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Limit analysis of plates and slabs using a meshless equilibrium formulation. (English) Zbl 1202.74019

Summary: A meshless Element-Free Galerkin (EFG) equilibrium formulation is proposed to compute the limit loads which can be sustained by plates and slabs. In the formulation pure moment fields are approximated using a moving least-squares technique, which means that the resulting fields are smooth over the entire problem domain. There is therefore no need to enforce continuity conditions at interfaces within the problem domain, which would be a key part of a comparable finite element formulation. The collocation method is used to enforce the strong form of the equilibrium equations and a stabilized conforming nodal integration scheme is introduced to eliminate numerical instability problems. The combination of the collocation method and the smoothing technique means that equilibrium only needs to be enforced at the nodes, and stable and accurate solutions can be obtained with minimal computational effort. The von Mises and Nielsen yield criteria which are used in the analysis of plates and slabs, respectively, are enforced by introducing second-order cone constraints, ensuring that the resulting optimization problem can be solved using efficient interior-point solvers. Finally, the efficacy of the procedure is demonstrated by applying it to various benchmark plate and slab problems.

MSC:

74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74S30 Other numerical methods in solid mechanics (MSC2010)
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References:

[1] Hodge, Numerical methods for the limit analysis of plates, Transactions of ASME, Journal of Applied Mechanics 35 pp 796– (1968) · Zbl 0172.52003
[2] Capsoni, Limit analysis of plates-a finite element formulation, Structural Engineering and Mechanics 8 pp 325– (1999) · Zbl 0921.73235
[3] Lubliner, Plasticity Theory (1990)
[4] Chan, The collapse load of reinforced concrete plates, International Journal for Numerical Methods in Engineering 5 pp 57– (1972)
[5] Anderheggen, Finite element limit analysis using linear programming, International Journal of Solids and Structures 8 pp 1413– (1972) · Zbl 0255.73045
[6] Faccioli, A finite element, linear programming methods for the limit analysis of thin plates, International Journal for Numerical Methods in Engineering 5 pp 311– (1973) · Zbl 0248.73031
[7] Munro, Yield line method by finite elements and linear programming, The Structural Engineer 56 pp 37– (1978)
[8] Save, North-Holland Series in Applied Mathematics and Mechanics, in: Plastic Analysis and Design of Plates, Shells and Disks (1997)
[9] Le, Limit analysis of plates using the EFG method and second-order cone programming, International Journal for Numerical Methods in Engineering 78 pp 1532– (2009) · Zbl 1171.74466
[10] Chen, Lower-bound limit analysis by using the EFG method and non-linear programming, International Journal for Numerical Methods in Engineering 74 pp 391– (2008)
[11] Krenk, Limit analysis and optimal design of plates with equilibrium elements, Journal of Engineering Mechanics 120 pp 1237– (1994)
[12] Krabbenhoft, Lower bound limit analysis of slabs with nonlinear yield criteria, Computers and Structures 80 pp 2043– (2002)
[13] Krabbenhoft, A general nonlinear optimization algorithm for lower bound limit analysis, International Journal for Numerical Methods in Engineering 56 pp 165– (2003) · Zbl 1116.74404
[14] Chen, A stabilized conforming nodal integration for Galerkin mesh-free methods, International Journal for Numerical Methods in Engineering 50 pp 435– (2001) · Zbl 1011.74081
[15] Sze, Stabilized conforming nodal integration: exactness and variational justification, Finite Elements in Analysis and Design 41 pp 147– (2004)
[16] Wang, Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation, Computer Methods in Applied Mechanics and Engineering 193 pp 1065– (2004) · Zbl 1060.74675
[17] Yoo, Stabilized conforming nodal integration in the natural-element method, International Journal for Numerical Methods in Engineering 60 pp 861– (2004) · Zbl 1060.74677
[18] 2008 http://www.mosek.com
[19] Nielsen, Limit Analysis of Reinforced Concrete Slabs (1964) · Zbl 0235.73015
[20] Nielsen, Limit Analysis and Concrete Plasticity (1998) · Zbl 0558.73024
[21] Wolfensberger, Traglast und Optimale Bemessung von Platten (1964)
[22] Belytschko, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering 37 pp 229– (1994) · Zbl 0796.73077
[23] Zienkiwicz, The Finite Element Method, Volume 1: The Basis (2000)
[24] Liu, An Introduction to Meshfree Methods and their Programming (2005)
[25] Chen, Meshless Methods in Solid Mechanics (2006) · Zbl 1106.74001
[26] Fraeijs de Veubeke, Displacement and equilibrium models in the finite element method, International Journal for Numerical Methods in Engineering 52 pp 287– (2001) · Zbl 0359.73007
[27] Fraeijs de Veubeke, Strain-energy bounds in finite element analysis by slab analogy, Journal of Strain Analysis 2 pp 265– (1967)
[28] Duflot, Dual analysis by a meshless method, Communications in Numerical Methods in Engineering 18 pp 621– (2002) · Zbl 1073.74641
[29] Chen, Regularization of material instabilities by meshfree approximations with intrinsic length scales, International Journal for Numerical Methods in Engineering 47 pp 1303– (2000) · Zbl 0987.74079
[30] Zhu, A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method, Computational Mechanics 21 pp 211– (1998) · Zbl 0947.74080
[31] Krabbenhoft, Formulation and solution of some plasticity problems as conic programs, International Journal of Solids and Structures 44 pp 1533– (2007)
[32] Ciria, Mesh adaptive computation of upper and lower bounds in limit analysis, International Journal for Numerical Methods in Engineering 75 pp 899– (2008) · Zbl 1195.74016
[33] Makrodimopoulos, Upper bound limit analysis using simplex strain elements and second-order cone programming, International Journal for Numerical and Analytical Methods in Geomechanics 31 pp 835– (2006) · Zbl 1196.74014
[34] Braestrup, Yield line theory and concrete plasticity, Magazine of Concrete Research 60 pp 549– (2008)
[35] Fox, Limit analysis for plates: the exact solution for a clamped square plate of isotropic homogeneous material obeying the square yield criterion and loaded by uniform pressure, Philosophical Transactions of the Royal Society of London, Series A 277 pp 121– (1974) · Zbl 0285.73016
[36] Andersen, Computing limit loads by minimizing a sum of norms, SIAM Journal on Scientific Computing 19 pp 1046– (1998) · Zbl 0924.73074
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