Fluid-structure models for dynamic studies of dam-water systems B. Tiliouine & A. Seghir Ecole Nationale Polytechnique, Algiers, Algiers, ALGERIA
Keywords: Keywords: Earthquake response, Finite Element Modeling, Dam flexibility, Water compressibility, Fluid-structure interaction, Hydrodynamic pressures. ABSTRACT: The performance of four different dam-reservoir finite element models, suitable for direct time domain analysis of the earthquake response of concrete gravity dams including dynamic fluid-structure interaction is investigated. Namely, these models, M1 to M4, are respectively: 1) the standard rigid dam-incompressible water model, 2) the flexible dam-incompressible water model, 3) the rigid dam-compressible water model and 4) the flexible dam-compressible water model. First, the discrete system of finite element equations resulting from a Galerkin variational formulation of the governing equations of the pressure and displacement fields are established for each model. Then, the distribution of the dynamic pressure coefficient at the upstream face of a typical concrete gravity dam and the hydrodynamic pressure time histories at its base, derived from the application of the four fluid-structure models have been determined. In addition, conclusions of engineering significance on the relative performance of the four mathematical models are given.
1 INTRODUCTION The dramatic consequences on life and property resulting from failure of large dams have led engineers to consider that these structures should withstand strong ground motions with no or only minor damage. This has provided a strong impetus for increased research, particularly for the development of new methods of dynamic analysis in seismic s tudies of concrete gravity dams. The evaluation of the import ant hydrodynamic forces that develop on the upstream face of a large dam during severe transient excitations has been the subject of numerous investigations, starting with Westergaard’s (1933) classical work. Since then, several contributions on the subject have appeared in the literature especially the fundamental works of Chopra and his coworkers (e.g. Chopra & Gupta 1981, Chopra & Hall 1982). These investigations have been particularly carried out in the frequency domain using rigid or finite element dam models along with a continuum approach based on analytical solutions for the wave equation governing the motion of water in reservoirs having rather simple spatial configurations (e.g. rectangular shapes or semi-infinite shapes with constant or linearly varying depths). This has contributed greatly to the understanding of the earthquake response of dam- reservoir systems and to the development of analysis procedures used to study their seismic behavior (Wylie 1975). However, the necessity to handle complex and irregular boundaries for which analytical solutions are not available on the one hand, and to consider the effects of material and geometrical non-linearities on the dynamic response to earthquake strong ground motions on the other hand, requires that the solution be expressed directly in the time domain. In this work, the relative performance of four different fluid-structure models suitable for direct time domain analysis analys is of the earthquake response of concrete gravity dam-reservoir systems is investigated. These models, conventionally designated herein by the symbols M1 to M4, are respectively: 1) the standard rigid dam-incompressible water model, 2) the flexible dam-incompressible water model, 3) the rigid dam-compressible water model and 4) the flexible dam-compressible water model. First, the discrete system of finite element equations resulting from a Galerkin variational formulation of the governing equations of the pressure and displacement fields are established for each model. Then, the distribution of the dynamic water press ure coefficient at the upstream face of
a typical concrete gravity dam and the hydrodynamic pressure time histories at its base derived from the application of the four fluid-structure models have been determined. Finally, on the basis of the numerical results obtained in this work, conclusions of engineering significance on the relative performance of the four fluid-structure models, suitable for direct time domain studies of dam-reservoir systems, are given.
2 MODELS FOR DYNAMIC STUDIES OF DAM-RESERVOIR SYSTEMS The evaluation of hydrodynamic forces and their effects on the response of concrete gravity dams depends on the numerical models used for the idealization of the dam and the reservoir subsystems. These models can be broadly categorized as a function of the assumptions retained as to the physical properties of the dam material and the reservoir fluid. In this paper, linearly elastic properties are assumed for the material of the dam and the water of the reservoir. The motion of the dam-reservoir system is considered as two dimensional and restricted to small amplitudes. The fluid is assumed to be inviscid and extends to infinity in the upstream direction. However, the effects of surface waves, water compressibility, dam flexibility, radiation damping at the upstream boundary of the reservoir and the slope of the upstream dam wall are considered in this work and constitute to some extent a departure from the conventional assumptions often made in direct time domain analysis of gravity dams including hydrodynamic interaction. We conveniently choose to retain a pressure field for the representation of the fluid action and a displacement field for the description of the dam behavior. The dam and the reservoir domains, illustrated in Figure 1, are thus modeled separately and it is demonstrated that the dynamic interaction forces linking the two subsystems at the dam-water interface are caused by hydrodynamic pressures from the fluid region acting on the upstream face of the dam, and the structural accelerations at the interface acting in turn on the reservoir.
Γ3
h
Σs − Γs −Γ1 Ωf
Γ1
Γ4
σ ij n j = pn i
Ωs
i,j = 1,2
Γs
Γ2
Figure 1. Dam-reservoir domain and distribution of hydrodynamic forces.
In general, the continuum fields equations for a flexible gravity dam fixed at its base, along with the appropriate boundary conditions can be summarized as follows:
σ ij, j + f i = ρ s &u& i
Ωs
(1)
σ ij n j = pn i
Γ1
(2)
σ ij n j = 0
∑s – Γ1 – Γs
(3)
Γs
(4)
ui
=0
In the above equations, the following notations have been used: Ωs, ∑s structure domain and its contour at equilibrium Γs base of dam structure Γ1 fluid-structure interface ρs mass density of dam structure ui, üi dam displacement and acceleration in the i th direction p hydrodynamic pressures on the upstream face ni the ith component of n (the outward unit normal) f i body forces in the ith direction σ ij stress tensor. Alternatively, it can be easily shown (Tiliouine & Seghir 1998) that a finite element discretization of the Galerkin variational formulation of the preceding equations yields the following system of second order differential equations:
&& + C U & MSU S + K S U = Fg + F p
(5)
in which MS, CS and K S are the classical finite element mass, damping and stiffness matrices of the dam structure. The global damping matrix for the dam is most effectively constructed by applying the concept of Rayleigh damping to the dam and may be computed from the equation:
CS = αM S + βK S
(6)
where α and β are proportionality constants selected to control the damping ratios of the lowest and highest modes expected to contribute significantly to the response. The unknown vector of basic nodal variable U represents the relative displacements at the nodal points of the finite element model of the dam to be analyzed and the symbol . denotes differentiation with respect to time. The forcing vector
&& Fg = −M S U g
(7)
contains the driving force components generated by the prescribed ground accelerations vector Üg applied at the structure nodal points. The additional forcing vector F p of hydrodynamic forces acting on the upstream face of the dam is related to the unknown vector of nodal pressures P, at the nodal points of the reservoir finite element model through the transformation matrix Q as follows:
F p = QP
(8)
with
∫
Q = − N TunN pdΓ Γ1
(9)
In the latter equation, Nu and Np represent respectively the finite element shape functions used for the interpolation of the dam displacement and the reservoir pressure field variables. This being the case, and in order to investigate the incidence of various assumptions on the response of concrete gravity dams subjected to seismic excitations, the hydrodynamic pressures to be used in Equation 8, will be computed according to one of the following four dam-reservoir models. 2.1 Model M1 (rigid dam-incompressible water model) This is the standard rigid dam-incompressible water model which may be considered as an extension of Westergaard’s problem. For this model, the governing equations for the displacement and hydrodynamic pressure fields are uncoupled. The continuous pressure field satisfies Laplace’s equation and the distribution of hydrodynamic pressures can be obtained from the following equations:
∇ 2 p = 0
Ωf
(10)
∂ p = −ρ f &u& gn ∂n
Γ1
(11)
∂ p =0 ∂n
Γ2+Γ4
(12)
∂ p 1 && =− p ∂n ρ f g
Γ3
(13)
In these equations, the physical parameters ρf and g represent respectively the fluid mass density and the gravitational constant. The symbols Γ2, Γ3 and Γ4 correspond respectively to the reservoir bottom, the mean surface at equilibrium and the finite element truncation boundaries. In Equation 11, the symbol ü ng designates the outward normal component of the prescribed ground acceleration vector applied to the wet solid particles at the dam-water interface. Alternatively, the corresponding discrete system of finite element equations i s:
&& + K P = q M FP F
(14)
where the assembled finite element “mass” (introduced by the surface wave effects) and “stiffness” matrices for the reservoir water subsystem, are respectively : 1
∫
N pT N p dΓ
(15)
∫
∇N TP ∇N p dΩ
(16)
MF =
Γ3
g
and
K F =
Ω f
The load vector q in Equation 14 is given by the expression:
q=−
∫
Γ1
ρ f u&& gn N pT dΓ
(17)
This model is not expected to give accurate results for large dams as dam flexibility and water compressibility have been omitted. Since, in general, surface wave effects are of minor importance and can be ignored for all practical purposes (Tiliouine & Seghir 1998, O’Connor & Boot 1988), the vector of hydrodynamic pressures acting at the nodal points of the water reservoir model can be directly estimated from:
P = −K −F1q
(18)
2.2 Model M2 (flexible dam-incompressible water model) This is the flexible dam-incompressible water model. The essential difference with respect to model M1 is that the flexibility of the dam is now taken into account. The mathematical formulation is slightly more complicated than that used for model M1 in the sense that the unknown basic continuous variables p and u are now coupled through the following boundary condition applied at the dam upstream face:
∂ p = −ρu&& n ∂n
Γ1
(19)
This equation involves the unknown outward normal accelerations ün of the wet solid particles at the fluid-solid interface and should not be confused with Equation 11.
Thus, the problem can now be cast in the following coupled form :
&& C S M S 0 U && + T Q M ρ F f P 0
0 U& K S − Q U Fg + = K F P 0 0 P& 0
(20)
It is immediately observed that the form of this global system of finite element equations is not symmetric. However, a direct step by step integration of the complete system is not necessary. As a matter of the fact, if surface waves are ignored from a practical point of view (which is a plausible assumption), the vector of nodal pressures in the water reservoir finite element model can be written as:
&& P = −ρ f K F−1Q T U
(21)
which after substitution in Equation 20 yields the uncoupled problem:
&& }+[C ]{U & }+[K ]{U}={F } M S + ρQK F−1Q T {U S S g
(22)
It should be noted that this problem differs from the classical problem of structural dynamics only by the added mass term. This may prove useful for practical applications since the solution can be implemented numerically with advantage, using standard commercial finite element packages. 2.3 Model M3 (rigid dam-compressible water model) This is the rigid dam-compressible water model. It may be considered as another extension of Westergaard’s model which explicitly includes, under certain restrictive assumptions, the effect of water compressibility. The mathematical formulation is similar to that presented for model M1 except that the hydrodynamic pressure field must now satisfy Helmoltz’s wave equation
∇ 2 p −
1 C2
&& = 0 p
and the boundary condition on
(23)
Γ4 has also be modified to take into account the radiation condition:
∂ p 1 = − p& C ∂n
Γ4
(24)
The parameter C being the velocity of sound in water. Alternatively, the hydrodynamic pressures can be computed from:
&& + C P& + K P = q MFP F F
(25)
Here, the global fluid mass matrix must be modified to account for both compressibility and surface wave effects, which leads to the following expression:
MF =
∫
Ωf
N pT
1 C
2
N p dΩ −
∫
Γ3
1
N pT N p dΓ g
(26)
It is seen that water compressibility introduces an additional inertial term but the pressure and the displacement fields are, as for model M1, still uncoupled. Nevertheless, a direct step by step integration scheme must now be utilized to deal effectively with the non-proportional damped system of finite element equations. 2.4 Model M4 (flexible dam-compressible water model) This is the flexible dam-compressible w ater model. Comparatively to the aforementioned models, it is the most comprehensive and most realistic fluid-structure model since the dam flexibility, water
compressibility and surface waves effects are directly taken into account. The seismic response in terms of dam displacements and hydrodynamic pressures in the reservoir should theoretically be more accurate than those obtained from models M1, M2 and M3. The mathematical formulation for the solution to the problem of determining the hydrodynamic pressure distribution within the reservoir is identical to that of model M3, except that the continuous pressure field equations are now governed by Helmoltz’s equation. Moreover the basic field variable u and p are coupled through Equation 19 that must be satisfied at the dam-water interface. It can then be shown that a standard Galerkin variational formulation of the coupled pressure and displacement fields yields the following coupled discrete system of finite elements equations:
&& C S 0 U & K S − Q U Fg M S 0 U + T && & + 0 K P = 0 C Q M ρ P F F 0 F P
(27)
where the “mass” matrix MF is again given by Equation 26. It is observed that this global matrix system of coupled second order differential equations is not symmetric and that a direct step by step integration scheme must again be utilized to deal effectively with this non-proportionally damped fluid-structure model.
3 PERFORMANCE OF THE FLUID-STRUCTURE MODELS 3.1 System properties and input data Based on the theoretical developments described in the preceding sections, a case study concerning the Oued-Fodda dam-water system (north-western Algeria) has been examined. Figure 2 shows the finite element mesh used to analyze the dynamic behavior of the dam-water reservoir system. Input data required for the dynamic analysis of this system consists of geometry, material properties of the dam and the water reservoir subsystems, boundary conditions as well as prescribed dynamic excitations. A detailed description of the relevant input data is given by Tiliouine & Seghir (1997). 4.6 m
m 0 . 1 0 1
m 4 . 6 9
65.5 m
300.0 m
Figure 2. Finite element mesh used for the Oued-Fodda dam-water system (dam upstream slope=10%).
The ground motions selected for this study were the horizontal components of the El-Asnam 1980 and Loma Preita 1989 earthquakes (Fig. 3).
1.4
amax=1.302 m/s² at t=1.60 sec.
0.7
) ² 0.0 s / m-0.7 ( s n-1.4 o i t 6.5 a r e 3.3 l e c c 0.0 A
(a)
amax=6.177 m/s² at t=2.60 sec. (b)
-3.3
-6.5 0
5
10
15
20
25
30
35
40
Time (sec.) Figure 3. Ground accelerations used for dynamic analysis of fluid-structure models. (a) El-Asnam 1980 Earthquake, (b) Loma Preita 1989 Earthquake.
3.2 Numerical results and discussion For the compressible water models, a direct step by step integration of the finite element system of differential equations expressed in geometrical coordinates has been used to deal effectively with the non proportionally damped dam-water systems. The dynamic response of the Oued-Fodda damwater system was determined in terms of various response quantities of interest including nodal displacements, element stresses, and nodal pressures within the reservoir. The distribution of the dynamic water pressure coefficient at the dam upstream face and the time histories at its base, derived from the application of the four fluid-structure models have been determined. A summary of the main numerical results are presented below in order to illustrate the relative performance of the proposed fluid-structure models. 3.2.1 Distribution of pressure coefficient Figure 4 shows the distribution of hydrodynamic pressures at the dam-water interface, normalized by the maximum hydrostatic pressure and resulting from the successive applications of the horizontal components of the 1980 El-Asnam and 1989 Loma Preita Earthquakes to the OuedFodda concrete gravity dam according to the four proposed fluid-structure models M1 to M4. In this figure, C p = p /ρf g Hr represent the pressure coefficient and y/H r , the reservoir height ratio of y, the distance above reservoir bottom, to H r , the total depth of reservoir. It should be noted that in this definition of C p, all p values are computed at a time equal to that of the maximum hydrodynamic pressure. It is clearly observed that the use of different numerical fluid-structure models may induce significant differences in hydrodynamic and structural response. In particular, in the case of an incompressible fluid, the hypothesis of a rigid dam is seen to lead to hydrodynamic pressures practically similar to those which would result from the assumption of a flexible structure. However, in the case of a compressible fluid, the rigid dam hypothesis is shown to lead clearly to lower pressure coefficients than those resulting from the more realistic flexible dam assumption. It may thus be concluded that dam flexibility does not have an important effect on hydrodynamic pressures for an incompressible fluid but it does induce a substantial increase of dynamic water pressures for a compressible fluid. These observations remain essentially valid regardless of the ground motions used for the dynamic of the dam-water system. In other words, the results obtained clearly indicate that water compressibility induces a substantial increase of hydrodynamic pressures for large flexible dams. The errors in neglecting compressibility in this case, especially under resonant response conditions may become quite large. The combined effects of fluid compressibility and dam flexibility may lead, because of a beating
behavior in response, to considerably large values of the hydrodynamic forces exerted on the upstream face of a large dam and hence to high response amplifications of the dam-reservoir system. Finally, it is interesting to note that results obtained using model M1 (the analogue of Westergaard’s model) significantly underestimate the hydrodynamic forces that would really results the application of the upper limiting case associated with model M4 (the theoretically more realistic flexible dam-compressible water model). Rigid dam-incompressible water model
(M1)
Flexible dam-incompressible water model (M2) Rigid dam-compressible water model
(M3)
Flexible dam-compressible water model fluide compressible, barrage flexible
(M4)
fluide compressible, barrage rigide
1.00
1.00
fluide incompressible, barrage rigide fluide incompressible, barrage flexible
0.75
0.75
r
r H / H / y
H / y
y
0.50
0.50
(a)
0.25
0.25
0.00 0.00
(b)
0.04
0.07
Cp 0.11
0.00 0.00
0.14
0.18
0.35
C p
0.52
0.70
C p
Figure 4. Pressure coefficient distribution using the four fluid-structure models. (a) El-Asnam 1980 Earthquake, (b) Loma Preita 1989 Earthquake.
3.2.2 Time histories of hydrodynamic pressures To further illustrate the relative performance of the four proposed fluid-structure models, time histories of response quantities of interest were performed for the total duration of the specified base earthquake accelerations represented in Figure 3. For the sake of brevity, only the response time histories of hydrodynamic pressures acting at the node located at the upstream face base of the Oued-Fodda dam will be discussed. A summary of the main results derived from the application of the Loma Preita 1989 earthquake to the dam-water system is presented in Table 1 below. Table 1. Maximum hydrodynamic pressures Pmax and time of occurrence T, derived from the four fluidstructure finite element models ( Loma Preita 1989 earthquake). Models
Dam
Reservoir water
M1 M2 M3 M4
Rigid Flexible Rigid Flexible
Incompressible Incompressible Compressible Compressible
Pmax (MPa) 0.22 −0.27 0.49 0.66
T (sec.) 2.60 3.10 2.66 3.26
Peak phase lag (sec.) 0.00 0.50 0.06 0.66
It is clearly seen that the combined effects of dam flexibility and water compressibility are the most critical in studying the response of a dam-water system. As may be expected, the phase lag expressed in terms of the difference between the time of maximum hydrodynamic pressure and that of peak earthquake ground acceleration, is practically insignificant for the rigid dam models.
The time histories of hydrodynamic pressures computed from the four fluid-structure finite element models are displayed in Figure 5. It is seen that the first ten (10) seconds appear to be the most critical for the response of dam-reservoir system. The results given by model M4, the flexible dam-compressible water model are much larger than those derived from the other models M 1-M3. For the flexible dam models, the frequency contents of the results obtained from models M2 and M4 are very similar but a large discrepancy is noted in terms of maximum pressure values. The phase lags become now relatively important for both models. The beating behavior in response observed in both flexible dam models is greatly amplified in model M4 due to the water compressibility effects. For the rigid dam models, no beating behavior is observed. The hydrodynamic pressures derived from model M3 are larger than those obtained from model M1 (again because of water compressibility effects). The time of peak hydrodynamic pressures for both models are very close to the peak ground acceleration of the applied earthquake ground motion. This is because of the rigid dam assumption used for both models to compute the hydrodynamic pressure field within the reservoir. 7.0E+5 3.5E+5
Pmax
0.22 MPa at t=2.60
=
(a) Model M1
0.0E+0 -3.5E+5 -7.0E+5 7.0E+5 3.5E+5
(b) Model M2
0.0E+0
pmax
) -3.5E+5 a P ( s -7.0E+5 e r u 7.0E+5 s s e r P 3.5E+5
pmax
=
=
–0.27 MPa at t=3.10 sec.
0.49 MPa at t=2.66 (c) Model M3
0.0E+0 -3.5E+5 -7.0E+5 7.0E+5
pmax
=
0.66 MPa at t=3.26
3.5E+5
(d) Model M4
0.0E+0 -3.5E+5 -7.0E+5 0
5
10
15
20
25
30
35
40
Time (sec.) Figure 5. Time histories of hydrodynamic pressures (MPa) derived from the four fluid-structure models.
4 CONCLUSION An investigation of the relative performance of four different fluid-structure finite element models suitable for direct time domain evaluation of hydrodynamic pressures in earthquake response studies of concrete gravity dam-reservoir systems has been presented. Namely, these are the standard rigid dam-incompressible water model (model M1), the flexible dam-incompressible water model (model M2), the rigid dam-compressible water model (model M3) and the flexible dam-compressible water model (model M4). A Galerkin variational formulation of the governing equations and the boundary conditions as well as the resulting discrete system of finite element equations have been established for each of the four proposed models. The results derived from the application of the four proposed dam-reservoir models to the Oued-Fodda gravity dam-reservoir system have clearly shown that the use of different numerical models leads to significant differences in hydrodynamic pressures and structural response. The main conclusions were that: models with incompressible water, M1 and M2, produced practically similar results and much lower dynamic pressure coefficients than should be expected in real situations. They should be discarded or utilized with care and only in the preliminary seismic design stage of concrete gravity dams with moderate heights. Model M3 was shown to yield higher and more realistic hydrodynamic pressures than models M1 and M2. It can be, because of its uncoupled mathematical form, used for most practical purposes. The more comprehensive model, M4, was shown to be capable of capturing the significant dynamic pressure amplifications caused by the beating behavior in response resulting from the combined effects of dam flexibility and water compressibility, especially near resonant response conditions of the dam-reservoir system. Its use is particularly recommended for the analysis of the earthquake response of large dams.
5 REFERENCES Chopra, A. K. & Gupta, S. 1981. Hydrodynamic and foundation interaction effects in earthquake response of a concrete gravity dam. Journal of the Structural Division, ASCE 578: 1899-1412. Chopra, A. K. & Hall, J. F. 1982. Two-dimensional dynamic analysis of concrete gravity and embankment dams including hydrodynamic effects. Earthquake Engineering and Structural Dynamics10: 305-332. O’Connor, J. P. F. & Boot, J. C. 1988. A solution procedure for the earthquake analysis of arch dam-reservoir systems with compressible water. Earthquake Engineering and Structural Dynamics16: 757-773. Tiliouine, B. & Seghir, A. 1997. Influence de l’interaction fluide-structure sur le comportement sismique du barrage de Oued-Fodda (Nord-Ouest Algérien). Actes du 1er Congrès Arabe de Mécanique, Damas, Syrie, 1-5 Juin. Tiliouine, B. & Seghir, A. 1998. Développement d’une technique de symétrisation du problème de vibrations des systèmes couplés fluide-structure. 6 eme Collogue Magrebin sur les Modèles Numeriques de l’Ingénieur . Tunis, Tunisie, 24-26 Novembre. (to appear). Westergaard, H. M. 1933. Water pressures on dams during earthquakes. Transactions, ASCE 98: 418-472. Wylie, E. B. 1975. Seismic response of reservoir-dam systems. Journal of the Hydraulics Division, 101: 403-419.
