable of tackling problems with large numbers of potential members (e.g. >100,000,000) is presented. Though the method draws on the linear programming technique of “column generation”, since layout optimization specific heuristics are employed it is presented as an iterative “member adding” method. The method requires a ground structure with minimal connectivity to be used in the first iteration; members are then added as required in subsequent iterations until the (provably) optimal solution is found.

The performance of unreinforced masonry walls subjected to low-velocity impacts: experiments

Abstract Impact tests on a total of 21 full-scale unreinforced free-standing brickwork and blockwork walls have been performed in the laboratory using novel drop hammer/rotating quadrant apparatus. The impact loading generated in the laboratory had similar basic characteristics to accidental vehicle impacts (i.e. similar peak impact force and duration). The main variables investigated were wall length, wall thickness, masonry unit type/strength, impact location and end conditions. A number of different failure modes were identified. Mortar bonded walls subjected to a mid-height, mid-length, out-of-plane impact loading were observed to resist the loading in two phases: initial elastic action until fracture occurred, followed by gross displacements, resisted by friction at the base and both out-of-plane and in-plane inertial forces. When weak masonry units were used front face vertical fracture lines either side of the impact location were observed; when stronger units were used front face diagonal fracture lines were observed. Walls impacted close to their ends were observed to be significantly less capable of resisting impact loadings.

Automatic yield-line analysis of slabs using discontinuity layout optimization.

The yield-line method of analysis is a long established and extremely effective means of estimating the maximum load sustainable by a slab or plate. However, although numerous attempts to automate the process of directly identifying the critical pattern of yield-lines have been made over the past few decades, to date none has proved capable of reliably analysing slabs of arbitrary geometry. Here, it is demonstrated that the discontinuity layout optimization (DLO) procedure can successfully be applied to such problems. The procedure involves discretization of the problem using nodes inter-connected by potential yield-line discontinuities, with the critical layout of these then identified using linear programming. The procedure is applied to various benchmark problems, demonstrating that highly accurate solutions can be obtained, and showing that DLO provides a truly systematic means of directly and reliably automatically identifying yield-line patterns. Finally, since the critical yield-line patterns for many problems are found to be quite complex in form, a means of automatically simplifying these is presented.

Optimum structure to carry a uniform load between pinned supports: exact analytical solution

Recent numerical evidence indicates that a parabolic funicular is not necessarily the optimal structural form to carry a uniform load between pinned supports. When the constituent material is capable of resisting equal limiting tensile and compressive stresses, a more efficient structure can be identified, comprising a central parabolic section and networks of truss bars emerging from the supports. In the current article, a precise geometry for this latter structure is identified, avoiding the inconsistencies that render the parabolic form non-optimal. Explicit analytical expressions for the geometry, stress and virtual-displacement fields within and above the structure are presented. Furthermore, a suitable displacement field below the structure is computed numerically and shown to satisfy the Michell–Hemp optimality criteria, hence formally establishing the global optimality of this new structural form.

The performance of unreinforced masonry walls subjected to low-velocity impacts: mechanism analysis

Abstract The development of a rigid-body mechanism analysis capable of estimating the likely response of an unreinforced masonry wall subjected to an out-of-plane impact loading is described. The method automatically identifies the out-of-plane sliding mechanism likely to be most critical for a particular impact location, from a library of five possible mechanisms. Initially, a fracture-line analysis is performed to estimate the out-of-plane force required to crack the wall. Subsequent gross displacements of the wall panels formed are calculated using an iterative dynamic analysis. The fracture energy associated with initial crack opening is included in the calculations and checks are also conducted to determine whether the mode of failure changes during the impact event. Additionally, the possibility that wall rocking can occur, either on its own or simultaneously with the out-of-plane sliding mechanisms is considered. Results from the mechanism analysis model are compared with the experimentally observed results described in a companion paper. It was found that reasonable agreement was obtained for most of the walls subjected to mid-wall impacts, provided ‘dynamic’ rather than quasi-static values for the masonry flexural strength were used in the analysis. Sensitivity studies indicated that wall thickness and the coefficient of base friction were particularly important parameters. The analysis can readily be modified to approximately simulate the interaction between a wall and moving vehicle and is currently being extended to deal with reinforced masonry walls.

Application of discontinuity layout optimization to three-dimensional plasticity problems

A new three-dimensional limit analysis formulation that uses the recently developed discontinuity layout optimization (DLO) procedure is described. With DLO, limit analysis problems are formulated purely in terms of discontinuities, which take the form of polygons when three-dimensional problems are involved. Efficient second-order cone programming techniques can be used to obtain solutions for problems involving Tresca and Mohr–Coulomb yield criteria. This allows traditional ‘upper bound’ translational collapse mechanisms to be identified automatically. A number of simple benchmark problems are considered, demonstrating that good results can be obtained even when coarse numerical discretizations are employed.

Potential Use of Structural Layout Optimization at the Conceptual Design Stage

Despite recent developments in computer-aided design in architecture, both in terms of form generation techniques and performance-based design tools, there still appears to be polarization between the ‘visual’ and the ‘technical’ elements of design. Two causes of this are discussed: long-standing tradition within the discipline and perception of design as primarily a visual exercise. Structural layout optimization is a technique which enables automatic identification of optimal arrangements of structural elements in frames. As the technique appears to have the potential to help reduce the polarization between the visual and the technical elements of design, it can be considered as an ‘integrative’ form generation tool. Applications of the technique are considered via three design examples, demonstrating both its potential and areas where refinement is required before it is suitable for application in practice.

Ultimate Limit State Design to Eurocode 7 using numerical methods

Summary Assessment of the Ultimate Limit State (ULS) in Eurocode 7 is to be carried out using “Design Approach 1” in the UK. In most cases this involves two design checks, one which primarily involves an “action factor” approach (Design Approach 1, Combination 1, termed DA1/1) and one which primarily involves a “material factor” approach (Design Approach 1, Combination 2, termed DA1/2). The latter is generally straightforward to implement in numerical analysis procedures, but the former is potentially more challenging. A survey of the current literature on Eurocode 7 indicates differences of opinion on how best to undertake DA1/1 checks (the same differences of opinion also apply to the “action/resistance factor” Design Approach 2, DA2, checks). This can lead to inconsistent application of Eurocode 7 when undertaking a numerical analysis, which in turn can lead to differences in the resulting design solutions. In this two-part paper, a simple and consistent methodology for undertaking “action/resistance factor” design checks using numerical methods is proposed in Part I, while in Part II the methodology is used to develop a general-purpose design procedure which is then applied to a number of example problems.

Automatic Yield-Line Analysis of Practical Slab Configurations via Discontinuity Layout Optimization

AbstractThe yield-line method provides a powerful means of rapidly estimating the ultimate load that can be carried by a reinforced concrete slab. The method can reveal hidden reserves of strength ...