Abstract
Structural optimization based on the shakedown theory is a powerful and promising technique. However, due to the nonlinearities of physical materials and the number of variable loads in real structures, it is computationally complex and time-consuming. To simplify the occurring non-linear, non-convex optimization problems, the paper suggests reducing the number of yield conditions. The so-called a yield criterion of the mean (integral yield condition) is analysed and explained in detail, which allows taking into account one yield condition for the entire finite element instead of multiple point-wise conditions. This approach shows promising results in numerical application to the optimization of a circular plate, considering a possibility of employing the yield criteria of the mean or pointwise yield conditions in different areas of the plate in particular. The methods applied are based on the assumptions of perfect plasticity and small deformations.
Similar content being viewed by others
References
Atkočiūnas J (2012) Optimal shakedown design of elastic–plastic structures. Vilnius Gediminas Technical University, Vilnius, Lithuania, 300 p. doi:10.3846/1240-S
Atkočiūnas J, Borkowski A, König JA (1981) Improved bounds for displacements at shakedown. Comput Methods Appl Mech Eng 28:365–376. doi:10.1016/0045-7825(81)90007-4
Atkočiūnas J, Kalanta S, Čirba S (1994) Integral yield conditions of an equilibrium finite element. In: Lithuanian computational mechanics seminar III (in Lithuanian). pp 7–10
Atkociunas J, Jarmolajeva E, Merkeviciute D (2004) Optimal shakedown loading for circular plates. Struct Multidiscip Optim 27:178–188. doi:10.1007/s00158-003-0308-5
Atkočiūnas J, Ulitinas T, Kalanta S, Blaževičius G (2015) An extended shakedown theory on an elastic–plastic spherical shell. Eng Struct 101:352–363. doi:10.1016/j.engstruct.2015.07.021
Belytschko T (1972) Plane stress shakedown analysis by finite elements. Int J Mech Sci 14:619–625. doi:10.1016/0020-7403(72)90061-6
Blaževičius G, Atkočiūnas J (2015a) Optimal shakedown design of steel framed structures according to standards. Ann Solid Struct Mech. doi:10.1007/s12356-015-0039-5
Blaževičius G, Atkočiūnas J (2015b) Optimal shakedown design of circular plates using a yield criterion of the mean. In: Mechanika’2015: proceedings of the 20th international scientific conference. Kaunas, Lithuania, pp 61–66
Blaževičius G, Rimkus L, Atkočiūnas J (2014) An improved method for optimal shakedown design of circular plates. Mechanika 20:390–394. doi:10.5755/j01.mech.20.4.6542
Bousshine L, Chaaba A, De Saxce G (2003) A new approach to shakedown analysis for non-standard elastoplastic material by the bipotential. Int J Plast 19:583–598. doi:10.1016/S0749-6419(01)00070-5
Capsoni A, Corradi L (1999) Limit analysis of plates-a finite element formulation. Struct Eng Mech 8:325–341. doi:10.12989/sem.1999.8.4.325
Casciaro R, Garcea G (2002) An iterative method for shakedown analysis. Comput Methods Appl Mech Eng 191:5761–5792. doi:10.1016/S0045-7825(02)00496-6
Chen S, Liu Y, Cen Z (2008) Lower bound shakedown analysis by using the element free Galerkin method and non-linear programming. Comput Methods Appl Mech Eng 197:3911–3921. doi:10.1016/j.cma.2008.03.009
Chinh PD (2003) Plastic collapse of a circular plate under cyclic loads. Int J Plast 19:547–559. doi:10.1016/S0749-6419(01)00078-X
Čyras A (1983) Mathematical models for the analysis and optimization of elastoplastic structures. Ellis Horwood, Chichester, 121 p
Franco JRQ, Ponter ARS (1997a) A general approximate technique for the finite element shakedown and limit analysis of axisymmetrical shells. Part 1: theory and fundamental relations. Int J Numer Methods Eng 40:3495–3513. doi:10.1002/(SICI)1097-0207(19971015)40:19<3495::AID-NME222>3.0.CO;2-U
Franco JRQ, Ponter ARS (1997b) A general approximate technique for the finite element shakedown and limit analysis of axisymmetrical shells. Part 2: numerical applications. Int J Numer Methods Eng 40:3515–3536. doi:10.1002/(SICI)1097-0207(19971015)40:19<3515::AID-NME223>3.0.CO;2-W
Gallagher RH (1975) Finite element analysis: fundamentals. Int J Numer Methods Eng 9:732. doi:10.1002/nme.1620090322
Giambanco F, Palizzolo L, Caffarelli A (2004) Computational procedures for plastic shakedown design of structures. Struct Multidiscip Optim 28:317–329. doi:10.1007/s00158-004-0402-3
Hung ND, König JA (1976) A finite element formulation for shakedown problems using a yield criterion of the mean. Comput Methods Appl Mech Eng 8:179–192
Hung ND, Morelle P (1990) Optimal plastic design and the development of practical software. In: Mathematical programming methods in structural plasticity. Springer Vienna, Vienna, pp 207–229
Kačianauskas R, Čyras A (1988) The integral yield criterion of finite elements and its application to limit analysis and optimization problems of thin-walled elastic–plastic structures. Comput Methods Appl Mech Eng 67:131–147. doi:10.1016/0045-7825(88)90121-1
Kączkowski Z (1980) Plyty obliczenia statyczne (Plates, Statical analysis) (in Polish), 2nd edn. Arkady, Warsaw, 467 p
Kaliszky S, Lógó J (1997) Optimal plastic limit and shake-down design of bar structures with constraints on plastic deformation. Eng Struct 19:19–27. doi:10.1016/S0141-0296(96)00066-1
Kaliszky S, Lógó J (2002) Layout and shape optimization of elastoplastic disks with bounds on deformation and displacement. Mech Struct Mach 30:177–192. doi:10.1081/SME-120003014
Koiter WT (1960) General theorems of elastic–plastic solids. In: Sneddon JN, Hill R (eds) Progress in solid mechanics. North-Holland, Amsterdam, pp 165–221
König JA (1984) Stability of the incremental collapse. In: Polizzotto C, Sawczuk A (eds) Inelastic structures under variable loads. Cogras, Palermo, pp 329–344
König JA (1987) Shakedown of elastic–plastic structures. Elsevier Science Ltd., Warsaw. doi:10.1016/B978-0-444-98979-6.50018-9, 224 p
Lellep J, Polikarpus J (2012) Optimal design of circular plates with internal supports. WSEAS Trans Math 11:222–232
Maier G (1969) Shakedown theory in perfect elastoplasticity with associated and nonassociated flow-laws: a finite element, linear programming approach. Meccanica 4:250–260. doi:10.1007/BF02133439
Merkevičiūtė D, Atkočiūnas J (2006) Optimal shakedown design of metal structures under stiffness and stability constraints. J Constr Steel Res 62:1270–1275. doi:10.1016/j.jcsr.2006.04.020
Mróz Z, Weichert D, Dorosz S (eds) (1995) Inelastic behaviour of structures under variable loads. Springer Netherlands, Dordrecht. doi:10.1007/978-94-011-0271-1, 502 p
Palizzolo L, Caffarelli A, Tabbuso P (2014) Minimum volume design of structures with constraints on ductility and stability. Eng Struct 68:47–56. doi:10.1016/j.engstruct.2014.02.025
Polizzotto C, Borino G, Caddemi S, Fuschi P (1991) Shakedown problems for material models with internal variables. Eur J Mech A-Solid 10:621–639
Simon J, Weichert D (2011) Shakedown analysis with multidimensional loading spaces. Comput Mech 49:477–485. doi:10.1007/s00466-011-0656-8
Simon J-W, Kreimeier M, Weichert D (2013) A selective strategy for shakedown analysis of engineering structures. Int J Numer Methods Eng 94:985–1014. doi:10.1002/nme.4476
Staat M, Heitzer M (eds) (2003) Numerical methods for limit and shakedown analysis - deterministic and probabilistic problems. NIC Series, vol. 15. John von Neumann Institute for Computing, Jülich, pp 282
Stein E, Zhang G, Mahnken R (1993) Shakedown analysis for perfectly plastic and kinematic hardening materials. In: CISM. Progress in computernal analysis or inelastic structures. Springer Verlag, Wien, pp 175–244
Szilard R (2004) Theories and applications of plate analysis: classical, numerical and engineering methods. John Wiley & Sons, New Jersey, 1056 p
Tin-Loi F (2000) Optimum shakedown design under residual displacement constraints. Struct Multidiscip Optim 19:130–139. doi:10.1007/s001580050093
Tran TN (2011) A dual algorithm for shakedown analysis of plate bending. Int J Numer Methods Eng 86:862–875. doi:10.1002/nme.3081
Venskus A, Kalanta S, Atkočiūnas J, Ulitinas T (2010) Integrated load optimization of elastic–plastic axisymmetric plates at shakedown. J Civ Eng Manag 16:203–208. doi:10.3846/jcem.2010.22
Weichert D, Maier G (eds) (2000) Inelastic analysis of structures under variable loads. Springer Netherlands, Dordrecht. doi:10.1007/978-94-010-9421-4, 396 p
Weichert D, Maier G (eds) (2002) Inelastic behaviour of structures under variable repeated loads. Springer Vienna, Vienna. doi:10.1007/978-3-7091-2558-8, 396 p
Weichert D, Ponter A (eds) (2009) Limit states of materials and structures. Springer Netherlands, Dordrecht. doi:10.1007/978-1-4020-9634-1, 305 p
Zhou S, Liu Y, Chen S (2012) Upper bound limit analysis of plates utilizing the C1 natural element method. Comput Mech 50:543–561. doi:10.1007/s00466-012-0688-8
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Blaževičius, G., Rimkus, L., Merkevičūtė, D. et al. Shakedown analysis of circular plates using a yield criterion of the mean. Struct Multidisc Optim 55, 25–36 (2017). https://doi.org/10.1007/s00158-016-1460-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-016-1460-z