STUDY OF DAM-RESERVOIR DYNAMIC INTERACTION USING VIBRATION TESTS ON A PHYSICAL MODEL

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1 STUDY OF DAM-RESERVOIR DYNAMIC INTERACTION USING VIBRATION TESTS ON A PHYSICAL MODEL Paulo Mendes, Instituto Superior de Engenharia de Lisboa, Portugal Sérgio Oliveira, Laboratório Nacional de Engenharia Civil, Portugal Abstract This paper presents the main results of a water-structure dynamic interaction experiment under ambient excitation and impact-hammer excitation. Experimental tests were conducted on a physical model of a wall clumped at the base and submitted to water pressure. The main dynamic parameters of this system (natural frequencies, mode shapes and modal damping) were identified for different water levels. Test results are compared with the numerical results of a 2D finite element model. Test results lead to the conclusion that the Young modulus to be used on structural dynamic analysis should be evaluated by ultrasonic tests. On the other hand, it is concluded that the water effect on the dynamic behaviour of the structural system can either be simulated using added water masses or water finite elements. 1 Introduction The dynamic behaviour of large concrete dams is very complex due to the interaction phenomena involving the components of a structural system which includes the dam, the foundation and the reservoir (Figure 1). The analysis of this system is usually carried out taking into account some simple conservative hypotheses which allow the safety assessment of the structure. However, the dynamic behaviour of large concrete dams can only be evaluated after understanding the damfoundation, dam-reservoir and foundation-reservoir interface coupled problems. Figure 1 Dam-foundation-reservoir system. Presently, when a dynamic behaviour analysis of a large concrete dam is carried out, some doubts prevail, namely when the numerical results (based on the usual structural assumptions) are compared with experimental results (analysed using modal identification techniques). These doubts

2 are usually related with the value of the Young modulus considered on numerical models and the reliability of added water masses formulation (in accordance with Westergaard formula) for simulation of water effect. Therefore, the main purposes of this study are to check the use of Young modulus value obtained from ultrasonic tests and to verify the reliability of added water masses or water finite elements. These verifications are based on the comparison of numerical and experimental results from the physical model presented in Figure 2. To evaluate the water influence on the main dynamic parameters (namely on the evolution of the 1 st natural frequency) a set of vibration tests were carried out on a physical model, for a series of water heights. A 2D finite element model was developed assuming two different hypotheses for water consideration: added water masses and water finite elements. 2 Physical Model A physical model of a concrete wall clumped at the base and submitted to water pressure was used to study the water-structure interaction. This model is used to simulate the dam-reservoir interaction phenomena. Figure 2 shows the geometric characteristics of the model D model plan view lateral view Figure 2 Main characteristics of the physical model. This laboratory model, made of concrete, was built over a plastic stuff in order to prevent the connection between the pavement and the model. It was then verified that the deformability of the model-pavement joint influenced the dynamic behaviour of the cantilever wall (due to undesirable movements at the model basis). To minimize these movements the model was connected to a reaction wall, as shown in Figures 5 and 6. However, due to constructive imperfections it has been difficult, until now, to solve this problem definitely. 3 Numerical Model A 2D finite element model was developed and implemented as a MatLab routine which allows both static and dynamic analysis. This routine makes it possible to simulate the hydrodynamic effect using added water masses or water finite elements. The main features of these formulations are described in this section. The damping effect was simulated using Rayleigh damping C = α M+β K where the damping matrix C is proportional to mass and stiffness matrices (M and K) [1, 2].

3 The developed algorithm is based on a time domain modal formulation using an analytic integration technique that is exact for history loads defined by linear branches. With this routine it is possible to choose some degrees of freedom in order to apply predefined history loads (e.g. random noise or impact loads). Finally, it is also possible to pick up some acceleration data from some degrees of freedom (defined by the user) and perform a simple modal identification based on peak picking technique (namely display an average normalized spectrum). This feature is used in order to compare numerical and experimental results. 3.1 Finite element mesh The 2D finite element mesh was developed using isoparametric 2 nd degree elements with 8 nodes, as shown in Figure 3. A Young modulus of E = 32.5GPa was used for the cantilever wall. This value of E was obtained experimentally from ultrasonic tests (Figure 4). For the model base an equivalent value of the Young modulus, Eeq = 70.0GPa, was used in order to take into account the stiffness induced by the lateral reservoir walls, which can not be considered in a simple 2D model. E=32.5 GPa E eq =70GPa Figure 3 Two-dimensional finite element mesh, using 8 nodes by element. Young modulus for dynamic analysis: ultrasonic tests In order to evaluate the concrete Young s modulus an ultrasonic test was carried out on a concrete cylindrical sample. The Young modulus was computed using the following formula from elastodynamics: ( + ν)( ν) ( 1 ν) E = v ρ where v is the measured velocity of pressure wave propagation in concrete ( v 3800 m / s ), 3 ρ= 24kN / m is the specific concrete mass and ν = 0.2 is the concrete Poisson s ratio. As referred above an average value of E 32.5GPa was obtained for E. The results presented in the following sections show that the value of E obtained from ultrasonic tests is definitely a suitable value to use in structural dynamic analysis. Values obtained from static tests are smaller and not suitable for this purpose.

4 Figure 4 Ultrasonic test performed on a concrete cylindrical sample in order to obtain the concrete Young s modulus. 3.2 Models for water effect simulation Added water masses Using Westergaard added water masses, the hydrodynamic pressure is considered on the boundary nodes that are in contact with water, and is given by: p =α u N i i i where p i and u N i are the hydrodynamic pressure and the normal acceleration on i node respectively, α i is the pressure coefficient computed according to Westergaard proposal [3]. However, in the finite element model we build a mass matrix, which is added to element mass matrices, obtained from: M = α L λ λ inm i i n m In the previous formula represent the unit normal direction vectors to the upstream face. Water finite elements L i is the influence length associated with i node, and finally λ n and The hydrodynamic pressure was also simulated using water finite elements that can be easily formulated writing the constitutive relationship for isotropic materials in terms of the bulk modulus, K, and the shear modulus, G [4]: v K v E = 31 2 E G = 21 ( ν ) ( + ν) where E is the Young modulus and ν is the Poisson ratio. In this case, the constitutive equation for plane strain hypothesis can be written as follows: 4 2 σ 11 Kv + G K 3 v G 0 3 ε Kv G K 3 v G 0 σ = + 3 ε22 σ G ε 12 For water finite elements it must be assumed that G 0 and Kv 2GPa (in accordance with the water pressure wave s propagation velocity of v = 1440m/s). For concrete the values of G = 13.54GPa and K v = GPa are considered. λ m

5 4 Experimental tests A set of vibration tests were carried out on a physical model build at ISEL. The data collected was used to perform the modal identification of the physical model (clumped wall) for different water levels. Due to the low ambient excitation that occurs in lab, an impact hammer was used to better excite the model, in order to improve the modal identification results. Figure 5 Physical model testing. Figure 6 Connection between the model and the reaction wall. As it can be seen in Figures 5 and 6 the model is connected to a reaction wall in order to guarantee an adequate restriction to longitudinal movements. The data acquisition system is composed by an OROS 35 8 channel, 8 accelerometers, cables and a lab top, as shown in Figure 5. In this paper we only evaluate the results obtained from the 6 accelerometers placed at a vertical line in the middle the outer face of the wall (Figure 5) the data collected with these accelerometers is enough for the analysis of the first 2D cantilever wall mode shapes. The acceleration measurements were recorded using 2048 Hz of sampling frequency during 5 minutes. 5 Results The main results obtained from numerical and experimental tests are shown in this section. Figure 7 shows the first four plane mode shapes and corresponding frequencies obtained numerically with reservoir empty. F 1 = Hz F 2 = Hz F 3 = Hz F 4 = Hz Figure 7 First plane mode shapes obtained numerically (without water). It must be emphasized, as referred above, that the identified frequencies for the first vibration modes show a good agreement with numerical results obtained considering the dynamic value of 32.5 GPa for the Young modulus (evaluated from ultrasonic tests). Figure 8 shows a comparison between the average normalized spectrums [5,6,7] obtained numerically and experimentally for three different situations: i) reservoir empty; ii) water height of 0.5m (half reservoir); and iii) water height of 1.0m (full reservoir).

6 Numerical results (water finite elements) Experimental results Empty reservoir Empty reservoir Amplitude((m/s 2 ) 2 /Hz) Amplitude ((m/s 2 ) 2 /Hz) Mode associated to an undesirable base movement -6-6 Water level 0.50 m Water level 0.50 m Amplitude((m/s 2 ) 2 /Hz) Amplitude ((m/s 2 ) 2 /Hz) -6-6 Water level 1.00 m Water level 1.00 m Amplitude((m/s 2 ) 2 /Hz) Amplitude ((m/s 2 ) 2 /Hz) -6-6 Figure 8 Average normalized power spectrums (0 to 24Hz). Figure 8 shows that there is a good agreement between the power spectrums computed numerically using the 2D finite element model and the power spectrums obtained experimentally. It can be seen that both, physical and numerical models, present a similar decrease of natural frequencies as the water level increases.

7 For high frequencies it appears that the experimental values of modal damping are higher than the corresponding damping values considered in the numerical model (Rayleigh damping formulation). For this reason it is difficult to identify the higher frequency modes from experimental data. The variation of the 1 st mode natural frequency with the water level is presented in Figure 9. In this case it can be seen that the water effect really influences the 1 st natural frequency values for water levels greater than 0.50m (half reservoir). The experimental results obtained for nine different water levels, from empty to full reservoir, show a good agreement with the numerical results, obtained considering the previously mentioned dynamic value for the concrete Young s modulus (32.5 GPa) and the both proposed methodologies for the water effect simulation: Westergaard formulation (scheme of added water masses) or water finite elements with null shear modulus. Frequency (Hz) Numerical results Added water masses Water F.E. Experimental results Water level (m) Figure 9 Comparison between numerical and test results, for 1 st frequency. It must be noted that for high water levels the numerical values of the first mode frequency are slightly smaller than the experimental frequency values. 6 Conclusions The concrete Young s modulus obtained from ultrasonic tests is a good estimation to consider in the development of reliable numerical models for dynamic analysis involving water-structure interaction phenomena. The two formulations used for the simulation of water effect under dynamic excitation (added water masses and water F.E.) are similar and both present a good agreement with the experimental results obtained for the variation of the 1 st natural frequency with the water level. The results presented in this paper show that some improvements on numerical and physical model will be useful. Namely, the Rayleigh damping formulation could be improved taking into account a split on the mass and stiffness matrices in order to separate the damping contribution of concrete and water. The physical model could also be improved in order to guarantee a rigid connection between the base of the model, the floor and the reaction wall. It would be interesting to use a new physical model with a thinner cantilever wall in order to obtain lower frequencies for the first vibration modes.

8 7 Acknowledgments Thanks are due to Eng. João Costa, Eng. Pedro Silva and Mr. António Fernandes for their technical support and to Eng. Luísa Braga for the text revision. We also thank FCT for providing financial support through the project Study of Evolutive Deterioration Processes in Concrete Dams. Safety Control over Time. 8 References [1] Chopra A. K.: Dynamics of Structures: Theory and Applications to Earthquake Engineering (2 nd Edition), Prentice Hall, USA. [2] Tedesco, J. W., McDougal, W. G., Ross, C. A.: Structural Dynamics: Theory and Applications. [3] Westergaard, H. M.: Water Pressures on Dams during Earthquakes, American Society of Civil Engineers, Transactions, vol. 98, pp , Discussion, pp [4] Zienkiewicz, O. C., Taylor, R. L.: The Finite Element Method McGraw Hill, London, UK. [5] Felber, A. J.: Development of Hybrid Bridge Evaluation System, PhD Thesis, University of British Columbia, Vancouver, Canada. [6] Bendat, J.S., and Piersol A.G.: Random Data: Analysis and Measurement Procedures (3 rd Edition), John Wiley & Sons, USA. [7] Brincker, R., Zhang, L., and Andersen P.: Modal Identification from Ambient Responses using Frequency Domain Decomposition, Proc. 18 th International Modal Analysis

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