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1、Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013,Article ID 709430,12 pageshttp:/dx.doi.org/10.1155/2013/709430Research ArticleComparison between Duncan and Changs EB Modeland the Generalized Plasticity Model in the Analysis ofa High Earth-Rockfill DamWeixin Dong,Liming Hu,Yu

2、Zhen Yu,and He LvState Key Laboratory of Hydro-Science and Engineering,Department of Hydraulic Engineering,Tsinghua University,Beijing 100084,ChinaCorrespondence should be addressed to Yu Zhen Yu;Received 4 June 2013;Revised 19 August 2013;Accepted 20 August 2013Academic Editor:Fayun LiangCopyright

3、2013 Weixin Dong et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use,distribution,and reproduction in any medium,provided the original work is properly cited.Nonlinear elastic model and elastoplastic model are two main kinds

4、of constitutive models of soil,which are widely used in thenumerical analyses of soil structure.In this study,Duncan and Changs EB model and the generalized plasticity model proposedby Pastor,Zienkiewicz,and Chan was discussed and applied to describe the stress-strain relationship of rockfill materi

5、als.Thetwo models were validated using the results of triaxial shear tests under different confining pressures.The comparisons betweenthe fittings of models and test data showed that the modified generalized plasticity model is capable of simulating the mechanicalbehaviours of rockfill materials.The

6、 modified generalized plasticity model was implemented into a finite element code to carry outstatic analyses of a high earth-rockfill dam in China.Nonlinear elastic analyses were also performed with Duncan and Changs EBmodel in the same program framework.The comparisons of FEM results and in situ m

7、onitoring data showed that the modifiedPZ-III model can give a better description of deformation of the earth-rockfill dam than Duncan and Changs EB model.1.IntroductionThe constitutive model of soil is the keystone in the finiteelement analyses of geotechnical structures.A suitable con-stitutive mo

8、del can simulate the stress-strain relationships ofsoils under static or dynamic conditions.Numerical analysis,especially for finite element method incorporated with soilconstitutive models,has played a very important role ingeotechnical analyses which always include complex bound-ary conditions,non

9、linearity of material,and geometry 1.Biot presented the famous three-dimensional consolida-tion theory based on the effective stress theory,equilibriumequation,and continuity condition 2.However,it is quitedifficulttogivethetheoreticalsolutionofBiotsconsolidationtheory except for few simple problems

10、.Up to the 1960s,with the rapid development of electronic computer andconstitutive models of soils,Biots consolidation theory wassuccessfullyimplementedinfiniteelementcodestostudythebehavior of geotechnical structures 3,4.So far,thousandsof constitutive models have been proposed,which can bemainly g

11、rouped in two categories:nonlinear elastic modelsand elastoplastic models.For nonlinear elastic model,the nonlinear characteristicofsoilstress-strainrelationshipisconsideredbysectionalizedlinearization.A typical nonlinear elastic model is Duncanand Changs Model 5,6,which has been widely used inthe n

12、umerical analyses of earth-rockfill dams,as the modelparameters are quite easy to be determined from conven-tionaltriaxialtests.And,alotofexperienceofapplicationhasbeen accumulated for this model.However,nonlinear elasticmodels also have some inherent limitations to represent thestress-strain charac

13、teristics of soils,such as shear-induceddilatancy and stress path dependency.Elastoplastic models would be very adequate in describ-ing many key features of soils.Classical elastoplastic modelsarebasedontheplasticincrementaltheorycomposedofyieldcondition,flowrule,andhardeninglaw.Inthe1950s,Druckeret

14、 al.(1957)7 suggested a cap yield surface controlled byvolumetric strain.Roscoe et al.8,9 proposed the conceptsof critical state line and state boundary surface,and then2Journal of Applied Mathematicsthey built the Original Cam Clay Model based on triaxialtests.Burland 10 suggested a different energ

15、y equationand then established the Modified Cam Clay Model.Sincethe establishment of Cam Clay Model,some other typesof elastoplastic constitutive models have also achieved greatdevelopment 1118.Among these models,the generalizedplasticity model 16,19,20 can simulate the static anddynamicmechanicalbe

16、haviorsofclaysandsands.Thismodelis very flexible and convenient to extend,as the complicatedyield or plastic potential surfaces need not to be specifiedexplicitly.And the model has been used successfully in thestatic or dynamic analyses of some geotechnical structures2124.Furthermore,basedontheframe

17、workofgeneralizedplasticity theory 16,some limitations of the original modelhave been solved 2528,such as pressure dependency,den-sification under cyclic loading.The details of the generalizedplasticity theory and the original and proposed modifiedPastor-Zienkiewicz-Chans models will be introduced i

18、n thesections below.However,littleexperiencehasasyetbeenaccumulatedinapplying the generalized plasticity model to the simulationof rockfill materials.And we know that rockfill material isquite different from sands in mechanical properties 2931.The rockfill material has large particle size and sharp

19、edgesandcorners,whichcanresultinremarkableparticlebreakageand change the shear-induced dilation 32,33.On the otherhand,though the generalized plasticity model has gainedgreat success in the modeling of soils,the application of thismodelinthelarge-scalefiniteelementanalysesofearthdamswas less reporte

20、d.Inthisstudy,theoriginalgeneralizedplasticitymodelwasmodifiedto considerthestress-strain relationshipsofrockfillmaterials,as most of previous studies focused on sandsand clays.Then,based on conventional triaxial test data,the model parameters for dam materials of the Nuozhaduhigh earth-rockfill dam

21、 in Southwest China are determined.Finally,the static simulation of this dam is carried out byusing a finite element code incorporating with Duncan andChangs EB model and the modified generalize plasticitymodel.Thecomparisonofnumericalresultsandinsitumon-itoringdataillustratestheadvantagesofmodified

22、generalizedplasticity model in the simulation of earth-rockfill dams.2.Constitutive Model DescriptionsTwo constitutive models of soils were used in the finiteelement analyses.One is the Duncan and Changs EB modelbelonging to nonlinear elastic model,the other one is thegeneralized plasticity model.2.

23、1.Duncan and Changs Model.Duncan and Changs model5 is a nonlinear elastic model,which has been widely usedin the geotechnical engineering,especially in the numericalanalyses of earth dams.It is attributed to Kondner 34who proposed the hyperbolic stress-strain function below todescribethedeviatoricst

24、ress-axialstraincurveobtainedfromtriaxial tests.Consider1 3=1+1,(1)in which and are model constants.In this constitutive model,the tangential Youngs modu-lusandtangentialbulkmodulusareusedtosimulatethenonlinear elastic response of soils,which are assumed to be=(3)(1 )2,=(3),(2)where is the atmospher

25、ic pressure,and are modulusnumbers,and are exponents determining the rate ofvariation of moduli with confining pressure,and is thefailure ratio with a invariable value less than 1.The Mohr-Coulomb failure criterion is adopted in themodel,and is a factor defined as shear stress level givenby=(1 sin)(

26、1 3)2 cos+23 sin.(3)In the unloading and reloading stage,the tangentialYoungs modulus is defined as=(3).(4)So far,the model has 8 parameters,.These parameters can be determined with a set ofconventional triaxial tests.In general,a curved Mohr-Coulomb failure envelop isadopted by setting =0 and letti

27、ng vary with confiningpressure according to=0 log(3).(5)Then parameters and are replaced by 0and.Although Duncan and Changs EB constitutive model isquite simple,it has gained significant success in geotechnicalengineering.On one hand,it is easy to obtain the modelparameters;on the other hand,much ex

28、perience has beenaccumulated.Nevertheless,it cannot incorporate dilatancywhichhasanimportantinfluenceinthemechanicalbehaviorof soils.And furthermore,it can only consider unloadingprocess in a crude way.2.2.Generalized Plasticity Theory and Its OriginalConstitutive Model2.2.1.Basic Theory.The general

29、ized plasticity theory wasproposed by Zienkiewicz and Mroz(1984)16 to model thebehaviors of sand under monotonic and cyclic loading.TheJournal of Applied Mathematics3key futures of this theory are that neither yield surface norplastic potential surface needs to be defined explicitly,andconsistencyla

30、wisnotrequiredtodetermineplasticmodulus.In the theory,thetotalstrain incrementis divided into elasticand plastic components.Consider=+,(6)where and=elastic and plastic strain increments,respectively.The relationship between strain and stress increments isexpressed as=D:,(7)where Dis the elastoplasti

31、c stiffness tensor given asD=DD:n/:n:D/+n:D:n/,(8)where D,n/,n,and/are elastic stiffness tensor,plastic flow direction vector,loading direction vector,andplastic modulus under loading or unloading conditions,respectively.Theloadingdirectionvectornisusedtojudgetheloadingand unloading conditions:n 0lo

32、ading,n=0neutral loading,n 1,=0,1,(17)respectively,where0,aremodelparametersandisthestress ratio from which unloading takes place.4Journal of Applied Mathematics01000200030004000500060001(%)1 3(kPa)0510153=300kPa3=700kPa3=1200kPa(a)01234?(%)3=300kPa3=700kPa3=1200kPa0510151(%)(b)Figure 1:Simulation o

33、f stress-strain relationships for Original PZ-III model.2.2.3.Modified Model.The Pastor-Zienkiewicz-Chan model(PZ-III for short)has gained considerable success in describ-ing the behavior of sands and clays under monotonic andcyclic loadings.But it still has some shortcomings to predictthestaticordy

34、namicrespondsofsands,especiallyforrockfillmaterials which are widely used in earth-rockfill dams.TheOriginal PZ-III model has serious limitation in reflectingpressure dependency of soils.Figure 1 shows the stress-strain relationships of a rockfillmaterial under drained conventional triaxial tests us

35、ing aset of parameters under different confining pressures,butPZ-III model gives the same 1-Vcurve,where 1,Vareaxial strain and volumetric strain,respectively.As confiningpressure ranges from 0kPa to several MPa for a rockfill damwith height of 200300m,the original PZ-III model cannotbe used to desc

36、ribe the mechanical behavior of rockfill dams.Some relations of the original model are modified to takeinto account the influence of confining pressure asV=0(),=0(),=0()(V+),(18)where 0and 0are elastic constants,and are modelparameters to consider the effect of pressure dependency.As sand behavior i

37、s dependent on densities or void ratio,a state pressure index,proposed by Wang et al.35 wasintroduced in the PZ-III model and(13)was modified as=V=(1+)(),(19)where is a model parameter and=/in which is the mean pressure at critical state.The critical state line isgiven by=log().(20)3.Nuozhadu Hydrop

38、ower ProjectNuozhadu hydropower project is located in the LancangRiver which is also named Mekong River in the down-stream in Yunnan Province,Southwest China,as shown inFigure 2(a).The installed capacity of the powerstation is5850MW.ThemostimportantpartofNuozhaduhydropowerprojectisthehighearth-rockf

39、illdamwithamaximumheightof 261.5m,which is the highest one with the same type inChina and the fourth highest in the world.The reservoir hasa storage capacity of 237.0 108m3,with the normal storagewater level of 812.5m and dead water level of 765m.Figure 3 shows the material zoning and constructionst

40、ages of the maximum cross-section.The elevation of theearth core bottom and the crest of the dam are 562.6m and824.1m,respectively.The dam crest has a longitudinal lengthof630mwithawidthof18m.Theupstreamanddownstreamslopes are at 1.9:1 and 1.8:1,respectively.The dam body iscomposed of several differ

41、ent types of materials.The shellsof upstream and downstream are composed of decomposedrock materials.Anti-seepage material in the earth core is claymixed with gravel.Adding gravel to the clay can improve thestrength of clay and reduce the arching effect between shellsand earth core.The gravel materi

42、al consists of fresh crushedstone of breccia and granite with a maximum diameter of150mm.In addition to these,the fine rockfill and filtermaterials are filled against the earth core to prevent the fineparticle from being washed away.The dam construction was started in 2008 and wascompleted at the en

43、d of 2012.Figure 2(c)shows the damJournal of Applied Mathematics5BurmaLaosChinaVietnamThailand(a)(b)(c)(d)Figure 2:Nuozhadu dam.(a)Nuozhadu dam location,(b)project blueprint,(c)Nuozhadu dam under construction,and(d)dam sitegeomorphology.under construction.Figure 3(b)demonstrates the practicalconstru

44、ction process.4.Experimental Validation ofModel ParametersThe modified PZ-III model was implemented in a finiteelement code which has been successfully used to analyzeearth dams with Duncan and Changs EB model and someother constitutive models.A set of triaxial test data was usedto make sure that th

45、e model has been incorporated into theFEM code accurately.The proposed generalized plasticity model totally needs17 parameters.The model parameters used in the computa-tion of the earth-rockfill dam were obtained by fitting thetriaxialtestresults.Drainedtriaxialtestsunderdifferentcon-fining pressure

46、s were conducted to test the rockfill materialsand mixed gravel clay,which are the main parts of the dambody.Duncan and Changs EB model parameters are shown inTable 1 andthemodified PZ-III modelparametersin Table 2.As shown in Figures 4,5,6,7,8,and 9,the modified PZ-III model presents a better abili

47、ty to simulate the mechanicsTable 1:Material parameters of Duncan and Changs EB model.MaterialRockfill IRockfill IIMixed gravel clay/55.8254.3339.30/12.2912.079.800.730.740.7714501360520550600250280025009000.300.430.420.130.080.25behaviorofrockfillmaterialsandmixedgravelclay,especiallyfor dilatancy.

48、With the reduction of confining pressure,the rockfill materials tend to dilate as the experimentalvolumetric strain curve shows.Especially for the rockfillmaterials under low confining pressure,negative volumetricstrain rapidly develops after a short stage of volumetriccontraction.Due to the intrins

49、ic limitation,Duncan andChangs EB model cannot simulate the dilatancy which is acrucial feature of rockfill materials.6Journal of Applied MathematicsUpstreamDownstreamRU1RU3F2F1RU2RD1RD2CofferdamEDF2F1RD3RU1/RD1:upstream/downstream rockfill zone IRU2/RD2:upstream/downstream rockfill zone IIRU3/RD3:u

50、pstream/downstream fine rockfillF1/F2:filter material zone I/IIED:clay mixed gravel:electromagnetism type settlement gauges900800700600500824.1658(a)812.52012.12.312011.05.312012.05.312008.02.152008.05.312008.05.312009.05.312009.05.312010.05.312010.05.312011.05.31(b)Figure 3:The maximum cross-sectio