Application of direct computing of one-way coupling technique in seismic analysis of concrete gravity dam

Quoc Cong TRINH*, Liao-jun ZHANG

College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, P. R. China

Application of direct computing of one-way coupling technique in seismic analysis of concrete gravity dam

Quoc Cong TRINH*, Liao-jun ZHANG

College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, P. R. China

A dam-reservoir system subjected to an earthquake is a nonlinear system, because the fluid equations are always nonlinear regardless of the linear or nonlinear model used for the dam body. Therefore, transient analysis is necessary. In this study, dam-reservoir interaction during earthquake excitation was modeled by utilizing coupled finite element equations based on the Eulerian approach. Direct computing of the one-way coupling technique was used to solve the coupled equations. This technique is based on a simple assumption that the fluid hydrodynamic pressure is applied to the dam body while the deformation of the dam has no influence on the water field. Seismic response analysis of the Sonla concrete gravity dam constructed in Sonla Province, Vietnam was carried out as a verification example. The results of the methodology introduced are in close agreement with results of the iterative method and the solution procedure is found to be less time-consuming than that of the iterative method. This method is very convenient and can be easily implemented in finite element programs with fluid-structure interaction modules.

dam-reservoir interaction; time domain; one-way coupling; Sonla Dam

1 Introduction

A dam-reservoir system is a fluid-structure interaction problem. This system can be categorized as a coupled field system in which two physical domains, fluid and structure, interact at the interface. Most dam-reservoir analyses are based on one of two approaches: the Eulerian approach or the Lagrangian approach (Olson and Bathe 1983). In the Eulerian approach, displacements are the variables in the solid domain and pressures are the variables in the fluid domain. In the Lagrangian approach, the behavior of the fluid and structure is expressed in terms of displacements. Seismic analyses of dam-reservoir systems have been carried out using the Eulerian and Lagrangian approaches (Singhal 1991; Calayir et al. 1996; Bayraktar et al. 1996).

Analysis of dam-reservoir systems subjected to earthquake loads is a complex problem. There are different approaches to the solution of the coupled field problem. Traditionally, the frequency domain approach is utilized to obtain the linear dynamic response of the dam (Chopra et al. 1980). However, it is not applicable to nonlinear problems and does not reflectthe dam behavior during an earthquake period. Another approach to determining the linear and nonlinear response of the dam-reservoir system is the added mass approach. This approach is computationally very efficient but may not be suitable for analysis of cracking in the dam structure (Ghaemian and Ghobarah 1999).

A dam-reservoir system subjected to a strong earthquake is likely to behave nonlinearly, even though the concrete material of the dam remains elastic. Therefore, a transient analysis of the structure interacting with a fluid and subjected to earthquake ground motion is necessary for a realistic analysis. In general, an algorithm based on the iterative solvers can be applied. However, these methods require a large amount of memory and computing time. In this study, a formulation of fluid systems based on the Eulerian approach was obtained using the finite element method. The coupled equation of water-dam interaction was solved by utilizing direct computing of a one-way coupling technique based on the simplifying assumption that deformation of the dam does not influence the water field. Following from this, the Sonla gravity dam was analyzed in the time history domain with this technique.

2 Coupled finite element equation of dam-reservoir system

Dam-reservoir interaction is represented by two coupled differential equations of the second order. The equations of the dam and the reservoir can be written as (Ghaemian and Ghobarah 1999)

whereMis the mass matrix,Cis the damping matrix, andKis the stiffness matrix of the dam body;G,C′, andK′ are the assembled matrices of the fluid domain;Qis the coupling matrix (Zienkiewicz and Taylor 2000);f1is the vector of static load;U,, andare the vectors of displacement, velocity and acceleration of the dam body, respectively;P,, andare the vectors of hydrodynamic pressure, the first and the second order of the time derivatives of vectorP, respectively;U˙˙′ is the ground acceleration;ρis the density of the fluid; andFis the vector of force due to acceleration at the dam-reservoir and reservoir-foundation boundaries and due to truncation at the far boundary.

3 Direct computing of one-way coupling technique for solving coupled equation

A direct integration scheme is used to find the displacement and hydrodynamic pressure at the end of timeti+1, using the displacement and hydrodynamic pressure at timeti. In 1959, Newmark developed a step-by-step integration method based on the following equations (Chopra 2001):

where Δtis the time step.γandβare the integration parameters determining the stability and accuracy characteristics of the method. Typical selections ofandare satisfactory from all points of view.

The coupled Eqs. (1) and (2) at timeti+1can be written as follows:

where

In this equation, the fluid and solid solution variables are fully coupled. In general, an iterative solution is used to solve the coupled equations. However, since the assumption is that deformation of the dam is minimal, its influence on hydrodynamic pressure can be ignored. Therefore, this coupled equation can be solved without using an iterative scheme.

From the assumption thatUi+1equals zero, based on Eq. (4),can be obtained:

4 Application in seismic analysis of Sonla Dam

The analysis of Sonla Dam was considered as a verification example. Sonla Dam is located on the Da River in Sonla Province, Vietnam. It is 130 m high and 95 m wide at the bottom. Analysis of dam-reservoir interaction has been implemented in ADINA (automatic dynamic incremental nonlinear analysis) using the one-way coupling algorithm described above. The accuracy and efficiency of this software has been tested with many numerical examples (Bathe 2002).

4.1 Selected model

Two cases of the Sonla concrete gravity dam with different reservoir levels were analyzed to demonstrate the applicability and accuracy of the proposed methods. In the first case, a full reservoir with a height of 122 m above the base was considered. In the second case, the water level of the reservoir was the dead water level with a height of 82 m above the base. The dam-reservoir system was analyzed using a two-dimensional model with a unit thickness that is well accepted for a typical gravity dam. The dam and the reservoir were respectively represented by 608 four-node and 425 four-node isoparametric quadrilateral solid and fluidfinite elements for the first case, and 608 four-node and 300 four-node isoparametric quadrilateral solid and fluid finite elements for the second case. A fluid-structure interaction boundary condition was applied upstream of the dam and at the bottom of the reservoir. Initial conditions were a hydrostatic load and self-weight of the dam. The massless foundation input model was used in this study (Bayraktar et al. 2005). The finite element mesh in the first case is shown in Fig. 1.

Fig. 1 Finite element mesh of Sonla Dam-reservoir-foundation system for case 1

In order to determine the hydrodynamic pressure on the dam due to horizontal ground motion with the assumption of a finite reservoir, the reservoir must be truncated at a reasonable distance. The Sommerfeld boundary condition, which is based on the assumption that, at a great distance from the dam face, the outgoing wave can be considered a plane wave, is commonly used. Hanna and Humar (1982), Humar and Roufaiel (1983), and Sharan (1986, 1987) used a radiation condition that models the loss of the outgoing wave over a wide range of excitation frequencies. In this analysis, the Sharan (1986) radiation boundary condition was applied at a distance from the dam equal to five times the dam height. Truncation of the boundary was equivalent to adding dampers and springs to absorb the outgoing waves.

4.2 Basic parameters and loads

The concrete was assumed to be homogeneous and isotropic. The modulus of elasticity, unit weight, and Poisson’s ratio of the concrete were taken to be 25.5 GPa, 2 449 kg/m3, and 0.167, respectively. The water was considered a compressible fluid with a weight density of 1020 kg/m3. The Rayleigh damping matrix was applied and the corresponding coefficients were determined such that equivalent damping for the frequencies close to the first and the third mode of vibration would be 5%.

The dam was subjected to a self-weight load, hydrostatic load, hydrodynamic load, and earthquake load. The seismic excitation considered in this study was an acceleration spectrum given by the Vietnam Institute for Building Science and Technology (2006). The artificial acceleration history curve was computed from the acceleration spectrum correlatively. The response spectrum at the location of Sonla Dam and artificial acceleration are shown in Fig. 2 and Fig. 3, respectively.

Fig. 2 Response spectrum

Fig. 3 Time history of acceleration

4.3 Analysis results

Two cases of the Sonla concrete gravity dam with different reservoir levels were analyzed. In the first case, a full reservoir was considered and the structure had a fundamental frequency of 2.08 Hz. The second case had a typical configuration of a concrete gravity dam with a fundamental frequency of 2.40 Hz and a dead water level.

For comparison purposes, the Sonla Dam model was analyzed using both the one-way and the iterative methods. The results for the first case are presented in Fig. 4, and the results for case 2 are presented in Fig. 5. The peakz-direction stress at the dam heel and the peak displacement at the dam crest for the two cases are shown in Table 1.

Fig. 4 Analysis results for case 1

Fig. 4 shows the time history for the horizontal component of displacement at the crest and the vertical-direction stress (z-direction stress) at the dam heel for the first case. The dam crest displacement was measured with respect to the base motion. As for verification, the model of Sonla Dam was also analyzed with the iterative method. It is observed that the histories of displacements andz-direction stress follow very similar trends. The horizontal displacement at the dam crest andz-direction stress at the dam heel of the two methods for the second case are presented in Fig. 5. In this case, excellent agreement is found between the results obtained from the proposed method and the iterative method. Table 1 shows that the maximum values of displacement at the dam crest in the one-way coupling method are different by less than 6%, while the maximum values of thez-direction stress at the dam heel are different by less than 8%. Therefore, it can be concluded that results of the present studyare in good agreement with the results of the iterative computing method. Table 1 also shows that the calculation time of the one-way coupling method is much less than that of the iterative method. The solution procedure of this method is found to be 12 times less time-consuming than that of the iterative method.

Fig. 5 Analysis results for case 2

Table 1 Comparison between results of one-way coupling method and iterative method

5 Conclusions

A technique has been proposed for the dynamic analysis of concrete gravity dams. Coupled equations of dam-reservoir interaction based on the Eulerian approach were obtained and solved by utilizing the one-way coupling technique. The analysis of Sonla Dam was considered as a verifying example. A two-dimensional model of the dam-reservoir-foundation system was analyzed. The results were also compared with the results of the iterative method. Overall, the main conclusions are as follows:

(1) The results of the present study are in close agreement with results of the iterative method.

(2) The methodology introduced takes less calculation time and computer memory than other iterative methods. This is very significant to the models using a large number of finite elements.

(3) The proposed method is very convenient and can be easily implemented in finite element programs with fluid-structure interaction modules.

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*Corresponding author (e-mail:quoccongdhtl@yahoo.com)

Received Nov. 6, 2009; accepted Feb. 22, 2010