Computers and Geotechnics 69 (2015) 462–473

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Review

Linear elastic and plastic-damage analyses of a concrete cut-off wallconstructed in deep overburden

http://dx.doi.org/10.1016/j.compgeo.2015.05.0150266-352X/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: School of Hydraulic Engineering, Dalian University of Technology, Dalian 116024, China.E-mail address: kongxj@dlut.edu.cn (X. Kong).

Xiang Yu a, Xianjing Kong a,b,⇑, Degao Zou a,b, Yang Zhou a,b, Zhiqiang Hu a,b

a School of Hydraulic Engineering, Dalian University of Technology, Dalian 116024, Chinab The State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China

a r t i c l e i n f o

Article history:Received 8 December 2014Received in revised form 29 May 2015Accepted 31 May 2015Available online 25 June 2015

Keywords:Asphalt concrete core damDeep overburdenConcrete cut-off wallPlastic-damage modelDamage degree

a b s t r a c t

Asphalt concrete core dams (ACCDs) are becoming more widely used worldwide. ACCDs with concretecut-off walls (for controlling foundation seepage) have been constructed in deep overburden. It is impor-tant to assess dam safety by analysing the stress and deformation behaviour of the concrete cut-off wall.In this study, a 3D finite element (FE) procedure was developed to simulate the dam construction andwater impounding processes of an ACCD. Rockfill/gravel materials were described using a Duncan–Chang model, and the interface between the concrete cut-off wall and the foundation gravel wasmodelled using interfacial elements that follow a tangential hyperbolic stress–strain model. The linearelastic and plastic-damage models were employed to model the concrete cut-off wall. Thestress-deformation behaviour and the damage distribution of the concrete cut-off wall were numericallysimulated and analysed. The results indicate that the plastic-damage model was more reliable than theelastic model in describing the mechanical behaviour of the concrete cut-off wall. The plastic-damagemodel can be used to evaluate the safety of concrete cut-off walls constructed in deep overburden.

� 2015 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4632. Constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

2.1. Plastic-damage model for concrete material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4632.2. Hyperbolic model for interface material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4642.3. Duncan–Chang model for rockfill/gravel material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

3. Dam FE model and material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

3.1. Basic information about the dam case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4653.2. FE mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4653.3. Material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

4. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

4.1. Deformation behaviour of the concrete cut-off wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4674.2. Elastic analysis results of the concrete cut-off wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4674.3. Plastic-damage analyses of the concrete cut-off wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4694.4. Constructing the cut-off wall with plastic concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

5. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

X. Yu et al. / Computers and Geotechnics 69 (2015) 462–473 463

1. Introduction

In the past half-century, over one hundred asphalt concretecore dams (ACCDs) have been built worldwide due to theadvanced waterproof and deformation-adaptive behaviour [1].As evidenced by past studies [2–6], China is experienced in theconstruction of ACCDs. A few ACCDs are over 100 m in height,such as the Maoping Xi Dam [7] and the Yele Dam [8]. TheQuxue dam (on the Shuoqu River, Sichuan Province), which iscurrently in the design stage, will be approximately 170 m tall[9]. However, the rapid development of dams in China hasresulted in ACCDs being built on overburden. Table 1 listsACCDs that have been built on overburden in China [10–12].Because of this, foundation seepage control is an important issueto address in ACCD construction.

A concrete cut-off wall is a popular and effective way to controlfoundation seepage and is an indispensable component of animpervious system. However, it is important to assess dam safetyby properly analysing the stress and deformation behaviour ofthe concrete cut-off wall. Past studies have adopted linear elasticmodels to describe the stress–strain relationship of concretecut-off walls [13–17]. However, many testing results have indi-cated that concrete exhibits nonlinear behaviour, such asmulti-axial strength and strain-softening properties [18–20],which cannot be characterized by linear elastic models.Moreover, a concrete cut-off wall built in deep overburden pre-sents a complex 3D stress state due to the forces of the dam body,the retained water and the foundation pressures. An advancedmodel is required to accurately describe these variable features.

The nonlinear behaviour of concrete at the macroscopic level isdependent on the formation of micro-cracks. Consequently, it isimportant to simulate crack initiation and propagation in the anal-ysis of concrete structures [21]. Hillerborg [22] proposed a theoret-ical crack model in which fracture energy [23–25] was firstintroduced. The crack was assumed to form as the stress reachedsufficient strength. The fracture energy was applied to controlthe propagation of the crack. A crack band model presented byBanzant and Oh [26] modelled the fracture as a blunt smearedcrack band. The fracture energy, uniaxial strength and width ofthe crack band were adopted to characterize the fracture proper-ties. Lubliner et al. [27] presented a plastic-damage model (i.e.,the Barcelona model) with consistent and physically relevant con-stitutive relations originating from plasticity theory and a scalardamage variable based on the fracture energy used to representdamage states. Lee and Fenves [28] proposed a modifiedBarcelona model (i.e., the Lee-Fenves model) in whichmultiple-hardening variables were applied to account for the dif-ferent damage states. They also derived a return-mapping

Table 1Basic information of ACCDs built on overburden.

Dam name Basin location Damheight(m)

Maximum overburdenthickness (m)

Yele Nanya River, SichuanProvince

125.5 400

Longtoushi Dadu River, SichuanProvince

70.0 70

Xiabandi Tashkurghan River,Xinjiang Province

78.0 148

Pangduo Lasalle River, TibetanProvince

72.0 200

Nierji Nenjiang River.Mongolia province

41.5 40

Huangjinping Dadu River, SichuanProvince

95.5 130

algorithm for a more convenient and efficient FE implementation[29]. In recent years, the Lee-Fenves model has been widely usedin the FE damage analysis of concrete dams [30–33]. This modelwas also employed by Dakoulas [34] to study the state of concreteslabs of concrete-faced rockfill dams.

The water tightness of the concrete cut-off wall is importantwith respect to the safety of dams built on deep overburden. Arealistic modelling of the nonlinear material behaviour is essential.In this study, a linear elastic model and the Lee-Fenves model wereemployed for the thorough analysis of a concrete cut-off wall onthe deep overburden of an ACCD. The dam/foundation deformationand the contact conditions of the impervious structure werenumerically simulated. A 3D FE program, GEODYNA, developedby Zou [35] and used in many studies [36–40], was also adoptedfor the multi-stage static analysis of ACCD construction and waterimpounding. The constitutive models presented in Section 2were incorporated into GEODYNA, and the performance of theLee-Fenves model and this program were validated. Thestress-deformation behaviour and the distribution of damage forthe concrete cut-off wall were analysed based on the numericalsimulations.

2. Constitutive model

2.1. Plastic-damage model for concrete material

The Lee-Fenves model was used to simulate thedamage-cracking of a concrete cut-off wall. The main componentsof this model are introduced here; a full description can be found in[21,28].

According to the theory of plasticity, the total strain e can bedecomposed into two parts:

e ¼ ee þ ep ð1Þ

where ee and ep are the elastic and the plastic components, respec-tively. Then, the stress–strain relationship can be written as

r ¼ ð1� DÞ�r ¼ ð1� DÞE0 : ðe� epÞ ð2Þ

where �r is the effective stress, E0 is the undamaged elastic stiffness,and D is a scalar degradation damage variable that representsdecreased elastic stiffness.

Lubliner [27] assumed that the total stress strength can bedescribed by the plastic strain, and the relationship is presented as

r@ðepÞ ¼ f @0½ð1þ a@Þ expð�b@epÞ � a@ expð�2b@epÞ� ð3Þ

where f@0 is initial yield strength, and a@ and b@ are constant. @ is astate variable, @ = t indicates uniaxial tension state, and the state ofuniaxial compression is @ = c.

The damage variable j@ represents the damage states and isdefined as

j@ ¼1g@

Z ep

0r@ðepÞdep; g@ ¼

Z 1

0r@ðepÞdep ð4Þ

where g@ is the dissipated energy density during the progression ofa micro-crack. The damage variable can also be derived from theratio of the fracture energy G@ [22] to the characteristic length l@(i.e., g@ = G@/l@) [41].

Two additional damage parameters are defined to indepen-dently represent the tensile and compressive behaviours and areonly functions of j:

D@ðj@Þ ¼ 1� 1a@

1þ a@ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/@ðj@Þ

p� �� �d@=b@

;

/@ðj@Þ ¼ 1þ a@ð2þ a@Þj@ ð5Þ

464 X. Yu et al. / Computers and Geotechnics 69 (2015) 462–473

In addition, a weight factor s is introduced to model the openingand closing behaviour of the crack. The degradation damage vari-able D in Eq. (2) can be expressed by

D ¼ 1� ð1� DcðjcÞÞð1� sDtðjtÞÞ ð6Þ

The yield function is defined by the effective stress and is givenby

Fð�r;jÞ ¼ 11� a

ðaI1 þffiffiffiffiffiffiffi3J2

pþ bðjÞh �̂rmaxiÞ � cðjÞ ð7Þ

where a and b are dimensionless parameters, �̂rmax is the maximumprincipal stress, I1 and J2 are stress invariants, c is the cohesionstrength, and h�i denotes the Macaulay bracket function. TheDrucker–Prager yield function and Mises yield function can beobtained by setting b = 0 and a = b = 0, respectively. Fig. 1 showsthe relationship between the yield functions.The normality plasticflow rule is applied as

_ep ¼ _kr�rUð�rÞ; Uð�rÞ ¼ffiffiffiffiffiffiffi2J2

pþ apI1 ð8Þ

where k is the plastic invariant, U is the plastic potential function,and ap is a parameter of dilatancy.

Fig. 1. Initial yield surface for different yield criterion.

0 1 2 3 4 5

Strain ε1 (mm/m)

0

5

10

15

20

25

30

35

40

Stre

ssσ 1

(MPa

)

1/1 (Experimental)1/0 (Experimental)Numerical

σ1/σ2

+σ 2

+σ 1

(a) Compressive case

Fig. 2. Numerical solution of uniaxial and biaxial

This plastic-damage model was implemented as a componentin GEODYNA. To validate the performance of this advanced modeland the FE program, existing uniaxial and biaxial loading experi-ments [19] were simulated. As shown in Fig. 2, the simulatedresults were compared with the reported results from the corre-sponding experiments. The numerical results from the simulationsgenerally agreed with the experimental data. The plastic-damagemodel was able to accurately describe the stress–strain behaviourof concrete under different stress states. The simulated resultsshowed that the plastic-damage constitutive model was success-fully incorporated into the analysis and was adequate for theplastic-damage analysis of concrete.

2.2. Hyperbolic model for interface material

Generally, the contact element should be set to simulate thecontact behaviour between two materials with distinctly differentdeformation properties. In this study, the Goodman contactelement [42] was applied between two materials with distinctbehaviour (concrete cut-off wall and foundation gravel) to examinethe contact behaviour, especially slipping. The relationshipbetween the force and displacement of the contact element wasreported in [37,38]. Although many models have been developedto express the stress–strain relationship of contact elements[43–48], the hyperbolic model proposed by Clough and Duncan[43] was adopted for the Goodman contact element. The stiffnessin the tangential of the contact element can be expressed by afew parameters obtained from shear tests. The detailed expres-sions of stiffness in different directions and the description of theemployed parameters of the hyperbolic model were reported in[37,38].

2.3. Duncan–Chang model for rockfill/gravel material

Although a general plastic model [49–52] would be better tosimulate the behaviour of rockfill/gravel, the lack of available test-ing data would be a disadvantage. The hyperbolic model proposedby Duncan and Chang [53] has been used extensively in the studyof embankment dams to establish the pre-seismic stress–strainstate, and the numerical results were consistent with the in situmeasurements [54]. The model accounts for the loading/unloadingstress path and the hyperbolic dependency of the elastic moduli onthe current stress state. The involved parameters can be readilydetermined from well-studied triaxial data despite the dilatancyand the plasticity not being well-expressed. In this study, theDuncan–Chang model was employed to describe the behaviour ofdam materials and foundation gravel. In addition, although the

Strain ε1 (mm/m)-0.5-0.4-0.3-0.2-0.10

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Stre

ssσ 1

(MPa

)

-1/-1 (Experimental)-1/ 0 (Experimental)Numerical

σ1/σ2

(b) Tensile case

loading compared with experimental result.

X. Yu et al. / Computers and Geotechnics 69 (2015) 462–473 465

creep behaviour of asphalt concrete is relatively clear, the size ofthe asphalt concrete core located in the centre of the dam is quitesmall compared with the dam body. Thus, the deformation of thecore will be controlled by the dam body, and the creep of the coremay hardly have an effect on the concrete cut-off wall. Thus, theasphalt concrete core was also studied using the Duncan–Changmodel [55–57].

3. Dam FE model and material parameters

3.1. Basic information about the dam case

In the feasibility study stage of a water control project used forpower generation, irrigation, and water storage, an ACCD with amaximum height of 56 m was designed to act as thewater-retaining structure. The elevation of the dam crest was2136 m, and the dam crest was 330 m in length and 10 m in width.Both the upstream and downstream dam slopes were 1/1.8 (verticalto horizontal). A 2 m wide road was set on the downstream slope atthe elevation level (EL) of 2115 m. The dam was also designed to bebuilt on a deep gravel overburden. To control foundation leakage, aconcrete cut-off wall was selected as the vertical anti-seepage mea-sure. The wall was designed with a thickness of 1 m and a maximumdepth of 87 m with 1 m inserted into the bedrock. The geologicalcross section of the river valley and the typical cross-section of thedam are shown in Figs. 3 and 4, respectively.

3.2. FE mesh

After a slight simplification (which did not affect the generalshape) to the slope of the valley shown in Fig. 3, a 3D FE mesh

Fig. 3. Geological cross sect

Fig. 4. Maximum cross section of dam (r: Bedrock; s: Overburden; t: Concrete C

was obtained, as shown in Fig. 5. The generated mesh contained230,220 elements, and 4730 elements were used to model theconcrete cut-off wall. Spatial 8-node isoparametric elements wereused to simulate the material mesh. To more accurately capturethe behaviour of the concrete cut-off wall, two layers of meshelements were used to model the wall thickness, and the Wilsonnonconforming element [58,59] was employed to reveal thebending behaviour more precisely. As shown in Fig. 6, the damconstruction and water impounding stages were step-by-stepand were simulated with 16 steps and 13 steps, respectively. Thelayer thickness was not greater than 5 m. Water was impoundedfrom ELs of 2080 m to 2130 m after dam construction was com-pleted. The water pressure was simulated as surface forces andwas applied to the impervious system, as shown in Fig. 7, demon-strating the FE mesh of the entire impervious system. The dam wasfixed in the x, y and z directions of the bottom boundary. Theboundary condition of the maximum cross-section was indicatedin Fig. 6.

3.3. Material parameters

According to the design information, the first three parameterslisted in Table 2 were used to perform a linear elastic analysis ofthe concrete cut-off wall. To describe the plastic-damage beha-viour of the concrete, certain parameters employed by this studywere determined based on earlier studies. Based on Ref. [31],2.5 MPa tensile strength and 325 N/m tensile fracture energy wereadopted. The maximum compressive strength was set as 10 timesthe tensile strength. The compression fracture energy was set to100 times the tensile value according to previous studies[21,27,31,60,61]. The characteristic length lr was computed from

ion of the river valley.

ut-off Wall; u: Dam Rockfill; v: Transition Layer; w: Asphalt Concrete Core.).

Fig. 5. 3D FE mesh of ACCD (element amount: 230,220; node amount: 236,004).

Fig. 6. Cross Section 1-1 and sketch of construction stages.

Core

Plinth

Cut-off Wall

2

2 2-2

Water Pressure

Interface

Water Pressure

Fig. 7. FE mesh of impervious system.

Table 2Parameters for concrete cut-off wall.

qsat (kg/m3) E (GPa) m ft0 (MPa) Gt (N/m) fc0 (MPa) fb0 (MPa) Gc (N/m)

2400 30 0.17 2.50 325 16.0 18.4 32,500

Table 3Parameters for interfaces.

Location k1 n u (�) Rf C (kPa)

Cut-off wall – Overburden 757 0.86 11.7 0.89 10.5Core – Dam 2022 0.63 32.4 0.83 19.5

466 X. Yu et al. / Computers and Geotechnics 69 (2015) 462–473

the size of an individual mesh element of the concrete cut-off wall.Table 2 lists the parameters for the FE analysis of the concretecut-off wall.

Slurry is typically used to stabilize trenches during the con-struction of concrete cut-off walls. Thus, a thin layer of slurry, gen-erally 2–3 cm, was designed between the concrete cut-off wall andthe foundation [62]. Fu and Zhang [63] conducted direct shear teststo study the contact behaviour between concrete and gravel with a3-cm-thick slurry; the results of the study are listed in Table 3. Inaddition, the asphalt concrete core is soft compared to the transi-tion layer. This contact behaviour was experimentally studied byZhang and Rao using a simple shear apparatus [64]. The parame-ters for the hyperbolic model were determined and are listed inTable 3.

The material properties of the rockfill/gravel and the asphaltconcrete are provided in Table 4. Respectively, cs and cd representthe submerged and dry unit weights. If the element was sub-merged, cs was used in the FE analysis; otherwise, cd was applied.The foundation gravel was always set under water, and the asphalt

Table 4Parameters for rockfill/gravel and asphalt concrete.

Material cs (KN/m3) cd (KN/m3) u0 (�) Du (�) C (kPa) K n Rf Kb m

Dam Rockfill 13.0 21.4 51.7 9.1 0.0 810 0.25 0.65 265 0.20Transition Layer 12.0 21.1 50.6 7.2 0.0 910 0.31 0.63 396 0.34Overburden Gravel 11.0 – 43.5 3.6 0.0 850 0.48 0.93 280 0.15Asphalt Concrete – 25.4 27.3 0.0 140 303 0.24 0.81 719 0.59

Table 5Parameters for plinth.

Material q (kg/m3) E (GPa) m

Plinth 2400 28 0.17

X. Yu et al. / Computers and Geotechnics 69 (2015) 462–473 467

concrete core was impervious. The upstream side was graduallysubmerged as the water level increased.

A plinth, usually constructed as a connecting structure betweenthe concrete cut-off wall and the asphalt concrete core, is a part ofthe impervious system. In this study, the plinth was consideredlinearly elastic, and its parameters are listed in Table 5.

4. Results and discussion

The analyses of the different stages are presented in thissection. ‘‘End of construction stage’’ indicates that the dam wasconstructed to the crest, and ‘‘Impounding completed stage’’implies that water was impounded to the normal water level.‘‘C.EL’’ and ‘‘I.EL’’ represent the elevation levels of the dam con-struction and water impounding, respectively. The tensile stresswas set as negative.

4.1. Deformation behaviour of the concrete cut-off wall

Fig. 8a and b presents the deformations at various locations ofthe concrete cut-off wall at different stages. As the dam was con-structed layer-by-layer, the wall deformations were indicated byvertical settlement (due to the successively increased dam weight).The water and soil pressures acting on the upstream and down-stream sides of the wall were identical so that the horizontal dis-placement was small. Figs. 9a and 10 illustrate the deformedshape and deformation vector of the wall, respectively, at the

-0.08

-0.06

-0.04

-0.02

0.00

50 100 150 200 250 300-0.04

0.00

0.04

0.08

0.12

0.16

0.20

Ver

tical

Set

tlem

ent/m

C.EL.2105m C.EL.2136m I.EL. 2115m I.EL. 2130m

Hor

izon

tal D

ispl

acem

ent/m

Axial Direction/Z(m)

(a) Deformation of the top

Fig. 8. Deformation at typical loca

end of construction stage. The deformation vector of the wallpointed to the central region of the valley due to the special shapeof the wall and the restrictive effect of the surrounding bedrock.Moreover, the deformation at the tip of the wall was seriouslyrestricted so that this section was in an unfavourable stress state.

With the rise of the water level, the vertical settlement of thewall gradually decreased due to the buoyancy of water acting onthe upstream shell material. The horizontal displacementincreased with the upstream water pressure. The deformed shapeof the wall at the impounding completed stage is shown inFig. 9b. The deformation mode was different from the constructionstage. Horizontal deformation and bending near the bedrock wereimmediately apparent. Sections in the downstream side of the wallwith significant bending were in a tensile stress state.

4.2. Elastic analysis results of the concrete cut-off wall

Figs. 11 and 12 illustrate the distribution of the minor principalstress in the concrete cut-off wall using a linear elastic model.Although the wall thickness was divided into two layers, the stressstates in both layers were nearly the same during dam construc-tion (based on the deformation behaviour). Consequently, the FEresults of the upstream layer are only presented for the construc-tion stage. Large portions of the wall were in compression stressstates, and the minor principal stress increased as the dam filled.However, as the maximum tensile stress increased, the affectedarea at the tip of the wall also increased. At the end of constructionstage, the maximum tension stress was over 10.0 MPa at the tip ofthe upper part, which exceeded the tension strength 2.5 MPa. Thelarge tensile stress, which appears to be unrealistic, was caused bythe restricted deformation and high elastic modulus of the wall.

As plotted in Fig. 12, different stress states were observed in thetwo layers of the wall during the water impounding stage. As thewater level rose, the maximum tensile stress and the tensile area

1980

2000

2020

2040

2060

2080

0.00 -0.02 -0.04 -0.06 -0.08 0.00 0.04 0.08 0.12 0.16 0.20

Ver

tical

Dir

ectio

n/Y

(m)

Vertical Settlement/m

C.EL.2105m C.EL.2136m I.EL. 2115m I.EL. 2130m

Horizontal Displacement/m

(b) Deformation of the middle

tions of concrete cut-off wall.

(m)

Y

Z

(m)

(a) End of construction stage (b) Impounding completed stage

Y

XZ X

Fig. 9. Spatial deformed shapes of concrete cut-off wall (Magnification: 100).

Y/m

200 220 240 260 280

2050

2060

2070

2080

Z/m

Fig. 10. Deformation vector of the concrete cut-off wall at the end of construction stage.

1.20.80.0-2.5-5.0-10.0

C.EL.2105 m

1.30.80.0-2.5-5.0-10.0

C.EL.2136 m

Fig. 11. Minor principal stress contour of elastic analysis for dam construction (Unit: MPa).

468 X. Yu et al. / Computers and Geotechnics 69 (2015) 462–473

decreased at the tip of the upper region on the upstream side of thewall due to the uplifting action of water. Meanwhile, the tensionregion expanded on the downstream side with serious bendingdue to the increased water pressure and the restraint of thebedrock.

As indicated by the deformation behaviour of the wall, thedevelopment of bending was noticeable during impounding. Tomore accurately capture the bending behaviour of the wall, thenonconforming element presented by Wilson [58] was employedto represent the wall. The minor principal stress of the down-stream side at two different water levels is shown in Fig. 13.Despite minimal change in the stress distribution rule, expansionin the tension range of the wall was observed based on a compar-ison with the result acquired using normal isoparametric elements.Moreover, the tensile stress increased in the original tensile zone.The wall exhibited more flexibility due to the adoption of the

nonconforming element. Furthermore, such elements can avoidfinite element shear locking and are theoretically more precise.Therefore, the Wilson element was adopted in the followinganalysis.

The spatial distribution of the internal forces of the wall at theimpounding completed stage is illustrated in Fig. 14. The directionand degree of bending can be determined from this figure. Asshown in Fig. 14a, there was a relatively larger moment, and theaxial force was tensile near the bank. The upstream side next tothe bedrock and downstream side where the axial force was almostzero were in a tensile state at the top of the wall. As indicated inFig. 14b, the bending at the central portion of the wall reached amaximum near the base, where the axial force was the largest.The compressive stress induced by the axial force was larger thanthe tensile stress caused by bending. Therefore, the base of the wallwas in a compression stress state.

1.41.00.0-2.5-5.0-9.0

I.EL.2115

1.51.00.0-2.5-5.0-9.0

I.EL.2130 m

(a) Upstream side

1.41.00.0-2.5-5.0-9.0

I.EL.2115

1.51.00.0-2.5-5.0-9.0

I.EL.2130 m

(b) Downstream side

Fig. 12. Minor principal stress contours of elastic analysis for water impounding (Unit: MPa).

1.41.00.0-2.5-5.0-9.0

I.EL.2115 m

1.51.00.0-2.5-5.0-9.0

I.EL.2130m

Fig. 13. Minor principal stress contours of concrete cut-off wall representing with nonconforming element (Unit: MPa).

-4.00

-2.00

0.00

2.00

4.00

-2.00

-1.00

0.00

1.00

2.00

Shea

r Fo

rce/

106 N

Mom

ent/1

06 N×

mA

xial

For

ce/1

07 N

50 100 150 200 250 300-2.00

-1.00

0.00

1.00

2.00

Axial Direction/Z(m)

x

z+ FN

Wall

x

z+ M

Wall

x

+ FS

Wall

z

1980

2000

2020

2040

2060

2080

-4 -2 0 2 4 6 8 0 1 2 3 4 5 -1 0 1 2 3 4 5

Ver

tical

Dir

ectio

n/Y

(m)

Moment/106N×m Axial Force/107N Shear Force/106N

z

Y

+FN

Wall

z

Y

+M

Wall

z

Y

+FS

Wall

(a) Internal force of the top (b) Internal force of the middle

Fig. 14. Internal force at typical locations of concrete cut-off wall at the impounding completed stage.

X. Yu et al. / Computers and Geotechnics 69 (2015) 462–473 469

4.3. Plastic-damage analyses of the concrete cut-off wall

Fig. 15 presents the minor principal stress obtained from theplastic-damage analyses of different stages. In contrast to theresults of the linear elastic analysis, the tensile area of the wall sig-nificantly decreased, and the maximum tension stress rarelyexceeded 2.0 MPa. Based on the yield criterion when the peakstrength was reached, the plastic-damage model was able todescribe the nonlinear behaviour of concrete and to account forthe evolution of damage and the stress release and redistribution.

Figs. 16 and 17 illustrate the tensile damage (denoted by dam-age variable rt) in the concrete cut-off wall during the constructionstages. During dam construction, the damage was localized at thetip of the upper part of the wall due to special deformation and

force characteristics. As shown in Fig. 17b, the damage graduallydeveloped with increasing water levels in the bending zone, andthe damage observed on the downstream side was more seriousthan on the upstream side, which was consistent with the bendingdirection. There was zero or only slight damage (rt less than 0.1)over large portions of the wall. On the downstream side, themaximum tensile damage reached 0.75 at the tip due to the tensilestress caused by relatively small deformations. The tensile damagereached 0.71 at a few positions of the downstream side, whichalso exhibited serious bending and small compression. Thelocalized damaged zone also demonstrated the ability of thisplastic-damage model to accurately represent strain softening. Inaddition, the distribution of the damage did not follow the patternsof the tensile region obtained by linear elastic analysis because the

1.20.80.50.0-0.5-1.0

C.EL.2105 m

1.30.80.50.0-0.5-1.0

C.EL.2136 m

(a) Dam construction stages (upstream side)

1.41.00.50.0-0.5-1.0

I.EL.2115 m

1.41.00.50.0-1.0-2.0

I.EL.2130 m

(b) Water impounding stages (downstream side)

Fig. 15. Minor principal stress contours of plastic-damage analysis (Unit: MPa).

0.700.550.400.200.10

C.EL.2105 m

0.700.550.400.200.10

C.EL.2136 m

Fig. 16. Evolution of tensile damage variable (rt) during dam construction.

0.700.550.400.200.10

I.EL.2115 m

0.700.550.400.200.10

I.EL.2130 m

(a) Upstream side

0.700.550.400.200.10

I.EL.2115 m

0.700.550.400.200.10

I.EL.2130 m

(b) Downstream side

Fig. 17. Evolution of tensile damage variable (rt) during water impounding.

24.022.020.016.010.04.0

C.EL.2136 m

24.022.020.016.010.04.0

C.EL.2136 m

(a) Elastic analysis result (b) Plastic-damage analysis result

Fig. 18. Maximum principal stress at the end of construction stage (Unit: MPa).

470 X. Yu et al. / Computers and Geotechnics 69 (2015) 462–473

yield criteria of this plastic-damage model was defined in a 3Dstress space and the stress presented in this paper was the averagestress of the Gauss points.

The distributions of the maximum principal stress andcompression damage at the end of construction stage are shownin Figs. 18 and 19. The stress of the wall rarely exceeded the

0.060.050.040.020.01

C.EL.2136 m

Fig. 19. Compression damage variable (rc) at the end of construction stage.

X. Yu et al. / Computers and Geotechnics 69 (2015) 462–473 471

compression strength (i.e., 25 MPa) according to the elasticanalysis. Furthermore, the stress distribution obtained by theplastic-damage analysis was minimally changed, even though thestress in some portions of the wall exceeded the initial yieldstrength (i.e., 16 MPa) because of the multi-axial strength and highfracture energy under compression for normal concrete. Theresults indicate that compression stress has little effect on thesafety of the wall.

Crack propagation may be a result of serious tensile damage. Toimprove the anti-seepage properties of the concrete cut-off wall,local reinforcement measures should be considered in the designstage. Replacing normal concrete with fibre-reinforced [65,66] orsteel-reinforced [67] concrete could be effective.

4.4. Constructing the cut-off wall with plastic concrete

Cut-off walls in deep overburden constructed with plasticconcrete have been successfully used in many dam engineeringprojects [68]. However, few applications of plastic concrete in deepoverburden can be found in China, and plastic concrete cut-offwalls have been most commonly used in cofferdam and embank-ment projects where the foundation and dam body have littleimpact on the wall. An elastic analysis was conducted by replacingthe construction material of the cut-off wall with plastic concrete.The material properties of the plastic concrete used for this analy-sis are as follows [69,70]: elastic modulus, E = 1.5 GPa; Poisson’sratio, v = 0.25; and density, q = 2200 kg/m3. The compressionstrength was assumed to be 5 MPa, and the tensile strength wasset to 10% of the compression strength.

The principal stress of the plastic concrete wall at differentstages is shown in Fig. 20. The area under tension decreased sub-stantially, and tensile stress only appeared at the top of the wallnear the bedrock due to the lower elastic modulus. The maximum

1.51.00.0-0.5-1.5-3.5

I.EL.2130 m

(a) Minimum Principal stress at the impounding complete stage

14.012.010.08.05.02.0

C.EL.2136 m

(b) Maximum Principal stress at the end of construction stage

Fig. 20. Principal stress of plastic concrete cut-off wall (Unit: MPa).

compressive stress also decreased from 25 MPa to 15 MPa.However, the plastic concrete had relatively low compressionstrength. Thus, the principal stress in large portions of the wallexceeded the strength. The ratio of the maximum compressivestress to the strength was approximately 3.0, which may inducecompression damage. If plastic concrete is adopted to constructthe cut-off wall in this project, the material parameter should becarefully designed, and its 3D stress state behaviour should bestudied. The building plans and methods may differ from thoseused for normal concrete to decrease the relatively large compres-sive stress.

5. Conclusions

In this study, the linear elastic and plastic-damage analyses of aconcrete cut-off wall with a height of 87 m in deep overburden ofan ACCD were performed using GEODYNA, a 3D FE program. Aplastic-damage model proposed by Lee-Fenves was used to con-duct the plastic-damage analysis. The existing monotonic uniaxialand biaxial loading tests of concrete were simulated, and the pre-dicted results were compared with experimental data from the lit-erature. Furthermore, the stress, deformation and damagebehaviour of the concrete cut-off wall during dam constructionand water impounding were investigated.

The deformation modes of the concrete cut-off wall were dis-tinct at different construction stages. During dam construction,the settlement at the centre was obvious. The deformation at thetip of the upper part of the wall was significantly restricted, whichmay give rise to an unfavourable stress state. As the water levelrose, the horizontal displacement significantly increased, and seri-ous bending occurred in the downstream side near the bedrockwhere high tensile stresses were observed.

The stress distribution from the linear elastic analysis can beexplained by the deformation mode and the distribution of internalforces at the different stages; nonconforming element should beadopted to capture the behaviour of the wall in the deep overbur-den more precisely. However, the linear elastic model is not able toreasonably express the nonlinear behaviour of concrete; thus, theobtained tensile stress of a few regions exceeded the tensilestrength. It is not accurate to rely on the linear elastic analysisresults to assess the safety of the concrete cut-off wall.

The tensile area in the concrete cut-off wall was significantlyreduced, and the maximum tensile stress rarely exceeded2.0 MPa, as computed by the plastic-damage model. Furthermore,the damage distribution indicated that most portions of the wallexhibited no damage or only slight tensile damage. The tensiledamage that did occur was mainly concentrated at the restrictedupper tip of the wall and at the bending locations on the down-stream side near the bedrock. The result also indicated that thecompression stress would have little effect on the safety of thewall. To improve the anti-seepage capability of the wall, particularengineering measures should be practiced at these positions. Theplastic-damage model reasonably predicated the nonlinear proper-ties of concrete in a 3D stress space, including strain softening,stress redistribution and damage accumulation. The numericalresults of the concrete cut-off wall using the plastic-damage modelwere more reasonable and useful for design and construction.

The reinforcement cage was not considered in this study, andthe use of remedial measures to prevent damage requires furtherresearch. Moreover, safety evaluations of concrete cut-off wallsutilizing plastic-damage analyses would be more meaningful ifthe relationship between the damage variables and concretepermeability was addressed. Furthermore, more efforts should bedevoted to investigate the nonlinear damage behaviour of plasticconcrete under different stress states.

472 X. Yu et al. / Computers and Geotechnics 69 (2015) 462–473

Acknowledgements

This research was supported by the Natural Science Foundation ofChina (Nos. 51138001, 51279025, 91215301) and the Program forNew Century Excellent Talents in University (No. NCET-12-0083).These financial supports are gratefully acknowledged.

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