Numerical Methods for the Limit Analysis of Plates

Hodge, Philip G.; Belytschko, Ted

doi:10.1115/1.3601308

Cited by 120 publications

(67 citation statements)

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“…Figure 6. Comparison between SQP and SOCP using SCNI solutions, the present result is lower than solutions in [36], [1] and [34] by 13.34%, 8.49% and 0.49% respectively. If a comparison is made in terms of the number of variables in the optimization problem, the present method using EFG has a significantly smaller number than mesh-based approaches; in the EFG method there is only one variable at each node while at least 3 are required when FEM is used (deflection and 2 rotation components) [34].…”

Section: Numerical Examplescontrasting

confidence: 39%

“…The only obvious drawback is that the high order shape functions used in EFG make a priori proof of the strict upper bound status of the solutions difficult (though these can potentially be checked a posteriori ). [1] 49.25 42.86 Lubliner [36] 52.01 -Capsoni and Corradi [34] 45.29 -Andersen et al (mixed element) [10] 44.13…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Computational limit analysis approaches based on the finite element method (FEM) are well-established. Significant contributions include [1][2][3][4]. Once the stress or displacement/velocity fields are approximated and the bound theorems applied, limit analysis becomes a problem of optimization involving either linear or nonlinear programming.…”

Section: Introductionmentioning

confidence: 99%

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Limit analysis of plates using the EFG method and second‐order cone programming

Gilbert

Askes

2009

Numerical Meth Engineering

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Published paperLe, Canh V., Gilbert, Matthew and Askes, Harm (2009) SUMMARYThe meshless Element-Free Galerkin (EFG) method is extended to allow computation of the limit load of plates. A kinematic formulation which involves approximating the displacement/velocity field using the moving least squares technique is developed. Only one displacement variable is required for each EFG node, ensuring that the total number of variables in the resulting optimization problem is kept to a minimum, with far fewer variables being required compared with finite element formulations. A stabilized conforming nodal integration scheme is extended to plastic plate bending problems. The evaluation of integrals at nodal points using curvature smoothing stabilization both keeps the size of the optimization problem small and also results in stable and accurate solutions. Difficulties imposing essential boundary conditions are overcome by enforcing directly displacements at the nodes. The formulation can be expressed as the problem of minimizing a sum of Euclidean norms subject to a set of equality constraints. This non-smooth minimization problem can be transformed into a form suitable for solution using Second-Order Cone Programming (SOCP). The procedure is applied to several benchmark problems and is found in practice to generate good upper bound solutions for benchmark problems.

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Section: Numerical Examplescontrasting

confidence: 39%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Limit analysis of plates using the EFG method and second‐order cone programming

Gilbert

Askes

2009

Numerical Meth Engineering

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“…in which qk β and d are actually Newton directions which assure that a suitable step along them will lead to a decrease of the objective function of the primal problem (18) and to an increase of the objective function of the objective function of the dual problem (12). Based on(25), (26) we can update the vectors of , , ik i qk β and  .…”

Section: Primal-dual Shakedown Algorithmmentioning

confidence: 99%

Shakedown Analysis of Plate Bending Under Stochastic Uncertainty by Chance Constrained Programming

Trần¹,

Tran²,

Matthies³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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“…Developing numerical analysis tools capable of modelling the ultimate behaviour of solids and structures in a rapid and direct manner has therefore become a priority; such methods are often termed 'direct methods', and are the subject of this themed issue. The second part of the issue includes five papers, which cover a wide range of direct methods and applications.In the first paper, Makrodimopoulos (2015) revisits the pioneering finite-element limit analysis formulation for thin plates devised by Hodge and Belytschko (1968). When first published in 1968, the computing facilities and optimisation algorithms available were rudimentary, affecting the quality of results that could be obtained.…”

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confidence: 99%

Editorial

Gilbert¹

2015

Proceedings of the Institution of Civil Engineers - Engineering

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To evaluate the performance of solids and structures, engineers have traditionally relied on methods of analysis based on either elastic or plastic theory in order to respectively model in-service and ultimate behaviour. In the case of elastic analysis, the availability of the stiffness method has meant that linear elastic simulations of solids and structures of any geometry can now be performed quickly and easily on a desktop PC. However, in the case of plastic analysis, computer-based formulations have proved more troublesome to develop. This has meant that engineers who wish to model plastic behaviour have had to rely either on hand calculations, perhaps automated in a simple computer program, or on significantly more complex incremental analysis methods (e.g. non-linear finite-element analysis). Developing numerical analysis tools capable of modelling the ultimate behaviour of solids and structures in a rapid and direct manner has therefore become a priority; such methods are often termed 'direct methods', and are the subject of this themed issue. The second part of the issue includes five papers, which cover a wide range of direct methods and applications.In the first paper, Makrodimopoulos (2015) revisits the pioneering finite-element limit analysis formulation for thin plates devised by Hodge and Belytschko (1968). When first published in 1968, the computing facilities and optimisation algorithms available were rudimentary, affecting the quality of results that could be obtained. However, in the intervening years, results from the paper have frequently been compared with those obtained by researchers employing modern computing facilities and optimisation algorithms. Here Makrodimopoulos (2015) seeks to assess how well Hodge and Belytschko's formulation performs when used in conjunction with modern computing facilities and optimisation algorithms. In fact, it is shown that the Hodge and Belytschko formulation compares favourably with a number of recently proposed methods, and it is also shown that it can be extended to model reinforced concrete slabs, by using the Neilsen failure criteria.In the paper by Dollerup et al. (2015), an optimisation-based design procedure for reinforced concrete slabs is proposed. The lack of suitable computer-based tools for the design of reinforced concrete slabs has led some practitioners to turn to elastic methods, even when ultimate limit state behaviour is of primary importance. This can lead to complex reinforcement layouts and/or to very conservative designs. Dollerup et al.(2015) describe a lower bound limit analysis/design formulation capable of treating multiple load cases, and demonstrate its efficacy by applying it to various slab design problems. It is shown that it can be used to identify slab designs with significantly reduced required moment capacities (and hence; for example, steel reinforcement).In the paper by Smith and Gilbert (2015), the discontinuity layout optimisation (DLO) procedure is applied to problems involving rotational failure in confined geometr...

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Numerical Methods for the Limit Analysis of Plates

Cited by 120 publications

References 0 publications

Limit analysis of plates using the EFG method and second‐order cone programming

Limit analysis of plates using the EFG method and second‐order cone programming

Shakedown Analysis of Plate Bending Under Stochastic Uncertainty by Chance Constrained Programming

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