ch

HC

Conic optimization

ericiterentThegraefc

2015 Elsevier Ltd. All rights reserved.

it anaasingicationroachecollap

optimization problem (for convenience, the term displacement ishere used as shorthand for displacement rate). Of several optimiza-tion techniques that have been developed to solve such thelarge-scale optimization problem, the primaldual interior-pointmethod presented in [12,13], which was implemented in commer-cial codes such as the Mosek software package [14], has been found

of nite elementte elementrein. It haies of the

smoothing technique and advantages of FEM, and thereforesolutions that are accurate and computational inexpensivmain idea of the smoothing technique is that elementalare determined by spatially averaging eld values using the diver-gence theorem. The resulting strains are constant over a smoothingcell, and hence problems involving integration can be treated in avery straightforward manner. Following this line of research, themain objective of this paper is to develop a displacement niteelement for plate in bending, which combines the original HCTconforming elements with the smoothing technique, ensuring that

Corresponding author.E-mail addresses: lvcanh@hcmiu.edu.vn (C.V. Le), cqthang@hcmiu.edu.vn

(T.Q. Chu).

Computers and Structures 152 (2015) 5965

Contents lists availab

Computers an

lseplates in bending, the conforming HCT elements [11] are common-ly utilized in practical engineering.

When the displacement/velocity elds are approximated andthe upper-bound theorem is applied, limit analysis becomes an

ing technique is then applied to the frameworkmethod (FEM), forming a class of smoothed niods, e.g. see Liu et al. [22], and references theshown that the method retains most properthttp://dx.doi.org/10.1016/j.compstruc.2015.02.0090045-7949/ 2015 Elsevier Ltd. All rights reserved.meth-s beenstrainyieldse. ThestrainsOwning to their advantages, numerical approaches based on boundtheorems and mathematical programming have been developedover past decades [38]. In the kinematic formulation, velocityelds must be discretized using continuous, discontinuous niteelements [9] or discontinuities-only [10]. Of several continuousdisplacement elements that have been developed for Krichhoff

therefore worthwhile to explore a range of alternative integrationtechniques that require a small number of integration points whileproviding accurate solutions.

In the effort to further advance meshfree methods, Chen et al.[21] have proposed a strain smoothing technique to stabilize adirect nodal integration in mesh-free methods. The strain smooth-Smoothing techniqueCS-HCT

1. Introduction

In recent years, yield design or limcrete slabs [1,2] has gained an increacceptance of perfect plasticity applfor concrete structures. Various appmate of the load required to causelysis of reinforced con-interest due to a widers in the new Euro-codes can be used to esti-se of such a structure.

to be more efcient and robust [1517,7,18,19,9,20]. Making use ofsuch an optimization technique, the plastic dissipation function orthe yield criterion must be cast in the form of a standard second-order cone constraint. However, it should be stressed that the sizeof the resulting optimization problem increases rapidly due toauxiliary variables introduced. The number of these auxiliary vari-ables often depends on the number of integration points used. It isCollapse loadReinforced concrete slabs

dure is applied to various reinforced concrete slab problems with arbitrary geometries and differentboundary conditions.A curvature smoothing HsiehCloughToof reinforced concrete slabs

Canh V. Le a,, Phuong H. Nguyen b, Thang Q. Chu aaDepartment of Civil Engineering, International University VNU HCMC, Viet Namb Faculty of Applied Mechanics and Civil Engineering, University of Technical Education

a r t i c l e i n f o

Article history:Received 11 September 2014Accepted 9 February 2015

Keywords:Limit-state

a b s t r a c t

This paper describes a numerned by Nielsen yield crCloughTocher (HCT) elemlish stabilized curvatures.the fact that only one intewas solved using a highly

journal homepage: www.eer element for yield design

MC, Viet Nam

al kinematic formulation for yield design of reinforced concrete slabs gov-ion. A cell-based smoothing technique is introduced to original Hsieh, ensuring that only rst derivatives of shape functions are needed to estab-size of the resulting optimization problem is reduced signicantly due totion point is needed per sub-element. The discrete optimization problemient primaldual interior point algorithm. The proposed numerical proce-

le at ScienceDirect

d Structures

vier .com/locate /compstruc

accurate solutions can be obtained with minimal computationalcost.

The layout of the paper is as follows. The next section willdescribe a kinematic formulation for yield design or limit analysisof reinforced concrete slabs governed by Nielsens yield criterion. Acurvature cell-based smoothed HCT element (CS-HCT) and associ-ated discrete kinematic formulation are described in Section 3.Numerical examples are provided in Section 4 to illustrate the per-formance of the proposed procedure.

2. Kinematic limit analysis of reinforced concrete slabs

Consider a thin rigid-perfectly plastic plate bounded by a curveenclosing a plane area X with kinematic boundary Cu and staticboundary Ct , and subjected to an out-of-plane load kq. The exactcollapse multiplier kexact can be determined by solving any of the

60 C.V. Le et al. / Computers and Stfollowing optimization problems [7]

kexact maxfk j 9m 2 B : am;u kFu;8u 2 Yg 1max

m2Bminu2C

am;u 2min

u2Cmaxm2B

am;u 3min

u2CDu; 4

where C fu 2 Y jFu 1g;Du maxm2Bam; u is the plasticdissipation rate, Y is a space of kinematically admissible velocityelds u;B is the yield condition and Fu; am;u are the externaland internal virtual work respectively given by

Fu ZXqudX 5

am;u ZXmT jdX

ZXmTr2udX 6

In this study, the yield criterion proposed by Nielsen [23,24] andWolfensberger [25], which is commonly known as Nielsens yieldcriterion, is used for the analysis of reinforced concrete slabs. Thecriterion is expressed as

mpx mxxmpy myyP m2xympx mxxmpy myyP m2xympx 6 mxx 6 mpxmpy 6 myy 6 mpy

7

where mpx and mpy are the negative yield moments in the x and y

directions, respectively, and similarly mpx and mpy are the positive

yield moments in the two directions. The constraints in (7) repre-sent a bi-conical yield surface, as shown in Fig. 1.

mxy

myy

mxxFig. 1. Yield criterion for reinforced concrete slabs (after Nielsen andWolfensberger[23,25,24]).The above relations are the intersection of two rotated quadrat-ic cones

bi Q im 2K3r ; i 1;2 8with

Q 1 1 0 00 1 00 0

2

p

264

375; b1

mpxmpy0

264

375;

Q 2 1 0 00 1 00 0

2

p

264

375; b2

mpxmpy0

264

375 9

The upper bound on the collapse load of reinforced concrete slabscan be then determined by the following mathematical program-ming [26]

k minDu RXdudX RX mpxjx mpyjy mpxjx mpyjx dX

s:t

jx ;jy ;2jxy 2K3rjx ;jy ;2jxy 2K3rjxjx jxjyjy jyjxy

2

pjxyjxy

u0 onCuFu1

8>>>>>>>>>>>>>>>>>>>>>>>:

10where the last two constraints enforce boundary conditions andunitary external work.

3. A curvature cell-based smoothed HCT element

3.1. Existing HCT element

First, the C1-conforming HCT element for bending plates isrecalled. The problem domain is discretized into triangular ele-

ments such as X X1 [X2 [ . . . [Xnel and Xi \X j ; i j. A tri-angular element is then subdivided into 3 sub-elements usingindividual cubic expansions over each sub-element as shown inFig. 2. The element has 12 degrees of freedom: the transverse dis-placements and 2 the rotation components at each corner node(wi; hxi @wi=@x ji; hyi @wi=@y ji; i 1; 2; 3) and normal rotationsat 3 mid-side nodes (hi @wi=@n ji; i 4; 5; 6).

The displacement expansion wk can be expressed in terms ofarea coordinates f f1; f2; f3 over each sub-triangle as

wkf Nke f Nk0 fF

qe; k 1;2;3 11

where the partitions Nke f and Nk0 f respectively represent theinterpolation functions associated with element displacements qeand internal nodal displacements and F is the matrix of eliminationobtained by applying compatible requirements at internal nodes 7,8, 9.

The curvatures can be then determined by

jk jkxxjkyyjkxy

2664

3775

Nke;xx Nk0;xxFNke;yy Nk0;yyFNke;xy Nk0;xyF

2664

3775qe

vke;xx

vke;yy

vke;xy

2664

3775qe 12

It should be noted that for accurate computation of compatible cur-

ructures 152 (2015) 5965vatures dened in Eq. (12) (at least) three Gauss points per sub-ele-ment are required to perform numerical integration of the plasticdissipation function. Consequently, the number of variables in the

CT e

d Strproblem involving second-order cone programming increasesrapidly. In the following, we will present a technique that requiresonly one integration point per sub-element.

3.2. Cell-based smoothed HCT element

In the smoothed nite element method (S-FEM), the problemdomain X is divided into a set of Ns non-overlap and no-gapsmoothing domains Xs such as X X1 [X2 [ . . . [XNs andXi \X j ; i j. Generally, such a division can be arbitrary.However, in practice it is usually performed based on elemententities, such as cells residing in elements, nodes, or edges of theelements for easy formulation, numerical treatments, implementa-tion, and efcient computation [22]. These smoothing domain canbe further divided into a set of sub-smoothing domains. For thesake of simplicity, here triangle elements and its sub-elementsare respectively chosen as smoothing and sub-smoothing domains.

Introducing such a cell-based smoothing technique, stabilizedcurvatures can be determined by

~jkxC ZXekC

jkx/xdX 13

where /x is a distribution function or a smoothing function that ispositive and normalized to unity:ZXekC

/xdX 1 14

For simplicity, the smoothing function / is taken as

/x 1=AkC ; x 2 XekC

0; otherwise

(15

where AkC is the area of the smoothing cell XekC that will be a sub-

element here. Substituting Eq. (15) into Eq. (13), and applying thedivergence theorem, one obtains the following equation

~ k1Z

k ~k

Fig. 2. H

C.V. Le et al. / Computers anj xC AC XekCj xdX ve xCqe 16

in which

~vke;abxC 1

2AkC

ZCk

vke;axnbx vke;bxnax

dC 17

where Ck is the boundary of XekC ;v

ke;ax Nke;ax Nk0;axF.

Performing numerical boundary integration, Eq. (17) can berewritten as

~vke;abxC 1

2AkC

X3j

L j vke;ax jGnbx jG vke;bx jGnax jG

18

where x jG is the Gauss point (mid-point) of boundary segment Cjk

which has length L j and outward surface normal nj.It is worth noting that curvatures determined by Eq. (16) areconstant over a smoothing cell, and hence only one integration willbe needed to perform numerical integration of a function of thesesmoothed curvatures. Moreover, the computation of the smoothedcurvatures is inexpensive due to only rst derivatives of shapefunctions needed. This is not the case for the original HCTs curva-tures where second derivatives involving.

3.3. CS-HCT based kinematic formulation

If compatible curvatures determined by Eq. (12) and numericalintegration are used, the plastic dissipation and the work done byapplied loads can be respectively expressed as

D Xnele1

X3k1

Xngj1

nj mpxj

x mpyjy mpxjx mpyjx

e;k;j

19

F Xnele1

X3k1

Xngj1

njpwke fj 20

where nel is the number of elements, ng 3 is the number of Gaussintegration points in each sub-element Ak; nj is the weighting fac-tor of the Gauss point fj.

When employing the smooth version of the curvatures ratherthan compatible curvatures, the plastic dissipation becomes

D Xnele1

X3k1

AkC mpxj

x mpyjy mpxjx mpyjx

e;k

21

Finally, the upper bound on the collapse load of reinforced con-crete slabs can be determined by the following optimizationproblem

k minXnele1

X3k1

AkC mpxjx mpyjy mpxjx mpyjx

e;k

jx ;jy ;2jxye;k 2K3r ; with k 1;2;3; e 1;2; . . . ;nel

3

8>>>>>>

lement.

uctures 152 (2015) 5965 61s:t

jx ;jy ;2jxye;k 2Kr~vke;xxqe jx jx e;k~vke;yyqe jy jy e;k~vke;xyqe

2

pjxy jxye;k

Aedv beq

>>>>>>>>>>>>>>>>>:

22where the matrix Aeq and vector beq are obtained from unitaryexternal work and boundary conditions, and v is the global dis-placement vector. This optimization problem is already in the formof a standard second order cone programming involving equalityand quadratic cone constraints. The total number of variablesincluding the global number of kinematic degrees of freedom sdofand auxiliary variables is Nvar sdof nel 3 6. The number of

second-order cone constraints is Ncone nel 3 2. These numbersare approximately three times smaller than those in the formula-tion using the usual Gauss integration with the same discretizationnite element mesh.

It should be noted that the collapse multiplier determined usingthe described procedure is not guaranteed to represent a strictupper-bound on the exact value because the smoothed curvatureswere used instead of the compatible ones. However, as the numer-ical discretization becomes increasingly ne one can expect toachieve an increasingly reliable approximation of the actualcollapse load multiplier.

4. Numerical examples

This section will investigate the performance of the proposedsolution procedure via a number of benchmark problems in whichanalytical and other numerical solutions are available. For all theexamples considered in the following parameters were assumed:

procedure. For simply supported slab, both HCT and CS-HCT ele-ments can provide extremely satisfactory solutions, with less than0.3% errors using a mesh of just 450 three-node triangle elements.

k e (%) var t (s) k e (%) var t (s)

10 10 48.43 13.02 11,163 1 48.28 12.68 3963 115 15 46.62 8.80 25,068 3 46.47 8.44 8868 120 20 45.71 6.69 44,523 93 45.58 6.38 15,723 225 25 45.19 5.46 69,528 249 45.08 5.21 24,528 330 30 44.83 4.60 100,083 460 44.81 4.58 35,283 5

e: the relative error; var: the number of variables; t: optimization CPU time.

1.1 1 0.9 0.8 0.7 0.60.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

log 1

0(R

elat

ive

erro

r in

colla

pse

load

)

log10(mesh size h)

HCTCSHCT

Fig. 4. Clamped square slab: convergence behaviour of HCT and CS-HCT elements.

Table 2Limit load factor of simply supported plate in comparison with other solutions mp

qL2

Authors Yield criterion Upper bound Lower bound

CS-HCT Nielsen 24.07 Bleyer et al. [9] T6b Nielsen 24.00 Bleyer et al. [9] H3 Nielsen 24.43Le et al. [29] Nielsen 23.996Le et al. [17,29] von Mises 25.01 24.98Hodge and Belytschko [30] von Mises 26.54 24.86Capsoni and Corradi [31] von Mises 25.02 Andersen et al. [32] von Mises 25.00

Table 3Limit load factor of clamped plate in comparison with other solutions mp

qL2

Authors Yield criterion Upper bound Lower bound

CS-HCT Nielsen 44.81 Bleyer et al. [9] T6b Nielsen 44.03 Bleyer et al. [9] H3 Nielsen 43.45Le et al. [29] Nielsen 42.83Krabbenhoft [33] Nielsen 42.82Maunder et al. [20] Nielsen 42.00Le et al. [17,29] von Mises 45.07 43.86Hodge and Belytschko [30] von Mises 49.25 42.86Capsoni and Corradi [31] von Mises 45.29 Andersen et al. [32] von Mises 44.13

62 C.V. Le et al. / Computers and StL/2 L/2

q

O N

N L/2

L/2

xlength L 10 m, plate thickness t = 0.1 m. The proposed numericalprocedure is developed in a Matlab environment (version 7.11) andthe resulting SOCP minimization problems are solved using Mosek(version 6.0) on a 2.50 GHz Intel i5 running Windows 7.

Square slabs with either simply supports or clamped on alledges and subjected to uniform out-of plane pressure loading wereconsidered. Owing to symmetry, only the upper-right quarter ofthe plate is modeled, see Fig. 3. It is assumed that the slab isisotropic with positive and negative yield moments mp in bothdirections (constant reinforcement). For this case, the Nielsen yieldcriterion may be represented as a square yield locus in the plane ofthe principal moments, and analytical solutions have beenidentied by Prager [27] as k 24mp=qL2 and by Fox [28] ask 42:851mp=qL2 for simply supported and clamped plates,respectively.

The efcacy of the proposed method was studied rst. Theclamped slab was solved using both HCT and CS-HCT elementswith various uniform meshes of N N (N 10;15;20;25;30).Computed collapse multipliers and computational cost are shownin Table 1. Convergence behaviour of HCT and CS-HCT elementsis also shown in Fig. 4. It can be observed that for all meshes solu-tions obtained using CS-HCT elements are lower (better) thanthose obtained using HCT elements, despite the fact that the num-ber of variables and optimization CPU time used in CS-HCT basedformulation are very much smaller than those used in HCT based

z

ytFig. 3. Square slab: geometry, loading and nite element mesh (N is the number ofnodes in each edge).Table 1Results for clamped square plates using HCT and CS-HCT elements.

Meshes HCT CS-HCT

ructures 152 (2015) 5965However, the number of variables in CS-FEM based formulation isrelatively small, with 8868 compared with 25,068 variables in HCTformulation.

d StrC.V. Le et al. / Computers anTables 2 and 3 compare solutions obtained using CS-FEM basedformulation with previously obtained solutions obtained usingother simulations. It is worth mentioning that upper bound col-lapse multipliers of slabs governed by Nielsen yield criterion wererarely reported, with the exception of those obtained by Bleyeret al. [9]. It can be seen that for simply supported slabs, the pro-posed CS-HCT element performs better than H3 element in [9]where discontinuities have been included. T3, T6 and T6b elementscan provide the exact bound when using a structured mesh alongthe diagonal direction. However, these elements seem to be verysensitive the mesh layout. For clamped slabs, the performance of

Fig. 5. Polygon slabs:

Fig. 6. Polygon slabs: plastic dissipation distribution a

Fig. 7. Arbitrary geometric slab with an eccentric rectangular cutout.uctures 152 (2015) 5965 63T6b and H3 is better than CS-HCT. This can be explained by the factthat the discontinuous T6b and H3 elements ensure its edges auto-matically oriented along the negative yield lines, and hence plasticdissipation along clamped boundaries can be easily produced.

Rectangular slabs (dimensions a b) under uniform pressureand different boundary conditions including free (F), simply sup-ported (S) and clamped (C) edges were also considered.Computed collapse multipliers were reported in Table 4, witha b 2. The plate with 3 clamped and 1 free edge was solvedusing 50 25 nodes using half symmetry whilst in the remainingcases quarter symmetry was used with 32 16 nodes. Again, itcan be seen that the CS-HCT elements can provide smaller (better)upper bound solutions than the HCT elements, despite the fact thatthe number of variables used in the CS-HCT based formulation isvery much smaller than that in the HCT-based formulation.

nite element mesh.

nd collapse multiplier (A is the area of the slab).

Fig. 8. Arbitrary geometric slab with an eccentric rectangular cutout: nite elementmesh.

d StTable 4Collapse limit load of rectangular slabs with various boundary conditions mpqab

Models Criterion CCCC SSSS CFCF CCCF

HCT Nielsen k 54.59 28.73 8.34 43.03

64 C.V. Le et al. / Computers anFurther illustration of the method can be made by examiningpolygon slabs of edge L with simply supported boundary condi-tions. Here numerical solutions were obtained by using regularmeshes of 1083 elements for triangle slab and 1152 elements forhexagon slab, as shown in Fig. 5. Plastic dissipation distributionand collapse multipliers for triangular and hexagon slabs areshown in Fig. 6.

var 56,979 56,979 56,979 69,489

CS-HCT Nielsen k 54.44 28.54 8.33 42.96var 20,115 20,115 20,115 24,489

Le et al. [17] von Mises 54.61 29.88 9.49 43.86Capsoni et al. [31] von Mises 29.88

Table 5Collapse load for the isotropic reinforcement slab with different negative momentcapacity.

mpmp

1 12

14

18

kmp 0.1420 0.1298 0.1233 0.1217

Fig. 9. Arbitrary geometric slab with an eccentric rectangular cutout: plasticdissipation distribution.

50510

1520

0

5

10

15

6

4

2

0

2

x 104

Fig. 10. Arbitrary geometric slab with an eccentric rectangular cutout: collapsemechanism.The last example involves an arbitrary geometric slab with aneccentric rectangular cutout, of the same geometry as examinedpreviously in [6]. The dimensions (in meter) and boundary condi-tions are shown in Fig. 7. The problem was solved using a meshof 1942 elements as shown in Fig. 8. First, isotropic reinforcementslab is considered. The inuence of the negative moment capacityon the collapse multiplier is illustrated in Table 5. It can be seenthat the collapse load factor decreases monotonically as the nega-tive moment capacity reduces. In the case of isotropic reinforce-ment with mp mp mp, the computed solution of 0:142mpis in good agreement with the solution of 0:148mp reported in[33]. Plastic dissipation distribution and collapse mechanism forthis case are also shown in Figs. 9 and 10. Orthotropic slab withunequally negative and positive moment capacity with the rein-forcement ratio mpy=mpx 0:5 was also studied. The collapse mul-tiplier of 0:086mpx was obtained.

5. Conclusions

A numerical limit analysis procedure for computation of upperbound on the collapse load of reinforced slabs has been described.The smoothing technique is combined with the original HCT ele-ments. The size of the resulting optimization is reduced signicant-ly, and accurate solutions can be obtained with minimalcomputational cost. Second-order cone programming solver,Mosek, has been used to produce approximated upper bounds onthe collapse load of several practical reinforced concrete slabs(although the procedure cannot be guaranteed to produce strictupper bound solutions, for the problems investigated solutionswere always higher than known exact solutions). It has beenshown in numerical examples that the proposed method can alsobe able to capture yield-line patterns arising from localized plasticdeformations for problems of arbitrary geometry. Including dis-continuities to the proposed CS-HCT elements may improve theaccuracy of the solutions, and this will be the subject of futureresearch. Furthermore, the CS-FEM can be extended to tackle var-ious problems such as shakedown analysis and dynamics of plates.

Acknowledgements

This research is funded by Vietnam National University Ho ChiMinh City (VNU-HCMC) under Grant No. B2014-28-01.

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C.V. Le et al. / Computers and Structures 152 (2015) 5965 65

A curvature smoothing HsiehCloughTocher element for yield design of reinforced concrete slabs1 Introduction2 Kinematic limit analysis of reinforced concrete slabs3 A curvature cell-based smoothed HCT element3.1 Existing HCT element3.2 Cell-based smoothed HCT element3.3 CS-HCT based kinematic formulation

4 Numerical examples5 ConclusionsAcknowledgementsReferences