Analysis of Longyangxia dam deformation based on seepage and creep coupling method.Dam failures A dam is a barrier across flowing water that obstructs, directs or slows down the flow, often creating a reservoir, lake or impoundments. Most dams have a section called a spillway or weir are mainly caused by cracks or failures 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. By use of visco-elastic constitutive constitutive /con·sti·tu·tive/ (kon-stich´u-tiv) produced constantly or in fixed amounts, regardless of environmental conditions or demand. models and finite element See FEA. solution method, the stress and seepage fields of foundation rocks are studied as a coupled system in this paper. Using this coupled models, the deformation doubts of the continuous displacement of the 13th dam section of the Longyangxia dam are analyzed and explained reasonably. INTRODUCTION The Longyangxia hydropower hy·dro·pow·er n. Hydroelectric power. gravity arch is the main part of the major hydropower generation and water resources utilization systems built along the Yellow River in China. However, continuous displacement towards the left bank at the 13th dam section puzzled both the dam safety administration and engineers. Using the seepage and creep coupling theory and finite element method, this paper represents the results of a research effort devoted to investigate its causes. The theoretical foundations of the theory and the FEM FEM Female FEM Finite Element Method FEM Feminine FEM Finite Element Model FEM Fédération Européenne des Métallurgistes (European Metalworkers' Federation) FEM Faculdade de Engenharia Mecânica (Brasil) formulations presented in this paper are based on works in Oda (1986), Ohnishi and Kobayashi (1993), Shen Shen, in the Bible, place, perhaps close to Bethel, near which Samuel set up the stone Ebenezer. et al. (2000) and Wu et al. (2001). BASIC EQUATIONS The balance differential equations expressed by displacements and general water heads can be given as [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] (1) where, G is the shear module, E is the Young's modulus Young's modulus [for Thomas Young], number representing (in pounds per square inch or dynes per square centimeter) the ratio of stress to strain for a wire or bar of a given substance. , v is the Poisson's ratio When a sample of material is stretched in one direction, it tends to get thinner in the other two directions. Poisson's ratio (ν,  ), named after Simeon Poisson, is a measure of this tendency. , is the density of water, [r.sub.c] is the
saturated density
of the concrete or foundation rock, h is the water head. In addition, [[nabla].sup.2] is the Laplace operator In mathematics and physics, the Laplace operator or Laplacian, denoted by  or , [[epsilon].sub.v] is the volume
strain, [[delta].sub.i](I = x,y,z) are displacement components and
[X.sub.0], [Y,sub.0], [Z.sub.0] are equivalent body force components
caused by initial strain ([[epsilon].sub.0]}.
To get equation q, we have four hypotheses: a) the dam and its rock base are isotropic Refers to properties that do not differ no matter which direction is measured. For example, an isotropic antenna radiates almost the same power in all directions. In practice, antennas cannot be 100% isotropic. continuum media in different areas. b) The seepage follows the Darcy 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. According to according to prep. 1. As stated or indicated by; on the authority of: according to historians. 2. In keeping with: according to instructions. 3. the law of mass conservation, the continuity equation of the water is derived as given in equation (2). For simplification, the equation is expressed along the main seepage directions denoted as coordinate axes x , and y z. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2) where [k] is the permeability tensor tensor, in mathematics, quantity that depends linearly on several vector variables and that varies covariantly with respect to some variables and contravariantly with respect to others when the coordinate axes are rotated (see Cartesian coordinates). , whose main components are [k.sub.x], [k.sub.y], and [k.sub.z]; n is the void ratio Void ratio, in materials science, is defined as the volume of voids in a mixture divided by the volume of solids. This figure is relevant in composites, in mining (particular with regard to the properties of tailings), and in soil science. , and [beta] is the compression parameter of the water. The basic equations for the coupling analysis of stress and seepage fields are composed of equations (1) and (2), whose boundary conditions include (a) displacement boundary condition {[dela]} = {[[delta].sub.0]} stress boundary condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (where k, l = 1,2,3), (c) water head boundary condition h = [h.sub.0], (d) seepage boundary condition -[k.sub.n] [partial derivative partial derivative In differential calculus, the derivative of a function of several variables with respect to change in just one of its variables. Partial derivatives are useful in analyzing surfaces for maximum and minimum points and give rise to partial differential ]h/[partial derivative]n = [q.sub.0], which should be specified according to the site conditions. For studying the viscous deformation caused by the creep of rock foundation upon time-dependent 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. The time-dependent deformation of the foundation rock, caused by loadings, is described by a Burgers model, which is composed of a Kelvin kelvin, abbr. K, official name in the International System of Units (SI) for the degree of temperature as measured on the Kelvin temperature scale. A unit of measurement of temperature. model and Maxwell model in series. The partial strain expression of Burgers model is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3) Specially, when [S.sub.ij] = [S.sub.ij0] is a constant value, and [[??].sub.ij] = 0, the equation (3) becomes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4) The visco-elastic strain of the Maxwell component will change into [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at time [t.sub.0]. If t = [t.sub.0] + [DELTA]t and the stress remain the same during the increment of [DELTA]t, the visco-elastic strain increment of Burgers model can be derived from equation (4) as given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5) where [E.sub.K], [[eta].sub.K], [[eta].sub.M] is the stretch (compression or shear) modulus and viscous parameters and [C] is the Poisson's ratio matrix, respectively. For the basic equations of coupled stress-flow analysis mentioned above, the finite element method is used to solve the coupled partial differential equations partial differential equation In mathematics, an equation that contains partial derivatives, expressing a process of change that depends on more than one independent variable. in this paper. The FEM solution scheme for the coupled equations of displacement and seepage fields is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6) where [theta Theta A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option. ] is the integration parameter, [[bar.X]]] is the general stiffness matrix, [K'] is the general coupling matrix, [S] is the general compression matrix, [[??]] is the general seepage matrix, {[F.sub.0]} is the equivalent nodal Having to do with nodes. See node. NODAL - Interpreted language implemented on Norsk Data's NORD-10 computers. Used by CERN and DESY high energy physics labs to control their accelerator hardware, PADAC and SEDAC. Included trackball input, graphics. load vector by the initial strain, is the equivalent nodal load vector, is the equivalent nodal discharge vector, and [{h}.sub.m+1] is the general water head vector at time [t.sub.m+1], respectively. If displacement increment {[DELTA][delta]} and super-static water pressure {[DELTA]p} are taken as the unknown quantities, and the full Hermit hermit [Gr.,=desert], one who lives in solitude, especially from ascetic motives. Hermits are known in many cultures. Permanent solitude was common in ancient Christian asceticism; St. Anthony of Egypt and St. Simeon Stylites were noted hermits. differences are adopted with [theta] = 1, the equation (6) can be re-written as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7) where {[R.sub.t]} refers to the part of loads balanced by the stresses related to the displacement happened before the time [t.sub.m], and are calculated by the following equation {[R.sub.t]} = [[t.sub.m].summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) over (l = 0)] [[[[bar.X].sub.t]].sub.l][{[DELTA][delta]}.sub.l] (8) where l is the number of calculating periods of time before the time [t.sub.m]. According to the incremental initial-strain method used in equation (7), 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 [DELTA]t, 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. ANALYSIS OF LONGYANGXIA DAM'S DEFORMATION The main part of Longyangxia dam is a gravity arch dam An arch dam is a thin, curved concrete or masonry dam structure which is curved upstream in plan so that the force of the water against it squeezes the arch, compressing and strengthening the structure and pushing it into the ground. of 396 meters in length, 29.2/80 meters in width (at the top/base), 2610 meters for DCL (1) (Digital Command Language) Digital's standard command language for the VMS operating system on its VAX series. (2) (Data Compression L and 178 meters in height. 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 In place. When something is "in situ," it is in its original location. 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 1. The material parameters are listed in Table 1. [FIGURE 1 OMITTED] 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 tangential tan·gen·tial also tan·gen·tal adj. 1. Of, relating to, or moving along or in the direction of a tangent. 2. Merely touching or slightly connected. 3. 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 2. It can be seen that: the calculated values are close to the measured ones, and he 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 13th 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. [FIGURE 2 OMITTED] The calculated results, which agreed well with the measured ones, clearly indicated that the main reason for the continuous displacement of Longyangxia dam towards the left bank after July 1989, was caused by the influence of the seepage-stress combined operations For the department of the British War Office during World War II, see . In the military, combined operations are operations conducted by forces of two or more allied nations acting together for the accomplishment of a single mission. See also
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. Thermal-hydraulic-mechanical coupling analysis of rock mass. In Hudson J. A. (ed), Comprehensive Rock Engineering, Pergamon Press: 191-208 Shen Zhenzhong. Xu Zhiying. & Luo Cui. 2000. Coupled analysis of viscoelasticity Viscoelasticity, also known as anelasticity, is the study of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like honey, resist shear flow and strain linearly with time when a stress is applied. stress field and seepage field for the Three Gorges The Three Gorges (Simplified Chinese: 三峡; Traditional Chinese: 三峽; Pinyin: Sānxiá [ dam's foundation. Engineering Mechanics. 17(1): 105-113. Wu Zhongru. Gu Chongshi. & Wu Xianghao. 2001. Theory and its applications of safety monitoring Safety Monitoring of a clinical trial is conducted by an independent physician with relevant expertise. This is accomplished by review of adverse event, immediately after they occur, with timely follow-up through resolution. of roller concrete dam. Science Press. GUO GUO Glavnoye Upravleniye Okhraneniya (Russian: Main Administration Protection, aka GUORF) HAIQING Geotechnical Institute, Hohai University  This article or section may be confusing or unclear for some readers.Please [improve the article] or discuss this issue on the talk page. , 1, Xikang Road Nanjing, Jiangsu, 210098, China GU CHANGCUN, XU WEIYA Geotechnical Institute, Hohai University, 1, Xikang Road Nanjing, Jiangsu, 210098, China
Table 1. Parameters of Longyangxia dam and foundation materials
Elastic constants
Density Poisson's
kg/ Modulus ratio
[m.sup.3] GPa [micro]
Dam concrete 2400 20 0.18
Rock 2580 m 2400 8 0.25
foundation ~ 2560 m
2560 m 2650 12 0.23
~ 2540 m
2540 m 2700 16 0.22
~ 2500 m
Below 2750 22 0.22
2400 m
[G.sub.4] * 2755 3 0.25
[F.sub.18] 2600 4.5 0.25
[F.sub.71],
[F.sub.73],
[F.sub.32],
[F.sub.67] 2600 3.2 0.25
[A.sub/2] +
[F.sub.120] 2600 5.4 0.25
Viscous constants
[E.sub.K] [[eta].sub.K] [E.sub.M]
GPa GPa x S GPa
Dam concrete 250 2.3 x 66.7
[10.sup.5]
Rock 2580 m 50 4.0 x 15.0
foundation ~ 2560 m [10.sup.4]
2560 m 20 3.6 x 20.0
~ 2540 m [10.sup.4]
2540 m 200 2.8 x 40.0
~ 2500 m [10.sup.4]
Below 300 2.0 x 51.0
2400 m [10.sup.5]
[G.sub.4] * 30 5.4 x 10.0
[10.sup.4]
[F.sub.18] 40 4.0 x 15.0
[10.sup.4]
[F.sub.71],
[F.sub.73],
[F.sub.32],
[F.sub.67] 30 5.4 x 10.0
[10.sup.4]
[A.sub/2] +
[F.sub.120] 40 4.0 x 15.0
[10.sup.4]
Seepage
parameters
[[eta].sub.M]
GPa x S cm/s
Dam concrete 1.5 x Dam 1.0 x
[10.sup.9] [10.sup.-7]
Rock 2580 m 8.5 x Rock 1.0 x
foundation ~ 2560 m [10.sup.9] [10.sup.-9]
2560 m 5.4 x Rock
~ 2540 m [10.sup.9]
2540 m 5.0 x Crack 1.0 x
~ 2500 m [10.sup.9] [10.sup.-4]
Below 1.8 x Crack
2400 m [10.sup.9]
[G.sub.4] * 9.5 x Curtain 1.0 x
[10.sup.9] [10.sup.-8]
[F.sub.18] 9.0 x Curtain
[10.sup.9]
[F.sub.71],
[F.sub.73],
[F.sub.32],
[F.sub.67] 9.5 x Drainage 1.0 x
[10.sup.9] [10.sup.-2]
[A.sub/2] +
[F.sub.120] 9.0 x Drainage
[10.sup.9]
Table 2. Calculated values of time-dependent viscous displacements (mm)
Date/
Elevation 1989-4-16 1190-4-16 1996-4-16 1999-12-31
2463.3 m 0.12 -0.28 -1.01 -1.75
2497 m 1.05 -0.44 -2.85 -3.82
2530 m 1.24 -0.79 -4.44 -5.97
2560 m 0.89 -0.61 -3.69 -5.48
2585 m 0.57 -0.36 -1.84 -2.26
2600 m 0.06 -0.31 -4.14 -5.49
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