Application of pseudo-symmetric technique in dynamic analysis of concrete gravity dams
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1 Application of pseudo-symmetric technique in dynamic analysis of concrete gravity dams V. Lotfi Department of Civil and Environmental Engineering, Amirkabir University, Iran Abstract A new approach is presented for seismic analysis of concrete gravity dams in the time domain. The technique is based on symmetric formulation of dam-reservoir systems utilizing pressure degrees of freedom in the fluid domain. The method is explained initially in relation to other available techniques. Following from this, the analysis of a typical concrete dam is considered as a verification example by utilizing a modified program based on this technique. The methodology introduced is very convenient and can be easily implemented in general-purpose finite element programs with regard to their fluid-structure interaction modules. 1 Introduction There has been extensive research in relation to Fluid-Structure Interaction problem encountered in dynamic analysis of concrete dams. Some of these studies are formulated very efficiently in frequency domain [l, 21. However, these methods are limited to linear dynamic analysis. Others, which have studied techniques suitable for time domain, are actually encountered with the solution of an unsymmetric coupled equations. In general, algorithms relying on unsymmetric solvers can be applied as the first alternative. However, the efficiency would be very low. There is also a technique available which transforms the coupled Fluid-Structure equations to symmetric relations by formulating the fluid elements in terms of velocity potential degrees of freedom [31. In this paper, a symmetric approach based on nodal pressures degrees of freedom [4], is used as the basis of the study. The method is modified and a technique is proposed that is referred to as Pseudo-Symmetric Approach. The procedure is described initially in relation to other available techniques. Meanwhile, the methodology is implemented in a special computer program "MAP-76" [5]. Subsequently, the analysis of Pine Flat Dam is considered as a verification
2 208 Fluid Structure Interaction 11 example. The results are discussed and they are also compared with a previous study from accuracy point of view. 2 Formulation Let us consider a general two-dimensional dam-reservoir system interacting internally due to ground excitations. The dam and water domain, are assumed to be discretized by plane solid and fluid finite elements, respectively. It is easily shown that the coupled equations of dam-reservoir system can be written in combined form as [4]: M, C, K in this relation represent mass, damping and stiffness matrices of the dam body, respectively. r is the vector of dam nodal relative displacements. a, denotes the vector of ground accelerations, and J is a matrix with each two rows equal to a 2 X 2 identity matrix (its columns correspond to unit rigid body motion in horizontal and vertical directions) [l, 41. Furthermore, G, L, H in this relation, are assembled matrices of fluid domain, and the matrix B is usually referred to as the interaction matrix 14, 61. Meanwhile, p is the vector of nodal pressures, and p, c denote mass density and pressure wave velocity of fluid. It is clear that the overall mass and stiffness matrices of the system are unsymmetric in the above relation. Therefore, this coupled equation is usually solved by using the scattered approach [4, 71. That is separating dam and reservoir equations and solving the problem by an iterative scheme. Although there is no need for unsymmetric solvers in this approach, the major drawback is that it requires iterations even for linear problems. Meanwhile, the algorithm is not appealing to be coded in a general-purpose finite element program, which considers the fluid-structure interaction problem as one of its special cases. Another remedy for avoiding unsymmetric solvers is to use velocity potentials instead of pressures as degrees of freedom in the water domain. Because in that case, one would obtain the following coupled equations for the system [3, 61 that replaces equation (1): represents the vector of nodal velocity potentials. Meanwhile, v, is the vector of ground velocities. It is easy to see that this relation can become symmetrical by simply multiplying the lower partitions by a factor of -p. In this case, the mass, damping and stiffness matrices of the system are all symmetric. This is the formulation applied in some of the general-purpose finite
3 Fluid Structure Interaction I1 209 element programs [3]. The disadvantage of this method is that vector of ground velocities are required to be calculated which could also introduce some error into the solution. Furthermore, the calculation of nodal velocity potential quantities, are less attractive for most engineers in comparison to nodal pressures. 2.1 Symmetric approach based on nodal pressures It is well known that one can obtain symmetric coupled equations even in the case where nodal pressures are used as degrees of freedom in the fluid domain [4]. However, the method is presented in a slightly different technique in this section, which can be utilized as the base of the Pseudo-Symmetric approach. The method commences by modifying the relation (l), such that the lower partitions of that matrix equation, is multiplied by a factor of - (pao)-l. The parameter a, utilized in this factor, is one of the coefficients obtained in Newmark's algorithm [4], defined in terms of Newmark's parameter a (a =0.25 in constant average acceleration method) and time increment At : Therefore, the coupled equations of dam-reservoir system would be: Or alternatively in a more compact form: - - It is obvious that matrices M, C and K can be written as sum of the symmetric and unsymmetric parts as below: It is noted from equation (4) that the damping matrix (-6) is totally symmetric, and the following relation also holds:
4 21 0 Fluid Structure Interaction 11 Applying Newrnark7s algorithm on equation (5), the effective stiffness matrix (K) and the force vector at instant n+l (R,,, ) would be: + (Ms +Mu )(ao % +a, Gn +a, ;n ) + CS (a, F, +a, pn +a, ) Substituting (7) into (g), it yields, Although total mass and stiffness matrices of the system are unsymmetric, it is clear from relation (10) that the effective stiffness matrix is symmetric. This means that the need for unsymmetric solver is eliminated. 2.2 Pseudo-symmetric approach Considering the coupled equations (1) of dam-reservoir system, it is noticed that unsymmetric terms in the total mass and stiffness matrices are related to B matrix or its transpose. This matrix is usually obtained by assemblage of contributing submatrices of interface elements located at fluid-solid contact, or even surfaces where fluid elements are adjacent to rigid or absorbing boundaries. However, to make it more efficient from programming point of view, one can eliminate these interface elements and consider its effect as part of adjacent fluid element matrices. In that case, matrices of the i& fluid element which contribute to the corresponding total mass, damping and stiffness matrices of the system would be generally as follows, respectively: In the proposed method, interface elements are excluded and their effects are considered as part of fluid element matrices as explained above. Furthermore, everything is made symmetric from the very beginning. This means that fluid element matrices are made symmetric by transferring the unsymmetric part of the stiffness matrix to the mass matrix while multiplying it by a factor of ail. Meanwhile, rows corresponding to the pressure degrees of freedom are multiplied by a factor of - (pao)-l. Therefore, when the total mass, damping and stiffness matrices of the system are formed, one obtains:
5 Fluid Structure Interaction I The total matrices of the system are different from corresponding matrices obtained in the previous method (relation (6)), and in this case all matrices are symmetric. However, the effective stiffness matrix, which results from applying Newmark's method becomes, Which is exactly the same as effective stiffness matrix resulted for the previous method (relation (10)). Therefore, the need for unsymmetric solver is once more eliminated. The formulation of the effective force vector in this method is written as follows: - = + (Ms +MU +M; );(a, 5 +a, Pn +a, ;n ) + CS (a, i;, +a, kn +a, In ) In this relation, a special operator * is utilized, which emphasizes that in this matrix operation, the stored terms corresponding to non-zero elements of M; must not participate. Therefore, the resulting effective force vector would become exactly as in relation (9), obtained for previous method. The location of these storages would be very clear in the total mass matrix, if all of the pressure degrees of freedom are stored at the end. The unhown vector calculated at each step of the Pseudo-Symmetric algorithm would definitely be the same as in previous method, since the effective stiffness and force vector are exactly the same. Meanwhile, all the matrices of fluid elements are symmetric like any solid elements, even though this was forced artificially at the element level. This approach is very convenient for general-purpose programs, as far as symmetric upper skyline storage scheme that can be utilized for total mass, damping and stiffness matrices. Furthermore, it takes advantage of symmetric solvers to obtain the unknowns at each time step. 3 Application on concrete gravity dams In this section, the analysis of Pine Flat Dam is considered as a verification example. This particular dam was selected, because many researchers have analyzed it based on different approaches [l, 81. The dam is m high, with the crest length of m and it is located on the King's River near Fresno, California. A special computer program "MAP-76" is used as the basis of the present study [5]. The program was modified by introducing two-dimensional fluid elements with pressure degrees of freedom into the program, as well as implementing the other required changes in the code based on the Pseudo-Symmetric methodology. All fluid element matrices were defined symmetrical artificially, as explained in the previous section.
6 21 2 Fluid Structure Interaction Finite element idealization Selected models Most researchers have considered the analysis of Pine Flat Dam based on a two dimensional model, which is well accepted for a typical gravity dam. That means, a section of Dam-Reservoir system is considered with unit thickness. The two dimensional elements of dam body is usually assumed in a state of plane stress, while fluid elements formulation is presuming plane strain conditions for fluid domain. In the present study, a two-dimensional model of Pine Flat Dam is considered (Figure 1). The model is based on normal practice idealization as mentioned above. The 8-node plane solid and fluid finite elements are used for the dam and reservoir domains, respectively. The model consists of a total of 439 nodes and 568 degrees of freedom and the mesh includes 40, 90 plane solid and fluid elements, respectively. Figure 1. Finite Element mesh of Dam-Reservoir system Basic parameters The concrete is assumed to be homogeneous and isotropic with the following basic properties: Elastic modulus E, = GPa Poisson's ratio v, = 0.20 Unit weight y, = 24.8 w/m3 The water is taken as compressible, inviscid fluid, with weight density of 10.0 kn/m3 and pressure wave velocity of mls. The water level is considered at the height of m above the base, similar to the reference [l]. The length of the reservoir discretized is 200 m, and Sornmerfeld boundary condition is applied at the upstream boundary, while the reservoir bottom condition is assumed completely reflective. Moreover, in the analysis carried out, the Rayleigh damping matrix is applied and the corresponding coefficients are determined such that equivalent damping for frequencies close to the first and third modes of vibration would be 5% of the critical damping Loading It should be mentioned that static loads (weight, hydrostatic pressures) are each visualized as being applied in one separate increment of time. Therefore, the
7 Fluid Structure Interaction I same time step of 0.01 second, which is chosen in dynamic analysis, is also considered as time increment of static loads application. It is noted that time for static analysis is just a convenient tool for applying the load sequentially, but it is obvious that inertia and damping effects are disregarded in the process. In this respect, the dead load is applied in one increment and hydrostatic pressures thereafter in another increment at negative range of time. At time zero, the actual dynamic analysis begins with the static displacements and stresses being applied as initial conditions. The dynamic excitation considered, is the S69E component of Taft earthquake records, which is applied in the horizontal direction. The time duration utilized, is 13 seconds. 3.2 Analysis results As mentioned in the previous section, a two-dimensional model of Pine Flat Dam is considered. The analysis is carried out based on the Pseudo-Symmetric methodology by utilizing the MAP-76 program. The model is analyzed, and results corresponding to the envelope of maximum tensile and compressive stresses are illustrated in Figures 2 and 3. Furthermore, time histories of some important quantities are presented in Figures 4 and 5. As for verification, some of the available results taken from the work of Chopra et a1 [l] are illustrated in Figure 6 for comparison purposes. However, it should be noticed that even though similar properties are used in both studies, there are still slight differences. In this reference, the dam is discretized by incompatible quadratic finite elements. Meanwhile, closed form solution is utilized for the effects of impounded water, which is derived for a region extending to infinity in the upstream direction, and also assuming a vertical boundary at the damreservoir interface. Furthermore, the analysis is carried out in frequency domain and hysteretic type of damping is employed. While in the present study, the solution is obtained in time domain and Rayleigh type of damping is utilized. Having the above comments in perspective, one can compare the envelope of maximum tensile stresses between the present study (Figure 2) and the reference (Figure 6a). It is observed that very good agreement exist for the magnitude and distribution of maximum tensile stresses in different regions. However, there seems to be some difference in the maximum value of tensile stress, which occurs at the dam heel. This discrepancy can be explained by the fact that, stress concentration is present in that location, and the reference is not providing additional contour lines close to this corner. The interpolated value in the present study results, which correspond to the location of maximum contour shown in the reference, would be roughly o, =2.35 Mpa. This shows a reasonable agreement (less than 5% difference) as far as the envelope of maximum tensile stresses is concerned. Additional result of the reference that can be compared, is the time history of horizontal displacement at dam crest. This is shown in Figure 6b and can be compared with the result of present work depicted in Figure 4a. It is observed that the history of displacements follow very similar trends and the maximum values are different by less than 2%. Therefore, it can be concluded that results of the present study are in good agreement with the reference [l].
8 21 4 Fluid Structure Interaction 11 Figure 2. Envelope of maximum tensile principal stresses (MPa) Figure 3. Envelope of maximum compressive principal stresses (MPa) Figure 4. Displacement histories at dam crest
9 DISPL - INCHES r&lo-w
10 21 6 Fluid Structure Interaction 11 4 Conclusions A technique is proposed for the seismic analysis of concrete gravity dams. The method is described, and the procedure is implemented in a special computer program "MAP-76". Meanwhile, analysis of Pine Flat Dam was considered as a verifying example. A two-dimensional model of dam-reservoir system is considered and analyzed. The results are also compared with the reference. Overall, the main conclusions obtained can be listed as follows: The results of present study are in close agreement with the reference. The small differences, which exist with respect to the reference, are in acceptable range, having in mind that there were some differences in the idealizations utilized as mentioned in the discussion of the results. The Pseudo-Symmetric approach is proved to be an effective technique for dynamic analysis of concrete gravity dams. The methodology introduced is very convenient and can be easily implemented in general-purpose finite element programs in regard to their fluid-structure interaction modules. Although the procedure was explained for linear dynamic analysis, it is easily expandable to nonlinear dynamic analysis of concrete dams, since it is carried out in time domain. References 1. Chopra, A. K., Chakrabarti, P., and Gupta, S., "Earthquake response of concrete gravity dams including hydrodynamic and foundation interaction effects", Report No. EERC-80101, University of California, Berkeley, Jan Fok, K.-L. and Chopra, A. K., "Earthquake analysis and response of concrete arch dams", Report No. EERC-85/07, University of California, Berkeley, July Olson, L. G. and Bathe, K. J., "Analysis of fluid-structure interactions. A direct symmetric coupled formulation based on the fluid velocity potential", Intern. Journal of Computers & Structures, Vol. 21, Nos. 112, Special Issue: Nonlinear Finite Element Analysis and ADINA, Zienkiewicz, O.C. and Taylor, R.L., "The Finite Element Method", 5th Edition, Butterworth-Heinemann, Oxford, UK, Lotfi, V., "MAP-76: A program for analysis of concrete dams", Amirkabir University of Technology, Tehran, Iran, Lotfi, V., "Dynamic analysis of concrete dams", Mahab Ghodss Consulting Engineers, Tehran, Iran, Cervera, M., Oliver, J. and Faria, R. "Seismic evaluation of concrete dams via continuum damage models", Journal of Earthquake Engineering & Structural Dynamics, Vol. 24, No. 9, Sept Feltrin, G., Wepf, D. and Bachmann, H., "Seismic cracking of concrete gravity dams", Journal of Dam Engineering, Vol. I, Issue 4, Oct
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