Numerical Assessment of the Deformation of CFRD Dams during Earthquakes

Transcription

1 The 1 th International Conference of International Association for Comuter Methods and Advances in Geomechanics (IACMAG) 1-6 October, 008 Goa, India Numerical Assessment of the Deformation of CFRD Dams during Earthquakes A. O. Sfriso LMNI, Faculty of Engineering, University of Buenos Aires, Argentina Keywords: CFRD dams, earthquakes, lasticity, seismic analysis. ABSTRACT: A numerical rocedure for the reliminary estimation of the earthquake-induced settlement of concrete face rockfill dams is resented. The method is based on a simle constitutive model that accounts for the key features of the behavior of coarse grained materials that affect the resonse of CFRDs, namely ressure deendent elasticity and a eak friction angle deendent on both ressure and void ratio. The actual earthquake design record is relaced by a simle sinusoidal base acceleration having an equivalent effect on the dam. The numerical model uses isotroic hyerelasticity and isotroic hardening/softening lasticity combined with stressdilatancy theory, and thus is alicable to geometries and materials where only monotonic lasticity is exected to occur, as is the case of CFRD dams. A case study is resented where the rocedure is alied to a CFRD 140 m high, located in Argentina and designed for a strong earthquake. The comuted settlement comares well with an analytical estimation and with a decouled numerical model of the same roblem. The main advantage of the roosed rocedure is that all material nonlinearities are accounted for by the constitutive equations, thus allowing for the usage of a simle mesh with no artificial sub-zonification. 1 Introduction Concrete Face Rockfill Dam (CFRD) engineering has reached a mature state in many asects, including foundation design, selection rocedures for construction materials, zonification criteria, comaction methods and construction rocedures for the concrete face. Design of CFRDs is, however, still based on exerience and engineering judgement (Cooke, 1984; Cooke, 1997; Núñez, 007b). When laced in areas of high seismicity, an assessment of the effects of strong earthquakes on the overall behavior of CFRDs is required. Performance rather than safety is the main concern, as it is widely acceted that the effect of seismic loading on CFRDs is lastic deformation and settlement but not a sloe failure in its classical sense (Cooke, 1984; Gazetas and Dakoulas, 199; Makdisi and Seed, 1978; Newmark, 1965; Seed, 1979). Freeboard allowance for earthquake-induced settlements may exceed 1% of the dam s height. In the current state of the art, this allowance cannot be significantly reduced by design efforts and is weakly deendent on the dam s geometry and materials (Núñez, 007b). Therefore, the roblem can be osed as follows: verify that the minimum freeboard is enough for your dam; if not, increase the freeboard or modify your design otherwise. The design of the concrete face for earthquake loading oses additional challenges. While the concrete face is just a water barrier and not a structural member of the dam, its structural behavior must be verified at design stage. To date, design efforts are limited to avoid in-lane comressive failure, to minimize cracking and to assure a deendable behavior of the concrete joints. Provisions are taken to avoid ore ressures to develo in the dam s body during and after strong seismic shaking (Cooke and Sherard, 1987; Sherard and Cooke, 1987). Methods used to estimate earthquake-induced dam deformation range from simle analytical tools (Newmark, 1965; Makdisi and Seed, 1978; Núñez, 007) to involved three dimensional (D) numerical models. While analytical tools are simle to use, they cannot take into account secial features of dam design, like zonification, existence of berms or non uniform sloes. On the other hand, the reliability of numerical methods heavily deends on the choice of the constitutive models and the selection of inut arameters. Moreover, D numerical models are too involved to be used at design stage. The goal is to advance in the develoment of simle methods to estimate earthquake-induced deformation of dams at design stage. Due to dam zonification, numerical methods aear to be the natural choice. Due to the uncertainties in material roerties, simle constitutive models should be adoted. Finally, uncertainties in earthquake loading favors the adotion of conservative, synthetic seismic records. Design is always followed by 4054

2 analyisis, where D nonlinear models can be erformed to account for construction rocesses, relevant features of construction materials and selected seismic records. For simlicity, in this aer only CFRDs resting on hard rock are considered. The suggested rocedure, however, can accomodate any foundation tye. Earthquake-induced deformation of dams.1 Sliding block concetual model The concetual roblem can be best understood with the aid of the sliding block model (Newmark, 1965). A rigid block rests on a lane having a block-surface friction angle φ and an inclination β < φ, see Fig. 1a. The block can resist a maximum static force T s arallel to the sloe of magnitude s ( [ φ β] [ φ] ) T = sin cos mg (1) where m is the mass of the block and g is the acceleration of gravity. Inertia forces aear when the system is accelerated to the right with an acceleration a λ g T =, see Fig. 1b. During the alication of this acceleration, s is reduced to a dynamic value T d of magnitude which can be negative if ( [ φ β] [ φ] λ [ φ β] [ φ] ) Td = sin cos cos cos mg () λ > λc, where λc = tan[ φ β] () g T acting during is a threshold acceleration coefficient. An acceleration λ large enough to roduce a negative d a eriod t the dislacement at the acceleration hase, and the dislacement of the braking hase Δ roduces a net downwards dislacement δ of the block on the base surface which is the sum of δ = 1 Td 1 Ts t tb m Δ + m Δ (4) where Δ tb = Td Ts Δ t is the duration of the braking hase. After some algebra, the final exression is cos[ φ β] λ δ = λg 1 Δt (5) cos[ φ] tan[ φ β] A series of random ulses acting on the system roduces a cumulative dislacement that is the sum of the contribution of the individual ulses of the series. All ulses having λ < λc roduce no effect. It is a common assumtion that negative ulses, acting leftwards in Fig. 1b, are not large enough to roduce uwards dislacements of the block (Newmark, 1965; Makdisi and Seed, 1978). This imlies that cyclic earthquake loading, however comlex it may be, only roduces monotonic lastic deformations to the system. φ mg T s φ mg T d T s β T d λmg β λg Fig. 1. Sliding block model. a) static stability. b) stability of the accelerated system.. Extension to dam s geometry A dam can be understood as a triangular body of earth-like material where sliding wedges can form with any shae and size, rovided limit state conditions are reached in the sliding surface. Moreover, an earthquake can be understood as a large series of ositive and negative acceleration ulses acting on the base of the dam and roagating through the dam s body (Makdisi and Seed, 1978; Gazetas and Dakoulas, 199). A given wedge has a articular λ c which deends on the inclination of the otential sliding surface of the wedge. Pulses having λ < λ c roduce no net dislacement between the wedge and the rest of the dam s body and are termed non effective ulses. Effective ositive ulses roduce no effect on the otential wedges at one side of the dam, while effective negative ulses roduce no effect on the other side. Total settlement of the dam is the contribution of 4055

3 the vertical comonent of the downwards dislacement of all wedges in both sides of the dam. For CFRDs, a full reservoir imlies that water ressure stabilizes the otential wedges on the ustream sloe. Therefore, for a given CFRD and earthquake, total settlement of the dam with emty reservoir is larger than the settlement of the dam with full reservoir (Uddin and Gazetas, 1995; Núñez, 007).. Equivalent earthquake The dam is a dynamic system that roagates stress waves through the dam body, roducing wave amlification, attenuation and interference that result in a comlex D resonse. The overall resonse of the dam deends on it s natural eriod comared with the dominant eriod of the earthquake. The natural eriod of the dam, in turn, deends on the geometry of the dam, the elastic roerties of the construction materials and the interaction between the dam and it s foundation. For triangular, uniform, elastic dams of infinite length, the natural eriod T can be comuted with (Gazetas and Dakoulas, 199) T =.61H vs (6) where H is the height of the dam vs = G ρ (7) is the shear wave velocity of the dam material, G is the shear modulus and ρ is the material density. For other dam shaes, Núñez (Núñez, 007) roosed a rocedure to comute T that can be condensed into H 1α c + 16( αb + αc) T =.61 (8) v + 6α + 9α s c c, l c is the crest and l b is the base length of the dam. where α c = lc H, α = l H b b An earthquake can be defined by it s magnitude M and it s eak ground acceleration PGA. A simle correlation between M and the effective number of ulses N is (Núñez, 007) N = M 1 (9) Therefore, the simlest dynamic loading that aroximates the effect of a real earthquake loading is a sinusoidal wave action of amlitude PGA, eriod T and duration t = NT. This acceleration record is too conservative, as PGA only occurs once in a given earthquake, Núñez (Núñez, 007) roosed to use a max. acceleration amax = ηpga with η deending on the earthquake tye in the range 0.8 < η < 0.9. Material roerties relevant for the analysis.1 Elasticity Dams are built by comaction of layers of rockfill, a fact that imairs orthotroic roerties to the material. The limited information on material roerties available at design stage, however, deems the alication of orthotroic elasticity imractical, and therefore isotroic elasticity is usually assumed. The shear modulus of the rockfill is often comuted with the exression (Hardin and Richart, 196; Kokusho, 1980) ( ) ce e G = cs ref 1+ e ref c and m are material arameters, e is void ratio, is mean ressure and where c s, e ref is a reference ressure usually adoted equal to atmosheric ressure. Elastic bulk modulus has a minor influence in most roblems involving CFRDs. It can be comuted with the shear modulus and an assumed Poisson s ratio in the range < ν < for dense gravels and rockfills. m (10). Peak friction angle It is widely recognized that the eak friction angle of granular materials deends on mean ressure and void ratio (De Beer, 1965; Marsal, 1967). Bolton (1986) introduced the exression ( [ ]) φ = φ + D Q ln 1KPa (11) tc c r where φ tc is the eak friction angle for triaxial comression, φ c is the critical state (CS) friction angle, D r is relative density and Q is a material arameter accounting for article strength. The deendence of φ tc on void ratio is rarely acknowledged for in dam design, as material arameters are determined for the secified comaction void ratio. An exression frequently used to estimate the eak friction angle of rockfills is (Duncan et al, 1980; Les, 1970) 4056

4 φ = φ Δφ log (1) tc 0 ref where φ 0 and Δ φ are material arameters. While Eqn. 1 roerly accounts for the reduction of φ tc with increasing ressure, it fails to cature its evolution with lastic strain, and therefore demands for the adotion of a set of arameters φ 0 and Δ φ consistent with the assumed level of material softening for a given seismic action. Sfriso (007) introduced the exression where A =, B = 1 and tc c r [ ] φ = φ A Dlogχ B (1) 0.4 ( r ref ) χ = e (14) is a measure of stress level and r is a material arameter accounting for article strength. Eqn. (1) is concetually equivalent to Eqn. (11) excet for the fact that χ is void-ratio deendent while the equivalent term [ ] ex Q is not. Deendency of χ on void ratio reflects the higher crushing resistance of articles for increasing number of contacts er article, i.e. at denser states. As ointed out by Bolton (1986), a minimum resure = 150KPa should be used in Eqn. (11) and subsequently in Eqn. (14) to avoid the use of an unsafe friction angle due to overestimation of dilatancy. Eqn. (1) yields φtc = φc (i.e. CS behavior) at a ressure 0.4 = r e ex B ( ADr) ref (15) Fig. a shows φ tc of Toyoura sand (Bolton, 1987) for different ressures and void ratios and the rediction by Eqn. (1) with r = 40 and φ c =.5. Fig. b shows the CS void ratio e c for Toyoura sand (Verdugo and Ishihara, 1996) and the rediction by Eqn. (15) with the same set of material arameters. The good agreement shown imlies that Eqn. (1) has the caability to reroduce that e ec imlies φtc φc φ tc e=0.86 e=0.78 e=0.70 e=0.66 Eqn [ KPa] e c Eqn [ KPa] Fig.. a: Peak friction angle as a function of mean ressure and void ratio, Toyoura sand. Predicted vs. exerimental results. b: critical state line for Toyoura sand. Predicted vs. exerimental results. A CFRD subjected to an earthquake is a lane strain roblem even for irregular valley shaes, an therefore a lane strain eak friction angle φs should be adoted. It is usual to use the estimation φs = 1.15φtc.. Stress-strain A vast literature exists on the alication of moduli reduction curves and daming coefficients to the decouled analysis of earthquake loading to dams (e.g. Duncan et al, 1980; Kokusho, 1980; Seed et al, 1984). For a couled analyisis, however, simle failure lasticity can account for much of the energy dissiation normally attributed to daming and nonlinear stress-strain resonse. A simle exercise can rove this assessment. A symmetric triangular dam 100m high with sloes 1.4 : 1 was subjected to a wave acceleration of PGA = 0.5 for a duration t = 6T =.4s and left free to oscillate for another t =.4s. The hyerbolic model available in Plaxis (Duncan and Chang, 1970; Brinkgreve, 00) was ref used with the following arameters: γ =.5 KN m, E ur = 70 MN m, m = 0.5, ν = 0., φ = c = 1KN m, 50. Parameters controlling the stress-strain resonse of the model were varied as shown in Table 1. For the meaning of the arameters of the hyerbolic model refer to (Brinkgreve, 00). Triangular 15 noded elements with an average size of 5 m were used along with an absorbent base boundary. δ is the vertical settlement of the crest of the dam at the end of the simulation. 4057

5 Table 1. Material arameters for the sensitivity analysis of the hyerbolic model. E 50 ref MN/m R f δ m It is worth noting that all results fall within δ = 1.98m ±.5%, desite the fact that the arameters controlling ref the stress-strain resonse, namely E 50 and R f where varied beyond their usual bounds. 4 A simle constitutive model for coarse grained granular materials 4.1 Formulation A simle elastolastic model for the reliminar assessment of earthquake-induced deformation of dams is resented. The model uses ressure deendent hyerelasticity and non-associative isotroic hardeningsoftening lasticity in the ost-failure regime. Hardening is due solely to the evolution of φ tc with void ratio. Material arameters are the already mentioned c s, c e, m, φ c, r along with e min and e max, the min. and max. void ratios used to comute D r. The state of the material is characterized by the Cauchy stress σ and the void ratio e. Standard additive decomosition of the infinitesimal strain tensor rate &ε is adoted. It holds e ε& = ε& + ε & (16) where where e &ε and s &ε are the elastic and lastic strain tensor rates. Stress-strain equation is of the form e σ = W s ε (17) σ,e of the form (Molenkam, 1988) Dr r Ws = + (18) W is a comlementary strain energy function of state variables { } ( )( ) G 1 m m 4 where r = r, r = s /, s= σ 1 and 1 is the second order unit tensor. G is comuted with Eqn. (10). Bulk modulus, as derived from Eqn. (18), is K = (19) Dr m 1 ( m) 4 r yielding a Poisson s ratio 0.05 < ν < 0. for 0< D r < 1. The yield function is a modified version of the Matsuoka-Nakai yield function of the form (Matsuoka and Nakai, 1974) 1 1 F = ( μ+ 6 ) r: r ( μ+ 9 ) r r: r μ = 0 (0) where () and () : denote simle and double contraction and μ = 8tan [ φtc ] (1) φ tc is comuted with Eqn. (1). The effect of the intermediate rincial stress on shear strength (i.e. the difference between φ tc and φ s ) is roerly accounted for by Eqn. (0). The lastic shear rate is of the form &ε & () = λm where & λ is a lastic multilier and m is the lastic strain rate unit tensor of the form m= m + β1 () d The deviatoric comonent of the flow direction in shear m d is given by d d d where n = n- 1 d ( n : 1 ) 1 and n = F, σ is the (outwards) normal to the yield function. β is comuted after stress-dilatancy theory (Rowe, 196) as (Sfriso, 007) G m = n n ( ) ( ) ( ( )) ( ( )) σ1md1+ σmd + Ncσmd σ1+ σ + Ncσ md + β > 0 β = σ1md1 + Nc σmd + σmd σ1 + Nc σ + σ md + β < 0 where { σ, σ, σ } and {,, } 1 m m m are the sorted eigenvalues of σ and d d1 d d m and (4) (5) 4058

6 Evolution of the state variable e is of the form ( 1 [ φ ]) ( 1 [ φ ]) N = + sin sin (6) c c c ( 1 ) & (7) e= + e & ε v The concet underlying this formulation is that the initial state is denser/looser than the critical state, and hence φtc φc. Plastic shearing roduces a change in void ratio which is obtained by time integration of Eqn. (7), where & ε v = βλ&. This change in void ratio affects φ tc and the aerture of the failure cone given by Eqn. (0). Continued shearing attracts the material state to the CS state, where Eqn. (1) yields φ tc = φ c and Eqn. (5) yields 0, i.e. a stationary condition at the critical state. β = The model was imlemmented in Plaxis as an user define constitutive model. A fully imlicit one ste backwardreturn algorithm was develoed for the time integration of the constitutive equations. 4. Validation While the lastic formulation is of standard design, the hyerelastic formulation is somewhat atyical and requires some sort of validation for the intended urose. To comly with this requirement, the natural eriod T of a number of symmetric triangular dams was comuted numerically. Sloe inclination and height were varied in the ranges B = { 1.4 : 1,1.6 : 1,1.8 : 1} and H = { 50m, 75m, 100m, 15m, 150m}. The models were excited with a short base ulse and left free to oscillate. A fix set of material arameters was used as follows: e min = 0.10, e max = 0.40, c s = 750, c e =.17, m = 0.50, and an unit weight γ = 7 KN m ( 1+ e). Three void ratios e = { 0.15, 0.0, 0.5} were used. The roduct of five heights, three sloes and three void ratios yields 45 runs. The natural eriod was determined as the inverse of the dominant frequency of the outut crest dislacements. Fig. shows the results, normalized by the rediction of Eqn. (6) for a shear wave velocity comuted with Eqn. (10) and a ressure = γ H equal to the weight of rockfill column above the baricenter of the dam. T 1.15 Tth 1.10 B=1.4; e=0.15 B=1.4; e=0.0 B=1.4; e=0.5 B=1.6; e=0.15 B=1.6; e=0.0 B=1.6; e=0.5 B=1.8; e=0.15 B=1.8; e=0.0 B=1.8; e= H m 150 Fig.. Numerically comuted natural eriods T normalized by T th given by Eqn (6). [ ] 5 Suggested design rocedure A rocedure for the reliminary estimation of the earthquake-induced settlement of CFRDs is as follows: I) Comute the natural eriod of the dam using Eqns (7) (8) (10) and = γ H ; ii) Comute the effective number of ulses N using Eqn. (9); iii) Build and load the model with a sine wave of amlitude 0.8PGA, eriod T and duration t = NT ; iv) Let the model oscillate freely for a time san t = NT ; and v) Obtain the crest settlement at the end of the simulation. For simlicity, no recommendations are included here w.r.t. numerical issues as tye and size of elements, boundary conditions, integration methods, or otherwise. 6 Case study Los Caracoles dam is a CFRD located in San Juan, Argentina, in an area of high seismicity. The main dimensions of the dam are: H = 140m, l c = 60m, l b = 10m. Ustream sloe is 1.5 : 1 and downstream sloe is 1.7 : 1 with two berms. The design earthquake has a magnitude M = 7.7 and a PGA = 1.0g. Material arameters, comuted or estimated with the information available at design stage (Bissio and Tejada, 006), are shown in Table. G av is the average shear modulus comuted using Eqn. (10) with = γ H. The mesh and zone identification is shown in Fig. 4. It is worth noting that a simle mesh is used because the constitutive model accounts for all material nonlinearities that would otherwise require sub-division of the mesh 4059

7 and the definition of a large set of zones with different material roerties. Table. Material roerties. Los Caracoles dam. e min e max c s c e m φ c r e G av [MPa ] B alluvium / colluvium D blasted greywacke L colluvium Riverbed aged alluvium v s [m/s] Fig. 4. Mesh and zonification, Los Caracoles Dam. The natural eriod comuted with Eqn. (8) is T = 0.56s. A sinusoidal acceleration of amlitude 0.8g, eriod T = 0.56s and duration t = 7T was alied horizontally at the base of the model. The model was then allowed to oscillate freely for another t = 7T. The resulting settlement is δ =.8m. This result can be comared with an analytical rediction of δ =.57m.9m (Núñez, 007) and a numerical estimation with an involved decouled analysis of δ =.1m (Bissio and Tejada, 006). One hour was needed to build and run the model and obtain the final results. 7 Conclusions Design of CFRDs for earthquake loading is mainly based on exerience and engineering judgement. Freeboard allowance for earthquake-induced settlements, however, must be verified at design stage. While methods used range from simle analytical tools to involved D models, simle numerical tools that account for dam zonification and irregular dam shaes can be used for convenience. The numerical rocedure resented is based on a simle constitutive model that includes ressure deendent elasticity in the re-failure regime and hardening-softening isotroic lasticity combined with stress-dilatancy theory in the ost-failure regime, and thus is alicable to geometries and materials where only monotonic lasticity is exected to occur, as is the case of CFRD dams. The actual earthquake design record is relaced by a simle sinusoidal base acceleration having an equivalent effect on the dam. For simlicity, only CFRDs resting on hard rock are considered. The suggested rocedure, however, can accomodate any foundation tye. A case study is resented where the rocedure is alied to a CFRD 140 m high, located in Argentina and designed for a strong earthquake. The comuted settlement comares well with an analytical estimation and with a decouled numerical model of the same roblem. The main advantage of the roosed rocedure is that all material nonlinearities are accounted for by the constitutive equations, thus allowing for the usage of a simle mesh with no artificial sub-zonification. 8 Acknowledgements E. Núñez shared his exerience on earth dams with the author and rovided comments to the rocedure described in this aer. G. Weber co-authored several asects of the imlemmentation of the constitutive model used. J. Laiún co-worked in the numerical simulations. Their contributions is gratefully acknowledged. 4060

8 9 References Bissio J.,Tejada, C Caracoles, análisis dinámico de la resa. Proc. IV Conf. CAP, Posadas (Argentina), Bolton M The strength and dilatancy of sands. Geotechnique 6(1), Bolton M The strength and dilatancy of sands, Discussion. Geotechnique 7(), Brinkgreve R. 00. Plaxis users manual. Balkema, Rotterdam (The Neetherlands). Cooke J Progress in rockfill dams. ASCE JGE, 110(10), Cooke J., Sherard J Concrete-face rockfill dam: II. Design. ASCE JGE, 11(10), Cooke J The concrete face rockfill dam. Proc. 17 USCOLD Lect., San Diego (USA), De Beer E Influence of the mean normal stress on the shearing resistance of sand. Proc. VI ICSMFE, Montreal (Canada), 1, Duncan J., Chang C Non-linear analysis of stress and strain in soil. ASCE JSMFD, 96(5), Duncan J., Byrne P., Wong K., Mabry P Strength, stress-strain and bulk modulus arameters for finite element analyses of stresses and movements in soil masses. Reort UCB/GT/80-01, Univ. of California at Berkeley. Gazetas G., Dakoulas P Seismic analysis and design of rockfill dams: State of the art. Soil Dyn. Earthq. Eng., 11(1), Hardin B., Richart F Elastic wave velocities in granular soils. ASCE JSMFD, 89(1), -65. Kokusho T Cyclic triaxial test of dynamic soil roerties for wide strain range. Soils and Foundations, 0(), Les T Review of Shearing Strength of Rockfill. ASCE JSMFD, 96(4), Makdisi F., Seed B Simlified rocedure for estimating dam and embankment earthquakes induced deformation. ASCE JGE, 104(7), Marsal R Large scale testing of rockfill materials, ASCE JSMFD, 9(), 7-4. Matsuoka H., Nakai T Stress-deformation and strength characteristics of soil under three different rincial stresses. Proc. Jaan Soc. of Civil Eng.,, Molenkam F A simle model for isotroic non-linear elasticity of frictional materials. IJNAMG., 1(5), Newmark N Effects of earthquakes on dams and embankments. Geotechnique 15(), Núñez E Behavior of coarse alluvium sloes subjected to earthquakes. Proc. XIII PCSMGE, Margarita (Venezuela), Núñez E. 007b. Uncertainties and aroximations in geotechnics. Proc. XIII PCSMGE, Margarita (Venezuela), 6-9. Rowe P The stress dilatancy relation for static equilibrium of an assembly of articles in contact. Proc. Royal Soc. London, 69, Seed H Considerations in the earthquake-resistant design of earth and rockfill dams. Geotechnique 9(), Seed H., Wong R., Idriss I. Tokimatu K Moduli and daming factors for dynamic analyses of cohesionless soils. Reort UCB/EERC-84/14, Univ. of California at Berkeley. Sfriso A A constitutive model for sands: Evaluation of redictive caability. Proc. XIII PCSMGE, Margarita (Venezuela), Sherard J. Cooke J Concrete-face rockfill dam: I. Assessment. ASCE JGE, 11(10), Uddin N., Gazetas G Dynamic resonse of CFRD to strong seismic excitation. ASCE JGE, 11(), Verdugo R., Ishihara K The steady state of sandy soils. Soils and Found. 6(),