A nonlinear programming approach, along with finite element implementation, has been developed to perform plastic limit analysis for materials under non-associated plastic flow, so the plastic stability condition of non-standard materials can be directly calculated. In the framework of Radenkovic’s theorems, a decoupled material model with non-associated plastic flow was introduced into kinematic limit analysis so that the classic limit theorems were extended for non-standard plastic flow materials, such as cohesive-frictional materials. A non-associated plastic dissipation power is derived, and a purely kinematic formulation is obtained for limit analysis. Based on the mathematical programming theory and the finite element method, the numerical implementation of kinematic limit analysis is formulated as a nonlinear programming problem subject only to one equality constraint. An extended direct iterative algorithm is proposed to solve the resulting programming problem. The developed method has a wide applicability for limit analysis. The effectiveness and efficiency of the proposed method are validated through numerical examples and the influence of non-associated plastic flow on stability conditions of structures is numerically investigated.
GaoY. On the complementary bounding theorems for limit analysis. Int J Solid Struct1988, 24: 545–556.
11.
ZouainNHerskovitsJBorgesLA. An iterative algorithm for limit analysis with nonlinear yield functions. Int J Solid Struct1993; 30: 1397–1417.
12.
ZhangYGLuMW. An algorithm for plastic limit analysis. Comput Meth Appl M1995; 126: 333–341.
13.
LiuYHCenZZXuBY. A numerical method for plastic limit analysis of 3-D structures. Int J Solid Struct1995, 32: 1645–1658.
14.
CapsoniACorradiL. A finite element formulation of the rigid-plastic limit analysis problem. Int J Numer Meth Eng1997, 40: 2063–2086.
15.
LiHXLiuYHFengXQ. Limit analysis of ductile composites based on homogenization theory. Proc R Soc Lond A2003, 459: 659–675.
16.
MackenzieDShiJBoyleJT. Finite element modelling for limit analysis by the elastic compensation method. Comput Struct1994; 51: 403–410.
17.
MackenzieDBoyleJTHamiltonR. The elastic compensation method of limit and shakedown analysis: a review. J Anal Eng Des2000; 35: 171–188.
18.
PonterARSCarterKF. Limit state solutions based upon linear elastic solutions with a spatially varying elastic modulus. Comput Meth Appl M1997; 140: 237–258.
19.
PonterARSFuschiPEngelhardtM. Limit analysis for a general class of yield conditions. Eur J Mech A Solids2000; 19: 401–421.
20.
ChenLLiuYYangP. Limit analysis of structures containing flaws based on a modified elastic compensation method. Eur J Mech A Solids2008; 27: 195–209.
21.
PastorJThaiTFrancescatoP. Interior point optimization and limit analysis: an application. Commun Num Meth Eng2003; 19: 779–785.
22.
PastorFLouteEPastorJ. Limit analysis and convex programming: a decomposition approach of the kinematic mixed method. Int J Numer Meth Eng2009; 78: 254–274.
23.
LiHXYuHS. Kinematic limit analysis of frictional materials using nonlinear programming. Int J Solid Struct2005; 42: 4058–4076.
24.
MakrodimopoulosAMartinCM. Lower bound limit analysis of cohesive-frictional materials using second-order cone programming. Int J Numer Meth Eng2006; 66: 604–634.
25.
MakrodimopoulosAMartinCM. Upper bound limit analysis using simplex strain elements and second-order cone programming. Int J Numer Anal Methods Geomech2007; 31: 835–865.
ZhangYGLuMW. Computational limit analysis of anisotropic axisymmetric shells. Int. J. Pressure Vessels Piping1994; 58: 283–287.
28.
CapsoniACorradiLVenaP. Limit analysis of orthotropic structures based on Hill’s yield condition. Int J Solid Struct2001; 38: 3945–3963.
29.
LiHXLiuYHFengXQ. Micro/macromechanical plastic limit analyses of composite materials and structures. Acta Mech Solida Sin2001; 14: 323–333.
30.
CorradiLVenaP. Limit analysis of orthotropic plates. Int J Plast2003; 19: 1543–1566.
31.
LiHXYuHS. Limit analysis of composite materials based on an ellipsoid yield criterion. Int J Plast2006; 22: 1962–1987.
32.
CapsoniACorradiLVenaP. Limit analysis of anisotropic structures based on the kinematic theorem. Int J Plast2001; 17: 1531–1549.
33.
CorradiLLuzziLVenaP. Finite element limit analysis of anisotropic structures. Comput Meth Appl M2006; 195: 5422–5436.
34.
FrancescatoPPastorJ. Lower and upper numerical bounds to the off-axis strength of unidirectional fiber-reinforced composite by limit analysis methods. Eur J Mech A Solids1997; 16: 213–234.
35.
ThaiTFrancescatoPPastorJ. Limit analysis of unidirectional porous media. Mech Res Commun1998; 25: 535–542.
36.
TrillatMPastorJ. Limit analysis and Gurson’s Model. Eur J Mech A Solids2005; 24: 800–819.
37.
PastorFThoréPLouteE. Convex optimization and limit analysis: application to Gurson and porous Drucker–Prager materials. Eng Fract Mech. 2008; 75: 1367–1383.
38.
PastorFLouteEPastorJ. Mixed method and convex optimization for limit analysis of homogeneous Gurson materials: a kinematical approach. Eur J Mech A Solids2009; 28: 25–35.
39.
DallotJSabK. Limit analysis of multi-layered plates, part I: the homogenized Love–Kirchhoff model. J Mech Phys Solids2008; 56: 561–580.
40.
DallotJSabK. Limit analysis of multi-layered plates, part II: shear effects. J Mech Phys Solids2008; 56: 581–612.
CocchettiGMaierG. Elastic-plastic and limit-state analyses of frames with softening plastic-hinge models by mathematical programming. Int J Solids Struct2003; 40: 7219–7244.
43.
Tin-LoiF. A GAMS model for the plastic limit analysis of plane frames. Appl Math Modelling1993; 17: 595–602.
44.
Tin-LoiFTangaramvongSXiaSH. Limit analysis of frames involving unilateral supports with frictional contact. Int J Mech Sci2007; 49: 454–465.
45.
Tin-LoiF. Plastic limit analysis of plane frames and grids using GAMS. Comput Struct1995; 54: 15–25.
46.
TangaramvongSTin-LoiF. Limit analysis of elastoplastic frames considering 2nd-order geometric nonlinearity and displacement constrains. Int J Mech Sci2009; 51: 179–191.
47.
ShiJBoyleJTMackenzieD. Approximate limit design of frames using elastic analysis. Comput Struct1996; 61: 495–501.
48.
ShenWQ. Limit analyses of plane frames with a penalty linear programming method. Comput Struct1995, 56: 687–695.
49.
MilaniGMilaniETralliA. Upper-bound limit analysis model for FRP-reinforced masonry curved structures, part I: unreinforced masonry failure surfaces. Comput Struct2009; 87: 1516–1533.
50.
MilaniGMilaniETralliA. Upper-bound limit analysis model for FRP-reinforced masonry curved structures, part II: structural analyses. Comput Struct2009; 87: 1534–1558.
51.
CavicchiAGambarottaL. Two-dimensional finite element upper-bound limit analysis of masonry bridges. Comput Struct2006; 84: 2316–2328.
GilbertMCasapullaCAhmedHM. Limit analysis of masonry block structures with non-associative frictional joints using linear programming. Comp Struct2006; 84: 873–887.
54.
MilaniGLourençoPBTralliA. Homogenised limit analysis of masonry walls, part I: failure surfaces. Comput Struct2006; 84: 166–180.
55.
MilaniGLourençoPBTralliA. Homogenised limit analysis of masonry walls, part II: structural examples. Comput Struct2006; 84: 181–195.
56.
MilaniGLourençoPB. A simplified homogenized limit analysis model for randomly assembled blocks out-of-plane loaded. Comput Struct2010; 88: 690–717.
57.
MihaiLA. A fixed-point approach to the limit load analysis of multibody structures with Coulomb friction. Comput Struct2010; 88: 859–869.
58.
LeuSY. Convergence analysis and validation of sequential limit analysis of plane-strain problems of the von Mises model with nonlinear isotropic hardening. Int J Numer Meth Eng2005, 64: 322–334.
59.
LeuSY. Analytical and numerical investigation of strain-hardening viscoplastic thick-walled cylinders under internal pressure by using sequential limit analysis. Comput Meth Appl M2007; 196: 2713–2722.
60.
LeuSY. Limit analysis of strain-hardening viscoplastic cylinders under internal pressure by using the velocity control: analytical and numerical investigation. Int J Mech Sci2008; 50: 1578–1585.
61.
LeuSY. Investigation of rotating hollow cylinders of strain-hardening viscoplastic materials by sequential limit analysis. Comput Meth Appl M2008; 197: 4858–4865.
62.
ZhangXLiuYZhaoY. Lower bound limit analysis by the symmetric Galerkin boundary element method and the complex method. Comput Meth Appl M2002; 191: 1967–1982.
63.
ZhangXLiuYCenZ. Boundary element methods for lower bound limit and shakedown analysis. Eng Anal Boundary Elements2004; 28: 905–917.
64.
ChenSLiuYCenZ. Lower-bound limit analysis by using the EFG method and nonlinear programming. Int J Numer Meth Eng2008; 74: 391–415.
65.
LeCGilbertMAskesH. Limit analysis of plates using the EFG method and second-order cone programming. Int J Numer Meth Eng2009; 78: 1532–1552.
RadenkovicD. Théorèmes limites pour un materiau de Coulomb à dilatation non standardisée. C R Acad Sci Paris1961; 252: 4103–4104.
68.
Josselin de JongG. Lower bound collapse theorem and lack of normality of strain rate to yield surface for soils. In: Proceedings of IUTAM, Symposium: Rheology and Soil Mechanics, Grenoble, France, 1964, pp. 69–78.
69.
PalmerAC. A limit theorem for materials with non-associated flow laws. J Mec1966; 5: 217–222.
70.
SacchiGSaveMA. A note on the limit loads of nonstandard materials. Meccanica1968; 3: 43–45.
71.
CollinsIF. The upper bound theorem for rigid plastic solids generalised to include Coulomb friction. J Mech Phys Solids1969; 17: 323–338.
72.
SalenconJ. Applications of the theory of plasticity in soil mechanics. Chichester: John Wiley & Sons, 1977.
73.
De SaxceGBousshineL. Limit analysis theorems for implicit standard materials: application to the unilateral contact with dry friction and the non-associated flow rules in soils and rocks. Int J Mech Sci1998; 40: 387–398.
74.
LublinerJ. Plasticity theory. New York: Macmillan, 1990.
75.
BazaraaMSSheraliHDShettyCM. Nonlinear programming: theory and algorithm. Hoboken, NJ: John Wiley & Sons, 2006.
76.
NocedalJWrightSJ. Numerical optimization. New York: Springer, 2006.
77.
HimmelblauDM. Applied nonlinear programming. New York: McGraw-Hill, 1972.
78.
PrandtlL. Über die Härte plastischer Körper. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse1920; 12: 74–85.
79.
HillR. A general theory of uniqueness and stability in elastic-plastic solids. J Mech Phys Solid1958; 6: 236–250.
80.
MaierG. A minimum principle for incremental elastoplasticity with non-associated flow laws. J Mech Phys Solid1970; 18: 319–330.
81.
RanieckiBBruhnsOT. Bounds to bifurcation stresses in solids with non-associated plastic flow law at finite strain. J Mech Phys Solid1981; 29: 153–172.
82.
MrozZ. Non-associated flow laws in plasticity. J Mec1963; 2: 21–42.
83.
RudnickiJWRiceJR. Conditions for the localization of deformation in pressure-sensitive dilatant materials. J Mech Phys Solid1975; 23: 371–394.
84.
FangSCGaoDYSheuRL. Global optimization for a class of fractional programming problems. J Global Optimization2009; 45: 337–353.
85.
GaoDY. Duality principles in nonconvex systems: theory, methods and applications. Dordrecht, The Netherlands: Kluwer Academic Publishers, 2000.
86.
GaoY. Panpenalty finite element programming for plastic limit analysis. Comput Struct1998; 28: 749–755.
