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DEVELOPMENTS AND APPLICATIONS IN COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru and M.L. Romero
Meshless Methods for Upper Bound and Lower Bound Limit Analysis of Thin Plates
C.V. Le, M. Gilbert and H. Askes
Department of Civil and Structural Engineering, University of Sheffield, United Kingdom
C.V. Le, M. Gilbert, H. Askes, "Meshless Methods for Upper Bound and Lower Bound Limit Analysis of Thin Plates", in B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru and M.L. Romero, (Editors), "Developments and Applications in Computational Structures Technology", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 6, pp 145-168, 2010. doi:10.4203/csets.25.6
Keywords: displacement and equilibrium element-free Galerkin models, meshfree methods, limit analysis, smoothing technique, second-order cone programming, plates.
Limit state techniques are used to design and assess the safety of engineering components and structures. Difficulties in performing incremental-iterative elasto-plastic analyses have motivated the development of numerical variants of classical 'limit analysis' methods, in which the limit load is identified directly.
Current research in the field of limit analysis is focussed on the development of numerical tools which are sufficiently efficient and robust to be used in engineering practice. This places demands on the numerical discretisation strategy adopted, as well as on the mathematical programming tools applied. Numerical procedures based on the finite element method (FEM) are particularly well-established. However, when finite elements are used, the solutions obtained can be highly sensitive to the geometry of the original mesh, particularly in the region of stress or displacement singularities. It is therefore worthwhile exploring a range of alternative methods. Here the element-free Galerkin (EFG) method is applied to limit analysis problems.
In the upper bound formulation, a moving least squares technique is used to approximate the displacement field, which involves only one displacement variable for each EFG node. The total number of variables in the resulting optimisation problem is therefore much smaller than when using finite element formulations involving compatible elements. On the other hand, in the lower bound formulation pure stress/moment fields are approximated using the moving least squares technique, ensuring that the resulting fields are smooth over the entire problem domain. This means that there is no need to enforce continuity conditions at interfaces within the problem domain, which would be a key part of a comparable equilibrium-based finite element formulation.
In order to increase the efficiency of the EFG method, the stabilised conforming nodal integration (SCNI) scheme can be extended to stabilise curvature rates in the upper bound formulation and equilibrium equations in the lower bound formulation. When using a SCNI scheme it is found that far fewer variables and constraints are needed in the optimisation problem than when using a more standard Gauss integration scheme.
Once the displacement or stress fields are approximated and the bound theorems applied, the underlying limit analysis problem becomes a problem of optimisation involving either linear or nonlinear programming. This research continues recent trends by combining second-order cone programming with displacement and equilibrium-based EFG models. The upper bound limit analysis problem for plates is formulated as a minimising the sum of norms problem, which is then cast as a second-order cone programming (SOCP) problem. In the lower bound formulation the von Mises yield criteria is enforced by introducing a second-order cone constraint, ensuring that the resulting optimization problem can be solved using efficient interior-point solvers.
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