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Non-linear response of the rockfill in a concrete-faced rockfill dam under seismic excitation

Non-linear response of the rockfill in a concrete-faced rockfill dam under seismic excitation Mathematical and Computer Modelling of Dynamical Systems, 2015 Vol. 21, No. 1, 77–101, http://dx.doi.org/10.1080/13873954.2014.898157 Non-linear response of the rockfill in a concrete-faced rockfill dam under seismic excitation a b Murat Emre Kartal * and Alemdar Bayraktar a b Department of Civil Engineering, Bülent Ecevit University, 67100 Zonguldak, Turkey; Department of Civil Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey (Received 28 July 2013; accepted 22 February 2014) This study investigates the non-linear response of the rockfill in a concrete-faced rockfill dam under seismic excitation for various concrete slab thicknesses from 30 to 100 cm thick. The effect of the friction in the concrete slab rockfill interface on the non-linear response of the rockfill is also considered in the numerical solutions. The friction contact in the concrete slab joints is provided with contact elements based on the Coulomb’s friction law. The hydrodynamic pressure of the reservoir water is considered by the fluid finite elements based on the Lagrangian approach. Geometrical and material non-linear behaviours of the dam-foundation-reservoir inter- action system are considered together in the finite element analyses. The Drucker– Prager model is used to obtain materially non-linear behaviour of the concrete slab. The multi linear kinematic hardening model is used for rockfill and foundation rock in the material non-linear analyses. Therefore, the uniaxial stress–strain relations of the rockfill and foundation rock are determined from the shear stress–shear strain relations of the gravel and rock, respectively. The principle compressive and tensile stresses of the rockfill are investigated along the horizontal section of the rockfill. According to numerical analyses, the most critical stresses occur in the upstream and downstream faces of the dam for the concrete slab thickness of 30 cm, while the optimum thickness for seismic performance is 58 cm. Keywords: concrete-faced rockfill dam; friction contact; geometrical and material non-linear behaviour; Lagrangian approach 1. Introduction Concrete-faced rockfill (CFR) dams prevent water penetration in the upstream face with concrete slab. Many researchers focused on the performance of concrete slab [1–17]. CFR dams are known to be safe under earthquake ground motions because they do not develop excess porewater pressures [11]. Nevertheless, excessive stresses can still occur in the rockfill under seismic excitations. Especially, tensile stresses may result in stability problems in the upstream and downstream faces of the dam. In addition to this, the observations indicate plastic deformations in the rockfill dams under seismic excitations. The rockfill is the main load-bearing system of a CFR dam and significantly affects the response of the concrete slab. Therefore, its seismic behaviour is very important and should be determined under ground motion effects. The CFR dams include dam-foundation-reservoir interaction problems. Also, CFR dams are exposed to hydrodynamic pressure on their face slab under earthquake effects. The response of the concrete slab depends upon the response of the rockfill, but there is *Corresponding author. Email: murat_emre_kartal@hotmail.com © 2014 Taylor & Francis 78 M.E. Kartal and A. Bayraktar little information related to the rockfill-concrete face dynamic interaction response. Recently, [1] investigated a CFR dam subjected to near-fault ground motion effects considering the rockfill response. This study focuses on the non-linear response of the rockfill in a CFR dam under seismic excitation. Numerical analyses include geometrically non-linear response of the dam to consider large deformations under ground motion effects. The Drucker–Prager model is used for concrete slab in the non-linear analyses. In addition to this, the multi- linear kinematic-hardening model is considered for the non-linear response of the rockfill and rock foundation [18,19]. The uniaxial stress–strain relation of the medium is required for this model. These relations are obtained from the normalized shear modulus–shear strain relation of the gravels [20] and rocks [21]. Both welded and friction contacts are considered in the interaction interfaces. The friction in the joints is modelled by the contact elements based on the Coulomb’s friction law. This study also considers hydro- dynamic pressure of the reservoir water. The reservoir water is modelled by the fluid finite elements based on the Lagrangian approach. The principle compressive and tensile stresses occurring in the rockfill along the largest cross section of the dam in valley direction under seismic excitation are presented and investigated in detail. All of the numerical analyses were performed using ANSYS finite element programme [22]. According to the numerical analyses, this study indicates which parameters should be considered for the optimum design rockfill of concrete-faced rockfill dams. 2. Dam-reservoir-foundation interaction by the Lagrangian approach The formulation of the fluid system based on the Lagrangian approach is presented as follows [21,22]. This approach assumes the fluid as linearly compressible, inviscid and irrotational. For a general two-dimensional fluid, pressure–volumetric strain relationships can be written in matrix form as follows, C 0 ε 11 v ¼ (1) 0C w z 22 z where P, C and ε are the pressures which are equal to mean stresses, the bulk modulus 11 v and the volumetric strains of the fluid, respectively. Since irrotationality of the fluid is considered like penalty methods [23,24], rotations and constraint parameters are included in the pressure–volumetric strain equation (Equation (1)) of the fluid. In this equation, P is the rotational stress, C is the constraint parameter and w is the rotation about the 22 z Cartesian axis z. In this study, the equations of motion of the fluid system are obtained using energy principles. Using the finite element approximation, the total strain energy of the fluid system may be written as, π ¼ U K U (2) e f f where U and K are the nodal displacement vector and the stiffness matrix of the fluid f f system, respectively. K is obtained by the sum of the stiffness matrices of the fluid elements as follows, Mathematical and Computer Modelling of Dynamical Systems 79 K ¼ K (3) e e e e K ¼ B C B dV f f f where C is the elasticity matrix consisting of diagonal terms in Equation (1). B is the strain-displacement matrix of the fluid element. An important behaviour of fluid systems is the ability to displace without a change in volume. For reservoir and storage tanks, this movement is known as sloshing waves in which the displacement is in the vertical direction. Using the finite element method, the increase in the potential energy of the system due to the free surface motion can be written as, π ¼ U S U (4) s f sf sf where U and S are the vertical nodal displacement vector and the stiffness matrix of the sf f free surface of the fluid system, respectively. S is obtained by the sum of the stiffness matrices of the free surface fluid elements as follows, S ¼ S = e e T (5) S ¼ ρ g h h dA f f s ; where h is the vector consisting of interpolation functions of the free surface fluid element. ρ and g are the mass density of the fluid and the acceleration due to gravity, respectively. Also, the kinetic energy of the system using the finite element method can be written as, _ _ T ¼ U M U (6) f f where U and M are the nodal velocity vector and the mass matrix of the fluid system, f f respectively. M is also obtained by the sum of the mass matrices of the fluid elements as follows, M ¼ M = e T e (7) M ¼ ρ H HdV f f ; where H is the matrix consisting of interpolation functions of the fluid element. If Equations (2), (4) and (6) are combined using the Lagrange’s equation, the following set of equations is obtained, M U þ K U ¼ R (8) f f f f where K , Ü , U and R are the system stiffness matrix including the free surface f f f stiffness, the nodal acceleration and displacement vectors and time-varying nodal force 80 M.E. Kartal and A. Bayraktar vector for the fluid system, respectively. In the formation of the fluid element matrices, reduced integration orders are used [25]. The equations of motion of the fluid system (Equation (8)) have a similar form as those of the structure system. To obtain the coupled equations of the fluid-structure system, the determination of the interface condition is required. Since the fluid is assumed to be inviscid, only the displacement in the normal direction to the interface is continuous at the interface of the system. Assuming that the structure has the positive face and the fluid has the negative face, the boundary condition at the fluid-structure interface is U ¼ U (9) n n where U is the normal component of the interface displacement [26]. Using the interface condition, the equation of motion of the coupled system to ground motion including damping effects are given by, € _ M U þ C U þ K U ¼ R (10) c c c c c c c in which M , C , and K are the mass, damping and stiffness matrices for the coupled c c c system, respectively. U , U , Ü and R are the vectors of the displacements, velocities, c c c c accelerations and external loads of the coupled system, respectively. 3. Contact mechanics Structural response is mostly dependent upon contact between discrete systems. These systems were modelled in the common nodes of the finite element models for many years. Those are named as welded contact. But, this modelling technique actually does not consider interaction of discrete systems. This may be achieved by a modelling technique which can consider separation and friction between discrete systems. For this purpose, contact elements were generated and developed till today. Contact elements may realize frictional contact behaviour by normal and tangential contact stiffness or by the maximum shear stress allowed in the implementation of a friction coefficient. 1 2 Contact problems may include small and large deformations. Consider X and X nodes on B bodies in Figure 1 which have different initial conditions. After deformations, 2 1 φ X ¼ φ X , these nodes come into same position in the Γ boundaries (Figure 1). a a a a Consider B elastic bodies, a = 1,2, Γ boundary of the B body consists of: Γ with Figure 1. Contact mechanism of discrete systems [27]. Mathematical and Computer Modelling of Dynamical Systems 81 a a 1 2 prescribed surface loads, Γ with prescribed displacements and Γ in which B and B u c bodies come into contact [27]. 3.1. Constraint formulation The mathematical condition for non-penetration is stated as gN  0 which precludes the 1 2 penetration of body B into body B . Here, gN is named as normal gap. When gN is equal to zero, contact occurs. In this case, the associated normal component p of the stress vector, 1 1 1 1 1 1 1 t ¼ σ n ¼ p n þ t a (11) N T β must be non-zero in the contact interface. In the above equation t tangential stress is zero in the case of frictionless contact. If the bodies come into contact, gN ¼ 0 and pN < 0 where pN is normal contact pressure. If there is a gap between the bodies, gN > 0 and pN ¼ 0. These are known as Hertz–Signorini–Moreau conditions [27]. gN  0; pN  0; pNgN ¼ 0 (12) In Equation (11) Cauchy theorem is given by Cauchy stress. Correspondingly, the stress vector can be written in two different ways for nominal stresses or P first Piola–Kirrchoff stress. t ¼ σ n or T ¼ PN (13) 3.2. Treatment of contact constraints Various methods can be used to combine the contact constraints into the variational formulation to obtain maximum and minimum boundary values. If the contact interface is known, the weak form, which is the integral form of a differential equation along the boundary conditions for a boundary value problem, can be written as equality [27]. 8 9 > > ð ð ð 2 < = γ γ γ γ γ γ τ : grad η dV  f : η dV  t : η dA þ c ¼ 0 (14) > > : ; γ¼1 γ γ γ B B Here, c is the contact contributions related to the active constraint set. η 2 V is named as test function or virtual displacement and which is zero at the boundary Γ where the γ γ γ deformations are prescribed. τ , f and t are the Kirchhoff stress, the body force of body γ γ B and the surface traction applied on the boundary of B , respectively. There are several different variants for the formulation of c. One of the methods which regularize the non-differentiable normal contact and friction terms is the Augmented Lagrangian Method used in this study. The main idea of this method is to combine either the penalty method or the constitutive interface laws with Lagrange multiplier methods. This method was applied to contact problems for frictionless contact [28,29] and then this approach was extended to large displacement contact problems including friction [30,31]. 82 M.E. Kartal and A. Bayraktar 4. Numerical model of a CFR dam 4.1. Torul dam The Torul CFR dam is located on the Harsit River, approximately 14 km northwest of Torul, Gumushane, Turkey (Figure 2). The dam construction was completed in 2007 by the General Directorate of State Hydraulic Works [32]. Reservoir is used for power 6 3 generation. The volume of the dam body is 4.6 × 10 m and the water area of the reservoir at the normal water level is 3.62 km . The annual total power generation capacity is 322.28 GW. The length of the dam crest is 320 m, the width of the dam crest is 12 m and the maximum height and base width of the dam are 142 m and 420 m, respectively. The maximum water level is 137.5 m. The thickness of the concrete slab is Figure 2. The view of Torul CFR dam [32]. (a) Upstream face; (b) Downstream face. Mathematical and Computer Modelling of Dynamical Systems 83 Figure 3. The largest cross section and the dimensions of the Torul CFR dam [32]. 0.3 m at the crest level and 0.7 m at foundation. The largest cross section and the dimensions of the dam are shown in Figure 3. 4.2. Material properties of Torul dam The Torul dam body consists of concrete face slab, transition zones (2A, 3A), rockfill zones (3B, 3C) and riprap (3D), respectively, from upstream to downstream as shown in Figure 3. These zones are arranged from thin granules to thick particles in the upstream–downstream direction. Spilite (below), limestone (middle) and vol- canic tufa (upper) exist in the foundation rock. The material properties of the dam andfoundationrockaregivenin Table 1. This study selects the Young’s modulus of 3C zone as 200 MPa [19] considering the rockfill is well graded, well compacted and consisting of materials with a high compressive modulus. The material properties of rockfill zones were chosen considering that elastic constant increases with maximum particle size for alluvial material while it decreases with maximum particle size for quarried material [33]. The cohesion and the angle of internal friction of theconcreteareassumedtobe2.50MPaand 30°, respectively. In addition, it has a tensile strength of 1.6 MPa and compressive strength of 20 MPa [34]. The bulk modulus and mass density of the reservoir water are 2.07 × 10 MPa and 1000 kg/m , respectively. Table 1. Material properties of Torul CFR dam. Material properties Young’s modulus Poisson’s Mass density 7 2 3 Material *D (mm) (10 kN/m ) ratio (kg/m ) max Concrete — 2.800 0.20 2395.5 2A (sifted rock or alluvium) 150 0.040 0.36 1880.0 3A (selected rock) 300 0.030 0.36 1870.0 3B (quarry rock) 600 0.025 0.32 1850.0 3C (quarry rock) 800 0.020 0.32 1850.0 3D (selected rock) 1000 0.018 0.26 1800.0 Foundation Soil (volcanic tufa) — 1.036 0.17 2732.9 Foundation Soil (limestone) — 1.206 0.18 2834.8 Foundation Soil (Spilite) — 1.387 0.18 2834.8 Note: *Maximum particle size. 84 M.E. Kartal and A. Bayraktar 4.3. Finite element model of Torul dam The two-dimensional finite element model including dam-foundation-reservoir interac- tion is shown in Figure 4. This model also includes the plinth. The solid elements used in the finite element model have four nodes and 2 × 2 integration points; the fluid elements have four nodes and 1 × 1 integration point. Element matrices are computed using the Gauss numerical integration technique [25]. The dam-foundation- Figure 4. Two-dimensional finite element model of Torul dam including impounded water. Table 2. Number of finite elements and nodal points in the 2D finite element model. Number of Number of Zones in finite element model finite elements nodal points Concrete slab 15 32 Plinth 18 30 2A Zone (Transition Zone) 14 720 3A Zone (Transition Zone) 57 3B Zone (Rockfill) 262 3C Zone (Rockfill) 312 3D Zone (Riprap) 15 Foundation Soil (Upper layer, Volcanic Tufa) 251 661 Foundation Soil (Middle layer, Limestone) 160 Foundation Soil (Bottom layer, Spilite) 160 Reservoir water 406 450 Concrete Slab-Rockfill Contact-Target Element Pair 15 32 Rockfill-Foundation Contact-Target Element Pair 44 90 Plinth-Foundation Contact-Target Element Pair 12 26 Concrete Slab-Plinth Contact-Target Element Pair 1 4 Total 1814 2045 Mathematical and Computer Modelling of Dynamical Systems 85 ≥ 0 Water Water Water CS 3A CS CS 2A 2A 2A 3A 3A 1. Joint with interface elements. 2. Joint with contact elements. (a) Model including (b) Model including friction contact. welded contact. Interface CS: Concrete slab 3D 2A: 2A Transition zone 3A: 3A Transition zone Concrete Face 3B: 3B Rockfill Zone Slab 3C: 3C Rockfill Zone 3A 3D: Riprap 2A 3B 3C I: Interface Element C: Contact Element T: Target Element Figure 5. Structural connections in joints. reservoir interaction model involves 47 numbers of couplings. The coupling length is set as 1 mm at the reservoir-dam and reservoir-foundation interfaces. The main objective of the couplings is to maintain equal displacements between the opposite nodes in the normal direction to the interface. The finite element model contains various joints. If each combination of the contact-target elements defined in the opposite surfaces is assumed as a contact pair, 72 contact pairs are defined in the joints of the dam. The numbers of finite elements and nodal points are given in Table 2. G/Gmax Damping Ratio (%) 1 20 0.8 16 0.6 12 0.4 8 0.2 4 0 0 0.0001 0.001 0.01 0.1 1 Shear Strain, γ (%) Figure 6. Normalized shear modulus–shear strain and damping ratio relationships for gravels [20]. G/Gmax Damping Ratio (%) 86 M.E. Kartal and A. Bayraktar G/Gmax Damping Ratio (%) 1 5 0.8 4 0.6 3 0.4 2 0.2 1 0 0 0.0001 0.0003 0.001 0.003 0.01 0.03 0.1 1 Shear Strain, γ (%) Figure 7. Normalized shear modulus–shear strain and damping ratio relationships for rocks [21]. 2A 3A 3B 3C 3D 0.0 0.1 0.2 0.3 0.4 Strain (%) Figure 8. The uniaxial stress–strain relationship for rockfill [35]. Spilite Volcanic Tuff Limestone 40,000 30,000 20,000 10,000 0.0 0.1 0.2 0.3 0.4 0.5 Strain (%) Figure 9. The uniaxial stress–strain relationship for foundation rock [35]. G/Gmax 2 2 Stress (kN/m ) Stress (kN/m ) Damping Ratio (%) Mathematical and Computer Modelling of Dynamical Systems 87 Table 3. Various thickness functions for concrete slab of CFR dams in China [36]. The thickness Concrete slab thickness (cm) functions for concrete slab Crest Bottom Dam-completed year T = 0.3 30.0 30.0 Hengshan-1992, Douyan-1995, Chusong-1999 T = 0.3 + 0.002H 30.0 58.4 Baiyun-1998, Da’ao-1999, Tianhuagping-1997 b D T = 0.3 + 0.003H 30.0 72.6 Baixi- 2001, Tankeng-2005, Xiaoshan-1997 b D T = 0.3 + 0.004H 30.0 86.8 Gouhou-1989 b D T = 0.3 + 0.8 m 30.0 110.0 Jiangpinghe – (221 m) Table 4. Numerical analysis cases. Response Geometrically Materially Cases Concrete slab Rockfill Soil Concrete slab Rockfill Soil Case 1 Non-linear Non-linear Non-linear Linear Linear Linear Case 2 Non-linear Non-linear Non-linear Linear Non-linear Non-linear Case 3 Non-linear Non-linear Non-linear Non-linear Non-linear Non-linear (a) t = 2.89 s a = 5.054 m/s –1 –2 –3 0369 12 15 Time (s) (b) t = 3.37 s 3 a = 3.232 m/s –1 –2 –3 0369 12 15 Time (s) Figure 10. 1992 Erzincan earthquake record. (a) The free-surface accelerogram (PEER, 2013); (b) The deconvolved accelerogram (Kartal, 2010). Acceleration (m/s ) Acceleration (m/s ) 88 M.E. Kartal and A. Bayraktar 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 11. The maximum principle stresses in the rockfill for welded contact and empty reservoir condition in Case 1. (a) The principle compression stresses; (b) The principle tensile stresses. 4.4. Structural connections in a CFR dam There are various joints in a CFR dam. The connections in these joints are generally modelled with welded and friction contact. Welded contact includes common nodes in the contact interface. Interface or contact elements are required for friction contact (Figure 5). Four or six noded plane interface elements provide friction behaviour with their transverse shear stiffness. Besides, two or three noded contact element pairs define friction. In this study, two noded surface-to-surface contact and target elements are used. If contact occurs, sliding appears depending on the maximum shear stress allowed and friction coefficient. Two contact surfaces can bear shear stress up to a certain extent across their interfaces before sliding in the Coulomb’s friction law. This is named as ‘stick’ case. Coulomb’s friction model is defined with ‘τ’ equivalent shear stress in which sliding begins as a part of ‘p’ contact pressure. This stress is, τ ¼ μp þ c (15) lim Stress (kN/m ) Stress (kN/m ) Mathematical and Computer Modelling of Dynamical Systems 89 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 12. The maximum principle stresses in the rockfill for welded contact and empty reservoir condition in Case 2. (a) The principle compression stresses; (b) The principle tensile stresses. where, τ is limit shear stress, ‘μ’ is friction coefficient and ‘c’ is contact cohesion [22]. lim Once equivalent shear stress exceeds τ , the contact and target surfaces move relatively lim in respect of each other. This is named as ‘sliding’. Friction coefficient of ‘0’ refers the frictionless contact problems. However, friction coefficient is ‘1’ for bonded surfaces. The other term, which is cohesion, has stress unit as shear stress and provides sliding resistance even with zero normal pressure. 4.5. Non-linear response of CFR dams The Drucker–Prager model is used for concrete slab in the materially non-linear analysis. Non-linear response of rockfill and foundation rock is determined by the multi-linear kinematic hardening model. In this method, a uniaxial stress–strain curve of the material is required. This curve can be determined by a shear modulus–shear strain relationship for rock and rockfill materials [20] produced by the best-fit hyper- bolic curve defining G/G versus cyclic shear strain relationship for gravel soils based max on testing by 15 investigators (Figure 6). This study considers the best curve produced Stress (kN/m ) Stress (kN/m ) 90 M.E. Kartal and A. Bayraktar 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 13. The maximum principle stresses in the rockfill for welded contact and empty reservoir condition in Case 3. (a) The principle compression stresses; (b) The principle tensile stresses. by Rollins, Evans, Diehl and Daily III [20] for rockfill. In addition, shear modulus– shear strain relation for rock soils obtained from experimental studies by Schnabel, Lysmer and Seed [21] is used for rock foundation (Figure 7). Using these curves, the uniaxial stress–strain curves for rockfill and foundation soil are determined as shown in Figures 8 and 9 [35]. ‘No separation’ contact model is preferred in the concrete slab-rockfill, rockfill- foundation and plinth-foundation interfaces. However, ‘standard’ contact model is pre- ferred in the concrete slab-plinth interface. In the standard contact model, the structural element behind the contact element may slide over and leaves from the structural element behind the target element. However, though the contact surface does not separate from the target surface, it may slide over the target surface in the no separation contact model. 4.6. Concrete slab thickness There are several thickness functions produced for concrete slab depending on the dam height (H ). Qian [36] compiled several thickness functions for concrete slab on CFR dams constructed in China. Table 3 gives some selected functions for concrete slab Stress (kN/m ) Stress (kN/m ) Mathematical and Computer Modelling of Dynamical Systems 91 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 14. The maximum principle stresses in the rockfill for welded contact and full reservoir condition in Case 1. (a) The principle compression stresses; (b) The principle tensile stresses. considered in this study [36]. The bottom thickness is obtained as 72.6 cm in Table 3 in line 3, but this is quite near the current bottom thickness of the concrete slab thickness of Torul dam. Therefore, the current thickness of 70 cm is considered in the numerical models. 4.7. Analysis cases In this study, finite element analyses are performed for three analytical cases which contain geometrical and material non-linear analyses as given in Table 4. These cases aim to reveal the effect of the material non-linear response of the dam body on the rockfill. Besides, the effect of the concrete slab thickness on the rockfill response is also investigated. This study almost fully ignores the response of concrete slab and concrete slab-rockfill interface. 4.8. Deconvolved ground motion model Free-field surface motions recorded during earthquakes reflect the characteristics of underlying soil layers at the recording site [37,38]. Real ground response problems Stress (kN/m ) Stress (kN/m ) 92 M.E. Kartal and A. Bayraktar 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 15. The maximum principle stresses in the rockfill for welded contact and full reservoir condition in Case 2. (a) The principle compression stresses; (b) The principle tensile stresses. usually involve soil deposits with layers of different stiffness and damping character- istics with boundaries at the elastic wave energy that will be reflected and/or transmitted [38]. Therefore, earthquake records must be deconvolved to the base of the rock foundation. The North–South component of the 1992 Erzincan earthquake with a peak ground acceleration (pga) of 0.515 g is selected for the analyses [39]. Torul dam is nearby the North Anatolian fault and a strong ground motion of which epicenter Erzincan occurred in 1992. In this study, the earthquake record is deconvolved to the base of the rock foundation considering three foundation layers using SHAKE91 [40]. The earthquake record obtained at the ground surface and the deconvolved earthquake record are shown in Figure 10. It is clearly seen that the pga of the deconvolved accelerogram is lower than the free-surface accelerogram. 4.9. Stresses This study investigated the principle compressive and tensile stresses that occurred in the rockfill under deconvolved ground motion for the selected three different cases and Stress (kN/m ) Stress (kN/m ) Mathematical and Computer Modelling of Dynamical Systems 93 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 16. The maximum principle stresses in the rockfill for welded contact and full reservoir condition in Case 3. (a) The principle compression stresses; (b) The principle tensile stresses. concrete slab thicknesses. The principle stress components are calculated for both welded and friction contacts in the joints of the CFR dam. 4.9.1. The principle stresses in the rockfill for welded contact in the joints The maximum principle stress components occurred in Section I-I (Figure 3)of the rockfill for welded contact in the joints and empty reservoir condition are shown in Figures 11–13 for Cases 1–3. In addition, Figures 14–16 show the maximum principle stress components occurred in I-I section of the rockfill for welded contact in the joints and full reservoir condition for Case 1, Case 2 and Case 3. The principle compressive stresses in the rockfill increase when getting close to the upstream face. However, the principle compressive stresses decrease as the concrete slab increases. The principle compressive stresses of the rockfill decrease in Case 2, in which the rockfill is assumed to be non-linear and the concrete slab is linear, as compared to those in Case 1. But, those significantly increase near the concrete slab-rockfill inter- face in Case 3, in which case the concrete slab and rockfill are non-linear. The Stress (kN/m ) Stress (kN/m ) 94 M.E. Kartal and A. Bayraktar 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 17. The maximum principle stresses in the rockfill for friction contact and empty reservoir condition in Case 1. (a) The principle compression stresses; (b) The principle tensile stresses. variation of the compressive stresses in 2A, 3A and 3C zones depends on the concrete slab thickness for welded contact. However, those are relatively close to each other in 3C and 3D zones. The principle tensile stresses of the rockfill decrease as the concrete slab increases. The differences in the principle tensile stresses are evident especially in Case 1 and those are clearly distinguished in 3C and 3D zones. The principle tensile stresses are obtained close to each other only in Case 3. While the principle compressive stresses of the rockfill decrease, the principle tensile stresses of the rockfill increase when getting close to the downstream face of the dam. The change of the principle stress components resembles each other throughout the horizontal section of the rockfill under empty and full reservoir conditions. The stresses in the rockfill under hydrodynamic pressure, however, are higher than those under empty reservoir condition. The stresses in the rockfill under hydro- dynamic pressure decrease as the concrete slab thickness increases. Numerical analyses reveal that the most critical stresses appear for the concrete slab thickness of 30 cm. Stress (kN/m ) Stress (kN/m ) Mathematical and Computer Modelling of Dynamical Systems 95 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 18. The maximum principle stresses in the rockfill for friction contact and empty reservoir condition in Case 2. (a) The principle compression stresses; (b) The principle tensile stresses. 4.9.2. The principle stresses in the rockfill for friction contact in the joints The maximum principle stress components occurred in the Section I-I (Figure 3)ofthe rockfill for friction contact in the joints and empty reservoir condition are shown in Figures 17–19 for Case 1, Case 2 and Case 3. In addition, Figures 20–22 show the maximum principle stress components occurred in the Section I-I of the rockfill for friction contact in the joints and full reservoir condition for Case 1, Case 2 and Case 3. The concrete slab can make a sliding motion over the rockfill when the friction is defined in the concrete slab-rockfill interface. This causes partially independent behaviours for the concrete slab and rockfill. Therefore, rockfill will make higher deformations under earth- quake ground motions because the concrete slab contributes less when friction is con- sidered at the concrete slab-rockfill interface. The higher deformations cause higher compressive stresses near the concrete slab-rockfill interface. The principle compressive stresses increase between 2 and 3.5 times as compared to those of the rockfill including welded contact in the joints. The principle compressive stresses decrease when getting close to the downstream of the dam. The concrete slab thickness does not considerably affect the compressive stresses of the rockfill in an empty reservoir condition when friction is considered. The principle stresses along the horizontal section decrease when Stress (kN/m ) Stress (kN/m ) 96 M.E. Kartal and A. Bayraktar 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 19. The maximum principle stresses in the rockfill for friction contact and empty reservoir condition in Case 3. (a) The principle compression stresses; (b) The principle tensile stresses. the non-linear material behaviour of the rockfill is considered in empty reservoir condi- tion. This decrease is clearly seen in the principle tensile stresses. The principle compressive stresses under hydrodynamic pressure clearly increase as compared to those of the dam including welded contact at the joints. This increase is actually less than that obtained under empty reservoir condition. However, the principle tensile stresses occurred under hydrodynamic pressure, obviously increases near the concrete slab-rockfill interface. The high principle tensile stresses may be on the unsafe side in this section which is commonly known as the cohesionless region. As seen under empty reservoir condition, the principle tensile stresses increase when getting close to the downstream face. The concrete slab thickness is not effective on the principle stresses of the rockfill when the reservoir is full. The decrease in the principle tensile stresses is very apparent for the higher values of the concrete slab thickness when the concrete slab material is modelled as linear. While the principle tensile stresses decrease near the upstream face, they generally increase along the horizontal section. However, the principle compressive stresses continuously decrease in the horizontal section. Numerical analyses give the most critical stresses in the rockfill for the concrete slab thickness of 30 cm. Stress (kN/m ) Stress (kN/m ) Mathematical and Computer Modelling of Dynamical Systems 97 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 20. The maximum principle stresses in the rockfill for friction contact and full reservoir condition in Case 1. (a) The principle compression stresses; (b) The principle tensile stresses. 5. Conclusions In this study, the non-linear response of the rockfill in a CFR dam-foundation-reservoir system to deconvolved ground motion was investigated for different slab thicknesses. Non-linear geometrical and material behaviour of the rockfill was considered in this study. The multi-linear kinematic hardening model was used for the rockfill and foundation rock. The Drucker–Prager model was used for the non-linear response of the concrete slab. The friction between the various interaction surfaces is considered with contact elements based on the Coulomb’s friction law. The hydrodynamic pressure of the reservoir water is considered with the fluid finite elements based on the Lagrangian approach. The principle compressive and tensile stresses of the rockfill from upstream to downstream along the horizontal section were investigated for empty and full reservoir conditions considering welded and friction contacts. The following conclusions can be deducted from the numerical analyses: The CFR dam, including the welded contact in the joint (Figures 11–16), indicate that (1) the maximum principle stress components in the rockfill decrease if the Stress (kN/m ) Stress (kN/m ) 98 M.E. Kartal and A. Bayraktar 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 21. The maximum principle stresses in the rockfill for friction contact and full reservoir condition in Case 2. (a) The principle compression stresses; (b) The principle tensile stresses. non-linear material response of the rockfill is considered in the solutions. (2) The principle stresses in the rockfill increases by the effect of the hydrodynamic pressure; however, the effect is limited. (3) The principle stresses also increase when decreasing the concrete slab thickness. The CFR dam, including friction contact in the joints (Figures 17–22), indicate that (1) the principle stresses decrease when getting close to the upstream face of the dam for each reservoir condition. (2) The principle tensile stresses also decrease with the materially non-linear response of the rockfill. (3) However, those increase by the effect of the hydrodynamic pressure near the upstream face. ● The comparison of the CFR dams, including welded and friction contacts in the joints (Figures 11–22), indicate that the principle stresses in the rockfill of the CFR dam including friction in the joints are bigger than those in the CFR dam ignoring friction. ● The change of the principle stresses in the rockfill is low for the concrete face slab thickness greater than 58 cm. Consequently, the use of face slab thickness greater than 58 cm may provide the optimum theoretical design for CFR dams for dynamic loading conditions. Stress (kN/m ) Stress (kN/m ) Mathematical and Computer Modelling of Dynamical Systems 99 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 22. The maximum principle stresses in the rockfill for friction contact and full reservoir condition in Case 3. (a) The principle compression stresses; (b) The principle tensile stresses. Beyond this study, if the vertical components of earthquakes should be taken into account in the analyses, the realistic earthquake effects on dams may be determined. In addition, depending on the foundation conditions surface wave effects may appear. 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Civil Eng. 32 (5) (2005), pp. 940–947. doi:10.1139/l05-055 Stress (kN/m ) Stress (kN/m ) 100 M.E. Kartal and A. Bayraktar [5] A. Bayraktar, M.E. Kartal, and H.B. Basaga, Reservoir effects on the earthquake performance evaluation of Torul CFR dam considering slippage between concrete slab and rockfill, Water Sci. Eng. 2 (1) (2009a), pp. 43–57. [6] A. Bayraktar, M.E. Kartal, and S. Adanur, The effect of concrete slab-rockfill interface behavior on the earthquake performance of a CFR dam, Int. J. Nonlin. Mech. 46 (1) (2011), pp. 35–46. doi:10.1016/j.ijnonlinmec.2010.07.001 [7] F.B. Guros, G.R. Thiers, T.R. Wathen, and C.E. Buckles, Seismic design of concrete-faced rockfill dams,in Proceedings of the 8th World Conference on Earthquake Engineering, Vol. 3, San Francisco, CA, Prentice-Hall, Englewood Cliffs, NJ, 1984, pp. 317–323. [8] G. Bureau, R.L. Volpe, W. Roth, and T. Udaka, Seismic analysis of concrete face rockfill dams, in Proceedings of the Symposium on Concrete Face Rockfill Dams – Design, Construction and Performance, ASCE, Detroit, MI, 1985, pp. 479–508. [9] G. Gazetas and P. Dakoulas, Seismic analysis and design of rockfill dams: state-of-the-art, Soil Dyn. Earthq. Eng. 11 (1) (1992), pp. 27–61. doi:10.1016/0267-7261(92)90024-8 [10] H.B. Seed, R.B. Seed, S.S. Lai, and B. Khamenehpour, Seismic design of concrete face rockfill dams,in Proceedings of the Symposium on Concrete Face Rockfill Dams-Design, Construction and Performance, ASCE, New York, 1985, pp. 459–478. [11] J.L. Sherard and J.B. Cooke, Concrete‐face rockfill dam: I. Assessment, J. Geotech. Eng. 113 (10) (1987), pp. 1096–1112. doi:10.1061/(ASCE)0733-9410(1987)113:10(1096) [12] M.E. Kartal, A. Bayraktar, and H.B. Başağa, Seismic failure probability of concrete slab on CFR dams with welded and friction contacts by response surface method, Soil Dyn. Earthq. Eng. 30 (11) (2010), pp. 1383–1399. doi:10.1016/j.soildyn.2010.06.013 [13] M.E. Kartal, A. Bayraktar, and H.B. Başağa, Nonlinear finite element reliability analysis of Concrete-Faced Rockfill (CFR) dams under static effects, Appl. Math. Model. 36 (11) (2012), pp. 5229–5248. doi:10.1016/j.apm.2011.12.004 [14] M.E. Kartal and A. Bayraktar, Non-linear earthquake response of CFR dam–reservoir–foun- dation systems, Math. Comp. Model Dyn. Syst. 19 (4) (2013), pp. 353–374. doi:10.1080/ 13873954.2012.759595 [15] N. Uddin, A dynamic analysis procedure for concrete-faced rockfill dams subjected to strong seismic excitation, Comput. Struct. 72 (1999), pp. 409–421. doi:10.1016/S0045-7949(99) 00011-5 [16] N. Uddin and G. Gazetas, Dynamic response of concrete-faced rockfill dams to strong seismic excitation, J. Geotech. Eng. ASCE 121 (2) (1995), pp. 185–197. doi:10.1061/(ASCE)0733- 9410(1995)121:2(185) [17] Y. Arici, Investigation of the cracking of CFRD face plates, Comp. Geotech. 38 (7) (2011), pp. 905–916. doi:10.1016/j.compgeo.2011.06.004 [18] S. Esmaeili, Three dimensional dynamic analysis of Concrete-Faced Rockfill Dams (CFRDs), in Proceedings of Symposium on 20 Years for Chinese CFRD Construction, Yichang, 19–26 September 2005, pp. 192–201. [19] S.M. Haeri and M. Karimi, Effects of dam foundation interaction and special variable ground motion on seismic response of Concrete Faced Rockfill Dam,in Proceedings of Symposium on 20 Years for Chinese CFRD Construction, Yichang, 19–26 September 2005, pp. 202–210. [20] K.M. Rollins, M.D. Evans, N.B. Diehl, and W.D. Daily III., Shear modulus and damping relationships for gravels, J. Geotech. Geoenviron. Eng. 124 (5) (1998), 396–405. doi:10.1061/ (ASCE)1090-0241(1998)124:5(396) [21] P.B. Schnabel, J. Lysmer, and H.B. Seed, SHAKE: A computer program for earthquake response analysis of horizontally layered sites. Report no. EERC-72/12, Earthquake Engineering Research Centre, University of California, Berkeley, CA, 1972. [22] ANSYS, Swanson Analysis Systems Inc., Engineering Simulation Software. Houston, PA, 2013; software available at http://www.ansys.com/ (Accessed 1 February 2014). [23] K.J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, London, Englewood Cliffs, NJ, 1996. [24] O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, Mc Graw-Hill, London, 1989. [25] E.L. Wilson and M. Khalvati, Finite elements for the dynamic analysis of fluid-solid systems, Int. J. Numer. Meth. Eng. 19 (11) (1983), pp. 1657–1668. doi:10.1002/nme.1620191105 [26] N. Akkas, H.U. Akay, and Ç. Yilmaz, Applicability of general-purpose finite element pro- grams in solid-fluid interaction problems, Comput. Struct. 10 (5) (1979), pp. 773–783. doi:10.1016/0045-7949(79)90041-5 Mathematical and Computer Modelling of Dynamical Systems 101 [27] P. Wriggers, Computational Contact Mechanics, 2nd ed., Springer-Verlag, Berlin, 2006. [28] N. Kikuchi and J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities And Finite Element Methods, SIAM, Philadelphia, PA, 1988. [29] P. Wriggers, J. Simo, and R. Taylor, Penalty and augmented Lagrangian formulations for contact problems,in Proceedings of the NUMETA Conference, J. Middleton and G. Pande, eds., Balkema, Rotterdam, 1985. [30] P. Alart and A. Curnier, A mixed formulation for frictional contact problems prone to Newton like solution methods, Comput. Method Appl. Mech. Eng. 92 (3) (1991), 353–375. doi:10.1016/0045-7825(91)90022-X [31] T.A. Laursen and J.C. Simo, Algorithmic symmetrization of Coulomb frictional problems using augmented Lagrangians, Comput. Method Appl. Mech. Eng. 108 (1–2) (1993), pp. 133–146. [32] DSI, General Directorate of State Hydraulic Works, The XXII Regional Directorate, Trabzon, [33] A. Varadarajan, K.G. Sharma, S.M. Abbas, and A.K. Dhawan, Constitutive model for rockfill materials and determination of material constants, Int. J. Geomech. 6 (4) (2006), pp. 226–237. doi:10.1061/(ASCE)1532-3641(2006)6:4(226) [34] TS500, Requirements for Design and Construction of Reinforced Concrete Structures, Turkish Standards Institution, Ankara, 2000. [35] M.E. Kartal, Reliability analysis of concrete-faced rockfill dams [In Turkish]. Ph.D. Thesis, Karadeniz Technical University, 2010. [36] C. Qian, Recent development of CFRD in China,in Proceedings of Symposium on 20 years for Chinese CFRD Construction, Yichang, 19–26 September 1969, pp. 8–15. [37] H.B. Seed, The influence of local soil conditions on earthquake damage, Proceedings, Soil Dynamics Specialty Session, Seventh International Conference of Soil Mechanics and Foundation Engineering, Mexico City, 1 January 1969. [38] S.L. Kramer, Geotechnical Earthquake Engineering, 1st ed., Prentice-Hall, Upper Saddle River, NJ, 1996. [39] PEER, Pacific Earthquake Engineering Research Centre, University of California, Berkeley, 2013. Available at: http://peer.berkeley.edu/smcat (Accessed 1 February 2014). [40] I.M. Idriss and J.I. Sun, SHAKE91: A Computer Program for Conducting Equivalent Linear Seismic Response Analyses of Horizontally Layered Soil Deposits, Center of Geotechnical Modeling Department of Civil & Environmental Engineering, University of California Davis, Davis, CA, 1992. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical and Computer Modelling of Dynamical Systems Taylor & Francis

Non-linear response of the rockfill in a concrete-faced rockfill dam under seismic excitation

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10.1080/13873954.2014.898157
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Mathematical and Computer Modelling of Dynamical Systems, 2015 Vol. 21, No. 1, 77–101, http://dx.doi.org/10.1080/13873954.2014.898157 Non-linear response of the rockfill in a concrete-faced rockfill dam under seismic excitation a b Murat Emre Kartal * and Alemdar Bayraktar a b Department of Civil Engineering, Bülent Ecevit University, 67100 Zonguldak, Turkey; Department of Civil Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey (Received 28 July 2013; accepted 22 February 2014) This study investigates the non-linear response of the rockfill in a concrete-faced rockfill dam under seismic excitation for various concrete slab thicknesses from 30 to 100 cm thick. The effect of the friction in the concrete slab rockfill interface on the non-linear response of the rockfill is also considered in the numerical solutions. The friction contact in the concrete slab joints is provided with contact elements based on the Coulomb’s friction law. The hydrodynamic pressure of the reservoir water is considered by the fluid finite elements based on the Lagrangian approach. Geometrical and material non-linear behaviours of the dam-foundation-reservoir inter- action system are considered together in the finite element analyses. The Drucker– Prager model is used to obtain materially non-linear behaviour of the concrete slab. The multi linear kinematic hardening model is used for rockfill and foundation rock in the material non-linear analyses. Therefore, the uniaxial stress–strain relations of the rockfill and foundation rock are determined from the shear stress–shear strain relations of the gravel and rock, respectively. The principle compressive and tensile stresses of the rockfill are investigated along the horizontal section of the rockfill. According to numerical analyses, the most critical stresses occur in the upstream and downstream faces of the dam for the concrete slab thickness of 30 cm, while the optimum thickness for seismic performance is 58 cm. Keywords: concrete-faced rockfill dam; friction contact; geometrical and material non-linear behaviour; Lagrangian approach 1. Introduction Concrete-faced rockfill (CFR) dams prevent water penetration in the upstream face with concrete slab. Many researchers focused on the performance of concrete slab [1–17]. CFR dams are known to be safe under earthquake ground motions because they do not develop excess porewater pressures [11]. Nevertheless, excessive stresses can still occur in the rockfill under seismic excitations. Especially, tensile stresses may result in stability problems in the upstream and downstream faces of the dam. In addition to this, the observations indicate plastic deformations in the rockfill dams under seismic excitations. The rockfill is the main load-bearing system of a CFR dam and significantly affects the response of the concrete slab. Therefore, its seismic behaviour is very important and should be determined under ground motion effects. The CFR dams include dam-foundation-reservoir interaction problems. Also, CFR dams are exposed to hydrodynamic pressure on their face slab under earthquake effects. The response of the concrete slab depends upon the response of the rockfill, but there is *Corresponding author. Email: murat_emre_kartal@hotmail.com © 2014 Taylor & Francis 78 M.E. Kartal and A. Bayraktar little information related to the rockfill-concrete face dynamic interaction response. Recently, [1] investigated a CFR dam subjected to near-fault ground motion effects considering the rockfill response. This study focuses on the non-linear response of the rockfill in a CFR dam under seismic excitation. Numerical analyses include geometrically non-linear response of the dam to consider large deformations under ground motion effects. The Drucker–Prager model is used for concrete slab in the non-linear analyses. In addition to this, the multi- linear kinematic-hardening model is considered for the non-linear response of the rockfill and rock foundation [18,19]. The uniaxial stress–strain relation of the medium is required for this model. These relations are obtained from the normalized shear modulus–shear strain relation of the gravels [20] and rocks [21]. Both welded and friction contacts are considered in the interaction interfaces. The friction in the joints is modelled by the contact elements based on the Coulomb’s friction law. This study also considers hydro- dynamic pressure of the reservoir water. The reservoir water is modelled by the fluid finite elements based on the Lagrangian approach. The principle compressive and tensile stresses occurring in the rockfill along the largest cross section of the dam in valley direction under seismic excitation are presented and investigated in detail. All of the numerical analyses were performed using ANSYS finite element programme [22]. According to the numerical analyses, this study indicates which parameters should be considered for the optimum design rockfill of concrete-faced rockfill dams. 2. Dam-reservoir-foundation interaction by the Lagrangian approach The formulation of the fluid system based on the Lagrangian approach is presented as follows [21,22]. This approach assumes the fluid as linearly compressible, inviscid and irrotational. For a general two-dimensional fluid, pressure–volumetric strain relationships can be written in matrix form as follows, C 0 ε 11 v ¼ (1) 0C w z 22 z where P, C and ε are the pressures which are equal to mean stresses, the bulk modulus 11 v and the volumetric strains of the fluid, respectively. Since irrotationality of the fluid is considered like penalty methods [23,24], rotations and constraint parameters are included in the pressure–volumetric strain equation (Equation (1)) of the fluid. In this equation, P is the rotational stress, C is the constraint parameter and w is the rotation about the 22 z Cartesian axis z. In this study, the equations of motion of the fluid system are obtained using energy principles. Using the finite element approximation, the total strain energy of the fluid system may be written as, π ¼ U K U (2) e f f where U and K are the nodal displacement vector and the stiffness matrix of the fluid f f system, respectively. K is obtained by the sum of the stiffness matrices of the fluid elements as follows, Mathematical and Computer Modelling of Dynamical Systems 79 K ¼ K (3) e e e e K ¼ B C B dV f f f where C is the elasticity matrix consisting of diagonal terms in Equation (1). B is the strain-displacement matrix of the fluid element. An important behaviour of fluid systems is the ability to displace without a change in volume. For reservoir and storage tanks, this movement is known as sloshing waves in which the displacement is in the vertical direction. Using the finite element method, the increase in the potential energy of the system due to the free surface motion can be written as, π ¼ U S U (4) s f sf sf where U and S are the vertical nodal displacement vector and the stiffness matrix of the sf f free surface of the fluid system, respectively. S is obtained by the sum of the stiffness matrices of the free surface fluid elements as follows, S ¼ S = e e T (5) S ¼ ρ g h h dA f f s ; where h is the vector consisting of interpolation functions of the free surface fluid element. ρ and g are the mass density of the fluid and the acceleration due to gravity, respectively. Also, the kinetic energy of the system using the finite element method can be written as, _ _ T ¼ U M U (6) f f where U and M are the nodal velocity vector and the mass matrix of the fluid system, f f respectively. M is also obtained by the sum of the mass matrices of the fluid elements as follows, M ¼ M = e T e (7) M ¼ ρ H HdV f f ; where H is the matrix consisting of interpolation functions of the fluid element. If Equations (2), (4) and (6) are combined using the Lagrange’s equation, the following set of equations is obtained, M U þ K U ¼ R (8) f f f f where K , Ü , U and R are the system stiffness matrix including the free surface f f f stiffness, the nodal acceleration and displacement vectors and time-varying nodal force 80 M.E. Kartal and A. Bayraktar vector for the fluid system, respectively. In the formation of the fluid element matrices, reduced integration orders are used [25]. The equations of motion of the fluid system (Equation (8)) have a similar form as those of the structure system. To obtain the coupled equations of the fluid-structure system, the determination of the interface condition is required. Since the fluid is assumed to be inviscid, only the displacement in the normal direction to the interface is continuous at the interface of the system. Assuming that the structure has the positive face and the fluid has the negative face, the boundary condition at the fluid-structure interface is U ¼ U (9) n n where U is the normal component of the interface displacement [26]. Using the interface condition, the equation of motion of the coupled system to ground motion including damping effects are given by, € _ M U þ C U þ K U ¼ R (10) c c c c c c c in which M , C , and K are the mass, damping and stiffness matrices for the coupled c c c system, respectively. U , U , Ü and R are the vectors of the displacements, velocities, c c c c accelerations and external loads of the coupled system, respectively. 3. Contact mechanics Structural response is mostly dependent upon contact between discrete systems. These systems were modelled in the common nodes of the finite element models for many years. Those are named as welded contact. But, this modelling technique actually does not consider interaction of discrete systems. This may be achieved by a modelling technique which can consider separation and friction between discrete systems. For this purpose, contact elements were generated and developed till today. Contact elements may realize frictional contact behaviour by normal and tangential contact stiffness or by the maximum shear stress allowed in the implementation of a friction coefficient. 1 2 Contact problems may include small and large deformations. Consider X and X nodes on B bodies in Figure 1 which have different initial conditions. After deformations, 2 1 φ X ¼ φ X , these nodes come into same position in the Γ boundaries (Figure 1). a a a a Consider B elastic bodies, a = 1,2, Γ boundary of the B body consists of: Γ with Figure 1. Contact mechanism of discrete systems [27]. Mathematical and Computer Modelling of Dynamical Systems 81 a a 1 2 prescribed surface loads, Γ with prescribed displacements and Γ in which B and B u c bodies come into contact [27]. 3.1. Constraint formulation The mathematical condition for non-penetration is stated as gN  0 which precludes the 1 2 penetration of body B into body B . Here, gN is named as normal gap. When gN is equal to zero, contact occurs. In this case, the associated normal component p of the stress vector, 1 1 1 1 1 1 1 t ¼ σ n ¼ p n þ t a (11) N T β must be non-zero in the contact interface. In the above equation t tangential stress is zero in the case of frictionless contact. If the bodies come into contact, gN ¼ 0 and pN < 0 where pN is normal contact pressure. If there is a gap between the bodies, gN > 0 and pN ¼ 0. These are known as Hertz–Signorini–Moreau conditions [27]. gN  0; pN  0; pNgN ¼ 0 (12) In Equation (11) Cauchy theorem is given by Cauchy stress. Correspondingly, the stress vector can be written in two different ways for nominal stresses or P first Piola–Kirrchoff stress. t ¼ σ n or T ¼ PN (13) 3.2. Treatment of contact constraints Various methods can be used to combine the contact constraints into the variational formulation to obtain maximum and minimum boundary values. If the contact interface is known, the weak form, which is the integral form of a differential equation along the boundary conditions for a boundary value problem, can be written as equality [27]. 8 9 > > ð ð ð 2 < = γ γ γ γ γ γ τ : grad η dV  f : η dV  t : η dA þ c ¼ 0 (14) > > : ; γ¼1 γ γ γ B B Here, c is the contact contributions related to the active constraint set. η 2 V is named as test function or virtual displacement and which is zero at the boundary Γ where the γ γ γ deformations are prescribed. τ , f and t are the Kirchhoff stress, the body force of body γ γ B and the surface traction applied on the boundary of B , respectively. There are several different variants for the formulation of c. One of the methods which regularize the non-differentiable normal contact and friction terms is the Augmented Lagrangian Method used in this study. The main idea of this method is to combine either the penalty method or the constitutive interface laws with Lagrange multiplier methods. This method was applied to contact problems for frictionless contact [28,29] and then this approach was extended to large displacement contact problems including friction [30,31]. 82 M.E. Kartal and A. Bayraktar 4. Numerical model of a CFR dam 4.1. Torul dam The Torul CFR dam is located on the Harsit River, approximately 14 km northwest of Torul, Gumushane, Turkey (Figure 2). The dam construction was completed in 2007 by the General Directorate of State Hydraulic Works [32]. Reservoir is used for power 6 3 generation. The volume of the dam body is 4.6 × 10 m and the water area of the reservoir at the normal water level is 3.62 km . The annual total power generation capacity is 322.28 GW. The length of the dam crest is 320 m, the width of the dam crest is 12 m and the maximum height and base width of the dam are 142 m and 420 m, respectively. The maximum water level is 137.5 m. The thickness of the concrete slab is Figure 2. The view of Torul CFR dam [32]. (a) Upstream face; (b) Downstream face. Mathematical and Computer Modelling of Dynamical Systems 83 Figure 3. The largest cross section and the dimensions of the Torul CFR dam [32]. 0.3 m at the crest level and 0.7 m at foundation. The largest cross section and the dimensions of the dam are shown in Figure 3. 4.2. Material properties of Torul dam The Torul dam body consists of concrete face slab, transition zones (2A, 3A), rockfill zones (3B, 3C) and riprap (3D), respectively, from upstream to downstream as shown in Figure 3. These zones are arranged from thin granules to thick particles in the upstream–downstream direction. Spilite (below), limestone (middle) and vol- canic tufa (upper) exist in the foundation rock. The material properties of the dam andfoundationrockaregivenin Table 1. This study selects the Young’s modulus of 3C zone as 200 MPa [19] considering the rockfill is well graded, well compacted and consisting of materials with a high compressive modulus. The material properties of rockfill zones were chosen considering that elastic constant increases with maximum particle size for alluvial material while it decreases with maximum particle size for quarried material [33]. The cohesion and the angle of internal friction of theconcreteareassumedtobe2.50MPaand 30°, respectively. In addition, it has a tensile strength of 1.6 MPa and compressive strength of 20 MPa [34]. The bulk modulus and mass density of the reservoir water are 2.07 × 10 MPa and 1000 kg/m , respectively. Table 1. Material properties of Torul CFR dam. Material properties Young’s modulus Poisson’s Mass density 7 2 3 Material *D (mm) (10 kN/m ) ratio (kg/m ) max Concrete — 2.800 0.20 2395.5 2A (sifted rock or alluvium) 150 0.040 0.36 1880.0 3A (selected rock) 300 0.030 0.36 1870.0 3B (quarry rock) 600 0.025 0.32 1850.0 3C (quarry rock) 800 0.020 0.32 1850.0 3D (selected rock) 1000 0.018 0.26 1800.0 Foundation Soil (volcanic tufa) — 1.036 0.17 2732.9 Foundation Soil (limestone) — 1.206 0.18 2834.8 Foundation Soil (Spilite) — 1.387 0.18 2834.8 Note: *Maximum particle size. 84 M.E. Kartal and A. Bayraktar 4.3. Finite element model of Torul dam The two-dimensional finite element model including dam-foundation-reservoir interac- tion is shown in Figure 4. This model also includes the plinth. The solid elements used in the finite element model have four nodes and 2 × 2 integration points; the fluid elements have four nodes and 1 × 1 integration point. Element matrices are computed using the Gauss numerical integration technique [25]. The dam-foundation- Figure 4. Two-dimensional finite element model of Torul dam including impounded water. Table 2. Number of finite elements and nodal points in the 2D finite element model. Number of Number of Zones in finite element model finite elements nodal points Concrete slab 15 32 Plinth 18 30 2A Zone (Transition Zone) 14 720 3A Zone (Transition Zone) 57 3B Zone (Rockfill) 262 3C Zone (Rockfill) 312 3D Zone (Riprap) 15 Foundation Soil (Upper layer, Volcanic Tufa) 251 661 Foundation Soil (Middle layer, Limestone) 160 Foundation Soil (Bottom layer, Spilite) 160 Reservoir water 406 450 Concrete Slab-Rockfill Contact-Target Element Pair 15 32 Rockfill-Foundation Contact-Target Element Pair 44 90 Plinth-Foundation Contact-Target Element Pair 12 26 Concrete Slab-Plinth Contact-Target Element Pair 1 4 Total 1814 2045 Mathematical and Computer Modelling of Dynamical Systems 85 ≥ 0 Water Water Water CS 3A CS CS 2A 2A 2A 3A 3A 1. Joint with interface elements. 2. Joint with contact elements. (a) Model including (b) Model including friction contact. welded contact. Interface CS: Concrete slab 3D 2A: 2A Transition zone 3A: 3A Transition zone Concrete Face 3B: 3B Rockfill Zone Slab 3C: 3C Rockfill Zone 3A 3D: Riprap 2A 3B 3C I: Interface Element C: Contact Element T: Target Element Figure 5. Structural connections in joints. reservoir interaction model involves 47 numbers of couplings. The coupling length is set as 1 mm at the reservoir-dam and reservoir-foundation interfaces. The main objective of the couplings is to maintain equal displacements between the opposite nodes in the normal direction to the interface. The finite element model contains various joints. If each combination of the contact-target elements defined in the opposite surfaces is assumed as a contact pair, 72 contact pairs are defined in the joints of the dam. The numbers of finite elements and nodal points are given in Table 2. G/Gmax Damping Ratio (%) 1 20 0.8 16 0.6 12 0.4 8 0.2 4 0 0 0.0001 0.001 0.01 0.1 1 Shear Strain, γ (%) Figure 6. Normalized shear modulus–shear strain and damping ratio relationships for gravels [20]. G/Gmax Damping Ratio (%) 86 M.E. Kartal and A. Bayraktar G/Gmax Damping Ratio (%) 1 5 0.8 4 0.6 3 0.4 2 0.2 1 0 0 0.0001 0.0003 0.001 0.003 0.01 0.03 0.1 1 Shear Strain, γ (%) Figure 7. Normalized shear modulus–shear strain and damping ratio relationships for rocks [21]. 2A 3A 3B 3C 3D 0.0 0.1 0.2 0.3 0.4 Strain (%) Figure 8. The uniaxial stress–strain relationship for rockfill [35]. Spilite Volcanic Tuff Limestone 40,000 30,000 20,000 10,000 0.0 0.1 0.2 0.3 0.4 0.5 Strain (%) Figure 9. The uniaxial stress–strain relationship for foundation rock [35]. G/Gmax 2 2 Stress (kN/m ) Stress (kN/m ) Damping Ratio (%) Mathematical and Computer Modelling of Dynamical Systems 87 Table 3. Various thickness functions for concrete slab of CFR dams in China [36]. The thickness Concrete slab thickness (cm) functions for concrete slab Crest Bottom Dam-completed year T = 0.3 30.0 30.0 Hengshan-1992, Douyan-1995, Chusong-1999 T = 0.3 + 0.002H 30.0 58.4 Baiyun-1998, Da’ao-1999, Tianhuagping-1997 b D T = 0.3 + 0.003H 30.0 72.6 Baixi- 2001, Tankeng-2005, Xiaoshan-1997 b D T = 0.3 + 0.004H 30.0 86.8 Gouhou-1989 b D T = 0.3 + 0.8 m 30.0 110.0 Jiangpinghe – (221 m) Table 4. Numerical analysis cases. Response Geometrically Materially Cases Concrete slab Rockfill Soil Concrete slab Rockfill Soil Case 1 Non-linear Non-linear Non-linear Linear Linear Linear Case 2 Non-linear Non-linear Non-linear Linear Non-linear Non-linear Case 3 Non-linear Non-linear Non-linear Non-linear Non-linear Non-linear (a) t = 2.89 s a = 5.054 m/s –1 –2 –3 0369 12 15 Time (s) (b) t = 3.37 s 3 a = 3.232 m/s –1 –2 –3 0369 12 15 Time (s) Figure 10. 1992 Erzincan earthquake record. (a) The free-surface accelerogram (PEER, 2013); (b) The deconvolved accelerogram (Kartal, 2010). Acceleration (m/s ) Acceleration (m/s ) 88 M.E. Kartal and A. Bayraktar 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 11. The maximum principle stresses in the rockfill for welded contact and empty reservoir condition in Case 1. (a) The principle compression stresses; (b) The principle tensile stresses. 4.4. Structural connections in a CFR dam There are various joints in a CFR dam. The connections in these joints are generally modelled with welded and friction contact. Welded contact includes common nodes in the contact interface. Interface or contact elements are required for friction contact (Figure 5). Four or six noded plane interface elements provide friction behaviour with their transverse shear stiffness. Besides, two or three noded contact element pairs define friction. In this study, two noded surface-to-surface contact and target elements are used. If contact occurs, sliding appears depending on the maximum shear stress allowed and friction coefficient. Two contact surfaces can bear shear stress up to a certain extent across their interfaces before sliding in the Coulomb’s friction law. This is named as ‘stick’ case. Coulomb’s friction model is defined with ‘τ’ equivalent shear stress in which sliding begins as a part of ‘p’ contact pressure. This stress is, τ ¼ μp þ c (15) lim Stress (kN/m ) Stress (kN/m ) Mathematical and Computer Modelling of Dynamical Systems 89 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 12. The maximum principle stresses in the rockfill for welded contact and empty reservoir condition in Case 2. (a) The principle compression stresses; (b) The principle tensile stresses. where, τ is limit shear stress, ‘μ’ is friction coefficient and ‘c’ is contact cohesion [22]. lim Once equivalent shear stress exceeds τ , the contact and target surfaces move relatively lim in respect of each other. This is named as ‘sliding’. Friction coefficient of ‘0’ refers the frictionless contact problems. However, friction coefficient is ‘1’ for bonded surfaces. The other term, which is cohesion, has stress unit as shear stress and provides sliding resistance even with zero normal pressure. 4.5. Non-linear response of CFR dams The Drucker–Prager model is used for concrete slab in the materially non-linear analysis. Non-linear response of rockfill and foundation rock is determined by the multi-linear kinematic hardening model. In this method, a uniaxial stress–strain curve of the material is required. This curve can be determined by a shear modulus–shear strain relationship for rock and rockfill materials [20] produced by the best-fit hyper- bolic curve defining G/G versus cyclic shear strain relationship for gravel soils based max on testing by 15 investigators (Figure 6). This study considers the best curve produced Stress (kN/m ) Stress (kN/m ) 90 M.E. Kartal and A. Bayraktar 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 13. The maximum principle stresses in the rockfill for welded contact and empty reservoir condition in Case 3. (a) The principle compression stresses; (b) The principle tensile stresses. by Rollins, Evans, Diehl and Daily III [20] for rockfill. In addition, shear modulus– shear strain relation for rock soils obtained from experimental studies by Schnabel, Lysmer and Seed [21] is used for rock foundation (Figure 7). Using these curves, the uniaxial stress–strain curves for rockfill and foundation soil are determined as shown in Figures 8 and 9 [35]. ‘No separation’ contact model is preferred in the concrete slab-rockfill, rockfill- foundation and plinth-foundation interfaces. However, ‘standard’ contact model is pre- ferred in the concrete slab-plinth interface. In the standard contact model, the structural element behind the contact element may slide over and leaves from the structural element behind the target element. However, though the contact surface does not separate from the target surface, it may slide over the target surface in the no separation contact model. 4.6. Concrete slab thickness There are several thickness functions produced for concrete slab depending on the dam height (H ). Qian [36] compiled several thickness functions for concrete slab on CFR dams constructed in China. Table 3 gives some selected functions for concrete slab Stress (kN/m ) Stress (kN/m ) Mathematical and Computer Modelling of Dynamical Systems 91 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 14. The maximum principle stresses in the rockfill for welded contact and full reservoir condition in Case 1. (a) The principle compression stresses; (b) The principle tensile stresses. considered in this study [36]. The bottom thickness is obtained as 72.6 cm in Table 3 in line 3, but this is quite near the current bottom thickness of the concrete slab thickness of Torul dam. Therefore, the current thickness of 70 cm is considered in the numerical models. 4.7. Analysis cases In this study, finite element analyses are performed for three analytical cases which contain geometrical and material non-linear analyses as given in Table 4. These cases aim to reveal the effect of the material non-linear response of the dam body on the rockfill. Besides, the effect of the concrete slab thickness on the rockfill response is also investigated. This study almost fully ignores the response of concrete slab and concrete slab-rockfill interface. 4.8. Deconvolved ground motion model Free-field surface motions recorded during earthquakes reflect the characteristics of underlying soil layers at the recording site [37,38]. Real ground response problems Stress (kN/m ) Stress (kN/m ) 92 M.E. Kartal and A. Bayraktar 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 15. The maximum principle stresses in the rockfill for welded contact and full reservoir condition in Case 2. (a) The principle compression stresses; (b) The principle tensile stresses. usually involve soil deposits with layers of different stiffness and damping character- istics with boundaries at the elastic wave energy that will be reflected and/or transmitted [38]. Therefore, earthquake records must be deconvolved to the base of the rock foundation. The North–South component of the 1992 Erzincan earthquake with a peak ground acceleration (pga) of 0.515 g is selected for the analyses [39]. Torul dam is nearby the North Anatolian fault and a strong ground motion of which epicenter Erzincan occurred in 1992. In this study, the earthquake record is deconvolved to the base of the rock foundation considering three foundation layers using SHAKE91 [40]. The earthquake record obtained at the ground surface and the deconvolved earthquake record are shown in Figure 10. It is clearly seen that the pga of the deconvolved accelerogram is lower than the free-surface accelerogram. 4.9. Stresses This study investigated the principle compressive and tensile stresses that occurred in the rockfill under deconvolved ground motion for the selected three different cases and Stress (kN/m ) Stress (kN/m ) Mathematical and Computer Modelling of Dynamical Systems 93 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 16. The maximum principle stresses in the rockfill for welded contact and full reservoir condition in Case 3. (a) The principle compression stresses; (b) The principle tensile stresses. concrete slab thicknesses. The principle stress components are calculated for both welded and friction contacts in the joints of the CFR dam. 4.9.1. The principle stresses in the rockfill for welded contact in the joints The maximum principle stress components occurred in Section I-I (Figure 3)of the rockfill for welded contact in the joints and empty reservoir condition are shown in Figures 11–13 for Cases 1–3. In addition, Figures 14–16 show the maximum principle stress components occurred in I-I section of the rockfill for welded contact in the joints and full reservoir condition for Case 1, Case 2 and Case 3. The principle compressive stresses in the rockfill increase when getting close to the upstream face. However, the principle compressive stresses decrease as the concrete slab increases. The principle compressive stresses of the rockfill decrease in Case 2, in which the rockfill is assumed to be non-linear and the concrete slab is linear, as compared to those in Case 1. But, those significantly increase near the concrete slab-rockfill inter- face in Case 3, in which case the concrete slab and rockfill are non-linear. The Stress (kN/m ) Stress (kN/m ) 94 M.E. Kartal and A. Bayraktar 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 17. The maximum principle stresses in the rockfill for friction contact and empty reservoir condition in Case 1. (a) The principle compression stresses; (b) The principle tensile stresses. variation of the compressive stresses in 2A, 3A and 3C zones depends on the concrete slab thickness for welded contact. However, those are relatively close to each other in 3C and 3D zones. The principle tensile stresses of the rockfill decrease as the concrete slab increases. The differences in the principle tensile stresses are evident especially in Case 1 and those are clearly distinguished in 3C and 3D zones. The principle tensile stresses are obtained close to each other only in Case 3. While the principle compressive stresses of the rockfill decrease, the principle tensile stresses of the rockfill increase when getting close to the downstream face of the dam. The change of the principle stress components resembles each other throughout the horizontal section of the rockfill under empty and full reservoir conditions. The stresses in the rockfill under hydrodynamic pressure, however, are higher than those under empty reservoir condition. The stresses in the rockfill under hydro- dynamic pressure decrease as the concrete slab thickness increases. Numerical analyses reveal that the most critical stresses appear for the concrete slab thickness of 30 cm. Stress (kN/m ) Stress (kN/m ) Mathematical and Computer Modelling of Dynamical Systems 95 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 18. The maximum principle stresses in the rockfill for friction contact and empty reservoir condition in Case 2. (a) The principle compression stresses; (b) The principle tensile stresses. 4.9.2. The principle stresses in the rockfill for friction contact in the joints The maximum principle stress components occurred in the Section I-I (Figure 3)ofthe rockfill for friction contact in the joints and empty reservoir condition are shown in Figures 17–19 for Case 1, Case 2 and Case 3. In addition, Figures 20–22 show the maximum principle stress components occurred in the Section I-I of the rockfill for friction contact in the joints and full reservoir condition for Case 1, Case 2 and Case 3. The concrete slab can make a sliding motion over the rockfill when the friction is defined in the concrete slab-rockfill interface. This causes partially independent behaviours for the concrete slab and rockfill. Therefore, rockfill will make higher deformations under earth- quake ground motions because the concrete slab contributes less when friction is con- sidered at the concrete slab-rockfill interface. The higher deformations cause higher compressive stresses near the concrete slab-rockfill interface. The principle compressive stresses increase between 2 and 3.5 times as compared to those of the rockfill including welded contact in the joints. The principle compressive stresses decrease when getting close to the downstream of the dam. The concrete slab thickness does not considerably affect the compressive stresses of the rockfill in an empty reservoir condition when friction is considered. The principle stresses along the horizontal section decrease when Stress (kN/m ) Stress (kN/m ) 96 M.E. Kartal and A. Bayraktar 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 19. The maximum principle stresses in the rockfill for friction contact and empty reservoir condition in Case 3. (a) The principle compression stresses; (b) The principle tensile stresses. the non-linear material behaviour of the rockfill is considered in empty reservoir condi- tion. This decrease is clearly seen in the principle tensile stresses. The principle compressive stresses under hydrodynamic pressure clearly increase as compared to those of the dam including welded contact at the joints. This increase is actually less than that obtained under empty reservoir condition. However, the principle tensile stresses occurred under hydrodynamic pressure, obviously increases near the concrete slab-rockfill interface. The high principle tensile stresses may be on the unsafe side in this section which is commonly known as the cohesionless region. As seen under empty reservoir condition, the principle tensile stresses increase when getting close to the downstream face. The concrete slab thickness is not effective on the principle stresses of the rockfill when the reservoir is full. The decrease in the principle tensile stresses is very apparent for the higher values of the concrete slab thickness when the concrete slab material is modelled as linear. While the principle tensile stresses decrease near the upstream face, they generally increase along the horizontal section. However, the principle compressive stresses continuously decrease in the horizontal section. Numerical analyses give the most critical stresses in the rockfill for the concrete slab thickness of 30 cm. Stress (kN/m ) Stress (kN/m ) Mathematical and Computer Modelling of Dynamical Systems 97 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 20. The maximum principle stresses in the rockfill for friction contact and full reservoir condition in Case 1. (a) The principle compression stresses; (b) The principle tensile stresses. 5. Conclusions In this study, the non-linear response of the rockfill in a CFR dam-foundation-reservoir system to deconvolved ground motion was investigated for different slab thicknesses. Non-linear geometrical and material behaviour of the rockfill was considered in this study. The multi-linear kinematic hardening model was used for the rockfill and foundation rock. The Drucker–Prager model was used for the non-linear response of the concrete slab. The friction between the various interaction surfaces is considered with contact elements based on the Coulomb’s friction law. The hydrodynamic pressure of the reservoir water is considered with the fluid finite elements based on the Lagrangian approach. The principle compressive and tensile stresses of the rockfill from upstream to downstream along the horizontal section were investigated for empty and full reservoir conditions considering welded and friction contacts. The following conclusions can be deducted from the numerical analyses: The CFR dam, including the welded contact in the joint (Figures 11–16), indicate that (1) the maximum principle stress components in the rockfill decrease if the Stress (kN/m ) Stress (kN/m ) 98 M.E. Kartal and A. Bayraktar 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 21. The maximum principle stresses in the rockfill for friction contact and full reservoir condition in Case 2. (a) The principle compression stresses; (b) The principle tensile stresses. non-linear material response of the rockfill is considered in the solutions. (2) The principle stresses in the rockfill increases by the effect of the hydrodynamic pressure; however, the effect is limited. (3) The principle stresses also increase when decreasing the concrete slab thickness. The CFR dam, including friction contact in the joints (Figures 17–22), indicate that (1) the principle stresses decrease when getting close to the upstream face of the dam for each reservoir condition. (2) The principle tensile stresses also decrease with the materially non-linear response of the rockfill. (3) However, those increase by the effect of the hydrodynamic pressure near the upstream face. ● The comparison of the CFR dams, including welded and friction contacts in the joints (Figures 11–22), indicate that the principle stresses in the rockfill of the CFR dam including friction in the joints are bigger than those in the CFR dam ignoring friction. ● The change of the principle stresses in the rockfill is low for the concrete face slab thickness greater than 58 cm. Consequently, the use of face slab thickness greater than 58 cm may provide the optimum theoretical design for CFR dams for dynamic loading conditions. Stress (kN/m ) Stress (kN/m ) Mathematical and Computer Modelling of Dynamical Systems 99 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (a) –200 –400 –600 –800 –1000 –1200 –1400 Upstream Downstream –1600 –1800 –2000 –200 –100 0 100 200 Horizontal Section (m) 30 cm 58.4 cm 70 cm 86.8 cm 110 cm (b) Upstream Downstream –200 –100 0 100 200 Horizontal Section (m) Figure 22. The maximum principle stresses in the rockfill for friction contact and full reservoir condition in Case 3. (a) The principle compression stresses; (b) The principle tensile stresses. Beyond this study, if the vertical components of earthquakes should be taken into account in the analyses, the realistic earthquake effects on dams may be determined. In addition, depending on the foundation conditions surface wave effects may appear. 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Journal

Mathematical and Computer Modelling of Dynamical Systems – Taylor & Francis

Published: Jan 2, 2015

Keywords: concrete-faced rockfill dam; friction contact; geometrical and material non-linear behaviour; Lagrangian approach

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