Li_Limit Analysis of Materials With Non-Associated Flow

Article

A nonlinear programming approachto limit analysis of non-associatedplastic flow materials

Mathematics and Mechanics of Solids18(5): 524542The Author(s) 2012Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1081286512444749mms.sagepub.com

Hua-Xiang LiDepartment of Civil Engineering, University of Nottingham, Nottingham, UK

Received 17 February 2012; accepted 18 March 2012

AbstractA nonlinear programming approach, along with finite element implementation, has been developed to perform plasticlimit analysis for materials under non-associated plastic flow, so the plastic stability condition of non-standard materialscan be directly calculated. In the framework of Radenkovics theorems, a decoupled material model with non-associatedplastic flow was introduced into kinematic limit analysis so that the classic limit theorems were extended for non-standardplastic flow materials, such as cohesive-frictional materials. A non-associated plastic dissipation power is derived, anda purely kinematic formulation is obtained for limit analysis. Based on the mathematical programming theory and thefinite element method, the numerical implementation of kinematic limit analysis is formulated as a nonlinear programmingproblem subject only to one equality constraint. An extended direct iterative algorithm is proposed to solve the resultingprogramming problem. The developed method has a wide applicability for limit analysis. The effectiveness and efficiencyof the proposed method are validated through numerical examples and the influence of non-associated plastic flow onstability conditions of structures is numerically investigated.

KeywordsNonlinear programming, limit analysis, non-associated plastic flow, finite element method, cohesive-frictional material

1. Introduction

Limit analysis is a direct method for stability analysis of structures subject to quasi-static loads. It can beused to calculate the plastic limit loads of structures in a direct way and provides a theoretical foundation forengineering design and integrity assessment of structures. Compared with the traditional incremental elastic-plastic method, which involves the calculation of the full history of load-deformation response to find thelimit load of a structure, limit analysis can obtain the collapse load by searching the critical failure pointand requires much less computational effort. Therefore, limit analysis provides a very rigorous method forthe stability analysis of structures. The limit analysis is based on two dual theorems, the static (lower-bound)limit theorem [1] and the kinematic (upper-bound) limit theorem [24]. Due to the complexity of engineeringproblems, numerical methods are often necessary to implement limit analysis. In the last several decades, manystudies have been devoted to the numerical implementation of limit analysis, especially when the mathematicalprogramming technique was introduced and the finite element method was applied. However, most of theseworks are focused on standard materials (rigid, perfectly plastic body with associated plastic flow).

A linear programming (LP) approach was first developed [59]. One advantage of this approach is that itenables solving large scale problems. The fundamental technique in this approach is to discretize the nonlinear

Corresponding author:Hua-Xiang Li, Department of Civil Engineering, University of Nottingham, Nottingham NG7 2RD, UK.Email: [email protected]

Li 525

yield surface and then use the linearized yield condition in limit analysis. Therefore, some additional constraintsand computational errors will be introduced. In the latter, a nonlinear programming (NLP) approach has beenproposed and widely used in limit analysis [10-15], where the nonlinear yield surface was not discretized anddirectly introduced into limit analysis.

Based on the elastic compensation method [16, 17], Ponter et al. [18, 19] developed a linear numericalmethod (the linear matching method) to implement kinematic limit analysis. The linear solutions are definedwith a spatially varying shear modulus, which provides a sequence of upper bounds to the limit load. A sim-ilar approach, the modified elastic compensation method, was presented by Chen et al. [20] to overcome thenumerical singularity and computational errors in the limit analysis of a structure containing flaws.

By introducing the interior point optimization method [21] into limit analysis, Pastor et al. [22] developeda decomposition approach to implement the kinematic limit analysis in the plane strain condition. A mixedkinematic formulation was finally obtained to calculate the dissipation power on discontinuous surfaces interms of velocity and stress.

It should be mentioned that almost all of the above works are based on limit analysis for isotropic, pressure-independent materials that obey the von Mises and Tresca yield criteria. To implement limit analysis for soilmaterials that obey a pressure-dependent yield rule (e.g. the MohrCoulomb or DruckerPrager yield criterion),Li and Yu [23] developed a nonlinear programming approach to implement the kinematic limit analysis for thesematerials, and they found that an upper bound to limit load can be obtained. Recently, Makrodimopoulos andMartin [24, 25] developed a second-order cone programming approach to implement both upper- and lower-bound limit analyses for soil materials. The MohrCoulomb yield criterion under the plane-strain conditionwas assumed.

There are only a few results about anisotropic limit analysis because it is much more difficult to introduce ananisotropic yield criterion into limit analysis. An early work was from Kao et al. [26], who theoretically studiedthe static and kinematic stability conditions of an orthotropic plate. Hills yield criterion was used and theplane-stress model was assumed. Zhang and Lu [27] presented a nonlinear programming approach to solve theupper-bound limit problem of axisymmetric shells, where a two-dimensional Hills yield condition was used.Recently, a more general nonlinear programming approach has been developed for Hills materials [2831] andfor the TsaiWu type of materials [23, 32, 33]. The three-dimensional model for kinematic limit analysis wasfully implemented.

As composite and inhomogeneous materials are increasingly used in engineering, limit analysis ofmicrostructures has drawn much attention in recent years. Francescato and Pastor [34] first presented a lin-ear programming approach with the homogenization technique to apply limit analysis to composites. A convexoptimization [3538] was recently proposed to apply limit analysis to porous materials using the Gurson model,where the homogenization technique for heterogeneous materials was applied. Based on the homogenizationtheory, Li et al. [15, 29] and Li and Yu [31] developed a nonlinear programming method to apply the kinematiclimit analysis for heterogeneous materials obeying the von Mises and ellipsoid yield criteria, where nonlinearyield surface was directly introduced into limit analysis without discretization. Dallot and Sab [39, 40] alsoused the homogenization technique to study the limit status of multilayered plates.

Another important application of limit analysis to engineering is to get specific solutions for some engineer-ing structures so that a complex engineering problem can be simplified and some special structural features canbe fully considered, e.g. for arches [41], for frames [4248], for masonry [4956], for friction contacts [57],and for cylinders [5861].

It should be noted that most of current numerical implementations of limit analysis are carried out byapplying the finite element method. However, Zhang et al. [62, 63] presented a symmetric Galerkin boundarymethod to numerically implement the lower-bound limit analysis. Chen et al. [64] and Le et al. [65] used theelement-free Galerkin (EFG) method to numerically perform limit analysis.

The type of the plastic flow is a key condition for limit analysis. Most of the current works on limit analysisare presented for standard materials (rigid, perfectly plastic model with associated plastic flow). However, itis well known that most soil materials are non-associated with plastic flow. Moreover, due to the diversity ofengineering materials, more and more materials are being found to have non-associated plastic flow. However,up to now, there are only a few preliminary studies on limit analysis for non-associated plastic flow materials.

526 Mathematics and Mechanics of Solids 18(5)

The early theoretical study of limit analysis for non-associated materials was first discussed by Drucker [66]and then later formulated by Radenkovic [67]. The further applications and modifications were presented byJosselin de Jong [68], Palmer [69], Sacchi and Save [70], Collins [71], and Salencon [72]. Recently, De Saxceand Bousshine [73] studied the stability condition of soils and rocks under non-associated plastic flow usingthe bipotential approach. A coupled upper-lower bound formulation was obtained to calculate the limit state ofimplicit standard materials.

The purpose of this paper is to develop a general nonlinear programming approach for limit analysis undernon-associated plastic flow. First, the plastic dissipation power is explicitly expressed in terms of kinematicadmissible velocity by introducing a convex non-associated plastic rule into a general yield criterion. Basedon the nonlinear programming technique, the kinematic limit analysis is formulated as a purely kinematicformulation and the corresponding finite element model is expressed as a nonlinear programming problemsubject only to one equality constraint. The objective function corresponding to plastic dissipation power is to beminimized, which can be solved by an extended direct iterative algorithm. Finally, some numerical examples aregiven to illustrate the validity, and the efficiency of the proposed approach and the influence of non-associatedplastic flow on limit loads are investigated.

2. A non-associated plastic-flow material

Classic limit analysis is based on the extreme theorem that is proposed for the rigid, perfectly plastic materialmodel with the associated plastic-flow rule. To capture more plastic behaviours of materials, such as cohesive-frictional materials, a non-associated plastic-flow rule is used in this paper. First, the yield criterion of materialsis mathematically expressed by a second-order polynomial, which can be used for most of current yield crite-ria, e.g., isotropic material (von Misess criterion), orthotropic material (Hills criterion), generally anisotropicmaterial (TsaiWu criterion), and pressure-dependent material (MohrCoulomb or DruckerPrager criterion).Then, a decoupled plastic-flow law is assumed to capture both associated and non-associated plastic flowof materials. The general expression of plastic-flow rule will be directly introduced into the kinematic limittheorem so that the effect of non-associated plastic flow on the stability condition of materials can be revealed.

For the simplicity of expression when the finite element method is used, the responses of the body to the load(true stress , true strain , and displacement u) are represented as column vectors, i.e. in the three-dimensionalmodel, = [11, 22, 33,

212,

223,

213]T, = [11, 22, 33,

212,

223,

213]T, and u= [u1, u2,

u3]T.

2.1. A general yield criterion

Most yield criteria (isotropic, anisotropic, or pressure-dependent) of materials can be expressed in a generalform as follows:

F( ) = TP + TQ 1 = 0 (1)where F( ) defines a yield function in terms of strength parameters. P and Q are coefficient matrices and arerelated to the strength properties of the material.

Generally speaking, the physical laws of materials must be objective, i.e. frame indifference. Therefore, inplasticity theory, the general yield condition should depend only on the invariants of a stress tensor. However,for a specific material on a determined topic, an explicit expression in terms of stress tensor is mostly used.Moreover, for finite element implementation, all discretized variables must be at the bottom level. So it is con-venient to use equation (1) as the yield expression from the start of the analysis. Then, the material coefficientmatrices P and Q in equation (1) will be determined from the physical law, which is originally expressed interms of the invariants of the stress tensor for any specific material.

For example, the MohrCoulomb and DruckerPrager yield criteria are often used for cohesive-frictionalmaterial, which is normally accompanied by a non-associated plastic-flow rule. And these criteria are mostlyexpressed as follows:

F(I1, J2) = 0I1 +J2c0 = 0 (2)

Li 527

where I1 is the first invariant of the stress tensor, J2 is the second invariant of the deviatory stress tensor, 0 andc0 are the strength parameters of the material.

The general yield equation (1), can be used for the above material, equation (2), when the coefficientmatrices P and Q are chosen as

P =

13203c20

1+6206c20

1+6206c20

0 0 0

1+6206c20

13203c20

1+6206c20

0 0 0

1+6206c20

1+6206c20

13203c20

0 0 0

0 0 0 12c20

0 0

0 0 0 0 12c20

0

0 0 0 0 0 12c20

(3)

Q =[

20c0

, 20c0 ,20c0

, 0, 0, 0]T

(4)

In a similar way, it is not difficult to determine the material coefficient matrices P and Q, when the general yieldequation (1), is used for any other material, e.g. the von Mises- or Hill-type material.

It should be noted that when Q = 0, the general yield equation (1) is reduced for a pressure-independentmaterial. And for a generally anisotropic material, such as TsaiWu-type material, the coefficient matrices Pand Q can be directly determined. For example, the quadratic TsaiWu criterion is normally expressed as thefollowing form

F( ) = Aijij + Bijklijkl 1 = 0 (5)where Aij and Bijkl (i, j, k, l = 1, 2, 3) are the material-strength parameters. The material-strength parameter Aijcan be experimentally determined by a pure tension/compression or shear test, and the material parameter Bijklreflects the interaction effects of stress components.

When the stresses are expressed in the column vector, the TsaiWu yield criterion, equation (5), has theexact same expression of equation (1), and it is not difficult to find the relation between the coefficients P andQ and the material parameters Aij and Bijkl.

2.2. Non-associated plastic-flow rule

When the external force is beyond the yield stress of a material, plastic flow and plastic deformation will occur.Accordingly, a plastic-flow rule is required to determine the pattern of deformation. Although for metal materi-als it is normally assumed that the plastic flow is associated, this assumption may conflict with the experimentalresults for some other materials, e.g. soil or granular materials. In practice, for soil or granular materials, anon-associated plastic flow is often adopted by using the dilation angle in the plastic-flow rule instead of thefriction angle used in the yield criterion.

The plastic-flow rule determines both the direction and magnitude of plastic strain-rate and is normallyexpressed by a plastic potential ( ), as in

p = ( )

(6)

where ( ) denotes a plastic potential function that resembles the yield function F( ), and is a non-negativeplastic proportionality factor. If the associated plastic flow is assumed for a material, ( ) will be equivalent toF( ). However, in this paper, the non-associated plastic flow is adopted, i.e.( ) = F( ), so that the developedmethod has a general applicability. A similar second-order polynomial is used in this paper to express the plasticpotential function

( ) = TPf + TQf 1 = 0 (7)


where Pf and Qf are the coefficient matrices of plastic flow, and Pf is symmetric.For a specific material, there may be an indirect way to determine the plastic potential function. For example,

for a material with MohrCoulomb or DruckerPrager yield criterion, the plastic flow can be determined bydecomposing the plastic potential into two partsthe deviatory plastic flow and the bulk plastic flow as

p = (

D(S)

S+

B(m)

mIB)

(8)

where D(S) and B(m) denote the deviatory- and bulk-plastic potential functions, respectively, S and m arethe deviatory and mean stresses, respectively, and the coefficient matrix IB = [1, 1, 1, 0, 0, 0]T.

Equation (8) is frequently used to express the non-associated plastic flow of cohesive-frictional materials,e.g. soils. Accordingly, the plastic strain-rate based on the non-associated plastic flow for the type of MohrCoulomb or DruckerPrager material can be calculated as

p = (S

c2+ 18(tan)

2m

c2IB)

(9)

Then, based on the equations (6) and (7), the coefficient matrices Pf and Qf in the plastic-flow rule can bedetermined. In this way, the matrix Pf can be ensured to be a positive definite or semi-definite.

2.3. Plastic dissipation power

According to the plastic-flow potential, equation (7), and plastic-flow rule, equation (6), the plastic strain-ratecan be expressed as

p = 2Pf + Qf (10)Then, the stress vector at the yield surface can be expressed in terms of the strain-rate as

= 12

(Pf )1p 12(Pf )1Qf (11)

Since Pf is a positive definite or semi-definite matrix, (Pf )1 can be uniquely determined when Pf is non-singular. However, the matrix Pf may be semi-singular when Pf is a positive semi-definite. For these materials,(Pf )1 can be approximately determined as

(Pf + I)1, where I is the identity matrix of the same order with

the matrix Pf and is a very small real number. Theoretically, the smaller is, the more accurate the numericalcalculation will be. However, due to the storage limitation of a computer, in case of data overflow in a numericalcomputation, there is a limit for the choice of the small value . In practice, the value , which is between106 pmin and 1012 pmin, can guarantee both numerical precision and safe data storage in a computer program,where pmin is the smallest diagonal component of the matrix Pf .

Since the stress at the yield surface must be satisfied with the yield criterion, as in equation (1), then it isnot difficult to find the non-negative plastic-flow factor by introducing equation (11) into the yield criterionin equation (1), and finally the non-negative plastic-flow factor can be obtained as

= (p)T(Pf )1P(Pf )1p

(p)T(Pf )1(4P + QQT)(Pf )1p + ((Qf )T(Pf )1P(Pf )1 QT(Pf )1)p(12)

Then, the plastic-dissipation power for a material obeying the yield criterion in equation (1) and the plastic-flow rule in equation (7) can be expressed as follows

D(p) = Tp=(

12 (P

f )1p 12 (Pf )1Qf)T

p

= 12 (p)T(Pf )1p 12 (Qf )T(Pf )1p(13)

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Finally, the plastic-dissipation power for a decoupled yieldplastic flow material model is expressed in termsof a pure strain-rate field without any stress fields. Once the kinematically admissible velocity is obtained, theplastic-dissipation power can then be determined.

3. Kinematic limit analysis based on non-associated plastic flow

The classic kinematic limit theorem for standard materials was first proposed by Hill [2] and Drucker [3, 4] todefine the stability condition of materials by which the plastic limit load of materials can be directly calculated.Later, the theorem was extended for non-associated plastic-flow materials by Radenkovic [67].

3.1. Classical kinematic limit theorem

The collapse load of a structure under quasi-static loading can be found by applying the kinematic theoremof limit analysis [24], which states that among all kinematically admissible velocities, the real one yields thelowest rate of plastic-dissipation power. In terms of the kinematically admissible velocities, the kinematic limittheorem can be expressed as follows

( t

tTud +VfTudv

)VD(p)dv (14)

where is the limit load multiplier, t is the reference load of external tractions, f is the reference load of bodyforces, u is the displacement velocity, p is the plastic strain rate,D(p) represents a function for the rate of theplastic-dissipation power in terms of the admissible strain rate p, the superscript * stands for a parametercorresponding to the kinematically admissible velocity, t denotes the traction boundary, and V represents thespace domain of the structure.

If the body force is omitted and the geometry-compatible conditions are introduced, the kinematic limittheorem represented in equation (14) can be re-expressed as the following

t

tTud V D(p)dvs.t. p = (u) in V

u = 0 on u(15)

where s.t. is the abbreviation of subject to, u denotes the displacement boundary, and (u) is a lineardifferential operator that defines the geometry-compatibility condition of deformation. The purpose of thekinematic limit analysis is to find the minimum, optimised limit multiplier with t being the limit load of thestructure.

3.2. Radenkovics theorems

The classic limit theorems of limit analysis are proposed based on the assumption of the rigid, perfectly plasticmodel with associated plastic flow. To apply them to non-standard materials, some special constraints mustbe imposed on the material model so that the bases of the limit theorems are still valid. It is not possiblenow to theoretically prove a sufficient and necessary condition on limit analysis for a general non-standardmaterial model. However, it is feasible and important to find a sufficient condition for a special kind of non-standard material. In order to extend limit analysis to non-standard materials such as non-associated plasticflow materials, Radenkovic proposed two theorems [67, 74] for further application of classic limit analysis tosoil materials.

Radenkovics first theorem states that the limit loading for a body made of a non-standard material isbounded from above by the limit loading for the standard material obeying the same yield criterion.

Radenkovics second theorem states that the limit loading for a body made of a non-standard material isbounded from below by the limit loading for the standard material obeying the yield criterion g( ) = 0, whichhas the following properties:


1. g( ) is a convex function.2. The surface g( ) = 0 lies entirely within the yield surface F( ) = 0.3. To any with F( ) = 0, there corresponds a such that p is normal to the surface g( ) = 0 at and

( )Tp 0.

Radenkovics first theorem defines an upper bound to the plastic limit load of a non-standard material, wherethe upper bound is the plastic limit load of a standard material that has the same yield condition as the non-standard material. Radenkovics second theorem defines a lower bound to the plastic limit load of a non-standardmaterial, where the lower bound is the plastic limit load of a standard material that has the same plastic flowpotential as the non-standard material. It is well known that limit analysis of standard materials provides a strictplastic limit load. Therefore, for a non-associated plastic-flow material, if its plastic-flow potential ( ) = 0is satisfied with the above three conditions, the limit load obtained from using both the non-associated plasticflow ( ) = 0 and the yield function F( ) = 0 should lie between the upper and lower bounds of Radenkovicsprediction on the two corresponding standard materials.

In practice, Radenkovics theorems are often used as a criterion to create a non-associated plastic-flowsurface for a non-standard material so that the sufficient condition of limit theorems for this kind of materialis satisfied. For examples, for MohrCoulomb or the DruckerPrager type of material, which is often used fornon-associated plastic-flow materials in civil engineering, the dilation angle in the plastic-flow rule must beless than the frictional angle in the yield condition. The non-associated plastic-flow rule, which is defined in theabove section and frequently used for cohesive-frictional materials, is based on this strict requirement so that theplastic-flow function is satisfied with the full conditions of g( ) in Radenkovics second theorem. Consequently,the kinematic limit analysis as given in equation (14) with the plastic dissipation power in equation (13) canobtain a kinematic limit solution for the non-associated plastic-flow material.

3.3. Nonlinear programming of kinematic limit analysis

The kinematic limit equation (15) provides a direct calculation of the upper bound ( t) to the limit load ofa structure. Based on the solution of the plastic-dissipation power as given in equation (13) for a non-standardmaterial, the kinematic limit analysis for a material with non-associated plastic flow can then be expressed asfollows

t

tTud V(

12 (

p)T(Pf)1p 12 (Qf)T(Pf)1p)dv

s.t. p = (u) in Vu = 0 on u

(16)

The kinematic limit analysis as given in equation (16) is actually an optimization problem subject to equalityconstraints. Based on the mathematical programming theory, the kinematic limit analysis, equation (16), canbe formulated as the following nonlinear programming equation:

= minp

V

(12 (

p)T(Pf)1p 12 (Qf)T(Pf)1p)

dv

s. t. t

tTud = 1p = (u) in Vu = 0 on u

(17)

When the plastic flow factor is found by equation (12), the kinematic limit analysis, as in equation (17), canthen be expressed in terms of purely kinematic variables as

= minp

V

((

(p)TRqp+T fp)(p)T(Pf)1p2(p)TRfp

12 (Qf)T(Pf)1p)

dv

s. t. t


(18)

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where the coefficient matrices Rq, Rf and T f are defined as

Rf = (Pf)1P(Pf)1 (19)Rq = (Pf)1(4P + QQT)(Pf)1 (20)T f = (Qf)TRf QT(Pf)1 (21)

Finally, the kinematic limit analysis under non-associated plastic flow is formulated as a minimum optimizationproblem subject to equality constraints. The objective function is nonlinear. For the purpose of a potentialiteration algorithm, a ratio factor f(p) is defined as follows

f = (p)T(Pf)1p

(p)TRfp(22)

Then, a reduced form for the kinematic limit analysis, equation (18), can be obtained as

= minp

V

(f

2 ((p)TRqp + T fp) 12 (Qf)T(Pf)1p

)dv

s. t. t


(23)

The ratio-factor f(p) is a solution/material-dependent variable that is related to the plastic-flow type of mate-rials. For a standard (associated plastic flow) material, due to P = Pf and Rf = (Pf)1, the ratio-factor f(p)will become 1.0 (f 1.0) for any deformation conditions.

4. Finite element modelling

The traditional displacement-based finite element method is used in this paper to implement the numericalcalculation for the kinematic limit analysis as given in equation (23). Based on the finite element technique, astructure is discretized by finite elements Ve(V =

Ne=1 Ve), and then the displacement velocity and strain rate

can be interpolated in terms of an unknown nodal displacement velocity vector:

ue(x) = Ne(x)e (24)e(x) = Be(x)e (25)

where, with reference to the eth finite element, e is the nodal displacement column vector, Ne is theinterpolation function, and Be is the strain function.

By means of the Gaussian integration technique, the objective function in equation (23) can be discretizedand expressed in terms of the nodal displacement velocity as follows

V

(f


)dv

=Ne=1

Ve

(f

2

((Bee

)TRq(Bee

)+ T f (Bee))

12 (Qf)T(Pf)1(Bee

))dv

=Ne=1

IGi=1

(e)i|J |i(

f

2

(Te(BTe)iRq (Be)i e + T f (Be)i e

) 12 (Qf)T(Pf)1 (Be)i e

)

=Ne=1

IGi=1

(e)i|J |i(

f

2

(Te (Ke)i e + (He)i e

) 12 (Ge)i e

)(26)

where (e)i is the Gaussian integral weight at the ith Gaussian integral point in the element e, |J |i is the deter-minant of the Jacobian matrix at the ith Gaussian integral point, IG is the number of Gaussian integral pointsin the finite element e, and

Ke = BTe RqBe (27)


He = T fBe (28)Ge = (Qf)T(Pf)1Be (29)

By introducing the transformation matrix of each element Ce, the nodal displacement velocity vector e foreach element can be expressed by the global nodal displacement velocity vector for the structure as

e = Ce (30)Then, equation (26) can be expressed in terms of the global displacement velocity as follows

V

(f


)dv

=Ne=1

IGi=1

(e)i|J |i(

f

2

(Te (Ke)i e + (He)i e

) 12 (Ge)i e

)

= iI

i|J |i(

f

2

(TKi + Hi

) 12Gi

)(31)

where I denotes the set of all Gaussian integral points of the FE-discretized structure, and

Ki = CTe (Ke)iCe (32)

Hi = (He)iCe (33)Gi = (Ge)iCe (34)

Accordingly, the normalization condition in the kinematic limit analysis as defined in equation (23) can bediscretized as

FT = 1 (35)where F is the column vector of the equivalent reference nodal loads.

Finally, the finite element modelling of the kinematic limit analysis can be expressed as the followingdiscretized, nonlinear programming problem:

= minp

iI

i|J |i(

f

2

(TKi + Hi

) 12Gi

)s. t. FT = 1

(36a, b)

After the displacement boundary condition is imposed by utilizing the conventional finite element technique,a minimum optimized upper bound to the plastic limit load multiplier of the structure can be obtained bysolving the above mathematical programming equation. The plastic limit load of the structure is given by F.

5. Numerical algorithm

The kinematic limit analysis as given in equation (36) is a nonlinear, minimum optimization problem subject toonly one equality constraint. It can be solved by a direct iterative algorithm that was developed [15, 23, 29, 31]for limit analysis of an associated plastic flow material. To extend it for non-associated plastic-flow materials,an additional iterative control factor needs to be introduced. The basic strategy of the numerical algorithm isgiven below. First, based on the mathematical programming theory, the equality constraint of the normalizationcondition in equation (36b) is introduced into the optimization problem by means of the Lagrangian method[75, 76]. As a result, an unconstrained minimum optimization problem is obtained as follows

(, LF) =iI

i|J |i(

f

2

(TKi + Hi

) 1

2Gi

)+ LF(1 FT) (37)

where LF are the Lagrangian multiplier.

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Then, according to the KuhnTucker stationary condition [75, 77], once applying

= 0, LF

= 0 toequation (37), the following set of equations is obtained to solve the kinematic limit analysis equation (36):

iI

i|J |i(

f

2

(Ki TKi

+ HTi)

12GTi)

LFF = 0FT = 1

(38a, b)

It is quite difficult to directly solve the set of equations (38a,b) because it is nonlinear and non-differentiableas well. An iteration technique is needed to linearize the equation set (38a,b) and to distinguish the non-differentiable areas. In order to implement an iteration technique, the set of equations (38a,b) is re-expressedas follows

iI

i|J |i(

f

2

(ICPKi + HTi

) 12GTi)

LFF = 0FT = 1

(39a, b)

where ICP is the coefficient parameter, which is defined by

ICP =(

TKi

)1(40)

and the superscript ICP indicates that ICP is an iteration control parameter.By solving the linearized equation (39a,b), the variable can be determined. Then the limit load multiplier

can be calculated as

=iI

i|J |i(

f

2

(TKi + Hi

) 1

2Gi

)(41)

In order to trigger an iteration to linearize equation (38a,b), two iteration seeds are defined as

(ICP

)0= 1.0 (42)

(f)0= 1.0 (43)

where the subscript 0 denotes that the variables are determined at step 0. During the iterative calculation, thesetwo iteration factors need to be updated at each iterative step.

Once the iteration starts, it will be repeated until the following convergence criteria are satisfied

|h+1 h||h+1| 1;

h+1 hh+1 2 (44a, b)where h is the number of the iterative step, and 1 and 2 are the computational error tolerances.

Once the convergence condition in equation (44) is attained, the limit load multiplier at the current iterativestep is the optimized solution to the real limit load. More details about the direct iterative algorithm can befound in the work of Li et al. [15, 23, 29, 31].

Mathematically speaking, equation (18) of kinematic limit analysis is a fractional programming subjected toequality constraints, and the finite element approximation will lead to the so-called sum-of-ratios problems,which have been studied extensively in global optimization for several decades [84]. The uniqueness of theglobal optimal solution is governed by the convexity of the problem. As it was indicated in [84], the fractionalprogramming problems are generally non-convex and may have multiple local optima. Therefore, such prob-lems are usually NP-hard. In order to solve such problems, a canonical duality theory has been proposed inFang et al. [84]. The direct iterative algorithm proposed in this paper is based on the strict requirements onthe material laws and purely kinematic analysis. The initial ratio factor f and the initial parameter ICP aretriggered based on the upper-bound kinematic assumption. If a new triggering mechanism is adopted, these twoiterative factors must be carefully chosen. A general theory for these kinds of nonlinear programming problems


can be found in Fang et al. and Gao [84, 85]. Moreover, for a similar problem to equation (23), Gao [86] pro-posed a pan-penalty method to solve the difficulty of the flat objective function. Fundamentally, the pan-penaltymethod is a mixed static-kinematic dual approach to plastic limit analysis. The motivation of this paper is todevelop a purely kinematic approach for plastic limit analysis, and therefore the corresponding direct iterativealgorithm can be regarded as a purely kinematic solution. If the static theorem of limit analysis is introducedinto the analysis, a similar solution to Gaos will be obtained.

6. Applications

Soil is a typical kind of non-associated plastic-flow material. Due to the theoretical complexity and numericaldifficulties in non-associated flow analysis, most of limit analyses of soil materials are still based on associatedplastic flow. The purpose of this section will give some preliminary investigation about how non-associatedplastic flow affects the plastic limit load of a structure.

6.1. Stability of a wedge

The first numerical application of the developed method is for the stability analysis of a simple wedge underuniform pressure, as shown in Figure 1. The plane-strain condition is assumed. The weightless soil materialobeys the MohrCoulomb yield criterion. When the wedge angle = 0, the solution becomes the stabilityanalysis of a half space under uniform pressure. First, the numerical simulation is implemented for = 0and the associated plastic flow ( = ) using the developed approach. The numerical results for this case arepresented in Figure 2, where ps denotes the limit pressure. The numerical results agree well with Prandtls exactsolution [78]. In order to investigate how non-associated plastic flow affects bearing capacity of a structure,more numerical simulations are implemented for different dilation angles ( = /2 and = 0), and thenumerical results are presented in Figure 3. It can be seen in Figure 3 that non-associated plastic flow bringsa decreased collapse load. The plastic flow behaviour has a significant effect on the bearing capacity of astructure, and the classic associated plastic flow may overestimate the limit load.

More numerical simulations are implemented for a wedge with the angle = 60. Due to the symmetry, itis only necessary to simulate half of the wedge, and the finite element mesh is shown in Figure 1. Numericalresults for both associated and non-associated plastic flow are presented in Figure 4, which further shows thatthe bearing capacity of a structure drops when the degree of non-associated plastic flow increases.

6.2. A layered thick-walled cylinder under internal pressure

Components in engineering structures, such as pipes or tunnels, can be simplified as thick-walled cylinders.In this section, the limit load of a two-layered, thick-walled cylinder under internal pressure is calculated byapplying the proposed approach. The size of the simulated cylinder here is defined as: the inner radius of thecylinder R0 = 2, the outer radius of the first layer R1 = 4, and the outer radius of the second layer R2 = 5 inan arbitrary unit. The outer layer of the two-layered cylinder is made of an anisotropic material that is assumedto obey the Hill yield criterion with associated plastic flow. The inner layer of the two-layered cylinder is madeof a pressure-dependent material that here is assumed to obey the DruckerPrager yield criterion with non-associated plastic flow. The plane-strain model is adopted for the numerical simulation. In the plane-strainmodel, the strength parameters 0 and c0 in the DruckerPrager criterion can be expressed by the friction angle and cohesion c in the MohrCoulomb criterion as follows

0 = tg9 + 12tg2 (45)

c0 = 3c9 + 12tg2 (46)

The materials of the two layers of the cylinder are chosen for the numerical simulation as follows:

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Figure 1. The finite element mesh of a half wedge under uniform pressure

Figure 2. The limit loads of the wedge under associated plastic flow ( = 0)

1) the inner, the DruckerPrager material: the cohesion c with various friction angles ( = 0 45)2) the outer, Hills strength material in terms of the cohesion c of the inner material: rr =4.0c, =3.2c,

zz =5.0c, and r=2.5c.

Due to the symmetry of the structure and force, only one quarter of the structure is to be numerically analyzed.The finite element mesh of one-quarter of the cylinder is plotted, as shown in Figure 5. The convergencetolerances in the numerical simulation are chosen as 1 = 2 = 103. Using the developed approach, the


Figure 3. The limit loads of the wedge under non-associated flow ( = 0)

Figure 4. The limit loads of the wedge ( = 60)

limit load of this cylinder subjected to internal pressure is numerically calculated, and the numerical results areshown in Figure 6. The numerical results further prove that the type of plastic flow has a dramatic effect on thebearing capacity, and non-associated plastic flow may have a smaller limit load than associated plastic flow.

Figures 7 and 8 further show how the non-associated plastic flow affects the bearing capacity and limitanalysis, where the degree of non-associated flow is defined by d=(-)/. As the degree of non-associatedplastic flow increases, the bearing capacity of structures dramatically decreases.

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Figure 5. The finite element mesh of one-quarter of the cylinder

Figure 6. The limit loads of the cylinder under internal pressure

7. Conclusion

A general nonlinear approach is proposed to implement kinematic limit analysis for materials obeying non-associated plastic flow. The dissipation power based on a general yield criterion with non-associated plastic


Figure 7. Effect of non-associated plastic flow on limit loads

Figure 8. Effect of non-associated plastic flow on limit loads

flow is explicitly expressed in terms of the kinematic admissible velocity, and the nonlinear yield surface isdirectly introduced into the kinematic limit theorem. Based on the mathematical programming theory, the finiteelement model of the kinematic limit analysis is finally formulated as a nonlinear programming problem subjectto only one equality constraint. Therefore, the computational effort of the proposed approach is very modest.The extended direct iterative algorithm can solve the resulting nonlinear programming problem. The numericalresults reveal that the associated plastic-flow rule may overestimate the bearing capacity of structures and anon-associated plastic-flow rule is necessary to calculate the plastic limit load of materials more accurately.The proposed approach provides a general methodology for the application of the classic limit theorem to amaterial with different plastic behaviours.

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It should be emphasized that for the MohrCoulomb or DruckerPrager types of soil materials, the non-associated plastic-flow function should have the same form as the yield function but with a dilation angle inthe plastic-flow rule less than the friction angle in the yield criterion. The full conditions of the non-associatedplastic flow, as stated in the Radenkovic theorems, must be satisfied, and then the limit load of the developedapproach in this paper can be regarded as a kinematic solution to the collapse load of structures.

However, it should be also emphasized that non-associated plastic flow violates the normality condition,and this may bring some theoretical difficulties in the implementation and application of limit analysis for thesekinds of materials. Due to the lack of extreme theorems for limit analysis of any non-standard materials, it isnot possible to theoretically prove that the kinematic limit analysis can still provide a strict upper bound to thetrue plastic limit load for non-associated plastic-flow materials. Nevertheless, based on the sufficient boundingconditions, the kinematic solution of limit analysis presented in this paper provides a quasi-upper bound to thetrue collapse load of structures.

Uniqueness of solution for boundary-value problems is a fundamental issue in plasticity analysis for non-associated plastic-flowmaterials. A pioneering work was presented by Hill [79], who first formulated a rigoroustheory of the uniqueness and stability condition for standard materials. Later, Maier [80] proposed a weak,yet sufficient, condition for the uniqueness of non-associated flow and working-hardening/softening materials,which involved Greens function of the infinitesimal elasticity theory. Raniecki and Bruhns [81] defined a one-parameter family of linear comparison solids and proposed a sufficient uniqueness condition for non-associatedplastic-flow materials. All of these fundamental works provide insight into uniqueness and the stability condi-tion of different plastic-flow materials. However, it is still hard to theoretically and rigorously prove the generalcondition of uniqueness for non-associated plastic-flow materials. This paper is based on the strict sufficientcondition of uniqueness for non-associated plastic flow as defined by the Radenkovic theorems. Palmer [69]also proposed a similar approach to construct a non-associated plastic potential so that a sufficient condition ofuniqueness can be obtained.

The non-associated plastic-flow law used in this paper is actually a three-parameter (cohesion, friction angle,and dilation angle) material model, which is frequently used for soil materials. The yield function and theplastic-potential function are separately expressed. However, according to thermodynamic principles, someconstraints are required to be imposed on the intrinsic relation between these two functions. More detailsabout this analysis can be found in the works of Palmer, Maier, Raniecki and Bruhns, and Mroz [69, 8082].These necessary conditions also construct the basis of the Radenkovic theorems. It should be noted that thereexist other kinds of non-standard plastic-flow laws, e.g. the fissure model proposed by Rudnicki and Rice [83]for dilatant materials. This is actually a non-coaxial plastic-flow theory and can be regarded as one kind ofgeneralized non-associated plastic-flow rule. In this fissure model, two modules were adopted, one of whichwas the traditional plastic-hardening modulus governing straight ahead stressing and another of which was ahardening modulus governing the tangential response to the stress increment. Then, the full plastic flow can bedecomposed into two partsthe coaxial and non-coaxial plastic flow. The methodology developed in this paperfor implementing kinematic limit analysis for non-associated flow materials can be extended for Rudnicki andRice non-coaxial plastic-flow materials, and this will be presented in future work.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Conflict of interest

None declared.

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Li_Limit Analysis of Materials With Non-Associated Flow

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