Abstract
A mathematical programming formulation of strain-driven path-following strategies to perform shakedown and limit analysis for perfectly elastoplastic materials in a FEM context, is presented. From the optimization point of view, standard arc–length strain driven elastoplastic analysis, recently extended to shakedown, are identified as particular decomposition strategies used to solve a proximal point algorithm applied to the static shakedown theorem that is then solved by means of a convergent sequence of safe states. The mathematical programming approach allows: a direct comparison with other nonlinear programming methods, simpler convergence proofs and duality to be exploited. Due to the unified approach in terms of total stresses, the strain driven algorithms become more effective and less nonlinear with respect to a self equilibrated stress formulation and easier to implement in existing codes performing elastoplastic analysis.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Zouain, N.: Shakedown and safety assessment. In: Encyclopedia of Computational Mechanics, vol. 2, pp. 291–334. Wiley, New York (2004)
Armero: Elastoplastic and viscoplastic deformations of solids and structures. In: Encyclopedia of Computational Mechanics, vol. 2. Wiley, New York (2004)
Casciaro, R., Garcea, G.: An iterative method for shakedown analysis. Comput. Methods Appl. Mech. Eng. 191, 5761–5792 (2002)
Garcea, G., Armentano, G., Petrolo, S., Casciaro, R.: Finite element shakedown analysis of two-dimensional structures. Int. J. Numer. Methods Eng. 63(8), 1174–1202 (2005)
Boyd, S.: Lecture Slides and Notes. Courses EE364a and EE364b, http://www.stanford.edu/class/ee364b/lectures.html
Nocedal, Wright, S.J.: Numerical Optimization. Springer, Philadelphia (1997)
Bertsekas: Nonlinear Programming. Athena Scientific, Nashua (2003)
Nemirovski, A., Todd, M.: Interior-point methods for optimization. Acta Numer. 17, 191–234 (2008)
Wright, M.H.: The interior-point revolution in optimization: history, recent developments, and lasting consequences. Bull. Am. Math. Soc. 42(1), 39–56 (2005)
Krabbenhoft, K., Lyamin, A.V., Sloan, S.W., Wriggers, P.: An interior-point algorithm for elastoplasticity. Int. J. Numer. Methods Eng. 69(3), 592–626 (2007)
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)
Pastor, F., Loute, E.: Solving limit analysis problems: an interior-point method. Commun. Numer. Methods Eng. 21(11), 631–642 (2005)
Vu, D.K., Yan, A.M., Nguyen-Dang, H.: A primal–dual algorithm for shakedown analysis of structures. Comput. Methods Appl. Mech. Eng. 193, 4663–4674 (2004)
Nguyen, A.D., Hachemi, A., Weichert, D.: Application of the interior-point method to shakedown analysis of pavements. Int. J. Numer. Methods Eng. 4(75), 414–439 (2008)
Makrodimopoulos, A.: Computational formulation of shakedown analysis as a conic quadratic optimization problem. Mech. Res. Commun. 33, 72–83 (2006)
Makrodimopoulos, A., Martin, C.M.: Lower: bound limit analysis of cohesive frictional materials using second-order cone programming. Int. J. Numer. Methods Eng. 66(4), 604–634 (2006)
Makrodimopoulos, A., Martin, C.M.: Upper bound limit analysis using simplex strain elements and second-order cone programming. Int. J. Numer. Anal. Methods Geomech. 31(6), 835–865 (2007)
Ha, C.D.: A generalization of the proximal point method. SIAM J. Control Optim. 28(3), 503–512 (1990)
Kaneko, Y., Ha, C.D.: A decomposition procedure for large-scale optimum plastic design problems. Int. J. Numer. Methods Eng. 19, 873–889 (1983)
Ponter, A.R.S., Martin, J.B.: Some extremal properties and energy theorems for inelastic materials and their relationship to the deformation theory of plasticity. Int. J. Mech. Phys. Solids 20(5), 281–300 (1972)
Liu, G.R., Nguyen-Thoi, T., Nguyen-Xuan, H.b., Lam, K.Y.: A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems. Comput. Struct. 87, 14–26 (2009)
Bilotta, A., Casciaro, R.: An hight–performance element for the analysis of 2D elastoplactic continua. Comput. Methods Appl. Mech. Eng. 196, 818–828 (2007)
Garcea, G., Trunfio, G.A., Casciaro, R.: Mixed formulation and locking in path following nonlinear analysis. Comput. Methods Appl. Mech. Eng. 165(1–4), 247–272 (1998)
A.S.: The MOSEK Optimization Tools Version 3.2 (Revision 8). User’s Manual and Reference Available from January (2005). http://www.mosek.com
Goldfarb, D., Idnani, A.: A numerically stable dual method for solving strictly convex quadratic programs. Math. Program. 27, 1–33 (1983)
Bathe, K.J.: Finite Element Procedures. Prentice-Hall, New York (1996)
Groß-Wedge, J.: On the numerical assessment of the safety factor of elastic-plastic structures under variable loading. Int. J. Mech. Sci. 39(4), 417–433 (1997)
Zouanin, N., Borges, L., Silveira, J.L.: An algorithm for shakedown analysis with nonlinear yield function. Comput. Methods Appl. Mech. Eng. 191, 2463–2481 (2002)
Stein, E., Zhang, G.: Shakedown with nonlinear strain–hardening including structural computation using finite element method. Int. J. Plast. 8, 1–31 (1992)
Zhang, X., Liu, Y., Cen, Z.: Boundary element methods for lower bound limit and shakedown analysis. Eng. Anal. Bound. Methods 28, 905–917 (2004)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Garcea, G., Leonetti, L. (2013). Decomposition Methods and Strain Driven Algorithms for Limit and Shakedown Analysis. 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_2
Download citation
DOI: https://doi.org/10.1007/978-94-007-5425-6_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-5424-9
Online ISBN: 978-94-007-5425-6
eBook Packages: EngineeringEngineering (R0)