Ji-he Zhng,Jin Wng,*,Li-sh Chi

aCollege ofWater Conservancy and Hydropower Engineering,Hohai University,Nanjing 210098,China

bLixi Barrage Irrigation Management Office of Guangzhou City,Guangzhou 510890,China

Factors influencing hysteresis characteristics of concrete dam deformation

Jia-he Zhanga,Jian Wanga,*,Li-sha Chaib

aCollege ofWater Conservancy and Hydropower Engineering,Hohai University,Nanjing 210098,China

bLixi Barrage Irrigation Management Office of Guangzhou City,Guangzhou 510890,China

Abstract

Thermal deformation of a concrete dam changes periodically,and its variation lagsbehind the air temperature variation.The lag,known as the hysteresis time,isgenerally attributed to the low velocity of heatconduction in concrete,but thisexplanation isnotentirely sufficient.In this paper,analytical solutions of displacementhysteresis time for a cantilever beam and an arch ring are derived.The influence of different factors on the displacement hysteresis timewas examined.A finite elementmodelwas used to verify the reliability of the theoretical analytical solutions.The follow ing conclusions are reached:(1)the hysteresis time of themean temperature is longer than that of the linearly distributed temperature difference；(2)the dam type has a large impact on the displacement hysteresis time,and the hysteresis time of the horizontal displacementof an arch dam is longer than thatof agravity dam；(3)the reservoirwater temperature variation lagsbehind of theair temperature variation,which intensifies the differences in the horizontal displacementhysteresis time between the gravity dam and the arch dam；(4)w ith a decrease in elevation,the horizontal displacement hysteresis time of a gravity dam tends to increase,whereas the horizontal displacement hysteresis time of an arch dam is likely to increase initially,and then decrease；and(5)along thew idth of the dam,the horizontal displacement hysteresis time of a gravity dam decreasesasawhole,while the horizontaldisplacementhysteresis time ofan arch dam is shorternear the center and longer near dam surfaces.

?2017 Hohai University.Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license(http:// creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords:Concrete dam；Displacement；Hysteresis；Temperature；Analytical solution

1.Introduction

The measurement and interpretation of dam deformation data isan importantpartof dam safety evaluation and hasbeen studied by various researchers(Li,1992；Wu et al.,2007a, 2007b；Zhang et al.,2008；Jesung et al.,2009；Gu et al., 2011).For a concrete dam,the total deformation is generally divided into various components,including a hydrostatic component,a thermal component,and a time-dependent component(or irreversible component)(Bonaldiet al.,1977；Fanelli and Giuseppetti,1982；Wu,2003；Gu and Wu,2006；Xu et al.,2014).The thermal component,which is the topic of this paper,is the deformation caused by the change of the outside temperature.

In the last decade,a few studies have explored the hysteresis characteristics of concrete dam deformation.Xu et al. (2012)pointed out that the variation of the thermal displacement component lags behind the variation of the air temperature,and this lag is known as the hysteresis time. According to previous interpretations,the reason for the hysteresis time is the low heat conduction velocity in concrete (Troxell and Davis,1956；Neville,1963；Li et al.,1996). However,in practice we find that the hysteresis time of the horizontal displacement of different damsmay differ significantly even if the dam w idth and thermal conductivities are sim ilar.The displacement hysteresis time of a narrow arch dam is often longer than thatof aw ide gravity dam.The low heatconduction velocity in concretealone cannot fully explainthis phenomenon.Zhang et al.(2015)derived analytical solutions of the hysteresis time of thermal deformation for a cantileverbeam and an arch ring and pointed out that the dam type has a large impacton the hysteresis time of dam thermal deformation.However,their study was only based on theoretical analysis and few factorswere considered.Therefore,it is necessary to study the factors influencing the displacement hysteresis time from additional perspectives.

This study mainly focused on the effects of dam type, reservoir water temperature,and spatial position on the displacementhysteresis time as determ ined w ith the analytical method and numericalsimulation.Some case studieshavealso been examined to further verify the validity of conclusions.

2.Quasi-stablem ean tem perature and equivalent linearly distributed tem perature difference in concrete dam

Since the heat conduction in a concrete dam is mainly determ ined by the temperature at the upstream and downstream surfaces of the dam,which can be approximated by one-dimensional heat conduction along the thickness of an infinite slab,this principle can be applied in the follow ing analysis.Assum ing that the thickness of the infinite slab isl, and the surface temperature changes according to the cosine function w ith different amplitudes and phases,the heat conduction equation can be expressed as

The boundary conditions are as follows:

whereTis the temperature,αis the thermal diffusivity,τis time,A1is the amplitude of the boundary temperature atx=l,A2is the amplitude of the boundary temperature atx=0,ωis the circular frequency of temperature change,andεis the hysteresis time of the boundary temperature atx=0 relative to the boundary temperature atx=l.

In thermal deformation and stress analysis,temperature is typically divided into three parts:the mean temperatureTm, linearly distributed temperature differenceTd,and nonlinearly distributed temperature differenceTn.For a free slab whose deformation is unconstrained,TmandTdw ill produce tensile deformation and rotation,respectively,whileTnw ill only cause thermal stress and have no effect on the deformation. Therefore,onlyTmandTdneed to be considered in the hysteresis analysis for concrete dams.They can be computed as follows(Zhu,1999):

According to Eqs.(2)and(3),θmandθdrepresent the hysteresis times of the mean temperatureTmand linearly distributed temperature differenceTd,respectively.For the sake of simplicity,we fi rstassume thatεin Eqs.(2)and(3)is 0(the effect ofεw ill be analyzed in section 4.2),and then calculate the relationship between the hysteresis timesθmand θdand the slab thicknesslfor different thermal diffusivitiesα, as shown in Fig.1,where the thermal diffusivity ranges from 0.05 to 0.15 m2/d,covering the common range of concrete thermal diffusivity.The circular frequency isω=2π/365. That is,the cycle of externalambient temperature is365 days. The follow ing can be seen in Fig.1:

(1)θm＞θdwhen the slab thickness is the same,indicating that the hysteresis time of the horizontal displacement of an arch dam is longer than that of a gravity dam,which w ill be discussed in detail below.

(2)The hysteresis timeθmof the mean temperature increases rapidly w ith the slab thickness,and then decreases slightly,and remains nearly unchanged at a value of 0.25π/ω when the slab thickness is greater than 15 m.

Fig.1.Relationship between slab thickness and hysteresis times of average temperature and linearly distributed temperature difference.

(3)The hysteresis timeθdof the linearly distributed temperature difference increaseswith the slab thicknessl.

3.Analytical solutions of disp lacem ent hysteresis time

For the sake of simplicity,we consider the horizontal displacement of a gravity dam and an arch dam,respectively, so as to derive the analytical solutions of the displacement hysteresis time and analyze their characteristics.

3.1.Analyticalsolutionofdisplacementhysteresistime for gravity dam

From a structural perspective,a gravity dam can be treated as a cantilever beam.Thus,a triangular cantilever beam is considered,w ith a heightofHand a bottom w idth ofB,and its deformation characteristics are calculated when one side is subjected to variable temperature,as shown in Fig.2.

According to the principal of virtual work,the total displacement of the gravity dam can be obtained by integrating the deformation for each m icro-segment:

whereuis the totaldisplacement；κ,ε,andγare the curvature, axial strain,and shear strain of the micro-segment,respectively；,andare the bendingmoment,axial force,and shear force of the m icro-segment caused by the virtual unit load,respectively；and dsis the area element.

As for a pointPw ith a height ofh,the horizontal displacementuxcaused by temperature load is calculated as follows:

wherenis the number of partitions,andTdi,zi,Li,andΔziare the linearly distributed temperature difference,height,dam w idth,andm icro-segmentheightof theith layer,respectively.

According to Eq.(12),the horizontal displacement of a gravity dam is proportional to the linearly distributed temperature difference.Substituting Eq.(3)into Eq.(12),and after trigonometric transformation(product to sum formula), Eq.(12)can be rew ritten as

Fig.2.Calculation of gravity dam deformation.

wherekdi,θdi,andεiare thekd,θd,andεvalues of theith micro-segment.

After the trigonometric transformation(sum to product formula)of Eq.(13),we can obtain

Eq.(18)provides the solution of the horizontal displacement hysteresis timeat pointP.

3.2.Analyticalsolutionofdisplacementhysteresistime for an arch dam

An arch dam is a spatial,statically indeterm inate structure. Its structural response can be divided into an arch effectand a beam effect,in which the arch effectplays the dom inant role. Accordingly,we take amonolayer arch ring to perform dam deformation analysis according to a pure archmethod.

According to Eq.(11)and the symmetricity conditions underwhich the rotation angle and the tangentialdisplacement at the arch crown are both zero,the axial forceNand the bending momentMat the arch crown caused by the temperature loadsTmandTdcan be obtained,and the horizontal displacement(in the radial direction)caused byTmandTdcan be computed.The horizontal displacement is

whereEandGare the elasticmodulus and the shearmodulus of the material,respectively；AandIare the area and the moment of inertia of the cross-section,respectively；φis half of the centralangleof the arch ring；Ris the radiusof the arch ring；k0is a dimensionless parameter related to the crosssection shape,which is 1.2 for a rectangular cross-section.

According to Eqs.(19)through(21),the horizontal displacement of the arch ring is proportional to the mean temperatureTm,but is independent of the linearly distributed temperature differenceTd.The shear displacement has the same characteristics.The horizontal displacement hysteresis time of the arch dam is identical to the hysteresis time of the mean temperatureTm,namelyθmin Eq.(2).

4.Factors influencing hysteresis time of dam thermal deformation

4.1.Influenceofdam typeon displacementhysteresistime

The displacement hysteresis time of an arch dam is generally longer than that of a gravity dam due to the difference in structural response.The horizontal displacement hysteresis time of a gravity dam depends on the linearly distributed temperature differenceTd,whereas the horizontal displacement hysteresis time of an arch dam largely depends on themean temperatureTm.

For the purpose of illustration,the horizontal displacement hysteresis times of a gravity dam and an arch dam w ith the same heightHof 100 m were calculated according to Eqs.(18)and(9),respectively,and the corresponding curves are shown in Fig.3.The bottom w idthsBof the gravity dam and arch dam were set at 70 m and 35 m,respectively,both crest w idths were 8 m,and the thermal diffusivity of the concrete was assumed to be 0.1 m2/d.Furthermore,εin Eqs.(2)and(3)was setat 0,i.e.,the effectof hysteresis time of the boundary temperaturewasnot considered(the effectof εon displacementhysteresis timewillbe analyzed in the next section).Thus,the values ofA1andA2did not affect the calculation results of hysteresis time.

Fig.3.Comparison of horizontal displacement hysteresis times of gravity dam and arch dam.

It can be seen from Fig.3 that the horizontal displacement hysteresis time of an arch dam is longer than thatof a gravity dam even if the w idth of the arch dam is less than that of a gravity dam.It is noted that the beam effectof an arch dam is not considered in this analysis.Actually,the horizontal displacementhysteresis time of an arch ringmightbe slightly reduced w ith consideration of the beam effect.However,since the arch effectplaysa dom inant role,the basic conclusion that the horizontal displacement hysteresis time of an arch dam is longer than that of a gravity dam is still true(this w ill be verified in subsection 4.3.2).

4.2.Influence ofwater temperature on displacement hysteresis time

In the previous analysis,we have assumed thatε=0 in Eqs.(2)and(3).In reality,for a dam in operation,the upstream surface is in contactw ith the reservoir water,and the variation of the reservoirwater temperature lagsbehind thatof the air temperature,i.e.,ε＞0.Therefore,it is necessary to further analyze the influence of the water temperature hysteresis time on the displacementhysteresis time of the gravity dam and the arch dam.

For thisanalysis,we stilluse the infinite slab in section 2 as an example,assum ing that the amplitudes of the water temperature and the air temperature areA2andA1,respectively, and the lag time isε.Through trigonometric transformation, Eqs.(2)and(3)can be rew ritten as

The variablesμmandμdreflect the influence of water temperature hysteresis on the displacement hysteresis time. The parametersμmandμdmainly depend onε,A1,andA2rather than the slab thicknessl.Fig.4 shows the relationshipsofεw ithμmandμdfor differentA1/A2values.In situ measurements show thatεis generally less than 90 days.Within this scope,μdismore sensitive to the change ofεthanμm.

Numerousmeasured dataof the reservoirwater temperature indicate thatA1＞A2andεis less than sixmonths,so,according to Eqs.(22)through(30),μm＞0 andμd＞0.Thismeans that, due to the influence of the reservoir water temperature,the hysteresis time of mean temperature increases fromθmto θm+μm,while the hysteresis time of the linearly distributed temperature difference decreases fromθdtoθd-μd.

With reference to theanalysisabove,itisconcluded that,due to the lag of thewater temperaturevariation behind theair temperature variation,the hysteresis time of the horizontal displacementofanarchdam isextended toθm+μm,whilethatof a gravity dam isshortened toθd-μdwhen the influenceofεis considered.That is,the hysteresis characteristic of reservoir water temperature intensifies the differences between the hysteresistimesofhorizontaldisplacementofgravityandarchdams.

4.3.Variation of displacement hysteresis time with elevation

A2andεin Eqs.(2)and(3)atdifferentelevationsinadam are different.Theelevationhofapointaffectstheupper lim itof the integral of the cantilever beam(for a gravity dam or for the beam effect of an arch dam)during the calculation of displacementhysteresis time.Therefore,itisnecessary to study the variation of displacementhysteresis timew ith elevation.

Fig.4.Relationships ofεw ithμmandμd.

4.3.1.Variation of displacementhysteresis timewith elevation in gravity dam

For the sake of simplicity,we assume thatεin Eqs.(2)and (3)is0；that is,the hysteresiseffectofwater temperature isnot considered provisionally.Two pointsP1andP2on the upstream dam surface are selected,as shown in Fig.5.

We fi rst calculate the horizontal displacementux1of the lower pointP1using Eq.(12):

According to the differential mean value theorem,the equation above can be substituted w ith the follow ing:

whereξ1∈[0,h1].

According to Eq.(3)and the trigonometric transformation,ux1can be expressed as

Sim ilarly,for the higher pointP2,the integration interval is divided into[0,h1]and[h1,h2],and follow ing the procedure sim ilar to the one described above,we obtain

Fig.5.Measuring point positions.

Since the hysteresis time of the linearly distributed temperature difference increases w ith the dam w idth andL(ξ2)＜L(ξ1),the horizontal displacement hysteresis timeθd2at pointP2is shorter than the horizontal displacement hysteresis timeθd1at pointP1,which means that the horizontal displacement hysteresis time increases gradually w ith a decrease in elevation.

Using the trigonometric transformation,Eq.(34)can be rew ritten as

Next,the variations in the amplitude and hysteresis time of water temperature are considered,and their influence on the displacement hysteresis time variation of a gravity dam w ith elevation is analyzed.According to numerousmeasured data of reservoir water temperature,the amplitude of reservoir water temperatureA2tends to decline w ith an increase in the water depthz′,whereas the hysteresis time of water temperatureεtends to increase w ith the water depth,as shown in Fig.6.

To consider the influenceof thehysteresisproperty ofwater temperature,Eq.(16)was used to calculate displacement hysteresis timesatdifferentdistancesbelow the dam top of the gravity dam introduced in section 4.1.Fig.7 shows that the hysteresis timeof thehorizontaldisplacementofagravity dam tends to increasew ith the distance below the dam top for both carryoverand non-carryover storage reservoirswhen thewater temperature hysteresis property is considered,which is consistentw ith the casew ithoutconsideration of thehysteresis property of reservoir water temperature.

Fig.6.Changes of A2andεw ith z′.

For the sake of simplicity,the finite elementmethod was used to analyze the displacementhysteresis time changeof the gravity dam w ith the distance below the dam top,and the water temperature on the upstream dam surface was assumed to be consistentw ith thatof a non-carryover storage reservoir.

Fig.8 shows the finite elementmesh of the gravity dam. Fig.9 shows thevariation curveof thehorizontaldisplacement hysteresis timeof thegravity dam obtained from theanalytical and FEM methods.

As shown in Fig.9,the finite element solution for the hysteresis time of the horizontal displacement of the gravity dam tends to increase w ith the distance below the dam top, which is consistentw ith the analytical solution.

4.3.2.Variation of displacement hysteresis time with elevation in arch dam

The change in displacement hysteresis time w ith elevation in an arch dam ismore complex than that in a gravity dam. This is because a gravity dam can be treated as a statically determinate structure(a cantilever beam),whereas an arch dam is a spatial,statically indeterm inate shell structure,which means that there aremore connections between each part of the arch dam.From a structural response standpoint,an arch dam can be separated into a series of independent arch rings and beams.The variation of the displacement hysteresis time of an arch dam w ith elevation has the same characteristics as those of both an arch ring and a beam(the latter is sim ilar to a gravity dam).Specifically,the distribution of the displacement hysteresis time is related to the shape of the arch dam,and is affected by thew idth-to-height ratio of the dam and thew idthto-height ratio of the river valley.

Fig.7.Variation of horizontal displacementhysteresis time of gravity dam w ith distance below dam top.

Fig.8.Finite elementmesh of gravity dam.

Two examples of single-curvature arch dams were analyzed.The dams,located in a rectangular river valley,had the same radiusand central angle.Themain shape parameters of the arch dam were as follows:the dam heightwas 100m, the centralanglewas 110°,the arch ring radiuswas70m,the crown w idth was 8m,and the bottom w idthswere 20m and 35m forw idth-to-height ratios of 0.20 and 0.35,respectively.

Three computation schemes were used for the analysis:

(a)The pure arch scheme,inwhich only the arch effectwas considered and the beam effectwas ignored.The layers of the arch ring were independentof each other.

(b)The pure beam scheme,in which only the beam effect was taken into account.In this scheme,a single beam in the arch crown profi lewas considered in the computation,and its hysteresis time calculation was sim ilar to that of a gravity dam.This scheme had no real significance,and wasmainly used for comparative purposes.

(c)Themonolith scheme,inwhich the dam was considered amonolith subject to both arch and beam effects.

Both the analyticalmethod and the finite elementmethod were used to analyze scheme(a)and scheme(b),whereas scheme(c)wasanalyzed using only the finite elementmethod because no analytical solution was available for scheme(c). The thermal diffusivities of the dam body and the bedrock were both set at 0.1 m2/d,and the water temperature on the upstream dam surfacewas assumed to be consistentwith that in a non-carryover storage reservoir.

Fig.10 shows the finite element mesh adopted in scheme(c),where the dam is a symmetric structure,and only half of the arch dam is considered in themodeling.Thismodel can bemodified and applied to scheme(a)by splitting the dam into independent layersof arch rings,orapplied to scheme(b) by only picking the grids located at the crown cantilever (Wang and Liu,2008).

Fig.11 shows the change in the horizontal displacement hysteresis timeofan arch crown in the threeschemeswhen the width-to-height ratios of arch dams are set at 0.20 and 0.35, respectively.

The follow ing can be seen from Fig.11:

(1)Theanalyticalsolutions for thepurearch schemeand the purebeam scheme areboth sim ilar to the results from the finite elementsolutions,which suggests that theanalyticalalgorithm of displacementhysteresis timeof thegravity dam and thearch ring proposed in the previoussection is reasonable.

(2)If the shape of the dam cross-section remains constant, the displacement hysteresis time of the pure beam is less than that of the pure arch,which verifies the conclusion drawn in the previous section that the dam type is one of the crucial factors affecting the displacement hysteresis time.

(3)When thew idth-to-height ratio of the dam is lower,i.e., when thecross-section of thearch dam isnarrower,the resultsof purearch analysismatch the resultsof the finiteelementmethod for amonolithic structuremore closely.This is consistentw ith the load-sharing characteristicsof themulti-arch-beam solution.

(4)When the monolith scheme is used,the horizontal displacementhysteresis time of an arch dam fi rst increasesand thendecreasesw ith theincreaseofthedistancebelow thedam top. This observation is based on analysis of a 100m-height singlecurvaturearch dam with a constant radiusand centralangleand located inarectangular rivervalley.Fordifferentheights,variable radii,and variable central angles,the displacement hysteresis curvesofhyperbolic arch damsare likely to be different.

(5)The finite element calculation suggests that,due to the constraint effect of the bedrock,the displacement hysteresistime near the base of both the gravity dam and the arch dam decreases slightly.

Fig.9.Variation of horizontal displacement hysteresis time w ith distance below gravity dam top calculated using differentmethods.

Fig.10.Finite elementmesh adopted in scheme(c)(B/H=0.35).

Fig.11.Variation of horizontal displacement hysteresis time w ith distance below arch dam top for differentw idth-to-height ratios.

4.4.Variation of displacementhysteresis timewith dam width

In the analysis described above,the measurement points were placed at the upstream dam surfacew ithout considering the displacementhysteresis time change along the dam w idth. In reality,due to the influence of transverse deformation,the displacement hysteresis time varies w ith the dam w idth as well.In this study,the effect of the dam w idth was analyzed w ith the finite elementmethod.

Fig.12(a)through 12(c)show the distribution of horizontal displacement hysteresis time along thew idth at different elevations for the gravity dam and the arch dam(w ith width-toheight ratios of 0.35 and 0.20).The computational parameters were the same as those described in subsection 4.3.The computation results indicate the follow ing:

Fig.12.Distribution of hysteresis time of dam horizontal displacement along dam w idth.

(1)Along the dam w idth,from upstream to downstream,the horizontal displacementhysteresis time of the gravity dam decreases,while thehorizontaldisplacementhysteresistimeof the arch dam is shorter near the center and longer near the dam surface.This isbecause the gravity dam deformation along the dam w idth isacombination of thedeflection deformation caused by beam rotation and the transverse expansion.The beam rotation and transverse expansion are dependenton the changes of the linearly distributed temperature difference and average temperature,respectively.The arch dam deformation along the dam width isacombinationof theexpansionof thearchaxisand the transverse expansion of the beam,w ith both components beingdependenton thechange inmean temperature.Inaddition, the influences of the shape of the dam cross-section and the gradualchangeof the reservoirwater temperaturew ith elevation also result in differences in horizontal displacementhysteresis timesof gravity and arch damsalong the dam w idth.

(2)The horizontal displacement hysteresis time near the dam bottom is less than that in the upper part,which is related to the constraint created by the friction between the dam and the bedrock.The displacementhysteresis time at the center of the dam base is longer than that at the upstream and downstream sides.This is because the dam bottom deformation is dominated by the tensile deformation along thew idth,which depends on the heat conduction velocity and time.Therefore, the horizontal displacementhysteresis time ata point far away from the dam surface is relatively long.

5.Case studies

In order to further verify the conclusions,especially the conclusion that dam type has a large impact on the hysteresis time of thermal deformation,different gravity and arch dams were analyzed,including the Shuikou Gravity Dam(whose maximum height is 101 m),Lijiaxia Hyperbolic Arch Dam (whose maximum height is 165 m),Baishan Gravity Arch Dam(whose maximum height is 149.5 m),and Chencun Gravity Arch Dam(whosemaximum height is 76.3m).There are relatively complete and continuous observation records of displacementand environmental factors for all of these dams.

Fig.13 shows the variation of hysteresis time of dam horizontal displacement w ith the elevation in the dam.The follow ing can be seen:

Fig.13.Distribution of hysteresis time of dam horizontal displacement.

(1)The horizontal displacementhysteresis time presentsan increasing tendency in the sequence of a gravity dam,gravity arch dam,and hyperbolic arch dam,which is consistentw ith the theoretical analysis.

(2)Thehorizontaldisplacementhysteresistimeincreasesw ith the distancebelow the dam top.Themain reason is that the dam body thickness and the hysteresis time of reservoir water temperature increasew ith thedistancebelow the dam top.However, due to theconstrainteffectof thebedrock,thehysteresistimenear thebaseof the dam decreasesslightly for some dams.

6.Conclusions

The thermal deformation of a concrete dam changes periodically and its variation lags behind the air temperature variation.To analyze and explain the causesof this lag,known as the hysteresis time of the dam displacement,an extensive investigation was performed.The conclusions can be summarized as follows:

(1)The dam type has a large impacton the hysteresis time of thermal deformation.In general,the hysteresis time of horizontal displacement for an arch dam is longer than thatof a gravity dam.This is because the hysteresis time of the former isdependenton themean temperatureof the dam body, while that of the latter is more dependent on the linearly distributed temperature difference.

(2)The water temperature variation in the reservoir lags behind the air temperature variation,which intensifies the differences between horizontal displacement hysteresis times of gravity and arch dams.

(3)W ith an increase in the distance below the dam top,the horizontaldisplacementhysteresistimeof thegravity dam tends to increase,whereas thehorizontaldisplacementhysteresistime of the arch dam is likely to increase fi rstand then decrease.In addition,due to the constraint of the bedrock,the horizontal displacementhysteresis time near the dam bottom is less than that in the upper partof the dam.

(4)A long the dam w idth,the hysteresis time of horizontal displacement of a gravity dam decreases from upstream to downstream,while the hysteresis time of horizontal displacement of an arch dam is shorter near the center and longer near the dam surfaces.

Finally,it is noted that this study only exam ined horizontal displacement of the dam.To study vertical displacements, sim ilar procedures can be followed.For example,the vertical displacementof a gravity dam is caused by verticalexpansion and contraction and itshysteresis time ismainly dependenton the hysteresis time of themean temperature,which is different from the hysteresis time for horizontal displacement.

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Received 22 April 2016；accepted 5 December 2016 Available online 14 March 2017

Thiswork was supported by the the National Natural Science Foundation of China(GrantNo.51679073)and a project funded by the Priority Academ ic Program Developmentof Jiangsu Higher Education Institutions.

*Corresponding author.

E-mail address:wang_jian@hhu.edu.cn(Jian Wang).

Peer review under responsibility of Hohai University.

http://dx.doi.org/10.1016/j.wse.2017.03.007

1674-2370/?2017 Hohai University.Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license(http:// creativecommons.org/licenses/by-nc-nd/4.0/).