Ren-kun Wang*,Lin Chen,Chong Zhang

PowerChina Chengdu Engineering Corporation Limited,Chengdu 610072,China

Abstract The 285.5 m-high Xiluodu Arch Dam is located in a seismic region along the Jinsha River in China,where the horizontal components of peak ground accelerations for design and checking earthquakes have been estimated to be 0.355g and 0.423g,respectively(g is the gravitational acceleration).The ground motion parameters of design and checking earthquakes are defined by exceedance probabilities of 2%over 100 years and 1%over 100 years,respectively.The dam shape was first selected and optimized through static analysis of the basic load combinations,and then adjusted after taking into account the seismic loads.The dam should be operational during and after the design earthquake with or without minor repairs,and maintain local and global stabilities during an extreme earthquake.Both linear elastic dynamic analysis and nonlinear dynamic analysis considering radiation damping,contraction joints,and material nonlinearity were conducted to assess the stress in the arch dam.The dynamic analysis shows that the maximum dynamic compressive stresses are less than the allowable levels,while the area with tensile stress over the limit is less than 15%of the dam surface and the maximum contraction openings range from 10 mm to 25 mm.The arch dam has sufficient earthquake-resistance capacity and meets the safety requirements.Nevertheless,steel reinforcement has been provided at the dam toe and in the zones of high tensile stress on the dam surface out of extra precaution.

Keywords:Seismic design;Nonlinear dynamic analysis;Dam shape optimization;Seismic strengthening;Xiluodu arch dam

1.Introduction

The importance of seismic safety for an ultra-high arch dam cannot be over-emphasized,as any potential failure of such a dam under extreme loadings,including seismic hazards,may cause a major disaster,with devastating economic and social consequences.It is a truly big challenge for dam designers to model the real behaviors of ultra-high arch dams under operational and seismic loadings and with realistic consideration of all the important practical factors and nonlinear features involved(Zhang et al.,2014).

The dam static safety criteria in China include limits for the allowable stresses and the factors of safety against sliding.These are similar to those used by the United Sates Bureau of Reclamation(USBR,1977),the United States Army Corps of Engineers(USACE,1994),the Federal Energy Regulatory Commission(FERC,1999),and the Australian National Committee on Large Dams(ANCOLD,2013).In contrast,there are no universally applicable codes and regulations for the seismic design of concrete arch dams;they are instead evaluated on a case-by-case basis(Jonker and Espandar,2014).

The effects of fluid-structure interaction on the seismic behavior of dams have been extensively studied(Ghaemian and Ghobarah,1998;Fahjan et al.,2003;Bouaanani and Lu,2009;Aftabi Sani and Lot fi,2010;Kalateh and Attarnejad,2011).The widely accepted approach based on the generalized Westergaard theory(Westergaard,1933;Mays and Roehm,1991)was employed in this study,in which the reservoir water was considered an additional concentrated mass spreading out over the dam.

Various types of foundation models may be considered in seismic analysis of concrete dams.The foundation can be considered rigid,massless,or massed.Zhang et al.(2009)found that stresses,displacements,and contraction joint openings in arch dams are significantly reduced both in linear and nonlinear analyses using the viscoelastic boundary model rather than the massless foundation model.In the seismic design of the Xiluodu Arch Dam,foundation rocks and faults,as well as the dam,were considered in the dam-foundation reservoir system.The faults were simulated as contact elements with a certain thickness and a system damping ratio of 0.05.

Contraction joints play an important role in both the static and seismic analysis of concrete arch dams(Ahmadi et al.,2001;Wang et al.,2013).In the safety evaluation of the arch dam,23 contraction joints were simulated using a joint model that could simulate the contact between two adjacent nodes in a three-dimensional(3D)domain.

Saouma et al.(2011)carried out a time-history finite element analysis of rock-structure interaction considering the lateral energy dissipation and the interaction between the farfield and the numerical model itself.Hariri-Ardebili et al.(2013)studied dynamic stability of a coupled reservoir-damfoundation system assuming infinite elements and viscous boundaries at the far- field of the massed foundation model.In seismic safety evaluation of the Xiluodu Arch Dam,a 3D frequency domain and infinite boundary elements were used to simulate the arch dam foundation and transfer the dynamic frequency domain stiffness to time domain parameters.

Thermal loads have a significant effect on the structural behavior of thin concrete arch dams(Leger et al.,1993),and have been considered in dam seismic design through investigation of the differences between the dam closure temperature and concrete temperatures in typical seasons during operation.

This paper presents the seismic design of an ultra-high arch dam and illustrates the evaluation of various safety criteria,including the allowable stresses,damage control range,and maximum opening of contraction joints,through linear response spectrum analysis,linear time-history analysis,and nonlinear time-history analysis.These analyses took into account the dam-foundation-reservoir dynamic interaction,reservoir compressibility effect,non-uniform seismic input,dissipation of seismic energy in an infinite foundation,nonlinear response of dam contraction joint opening and closing,and individual or coupled effects of the factors mentioned above.As usual for seismic analysis of arch dams(Moradloo et al.,2008),geometric nonlinearity was not taken into consideration in this study.

2.Design scheme and approach

2.1.Project overview

Located in the Xiluodu Gorge on the Jinsha River that demarcates Leibo County in Sichuan Province and Yongshan County in Yunnan Province,the Xiluodu Power Station is a large project built primarily for electricity generation,as well as flood control,silt blocking,and navigation improvement.The height of 285.5 m makes the concrete double-curvature arch dam one of the world's highest dams.With an installed capacity of 13860 MW,the power station ranks just behind the Three Gorges Power Station in China.The power station lies in the center of the Leibo-Yongshan synclinal land mass,a relatively intact and steady tectonic unit with a stiff and integral base.After the great Wenchuan Earthquake of May 12,2008,according to the China Earthquake Administration,the horizontal earthquake component of the design peak ground accelerations(PGA)on the rock surface at the dam site corresponding to design(exceedance probability of 2%over 100 years)and checking(exceedance probability of 1%over 100 years)earthquakes were estimated to be 0.355gand 0.423g, respectively, wheregis the gravitational acceleration.

The dam foundation consists of basalt layers of multiple eruptions with an integral blocky structure.The basalt layers have high strength in general,except for the gently sloped dislocation zones between and within the layers.The concrete arch dam is constructed on rocks of different levels varying in elevation.The top 50-m zone of the foundation is mainly located on the rocks of level III2(Wang,2016),weakly weathered and unloaded with deformation modulus ranging from 6 to 8 GPa.The middle 130-m zone of the foundation is mainly composed of weakly weathered rocks of level III1,whose deformation modulus ranges from 11 to 12 GPa without unloading.The foundation at lower elevations and the riverbed is located on rocks of level III1 and,partly,level II with a deformation modulus higher than 16 GPa.

The symmetric narrow U-shaped dam base is mainly composed of multistage-erupted basalts,containing 14 flow rock layers with a total thickness of 550 m.Although interformational disturbed belts have developed,the rock is massive and has high strength as a whole,suitable for construction of a high dam with a large reservoir or an underground cavern.

With a height of 285.5 m and a crest length of 681.5 m(with a length-height ratio of 2.39),this arch dam supports a hydraulic thrust of 1.4×107t in total.25 orifices are used in four different layers in the dam body,including seven surface spillways,eight deep orifices,and ten openings for river diversion,to satisfy large flood discharge requirements.With most openings embedded,the Xiluodu ultra-high arch dam body is regarded as the most complicated hydraulic structure in the world at this time.

2.2.Loads and combinations

2.2.1.Dead load

Dead load includes the weight of both the concrete and appurtenant structures(gates,bridges,and outlet works).The weight has been loaded incrementally according to the construction procedures.

2.2.2.Temperature load

The temperature load is caused by the differences between the dam closure temperature and concrete temperatures during the operation of the dam.It includes summer and winter temperature loads and is shown in Table 1.

2.2.3.Hydrostatic loads

The hydrostatic loads include reservoir and tailwater loads as well as silt load.Reservoir and tailwater loads were considered at the normal, flood,and low(dead)water levels.The silt load was considered to correspond to the sediment depth.

2.2.4.Hydrodynamic loads

The dynamic interaction between the reservoir and the dam during an earthquake was considered a lumped mass as determined by the generalized Westgaard theory of added mass(the same method was used throughout the study unless otherwise specified)to model incompressible fluid.Subsequent calculations have the same value unless otherwise specified.

2.2.5.Earthquake load

The standard seismic response spectrum,specified in theCode for Seismic Design of Hydraulic Structures(DL 5073-2000),was used in design and checking earthquakes.As the dam site belongs to the Class I type,the characteristic period is 0.2 s,and the maximum response spectrum βmaxis 2.5.The normalized acceleration time histories used in the dynamic analysis of the Xiluodu Arch Dam are shown in Fig.1.The maximum peak value of the vertical components of ground accelerations for the earthquake was 2/3 of the horizontal components.

2.2.6.Load combinations

In this study,load combinations included two groups:one combined all the static loads and the other took into account the effects of earthquakes.

Fig.1.Normalized acceleration time histories of seismic wave.

The static load combinations consisted of(1)case 0-1:reservoir and tailwater loads at the normal water level,dam gravity,silt,and temperature load;and(2)case 0-2:reservoir and tailwater loads at the dead water level,dam gravity,silt,and temperature load.

The dynamic load combinations consisted of(1)case 1-1:case 0-1,together with the design earthquake load;(2)case 1-2:case 0-2,together with the design earthquake load;(3)case 2-1:case 0-1,together with the checking earthquake load;and(4)case 2-2:case 0-2,together with the checking earthquake load.

Table 1 Temperature loads in linear elastic analysis.

3.Static analysis of arch dam

3.1.Dam shape and concrete zoning

As controlled by the allowable compressive stress of 9 MPa and tensile stresses of 1.2 MPa(upstream surface)and 1.5 MPa(downstream surface)provided by the trial-load method in accordance with the code,and constrained by the safety factors(3.5 with the shear friction equation and 1.3 with the pure-friction equation)against sliding of the base,the shape of the arch dam is a parabolic double-curvature dam with optimized design and safety proofing.The characteristic parameters for the shape are given in Table 2.The dam concrete is designed as described in Table 3 and Fig.2.The dynamic modulus of concrete is 1.3 times the static modulus of elasticity.As there are no standard control standards in the code,the FEM stress results are comprehensively evaluated corresponding to the allowable stresses for the trial-load method.

3.2.Static analysis results

The performance characteristics and overload capacity of the arch dam subjected to the basic load combinations were analyzed using the trial-load method,the linear and nonlinear finite element method(FEM),and geo-mechanical model tests.

Table 2 Characteristic parameters of shape of Xiluodu Arch Dam.

According to the results of the trial-load method,most of the area of the dam body is in compression with a small tensile area distributed close to the foundation.The distributions of stress and displacement of dam surfaces are subject to general laws,and the maximum principal compressive stress is 8.96 MPa and tensile stress is 1.09 MPa.

The linear FEM results show dam stress distribution similar to that obtained from the trial-load method and both methods demonstrate compressive states on dam surfaces when FEM takes into consideration rock mass quality classifications,faults,dam toe reinforcements,and uses thin-layer elements with Gaussian integral stress around the foundation(an area within 1/50 dam height)to reduce the impacts of stress concentration at dam body edges.On the upstream surface the maximum principal compressive stress of around 6 MPa appears in the middle area,and on the downstream surface this value turns into 17 MPa near the dam heel.The maximum principal tensile stress on the upstream surface appears near the arch abutment at low elevation,and is 3 MPa,corresponding to the normal water level,whereas the tensile stress on the downstream surface appears at the high-elevation arch abutment,and is about 2 MPa,corresponding to the dead water level.

The sliding stability analysis of the dam base contains shear friction calculations for large and stair-stepped blocks,as well as small ones.The results show that safety factors for all assumed blocks satisfy the design requirement except that the shear friction safety factor for the block near the bottom faults is lower than 3.5.

The 3D nonlinear analysis of the arch dam was performed with the FEM program TFINE developed by Tsinghua University.Like the linear analysis,the nonlinear analysis contains dam toe reinforcement,concrete replacement at the dam foundation,and weak rocks like faults.Under the basic load combination corresponding to the normal water level,the distributions of stress and displacement agree with results of the linear analysis,and the dam is in a uniform compressive state.The maximum principal compressive stress,which is about 15-16 MPa,appears at the arch abutment on the downstream surface.The maximum principal tensile stress is 1.13 MPa,appearing at the right side of middle elevation of the arch crown on the downstream surface.

Table 3 Mechanical and thermal parameters of dam concrete used in Xiluodu Arch Dam.

Fig.2.Concrete zones on downstream surface(units:m).

The geo-mechanical model test(Zhou et al.,2008)of the arch dam was performed to simulate the rock categories and faults in excavation and to explore the dam's overload capacity by increasing the density of water under the condition of normal water level.The overload capacity could be evaluated by the following three factors:K1,K2,andK3.K1is the crackinitiation overload safety factor.The overload ofK1P0,in whichP0is the basic load combination corresponding to the normal water level,corresponds to the moment when the first crack appears in a physical model test or to the moment when the range of yield zones reaches 1/6 of the arch dam thickness in the nonlinear FEM analysis.K2is the quasi-elastic overload safety factor.As the overload increases further,the crack propagates in the physical model test,while the yield zones continue to expand in the nonlinear FEM simulation.The structure is stable overall,and the dam displacements remain in linear correlation with the load until the overload exceedsK2P0.K3is the ultimate overload safety factor.The overload ofK3P0corresponds to the occurrence of a large number of cracks throughout most of the dam or foundation,and the dam becomes unstable overall in the physical model test.In the nonlinear FEM calculation,K3P0coincides with the time when the nonlinear solution does not converge.

The results show that the safety factorK1of the Xiluodu Arch Dam ranges from 2.0 to 2.5,while the nonlinear deformation safety factorK2is between 5.0 and 6.0.

4.Dynamic design scheme and approach

Considering the fact that the scale of the Xiluodu Arch Dam exceeds the scope of the current design code(DL 5073-2000)and that the earthquake safety presents a challenge due to the high seismicity at the site,some key technological problems in the design procedure of the Xiluodu project arise:(1)the dynamic performance properties of the dam,(2)reliable assessment of the seismic safety capacity of the dam,and(3)the design of earthquake-resistance measures in order to ensure the dam's long-term safe operation.

For the seismic design of the Xiluodu Arch Dam,based on conventional static and dynamic analyses,through full use of sophisticated numerical analysis and physical experimental methods,a comprehensive design approach for the dam has been proposed and implemented for the first time.Furthermore,through comparison with other high arch dams,a systematic evaluation method and design criteria for seismic strengthening for ultra-high arch dams have been established.Using these approaches,the Xiluodu Arch Dam aseismic measures have been designed and completed to further ensure the seismic safety of the dam,solving the key technical problems of high dam seismic design.

Through cooperation with the China Institute of Water Resources and Hydropower Research,Tsinghua University,and Dalian University of Technology,the following approaches have been developed:

(1)The dam shape can be designed with static mechanics and checked with dynamic mechanics(Wang,2016).The structural design is first carried out by matching the fundamental load combination,then checking the earthquake resistance capacity in the case of an earthquake,and finally adjusting the shape of the dam or adding strengthening measures.

(2)The seismic design can be implemented through analysis with multiple methods,verification with multiple approaches,and evaluation on multiple scales,as well as the project analogy analysis,to help comprehensively analyze the mechanical characteristics of the dam,and properly evaluate the earthquake-resistance capacity of the dam.This includes research on code methods and modern simulation approaches.Code methods mainly refer to the rigid limit equilibrium method for dynamic stability analysis of the dam base,the dynamic trial-load method,and the linear elastic dynamic FEM.Modern simulation approaches mainly include nonlinear dynamic FEM considering radiation damping,contraction opening,material nonlinearity,and large-scale shaking table overload testing.

(3)Feasible earthquake-strengthening measures should be presented to ensure that the dam will withstand the design earthquake and not fall in an extreme event.This means that cracks may appear when the dam encounters the design earthquake but will not affect the dam's normal function of water retaining and can be fixed through measures such as grouting.When the dam suffers an extreme earthquake such as the checking earthquake,it can still safely hold the water even with apparent cracks(Chen,2012).In the latter case,the numerical calculation should be kept stable and the yield area should not run through the dam body(Zhang et al.,2009,2014).

Combined with the project analogy analysis,the earthquake-resistance capacity of the dam and potential weak parts can be comprehensively evaluated and proper strengthening measures can be taken.

5.Dynamic analysis for arch dam in codes

In accordance with DL 5073-2000,the controlling allowable stress of the Xiluodu Arch Dam based on the dynamic trial-load method is given in Table 4.The dynamic stability safety factorKcdobtained through the limit equilibrium method using the shear-friction formula is larger than 1.31.

Table 4 Dynamic stress control criterion for Xiluodu Arch Dam obtained through trialload method.

5.1.Dynamic analysis by trial-load method

Responses of the dam structure under four dynamic load combinations were analyzed with the dynamic trial-load method,showing that the dynamic stress distributions of the dam body are similar to one another except for the difference in magnitudes,corresponding to different water levels.

Load case 1-1 and case 2-1 are the controlling cases for the maximum principal compressive stress.The shaking caused the zones near the crest and arch abutment to be with the maximum principal compressive stress under earthquake conditions.The maximum compressive stresses on the upstream surface,which appear at the top arch,are 11.23 MPa and 12.89 MPa for design and checking earthquakes,respectively.On the downstream surface the values are 11.86 MPa and 12.38 MPa,respectively.During both design and checking earthquakes the stresses satisfy the criterion of dynamic allowable compressive stress of concrete.The maximum radial dynamic displacements of the dam are 10.59 cm and 12.81 cm for design and checking earthquakes,respectively.

Load case 1-2 and case 2-2 are the controlling cases for the maximum principal tensile stress.The shaking causes the zones near the crest and arch abutment at middle elevations to be in tension,with a tensile stress as high as 3.0-7.0 MPa.The extreme values at the crest are 6.92 MPa and 8.37 MPa for design and checking earthquakes,respectively.Areas where the tensile stresses are higher than controlling criterions account for 10%and 17%of the whole dam surface during design and checking earthquakes,respectively.

Contours of maximum principal dynamic stresses due to the design earthquake computed by the trial-load method are shown in Fig.3.Table 5 shows the locations of extreme principal stress values in different cases.

Fig.3.Contours of principal stresses during design earthquake obtained through dynamic trial-load method.

Table 5 Maximum principal stresses of dam surfaces obtained through dynamic trial-load method.

5.2.Linear elastic finite element analysis of arch dam

The arch dam analysis program(ADAP,Clough et al.,1973)was used to analyze the linear elastic dynamic responses of the Xiluodu Arch Dam.Results obtained from dynamic FEM are consistent with those from the trial-load method.As affected by the stress concentration,extreme stress values close to the dam edges obtained through FEM are higher than those obtained through the trial-load method,while those in the other areas,especially in the central area,are significantly lower.

During design and checking earthquakes,most compressive stresses of the dam can satisfy the design requirements except for some areas slightly over the limit near the base.The areas over the limit of tensile stress are mainly distributed close to the dam heel and the central part of the dam on the downstream surface,accounting for 10%and 15%of the total dam surface area for design and checking earthquakes,respectively.The maximum radial dynamic displacement of the dam is also slightly reduced,as compared with that obtained through the trial-load method.

Above all,results of the FEM and trial-load method show that the dam is mainly in a pressure-arch state with the largest compressive stress appearing near the dam toe and foundation areas.On the other hand,tensile stresses are relatively high in the upper crown cantilever and arch abutments on the upstream surface,and the central part of the dam on the downstream surface.

Table 6 gives the maximum values of principal stresses in different load cases computed by linear elastic FEM and Fig.4 shows the principal stress contours during the design earthquake.

5.3.Rigid limit equilibrium method for dynamic stability analysis of dam base

The dynamic stability analysis of dam abutments was performed using the shear rupture formula considering different combination coefficients of the earthquake and the working condition of the grout curtain.The dynamic anti-slide safety factors of controlling blocks at two abutments under conditions of design and checking earthquakes are given in Table 7.All the factors are larger than 1.31,satisfying the requirements of the code.

Table 6 Maximum principal stresses for different load cases obtained through linear elastic FEM.

Fig.4.Contours of principal stresses during design earthquake obtained through linear elastic FEM.

6.Nonlinear dynamic analysis of arch dam

6.1.Methods and model

The nonlinear wave propagation analysis program(NMPAP),developed by the China Institute of Water Resources and Hydropower Research,divides the computation area into the nonlinear near- field domain and the linear artificial boundary domain by decoupling the near- field wave propagation equations(Chen,2011).This program considers the dam,foundation rock,and reservoir water as an integrated wave propagation model,with boundary constraints on contact interfaces,and solves the equations in the time domain using the explicit FEM.Thus,the numerical stability depends on the minimum element size,wave speed,and time step in the calculation.This program can also simulate the sliding surface of blocks at abutments with the Mohr-Coulomb constitutive relations in a similar way of simulating joints in the dam body,using binodal dynamic contact boundaries.Using the generalized Westergaard theory,the reservoir water-added mass is applied to the corresponding nodes after diagonalization.

The nonlinear FEM model of the Xiluodu Arch Dam contains the dam body and foundation rocks,as well as 23 joints and faults.The faults are simulated with contact elements with a certain thickness.The system damping ratio is set to 0.05.The whole model ranges are 1200 m,1600 m,and 900 m along the longitudinal,transversal,and vertical directions,respectively,and the model contains 27727 nodes and 80000 degrees of freedom.Fig.5 and Fig.6 illustrate the mesh and joints,respectively.

By taking into account the topographic and geological conditions at the dam site and the effects of free- field seismic inputs,joint opening,and radiation damping,the nonlinear dynamic FEM sees the dam-foundation as a whole subject,in order to study the dynamic strength safety of sliding planes in the foundation and the dam concrete,and the dynamic anti-sliding safety of the controlling sliding blocks,and to analyze the possible failure mechanisms and the earthquake-resistance capacity of the highly nonlinear dam-foundation system.

Fig.5.Finite element mesh of dam-foundation system.

Fig.6.Contraction joints simulated in FEM(numbers indicate different joints).

6.2.Results

Load case 1-2 and case 2-2 control the joint opening of the dam,in which the largest openings reach 29.94 mm and 28.75 mm,respectively,near the top of the crown cantilever,almost twice as much as in case 1-1 and case 2-1.Due to the consideration of various nonlinear factors,the displacement of the dam body from nonlinear dynamic analysis is less than those from the trial-load method and linear FEM.

Table 7 Dynamic anti-slide safety factors of controlling blocks at abutments.

The principal compressive stress distribution from nonlinear analysis is similar to the linear elastic finite element analysis result,accompanied by a stress reduction ranging from 15%to 30%.In case 1-1 and case 2-1,the maximum compressive stresses are 12.23 MPa and 13.24 MPa,respectively,which means that the dynamic compressive stresses fully satisfy the controlling criterion.

The difference between tensile stresses of the nonlinear and linear models is clear when different dam locations and different working conditions are compared.When the damisshaken by an earthquake at the normal water level,the tensile stress magnitude and over-the-limit range near the upstream dam toe from nonlinear analysis both decrease as compared to linear analysis results,whereas the tensile stress at the arch abutment on the downstream surface increases slightly.When the dam is shaken by an earthquake at the dead water level,the tensile stress magnitude and over-the-limit range at the middle-upper part of the upstream surface from nonlinear analysis significantly increase as compared to the linear elastic results.As for the stresses on the downstream surface,the magnitude increases while the over-the-limit range decreases due to the effects of faults in the middle-lower arch abutment area.

The maximum values of comprehensive static and dynamic tensile stresses during design and checking earthquakes are 9.71 MPa and 11.43 MPa,respectively,occurring at the same position where weak rock of shear-belts can be found at high elevations nearby.The local slip of the belt during an earthquake leads to the increase of the stress of the adjacent local dam area.

As the computations indicated,there are displacements along the simulated dislocation zones between and within the layers during and after the earthquake,and the displacements were larger in the load cases of high water level(case 1-1 and case 2-1).The maximum values occurred in dislocation zones,23.35 mm during the design earthquake and 27.84 mm during the checking earthquake.Moreover,the residual displacements after the earthquake took place in the same area were 17.56 mm(design earthquake)and 18.39 mm(checking earthquake).Although the displacements or residual slip occurred in dislocation zones,they did not have significant effects on the static dynamic response and safety of the dam.

Table 8 provides the maximum stress values,locations,and joint opening magnitudes in different cases from nonlinear dynamic analysis.Fig.7 shows the contours of principal stresses during the design earthquake.

7.Dynamic analysis for arch dam-foundation interaction in time domain

7.1.Method and model

The dynamic analysis model(Zhang et al.,1995;Wang et al.,2013)of dam-foundation interaction in time domain was introduced by Tsinghua University.This model uses a 3D frequency domain or an infinite boundary element to simulate the arch dam-foundation system and transfers the dynamic frequency domain stiffness to time domain parameters,through which the finite element dam model is coupled and the integrated model of finite elements,boundary elements(Shi et al.,2013),and infinite boundary elements of the arch dam-foundation in the time domain is formed.

Table 8 Maximum principal stresses of dam body and contraction joint opening.

Fig.7.Contours of maximum principal stresses during design earthquake.

7.2.Analysis results

The time domain arch dam-foundation interaction model was used to calculate the dynamic responses of the dam body in cases 1-1 and 1-2.The 27 joints,foundation radiation damping ratio of 0.05,dynamic water pressure,and uniform free- field seismic input were applied in this model with the assumption of equivalent homogenous rock material of bulk density of 2700 kg/m3.The standard response spectrum in the code was used as the seismic input load.Table 9 demonstrates the dynamic response results for different water levels during the design earthquake.

The stress peak values,including tensile and compressive values in both the arch and cantilever directions,all satisfy the controlling criterion for the design earthquake.The maximum tensile stress is about 3 MPa at the position of 1/4 of the arch at the middle elevation on the downstream surface along the cantilever direction(Fig.8(a));the maximum compressive stress is about 10 MPa at the low elevation on the downstream surface(Fig.8(b)),also along the cantilever direction.The maximum radial displacement is 8.54 cm(Fig.9(a))and the joint opening is about 9.36 mm(Fig.9(b)).

After taking into account the influence of foundation radiation damping and contraction joint opening,the stress level of the dam,especially the tensile stress on the upstream surface, decreased,most of which were released by using the time domain dynamic coupling method.The high tensile stress zone is mainly located on the left and right sides of 1/4 arch portions at middle elevation of the dam on the downstream surface,while the tensile stress zone determined through the linear FEM is mainly concentrated at the dam crest and in the low dam portions near the foundation.

Table 9 Maximum values of dynamic responses of arch dam during design earthquake.

Fig.8.Non-concurrent envelope of stress along cantilever direction in load case 1-1.

Fig.9.Dynamic response of arch dam during design earthquake.

8.Large-scale shaking table tests

The China Institute of Water Resources and Hydropower Research carried out a dynamic model test using a 5 m×5 m shaking table,which included simulations of radiation damping using artificial transmitting boundaries,geological structures,and topography,as well as sliding blocks at the dam site and vicinal areas,interactions between the dam body and reservoir water,and seven joints in the dam body,with a model scale of 1:300 as shown in Fig.10.Specific research subjects included dynamic responses to different earthquake over loadings,the earthquake level that leads to initial cracks,the development of damage,and the stability of blocks.Earthquake-resistance safety and the limit of earthquake-resistance capacity(Wang et al.,2014)can be deeply examined by precisely controlling the earthquake input to the shaking table.

Table 10 lists the instances of visible damage to the dam after vibration in various working conditions.All the instances of damage or cracks are shown in Fig.11,except some damages whose initial time of occurrence cannot be identified.

The results show that,during the design earthquake,no apparent damage was evident and strains observed at all points were in the elastic range;under the acceleration of 1.68 times the design earthquake(1.4 times the checking earthquake),initial cracks appeared along the cantilever direction between the right two joints,13 cm to the model dam crest on the downstream surface.As the earthquake level was further increased,cracks extended in the dam body.Under the condition of 4.88 times the design earthquake(4.1 times the checking earthquake),multiple instances of damage were observed in the dam body.Meanwhile,small residual displacements were detected from blocks at two abutments,which could still remain stable without wholly sliding when the shaking stopped.From all of the above it can be inferred that the dam foundation interface and central zone of the downstream surface are critical areas for the earthquake resistance capacity of the dam.

Fig.10.Schematic view of model adopted for shaking table tests.

Table 10 Dam damage in shaking table test.

Model testing and numerical analysis are two important methods of seismic safety research for arch dams.Due to differences in simulation conditions,material values,structural simulations,etc.,and a shortage of physical test results,comparison between the results of model testing and numerical analysis is difficult.However,the overall pattern of dam dynamic behavior obtained from the two methods is consistent,indicating that the Xiluodu Arch Dam has a high degree of seismic resistance.The comparatively weak parts of the dam are the foundation confinement zones and the middle high-elevation zones on the downstream surface.

9.Earthquake-resistance measures

From the static and dynamic analyses described above,it can be concluded that the dam base is stable and the structural concrete has a safety margin in dynamic compression and static tension except for some areas over the limit of tensile stress in earthquakes.Enveloping all the areas of tensile stresses with multiple methods,zones including the foundation vicinity of dam surface,and central and upper parts of the dam body,especially the surface and central holes,may face the risk of cracking.Thus,the following earthquake-resistance measures were implemented:

(1)Concrete zoning was rationally set according to the magnitudes and distribution patterns of compressive and tensile stresses.Specifically,C18040 concrete was used in the vicinity of the foundation and central areas with openings,C18035 concrete was used in the middle-lower dam body and outlet works areas,and C18030 concrete was used in the middle-upper dam body areas near the two abutments.

Fig.11.Locations of dam damage in shaking table test.

(2)Fillets,with the thickness ranging from 3 to 6 m and height ranging from 15 to 25 m,were added at the dam toe on the downstream surface,in order to reduce the compressive stress concentration at the dam toe area and enhance the earthquake-resistance capacity of the structure.Anchor cable was used near the dam toe area,especially in places where faults and joint fissures developed,to reinforce the stability of the foundation(Lin et al.,2009).

(3)Reinforcing mesh was paved on the dam surface at high tensile stress areas(Pan et al.,2007;Long et al.,2011)to control the development of cracking of concrete.The surface reinforcement zones can be seen in Fig.12,which is composed of staggered arch reinforcements using HRB335 with the diameter of 28 mm and interval of 500 mm(avoiding transversal joints) and cantilever reinforcements using HRB335 with the diameter of 32 mm and interval of 300 mm(HRB means hot rolled ribbed bars)in Zone II and zone III,and arch reinforcements with the diameter of 32 mm and interval of 300 mm(avoiding transversal joints)and cantilever reinforcements with the diameter of 36 mm and interval of 250 mm in Zone I.

(4)Shear keys along contraction joints,even with joint openings of several centimeters due to the earthquake,can ensure sufficient shear strength.In addition,copper and rubber water-stops were utilized to prevent joints from breaking during an earthquake(Fig.13).

Fig.12.Distribution of reinforcement on dam surfaces.

Fig.13.Shear keys and upstream water-stop(units:m).

(5)The reinforcement at outlet works and gate piers is strengthened.Steel liners are used at openings and spillways to prevent hydraulic fractures.

10.Conclusions

On the basis of the seismic design of the Xiluodu Arch Dam,with full use of the modern numerical and experimental analysis methods,a comprehensive scheme of design and various approaches consisting of analysis with multiple methods,verification with multiple techniques and evaluation on multiple scales,as well as dam staging dynamic safety criteria for withstanding the design earthquake and avoiding failure in extreme situations,have been put forward for seismic design of an ultra-high arch dam.Using these means,the mechanical characteristics of a dam during earthquakes and the earthquake-resistance capacity of the Xiluodu Arch Dam have been systematically analyzed.Reasonable and feasible seismic measures have been proposed and implemented.The main conclusions of this study are as follows:

(1)The dynamic stability of blocks at two abutments of the arch dam has been ensured.

(2)As both the dynamic trial-load method and linear elastic FEM indicated,the compressive stresses during design and checking earthquakes can satisfy the design requirements,and only in areas at the high arch crown and middle-elevation arch abutments are the tensile stresses over the limit.These areas take up less than 15%of the dam surface.

(3)Taking the effects of radiation damping and contraction joint opening into consideration,the nonlinear analysis shows that the area in tension and tensile stress both decrease,with the maximum joint opening of 10-25 mm.According to large-scale dynamic shaking table tests,the conclusion can be drawn that the Xiluodu Arch Dam provides adequate earthquake-resistance capacity.

(4)Based on the comprehensive seismic safety evaluation,systematic aseismic measures,including steel reinforcement at the dam toe and in the zones of high tensile stresses at the dam surface,as well as shear keys,have been put forward and employed.