Rong He School of Civil Engineering and Communication, North China University of Water Resources and Electric Power, Zhengzhou, , China

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1 doi: / Dynamic Characteristics Analysis of Wide Well Ditch Aqueduct Wei He * School of Civil Engineering and Communication, North China University of Water Resources and Electric Power, Zhengzhou, , China *Corresponding author( hwhr123@sina.com) Xu Liu School of Civil Engineering and Communication, North China University of Water Resources and Electric Power, Zhengzhou, , China Rong He School of Civil Engineering and Communication, North China University of Water Resources and Electric Power, Zhengzhou, , China Abstract As a water conveyance structure, aqueduct plays an important role in large-scale water conservancy project. A three-dimensional finite element model is established by the finite element software ANSYS and the dynamic characteristics of aqueduct is analyzed based on the model. The additional mass method is used to simulate fluid-solid coupling and calculate the nature frequencies and vibration modes of aqueduct under different water depth conditions. The results show that the effect of water on aqueduct natural frequency is obvious, and the natural frequency decreases with the increases of water depth. In addition, when the structure is in vibration, the groove holder, the bearing and the connecting place are deformed. Where the stress is concentrated, it is easy to be destroyed. The transverse stiffness of wide wells ditch aqueduct is so weak that some appropriate measures should be taken to improve the bearing capacity and local stability. Key words: Finite element method, Added mass, Dynamic analysis, Modal analysis. 1. INTRODUCTION The aqueduct is an important hydraulic engineering structure, along with the development of water conservancy case, especially in the modern construction of South-to-North Water Transfer Project, the aqueduct structure has been widely used in this engineering, and plays an indispensable role (Li and Chen, 2011). Design of aqueduct structure rationality, applicability, economy and scientific, has become a key problem in inter-basin water transfer project. At present, domestic and foreign scholars have done a lot of research on the dynamic characteristics of the aqueduct. A new method using ordinary differential equation (ODE) solver is applied in order to ease the dynamic analysis of tall-pier aqueduct. By simplifying the boundary conditions associated with fluid-structure interaction at pier top and considering the pier as a continuous beam instead of discretized elements, this paper converts the dynamic analysis of tall-pier aqueduct into standard forms of ODE Eigen problem and general boundary value problem, which can be solved by ODE solver based on the presented ODE eigenvalue algorithm, modal superposition and Newmark-semi-discretization method. For illustration, two numerical examples are utilized to show the feasibility of this method to obtain dynamic characteristic, elastic seismic response and nonlinear vibration isolation response of tall-pier aqueduct. The results show that compared to the finite element method, the model and method in this paper is not only reliable, but time-saving (Hu and You, 2010). R Hamilto and K Baldwin identifies a novel relationship between cerebrospinal fluid (CSF) stroke volume through the cerebral aqueduct and the characteristic peaks of the intracranial pulse (ICP) waveform. ICP waveform analysis has become much more advanced in recent years; however, clinical practice remains restricted to mean ICP, mainly due to the lack of physiological understanding of the ICP waveform. In the study, it shows the width of the second peak (ICP-Wi2) of an ICP pulse wave is positively related to the volume of CSF movement through the cerebral aqueduct. This finding is an initial step in bridging the gap between ICP waveform morphology research and clinical practice (Hamilto and Baldwin, 2012). Because of the three-dimensional finite element analysis can reflect the whole mechanical characteristics of the structure, and more easily to process complex boundary conditions and structure form, making a reasonable correction to the structure design results calculate based on planar structure. At present, most of the domestic scholars in the study of prestressed concrete aqueduct is using the equivalent load method, but this method 63

2 cannot simulate the prestressed reinforcement element, the stress analysis results have a great error with the actual value. Therefore, a three-dimensional finite element model of the aqueduct is established using general finite element software ANSYS in this paper, in considering the prestressed, the vibration characteristics of aqueduct is calculated under the conditions of space level, semi tank level, design level and bank-full level during the operational period. The additional mass method is used to simplify the dynamic interaction between water and aqueduct wall and considering the effect of interaction between fluid and aqueduct structure to the dynamic characteristics of the aqueduct, natural frequency and vibration mode of the aqueduct is calculated in different water depth conditions, the results could provide suggestions for the design of the aqueduct. 2. ENGINEERING SITUATION The wide wells ditch aqueduct is located in TongXinYingShui engineering channel in NingXia of which the design flow is 4.0m 3 /s, longitudinal slope is 1/1000, and full length is 260m. This is a double cantilever beam type aqueduct, U type groove body is reinforced concrete structure, which is U type. Each slot is 20 meters long, the two ends of the groove are lay on the pier foundation consist of the brick and stone. The concrete density is 2500kg/m 3, the elastic modulus is 3E10Pa, and the Poisson's ratio is The density of the reinforcement is 7800kg/m 3, the elastic modulus is 2E11Pa, and the Poisson's ratio is 0.3 (Zhao and Li, 1999). 3. CALCULATION MODEL 3.1. Theoretical Calculation and Analysis According to the basic theory of structural dynamics, the undamped free vibration equation of aqueduct is as Mu Ku 0 (1) where M is mass matrix, K is stiffness matrix, u is displacement vector, and u is acceleration vector. Assumed structural displacement form is, u sin( t ) (2) where is natural frequency of structure, is the main vibration type of structure, 2 u sin( t ) (3) Substitute (2) (3) in (1), it will be obtain the characteristic equation of structural vibration, 2 ( K M ) 0 (4) In the process of dynamic characteristic analysis of finite element model, the modal analysis method is usually used to obtain the natural frequency and main vibration mode. These parameters are the important parameters of the structure under dynamic loads, and also the basis of other types of dynamic analysis of the structure. Modal analysis is a modern method to study the dynamic characteristics of structures, and it is the application of the system identification method in the field of engineering vibration. Modal analysis transform physical coordinate into modal coordinate system in differential equations of vibration of linear fixed system, the equations are decoupled, become a group of independent equations described by modal coordinates and modal parameters, in order to calculate the modal parameters of the system. The transformation matrix of coordinate transformation is the modal matrix, each column is mode shape. The final goal of modal analysis is to identify the modal parameters of the system, and provide the basis for the analysis of the vibration characteristics of the structural system, the fault diagnosis and prediction of vibration and the optimization of the dynamic characteristics of the structure Finite Element Model The units use finite element model are SOLID65, LINK 8 and MASS 21, SOLID65 unit is use for 3D model with or without steel bars, each unit has 8 nodes, each node has 3 degrees of freedom, namely x, y, Z three direction line displacement, this model has crack and crush function, it can be defined the reinforced condition on the three directions, also could simulate the stretch, compression, plastic deformation and creep of steel bar. The prestressed reinforced bar is simulated by LINK8 unit, the 3D bar element is tension and compression unit along axial direction, each node has 3 degrees of freedom can be used to simulate the truss, cable, rod, spring and so on. Boundary conditions for one side of aqueduct is roller constraint, constraint vertical and lateral displacement; the other side is articulated constraint, constraint vertical, longitudinal and transverse displacement. Finite element model is shown in Figure 1, it has nodes, units, and the unit length is 0.08m. 64

3 In the model, the position of the whole coordinate system is as follows: X- horizontal direction; Y- vertical direction; Z- direction of flow along the aqueduct. The model considers the effect of ordinary steel bar and prestressed reinforcement. Among them, the common reinforcement is simulated by the integral method, and the prestressed reinforcement is simulated by the separate model, without considering the bond slip between the steel bar and the concrete. The longitudinal prestressed reinforcement unit is shown in figure 2. Figure 1. Finite element model Figure 2. Longitudinal prestressed reinforcement unit 4. ANALYSIS OF CALCULATION RESULTS 4.1. Static Analysis In order to verify whether the aqueduct strength meets the requirements or not during operation period, the deformation and stress of aqueduct is analyzed in bank-full water conditions at first. Figure 3. Z direction stress in bank-full level(pa) 65

4 Figure 4. Z direction stress curve at bottom of aqueduct in bank-full level Figure 5. First main stress in bank-full level Figure 6. First main stress curve at bottom of aqueduct in bank-full level Through the above analysis results: aqueduct in bank-full water condition, the maximum deformation is 0.54mm, meet the requirement of deformation, the largest tension of aqueduct is 1.28MPa, less than concrete axial tensile strength design value 1.43MPa of C30, the largest aqueduct pressure is 4.63MPa, less than concrete axial compressive strength design value 14.3MPa of C30, aqueduct strength is meet the requirement in the bank-full level condition Dynamic Analysis Table 1 gives the first ten order vibration frequency of aqueduct when empty slot state, half tank level, design level and the bank-full water. Table 1. The first 10 step natural frequencies of aqueduct (Hz) Order Empty Semi tank Design Bank-full 66

5 tank level level level level The first mode is the main mode, which has an important meaning for the design of aqueduct (Bai, Li and Guo, 2008), from table 1 we can obtain, the empty slot, half tank level, design level, the bank-full water natural frequencies were , , , Influence of dynamic characteristics of water to aqueduct is obvious, the self-vibration frequency decreases with the increase of water depth. That is, the higher the quality, the lower the frequency, in agreement with the theoretical formula, so when analysis the dynamic characteristic of large aqueduct, must take the role of water into account. Table 2 gives the first five order vibration modes and displacement maps under the conditions of space level and design level. Table 2. The first 5 order natural frequencies and displacement of aqueduct (Hz) order The condition of space level The condition of design level

6 The first vibration frequency of the space level is Hz, the maximum displacement is mm, the aqueduct under anhydrous condition, the first mode is the main mode, and vibration model is transverse vibration. The displacement map indicates that the upper part of pull rod relative displacement is relatively large, the lower part relative displacement is relatively small, the reason is only a few pull rod at aqueduct body, the transverse system stiffness is the weakest in all directions, so this kind of vibration mode occurs first. The second order vibration frequency is Hz, the maximum displacement is 5.082mm, second vibration mode is longitudinal vibration, from the map can clearly see the whole aqueduct translate along the longitudinal direction, namely the second order vibration is aqueduct support vibration, from the displacement map can summarize that the longitudinal displacement of the aqueduct is asymmetric, because one side of the aqueduct is constraint by vertical direction, and the other side is not. The third order vibration frequency is Hz, the maximum displacement is mm, the third vibration is mainly torsional vibration in the horizontal plane, the frequency is significantly greater than the first two frequencies, was gradually increased, which is consistent with the actual situation. The fourth order vibration frequency is Hz, the maximum displacement is mm, it is a higher mode, is a combination of bending and torsion vibration, the vibration is more complex and the rod has a corresponding deformation, it also has expansion, bending and torsion. The fifth order vibration frequency is Hz, the maximum displacement is mm, the fifth mode is mainly horizontal vibration modes and is close to sine wave, span and cantilever aqueduct relative displacement is small, because the upper rod structure stiffness is relatively small, the lower part of aqueduct is reinforced concrete structure, so the stiffness is relative large, the upper part deformation is large and the lower part deformation is small, in line with the actual structure of the dynamic characteristics (Li and Wang, 2009; Xu and Wang, 1999; Wang, Shen and Liu, 2013; Zhang, Lu and Li, 2011). The vibration frequency of the design level is Hz, the maximum displacement is 3.038mm, in the depth of design level, it is same with the empty level that the first mode is the main mode of vibration, comparing vibration characteristics with the drying aqueduct, the first mode is change, the main reason is the whole quality increase after add water in the aqueduct, so the vibration state change. The second order vibration frequency is Hz, the maximum displacement is mm, maximum displacement occurs in the middle of the span. The third order vibration frequency is Hz, the maximum displacement is 7.945mm, which is similar to the third order of the dry mode, but the vibration frequency and the maximum displacement are significantly reduced compared with the third order mode. The fourth order vibration frequency is 14.40Hz, the maximum displacement is 8.667mm. The tank body appear obvious vertical vibration, in addition to the upper part of the rod appear longitudinal vibration along the transverse direction, this is consistent with our previous conclusion that less rigidity is easier to be excited. The fifth order vibration frequency is Hz, the maximum displacement is mm, the vibration mainly in the middle span of aqueduct, also has a slight twist deformation, it is shows that torsional stiffness of the aqueduct is relative large. 5. CONCLUSIONS A three-dimensional finite element model of wide wells ditch aqueduct is established with ANSYS. Through analyze this model conducts the dynamic characteristics of the aqueduct. 1)The first two modes of vibration are mainly single direction, vibration shape is more and more complex with the increase of order, it is often vibrated at all directions, accompany with torsion movement, but mainly vibration is along transverse direction, it shows that lateral stiffness is weak. 2) Water has a great effect on the aqueduct dynamic characteristics, the natural vibration frequency decreases with depth increases, which is the more massive, the lower the frequency, this is because with the increase of water depth, the mass matrix system increases, so the dynamic analysis of large aqueduct must consider the role of water, build reasonable calculation model including the water. 3) In this study, the main vibration mode is changed after add water quality, there is almost no change when the water level increase. 4) When the structure is in vibration, the groove holder, the bearing and the connecting place are deformed. Prone to stress concentration, so easy to destroy. U type aqueduct wall transverse stiffness is weak, easily deformed, appropriate measures should be taken to improve the bearing capacity and local stability. 5) When the aqueduct in the actual operation period, all parts of the aqueduct stress situation is complex, many factors are difficult to predict, therefore, starting from the operation and maintenance angle should be tracked dynamic analysis. 68

7 ACKNOWLEDGEMENTS This work was supported by Basic and Advanced Technology Research Project of Henan Province (No ), China, Science and Technology Project of Zhengzhou (No.131PPTGG410-13), and Science and Technology Project of Education Department of Henan Province (No. 17B130001). REFERENCES Zhao, Y., Zhao, P., Li, S.Y. (1999) The Finite Element Analysis of Large Prestressed Concrete Box Aqueduct Structure, Journal of Yangtze River Scientific Research Institute, 16 (2), pp Xu, J.G., Wang, B. (1999) Finite Element Analysis of Aqueduct Structure Dynamic Performance, Zhengzhou Industrial University, 20(2), pp Bai, X.B., Li, Y.H., Guo, R.Q., Nie, J.R. (2008) Dynamic Characteristics Analysis of Shuangji River Aqueduct of South-to-North Water Diversion Project, Yangtze River, 39(8), pp Li, Y.J., Wang, H. (2009) Analysis of Large Aqueduct Structure Vibration Characteristics, People s Long Press, 40(5), pp Hu, S., You, R., Niu, Z. (2010) Dynamic Analysis of Tall-Pier Aqueduct Using ODE Solver, Third International Joint Conference on Computational Science & Optimization, 41(1), pp Li, X., Chen, H.S. (2011) Current Actuation and Trend of Development of Domestic and Foreign Aqueduct, The Agricultural Science and Technology and Equipment, 30(12), pp Zhang, H.W., Lu, L., Li, B. (2011) Aqueduct Structure Modal Analysis, Industry Technology, 40 (3), pp Hamilton,R., Baldwin,K., Fuller, J. (2012) Intracranial pressure pulse waveform correlates with aqueduct cerebrospinal fluid stroke volume, Journal of Applied Physiology, 113(10), pp Wang, L.Y., Shen, T.S., Wei, D.H. (2013) Vibration Test and Dynamic Characteristics Analysis of the Aqueduct, Water Conservancy and Hydropower Technology, 44 (11), pp