research article comparison between duncan and chang s eb...

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


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


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


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

()


(b)

Figure 4 Comparison between fittings of Duncan and Changrsquos EB model and experimental triaxial tests results for rockfill material I


0

2000

4000

6000

8000

10000

300 kPa 900 kPa1500 kPa 2500 kPaModified PZ

0 5 10 15

1205901minus1205903

(kPa

)

1205761 ()

(a)


01234

1205761 ()0 5 10 15

120576

()


(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


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


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


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


(c)


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


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


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


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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



Consider

119889120576 = 119889120576119890+ 119889120576119901 (6)



119889120590 = D119890119901 119889120576 (7)


D119890119901 = D119890 minusD119890 n

119892119871119880 n119879 D119890

119867119871119880+ n119879 D119890 n

119892119871119880

(8)

where D119890 n119892119871119880

n and 119867119871119880



119889120590119879


119889120590119879


119889120590119879


(9)




1198891199011015840= 119870119890V119889120576119890

V 119889119902 = 3119866119890119904119889120576119890

119904 (10)



119870119890V = 119870119890119904119900

1199011015840

119901119900

119866119890119904= 119866119890119904119900

1199011015840

119901119900

(11)

where119870119890119904119900 119866119890119904119900 and 119901



n and 119867119871119880



n119892119871= (

119889119892

radic1 + 1198892

119892

1

radic1 + 1198892

119892

)

119879

n = (119889119891

radic1 + 1198892

119891

1

radic1 + 1198892

119891

)

119879

(12)


follows

119889119892=

119889120576119901

V

119889120576119901

119904

= (1 + 120572119892) (119872119892minus 120578) (13)


119889119891= (1 + 120572

119891) (119872119891minus 120578) (14)


119892119872119891is equal




is defined as

n119892119880= (minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119892

radic1 + 1198892

119892

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

radic1 + 1198892

119892

)

119879

(15)


119867119871= 11986701199011015840119867119891(119867V + 119867119904)119867119863119872 (16)

where119867119891= (1 minus 120578120578


119891= (1+


119867119904= 12057301205731exp(minus120573




119891120578(1 + 120572

119891)119872119891](minus1120572)

119891which



defined as

119867119880= 1198671199060(

119872119892

120578119906

)

120574119906

119872119892

120578

gt 1

119867119880= 1198671199060

119872119892

120578

le 1

(17)


119906is the



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)




1-120576V curve where 120576

1 120576V are



119870119890V = 1198701198900119901119886(

1199011015840

119901119886

)

119898

119866119890119904= 1198661198900119901119886(

1199011015840

119901119886

)

119899

119867119871= 1198670119901119886(

1199011015840

119901119886

)

119898

119867119891(119867V + 119867119904)119867119863119872

(18)

where 1198701198900and 119866





119889119892=

119889120576119901

V

119889120576119901

119904

= (1 + 120572119892) (119872119892119868119901

119898119901

minus 120578) (19)


119901= 119901119901

119888in which 119901

119888


119890119888= Γ minus 120582 log (119901

119888) (20)






BurmaLaos

China

VietnamThailand

(a) (b)

(c) (d)









073 074 077119870 1450 1360 520119870119887

550 600 250119870119906119903

2800 2500 900119899 030 043 042119898 013 008 025



Upstream Downstream

RU1RU3F2F1

RU2 RD1

RD2

Cofferdam ED

F2F1

RD3




900

800

700

600

500

8241

658

(a)

8125 20121231

20110531sim20120531

20080215sim20080531 20080531sim20090531

20090531sim20100531

20100531sim20110531

(b)


0

2000

4000

6000

8000

10000


0 5 10 15

1205901minus1205903

(kPa

)

1205761 ()

(a)


01234

1205761 ()0 5 10 15

120576

()


(b)



0

2000

4000

6000

8000

10000


0 5 10 15

1205901minus1205903

(kPa

)

1205761 ()

(a)


01234

1205761 ()0 5 10 15

120576

()


(b)


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)


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)



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)


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)










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)


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)













550

600

650

700

750

800

850El

evat

ion

(m)

Settlement (mm)


minus1000 0 1000 2000 3000 4000




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


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


(c)




6 Conclusions






Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Consider

119889120576 = 119889120576119890+ 119889120576119901 (6)



119889120590 = D119890119901 119889120576 (7)


D119890119901 = D119890 minusD119890 n

119892119871119880 n119879 D119890

119867119871119880+ n119879 D119890 n

119892119871119880

(8)

where D119890 n119892119871119880

n and 119867119871119880



119889120590119879


119889120590119879


119889120590119879


(9)




1198891199011015840= 119870119890V119889120576119890

V 119889119902 = 3119866119890119904119889120576119890

119904 (10)



119870119890V = 119870119890119904119900

1199011015840

119901119900

119866119890119904= 119866119890119904119900

1199011015840

119901119900

(11)

where119870119890119904119900 119866119890119904119900 and 119901



n and 119867119871119880



n119892119871= (

119889119892

radic1 + 1198892

119892

1

radic1 + 1198892

119892

)

119879

n = (119889119891

radic1 + 1198892

119891

1

radic1 + 1198892

119891

)

119879

(12)


follows

119889119892=

119889120576119901

V

119889120576119901

119904

= (1 + 120572119892) (119872119892minus 120578) (13)


119889119891= (1 + 120572

119891) (119872119891minus 120578) (14)


119892119872119891is equal




is defined as

n119892119880= (minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119892

radic1 + 1198892

119892

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

radic1 + 1198892

119892

)

119879

(15)


119867119871= 11986701199011015840119867119891(119867V + 119867119904)119867119863119872 (16)

where119867119891= (1 minus 120578120578


119891= (1+


119867119904= 12057301205731exp(minus120573




119891120578(1 + 120572

119891)119872119891](minus1120572)

119891which



defined as

119867119880= 1198671199060(

119872119892

120578119906

)

120574119906

119872119892

120578

gt 1

119867119880= 1198671199060

119872119892

120578

le 1

(17)


119906is the



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)




1-120576V curve where 120576

1 120576V are



119870119890V = 1198701198900119901119886(

1199011015840

119901119886

)

119898

119866119890119904= 1198661198900119901119886(

1199011015840

119901119886

)

119899

119867119871= 1198670119901119886(

1199011015840

119901119886

)

119898

119867119891(119867V + 119867119904)119867119863119872

(18)

where 1198701198900and 119866





119889119892=

119889120576119901

V

119889120576119901

119904

= (1 + 120572119892) (119872119892119868119901

119898119901

minus 120578) (19)


119901= 119901119901

119888in which 119901

119888


119890119888= Γ minus 120582 log (119901

119888) (20)






BurmaLaos

China

VietnamThailand

(a) (b)

(c) (d)









073 074 077119870 1450 1360 520119870119887

550 600 250119870119906119903

2800 2500 900119899 030 043 042119898 013 008 025



Upstream Downstream

RU1RU3F2F1

RU2 RD1

RD2

Cofferdam ED

F2F1

RD3




900

800

700

600

500

8241

658

(a)

8125 20121231

20110531sim20120531

20080215sim20080531 20080531sim20090531

20090531sim20100531

20100531sim20110531

(b)


0

2000

4000

6000

8000

10000


0 5 10 15

1205901minus1205903

(kPa

)

1205761 ()

(a)


01234

1205761 ()0 5 10 15

120576

()


(b)



0

2000

4000

6000

8000

10000


0 5 10 15

1205901minus1205903

(kPa

)

1205761 ()

(a)


01234

1205761 ()0 5 10 15

120576

()


(b)


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)


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)



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)


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)










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)


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)













550

600

650

700

750

800

850El

evat

ion

(m)

Settlement (mm)


minus1000 0 1000 2000 3000 4000




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


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


(c)




6 Conclusions






Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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)




1-120576V curve where 120576

1 120576V are



119870119890V = 1198701198900119901119886(

1199011015840

119901119886

)

119898

119866119890119904= 1198661198900119901119886(

1199011015840

119901119886

)

119899

119867119871= 1198670119901119886(

1199011015840

119901119886

)

119898

119867119891(119867V + 119867119904)119867119863119872

(18)

where 1198701198900and 119866





119889119892=

119889120576119901

V

119889120576119901

119904

= (1 + 120572119892) (119872119892119868119901

119898119901

minus 120578) (19)


119901= 119901119901

119888in which 119901

119888


119890119888= Γ minus 120582 log (119901

119888) (20)






BurmaLaos

China

VietnamThailand

(a) (b)

(c) (d)









073 074 077119870 1450 1360 520119870119887

550 600 250119870119906119903

2800 2500 900119899 030 043 042119898 013 008 025



Upstream Downstream

RU1RU3F2F1

RU2 RD1

RD2

Cofferdam ED

F2F1

RD3




900

800

700

600

500

8241

658

(a)

8125 20121231

20110531sim20120531

20080215sim20080531 20080531sim20090531

20090531sim20100531

20100531sim20110531

(b)


0

2000

4000

6000

8000

10000


0 5 10 15

1205901minus1205903

(kPa

)

1205761 ()

(a)


01234

1205761 ()0 5 10 15

120576

()


(b)



0

2000

4000

6000

8000

10000


0 5 10 15

1205901minus1205903

(kPa

)

1205761 ()

(a)


01234

1205761 ()0 5 10 15

120576

()


(b)


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)


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)



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)


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)










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)


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)













550

600

650

700

750

800

850El

evat

ion

(m)

Settlement (mm)


minus1000 0 1000 2000 3000 4000




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


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


(c)




6 Conclusions






Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










BurmaLaos

China

VietnamThailand

(a) (b)

(c) (d)









073 074 077119870 1450 1360 520119870119887

550 600 250119870119906119903

2800 2500 900119899 030 043 042119898 013 008 025



Upstream Downstream

RU1RU3F2F1

RU2 RD1

RD2

Cofferdam ED

F2F1

RD3




900

800

700

600

500

8241

658

(a)

8125 20121231

20110531sim20120531

20080215sim20080531 20080531sim20090531

20090531sim20100531

20100531sim20110531

(b)


0

2000

4000

6000

8000

10000


0 5 10 15

1205901minus1205903

(kPa

)

1205761 ()

(a)


01234

1205761 ()0 5 10 15

120576

()


(b)



0

2000

4000

6000

8000

10000


0 5 10 15

1205901minus1205903

(kPa

)

1205761 ()

(a)


01234

1205761 ()0 5 10 15

120576

()


(b)


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)


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)



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)


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)










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)


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)













550

600

650

700

750

800

850El

evat

ion

(m)

Settlement (mm)


minus1000 0 1000 2000 3000 4000




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


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


(c)




6 Conclusions






Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Upstream Downstream

RU1RU3F2F1

RU2 RD1

RD2

Cofferdam ED

F2F1

RD3




900

800

700

600

500

8241

658

(a)

8125 20121231

20110531sim20120531

20080215sim20080531 20080531sim20090531

20090531sim20100531

20100531sim20110531

(b)


0

2000

4000

6000

8000

10000


0 5 10 15

1205901minus1205903

(kPa

)

1205761 ()

(a)


01234

1205761 ()0 5 10 15

120576

()


(b)



0

2000

4000

6000

8000

10000


0 5 10 15

1205901minus1205903

(kPa

)

1205761 ()

(a)


01234

1205761 ()0 5 10 15

120576

()


(b)


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)


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)



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)


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)










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)


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)













550

600

650

700

750

800

850El

evat

ion

(m)

Settlement (mm)


minus1000 0 1000 2000 3000 4000




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


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


(c)




6 Conclusions






Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

2000

4000

6000

8000

10000


0 5 10 15

1205901minus1205903

(kPa

)

1205761 ()

(a)


01234

1205761 ()0 5 10 15

120576

()


(b)


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)


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)



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)


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)










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)


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)













550

600

650

700

750

800

850El

evat

ion

(m)

Settlement (mm)


minus1000 0 1000 2000 3000 4000




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


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


(c)




6 Conclusions






Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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)


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)










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)


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)













550

600

650

700

750

800

850El

evat

ion

(m)

Settlement (mm)


minus1000 0 1000 2000 3000 4000




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


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


(c)




6 Conclusions






Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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)


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)













550

600

650

700

750

800

850El

evat

ion

(m)

Settlement (mm)


minus1000 0 1000 2000 3000 4000




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


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


(c)




6 Conclusions






Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










550

600

650

700

750

800

850El

evat

ion

(m)

Settlement (mm)


minus1000 0 1000 2000 3000 4000




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


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


(c)




6 Conclusions






Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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