- Email: [email protected]
Study on Coupling Influences of Concrete Dam Foundation Seepage, Stress, and Creep on Structure Behaviors of Dam Body
753 STUDY ON COUPLING INFLUENCES OF CONCRETE DAM FOUNDATION SEEPAGE, STRESS, AND CREEP ON STRUCTURE BEHAVIORS OF DAM BODY Guo Halqing, Xu Weiya*, Wu Zhongru HoHai University, 210098, Nanjing, P. R. China Abstract: Accidents of some arch dams show that dam failures are mainly caused by crack or failure of their foundation rocks, which are directly related to water seepage in the rock. This kind of fluid-rock interaction has an important influence on deformation and stress characters of the dam-rock system. In this paper, the stress and flow fields of dams and their foundation rocks are studied as a coupled system, using visco-elastic constitutive models and finite element solution method. The developed models and FEM technique were applied for analysing the continuous displacement of the 13* dam section of the Longyangxia Dam, and the calculated results agree well with the measured ones.
1. INTRODUCTION The Longyangxia hydropower project is one of the major hydropower generation and water resources utilization systems built in Yellow River China. Continuous displacement towards the left bank at the 13'*' dam section has been observed and monitored. This has become a major concern to both the dam safety administration and engineers. The paper represents the results of a research effort devoted to investigate its causes. Recognizing the importance of the coupled hydro-mechanical effects on the performance of civil engineering structures involving fractured rocks, the stress-flow coupling mechanism of the dam-foundation system at Longyangxia site was simulated using a three-dimensional Finite Element code, supported by two visco-elastic constitutive models to represent the time-dependent material behaviour of the dam concrete and the foundation rock. The calculated results were concord with the measured ones and helped to interpret the causes of this continuous displacement at the 13*^ dam section of the Longyangxia hydropower project, towards the left bank. The theoretical foundations of the theory and the FEM formulations presented in this paper are based on works in Zienkiewicz and Cormeau (1974), Oda (1986), Ohnishi et al. (1985), Ohnishi and Kobayashi (1993), Shen et al. (2000) and Wu et al. (2001).
2. THE MATHEMATICAL THEORY 2.1 Basic equations of stress and seepage coupiing The main hypothesis of dam concrete and its foundation rock are as follows: a) the dam and its rock foundation are isotropic continuum media in
different areas, b) The seepage follows the Darcy's law. c) The grain skeletons' deformations of dam concrete and foundation rock are ignored, d) The deformations of dam concrete and its foundation rock are mainly caused by deformations of void spaces and cracks between grain skeletons mentioned above. If the dam and its base rock are affected only by gravity and seepage force, which are considered as body forces, and denoting that the tensile stress is positive, the balance differential equations expressed by displacements and general water heads can be given as
GV^S^-^-^-r..^^X„=0 ' \-2v dx ** dx (1)
'
l-2v dy ''dy ' G de^___^ a^ + Z, + r - r =0 GV'S^ + l-2v" az ^''dz where, G is the shear module, E is the Young's modulus, V is the Poisson's ratio, r^ is the density of water, r^ is the saturated density of the concrete or foundation rock, h is the water head. In addition, V^ is the Laplace operator, e^ is the volume strain, S^{i = X, y, z) are displacement components and XQ , KQ , ZQ are equivalent body force components caused by initial strain {SQ} . According to the law of mass conservation, the continuity equation of the water are derived as given in eq. (2). For simplify, the equation ie expressed along the main seepage directions denoted as coordinate axes x, y , z .
754 A.(k—^ —(k—) —(k—\-^ dx "^ dx dy ^ dy dz ^ dz
dt
/?— (2) ** dt
where [k] is the permeability tensor, whose main components are k^ , ky , and k, \ n is the void
^.=^-zk^iI/.-P[f('-'o)W' 277, J" m
(3)
exp[—Ht-to)]
ratio, and P is the compression parameter of the water. The basic equations for the coupHng analysis of stress and seepage fields are composed of equations (1) and (2), whose boundary conditions include (a) displacement boundary condition {J) = {^o) ' (^) stress boundary condition cr^^m, = a^ (where ^,/ = 1,2,3), (c) water head boundary condition h = h^ , (d) seepage boundary condition
where E^, rj^, rj^ are the tension (compress, or shear) module and viscous parameters, respectively, and 5, is the deviatoric stress tensor,
For a special case, when 5^ = 5,^0, equation (3) is changed to
-k„ — = ^0' ^hich should be specified according 2£.
to the site conditions.
2.2. The constitutive modeis For studying the viscous deformation caused by the creep of dam and its base rock under timedependent loading, different visco-elastic constitutive models are developed to identify the most suitable models and parameters for more accurate simulation of the time-dependent deformation of the dam-foundation system.
+ Z^y.o*exp -^it-t,)
2E,
(4)
l-exp[—-(t-t^)]
If the viscous strain of Kelvin model is le^ ^ \ at f^ ^ / = /Q + A/ , and the stress remains constant during the Ar time increment, the viscous strain increment of the generalized Kelvin model during At can be derived from equation (4) as
2.2.1 The visco-elastic constitutive model of the dam concrete The creep of dam concrete includes two parts: instantaneous elastic deformation and viscous deformation. A generalized Kelvin model consisting of two standard Kelvin model in series is used to describe the time-dependent deformation of the dam concrete as shown in Figure 1.
Figure 1. Sketch map of the generalized Kelvin model The model's strain is the sum of the initial elastic strain and the strains of the two Kelvin models, so that the partial strain e„ of the generalized Kelvin model is expressed as
{Af,.),=i
l-exp(-
[C]M-KtK
(5)
where [C] is the Poisson's ratio matrix. 2.2.2 The visco-elastic constitutive model of the foundation rock The time-dependent deformation of the foundation rock, caused by loadings, is described by a Burgers model, which is composed of a Kelvin model and Maxwell model in series, as showed in Figure 2. The partial strain expression of Burgers model is
^-=r-^ir/'>^'"'ff"^""f *'• exp^
^^M
(^-^0)
^ij.OM '
5,, J/
755 Specially, when s-j = S-JQ is a constant value, and
[KY{^8]-[K'\{h]„,,=[^F,]-[K']{h]^ [K'Y{^5]^{[S]^e^t[k])[h)^,,=lsI{^{^F,]^{F,]^)
S-j = 0 , the equation (6) becomes
+ ([5]-(l-^)A/[^]){/iL (9)
Figure 2. Sketch map of the Burgers model
(7)
The visco-elastic strain of the Maxwell component will change into {^^ M) ^^ ^'"^^ ^o • I^
where 0 is the integration parameter, [/^l is the general stiffness matrix, [K'] is the general coupling matrix, [S] is the general compression matrix, [^j is the general seepage matrix, [F^] is the equivalent nodal load vector by the initial strain, {F,} is the equivalent nodal load vector, [F^] is the equivalent nodal discharge vector, and {/z}^^, is the general water head vector at time /^^,, respectively. If displacement increment [is.S) and super-static water pressure {A^} are taken as the unknown quantities, and the full Hermit differences are adopted with ^ = 1 , the equation (9) can be rewritten as
r = ^0 + Ar, and the stress remain the same during the increment of Ar , the viscoelastic strain increment of Burgers model can be derived from equation (7) as given by
{Afv),=
l-exp(-^A/) .AT
;
(8)
+ —[C]{C7}
where E^, rj^, t]^ is the stretch (compression or shear) modulus and viscous parameters and [C] is the Poisson's ratio matrix, respectively.
2.3 The finite element method For the basic equations of coupled stress-flow analysis mentioned above, it is very difficult to solve them in closed-form. The transposition method of progression and integration can only be applied for problems of boundary value problems of simple geometry and boundary conditions. Therefore the finite element method (FEM) is used to solve the coupled partial differential equations in this paper. 2.3.1 The basics of the FEM formulation The FEM solution scheme for the coupled equations of displacement and seepage fields is
[K^{^S)-[K']{^p)
= {R)-{R^
[K'Y{^S]-i{S]^^t[K]){^p]
(10)
= o_
where {R^ ] refers to the part of loads balanced by the stresses related to the displacement happened before the time /^ , and are calculated by the following equation {/?,} = Z[/^J,{AcJ},
(11)
where / is the number of calculating periods of time before the time t^. 2.3.2 The calculation process According to the incremental initial-strain method used in equation (9), the initial stress field and seepage field of dam-foundation system need to be created by iterations starting from the beginning of the time-marching process. The element stress remains unchanged during the subsequent time step Ar , so that the seepage coefficients, which are decided by the state of stress at the beginning of the time step, remain the same value. They change step by step with loads increasing gradually at each time step. The water heads at the end of every time step are the initial ones of the next time step.
756 In terms of the magnitude of the time step Ar, two factors need to be considered. One is to guarantee the numerical stability of the viscous initial strain iteration by choosing a small value of At and the other is to ensure that the coefficients matrix not to be morbid by making the Ar value adequately large. It be estimated by the following formula Ar =
(12)
k
4
-(K + ^G) where L is the linear dimension of the elements, k is the seepage coefficient, and /i , G are bulk modulus and shear modulus, respectively. The appropriate time step Ar should be determined by trial and error. Generally speaking, the state of stress field varies mostly at the beginning of a loading step, so that the time step should be small at the beginning of the loading, and gradually increases in order to speed up the calculations.
2.4 The mechanisms of the damfoundation interaction for creep analysis A dam and its rock foundation are the two integral parts of a system. Different types of dams have different interaction behaviours between the two parts. Project practices show that for concrete gravity dams, the rock foundations have significant influence on the stress behaviours of the concrete dams up to one-third of the dam height from the base. The effect is small in other parts. For arch dams, on the other hand, the rock foundations have significant influences on the stress and behaviour of the whole dam body. For both gravity and arch dams, their rock foundations affect significantly the deformations of the concrete dams. Therefore, for stress analysis of the dam-foundation system of gravity dams, the series combination creep model is applied for the first one-third of 1/3 dam height from the base and the parallel combination creep model is applied to the rest. In terms of strain, this relationship can be expressed as
l = i- -1
for the rest, where e^ and e^ are the strains of dam concrete and foundation rock, respectively. The e^ can be obtained from equation (3) or (4) and s^ from equation (6) or (7).
3. AN APPLICATION EXAMPLE The above theory and FEM code were applied to analyse the continuous displacements of the 13 dam section of Longyangxia dam from July 1989, as an example. The main part of Longyangxia dam is a gravity arch dam of 396 meters in length, 29.2/80 meters in width (at the top/base), 2610 meters for DCL and 178 meters in height.
3.1 The FEM model and conditions 3.1.1 The FEM model and material parameters The dimension of the FEM model is 540 m in both length and width and 360 m in height. When dividing the FEM mesh, the geological structures in the foundation rock and their engineering treatment measures were taken into account. While laying out the element nodes, the locations of the in situ measuring points of displacement, temperature and stresses are considered. The FEM mesh, which is consisted of 21,189 eight-node-hexahedron elements with 24,873 nodes, is illustrated in Figure 3. The material parameters are listed in Table 1.
(13)
for the first one-third part of the dam body and e = £, + e,
(14)
Figure 3. The FEM mesh of the Longyangxia dam model
757 Table 1. Parameters of Longyangxia dam and foundation materials ^\^CIassification Location
^\^^
Density
Viscous constants
Elastic constants
kg/m^ Modulus
Seepage parameters
(GPa)
Poisson 's ratio ^
2400
20
0.18
250 2.3x10^ 66.7^1.15x10^ Dam
2580 m ~ 2560 m 2650
8
0.25
50 4.0x10' 15.0 8.5x10^
2560 m - 2540 m 2700
12
0.23
200p.6xloi20.0 5.4x10'
2540 m ~ 2500 m 2750
16
0.22
200p.8xlO' 40.0 5.0x10'
Dam body (concrete)
GPa GPa S GPa GPa S
Rock
Below 2400 m
2755
22
0.22
300p.OxlO^ 51.0 1.8x10'
foundation
G4*
2600
3
0.25
30 5.4x10' 10.0 9.5x10'
F,8
2600
4.5
0.25
40 4.0x10'' 15.0 9.0x10'
F71, F73, F32, F67
2600
3.2
0.25
30 5.4x10^ lO.q 9.5x10^
A2+F120
2600
5.4
0.25
40 4.0x10' 15.0 9.0x10'
cm/s l.OxlO'
Rock
1.0x10"'
Crack
l.OxlO"'*
Curtain 1.0x10"^
Drainage l.OxlO"
*G4, F18, F71, F73, F32, F67, and A2+Fi2oare geological structures and cracks in Longyangxia dam foundation.
3.1.2 Water level variations as timedependent loading cases in the calculation
3.2 Calculation and analyses of tangential displacements of typical days
From April 16, 1990 to May 1992, the water level of Longyangxia reservoir dropped from 2575.04 m to 2533.15 m. From May 1992 to December 1994, the water level rose from 2533.15 m to 2577.58 m. From January 1995 to July 1998, the water level dropped from 2577.58 m to 2533.54 m again. After July 1998, the water level rose from 2533.54 m to 2581.08 m. These water level variations were used for deriving the water head loading conditions.
The FEM modelling was applied for the simulating the dam behaviour with coupled stressflow effects under different loading cases according to the water level variations. The tangential displacements of Longyangxia dam's typical dam section on April 16, 1989, April 16, 1990, and April 16, 1996, and December 31, 1999 were calculated and compared with measured data. The results are listed in Table 2 and illustrated in Figure 4.
• 1989-4-16Sur.
'1990-4-16Sur.
-1996-4-leSur.
-1989-4-16C&I.
' 1990-4-16C&I.
" 1996-4-16C&I.
Figure 4. Survey values and calculation values ofgeneral displacement of the 1J dam section
758
^\^
Table 2. Measured and calculated values of time-dependent displacements (mm) 1996-4-16 1999-12-31 1989-4-16 1990-4-16 Time
Points ^ \ ^
Measured|Calculated|Measured|Calculated|Measured| Calculated iMeasuredl Calculated
of measurments'v^ 2463.3 m 2497 m 2530 m 2560 m 2585 m 2600 m
Values
Values*
values
Values
Values
Values
Values
Values
-~ -----
0.12
-0.39
-0.28
-1.15
-1.01
-1.79
-1.75
1.05
-0.33
-0.44
-2.70
-2.85
-3.95
-3.82
1.24
--
-0.79
-4.81
-4.44
-5.86
-5.97
-3.69
-5.27
-5.48
0.89
-0.68
-0.61
-3.88
0.57
-0.49
-0.36
-1.93
-1.84
-2.17
-2.26
0.06
-0.55
-0.31
-4.24
-4.14
-5.60
-5.49
It can be seen from Table 2 and Fig. 4 that: (1) the calculated values are close to the measured ones, and their respective trends of change are very similar as well. (2) The calculated tangential displacements above the 2500m level on April 16, 1996 are as follows: -3.04mm, -6.91mm, -6.33mm, -5.77mm, and -4.68mm, respectively. These data show that the 13* dam section moved towards the left bank. This was caused by the continuing high water level, about 100m ~ 110m higher than adjacent uplift pressure, and the creep displacements of dam body and rock foundation.
4. CONCLUSIONS (1) The time-dependent constitutive models of the dam concrete and foundation rocks for the Longyangxia hydropower complex were established. (2) The three-dimensional FEM model with the established constitutive models and realistic time-variations of reservoir water level as loading cases were successfully applied to simulate the coupled stress-flow behaviour of the damfoundation system. (3) The calculated results, which agreed well with the measured ones, clearly indicated that the main reason for the continuous displacement towards the left bank after July 1989, was caused by the influence of the seepage-stress coupling of Longyangxia gravity arch dam on its typical dam section's tangential displacements.
ACKNOWLEDGEMENT The authors would like to thank the Natural Science Foundation of China for their financial supports Fund 50128908.
REFERENCES Oda, M. 1986. An equivalent continuum model for coupled stress and fluid flow analysis in jointed rock massed. Water Resource Research 22(13): 1845 - 1856 Ohnishi, Y. & Kabayashi, A. 1993. Thermalhydraulic-mechanical coupling analysis of rock mass. In Hudson J. A. (ed). Comprehensive Rock Engineering, Pergamon Press. 191 - 208 Ohnishi, Y. Shibata, H. & Kabayashi, A. 1985. Development offinite element code for analysis of coupled thermo-hydro-mechanical behaviours of saturated-unsaturated medium. Int. Symp. On Coupled Process Affecting the Performance of a Nuclear Waste Repository. Shen Zhenzhong. Xu Zhiying. & Luo Cui. 2000. Coupled analysis of viscoelasticity stress field and seepage field for the Three Gorges dam's foundation. Engineering Mechanics. 17(1): 105 -113. Wu Zhongru. Gu Chongshi. & Wu Xianghao. 2(X)1. Theory and its applications of safety monitoring of roller concrete dam. Science Press. Zienkiewicz, O. C. & Cormeau, I. C. 1974. Viscoplasticity, plasticity and creep in elastic solids. Int. J. Numer Methods, Eng. 8: 821 - 845.
1. INTRODUCTION The Longyangxia hydropower project is one of the major hydropower generation and water resources utilization systems built in Yellow River China. Continuous displacement towards the left bank at the 13'*' dam section has been observed and monitored. This has become a major concern to both the dam safety administration and engineers. The paper represents the results of a research effort devoted to investigate its causes. Recognizing the importance of the coupled hydro-mechanical effects on the performance of civil engineering structures involving fractured rocks, the stress-flow coupling mechanism of the dam-foundation system at Longyangxia site was simulated using a three-dimensional Finite Element code, supported by two visco-elastic constitutive models to represent the time-dependent material behaviour of the dam concrete and the foundation rock. The calculated results were concord with the measured ones and helped to interpret the causes of this continuous displacement at the 13*^ dam section of the Longyangxia hydropower project, towards the left bank. The theoretical foundations of the theory and the FEM formulations presented in this paper are based on works in Zienkiewicz and Cormeau (1974), Oda (1986), Ohnishi et al. (1985), Ohnishi and Kobayashi (1993), Shen et al. (2000) and Wu et al. (2001).
2. THE MATHEMATICAL THEORY 2.1 Basic equations of stress and seepage coupiing The main hypothesis of dam concrete and its foundation rock are as follows: a) the dam and its rock foundation are isotropic continuum media in
different areas, b) The seepage follows the Darcy's law. c) The grain skeletons' deformations of dam concrete and foundation rock are ignored, d) The deformations of dam concrete and its foundation rock are mainly caused by deformations of void spaces and cracks between grain skeletons mentioned above. If the dam and its base rock are affected only by gravity and seepage force, which are considered as body forces, and denoting that the tensile stress is positive, the balance differential equations expressed by displacements and general water heads can be given as
GV^S^-^-^-r..^^X„=0 ' \-2v dx ** dx (1)
'
l-2v dy ''dy ' G de^___^ a^ + Z, + r - r =0 GV'S^ + l-2v" az ^''dz where, G is the shear module, E is the Young's modulus, V is the Poisson's ratio, r^ is the density of water, r^ is the saturated density of the concrete or foundation rock, h is the water head. In addition, V^ is the Laplace operator, e^ is the volume strain, S^{i = X, y, z) are displacement components and XQ , KQ , ZQ are equivalent body force components caused by initial strain {SQ} . According to the law of mass conservation, the continuity equation of the water are derived as given in eq. (2). For simplify, the equation ie expressed along the main seepage directions denoted as coordinate axes x, y , z .
754 A.(k—^ —(k—) —(k—\-^ dx "^ dx dy ^ dy dz ^ dz
dt
/?— (2) ** dt
where [k] is the permeability tensor, whose main components are k^ , ky , and k, \ n is the void
^.=^-zk^iI/.-P[f('-'o)W' 277, J" m
(3)
exp[—Ht-to)]
ratio, and P is the compression parameter of the water. The basic equations for the coupHng analysis of stress and seepage fields are composed of equations (1) and (2), whose boundary conditions include (a) displacement boundary condition {J) = {^o) ' (^) stress boundary condition cr^^m, = a^ (where ^,/ = 1,2,3), (c) water head boundary condition h = h^ , (d) seepage boundary condition
where E^, rj^, rj^ are the tension (compress, or shear) module and viscous parameters, respectively, and 5, is the deviatoric stress tensor,
For a special case, when 5^ = 5,^0, equation (3) is changed to
-k„ — = ^0' ^hich should be specified according 2£.
to the site conditions.
2.2. The constitutive modeis For studying the viscous deformation caused by the creep of dam and its base rock under timedependent loading, different visco-elastic constitutive models are developed to identify the most suitable models and parameters for more accurate simulation of the time-dependent deformation of the dam-foundation system.
+ Z^y.o*exp -^it-t,)
2E,
(4)
l-exp[—-(t-t^)]
If the viscous strain of Kelvin model is le^ ^ \ at f^ ^ / = /Q + A/ , and the stress remains constant during the Ar time increment, the viscous strain increment of the generalized Kelvin model during At can be derived from equation (4) as
2.2.1 The visco-elastic constitutive model of the dam concrete The creep of dam concrete includes two parts: instantaneous elastic deformation and viscous deformation. A generalized Kelvin model consisting of two standard Kelvin model in series is used to describe the time-dependent deformation of the dam concrete as shown in Figure 1.
Figure 1. Sketch map of the generalized Kelvin model The model's strain is the sum of the initial elastic strain and the strains of the two Kelvin models, so that the partial strain e„ of the generalized Kelvin model is expressed as
{Af,.),=i
l-exp(-
[C]M-KtK
(5)
where [C] is the Poisson's ratio matrix. 2.2.2 The visco-elastic constitutive model of the foundation rock The time-dependent deformation of the foundation rock, caused by loadings, is described by a Burgers model, which is composed of a Kelvin model and Maxwell model in series, as showed in Figure 2. The partial strain expression of Burgers model is
^-=r-^ir/'>^'"'ff"^""f *'• exp^
^^M
(^-^0)
^ij.OM '
5,, J/
755 Specially, when s-j = S-JQ is a constant value, and
[KY{^8]-[K'\{h]„,,=[^F,]-[K']{h]^ [K'Y{^5]^{[S]^e^t[k])[h)^,,=lsI{^{^F,]^{F,]^)
S-j = 0 , the equation (6) becomes
+ ([5]-(l-^)A/[^]){/iL (9)
Figure 2. Sketch map of the Burgers model
(7)
The visco-elastic strain of the Maxwell component will change into {^^ M) ^^ ^'"^^ ^o • I^
where 0 is the integration parameter, [/^l is the general stiffness matrix, [K'] is the general coupling matrix, [S] is the general compression matrix, [^j is the general seepage matrix, [F^] is the equivalent nodal load vector by the initial strain, {F,} is the equivalent nodal load vector, [F^] is the equivalent nodal discharge vector, and {/z}^^, is the general water head vector at time /^^,, respectively. If displacement increment [is.S) and super-static water pressure {A^} are taken as the unknown quantities, and the full Hermit differences are adopted with ^ = 1 , the equation (9) can be rewritten as
r = ^0 + Ar, and the stress remain the same during the increment of Ar , the viscoelastic strain increment of Burgers model can be derived from equation (7) as given by
{Afv),=
l-exp(-^A/) .AT
;
(8)
+ —[C]{C7}
where E^, rj^, t]^ is the stretch (compression or shear) modulus and viscous parameters and [C] is the Poisson's ratio matrix, respectively.
2.3 The finite element method For the basic equations of coupled stress-flow analysis mentioned above, it is very difficult to solve them in closed-form. The transposition method of progression and integration can only be applied for problems of boundary value problems of simple geometry and boundary conditions. Therefore the finite element method (FEM) is used to solve the coupled partial differential equations in this paper. 2.3.1 The basics of the FEM formulation The FEM solution scheme for the coupled equations of displacement and seepage fields is
[K^{^S)-[K']{^p)
= {R)-{R^
[K'Y{^S]-i{S]^^t[K]){^p]
(10)
= o_
where {R^ ] refers to the part of loads balanced by the stresses related to the displacement happened before the time /^ , and are calculated by the following equation {/?,} = Z[/^J,{AcJ},
(11)
where / is the number of calculating periods of time before the time t^. 2.3.2 The calculation process According to the incremental initial-strain method used in equation (9), the initial stress field and seepage field of dam-foundation system need to be created by iterations starting from the beginning of the time-marching process. The element stress remains unchanged during the subsequent time step Ar , so that the seepage coefficients, which are decided by the state of stress at the beginning of the time step, remain the same value. They change step by step with loads increasing gradually at each time step. The water heads at the end of every time step are the initial ones of the next time step.
756 In terms of the magnitude of the time step Ar, two factors need to be considered. One is to guarantee the numerical stability of the viscous initial strain iteration by choosing a small value of At and the other is to ensure that the coefficients matrix not to be morbid by making the Ar value adequately large. It be estimated by the following formula Ar =
(12)
k
4
-(K + ^G) where L is the linear dimension of the elements, k is the seepage coefficient, and /i , G are bulk modulus and shear modulus, respectively. The appropriate time step Ar should be determined by trial and error. Generally speaking, the state of stress field varies mostly at the beginning of a loading step, so that the time step should be small at the beginning of the loading, and gradually increases in order to speed up the calculations.
2.4 The mechanisms of the damfoundation interaction for creep analysis A dam and its rock foundation are the two integral parts of a system. Different types of dams have different interaction behaviours between the two parts. Project practices show that for concrete gravity dams, the rock foundations have significant influence on the stress behaviours of the concrete dams up to one-third of the dam height from the base. The effect is small in other parts. For arch dams, on the other hand, the rock foundations have significant influences on the stress and behaviour of the whole dam body. For both gravity and arch dams, their rock foundations affect significantly the deformations of the concrete dams. Therefore, for stress analysis of the dam-foundation system of gravity dams, the series combination creep model is applied for the first one-third of 1/3 dam height from the base and the parallel combination creep model is applied to the rest. In terms of strain, this relationship can be expressed as
l = i- -1
for the rest, where e^ and e^ are the strains of dam concrete and foundation rock, respectively. The e^ can be obtained from equation (3) or (4) and s^ from equation (6) or (7).
3. AN APPLICATION EXAMPLE The above theory and FEM code were applied to analyse the continuous displacements of the 13 dam section of Longyangxia dam from July 1989, as an example. The main part of Longyangxia dam is a gravity arch dam of 396 meters in length, 29.2/80 meters in width (at the top/base), 2610 meters for DCL and 178 meters in height.
3.1 The FEM model and conditions 3.1.1 The FEM model and material parameters The dimension of the FEM model is 540 m in both length and width and 360 m in height. When dividing the FEM mesh, the geological structures in the foundation rock and their engineering treatment measures were taken into account. While laying out the element nodes, the locations of the in situ measuring points of displacement, temperature and stresses are considered. The FEM mesh, which is consisted of 21,189 eight-node-hexahedron elements with 24,873 nodes, is illustrated in Figure 3. The material parameters are listed in Table 1.
(13)
for the first one-third part of the dam body and e = £, + e,
(14)
Figure 3. The FEM mesh of the Longyangxia dam model
757 Table 1. Parameters of Longyangxia dam and foundation materials ^\^CIassification Location
^\^^
Density
Viscous constants
Elastic constants
kg/m^ Modulus
Seepage parameters
(GPa)
Poisson 's ratio ^
2400
20
0.18
250 2.3x10^ 66.7^1.15x10^ Dam
2580 m ~ 2560 m 2650
8
0.25
50 4.0x10' 15.0 8.5x10^
2560 m - 2540 m 2700
12
0.23
200p.6xloi20.0 5.4x10'
2540 m ~ 2500 m 2750
16
0.22
200p.8xlO' 40.0 5.0x10'
Dam body (concrete)
GPa GPa S GPa GPa S
Rock
Below 2400 m
2755
22
0.22
300p.OxlO^ 51.0 1.8x10'
foundation
G4*
2600
3
0.25
30 5.4x10' 10.0 9.5x10'
F,8
2600
4.5
0.25
40 4.0x10'' 15.0 9.0x10'
F71, F73, F32, F67
2600
3.2
0.25
30 5.4x10^ lO.q 9.5x10^
A2+F120
2600
5.4
0.25
40 4.0x10' 15.0 9.0x10'
cm/s l.OxlO'
Rock
1.0x10"'
Crack
l.OxlO"'*
Curtain 1.0x10"^
Drainage l.OxlO"
*G4, F18, F71, F73, F32, F67, and A2+Fi2oare geological structures and cracks in Longyangxia dam foundation.
3.1.2 Water level variations as timedependent loading cases in the calculation
3.2 Calculation and analyses of tangential displacements of typical days
From April 16, 1990 to May 1992, the water level of Longyangxia reservoir dropped from 2575.04 m to 2533.15 m. From May 1992 to December 1994, the water level rose from 2533.15 m to 2577.58 m. From January 1995 to July 1998, the water level dropped from 2577.58 m to 2533.54 m again. After July 1998, the water level rose from 2533.54 m to 2581.08 m. These water level variations were used for deriving the water head loading conditions.
The FEM modelling was applied for the simulating the dam behaviour with coupled stressflow effects under different loading cases according to the water level variations. The tangential displacements of Longyangxia dam's typical dam section on April 16, 1989, April 16, 1990, and April 16, 1996, and December 31, 1999 were calculated and compared with measured data. The results are listed in Table 2 and illustrated in Figure 4.
• 1989-4-16Sur.
'1990-4-16Sur.
-1996-4-leSur.
-1989-4-16C&I.
' 1990-4-16C&I.
" 1996-4-16C&I.
Figure 4. Survey values and calculation values ofgeneral displacement of the 1J dam section
758
^\^
Table 2. Measured and calculated values of time-dependent displacements (mm) 1996-4-16 1999-12-31 1989-4-16 1990-4-16 Time
Points ^ \ ^
Measured|Calculated|Measured|Calculated|Measured| Calculated iMeasuredl Calculated
of measurments'v^ 2463.3 m 2497 m 2530 m 2560 m 2585 m 2600 m
Values
Values*
values
Values
Values
Values
Values
Values
-~ -----
0.12
-0.39
-0.28
-1.15
-1.01
-1.79
-1.75
1.05
-0.33
-0.44
-2.70
-2.85
-3.95
-3.82
1.24
--
-0.79
-4.81
-4.44
-5.86
-5.97
-3.69
-5.27
-5.48
0.89
-0.68
-0.61
-3.88
0.57
-0.49
-0.36
-1.93
-1.84
-2.17
-2.26
0.06
-0.55
-0.31
-4.24
-4.14
-5.60
-5.49
It can be seen from Table 2 and Fig. 4 that: (1) the calculated values are close to the measured ones, and their respective trends of change are very similar as well. (2) The calculated tangential displacements above the 2500m level on April 16, 1996 are as follows: -3.04mm, -6.91mm, -6.33mm, -5.77mm, and -4.68mm, respectively. These data show that the 13* dam section moved towards the left bank. This was caused by the continuing high water level, about 100m ~ 110m higher than adjacent uplift pressure, and the creep displacements of dam body and rock foundation.
4. CONCLUSIONS (1) The time-dependent constitutive models of the dam concrete and foundation rocks for the Longyangxia hydropower complex were established. (2) The three-dimensional FEM model with the established constitutive models and realistic time-variations of reservoir water level as loading cases were successfully applied to simulate the coupled stress-flow behaviour of the damfoundation system. (3) The calculated results, which agreed well with the measured ones, clearly indicated that the main reason for the continuous displacement towards the left bank after July 1989, was caused by the influence of the seepage-stress coupling of Longyangxia gravity arch dam on its typical dam section's tangential displacements.
ACKNOWLEDGEMENT The authors would like to thank the Natural Science Foundation of China for their financial supports Fund 50128908.
REFERENCES Oda, M. 1986. An equivalent continuum model for coupled stress and fluid flow analysis in jointed rock massed. Water Resource Research 22(13): 1845 - 1856 Ohnishi, Y. & Kabayashi, A. 1993. Thermalhydraulic-mechanical coupling analysis of rock mass. In Hudson J. A. (ed). Comprehensive Rock Engineering, Pergamon Press. 191 - 208 Ohnishi, Y. Shibata, H. & Kabayashi, A. 1985. Development offinite element code for analysis of coupled thermo-hydro-mechanical behaviours of saturated-unsaturated medium. Int. Symp. On Coupled Process Affecting the Performance of a Nuclear Waste Repository. Shen Zhenzhong. Xu Zhiying. & Luo Cui. 2000. Coupled analysis of viscoelasticity stress field and seepage field for the Three Gorges dam's foundation. Engineering Mechanics. 17(1): 105 -113. Wu Zhongru. Gu Chongshi. & Wu Xianghao. 2(X)1. Theory and its applications of safety monitoring of roller concrete dam. Science Press. Zienkiewicz, O. C. & Cormeau, I. C. 1974. Viscoplasticity, plasticity and creep in elastic solids. Int. J. Numer Methods, Eng. 8: 821 - 845.