Abstract
This paper deals with the application of a new algorithm of probabilistic limit and shakedown analysis for 2D structures, in which the loading and strength of the material are to be considered as random variables. The procedure involves a deterministic shakedown analysis for each probabilistic iteration, which is based on the primal-dual approach and the edge-based smoothed finite element method (ES-FEM). The limit state function separating the safe and failure regions is defined directly as the difference between the obtained shakedown load factor and the current load factor. A Sequential Quadratic Programming (SQP) is implemented for finding the design point. Sensitivity analyses are performed numerically from a mathematical model and the probability of failure is calculated by the First Order Reliability Method. Because of use of constant smoothing functions in the ES-FEM, only one Gaussian point is required for each smoothing domain ensuring that the total number of variables in the resulting optimization problem is kept to a minimum compared with standard finite element formulation. Numerical examples are presented to show the validity and effectiveness of the present method.
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References
Bjerager, P.: Plastic systems reliability by LP and FORM. Comput. Struct. 31(2), 187–196 (1989)
Heitzer, M., Staat, M.: FEM-computation of load carrying capacity of highly loaded passive components by direct methods. Nucl. Eng. Des. 193(3), 349–358 (1999)
Heitzer, M., Staat, M.: Reliability analysis of elasto-plastic structures under variable loads. In: Maier, G., Weichert, D. (eds.) Inelastic Analysis of Structures Under Variable Loads: Theory and Engineering Applications, pp. 269–288. Kluwer Academic, Dordrecht (2000)
Krabbenhoft, K., Damkilde, L.: A general non-linear optimization algorithm for lower bound limit analysis. Int. J. Numer. Methods Eng. 56(2), 165–184 (2003)
Liu, G.R., Dai, K.Y., Nguyen, T.T.: A smoothed finite element for mechanics problems. Comput. Mech. 39(6), 859–877 (2007)
Liu, G.R., Nguyen-Thoi, T., Lam, K.Y.: An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids. J. Sound Vib. 320(4–5), 1100–1130 (2009)
Rackwitz, R.: Zuverlässigkeit und Lasten im Konstruktiven Ingenieurbau. Course Notes, Technische Universität, München (1993–2004)
Staat, M., Heitzer, M.: Probabilistic limit and shakedown problems. In: Staat, M., Heitzer, M. (eds.) Numerical Methods for Limit and Shakedown Analysis, Deterministic and Probabilistic Problems, NIC–Series, vol. 15, pp. 217–268. John von Neumann Institute for Computing, Jülich (2003). http://www.fz-juelich.de/nic-series/volume15/nic-series-band15.pdf
Tran, T.N., Kreißig, R., Vu, D.K., Staat, M.: Upper bound limit and shakedown analysis of shells using the exact Ilyushin yield surface. Comput. Struct. 86(17–18), 1683–1695 (2008)
Tran, T.N., Kreißig, R., Staat, M.: Probabilistic limit and shakedown analysis of thin shells. Struct. Saf. 31(1), 1–18 (2009)
Tran, T.N., Liu, G.R., Nguyen-Xuan, H., Nguyen-Thoi, T.: An edge-based smoothed finite element method for primal-dual shakedown analysis of structures. Int. J. Numer. Methods Eng. 82(7), 917–938 (2010)
Tran, T.N., Liu, G.R.: Probabilistic primal-dual shakedown analysis of structures using the edge-based smoothed finite element method. Int. J. Numer. Methods Eng. (2012, submitted)
Tran, T.N., Staat, M.: Shakedown analysis of two dimensional structures by an edge-based smoothed finite element method. In: ECCM 2010, IV European Conference on Computational Mechanics, Paris, France, May 16–21 (2010). http://www.eccm2010.org/complet/fullpaper_1042.pdf
Vu, D.K.: Dual limit and shakedown analysis of structures. Dissertation, Université de Liège, Belgium (2001)
Yan, A.M., Nguyen-Dang, H.: Kinematical shakedown analysis with temperature-dependent yield stress. Int. J. Numer. Methods Eng. 50(5), 1145–1168 (2001)
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Trân, T.N., Staat, M. (2013). An Edge-Based Smoothed Finite Element Method for Primal-Dual Shakedown Analysis of Structures Under Uncertainties. In: de Saxcé, G., Oueslati, A., Charkaluk, E., Tritsch, JB. (eds) Limit State of Materials and Structures. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5425-6_5
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DOI: https://doi.org/10.1007/978-94-007-5425-6_5
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