Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 709430 12 pageshttpdxdoiorg1011552013709430
Research ArticleComparison between Duncan and Changrsquos EB Modeland the Generalized Plasticity Model in the Analysis ofa High Earth-Rockfill Dam
Weixin Dong Liming Hu Yu Zhen Yu and He Lv
State Key Laboratory of Hydro-Science and Engineering Department of Hydraulic Engineering Tsinghua UniversityBeijing 100084 China
Correspondence should be addressed to Yu Zhen Yu yuyuzhentsinghuaeducn
Received 4 June 2013 Revised 19 August 2013 Accepted 20 August 2013
Academic Editor Fayun Liang
Copyright copy 2013 Weixin Dong et al This is an open access article distributed under the Creative Commons Attribution Licensewhich 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 of constitutive models of soil which are widely used in thenumerical analyses of soil structure In this study Duncan and Changrsquos EB model and the generalized plasticity model proposedby Pastor Zienkiewicz and Chan was discussed and applied to describe the stress-strain relationship of rockfill materials 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 Themodified 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 Changrsquos EBmodel in the same program framework The comparisons of FEM results and in situ monitoring data showed that the modifiedPZ-III model can give a better description of deformation of the earth-rockfill dam than Duncan and Changrsquos EB model
1 Introduction
The constitutive model of soil is the keystone in the finiteelement analyses of geotechnical structures A suitable con-stitutive model can simulate the stress-strain relationships ofsoils under static or dynamic conditions Numerical analysisespecially for finite element method incorporated with soilconstitutive models has played a very important role ingeotechnical analyses which always include complex bound-ary conditions nonlinearity 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 quitedifficult to give the theoretical solution of Biotrsquos consolidationtheory except for few simple problems Up to the 1960swith the rapid development of electronic computer andconstitutive models of soils Biotrsquos consolidation theory wassuccessfully implemented in finite element codes to study thebehavior of geotechnical structures [3 4] So far thousandsof constitutive models have been proposed which can be
mainly grouped in two categories nonlinear elastic modelsand elastoplastic models
For nonlinear elastic model the nonlinear characteristicof soil stress-strain relationship is considered by sectionalizedlinearization A typical nonlinear elastic model is Duncanand Changrsquos Model [5 6] which has been widely used inthe numerical analyses of earth-rockfill dams as the modelparameters are quite easy to be determined from conven-tional triaxial tests And a lot of experience of application hasbeen accumulated for this model However nonlinear elasticmodels also have some inherent limitations to represent thestress-strain characteristics 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 modelsare based on the plastic incremental theory composed of yieldcondition flow rule and hardening law In the 1950s Druckeret 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 then
2 Journal of Applied Mathematics
they built the Original Cam Clay Model based on triaxialtests Burland [10] suggested a different energy equationand then established the Modified Cam Clay Model Sincethe establishment of Cam Clay Model some other typesof elastoplastic constitutive models have also achieved greatdevelopment [11ndash18] Among these models the generalizedplasticity model [16 19 20] can simulate the static anddynamicmechanical behaviors of clays and sandsThismodelis 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 structures[21ndash24] Furthermore based on the framework of generalizedplasticity theory [16] some limitations of the original modelhave been solved [25ndash28] such as pressure dependency den-sification under cyclic loading The details of the generalizedplasticity theory and the original and proposed modifiedPastor-Zienkiewicz-Chanrsquos models will be introduced in thesections below
However little experience has as yet been accumulated inapplying the generalized plasticity model to the simulationof rockfill materials And we know that rockfill material isquite different from sands in mechanical properties [29ndash31]The rockfill material has large particle size and sharp edgesand corners which can result in remarkable particle breakageand 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 thismodel in the large-scale finite element analyses of earth damswas less reported
In this study the original generalized plasticitymodel wasmodified to consider the stress-strain relationships of rockfillmaterials as most of previous studies focused on sandsand clays Then based on conventional triaxial test datathe model parameters for dam materials of the Nuozhaduhigh earth-rockfill dam in Southwest China are determinedFinally the static simulation of this dam is carried out byusing a finite element code incorporating with Duncan andChangrsquos EB model and the modified generalize plasticitymodelThe comparison of numerical results and in situmon-itoring data illustrates the advantages ofmodified generalizedplasticity model in the simulation of earth-rockfill dams
2 Constitutive Model Descriptions
Two constitutive models of soils were used in the finiteelement analyses One is the Duncan and Changrsquos EB modelbelonging to nonlinear elastic model the other one is thegeneralized plasticity model
21 Duncan and Changrsquos Model Duncan and Changrsquos model[5] 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 [34]who proposed the hyperbolic stress-strain function below todescribe the deviatoric stress-axial strain curve obtained fromtriaxial tests
Consider
1205901minus 1205903=
1205761
119886 + 1198871205761
(1)
in which 119886 and 119887 are model constantsIn this constitutive model the tangential Youngrsquos modu-
lus119864119905and tangential bulkmodulus119861
119905are used to simulate the
nonlinear elastic response of soils which are assumed to be
119864119905= 119870119875119886(
1205903
119875119886
)
119899
(1 minus 119877119891119878119897)
2
119861119905= 119870119887119875119886(
1205903
119875119886
)
119898
(2)
where 119875119886is the atmospheric pressure119870 and119870
119887are modulus
numbers 119899 and 119898 are exponents determining the rate ofvariation of moduli with confining pressure and 119877
119891is the
failure ratio with a invariable value less than 1The Mohr-Coulomb failure criterion is adopted in the
model and 119878119897is a factor defined as shear stress level given
by
119878119897=
(1 minus sin120601) (1205901minus 1205903)
2119888 sdot cos120601 + 21205903sdot sin120601
(3)
In the unloading and reloading stage the tangentialYoungrsquos modulus is defined as
119864119906119903= 119870119906119903119875119886(
1205903
119875119886
)
119899
(4)
So far the model has 8 parameters 119888 120593 119870 119870119906119903 119899 119877119891
119870119887 119898 These parameters can be determined with a set of
conventional triaxial testsIn general a curved Mohr-Coulomb failure envelop is
adopted by setting 119888 = 0 and letting 120593 vary with confiningpressure according to
120593 = 1205930minus Δ120593 log(
1205903
119875119886
) (5)
Then parameters 119888 and 120593 are replaced by 1205930and Δ120593
Although Duncan and Changrsquos 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 experience has beenaccumulated Nevertheless it cannot incorporate dilatancywhich has an important influence in themechanical behaviorof soils And furthermore it can only consider unloadingprocess in a crude way
22 Generalized Plasticity Theory and Its OriginalConstitutive Model
221 Basic Theory The generalized plasticity theory wasproposed by Zienkiewicz and Mroz (1984) [16] to model thebehaviors of sand under monotonic and cyclic loading The
Journal of Applied Mathematics 3
key futures of this theory are that neither yield surface norplastic potential surface needs to be defined explicitly andconsistency law is not required to determine plastic modulusIn the theory the total strain increment is divided into elasticand plastic components
Consider
119889120576 = 119889120576119890+ 119889120576119901 (6)
where 119889120576119890 and 119889120576119901 = elastic and plastic strain incrementsrespectively
The relationship between strain and stress increments isexpressed as
119889120590 = D119890119901 119889120576 (7)
whereD119890119901 is the elastoplastic stiffness tensor given as
D119890119901 = D119890 minusD119890 n
119892119871119880 n119879 D119890
119867119871119880+ n119879 D119890 n
119892119871119880
(8)
where D119890 n119892119871119880
n and 119867119871119880
are elastic stiffness tensorplastic flow direction vector loading direction vector andplastic modulus under loading or unloading conditionsrespectively
The loading direction vectorn is used to judge the loadingand unloading conditions
119889120590119879
119890sdot n gt 0 loading
119889120590119879
119890sdot n = 0 neutral loading
119889120590119879
119890sdot n lt 0 unloading
(9)
Then the elastoplastic stiffness tensor D119890119901 can beobtained corresponding to the loading and unloading con-ditions
In the framework of generalized plasticity theory all thecomponents of the elastoplastic constitutive matrix are deter-mined by the current state of stress and loadingunloadingcondition
222 Pastor-Zienkiewicz-Chan Model This model was pre-sented by Pastor et al [19] The relationships between elasticvolumetric and shear strain increments and stress incrementsare defined as
1198891199011015840= 119870119890V119889120576119890
V 119889119902 = 3119866119890119904119889120576119890
119904 (10)
where 119870119890V 119866119890119904 are tangential bulk and shear moduli respec-
tively and they are assumed to be
119870119890V = 119870119890119904119900
1199011015840
119901119900
119866119890119904= 119866119890119904119900
1199011015840
119901119900
(11)
where119870119890119904119900 119866119890119904119900 and 119901
119900are model parameters
In order to determine the plastic stiffness tensor variablesn119892119871119880
n and 119867119871119880
need to be defined n119892119871119880
and n areexpressed as follows
n119892119871= (
119889119892
radic1 + 1198892
119892
1
radic1 + 1198892
119892
)
119879
n = (119889119891
radic1 + 1198892
119891
1
radic1 + 1198892
119891
)
119879
(12)
The dilatancy 119889119892and stress ratio 120578 = 119902119901 are related as
follows
119889119892=
119889120576119901
V
119889120576119901
119904
= (1 + 120572119892) (119872119892minus 120578) (13)
And 119889119891has a similar expression as
119889119891= (1 + 120572
119891) (119872119891minus 120578) (14)
where 120572119891 120572119892are model parameters and 119872
119892119872119891is equal
to relative density If 119889119891= 119889119892 associated flow rule is used
otherwise nonassociated flow rule is usedIn the case of unloading the unloading plastic flow
direction vector n119892119880
is defined as
n119892119880= (minus
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119889119892
radic1 + 1198892
119892
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1
radic1 + 1198892
119892
)
119879
(15)
The loading plastic modulus119867119871is proposed as
119867119871= 11986701199011015840119867119891(119867V + 119867119904)119867119863119872 (16)
where119867119891= (1 minus 120578120578
119891)4 limits the possible state and 120578
119891= (1+
1120572119891)119872119891119867V = 1minus120578119872119892 accounts for phase transformation
119867119904= 12057301205731exp(minus120573
0120585) considers soil degradation and 120585 is the
accumulated plastic shear strain119867119863119872= (120589MAX120589)
120574 accountsfor past history and 120589 = 119901[1 minus 120572
119891120578(1 + 120572
119891)119872119891](minus1120572)
119891which
is the mobilized stress function and 1198670 1205730 1205731 120574 are model
parametersUnder unloading condition the plastic modulus is
defined as
119867119880= 1198671199060(
119872119892
120578119906
)
120574119906
119872119892
120578
gt 1
119867119880= 1198671199060
119872119892
120578
le 1
(17)
respectively where1198671199060 120574119906aremodel parameters and 120578
119906is the
stress ratio from which unloading takes place
4 Journal of Applied Mathematics
0
1000
2000
3000
4000
5000
6000
1205761 ()
1205901minus1205903
(kPa
)
0 5 10 15
1205903 = 300kPa1205903 = 700kPa1205903 = 1200 kPa
(a)
0
1
2
3
4
120576
()
1205903 = 300kPa1205903 = 700kPa1205903 = 1200 kPa
0 5 10 151205761 ()
(b)
Figure 1 Simulation of stress-strain relationships for Original PZ-III model
223 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 predictthe static or dynamic responds of sands especially for rockfillmaterials 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 using aset of parameters under different confining pressures butPZ-III model gives the same 120576
1-120576V curve where 120576
1 120576V are
axial strain and volumetric strain respectively As confiningpressure ranges from 0 kPa to several MPa for a rockfill damwith height of 200ndash300m the original PZ-III model cannotbe used to describe the mechanical behavior of rockfill dams
Some relations of the original model are modified to takeinto account the influence of confining pressure as
119870119890V = 1198701198900119901119886(
1199011015840
119901119886
)
119898
119866119890119904= 1198661198900119901119886(
1199011015840
119901119886
)
119899
119867119871= 1198670119901119886(
1199011015840
119901119886
)
119898
119867119891(119867V + 119867119904)119867119863119872
(18)
where 1198701198900and 119866
1198900are elastic constants 119898 and 119899 are model
parameters to consider the effect of pressure dependencyAs sand behavior is dependent on densities or void ratio
a state pressure index 119868119901 proposed by Wang et al [35] was
introduced in the PZ-III model and (13) was modified as
119889119892=
119889120576119901
V
119889120576119901
119904
= (1 + 120572119892) (119872119892119868119901
119898119901
minus 120578) (19)
where 119898119901is a model parameter and 119868
119901= 119901119901
119888in which 119901
119888
is the mean pressure at critical state The critical state line isgiven by
119890119888= Γ minus 120582 log (119901
119888) (20)
3 Nuozhadu Hydropower Project
Nuozhadu 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 is5850MWThemost important part ofNuozhaduhydropowerproject is the high earth-rockfill damwith amaximumheightof 2615m which is the highest one with the same type inChina and the fourth highest in the world The reservoir hasa storage capacity of 2370 times 108m3 with the normal storagewater level of 8125m and dead water level of 765m
Figure 3 shows the material zoning and constructionstages of the maximum cross-section The elevation of theearth core bottom and the crest of the dam are 5626m and8241m respectively The dam crest has a longitudinal lengthof 630mwith a width of 18mThe upstream and downstreamslopes are at 19 1 and 18 1 respectively The dam body iscomposed of several different 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 material 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 end of 2012 Figure 2(c) shows the dam
Journal of Applied Mathematics 5
BurmaLaos
China
VietnamThailand
(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 practicalconstruction process
4 Experimental Validation ofModel Parameters
The modified PZ-III model was implemented in a finiteelement code which has been successfully used to analyzeearth dams with Duncan and Changrsquos EB model and someother constitutive models A set of triaxial test data was usedto make sure that the 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 thetriaxial test results Drained triaxial tests under different con-fining pressures were conducted to test the rockfill materialsand mixed gravel clay which are the main parts of the dambody
Duncan and Changrsquos EB model parameters are shown inTable 1 and the modified PZ-III model parameters in Table 2As shown in Figures 4 5 6 7 8 and 9 the modified PZ-III model presents a better ability to simulate the mechanics
Table 1 Material parameters of Duncan and Changrsquos EB model
Material Rockfill I Rockfill II Mixed gravel clay120593∘ 5582 5433 3930Δ120593∘ 1229 1207 980119877119891
073 074 077119870 1450 1360 520119870119887
550 600 250119870119906119903
2800 2500 900119899 030 043 042119898 013 008 025
behavior of rockfillmaterials andmixed gravel clay especiallyfor dilatancy With the reduction of confining pressurethe 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 intrinsic limitation Duncan andChangrsquos EB model cannot simulate the dilatancy which is acrucial feature of rockfill materials
6 Journal of Applied Mathematics
Upstream Downstream
RU1RU3F2F1
RU2 RD1
RD2
Cofferdam ED
F2F1
RD3
RU1RD1 upstreamdownstream rockfill zone IRU2RD2 upstreamdownstream rockfill zone IIRU3RD3 upstreamdownstream fine rockfill
F1F2 filter material zone IIIED clay mixed gravel
electromagnetism type settlement gauges
900
800
700
600
500
8241
658
(a)
8125 20121231
20110531sim20120531
20080215sim20080531 20080531sim20090531
20090531sim20100531
20100531sim20110531
(b)
Figure 3 The maximum cross-section (a) Material zoning and (b) construction stage
0
2000
4000
6000
8000
10000
300 kPa 900 kPa1500 kPa 2500 kPaDuncan-Chang EB
0 5 10 15
1205901minus1205903
(kPa
)
1205761 ()
(a)
minus4minus3minus2minus1
01234
1205761 ()0 5 10 15
120576
()
300 kPa 900 kPa1500 kPa 2500 kPaDuncan-Chang EB
(b)
Figure 4 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for rockfill material I
Journal of Applied Mathematics 7
0
2000
4000
6000
8000
10000
300 kPa 900 kPa1500 kPa 2500 kPaModified PZ
0 5 10 15
1205901minus1205903
(kPa
)
1205761 ()
(a)
minus4minus3minus2minus1
01234
1205761 ()0 5 10 15
120576
()
300 kPa 900 kPa1500 kPa 2500 kPaModified PZ
(b)
Figure 5 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for rockfill material I
Duncan-Chang EB
0 5 10 15
1205761 ()
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0
2000
4000
6000
8000
10000
1205901minus1205903
(kPa
)
(a)
0 5 10 151205761 ()
minus3
minus2
minus1
0
1
2
3
120576
()
Duncan-Chang EB
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 6 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for rockfill material II
Modified PZ
0 5 10 151205761 ()
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0
2000
4000
6000
8000
10000
1205901minus1205903
(kPa
)
(a)
0 5 10 151205761 ()
minus3
minus2
minus1
0
1
2
3
120576
()
Modified PZ
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 7 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for rockfill material II
8 Journal of Applied Mathematics
0
2000
4000
6000
Duncan-Chang EB
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3 0 5 10 151205761 ()
120576
()
Duncan-Chang EB
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 8 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for clay
0
2000
4000
6000
Modified PZ
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3
Modified PZ
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0 5 10 151205761 ()
120576
()
(b)
Figure 9 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for clay
Figure 10 3D FEMmesh of Nuozhadu dam
5 Three-Dimensional Finite Element Analyses
51 Computation Model The numerical analyses were per-formed to simulate the performance of the dam duringconstruction and impounding periods with effective stressfinite element analysis
First the 2D finite element mesh of the maximum cross-section of the dam was discretized according to the materialzoning and construction design (see Figure 3) Then the 2Dmesh was extended to 3D mesh in accordance with contourline of the river valley Figure 10 shows the 3D mesh ofthe Nuozhadu dam with 8095 brick and degenerated brickelements and 8340 nodes
The numerical simulations contain two stages filling andimpounding During the filling stage the dam body mainlysubjects to body weight Then at the end of constructionupstream water level goes up to the normal storage waterlevel The interaction between pore water and soil skeletonwas considered through the whole numerical computation
52 Results and Analyses
521 Numerical Results Analyses Figures 11 and 12 show thenumerical results of finite element analyses with Duncanand Changrsquos EB model and the modified PZ-III modelrespectively
Journal of Applied Mathematics 9
1
070503
09
01
(a)
0
05
0
minus1
minus15
minus24
minus05
minus2
(b)
051
152 25
3 353
252
151
05
(c)
0
03
05 1
13
minus02
(d)
Figure 11 Displacement and stress contour of the maximum section for Duncan and Changrsquos EB model (a) displacement along river (m)(b) vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
0706
06
050403
0201
0
02
minus02
01
(a)
minus25minus29
minus05minus15
minus1
minus2
(b)
05
1 152 3
3 435
3
25 215
105
(c)
01
05
1 15
0
(d)
Figure 12 Displacement and stress contour of the maximum section for the modified PZ-III model (a) displacement along river (m) (b)vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
Through the comparison and analysis of the numericalresults (Figures 11 and 12) we can find some similarities anddifferences for these two models
On one hand we can see many similar places in thedistributions of displacements and stresses
(1) After the reservoir impounding due to the hugewaterpressure on upstream dam horizontal displacementdevelops toward the downstream and the largestdisplacement is about 105m for EBmodel and 074mfor modified PZ-III model
(2) Themaximumsettlement occurs in themiddle of corewall due to lower modulus of clayey soil
(3) Because of the tremendous differences of modulusbetween rockfill material and clayey soil there existsobvious arching effect in the core wall
(4) Effective stress in upstream shell is less than thedownstream shell due to the water pressure in theupstream shell
On the other hand some differences also exist whichillustrate the advantages of modified PZ-III model
(1) After the reservoir is impounded upward displace-ment as large as 07m (see Figure 11(b)) developson the upstream shell near dam crest for EB modeland nearly 0m for modified PZ-III model (seeFigure 12(b)) In fact monitoring data of practicalengineering projects shows that no large upwarddisplacement happened after impounding This isdue to its weakness of EB model to distinguish theloading and unloading condition during the waterimpounding
(2) In the distribution of minor principle stress (Figures11(d) and 12(d)) negative stress (ie tensile stress)occurs in the upstream shell for EB model whereasvery little tensile stress exists for modified PZ-IIImodel As we know rockfill material is a typical kindof cohesionless coarse-grained soil which means thatit has no tensile strength Therefore the existence oflarge area of tensile stress in the upstream shell isunreasonable
522 Comparison between Numerical and In Situ MonitoringData Settlement is a key indicator to assess the safety of an
10 Journal of Applied Mathematics
550
600
650
700
750
800
850El
evat
ion
(m)
Settlement (mm)
In-situEBModified PZ
minus1000 0 1000 2000 3000 4000
Figure 13 Comparison between in situ monitoring settlement andFEM results
Table 2 Material parameters of the modified PZ-III model
Material Rockfill I Rockfill II Mixed gravel clay1198700
500 1000 3001198660
1500 3000 900119898 050 050 050119899 050 050 050120572119891
045 045 045120572119892
045 045 045119872119891119888
105 090 060119872119892119888
160 135 1101205730
000 000 0001205731
000 000 000Γ 034 031 034120582 010 009 003119898119901
035 040 001198670
800 1200 900120574 5 5 5120574119906
5 5 51198671199060MPa 9 9 10
earth dam Figures 13 and 14 show the in situmonitoring dataand FEM results of settlement in themaximum cross-sectionThe in situ data were obtained from electromagnetism typesettlement gaugeswhichwere embedded during constructionin the dam (as shown in Figure 3(a)) Through the compar-isons of in situmonitoring and numerical results we can seethat the modified PZ-III model gave a better prediction thanthe EB model However as deformation induced by wetting
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 655 m
2010
11
2010
61
0
2010
11
17
2011
42
6
2011
10
3
2012
31
1
2012
81
8
(a)
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 701 m
2010
91
2011
22
8
2011
82
7
2012
22
3
2012
82
1
(b)
Settl
emen
t (m
m)
0
500
1000
1500
2000
2500
Time
Elevation 751 m
2011
10
1
2011
12
20
2012
39
2012
52
8
2012
81
6
In-situEBModified PZ
(c)
Figure 14 Comparison between in situ monitoring settlement andFEM results
of rockfill materials was not considered the FEM result ofsettlement was below than the in situmonitoring data
As an elastoplastic model the PZ-III model is capableof representing the mechanical behavior of soils better thannonlinear elastic model such as Duncan and Changrsquos EBmodel And the above finite element analyses also proved it
6 Conclusions
This paper presents a modified PZ-III model based on thegeneralized theory and original Pastor-Zienkiewicz-Chan
Journal of Applied Mathematics 11
model to simulate the stress-strain relationship of rockfillmaterials
Triaxial test results of the filling materials of Nuozhadudamwere used to validate the proposedmodel and determinethe model parameters of Duncan and Changrsquos EB model andthe modified PZ-III model respectively The simulations oftriaxial stress-strain response show that the modified PZ-III model is capable of representing the key features ofcohesionless soil such as nonlinearity dilatancy and pressuredependency
The proposed model has been incorporated into a finiteelement code to simulate the static response of a high earth-rockfill dam in China The results were compared with thoseof Duncan and Changrsquos EB model The two set of resultshave both similarities and differences and the differencesillustrate the advantages of the modified PZ-III model Thecomparisons of FEM results and in situ monitoring datashowed that the modified PZ-III model can give a betterdescription of deformation of the earth-rockfill dam thanDuncan and Changrsquos EB model
Acknowledgments
This work was supported by the National Nature ScienceFoundation of China (51179092) and the State Key Laboratoryof Hydroscience and Engineering Project (2012-KY-02 and2013-KY-4)
References
[1] J M Duncan ldquoState of the art limit equilibrium and finite-element analysis of slopesrdquo Journal of Geotechnical and Geoen-vironmental Engineering vol 122 no 7 pp 577ndash596 1996
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] R S Sandhu and E L Wilson ldquoFinite element analysis ofseepage in elastic mediardquo Journal of the Engineering MechanicsDivision vol 95 no 3 pp 641ndash652 1969
[4] J T Christian and J W Boehmer ldquoPlane strain consolidationby finite elementsrdquo Journal of Soil Mechanics amp FoundationsDivision vol 96 no 4 pp 1435ndash1457 1970
[5] JMDuncan andC-Y Chang ldquoNonlinear analysis of stress andstrain in soilsrdquo Journal of the Soil Mechanics and FoundationsDivision vol 96 no 5 pp 1629ndash1653 1970
[6] J M Duncan P M Byrne K SWong and P Mabry ldquoStrengthstress-strain and bulk modulus parameters for finite elementanalyses of stresses and movements in soil massesrdquo Tech RepUCBGT80-01 University of California Berkeley Calif USA1980
[7] D C Drucker R E Gibson and D J Henkel ldquoSoil mechanicsand work-hardening theories of plasticityrdquo Transactions of theAmerican Society of Civil Engineers vol 122 pp 338ndash346 1957
[8] K Roscoe A Schofield andCWroth ldquoOn the yielding of soilsrdquoGeotechnique vol 8 no 1 pp 22ndash53 1958
[9] K Roscoe A Schofield and A Thurairajah ldquoYielding of claysin states wetter than criticalrdquo Geotechnique vol 13 no 3 pp211ndash240 1963
[10] J Burland ldquoCorrespondence on lsquoThe yielding and dilation ofclayrsquordquo Geotechnique vol 15 pp 211ndash214 1965
[11] P V Lade and J M Duncan ldquoElastoplastic stress-strain theoryfor cohesionless soilrdquo Journal of the Geotechnical EngineeringDivision vol 101 no 10 pp 1037ndash1053 1975
[12] I S Sandler F L DiMaggio and G Y Baladi ldquoGeneralizedcap model for geological materialsrdquo Journal of the GeotechnicalEngineering Division vol 102 no 7 pp 683ndash699 1976
[13] X-S Li Y F Dafalias and Z-L Wang ldquoState-dependent dila-tancy in critical-state constitutive modelling of sandrdquoCanadianGeotechnical Journal vol 36 no 4 pp 599ndash611 1999
[14] Y-P Yao and D Sun ldquoApplication of Ladersquos criterion to Cam-clay modelrdquo Journal of Engineering Mechanics vol 126 no 1pp 112ndash119 2000
[15] G Y Baladi and B Rohani ldquoElastic-plastic model for saturatedsandrdquo Journal of the Geotechnical Engineering Division vol 105no 4 pp 465ndash480 1979
[16] O Zienkiewicz and Z Mroz ldquoGeneralized plasticity formu-lation and applications to geomechanicsrdquo in Mechanics ofEngineering Materials C S Desai and R H Gallagher Eds pp655ndash679 John Wiley amp Sons New York NY USA 1984
[17] C S Desai and M O Faruque ldquoConstitutive model forgeological materialsrdquo Journal of Engineering Mechanics vol 110no 9 pp 1391ndash1408 1984
[18] S B R Murthy A Vatsala and T S Nagaraj ldquoRevised Cam-clay modelrdquo Journal of Geotechnical Engineering vol 117 no 6pp 851ndash871 1991
[19] M Pastor O C Zienkiewicz and A H C Chan ldquoGeneralizedplasticity and the modelling of soil behaviourrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 14 no 3 pp 151ndash190 1990
[20] ZMroz andO Zienkiewicz ldquoUniform formulation of constitu-tive equations for clays and sandsrdquo inMechanics of EngineeringMaterials C S Desai and R H Gallangher Eds pp 415ndash449John Wiley amp Sons New York NY USA 1984
[21] G Wang and J-M Zhang ldquoDynamic consolidation finiteelement analysis of a sediment-protecting dyke under oceanwave loadingrdquo Rock and Soil Mechanics vol 27 no 4 pp 555ndash560 2006
[22] MAlyamiMRouainia and SMWilkinson ldquoNumerical anal-ysis of deformation behaviour of quay walls under earthquakeloadingrdquo Soil Dynamics and Earthquake Engineering vol 29 no3 pp 525ndash536 2009
[23] H Li P Manuel and T Li ldquoApplication of an generalizedplasticity model to ultra-high rockfill damrdquo in Proceedingsof the 12th International Conference on Engineering ScienceConstruction and Operations in Challenging EnvironmentsmdashEarth and Space pp 385ndash398 Honolulu Hawaii USA March2010
[24] T Li and H Zhang ldquoDynamic parameter verification of P-Z model and its application of dynamic analysis on rockfilldamrdquo in Proceedings of the 12th International Conference onEngineering Science Construction and Operations in Challeng-ing EnvironmentsmdashEarth and Space pp 2706ndash2713 HonoluluHawaii USA March 2010
[25] M Pastor ldquoA generalized plasticity model for anisotropicbehaviour of sandrdquoComputer Methods and Advances in Geome-chanics vol 1 pp 661ndash668 1991
[26] G Bolzon B A Schrefler and O C Zienkiewicz ldquoElastoplasticsoil constitutive laws generalized to partially saturated statesrdquoGeotechnique vol 46 no 2 pp 279ndash289 1996
[27] H I Ling and H Liu ldquoPressure-level dependency and densifi-cation behavior of sand through generalized plasticity modelrdquo
12 Journal of Applied Mathematics
Journal of Engineering Mechanics vol 129 no 8 pp 851ndash8602003
[28] H I Ling and S Yang ldquoUnified sand model based on thecritical state and generalized plasticityrdquo Journal of EngineeringMechanics vol 132 no 12 pp 1380ndash1391 2006
[29] N D Marschi C K Chan and H B Seed ldquoEvaluation ofproperties of rockfill materialsrdquo Journal of the Soil Mechanicsand Foundations Division vol 98 no 1 pp 95ndash114 1972
[30] R J Marsal ldquoLarge scale testing of rockfill materialsrdquo Journal ofthe Soil Mechanics and Foundations Division vol 93 no 2 pp27ndash43 1967
[31] R JMarsal ldquoMechanical properties of rockfillrdquo in EmbankmentDam Engineering pp 109ndash200 John Wiley amp Sons New YorkNY USA 1973
[32] P V Lade J A Yamamuro and P A Bopp ldquoSignificance ofparticle crushing in granular materialsrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 122 no 4 pp 309ndash3161996
[33] BOHardin ldquoCrushing of soil particlesrdquo Journal of GeotechnicalEngineering vol 111 no 10 pp 1177ndash1192 1985
[34] R L Kondner ldquoHyperbolic stress-strain response cohesivesoilsrdquo Journal of the Soil Mechanics and Foundations Divisionvol 89 no 1 pp 115ndash143 1963
[35] Z-LWang Y F Dafalias X-S Li and F I Makdisi ldquoState pres-sure index for modeling sand behaviorrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 128 no 6 pp 511ndash5192002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
they built the Original Cam Clay Model based on triaxialtests Burland [10] suggested a different energy equationand then established the Modified Cam Clay Model Sincethe establishment of Cam Clay Model some other typesof elastoplastic constitutive models have also achieved greatdevelopment [11ndash18] Among these models the generalizedplasticity model [16 19 20] can simulate the static anddynamicmechanical behaviors of clays and sandsThismodelis 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 structures[21ndash24] Furthermore based on the framework of generalizedplasticity theory [16] some limitations of the original modelhave been solved [25ndash28] such as pressure dependency den-sification under cyclic loading The details of the generalizedplasticity theory and the original and proposed modifiedPastor-Zienkiewicz-Chanrsquos models will be introduced in thesections below
However little experience has as yet been accumulated inapplying the generalized plasticity model to the simulationof rockfill materials And we know that rockfill material isquite different from sands in mechanical properties [29ndash31]The rockfill material has large particle size and sharp edgesand corners which can result in remarkable particle breakageand 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 thismodel in the large-scale finite element analyses of earth damswas less reported
In this study the original generalized plasticitymodel wasmodified to consider the stress-strain relationships of rockfillmaterials as most of previous studies focused on sandsand clays Then based on conventional triaxial test datathe model parameters for dam materials of the Nuozhaduhigh earth-rockfill dam in Southwest China are determinedFinally the static simulation of this dam is carried out byusing a finite element code incorporating with Duncan andChangrsquos EB model and the modified generalize plasticitymodelThe comparison of numerical results and in situmon-itoring data illustrates the advantages ofmodified generalizedplasticity model in the simulation of earth-rockfill dams
2 Constitutive Model Descriptions
Two constitutive models of soils were used in the finiteelement analyses One is the Duncan and Changrsquos EB modelbelonging to nonlinear elastic model the other one is thegeneralized plasticity model
21 Duncan and Changrsquos Model Duncan and Changrsquos model[5] 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 [34]who proposed the hyperbolic stress-strain function below todescribe the deviatoric stress-axial strain curve obtained fromtriaxial tests
Consider
1205901minus 1205903=
1205761
119886 + 1198871205761
(1)
in which 119886 and 119887 are model constantsIn this constitutive model the tangential Youngrsquos modu-
lus119864119905and tangential bulkmodulus119861
119905are used to simulate the
nonlinear elastic response of soils which are assumed to be
119864119905= 119870119875119886(
1205903
119875119886
)
119899
(1 minus 119877119891119878119897)
2
119861119905= 119870119887119875119886(
1205903
119875119886
)
119898
(2)
where 119875119886is the atmospheric pressure119870 and119870
119887are modulus
numbers 119899 and 119898 are exponents determining the rate ofvariation of moduli with confining pressure and 119877
119891is the
failure ratio with a invariable value less than 1The Mohr-Coulomb failure criterion is adopted in the
model and 119878119897is a factor defined as shear stress level given
by
119878119897=
(1 minus sin120601) (1205901minus 1205903)
2119888 sdot cos120601 + 21205903sdot sin120601
(3)
In the unloading and reloading stage the tangentialYoungrsquos modulus is defined as
119864119906119903= 119870119906119903119875119886(
1205903
119875119886
)
119899
(4)
So far the model has 8 parameters 119888 120593 119870 119870119906119903 119899 119877119891
119870119887 119898 These parameters can be determined with a set of
conventional triaxial testsIn general a curved Mohr-Coulomb failure envelop is
adopted by setting 119888 = 0 and letting 120593 vary with confiningpressure according to
120593 = 1205930minus Δ120593 log(
1205903
119875119886
) (5)
Then parameters 119888 and 120593 are replaced by 1205930and Δ120593
Although Duncan and Changrsquos 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 experience has beenaccumulated Nevertheless it cannot incorporate dilatancywhich has an important influence in themechanical behaviorof soils And furthermore it can only consider unloadingprocess in a crude way
22 Generalized Plasticity Theory and Its OriginalConstitutive Model
221 Basic Theory The generalized plasticity theory wasproposed by Zienkiewicz and Mroz (1984) [16] to model thebehaviors of sand under monotonic and cyclic loading The
Journal of Applied Mathematics 3
key futures of this theory are that neither yield surface norplastic potential surface needs to be defined explicitly andconsistency law is not required to determine plastic modulusIn the theory the total strain increment is divided into elasticand plastic components
Consider
119889120576 = 119889120576119890+ 119889120576119901 (6)
where 119889120576119890 and 119889120576119901 = elastic and plastic strain incrementsrespectively
The relationship between strain and stress increments isexpressed as
119889120590 = D119890119901 119889120576 (7)
whereD119890119901 is the elastoplastic stiffness tensor given as
D119890119901 = D119890 minusD119890 n
119892119871119880 n119879 D119890
119867119871119880+ n119879 D119890 n
119892119871119880
(8)
where D119890 n119892119871119880
n and 119867119871119880
are elastic stiffness tensorplastic flow direction vector loading direction vector andplastic modulus under loading or unloading conditionsrespectively
The loading direction vectorn is used to judge the loadingand unloading conditions
119889120590119879
119890sdot n gt 0 loading
119889120590119879
119890sdot n = 0 neutral loading
119889120590119879
119890sdot n lt 0 unloading
(9)
Then the elastoplastic stiffness tensor D119890119901 can beobtained corresponding to the loading and unloading con-ditions
In the framework of generalized plasticity theory all thecomponents of the elastoplastic constitutive matrix are deter-mined by the current state of stress and loadingunloadingcondition
222 Pastor-Zienkiewicz-Chan Model This model was pre-sented by Pastor et al [19] The relationships between elasticvolumetric and shear strain increments and stress incrementsare defined as
1198891199011015840= 119870119890V119889120576119890
V 119889119902 = 3119866119890119904119889120576119890
119904 (10)
where 119870119890V 119866119890119904 are tangential bulk and shear moduli respec-
tively and they are assumed to be
119870119890V = 119870119890119904119900
1199011015840
119901119900
119866119890119904= 119866119890119904119900
1199011015840
119901119900
(11)
where119870119890119904119900 119866119890119904119900 and 119901
119900are model parameters
In order to determine the plastic stiffness tensor variablesn119892119871119880
n and 119867119871119880
need to be defined n119892119871119880
and n areexpressed as follows
n119892119871= (
119889119892
radic1 + 1198892
119892
1
radic1 + 1198892
119892
)
119879
n = (119889119891
radic1 + 1198892
119891
1
radic1 + 1198892
119891
)
119879
(12)
The dilatancy 119889119892and stress ratio 120578 = 119902119901 are related as
follows
119889119892=
119889120576119901
V
119889120576119901
119904
= (1 + 120572119892) (119872119892minus 120578) (13)
And 119889119891has a similar expression as
119889119891= (1 + 120572
119891) (119872119891minus 120578) (14)
where 120572119891 120572119892are model parameters and 119872
119892119872119891is equal
to relative density If 119889119891= 119889119892 associated flow rule is used
otherwise nonassociated flow rule is usedIn the case of unloading the unloading plastic flow
direction vector n119892119880
is defined as
n119892119880= (minus
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119889119892
radic1 + 1198892
119892
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1
radic1 + 1198892
119892
)
119879
(15)
The loading plastic modulus119867119871is proposed as
119867119871= 11986701199011015840119867119891(119867V + 119867119904)119867119863119872 (16)
where119867119891= (1 minus 120578120578
119891)4 limits the possible state and 120578
119891= (1+
1120572119891)119872119891119867V = 1minus120578119872119892 accounts for phase transformation
119867119904= 12057301205731exp(minus120573
0120585) considers soil degradation and 120585 is the
accumulated plastic shear strain119867119863119872= (120589MAX120589)
120574 accountsfor past history and 120589 = 119901[1 minus 120572
119891120578(1 + 120572
119891)119872119891](minus1120572)
119891which
is the mobilized stress function and 1198670 1205730 1205731 120574 are model
parametersUnder unloading condition the plastic modulus is
defined as
119867119880= 1198671199060(
119872119892
120578119906
)
120574119906
119872119892
120578
gt 1
119867119880= 1198671199060
119872119892
120578
le 1
(17)
respectively where1198671199060 120574119906aremodel parameters and 120578
119906is the
stress ratio from which unloading takes place
4 Journal of Applied Mathematics
0
1000
2000
3000
4000
5000
6000
1205761 ()
1205901minus1205903
(kPa
)
0 5 10 15
1205903 = 300kPa1205903 = 700kPa1205903 = 1200 kPa
(a)
0
1
2
3
4
120576
()
1205903 = 300kPa1205903 = 700kPa1205903 = 1200 kPa
0 5 10 151205761 ()
(b)
Figure 1 Simulation of stress-strain relationships for Original PZ-III model
223 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 predictthe static or dynamic responds of sands especially for rockfillmaterials 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 using aset of parameters under different confining pressures butPZ-III model gives the same 120576
1-120576V curve where 120576
1 120576V are
axial strain and volumetric strain respectively As confiningpressure ranges from 0 kPa to several MPa for a rockfill damwith height of 200ndash300m the original PZ-III model cannotbe used to describe the mechanical behavior of rockfill dams
Some relations of the original model are modified to takeinto account the influence of confining pressure as
119870119890V = 1198701198900119901119886(
1199011015840
119901119886
)
119898
119866119890119904= 1198661198900119901119886(
1199011015840
119901119886
)
119899
119867119871= 1198670119901119886(
1199011015840
119901119886
)
119898
119867119891(119867V + 119867119904)119867119863119872
(18)
where 1198701198900and 119866
1198900are elastic constants 119898 and 119899 are model
parameters to consider the effect of pressure dependencyAs sand behavior is dependent on densities or void ratio
a state pressure index 119868119901 proposed by Wang et al [35] was
introduced in the PZ-III model and (13) was modified as
119889119892=
119889120576119901
V
119889120576119901
119904
= (1 + 120572119892) (119872119892119868119901
119898119901
minus 120578) (19)
where 119898119901is a model parameter and 119868
119901= 119901119901
119888in which 119901
119888
is the mean pressure at critical state The critical state line isgiven by
119890119888= Γ minus 120582 log (119901
119888) (20)
3 Nuozhadu Hydropower Project
Nuozhadu 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 is5850MWThemost important part ofNuozhaduhydropowerproject is the high earth-rockfill damwith amaximumheightof 2615m which is the highest one with the same type inChina and the fourth highest in the world The reservoir hasa storage capacity of 2370 times 108m3 with the normal storagewater level of 8125m and dead water level of 765m
Figure 3 shows the material zoning and constructionstages of the maximum cross-section The elevation of theearth core bottom and the crest of the dam are 5626m and8241m respectively The dam crest has a longitudinal lengthof 630mwith a width of 18mThe upstream and downstreamslopes are at 19 1 and 18 1 respectively The dam body iscomposed of several different 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 material 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 end of 2012 Figure 2(c) shows the dam
Journal of Applied Mathematics 5
BurmaLaos
China
VietnamThailand
(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 practicalconstruction process
4 Experimental Validation ofModel Parameters
The modified PZ-III model was implemented in a finiteelement code which has been successfully used to analyzeearth dams with Duncan and Changrsquos EB model and someother constitutive models A set of triaxial test data was usedto make sure that the 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 thetriaxial test results Drained triaxial tests under different con-fining pressures were conducted to test the rockfill materialsand mixed gravel clay which are the main parts of the dambody
Duncan and Changrsquos EB model parameters are shown inTable 1 and the modified PZ-III model parameters in Table 2As shown in Figures 4 5 6 7 8 and 9 the modified PZ-III model presents a better ability to simulate the mechanics
Table 1 Material parameters of Duncan and Changrsquos EB model
Material Rockfill I Rockfill II Mixed gravel clay120593∘ 5582 5433 3930Δ120593∘ 1229 1207 980119877119891
073 074 077119870 1450 1360 520119870119887
550 600 250119870119906119903
2800 2500 900119899 030 043 042119898 013 008 025
behavior of rockfillmaterials andmixed gravel clay especiallyfor dilatancy With the reduction of confining pressurethe 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 intrinsic limitation Duncan andChangrsquos EB model cannot simulate the dilatancy which is acrucial feature of rockfill materials
6 Journal of Applied Mathematics
Upstream Downstream
RU1RU3F2F1
RU2 RD1
RD2
Cofferdam ED
F2F1
RD3
RU1RD1 upstreamdownstream rockfill zone IRU2RD2 upstreamdownstream rockfill zone IIRU3RD3 upstreamdownstream fine rockfill
F1F2 filter material zone IIIED clay mixed gravel
electromagnetism type settlement gauges
900
800
700
600
500
8241
658
(a)
8125 20121231
20110531sim20120531
20080215sim20080531 20080531sim20090531
20090531sim20100531
20100531sim20110531
(b)
Figure 3 The maximum cross-section (a) Material zoning and (b) construction stage
0
2000
4000
6000
8000
10000
300 kPa 900 kPa1500 kPa 2500 kPaDuncan-Chang EB
0 5 10 15
1205901minus1205903
(kPa
)
1205761 ()
(a)
minus4minus3minus2minus1
01234
1205761 ()0 5 10 15
120576
()
300 kPa 900 kPa1500 kPa 2500 kPaDuncan-Chang EB
(b)
Figure 4 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for rockfill material I
Journal of Applied Mathematics 7
0
2000
4000
6000
8000
10000
300 kPa 900 kPa1500 kPa 2500 kPaModified PZ
0 5 10 15
1205901minus1205903
(kPa
)
1205761 ()
(a)
minus4minus3minus2minus1
01234
1205761 ()0 5 10 15
120576
()
300 kPa 900 kPa1500 kPa 2500 kPaModified PZ
(b)
Figure 5 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for rockfill material I
Duncan-Chang EB
0 5 10 15
1205761 ()
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0
2000
4000
6000
8000
10000
1205901minus1205903
(kPa
)
(a)
0 5 10 151205761 ()
minus3
minus2
minus1
0
1
2
3
120576
()
Duncan-Chang EB
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 6 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for rockfill material II
Modified PZ
0 5 10 151205761 ()
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0
2000
4000
6000
8000
10000
1205901minus1205903
(kPa
)
(a)
0 5 10 151205761 ()
minus3
minus2
minus1
0
1
2
3
120576
()
Modified PZ
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 7 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for rockfill material II
8 Journal of Applied Mathematics
0
2000
4000
6000
Duncan-Chang EB
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3 0 5 10 151205761 ()
120576
()
Duncan-Chang EB
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 8 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for clay
0
2000
4000
6000
Modified PZ
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3
Modified PZ
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0 5 10 151205761 ()
120576
()
(b)
Figure 9 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for clay
Figure 10 3D FEMmesh of Nuozhadu dam
5 Three-Dimensional Finite Element Analyses
51 Computation Model The numerical analyses were per-formed to simulate the performance of the dam duringconstruction and impounding periods with effective stressfinite element analysis
First the 2D finite element mesh of the maximum cross-section of the dam was discretized according to the materialzoning and construction design (see Figure 3) Then the 2Dmesh was extended to 3D mesh in accordance with contourline of the river valley Figure 10 shows the 3D mesh ofthe Nuozhadu dam with 8095 brick and degenerated brickelements and 8340 nodes
The numerical simulations contain two stages filling andimpounding During the filling stage the dam body mainlysubjects to body weight Then at the end of constructionupstream water level goes up to the normal storage waterlevel The interaction between pore water and soil skeletonwas considered through the whole numerical computation
52 Results and Analyses
521 Numerical Results Analyses Figures 11 and 12 show thenumerical results of finite element analyses with Duncanand Changrsquos EB model and the modified PZ-III modelrespectively
Journal of Applied Mathematics 9
1
070503
09
01
(a)
0
05
0
minus1
minus15
minus24
minus05
minus2
(b)
051
152 25
3 353
252
151
05
(c)
0
03
05 1
13
minus02
(d)
Figure 11 Displacement and stress contour of the maximum section for Duncan and Changrsquos EB model (a) displacement along river (m)(b) vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
0706
06
050403
0201
0
02
minus02
01
(a)
minus25minus29
minus05minus15
minus1
minus2
(b)
05
1 152 3
3 435
3
25 215
105
(c)
01
05
1 15
0
(d)
Figure 12 Displacement and stress contour of the maximum section for the modified PZ-III model (a) displacement along river (m) (b)vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
Through the comparison and analysis of the numericalresults (Figures 11 and 12) we can find some similarities anddifferences for these two models
On one hand we can see many similar places in thedistributions of displacements and stresses
(1) After the reservoir impounding due to the hugewaterpressure on upstream dam horizontal displacementdevelops toward the downstream and the largestdisplacement is about 105m for EBmodel and 074mfor modified PZ-III model
(2) Themaximumsettlement occurs in themiddle of corewall due to lower modulus of clayey soil
(3) Because of the tremendous differences of modulusbetween rockfill material and clayey soil there existsobvious arching effect in the core wall
(4) Effective stress in upstream shell is less than thedownstream shell due to the water pressure in theupstream shell
On the other hand some differences also exist whichillustrate the advantages of modified PZ-III model
(1) After the reservoir is impounded upward displace-ment as large as 07m (see Figure 11(b)) developson the upstream shell near dam crest for EB modeland nearly 0m for modified PZ-III model (seeFigure 12(b)) In fact monitoring data of practicalengineering projects shows that no large upwarddisplacement happened after impounding This isdue to its weakness of EB model to distinguish theloading and unloading condition during the waterimpounding
(2) In the distribution of minor principle stress (Figures11(d) and 12(d)) negative stress (ie tensile stress)occurs in the upstream shell for EB model whereasvery little tensile stress exists for modified PZ-IIImodel As we know rockfill material is a typical kindof cohesionless coarse-grained soil which means thatit has no tensile strength Therefore the existence oflarge area of tensile stress in the upstream shell isunreasonable
522 Comparison between Numerical and In Situ MonitoringData Settlement is a key indicator to assess the safety of an
10 Journal of Applied Mathematics
550
600
650
700
750
800
850El
evat
ion
(m)
Settlement (mm)
In-situEBModified PZ
minus1000 0 1000 2000 3000 4000
Figure 13 Comparison between in situ monitoring settlement andFEM results
Table 2 Material parameters of the modified PZ-III model
Material Rockfill I Rockfill II Mixed gravel clay1198700
500 1000 3001198660
1500 3000 900119898 050 050 050119899 050 050 050120572119891
045 045 045120572119892
045 045 045119872119891119888
105 090 060119872119892119888
160 135 1101205730
000 000 0001205731
000 000 000Γ 034 031 034120582 010 009 003119898119901
035 040 001198670
800 1200 900120574 5 5 5120574119906
5 5 51198671199060MPa 9 9 10
earth dam Figures 13 and 14 show the in situmonitoring dataand FEM results of settlement in themaximum cross-sectionThe in situ data were obtained from electromagnetism typesettlement gaugeswhichwere embedded during constructionin the dam (as shown in Figure 3(a)) Through the compar-isons of in situmonitoring and numerical results we can seethat the modified PZ-III model gave a better prediction thanthe EB model However as deformation induced by wetting
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 655 m
2010
11
2010
61
0
2010
11
17
2011
42
6
2011
10
3
2012
31
1
2012
81
8
(a)
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 701 m
2010
91
2011
22
8
2011
82
7
2012
22
3
2012
82
1
(b)
Settl
emen
t (m
m)
0
500
1000
1500
2000
2500
Time
Elevation 751 m
2011
10
1
2011
12
20
2012
39
2012
52
8
2012
81
6
In-situEBModified PZ
(c)
Figure 14 Comparison between in situ monitoring settlement andFEM results
of rockfill materials was not considered the FEM result ofsettlement was below than the in situmonitoring data
As an elastoplastic model the PZ-III model is capableof representing the mechanical behavior of soils better thannonlinear elastic model such as Duncan and Changrsquos EBmodel And the above finite element analyses also proved it
6 Conclusions
This paper presents a modified PZ-III model based on thegeneralized theory and original Pastor-Zienkiewicz-Chan
Journal of Applied Mathematics 11
model to simulate the stress-strain relationship of rockfillmaterials
Triaxial test results of the filling materials of Nuozhadudamwere used to validate the proposedmodel and determinethe model parameters of Duncan and Changrsquos EB model andthe modified PZ-III model respectively The simulations oftriaxial stress-strain response show that the modified PZ-III model is capable of representing the key features ofcohesionless soil such as nonlinearity dilatancy and pressuredependency
The proposed model has been incorporated into a finiteelement code to simulate the static response of a high earth-rockfill dam in China The results were compared with thoseof Duncan and Changrsquos EB model The two set of resultshave both similarities and differences and the differencesillustrate the advantages of the modified PZ-III model Thecomparisons of FEM results and in situ monitoring datashowed that the modified PZ-III model can give a betterdescription of deformation of the earth-rockfill dam thanDuncan and Changrsquos EB model
Acknowledgments
This work was supported by the National Nature ScienceFoundation of China (51179092) and the State Key Laboratoryof Hydroscience and Engineering Project (2012-KY-02 and2013-KY-4)
References
[1] J M Duncan ldquoState of the art limit equilibrium and finite-element analysis of slopesrdquo Journal of Geotechnical and Geoen-vironmental Engineering vol 122 no 7 pp 577ndash596 1996
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] R S Sandhu and E L Wilson ldquoFinite element analysis ofseepage in elastic mediardquo Journal of the Engineering MechanicsDivision vol 95 no 3 pp 641ndash652 1969
[4] J T Christian and J W Boehmer ldquoPlane strain consolidationby finite elementsrdquo Journal of Soil Mechanics amp FoundationsDivision vol 96 no 4 pp 1435ndash1457 1970
[5] JMDuncan andC-Y Chang ldquoNonlinear analysis of stress andstrain in soilsrdquo Journal of the Soil Mechanics and FoundationsDivision vol 96 no 5 pp 1629ndash1653 1970
[6] J M Duncan P M Byrne K SWong and P Mabry ldquoStrengthstress-strain and bulk modulus parameters for finite elementanalyses of stresses and movements in soil massesrdquo Tech RepUCBGT80-01 University of California Berkeley Calif USA1980
[7] D C Drucker R E Gibson and D J Henkel ldquoSoil mechanicsand work-hardening theories of plasticityrdquo Transactions of theAmerican Society of Civil Engineers vol 122 pp 338ndash346 1957
[8] K Roscoe A Schofield andCWroth ldquoOn the yielding of soilsrdquoGeotechnique vol 8 no 1 pp 22ndash53 1958
[9] K Roscoe A Schofield and A Thurairajah ldquoYielding of claysin states wetter than criticalrdquo Geotechnique vol 13 no 3 pp211ndash240 1963
[10] J Burland ldquoCorrespondence on lsquoThe yielding and dilation ofclayrsquordquo Geotechnique vol 15 pp 211ndash214 1965
[11] P V Lade and J M Duncan ldquoElastoplastic stress-strain theoryfor cohesionless soilrdquo Journal of the Geotechnical EngineeringDivision vol 101 no 10 pp 1037ndash1053 1975
[12] I S Sandler F L DiMaggio and G Y Baladi ldquoGeneralizedcap model for geological materialsrdquo Journal of the GeotechnicalEngineering Division vol 102 no 7 pp 683ndash699 1976
[13] X-S Li Y F Dafalias and Z-L Wang ldquoState-dependent dila-tancy in critical-state constitutive modelling of sandrdquoCanadianGeotechnical Journal vol 36 no 4 pp 599ndash611 1999
[14] Y-P Yao and D Sun ldquoApplication of Ladersquos criterion to Cam-clay modelrdquo Journal of Engineering Mechanics vol 126 no 1pp 112ndash119 2000
[15] G Y Baladi and B Rohani ldquoElastic-plastic model for saturatedsandrdquo Journal of the Geotechnical Engineering Division vol 105no 4 pp 465ndash480 1979
[16] O Zienkiewicz and Z Mroz ldquoGeneralized plasticity formu-lation and applications to geomechanicsrdquo in Mechanics ofEngineering Materials C S Desai and R H Gallagher Eds pp655ndash679 John Wiley amp Sons New York NY USA 1984
[17] C S Desai and M O Faruque ldquoConstitutive model forgeological materialsrdquo Journal of Engineering Mechanics vol 110no 9 pp 1391ndash1408 1984
[18] S B R Murthy A Vatsala and T S Nagaraj ldquoRevised Cam-clay modelrdquo Journal of Geotechnical Engineering vol 117 no 6pp 851ndash871 1991
[19] M Pastor O C Zienkiewicz and A H C Chan ldquoGeneralizedplasticity and the modelling of soil behaviourrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 14 no 3 pp 151ndash190 1990
[20] ZMroz andO Zienkiewicz ldquoUniform formulation of constitu-tive equations for clays and sandsrdquo inMechanics of EngineeringMaterials C S Desai and R H Gallangher Eds pp 415ndash449John Wiley amp Sons New York NY USA 1984
[21] G Wang and J-M Zhang ldquoDynamic consolidation finiteelement analysis of a sediment-protecting dyke under oceanwave loadingrdquo Rock and Soil Mechanics vol 27 no 4 pp 555ndash560 2006
[22] MAlyamiMRouainia and SMWilkinson ldquoNumerical anal-ysis of deformation behaviour of quay walls under earthquakeloadingrdquo Soil Dynamics and Earthquake Engineering vol 29 no3 pp 525ndash536 2009
[23] H Li P Manuel and T Li ldquoApplication of an generalizedplasticity model to ultra-high rockfill damrdquo in Proceedingsof the 12th International Conference on Engineering ScienceConstruction and Operations in Challenging EnvironmentsmdashEarth and Space pp 385ndash398 Honolulu Hawaii USA March2010
[24] T Li and H Zhang ldquoDynamic parameter verification of P-Z model and its application of dynamic analysis on rockfilldamrdquo in Proceedings of the 12th International Conference onEngineering Science Construction and Operations in Challeng-ing EnvironmentsmdashEarth and Space pp 2706ndash2713 HonoluluHawaii USA March 2010
[25] M Pastor ldquoA generalized plasticity model for anisotropicbehaviour of sandrdquoComputer Methods and Advances in Geome-chanics vol 1 pp 661ndash668 1991
[26] G Bolzon B A Schrefler and O C Zienkiewicz ldquoElastoplasticsoil constitutive laws generalized to partially saturated statesrdquoGeotechnique vol 46 no 2 pp 279ndash289 1996
[27] H I Ling and H Liu ldquoPressure-level dependency and densifi-cation behavior of sand through generalized plasticity modelrdquo
12 Journal of Applied Mathematics
Journal of Engineering Mechanics vol 129 no 8 pp 851ndash8602003
[28] H I Ling and S Yang ldquoUnified sand model based on thecritical state and generalized plasticityrdquo Journal of EngineeringMechanics vol 132 no 12 pp 1380ndash1391 2006
[29] N D Marschi C K Chan and H B Seed ldquoEvaluation ofproperties of rockfill materialsrdquo Journal of the Soil Mechanicsand Foundations Division vol 98 no 1 pp 95ndash114 1972
[30] R J Marsal ldquoLarge scale testing of rockfill materialsrdquo Journal ofthe Soil Mechanics and Foundations Division vol 93 no 2 pp27ndash43 1967
[31] R JMarsal ldquoMechanical properties of rockfillrdquo in EmbankmentDam Engineering pp 109ndash200 John Wiley amp Sons New YorkNY USA 1973
[32] P V Lade J A Yamamuro and P A Bopp ldquoSignificance ofparticle crushing in granular materialsrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 122 no 4 pp 309ndash3161996
[33] BOHardin ldquoCrushing of soil particlesrdquo Journal of GeotechnicalEngineering vol 111 no 10 pp 1177ndash1192 1985
[34] R L Kondner ldquoHyperbolic stress-strain response cohesivesoilsrdquo Journal of the Soil Mechanics and Foundations Divisionvol 89 no 1 pp 115ndash143 1963
[35] Z-LWang Y F Dafalias X-S Li and F I Makdisi ldquoState pres-sure index for modeling sand behaviorrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 128 no 6 pp 511ndash5192002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
key futures of this theory are that neither yield surface norplastic potential surface needs to be defined explicitly andconsistency law is not required to determine plastic modulusIn the theory the total strain increment is divided into elasticand plastic components
Consider
119889120576 = 119889120576119890+ 119889120576119901 (6)
where 119889120576119890 and 119889120576119901 = elastic and plastic strain incrementsrespectively
The relationship between strain and stress increments isexpressed as
119889120590 = D119890119901 119889120576 (7)
whereD119890119901 is the elastoplastic stiffness tensor given as
D119890119901 = D119890 minusD119890 n
119892119871119880 n119879 D119890
119867119871119880+ n119879 D119890 n
119892119871119880
(8)
where D119890 n119892119871119880
n and 119867119871119880
are elastic stiffness tensorplastic flow direction vector loading direction vector andplastic modulus under loading or unloading conditionsrespectively
The loading direction vectorn is used to judge the loadingand unloading conditions
119889120590119879
119890sdot n gt 0 loading
119889120590119879
119890sdot n = 0 neutral loading
119889120590119879
119890sdot n lt 0 unloading
(9)
Then the elastoplastic stiffness tensor D119890119901 can beobtained corresponding to the loading and unloading con-ditions
In the framework of generalized plasticity theory all thecomponents of the elastoplastic constitutive matrix are deter-mined by the current state of stress and loadingunloadingcondition
222 Pastor-Zienkiewicz-Chan Model This model was pre-sented by Pastor et al [19] The relationships between elasticvolumetric and shear strain increments and stress incrementsare defined as
1198891199011015840= 119870119890V119889120576119890
V 119889119902 = 3119866119890119904119889120576119890
119904 (10)
where 119870119890V 119866119890119904 are tangential bulk and shear moduli respec-
tively and they are assumed to be
119870119890V = 119870119890119904119900
1199011015840
119901119900
119866119890119904= 119866119890119904119900
1199011015840
119901119900
(11)
where119870119890119904119900 119866119890119904119900 and 119901
119900are model parameters
In order to determine the plastic stiffness tensor variablesn119892119871119880
n and 119867119871119880
need to be defined n119892119871119880
and n areexpressed as follows
n119892119871= (
119889119892
radic1 + 1198892
119892
1
radic1 + 1198892
119892
)
119879
n = (119889119891
radic1 + 1198892
119891
1
radic1 + 1198892
119891
)
119879
(12)
The dilatancy 119889119892and stress ratio 120578 = 119902119901 are related as
follows
119889119892=
119889120576119901
V
119889120576119901
119904
= (1 + 120572119892) (119872119892minus 120578) (13)
And 119889119891has a similar expression as
119889119891= (1 + 120572
119891) (119872119891minus 120578) (14)
where 120572119891 120572119892are model parameters and 119872
119892119872119891is equal
to relative density If 119889119891= 119889119892 associated flow rule is used
otherwise nonassociated flow rule is usedIn the case of unloading the unloading plastic flow
direction vector n119892119880
is defined as
n119892119880= (minus
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119889119892
radic1 + 1198892
119892
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1
radic1 + 1198892
119892
)
119879
(15)
The loading plastic modulus119867119871is proposed as
119867119871= 11986701199011015840119867119891(119867V + 119867119904)119867119863119872 (16)
where119867119891= (1 minus 120578120578
119891)4 limits the possible state and 120578
119891= (1+
1120572119891)119872119891119867V = 1minus120578119872119892 accounts for phase transformation
119867119904= 12057301205731exp(minus120573
0120585) considers soil degradation and 120585 is the
accumulated plastic shear strain119867119863119872= (120589MAX120589)
120574 accountsfor past history and 120589 = 119901[1 minus 120572
119891120578(1 + 120572
119891)119872119891](minus1120572)
119891which
is the mobilized stress function and 1198670 1205730 1205731 120574 are model
parametersUnder unloading condition the plastic modulus is
defined as
119867119880= 1198671199060(
119872119892
120578119906
)
120574119906
119872119892
120578
gt 1
119867119880= 1198671199060
119872119892
120578
le 1
(17)
respectively where1198671199060 120574119906aremodel parameters and 120578
119906is the
stress ratio from which unloading takes place
4 Journal of Applied Mathematics
0
1000
2000
3000
4000
5000
6000
1205761 ()
1205901minus1205903
(kPa
)
0 5 10 15
1205903 = 300kPa1205903 = 700kPa1205903 = 1200 kPa
(a)
0
1
2
3
4
120576
()
1205903 = 300kPa1205903 = 700kPa1205903 = 1200 kPa
0 5 10 151205761 ()
(b)
Figure 1 Simulation of stress-strain relationships for Original PZ-III model
223 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 predictthe static or dynamic responds of sands especially for rockfillmaterials 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 using aset of parameters under different confining pressures butPZ-III model gives the same 120576
1-120576V curve where 120576
1 120576V are
axial strain and volumetric strain respectively As confiningpressure ranges from 0 kPa to several MPa for a rockfill damwith height of 200ndash300m the original PZ-III model cannotbe used to describe the mechanical behavior of rockfill dams
Some relations of the original model are modified to takeinto account the influence of confining pressure as
119870119890V = 1198701198900119901119886(
1199011015840
119901119886
)
119898
119866119890119904= 1198661198900119901119886(
1199011015840
119901119886
)
119899
119867119871= 1198670119901119886(
1199011015840
119901119886
)
119898
119867119891(119867V + 119867119904)119867119863119872
(18)
where 1198701198900and 119866
1198900are elastic constants 119898 and 119899 are model
parameters to consider the effect of pressure dependencyAs sand behavior is dependent on densities or void ratio
a state pressure index 119868119901 proposed by Wang et al [35] was
introduced in the PZ-III model and (13) was modified as
119889119892=
119889120576119901
V
119889120576119901
119904
= (1 + 120572119892) (119872119892119868119901
119898119901
minus 120578) (19)
where 119898119901is a model parameter and 119868
119901= 119901119901
119888in which 119901
119888
is the mean pressure at critical state The critical state line isgiven by
119890119888= Γ minus 120582 log (119901
119888) (20)
3 Nuozhadu Hydropower Project
Nuozhadu 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 is5850MWThemost important part ofNuozhaduhydropowerproject is the high earth-rockfill damwith amaximumheightof 2615m which is the highest one with the same type inChina and the fourth highest in the world The reservoir hasa storage capacity of 2370 times 108m3 with the normal storagewater level of 8125m and dead water level of 765m
Figure 3 shows the material zoning and constructionstages of the maximum cross-section The elevation of theearth core bottom and the crest of the dam are 5626m and8241m respectively The dam crest has a longitudinal lengthof 630mwith a width of 18mThe upstream and downstreamslopes are at 19 1 and 18 1 respectively The dam body iscomposed of several different 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 material 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 end of 2012 Figure 2(c) shows the dam
Journal of Applied Mathematics 5
BurmaLaos
China
VietnamThailand
(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 practicalconstruction process
4 Experimental Validation ofModel Parameters
The modified PZ-III model was implemented in a finiteelement code which has been successfully used to analyzeearth dams with Duncan and Changrsquos EB model and someother constitutive models A set of triaxial test data was usedto make sure that the 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 thetriaxial test results Drained triaxial tests under different con-fining pressures were conducted to test the rockfill materialsand mixed gravel clay which are the main parts of the dambody
Duncan and Changrsquos EB model parameters are shown inTable 1 and the modified PZ-III model parameters in Table 2As shown in Figures 4 5 6 7 8 and 9 the modified PZ-III model presents a better ability to simulate the mechanics
Table 1 Material parameters of Duncan and Changrsquos EB model
Material Rockfill I Rockfill II Mixed gravel clay120593∘ 5582 5433 3930Δ120593∘ 1229 1207 980119877119891
073 074 077119870 1450 1360 520119870119887
550 600 250119870119906119903
2800 2500 900119899 030 043 042119898 013 008 025
behavior of rockfillmaterials andmixed gravel clay especiallyfor dilatancy With the reduction of confining pressurethe 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 intrinsic limitation Duncan andChangrsquos EB model cannot simulate the dilatancy which is acrucial feature of rockfill materials
6 Journal of Applied Mathematics
Upstream Downstream
RU1RU3F2F1
RU2 RD1
RD2
Cofferdam ED
F2F1
RD3
RU1RD1 upstreamdownstream rockfill zone IRU2RD2 upstreamdownstream rockfill zone IIRU3RD3 upstreamdownstream fine rockfill
F1F2 filter material zone IIIED clay mixed gravel
electromagnetism type settlement gauges
900
800
700
600
500
8241
658
(a)
8125 20121231
20110531sim20120531
20080215sim20080531 20080531sim20090531
20090531sim20100531
20100531sim20110531
(b)
Figure 3 The maximum cross-section (a) Material zoning and (b) construction stage
0
2000
4000
6000
8000
10000
300 kPa 900 kPa1500 kPa 2500 kPaDuncan-Chang EB
0 5 10 15
1205901minus1205903
(kPa
)
1205761 ()
(a)
minus4minus3minus2minus1
01234
1205761 ()0 5 10 15
120576
()
300 kPa 900 kPa1500 kPa 2500 kPaDuncan-Chang EB
(b)
Figure 4 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for rockfill material I
Journal of Applied Mathematics 7
0
2000
4000
6000
8000
10000
300 kPa 900 kPa1500 kPa 2500 kPaModified PZ
0 5 10 15
1205901minus1205903
(kPa
)
1205761 ()
(a)
minus4minus3minus2minus1
01234
1205761 ()0 5 10 15
120576
()
300 kPa 900 kPa1500 kPa 2500 kPaModified PZ
(b)
Figure 5 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for rockfill material I
Duncan-Chang EB
0 5 10 15
1205761 ()
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0
2000
4000
6000
8000
10000
1205901minus1205903
(kPa
)
(a)
0 5 10 151205761 ()
minus3
minus2
minus1
0
1
2
3
120576
()
Duncan-Chang EB
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 6 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for rockfill material II
Modified PZ
0 5 10 151205761 ()
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0
2000
4000
6000
8000
10000
1205901minus1205903
(kPa
)
(a)
0 5 10 151205761 ()
minus3
minus2
minus1
0
1
2
3
120576
()
Modified PZ
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 7 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for rockfill material II
8 Journal of Applied Mathematics
0
2000
4000
6000
Duncan-Chang EB
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3 0 5 10 151205761 ()
120576
()
Duncan-Chang EB
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 8 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for clay
0
2000
4000
6000
Modified PZ
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3
Modified PZ
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0 5 10 151205761 ()
120576
()
(b)
Figure 9 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for clay
Figure 10 3D FEMmesh of Nuozhadu dam
5 Three-Dimensional Finite Element Analyses
51 Computation Model The numerical analyses were per-formed to simulate the performance of the dam duringconstruction and impounding periods with effective stressfinite element analysis
First the 2D finite element mesh of the maximum cross-section of the dam was discretized according to the materialzoning and construction design (see Figure 3) Then the 2Dmesh was extended to 3D mesh in accordance with contourline of the river valley Figure 10 shows the 3D mesh ofthe Nuozhadu dam with 8095 brick and degenerated brickelements and 8340 nodes
The numerical simulations contain two stages filling andimpounding During the filling stage the dam body mainlysubjects to body weight Then at the end of constructionupstream water level goes up to the normal storage waterlevel The interaction between pore water and soil skeletonwas considered through the whole numerical computation
52 Results and Analyses
521 Numerical Results Analyses Figures 11 and 12 show thenumerical results of finite element analyses with Duncanand Changrsquos EB model and the modified PZ-III modelrespectively
Journal of Applied Mathematics 9
1
070503
09
01
(a)
0
05
0
minus1
minus15
minus24
minus05
minus2
(b)
051
152 25
3 353
252
151
05
(c)
0
03
05 1
13
minus02
(d)
Figure 11 Displacement and stress contour of the maximum section for Duncan and Changrsquos EB model (a) displacement along river (m)(b) vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
0706
06
050403
0201
0
02
minus02
01
(a)
minus25minus29
minus05minus15
minus1
minus2
(b)
05
1 152 3
3 435
3
25 215
105
(c)
01
05
1 15
0
(d)
Figure 12 Displacement and stress contour of the maximum section for the modified PZ-III model (a) displacement along river (m) (b)vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
Through the comparison and analysis of the numericalresults (Figures 11 and 12) we can find some similarities anddifferences for these two models
On one hand we can see many similar places in thedistributions of displacements and stresses
(1) After the reservoir impounding due to the hugewaterpressure on upstream dam horizontal displacementdevelops toward the downstream and the largestdisplacement is about 105m for EBmodel and 074mfor modified PZ-III model
(2) Themaximumsettlement occurs in themiddle of corewall due to lower modulus of clayey soil
(3) Because of the tremendous differences of modulusbetween rockfill material and clayey soil there existsobvious arching effect in the core wall
(4) Effective stress in upstream shell is less than thedownstream shell due to the water pressure in theupstream shell
On the other hand some differences also exist whichillustrate the advantages of modified PZ-III model
(1) After the reservoir is impounded upward displace-ment as large as 07m (see Figure 11(b)) developson the upstream shell near dam crest for EB modeland nearly 0m for modified PZ-III model (seeFigure 12(b)) In fact monitoring data of practicalengineering projects shows that no large upwarddisplacement happened after impounding This isdue to its weakness of EB model to distinguish theloading and unloading condition during the waterimpounding
(2) In the distribution of minor principle stress (Figures11(d) and 12(d)) negative stress (ie tensile stress)occurs in the upstream shell for EB model whereasvery little tensile stress exists for modified PZ-IIImodel As we know rockfill material is a typical kindof cohesionless coarse-grained soil which means thatit has no tensile strength Therefore the existence oflarge area of tensile stress in the upstream shell isunreasonable
522 Comparison between Numerical and In Situ MonitoringData Settlement is a key indicator to assess the safety of an
10 Journal of Applied Mathematics
550
600
650
700
750
800
850El
evat
ion
(m)
Settlement (mm)
In-situEBModified PZ
minus1000 0 1000 2000 3000 4000
Figure 13 Comparison between in situ monitoring settlement andFEM results
Table 2 Material parameters of the modified PZ-III model
Material Rockfill I Rockfill II Mixed gravel clay1198700
500 1000 3001198660
1500 3000 900119898 050 050 050119899 050 050 050120572119891
045 045 045120572119892
045 045 045119872119891119888
105 090 060119872119892119888
160 135 1101205730
000 000 0001205731
000 000 000Γ 034 031 034120582 010 009 003119898119901
035 040 001198670
800 1200 900120574 5 5 5120574119906
5 5 51198671199060MPa 9 9 10
earth dam Figures 13 and 14 show the in situmonitoring dataand FEM results of settlement in themaximum cross-sectionThe in situ data were obtained from electromagnetism typesettlement gaugeswhichwere embedded during constructionin the dam (as shown in Figure 3(a)) Through the compar-isons of in situmonitoring and numerical results we can seethat the modified PZ-III model gave a better prediction thanthe EB model However as deformation induced by wetting
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 655 m
2010
11
2010
61
0
2010
11
17
2011
42
6
2011
10
3
2012
31
1
2012
81
8
(a)
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 701 m
2010
91
2011
22
8
2011
82
7
2012
22
3
2012
82
1
(b)
Settl
emen
t (m
m)
0
500
1000
1500
2000
2500
Time
Elevation 751 m
2011
10
1
2011
12
20
2012
39
2012
52
8
2012
81
6
In-situEBModified PZ
(c)
Figure 14 Comparison between in situ monitoring settlement andFEM results
of rockfill materials was not considered the FEM result ofsettlement was below than the in situmonitoring data
As an elastoplastic model the PZ-III model is capableof representing the mechanical behavior of soils better thannonlinear elastic model such as Duncan and Changrsquos EBmodel And the above finite element analyses also proved it
6 Conclusions
This paper presents a modified PZ-III model based on thegeneralized theory and original Pastor-Zienkiewicz-Chan
Journal of Applied Mathematics 11
model to simulate the stress-strain relationship of rockfillmaterials
Triaxial test results of the filling materials of Nuozhadudamwere used to validate the proposedmodel and determinethe model parameters of Duncan and Changrsquos EB model andthe modified PZ-III model respectively The simulations oftriaxial stress-strain response show that the modified PZ-III model is capable of representing the key features ofcohesionless soil such as nonlinearity dilatancy and pressuredependency
The proposed model has been incorporated into a finiteelement code to simulate the static response of a high earth-rockfill dam in China The results were compared with thoseof Duncan and Changrsquos EB model The two set of resultshave both similarities and differences and the differencesillustrate the advantages of the modified PZ-III model Thecomparisons of FEM results and in situ monitoring datashowed that the modified PZ-III model can give a betterdescription of deformation of the earth-rockfill dam thanDuncan and Changrsquos EB model
Acknowledgments
This work was supported by the National Nature ScienceFoundation of China (51179092) and the State Key Laboratoryof Hydroscience and Engineering Project (2012-KY-02 and2013-KY-4)
References
[1] J M Duncan ldquoState of the art limit equilibrium and finite-element analysis of slopesrdquo Journal of Geotechnical and Geoen-vironmental Engineering vol 122 no 7 pp 577ndash596 1996
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] R S Sandhu and E L Wilson ldquoFinite element analysis ofseepage in elastic mediardquo Journal of the Engineering MechanicsDivision vol 95 no 3 pp 641ndash652 1969
[4] J T Christian and J W Boehmer ldquoPlane strain consolidationby finite elementsrdquo Journal of Soil Mechanics amp FoundationsDivision vol 96 no 4 pp 1435ndash1457 1970
[5] JMDuncan andC-Y Chang ldquoNonlinear analysis of stress andstrain in soilsrdquo Journal of the Soil Mechanics and FoundationsDivision vol 96 no 5 pp 1629ndash1653 1970
[6] J M Duncan P M Byrne K SWong and P Mabry ldquoStrengthstress-strain and bulk modulus parameters for finite elementanalyses of stresses and movements in soil massesrdquo Tech RepUCBGT80-01 University of California Berkeley Calif USA1980
[7] D C Drucker R E Gibson and D J Henkel ldquoSoil mechanicsand work-hardening theories of plasticityrdquo Transactions of theAmerican Society of Civil Engineers vol 122 pp 338ndash346 1957
[8] K Roscoe A Schofield andCWroth ldquoOn the yielding of soilsrdquoGeotechnique vol 8 no 1 pp 22ndash53 1958
[9] K Roscoe A Schofield and A Thurairajah ldquoYielding of claysin states wetter than criticalrdquo Geotechnique vol 13 no 3 pp211ndash240 1963
[10] J Burland ldquoCorrespondence on lsquoThe yielding and dilation ofclayrsquordquo Geotechnique vol 15 pp 211ndash214 1965
[11] P V Lade and J M Duncan ldquoElastoplastic stress-strain theoryfor cohesionless soilrdquo Journal of the Geotechnical EngineeringDivision vol 101 no 10 pp 1037ndash1053 1975
[12] I S Sandler F L DiMaggio and G Y Baladi ldquoGeneralizedcap model for geological materialsrdquo Journal of the GeotechnicalEngineering Division vol 102 no 7 pp 683ndash699 1976
[13] X-S Li Y F Dafalias and Z-L Wang ldquoState-dependent dila-tancy in critical-state constitutive modelling of sandrdquoCanadianGeotechnical Journal vol 36 no 4 pp 599ndash611 1999
[14] Y-P Yao and D Sun ldquoApplication of Ladersquos criterion to Cam-clay modelrdquo Journal of Engineering Mechanics vol 126 no 1pp 112ndash119 2000
[15] G Y Baladi and B Rohani ldquoElastic-plastic model for saturatedsandrdquo Journal of the Geotechnical Engineering Division vol 105no 4 pp 465ndash480 1979
[16] O Zienkiewicz and Z Mroz ldquoGeneralized plasticity formu-lation and applications to geomechanicsrdquo in Mechanics ofEngineering Materials C S Desai and R H Gallagher Eds pp655ndash679 John Wiley amp Sons New York NY USA 1984
[17] C S Desai and M O Faruque ldquoConstitutive model forgeological materialsrdquo Journal of Engineering Mechanics vol 110no 9 pp 1391ndash1408 1984
[18] S B R Murthy A Vatsala and T S Nagaraj ldquoRevised Cam-clay modelrdquo Journal of Geotechnical Engineering vol 117 no 6pp 851ndash871 1991
[19] M Pastor O C Zienkiewicz and A H C Chan ldquoGeneralizedplasticity and the modelling of soil behaviourrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 14 no 3 pp 151ndash190 1990
[20] ZMroz andO Zienkiewicz ldquoUniform formulation of constitu-tive equations for clays and sandsrdquo inMechanics of EngineeringMaterials C S Desai and R H Gallangher Eds pp 415ndash449John Wiley amp Sons New York NY USA 1984
[21] G Wang and J-M Zhang ldquoDynamic consolidation finiteelement analysis of a sediment-protecting dyke under oceanwave loadingrdquo Rock and Soil Mechanics vol 27 no 4 pp 555ndash560 2006
[22] MAlyamiMRouainia and SMWilkinson ldquoNumerical anal-ysis of deformation behaviour of quay walls under earthquakeloadingrdquo Soil Dynamics and Earthquake Engineering vol 29 no3 pp 525ndash536 2009
[23] H Li P Manuel and T Li ldquoApplication of an generalizedplasticity model to ultra-high rockfill damrdquo in Proceedingsof the 12th International Conference on Engineering ScienceConstruction and Operations in Challenging EnvironmentsmdashEarth and Space pp 385ndash398 Honolulu Hawaii USA March2010
[24] T Li and H Zhang ldquoDynamic parameter verification of P-Z model and its application of dynamic analysis on rockfilldamrdquo in Proceedings of the 12th International Conference onEngineering Science Construction and Operations in Challeng-ing EnvironmentsmdashEarth and Space pp 2706ndash2713 HonoluluHawaii USA March 2010
[25] M Pastor ldquoA generalized plasticity model for anisotropicbehaviour of sandrdquoComputer Methods and Advances in Geome-chanics vol 1 pp 661ndash668 1991
[26] G Bolzon B A Schrefler and O C Zienkiewicz ldquoElastoplasticsoil constitutive laws generalized to partially saturated statesrdquoGeotechnique vol 46 no 2 pp 279ndash289 1996
[27] H I Ling and H Liu ldquoPressure-level dependency and densifi-cation behavior of sand through generalized plasticity modelrdquo
12 Journal of Applied Mathematics
Journal of Engineering Mechanics vol 129 no 8 pp 851ndash8602003
[28] H I Ling and S Yang ldquoUnified sand model based on thecritical state and generalized plasticityrdquo Journal of EngineeringMechanics vol 132 no 12 pp 1380ndash1391 2006
[29] N D Marschi C K Chan and H B Seed ldquoEvaluation ofproperties of rockfill materialsrdquo Journal of the Soil Mechanicsand Foundations Division vol 98 no 1 pp 95ndash114 1972
[30] R J Marsal ldquoLarge scale testing of rockfill materialsrdquo Journal ofthe Soil Mechanics and Foundations Division vol 93 no 2 pp27ndash43 1967
[31] R JMarsal ldquoMechanical properties of rockfillrdquo in EmbankmentDam Engineering pp 109ndash200 John Wiley amp Sons New YorkNY USA 1973
[32] P V Lade J A Yamamuro and P A Bopp ldquoSignificance ofparticle crushing in granular materialsrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 122 no 4 pp 309ndash3161996
[33] BOHardin ldquoCrushing of soil particlesrdquo Journal of GeotechnicalEngineering vol 111 no 10 pp 1177ndash1192 1985
[34] R L Kondner ldquoHyperbolic stress-strain response cohesivesoilsrdquo Journal of the Soil Mechanics and Foundations Divisionvol 89 no 1 pp 115ndash143 1963
[35] Z-LWang Y F Dafalias X-S Li and F I Makdisi ldquoState pres-sure index for modeling sand behaviorrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 128 no 6 pp 511ndash5192002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
0
1000
2000
3000
4000
5000
6000
1205761 ()
1205901minus1205903
(kPa
)
0 5 10 15
1205903 = 300kPa1205903 = 700kPa1205903 = 1200 kPa
(a)
0
1
2
3
4
120576
()
1205903 = 300kPa1205903 = 700kPa1205903 = 1200 kPa
0 5 10 151205761 ()
(b)
Figure 1 Simulation of stress-strain relationships for Original PZ-III model
223 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 predictthe static or dynamic responds of sands especially for rockfillmaterials 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 using aset of parameters under different confining pressures butPZ-III model gives the same 120576
1-120576V curve where 120576
1 120576V are
axial strain and volumetric strain respectively As confiningpressure ranges from 0 kPa to several MPa for a rockfill damwith height of 200ndash300m the original PZ-III model cannotbe used to describe the mechanical behavior of rockfill dams
Some relations of the original model are modified to takeinto account the influence of confining pressure as
119870119890V = 1198701198900119901119886(
1199011015840
119901119886
)
119898
119866119890119904= 1198661198900119901119886(
1199011015840
119901119886
)
119899
119867119871= 1198670119901119886(
1199011015840
119901119886
)
119898
119867119891(119867V + 119867119904)119867119863119872
(18)
where 1198701198900and 119866
1198900are elastic constants 119898 and 119899 are model
parameters to consider the effect of pressure dependencyAs sand behavior is dependent on densities or void ratio
a state pressure index 119868119901 proposed by Wang et al [35] was
introduced in the PZ-III model and (13) was modified as
119889119892=
119889120576119901
V
119889120576119901
119904
= (1 + 120572119892) (119872119892119868119901
119898119901
minus 120578) (19)
where 119898119901is a model parameter and 119868
119901= 119901119901
119888in which 119901
119888
is the mean pressure at critical state The critical state line isgiven by
119890119888= Γ minus 120582 log (119901
119888) (20)
3 Nuozhadu Hydropower Project
Nuozhadu 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 is5850MWThemost important part ofNuozhaduhydropowerproject is the high earth-rockfill damwith amaximumheightof 2615m which is the highest one with the same type inChina and the fourth highest in the world The reservoir hasa storage capacity of 2370 times 108m3 with the normal storagewater level of 8125m and dead water level of 765m
Figure 3 shows the material zoning and constructionstages of the maximum cross-section The elevation of theearth core bottom and the crest of the dam are 5626m and8241m respectively The dam crest has a longitudinal lengthof 630mwith a width of 18mThe upstream and downstreamslopes are at 19 1 and 18 1 respectively The dam body iscomposed of several different 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 material 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 end of 2012 Figure 2(c) shows the dam
Journal of Applied Mathematics 5
BurmaLaos
China
VietnamThailand
(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 practicalconstruction process
4 Experimental Validation ofModel Parameters
The modified PZ-III model was implemented in a finiteelement code which has been successfully used to analyzeearth dams with Duncan and Changrsquos EB model and someother constitutive models A set of triaxial test data was usedto make sure that the 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 thetriaxial test results Drained triaxial tests under different con-fining pressures were conducted to test the rockfill materialsand mixed gravel clay which are the main parts of the dambody
Duncan and Changrsquos EB model parameters are shown inTable 1 and the modified PZ-III model parameters in Table 2As shown in Figures 4 5 6 7 8 and 9 the modified PZ-III model presents a better ability to simulate the mechanics
Table 1 Material parameters of Duncan and Changrsquos EB model
Material Rockfill I Rockfill II Mixed gravel clay120593∘ 5582 5433 3930Δ120593∘ 1229 1207 980119877119891
073 074 077119870 1450 1360 520119870119887
550 600 250119870119906119903
2800 2500 900119899 030 043 042119898 013 008 025
behavior of rockfillmaterials andmixed gravel clay especiallyfor dilatancy With the reduction of confining pressurethe 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 intrinsic limitation Duncan andChangrsquos EB model cannot simulate the dilatancy which is acrucial feature of rockfill materials
6 Journal of Applied Mathematics
Upstream Downstream
RU1RU3F2F1
RU2 RD1
RD2
Cofferdam ED
F2F1
RD3
RU1RD1 upstreamdownstream rockfill zone IRU2RD2 upstreamdownstream rockfill zone IIRU3RD3 upstreamdownstream fine rockfill
F1F2 filter material zone IIIED clay mixed gravel
electromagnetism type settlement gauges
900
800
700
600
500
8241
658
(a)
8125 20121231
20110531sim20120531
20080215sim20080531 20080531sim20090531
20090531sim20100531
20100531sim20110531
(b)
Figure 3 The maximum cross-section (a) Material zoning and (b) construction stage
0
2000
4000
6000
8000
10000
300 kPa 900 kPa1500 kPa 2500 kPaDuncan-Chang EB
0 5 10 15
1205901minus1205903
(kPa
)
1205761 ()
(a)
minus4minus3minus2minus1
01234
1205761 ()0 5 10 15
120576
()
300 kPa 900 kPa1500 kPa 2500 kPaDuncan-Chang EB
(b)
Figure 4 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for rockfill material I
Journal of Applied Mathematics 7
0
2000
4000
6000
8000
10000
300 kPa 900 kPa1500 kPa 2500 kPaModified PZ
0 5 10 15
1205901minus1205903
(kPa
)
1205761 ()
(a)
minus4minus3minus2minus1
01234
1205761 ()0 5 10 15
120576
()
300 kPa 900 kPa1500 kPa 2500 kPaModified PZ
(b)
Figure 5 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for rockfill material I
Duncan-Chang EB
0 5 10 15
1205761 ()
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0
2000
4000
6000
8000
10000
1205901minus1205903
(kPa
)
(a)
0 5 10 151205761 ()
minus3
minus2
minus1
0
1
2
3
120576
()
Duncan-Chang EB
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 6 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for rockfill material II
Modified PZ
0 5 10 151205761 ()
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0
2000
4000
6000
8000
10000
1205901minus1205903
(kPa
)
(a)
0 5 10 151205761 ()
minus3
minus2
minus1
0
1
2
3
120576
()
Modified PZ
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 7 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for rockfill material II
8 Journal of Applied Mathematics
0
2000
4000
6000
Duncan-Chang EB
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3 0 5 10 151205761 ()
120576
()
Duncan-Chang EB
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 8 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for clay
0
2000
4000
6000
Modified PZ
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3
Modified PZ
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0 5 10 151205761 ()
120576
()
(b)
Figure 9 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for clay
Figure 10 3D FEMmesh of Nuozhadu dam
5 Three-Dimensional Finite Element Analyses
51 Computation Model The numerical analyses were per-formed to simulate the performance of the dam duringconstruction and impounding periods with effective stressfinite element analysis
First the 2D finite element mesh of the maximum cross-section of the dam was discretized according to the materialzoning and construction design (see Figure 3) Then the 2Dmesh was extended to 3D mesh in accordance with contourline of the river valley Figure 10 shows the 3D mesh ofthe Nuozhadu dam with 8095 brick and degenerated brickelements and 8340 nodes
The numerical simulations contain two stages filling andimpounding During the filling stage the dam body mainlysubjects to body weight Then at the end of constructionupstream water level goes up to the normal storage waterlevel The interaction between pore water and soil skeletonwas considered through the whole numerical computation
52 Results and Analyses
521 Numerical Results Analyses Figures 11 and 12 show thenumerical results of finite element analyses with Duncanand Changrsquos EB model and the modified PZ-III modelrespectively
Journal of Applied Mathematics 9
1
070503
09
01
(a)
0
05
0
minus1
minus15
minus24
minus05
minus2
(b)
051
152 25
3 353
252
151
05
(c)
0
03
05 1
13
minus02
(d)
Figure 11 Displacement and stress contour of the maximum section for Duncan and Changrsquos EB model (a) displacement along river (m)(b) vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
0706
06
050403
0201
0
02
minus02
01
(a)
minus25minus29
minus05minus15
minus1
minus2
(b)
05
1 152 3
3 435
3
25 215
105
(c)
01
05
1 15
0
(d)
Figure 12 Displacement and stress contour of the maximum section for the modified PZ-III model (a) displacement along river (m) (b)vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
Through the comparison and analysis of the numericalresults (Figures 11 and 12) we can find some similarities anddifferences for these two models
On one hand we can see many similar places in thedistributions of displacements and stresses
(1) After the reservoir impounding due to the hugewaterpressure on upstream dam horizontal displacementdevelops toward the downstream and the largestdisplacement is about 105m for EBmodel and 074mfor modified PZ-III model
(2) Themaximumsettlement occurs in themiddle of corewall due to lower modulus of clayey soil
(3) Because of the tremendous differences of modulusbetween rockfill material and clayey soil there existsobvious arching effect in the core wall
(4) Effective stress in upstream shell is less than thedownstream shell due to the water pressure in theupstream shell
On the other hand some differences also exist whichillustrate the advantages of modified PZ-III model
(1) After the reservoir is impounded upward displace-ment as large as 07m (see Figure 11(b)) developson the upstream shell near dam crest for EB modeland nearly 0m for modified PZ-III model (seeFigure 12(b)) In fact monitoring data of practicalengineering projects shows that no large upwarddisplacement happened after impounding This isdue to its weakness of EB model to distinguish theloading and unloading condition during the waterimpounding
(2) In the distribution of minor principle stress (Figures11(d) and 12(d)) negative stress (ie tensile stress)occurs in the upstream shell for EB model whereasvery little tensile stress exists for modified PZ-IIImodel As we know rockfill material is a typical kindof cohesionless coarse-grained soil which means thatit has no tensile strength Therefore the existence oflarge area of tensile stress in the upstream shell isunreasonable
522 Comparison between Numerical and In Situ MonitoringData Settlement is a key indicator to assess the safety of an
10 Journal of Applied Mathematics
550
600
650
700
750
800
850El
evat
ion
(m)
Settlement (mm)
In-situEBModified PZ
minus1000 0 1000 2000 3000 4000
Figure 13 Comparison between in situ monitoring settlement andFEM results
Table 2 Material parameters of the modified PZ-III model
Material Rockfill I Rockfill II Mixed gravel clay1198700
500 1000 3001198660
1500 3000 900119898 050 050 050119899 050 050 050120572119891
045 045 045120572119892
045 045 045119872119891119888
105 090 060119872119892119888
160 135 1101205730
000 000 0001205731
000 000 000Γ 034 031 034120582 010 009 003119898119901
035 040 001198670
800 1200 900120574 5 5 5120574119906
5 5 51198671199060MPa 9 9 10
earth dam Figures 13 and 14 show the in situmonitoring dataand FEM results of settlement in themaximum cross-sectionThe in situ data were obtained from electromagnetism typesettlement gaugeswhichwere embedded during constructionin the dam (as shown in Figure 3(a)) Through the compar-isons of in situmonitoring and numerical results we can seethat the modified PZ-III model gave a better prediction thanthe EB model However as deformation induced by wetting
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 655 m
2010
11
2010
61
0
2010
11
17
2011
42
6
2011
10
3
2012
31
1
2012
81
8
(a)
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 701 m
2010
91
2011
22
8
2011
82
7
2012
22
3
2012
82
1
(b)
Settl
emen
t (m
m)
0
500
1000
1500
2000
2500
Time
Elevation 751 m
2011
10
1
2011
12
20
2012
39
2012
52
8
2012
81
6
In-situEBModified PZ
(c)
Figure 14 Comparison between in situ monitoring settlement andFEM results
of rockfill materials was not considered the FEM result ofsettlement was below than the in situmonitoring data
As an elastoplastic model the PZ-III model is capableof representing the mechanical behavior of soils better thannonlinear elastic model such as Duncan and Changrsquos EBmodel And the above finite element analyses also proved it
6 Conclusions
This paper presents a modified PZ-III model based on thegeneralized theory and original Pastor-Zienkiewicz-Chan
Journal of Applied Mathematics 11
model to simulate the stress-strain relationship of rockfillmaterials
Triaxial test results of the filling materials of Nuozhadudamwere used to validate the proposedmodel and determinethe model parameters of Duncan and Changrsquos EB model andthe modified PZ-III model respectively The simulations oftriaxial stress-strain response show that the modified PZ-III model is capable of representing the key features ofcohesionless soil such as nonlinearity dilatancy and pressuredependency
The proposed model has been incorporated into a finiteelement code to simulate the static response of a high earth-rockfill dam in China The results were compared with thoseof Duncan and Changrsquos EB model The two set of resultshave both similarities and differences and the differencesillustrate the advantages of the modified PZ-III model Thecomparisons of FEM results and in situ monitoring datashowed that the modified PZ-III model can give a betterdescription of deformation of the earth-rockfill dam thanDuncan and Changrsquos EB model
Acknowledgments
This work was supported by the National Nature ScienceFoundation of China (51179092) and the State Key Laboratoryof Hydroscience and Engineering Project (2012-KY-02 and2013-KY-4)
References
[1] J M Duncan ldquoState of the art limit equilibrium and finite-element analysis of slopesrdquo Journal of Geotechnical and Geoen-vironmental Engineering vol 122 no 7 pp 577ndash596 1996
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] R S Sandhu and E L Wilson ldquoFinite element analysis ofseepage in elastic mediardquo Journal of the Engineering MechanicsDivision vol 95 no 3 pp 641ndash652 1969
[4] J T Christian and J W Boehmer ldquoPlane strain consolidationby finite elementsrdquo Journal of Soil Mechanics amp FoundationsDivision vol 96 no 4 pp 1435ndash1457 1970
[5] JMDuncan andC-Y Chang ldquoNonlinear analysis of stress andstrain in soilsrdquo Journal of the Soil Mechanics and FoundationsDivision vol 96 no 5 pp 1629ndash1653 1970
[6] J M Duncan P M Byrne K SWong and P Mabry ldquoStrengthstress-strain and bulk modulus parameters for finite elementanalyses of stresses and movements in soil massesrdquo Tech RepUCBGT80-01 University of California Berkeley Calif USA1980
[7] D C Drucker R E Gibson and D J Henkel ldquoSoil mechanicsand work-hardening theories of plasticityrdquo Transactions of theAmerican Society of Civil Engineers vol 122 pp 338ndash346 1957
[8] K Roscoe A Schofield andCWroth ldquoOn the yielding of soilsrdquoGeotechnique vol 8 no 1 pp 22ndash53 1958
[9] K Roscoe A Schofield and A Thurairajah ldquoYielding of claysin states wetter than criticalrdquo Geotechnique vol 13 no 3 pp211ndash240 1963
[10] J Burland ldquoCorrespondence on lsquoThe yielding and dilation ofclayrsquordquo Geotechnique vol 15 pp 211ndash214 1965
[11] P V Lade and J M Duncan ldquoElastoplastic stress-strain theoryfor cohesionless soilrdquo Journal of the Geotechnical EngineeringDivision vol 101 no 10 pp 1037ndash1053 1975
[12] I S Sandler F L DiMaggio and G Y Baladi ldquoGeneralizedcap model for geological materialsrdquo Journal of the GeotechnicalEngineering Division vol 102 no 7 pp 683ndash699 1976
[13] X-S Li Y F Dafalias and Z-L Wang ldquoState-dependent dila-tancy in critical-state constitutive modelling of sandrdquoCanadianGeotechnical Journal vol 36 no 4 pp 599ndash611 1999
[14] Y-P Yao and D Sun ldquoApplication of Ladersquos criterion to Cam-clay modelrdquo Journal of Engineering Mechanics vol 126 no 1pp 112ndash119 2000
[15] G Y Baladi and B Rohani ldquoElastic-plastic model for saturatedsandrdquo Journal of the Geotechnical Engineering Division vol 105no 4 pp 465ndash480 1979
[16] O Zienkiewicz and Z Mroz ldquoGeneralized plasticity formu-lation and applications to geomechanicsrdquo in Mechanics ofEngineering Materials C S Desai and R H Gallagher Eds pp655ndash679 John Wiley amp Sons New York NY USA 1984
[17] C S Desai and M O Faruque ldquoConstitutive model forgeological materialsrdquo Journal of Engineering Mechanics vol 110no 9 pp 1391ndash1408 1984
[18] S B R Murthy A Vatsala and T S Nagaraj ldquoRevised Cam-clay modelrdquo Journal of Geotechnical Engineering vol 117 no 6pp 851ndash871 1991
[19] M Pastor O C Zienkiewicz and A H C Chan ldquoGeneralizedplasticity and the modelling of soil behaviourrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 14 no 3 pp 151ndash190 1990
[20] ZMroz andO Zienkiewicz ldquoUniform formulation of constitu-tive equations for clays and sandsrdquo inMechanics of EngineeringMaterials C S Desai and R H Gallangher Eds pp 415ndash449John Wiley amp Sons New York NY USA 1984
[21] G Wang and J-M Zhang ldquoDynamic consolidation finiteelement analysis of a sediment-protecting dyke under oceanwave loadingrdquo Rock and Soil Mechanics vol 27 no 4 pp 555ndash560 2006
[22] MAlyamiMRouainia and SMWilkinson ldquoNumerical anal-ysis of deformation behaviour of quay walls under earthquakeloadingrdquo Soil Dynamics and Earthquake Engineering vol 29 no3 pp 525ndash536 2009
[23] H Li P Manuel and T Li ldquoApplication of an generalizedplasticity model to ultra-high rockfill damrdquo in Proceedingsof the 12th International Conference on Engineering ScienceConstruction and Operations in Challenging EnvironmentsmdashEarth and Space pp 385ndash398 Honolulu Hawaii USA March2010
[24] T Li and H Zhang ldquoDynamic parameter verification of P-Z model and its application of dynamic analysis on rockfilldamrdquo in Proceedings of the 12th International Conference onEngineering Science Construction and Operations in Challeng-ing EnvironmentsmdashEarth and Space pp 2706ndash2713 HonoluluHawaii USA March 2010
[25] M Pastor ldquoA generalized plasticity model for anisotropicbehaviour of sandrdquoComputer Methods and Advances in Geome-chanics vol 1 pp 661ndash668 1991
[26] G Bolzon B A Schrefler and O C Zienkiewicz ldquoElastoplasticsoil constitutive laws generalized to partially saturated statesrdquoGeotechnique vol 46 no 2 pp 279ndash289 1996
[27] H I Ling and H Liu ldquoPressure-level dependency and densifi-cation behavior of sand through generalized plasticity modelrdquo
12 Journal of Applied Mathematics
Journal of Engineering Mechanics vol 129 no 8 pp 851ndash8602003
[28] H I Ling and S Yang ldquoUnified sand model based on thecritical state and generalized plasticityrdquo Journal of EngineeringMechanics vol 132 no 12 pp 1380ndash1391 2006
[29] N D Marschi C K Chan and H B Seed ldquoEvaluation ofproperties of rockfill materialsrdquo Journal of the Soil Mechanicsand Foundations Division vol 98 no 1 pp 95ndash114 1972
[30] R J Marsal ldquoLarge scale testing of rockfill materialsrdquo Journal ofthe Soil Mechanics and Foundations Division vol 93 no 2 pp27ndash43 1967
[31] R JMarsal ldquoMechanical properties of rockfillrdquo in EmbankmentDam Engineering pp 109ndash200 John Wiley amp Sons New YorkNY USA 1973
[32] P V Lade J A Yamamuro and P A Bopp ldquoSignificance ofparticle crushing in granular materialsrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 122 no 4 pp 309ndash3161996
[33] BOHardin ldquoCrushing of soil particlesrdquo Journal of GeotechnicalEngineering vol 111 no 10 pp 1177ndash1192 1985
[34] R L Kondner ldquoHyperbolic stress-strain response cohesivesoilsrdquo Journal of the Soil Mechanics and Foundations Divisionvol 89 no 1 pp 115ndash143 1963
[35] Z-LWang Y F Dafalias X-S Li and F I Makdisi ldquoState pres-sure index for modeling sand behaviorrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 128 no 6 pp 511ndash5192002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
BurmaLaos
China
VietnamThailand
(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 practicalconstruction process
4 Experimental Validation ofModel Parameters
The modified PZ-III model was implemented in a finiteelement code which has been successfully used to analyzeearth dams with Duncan and Changrsquos EB model and someother constitutive models A set of triaxial test data was usedto make sure that the 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 thetriaxial test results Drained triaxial tests under different con-fining pressures were conducted to test the rockfill materialsand mixed gravel clay which are the main parts of the dambody
Duncan and Changrsquos EB model parameters are shown inTable 1 and the modified PZ-III model parameters in Table 2As shown in Figures 4 5 6 7 8 and 9 the modified PZ-III model presents a better ability to simulate the mechanics
Table 1 Material parameters of Duncan and Changrsquos EB model
Material Rockfill I Rockfill II Mixed gravel clay120593∘ 5582 5433 3930Δ120593∘ 1229 1207 980119877119891
073 074 077119870 1450 1360 520119870119887
550 600 250119870119906119903
2800 2500 900119899 030 043 042119898 013 008 025
behavior of rockfillmaterials andmixed gravel clay especiallyfor dilatancy With the reduction of confining pressurethe 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 intrinsic limitation Duncan andChangrsquos EB model cannot simulate the dilatancy which is acrucial feature of rockfill materials
6 Journal of Applied Mathematics
Upstream Downstream
RU1RU3F2F1
RU2 RD1
RD2
Cofferdam ED
F2F1
RD3
RU1RD1 upstreamdownstream rockfill zone IRU2RD2 upstreamdownstream rockfill zone IIRU3RD3 upstreamdownstream fine rockfill
F1F2 filter material zone IIIED clay mixed gravel
electromagnetism type settlement gauges
900
800
700
600
500
8241
658
(a)
8125 20121231
20110531sim20120531
20080215sim20080531 20080531sim20090531
20090531sim20100531
20100531sim20110531
(b)
Figure 3 The maximum cross-section (a) Material zoning and (b) construction stage
0
2000
4000
6000
8000
10000
300 kPa 900 kPa1500 kPa 2500 kPaDuncan-Chang EB
0 5 10 15
1205901minus1205903
(kPa
)
1205761 ()
(a)
minus4minus3minus2minus1
01234
1205761 ()0 5 10 15
120576
()
300 kPa 900 kPa1500 kPa 2500 kPaDuncan-Chang EB
(b)
Figure 4 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for rockfill material I
Journal of Applied Mathematics 7
0
2000
4000
6000
8000
10000
300 kPa 900 kPa1500 kPa 2500 kPaModified PZ
0 5 10 15
1205901minus1205903
(kPa
)
1205761 ()
(a)
minus4minus3minus2minus1
01234
1205761 ()0 5 10 15
120576
()
300 kPa 900 kPa1500 kPa 2500 kPaModified PZ
(b)
Figure 5 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for rockfill material I
Duncan-Chang EB
0 5 10 15
1205761 ()
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0
2000
4000
6000
8000
10000
1205901minus1205903
(kPa
)
(a)
0 5 10 151205761 ()
minus3
minus2
minus1
0
1
2
3
120576
()
Duncan-Chang EB
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 6 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for rockfill material II
Modified PZ
0 5 10 151205761 ()
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0
2000
4000
6000
8000
10000
1205901minus1205903
(kPa
)
(a)
0 5 10 151205761 ()
minus3
minus2
minus1
0
1
2
3
120576
()
Modified PZ
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 7 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for rockfill material II
8 Journal of Applied Mathematics
0
2000
4000
6000
Duncan-Chang EB
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3 0 5 10 151205761 ()
120576
()
Duncan-Chang EB
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 8 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for clay
0
2000
4000
6000
Modified PZ
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3
Modified PZ
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0 5 10 151205761 ()
120576
()
(b)
Figure 9 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for clay
Figure 10 3D FEMmesh of Nuozhadu dam
5 Three-Dimensional Finite Element Analyses
51 Computation Model The numerical analyses were per-formed to simulate the performance of the dam duringconstruction and impounding periods with effective stressfinite element analysis
First the 2D finite element mesh of the maximum cross-section of the dam was discretized according to the materialzoning and construction design (see Figure 3) Then the 2Dmesh was extended to 3D mesh in accordance with contourline of the river valley Figure 10 shows the 3D mesh ofthe Nuozhadu dam with 8095 brick and degenerated brickelements and 8340 nodes
The numerical simulations contain two stages filling andimpounding During the filling stage the dam body mainlysubjects to body weight Then at the end of constructionupstream water level goes up to the normal storage waterlevel The interaction between pore water and soil skeletonwas considered through the whole numerical computation
52 Results and Analyses
521 Numerical Results Analyses Figures 11 and 12 show thenumerical results of finite element analyses with Duncanand Changrsquos EB model and the modified PZ-III modelrespectively
Journal of Applied Mathematics 9
1
070503
09
01
(a)
0
05
0
minus1
minus15
minus24
minus05
minus2
(b)
051
152 25
3 353
252
151
05
(c)
0
03
05 1
13
minus02
(d)
Figure 11 Displacement and stress contour of the maximum section for Duncan and Changrsquos EB model (a) displacement along river (m)(b) vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
0706
06
050403
0201
0
02
minus02
01
(a)
minus25minus29
minus05minus15
minus1
minus2
(b)
05
1 152 3
3 435
3
25 215
105
(c)
01
05
1 15
0
(d)
Figure 12 Displacement and stress contour of the maximum section for the modified PZ-III model (a) displacement along river (m) (b)vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
Through the comparison and analysis of the numericalresults (Figures 11 and 12) we can find some similarities anddifferences for these two models
On one hand we can see many similar places in thedistributions of displacements and stresses
(1) After the reservoir impounding due to the hugewaterpressure on upstream dam horizontal displacementdevelops toward the downstream and the largestdisplacement is about 105m for EBmodel and 074mfor modified PZ-III model
(2) Themaximumsettlement occurs in themiddle of corewall due to lower modulus of clayey soil
(3) Because of the tremendous differences of modulusbetween rockfill material and clayey soil there existsobvious arching effect in the core wall
(4) Effective stress in upstream shell is less than thedownstream shell due to the water pressure in theupstream shell
On the other hand some differences also exist whichillustrate the advantages of modified PZ-III model
(1) After the reservoir is impounded upward displace-ment as large as 07m (see Figure 11(b)) developson the upstream shell near dam crest for EB modeland nearly 0m for modified PZ-III model (seeFigure 12(b)) In fact monitoring data of practicalengineering projects shows that no large upwarddisplacement happened after impounding This isdue to its weakness of EB model to distinguish theloading and unloading condition during the waterimpounding
(2) In the distribution of minor principle stress (Figures11(d) and 12(d)) negative stress (ie tensile stress)occurs in the upstream shell for EB model whereasvery little tensile stress exists for modified PZ-IIImodel As we know rockfill material is a typical kindof cohesionless coarse-grained soil which means thatit has no tensile strength Therefore the existence oflarge area of tensile stress in the upstream shell isunreasonable
522 Comparison between Numerical and In Situ MonitoringData Settlement is a key indicator to assess the safety of an
10 Journal of Applied Mathematics
550
600
650
700
750
800
850El
evat
ion
(m)
Settlement (mm)
In-situEBModified PZ
minus1000 0 1000 2000 3000 4000
Figure 13 Comparison between in situ monitoring settlement andFEM results
Table 2 Material parameters of the modified PZ-III model
Material Rockfill I Rockfill II Mixed gravel clay1198700
500 1000 3001198660
1500 3000 900119898 050 050 050119899 050 050 050120572119891
045 045 045120572119892
045 045 045119872119891119888
105 090 060119872119892119888
160 135 1101205730
000 000 0001205731
000 000 000Γ 034 031 034120582 010 009 003119898119901
035 040 001198670
800 1200 900120574 5 5 5120574119906
5 5 51198671199060MPa 9 9 10
earth dam Figures 13 and 14 show the in situmonitoring dataand FEM results of settlement in themaximum cross-sectionThe in situ data were obtained from electromagnetism typesettlement gaugeswhichwere embedded during constructionin the dam (as shown in Figure 3(a)) Through the compar-isons of in situmonitoring and numerical results we can seethat the modified PZ-III model gave a better prediction thanthe EB model However as deformation induced by wetting
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 655 m
2010
11
2010
61
0
2010
11
17
2011
42
6
2011
10
3
2012
31
1
2012
81
8
(a)
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 701 m
2010
91
2011
22
8
2011
82
7
2012
22
3
2012
82
1
(b)
Settl
emen
t (m
m)
0
500
1000
1500
2000
2500
Time
Elevation 751 m
2011
10
1
2011
12
20
2012
39
2012
52
8
2012
81
6
In-situEBModified PZ
(c)
Figure 14 Comparison between in situ monitoring settlement andFEM results
of rockfill materials was not considered the FEM result ofsettlement was below than the in situmonitoring data
As an elastoplastic model the PZ-III model is capableof representing the mechanical behavior of soils better thannonlinear elastic model such as Duncan and Changrsquos EBmodel And the above finite element analyses also proved it
6 Conclusions
This paper presents a modified PZ-III model based on thegeneralized theory and original Pastor-Zienkiewicz-Chan
Journal of Applied Mathematics 11
model to simulate the stress-strain relationship of rockfillmaterials
Triaxial test results of the filling materials of Nuozhadudamwere used to validate the proposedmodel and determinethe model parameters of Duncan and Changrsquos EB model andthe modified PZ-III model respectively The simulations oftriaxial stress-strain response show that the modified PZ-III model is capable of representing the key features ofcohesionless soil such as nonlinearity dilatancy and pressuredependency
The proposed model has been incorporated into a finiteelement code to simulate the static response of a high earth-rockfill dam in China The results were compared with thoseof Duncan and Changrsquos EB model The two set of resultshave both similarities and differences and the differencesillustrate the advantages of the modified PZ-III model Thecomparisons of FEM results and in situ monitoring datashowed that the modified PZ-III model can give a betterdescription of deformation of the earth-rockfill dam thanDuncan and Changrsquos EB model
Acknowledgments
This work was supported by the National Nature ScienceFoundation of China (51179092) and the State Key Laboratoryof Hydroscience and Engineering Project (2012-KY-02 and2013-KY-4)
References
[1] J M Duncan ldquoState of the art limit equilibrium and finite-element analysis of slopesrdquo Journal of Geotechnical and Geoen-vironmental Engineering vol 122 no 7 pp 577ndash596 1996
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] R S Sandhu and E L Wilson ldquoFinite element analysis ofseepage in elastic mediardquo Journal of the Engineering MechanicsDivision vol 95 no 3 pp 641ndash652 1969
[4] J T Christian and J W Boehmer ldquoPlane strain consolidationby finite elementsrdquo Journal of Soil Mechanics amp FoundationsDivision vol 96 no 4 pp 1435ndash1457 1970
[5] JMDuncan andC-Y Chang ldquoNonlinear analysis of stress andstrain in soilsrdquo Journal of the Soil Mechanics and FoundationsDivision vol 96 no 5 pp 1629ndash1653 1970
[6] J M Duncan P M Byrne K SWong and P Mabry ldquoStrengthstress-strain and bulk modulus parameters for finite elementanalyses of stresses and movements in soil massesrdquo Tech RepUCBGT80-01 University of California Berkeley Calif USA1980
[7] D C Drucker R E Gibson and D J Henkel ldquoSoil mechanicsand work-hardening theories of plasticityrdquo Transactions of theAmerican Society of Civil Engineers vol 122 pp 338ndash346 1957
[8] K Roscoe A Schofield andCWroth ldquoOn the yielding of soilsrdquoGeotechnique vol 8 no 1 pp 22ndash53 1958
[9] K Roscoe A Schofield and A Thurairajah ldquoYielding of claysin states wetter than criticalrdquo Geotechnique vol 13 no 3 pp211ndash240 1963
[10] J Burland ldquoCorrespondence on lsquoThe yielding and dilation ofclayrsquordquo Geotechnique vol 15 pp 211ndash214 1965
[11] P V Lade and J M Duncan ldquoElastoplastic stress-strain theoryfor cohesionless soilrdquo Journal of the Geotechnical EngineeringDivision vol 101 no 10 pp 1037ndash1053 1975
[12] I S Sandler F L DiMaggio and G Y Baladi ldquoGeneralizedcap model for geological materialsrdquo Journal of the GeotechnicalEngineering Division vol 102 no 7 pp 683ndash699 1976
[13] X-S Li Y F Dafalias and Z-L Wang ldquoState-dependent dila-tancy in critical-state constitutive modelling of sandrdquoCanadianGeotechnical Journal vol 36 no 4 pp 599ndash611 1999
[14] Y-P Yao and D Sun ldquoApplication of Ladersquos criterion to Cam-clay modelrdquo Journal of Engineering Mechanics vol 126 no 1pp 112ndash119 2000
[15] G Y Baladi and B Rohani ldquoElastic-plastic model for saturatedsandrdquo Journal of the Geotechnical Engineering Division vol 105no 4 pp 465ndash480 1979
[16] O Zienkiewicz and Z Mroz ldquoGeneralized plasticity formu-lation and applications to geomechanicsrdquo in Mechanics ofEngineering Materials C S Desai and R H Gallagher Eds pp655ndash679 John Wiley amp Sons New York NY USA 1984
[17] C S Desai and M O Faruque ldquoConstitutive model forgeological materialsrdquo Journal of Engineering Mechanics vol 110no 9 pp 1391ndash1408 1984
[18] S B R Murthy A Vatsala and T S Nagaraj ldquoRevised Cam-clay modelrdquo Journal of Geotechnical Engineering vol 117 no 6pp 851ndash871 1991
[19] M Pastor O C Zienkiewicz and A H C Chan ldquoGeneralizedplasticity and the modelling of soil behaviourrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 14 no 3 pp 151ndash190 1990
[20] ZMroz andO Zienkiewicz ldquoUniform formulation of constitu-tive equations for clays and sandsrdquo inMechanics of EngineeringMaterials C S Desai and R H Gallangher Eds pp 415ndash449John Wiley amp Sons New York NY USA 1984
[21] G Wang and J-M Zhang ldquoDynamic consolidation finiteelement analysis of a sediment-protecting dyke under oceanwave loadingrdquo Rock and Soil Mechanics vol 27 no 4 pp 555ndash560 2006
[22] MAlyamiMRouainia and SMWilkinson ldquoNumerical anal-ysis of deformation behaviour of quay walls under earthquakeloadingrdquo Soil Dynamics and Earthquake Engineering vol 29 no3 pp 525ndash536 2009
[23] H Li P Manuel and T Li ldquoApplication of an generalizedplasticity model to ultra-high rockfill damrdquo in Proceedingsof the 12th International Conference on Engineering ScienceConstruction and Operations in Challenging EnvironmentsmdashEarth and Space pp 385ndash398 Honolulu Hawaii USA March2010
[24] T Li and H Zhang ldquoDynamic parameter verification of P-Z model and its application of dynamic analysis on rockfilldamrdquo in Proceedings of the 12th International Conference onEngineering Science Construction and Operations in Challeng-ing EnvironmentsmdashEarth and Space pp 2706ndash2713 HonoluluHawaii USA March 2010
[25] M Pastor ldquoA generalized plasticity model for anisotropicbehaviour of sandrdquoComputer Methods and Advances in Geome-chanics vol 1 pp 661ndash668 1991
[26] G Bolzon B A Schrefler and O C Zienkiewicz ldquoElastoplasticsoil constitutive laws generalized to partially saturated statesrdquoGeotechnique vol 46 no 2 pp 279ndash289 1996
[27] H I Ling and H Liu ldquoPressure-level dependency and densifi-cation behavior of sand through generalized plasticity modelrdquo
12 Journal of Applied Mathematics
Journal of Engineering Mechanics vol 129 no 8 pp 851ndash8602003
[28] H I Ling and S Yang ldquoUnified sand model based on thecritical state and generalized plasticityrdquo Journal of EngineeringMechanics vol 132 no 12 pp 1380ndash1391 2006
[29] N D Marschi C K Chan and H B Seed ldquoEvaluation ofproperties of rockfill materialsrdquo Journal of the Soil Mechanicsand Foundations Division vol 98 no 1 pp 95ndash114 1972
[30] R J Marsal ldquoLarge scale testing of rockfill materialsrdquo Journal ofthe Soil Mechanics and Foundations Division vol 93 no 2 pp27ndash43 1967
[31] R JMarsal ldquoMechanical properties of rockfillrdquo in EmbankmentDam Engineering pp 109ndash200 John Wiley amp Sons New YorkNY USA 1973
[32] P V Lade J A Yamamuro and P A Bopp ldquoSignificance ofparticle crushing in granular materialsrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 122 no 4 pp 309ndash3161996
[33] BOHardin ldquoCrushing of soil particlesrdquo Journal of GeotechnicalEngineering vol 111 no 10 pp 1177ndash1192 1985
[34] R L Kondner ldquoHyperbolic stress-strain response cohesivesoilsrdquo Journal of the Soil Mechanics and Foundations Divisionvol 89 no 1 pp 115ndash143 1963
[35] Z-LWang Y F Dafalias X-S Li and F I Makdisi ldquoState pres-sure index for modeling sand behaviorrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 128 no 6 pp 511ndash5192002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
Upstream Downstream
RU1RU3F2F1
RU2 RD1
RD2
Cofferdam ED
F2F1
RD3
RU1RD1 upstreamdownstream rockfill zone IRU2RD2 upstreamdownstream rockfill zone IIRU3RD3 upstreamdownstream fine rockfill
F1F2 filter material zone IIIED clay mixed gravel
electromagnetism type settlement gauges
900
800
700
600
500
8241
658
(a)
8125 20121231
20110531sim20120531
20080215sim20080531 20080531sim20090531
20090531sim20100531
20100531sim20110531
(b)
Figure 3 The maximum cross-section (a) Material zoning and (b) construction stage
0
2000
4000
6000
8000
10000
300 kPa 900 kPa1500 kPa 2500 kPaDuncan-Chang EB
0 5 10 15
1205901minus1205903
(kPa
)
1205761 ()
(a)
minus4minus3minus2minus1
01234
1205761 ()0 5 10 15
120576
()
300 kPa 900 kPa1500 kPa 2500 kPaDuncan-Chang EB
(b)
Figure 4 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for rockfill material I
Journal of Applied Mathematics 7
0
2000
4000
6000
8000
10000
300 kPa 900 kPa1500 kPa 2500 kPaModified PZ
0 5 10 15
1205901minus1205903
(kPa
)
1205761 ()
(a)
minus4minus3minus2minus1
01234
1205761 ()0 5 10 15
120576
()
300 kPa 900 kPa1500 kPa 2500 kPaModified PZ
(b)
Figure 5 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for rockfill material I
Duncan-Chang EB
0 5 10 15
1205761 ()
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0
2000
4000
6000
8000
10000
1205901minus1205903
(kPa
)
(a)
0 5 10 151205761 ()
minus3
minus2
minus1
0
1
2
3
120576
()
Duncan-Chang EB
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 6 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for rockfill material II
Modified PZ
0 5 10 151205761 ()
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0
2000
4000
6000
8000
10000
1205901minus1205903
(kPa
)
(a)
0 5 10 151205761 ()
minus3
minus2
minus1
0
1
2
3
120576
()
Modified PZ
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 7 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for rockfill material II
8 Journal of Applied Mathematics
0
2000
4000
6000
Duncan-Chang EB
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3 0 5 10 151205761 ()
120576
()
Duncan-Chang EB
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 8 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for clay
0
2000
4000
6000
Modified PZ
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3
Modified PZ
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0 5 10 151205761 ()
120576
()
(b)
Figure 9 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for clay
Figure 10 3D FEMmesh of Nuozhadu dam
5 Three-Dimensional Finite Element Analyses
51 Computation Model The numerical analyses were per-formed to simulate the performance of the dam duringconstruction and impounding periods with effective stressfinite element analysis
First the 2D finite element mesh of the maximum cross-section of the dam was discretized according to the materialzoning and construction design (see Figure 3) Then the 2Dmesh was extended to 3D mesh in accordance with contourline of the river valley Figure 10 shows the 3D mesh ofthe Nuozhadu dam with 8095 brick and degenerated brickelements and 8340 nodes
The numerical simulations contain two stages filling andimpounding During the filling stage the dam body mainlysubjects to body weight Then at the end of constructionupstream water level goes up to the normal storage waterlevel The interaction between pore water and soil skeletonwas considered through the whole numerical computation
52 Results and Analyses
521 Numerical Results Analyses Figures 11 and 12 show thenumerical results of finite element analyses with Duncanand Changrsquos EB model and the modified PZ-III modelrespectively
Journal of Applied Mathematics 9
1
070503
09
01
(a)
0
05
0
minus1
minus15
minus24
minus05
minus2
(b)
051
152 25
3 353
252
151
05
(c)
0
03
05 1
13
minus02
(d)
Figure 11 Displacement and stress contour of the maximum section for Duncan and Changrsquos EB model (a) displacement along river (m)(b) vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
0706
06
050403
0201
0
02
minus02
01
(a)
minus25minus29
minus05minus15
minus1
minus2
(b)
05
1 152 3
3 435
3
25 215
105
(c)
01
05
1 15
0
(d)
Figure 12 Displacement and stress contour of the maximum section for the modified PZ-III model (a) displacement along river (m) (b)vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
Through the comparison and analysis of the numericalresults (Figures 11 and 12) we can find some similarities anddifferences for these two models
On one hand we can see many similar places in thedistributions of displacements and stresses
(1) After the reservoir impounding due to the hugewaterpressure on upstream dam horizontal displacementdevelops toward the downstream and the largestdisplacement is about 105m for EBmodel and 074mfor modified PZ-III model
(2) Themaximumsettlement occurs in themiddle of corewall due to lower modulus of clayey soil
(3) Because of the tremendous differences of modulusbetween rockfill material and clayey soil there existsobvious arching effect in the core wall
(4) Effective stress in upstream shell is less than thedownstream shell due to the water pressure in theupstream shell
On the other hand some differences also exist whichillustrate the advantages of modified PZ-III model
(1) After the reservoir is impounded upward displace-ment as large as 07m (see Figure 11(b)) developson the upstream shell near dam crest for EB modeland nearly 0m for modified PZ-III model (seeFigure 12(b)) In fact monitoring data of practicalengineering projects shows that no large upwarddisplacement happened after impounding This isdue to its weakness of EB model to distinguish theloading and unloading condition during the waterimpounding
(2) In the distribution of minor principle stress (Figures11(d) and 12(d)) negative stress (ie tensile stress)occurs in the upstream shell for EB model whereasvery little tensile stress exists for modified PZ-IIImodel As we know rockfill material is a typical kindof cohesionless coarse-grained soil which means thatit has no tensile strength Therefore the existence oflarge area of tensile stress in the upstream shell isunreasonable
522 Comparison between Numerical and In Situ MonitoringData Settlement is a key indicator to assess the safety of an
10 Journal of Applied Mathematics
550
600
650
700
750
800
850El
evat
ion
(m)
Settlement (mm)
In-situEBModified PZ
minus1000 0 1000 2000 3000 4000
Figure 13 Comparison between in situ monitoring settlement andFEM results
Table 2 Material parameters of the modified PZ-III model
Material Rockfill I Rockfill II Mixed gravel clay1198700
500 1000 3001198660
1500 3000 900119898 050 050 050119899 050 050 050120572119891
045 045 045120572119892
045 045 045119872119891119888
105 090 060119872119892119888
160 135 1101205730
000 000 0001205731
000 000 000Γ 034 031 034120582 010 009 003119898119901
035 040 001198670
800 1200 900120574 5 5 5120574119906
5 5 51198671199060MPa 9 9 10
earth dam Figures 13 and 14 show the in situmonitoring dataand FEM results of settlement in themaximum cross-sectionThe in situ data were obtained from electromagnetism typesettlement gaugeswhichwere embedded during constructionin the dam (as shown in Figure 3(a)) Through the compar-isons of in situmonitoring and numerical results we can seethat the modified PZ-III model gave a better prediction thanthe EB model However as deformation induced by wetting
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 655 m
2010
11
2010
61
0
2010
11
17
2011
42
6
2011
10
3
2012
31
1
2012
81
8
(a)
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 701 m
2010
91
2011
22
8
2011
82
7
2012
22
3
2012
82
1
(b)
Settl
emen
t (m
m)
0
500
1000
1500
2000
2500
Time
Elevation 751 m
2011
10
1
2011
12
20
2012
39
2012
52
8
2012
81
6
In-situEBModified PZ
(c)
Figure 14 Comparison between in situ monitoring settlement andFEM results
of rockfill materials was not considered the FEM result ofsettlement was below than the in situmonitoring data
As an elastoplastic model the PZ-III model is capableof representing the mechanical behavior of soils better thannonlinear elastic model such as Duncan and Changrsquos EBmodel And the above finite element analyses also proved it
6 Conclusions
This paper presents a modified PZ-III model based on thegeneralized theory and original Pastor-Zienkiewicz-Chan
Journal of Applied Mathematics 11
model to simulate the stress-strain relationship of rockfillmaterials
Triaxial test results of the filling materials of Nuozhadudamwere used to validate the proposedmodel and determinethe model parameters of Duncan and Changrsquos EB model andthe modified PZ-III model respectively The simulations oftriaxial stress-strain response show that the modified PZ-III model is capable of representing the key features ofcohesionless soil such as nonlinearity dilatancy and pressuredependency
The proposed model has been incorporated into a finiteelement code to simulate the static response of a high earth-rockfill dam in China The results were compared with thoseof Duncan and Changrsquos EB model The two set of resultshave both similarities and differences and the differencesillustrate the advantages of the modified PZ-III model Thecomparisons of FEM results and in situ monitoring datashowed that the modified PZ-III model can give a betterdescription of deformation of the earth-rockfill dam thanDuncan and Changrsquos EB model
Acknowledgments
This work was supported by the National Nature ScienceFoundation of China (51179092) and the State Key Laboratoryof Hydroscience and Engineering Project (2012-KY-02 and2013-KY-4)
References
[1] J M Duncan ldquoState of the art limit equilibrium and finite-element analysis of slopesrdquo Journal of Geotechnical and Geoen-vironmental Engineering vol 122 no 7 pp 577ndash596 1996
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] R S Sandhu and E L Wilson ldquoFinite element analysis ofseepage in elastic mediardquo Journal of the Engineering MechanicsDivision vol 95 no 3 pp 641ndash652 1969
[4] J T Christian and J W Boehmer ldquoPlane strain consolidationby finite elementsrdquo Journal of Soil Mechanics amp FoundationsDivision vol 96 no 4 pp 1435ndash1457 1970
[5] JMDuncan andC-Y Chang ldquoNonlinear analysis of stress andstrain in soilsrdquo Journal of the Soil Mechanics and FoundationsDivision vol 96 no 5 pp 1629ndash1653 1970
[6] J M Duncan P M Byrne K SWong and P Mabry ldquoStrengthstress-strain and bulk modulus parameters for finite elementanalyses of stresses and movements in soil massesrdquo Tech RepUCBGT80-01 University of California Berkeley Calif USA1980
[7] D C Drucker R E Gibson and D J Henkel ldquoSoil mechanicsand work-hardening theories of plasticityrdquo Transactions of theAmerican Society of Civil Engineers vol 122 pp 338ndash346 1957
[8] K Roscoe A Schofield andCWroth ldquoOn the yielding of soilsrdquoGeotechnique vol 8 no 1 pp 22ndash53 1958
[9] K Roscoe A Schofield and A Thurairajah ldquoYielding of claysin states wetter than criticalrdquo Geotechnique vol 13 no 3 pp211ndash240 1963
[10] J Burland ldquoCorrespondence on lsquoThe yielding and dilation ofclayrsquordquo Geotechnique vol 15 pp 211ndash214 1965
[11] P V Lade and J M Duncan ldquoElastoplastic stress-strain theoryfor cohesionless soilrdquo Journal of the Geotechnical EngineeringDivision vol 101 no 10 pp 1037ndash1053 1975
[12] I S Sandler F L DiMaggio and G Y Baladi ldquoGeneralizedcap model for geological materialsrdquo Journal of the GeotechnicalEngineering Division vol 102 no 7 pp 683ndash699 1976
[13] X-S Li Y F Dafalias and Z-L Wang ldquoState-dependent dila-tancy in critical-state constitutive modelling of sandrdquoCanadianGeotechnical Journal vol 36 no 4 pp 599ndash611 1999
[14] Y-P Yao and D Sun ldquoApplication of Ladersquos criterion to Cam-clay modelrdquo Journal of Engineering Mechanics vol 126 no 1pp 112ndash119 2000
[15] G Y Baladi and B Rohani ldquoElastic-plastic model for saturatedsandrdquo Journal of the Geotechnical Engineering Division vol 105no 4 pp 465ndash480 1979
[16] O Zienkiewicz and Z Mroz ldquoGeneralized plasticity formu-lation and applications to geomechanicsrdquo in Mechanics ofEngineering Materials C S Desai and R H Gallagher Eds pp655ndash679 John Wiley amp Sons New York NY USA 1984
[17] C S Desai and M O Faruque ldquoConstitutive model forgeological materialsrdquo Journal of Engineering Mechanics vol 110no 9 pp 1391ndash1408 1984
[18] S B R Murthy A Vatsala and T S Nagaraj ldquoRevised Cam-clay modelrdquo Journal of Geotechnical Engineering vol 117 no 6pp 851ndash871 1991
[19] M Pastor O C Zienkiewicz and A H C Chan ldquoGeneralizedplasticity and the modelling of soil behaviourrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 14 no 3 pp 151ndash190 1990
[20] ZMroz andO Zienkiewicz ldquoUniform formulation of constitu-tive equations for clays and sandsrdquo inMechanics of EngineeringMaterials C S Desai and R H Gallangher Eds pp 415ndash449John Wiley amp Sons New York NY USA 1984
[21] G Wang and J-M Zhang ldquoDynamic consolidation finiteelement analysis of a sediment-protecting dyke under oceanwave loadingrdquo Rock and Soil Mechanics vol 27 no 4 pp 555ndash560 2006
[22] MAlyamiMRouainia and SMWilkinson ldquoNumerical anal-ysis of deformation behaviour of quay walls under earthquakeloadingrdquo Soil Dynamics and Earthquake Engineering vol 29 no3 pp 525ndash536 2009
[23] H Li P Manuel and T Li ldquoApplication of an generalizedplasticity model to ultra-high rockfill damrdquo in Proceedingsof the 12th International Conference on Engineering ScienceConstruction and Operations in Challenging EnvironmentsmdashEarth and Space pp 385ndash398 Honolulu Hawaii USA March2010
[24] T Li and H Zhang ldquoDynamic parameter verification of P-Z model and its application of dynamic analysis on rockfilldamrdquo in Proceedings of the 12th International Conference onEngineering Science Construction and Operations in Challeng-ing EnvironmentsmdashEarth and Space pp 2706ndash2713 HonoluluHawaii USA March 2010
[25] M Pastor ldquoA generalized plasticity model for anisotropicbehaviour of sandrdquoComputer Methods and Advances in Geome-chanics vol 1 pp 661ndash668 1991
[26] G Bolzon B A Schrefler and O C Zienkiewicz ldquoElastoplasticsoil constitutive laws generalized to partially saturated statesrdquoGeotechnique vol 46 no 2 pp 279ndash289 1996
[27] H I Ling and H Liu ldquoPressure-level dependency and densifi-cation behavior of sand through generalized plasticity modelrdquo
12 Journal of Applied Mathematics
Journal of Engineering Mechanics vol 129 no 8 pp 851ndash8602003
[28] H I Ling and S Yang ldquoUnified sand model based on thecritical state and generalized plasticityrdquo Journal of EngineeringMechanics vol 132 no 12 pp 1380ndash1391 2006
[29] N D Marschi C K Chan and H B Seed ldquoEvaluation ofproperties of rockfill materialsrdquo Journal of the Soil Mechanicsand Foundations Division vol 98 no 1 pp 95ndash114 1972
[30] R J Marsal ldquoLarge scale testing of rockfill materialsrdquo Journal ofthe Soil Mechanics and Foundations Division vol 93 no 2 pp27ndash43 1967
[31] R JMarsal ldquoMechanical properties of rockfillrdquo in EmbankmentDam Engineering pp 109ndash200 John Wiley amp Sons New YorkNY USA 1973
[32] P V Lade J A Yamamuro and P A Bopp ldquoSignificance ofparticle crushing in granular materialsrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 122 no 4 pp 309ndash3161996
[33] BOHardin ldquoCrushing of soil particlesrdquo Journal of GeotechnicalEngineering vol 111 no 10 pp 1177ndash1192 1985
[34] R L Kondner ldquoHyperbolic stress-strain response cohesivesoilsrdquo Journal of the Soil Mechanics and Foundations Divisionvol 89 no 1 pp 115ndash143 1963
[35] Z-LWang Y F Dafalias X-S Li and F I Makdisi ldquoState pres-sure index for modeling sand behaviorrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 128 no 6 pp 511ndash5192002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
0
2000
4000
6000
8000
10000
300 kPa 900 kPa1500 kPa 2500 kPaModified PZ
0 5 10 15
1205901minus1205903
(kPa
)
1205761 ()
(a)
minus4minus3minus2minus1
01234
1205761 ()0 5 10 15
120576
()
300 kPa 900 kPa1500 kPa 2500 kPaModified PZ
(b)
Figure 5 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for rockfill material I
Duncan-Chang EB
0 5 10 15
1205761 ()
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0
2000
4000
6000
8000
10000
1205901minus1205903
(kPa
)
(a)
0 5 10 151205761 ()
minus3
minus2
minus1
0
1
2
3
120576
()
Duncan-Chang EB
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 6 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for rockfill material II
Modified PZ
0 5 10 151205761 ()
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0
2000
4000
6000
8000
10000
1205901minus1205903
(kPa
)
(a)
0 5 10 151205761 ()
minus3
minus2
minus1
0
1
2
3
120576
()
Modified PZ
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 7 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for rockfill material II
8 Journal of Applied Mathematics
0
2000
4000
6000
Duncan-Chang EB
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3 0 5 10 151205761 ()
120576
()
Duncan-Chang EB
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 8 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for clay
0
2000
4000
6000
Modified PZ
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3
Modified PZ
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0 5 10 151205761 ()
120576
()
(b)
Figure 9 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for clay
Figure 10 3D FEMmesh of Nuozhadu dam
5 Three-Dimensional Finite Element Analyses
51 Computation Model The numerical analyses were per-formed to simulate the performance of the dam duringconstruction and impounding periods with effective stressfinite element analysis
First the 2D finite element mesh of the maximum cross-section of the dam was discretized according to the materialzoning and construction design (see Figure 3) Then the 2Dmesh was extended to 3D mesh in accordance with contourline of the river valley Figure 10 shows the 3D mesh ofthe Nuozhadu dam with 8095 brick and degenerated brickelements and 8340 nodes
The numerical simulations contain two stages filling andimpounding During the filling stage the dam body mainlysubjects to body weight Then at the end of constructionupstream water level goes up to the normal storage waterlevel The interaction between pore water and soil skeletonwas considered through the whole numerical computation
52 Results and Analyses
521 Numerical Results Analyses Figures 11 and 12 show thenumerical results of finite element analyses with Duncanand Changrsquos EB model and the modified PZ-III modelrespectively
Journal of Applied Mathematics 9
1
070503
09
01
(a)
0
05
0
minus1
minus15
minus24
minus05
minus2
(b)
051
152 25
3 353
252
151
05
(c)
0
03
05 1
13
minus02
(d)
Figure 11 Displacement and stress contour of the maximum section for Duncan and Changrsquos EB model (a) displacement along river (m)(b) vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
0706
06
050403
0201
0
02
minus02
01
(a)
minus25minus29
minus05minus15
minus1
minus2
(b)
05
1 152 3
3 435
3
25 215
105
(c)
01
05
1 15
0
(d)
Figure 12 Displacement and stress contour of the maximum section for the modified PZ-III model (a) displacement along river (m) (b)vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
Through the comparison and analysis of the numericalresults (Figures 11 and 12) we can find some similarities anddifferences for these two models
On one hand we can see many similar places in thedistributions of displacements and stresses
(1) After the reservoir impounding due to the hugewaterpressure on upstream dam horizontal displacementdevelops toward the downstream and the largestdisplacement is about 105m for EBmodel and 074mfor modified PZ-III model
(2) Themaximumsettlement occurs in themiddle of corewall due to lower modulus of clayey soil
(3) Because of the tremendous differences of modulusbetween rockfill material and clayey soil there existsobvious arching effect in the core wall
(4) Effective stress in upstream shell is less than thedownstream shell due to the water pressure in theupstream shell
On the other hand some differences also exist whichillustrate the advantages of modified PZ-III model
(1) After the reservoir is impounded upward displace-ment as large as 07m (see Figure 11(b)) developson the upstream shell near dam crest for EB modeland nearly 0m for modified PZ-III model (seeFigure 12(b)) In fact monitoring data of practicalengineering projects shows that no large upwarddisplacement happened after impounding This isdue to its weakness of EB model to distinguish theloading and unloading condition during the waterimpounding
(2) In the distribution of minor principle stress (Figures11(d) and 12(d)) negative stress (ie tensile stress)occurs in the upstream shell for EB model whereasvery little tensile stress exists for modified PZ-IIImodel As we know rockfill material is a typical kindof cohesionless coarse-grained soil which means thatit has no tensile strength Therefore the existence oflarge area of tensile stress in the upstream shell isunreasonable
522 Comparison between Numerical and In Situ MonitoringData Settlement is a key indicator to assess the safety of an
10 Journal of Applied Mathematics
550
600
650
700
750
800
850El
evat
ion
(m)
Settlement (mm)
In-situEBModified PZ
minus1000 0 1000 2000 3000 4000
Figure 13 Comparison between in situ monitoring settlement andFEM results
Table 2 Material parameters of the modified PZ-III model
Material Rockfill I Rockfill II Mixed gravel clay1198700
500 1000 3001198660
1500 3000 900119898 050 050 050119899 050 050 050120572119891
045 045 045120572119892
045 045 045119872119891119888
105 090 060119872119892119888
160 135 1101205730
000 000 0001205731
000 000 000Γ 034 031 034120582 010 009 003119898119901
035 040 001198670
800 1200 900120574 5 5 5120574119906
5 5 51198671199060MPa 9 9 10
earth dam Figures 13 and 14 show the in situmonitoring dataand FEM results of settlement in themaximum cross-sectionThe in situ data were obtained from electromagnetism typesettlement gaugeswhichwere embedded during constructionin the dam (as shown in Figure 3(a)) Through the compar-isons of in situmonitoring and numerical results we can seethat the modified PZ-III model gave a better prediction thanthe EB model However as deformation induced by wetting
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 655 m
2010
11
2010
61
0
2010
11
17
2011
42
6
2011
10
3
2012
31
1
2012
81
8
(a)
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 701 m
2010
91
2011
22
8
2011
82
7
2012
22
3
2012
82
1
(b)
Settl
emen
t (m
m)
0
500
1000
1500
2000
2500
Time
Elevation 751 m
2011
10
1
2011
12
20
2012
39
2012
52
8
2012
81
6
In-situEBModified PZ
(c)
Figure 14 Comparison between in situ monitoring settlement andFEM results
of rockfill materials was not considered the FEM result ofsettlement was below than the in situmonitoring data
As an elastoplastic model the PZ-III model is capableof representing the mechanical behavior of soils better thannonlinear elastic model such as Duncan and Changrsquos EBmodel And the above finite element analyses also proved it
6 Conclusions
This paper presents a modified PZ-III model based on thegeneralized theory and original Pastor-Zienkiewicz-Chan
Journal of Applied Mathematics 11
model to simulate the stress-strain relationship of rockfillmaterials
Triaxial test results of the filling materials of Nuozhadudamwere used to validate the proposedmodel and determinethe model parameters of Duncan and Changrsquos EB model andthe modified PZ-III model respectively The simulations oftriaxial stress-strain response show that the modified PZ-III model is capable of representing the key features ofcohesionless soil such as nonlinearity dilatancy and pressuredependency
The proposed model has been incorporated into a finiteelement code to simulate the static response of a high earth-rockfill dam in China The results were compared with thoseof Duncan and Changrsquos EB model The two set of resultshave both similarities and differences and the differencesillustrate the advantages of the modified PZ-III model Thecomparisons of FEM results and in situ monitoring datashowed that the modified PZ-III model can give a betterdescription of deformation of the earth-rockfill dam thanDuncan and Changrsquos EB model
Acknowledgments
This work was supported by the National Nature ScienceFoundation of China (51179092) and the State Key Laboratoryof Hydroscience and Engineering Project (2012-KY-02 and2013-KY-4)
References
[1] J M Duncan ldquoState of the art limit equilibrium and finite-element analysis of slopesrdquo Journal of Geotechnical and Geoen-vironmental Engineering vol 122 no 7 pp 577ndash596 1996
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] R S Sandhu and E L Wilson ldquoFinite element analysis ofseepage in elastic mediardquo Journal of the Engineering MechanicsDivision vol 95 no 3 pp 641ndash652 1969
[4] J T Christian and J W Boehmer ldquoPlane strain consolidationby finite elementsrdquo Journal of Soil Mechanics amp FoundationsDivision vol 96 no 4 pp 1435ndash1457 1970
[5] JMDuncan andC-Y Chang ldquoNonlinear analysis of stress andstrain in soilsrdquo Journal of the Soil Mechanics and FoundationsDivision vol 96 no 5 pp 1629ndash1653 1970
[6] J M Duncan P M Byrne K SWong and P Mabry ldquoStrengthstress-strain and bulk modulus parameters for finite elementanalyses of stresses and movements in soil massesrdquo Tech RepUCBGT80-01 University of California Berkeley Calif USA1980
[7] D C Drucker R E Gibson and D J Henkel ldquoSoil mechanicsand work-hardening theories of plasticityrdquo Transactions of theAmerican Society of Civil Engineers vol 122 pp 338ndash346 1957
[8] K Roscoe A Schofield andCWroth ldquoOn the yielding of soilsrdquoGeotechnique vol 8 no 1 pp 22ndash53 1958
[9] K Roscoe A Schofield and A Thurairajah ldquoYielding of claysin states wetter than criticalrdquo Geotechnique vol 13 no 3 pp211ndash240 1963
[10] J Burland ldquoCorrespondence on lsquoThe yielding and dilation ofclayrsquordquo Geotechnique vol 15 pp 211ndash214 1965
[11] P V Lade and J M Duncan ldquoElastoplastic stress-strain theoryfor cohesionless soilrdquo Journal of the Geotechnical EngineeringDivision vol 101 no 10 pp 1037ndash1053 1975
[12] I S Sandler F L DiMaggio and G Y Baladi ldquoGeneralizedcap model for geological materialsrdquo Journal of the GeotechnicalEngineering Division vol 102 no 7 pp 683ndash699 1976
[13] X-S Li Y F Dafalias and Z-L Wang ldquoState-dependent dila-tancy in critical-state constitutive modelling of sandrdquoCanadianGeotechnical Journal vol 36 no 4 pp 599ndash611 1999
[14] Y-P Yao and D Sun ldquoApplication of Ladersquos criterion to Cam-clay modelrdquo Journal of Engineering Mechanics vol 126 no 1pp 112ndash119 2000
[15] G Y Baladi and B Rohani ldquoElastic-plastic model for saturatedsandrdquo Journal of the Geotechnical Engineering Division vol 105no 4 pp 465ndash480 1979
[16] O Zienkiewicz and Z Mroz ldquoGeneralized plasticity formu-lation and applications to geomechanicsrdquo in Mechanics ofEngineering Materials C S Desai and R H Gallagher Eds pp655ndash679 John Wiley amp Sons New York NY USA 1984
[17] C S Desai and M O Faruque ldquoConstitutive model forgeological materialsrdquo Journal of Engineering Mechanics vol 110no 9 pp 1391ndash1408 1984
[18] S B R Murthy A Vatsala and T S Nagaraj ldquoRevised Cam-clay modelrdquo Journal of Geotechnical Engineering vol 117 no 6pp 851ndash871 1991
[19] M Pastor O C Zienkiewicz and A H C Chan ldquoGeneralizedplasticity and the modelling of soil behaviourrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 14 no 3 pp 151ndash190 1990
[20] ZMroz andO Zienkiewicz ldquoUniform formulation of constitu-tive equations for clays and sandsrdquo inMechanics of EngineeringMaterials C S Desai and R H Gallangher Eds pp 415ndash449John Wiley amp Sons New York NY USA 1984
[21] G Wang and J-M Zhang ldquoDynamic consolidation finiteelement analysis of a sediment-protecting dyke under oceanwave loadingrdquo Rock and Soil Mechanics vol 27 no 4 pp 555ndash560 2006
[22] MAlyamiMRouainia and SMWilkinson ldquoNumerical anal-ysis of deformation behaviour of quay walls under earthquakeloadingrdquo Soil Dynamics and Earthquake Engineering vol 29 no3 pp 525ndash536 2009
[23] H Li P Manuel and T Li ldquoApplication of an generalizedplasticity model to ultra-high rockfill damrdquo in Proceedingsof the 12th International Conference on Engineering ScienceConstruction and Operations in Challenging EnvironmentsmdashEarth and Space pp 385ndash398 Honolulu Hawaii USA March2010
[24] T Li and H Zhang ldquoDynamic parameter verification of P-Z model and its application of dynamic analysis on rockfilldamrdquo in Proceedings of the 12th International Conference onEngineering Science Construction and Operations in Challeng-ing EnvironmentsmdashEarth and Space pp 2706ndash2713 HonoluluHawaii USA March 2010
[25] M Pastor ldquoA generalized plasticity model for anisotropicbehaviour of sandrdquoComputer Methods and Advances in Geome-chanics vol 1 pp 661ndash668 1991
[26] G Bolzon B A Schrefler and O C Zienkiewicz ldquoElastoplasticsoil constitutive laws generalized to partially saturated statesrdquoGeotechnique vol 46 no 2 pp 279ndash289 1996
[27] H I Ling and H Liu ldquoPressure-level dependency and densifi-cation behavior of sand through generalized plasticity modelrdquo
12 Journal of Applied Mathematics
Journal of Engineering Mechanics vol 129 no 8 pp 851ndash8602003
[28] H I Ling and S Yang ldquoUnified sand model based on thecritical state and generalized plasticityrdquo Journal of EngineeringMechanics vol 132 no 12 pp 1380ndash1391 2006
[29] N D Marschi C K Chan and H B Seed ldquoEvaluation ofproperties of rockfill materialsrdquo Journal of the Soil Mechanicsand Foundations Division vol 98 no 1 pp 95ndash114 1972
[30] R J Marsal ldquoLarge scale testing of rockfill materialsrdquo Journal ofthe Soil Mechanics and Foundations Division vol 93 no 2 pp27ndash43 1967
[31] R JMarsal ldquoMechanical properties of rockfillrdquo in EmbankmentDam Engineering pp 109ndash200 John Wiley amp Sons New YorkNY USA 1973
[32] P V Lade J A Yamamuro and P A Bopp ldquoSignificance ofparticle crushing in granular materialsrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 122 no 4 pp 309ndash3161996
[33] BOHardin ldquoCrushing of soil particlesrdquo Journal of GeotechnicalEngineering vol 111 no 10 pp 1177ndash1192 1985
[34] R L Kondner ldquoHyperbolic stress-strain response cohesivesoilsrdquo Journal of the Soil Mechanics and Foundations Divisionvol 89 no 1 pp 115ndash143 1963
[35] Z-LWang Y F Dafalias X-S Li and F I Makdisi ldquoState pres-sure index for modeling sand behaviorrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 128 no 6 pp 511ndash5192002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
0
2000
4000
6000
Duncan-Chang EB
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3 0 5 10 151205761 ()
120576
()
Duncan-Chang EB
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(b)
Figure 8 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for clay
0
2000
4000
6000
Modified PZ
0 5 10 15
1205761 ()
1205901minus1205903
(kPa
)
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
(a)
0
1
2
3
Modified PZ
1205903 = 300kPa 1205903 = 900kPa1205903 = 1500 kPa 1205903 = 2500 kPa
0 5 10 151205761 ()
120576
()
(b)
Figure 9 Comparison between fittings of the modified PZ-III model and experimental triaxial tests results for clay
Figure 10 3D FEMmesh of Nuozhadu dam
5 Three-Dimensional Finite Element Analyses
51 Computation Model The numerical analyses were per-formed to simulate the performance of the dam duringconstruction and impounding periods with effective stressfinite element analysis
First the 2D finite element mesh of the maximum cross-section of the dam was discretized according to the materialzoning and construction design (see Figure 3) Then the 2Dmesh was extended to 3D mesh in accordance with contourline of the river valley Figure 10 shows the 3D mesh ofthe Nuozhadu dam with 8095 brick and degenerated brickelements and 8340 nodes
The numerical simulations contain two stages filling andimpounding During the filling stage the dam body mainlysubjects to body weight Then at the end of constructionupstream water level goes up to the normal storage waterlevel The interaction between pore water and soil skeletonwas considered through the whole numerical computation
52 Results and Analyses
521 Numerical Results Analyses Figures 11 and 12 show thenumerical results of finite element analyses with Duncanand Changrsquos EB model and the modified PZ-III modelrespectively
Journal of Applied Mathematics 9
1
070503
09
01
(a)
0
05
0
minus1
minus15
minus24
minus05
minus2
(b)
051
152 25
3 353
252
151
05
(c)
0
03
05 1
13
minus02
(d)
Figure 11 Displacement and stress contour of the maximum section for Duncan and Changrsquos EB model (a) displacement along river (m)(b) vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
0706
06
050403
0201
0
02
minus02
01
(a)
minus25minus29
minus05minus15
minus1
minus2
(b)
05
1 152 3
3 435
3
25 215
105
(c)
01
05
1 15
0
(d)
Figure 12 Displacement and stress contour of the maximum section for the modified PZ-III model (a) displacement along river (m) (b)vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
Through the comparison and analysis of the numericalresults (Figures 11 and 12) we can find some similarities anddifferences for these two models
On one hand we can see many similar places in thedistributions of displacements and stresses
(1) After the reservoir impounding due to the hugewaterpressure on upstream dam horizontal displacementdevelops toward the downstream and the largestdisplacement is about 105m for EBmodel and 074mfor modified PZ-III model
(2) Themaximumsettlement occurs in themiddle of corewall due to lower modulus of clayey soil
(3) Because of the tremendous differences of modulusbetween rockfill material and clayey soil there existsobvious arching effect in the core wall
(4) Effective stress in upstream shell is less than thedownstream shell due to the water pressure in theupstream shell
On the other hand some differences also exist whichillustrate the advantages of modified PZ-III model
(1) After the reservoir is impounded upward displace-ment as large as 07m (see Figure 11(b)) developson the upstream shell near dam crest for EB modeland nearly 0m for modified PZ-III model (seeFigure 12(b)) In fact monitoring data of practicalengineering projects shows that no large upwarddisplacement happened after impounding This isdue to its weakness of EB model to distinguish theloading and unloading condition during the waterimpounding
(2) In the distribution of minor principle stress (Figures11(d) and 12(d)) negative stress (ie tensile stress)occurs in the upstream shell for EB model whereasvery little tensile stress exists for modified PZ-IIImodel As we know rockfill material is a typical kindof cohesionless coarse-grained soil which means thatit has no tensile strength Therefore the existence oflarge area of tensile stress in the upstream shell isunreasonable
522 Comparison between Numerical and In Situ MonitoringData Settlement is a key indicator to assess the safety of an
10 Journal of Applied Mathematics
550
600
650
700
750
800
850El
evat
ion
(m)
Settlement (mm)
In-situEBModified PZ
minus1000 0 1000 2000 3000 4000
Figure 13 Comparison between in situ monitoring settlement andFEM results
Table 2 Material parameters of the modified PZ-III model
Material Rockfill I Rockfill II Mixed gravel clay1198700
500 1000 3001198660
1500 3000 900119898 050 050 050119899 050 050 050120572119891
045 045 045120572119892
045 045 045119872119891119888
105 090 060119872119892119888
160 135 1101205730
000 000 0001205731
000 000 000Γ 034 031 034120582 010 009 003119898119901
035 040 001198670
800 1200 900120574 5 5 5120574119906
5 5 51198671199060MPa 9 9 10
earth dam Figures 13 and 14 show the in situmonitoring dataand FEM results of settlement in themaximum cross-sectionThe in situ data were obtained from electromagnetism typesettlement gaugeswhichwere embedded during constructionin the dam (as shown in Figure 3(a)) Through the compar-isons of in situmonitoring and numerical results we can seethat the modified PZ-III model gave a better prediction thanthe EB model However as deformation induced by wetting
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 655 m
2010
11
2010
61
0
2010
11
17
2011
42
6
2011
10
3
2012
31
1
2012
81
8
(a)
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 701 m
2010
91
2011
22
8
2011
82
7
2012
22
3
2012
82
1
(b)
Settl
emen
t (m
m)
0
500
1000
1500
2000
2500
Time
Elevation 751 m
2011
10
1
2011
12
20
2012
39
2012
52
8
2012
81
6
In-situEBModified PZ
(c)
Figure 14 Comparison between in situ monitoring settlement andFEM results
of rockfill materials was not considered the FEM result ofsettlement was below than the in situmonitoring data
As an elastoplastic model the PZ-III model is capableof representing the mechanical behavior of soils better thannonlinear elastic model such as Duncan and Changrsquos EBmodel And the above finite element analyses also proved it
6 Conclusions
This paper presents a modified PZ-III model based on thegeneralized theory and original Pastor-Zienkiewicz-Chan
Journal of Applied Mathematics 11
model to simulate the stress-strain relationship of rockfillmaterials
Triaxial test results of the filling materials of Nuozhadudamwere used to validate the proposedmodel and determinethe model parameters of Duncan and Changrsquos EB model andthe modified PZ-III model respectively The simulations oftriaxial stress-strain response show that the modified PZ-III model is capable of representing the key features ofcohesionless soil such as nonlinearity dilatancy and pressuredependency
The proposed model has been incorporated into a finiteelement code to simulate the static response of a high earth-rockfill dam in China The results were compared with thoseof Duncan and Changrsquos EB model The two set of resultshave both similarities and differences and the differencesillustrate the advantages of the modified PZ-III model Thecomparisons of FEM results and in situ monitoring datashowed that the modified PZ-III model can give a betterdescription of deformation of the earth-rockfill dam thanDuncan and Changrsquos EB model
Acknowledgments
This work was supported by the National Nature ScienceFoundation of China (51179092) and the State Key Laboratoryof Hydroscience and Engineering Project (2012-KY-02 and2013-KY-4)
References
[1] J M Duncan ldquoState of the art limit equilibrium and finite-element analysis of slopesrdquo Journal of Geotechnical and Geoen-vironmental Engineering vol 122 no 7 pp 577ndash596 1996
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] R S Sandhu and E L Wilson ldquoFinite element analysis ofseepage in elastic mediardquo Journal of the Engineering MechanicsDivision vol 95 no 3 pp 641ndash652 1969
[4] J T Christian and J W Boehmer ldquoPlane strain consolidationby finite elementsrdquo Journal of Soil Mechanics amp FoundationsDivision vol 96 no 4 pp 1435ndash1457 1970
[5] JMDuncan andC-Y Chang ldquoNonlinear analysis of stress andstrain in soilsrdquo Journal of the Soil Mechanics and FoundationsDivision vol 96 no 5 pp 1629ndash1653 1970
[6] J M Duncan P M Byrne K SWong and P Mabry ldquoStrengthstress-strain and bulk modulus parameters for finite elementanalyses of stresses and movements in soil massesrdquo Tech RepUCBGT80-01 University of California Berkeley Calif USA1980
[7] D C Drucker R E Gibson and D J Henkel ldquoSoil mechanicsand work-hardening theories of plasticityrdquo Transactions of theAmerican Society of Civil Engineers vol 122 pp 338ndash346 1957
[8] K Roscoe A Schofield andCWroth ldquoOn the yielding of soilsrdquoGeotechnique vol 8 no 1 pp 22ndash53 1958
[9] K Roscoe A Schofield and A Thurairajah ldquoYielding of claysin states wetter than criticalrdquo Geotechnique vol 13 no 3 pp211ndash240 1963
[10] J Burland ldquoCorrespondence on lsquoThe yielding and dilation ofclayrsquordquo Geotechnique vol 15 pp 211ndash214 1965
[11] P V Lade and J M Duncan ldquoElastoplastic stress-strain theoryfor cohesionless soilrdquo Journal of the Geotechnical EngineeringDivision vol 101 no 10 pp 1037ndash1053 1975
[12] I S Sandler F L DiMaggio and G Y Baladi ldquoGeneralizedcap model for geological materialsrdquo Journal of the GeotechnicalEngineering Division vol 102 no 7 pp 683ndash699 1976
[13] X-S Li Y F Dafalias and Z-L Wang ldquoState-dependent dila-tancy in critical-state constitutive modelling of sandrdquoCanadianGeotechnical Journal vol 36 no 4 pp 599ndash611 1999
[14] Y-P Yao and D Sun ldquoApplication of Ladersquos criterion to Cam-clay modelrdquo Journal of Engineering Mechanics vol 126 no 1pp 112ndash119 2000
[15] G Y Baladi and B Rohani ldquoElastic-plastic model for saturatedsandrdquo Journal of the Geotechnical Engineering Division vol 105no 4 pp 465ndash480 1979
[16] O Zienkiewicz and Z Mroz ldquoGeneralized plasticity formu-lation and applications to geomechanicsrdquo in Mechanics ofEngineering Materials C S Desai and R H Gallagher Eds pp655ndash679 John Wiley amp Sons New York NY USA 1984
[17] C S Desai and M O Faruque ldquoConstitutive model forgeological materialsrdquo Journal of Engineering Mechanics vol 110no 9 pp 1391ndash1408 1984
[18] S B R Murthy A Vatsala and T S Nagaraj ldquoRevised Cam-clay modelrdquo Journal of Geotechnical Engineering vol 117 no 6pp 851ndash871 1991
[19] M Pastor O C Zienkiewicz and A H C Chan ldquoGeneralizedplasticity and the modelling of soil behaviourrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 14 no 3 pp 151ndash190 1990
[20] ZMroz andO Zienkiewicz ldquoUniform formulation of constitu-tive equations for clays and sandsrdquo inMechanics of EngineeringMaterials C S Desai and R H Gallangher Eds pp 415ndash449John Wiley amp Sons New York NY USA 1984
[21] G Wang and J-M Zhang ldquoDynamic consolidation finiteelement analysis of a sediment-protecting dyke under oceanwave loadingrdquo Rock and Soil Mechanics vol 27 no 4 pp 555ndash560 2006
[22] MAlyamiMRouainia and SMWilkinson ldquoNumerical anal-ysis of deformation behaviour of quay walls under earthquakeloadingrdquo Soil Dynamics and Earthquake Engineering vol 29 no3 pp 525ndash536 2009
[23] H Li P Manuel and T Li ldquoApplication of an generalizedplasticity model to ultra-high rockfill damrdquo in Proceedingsof the 12th International Conference on Engineering ScienceConstruction and Operations in Challenging EnvironmentsmdashEarth and Space pp 385ndash398 Honolulu Hawaii USA March2010
[24] T Li and H Zhang ldquoDynamic parameter verification of P-Z model and its application of dynamic analysis on rockfilldamrdquo in Proceedings of the 12th International Conference onEngineering Science Construction and Operations in Challeng-ing EnvironmentsmdashEarth and Space pp 2706ndash2713 HonoluluHawaii USA March 2010
[25] M Pastor ldquoA generalized plasticity model for anisotropicbehaviour of sandrdquoComputer Methods and Advances in Geome-chanics vol 1 pp 661ndash668 1991
[26] G Bolzon B A Schrefler and O C Zienkiewicz ldquoElastoplasticsoil constitutive laws generalized to partially saturated statesrdquoGeotechnique vol 46 no 2 pp 279ndash289 1996
[27] H I Ling and H Liu ldquoPressure-level dependency and densifi-cation behavior of sand through generalized plasticity modelrdquo
12 Journal of Applied Mathematics
Journal of Engineering Mechanics vol 129 no 8 pp 851ndash8602003
[28] H I Ling and S Yang ldquoUnified sand model based on thecritical state and generalized plasticityrdquo Journal of EngineeringMechanics vol 132 no 12 pp 1380ndash1391 2006
[29] N D Marschi C K Chan and H B Seed ldquoEvaluation ofproperties of rockfill materialsrdquo Journal of the Soil Mechanicsand Foundations Division vol 98 no 1 pp 95ndash114 1972
[30] R J Marsal ldquoLarge scale testing of rockfill materialsrdquo Journal ofthe Soil Mechanics and Foundations Division vol 93 no 2 pp27ndash43 1967
[31] R JMarsal ldquoMechanical properties of rockfillrdquo in EmbankmentDam Engineering pp 109ndash200 John Wiley amp Sons New YorkNY USA 1973
[32] P V Lade J A Yamamuro and P A Bopp ldquoSignificance ofparticle crushing in granular materialsrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 122 no 4 pp 309ndash3161996
[33] BOHardin ldquoCrushing of soil particlesrdquo Journal of GeotechnicalEngineering vol 111 no 10 pp 1177ndash1192 1985
[34] R L Kondner ldquoHyperbolic stress-strain response cohesivesoilsrdquo Journal of the Soil Mechanics and Foundations Divisionvol 89 no 1 pp 115ndash143 1963
[35] Z-LWang Y F Dafalias X-S Li and F I Makdisi ldquoState pres-sure index for modeling sand behaviorrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 128 no 6 pp 511ndash5192002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 9
1
070503
09
01
(a)
0
05
0
minus1
minus15
minus24
minus05
minus2
(b)
051
152 25
3 353
252
151
05
(c)
0
03
05 1
13
minus02
(d)
Figure 11 Displacement and stress contour of the maximum section for Duncan and Changrsquos EB model (a) displacement along river (m)(b) vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
0706
06
050403
0201
0
02
minus02
01
(a)
minus25minus29
minus05minus15
minus1
minus2
(b)
05
1 152 3
3 435
3
25 215
105
(c)
01
05
1 15
0
(d)
Figure 12 Displacement and stress contour of the maximum section for the modified PZ-III model (a) displacement along river (m) (b)vertical displacement (m) (c) major principle stress (MPa) and (d) minor principle stress (MPa)
Through the comparison and analysis of the numericalresults (Figures 11 and 12) we can find some similarities anddifferences for these two models
On one hand we can see many similar places in thedistributions of displacements and stresses
(1) After the reservoir impounding due to the hugewaterpressure on upstream dam horizontal displacementdevelops toward the downstream and the largestdisplacement is about 105m for EBmodel and 074mfor modified PZ-III model
(2) Themaximumsettlement occurs in themiddle of corewall due to lower modulus of clayey soil
(3) Because of the tremendous differences of modulusbetween rockfill material and clayey soil there existsobvious arching effect in the core wall
(4) Effective stress in upstream shell is less than thedownstream shell due to the water pressure in theupstream shell
On the other hand some differences also exist whichillustrate the advantages of modified PZ-III model
(1) After the reservoir is impounded upward displace-ment as large as 07m (see Figure 11(b)) developson the upstream shell near dam crest for EB modeland nearly 0m for modified PZ-III model (seeFigure 12(b)) In fact monitoring data of practicalengineering projects shows that no large upwarddisplacement happened after impounding This isdue to its weakness of EB model to distinguish theloading and unloading condition during the waterimpounding
(2) In the distribution of minor principle stress (Figures11(d) and 12(d)) negative stress (ie tensile stress)occurs in the upstream shell for EB model whereasvery little tensile stress exists for modified PZ-IIImodel As we know rockfill material is a typical kindof cohesionless coarse-grained soil which means thatit has no tensile strength Therefore the existence oflarge area of tensile stress in the upstream shell isunreasonable
522 Comparison between Numerical and In Situ MonitoringData Settlement is a key indicator to assess the safety of an
10 Journal of Applied Mathematics
550
600
650
700
750
800
850El
evat
ion
(m)
Settlement (mm)
In-situEBModified PZ
minus1000 0 1000 2000 3000 4000
Figure 13 Comparison between in situ monitoring settlement andFEM results
Table 2 Material parameters of the modified PZ-III model
Material Rockfill I Rockfill II Mixed gravel clay1198700
500 1000 3001198660
1500 3000 900119898 050 050 050119899 050 050 050120572119891
045 045 045120572119892
045 045 045119872119891119888
105 090 060119872119892119888
160 135 1101205730
000 000 0001205731
000 000 000Γ 034 031 034120582 010 009 003119898119901
035 040 001198670
800 1200 900120574 5 5 5120574119906
5 5 51198671199060MPa 9 9 10
earth dam Figures 13 and 14 show the in situmonitoring dataand FEM results of settlement in themaximum cross-sectionThe in situ data were obtained from electromagnetism typesettlement gaugeswhichwere embedded during constructionin the dam (as shown in Figure 3(a)) Through the compar-isons of in situmonitoring and numerical results we can seethat the modified PZ-III model gave a better prediction thanthe EB model However as deformation induced by wetting
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 655 m
2010
11
2010
61
0
2010
11
17
2011
42
6
2011
10
3
2012
31
1
2012
81
8
(a)
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 701 m
2010
91
2011
22
8
2011
82
7
2012
22
3
2012
82
1
(b)
Settl
emen
t (m
m)
0
500
1000
1500
2000
2500
Time
Elevation 751 m
2011
10
1
2011
12
20
2012
39
2012
52
8
2012
81
6
In-situEBModified PZ
(c)
Figure 14 Comparison between in situ monitoring settlement andFEM results
of rockfill materials was not considered the FEM result ofsettlement was below than the in situmonitoring data
As an elastoplastic model the PZ-III model is capableof representing the mechanical behavior of soils better thannonlinear elastic model such as Duncan and Changrsquos EBmodel And the above finite element analyses also proved it
6 Conclusions
This paper presents a modified PZ-III model based on thegeneralized theory and original Pastor-Zienkiewicz-Chan
Journal of Applied Mathematics 11
model to simulate the stress-strain relationship of rockfillmaterials
Triaxial test results of the filling materials of Nuozhadudamwere used to validate the proposedmodel and determinethe model parameters of Duncan and Changrsquos EB model andthe modified PZ-III model respectively The simulations oftriaxial stress-strain response show that the modified PZ-III model is capable of representing the key features ofcohesionless soil such as nonlinearity dilatancy and pressuredependency
The proposed model has been incorporated into a finiteelement code to simulate the static response of a high earth-rockfill dam in China The results were compared with thoseof Duncan and Changrsquos EB model The two set of resultshave both similarities and differences and the differencesillustrate the advantages of the modified PZ-III model Thecomparisons of FEM results and in situ monitoring datashowed that the modified PZ-III model can give a betterdescription of deformation of the earth-rockfill dam thanDuncan and Changrsquos EB model
Acknowledgments
This work was supported by the National Nature ScienceFoundation of China (51179092) and the State Key Laboratoryof Hydroscience and Engineering Project (2012-KY-02 and2013-KY-4)
References
[1] J M Duncan ldquoState of the art limit equilibrium and finite-element analysis of slopesrdquo Journal of Geotechnical and Geoen-vironmental Engineering vol 122 no 7 pp 577ndash596 1996
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] R S Sandhu and E L Wilson ldquoFinite element analysis ofseepage in elastic mediardquo Journal of the Engineering MechanicsDivision vol 95 no 3 pp 641ndash652 1969
[4] J T Christian and J W Boehmer ldquoPlane strain consolidationby finite elementsrdquo Journal of Soil Mechanics amp FoundationsDivision vol 96 no 4 pp 1435ndash1457 1970
[5] JMDuncan andC-Y Chang ldquoNonlinear analysis of stress andstrain in soilsrdquo Journal of the Soil Mechanics and FoundationsDivision vol 96 no 5 pp 1629ndash1653 1970
[6] J M Duncan P M Byrne K SWong and P Mabry ldquoStrengthstress-strain and bulk modulus parameters for finite elementanalyses of stresses and movements in soil massesrdquo Tech RepUCBGT80-01 University of California Berkeley Calif USA1980
[7] D C Drucker R E Gibson and D J Henkel ldquoSoil mechanicsand work-hardening theories of plasticityrdquo Transactions of theAmerican Society of Civil Engineers vol 122 pp 338ndash346 1957
[8] K Roscoe A Schofield andCWroth ldquoOn the yielding of soilsrdquoGeotechnique vol 8 no 1 pp 22ndash53 1958
[9] K Roscoe A Schofield and A Thurairajah ldquoYielding of claysin states wetter than criticalrdquo Geotechnique vol 13 no 3 pp211ndash240 1963
[10] J Burland ldquoCorrespondence on lsquoThe yielding and dilation ofclayrsquordquo Geotechnique vol 15 pp 211ndash214 1965
[11] P V Lade and J M Duncan ldquoElastoplastic stress-strain theoryfor cohesionless soilrdquo Journal of the Geotechnical EngineeringDivision vol 101 no 10 pp 1037ndash1053 1975
[12] I S Sandler F L DiMaggio and G Y Baladi ldquoGeneralizedcap model for geological materialsrdquo Journal of the GeotechnicalEngineering Division vol 102 no 7 pp 683ndash699 1976
[13] X-S Li Y F Dafalias and Z-L Wang ldquoState-dependent dila-tancy in critical-state constitutive modelling of sandrdquoCanadianGeotechnical Journal vol 36 no 4 pp 599ndash611 1999
[14] Y-P Yao and D Sun ldquoApplication of Ladersquos criterion to Cam-clay modelrdquo Journal of Engineering Mechanics vol 126 no 1pp 112ndash119 2000
[15] G Y Baladi and B Rohani ldquoElastic-plastic model for saturatedsandrdquo Journal of the Geotechnical Engineering Division vol 105no 4 pp 465ndash480 1979
[16] O Zienkiewicz and Z Mroz ldquoGeneralized plasticity formu-lation and applications to geomechanicsrdquo in Mechanics ofEngineering Materials C S Desai and R H Gallagher Eds pp655ndash679 John Wiley amp Sons New York NY USA 1984
[17] C S Desai and M O Faruque ldquoConstitutive model forgeological materialsrdquo Journal of Engineering Mechanics vol 110no 9 pp 1391ndash1408 1984
[18] S B R Murthy A Vatsala and T S Nagaraj ldquoRevised Cam-clay modelrdquo Journal of Geotechnical Engineering vol 117 no 6pp 851ndash871 1991
[19] M Pastor O C Zienkiewicz and A H C Chan ldquoGeneralizedplasticity and the modelling of soil behaviourrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 14 no 3 pp 151ndash190 1990
[20] ZMroz andO Zienkiewicz ldquoUniform formulation of constitu-tive equations for clays and sandsrdquo inMechanics of EngineeringMaterials C S Desai and R H Gallangher Eds pp 415ndash449John Wiley amp Sons New York NY USA 1984
[21] G Wang and J-M Zhang ldquoDynamic consolidation finiteelement analysis of a sediment-protecting dyke under oceanwave loadingrdquo Rock and Soil Mechanics vol 27 no 4 pp 555ndash560 2006
[22] MAlyamiMRouainia and SMWilkinson ldquoNumerical anal-ysis of deformation behaviour of quay walls under earthquakeloadingrdquo Soil Dynamics and Earthquake Engineering vol 29 no3 pp 525ndash536 2009
[23] H Li P Manuel and T Li ldquoApplication of an generalizedplasticity model to ultra-high rockfill damrdquo in Proceedingsof the 12th International Conference on Engineering ScienceConstruction and Operations in Challenging EnvironmentsmdashEarth and Space pp 385ndash398 Honolulu Hawaii USA March2010
[24] T Li and H Zhang ldquoDynamic parameter verification of P-Z model and its application of dynamic analysis on rockfilldamrdquo in Proceedings of the 12th International Conference onEngineering Science Construction and Operations in Challeng-ing EnvironmentsmdashEarth and Space pp 2706ndash2713 HonoluluHawaii USA March 2010
[25] M Pastor ldquoA generalized plasticity model for anisotropicbehaviour of sandrdquoComputer Methods and Advances in Geome-chanics vol 1 pp 661ndash668 1991
[26] G Bolzon B A Schrefler and O C Zienkiewicz ldquoElastoplasticsoil constitutive laws generalized to partially saturated statesrdquoGeotechnique vol 46 no 2 pp 279ndash289 1996
[27] H I Ling and H Liu ldquoPressure-level dependency and densifi-cation behavior of sand through generalized plasticity modelrdquo
12 Journal of Applied Mathematics
Journal of Engineering Mechanics vol 129 no 8 pp 851ndash8602003
[28] H I Ling and S Yang ldquoUnified sand model based on thecritical state and generalized plasticityrdquo Journal of EngineeringMechanics vol 132 no 12 pp 1380ndash1391 2006
[29] N D Marschi C K Chan and H B Seed ldquoEvaluation ofproperties of rockfill materialsrdquo Journal of the Soil Mechanicsand Foundations Division vol 98 no 1 pp 95ndash114 1972
[30] R J Marsal ldquoLarge scale testing of rockfill materialsrdquo Journal ofthe Soil Mechanics and Foundations Division vol 93 no 2 pp27ndash43 1967
[31] R JMarsal ldquoMechanical properties of rockfillrdquo in EmbankmentDam Engineering pp 109ndash200 John Wiley amp Sons New YorkNY USA 1973
[32] P V Lade J A Yamamuro and P A Bopp ldquoSignificance ofparticle crushing in granular materialsrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 122 no 4 pp 309ndash3161996
[33] BOHardin ldquoCrushing of soil particlesrdquo Journal of GeotechnicalEngineering vol 111 no 10 pp 1177ndash1192 1985
[34] R L Kondner ldquoHyperbolic stress-strain response cohesivesoilsrdquo Journal of the Soil Mechanics and Foundations Divisionvol 89 no 1 pp 115ndash143 1963
[35] Z-LWang Y F Dafalias X-S Li and F I Makdisi ldquoState pres-sure index for modeling sand behaviorrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 128 no 6 pp 511ndash5192002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Applied Mathematics
550
600
650
700
750
800
850El
evat
ion
(m)
Settlement (mm)
In-situEBModified PZ
minus1000 0 1000 2000 3000 4000
Figure 13 Comparison between in situ monitoring settlement andFEM results
Table 2 Material parameters of the modified PZ-III model
Material Rockfill I Rockfill II Mixed gravel clay1198700
500 1000 3001198660
1500 3000 900119898 050 050 050119899 050 050 050120572119891
045 045 045120572119892
045 045 045119872119891119888
105 090 060119872119892119888
160 135 1101205730
000 000 0001205731
000 000 000Γ 034 031 034120582 010 009 003119898119901
035 040 001198670
800 1200 900120574 5 5 5120574119906
5 5 51198671199060MPa 9 9 10
earth dam Figures 13 and 14 show the in situmonitoring dataand FEM results of settlement in themaximum cross-sectionThe in situ data were obtained from electromagnetism typesettlement gaugeswhichwere embedded during constructionin the dam (as shown in Figure 3(a)) Through the compar-isons of in situmonitoring and numerical results we can seethat the modified PZ-III model gave a better prediction thanthe EB model However as deformation induced by wetting
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 655 m
2010
11
2010
61
0
2010
11
17
2011
42
6
2011
10
3
2012
31
1
2012
81
8
(a)
0500
100015002000250030003500
Settl
emen
t (m
m)
Time
Elevation 701 m
2010
91
2011
22
8
2011
82
7
2012
22
3
2012
82
1
(b)
Settl
emen
t (m
m)
0
500
1000
1500
2000
2500
Time
Elevation 751 m
2011
10
1
2011
12
20
2012
39
2012
52
8
2012
81
6
In-situEBModified PZ
(c)
Figure 14 Comparison between in situ monitoring settlement andFEM results
of rockfill materials was not considered the FEM result ofsettlement was below than the in situmonitoring data
As an elastoplastic model the PZ-III model is capableof representing the mechanical behavior of soils better thannonlinear elastic model such as Duncan and Changrsquos EBmodel And the above finite element analyses also proved it
6 Conclusions
This paper presents a modified PZ-III model based on thegeneralized theory and original Pastor-Zienkiewicz-Chan
Journal of Applied Mathematics 11
model to simulate the stress-strain relationship of rockfillmaterials
Triaxial test results of the filling materials of Nuozhadudamwere used to validate the proposedmodel and determinethe model parameters of Duncan and Changrsquos EB model andthe modified PZ-III model respectively The simulations oftriaxial stress-strain response show that the modified PZ-III model is capable of representing the key features ofcohesionless soil such as nonlinearity dilatancy and pressuredependency
The proposed model has been incorporated into a finiteelement code to simulate the static response of a high earth-rockfill dam in China The results were compared with thoseof Duncan and Changrsquos EB model The two set of resultshave both similarities and differences and the differencesillustrate the advantages of the modified PZ-III model Thecomparisons of FEM results and in situ monitoring datashowed that the modified PZ-III model can give a betterdescription of deformation of the earth-rockfill dam thanDuncan and Changrsquos EB model
Acknowledgments
This work was supported by the National Nature ScienceFoundation of China (51179092) and the State Key Laboratoryof Hydroscience and Engineering Project (2012-KY-02 and2013-KY-4)
References
[1] J M Duncan ldquoState of the art limit equilibrium and finite-element analysis of slopesrdquo Journal of Geotechnical and Geoen-vironmental Engineering vol 122 no 7 pp 577ndash596 1996
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] R S Sandhu and E L Wilson ldquoFinite element analysis ofseepage in elastic mediardquo Journal of the Engineering MechanicsDivision vol 95 no 3 pp 641ndash652 1969
[4] J T Christian and J W Boehmer ldquoPlane strain consolidationby finite elementsrdquo Journal of Soil Mechanics amp FoundationsDivision vol 96 no 4 pp 1435ndash1457 1970
[5] JMDuncan andC-Y Chang ldquoNonlinear analysis of stress andstrain in soilsrdquo Journal of the Soil Mechanics and FoundationsDivision vol 96 no 5 pp 1629ndash1653 1970
[6] J M Duncan P M Byrne K SWong and P Mabry ldquoStrengthstress-strain and bulk modulus parameters for finite elementanalyses of stresses and movements in soil massesrdquo Tech RepUCBGT80-01 University of California Berkeley Calif USA1980
[7] D C Drucker R E Gibson and D J Henkel ldquoSoil mechanicsand work-hardening theories of plasticityrdquo Transactions of theAmerican Society of Civil Engineers vol 122 pp 338ndash346 1957
[8] K Roscoe A Schofield andCWroth ldquoOn the yielding of soilsrdquoGeotechnique vol 8 no 1 pp 22ndash53 1958
[9] K Roscoe A Schofield and A Thurairajah ldquoYielding of claysin states wetter than criticalrdquo Geotechnique vol 13 no 3 pp211ndash240 1963
[10] J Burland ldquoCorrespondence on lsquoThe yielding and dilation ofclayrsquordquo Geotechnique vol 15 pp 211ndash214 1965
[11] P V Lade and J M Duncan ldquoElastoplastic stress-strain theoryfor cohesionless soilrdquo Journal of the Geotechnical EngineeringDivision vol 101 no 10 pp 1037ndash1053 1975
[12] I S Sandler F L DiMaggio and G Y Baladi ldquoGeneralizedcap model for geological materialsrdquo Journal of the GeotechnicalEngineering Division vol 102 no 7 pp 683ndash699 1976
[13] X-S Li Y F Dafalias and Z-L Wang ldquoState-dependent dila-tancy in critical-state constitutive modelling of sandrdquoCanadianGeotechnical Journal vol 36 no 4 pp 599ndash611 1999
[14] Y-P Yao and D Sun ldquoApplication of Ladersquos criterion to Cam-clay modelrdquo Journal of Engineering Mechanics vol 126 no 1pp 112ndash119 2000
[15] G Y Baladi and B Rohani ldquoElastic-plastic model for saturatedsandrdquo Journal of the Geotechnical Engineering Division vol 105no 4 pp 465ndash480 1979
[16] O Zienkiewicz and Z Mroz ldquoGeneralized plasticity formu-lation and applications to geomechanicsrdquo in Mechanics ofEngineering Materials C S Desai and R H Gallagher Eds pp655ndash679 John Wiley amp Sons New York NY USA 1984
[17] C S Desai and M O Faruque ldquoConstitutive model forgeological materialsrdquo Journal of Engineering Mechanics vol 110no 9 pp 1391ndash1408 1984
[18] S B R Murthy A Vatsala and T S Nagaraj ldquoRevised Cam-clay modelrdquo Journal of Geotechnical Engineering vol 117 no 6pp 851ndash871 1991
[19] M Pastor O C Zienkiewicz and A H C Chan ldquoGeneralizedplasticity and the modelling of soil behaviourrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 14 no 3 pp 151ndash190 1990
[20] ZMroz andO Zienkiewicz ldquoUniform formulation of constitu-tive equations for clays and sandsrdquo inMechanics of EngineeringMaterials C S Desai and R H Gallangher Eds pp 415ndash449John Wiley amp Sons New York NY USA 1984
[21] G Wang and J-M Zhang ldquoDynamic consolidation finiteelement analysis of a sediment-protecting dyke under oceanwave loadingrdquo Rock and Soil Mechanics vol 27 no 4 pp 555ndash560 2006
[22] MAlyamiMRouainia and SMWilkinson ldquoNumerical anal-ysis of deformation behaviour of quay walls under earthquakeloadingrdquo Soil Dynamics and Earthquake Engineering vol 29 no3 pp 525ndash536 2009
[23] H Li P Manuel and T Li ldquoApplication of an generalizedplasticity model to ultra-high rockfill damrdquo in Proceedingsof the 12th International Conference on Engineering ScienceConstruction and Operations in Challenging EnvironmentsmdashEarth and Space pp 385ndash398 Honolulu Hawaii USA March2010
[24] T Li and H Zhang ldquoDynamic parameter verification of P-Z model and its application of dynamic analysis on rockfilldamrdquo in Proceedings of the 12th International Conference onEngineering Science Construction and Operations in Challeng-ing EnvironmentsmdashEarth and Space pp 2706ndash2713 HonoluluHawaii USA March 2010
[25] M Pastor ldquoA generalized plasticity model for anisotropicbehaviour of sandrdquoComputer Methods and Advances in Geome-chanics vol 1 pp 661ndash668 1991
[26] G Bolzon B A Schrefler and O C Zienkiewicz ldquoElastoplasticsoil constitutive laws generalized to partially saturated statesrdquoGeotechnique vol 46 no 2 pp 279ndash289 1996
[27] H I Ling and H Liu ldquoPressure-level dependency and densifi-cation behavior of sand through generalized plasticity modelrdquo
12 Journal of Applied Mathematics
Journal of Engineering Mechanics vol 129 no 8 pp 851ndash8602003
[28] H I Ling and S Yang ldquoUnified sand model based on thecritical state and generalized plasticityrdquo Journal of EngineeringMechanics vol 132 no 12 pp 1380ndash1391 2006
[29] N D Marschi C K Chan and H B Seed ldquoEvaluation ofproperties of rockfill materialsrdquo Journal of the Soil Mechanicsand Foundations Division vol 98 no 1 pp 95ndash114 1972
[30] R J Marsal ldquoLarge scale testing of rockfill materialsrdquo Journal ofthe Soil Mechanics and Foundations Division vol 93 no 2 pp27ndash43 1967
[31] R JMarsal ldquoMechanical properties of rockfillrdquo in EmbankmentDam Engineering pp 109ndash200 John Wiley amp Sons New YorkNY USA 1973
[32] P V Lade J A Yamamuro and P A Bopp ldquoSignificance ofparticle crushing in granular materialsrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 122 no 4 pp 309ndash3161996
[33] BOHardin ldquoCrushing of soil particlesrdquo Journal of GeotechnicalEngineering vol 111 no 10 pp 1177ndash1192 1985
[34] R L Kondner ldquoHyperbolic stress-strain response cohesivesoilsrdquo Journal of the Soil Mechanics and Foundations Divisionvol 89 no 1 pp 115ndash143 1963
[35] Z-LWang Y F Dafalias X-S Li and F I Makdisi ldquoState pres-sure index for modeling sand behaviorrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 128 no 6 pp 511ndash5192002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 11
model to simulate the stress-strain relationship of rockfillmaterials
Triaxial test results of the filling materials of Nuozhadudamwere used to validate the proposedmodel and determinethe model parameters of Duncan and Changrsquos EB model andthe modified PZ-III model respectively The simulations oftriaxial stress-strain response show that the modified PZ-III model is capable of representing the key features ofcohesionless soil such as nonlinearity dilatancy and pressuredependency
The proposed model has been incorporated into a finiteelement code to simulate the static response of a high earth-rockfill dam in China The results were compared with thoseof Duncan and Changrsquos EB model The two set of resultshave both similarities and differences and the differencesillustrate the advantages of the modified PZ-III model Thecomparisons of FEM results and in situ monitoring datashowed that the modified PZ-III model can give a betterdescription of deformation of the earth-rockfill dam thanDuncan and Changrsquos EB model
Acknowledgments
This work was supported by the National Nature ScienceFoundation of China (51179092) and the State Key Laboratoryof Hydroscience and Engineering Project (2012-KY-02 and2013-KY-4)
References
[1] J M Duncan ldquoState of the art limit equilibrium and finite-element analysis of slopesrdquo Journal of Geotechnical and Geoen-vironmental Engineering vol 122 no 7 pp 577ndash596 1996
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] R S Sandhu and E L Wilson ldquoFinite element analysis ofseepage in elastic mediardquo Journal of the Engineering MechanicsDivision vol 95 no 3 pp 641ndash652 1969
[4] J T Christian and J W Boehmer ldquoPlane strain consolidationby finite elementsrdquo Journal of Soil Mechanics amp FoundationsDivision vol 96 no 4 pp 1435ndash1457 1970
[5] JMDuncan andC-Y Chang ldquoNonlinear analysis of stress andstrain in soilsrdquo Journal of the Soil Mechanics and FoundationsDivision vol 96 no 5 pp 1629ndash1653 1970
[6] J M Duncan P M Byrne K SWong and P Mabry ldquoStrengthstress-strain and bulk modulus parameters for finite elementanalyses of stresses and movements in soil massesrdquo Tech RepUCBGT80-01 University of California Berkeley Calif USA1980
[7] D C Drucker R E Gibson and D J Henkel ldquoSoil mechanicsand work-hardening theories of plasticityrdquo Transactions of theAmerican Society of Civil Engineers vol 122 pp 338ndash346 1957
[8] K Roscoe A Schofield andCWroth ldquoOn the yielding of soilsrdquoGeotechnique vol 8 no 1 pp 22ndash53 1958
[9] K Roscoe A Schofield and A Thurairajah ldquoYielding of claysin states wetter than criticalrdquo Geotechnique vol 13 no 3 pp211ndash240 1963
[10] J Burland ldquoCorrespondence on lsquoThe yielding and dilation ofclayrsquordquo Geotechnique vol 15 pp 211ndash214 1965
[11] P V Lade and J M Duncan ldquoElastoplastic stress-strain theoryfor cohesionless soilrdquo Journal of the Geotechnical EngineeringDivision vol 101 no 10 pp 1037ndash1053 1975
[12] I S Sandler F L DiMaggio and G Y Baladi ldquoGeneralizedcap model for geological materialsrdquo Journal of the GeotechnicalEngineering Division vol 102 no 7 pp 683ndash699 1976
[13] X-S Li Y F Dafalias and Z-L Wang ldquoState-dependent dila-tancy in critical-state constitutive modelling of sandrdquoCanadianGeotechnical Journal vol 36 no 4 pp 599ndash611 1999
[14] Y-P Yao and D Sun ldquoApplication of Ladersquos criterion to Cam-clay modelrdquo Journal of Engineering Mechanics vol 126 no 1pp 112ndash119 2000
[15] G Y Baladi and B Rohani ldquoElastic-plastic model for saturatedsandrdquo Journal of the Geotechnical Engineering Division vol 105no 4 pp 465ndash480 1979
[16] O Zienkiewicz and Z Mroz ldquoGeneralized plasticity formu-lation and applications to geomechanicsrdquo in Mechanics ofEngineering Materials C S Desai and R H Gallagher Eds pp655ndash679 John Wiley amp Sons New York NY USA 1984
[17] C S Desai and M O Faruque ldquoConstitutive model forgeological materialsrdquo Journal of Engineering Mechanics vol 110no 9 pp 1391ndash1408 1984
[18] S B R Murthy A Vatsala and T S Nagaraj ldquoRevised Cam-clay modelrdquo Journal of Geotechnical Engineering vol 117 no 6pp 851ndash871 1991
[19] M Pastor O C Zienkiewicz and A H C Chan ldquoGeneralizedplasticity and the modelling of soil behaviourrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 14 no 3 pp 151ndash190 1990
[20] ZMroz andO Zienkiewicz ldquoUniform formulation of constitu-tive equations for clays and sandsrdquo inMechanics of EngineeringMaterials C S Desai and R H Gallangher Eds pp 415ndash449John Wiley amp Sons New York NY USA 1984
[21] G Wang and J-M Zhang ldquoDynamic consolidation finiteelement analysis of a sediment-protecting dyke under oceanwave loadingrdquo Rock and Soil Mechanics vol 27 no 4 pp 555ndash560 2006
[22] MAlyamiMRouainia and SMWilkinson ldquoNumerical anal-ysis of deformation behaviour of quay walls under earthquakeloadingrdquo Soil Dynamics and Earthquake Engineering vol 29 no3 pp 525ndash536 2009
[23] H Li P Manuel and T Li ldquoApplication of an generalizedplasticity model to ultra-high rockfill damrdquo in Proceedingsof the 12th International Conference on Engineering ScienceConstruction and Operations in Challenging EnvironmentsmdashEarth and Space pp 385ndash398 Honolulu Hawaii USA March2010
[24] T Li and H Zhang ldquoDynamic parameter verification of P-Z model and its application of dynamic analysis on rockfilldamrdquo in Proceedings of the 12th International Conference onEngineering Science Construction and Operations in Challeng-ing EnvironmentsmdashEarth and Space pp 2706ndash2713 HonoluluHawaii USA March 2010
[25] M Pastor ldquoA generalized plasticity model for anisotropicbehaviour of sandrdquoComputer Methods and Advances in Geome-chanics vol 1 pp 661ndash668 1991
[26] G Bolzon B A Schrefler and O C Zienkiewicz ldquoElastoplasticsoil constitutive laws generalized to partially saturated statesrdquoGeotechnique vol 46 no 2 pp 279ndash289 1996
[27] H I Ling and H Liu ldquoPressure-level dependency and densifi-cation behavior of sand through generalized plasticity modelrdquo
12 Journal of Applied Mathematics
Journal of Engineering Mechanics vol 129 no 8 pp 851ndash8602003
[28] H I Ling and S Yang ldquoUnified sand model based on thecritical state and generalized plasticityrdquo Journal of EngineeringMechanics vol 132 no 12 pp 1380ndash1391 2006
[29] N D Marschi C K Chan and H B Seed ldquoEvaluation ofproperties of rockfill materialsrdquo Journal of the Soil Mechanicsand Foundations Division vol 98 no 1 pp 95ndash114 1972
[30] R J Marsal ldquoLarge scale testing of rockfill materialsrdquo Journal ofthe Soil Mechanics and Foundations Division vol 93 no 2 pp27ndash43 1967
[31] R JMarsal ldquoMechanical properties of rockfillrdquo in EmbankmentDam Engineering pp 109ndash200 John Wiley amp Sons New YorkNY USA 1973
[32] P V Lade J A Yamamuro and P A Bopp ldquoSignificance ofparticle crushing in granular materialsrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 122 no 4 pp 309ndash3161996
[33] BOHardin ldquoCrushing of soil particlesrdquo Journal of GeotechnicalEngineering vol 111 no 10 pp 1177ndash1192 1985
[34] R L Kondner ldquoHyperbolic stress-strain response cohesivesoilsrdquo Journal of the Soil Mechanics and Foundations Divisionvol 89 no 1 pp 115ndash143 1963
[35] Z-LWang Y F Dafalias X-S Li and F I Makdisi ldquoState pres-sure index for modeling sand behaviorrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 128 no 6 pp 511ndash5192002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Journal of Applied Mathematics
Journal of Engineering Mechanics vol 129 no 8 pp 851ndash8602003
[28] H I Ling and S Yang ldquoUnified sand model based on thecritical state and generalized plasticityrdquo Journal of EngineeringMechanics vol 132 no 12 pp 1380ndash1391 2006
[29] N D Marschi C K Chan and H B Seed ldquoEvaluation ofproperties of rockfill materialsrdquo Journal of the Soil Mechanicsand Foundations Division vol 98 no 1 pp 95ndash114 1972
[30] R J Marsal ldquoLarge scale testing of rockfill materialsrdquo Journal ofthe Soil Mechanics and Foundations Division vol 93 no 2 pp27ndash43 1967
[31] R JMarsal ldquoMechanical properties of rockfillrdquo in EmbankmentDam Engineering pp 109ndash200 John Wiley amp Sons New YorkNY USA 1973
[32] P V Lade J A Yamamuro and P A Bopp ldquoSignificance ofparticle crushing in granular materialsrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 122 no 4 pp 309ndash3161996
[33] BOHardin ldquoCrushing of soil particlesrdquo Journal of GeotechnicalEngineering vol 111 no 10 pp 1177ndash1192 1985
[34] R L Kondner ldquoHyperbolic stress-strain response cohesivesoilsrdquo Journal of the Soil Mechanics and Foundations Divisionvol 89 no 1 pp 115ndash143 1963
[35] Z-LWang Y F Dafalias X-S Li and F I Makdisi ldquoState pres-sure index for modeling sand behaviorrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 128 no 6 pp 511ndash5192002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of