DYNAMIC ANALYSIS OF ARCH DAMS: EFFECT OF THERMAL LOADING by Ryhane Moghadas Jafari A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE COLLEGE OF GRADUATE STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) January 2016 © Ryhane Moghadas Jafari, 2016 ii Abstract Conducting a case study, this thesis investigates the dynamic behavior of Karun IV arch dam (Iran) under the effect of maximum feasible thermal input. A three-dimensional finite element model is created using ANSYS software. The thesis determines the impact of thermal loadings under two scenarios of normal and minimum water levels. The dam-foundation, dam-water and water-foundation interactions are considered in the modeling to accurately capture its dynamic response. Furthermore, in addition to the water compressibility, appropriate wave absorbing boundaries are used for the reservoir far end and bottom and for the foundation. Stability, static, modal, and thermal analyses are conducted as initial conditions. Using three orthogonal earthquake components, linear dynamic time history analyses with and without thermal loading are performed. The comparison of the results verified that the application of thermal effects on the dynamic analysis increases the maximum tension, changes the maximum pressure, and decreases the displacements in the dam body. This study concludes that thermal loading must be considered in the dynamic analysis of an arch dam as it can worsen the tensile cracks. iii Table of Contents Abstract .................................................................................................................................... ii Table of Contents ................................................................................................................... iii List of Tables .......................................................................................................................... vi List of Figures ........................................................................................................................ vii List of Abbreviations ............................................................................................................. ix List of Symbols ........................................................................................................................ x Acknowledgements .............................................................................................................. xiii Dedication ............................................................................................................................. xiv Chapter 1 Introduction........................................................................................................... 1 1.1 Research Motivation ............................................................................................................. 1 1.2 Research Objective, Assumptions and Methodology ........................................................... 2 1.3 Thesis Outline ....................................................................................................................... 3 Chapter 2 Literature Review ................................................................................................. 4 2.1 Seismic Failure Mechanisms of Arch Dams and Their Affecting Factors............................ 4 2.2 Necessity of Seismic Analysis of Arch Dams ....................................................................... 5 2.3 Thermal Loading in Safety Analysis of Arch Dams ............................................................. 5 2.4 Literature Review .................................................................................................................. 7 Chapter 3 Governing Equations and Boundary Conditions ............................................ 10 3.1 Damping .............................................................................................................................. 10 3.1.1 Material Damping Matrix ............................................................................................... 10 3.2 Finite Element Equation of the Dam Body ......................................................................... 11 3.3 Equation of Motion for the Reservoir ................................................................................. 11 3.4 Finite Element Equation of the Reservoir ........................................................................... 12 3.5 Dam-Reservoir Interaction .................................................................................................. 13 3.6 Finite Element Equation of the Coupled Dam-Reservoir ................................................... 13 3.6.1 Effect of Soil-Structure Interaction................................................................................. 13 3.6.2 Direct Method ................................................................................................................. 14 iv 3.7 Effect of Foundation Mass .................................................................................................. 15 3.8 Thermal Boundary Conditions ............................................................................................ 16 3.9 Reservoir Boundary Conditions .......................................................................................... 16 3.9.1 Reservoir Free Level Boundary Condition ..................................................................... 16 3.9.2 Dam-Reservoir Boundary Condition .............................................................................. 17 3.9.3 Reservoir-Foundation Boundary Condition .................................................................... 17 3.9.4 Far End of the Reservoir Boundary Condition ............................................................... 18 3.10 Semi-Infinite Foundation Boundary Condition .................................................................. 20 Chapter 4 Modeling of the Dam Body, Reservoir, and Foundation ................................ 21 4.1 Geometric Characteristics of the Dam ................................................................................ 21 4.1.1 Concrete Characteristics ................................................................................................. 21 4.2 Water Specifications ........................................................................................................... 24 4.3 Foundation Materials Specifications ................................................................................... 24 4.4 Specifications of the Finite Element Model of the Dam, Reservoir, and Foundation ........ 24 4.4.1 Finite Element Model of the Dam Body ......................................................................... 25 4.4.1.1 Solid45 Element ..................................................................................................... 26 4.4.1.2 Solid70 Element ..................................................................................................... 26 4.4.2 Finite Element Model of the Reservoir........................................................................... 26 4.4.2.1 Fluid30 Element ..................................................................................................... 27 4.4.3 Finite Element Model of the Foundation ........................................................................ 28 4.4.3.1 Combine14 Element ............................................................................................... 29 Chapter 5 Loadings, Analyses and Results......................................................................... 30 5.1 Types of Loading ................................................................................................................ 30 5.2 Seismic Loading .................................................................................................................. 30 5.2.1 Earthquake Frequency Content ....................................................................................... 31 5.2.2 Earthquake Duration ....................................................................................................... 31 5.3 Thermal Analysis ................................................................................................................ 36 5.4 Stability Analysis ................................................................................................................ 38 5.5 Static Analysis .................................................................................................................... 39 5.6 Modal Analysis ................................................................................................................... 41 5.7 Dynamic Analysis ............................................................................................................... 42 5.7.1 Dynamic Analysis without Thermal Loading ................................................................. 43 5.7.2 Dynamic Analysis with Thermal Loading ...................................................................... 50 v Chapter 6 Conclusion ........................................................................................................... 58 6.1 Conclusions ......................................................................................................................... 58 6.2 Recommendations ............................................................................................................... 60 References .............................................................................................................................. 61 Appendix A: Permission Request ........................................................................................ 66 vi List of Tables Table 1 Basic information about the dam (Iran Water & Power Resources Development Co., 2004) ............................................................................................................... 21 Table 2 Mechanical and thermal properties of the dam concrete (Iran Water & Power Resources Development Co., 2004) ....................................................................... 23 Table 3 Natural vibration frequencies for the first fifteen vibration modes of the system .. 41 vii List of Figures Figure 1 Research overview ................................................................................................... 3 Figure 2 Heat sources and heat transfer processes of an arch dam (adopted from Sheibany & Ghaemian, 2006, with permission from ASCE) ................................ 6 Figure 3 Complete model of the dam-reservoir-foundation ................................................ 24 Figure 4 Format of the finite element grid of the dam body ................................................ 25 Figure 5 Analytical model of the reservoir .......................................................................... 27 Figure 6 Analytical model of the foundation rock ............................................................... 29 Figure 7 X-direction record of the earthquake ..................................................................... 32 Figure 8 Y-direction record of the earthquake ..................................................................... 33 Figure 9 Z-direction record of the earthquake ..................................................................... 34 Figure 10 Fourier Transform Amplitude of X, Y, and Z components of the earthquake .... 35 Figure 11 Curve of the energy released during the earthquake for X component ............... 35 Figure 12 Thermal input in the crown cantilever................................................................. 36 Figure 13 Temperature distribution in the dam body for the summer and winter critical times ..................................................................................................................... 37 Figure 14 Distribution of the maximum principal stress (S1) and minimum principal stress (S3) for downstream and upstream faces in the stability analysis ............. 38 Figure 15 Distribution of the X direction displacement (m) in the stability analysis .......... 39 Figure 16 Distribution of S1 and S3 for upstream and downstream faces in static analysis ................................................................................................................ 40 Figure 17 Distribution of X direction displacement (m) in static analysis .......................... 40 Figure 18 First seven vibration modes of the dam body ...................................................... 42 Figure 19 Nodes used for consideration of dynamic analyses results on the dam upstream and downstream faces .......................................................................... 43 Figure 20 S1 and S3 distribution envelops for the dam upstream and downstream faces in state (a) ............................................................................................................ 44 Figure 21 Time-history curve for the dam crest displacement in X direction in state (a) ... 45 Figure 22 Time-history curves of S1 for the selected nodes in the dam upstream face in state (a) ................................................................................................................. 45 viii Figure 23 Time-history curves of S1 for the selected nodes in the dam downstream face in state (a) ............................................................................................................ 46 Figure 24 S1 and S3 distribution envelops for the dam upstream and downstream in state (b) ......................................................................................................................... 47 Figure 25 Time-history curve for the dam crest displacement in X direction in state (b) ... 48 Figure 26 Time-history curves of S1 for the selected nodes in the dam upstream face in state (b) ................................................................................................................ 48 Figure 27 Time-history curves of S1 for the selected nodes in the dam downstream face in state (b) ............................................................................................................ 49 Figure 28 S1 and S3 distribution envelops for the dam upstream and downstream faces in state (c) ............................................................................................................ 51 Figure 29 Time-history curve for the dam crest displacement in X direction in state (c) ... 52 Figure 30 Time-history curves of S1 for the selected nodes in the dam upstream face in state (c) ................................................................................................................. 52 Figure 31 Time-history curves of S1 for the selected nodes in the dam upstream face in state (c) ................................................................................................................. 53 Figure 32 S1 and S3 distribution envelops for the dam upstream and downstream faces in state (d) ............................................................................................................ 54 Figure 33 Time-history curve for the dam crest displacement in X direction in state (d) ... 55 Figure 34 Time-history curves of S1 for the selected nodes in the dam upstream face in state (d) ................................................................................................................ 55 Figure 35 Time-history curves of S1 for the selected nodes in the dam upstream face in state (d) ................................................................................................................ 56 ix List of Abbreviations 3D Three-Dimensional FEM Finite Element Model CEA Canadian Electricity Association USACE U.S. Army Corps of Engineers DBE Design Basis Earthquake MCE Maximum Credible Earthquake NWL Normal Water Level MWL Minimum Water Level PREF Reference pressure x List of Symbols 𝜌 Considered medium density 𝐶 Specific heat 𝑇 Temperature {𝑉} Mass transport heat flow ∇ Divergence ∇° Gradient 𝑡 Time [𝐷] Conductivity matrix 𝑣𝑜𝑙 Element volume {𝑞} Thermal flux vector 𝑆2 Concrete dam exposed surfaces subject to thermal flows 𝑆3 Concrete dam exposed surfaces subject to convection ℎ𝑓 Heat transfer coefficient 𝑇𝐵 Temperature of the adjacent fluid 𝑞 Heat production rate per unit volume [𝑀] Mass matrix [𝐶] Damping matrix [𝐾] Stiffness matrix 𝛼 Mass coefficients matrix 𝛽 Stiffness coefficients matrix 𝑖, 𝑗 Dominant vibrational modes of the system 𝜉 Damping ratio 𝜔𝑖, 𝜔𝑗 ith and jth natural frequencies 𝜔 Frequency of dominant vibration mode of the system {?̈?} Acceleration of the structural nodes relative to the ground {?̇?} Nodal acceleration vector {𝑈} Nodal displacement vector {𝑓} Hydrodynamic pressure vector xi {𝑓1} Resultant vector of the other forces on the structure {?̈?𝑔} Ground acceleration (earthquake) in the base of the structure Ρ Fluid pressure 𝐶 Velocity of fluid pressure wave (sound wave) 𝐾 Bulk modulus [𝐺] Reservoir mass matrix {𝑃} Reservoir hydrodynamic pressure vector [?́?] Damping matrix of the reservoir [?́?] Reservoir stiffness matrix {𝐹} Pseudo force vector in the fluid due to the boundary conditions in the reservoir surfaces [𝑄] Coupled matrix of the dam-reservoir interface which converts the reservoir pressure into nodal forces on the structure {𝐹𝐹1} Force vector related to the acceleration of the dam-reservoir and reservoir-foundation {𝐹2} Force vector due to the ground acceleration in the dam-reservoir boundary and the total acceleration in the other boundaries [𝐷] Conductivity matrix [𝐻] Pseudo rigidity matrix ℎ Reservoir depth {𝐹1} Force vector of volumetric and hydrostatic forces 𝐶𝐿 Longitudinal wave speed in the foundation ambient 𝑎° Dimensionless frequency 𝑔 Acceleration of gravity Ζ Vertical axis with the center at the water surface n Perpendicular vector to the surface unit in the dam-reservoir interface ans Dam structure acceleration along n vector 𝜌𝑟 Density of the materials in the reservoir bottom 𝐶𝑟 Longitudinal wave velocity in the reservoir bottom materials 𝐸𝑟 Modulus of elasticity of the reservoir bottom materials xii 𝑎𝑛 (𝑡) Acceleration of the reservoir bottom in the time 𝑡 𝐸𝑐 Concrete modulus of elasticity 𝑓𝑐 90-day compressive strength of the concrete 𝑓𝑡 Tensile strength of the concrete 𝜐 Poisson’s ratio 𝛼 Coefficient of thermal expansion 𝐸𝑓 Deformation modulus of foundation rock mass 𝑆1 Maximum principal stress 𝑆3 Minimum principal stress 𝐷𝑆 Downstream face 𝑈𝑆 Upstream face 𝑈𝑥 Displacement of the dam crest in 𝑥 direction xiii Acknowledgements I appreciate the efforts of my supervisor, Dr. Bahman Naser. I owe special thanks to my parents because of their whole life supports and inspiring. I thank the faculty, staff, and my fellow students at the UBCO, who encouraged me to continue my work in this field. xiv Dedication 1 Chapter 1 Introduction Dams play an important role in the optimal use of surface water resources in dry or semi-dry countries, also for making clean hydropower energy in the world. Dam failure makes disastrous life and property losses as it has been observed in the history. Therefore, structural dam safety is very important and critical. A realistic consideration of all factors affecting dam safety while designing new projects or updating safety evaluations of existing dams can improve their efficient operation, particularly in arch dams. Comparing to the other dams (e.g. gravity and rock fill dams) arch dams indicate a more complicated behavior due to their three-dimensional thin arch structure, the mechanism of load transfer to the supports, and the operation of contraction joints. According to statistics, arch dams display somewhat worse performance over time compared to other type of dams (Regan, 2010). As various loading conditions and interactions among different media (water, dam, foundation, and ambient environment) may affect their performance more significantly, this dissertation focuses on arch dams. 1.1 Research Motivation Arch dams are under static, dynamic and thermal loadings. Static loads include the self-weight and hydrostatic pressure. Dynamic loads include the impact force of the earthquake on the structure and the hydrodynamic pressure of the water behind the dam. Thermal effects have been reported as the main causes of the deterioration in concrete dams. Freeze-thaw cycles covered 19% of cases, while temperature changes were 9% of cases (ICOLD, 1984). In regions with high difference between seasonal ambient air temperature, (e.g. in some areas in Canada the temperature changes between summer and winter reaches to 45°C), the associated thermal stresses may go above the allowable tensile strength of concrete and that may eventually cause tensile failures (Daoud et al., 1997). Previous researchers and designers considered thermal input as a predetermined assumption for their calculations. According to their point of view, arch dams are very sensitive to the variation of ambient temperature due to their special geometries and shapes. Therefore, temperature changes within the dam body and their related thermal stresses should be included as initial conditions in any safety analysis of arch dams (Agullo et al., 1996; 2 Daoud et al., 1997; Sheibany & Ghaemian, 2006; Mirzabozorg et al., 2014). This research tests and verifies the necessity of this assumption for the case of Karun IV arch dam (Iran). This is done by creating an advanced three-dimensional (3D) finite element model (FEM) of the dam. 1.2 Research Objective, Assumptions and Methodology As overall objective, this study aims at investigating the significance of thermal loading on the dynamic response of double curvature Karun IV arch dam. This objective is achieved through the following sets of goals: Stability Analysis: to verify dam stability under its weight; Static Analyses: to study dam stability under full static loading conditions as well as an initial condition for dynamic analyses; Thermal Analyses: to find temperature distribution over the dam body; Modal Analysis: to determine the natural mode shapes and frequencies of the dam; and Dynamic Analyses: to create a comprehensive dynamic response of the dam under all loading conditions. The following assumptions were adopted in this project: The reservoir is modeled with normal and minimum and water levels; Water is compressible; Ignoring the hydration heat, the research studies only thermal loads due to ambient temperature change; Dynamic analysis considers horizontal as well as vertical components of earthquake; Foundation is assumed to be massive; Dam-reservoir, dam-foundation and reservoir-foundation interactions are considered; and Absorbing boundaries are located on the bottom, walls, and far end of the reservoir, and around the foundation as well. Figure 1 shows the steps for research methodology. A 3D FEM of the dam is provided by the ANSYS software. Thermal, stability, static, and modal analyses are done as initial 3 conditions. Finally, linear dynamic time-history analyses, without and with thermal loading, are performed and the results are compared. Figure 1 Research overview 1.3 Thesis Outline This research is reported in six chapters. Chapter 1 is an introduction to the research problem. Chapter 2 briefly reviews the relevant background and literature. Chapter 3 explains briefly about governing equations and boundary conditions of the model. The characteristics of the FEM of the dam body, reservoir, and foundation are discussed in Chapter 4. Chapter 5 presents loading conditions and provides the analyses scenarios and their results. Chapter 6 concludes the thesis by highlighting the key findings and recommending future research. 4 Chapter 2 Literature Review 2.1 Seismic Failure Mechanisms of Arch Dams and Their Affecting Factors According to the previous observations and experiences, the probable mechanism of damage of an arch dam under the earthquake can be classified as follows: 1- Under seismic loads, tensile stresses in some points of the dam body exceed the limit, making cracks in those points. Vertical joints are made of lower resistance materials compared to the concrete. Therefore, due to an increase in the tensile stresses, the vertical joints between the blocks and also the horizontal seams open. This problem leads to the loss of integrity of the dam body and changes the force transmission system from arch to cantilever. If the cracked blocks do not withstand additional loads, they will break and result in the general or partial damage of the dam. 2- Sliding of the masses around the valley and over the reservoir on the dam body, sliding of the masses over the reservoir into the reservoir, and mud displacement beneath the reservoir can create damage in part of the dam or powerful waves in the reservoir. These waves can subsequently result in the overflow of the reservoir water on the dam crest, overpass, destruction of the spillways and the side structures, and ultimately the destruction of the dam. 3- Earthquake can cause displacement of the fault, landslide or cracks in the stony foundation of the dam. These can cause probable damage in the grout curtain, which causes an increase in water leakage and in pore-water pressure in foundation. Eventually, the problems in the foundation lead to the dam slide and damage (Tatalovich, 1998). Some of the factors that influence the seismic response of an arch dam are as follows (Zhou et al., 2000): Characteristics and intensity of the earthquake, waveforms, and vibration modes, and their changes. Interaction of the dam body with the foundation material and the reservoir water; interaction of the foundation with the reservoir water. Features of the existing materials; changes in the features of dam materials (material nonlinearity). Cracking or opening-closing of cracks in the dam body, foundation and joints; relative 5 displacement of vertical or horizontal joints of the dam body; joint sliding and compression (geometric nonlinearity). Type of computer modeling (in terms of meshing and other characteristics), and Reservoir water level. 2.2 Necessity of Seismic Analysis of Arch Dams Because of severity and intensity of dynamic forces, dynamic analysis and design of arch dams are essential. According to the statistic declared by CEA (Canadian Electricity Association) in 1990, USACE (U.S. Army Corps of Engineers, 1995), and Knight and Mason (1992), many arch concrete dams were affected by severe earthquakes such as Pacoima and Hoover (United States), Kurobe (Japan), Monteynard (France), Maina Sarris (Italy), Kariba (Zimbabwe), and Rapel (Chile). Among them, Rapel and Pacoima dams were damaged (Tinawi et al., 2000). Furthermore, according to the statistics declared by Serafim and Olivera regarding the information collected since 1987 to 1994 from 1527 existing dams, 42 dams were subjected to earthquake and 8 dams were damaged (Serafim & Oliveira, 1987). The mentioned statistics and studies have indicated that the previous design methods, contrary to the expectations, have not been conservative. It should be taken into consideration that due to the long return period for large earthquakes, many of the existing dams in the areas of high seismicity have not experienced yet the predicted earthquakes by the seismicity studies. Therefore, the catastrophic losses due to the failure of dams indicate the importance of updated seismic analysis for arch dams and re-examination of the behavior of these structures. It should be done through using up-to-date computational models and analysis techniques, and consequently evaluating the stress response, resistance, and movement of dam-reservoir-foundation system as well. 2.3 Thermal Loading in Safety Analysis of Arch Dams Arch dams are thin, arched and fixed in foundation and abutments. This special condition makes them sensitive to the variation of the ambient temperature. Figure 2 shows the general heat sources and thermal boundary conditions in the operational phase of an arch dam. The boundary conditions at the interface of concrete and air are solar radiation, concrete-air convection, and concrete radiation to the air (Sheibany & Ghaemian, 2006). Air and water 6 temperature are obtained from meteorological reports. The concrete and water temperature are the same at the concrete and water interface (Sheibany & Ghaemian, 2006). Figure 2 Heat sources and heat transfer processes of an arch dam (adopted from Sheibany & Ghaemian, 2006, with permission from ASCE) Heat transfer is taken into consideration in construction as well as operational phases of a concrete dam (U. S. Army Corps of Engineers, 1994; Sheibany & Ghaemian, 2006; Li et al., 2014). Heat of operation is also considered at least from two aspects; first, the effects of long term changes in the ambient temperature and second, the effects of temperature distribution in the dam body at certain times (i.e., after the peak of summer heat and the peak of winter cold). Regarding the first aspect, according to literature, stresses caused by annual temperature variation have a significant effect on reducing the strength and durability of arch dams throughout the year, especially in the areas with high thermal gradients. Moreover, the thermal response of concrete dams affects some properties of concrete mixture such as creep, thermo-elastic properties and alkali aggregate reactions (Sheibany & Ghaemian, 2006; Li et al., 2014). Thermal cracks are not directly responsible for the instability of arch dams. Rather, due to the weathering, water and sediments penetration, and the permanent cycles of melting- 7 freezing, the cracks gradually broaden and cause more destruction in the dam concrete. Therefore, an aged-dam affected by an earthquake at a specific moment has initial stress distribution due to the heat and also a series of thermal cracking and demolition. Consequently, earthquake and hydrodynamic forces exerted on the dam, together with already existing gravity and hydrostatic loads, can extend cracks and cause general instability of the dam (Javanmardi et al., 2005). 2.4 Literature Review In the realm of investigating temperature distribution and thermal effects on dams, Agullo and Aguado (1995) proposed a one-dimensional simple explicit finite-difference scheme by developing a simple analytical formula. This simple model predicted the thermal behavior of the dam body based on the dam height, at various sections for different given heights and changeable thicknesses, at any moment. The scheme considered concrete thermal variables, dam site and geometry, as well as environmental function of the dam. According to their research, annual mean temperature of the ambient, water, and the total daily solar radiation at the site, mainly influences the mean temperature of the section. The section thickness, annual range of the ambient and water temperatures affects the annual range of the mean temperature of the desired section. Between all those stimuli, the solar radiation mainly affects the temperature of each layer. An analytical method using unidirectional heat transfer was proposed by Zhang and Gargaa (1996). This model determined the temperature distribution for the thermal shock scenarios happening on mass concrete structures. By making an analytical formula based on superposition method for sinusoidal and triangular air temperature loading, the most temperature gradient and consequently stresses were observed near the exposed surface in a narrow area. They found the concrete properties and the coefficient of heat transfer should be changed to reduce the stress concentration. Meyer and Mouvet (1995) made a three-dimensional finite element model for Vieux–Emosson arch-gravity dam in Switzerland. Using the numerical analysis, they calculated the temperature gradients and consequent thermal stresses, strains, and displacements. They divided the downstream face into three areas and gave each area a different absorption of solar radiation. According to their results, which were not compatible with obtained 8 instrumental data, the thermal expansion coefficient has an important role on the deformability of the dam while the rock and the concrete module of elasticity are not significant. Leger et al. (1993) carried out a two-dimensionally modeled numerical analysis for finding the temperature-affected area in a concrete gravity dam with the assumptions of no horizontal heat transfer and similar boundary conditions for different cross sections. Because of the plain and non-concave downstream surface of the gravity dams, there is no change in the solar radiation exposure. According to the results, the temperature gradient is created close to the exposed surface resulting in the tensile stresses. The consequent surface cracks are harmful just because of freezing and thawing cycles of penetrated water, but do not make the dam instable. Daoud et al. (1997) proposed a somehow comprehensive two-dimensional finite element numerical analysis for gravity dams with periodic temperature field. They considered the assumptions of past researchers including solar radiation, air temperature variations, temperature gradients, as well as their own new ones including snow cover, reservoir ice formation, and conductivity change in the saturated part and unsaturated part of the dam. According to their observations, despite of a change in the related thermal conductivities by just 8%, temperature gradient varies significantly along with the interface of the saturated and unsaturated parts. Furthermore, they found thermal degradation happens nearly in 1 meter region from the open surface of the downstream. Besides Leger et al. (1993) and Daoud et al. (1997), who made a simplified variation pattern of the temperature profile of the reservoir, Bofang (1997) also suggested a boundary condition of water temperature variations for deep reservoirs of concrete dams using an analytical formula (Mirzabozorg & Varmazyari, 2009). Sheibany and ghaemian (2006) predicted the thermal gradient and the consequent thermal stress distribution in the operational phase of an arch dam. They made a three-dimensional finite element model and considered experimentally realistic air temperature and additional assumptions including experimentally realistic reservoir temperature and the changing share of solar radiation over the exposed surface of the dam. They applied the effects of diffuse and beam radiation, surface azimuth, sun declination, surface slope, latitude, and the water and ground reflectivity. They did not consider the foundation 9 temperature in the model. They neglected the effect of temperature on the mechanical and thermal properties of the concrete. The thermal properties were considered uniform and isotropic. The hydration heat of cement after construction was not applied. At the concrete-air interface, solar radiation, concrete-air convection, and concrete radiation to the air were the assumed boundary conditions. At the concrete-water interface, no convection and radiation, and as a result, similar water and concrete temperature were considered. Mirzabozorg et al. (2014) considered the solar radiation, air, and water temperature effects on the thermal analysis of dams with finite element modeling methods. Wang et al. (2011) made a thermal dynamic analysis of a concrete gravity dam using a three-dimensional finite element model, the application of iteration method, and cooling pipe discrete. They analyzed the transient temperature field and their distributions. They used sensible temperature control measures as reference. Their suggested temperature control measures effectively resulted into controlling the temperature and preventing the cracks. Hariri-Ardebili & Kianoush (2014) presented an integrative nonlinear seismic safety analysis of a calibrated model of a high arch dam. They carried out a static and thermal calibration procedure, followed by a nonlinear dynamic analysis on the resulted model. From the assumption side, their research was realistic as they considered mass concrete cracking, geometric nonlinearity, joint opening and closing, the effect of water pressure and penetration inside the joints. According to their findings, the loaded dam extensively cracked and joints slid under the applied assumptions. Liu et al. (2015) also considered thermal effects in the mass concrete in the presence of a pipe cooling system. They made a heat-fluid coupling model and analyzed a high arch dam monolith with that during the construction period. They considered different factors including thermal characteristics of the material, cooling pipe system and schedule, and real climate condition. They found this method effective on modeling the thermal field of complex mass concrete structures including cooling pipe systems. 10 Chapter 3 Governing Equations and Boundary Conditions This chapter explains the relevant governing equations used by the software to model the thermal and seismic analysis, considering dam-foundation-reservoir interactions, and the boundary conditions for those in an arch dam. 3.1 Damping Damping has an important effect on dynamic response of arch dams and includes material damping (viscous damping), frictional damping (Coulomb), and radiation damping (geometric). The associated energy loss originates from different sources including concrete arch structure, foundation rock and reservoir water. Energy dissipation in the arch structure is due to the internal friction of concrete materials and construction joints (frictional damping). The factors which cause earthquake energy loss in the foundation rock are expansion of the elastic waves from the dam body to the distant in foundation (radiation damping) and the hysteretic due to sliding of cracks and joints in the foundation rock (frictional damping). In the reservoir, diffraction of hydrodynamic pressure waves into the reservoir bottom material and expansion of pressure waves toward the reservoir upstream (radiation damping) create damping (U. S. Army Corps of Engineers, 1994). 3.1.1 Material Damping Matrix The seismic response of arch dams is determined based on linear elastic dynamic analysis. The dynamic analysis of the present model is made in time history analysis method. The direct integration model is used by software for step-by-step numerical integration in order to solve equations of motion. In direct integration method, it is necessary to determine explicit damping matrix. Rayleigh damping method is employed to create the damping matrix (Chopra, 1967): [𝐶] = 𝛼[𝑀] + 𝛽[𝐾] 3.1 where [𝑀] is the mass matrix, [𝐶] is the damping matrix, [𝐾] is the stiffness matrix, and 𝛼 and 𝛽 are the mass and stiffness coefficients matrices, respectively. They can be written as: 11 { 𝛼 = 𝜉2𝜔𝑖𝜔𝑗𝜔𝑖 + 𝜔𝑗𝛽 = 𝜉2𝜔𝑖 + 𝜔𝑗 3.2 where 𝑖 and 𝑗 are the dominant vibrational modes of the system, 𝜉 is the damping ratio, 𝜔𝑖 𝑎𝑛𝑑 𝜔𝑗 are the ith and jth natural frequencies. After determination of 𝛼 and 𝛽, damping matrix can be determined by the above equation (Chopra, 1967). Because of the severe and complicated effect of mass matrix coefficient on the response of dam-reservoir-foundation system in arch dams, 𝛼 is considered zero. Therefore, damping matrix and stiffness matrix coefficient is calculated as follows: [𝐶] = 𝛽[𝐾] 3.3 𝛽 = 𝜉 (2𝜔) 3.4 where 𝜔 is the frequency of dominant vibration mode of the system (Chopra, 1967). 3.2 Finite Element Equation of the Dam Body During an earthquake, dam body is modeled as a Multi-Degree-of-Freedom (MDF) system. To stimulate the earthquake in all directions, we will have: [𝑀]{?̈?} + [𝐶]{?̇?} + [𝐾]{𝑈} = {𝑓} + {𝑓1} − [𝑀]{?̈?𝑔} 3.5 where {?̈?} is the acceleration of the structural nodes relative to the ground, {?̇?} is the nodal acceleration vector, {𝑈} is the nodal displacement vector, {𝑓} is the hydrodynamic pressure vector, {𝑓1} is the resultant of the other forces on the structure, and {?̈?𝑔} is the ground acceleration (earthquake) in the base of the structure. This equation is the final finite element form of the equation of motion for the dam body (Ghaemian & Ghobarah, 1999). 3.3 Equation of Motion for the Reservoir By placing Stoke’s Viscosity Low in the linear momentum equation for a Newtonian fluid, with the assumption of constant density and viscosity of the fluid, the small amplitude motion for the reservoir, linear and non-rotational compressible fluid, and the application of 12 the continuity equation, the final equation for the reservoir is obtained as follows (White & Corfield, 2006): 𝛻2𝑃 =1𝐶2𝜕2𝑃𝜕𝑡2 3.6 where Ρ is fluid pressure, 𝐶 = √𝐾 𝜌⁄ is the velocity of fluid pressure wave (sound wave), 𝐾 is called bulk modulus, and 𝜌 is the water density. Given the fluid compressibility, the amount of added mass and the energy loss caused by fluid during earthquake change in the time domain. Therefore, there will be more coordination between the actual behavior of the dam and mathematical models (DeSalvo & Swanson, 2006). 3.4 Finite Element Equation of the Reservoir By application of the obtained boundary condition for the reservoir and weak Galerkin method, the final equation used by the software is: [𝐺]{?̈?} + [?́?]{?̇?} + [?́?]{𝑃} = {𝐹} − 𝜌[𝑄]𝑇({?̈?𝑔} + {?̈?}) = {𝐹𝐹1}= {𝐹2} − 𝜌[𝑄]𝑇{?̈?} 3.7 and: |[?́?] =[𝐷]𝐶[?́?] = [𝐻] +𝜋2ℎ[𝐷] 3.8 where [𝐺] is the reservoir mass matrix, {𝑃} is the reservoir hydrodynamic pressure vector, [?́?] is the damping matrix of the reservoir, [?́?] is the reservoir stiffness matrix, {𝐹} is the pseudo force vector in the fluid due to the boundary conditions in the reservoir surfaces, 𝜌 is the water density, [𝑄] is the coupled matrix of the dam-reservoir interface which converts the reservoir pressure into nodal forces on the structure, {𝐹𝐹1} is the force component related to the acceleration of the dam-reservoir and reservoir-foundation, {𝐹2} vector represents the forces due to the ground acceleration in the dam-reservoir boundary and the total acceleration in the other boundaries, [𝐷] is the conductivity matrix, [𝐻] is the pseudo rigidity matrix, and ℎ is the reservoir depth (Ghaemian & Ghobarah, 1999; DeSalvo & Swanson, 2006). 13 3.5 Dam-Reservoir Interaction During an earthquake, the main loading factor on the dam structure is the inertia exerted on the upstream face of the dam which stems from earth movement and the hydrodynamic force of the reservoir fluid. Because of the seismic motion of the abutments around the reservoir, the pressure waves created by the earthquakes, which travel to the reservoir upstream, remove part of the kinetic energy of the system from the environment. In addition, the deformations or vibrations of the dam body during earthquake affect the hydrodynamic pressures created in the fluid adjacent to the dam body (Chopra, 1967). Therefore, from the above, it can be concluded that the temporal response of both the dam and the reservoir subsystems are dependent to each other and must be assessed simultaneously (U. S. Army Corps of Engineers, 2003). 3.6 Finite Element Equation of the Coupled Dam-Reservoir According to the equations for the dam body and the reservoir, the coupled equation for the dam-reservoir can be written as follows (Ghaemian & Ghobarah, 1999): |[𝑀]{?̈?} + [𝐶]{?̇?} + [𝐾]{𝑈} = {𝐹1} − [𝑀]{?̈?𝑔} + [𝑄]{𝑃} = {𝐹1} + [𝑄]{𝑃}[𝐺]{?̈?} + [?́?]{?̇?} + [?́?]{𝑃} = {𝐹} − 𝜌[𝑄]𝑇({?̈?𝑔} + {?̈?}) = {𝐹2} − 𝜌[𝑄]𝑇{?̈?} 3.9 where {𝐹1} is the vector of volumetric and hydrostatic forces. Other symbols are already introduced. 3.6.1 Effect of Soil-Structure Interaction Another important factor in modeling of dam-reservoir-foundation systems and their seismic loading is the interaction of foundation with the structure. In the analysis of dam-foundation structure interactions, earthquake stimulates a complex dynamic system. The effect of this interaction should be considered in the amount of displacements and stresses, produced under the static and dynamic loadings in the large structures, particularly in dams (Iran Water & Power Resources Development Co., 2004). In general, soil environment comprises of irregular bounded medium and regular unbounded medium. Irregular bounded medium is in the vicinity of the structure and its dimensions change with the type of the problem. It is ignored in some common studies on the soil- structure interaction. The combination of structure and irregular bounded medium is 14 called generalized structure which allows for non-linear behavior. The contact surface of generalized structure with regular unbounded medium (semi-infinite) is called environment- generalized structure contact surface (Wolf & Song, 1996). In order to do the numerical analysis of the semi-infinite part of the soil, a surface (boundary) called interaction horizon is selected that delimits the structure. The characteristics of the nodes in interaction horizon determine the main properties of the unbounded area located outside this surface. The numerical dynamic model contains the nodes at or above the interaction horizon. There are different methods in the analysis of soil-structure interaction to consider the place of interaction horizon, namely direct solution, substructure method, and hybrid method (Bathe & Wilson, 1967; Wolf, 1988; Rixen et al., 1998). Each one has its own specific applications. In the first method called substructure method, interaction horizon is the contact surface of the soil and the generalized structure. In the second method called direct method, interaction horizon can be considered as an artificial boundary and soil is modeled in this boundary (Wolf, 1988). Direct method is used as the modeling assumption in this study. 3.6.2 Direct Method In the direct method, the (linear) soil is modeled from the vicinity of soil- generalized structure contact surface up to artificial boundary. Since the unbounded area of soil can be covered by a limited number of elements with limited sizes, there should be proper boundary conditions indicating the properties of the removed soil for the part of the soil which is cut in interaction horizon. As well as modeling of the soil infinite rigidity, this artificial boundary should act as an absorbing boundary and should prevent the re-reflection of the seismic waves reflected from the structure. Finally, this boundary condition should create a unique and stable solution for the problem. If the artificial boundary is located in a considerable distance from the structure, then the application of approximate boundary conditions is enough for accurate results. Therefore, the local boundary conditions which are independent of the frequency, such as viscous damper, can be selected. Although in the direct method the overall dynamic system is larger than the substructure method, however, direct method can be a suitable choice for the time domain analysis (Wolf, 1988). 15 It is also important to determine the dimensions of the environment and the size of used network in this method. The dimensions of the environment of foundation should be selected so large that the effect of artificial boundaries on the stresses and displacements responses of the structure becomes negligible. This means that the final solution should simulate the response of unlimited environment even if there is no boundary condition (Wolf, 1988). According to the arch dam design manual of USACE (1994), the minimum length in each side of the foundation should be 2 to 3 times of the dam height. 3.7 Effect of Foundation Mass If the foundation has mass, the seismic waves can propagate in foundation and so the volume of calculations increases. In addition, special measures should be applied on the foundation boundaries. In foundation modeling of the arch dams, it is common that the foundation mass and the resulted propagation of seismic waves are neglected. In this case, only the rigidity of the foundation is considered. As a result, the seismic waves reach to the dam structure as they are applied to the foundation boundaries. Moreover, the reflection problem does not happen in the foundation boundaries and no part of the seismic waves is absorbed by the foundation (Tehrani et al., 2005). If the frequency of the earthquake applied on the structure is high, or the speed of the shear waves in the area below the dam is low because of the low rigidity of soil, foundation should be modeled as massed. In this case, seismic waves are propagated into the environment. Therefore, to prevent a return of waves after they hit the foundation boundaries, the wave absorber boundaries on the walls and bottom of the foundation must be modeled (Tehrani et al., 2005). To consider the necessity of modeling of massed foundation, shear frequency should be calculated as (Wolf, 1988): 𝐶 =𝑎°√𝑎°2 − 1𝐶𝐿 3.10 where 𝐶 in here is the phase speed, 𝐶𝐿 is the longitudinal wave speed in the foundation ambient, and 𝑎° is the dimensionless frequency. Shear frequency happens in 𝑎° = 1 which the wave disturbance do not propagate. In frequencies less than shear frequency the wave motion does not spread but is exponentially reduced. Therefore, if 𝑎° > 1 waves are propagative (Wolf, 1988). Here the foundation mass is considered. 16 3.8 Thermal Boundary Conditions The environment temperature of the dam site is obtained from the meteorological reports (Iran Water & Power Resources Development Co., 2004). The effect of solar radiation on the temperature increase of the dam surface is +5 °C in winter and +2 °C in summer, based on the quantities proposed by Stucky and Derron (1957) which is rather conservative. Using the values of summer maximum points and winter minimum points of the water temperature annual fluctuation curves at different depths (Iran Water & Power Resources Development Co., 2004), the two equations for the maximum summer and minimum winter boundary conditions below the water level in the dam upstream face are obtained. The injection temperature for the dam vertical joints is considered 17 °C. The temperature of foundation increases from the surface to the depth, i.e. +3 °C increase in temperature in each 100 meters. However, the heat exchange between dam body and foundation is ignored. Because the dam-foundation interface is negligible relative to the other dam boundaries, the depth of the foundation is infinite, and the distribution of the temperature in foundation is independent of the dam body (Sheibany & Ghaemian, 2006). 3.9 Reservoir Boundary Conditions In general, as well as the spatial boundary condition (geometric), there are temporary boundary conditions that indicate the desired variable state in a particular time. This temporary condition is called initial conditions. In the considered reservoir, there is a spatial geometric boundary condition that contains the boundary condition of the dam-reservoir, reservoir-foundation, open surface, and far end of the reservoir (Ghaemian & Ghobarah, 1999). 3.9.1 Reservoir Free Level Boundary Condition The approximate boundary condition of water surface with simplified assumptions and concerning gravity waves (surface waves with low surface tension) can be written as follows (Ghaemian & Ghobarah, 1999): 1𝑔𝜕2𝑃𝜕𝑡2+𝜕𝑃𝜕𝑧= 0 (𝑊ℎ𝑒𝑛 𝑍 = 0) 3.11 17 where Ρ is hydrodynamic pressure, 𝑔 is the acceleration of gravity, 𝑡 is the time, and Ζ is the vertical axis with the center at the water surface. The above equation is usually converted to the following equation because it is possible to ignore the shallow waves of the reservoir in concrete dams. 𝛲 = 0 (𝑤ℎ𝑒𝑛 𝑍 = 0) 3.12 In addition, to apply the Eq. 3.11, the natural frequency of the dam structure should be different and away from the natural frequency of the reservoir surface waves. Since the natural frequency of the structure is much more than the natural frequency of the reservoir surface waves (0.01– 0.1 𝐻𝑟𝑧), Eq. 3.11 is a correct assumption (Ghaemian & Ghobarah, 1999). 3.9.2 Dam-Reservoir Boundary Condition It is clear that due to the impermeable surface of the concrete dam, there must not be any flow, or its subsequent relative velocity, perpendicular to the fluid-structure interface. Therefore, the calculations will be: 𝜕𝑃𝜕𝑛= −𝜌𝑎𝑛𝑠 3.13 In this equation, n is the perpendicular vector to the surface unit in the dam-reservoir interface and ans is the dam structure acceleration along n vector (Ghaemian & Ghobarah, 1999). 3.9.3 Reservoir-Foundation Boundary Condition Studies have shown that the consideration of interaction of the reservoir with the rock mass of the foundation and application of the earthquake to the walls and bottom of the reservoir in analytical model increase the dam response. In addition, the water pressure on the bottom and side walls of the reservoir causes displacement of the stone walls of the valley that leads to the limited displacement of the dam body toward upstream (Iran Water & Power Resources Development Co., 2004). Therefore, the interaction of the foundation with the reservoir is also taken into consideration in this study. 18 If there is no wave absorption or water penetration in the reservoir bottom (containing the bed rock and the reservoir lateral supports), the same dam-reservoir boundary condition can also be used for this part. While there are sediments in the reservoir bottom, they absorb part of the incoming waves. Therefore, a new type of the boundary condition equation is needed. In general, the boundary condition of the reservoir bottom connects the hydrodynamic pressure to the sum of the vertical acceleration, and the acceleration of the interaction between reservoir water and the reservoir bottom materials. Furthermore, with considering the propagation of just the vertical dilatational waves in the reservoir sediments by the hydrodynamic pressure, only the vertical interaction between water and the reservoir bottom materials are taken into consideration. Finally, applying Helmohltz equations, D’alembert’s solution for wave propagation equation, and the deletion of the component of the returning wave into the reservoir, the relation for the reservoir bottom boundary condition is derived as follows (Fok et al., 1996): 𝜕𝑃(0, 𝑡)𝜕𝑛−𝐾𝐶𝜕𝑃(0, 𝑡)𝜕𝑡= −𝜌𝑎𝑛 (𝑡) 3.14 In this equation, 𝑃(0, 𝑡) means hydrodynamic pressure in zero level and perpendicular to the reservoir bottom at the moment t. Therefore, 𝜕𝑃(0, 𝑡)/𝜕𝑛 will be the pressure gradient component in the reservoir bottom. Moreover, we will have the equation 𝐾 = 𝐶𝜌 𝐶𝑟𝜌𝑟 ⁄ . In this equation, 𝜌 is the density of the fluid of the reservoir, 𝜌𝑟 is the density of the materials in the reservoir bottom and 𝐶𝑟 = √𝐸𝑟 𝜌𝑟⁄ is the longitudinal wave velocity in the reservoir bottom materials. 𝐸𝑟 is the modulus of elasticity of the reservoir bottom materials and 𝑎𝑛 (𝑡) is the acceleration of the reservoir bottom (earthquake) in the time 𝑡 and perpendicular to the reservoir bottom. 𝐾 𝐶⁄ is the main factor which determines the effects of the absorption of hydrodynamic pressure waves in the reservoir bottom (Fok et al., 1996). 3.9.4 Far End of the Reservoir Boundary Condition The real size of the dam reservoir is very big. Therefore, in reality, the waves produced in a reservoir during earthquake, which move towards the far end of the reservoir, are attenuated in the way. As a result, the far end of the reservoir should be modeled as a completely absorbing boundary for the waves hitting the far end. To date, many researchers have 19 investigated on the creation of a truncated boundary for finite element modeling of the unlimited reservoirs, including the boundary condition obtained by Sommerfeld (Humar & Roufaiel, 1983). This boundary condition is based on the assumption that water waves are propagated in a flat form far away from the upstream face of the dam. According to Humar and Roufaiel (1983), Sommerfeld radiative damping condition for the stimulation frequencies between the first and the second natural frequencies of the reservoir ( 𝜔1 and 𝜔2 respectively) has not been a correct estimation. They created a new boundary condition that has indicated the energy loss of the waves in a wide range of frequencies and with the better results. But it is not precise enough for 𝜔 > 𝜔2 . Moreover, the previous relationships were for incompressible fluid, rectangular reservoir and rigid dam. Sharan (1985 & 1987), also performed different researches with different assumptions for the reservoir far end boundary condition. In the first step in 1985, he presented an equation for the radiation boundary condition of the submerged structure surrounded by the fluid with infinite compressibility. It is valid for all types of fluid-structure contact surface geometries. Finally, in 1987, he presented the damper boundary condition of radiation waves with the assumption of time domain analysis, fluid compressibility and small-amplitude waves for submerged structure in infinite fluid. It is valid for a wide range of stimulation frequencies. However, it is not helpful when the stimulation frequency is close to the natural frequency of the fluid fluctuations. Moreover, when in high stimulation frequency, the reservoir length tends to infinity, lack of convergence is observed. However, because there are not many hydrodynamic forces in high stimulation frequency, the resulting error in hydrodynamic pressure can be ignored (Sharan, 1987). To model this boundary condition in the software, like the reservoir bottom, boundary absorption coefficient can be used. In this case, for modeling a reservoir, if we assume that due to the energy loss, the hydrodynamic pressure in the end of the reservoir is zero, it is sufficient for the far end boundary to be about twice of the reservoir depth away from the upstream face. This assumption is equivalent to the application of a number of pressure wave dampers in the far end (Ghaemian & Ghobarah, 1998). 20 3.10 Semi-Infinite Foundation Boundary Condition After choosing the direct method to model the foundation environment and the soil-structure interaction, there should be some lateral boundary conditions which absorb the reflecting seismic waves back from the structure. The usual modeling methods for semi-infinite environment boundary condition which have been in local time and space coordinates and are independent of loading frequency, according to the increase in their accuracy, are viscous boundary condition (Wolf, 1988), conical model (Wolf, 1988), multi-directional boundary condition (Wolf & Song, 1995), doubly asymptotic boundary condition (Wolf & Song, 1995) and doubly asymptotic multi-directional boundary condition (Wolf & Song, 1995). The accuracy and the type of a given question determine the proper boundary condition. Here the viscous boundary condition has been used. In viscous boundary condition, presented by Lysmer in 1969, wave propagation is assumed one-dimensional. The important point about viscous boundary is that only the waves perpendicular to the boundary area are absorbed completely by a viscous damper and this is the main weakness of this method. In three-dimensional models, such as the three-dimensional model of the foundation in this study, while the semi-finite boundary is far enough from soil-structure contact surface, it can be roughly assumed that the wave propagation is one-dimensional and is perpendicular to the surface of the boundary (Lysmer & Kuhlemeyer, 1969). 21 Chapter 4 Modeling of the Dam Body, Reservoir, and Foundation In this chapter, the characteristics of the finite element modeling of the dam body, reservoir, and foundation are discussed. 4.1 Geometric Characteristics of the Dam In this dissertation, Karun IV is studied as a test case. It is the highest concrete double-curvature arch dam among other existing dams in Iran (Iran Water & Power Resources Development Co., 2004) and the 17th tallest dam in the world (List25 LLC., 2014). The dam was built and has been operating in the south-west of Iran on the Karun River. The dam site canyon is a nonsymmetrical V with steeper slope on the left. The dam was built with the aim of electric power supply, flood control, and water supply for downstream farms and industries (Iran Water & Power Resources Development Co., 2004). Table 1 indicates the main characteristics of the dam. Table 1 Basic information about the dam (Iran Water & Power Resources Development Co., 2004) Width of the dam foundation 37 (𝑚) Width of the dam crest 7 (𝑚) Maximum height of dam construction 230 (𝑚) Length of the dam crest 440 (𝑚) Normal level of operation +1025 (𝑚𝑎𝑠𝑙) Minimum level of operation +996(𝑚𝑎𝑠𝑙) Reservoir volume 2,300,000,000 (𝑚3) 4.1.1 Concrete Characteristics Generally, in the analyses of critical infrastructures such as large dams, the specific features of the applied materials should be considered. The features of the concrete used in the dam body are described as follows: The controlling criterion is the 90-day compressive strength of D25 category concrete which forms the main mass of the dam body. The rate of loading in the seismic loading 22 conditions creates an average of 31% increase in the compressive strength of the concrete (Raphael, 1984). Concrete modulus of elasticity is determined as the secant modulus in the static surface tension of 0.4𝑓𝑐 as: 𝐸𝑐 = 4.73𝑓𝑐0.5 4.1 where 𝐸𝑐 is the modulus of elasticity of concrete and 𝑓𝑐 is the 90-day compressive strength of the concrete. The modulus of elasticity of the concrete is increased up to 25% during seismic loading in the normal condition. Tensile strength of the concrete is an important factor in assessing the sustainability of the large concrete structures such as arch dams under the static and dynamic loading. Based on the results of the tests carried out on 12000 samples of concrete to find out a proper relation between the compressive strength and tensile stress of the concrete, Raphael (1984), suggested the following formulas to determine the tensile strength of the concrete: The actual tensile strength of the concrete under the static loading is calculated by the following equation: 𝑓𝑡 = 0.33𝑓𝑐23 4.2 The nominal tensile strength of the concrete under the static loading that can be directly compared with the results of the linear analysis is: 𝑓𝑡 = 0.44𝑓𝑐23 4.3 The actual tensile strength of the concrete under the dynamic loading is: 𝑓𝑡 = 0.5𝑓𝑐23 4.4 The nominal tensile strength of the concrete under dynamic loading, which can be directly compared with tensile stresses obtained from the linear elastic analysis, is: 𝑓𝑡 = 0.66𝑓𝑐23 4.5 23 For the static loads and Design Basis Earthquake (DBE) load, which its probability of occurrence is high in the early life of the structure, it is reasonable to judge on the results based on the 90-day strength of the concrete. Also for loads with a very low probability of occurrence in the early life of the structure, such as Maximum Credible Earthquake (MCE), it is better to compare the resulting stresses with the strength of the concrete with a life of 10 years or more. Due to the absence of an equation to calculate the compressive strength of the concrete in this life span, the one-year compressive strength of the concrete is applied. According to the results of the experiments carried out on the dams by USBR, for the concretes with 𝑓𝑐 ≥ 25 𝑀𝑃𝑎, the increase in the one-year compressive strength relative to the 90-day compressive strength will be 26% (U. S. Army Corps of Engineers, 1994). The values for the mechanical and thermal properties of the dam concrete, which are applied in the numerical model, are displayed in Table 2. Table 2 Mechanical and thermal properties of the dam concrete (Iran Water & Power Resources Development Co., 2004) Characteristic Quantity Density (𝜌) 2450 (𝐾𝑔 𝑚3⁄ ) Poisson’s ratio (𝜐) 0.2 Static modulus of elasticity (based on one-year 𝑓𝑐) 25.9 (𝐺𝑃𝑎) Dynamic modulus of elasticity (based on one-year 𝑓𝑐) 25.5 (𝐺𝑃𝑎) Static one-year compressive strength 30 (𝑀𝑃𝑎) Dynamic one-year compressive strength 39 (𝑀𝑃𝑎) Static nominal tensile strength (based on one-year 𝑓𝑐) 4.25 (𝑀𝑃𝑎) Dynamic nominal tensile strength (based on one-year 𝑓𝑐) 6.37 (𝑀𝑃𝑎) Coefficient of thermal expansion (𝛼) 8E − 6 (1 °𝐾)⁄ 24 4.2 Water Specifications To determine the hydrodynamic and hydrostatic pressures on the dam, the necessary characteristics should be identified. Accordingly, density (), bulk modulus (K), and speed of sound in the water (C) are set at 1000 𝐾𝑔 𝑚3⁄ , 2131 𝑀𝑃𝑎, and 1460 𝑚/𝑠 respectively. 4.3 Foundation Materials Specifications The intended characteristics for the modeled foundation rock of the desired dam are density, Poisson’s ratio, and static and dynamic deformation modulus of foundation rock mass. They are set as 25 𝐾𝑁 𝑚3⁄ , 0.25, 1.5 𝐺𝑃𝑎, and 15 𝐺𝑃𝑎, respectively. As foundations are made of layers with various properties, these characteristics are related to the dominant texture of the foundation rock mass. . 4.4 Specifications of the Finite Element Model of the Dam, Reservoir, and Foundation The model of the dam consists of the three main sections including dam, reservoir, and foundation; as displayed in Figure 3. Figure 3 Complete model of the dam-reservoir-foundation 25 4.4.1 Finite Element Model of the Dam Body To model the dam body, the geometry of arch dam is modeled through 8-node Solid45 elements for structural analysis, based on the specifications of the project. Concrete material properties and the interaction of the dam-foundation and dam-reservoir interface are taken into account. The dam body contains 642 elements which are arrayed in the three 214-element layers. Each node includes three displacement degrees of freedom. By this assembly, the model can simulate the flexural behavior of the dam and the nonlinear heat distribution. In order to carry out the thermal analysis, the Solid45 elements are replaced by Solid70 elements which include one thermal degree of freedom per node. Figure 4 displays the three-dimensional model of the main dam body. Figure 4 Format of the finite element grid of the dam body Since the surface area of the dam which is in contact with the foundation, especially in the central part of the dam body, is very important, it is attempted to use tetragonal prismatic elements adjacent to the foundation as much as possible. The dam body is assumed fully involved with the rock mass of foundation. With respect to the proper adhesion of concrete to the cleaned surface of the rock mass, this assumption is reasonable. 26 4.4.1.1 Solid45 Element This element allows modeling of the three-dimensional solid structures even with complex and irregular shapes. It is a structural 8-node volumetric element with three displacement degrees of freedom in 𝑥, 𝑦, and 𝑧 directions for each node. In the analytical model with Solid45, the plastic behavior for materials, creep, swelling, stress stiffening, and large stresses and strains can be considered (DeSalvo & Swanson, 2006). The characteristics of materials can be considered orthotropic in the direction of the element axis. The number of nodes, material properties, surface loads (e.g. the pressure on the surface), volumetric loads on the nodes, and specific behavioral characteristics (e.g. plasticity and those mentioned above) are the inputs for this element (DeSalvo & Swanson, 2006). 4.4.1.2 Solid70 Element This element has the ability for three-dimensional heat transfer and is used in transient or steady-state thermal analysis. This element encompasses 8 nodes with one thermal degree of freedom per node and orthotropic material characteristics. This element has also the possibility to consider the mass transport heat flow in a field of constant velocity. If the structural analysis of the model with Solid70 element is required, this element should be replaced by another element (e.g. Soild45). In addition, this element can model the non-linear steady state fluid flow in the porous media. Specific heat and density are not considered for the analysis of steady state. As the input of analysis, convection or heat flux, and radiation can be applied on the element surfaces as surface loads. The rate of heat generation can be applied on the nodes as element body loads (DeSalvo & Swanson, 2006). 4.4.2 Finite Element Model of the Reservoir Reservoir model is roughly in the shape of a half lying cylinder with the maximum 223 m radius (height), 860 m length (about four times of the height of the dam) and 350 m diameter (width) that is formed from a maximum of 6206 8-node Fluid30 element. The number of elements in the reservoir varies according to the levels of water which are Normal Water Level (NWL), and Minimum Water Level (MWL) (Figure 5). 27 Figure 5 Analytical model of the reservoir To model the reservoir, compressibility of water, the interaction of dam-foundation, and the interaction of dam-reservoir interface are considered. Furthermore, the sediments on the floor and walls of the reservoir and the long distance between the dam and the far end of the reservoir cause the damping of seismic waves in the floor, walls, and the far end of the reservoir. Accordingly, wave absorption coefficients are considered for the floor, wall and the reservoir far end elements. It should be noted that the 8-node elements of the dam body and the 8-node elements of the fluid are completely compatible with each other to model the interaction of fluid-structure system. In the finite element model of the reservoir, as the water does not tolerate shear stresses, there are no or very little of these stress distributions. Therefore, water should be able to slide freely on the surface of the dam in the interface of the dam-reservoir, or on the rock mass of the foundation in the interface of reservoir-foundation. 4.4.2.1 Fluid30 Element This element is used for the problems of fluid-structure interaction and to model the fluid environment. The main application of this element is to model the acoustic wave propagation and the dynamics of submerged structures. The major equation for acoustic 28 fluid, namely three-dimensional wave equation, is determined regarding the interaction of the acoustic pressure and structure movement in the interface. This element has 8 nodes and contains 4 degrees of freedom per node, including three displacement in 𝑧, 𝑥, and 𝑦 directions and one for pressure. Nevertheless, the nodal displacements are only applicable in the fluid-structure interface nodes. This element can consider the damping of sound absorbing materials in the fluid-structure interface (DeSalvo & Swanson, 2006). In addition, this element contains a reference pressure (PREF) and the isotropic material properties. Reference pressure is applied to determine the pressure level of sound wave in the element. The dissipative effect of fluid viscosity is not considered in this element. However, the sound absorption in the element interface is considered by creating a damping matrix. It is based on the area of the contact surfaces and the boundary admittance. The characteristics of fluid include fluid density, the speed of sound in it, and the sound absorption in the interface and boundary surfaces. In this element, the equations for wave propagation are solved by assuming the compressible fluid, inviscid fluid, no mean flow of fluid, uniform density and pressure of the fluid, and relatively small acoustic pressure (DeSalvo & Swanson, 2006). 4.4.3 Finite Element Model of the Foundation The interaction of dam-foundation interface has been considered while modeling the foundation. In terms of the geometry, foundation resembles a rectangular cube that dam body and hypothetical valley are removed from the upper surface of it. The model dimensions are 1240 𝑚 by 580 𝑚 by 1090 𝑚 corresponding to the length, height, and width, respectively. It contains 3920 Solid45 elements (Figure 6). To absorb the returning earthquake waves, which are caused by the movement of seismic waves in the massed foundation, the viscous boundary condition is used. For this purpose, a number of Combine14 spring-damper elements are used. They are connected to the nodes around the foundation orthogonally in three directions. The vertical dampers absorb the longitudinal waves. The dampers tangent on the surface absorb shear waves. The bottom of the foundation is completely fixed. 29 Figure 6 Analytical model of the foundation rock 4.4.3.1 Combine14 Element This element has the ability of longitudinal or torsional behaviors with one, two, and three-dimensional applications. The option of longitudinal spring-damper is a uniaxial tensile-compressive element. In each node, it has the maximum of three degrees of freedom in 𝑥, 𝑦, and 𝑧 directions. In this case, there is no bending or torsion. The option of a torsional spring-damper is a totally rotational element with three rotation degrees of freedom around 𝑥, 𝑦, and 𝑧 axes at each node. In this case, there is no axial or flexural load. The spring-damper element is considered massless. Mass, if required, is applied by a proper mass element. The spring or damper capabilities can be deleted from the element (DeSalvo & Swanson, 2006). 30 Chapter 5 Loadings, Analyses and Results This chapter considers the loading characteristics, procedures and results of the different types of analyses as well as the different factors affecting the behavior of arch dams. It has been attempted that the considered assumptions match the real natural conditions as much as possible. 5.1 Types of Loading In this research, according to the preformed analysis, the combinations of the self-weight, hydrostatic and hydrodynamic loads of the reservoir, seismic and thermal nodal loads are applied. It should be mentioned that the critical static loading are combined of water pressure at minimum reservoir level and thermal loading in summer critical time. It also includes water pressure at maximum reservoir level and the thermal loading in winter critical time. The results of the thermal loading are mentioned for these two states. In the usual and thin arch dams, the uplift pressure has little effect and usually is not considered in the stress analysis. The effect of hydrodynamic pressure of the reservoir in thin and light arch dams is significant and critical. Because in such structures the added mass of the reservoir due to the earthquake comparing to the self-weight is high. So, the response is extremely affected by the dam-reservoir interaction. In thermal loading, the concrete temperatures in the different nodes of the dam body are obtained from the thermal analyses. They are applied to the analytical model as nodal loads. Seismic loading will be briefly described in continuation. 5.2 Seismic Loading Considering site conditions, geotechnical properties, earth material, intensity and duration of the earthquakes at different dam sites and comparing the records across the world with similar geological conditions, the San Fernando earthquake of February 1971 was selected (California Institute of Technology (Caltech), 2013). This earthquake has the magnitude of 6.6 on the Richter scale and happened at the approximate distance of 25 𝐾𝑚 from the fault line and on the rigid soil. The maximum ground acceleration between the horizontal components of the earthquake is 0.324𝑔, used along the river (x direction). The different components of the earthquake acceleration record are displayed in Figure 7 to Figure 9. 31 5.2.1 Earthquake Frequency Content Frequency content is the most important characteristic of an earthquake. If the dominant frequency of the earthquake is equal to the frequency of the main vibration mode of the structure, the resonance occurs. In this study, to determine the dominant frequency of the earthquake, SeismoSignal software (Seismosoft Ltd., 2013), which processes the earthquake information, is employed. This software calculates the dominant frequency of the earthquake through Fourier Transform Amplitude method (Yilmaz, 2001). The results are displayed in Figure 10. Since the main direction of impulse in the dam body is 𝑥, the dominant earthquake frequency is determined according to the 𝑥 component of the acceleration record that is equal to 2.93 𝐻𝑧. 5.2.2 Earthquake Duration The amount of energy applied to the structure has direct relationship to the duration of the earthquake. Higher earthquake duration means more energy on the structure. Analysis of the structural systems such as arch dam-reservoir-foundation is time consuming. Due to computational burden, rather than applying the whole earthquake time history record in the analysis, the period and the related time history record in which the main energy of the earthquake happens, are determined and applied on the structure. There are different methods for the calculation of the earthquake duration such as Trifunac – Brady, Bolt – Beracket and Maccann – Shah (Arias, 1970). This research employs SeismoSignal software which uses Trifunac – Brady method (Seismosoft Ltd., 2013) to calculate earthquake duration. In this method, the earthquake duration is when 5 to 95% of the total earthquake energy is applied on the structure. Figure 11 illustrates an earthquake significant duration curve for the x component of the earthquake. The significant duration for the earthquake is 15𝑠 in x direction (from second 1 to 16), 13𝑠 in y direction (from second 1 to 14) and 15.5𝑠 in z direction (from second 1 to 16.5). To be in the safe side, the first 18 seconds of the earthquake record is applied in three directions. 32 Acceleration(g) Velocity(m/sec) Displacement(m) Acceleration Response Spectrum(g) Figure 7 X-direction record of the earthquake -0.2-0.100.10.20.30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Time(sec)Acceleration(g)-0.15-0.1-0.0500.050.10.150 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Time(sec)Velosity(m/sec)-0.025-0.02-0.015-0.01-0.00500.0050.010.0150 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Time(sec)Displacement(m)00.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.9510 1 2 3 4Period(sec)Response Acceleration(g)Damp 5% 33 Acceleration(g) Velocity(m/sec) Displacement(m) Acceleration Response Spectrum(g) Figure 8 Y-direction record of the earthquake -0.2-0.15-0.1-0.0500.050.10.150 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Time(sec)Acceleration(g)-0.08-0.06-0.04-0.0200.020.040.060.080 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Time(sec)Velocity(m/sec)-0.015-0.01-0.00500.0050.010.0150 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Time(sec)Displacement(m)00.050.10.150.20.250.30.350.40.450.50.550.60.650.70 1 2 3 4Period(sec)Response Acceleration(g)Damp 5% 34 Acceleration(g) Velocity(m/sec) Displacement(m) Acceleration Response Spectrum(g) Figure 9 Z-direction record of the earthquake -0.25-0.2-0.15-0.1-0.0500.050.10.150.20.250.30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Time(sec)Acceleration(g)-0.15-0.1-0.0500.050.10.150.20.250.30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Time(sec)Velosity(m/sec)-0.05-0.04-0.03-0.02-0.0100.010.020 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Time(sec)Displacement(m)00.050.10.150.20.250.30.350.40.450.50.550.60.650.70 1 2 3 4Period(sec)Response Acceleration(g)Damp 5% 35 Fourier Amplitude Fourier Amplitude Y component X component Fourier Amplitude Z component Figure 10 Fourier Transform Amplitude of X, Y, and Z components of the earthquake Arias Intensity (%) Time (Sec) Figure 11 Curve of the energy released during the earthquake for X component 00.010.020.030.040.050.060.070.080.090.10.110.0001 0.001 0.01 0.1 1 10 100Frequency(Hz)Fourier Amplitude00.010.020.030.040.050.060.070.080.090.10.110.120.130.140.150.160.170.0001 0.001 0.01 0.1 1 10 100Frequency(Hz)Fourier Amplitude00.010.020.030.040.050.060.070.080.090.10.110.120.130.140.150.160.170.180.190.20.0001 0.001 0.01 0.1 1 10 100Frequency(Hz)Fourier Amplitude 36 5.3 Thermal Analysis The temperature distribution and its related thermal quantities in the whole system or in part of it are calculated by thermal analysis. The usual thermal quantities contain temperature distribution, lost or gained heat, thermal gradient and thermal flux. Generally, thermal analysis aims to determine the thermal loading of the target system, to analyze and calculate thermal stresses (DeSalvo & Swanson, 2006). The thermal input in the crown cantilever and the results of thermal analyses of the upstream, downstream and central cantilever are presented in Figure 12 and Figure 13, respectively. Temperature variations are visible through the central cantilever profile. For winter time and NWL, approximate changes are from 9 to 17 degrees on the foundation level and from 12 to 16 degrees on the crest level. For summer time and MWL, temperature changes from 9 to 32 degrees on the foundation level and from 28 to 32 degrees on the crest level. Figure 12 Thermal input in the crown cantilever 37 DOWNSTREAM UPSTREAM WINTER (º C), NWL SUMMER (º C), MWL Figure 13 Temperature distribution in the dam body for the summer and winter critical times 38 5.4 Stability Analysis To check the stability and liability of the built model, static analysis is performed just on the dam-massless foundation system without water. This analysis aims to consider the dam stability under the dam weight before the application of operational loading. Figure 14 illustrates the distribution of the maximum principal stress (𝑆1) and minimum principal stress (𝑆3) for downstream and upstream faces. Figure 15 displays the resulting displacement in 𝑥 direction. DOWNSTREAM UPSTREAM S3 (N/m2) S1 (N/m2) Figure 14 Distribution of the maximum principal stress (S1) and minimum principal stress (S3) for downstream and upstream faces in the stability analysis -.647E+07-.575E+07-.503E+07-.431E+07-.358E+07-.286E+07-.214E+07-.141E+07-69195730748-.110E+07-598140-94194409752913698.142E+07.192E+07.243E+07.293E+07.343E+07-.647E+07-.575E+07-.503E+07-.431E+07-.358E+07-.286E+07-.214E+07-.141E+07-69195730748-.110E+07-598140-94194409752913698.142E+07.192E+07.243E+07.293E+07.343E+07 39 Figure 15 Distribution of the X direction displacement (m) in the stability analysis According to the results, the maximum tension happens at the top corners of the upstream and downstream. The maximum pressure happens around the dam heel. Comparing to the allowable tensile and compressive strengths, the results are in acceptable range indicating no failure for the dam. 5.5 Static Analysis In this stage, the static analysis is done on the complete model of the dam-foundation-reservoir under the dam and water self-weight, and the hydrostatic pressure. The results obtained from the static analysis and thermal loading are utilized as the initial conditions for the dynamic analysis. The distribution of 𝑆1 and 𝑆3 stresses for the upstream and downstream faces in maximum water level condition is displayed in Figure 16. Furthermore, the distribution of displacement in 𝑥 direction (upstream-downstream) in the dam body is shown in Figure 17. The maximum tensile areas are around the dam heel. The maximum compressive areas are along the walls and tow in the downstream, and in the middle and upper parts of the upstream face. All stresses are in the safe zone except for a narrow strip in the dam heel. -.016504-.014561-.012617-.010674-.008731-.006788-.004845-.002901-.958E-03.985E-03 40 DOWNSTREAM UPSTREAM S3 (N/m2) S1 (N/m2) Figure 16 Distribution of S1 and S3 for upstream and downstream faces in static analysis Figure 17 Distribution of X direction displacement (m) in static analysis -.863E+07-.757E+07-.651E+07-.544E+07-.438E+07-.332E+07-.226E+07-.119E+07-132002930262 -.256E+07-.142E+07-272490871817.202E+07.316E+07.430E+07.545E+07.659E+07.774E+07 -.863E+07-.757E+07-.651E+07-.544E+07-.438E+07-.332E+07-.226E+07-.119E+07-132002930262 -.256E+07-.142E+07-272490871817.202E+07.316E+07.430E+07.545E+07.659E+07.774E+07 0.007295.01459.021885.02918.036475.04377.051065.05836.065655 41 5.6 Modal Analysis The analysis of Rayleigh damping coefficients is a prerequisite for the dynamic analysis. These coefficients themselves are dependent on the dominant vibration frequency of the system. The natural frequencies and vibration mode shapes of the system can be determined through modal analysis (Clough & Penzin, 1993). Due to the asymmetry of the mass system of the fluid element, the determination of the vibration frequencies of the system in the presence of water is very complicated. Therefore, modal analysis is performed without reservoir elements. In dynamic analysis, because the dynamic modulus of elasticity applies to the system, modal analysis is also carried out using dynamic modulus of elasticity. The results of the performed analysis with the first fifteen natural frequencies are presented in the Table 3. In addition, the shapes of the first seven vibration modes for the dam are presented in Figure 18. Table 3 Natural vibration frequencies for the first fifteen vibration modes of the system Natural vibration frequency (Hrz) Vibration mode number 0.583035 1 0.616217 2 0.644124 3 0.736706 4 0.760074 5 0.809045 6 0.978767 7 0.998337 8 1.06159 9 1.07612 10 1.19599 11 1.21912 12 1.22859 13 1.28714 14 1.32452 15 42 mode 2 mode 1 mode 4 mode 3 mode 6 mode 5 mode 7 Figure 18 First seven vibration modes of the dam body 5.7 Dynamic Analysis To consider the dynamic response of the structures under any kind of the time-varying loads, the transient dynamic analysis is applied. In this procedure, the system equations of motion are analyzed as a set of static equilibrium equations, considering inertial and damping forces at any given time. The Newmark time integration method is used by the software to solve the equations of motion (DeSalvo & Swanson, 2006). Generally, there are three full transient dynamic analysis methods, namely direct method (Bathe & Wilson, Stability and accuracy analysis of direct integration methods, 1973), reduced method (Huang & Huang, 2015), and mode superposition method (Kuran & Özgüven, 1996). The method used in this study is direct integration method, in which the matrices of mass, stiffness and damping of the whole system is used in the analysis. However, the other methods do not include the whole system 43 in the analysis. Direct method is assumed stronger than the other methods for full transient dynamic analysis, though the time spent in this method is more than others (DeSalvo & Swanson, 2006). 5.7.1 Dynamic Analysis without Thermal Loading In the beginning of each seismic analysis, a static analysis under the self-weight and hydrostatic pressure is performed to determine the initial conditions of the system. Then, the earthquake time-history record is applied to the model in three directions. Dynamic analyses are carried out in the two states of normal and minimum water levels. To consider the results of the dynamic analyses, 𝑆1 and 𝑆3 distribution envelopes for the upstream and downstream faces, the time-history curve of the dam crest displacement in X direction, and the time-history curve of 𝑆1 variations for 4 nodal points on the upstream and 4 nodal points on the downstream are presented as shown in Figure 19. These points are selected as they have been considered as part of the vulnerable areas to the maximum stresses. Figure 19 Nodes used for consideration of dynamic analyses results on the dam upstream and downstream faces In brief, this part of the research performs linear dynamic analysis without thermal loading in the two situations of a) with the reservoir NWL and b) with the reservoir MWL. The results of these analyses are illustrated in Figure 20 to Figure 27. 44 Figure 20 S1 and S3 distribution envelops for the dam upstream and downstream faces in state (a) S1(MPa)-MAX-DS5.941875.553755.165624.77754.389374.001253.613123.2252.836872.448752.060621.67251.284370.896250.508125Frame 001 20 Apr 2009 DAM BODY CONTOURS1(MPa)-MAX-US6.6356.155.6655.184.6954.213.7253.242.7552.271.7851.30.8150.33-0.155Frame 001 20 Apr 2009 DAM BODY CONTOURS3(MPa)-MIN-DS-2.5075-3.055-3.6025-4.15-4.6975-5.245-5.7925-6.34-6.8875-7.435-7.9825-8.53-9.0775-9.625-10.1725Frame 001 20 Apr 2009 DAM BODY CONTOURS3(MPa)-MIN-US-1.5925-2.055-2.5175-2.98-3.4425-3.905-4.3675-4.83-5.2925-5.755-6.2175-6.68-7.1425-7.605-8.0675Frame 001 20 Apr 2009 DAM BODY CONTOUR 45 NODE14-UX(m) TIME(Sec) Figure 21 Time-history curve for the dam crest displacement in X direction in state (a) NODE748-S1(Pa) NODE750-S1(Pa) TIME(Sec) TIME(Sec) NODE787-S1(Pa) NODE803-S1(Pa) TIME(Sec) TIME(Sec) Figure 22 Time-history curves of S1 for the selected nodes in the dam upstream face in state (a) -0.02-0.0100.010.020.030.040.050.060.070.080 2 4 6 8 10 12 14 16 18Time(sec)UX(m)-Node14-1.00E+050.00E+001.00E+052.00E+053.00E+054.00E+055.00E+056.00E+057.00E+058.00E+050 3 6 9 12 15 18Time(sec)S1(N/m2)-Node748-1.00E+050.00E+001.00E+052.00E+053.00E+054.00E+055.00E+056.00E+050 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node750-4.00E+05-2.00E+050.00E+002.00E+054.00E+056.00E+058.00E+051.00E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node787-2.00E+050.00E+002.00E+054.00E+056.00E+058.00E+051.00E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node803 46 NODE64-S1(Pa) TIME(Sec) NODE66-S1(Pa) TIME(Sec) NODE103-S1(Pa) TIME(Sec) NODE119-S1(Pa) TIME(Sec) Figure 23 Time-history curves of S1 for the selected nodes in the dam downstream face in state (a) -5.00E+050.00E+005.00E+051.00E+061.50E+062.00E+062.50E+063.00E+060 2 4 6 8 10 12 14 16 18Time(sec) 20bS1(N/m2)-Node64 24b/22b`-5.00E+050.00E+005.00E+051.00E+061.50E+062.00E+062.50E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node66-1.00E+06-5.00E+050.00E+005.00E+051.00E+061.50E+062.00E+062.50E+060 3 6 9 12 15 18Time(sec)S1(N/m2)-Node103-1.00E+06-5.00E+050.00E+005.00E+051.00E+061.50E+062.00E+062.50E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node119 47 Figure 24 S1 and S3 distribution envelops for the dam upstream and downstream in state (b) S1(MPa)-MAX-DS5.136254.81254.488754.1653.841253.51753.193752.872.546252.22251.898751.5751.251250.92750.60375Frame 001 22 Apr 2009 DAM BODY CONTOURS1(MPa)-MAX-US6.6356.155.6655.184.6954.213.7253.242.7552.271.7851.30.8150.33-0.155Frame 001 22 Apr 2009 DAM BODY CONTOURS3(MPa)-MIN-DS-2.24375-2.6775-3.11125-3.545-3.97875-4.4125-4.84625-5.28-5.71375-6.1475-6.58125-7.015-7.44875-7.8825-8.31625Frame 001 22 Apr 2009 DAM BODY CONTOURS3(MPa)-MIN-US-2.03562-2.40125-2.76687-3.1325-3.49812-3.86375-4.22937-4.595-4.96062-5.32625-5.69187-6.0575-6.42312-6.78875-7.15437Frame 001 22 Apr 2009 DAM BODY CONTOUR 48 NODE14-UX(m) TIME(Sec) Figure 25 Time-history curve for the dam crest displacement in X direction in state (b) NODE748-S1(Pa) NODE750-S1(Pa) TIME(Sec) TIME(Sec) NODE787-S1(Pa) NODE803-S1(Pa) TIME(Sec) TIME(Sec) Figure 26 Time-history curves of S1 for the selected nodes in the dam upstream face in state (b) -0.02-0.0100.010.020.030.040.050.060.070.080 2 4 6 8 10 12 14 16 18Time(sec)UX(m)-Node14-1.00E+054.00E+059.00E+051.40E+061.90E+062.40E+062.90E+060 3 6 9 12 15 18Time(sec)S1(N/m2)-Node748-5.00E+050.00E+005.00E+051.00E+061.50E+062.00E+062.50E+063.00E+063.50E+064.00E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node750-4.00E+05-2.00E+050.00E+002.00E+054.00E+056.00E+058.00E+051.00E+061.20E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node787-2.00E+050.00E+002.00E+054.00E+056.00E+058.00E+051.00E+061.20E+061.40E+061.60E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node803 49 NODE64-S1(Pa) TIME(Sec) NODE66-S1(Pa) TIME(Sec) NODE103-S1(Pa) TIME(Sec) NODE119-S1(Pa) TIME(Sec) Figure 27 Time-history curves of S1 for the selected nodes in the dam downstream face in state (b) -1.00E+060.00E+001.00E+062.00E+063.00E+064.00E+065.00E+060 2 4 6 8 10 12 14 16 18Time(sec) 20bS1(N/m2)-Node64 24b/22b`-5.00E+050.00E+005.00E+051.00E+061.50E+062.00E+062.50E+063.00E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node66-1.00E+06-5.00E+050.00E+005.00E+051.00E+061.50E+062.00E+062.50E+063.00E+063.50E+060 3 6 9 12 15 18Time(sec)S1(N/m2)-Node103-5.00E+050.00E+005.00E+051.00E+061.50E+062.00E+062.50E+063.00E+063.50E+064.00E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node119 50 According to the results of state (a), the maximum tensile areas are the dam top corners and around the heel and tow, in the upstream and downstream faces, and also in the upper part of the downstream near the crest. The maximum compressive areas are along the walls, near the tow, and around the top middle of the downstream, and also along the middle of the crest in the upstream. Comparing to the allowable tensile and compressive strengths, all areas are in the acceptable range, unless for the upstream top left and right corners where tensile failure happens. The same dynamic analysis was conducted for state (b). The maximum tensile areas are top corners of the dam in the upstream and downstream especially in the upstream right corner, in the middle of the crest in the downstream and especially in the upstream, and around the heel and tow. The maximum compressive areas are along the walls, tow, and around the top middle of the downstream, and along the middle of the crest in upstream. Comparing to the allowable tensile and compressive strengths, all areas are in the safe zone, unless for the upstream top right corner where tensile failure can happen. 5.7.2 Dynamic Analysis with Thermal Loading In this stage, in addition to the previously mentioned loadings, the nodal temperatures are also applied on the dam body in the initial static analysis. Then, in the presence of initial thermal and static stresses, the linear dynamic analysis is performed in two states of c) in the winter critical time with the reservoir NWL and d) in the summer critical time with the reservoir MWL. The related results are represented in the Figure 28 to Figure 35. 51 Figure 28 S1 and S3 distribution envelops for the dam upstream and downstream faces in state (c) S1(MPa)-MAX-DS6.271885.833755.395634.95754.519384.081253.643133.2052.766882.328751.890631.45251.014370.576250.138125Frame 001 04 May 2009 DAM BODY CONTOURS1(MPa)-MAX-US8.141887.563756.985636.40755.829385.251254.673134.0953.516882.938752.360631.78251.204380.626250.048125Frame 001 04 May 2009 DAM BODY CONTOURS3(MPa)-MIN-DS-2.08375-2.6475-3.21125-3.775-4.33875-4.9025-5.46625-6.03-6.59375-7.1575-7.72125-8.285-8.84875-9.4125-9.97625Frame 001 04 May 2009 DAM BODY CONTOURS3(MPa)-MIN-US-0.815625-1.20125-1.58687-1.9725-2.35812-2.74375-3.12937-3.515-3.90062-4.28625-4.67187-5.0575-5.44312-5.82875-6.21437Frame 001 04 May 2009 DAM BODY CONTOUR 52 NODE14-UX(m) TIME(Sec) Figure 29 Time-history curve for the dam crest displacement in X direction in state (c) NODE748-S1(Pa) NODE750-S1(Pa) TIME(Sec) TIME(Sec) NODE787-S1(Pa) NODE803-S1(Pa) TIME(Sec) TIME(Sec) Figure 30 Time-history curves of S1 for the selected nodes in the dam upstream face in state (c) -0.0100.010.020.030.040.050.060.070.080 2 4 6 8 10 12 14 16 18Time(sec)UX(m)-Node14-1.00E+051.00E+053.00E+055.00E+057.00E+059.00E+050 3 6 9 12 15 18Time(sec)S1(N/m2)-Node748-1.00E+050.00E+001.00E+052.00E+053.00E+054.00E+055.00E+056.00E+057.00E+058.00E+050 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node7501.00E+053.00E+055.00E+057.00E+059.00E+051.10E+061.30E+061.50E+061.70E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node7870.00E+003.00E+056.00E+059.00E+051.20E+061.50E+061.80E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node803 53 NODE64-S1(Pa) TIME(Sec) NODE66-S1(Pa) TIME(Sec) NODE103-S1(Pa) TIME(Sec) NODE119-S1(Pa) TIME(Sec) Figure 31 Time-history curves of S1 for the selected nodes in the dam upstream face in state (c) -5.00E+050.00E+005.00E+051.00E+061.50E+062.00E+062.50E+063.00E+060 2 4 6 8 10 12 14 16 18Time(sec) 20bS1(N/m2)-Node64 24b/22b`-5.00E+050.00E+005.00E+051.00E+061.50E+062.00E+062.50E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node66-1.00E+06-5.00E+050.00E+005.00E+051.00E+061.50E+062.00E+060 3 6 9 12 15 18Time(sec)S1(N/m2)-Node103-1.00E+06-5.00E+050.00E+005.00E+051.00E+061.50E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node119 54 Figure 32 S1 and S3 distribution envelops for the dam upstream and downstream faces in state (d) S1(MPa)-MAX-DS4.68254.2853.88753.493.09252.6952.29751.91.50251.1050.70750.31-0.0875-0.485-0.8825Frame 001 05 May 2009 DAM BODY CONTOURS1(MPa)-MAX-US6.89756.4355.97255.515.04754.5854.12253.663.19752.7352.27251.811.34750.8850.4225Frame 001 05 May 2009 DAM BODY CONTOURS3(MPa)-MIN-DS-3.11437-3.61875-4.12312-4.6275-5.13188-5.63625-6.14063-6.645-7.14938-7.65375-8.15813-8.6625-9.16688-9.67125-10.1756Frame 001 05 May 2009 DAM BODY CONTOURS3(MPa)-MIN-US-0.815625-1.20125-1.58687-1.9725-2.35812-2.74375-3.12937-3.515-3.90062-4.28625-4.67187-5.0575-5.44312-5.82875-6.21437Frame 001 05 May 2009 DAM BODY CONTOUR 55 NODE14-UX(m) TIME(Sec) Figure 33 Time-history curve for the dam crest displacement in X direction in state (d) NODE748-S1(Pa) NODE750-S1(Pa) TIME(Sec) TIME(Sec) NODE787-S1(Pa) NODE803-S1(Pa) TIME(Sec) TIME(Sec) Figure 34 Time-history curves of S1 for the selected nodes in the dam upstream face in state (d) -0.08-0.06-0.04-0.0200.020.040 2 4 6 8 10 12 14 16 18Time(sec)UX(m)-Node14-5.00E+050.00E+005.00E+051.00E+061.50E+062.00E+062.50E+063.00E+063.50E+064.00E+060 3 6 9 12 15 18Time(sec)S1(N/m2)-Node748-5.00E+055.00E+051.50E+062.50E+063.50E+064.50E+065.50E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node7501.00E+053.00E+055.00E+057.00E+059.00E+051.10E+061.30E+061.50E+061.70E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node7870.00E+003.00E+056.00E+059.00E+051.20E+061.50E+061.80E+062.10E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node803 56 NODE64-S1(Pa) TIME(Sec) NODE66-S1(Pa) TIME(Sec) NODE103-S1(Pa) TIME(Sec) NODE119-S1(Pa) TIME(Sec) Figure 35 Time-history curves of S1 for the selected nodes in the dam upstream face in state (d) 0.00E+005.00E+051.00E+061.50E+062.00E+062.50E+063.00E+063.50E+064.00E+064.50E+060 2 4 6 8 10 12 14 16 18Time(sec) 20bS1(N/m2)-Node64 24b/22b`0.00E+002.00E+054.00E+056.00E+058.00E+051.00E+061.20E+061.40E+061.60E+061.80E+062.00E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node66-1.50E+06-5.00E+055.00E+051.50E+062.50E+063.50E+064.50E+060 3 6 9 12 15 18Time(sec)S1(N/m2)-Node103-1.50E+06-5.00E+055.00E+051.50E+062.50E+063.50E+064.50E+060 2 4 6 8 10 12 14 16 18Time(sec)S1(N/m2)-Node119 57 According to the results of state (c), the maximum tensile areas are the dam top corners and around the heel and tow, in the upstream and downstream faces, and also in the upper part of the downstream near the crest. The maximum compressive areas are along the walls, near the tow, and around the top middle of the downstream, and also along the middle of the crest in the upstream. Comparing to the allowable tensile and compressive strengths, all areas are in the acceptable range, except for the upstream top left and right corners where more intense tensile failure happens. The same dynamic analysis was done for state (d). The maximum tensile areas are near top corners of the dam in the upstream and downstream especially in the upstream right corner, in the middle of the crest in the downstream and especially in the upstream, and around the tow and mostly the heel. The maximum compressive areas are around the tow and middle of the downstream face and along the middle of the crest and at the top corners in the upstream. Comparing to the allowable tensile and compressive strengths, all areas are in the safe zone, except for the upstream top right corner and middle of the crest, where more intense tensile failure can happen. By adding thermal effects, the max tensile and compressive areas are nearly the same, but the stress distribution, levels, and failure points are somehow different. 58 Chapter 6 Conclusion 6.1 Conclusions The comparison of the results of the dynamic analyses with and without thermal loading, i.e. comparing the state (a) with the state (c) as well as comparing the state (b) with the state (d) indicates that: In the dynamic analysis with thermal loading in the winter critical time and NWL (state c): 1. The overall values for the maximum tensile S1 (the upper limit of S1 contour) in the upstream and downstream of the dam are more than the maximum tensile S1 in the dynamic analysis without thermal loading (state a). 2. The values for the minimum tensile S1 (the lower limit of S1 contour) are generally more in the upstream and less in the downstream, compared with the minimum tensile S1 in the dynamic analysis without thermal loading (state a). 3. The values for the maximum compressive S3 and minimum compressive S3 are generally less than the similar parameters in the dynamic analysis without thermal loading (state a). In the dynamic analysis with thermal loading in the summer critical time and MWL (state d): 1. The values for the maximum tensile S1 (the upper limit of S1 contour) are generally more in the upstream and less in the downstream compared with the maximum tensile S1 in the dynamic analysis without thermal loading (state b). 2. The values for the minimum tensile S1 (the lower limit of S1 contour) are generally more in the upstream and less in the downstream, compared with the minimum tensile S1 in the dynamic analysis without thermal loading (state b). 3. The values for the maximum compressive S3 are generally less in the upstream and downstream and the values for the minimum compressive S3 are more in the downstream and less in the upstream compared to the similar parameters in the dynamic analyses without thermal loading (state b). Overall, in the dynamic analysis with thermal loading, in winter critical time and NWL (state c), S1 values for the downstream face, except for the right and left corners of the 59 crest, are less and for the upstream face, except for the heel, are more than the similar values in cases without thermal loading (state a). In the dynamic analysis with thermal loading in the summer critical time and MWL (state d), S1 values for the downstream face, except for the upper right and left corners and the toe, is less, and for the upstream face, is more than the cases without thermal loading (state b). In the stress distribution envelop for the dam upstream and downstream faces in the dynamic analysis with thermal loading at winter critical time and NWL (state c), S3 values, except for the toe and side of the abutments of downstream, are mostly less than the corresponding values in the dynamic analysis without thermal loading (state a). In the dynamic analysis with thermal loading at summer critical time and MWL (state d), S3 values are mostly more for the downstream face, except for the upper left and right corners and slightly higher than the toe than the corresponding values in the dynamic analysis without thermal loading (state b). S3 values are mostly less for the upstream face, except for the middle and upper left corner, than the state (b). The above results indicate that, in overall, the application of thermal loading increases the tension values in 50% of cases and decrease the compression values in 75% of cases in the dam body. The maximum displacement of the dam crest in x direction (Ux) toward the downstream in the central cantilever and downstream face due to the thermal loading at winter critical time and NWL (state c) is higher than Ux produced at similar times in the situations without thermal loading. Also, Ux due to the thermal loading at summer critical time and MWL (state d) is less than Ux produced at similar times in the situations without thermal loading. Therefore, it can be concluded that, due to the thermal loading, the difference between the maximum and minimum displacement remains constant in each diagram comparing to the cases without thermal loading. But, the dam crest in the central cantilever moves toward upstream in 75% of cases during thermal loading. S1 values for the selected nodes at winter critical time and NWL (state c) in upstream and downstream faces are respectively more and less than the S1 values in the analysis without thermal loading. Finally, this research revealed that the application of thermal loading in the dynamic analysis of an arch dam is necessary. Thermal effects changed the results of dynamic 60 analyses. They increased the maximum tension, changed the maximum pressure, and decreased the displacements. Since the effects of various states of heat distribution on the values of compressive and tensile stresses in the dam body are different, the critical thermal states should be applied on the dam and their effects should be examined separately. The comparison of the time history of the displacement of the dam crest in the above states indicates that the displacement of the dam crest while applying thermal loading is less than its displacement while there is no thermal loading. 6.2 Recommendations The following suggestions are recommended for more realistic studies in the future: Study the reservoir-foundation interaction effects on the static and dynamic response of the arch dams more precisely. 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State of temperature and thermal stress in mass concrete structures subjected to thermal shock. Dam Engineering, 7, 336-350. Zhou, J., Lin, G., Zhu, T., Jefferson, A. D., & Williams, F. W. (2000). Experimental investigations into seismic failure of high arch dams. Journal of Structural Engineering, 126(8), 926-935. 66 Appendix A: Permission Request Dear Ryhane Moghadas Jafari: Permission is granted for you to reuse figure 2 from "Effects of Environmental Action on Thermal Stress Analysis of Karaj Concrete Arch Dam" for your Thesis. A full credit line must be added to the material being reprinted. For reuse in non-ASCE publications, add the words "With permission from ASCE" to your source citation. For Intranet posting, add the following additional notice: "This material may be downloaded for personal use only. 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Dynamic analysis of arch dams : effect of thermal loading Moghadas Jafari, Ryhane 2016
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Title | Dynamic analysis of arch dams : effect of thermal loading |
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Moghadas Jafari, Ryhane |
Publisher | University of British Columbia |
Date Issued | 2016 |
Description | Conducting a case study, this thesis investigates the dynamic behavior of Karun IV arch dam (Iran) under the effect of maximum feasible thermal input. A three-dimensional finite element model is created using ANSYS software. The thesis determines the impact of thermal loadings under two scenarios of normal and minimum water levels. The dam-foundation, dam-water and water-foundation interactions are considered in the modeling to accurately capture its dynamic response. Furthermore, in addition to the water compressibility, appropriate wave absorbing boundaries are used for the reservoir far end and bottom and for the foundation. Stability, static, modal, and thermal analyses are conducted as initial conditions. Using three orthogonal earthquake components, linear dynamic time history analyses with and without thermal loading are performed. The comparison of the results verified that the application of thermal effects on the dynamic analysis increases the maximum tension, changes the maximum pressure, and decreases the displacements in the dam body. This study concludes that thermal loading must be considered in the dynamic analysis of an arch dam as it can worsen the tensile cracks. |
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Thesis/Dissertation |
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Language | eng |
Date Available | 2016-01-20 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivs 2.5 Canada |
DOI | 10.14288/1.0223562 |
URI | http://hdl.handle.net/2429/56585 |
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Master of Applied Science - MASc |
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Civil Engineering |
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Applied Science, Faculty of Engineering, School of (Okanagan) |
Degree Grantor | University of British Columbia |
GraduationDate | 2016-02 |
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UBCO |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/2.5/ca/ |
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