Frequency Domain Analysis of Concrete Gravity Dam-Reservoir Systems by Wavenumber Approach

Transcription

1 Frequeny Domain Analysis of Conrete Gravity Dam-Reservoir Systems by Wavenumber Approah V. Lotfi & A. Samii Department of Civil and Environmental Engineering, Amirkabir University of Tehnology, Tehran, ran SUMMARY: Dynami analysis of onrete gravity dam-reservoir systems an be arried out in frequeny or time domain. t is well-known that the rigorous approah for solving this problem relies heavily on employing a twodimensional semi-infinite fluid element. The hyper-element is formulated in frequeny domain and its appliation in this field has led to many espeial purpose programs, whih were demanding from programming point of view. n this study, a tehnique is proposed for dynami analysis of dam-reservoir systems in the ontext of pure finite element programming whih is referred to as the Wavenumber approah. n this tehnique, the wavenumber ondition is imposed on the trunation boundary or the upstream fae of the near-field water domain. The method is initially desribed. Subsequently, the response of an idealized triangular dam-reservoir system is obtained by this approah, and the results are ompared against the exat response. Based on this investigation, it is onluded that this approah an be envisaged as a great substitute for the rigorous type of analysis. Keywords: onrete gravity dams, wavenumber, absorbing boundary onditions, trunation boundary 1. NTRODUCTON Dynami analysis of onrete gravity dam-reservoir systems an be arried out rigorously by FE-(FE- HE) method in the frequeny domain. This means that the dam is disretized by plane solid finite elements, while, the reservoir is divided into two parts, a near-field region (usually an irregular shape) in the viinity of the dam and a far-field part (assuming uniform depth) whih extends to infinity in the upstream diretion. The former region is disretized by plane fluid finite elements and the latter part is modeled by a two-dimensional fluid hyper-element (Hall and Chopra 198, Waas 197). t is well-known that employing fluid hyper-elements would lead to the exat solution of the problem. However, it is formulated in the frequeny domain and its appliation in this field has led to many espeial purpose programs whih were demanding from programming point of view. On the other hand, engineers have often tried to solve this problem in the ontext of pure finite element programming (FE-FE method of analysis). n this approah, an often simplified ondition is imposed on the trunation boundary or the upstream fae of the near-field water domain. Thus, the fluid hyper-element is atually exluded from the model. Some of these widely used simplified onditions (Sommerfeld 1949, Sharan 1987), may result in signifiant errors if the reservoir length is small, and it might lead to high omputational ost if the trunation boundary is loated at far distanes. The main advantage of these onditions is that it an be readily used for time domain analysis. Thus, they are vastly employed in nonlinear seismi analysis of onrete dams. Of ourse, there have also been many researhes in the last three deades to develop more aurate absorbing boundary onditions to be applied for similar fluid-struture or soil-struture interation problems. Perfetly mathed layer (Berenger 1994, Chew and Weedon 1994, Basu and Chopra 003) and, high-order non-refleting boundary ondition (Higdon 1986, Givoli and Neta 003, Hagstrom and Warbuton 004, Givoli, et al. 006) are among the two main popular groups of methods whih

2 researhers have applied in their attempts. t is emphasized that these tehniques have beome very popular in reent years due to the fat that they ould be applied in time domain as well as the frequeny domain. However, it should be realized that they are not that attrative in the frequeny domain. This is due to the fat that they are not very simple to be used and more importantly, they are ompared with the hyper-element alternative whih produes exat results no matter how small the reservoir near-field length is onsidered. n the present study, the FE-FE analysis tehnique is hosen as the basis of a proposed method for dynami analysis of onrete dam-reservoir system in the frequeny domain, whih is referred to as the Wavenumber approah. The method is simply applying an absorbing boundary ondition on the trunation boundary whih is referred to as the Wavenumber ondition. t is as simple as employing Sommerfeld or Sharan ondition on the trunation boundary. The method of analysis is initially explained. Subsequently, the response of an idealized triangular dam is studied due to horizontal ground motion based on this tehnique. A sensitivity analysis is arried out and the parameter varied, is the length of reservoir near-field region. t is mainly onluded that the Wavenumber approah is very aurate. t is also ideal from programming point of view due to the loal nature of wavenumber ondition imposed on trunation boundary. Moreover, it an be envisaged as a great substitute for the rigorous FE-(FE-HE) type of analysis in the frequeny domain whih is heavily relying on a fluid hyper-element as its main ore.. METHOD OF ANALYSS As mentioned, the analysis tehnique utilized in this study is based on the FE-FE method, whih is appliable for a general onrete gravity dam-reservoir system. The oupled equations an be obtained by onsidering eah region separately and then ombine the resulting equations..1 Dam Body Conentrating on the strutural part, the dynami behavior of the dam is desribed by the well-known equation of strutural dynamis (Zienkiewiz and Taylor 000): M&&r +C&r +K r = M Ja g + B T P (1) where Μ, C and K in this relation represent the mass, damping and stiffness matries of the dam body. Moreover, r is the vetor of nodal relative displaements, J is a matrix with eah two rows equal to a identity matrix (its olumns orrespond to a unit horizontal and vertial rigid body motion), and a denotes the vetor of ground aelerations. Furthermore, B is a matrix whih relates g vetors of hydrodynami pressures (i.e., P ), and its equivalent nodal fores. Let us now onsider harmoni exitation with frequeny ω, and limit the present study to the horizontal ground motion only. t is well known that the response will also behave harmoni (i.e., r( t ) = r( ω ) e iωt ( ) ). Thus, Eq. (1) an be expressed as: ω M + (1 + β i) K r= M J a + B P () h T g n this relation, it is assumed that the damping matrix of the dam is of hystereti type. This means: C = ( β / ω ) K (3) Moreover, it should be emphasized that the supersript h on the aeleration vetor refers to the horizontal type of exitation. That is: x h a g a g = (4) 0

3 . Water Domain Assuming water to be linearly ompressible and negleting its visosity, its small irrotational motion (Fig. 1) is governed by the wave equation (Chopra 1967, Chopra, et al. 1980): p x + p y 1 &&p = 0 in Ω and D (5) where p is the hydrodynami pressure and, is the pressure wave veloity in water. Figure 1. Shemati view of a typial dam-reservoir system. The near-field reservoir domain Ω, the trunation boundary Γ and the far-field region D (exluded in the FE-FE type of analysis) One an apply the weighted residual approah to this equation and impose all boundary onditions in expliit form exept for the trunation boundary (i.e., Γ ), to obtain the finite element equation of the fluid domain, whih may be written as: G e &&P e e + ql &P e + H e P e = R e (t) B e &&r e B e J e h a g (6) This is the well-known equation of the fluid domain and the exat definitions of the matries involved an be found elsewhere in details (Lotfi and Samii 011). t is worthwhile to emphasize that the supersript ( e ) states that these matries are related to the element level. Moreover, L is a matrix, whih orresponds to the absorption of energy at reservoir s bed (i.e., Γ ). t is reminded that the admittane or damping oeffiient q whih is fatored out from this matrix may also be related to a more meaningful wave refletion oeffiient α (Fenves and Chopra 1985): 1 q α = (7) 1 + q whih is defined as the ratio of the amplitude of the refleted hydrodynami pressure wave to the amplitude of a vertially propagating pressure wave inident on the reservoir bottom. For a fully refletive reservoir bottom ondition, α is equal to 1 whih leads to q = 0. Furthermore, R e is part of the right hand side vetor orresponding to the trunation boundary. This boundary integral has the following original form: R e = 1 N Γ ρ ( n p)dγ e (8) with N being the vetor of fluid element s shape funtions, ρ is the water density and n denotes the outward (with respet to fluid region) perpendiular diretion. t is emphasized that the diretional derivative needs to be defined by imposing an appropriate ondition. n p e

4 The equivalent form of equation (6) in the frequeny domain would be: h ω G e P e + i ω q L e P e + H e P e = R e ( ω) + ω B e r e B e J e a (9) g By assembling the element equations, one would obtain the overall FE equation of the fluid domain: h ω G P + iω q L P + H P = R + ω Br B J a (10) n this equation,.3 Dam-reservoir system g R is obtained by assembling the boundary integrals of Eq. (8) on Γ. The neessary equations for both dam and reservoir domains were developed in the previous setions. Thus, ombining the main relations () and (10) would result in the FE equations of the oupled damreservoir system in its initial form for the frequeny domain: T h ω M + (1 + β i) K B r M J a g = h (11) ω B ( ω G + i ωq L ( + H) P B J ag + R ) t is noted from the above equation that the vetor R still needs to be expliitly defined. This is related to the trunated boundary Γ whih will be disussed below..4 Modifiation due to trunation boundary ontribution The effet of trunation boundary will be treated in this setion. For this purpose, let us now assume that this boundary (i.e., Γ ) is vertial (i.e. along y-diretion) and onsider a harmoni plane wave with unit amplitude and frequeny ω propagating along a diretion whih makes an angle θ with negative x-diretion. This may be written in many different forms suh as: p = e i( k x + λ y + ωt ) (1a) p = e (i ω ) (os θ ) x + (sin θ ) y + t (1b) With the following relations being valid: ω k = osθ (13a) ω λ = sin θ (13b) ω k + λ = (13) t is easily verified that the following ondition is appropriate for the trunated boundary based on the assumed traveling wave (i.e, Eq. (1a)): p x i k p = 0 (14)

5 Employing (14) in (8), it yields: e e e R = (i k ) L P (15) with the following definition: e 1 T e L = N N dγ (16) e ρ Γ Assembling R e for all fluid elements adjaent to trunation boundary leads to: R = (i k ) L P (17) This an now be substituted in (11) to obtain the FE equations of the oupled dam-reservoir system in its final form for the frequeny domain: T h ω M + (1 + β i) K B r M J a g = h (18) B ω ( ω G + ik L ( + iωq L + H) P ω B J ag ) t is also notied that the lower matrix equation of (18) is multiplied by ω in this proess to obtain a symmetri dynami stiffness matrix for the dam-reservoir system..5 Defining parameter k The major remaining onept is the determination of parameter k whih will be disussed in this setion. There are different available options (Lotfi and Samii 011). However, before explaining about the adopted strategy, it is worthwhile to review some salient aspets on the exat analytial solution available for the domain D (Fig. 1). t is reminded that this domain is atually eliminated from our problem. This is a regular semi-infinite region with onstant depth H extending to infinity in the upstream diretion. t is also assumed presently that the base of this region is ompletely refletive (i.e, α = 1or q = 0 ) and we are onsidering merely horizontal ground exitation. Under these irumstanes, the exat solution for this region may be written as follows (Chopra, et al. 1980): i( k j x + ωt ) p( x, y, t ) = B j os( λ j y ) e (19) j = 1 t is noted that the solution is omposed of different modes, and amplitude B j depends on the existing onditions on the downstream fae of that region. Moreover, parameters λ j and k j are defined as: ( j 1) π λ j = (0a) H ω k j + λ j = (0b) Herein, H represents the water depth. Based on (0b), one an atually determine the j-th wavenumber) as follows: k = ± j i k j (referred to as ω λ j (1)

6 Although, there are two options in this definition, the negative sign is merely admissible for a semiinfinite region extending to infinity in the negative x-diretion as in our present ase. This is due to the fat we are only interested with evanesent modes (imaginary wavenumber) or traveling modes (real wavenumber) whih are deaying or propagating towards the upstream diretion. t is also noted that the j-th wavenumber ( k ) beomes zero at a ut-off frequeny referred to as the j-th natural j r frequeny of the reservoir (i.e., ω j ). This is obtained by substituting (0a) in (1) under that ondition whih results in: r ( j 1) π ω j = () H Eq. (1) with the admissible negative sign, may also be written as: ω i ( j 1) Ω k j = 1 ( 1) Ω j with the help of dimensionless frequeny Ω : ω Ω = ω r 1 (3) (4) Let us now desribe the strategy adopted in the Wavenumber approah for the seletion of parameter k in Eq. (18): The proposed method is to define k based on different wavenumbers (i.e., by utilizing Eq. (3)) for various frequeny ranges. n partiular, use the following plan: 1 ; r k = k for 0 Ω 3 (0 ) or ω ω (5a) ; r r k = k ( 1) Ω ( + 1) ( < + 1) j for j j or ω j ω ω j and j (5b) n other words, employ the most appropriate wavenumber for eah frequeny range. This is shown to be true in details elsewhere (Lotfi and Samii 011). 3. MODELNG AND BASC PARAMETERS The introdued methodology is employed to analyze an idealized dam-reservoir system. The details about modeling aspets suh as disretization, basi parameters and the assumptions adopted are summarized in this setion. 3.1 Models An idealized triangular dam with vertial upstream fae and a downstream slope of 1:0.8 is onsidered on a rigid base. The dam is disretized by 0 isoparametri 8-node plane-solid finite elements. As for the water domain, two strategies are adopted (Fig. ). For the FE-FE method of analysis whih is our main proedure, only the reservoir near-field is disretized and the Wavenumber ondition is employed on the upstream trunation boundary aording to the strategy disussed. The length of this near-field region is denoted by L and water depth is referred to as H. Three ases are onsidered. These are in partiular the L/H values of 0., 1 and 3 whih represent low, moderate and high

7 reservoir lengths. This region is disretized by 5, 5 and 75 isoparametri 8-node plane-fluid finite elements for three above-mentioned L/H values, respetively. For the FE-(FE-HE) method of analysis, the reservoir domain is divided into two regions. The nearfield region is disretized by fluid finite elements, and the far-field is treated by a fluid hyper-element. Of ourse, it should be emphasized that this option is merely utilized to obtain the exat solution (Lotfi 001). Moreover, it is well-known that the results are not sensitive in this ase to the length of the reservoir near-field region or L/H value. (a) (b) Figure. The dam-reservoir disretization for L/H=1; (a) FE-FE Model, (b) FE-(FE-HE) Model 3. Basi Parameters The dam body is assumed to be homogeneous and isotropi with linearly visoelasti properties for mass onrete: Elasti modulus ( E d ) = 7.5 GPa Poison's ratio = 0. 3 Unit weight = 4.8 kn/m Hystereti damping fator ( β d ) = 0.05 The impounded water is taken as invisid and ompressible fluid with unit weight equal to kn/m, and pressure wave veloity = 1440 m/se.

8 4. RESULTS t should be emphasized that all results presented herein, are obtained by the FE-FE method disussed, by imposing wavenumber ondition on the trunation boundary. Moreover, the response for eah ase is ompared against the exat result obtained by FE-(FE-HE) analysis tehnique. As mentioned previously, three ases are onsidered. These are in partiular the L/H values of 0., 1 and 3 whih represent low, moderate and high reservoir lengths. The results for these three ases are presented in Fig. 3. The response illustrated in eah ase is the transfer funtion for the horizontal aeleration at dam rest with respet to horizontal ground aeleration. t is noted that they are plotted versus the dimensionless frequeny. The normalization of exitation frequeny is arried out with respet toω 1, whih is defined as the natural frequeny of the dam with an empty reservoir on rigid foundation. a) L/H=0. b) L/H=0. ) L/H=3 Figure 3. Horizontal aeleration at dam rest due to horizontal ground motion Let us first onsider the very low reservoir length (i.e., L/H=0. ), whih is a hallenging test for examining any type of absorbing boundary ondition. t is observed that the response is relatively lose to the exat response in most frequeny ranges (Fig. 3a). However, there exist errors in the range of 5% at the major peaks of the response. Of ourse, this is still believed to be a remarkable result for suh a hallenging test. For the moderate and high reservoir lengths onsidered (i.e., L/H=1, 3), it is notied that the response agrees very well with the exat response for both ases (Figs. 3b and 3). Moreover, there

9 are no signs of distortions in the response for the high reservoir length, ontrary to what is notied for other usual alternative absorbing boundary onditions suh as Sommerfeld and Sharan onditions (Lotfi and Samii 011). 5. CONCLUSONS The formulation based on FE-FE proedure for dynami analysis of onrete dam-reservoir systems, was reviewed. Moreover, the Wavenumber approah was disussed in that ontext. A speial purpose finite element program was enhaned for this investigation. Thereafter, the response of an idealized triangular dam was studied due to horizontal ground motion for that approah. Overall, the main onlusions obtained by the present study an be listed as follows: t is notied that the response agrees very well in omparison with the exat response for the moderate as well as high reservoir lengths. However, errors in the range of 5% are notied at the major peaks of the response for the low reservoir length ( L/H=0. ), whih is still believed to be a remarkable result for suh a hallenging test. There are no signs of distortions in the response under any irumstanes. Obviously, the main disadvantage of this ondition is that it annot be utilized in time domain, and it is only suitable for frequeny domain. The Wavenumber approah is ideal from programming point of view due to the loal nature of Wavenumber ondition imposed on trunation boundary. t an also be envisaged as a great substitute for the rigorous FE-(FE-HE) type of analysis whih is heavily relying on a hyper-element as its main ore. t is undeniable that the rigorous approah is signifiantly more ompliated from programming point of view and also muh more omputationally expensive. REFERENCES Basu, U. and A.K. Chopra (003). Perfetly mathed layers for time-harmoni elastodynamis of unbounded domains: theory and finite-element implementation. Computer Methods in Applied Mehanis and Engineering, 19(11-1): p Berenger, J.P. (1994) A perfetly mathed layer for the absorption of eletromagneti waves. J. Computat. Phys., 114 (): p Chew, W.C. and W.H. Weedon (1994). A 3D perfetly mathed medium from modified Maxwell's equations with strethed oordinates, Mirow. Opt. Tehnol. Lett., 7(13): p Chopra, A.K. (1967). Hydrodynami pressure on dams during earthquake. Journal of the Engineering Mehanis Division, ASCE, 93: p Chopra, A.K., P. Chakrabarti and S. Gupta (1980). Earthquake response of onrete gravity dams inluding hydrodynami and foundation interation effets. Report No. EERC-80/01, University of California, Berkeley. Fenves, G. and A.K. Chopra (1985). Effets of reservoir bottom absorption and dam-water-foundation interation on frequeny response funtions for onrete gravity dams. Earth. Eng. Strut. Dyn., 13: p Givoli, D. and B. Neta (003). High order non-refleting boundary sheme for time-dependent waves. Journal of Computational Physis, 186: p Givoli, D., T. Hagstrom and. Patlashenko (006). Finite-element formulation with high-order absorbing onditions for time-dependent waves. Computational Methods in Applied Mehanis and Engineering, 195: p Hagstrom, T. and T. Warburton (004). A new auxiliary variable formulation of high order loal radiation boundary ondition: orner ompatibility onditions and extensions to first-order systems. Wave Motion, 39: p Hall, J.F. and A.K. Chopra (198). Two-dimensional dynami analysis of onrete gravity and embankment dams inluding hydrodynami effets. Earthquake Engineering and Strutural Dynamis, 10: p Higdon, R.L. (1986). Absorbing boundary onditions for differene approximations to the multi-dimensional wave equation. Mathematis of Computation, 176: p

10 Lotfi, V., Frequeny domain analysis of gravity dams inluding hydrodynami effets. Dam Engineering, (1): p Lotfi, V. and Samii, A. (011). Dynami analysis of onrete gravity dam-reservoir systems by Wavenumber approah in the frequeny domain., Submitted to the Journal of Earthquakes and Strutures. Sharan, S.K. (1987). Time domain analysis of infinite fluid vibration. nternational Journal for Numerial Methods in Engineering, 4: p Sommerfeld, A. (1949). Partial differential equations in physis. Aademi press, NY. Waas, G. (197). Linear two-dimensional analysis of soil dynamis problems in semi-infinite layered media. Ph.D. Dissertation, University of California, Berkeley, California. Zienkiewiz, O.C. and R.L. Taylor (000). The finite element method. Butterworth-Heinmann.

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