Static and seismic design of Dry Stone Retaining Walls (DSRWs) following Eurocode standards

1  StaticandseismicdesignofDryStoneRetainingWalls(DSRWs)followingEurocode1 standards2 NathanaëlSAVALLE1,2a*,ChristineMONCHAL3b,EricVINCENS4c,StenFORCIOLI3dandPauloB.LOURENÇO1e 3 1UniversityofMinho,ISISE,DepartmentofCivilEngineering,Guimarães,Portugal4 2UniversitéClermontAuvergne,ClermontAuvergneINP,CNRS,InstitutPascal,F‐63000Clermont–Ferrand,5 France6 3Géolithe–Grenoble:181ruedesBécasses38920Crolleshttp://www.geolithe.fr/ingenieurs‐conseils/7 4EcoleCentraleLyon–LTDSUMR5513:36AvenueGuydeCollongue69134EcullyCedex8 a[email protected],*correspondingauthor9 b[email protected]10 ceric.vincens@ec‐lyon.fr11 d[email protected]12 e[email protected]13 Abstract14 DryStoneRetainingWallsarestructuresmadeofrubblestonesassembledwithoutmortarandhavebeen15 presentworldwideforcenturies.Today,theystillconstituteanattractivealternativetobuildingtechniques16 involvinghigherembodiedenergy,suchasreinforcedconcretewalls.Thisstudyusesapseudo‐staticapproach17 togivedesignrecommendationstomaintainthisbuiltheritageandallowitsmodernconstruction.Bothnon‐18 seismic(Eurocode7)andseismic(Eurocode8)casesareaddressed.Thepresentworkconfirmsthataseismic19 designisnotcriticalandisthereforenotrequiredforzoneswithadesignaccelerationbelow0.05g.Inaddition,20 thisworkhighlightsthesignificantpositiveeffectofthestonebedinclinationandtheinternalwallfacebatter.21 Finally,dependingonthewallsiteconditionsandtheseismiczoneassociatedwiththeproject,generaldesign22 recommendationsaregiventooptimisethevolumeofstonesused,whichareillustratedinthecaseofFrance.23 Theserecommendationsbasedonpseudo‐staticanalysesarealreadyusableinpracticeforlowtomoderate24 seismicareasastherequiredretainingwalldimensionscanbeeasilyimplementedon‐site.Inaddition,itisalso25 shownthattheactualFrenchrecommendationsforthesewallsfullycomplywithEurocode7.26 Keywords27 Masonry,Drystone,Retainingwalls,Pseudo‐static,Earthquakes,Coulomb'swedge,Standards28 2  Introduction29 Drystonestructureshavebeenbuiltinmostregionsoftheworld,sometimesshapingtypicalandvaluable30 landscapes.Thesevernacularstructuresaremadeofrubblestonescarefullyassembledbyhandandwithout31 mortar.DryStoneRetainingWalls(DSRWs)arelikelytoconstitutethemostrepresentativepartofthisbuilt32 heritage,allowingagriculturalactivitiesonterracesandtrafficonruralroadsinmountainousorslopedareas.33 Therefore,DSRWsplayanessentialeconomicroleintheseregionsthatbenefitlessfromglobalisationand34 major investments. Inaddition, they alsohold a highcultural value, sometimes labelledby UNESCO(e.g.,35 Douro's Valley in Portugal or the Lavaux's Terraces in Switzerland). In fact, the art of dry stone walling,36 knowledgeandtechniquesweredesignatedasIntangibleCulturalHeritageofHumanitybyUNESCOin2018.37 However,thesestructureshaveoftenfacedalackofmaintenanceinrecentdecadesandrequireurgentrepair.38 GiventheneedtopreserveandrepairoldDSRWs,severalresearchstudieshavebeenconductedmainlyin39 Europe.Experimentalworks[1]–[3],analytical[4]–[7]andnumericalstudies[8]–[15]focusedonthestatic40 mechanicalbehaviourof2DslopedDSRWs,whileotherstudiesinvestigatedthe3Dmechanicalbehaviourof41 thesewallsincaseofaconcentratedtrafficload[16]–[19].InFrance,theseresearchesledtotwopractical42 handbooksthatincludedesignrulesforDSRWsretainingslopes.Thesearevalidforanycountrywithsimilar43 buildingtechniques,whichcanbefoundworldwide[20],[21].However,eventhoughtherecommendations44 areusedinpracticeandrecognisedbythedrystonemasonryandcivilengineeringcommunities,theydonot45 considerseismicaction.OnlyafewstudycaseshavebeeninvestigatedaccordingtothepastFrenchseismic46 recommendations[22],[23].Moreover,thevalidationoftherecommendationsaccordingtoEurocode7[24]–47 [26]hasnotbeeninvestigatedexhaustively,evenifpartlyconsideredinthelatestDSRWsFrenchhandbook48 [21].49 ToaddresstheseismicdesignofDSRWsinslopes,theauthorsdevelopedapseudo‐staticanalyticaltoolbased50 onCoulomb'swedgetheory,whichwasvalidatedbypseudo‐staticscaled‐downlaboratoryexperiments[27].51 Thefirstsectionofthispaperrevisedtheanalyticalmethod,whilethesecondsectionprovidesacomparison52 3  withthecurrentstandardsongeotechnicalengineeringandseismicengineering,respectively,Eurocode7&853 [24]–[26],[28],[29].Finally,followingtheEurocodes,recommendationsaregivenfordesigningDSRWsinareas54 rangingfromverylowtohighseismicity.55 Analyticalmethod 56 Theanalyticalmethodreliesonthelimit‐equilibriumtheoryunderplanestrainconditions(Figure1).TheDRSW57 ischaracterisedbyaheightH,abasewidthB,anexternalslopetotheverticalof  v andaninternalslopetothe58 verticalof  m .Thebedinclinationofthewallisreferredtoas  andthebackfillslopeas  .Contrarytothestatic59 limit‐equilibrium approach of Villemus [30], the analytical method includes seismic forces modelled as60 equivalenthorizontalpseudo‐staticactions.Briefly,thepseudo‐staticequilibriumofaCoulomb'swedgeofsoil61 isfirstcomputedtoobtainthepseudo‐staticactiveearthpressure[31]–[33].Thewall'sequilibriumisthen62 computed,statingthepossibletypesoffailure:aninternalslidingortopplingmode[34].63 64 Figure1:DSRWwithitsgeometricparameterisationa):actualbackfill‐wallsystem;b):modelledbackfill‐wallsystem.Thepresentsignconvention65 makestheinternalbatter(λ m )negative.66 Figure2describesthebackfill‐wallsystem,withallgeometricalandmechanicalparametersclarified.Thewall67 ischaracterisedbyahomogeneousmedium(Figure1)withitshomogeneousunitweight γ rw (accountingfor68 voidsbetweenstones) and jointfrictionalangle φ rw .Similarly,the backfillismodelledas a homogeneous69 medium described by its unit weight γ f , cohesion C f andfrictionangle φ f . Compared to other retaining70 4  structures,theparticularityofDSRWsliesinafailurelinedevelopingthroughthedryjoints,whichismodelled71 asanequivalentstraightfailurelinewithaninclination ω (Figure1).Inpractice,theinclinationofthisfailure72 linefromthebedjointsislimitedbyamaximumvalueof20°,see[27].Thisinclinationisalsodifferentfrom73 thehomogenisedinclination θ ofthefailurelinecrossingthebackfill,whichmainlydependsonmaterial(soil74 frictionandcohesion),geometrical(slopeofthebackfill)andseismic(pseudo‐staticaccelerations)parameters,75 asexplainedbelow.76 Themechanicalsystemhasthreeunknowns( θ , ω andhg)thatshouldbedeterminedtocomputetheearth77 pressure F δ .AccordingtotheCoulombsoil'swedgetheory,foreachcombinationoftheseparameters,thelimit78 equilibriumofthesoil'swedge D1D2D3canbecalculated(Figure2b‐c)toevaluatetheactiveearthpressureF δ . 79 Inparticular,theweightPfofthesoilisproportionaltothewedgearea(triangleD1D2D3)andisappliedatthe80 gravity centre of the triangle D1D2D3.Similarly,thepseudo‐staticaction(inertialforceduetothe seismic81 motion)Ffisalsoproportionaltothewedgeareaandappliedatitsgravitycentre.Thebackfillfrictionalreaction82 Rφapplicationpointandintensityareunknown,butitsorientationisgivenbythebackfillfrictionangle φ fsince83 thelimit‐equilibriumisassumed.Similarly,theorientation,yetnottheapplicationpoint,ofthebackfillcohesive84 reactionRCisknown.Regardingthebackfill‐wallinterface,theinterfacecohesivereactionRCinthasanunknown85 point of application and a known orientation. The intensities of the cohesive forces (RC and RCint) are86 proportionaltothecohesivestrengthandthelengthoftheinterface(D1D3andD1D2).Hereafter,theinterface87 cohesive strength (RCint) always equals zero, as the drain directly behind a DSRW is usually made of dry88 cohesivelessgravel.89 Finally,theearthpressureFδhasaknownorientation δ (internalfaceofthewall)andapplicationpointbut90 unknownintensity.Theintensityisdeducedfromthemechanicalequilibriumofthesoil’swedge(Figure2c).In91 theabsenceofpseudo‐staticactionFfandcohesiveresistance,theapplicationpointoftheearthpressureis92 located at one‐third of the height of the retaining structure. Then, adding cohesive effects decrease the93 5  applicationpointheightwhileaddingapseudo‐staticactionincreasesitsheight.Thereadercanrefertothe94 literaturefordeeperinsightsintothelocationoftheapplicationpointofearthpressureinthiscase[34]–[36].95 Theanalyticalmethodalsoaccountsforthetensilecracksthatclassicallyoccuratthetopofcohesivebackfill96 andreducethecohesiveforces(R C andR Cint ).Inaddition,inthepresentwork,thepresenceofdeadloadson97 topofthebackfillandsaturatedretainedbackfill[34]canbeaccountedfor,yetnotdescribedhereforbrevity.98 99 Figure2:Parametrisationofthemechanicalsystemandgeometricalequilibriumofthesoil’swedgeinordertocomputetheearthpressureF δ. 󰇟34󰇠 100 Inasecondstageofthecalculation,theequilibriumofthewallitselfiscomputed(Figure3)includingitsown101 weightandthepseudo‐statichorizontalaction,appliedatthecentreofgravityofthestudiedportionofwall102 A 2 A 3 D 1 E,correspondingtothepartabovethefailurelineD 1 E.TheinterfaceactionsF δ andR Cint (aswellastheir103 line of actions) are derived from the previous stage of calculation. Subsequently, the wall equilibrium is104 computedintheXandYdirectionstoevaluatethebasereactionR b .Theequilibriumintermsofmomentum105 givestheapplicationpointofR b .Finally,thestabilityofwallportionA 2 A 3 D 1 Eiscomputedconsideringatoppling106 modeoffailure(e.g.applicationpointinsidethewall,i.e.l b >0,oranyothercriteriadefinedinthefollowing107 sections)oraslidingmodeoffailure.Forthislastpoint,thebasereactionR b isprojectedontheplanedefined108 bytheorientationofthebedjoints(axesX s andY s inFigure3),andthenMohr‐Coulombcriterionischecked.109 6  Notethatthebedjoints'orientationisupdatedbecauseofthepossibleinternalrotationofstonesinsideDSRWs110 (see[34]formoredetails).Finally,theunknownsofthesystem( θ , ω andh g )areoptimisedforeachfailure111 modetofindthemostcriticalsituationforthecriterionchecked.SeveraliterationsinvolvingthebasewidthB112 ofthewallallowstoidentifytheminimumBvaluethatbarelysatisfiesthestabilitycriteria.113 114 Figure3:Equilibriumofthedrystoneretainingwall[34]115 Themethodhasbeenvalidatedonscaledpseudo‐staticexperimentsonatiltingtable,usingdryjointbrick116 retainingwallsretainingasandybackfill[27].Figure4givestheresults,showingthatthedevelopedpseudo‐117 staticapproachisaspreciseasmoresophisticatedDiscreteElementModelling(DEM)simulations[37].118 119 Figure4:ComparisonofDEMsimulation,analyticalsimulationandexperimentaltiltingtests[37]120 7  Asadditionalvalidation,theanalyticalmethodwasusedtomodeltwosetsofexperimentalcampaignscarried121 outonfull‐scaleDSRWswith1)ahydrostaticload[1];2)adrybackfillload,asdisplayedinFigure5[2],[3].122 Table1describesthegeometricandmechanicalparametersoftheexperiments,alongwiththeanalytical123 results,whichareinexcellentagreementforbothcampaigns.Moreover,thedevelopedanalyticalapproach124 providesasimilarlevelofaccuracytotheresultsofVillemusetal.[1]andColasetal.[6],[38].125 126 Figure5:ExperimentaltopplingfailureobtainedbyColas[39]127 Table1:Parametersofthefull‐scaleDSRWsexperimentalcampaignsfrom[1]–[3].Experimentalandanalyticalresultsarealsogiven.128 Name* V1l V2l V3l V4l V5s C1g† C2s C3s C4l Geometricalparameters HeightH(m) 2 1.95 4 2 4.25 2.5 2.5 2.5 2.5 BasewidthB(m) 0.9 0.91 1.8 0.9 1.8 0.6 0.6 0.7 0.65 Externalbatterλ v (%) 15 0 15 12 15 6 6 6 6 Bedsinclinationα(°) 0 0 0 4 8.5 3.4 3.4 9.1 9.1 Backfillslopeβ(°) NA NA NA NA NA 26.4 31.7 32.6 34.9 Mechanicalparameters Wallweight(kN/m 3 ) 15.4 14.9 15.7 15.7 18.0 21.0 20.0 20.0 21.8 Stonefriction(°) 36 36 36 36 28.5 27 25 25 35 Internalrotation(°)‡ 5 5 5 5 5 5 5 5 5 Soil weight γ r (kN/m 3 ) NA NA NA NA NA 14.9 14.9 14.9 14.9 Soilfrictionϕ r (°) NA NA NA NA NA 37.7 37.7 37.7 37.7 Experimentalresults Criticalheight(m) 1.74 1.9 3.37 1.94 3.62 >2.1 7 2.41 2.96 2.95 Failuremode(S/T) S S/T S S/T S NA S/T T T Analyticalresults Criticalheight(m) 1.73 1.84 3.48 1.86 3.68 2.86 2.69 3.02 2.82 Failuremode(S/T)§ S S/T S S/T S T S/T T T 8  Differencetoexp. ‐1% ‐3% +3%‐4%+2%NA+11 %+2%‐4% *VreferstoVillemus[1]withhydrostaticloadingandCtoColas[2],[3]withbackfill loading,whilelreferstolimestoneblocks,stoschistblocksandgtograniteblocks. †Thisexperimentfailed.However,thewallresistedatleastaloadingcorrespondingtoa backfillheightof2.17m. ‡Thisinformaonisbasedonexperimentalresults,usingadefaultvalue.Detailscanbe foundin[1]–[3]fortheexperimentsandin[34],[40]fortheanalyticalmethod. §Acombinedsliding‐overturningfailurehasbeendefinedifthecriticaltheoreticalheights ofthetwofailuremodeswerewithinarangeof±5%.  DesignofDSRWsfollowingEurocodes129 ThepresentsectionaimsatdesigningDSRWsusingtheanalyticalmethodandthepartialsafetyfactors(actions,130 materialpropertiesandresistance)fromEurocode7and8[24]–[26],[28].First,itisemphasisedthatina131 seismiccontext,apseudo‐staticmethod(liketheonepresentedabove)willneveraccuratelypredictthetrue132 timeevolutionofthedynamicresponseorresistanceofarealDSRWduringanearthquake.Therefore,itisonly133 usedasasimplifieddesignmethodproposedbyEurocode8togivefastseismicassessmentofretainingwalls.134 Nopartialsafetyfactorrelatedtothemethodisconsideredsincenoconstantbiashasbeenfoundinthe135 validationprocessesbetweentheoreticalandexperimentalresults.Moreover,onlythewall'sinternalsliding136 andtopplingfailuresareconsidered:thebearingcapacityofthefoundationsoilatultimateorserviceability137 limitstatesareassumednottobereached.Similarly,thepassivesoilisconsideredinfinitelyrigid,whichis138 reasonable according to Alejano et al. [5]. Finally, liquefaction and the failure of the entire soil slopeare139 disregarded.Table2presentsthesafetyfactorsusedforthecomputationsaswellasthosefromtheFrench140 professionalrules,whicharesimilar[21].141 Regarding non‐seismic verifications, the Ultimate Limit State Equilibrium (ULS EQU), the142 Structural/Geotechnical Ultimate Limit State with the second approach (ULS STR/GEO), see [41], and the143 ServiceabilityLimitState(SLS)areexamined.TheUltimateLimitState(ULSSEISM)isappliedfortheseismic144 verification.145 9  Table2:Partialsafetyfactorsfornon‐seismicandseismicsafetyverifications;AnnexAoftheEurocode7[24]andENTPE(Eds.)etal.[21].146  Eurocode7[24] French professional rules[21] ULS SLS  EQU STR/GEO SEISM Safetyfactorsforactions Favourableweightactionsfactor(γG,fav) 0.9 1 1 1 1 Unfavourableweightactionsfactor(γG, unfav) 1.1 1.35 1 1 1.35 Safetyfactorsformaterialproperties  Drainedsoilfrictionanglefactor(γφ') 1.25 1 1.25 1 1 Drainedsoilcohesionfactor(γC') 1.25 1 1.25 1 1.25 Safetyfactorsforresistances Slidingfactor(γR,h) 1 1.1 1 NA 1 Topplingfactor(γR,v) 1NA1NA1 Eccentricityfactor(1–2e/B) NA 1/15 NA 1/2 1 Modelresistantfactor(γR,d) 1 1 1 1 1.2 147 ForthetopplingverificationoftheSLSandtheULSSTR/GEO,thedesigncriteriontosatisfycorrespondstothe148 maximumeccentricity(notede)ofthetransmittedloadthroughthewall,asstatedinEurocode7[25],[26].In149 thecontextoftheULSEQUandSEISMverifications,thepartialsafetyfactorsγm(γφ'andγC')forthematerials150 areonlyappliedtothebackfillsoilpropertiesandnottothefrictionbetweenblocks(asmentionedin[41]–151 [43].Thefrictionbetweenblocksislinkedtothewallresistance,thustotheresistancesafetyfactorγR.Asa152 consequence,applyingasafetyfactorfortheblockmaterial(block‐blockfriction)wouldpenalisetwicethe153 sameparameters,whichisnotinagreementwiththeframeworkofEurocodes. Moreover,intheseismic154 verification(ULSSEISM),thesafetyfactorsformaterialsarelinkedtothedegradationoftheshearstrengthof155 soilsathighstrainand/orinthepresenceofporepressures.Theseareunlikelytooccurfordryblock‐block156 joints[42]–[44].157 Non‐seismiccase158 Computations discarding the seismic action are carried out on five walls, whose main characteristics are159 presentedinTable3.ThewallsarebuiltofstoneswithgeologicalnaturesrepresentativeofEuropeanand,in160 particularFrench,geology(molasseormarlsandstone,schist,limestoneandgranite).Theyincludetypical161 16  accelerations larger than 0.25g, due to the inherent approximations of the pseudo‐static approach, the249 maximumextra‐widthsexceed100%ofthenon‐seismicdesignwhichseemsnonreasonableinpractice.For250 thesecases,thepseudo‐staticmodellingapproachmayalsobeinadequate,giventheweakconsiderationof251 dynamicsandthefactthatwalldisintegrationisignored.252 Table4:Maximumextra‐widthsforDSRWsrequiredbyapseudo‐staticseismicdesigncomparedtoanon‐seismicdesigndependingonthereference253 acceleration(a gR ).TheanalysisonlycoversDSRWsbelongingtothenormalclassofimportance(γ I ≤1.0).Accordingtotheseismichazardmapof254 Europeancountries,thesecondcolumngivesthecorrespondingregionsofeachreferenceacceleration.ThereaderisreferredtoFigure7andFigure255 8.256 In257 conclusion,thesystematicuseofastonebedinclinationαisrecommended.Wheneverpossible,anexternal258 batterequaltothestonebedinclinationtofacilitatetheconstructionprocessisalsosuggested.Forlowto259 mediumseismicregions(agR<0.11g),astonebedinclinationαof10%issuitable,whereas,forlargerseismic260 hazards(agR≅0.16‐0.2g),avalueof20%canhelptoreducetherequiredextra‐widthssignificantly.261 Frenchcasestudy262 ThissectionprovidesamoredetailedcasestudyforFrance,chosenasanillustratingexampleoflowtomedium263 seismicityEuropeancountries.Thesectionhelpstounderstandthetrendsandpracticalseismicdesignfor264 DSRWs,dependingonthreedifferentbuttypicalcasesforthebackfill‐wallcondition.Thefirstcase(MAX)265 correspondsto thepreviousEuropean study, i.e., large amplificationforthesoil(S=1.8)andastandard266 construction(importancefactorγI=1.0).Thesecond(ROCK)representsthemostcriticalcasewhentheDSRW267 isdirectlyfoundedonthebedrock(S=1.0),consideredareferencecaseforfoundationconditions.Thelastone268 (RURAL)correspondstowallsoflessimportance(γI=0.8),builtfarfromanyroadorbuilding.Inpractice,269 17  accordingtotheFrenchregulations,thesespecificcases(RURAL)arenotsubjectedtoseismicregulations.270 However,thisworkgivesreferencevalues,whicharehelpfulfor DSRWs stakeholders. These three271 configurationsareanalysedaccordingtothefourseismiczonesofmetropolitanFrance(agR=0.04,0.07,0.11272 and0.16g),leadingtoatotalof12casestudies(Table5).273 Table5:Horizontalseismicdesignaccelerations𝑎fordifferentcriticalcases(asaproportionof 𝑔  9.81𝑚. 𝑠).Theverticalaccelerationavis274 systematicallytakenequalto0.5*ah.275 a gR MAX RURAL ROCK Formulasusedtocomputeah ‐ ah,max=agR*1.20 ah,rural=agR*0.96 ah,rock=agR*0.67 VeryLowSeismiczone(S1) 0.04g 0.05g 0.04g 0.03g LowSeismiczone(S2) 0.07g 0.09g 0.07g 0.05g ModerateSeismiczone(S3) 0.11g 0.13g 0.11g 0.07g MeanSeismiczone(S4) 0.16g 0.20g 0.16g 0.11g 276 Table6sumsupthemaximum(andmean)extra‐widths(amongthedifferentwallsandstonestype)obtained277 foreachseismicsituationdependingonthestonebedinclinations.InseismiczonesS1toS3,onecanagain278 notethatbedinclinationαdramaticallyimpactstheresults;however,abedinclinationof20%doesnotprovide279 asignificantincreaseinresistancecomparedwithaninclinationof10%(seealsoFigure8b‐c).Therefore,as280 usualintheSouthofFrance,aninclinationofthestonebedof10%isrecommended:seismicdesignrequires281 nomorethan40%extra‐width(comparedtoanon‐seismicdesign).InseismiczoneS4,wallsbuiltwithastone282 bed inclination of 10% require an extra‐width of 90%, which induces substantial extra costs for the wall283 construction.Inthiscase,one shouldeitheruse a steeperbedinclinationorconductaspecific analytical284 computationtooptimisethegeometryoftheDSRW.However,ifthewallisbuiltfarawayfromroadsand285 buildings(RURAL)andwithaninclinationbedof10%,themaximumexpectedextra‐widthonlyreaches50%.286 Onthecontrary,ifthewallisdirectlyfoundedonthebedrock(ROCK),theseismicrequiredextra‐widthdrops287 toamaximumof30%.Ifbothconditionsarefulfilled(RURAL&ROCK),therequiredextra‐widthsdonotexceed288 20%(casenotaddressedinTable6).Finally,thegeneralrecommendationsofTable6(maximumvalues)can289 18  be readily used for practical non‐seismic and seismic design of DSRWs without requiring more detailed290 computations.291 Table6:Influenceofthestonebedinclinationαontheextrawidthrequiredforseismicdesign.Maximumvalues,togetherwithaveragevaluesin292 parentheses,aregiven.293 294 Asalreadynoted,specificanalyticalcomputationsshouldbecarriedoutforthesituationMAX‐S4ormore295 criticalseismicimplantationsinsteadofusingthegeneralapproachwiththemaximumvaluesdisplayedinTable296 6.Morespecifically,oneshouldpayattentiontospecificparametersoftheDSRWinthedesignthatplaysa297 criticalroleintheseismiccomputation(seefulldetailsin[40]).Apartfromthepositiveinfluenceofthestone298 bedinclinationandthenegativeeffectoftheretainedslopeanglealreadyhighlighted,apositiveinternalwall299 batterisrecommended(thisisthecaseofaself‐stablewall).Tosupportthisrecommendation,Walls1to5300 (withvariousstonetypesandbedinclinations)havebeendesignedtowithstandaspecificseismicacceleration301 (0.05g,0.10g…and0.35g).Inasecondstep,eachwallsectiongeometryhasbeenmodified,addinganinternal302 batter(λ m =5%,10%and15%)butkeepingthesamesurfaceareaasbefore.Itmeansthatthetotalvolumeof303 stonesusedinthatcaseisthesamebutthatthegeometryofthewallsectionisdifferent(i.e.,withalarger304 width at the base). Finally, the maximum acceleration withstood by the walls with an internal batter is305 comparedtothemaximumaccelerationfoundforthosewithoutthebatter(Figure9).Thecurvescorrespond306 totheaveragevaluesfoundthroughoutthedifferentwallsandstones.Onlypositivevalueshavebeenfound,307 meaningasystematicimprovementoftheseismicresistancewhenaddinganinternalbatter.Thisimprovement308 isparticularlysignificantforwallswithnon‐zerostonebedinclinationandlow(S2)tomoderate(S3)seismic309 hazardregions.310 19  311 Figure9:Effectoftheinternalbatterλ m (keepingthesameareaforthewallsection)ontheseismicresistanceofaDSRWfordifferentbedinclinations312 a)α=0%;b)α=10%;c)α=20%.313 Toconclude,inzoneS2,noDSRWextra‐widthisrequiredtofulfilaseismicdesignifbothaninclinationbedof314 10%andaninternalbatterofatleast10%areused.Thismeansthatanadequatechoiceforthewallgeometry315 compensatesfortherequiredextraresistancerequiredtosatisfyaseismicdesignincaseoflowseismicity.316 Conclusions 317 Thepresentstudyaddressesnon‐seismicandseismicdesignsofDryStoneRetainingWalls(DSRWs)according318 to the European standards (Eurocodes) used for conventional retaining walls while proposing an adapted319 methodology (e.g. including internal failure of DSRW). It has been shown that the current French320 recommendationsforthenon‐seismicdesignofDSRWscomplywithEurocode7(geotechnicalengineering)321 20  standardsbeingslightlymoreconservativethanthelatter.Moreover,inverylowseismiczones(ah<0.05g),322 non‐seismiclimitstatesarethemostcriticalstatesforthedesignofDSRWs,whichconfirmsthataseismic323 designisnotrequiredintheseregions[29].Thisisgenerallynotthecaseinzonesofhigherseismicitywhere324 theseismicdesignaccordingtoEurocode8(seismicengineering)standardisalmostalwaysthemostcritical.325 ThestudyrevealeddifferentgeometricaloptimisationoptionsforDSRWs.Itishighlyrecommendedtouse326 systematically:i)abedinclinationofatleast10%;ii)aflatretainedbackfill;iii)andapositiveinternalbatterof327 at least 10%. These three geometric parameters have a significantimpactonseismicdesign.Ifthese328 recommendationsarefollowedinlowseismiczones(uptoah=0.08g),thereisalsonoincreaseindimensions329 tofulfilaseismicdesign.330 Thepseudo‐staticapproachgenerallygivespracticalglobalrecommendationsforlowtomoderateseismic331 hazardzones(uptoah=0.2g).Inaddition,aspecificgeometricallyoptimised(asstatedabove)pseudo‐static332 designcanstillproduceaffordablerecommendationsinmorecriticalcases(uptoah=0.3g).However,for333 higherdesignaccelerationorparticularlycriticalcases,thepseudo‐staticapproachfortheseismicdesignleads334 tovalueshigherthan50%fortheextra‐widths.Inthiscase,dynamic time history computations are335 recommendedtoobtainmoreaccurateresultsthataccountforwalldisintegrationfailure.Indeed,thisfailure336 modemaybecriticalforhighseismicityareas,particularlyifcombinedwithpoorexecutionconditions.337 Finally,asanexampleillustratingEuropeancountries,Franceisusedasacasestudyofseismicdesignapplied338 toDSRWsinlowtomoderateseismicityareas.InmetropolitanFrance,wheremanyDSRWscanbefound,the339 expectedextra‐widthprovidedbyseismicdesigndoesnotexceed40%forwallslocatedinlowseismiczones340 presentingastonebedinclinationof10%,incasethefoundationisnotonthebedrock.Inthesameconditions,341 wallsbuiltinmoderateseismichazardzonesneedeitherastonebedinclinationof20%oraspecificanalytical342 computationtooptimisethesection.Finally,wallsdirectlyfoundedonthebedrock,inthecaseoflowand343 moderateseismicityareas,requireamaximumof30%extrawidthtofulfilaseismicdesign.344 21  Funding345 ThisworkwaspartlyfinancedbyFCT/MCTESthroughnationalfunds(PIDDAC)undertheR&DUnitInstitute346 forSustainabilityandInnovationinStructuralEngineering(ISISE),underreferenceUIDB/04029/2020.This347 studyhasalsobeenpartlyfundedbytheSTAND4HERITAGEproject(NewStandardsforSeismicAssessmentof348 Built Cultural Heritage) that has received funding from the European Research Council (ERC) under the349 EuropeanUnion’sHorizon2020researchandinnovationprogramme(GrantagreementNo.833123),asan350 AdvancedGrant.Inaddition,theauthorswanttoacknowledgetheFrenchMinistryofEducationandResearch351 fortheirfinancialsupportthroughaPhDgrantattributedtothefirstauthor.Theopinionsandconclusions352 presentedinthispaperarethoseoftheauthorsanddonotnecessarilyreflecttheviewsofthesponsoring353 organisations.354 Competinginterest355 Theauthorsdeclaretherearenocompetinginterests.356 Authors'contribution357 NS: Investigation, Formal Analysis, Visualisation, Writing – original draft; CM: Methodology, Resources,358 Investigation; EV: Funding acquisition, Writing – original draft, Methodology, Supervision; SF: Resources,359 Writing–review&editing,Methodology;PBL:Supervision,Fundingacquisition,Writing–review&editing.360 Dataavailability361 Thedataarisingfromtheanalyticalsimulationspresentedinthepaperisnotavailablepublicly.362 References363 [1] B.Villemus,J.‐C.Morel,andC.Boutin,‘Experimentalassessmentofdrystoneretainingwallstabilityon364 a rigid foundation’, 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