Limit Analysis of Plates and Slabs Using a Meshless Equilibrium Formulation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng (2010)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2887

Limit analysis of plates and slabs using a meshless equilibriumformulation

Canh V. Le, Matthew Gilbert, and Harm Askes

Department of Civil and Structural Engineering, The University of Sheffield, Sheffield, S1 3JD, U.K.

SUMMARY

A meshless Element-Free Galerkin (EFG) equilibrium formulation is proposed to compute the limit loadswhich can be sustained by plates and slabs. In the formulation pure moment fields are approximatedusing a moving least-squares technique, which means that the resulting fields are smooth over the entireproblem domain. There is therefore no need to enforce continuity conditions at interfaces within theproblem domain, which would be a key part of a comparable finite element formulation. The collocationmethod is used to enforce the strong form of the equilibrium equations and a stabilized conformingnodal integration scheme is introduced to eliminate numerical instability problems. The combination ofthe collocation method and the smoothing technique means that equilibrium only needs to be enforcedat the nodes, and stable and accurate solutions can be obtained with minimal computational effort. Thevon Mises and Nielsen yield criteria which are used in the analysis of plates and slabs, respectively, areenforced by introducing second-order cone constraints, ensuring that the resulting optimization problemcan be solved using efficient interior-point solvers. Finally, the efficacy of the procedure is demonstratedby applying it to various benchmark plate and slab problems. Copyright q 2010 John Wiley & Sons, Ltd.

Received 17 July 2009; Revised 29 January 2010; Accepted 5 February 2010

KEY WORDS: limit analysis; meshless methods; EFG; equilibrium model; second-order coneprogramming; plates; slabs

1. INTRODUCTION

The fundamental theorems of limit analysis can be used to provide upper- and lower-boundestimates of the load required to cause collapse of a body or structure. If a suitable approximationfor the displacement field is used, and the kinematic theorem is applied, an upper-bound on theexact limit load can be obtained. Alternatively, if a suitable approximation for the stress field is

Correspondence to: Matthew Gilbert, Department of Civil and Structural Engineering, University of Sheffield,Sheffield, S1 3JD, U.K.

E-mail: [email protected]

Contract/grant sponsor: Ho Chi Minh City GovernmentContract/grant sponsor: The University of SheffieldContract/grant sponsor: EPSRC; contract/grant number: GR/S53329/01

Copyright q 2010 John Wiley & Sons, Ltd.

C. V. LE, M. GILBERT AND H. ASKES

used, and the static theorem is applied, a lower-bound can be obtained. Applying these theoremsand using a finite element discretization, numerical procedures have been developed to performthe limit analysis of perfectly plastic plates and slabs, e.g. [17]. These procedures can providegood bounds on the exact collapse load (or load multiplier), with the results in [1] remarkablyproviding the best lower bounds available for plate problems for many years [8]. However, solutionsobtained from finite element-based computational limit analysis procedures can be very sensitiveto mesh geometry, particularly for problems that contain strong singularities in the stress and/ordisplacement fields. It is therefore worthwhile to explore a range of alternative methods. Recently,Le et al. [9] proposed a numerical kinematic formulation using the Element-Free Galerkin (EFG)method and second-order cone programming (SOCP) to furnish good (approximate) upper-boundsolutions for Kirchhoff plate problems governed by the von Mises failure criterion. It has also beendemonstrated [9, 10] that the EFG method is in general well suited for limit analysis problems,allowing accurate solutions to be obtained with relatively few nodes. Following this line of research,the main objective of this paper is to develop an equilibrium formulation which combines the EFGmethod with SOCP to obtain accurate solutions for both plate and slab problems.

In a static equilibrium formulation, the stress/moment fields are generally chosen so that the equi-librium equations, boundary conditions and continuity requirements are met for all feasible valuesof the problem variables. In Chen et al. [10], a self-equilibrium stress basis vector at each Gaussianpoint is calculated by solving the equivalent weak form of the equilibrium equations. The self-equilibrium stress field is then obtained by a linear combination of several self-equilibrium stressbasis vectors which are generated by considering the differences between intermediate stressesduring the elasto-plastic equilibrium iteration. However, the stress field obtained is not guaranteedto be statically admissible as it is derived from an approximated virtual displacement field bysolving the weak form of the equilibrium equation. In contrast, in this paper the stress/momentfields will be constructed by using a moving least-squares approximation. It is well known thatthe field obtained when using this technique is smooth over the entire problem domain. Thereis therefore no need to enforce continuity conditions at interfaces within the problem domain,which would be a key part of a comparable finite element formulation. Furthermore, consideringthe various triangular finite elements previously proposed for use in equilibrium limit analysisformulations for plates and slabs in [1113], it can be observed that the use of such elements doesnot ensure that equilibrium is exactly enforced when a uniformly distributed pressure loading isinvolved. This is because the equilibrium equation 2m+q =0 does not hold when constant orlinear moment fields are assumed, and at least quadratic moment fields must be employed to ensurethat equilibrium is fully enforced. This requirement can however be met by the use of high-orderEFG shape functions.

In the framework of meshfree methods, it is advantageous if the problem under considerationcan be solved by evaluating quantities at the nodes only. For problems involving integration, astabilized conforming nodal integration (SCNI) scheme proposed in [14] is an effective option. Themain idea of the scheme is that nodal values are determined by spatially averaging field values usingthe divergence theorem. The scheme has been applied successfully to various analysis problems[9, 1517]. It is shown that, when the SCNI scheme is applied, the solutions obtained are accurateand stable, and the computational cost is much lower than when using Gauss integration. Owing tothese advantages the scheme will be used here to stabilize the moment derivatives. Using smoothedsecond derivatives of the moment field at nodes, the equilibrium equations only need to be fulfilledat the nodes. Furthermore, properties of Voronoi cells (which are representative nodal domainsused in the smoothing scheme) can be used as a reference when enforcing the yield criteria.

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2010)DOI: 10.1002/nme

LIMIT ANALYSIS OF PLATES AND SLABS

The aim of this paper is to present an EFG-based equilibrium limit analysis formulation forapplication to rigid-perfectly plastic plates and slabs, governed by the von Mises and Nielsenyield criteria, respectively. The Kirchhoff moment field is approximated by using the movingleast-squares technique and nodal collocation is used to impose boundary conditions. The yieldcriteria are cast in terms of conic constraints, allowing the limit analysis problem to be posedas a standard SOCP problem which can be solved using highly efficient solvers [18]. Finally, inorder to test the performance of the method, several benchmark plate and slab examples from theliterature are investigated.

2. LIMIT ANALYSIS OF PLATESEQUILIBRIUM FORMULATION

A lower-bound solution to the problem involving a rigid-perfectly plastic plate or slab can beobtained by using the static theorem of plasticity, which states that a moment field is statically andplastically admissible if (i) equilibrium and boundary conditions are fully satisfied, and (ii) the yieldcondition is not violated anywhere. The exact plastic collapse load multiplier, p , is the largestvalue among a set of lower-bound multipliers, , corresponding to any statically and plasticallyadmissible moment distribution [1, 4]. The moment field is denoted as m=[mxx myy mxy]T and isconstrained to belong to the domain

B={m|(m)0}, (1)in which the so-called yield function (m) is convex.

In this study, the yield criterion proposed by Nielsen [19, 20] and Wolfensberger [21], whichis commonly known as Nielsens yield criterion, is used for the analysis of reinforced concreteslabs. The criterion is expressed as

(m+px mxx)(m+pymyy) m2xy(mpx +mxx)(mpy+myy) m2xy

mpxmxx m+pxmpymyy m+py

(2)

where mpx and mpy are the negative yield moments in the x and y directions, respectively, andsimilarly m+px and m+py are the positive yield moments in the two directions. The constraints in (2)represent a bi-conical yield surface, as shown in Figure 1.

For steel plates, the von Mises failure criterion is often used and can be expressed as

(m)=

mTPmm p (3)where m p =0t2/4 is the plastic moment of resistance per unit width of a plate of thickness t andyield stress 0, and where

P= 12

2 1 01 2 00 0 6

(4)



mxy

myy

mxx

Figure 1. Yield criterion for reinforced concrete slabs (after Nielsen and Wolfensberger [1921]).

y

z q

x

mxx

mxy

myy

myx

mxxmxy

myymyx

Figure 2. Plate sign conventions.

Following the sign convention given in Figure 2, the lower-bound limit analysis of plate/slabproblems can be expressed in the form of a mathematical programming problem, as

= max (5a)s.t 2m+q =0 (5b)

mB (5c)where is the numerically computed load multiplier, q is the pressure load,

2 ={

2

x22

y22

2

xy

}

and the moment field m must also satisfy appropriate boundary conditions.

3. THE EFG EQUILIBRIUM MODEL

3.1. Moving least-squares approximation

Whereas in the kinematic formulation the displacement field is approximated, here the momentfield needs to be approximated. By using the moving least-squares technique [22], which is theCopyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2010)

DOI: 10.1002/nme


most frequently used approximation in meshless methods, approximations of these moment fieldscan be expressed as

mh(x)=

mhxx

mhyy

mhxy

= n

I=1I (x)

mxxI

myyI

mxyI

(6)

in which

I (x)= pT(x)A1(x)BI (x) (7)

A(x)=n

I=1wI (x)pT(xI )p(xI ) (8)

BI (x)=wI (x)p(xI ) (9)where n is the number of nodes; p(x) is a set of basis functions; wI (x) is a weight functionassociated with node I . In this work, an isotropic quartic spline function is used, which is given by

wI (x)={

16s2+8s33s4 if s10 if s>1

(10)

with s =xxI/RI , where RI is the support radius of node I and determined byRI =h I (11)

where is the dimensionless size of influence domain and h I is the nodal spacing when nodes aredistributed regularly, or the maximum distance to neighbouring nodes when nodes are distributedirregularly (Figure 3). The maximum distance is determined by

h I =max{dJ :dJ = PI PJ ,PJ NI } (12)where

NI = {PJ :V (PJ )V (PI ) =}= {p1, p2, p3, p4, p5, p6, p7} (13)

in which V (PI ) is the Voronoi cell of particle PI .

3.2. Stabilized equilibrium equation

The equilibrium equations are frequently treated in one of two ways in numerical procedures: (i)equilibrium is enforced at nodes in the problem domain and also at boundaries (using the collo-cation method), or (ii) the equilibrium equations are transformed into the equivalent weak-form(involving integrals), using the so-called weighted residual method [23, 24]. The former methodis simple and fast, but it has been reported to suffer from numerical stability problems [24, 25].In contrast, formulations which use the weak-form can usually produce a stable set of discretizedsystem equations, in turn leading to accurate solutions. Finite element-based formulations havebeen developed by several authors [26, 27]. Considering meshless methods, an equilibrium modelCopyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2010)

DOI: 10.1002/nme


RI

= 4

= 3

hI

hI

I

(a) (b)

Figure 3. Sizes of influence domain: (a) regular nodal layout and (b) irregular nodal layout.

for elastostatic problems was first introduced in [28], where stress fields were expressed by meansof an Airy stress function, approximated using the moving least-squares method. Alternatively theself-equilibrium stress field can be calculated by using an assumed displacement field and solvingthe weak form of the equilibrium equations [10]. However, here an alternative EFG equilibriumformulation in which the collocation method is used in combination with a smoothing techniqueis proposed.

A strain smoothing method was first presented in [29] for the regularization of material insta-bilities. The strain smoothing method was then modified to allow stabilization in nodal integrationschemes, leading to the so-called SCNI scheme [14]. The SCNI scheme was then successfullyapplied to both elastic analysis [15, 16] and plastic analysis [9] problems. It has been shown thatthe SCNI scheme results in an efficient and truly mesh-free method, and also numerically stablesolutions. This smoothing technique will now be adapted in order to stabilize problems involvingbending moment derivatives as follows:

mh,(x j )=

j

mh,(x)(x,xx j)d (14)

where mh, is the smoothed value of the second-derivative of moment m

h, at node j , and is a

distribution (or smoothing) function that has to satisfy the following properties [17, 29]:

0 and jd=1 (15)

For simplicity, the function is assumed to be a piecewise constant function and is given by

(x,xx j)={

1/a j , x j0, x / j

(16)

where a j is the area of the representative domain of node j , as shown in Figure 4.



Figure 4. Geometry of a representative nodal domain.

Substituting Equation (16) into Equation (14), and applying the divergence theorem, thefollowing expressions can be derived:

mh,(x j)=1a j

j

mh,(x)d

= 12a j

j(mh,(x)n(x)+mh,(x)n(x))d (17)

where j is the boundary of the representative domain j .Now introducing a moving least-squares approximation of the moment fields, the smooth version

of the moment second-derivative can be expressed as

mh,(x j )=n

I=1I,(x j)mI (18)

with

I,(x j )= 12a j j(I,(x j )n(x)+I,(x j )n(x))d

= 14a j

nsk=1

(nk lk +nk+1 lk+1)I,(xk+1j )

+ 14a j

nsk=1

(nklk +nk+1 lk+1)I,(xk+1j ) (19)

where is the smoothed version of ; ns is the number of segments of a Voronoi nodal domain j as shown in Figure 4; xkj and x

k+1j are the coordinates of the two end points of boundary

segment kj which has length lk and outward surface normal nk .



With the use of the smoothed value mh, the equilibrium equation can be enforced at n nodes,

and Equation (5b) can be rewritten asA1m1+A2m2+A3m3+qI=0 (20)

where

IT = [1 1 . . . 1]1n (21)

A1 =

. . . . . . . . . . . .

1,xx (x j ) 2,xx (x j ) . . . n,xx (x j )

. . . . . . . . . . . .

nn

(22)

A2 =

. . . . . . . . . . . .

1,yy(x j ) 2,yy(x j ) . . . n,yy(x j)

. . . . . . . . . . . .

nn

(23)

A3 =

. . . . . . . . . . . .

21,xy (x j) 22,xy(x j ) . . . 2n,xy(x j ). . . . . . . . . . . .

nn

(24)

m1 = [mxx1 . . . mxxn]T (25)m2 = [myy1 . . . myyn]T (26)m3 = [mxy1 . . . mxyn]T (27)

It is important to note that when the smoothing technique is used, the equilibrium equationis fulfilled at nodes only (unlike in [10] where the equilibrium equation is transformed into theequivalent weak form and enforced at Gauss points).

3.3. Enforcement of boundary conditionsIt should be borne in mind that the quantities mxxI, myyI and mxyI in Equation (6) are fictitiousnodal values, rather than actual moments acting at the nodes. This unfortunately complicatesmatters when seeking to enforce boundary conditions. One way of addressing this is to use thecollocation method proposed in [30]. Let Mn , Mnt and Qn denote the normal bending moment,twisting moment and transverse shear force at node xb on a free unloaded edge, where its normalvector n forms an angle with the x-axis. Conditions for this boundary can be expressed as

Mn mhxxc2+mhyys2+2mhxycs=0Mnt (mhyymhxx)cs+2mhxy(c2s2)=0Qn Qxc+ Qys=0

(28)

where c=cos and s= sin. It is important to note that smoothed moment derivatives are used inthe equilibrium Equation (20). Consequently, the shear forces Qx and Qy must also be calculated



from these smoothed moment derivatives as follows:

Qx = mxxx +mxyy

(29)

Qy = mxyx +myyy

(30)

Introducing a moving least-squares approximation of the moment field, Equation (28) can berewritten as

B1m1+B2m2+B3m3 =0 (31)where

B1 =

. . . . . . . . . . . .

c21(xi ) c22(xi ) . . . c

2n(xi)

. . . . . . . . . . . .

cs1(xi) cs2(xi ) . . . csn(xi ). . . . . . . . . . . .

c1,x (xi ) c2,x(xi ) . . . cn,x(xi )

. . . . . . . . . . . .

(32)

B2 =

. . . . . . . . . . . .

s21(xi ) s22(xi ) . . . s

2n(xi )

. . . . . . . . . . . .

cs1(xi ) cs2(xi) . . . csn(xi )

. . . . . . . . . . . .

s1,y(xi ) s2,y(xi ) . . . sn,y(xi )

. . . . . . . . . . . .

(33)

B3 =

. . . . . . . . . . . .

2cs1(xi) 2cs2(xi ) . . . 2csn(xi ). . . . . . . . . . . .

2(c2s2)1(xi) 2(c2s2)2(xi ) . . . 2(c2s2)n(xi ). . . . . . . . . . . .

|s1,x +c1,y|xi |s1,x +c2,y |xi . . . |s1,x +cn,y |xi. . . . . . . . . . . .

(34)

where i =1,2, . . .,nb, and where nb is the number of nodes with boundary conditions.Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2010)

DOI: 10.1002/nme


4. SECOND-ORDER CONE PROGRAMMING (SOCP)

It was recognized in [31] that most commonly used yield criteria can be cast in the form ofconic constraints, and optimization problems involving such constraints can be solved using highlyefficient solvers [18]. Consequently, several numerical limit analysis procedures which involve theuse of cone programming techniques have been reported recently [9, 32, 33]. This paper continuesthis trend by combining SOCP with the presented EFG equilibrium model.

There are two types of second-order cone (also known as Lorentz or ice cream cones) ingeneral use. The first is the standard quadratic cone, defined as

Lq ={

xRm |x1

mj=2

x2j =x2mL2}

(35)

and the second is the rotated quadratic cone, defined as

Lr ={

xRm+2|x1x2m+2j=3

x2j =x3m+22L2, x1, x20}

(36)

In the following sections, the Nielsen and von Mises yield criteria will be formulated usingrotated and standard quadratic cones, respectively.

4.1. The Nielsen yield criterion

Introducing additional problem variables as follows:

q+ =

+1+2+3

=

m+pxmhxxm+pymhyy

2mhxy

=D+m+d+ (37)

q =

123

=

mpx+mhxxmpy+mhyy

2mhxy

=Dm+d (38)

where

D+=

1 0 00 1 00 0

2

; d+=

m+pxm+py

0

; D=

1 0 0

0 1 0

0 0

2

; d=

mpxmpy

0

(39)

the relations in Equation (2) are the intersection of two rotated cones and are expressed asq+ L+r , L+r = {q+R3|2+1 +2 (+3 )2,+1 ,+2 0} (40)q Lr , Lr = {qR3|21 2 (3 )2,1 ,2 0} (41)



4.2. The von Mises yield criterion

In order to represent the von Mises criterion as a second-order cone constraint, Equation (3) isfirst rewritten in terms of the L2 norm as

(m)=JT mL2 m p (42)where J is the so-called Cholesky factor of P

J= 12

2 0 0

1 3 00 0 2

3

(43)

By applying the following transformation of the moment variables m

q24 =2

3

4

=JTm=

mhxx12

mhyy

32

mhyy

3mhxy

(44)

and defining 1 =m p, constraint (1) can be cast in terms of a second-order cone constraint asfollows:

BLq ={qR4|1q24L2 =22+23+24,1 =m p} (45)

where Lq is a four-dimensional quadratic cone and qT ={1234}.Using a moving least-squares approximation of the moment fields, the vector q24 is evaluated

at point xk and expressed as

qk24 =CkM (46)where M=[m1 m2 m3]T and

Ck =

CT1k 12

CT2k 0

0

32

CT2k 0

0 0

3CT3k

(47)

withCT1k = [1,xx (xk) 2,xx (xk) . . . n,xx (xk)]CT2k = [1,yy(xk) 2,yy(xk) . . . n,yy(xk)]CT3k = [1,xy(xk) 2,xy(xk) . . . n,xy(xk)]

(48)

Consequently, the quadratic cone at point xk is

Lkq ={qk R4|1qk24L2,1 =m p} (49)



extra yield points

nodal yield points

Figure 5. Locations of yield points (at nodes and elsewhere within Voronoi cells).

4.3. Limit analysis formulationThe limit analysis formulation can now be expressed in the form of a standard SOCP problem as

= max

s.t

A1m1+A2m2+A3m3+qI=0B1m1+B2m2+B3m3 =0qk Lk, k =1,2, . . .,np

(50)

where np is the number of yield points and q is defined by Equations (37), (38) or (46), dependingon the yield criteria used. Using the existing Voronoi cell geometry, the yield condition canconveniently be enforced at vertex points within Voronoi cells, as well as at nodes, as indicated inFigure 5.

It should be emphasized that the collapse multiplier determined using the described procedureis not guaranteed to represent a strict lower-bound on the exact value. This is because the smoothedmoment derivative field may not fully satisfy equilibrium conditions everywhere in the domain,and because the yield condition is only enforced at a limited number of points. However, asthe numerical discretization becomes increasingly fine one can expect to achieve an increasinglyreliable approximation of the actual collapse load multiplier.

5. NUMERICAL EXAMPLES

The performance of the limit analysis procedure described will now be tested by examining anumber of benchmark plate and slab problems for which upper- and/or lower-bound solutionshave previously been reported in the literature. For all the examples considered uniform out-of-plane pressure loading was applied and the reference length L was taken as 10 m in the numericalsimulations. Problems were set up using MATLAB and the Mosek version 5.0 optimization solver[18] was used to obtain all solutions. Note that for convenience in each case the whole plate orslab problem has been solved, obviating the need to consider symmetry boundary conditions.



N Nx

L

L

q

x

y

t

Figure 6. Clamped square slab subject to a uniform pressure load.

Table I. Clamped square slab: variation of collapse load multiplier with level of nodal refinement, N .

N N 1010 1515 2020 3030 4040Collapse multiplier (m p/qL2) 42.33 42.67 42.73 42.80 42.83Error (%) 1.22 0.42 0.28 0.12 0.05

5.1. Reinforced concrete slab examplesThe first example comprises a clamped square slab, as shown in Figure 6, which has been inves-tigated numerically by Chan [4], Krabbenhoft and Damkilde [12, 13] and Krenk et al. [11]. It isassumed that the slab is isotropic with positive and negative yield moments m p in both directions(constant reinforcement). For this case, the yield criterion (2) may be represented as a square yieldlocus in the plane of the principal moments [31, 34], and the exact solution has been identified byFox [35] as

p =42.851 m pq L2 (51)

The problem has been solved using N N nodes uniformly distributed across the whole slab.Taking the domain of influence, , as 3, solutions were obtained for various values of N , as shownin Table I.

It can be observed from Table I that close estimates of the exact solution can be obtained evenwhen only a moderate number of nodes are used. For the finest nodal discretization used (4040nodes), the solution obtained is very close (within 0.05%) to the exact solution. Furthermore,although it has been pointed out that the procedure cannot be guaranteed to provide strict lower-bound solutions, it is clear that all the solutions obtained are below the exact value.

The relationship between the computed collapse load multiplier and the size of the influencedomain, governed by the parameter , is illustrated in Figure 7. It can be observed that, when is taken to be larger than 3, a higher (i.e. improved) computed load multiplier can sometimes beobtained. However, since the computational cost increases with the size of the influence domain,



2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.242.2

42.4

42.6

42.8

43

42.642.64

42.76

42.7742.83

42.82

42.831

42.833

42.832

42.83

40 40 nodes

Figure 7. Clamped square slab: normalized collapse load multiplier vs size of the influence domain, (dotted line represents exact solution of Equation (51)).

(a) (b) (c)

Figure 8. Clamped square slab: moment distributions: (a) mxx/m p; (b) myy/m p; and (c) mxy/m p.

a reasonable compromise between accuracy and computational cost can be achieved when istaken as 3, as will be the case for all problems considered henceforth.

Compared with the results obtained by the previous workers, the best solution obtained using thepresent procedure is significantly higher than that obtained in [4, 11] (41.78 and 42.32, respectively),and slightly higher than the solution obtained in [13] (42.82), despite the fact that the number ofnodes used here is significant smaller than in [13] (4040 nodes compared with 101101 nodesfor the whole slab). The moment distributions at collapse are shown in Figure 8.

A simply supported isotropic square slab will be considered next. For this case the exact collapseload multiplier was given in [8] as p =24m p/q L2. When 2020 nodes were used to model theslab, the corresponding normalized collapse multiplier was found to be 23.996, which is clearlyin excellent agreement with the exact solution.



Figure 9. Clamped circular slab: nodal discretization and Voronoi cells.

Table II. Clamped and simply supported square plates: results for different levels of nodal refinement, N .

Clamped Simply supported

N N (m p/qL2) CPU time (s) (m p/qL2) CPU time (s)1414 43.2467 30 24.8554 421818 43.5364 107 24.9175 972222 43.6961 304 24.9462 2263030 43.8562 952 24.9766 882Time taken to solve on a 2.8 GHz Pentium 4 PC running Microsoft XP (Mosek time only).

The method will next be applied to a clamped isotropic circular slab subjected to a uniformpressure loading. The exact collapse multiplier was given in [8] as p =12m p/q R2, where R isthe slab radius. The problem was solved using 49 nodes laid out radially across the slab, as shownin Figure 9. A normalized solution of 11.89 was obtained, which is just 0.9% lower than the exactcollapse multiplier.

5.2. Metal plate examples

Square steel plates with either clamped or simply supports on all edges will now be considered;these have also been investigated by Hodge and Belytschko [1], Andersen et al. [36], Capsoniand Corradi [2] and more recently by Le et al. [9]. The problems were solved using N N nodesuniformly distributed across the whole plate and in all numerical simulations the plate thicknesswas taken as t =0.1 m. The solutions and CPU times are shown in Table II for various levels ofnodal refinement.

Table III compares normalized solutions obtained using the present method with upper- andlower-bound solutions that have been previously reported in the literature. It can be observed thatthe solutions obtained using the present method are in good agreement with the previous results.For both clamped and simply supported plate problems, the solutions obtained here are higherthan the best lower-bounds obtained in [1] (2.33% in the case of the clamped plate and 0.48%in the case of the simply supported plate). Together with [9], this indicates that numerical limitCopyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2010)

DOI: 10.1002/nme


Table III. Clamped and simply supported square plates: comparison with literature results.

Clamped Simply supported

Lower-bound Upper-bound Lower-bound Upper-boundAuthors (LB) (UB) (LB) UB)Present method 43.86 24.98 Le et al. [9] (EFG method) 45.07 25.01Hodge and Belytschko [1] 42.86 49.25 24.86 26.54Lubliner [3] 52.01 23.81 27.71Capsoni and Corradi [2] 45.29 25.02Andersen et al. [36] 44.13 25.00Methods produce approximate rather than rigorous lower- or upper-bound solutions.Mixed elements were used.

-1

-0.5

0

0.5

1

(a) (b) (c)

Figure 10. Clamped square plate: moment distributions: (a) mxx/m p; (b) myy/m p; and (c) mxy/m p .

analysis procedures which use the EFG method are capable of producing good results. Note thatin the case of the simply supported square plate, the mean value of the lower-bound obtained hereand the upper-bound obtained in [9] is 24.995, which is evidently in excellent agreement with thesolution obtained by Andersen et al. [36]. Moment distributions at collapse for these plates areshown in Figures 10 and 11.

It is also of interest to examine the form of the displacement fields derived from the dual valuesof the equilibrium constraints of (20). These fields can usefully be displayed in the form of inferredcollapse mechanisms. Figure 12 compares the collapse mechanisms derived for the square slaband square plate problems, for consistency using a numerical discretization comprising 4040nodes in all cases. It is apparent that the smooth form of the von Mises failure criteria gives riseto smooth deformed plate profiles, in contrast to the discontinuous deformed slab profiles whichresult from the use of the discontinuous Neilsen yield criteria. (When simple supports are involvedthe discontinuities in the deformed slab profile can be observed to coincide with the locationsof the diagonal yield lines present in the collapse mechanism which corresponds to the exactanalytical solution.)

Rectangular plates (dimensions ab) with different boundary conditions will be considerednext. All problems here were solved using 6030 nodes with a =2b=10 m. In case of the plateCopyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2010)

DOI: 10.1002/nme


-1

-0.5

0

0.5

1

(a) (b) (c)

Figure 11. Simply supported square plate: moment distributions: (a) mxx/m p; (b) myy/m p; and (c) mxy/m p .

(a)

(c)

(b)

(d)

Figure 12. Square slabs and plates: derived collapse mechanisms (inverted for sake of clarity): (a) simplysupported slab; (b) clamped slab; (c) simply supported plate; and (d) clamped plate.

with 3 clamped boundaries and 1 free edge, note that the free edge has length b. Collapse loadmultipliers are shown in Table IV. It can be observed from the table that the gaps between the(approximate) lower-bounds found in this paper and the (approximate) upper-bounds in [9] arenarrow, particularly for the case of a rectangular plate with simply supported boundaries.

Moment distributions at collapse are shown in Figure 13. In plates containing free boundaries itcan be observed that the moments oscillate slightly close to a free edge, as shown in Figure 13(c).This may be explained by the fact that average values of the shear forces were used to enforce theshear boundary condition. However, if the shear forces are computed using the actual values of



Table IV. Rectangular plates: collapse loads multipliers with various boundary conditions (m p/qab).

Models Clamped Simply supported 3 clamped, 1 free

Present results 53.43 29.85 43.11Le et al. [9] 54.61 29.88 43.86Capsoni et al. [2](UB) 29.88

(a)

(b)

(c)

Figure 13. Rectangular plates: moment distributions: (a) clamped; (b) simplysupported; and (c) 3 clamped+1 free.

the moment derivatives (rather than the smoothed values given in Equations (29) and (30)), largeroscillations result.

6. CONCLUSIONS

An Element-Free Galerkin (EFG)-based equilibrium limit analysis formulation has been proposed.This uses a moving least-squares approximation of the moment field, which means that the resultingfield is smooth over the entire problem domain. The collocation method is used in combination withthe stabilized conforming nodal integration (SCNI) scheme to ensure that equilibrium needs onlyto be enforced at nodes. The Nielsen and von Mises yield criteria are formulated as second-ordercones so that the underlying limit analysis problem becomes a standard SOCP problem, which canbe solved efficiently using primaldual interior point solvers. Although the procedure cannot beguaranteed to produce strict lower bound solutions, for the plate and slab problems investigated



solutions were in practice always lower than known exact solutions, and higher than (improved cf.)existing lower-bound numerical solutions in the literature.

ACKNOWLEDGEMENTS

This research has been sponsored by the Ho Chi Minh City Government (300 Masters & Doctors Project)and the University of Sheffield. Matthew Gilbert also acknowledges the support of EPSRC under grantreference GR/S53329/01.

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Limit Analysis of Plates and Slabs Using a Meshless Equilibrium Formulation

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