Seismic response analysis of arch dam-water-rock foundation systems

Vol.3, No.2EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATIONDecember, 2004Article ID: 1671-3664(2004)02-0283-09Seismic response analysis of arch dam-water-rock foundation systemsDu Xiuli (杜修力)”and Wang Jinting (王进廷)F:1. Bejig key laboraory of earthquake engineering and srcural rerofi., Beijing University of Tehnology, Bejing 10022 China2. Depurtment of Hydraulie Engineering, Tsinghua Universiy, Beiing 10084, ChinaAbstract: The eicet of water comresibility on the seismie responses of arch dams is not well understood. In thispaper, a numerical model is developed with rigorous representation of the dynamic interaction between arch dam-water-rock foundation. The model is applied to the seismic response analysis of an arch dam with a height of 292m designed to aseismic intensity of IX. It is shown that consideration of the water compressibility clearly decreases the stress responses atkey positions of the dam, while the added mass model gives a conservative estimate.Keywords: seismic response; arch dam; water comresibility; dam-water foundationo interaction; added mass model;finite clement method1 Introductionconsider the effect of water compressibility, Fokand Chopra (1986) developed a three-dimensionalIn the . scismic response arialysis of arch dams,finite element substructure model for analyzing theimportant factors should be considered such as: a) thedynamic interaction of arch dam-water-foundationdynamic interaction of arch dam-rock foundation, b) thesystems in the frequency domain. In this model,dynamic interaction of compressible water-arch dam, c)the rock foundation was assumed to be massless,the dynamic interaction of compressible water- sediment-and the interaction between compressible waterfoundation, d) the opening-closing of contraction joints,and the rock foundation was represented usinge) traveling wave, and ) the efect of local topographicalan absorptive boundary, in which the absorptiveand geological behavior of rock foundations. However,boundary characteristics were evaluated according toto the author 's knowledge, there is no model that takes allthe rescarcher 's experiences based on the geologicalthese factors into account. To reasonably and rigorouslyconditions of the dam foundation. However, theanalyze the seismic responses of arch dams, an effectivemassless foundation was unable to reasonablynumerical procedure is still needed.simulate the dynamic interaction between theBased on the finite element method, Clough (1960)foundation-structure and the traveling wave effect.developed one of the earliest computer programsMoreover, the complexity of the geometry and thefor scismic analysis of arch dams, ADAP, in whichgeology of the dam site may cause some difficultiesthe foundation was modeled as a massless elasticin adequately evaluating the absorptive coefficient.solid with the rigid boundary, and the hydrodynamicTo overcome these limitations, the Fok and Chopraeffect was not included. On the basis of ADAP, Kuomodel was extended to include the effct of inertia and(1982) and Ghanaat and Clough (1989) considered thedamping of rock foundation by discretizing the rockhydrodynamic effect with added mass approximationfoundation using the boundary element method givenand devcloped the computer prograrm EADAP. Toby Tan and Chopra (1995a, 1995b). Dominguez andMaeso (1993) presented a three-dimensional boundaryCorrespondence to: Wang Jinting, Department of Hydraulicelement model for analyzing the dynamic interactionEngincering, Tsinghua University, Beijing 100084, Chinabetween arch dam-compressible water-foundation.Tel: 86- 10-62792830; Fax: 86- 10-62782159Moreover, it should be noted that all theE-mail: wangit@tsinghua.edu.cnabovementioned methods are in the frequency domain.Professor; IAssistant ProfessorDu et al, (1996) presented a procedure for analyzingSponsored by: National Natural Science Foundation of China forthe nonlinear seismic responses of arch dam-foundationDistinguished Young Scholar of China Under Grantby comhinina the exnlicit finite element method andNo.50325826; National Natural Science Foundation 0the tr中国煤化工the time domain.China Under Grant No. 50309005; Science & TechnologyThey_effcct of faults inDevelopment Project of Education Committce of Beiingrock fMYHC N M H Gof inomogereityUnder Grant No. KM2003 10005017of the rock foundation and the wave path. However, inReceived date: 2004-05-11; Accepted date: 2004-09-26284EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATIONVol.3their study, the effect of bhydrodynamic pressure wasin which, O1 is time step. the superscriptsP- 1, P and P+I .performed by using the added mass model.denote the time (P- l)O1, POt and (P+ l)Qr, respectively.In this paper, a procedure to analyze the seismicThe formulations introduced above are conditionalresponse of arch dam-compressible reservoir-foundationstable if the time step Ot satisfiessystems is developed on the basis of the Du et al. (1996)method. Using Xiaowan arch dam as an example, the△r≤min(ORlin /Ce ,OR. 1c,. ,OR /cm )reffect of water compressibility on the seismic responsesof arch dams is studied.5)2 Fundamental analysis methodin which ORmin and Cmax are the minimum elementsize and maximum wave velocity of the medium,2.1 Arch dam- foundation regionrespectively, the superscript m is the number of mediaof the system considered. and y is a constant cofficientFor regions of the dam-foundation summed to berelated to damping ratio where its determination is givenlinear elastic media, the dynamic motion equation isby Du et al, (2000). .In general, in order to obtain solutions with enoughaccuracy, it is required(h+ u)i,m + uIL,= pi, .(1)AR≤(1/8~1/12)2mm(6)where the subscript I refers to the arch dam-foundationregion; λ and μ are Lame constants; p, is the massdensity; i and j (i, j=1,2,3) are Cartesian coordinatewhere, ORmx = max(AR'x , R.AR"x ), and 2mimncomponents; and u and i are the displacement andis the minimum wavelength of incident waves.acceleration, respectively.Discretizing the equation (1) from the explicit finite2.2 Reservoir regionelement method and introducing the Rayleigh dampingterms may lead to the following explicit FEM equationThe viscous coefficient of the compressibleimpounded water is very small and in general iassumed to be zero. The wave equation of non-viscousmiin tCiumtinj + kimyWm =fu(2)compressible fAuid takes the formwhere, the subscript 1, n are the node code; the repeatedPin = K12,(7)subscript n is summed over its ranges (n = .2..n),n。is the nodal number related to the node l; m, is thewhere the subscript 2 refers to the reservoir; and P, andsolid mass of the node I, in is the solid acceleration ofK denote the mass density and the bulk modulus of thei direction at the node 1, u1m and un are respectivelyfuid, respectively.velocity and displacement components in j directionBut to avoid instability of the multi-transmittingat the solid node n, fi is the extemal force exertingartificial boundary over the range of high frequencyon the solid node l in the i direction; kKhim and Cim are(Seen in Ref. by Liao, Z. P. (1996)), very lttle dampingrespectively the solid sifness and damping cofficientsin proportion to stiffness is necessary. For compressibleofj direction at the node n on i direction at the node 1.Applying the numerical intcgration formulationimpounded water with viscous, the explicit finitepresented by Du and Wang (2000), the displacement andelement motion of equation of i direction at any node Iin the discrete model domain can be written asvelocity responses can be expressed asmjign +rmgi2m +8kmUmw=fzu(8)u=. =号m +uhi +On"" -气where, the subscript m are the node code; the repeatedsubscript m is summed over its ranges (m = .,2..m),m, is the nodal number related to the node l; m21 is thefuid mass of the node l, is is the fuid acceleration ofi dirction _at the node I, irmwn and U2m are respectively溜=号m "+一(un" -u“")-中国煤化ponents in j directionExtemal force exertingmi (Nki +im)u" +3-micmlg (4);YHCNMHGtion;kzmandC2nmarerespectively the tlund stittness and damping coefficientsNo.2Du Xiuli et al: Seisimic response analysis of arch dam-water-rock foundatin systemsofj direction at the node m on i direction at the node 1.Considering both computing effort and stability,where the subscriptk= x, y.Wang and Du (2002) developed an explicit computingalgorithm that is efective for the infnitesimal damping2.4 Artificial boundary and incident seismic wavesproblem and correspondingly, the displacement andvelocity can be expressed as:Many types of artificial boundaries have beenproposed to simulate the outgoing wave toward artificialboundaries, for example, the viscous damping boundary(Lysmer and Kuhlemeyer, 1969), the absorbing boundary(ABC) (Clayton and Engquist, 1977), and the multi-transmitting boundary (MTB) (Liao et al, 1984). Theyare all available for seismic response analysis in the timedomain and may be used to explicitly solve responsesi= 3u5 -4u% +盟(10)of the artificial boundaries. In addition, combining the201artificial boundary with the explicit FEM is an importantimprovement in solving large and complicated nonlinear2.3 Interface between reservoir-rock foundation andwave motion problems. Furthermore, the simulationaccuracy of the last two boundaries can be increased byreservoir-arch damincluding high-order terms. However, the MTB proposedAssuming axis z is normal and x and y are tangentialby Liao et al,, was formulated based on the generalto the interface, similarly, the displacement and velocitysingle-side wave motion, while the ABC proposed byClayton et al. was based on the wave motion equations.can be expressed asTherefore, the former is more widely used than the later.In this paper, the MTB is used and an artificial dampingin proportion to the stifiness (Liao, 1996) is introducedto eliminate the unstable phenomenon that happens(2m, +m)+[m,(use -u5 )+ 20m,i +when the MTB is applied.In order to separate the outgoing wave from thetotal wave motion field in the artificial boundary region,orkhnulnn + orsmo,mJ2m +m,)(11)the total motion field,, is divided into two parts: theinput field motion“ I of semi-infinite space undervertica! incident seismic waves, and the scatteringuh# =uht'(12)wave field motion u, induced by the structure base,local topography, inhomogeneous and semi-infinitespace surface, etc. The scattering wave field motion, u,at artificial boundary nodes is calculated by the muti-u! =1m + srila -0i mhmnunul"transmitting formula stated above. The input earthquakemotions at the artificial boundary regions are determined-- sm,;Cnan (2i" -u", +%")(13)through the analysis of the one -dimensional elastic wavepropagation in the rock foundation system. Due to theexistence of the structure, non-homnogeneous media,and local iregular topography in the interior region,the motions of the interface between the arch dam androck foundation are not uniform. Hence, the efect of(14)traveling waves in the interior region can be simulatedin the proposed model.(15)For waves propagating parallel to the boundary, themulti-transmitting formula (MTF) is no longer effective.In this respect, the revised MTF given by Liao and Li(1995) has the advantage over the MTF and is used in的“=-(h”-n )--mcml(n" -")-this paper. .O(16)3 Application to seismic response analysis of，mknmnum中国煤化工3.1TYHCNMHG(17)20tThe proposed procedure is used to calculate the286EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATIONVol.3seismic response of a double-curvature concreteThe finite element model of the dam-rock foundationarch dam known as the Xiaowan arch dam locatedsystem developed by Du et al. (2000) is adopted in thisin the upper reaches of the Lancang River in Yunnanpaper. There are 20,107 clements with 22,878 nodes.Province, China. It has a height of 292m, a crestThe sketch of the finite element discrete model withoutwidth of 12m, and bottom width of 73m. The designthe artificial boundary region is shown in Fig.2.Givenearthquake intensity for the arch dam is IX with athe dam-foundation system mesh, the reservoir meshhorizontal peak acccleration of 0.308g, while theis determined. The reservoir has two water levels: avertical peak acccleration is two-thirds of the horizontalnormal level of 1240m (water depth of 287m) and theacceleration. The canyon of the dam site is a V-shape :usually encountered lower level of 1181m (water depthtype, with a width of 935m at the dam crest elevationof 228m). For the normal water level, there are 971and an average sloping angle of 40-42 degrees at bothelements with 1201 nodes, including 334 nodes at theside hills. Artificially synthesized earthquake wavesinterface between the reservoir and the dam-foundationshown in Fig.I were used as the incident seismicin the reservoir mesh. For the lower water level, there arewaves in the free field, where both the stream-wise and583 elements with 756 nodes, including 249 nodes at thethe cross-stream components were S waves, and theinterface in the reservoir mesh.vertical component was a P wave.3.1.1 Material parameters3.3 Case studies and numerical resultsThe material parameters of concrete of the dam are:the dynamic module E.=27300MPa, the mass densityThree cases were studied: empty reservoir, reservoirp-=2400kg/m', and the Poisson's ratio v_=0.189. Thewith a low water level, and a reservoir with a normalrock foundation material is subdivided into 21 types (seeevel. The effect of the water compressibility wastable 1). Material damping is assumed to be Rayleighexplored by using both the compressible reservoir modeldamping, in which the two damping ratios associatedand the added mass model. In addition, the normal waterwith the frequencies of IHz and 15Hz are assumed tolevel and the lower water conditions were considered.be 5%.3.3.1 Seismic responses to stream-wise incident seismicwaves3.2 Finite element discretizationIn order to have a clear understanding of the water1.51.5,1.01.0|当1.00.5.5三0.50.0-0.5-1.00246810121524681012-1524681012Time (s)(a) Cross-stream(b) Stream(c) VerticalFig. 1 Atifially synthesized earthquake wavesTable 1 Material parameters of foundationTypeMass densityDynamic modulus Poisson's ratioMass ensityDynamic modulusPoisson'sNo.Pr (kg/m)E,(MPa)lo.ratio v,2630325000.220012221000.2600260000.25003715000.30001900 .455000.3000 .14299000.23001900182000.3100;5200050700160.265017286000.240018338000.210013260中国煤化工E700390000.35000YHCNMHG4001104000.2900No.2Du Xiuli et al: Seisimic response analysis of arch dam-water-rock foundation systems287compressibility effect on the vibration performance ofthe3.3.2 Seismic responses to three directional seismic .arch dam, scismic frequency responses of the arch damwavessubjected to stream-wise incident earthquake motionWhile the arch dam was simultaneously subjected to(see Fig.l(b)) are shown in Fig.3, where the verticalthree-dircctional (stream-wise, cross-stream and vertical)axis represents the ratio of the amplitude of accelerationearthquake waves, the absolute maximum responses ofresponses of the dam to the amplitude of incidentdisplacement and acceleration, the maximum tensile andacceleration. It is shown from Fig.3 that the existence ofcompressive stresses at the some key positions on thewater decreases the predominant frequencies of the dam-upstream surface are shown in Figs.4-7, in which the firstfoundation-reservoir system. The responses obtainedletters A, B, C and D appearing on the horizontal axis A I,from the compressible model evidently differ from thoseA2,A3, BI, B2, B3, CI, C2, C3, DI, D2, D3 representfrom the added mass model, and the difference betweenthe points A, B, c and D, while the second I, 2 and 3the two models varies with the position of the nodes. Forrepresent the cross-stream, stream-wise and verticalsome positions, the cormpressible model may yield largerdirections, respectively. Figures.8~Fig.10 illustrate theresponses than the added model, and for other positions,displacement, acceleration and stress time histories atthe opposite is true.the arch crowns on the upstream face of the dam.Point A - Crest crown on the upstreamPoint B一Right one-quarter crest on the upstreamPoint C一Left one-quarter crest on the upstreamPoint D一Middle height of the crown cantileveron the upstreamFig. 2 Finite clement discretization of the arch dam-foundation system5030- Empty----- Added mass25 t10 t- - .Compressible- - Compressible20 t亘3020卜0t0I4Frequency (Hz)(a) PointA(b) Point B0「20一EmptyEmpty---- Added mass" ---”Added mass- - -Compressible15一- -Compressibld20|5trer中国煤化工，(C) Point cYHCNMHGFig. 3 Frequency transfer functions of acceleration responses to upstream-wise incident waves288EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATIONFigure 4 shows that the absolute maximumlarger decreases (up to 20%) in the arch tensile stressdisplacement of the dam obtained by the added massat the arch crown crest is observed than for the lowermodel are almost always larger than those obtained bywater level. Particularly, for the middle and upperthe compressible water model, especially for the crowncantilever tensile stresses of the arch crown cantilever,cantilever in the upstream direction for normal waterthey are the key to the seismic safety of arch dam as thelevel. The same trend is shown in the lower level case,water compressibility causes a decrease of about 45%.although the differences become smaller. The sameTherefore, it can be concluded that the added mass modelconclusion can be drawn from accclerations given byoverestimates the stress responses at some key positionsFig. 5 for normal water level. However, the dferencesof the arch dam and it is inadequat for design. As is wellbetween the two models are very small. The accelerationknown, the concrete tensile strength is much less thanresponscs of crown cantilever at the upstream directionits compressive strength. Engineering experience showsare slightly different for lower water level. It is shownthat the cantilever tensile stresses at above one-third ofin Figs 6 and 7 that both the maximum tensile and thethe upper part of the arch crown cantilever are the keycompressive stresses obtained by the added mass modelto controlling the seismic safety design of arch dams. Inare larger than those obtained by the compressible waterfact, due to the existence of contraction joints, the actualmodel. On the contrary, in the upstream direction, thearch tensile stresses at the middle and upper part of themaximum compressive and tensile stresses obtainedarch crown would be less than those calculated in theby the compressible water model are larger than thosefigures.obtained by the added mass model except for thestresses at the arch crown crest for lower water level.4 ConclusionFor the middle and upper part of the arch crown, thearch stress and the cantilever stress obtained by theA 3D explicit finite element method combinedcompressible water model are obviously smaller thanwith the multi-transmitting boundary proposed bythose by the added mass model. For normal water level,Liao et al. was applied to the analysis of the seismic十Added mass0.12+ Added mass0.10世Compressible甘Compressible0.080.08 I0.060.040.020.00A142 A3 BIB2 B3 CIC2 C3DI D2 D3A1A2 A3 B1B2 B3 CIC2 C3Dl D2 D3Position(a) Normal water level(b) Lower water levelFig. 4 Absolute maximum displacement responses12.00E 12.0010.008.006.004.00一Added mass2.00# CompressibleCompressible言. 0.00A142 A3 BIB2 B3 C1C2 C3DI D2 D3中国煤化工C3DI D2 D3MHCNMHGvelFig. 5 Absolute maximum acceleration responsesNo.2Du Xiuli e1 al: Seisimic response analysis of arch dam-water-rock foundation systems289response of an arch dam. The method considered theof the arch dam, which are key to its seismic safety. The ;interaction between dam-water-rock foundation anddecrease in the cantilever tensile stress at the middle andthe effect of waves traveling and focused on the effectupper areas of the crown cantilever is about 45% and inof the compressibility of the reservoir. The numericalthe arch tensile stress at the arch crown, it is 20% greater.results show that the water compressibility significantlyIt is also shown that by considering the compressibilitydecreases the dynamic stress responses at some positionsof impounded water, the model provides a more合5.00+ Added mass4.50+4.50 ↓甘Compressible+ Compressible4.003.50员3.s0 '3.002.502.001.501.000.500.00A1A2 A3 B1B2 B3 C1C2 C3DI D2 D3A1A2 A3 BIB2 B3 C1C2 C3D1 D2 D3Position(a) Normal water level(b) Lower water levelFig. 6 Maximum tensile stress5.00 「4.50十Added massE 4.50Compressible4.00 ，2.00 .0.50 t0.00 lA1A2 A3 B1B2 B3 CIC2 C3DI D2 D3A1A2 A3 BIB2 B3 CIC2 C3DI D2 D3Fig. 7 Maximum compressive stress0.120.12「-Added mass一Added mass0.08“”"Compressiblc--Compressiblc0.04rMM-0.04-0.08-0.12中国煤化工68 10Time (s)MYHCNMHGrlevelFig. 8 Displacement time histries at the crest crown of upstream surface under stream-wise incident waves290EARTHQUAKE ENGINEERING ANDENGINEERING VIBRATIONVol.32↑「-Added mass 12「一Added mass--Compressible6810-12 (8 10Time (s)(a) Normnal water level(b) Lower water levelHig. 9 Acceleration time histories at the crest crown of upstream surface under stream-wise incident waves2.0一Added mass ]” -Added mass1..“Compressibld--Compressiblc1.00.0 nMHtW0.0另-0.5-1.0-1.5-2.00)02(a) Normal water levelFig. 10 Stress timc histories at the erest crown of upstream surface under vertical incident wavesappropriate assessment of the seismic safety of archBSSA. 67(6): 1529-1540.dams than the added mass modcl. The added mass modclClough RW (1960), “The Finite Element Method ino:vercstimates the dynamic stress responses at somc keyPlanc Stress Analysis," Pro. 2m1 ASME Conference onpositions lo the carhquakc-resistance of the arch dam.Electronic Computation, Pittsburgh, Pa.Previous investigations showed that the cenergy radiationDomingucz J and MacsoO(1993), "Earthquake Analysisof the foundation decrcases the arch tensile stress aboutof Arch Dams: Dam1-water- foundation Interaction,'40% at the upper part of the arch, and has no effect onJEng. Mech.. ASCE, 119(3): 513-530.the cantilever tensile stresscs. Tihe present study showsthat the cffeet of the water compressibility has the saneDu Xiuli, Chen Houqun and Hou shunzai (1996),“Three-importance as that of the cfcct of energy radiation of thedimensional Non-linear Seismic Analysis of Archfoundation. Morcover, water compressibility would beSystem," Earthquuke Engineering and Engineeringmore important, if the effect of closing-opening of theVibration, 16(3): 39-47. (in Chinese)contraction joints is included.Du Xiuli and Wang Jinting (2000), “An ExplicitIt should be noted that the above conclusions areDifference Formulation of Dynamic Responseobtained from a limited scismic response analysisCalculation of Elastic Structure with Damping,"of only the Xiuowan arch dam subjected to only oncEngineering Mechanics, 17(5): 37-43. (in Chinese)artificial seismic wave. Further studies tu verify theseDu Xiuli, Zhang Yanhong, Zhang Boyan (2000),findings ftor broader applications are needed."Nonlincar Sceismic Response Analysis of Arch Dam-Foundation Systems," Theories and Applications ofReferencesSir中国煤化工LY. 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