Stability Assessment of the Three-Gorges Dam Foundation

ARTICLE IN PRESS

International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652

Stabilityassessment of the Three-Gorges Dam foundation, China, using physical and numerical modeling—Part II: numerical modeling Jian Liua,b,*, Xia-Ting Fenga, Xiu-Li Dingb a Institute of Rock and Soil Mechanics, The Chinese Academy of Sciences, Xiaohongshan, Hubei, Wuhan 430071, China b Yangtze River Scientiﬁc Research Institute, Hubei, Wuhan 430071, China Accepted 31 March 2003

Abstract

Sliding in a dam foundation along potential sliding paths is generallycaused bytwo kinds of external factors: one is the overloading of the designed upstream hydrostatic load due to flooding; and the other is the gradual degradation of the shear strength of joints due to seepage, deformation, damage, and geochemical reactions between water and joint surface minerals. Based on the conceptualized geomechanical model of the Three-Gorges Dam, described in the Part I paper, in this Part II paper the limit equilibrium method and finite element method are used to studythe effects of gradual degradation of the shear strength of joints on the stabilityof the dam foundation. The numerical modeling focuses on the stabilityconditions of the no. 3 powerhouse-dam section which are estimated to be the most critical. The constraint influences from the adjacent no. 2 and no. 4 powerhouse-dam sections are also included. The failure mechanisms, factors of safetyand critical displacements of these dam sections are derived numericallyas the measures for stabilityevaluation. The factor of safetyis defined as the ratio between the combined shear strength of joints and intact rock bridges, and the mean shear stress along potential sliding path required for limit equilibrium under the designed external loads. All the results obtained from these different numerical models, together with the results of physical model tests as presented in Part I, are compared in this paper. The comparisons show that both the numerical modeling and physical modeling results support each other and demonstrate the stabilityof the Three-Gorges Dam foundation as designed. Nevertheless, considering the overall engineering and social–economical importance of the Three-Gorges Dam complex, some additional treatment and reinforcement measures are recommended in this paper. r 2003 Elsevier Ltd. All rights reserved.

Keywords: Three-Gorges Dam; Stabilityassessment; Numerical modeling; Limit equilibrium analyses;Finite element method; Gentlydipping joints; Rock bridges

1. Introduction the shear strength of joints byphysicalmodel tests, to compensate the determination of the overall factor of To assess the stabilityof the Three-Gorges Dam safetyof the dam sections, and to understand and foundation, comprehensive physical modeling was con- estimate the uncertainties involved in physical model ducted. The geological features, conceptualized geome- testing process [1], the numerical modeling approach chanical models of the critical dam sections, mechanical was used to complete the stabilitystudyof the Three- parameters of the rock mass and joint, as well as the Gorges Dam foundation. This Part II paper reports the physical modeling of the foundation stability were results of the numerical modeling works, and discusses studied and are presented in the companion Part I the overall evaluation of the stabilityof the Three- paper (Figs. 1–4) [1]. However, considering the difﬁcul- Gorges Dam foundation. ties of simulating the effects of gradual degradation of Under static loadings, the potential failure mechanisms that are normallyinvestigated for a concrete dam foundation include four major modes: (i) sliding, *Corresponding author. Institute of Rock and Soil Mechanics, The Chinese Academyof Science, Xiaohongshan, Wuhan, Hubei 430071, (ii) deﬁcient bearing capacityof foundation, (iii) non- China. Tel.: +86-27-871-97913; fax: +86-27-871-97386. uniform settlement, and (iv) uplift. In the case of E-mail address: [email protected] (J. Liu). the Three-Gorges Dam foundation, the geological

1365-1609/03/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S1365-1609(03)00056-X ARTICLE IN PRESS 634 J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652

Nomenclature fi friction coefficients of the assumed sliding surfaces of a rigid block in a limit equilibrium K factor of safety analysis SG total vertical load acting on the foundation ci cohesion of the assumed sliding surfaces of surface of the dam rigid blocks in a limit equilibrium analysis Gz self-weight of the powerhouse li lengths of the assumed sliding surfaces of a P1; P2 upstream and downstream horizontal loads rigid block in a limit equilibrium analysis acting on the dam structure, respectively sx; sy; sz SM total moment along the foundation surface of normal stress components along x; y; z axial the dam directions, respectively 0 sn the vertical stress at the upstream starting f The friction coefficient point of the dam foundation surface c cohesion 00 sn the vertical stress at the downstream ending f friction angle max point of the dam foundation surface tn the maximum shear stress of joints B the width of the dam section sn normal stress across joints Ki the factor of safetyagainst sliding of a specific ks the shear stiffness of joints rigid block in a limit equilibrium analysis, kn the normal stiffness of joints subscript i is the number of rigid blocks E elastic modulus of intact rock Wi vertical load acting on a rigid block in a limit n Poisson’s ratio of intact rock equilibrium analysis si mean value of normal stress components at Ui uplift force acting on a rigid block in a limit Gauss points of a specific joint element in equilibrium analysis FEM Hi horizontal force acting on a rigid block in a ti mean value of shear stress components direct- limit equilibrium analysis ing downstream along the joint surfaces at ai dip angle of an assumed sliding surface of a Gauss points of a specific joint element in rigid block in a limit equilibrium analysis FEM Ri horizontal resistant force between two adja- Ai area of surface of a specific joint element in cent rigid blocks in a limit equilibrium analysis FEM

Harbin

Urumqi

Shenyang Hohhot Beijing

ver i

llow The Three Gorges Project Ye

River Lanzhou Yellow Yangzte Xi'an Nanjing River Shanghai Wuhan Chengdu Lhasa

River Chengqing

Yangzte Taiwan Kunming Guangzhou

Nanning

Haikou

Fig. 1. Location of the Three-Gorges Project in China. ARTICLE IN PRESS J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652 635

Assemblage section 1 2 3 4 5 6 7 8 10 9 11 12 13 14

The numbered left powerhouse dam sections

P erm N ane nt ship lock S hip lift

Te mp or ary Left power plant sh ipl ock

Spillway dam

plant er gtze Riv Right power Yan

1 km Scale

Fig. 2. Layout of the Three-Gorges Project and the numbered left powerhouse-dam sections.

Fig. 3. Upstream view of the Three-Gorges Project under construction. investigations, together with the laboratoryand in situ permeabilityand high strength. The faults and joints tests for the properties of the rock mass, revealed that in the foundation have thin thickness with well- the dam foundation mainlycomprises plagioclase cemented tectonite. The joints are mostlynot persistent, granite that is intact, homogeneous, and of low even in the regions where the gentlydipping joints are ARTICLE IN PRESS 636 J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652

Fig. 4. Downstream view of the Three-Gorges Project under construction. most developed: no deterministic and through-going demonstrated that the gentlydipping joints are the sliding paths in the rock mass appear, due to the most important factor governing the foundation stabi- existence of the rock bridges [1]. For this reason, the lity, and the foundation of the no. 3 powerhouse-dam other three modes could be excluded from these section is the most critical for the stabilityof the Three- potential failure mechanisms, and thereby, the modeling Gorges Dam [1]. On the other hand, the foundation works could focus on the stabilityagainst sliding along rocks are actuallynot cut into anycomplete blocks by potential shearing paths comprised of gentlydipping the joints due to the existence of rock bridges; thus, the joints and intact rock bridges. finite element method is more suitable than the The sliding in a dam foundation is usuallycaused by distinct element method (DEM) or discontinuous two external factors: one is the overloading bythe deformation analyses (DDA) [3]. For this reason, this designed upstream hydrostatic load applied on a dam study, as Part II of the modeling works for the stability due to flooding; and the other is the gradual degradation studyof the Three-Gorges Dam, was performed using of the shear strength of joints or other potential shearing both limit equilibrium analyses and the finite element paths due to water pressure, seepage, damage of joint method (including both 2-D and 3-D analyses), based surfaces, and geochemical reactions between water and on the same geomechanical model as that used in joint surface minerals. Correspondingly, there should be physical model tests (see Figs. 5–8 and Table 1). The two alternative definitions of the factor of safetyas the numerical modeling focuses on the stabilityconditions index of stability. In the first case, the factor of safety is of the no. 3 dam section, but the constraint influences defined as the ratio between the maximum external load from the adjacent no. 2 and no. 4 sections are also causing the sliding instabilityof the jointed rock mass included. and the designed load applied to the dam; while, in the To identifythe failure mechanism and derive the second case, it is defined as the ratio between the factor of safety, in the case of limit equilibrium analyses, comprehensive shear strength of joints and rock bridges the rock bridges were assumed as fictitious joints; the along potential sliding paths and the mean shear stress factor of safetywas computed as the ratio between the required for limit equilibrium under the designed resistant force and the driving force along a specific external loads [2]. To investigate the first sliding mode sliding path formed byboth natural and fictitious joints. for the Three-Gorges Dam foundation, physical model In the 2-D elasto-plastic FEM modeling, the friction tests were performed and proved to be an effective coefficients and cohesions of the joints, rock bridges and method (see Part I of this work [1]). However, it is foundation surfaces were decreased simultaneouslyand difficult for the physical modeling techniques to consider proportionallyuntil the convergence could not be the effects of the shear strength reduction of joints. reached while the loading on the dam remains un- Therefore, using numerical modeling to continue the changed. The factor of safetyis obtained as the ratio stabilitystudyof the dam foundation is necessary.In the between the initial value and final decreased value of the meantime, the numerical results can also be used to shear strengths of the potential sliding paths. In the 3-D check the critical failure mechanism derived from the FEM modeling, the intact rock and concrete were physical model tests. modeled as elastic materials, while the joints and rock Both the conceptualization of geomechanical condi- bridges were simulated as elasto-plastic media. The tions and the results of physical model tests have factor of safetywas calculated as the ratio between the ARTICLE IN PRESS J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652 637

Fig. 5. Stereographic projection of the joints in the foundation of the most critical no. 3 powerhouse-dam section of the Three-Gorges Dam.

Fig. 6. Geological model of the no. 3 left powerhouse-dam section of the Three-Gorges Dam. weighted mean of the shear strengths of all joint designed load conditions. Following this, all the results elements along potential sliding paths and the weighted derived from the different numerical models and the mean of the shear stresses of these elements under the physical model tests are compared. ARTICLE IN PRESS 638 J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652

Fig. 7. Geological model of the no. 2 left powerhouse-dam section of the Three-Gorges Dam.

Fig. 8. Geological model of the no. 4 left powerhouse-dam section of the Three-Gorges Dam.

2. Limit equilibrium analyses vertical ﬁctitious interfaces, and the factor of safetyis computed byestablishing the static limit equilibrium 2.1. Methodology conditions based on some assumed failure mechanisms [4,5]. This can be illustrated in Fig. 9 for an example of In the limit equilibrium method, the rock mass is the assumed sliding path ABCDFGHZ in the founda- generallydivided into a number of rigid blocks with tion of the critical no. 3 powerhouse-dam section. In this ARTICLE IN PRESS J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652 639

Table 1 Mechanical parameters of the foundation rock and joint plane used in the stabilitystudies

Type Compressive Density Deformation Poisson’s Friction Cohesion strength (MPa) (kN/m3) moduli (GPa) ratio angle (deg) (MPa)

Plagioclase granite 100 27.0 35 0.20 59.6 2.0 Joint planes — — — — 35 0.2 Dam concrete 200 24.5 26 0.167 48 3.0 Foundation surface — — — — 48 1.3

Fig. 9. Illustration of limit equilibrium analyses of the most critical sliding path ABCDFGHZ in the foundation of the no. 3 powerhouse-dam section of the Three-Gorges Dam.

analysis, the calculation of the factors of safety ðKÞ (1) Calculation of the total vertical load SG acting on along this speciﬁc sliding path can be determined the foundation surface and the self-weight ðGzÞ of through the following steps: the powerhouse. ARTICLE IN PRESS 640 J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652

represents the total vertical load acting on the foundation surface; SM is the total moment along the foundation surface; and B denotes the width of the dam section. 2 (4) For the analyses, dividing the rock mass into six rigid blocks with vertical interfaces (see Fig. 9) according to the geometryof the assumed sliding path. Dam (5) Calculation of the vertical loads Wi acting on the above six rigid blocks according to the vertical stress distributions along the foundation, as derived at Step 3. 1 x Q (6) According to limit equilibrium theory, if the shear ΣM forces along the ﬁctitious interfaces between the ΣG rigid blocks are not considered, the factor of safety y ðK Þ against sliding of each rigid block maybe (a) i calculated byusing the following equations:

f1ðW1 À U1Þþc1l1 þ f2ðW2 À U2Þþc2l2 K1 ¼ ; ð3Þ H1 38.3m f ðW cos a À H sin a þ R sin a À U Þþc l B K ¼ 3 3 1 2 1 1 1 3 3 3; 2 W sin a þ H cos a À R cos a (b) 3 1 2 1 1 1

f3ðW4cos a2 À H3sin a2 þ R2sin a2 À R1sin a2 À U4Þþc3l4 K3 ¼ ; W4sin a2 þ H3cos a2 À R2cos a2 þ R1cos a2 B ′ ð5Þ n ′′ n f3ðW5cos a3 þ R3sin a3 À R2sin a3 À U5Þþc3l5 K4 ¼ ; W5sin a3 þ R2cos a3 À R3cos a3 (c) ð6Þ

B f3½W6cos a4 À R3sin a4 þ R4cos ðb À a4ÞÀU6þc3l6 K5 ¼ ; τ′′ W6sin a4 þ R3cos a4 À R4sin ðb À a4Þ ð7Þ τ′ (d) f4ðGp À R4cos b À U7Þþc4l7 Fig. 10. Illustration of calculation of the vertical stress distribution K6 ¼ ; ð8Þ along the foundation surface for limit equilibrium analyses: (a) cross R4sin b À P2 section of dam section, (b) plan of foundation surface, (c) distribution of normal stresses along the foundation line, and (d) distribution of where W1; W2; W3; W4; W5; W6 are the vertical loads, shear stresses along the foundation line. and U1; U2; U3; U4; U5; U6; U7 are the uplift forces acting on the no. 1–6 rigid blocks, respectively; Gz represents (2) Calculation of the total horizontal loads acting on the weight of the powerhouse. P1 is the total horizontal the upstream surface (P1) and downstream surface load acting on the upstream surface of the dam ðP2Þ of the dam structure, respectively. structure; P2 is the total horizontal load acting on the (3) Calculation of the vertical stress distribution along downstream surface of the dam structure; H1; H2; H3 are the foundation byusing the Eqs. (1) and (2). The the horizontal forces distributed on the no. 1–no. 3 rigid dam is eccentricallycompressed and this distribu- blocks, respectively; a1; a2; a3; a4 denote the dip angles of tion could be assumed as linear [6] (see Fig. 10). the assumed sliding surface of the rigid blocks; SG 6SM R1; R2; R3; R4 are the horizontal resistant forces between s0 ¼ À ; ð1Þ n B B2 two adjacent rigid blocks; b is the intersection angle between the vertical direction and R4; f1; f2; f3; f4 are the 00 SG 6SM friction coefﬁcients of the assumed sliding surfaces of sn ¼ þ ; ð2Þ B B2 the rigid blocks, which is derived as the weighted means 0 00 where sn and sn represent the vertical stresses at the of those of both the joints and rock bridges. c1; c2; c3; c4 upstream starting point and downstream ending are the cohesions of the assumed sliding surfaces of the point of the foundation surface, respectively; SG rigid blocks, which are also derived as the weighted ARTICLE IN PRESS J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652 641

Table 2 Safetyfactors of the typicalsliding paths of no. 3 left powerhouse-dam section of the Three-Gorges Dam calculated bylimit equilibrium methods

Sliding paths ABCDE PBCDE ABCDFGH ABCDFGHZ* KNO KLME IH JH ICDE Safetyfactors 3.21 3.37 3.08 4.17 * 3.85 3.99 3.82 5.15 4.99

Note: Asterisk denotes that the safetyfactor was computed when the powerhouse effects were considered; If there is no asterisk, the safetyfactor was computed when the powerhouse effects were not considered. means of those of both the joints and rock bridges. assumed sliding paths were taken into account. In fact, l1; l2; l3; l4 are the lengths of the assumed sliding surfaces the uplift forces should be low due to the existence of the of the rigid blocks. rock bridges. Onlyif all the divided rigid blocks reach the limit force-equilibrium state simultaneously, can the sliding along the assumed sliding path ABCDFGH happen. In 3. Finite element analyses other words, when the sliding occurs, the factors of safetyof all the rigid blocks equal each other. This can In the ﬁnite element analyses, 2-D numerical model- be described in Eq. (9). ing was ﬁrstlyconducted and focused on the stability

K1 ¼ K2 ¼ K3 ¼ K4 ¼ K5 ¼ K6 ¼ K: ð9Þ conditions of the most critical no. 3 powerhouse-dam section. Then, 3-D numerical modeling was performed In addition, there is the following relationship to be used to studythe constraint inﬂuences on the no. 3 section for the calculation: from the adjacent no. 2 and no. 4 powerhouse-dam

P1 ¼ H1 þ H2 þ H3 ð10Þ sections. and 3.1. 2-D numerical modeling of the single no. 3 t2 H2 ¼ ðP1 À H1Þ; ð11Þ powerhouse-dam section t2 þ t3

t3 2-D numerical modeling was performed to analyze the H3 ¼ ðP1 À H1Þ; ð12Þ t2 þ t3 foundation stabilityof the no. 3 powerhouse-dam section, using an elasto-plastic FEM approach. The where t ; t are the mean shear stresses along the 2 3 behaviors of rock and concrete were simulated byuse of foundation surface distributed on the no. 2 and no. 3 linear solid isoparametric elements, and the behaviors of rigid blocks. rock bridges, joints and foundation surfaces were Finally, by substituting Eqs. (9)–(12) into Eq. (3)–(8) modeled using quadratic joint elements. The mechanical and solving these substituted equations byiteration, we parameters of the rock, joints, foundation surfaces, as could derive the factor of safety K: well as concretes used in this numerical modeling, are the same as those adopted in the physical modeling (see 2.2. Results of limit equilibrium analyses Table 1). Considering the Drucker–Prager failure criterion The factors of safetywere derived for all the potential yields a smooth failure surface and is quite convenient sliding paths of the no. 3 powerhouse-dam section by for finite element analyses [7,8], it was adopted as the using the above methodology. As shown in Table 2, plastic potential for the intact rock blocks and concrete when the powerhouse effect, (which is defined as the structure, which can be represented as resistant effect against sliding provided byintegrating pffiffiffiffiffi the powerhouse with the dam bodythrough down- F ¼ aI1 þ J2 À o ¼ 0; ð13Þ stream slope surface, see Fig. 6) was not taken into where account, the factor of safetyof sliding path ABCDFGH is 3.08, the lowest among all these assumed sliding I1 ¼ sx þ sy þ sz; ð14Þ paths. Since the foundation stabilitywould be governed bythe sliding path with the lowest factor of safety,when 1 2 2 J2 ¼ 6½ðsx À syÞ þðsy À szÞ the powerhouse effects were not taken into account, this þðs À s Þ2 þ 6ðt2 þ t2 þ t2 Þ; ð15Þ lowest factor of safetycan be defined as the factor of x z xy yz zx safetyof the dam foundation. However, the factor of f safetyof sliding path ABCDFGHZ in the foundation a ¼ ; ð16Þ 1=2 of the no. 3 dam section is 4.17 when the powerhouse ð9 þ 12f 2Þ effects are considered (see Table 2). Note that the above 3c factor of safetyderived should be considered as o ¼ ; ð17Þ 1=2 conservative because the full uplift forces along the ð9 þ 12f 2Þ ARTICLE IN PRESS 642 J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652

Fig. 11. 2-D FEM mesh for the stabilityanalysesof the no. 3 powerhouse-dam section.

Among them, sx; sy; sz are the normal stress boundaryconditions, the normal displacement con- components, respectively, in the x; y; z directions; f is straints were applied to the upstream and downstream the friction coefficient; and c is the cohesion value of the boundaries of the numerical model, while a fixed intact rock. displacement constraint is applied to the bottom The Mohr–Coulomb criterion was considered as the boundary. Note that this 2-D modeling was considered failure criterion of joint elements, which can be as plane strain conditions and the constraint effects represented as induced bythe adjacent foundations of the no. 2 and max no. 4 powerhouse-dam sections are ignored. The FEM t ¼ c À sn tan f; ð18Þ n mesh for the no. 3 section model is illustrated in Fig. 11. max where tn ; sn and f; respectively, represent the To investigate the failure mechanism and determine maximum shear stress, normal stress and friction angle the factor of safetyof this dam section, the numerical of the joints. The shear stiffness and normal stiffness of computation was carried out in the following steps. In joints are determined bythe slopes of the stress–strain the first step, the self-weights of the rock mass and curves in laboratorytests [9–12]. The shear stiffness ks concrete structure were considered. Then, the normal and normal stiffness kn of rock bridges are calculated external load combination, including the upstream using the mechanical properties of intact rocks, accord- hydrostatic load, was applied in five increment steps, ing to the following Eqs. (19) and (20) [13,14]: each being considered as 20% of the total. Afterwards, E while keeping the applied load conditions unchanged, k ¼ G ¼ ; ð19Þ s 2ð1 þ nÞ the values of shear strengths of the joints, rock bridges and foundation surfaces, including friction coefficient 3 ðf Þ; cohesion ðcÞ; shear stiffness ðksÞ as well as normal kn ¼ k þ G; ð20Þ 4 stiffness ðknÞ; were simultaneouslyand progressively where decreased according to an identical percentage. Mean- while, to examine the detailed development process of E k ¼ : ð21Þ the yielding zones and failure occurrences, these ð À Þ 3 1 2n identical percentages were defined within a range of less The symbol G is the shear modulus, E the elastic than 10% of the initial parameter values. The computa- modulus and n the Poisson’s ratio. tion continued until a divergence occurs. This diver- The computational model of the dam foundation has gence of numerical modeling maybe regarded as the a size of 160 m in depth, 95 m in length directing losing of static force-equilibrium state and thus could be upstream from the dam heel and 220 m in length taken as the failure conditions. Consequently, the factor directing downstream from the dam toe. For the of safetyof the dam is generallydefined as the ratio ARTICLE IN PRESS J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652 643

Table 3 Displacements of keypoints of no. 3 dam section under normal loading combination derived from the 2-D numerical modeling

Displacements of dam top (mm) Displacements of dam heel (mm) Displacements of dam toe (mm)

Horizontal Vertical Horizontal Vertical Horizontal Vertical

10.94 À8.32 1.76 À0.55 1.63 À5.23

Legend: Legend:

Mesh Line of FEM Mesh Line of FEM

Yield element Yield element

A K A K I I L N L N B C B C J D M J D M F O F O E E G G H H

(a) (b)

Legend: Legend:

Mesh Line of FEM Mesh Line of FEM

Yield element Yield element

A K A K I I L N L N B C B C J D M J D M F O F O E E G G H H

(c) (d) Fig. 12. Development of yielding elements due to the gradual reduction of the shear strengths of assumed sliding paths derived from 2-D finite element analyses of the no. 3 powerhouse-dam section of the Three-Gorges Dam. Distribution of yielding elements when friction coefficient (f ), cohesion (c), shear stiffness (ks) and normal stiffness (kn) along the assumed sliding paths are simultaneouslydecreased by(a) 80%, (b) 60%, (c) 40%, and (d) 26% of their initial values. between the initial values and final decreased values of foundation of the no. 3 powerhouse-dam section is the shear strengths of the potential sliding paths. stable. As shown in Table 3, the numerical results show that Fig. 12 illustrates several typical stages of the yield- under the normal loading combination and initial shear ing element development in the simulated domain strength values, the horizontal displacements at the dam following the gradual decrease of the initial shear heel and toe are both less than 2.0 mm. Besides, there are strength values during the computing process. When not anyyieldingzones occurring in the simulated the initial values of f ; c; ks; kn of the joints and domain. This reveals that for this designed normal rock bridges were simultaneouslydecreased by26%, loading combination and initial strength values, the the yielding zones were developed through the ARTICLE IN PRESS 644 J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652

Z Y

Fig. 13. 3-D FEM mesh for the stabilityanalysesof the combined no. 2, 3, 4 powerhouse-dam sections.

Table 4 Displacements of keypoints of no. 3 dam section under normal loading combination derived from the 3-D numerical modeling

Displacements of dam top (mm) Displacements of dam heel (mm) Displacements of dam toe (mm)

Horizontal Vertical Horizontal Vertical Horizontal Vertical

3.85 À7.81 1.62 À4.35 1.52 À4.65

sliding paths of ABCDFGH; and then ABCDEH: dam sections), 109.5 m in length directing upstream Manyother joint elements also behave plastically. from the dam heel and 311.5 m in length directing Subsequently, convergence cannot be reached. It was downstream from the dam toe, respectively. For the therefore concluded that starting at this moment, the boundaryconditions, the normal displacement con- static force-equilibrium along the potential sliding straints were applied to the upstream and downstream path of ABCDFGHZ was lost and dam sliding boundaries of the model, while a fixed displacement would happen. The factor of safetywas calculated constraint was applied to the bottom boundary. The left as 3.85. and right sides were left as free surfaces. Fig. 13 depicts the 3-D model mesh. 3.2. 3-D numerical modeling of associated no. 2, 3, 4 Similar to the 2-D numerical modeling, 3-D linear powerhouse-dam sections solid isoparametric elements were used to simulate the rock and dam bodywhile quadratic solid joint elements The factor of safetyof the no. 3 powerhouse-dam were adopted to model the joints and rock bridges. The section derived from the above 2-D numerical modeling Drucker–Prager and Mohr–Coulomb criteria were maybe regarded as over-conservative since the con- considered as the failure criteria for the above two types straint effects on the stabilityof the no. 3 section byits of the elements. The input mechanical parameter values adjacent no. 2 and no. 4 sections were not taken into and the normal load combination were the same as account. This is the main reason and objective of the those used in the 2-D numerical modeling. In terms of 3-D FEM modeling. the computational steps, self-weights of the rock mass The domain of this 3-D FEM model included the and concrete structures were loaded first, and then the three dam sections, powerhouse basements and dam normal external loads were applied. foundations, with a model size of 205 m in depth, For the 3-D calculations, the overall factor of safety 114.9 m in width (the same as the total width of the three K along a specific sliding path was calculated bythe ARTICLE IN PRESS J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652 645

Fig. 14. Distributions of horizontal displacements of the dam center-line section of the no. 3 powerhouse-dam section derived from 3-D FEM. following equation: As shown in Table 4, the 3-D numerical results show that under the normal loading combination, the P P f s A þ CA horizontal displacements at the dam heel and toe are K ¼ iPi i; ð22Þ t A both less than 2.0 mm. Figs. 14 and 15 illustrate the i i distributions of horizontal and vertical displacements of the dam, in a center-line section of the no. 3 power- where si is the mean value of the normal stress house-dam section; Figs. 16 and 17 give the distributions components at the Gauss points of a specific joint (or of maximum and minimum principal stresses in the rock bridge) element; ti is the mean value of shear stress same section. components directing downstream along the joint The factors of safetyalong some critical potential surfaces at the Gauss points; Ai represents the area of sliding paths have been computed byusing Eq. (22). The the surface of a specific joint element; and subscript i comparisons between them demonstrated that the denotes the numbers of the joint/bridge element of the potential sliding paths with the lowest factors of safety potential sliding paths. are, respectively, ABCDFGH for no. 3 dam section, Eq. (22) represents the limit equilibrium state between ABCD for no. 2 dam section and ABCD for no. 4 dam the shear force and shear strength along a potential section (see the geological models as illustrated in Figs. 6– sliding path. As such, this definition of factor of safety 8). Table 5 lists these lowest factors of safetyand the reflects the margin of the shear strengths against sliding corresponding potential sliding paths. The factor of along the specific sliding path, similar to that used in the safetyof the no. 3 dam section is less than that of the 2-D numerical modeling. no. 2 and no. 4 dam sections, demonstrating that the ARTICLE IN PRESS 646 J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652

Fig. 15. Distributions of vertical displacements of the dam center-line section of the no. 3 powerhouse-dam section derived from 3-D FEM. stabilityof the no. 3 section is poorest among the dam 4.1. Displacements sections and the constraint effects induced bythe adjacent no. 2 and no. 4 dam sections are important to the Table 6 compares the experimental and FEM results foundation stabilityof the no. 3 dam section. The lowest of the horizontal and vertical displacements at dam top, value of the factor of safetyfor no. 3 section is 4.37. heel, and toe of the no. 3 powerhouse-dam section. For both 2-D and 3-D cases, the experimental results of displacements agree fairlywell with the FEM results under the same normal loading conditions. Both the 4. Comparisons and discussions experimental and FEM results reveal that horizontal displacements downstream, in general, agree well with In total, two physical model tests (the 2-D and 3-D magnitude values of 2.0 mm or smaller at the dam heel physical model tests) and three computational modeling and toe. There are two major reasons for this. One is (the limit equilibrium analyses, 2-D and 3-D ﬁnite because the geomechanical model, initial and loading element analyses) have been carried out. Altogether, conditions, and boundaryconditions used for the 2-D/3- these studies mainlyfocused on the foundation stability D physical model test and FEM are identical; another of the most critical no. 3 powerhouse-dam section and a lies in the fact that the system behaves elastically in common geomechanical model was used. The results general and the plastic deformations of intact rocks and from these different approaches are compared and concrete have not happened under the normal loading discussed below. combination. ARTICLE IN PRESS J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652 647

Fig. 16. Distributions of maximum principal stresses of the dam center-line section of the no. 3 powerhouse-dam section derived from 3-D FEM.

On the other hand, the horizontal and vertical and concrete structure of the no. 3 powerhouse- displacements at the dam top, heel, and toe obtained dam section behave elastically. Both physical and from the 2-D modeling (for both physical model tests numerical models indicated that the path ABCDFGH and FEM modeling) are larger as compared to the 3-D is the most critical potential sliding path and results. It is mainlybecause of the more conservative plays a dominant role in the foundation stability for natures of the 2-D models compared to the 3-D ones, the no. 3 section. In both the 2-D and 3-D physical with the main differences being in the boundary model tests, this dominant sliding path extends conditions. In the 2-D cases, the left and right sides of into aA and bA paths (see Part I of this paper [1])of the foundation of the no. 3 powerhouse-dam section the dam body, and thus suggest that the strengths were modeled as the free surfaces; while, in the 3-D of the concrete structure playa positive role in the cases, the constraint effects applied bythe adjacent no. 2 stabilityagainst sliding (refer to Fig. 18). However, and no. 4 dam sections were included, thus resulting in in the numerical analyses, this critical sliding path the smaller deformations and improved stabilityof the connects with the foundation surfaces, which was no. 3 dam section. assumed as a part of this potential sliding path in the computations. In the physical model test, the slide paths 4.2. Failure mechanisms and failure of the dam were obtained without such assumptions during the process of loading when the As aforementioned, both the numerical simu- similarityrequirements were met in advance. This can be lations and physical model tests demonstrate that regarded as the limitations of the numerical modeling under normal loading conditions, the foundation approaches. ARTICLE IN PRESS 648 J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652

Fig. 17. Distributions of minimum principal stresses of the dam center-line section of the no. 3 powerhouse-dam section derived from 3-D FEM.

Table 5 3-D experimental and FEM results, which are 4.0 and Safetyfactors of the most critical sliding paths of no. 2, 3, 4 left 4.37, respectively. In addition, the factor of safety powerhouse-dam section of the Three-Gorges Dam calculated by3-D computed bythe limit equilibrium method is 3.08/4.17 FEM whether the effects of powerhouse are taken into Dam sections Most critical sliding paths Safetyfactors account or not. No. 2 dam section ABCD 5.88 There maybe four major reasons for the discrepan- No. 3 dam section ABCDFGH 4.37 cies. No. 4 dam section ABCD 5.53 * The difference between the factors of safetyfrom the 2-D and 3-D physical/numerical modeling are mainly resulting from the different boundaryconditions, which are the assumed free surface conditions for the 4.3. Factor of safety left and right sides of the no. 3 dam section model in the 2-D models, and were then constrained bythe Table 7 shows the agreements and discrepancies in the adjacent no. 2 and 4 sections for the 3-D models. factors of safetyof the no. 3 powerhouse-dam section * The second one lies in the different deﬁnitions of the derived from the diverse approaches. When the power- factor of safety. In the physical model tests, it is house effects are taken into account, the factor of safety deﬁned as the ratio between the maximum external from the 2-D physical model test and FEM are 3.5 and load that is able to induce the sliding instabilityof the 3.85, respectively, both being lower as compared to the dam and the actual load designed to act on the ARTICLE IN PRESS J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652 649

Table 6 Comparisons among the FEM and experimental results of the horizontal and vertical displacements at dam top, heel and toe of no. 3 powerhouse- dam section

Type Displacements of dam top (mm) Displacements of dam heel (mm) Displacements of dam toe (mm)

Horizontal Vertical Horizontal Vertical Horizontal Vertical

2-D FEM 10.94 À8.32 1.76 À0.55 1.63 À5.23 3-D FEM 3.85 À7.81 1.62 À4.35 1.52 À4.65 2-D physical Modeling 14.85 3.30 2.10 1.13 2.10 À0.37 3-D physical Modeling 11.74 2.1 1.85 0.95 1.20 À1.32

Fig. 18. Sketch of fracture traces on the model surface of the no. 3 powerhouse-dam section observed from the 2-D physical model test.

Table 7 Comparisons among the safetyfactors of no. 3 powerhouse-dam section derived from different analysismethods

2-D FEM 3-D FEM Limit equilibrium method 2-D physical modeling 3-D physical modeling

Safetyfactors 3.85 4.37 4.17 3.5 4.0

structure. However, in the numerical modeling, it is * Finally, the size effects probably also contributed to defined as the factor bywhich the shear strength these differences of the factors of safety. parameters along the potential sliding planes maybe reduced in order to bring the dam foundation into a state of limiting equilibrium. 5. Overall assessments * The third reason comes from the different simula- tions of the constitutive models and material proper- The design and construction of the Three-Gorges ties. Theywere simulated bysimilarityof model Dam is unique. Its stabilityassessment has to be materials in the physical model test, and by consti- considered in terms of the particular circumstances of tutive equations and associated material properties in the site geological setting, properties and structures of the FEM modeling, which were not taken into rock mass, designed loads, and end uses for which it is account in the limit equilibrium method. intended. The stabilityassessment depends upon the ARTICLE IN PRESS 650 J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652 level of confidence that the designers/researchers have in model test and 4.0 in the 3-D physical model test. The the shear strength parameters and the locations of the factor of safetyin the numerical modeling is defined potential failure surface. As presented in Part I of this as the factor bywhich the shear strength parameters combined paper, the geological settings and structures maybe reduced in order to bring the dam foundation of the rock mass were adequatelyunderstood through into a state of limiting equilibrium, which indicates numerous detailed geological investigations, and the the another important failure mode due to the properties of the rocks and discontinuities were identi- gradual degradation and the uncertainties of the fied bymanyof the field and laboratorytests. Following shear strengths of discontinuities. The factor of safety this, a geomechanical model was established byrational computed bynumerical modeling are 3.85 in 2-D conceptualization. All these investigations provided a FEM, 4.37 in 3-D FEM and 4.17 in Limit Equili- reliable basis for the stabilityanalyses. Despite some brium Method. In addition, the factors of safety differences in the results from the various approaches, derived from both the 2-D physical modeling and both physical modeling and numerical modeling sup- FEM are more conservative for the design than those ported each other and produced complementaryresults, from the corresponding 3-D analyses because the suggesting a stable dam foundation. The main assess- mutual constraint effects between the adjacent dam ment conclusions can therefore be made as follows. section foundations were not taken into account in the 2-D analyses. * Under normal loading conditions, the concrete structure and foundation of the no. 3 dam section As Hoek pointed out [15], there are no simple behave elasticallyand the deformations along the universal rules for design acceptabilitynor are there foundation surfaces are acceptable. Thus, the con- standard factors of safety, which can be used to cerned dam foundation is stable under the designed guarantee that a rock structure will be safe and that it normal loading conditions. will perform adequately. A safe and economical solution * The failure mechanisms are complicated. All the should be compatible with all the constraints that apply results suggested that in the foundation of the no. 3 to the dam and based upon engineering judgment powerhouse-dam section, ABCDFGH is the domi- guided bypractical and theoretical studies such as nant sliding path, which, on the other hand, is stabilityor deformation analyses [15]. Based on resisted byboth the upper concrete structure and the numerous research results, the Chinese Design Criterion downstream powerhouse basement. Considering of Concrete GravityDam (in which the factor of safety the limitation in all the numerical models that the against sliding of concrete gravitydam must be more foundation surfaces were pre-assumed as a part of the than 3.0), the experience from design and construction most critical potential sliding surface, thus leading to of manysimilar projects in China, as well as the the AK section as the expanded part of ABCDFGH particular importance of the Three-Gorges Project, the sliding path (Fig. 12(d)), the expanded paths aA and Technical Committee of the Three-Gorges Project bA in the dam concrete revealed in the physical decided that the factor of safetyagainst sliding of the modeling should be more rational (Fig. 18). In Three-Gorges Dam must be higher than 3.0 [16–19]. addition, because the shear strengths along the Therefore, it can be concluded that the stabilityof the interface between the powerhouse basement and the no. 3 powerhouse-dam section can meet the safety foundation are lower than that of the basement requirements because all the factors of safetyderived concrete, as revealed byboth the numerical and from the multiple approaches are more than 3.0. In physical modeling, the dominant sliding path addition, since the foundation of the no. 3 powerhouse- ABCDFGH should be more easilyextended down- dam section is the most critical in terms of stability stream along the interface HZ. Therefore, the most against sliding failure (as demonstrated in Part I of this critical sliding paths of the no. 3 section should be paper), a conclusion can be drawn that the global aABCDFGHZ and bABCDFGHZ: stabilityof the Three-Gorges Dam can also satisfythe * In this study, two definitions of the factor of safety safetyrequirements. were, respectively, used in physical and numerical modeling. The factor of safetyin the physical modeling is defined as the ratio between the 6. Treatment and reinforcement measures maximum external load inducing the start of sliding instabilityof the dam foundation and the upstream Considering the importance of the Three-Gorges hydrostatic load applied to the dam, which reflects Project and the potential failure mechanisms revealed the failure mode due to external overload and the bythe physicaland numerical modeling, the following uncertainties of the upstream hydrostatic load. The treatment and reinforcement measures for the founda- experimental results show that the failure load is 3.5 tion of the no. 3 powerhouse-dam section were times that of the designed load in the 2-D physical suggested and implemented. Some adjustments of these ARTICLE IN PRESS

J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652 651 ∆ 90.0m

Shotcrete T=50cm Downstream slope Bolting System L=8 m

Φ 25mm @2.5 ×2.5m

∆ 81.7m Random Bolts L=5 m ∆ 75.0m

Random Bolts L=5 m ∆ 65.0m

Legend: Bolting System L=8 m T: Thickness of Shotcrete Φ [email protected]× 3.0m L: Length of Bolt Shotcrete T=7cm

Φ: Diameter of Bolt ∆ @ : Spacing of Bolt Bolting System L=5 m 51.0m Φ 25mm@ 3.0×3.0m

Fig. 19. Design scheme recommended for reinforcing the foundation of some critical dam sections of the Three-Gorges Dam.

measures mayalso be appropriate for other dam (6) During the excavation of the foundation, it is sections. important to minimize the damage in the rock mass caused byblasting. (1) With the consideration that the mutual constraint effect between the adjacent foundations plays a helpful role in enhancing the global sliding-resistant 7. Conclusions capacities of these dam sections, it is suggested to set up keyseating and grout between the This paper has benefited from a long-term research transverse joints of the concrete of the no. 1–5 effort for the Three-Gorges Dam, which has involved dam sections. the geomechanical conceptualization, 2-D and 3-D finite (2) Tightlyintegrating the foundation with the power- element analyses, limit equilibrium analyses, 2-D and 3- house basement would make the basement more D physical model tests, the definition and computation resistance to the sliding for most of the potential of factors of safety, the comprehensive assessment of the sliding paths. Thus, joint grouting for the interfaces stability, and, finally, the treatment and reinforcement between the excavated downstream slope and the schemes. Some conclusions of overall importance upstream surface of the powerhouse basement will reached during this studyare: enhance the stabilityof this dam section. (3) To ensure the local stabilityof the downstream (1) Comprehensive assessment in terms of the factor of slope, bolting support should be adopted along its safety, deformations and failure mechanisms of the downstream free surface. Fig. 19 shows the no. 3 powerhouse-dam section demonstrate that the recommended reinforcement scheme for the down- stabilityagainst sliding of this critical dam section stream slope with free surface. can meet the safetyrequirements. Furthermore, (4) Although the grouting curtain can effectively since both the geological and geomechanical site decrease the seepage uplift, the remaining seepage characterization and numerical and physical mod- uplift, which was not fullytaken into account in this eling have revealed that the foundation of the no. 3 study, would play a disadvantageous influence on section is the most critical in terms of stability the sliding-resistant capacityof the rock mass. against sliding failure, it can be concluded that the Thus, a closed drain and pumping system should be global stabilityof the Three-Gorges Dam founda- constructed in the foundation. tion also can be ensured. (5) It is necessaryto implement grouting treatment for (2) The integration of multiple methods is shown to be the shallow gentle dipping joints. a more effective approach for the stabilityanalyses ARTICLE IN PRESS 652 J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 633–652

of the Three-Gorges Dam foundation since no [3] Jing L. Formulation of discontinuous deformation analysis satisfactoryindividual method can be used alone to (DDA)—an implicit discrete element model for block systems. solve the problem. Mutual validation and comple- Eng Geol 1998;49:371–81. [4] Duncan JM. State of the art: limit equilibrium and finite mentaryresults derived from various methods make element analysis of slopes. J Geotech Eng ASCE, the stabilityassessment more reliable and rational. 1996;122(7):577–96. (3) Considering the particular importance of the Three- [5] Chen ZY, Wang XG, C, Haberfield C, Yin JH, Wang YJ. Gorges Project, some treatment and reinforcement A three-dimensional slope stabilityanalysismethod using the measures for the foundations of some typical dam upper bound theorem—Part I: theoryand methods. Int J Rock Mech Min Sci Geomech Abstr 2001;38(3):369–78. sections are necessary. Moreover, according to the [6] Shu YF. Mechanics of materials. Beijing: Chinese Higher failure mechanisms revealed, the treatments and Education Publishers, 1984. p. 289–305 [in Chinese]. reinforcements recommended in this paper were [7] Dhawan KR, Singh DN, Gupta ID. 2D and 3D finite element implemented and should effectivelyincrease the analysis of underground openings in an inhomogeneous rock foundation stabilityof these dam sections. mass. Int J Rock Mech Min Sci 2002;39(2):217–27. [8] Kovari K. The elasto-plastic analysis in the design practice of (4) The systematic methodologies presented in this paper underground openings. London: Wiley; 1977. could be expanded into stabilitystudies for other dam [9] Liu XZ. Experimental studyon stiffness characteristics of rock engineering with similar geomechanical conditions. mass structural face in TGP. J Yangtze River Sci Res Inst, Wuhan, China 1998;15(2):25–7 [in Chinese]. [10] Dong XC. Experimental studyon rock mechanical properties at Sandouping dam site in the Three-Gorges Project. J Yangtze River Sci Res Inst, Wuhan, China, 1992;(Suppl.)A06:13–21 [in Acknowledgements Chinese]. [11] Tian Y. Studyon mechanical properties of rock mass of the The paper has been financiallysupported bythe Three- Three-Gorges Project. J Yangtze River Sci Res Inst, Wuhan, Gorges Development Corporation, Changjiang Water China, 1992;(Suppl.)A06:72–6 [in Chinese] [12] Ding XL, Liu J, Liu XZ. Experimental studyon creep behaviors Resources Commission, the Special Funds for Major of hard structural plane in TGP’s permanent lock regions. J State Basic Research Project under Grant no. Yangtze River Sci Res Inst, Wuhan, China 2000;17(4):30–4 [in 2002CB412708, the Pilot Project of Knowledge Innovation Chinese]. Program of Chinese Academyof Sciences under Grant no. [13] Zhou WY. Senior rock mechanics. Beijing: Chinese Water KGCX2-SW-302-02, and National Natural Science Foun- Conservancyand Hydro-electricPower Publishers, 1990. p. 53– 6 [in Chinese]. dation of China under Grant no. 59939190. The authors [14] Sun GR, Yin YQ, Qian ZG. Studyon bearing capacityof wish to thank all colleagues in the Department of concrete gravitydam. J HydraulEng 2001;4:15–20 [in Chinese]. Structures & Materials, and Rock Foundation Depart- [15] Hoek E. Practical rock engineering, Course Notes on websi- ment, Yangtze River Scientific Research Institute, for their te:http://www.rocscience/hoek/PracticalRockEngineering.asp, valuable contributions to various parts of the works 2000 [Chapter 2, p. 18–25]. [16] Ministryof Water Conservancyand Hydro-electricPower. described in the paper, particularlyProfessors Gong DL5108–1999 Chinese design criterion of concrete gravitydam. Zhaoxiong Chen Jin and Sheng Qian. The authors also Beijing: Chinese Water Conservancyand Hydro-electricPower grateful to Prof. Lanru Jing, Royal Institute of Technol- Publishers [in Chinese]. ogy, Stockholm, Sweden, and Prof. J.A. Hudson for [17] Yue DY, Jiang WQ, Liu JS. 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