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Effect of Surcharge on Active Earth Pressure inReinforced Retaining Walls: Application of Analytical

Calculation on a Case StudyAli Ghanbari, Mahyar Taheri, Cyrus Jedari, Mhatab Defan Azari

To cite this version:Ali Ghanbari, Mahyar Taheri, Cyrus Jedari, Mhatab Defan Azari. Effect of Surcharge on ActiveEarth Pressure in Reinforced Retaining Walls: Application of Analytical Calculation on a Case Study. 2018. �hal-01717161�

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EffectofSurchargeonActiveEarthPressureinReinforcedRetainingWalls:ApplicationofAnalyticalCalculationonaCaseStudy

AliGhanbari1,MahyarTaheri2,CyrusJedari2,MahtabDelfanAzari3.

Engineering,KharazmiUniversityCivilofDepartmentrofessor,P1

UniversityKharazmi,EngineeringCivilofDepartment,ResearchAssistant2,ShahidEngineeringWaterandEnvironmentalDepartmentofResearchstudent,3

BeheshtiUniversityABSTRACT: Inmany retainingwalls problem, it is necessary to determine additionalearthpressureproducedbysurchargeloadsactingonthesoilsurfacebehindthewall.In this study, based on analyticalmethods, previous formulation of horizontal slicesmethod are improved and a new formulationhas beenproposed for estimating theeffectofsurchargeonactiveearthpressureofretainingwallswithcohesive-frictionalbackfill.Acomparisonoftheanalyticalresultsobtainedfromtheproposedmethodwiththose of previous research and also a comparison between currentmethod and 11meterswallasacasestudyisproposed.Itwasconcludedthattheanalyticalprocedureproposed reliablycalculates theactiveearthpressuredue to surcharge in reinforcedwallwithcohesive-frictionalsoil.Notations:BasicSIunitsaregiveninparentheses.

Ai Areaofithslice(m2)c Cohesionofsoil(Pa)H Heightofwall(m)Hi Horizontalforceattopofithslice(N/m)Hi+1 Horizontalforceatbottomofithslice(N/m)hi Heightofithslice(m)Ni Normalforceonfailuresurfaceforithslice(N/m)n Numerofhorizontalslices(dimensionless)P Resultantactiveearthpressure(N/m)Pi Netforceonwallofeachslice(N/m)pi Activeearthpressureonwallforithslice(Pa)Si Shearforceonfailuresurfaceforithslice(N/m)d Distanceofsurchargefromthewall(m)L Lengthoffailurewedge(m)q Linearsurcharge(kN/m)

Kaq coefficientofactivepressurefromthesurcharge(dimensionless)

Paq Resultantlateralearthpressurefromthesurcharge(N/m)

paq Lateralearthpressurefromthesurcharge(Pa)

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Ti Tensileforceinreinforcementforithslice(N/m)ΣTi Totaltensileforceinthereinforcements(N/m)

Vi Normalforceattopofithslice(N/m)Vi+1 Normalforceatbottomofithslice(N/m)Wi Weightofithslice(N/m)XV1 Horizontaldistanceoflinesurchargefromwall(m)XVi HorizontaldistanceofVifromwall(m)XVi+1 HorizontaldistanceofVi+1fromwall(m)XGi HorizontaldistanceofWifromwall(m) f Angleofinternalfrictionofsoil(degrees)b Anglebetweenfailuresurfaceandhorizontalplan(degrees)λi Ratioofshearstresstoshearstrengthforithslice(dimensionless)𝛿 Frictionanglebetweenwallandbackfillsoil(degrees)

𝛾 Totalunitweight(N/m3)

𝜏$ Shearstrengthofsoil(Pa)

𝜏% Mobilizedshearstress(Pa)

1. IntroductionRetainingwallsareoneofthemostcommonsoil-retainingstructures.Thesewallshaveconsiderableflexibilityagainstearthquakeloadsandarealsolesssensitivetosettlement(Tatsuokaetal.,1997).Recently,however,thenecessityhasbeenfeltforanappropriatemethod to calculate the effect of a surcharge on the active earth pressure behindretainingwalls.Various research methods have been applied to the study of reinforced fill andconventionalwalls and slopes in recent years. Studies of reinforced soil include theexperimentalstudyofstructures(Kazimierowicz-Frankowska,2005;LeeandWu,2004;Yoo,2004;Chenetal.,2007;Rowe,2005;WonandKim,2007;El-EmamandBathurst,2007;Roweetal.,2007;JonesandClarke,2007;Sabermahanietal.,2008;LathaandKrishna,2008;Yangetal.,2009;Roweetal.,2009)andnumericalanalysis(RoweandHo, 1998; Hatami and Bathurst, 2000; Huang and Wu, 2006). Additional analyticalmodelsincludelimitanalysis(Porbahaetal.,2000),limitequilibriumandthehorizontalslicesmethod(Shahgholietal.,2001;BakerandKlein,2004;Nourietal.,2006,2008;Shekarianetal.,2008;Vieiraetal.,2011).Traditionalmethods for calculating activeearthpressure first consider the theoryofelasticityandthencalculatetheeffectofasurchargeonthebackfill.Finally,theresultsare added to an analytical solution obtained separately that does not consider asurcharge.Using large scaleexperimental setups,Gerber (1929)andSpangler (1938)proposedthefollowingequationtocalculatethelateralearthpressurecausedbyapointloadonaretainingwall:𝜎' =

).++,-.

%./.

%.0/. 1 (1)

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whereQistheconcentratedloadactingonthewall,Histheheightofthewall,misthehorizontal distance of the surcharge from thewall andn is the depth at which thehorizontalstressduetothesurchargehasbeencalculated.Mishra(1980)andJarquio(1981)calculatedthe lateralpressurecausedbystrip loadsurchargesbasedonthetheoryofelasticity.AccordingtothecorrelationproposedbyJarquio(1981),thehorizontalstressatdepthzonaretainingwallwithaheightHcouldbecalculatedas:

𝜎' =2-𝛽 − 𝑠𝑖𝑛 𝛽 𝑐𝑜𝑠 2𝛼 (2)

Anglesαandbweredeterminedforthedistanceofthesurchargefromthewall.BygeneralizingCoulomb’s formula,Motta (1994)obtainedtheactiveearthpressureactingonawallbyauniformly-distributedsurcharge.GeorgiadisandAnagnostopoulos(1998)comparedtheresultsofpressureappliedtosheetpilesunderstriploadingwiththose obtained from the distribution of the elastic stresses technique, the 45°approximation, and the Coulombmethod. They observed that values for the elasticmethodshowasignificantdifferencefromtherealdata.Caltabianoetal.(2000)consideredtheeffectofasurchargeandangleoffailurewedgeon the seismic stability of retaining walls. They approximated the effect of anearthquake pseudo-statically using the method proposed by Okabe (1926) andMononobe and Matsuo (1929). They also studied the effect of distance from andstrength of the surcharge. Kim and Baker (2002) studied the effect of a live trafficsurcharge on retainingwalls.Using the equivalent bendingmoment technique, theyproposedananalyticalapproachtocalculatetheactivehorizontalpressureonretainingwalls.Greco(1999,2003,2005and2006)appliedCoulomb’stheorytoinvestigatetheeffectofstripsurchargesonactiveearthpressureinretainingwallsandproposedananalyticalmethodtocalculatethepressuremagnitudeandactingpointoftheresultantforce.In addition to these researchers, AASHTO (2007), USACE (2005) and Beton Kalander(WihelmErnestandSohn,2002)havealsoproposedanalyticalmethodstoincludetheeffectofsurchargeincalculations.Thisstudyproposesacomparisonbetweenformulationbasedonthehorizontalslicesmethod for calculating active earth pressure caused by a line surcharge on rigidreinforcedandconventionalwallsandaCasestudyfora11meterswall.

2. Horizontalslicesmethod Thehorizontalslicesmethod(HSM)wasfirstproposedbyShahgholietal.(2001).Thistechniquewasoriginallydevelopedforreinforcedsoilslopesanddividedthemass

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ofsoilabovethefailurelineintoanumberofhorizontalslices.ComprehensiveequationsandtheunknownsofHSMforassessingreinforcedslopeswereproposedbyNourietal.(2006,2008).Azadetal. (2008)usedHSMto investigateactivepressuredistributionalongtheheightofawallandcalculatedtheangleoffailurewedgeintheactivestate.Shekarianet al. (2008)determined theactive staticpressureon reinforced retainingwallsinfrictionalandcohesivesoilsusingHSM.AhmadabadiandGhanbari(2009)andGhanbariandAhmadabadi (2010a,2010b,2010c)usedHSMtodetermine theactivepressureonreinforcedfillwallswithcohesivefrictionalbackfills.GhanbariandTaheri(2012)usedHSM inorder tocalculate theeffectofa linersurchargeonactiveearthpressureinreinforcedretainingwalls.

3. SummaryofcalculatingtheeffectoflinesurchargeonconventionalwallsbyHSM

Forreinforcedretainingwalls,thetensileforceofreinforcementsmustbeaddedtotheunknowns.Inadditiontotherecommendedequations,notherequationsarerequiredtoobtaintheseunknowns.AccordingtoAhmadabadiandGhanbari(2009),itisassumedthat the shear stress in each slice is a constant fraction of the shear strength. Theyassumed that the average shear stress along each slice is a coefficient of the shearstrengthofthesoilintheyieldcondition.Thiscoefficientisalwayslessthanunityandisshownasλi,ortheithslice.Thus: Toimplementtheproposedmethod,thefollowingassumptionsweremade:

• Theverticalstressactingonthefirstsliceequalsthesurchargeforcedividedbyitssurface.Thisverticalstressisassumedtobeunknowninotherslices.

• Aplanarfailuresurfaceisassumed.• Analysesarebasedonthelimitequilibriumtheorem.• Thesoilmassishomogeneousinallcalculations.• Thefailuresurfacepassesthroughthetoeofthewall.• Theshearforcesbetweenslicesareunequal 𝐻= ≠ H=0) .• ThepointofactionoftheNiforceisatthelowermiddleofeachslice.• TheactingpointofthePiforceisattheuppermiddleofeachslice.

TheequationsandunknownsforthecompleteformulationoftheproposedmethodarepresentedinTable1.Fig.1showsaretainingwallwithabackfilldividedintohorizontalslicesandafailurewedgeofanglebtothehorizontalaxis.Theappliedforcesontheithslicearealsoshown. TheheightofeachslicecanbecalculatedbyEq.(3),giventhatthebackfillisdividedintonhorizontalslicesofequalheight.h= =

AB (3)

ParametersXvi,Xvi+1andXgiinFig.1arethedistancesofverticalforcesfromthewallobtainedfromthefollowingequations:

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𝑋DE =FG

H IJ/ K+

FMNMOGPQ

H IJ/ K (4)

𝑋DE0) =FM

NMOGPQ

H IJ/ K (5)

𝑋RE = 𝑋DE + 𝑋DE0) /2 (6)

ThefirstthreeequationsinTable1are:

𝐹' = 0 ⟹𝑆E 𝑐𝑜𝑠 𝛽 − 𝑁E 𝑠𝑖𝑛 𝛽 + 𝑃E 𝑐𝑜𝑠 𝛿 + 𝐻E − 𝐻E0) = 0 (7)

𝐹[ = 0 ⟹𝑆E 𝑠𝑖𝑛 𝛽 + 𝑁E 𝑐𝑜𝑠 𝛽 + 𝑃E 𝑠𝑖𝑛 𝛿 − 𝑉E + 𝑉E0) −𝑊E = 0 (8)

𝑀_ = 0 ⟹−𝑉E𝑋DE + 𝑉E0)𝑋DE0) − 𝑊E𝑋RE +`�

aE/ K− 𝑃E 𝑐𝑜𝑠 𝛿 × ℎd + ℎd/2/

deE0) +𝐻E0) ℎd/deE0) − 𝐻E ℎd/

de) = 0 (9)

Wiistheweightoftheithslicecalculatedas:

𝑊= = 𝐴E×𝛾×1 (10)

4. HSMfortheeffectoflinesurchargeonreinforcedfillwalls

Forreinforcedretainingwalls,thetensileforceofreinforcementsmustbeaddedto theunknowns. Inaddition to the recommendedequations,notherequationsarerequiredtoobtaintheseunknowns.AccordingtoAhmadabadiandGhanbari(2009),itisassumedthattheshearstressineachsliceisaconstantfractionoftheshearstrength.Theyassumedthattheaverageshearstressalongeachsliceisacoefficientoftheshearstrengthofthesoilintheyieldcondition.Thiscoefficientisalwayslessthanunityandisshownasλi,ortheithslice.Thus:𝐻E = 𝑉E tan f+ 𝑐 𝜆E (11)

Eq.(11)wasaddedtothefivepreviousequationstocalculatetheactiveearthpressureforreinforcedwallsundergoingsurchargeloads.Itisassumedthatvariationsinverticalforcesactingoneachslice(∆Vi)arethesameundertheeffectofasurchargeforbothreinforcedandunreinforcedconditions.∆𝑉E mn=Bopmqnr = ∆𝑉E sBmn=Bopmqnr (12)

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Fig.1.Schematicviewofanalyzedwallandequilibriumofforcesinithslice.Table1.EquationsandunknownsforHSMcalculatingtheeffectofsurchargeonreinforcedfillwalls.

Number Equations Number Unknowns

n 𝐹' = 0foreachslice

n Hi

Inter-sliceshearforce

1

n 𝐹[ = 0foreachslice

n Ni

Normalforcesatbaseofeachslice

2

n 𝑀_ = 0foreachslice

n Si

Shearforcesatbaseofeachslice

3

n Si=Ni(tanf)+

cforeachslice

n PiNetforceonwall4

nτm=λτfforeachslicen

TiTensileforcesof

eachslice

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5nSummation5nSummation

5. Resultsforconventionalwalls AnanalysisofthewallisshowninFig.1andtheresultsarecomparedwiththoseofresearcherslistedinTable2.Inthiscondition,theresultsoftheproposedformulationareingoodagreementwiththoseofotherresearchers.

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Fig.2.Variationoflateralpressurefromalinesurchargealongheightofwall(𝑐 = 0 t`

%. , f = 30°, 𝛿 = 10°).

Fig.3.Variationofnetforceofreinforcementfromlinesurchargealongheightofwall(𝑐 =0 t`%. , f = 30°, 𝛿 = 10°).

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4

Height(m

)

Lateralpressureduetosurcharge(kN/m2)

q=50kN/m,d=2m q=50kN/m,d=4m q=100kN/m,d=2m q=100kN/m,d=4m

0

1

2

3

4

5

6

7

8

9

10

0 2 4 6 8 10

Height(m

)

Netforceofreinforcmentduetosurcharge(kN/m)

q=50kN/m,d=2m

q=50kN/m,d=4m

q=100kN/m,d=2m

q=100kN/m,d=4m

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Table2.Comparisonofreinforcingforcedandpressureactingonawallforproposedmethodandotherresearch.

c=0kN/m2

Proposedmethod

AhmadabadiandGhanbari

(2009)

AASHTO(2007)

f=20°ΣTi

(kN/m)432.0 442.7 440.0

P(kN/m) 51.4 43.0 50.3

f=30°ΣTi

(kN/m)262.0 294.8 303.8

P(kN/m) 41.6 35.5 30.1

6. Proposedformulaforcalculatingactiveearthpressureandreinforcingforcesfromalinesurcharge

AccordingtoGhanbariandTaheri(2012)Anewanalyticalformulatocalculatethelateralearthpressureinretainingwallswithfrictionalbackfillisproposed. .TheproposeddistributionoflateralpressurealongtheheightofareinforcedwallundertheeffectofalinesurchargeisshowninFig.4. Theproposedformulaforcalculatingthemaximumlateralpressurefromalinesurchargeonareinforcedwallwithcohesionlessbackfillisasfollows:

Fig.4.Proposeddistributionforlateralearthpressuresubjecttolinesurcharge.

𝑃) =x.)y2-.z

)0{|B d{|BK}{|B f

(13)

ℎ = 0.15𝐻 (14)

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7. ComparisonbetweenProposedformulaforcalculatingreinforcingforcesfromalinesurchargeandcasestudyinTehranbyPLAXIS

PLAXIS isafiniteelementbasesoftwarewhich isusedtoanalyzethestabilityand

displacementofgeotechnicalproblems.Inthissoftware,byusingsimplegraphicalmethods,modelcanbemadeandaftersomecalculation,resultswithcomprehensivedetailsareaccessible. Inordertodomodeling,anapplicablematerialmodelmustassigntosoilandwallcollectionandMohr-coulombisacceptablematerialbehaviorwhichisusedtogeneratethemodel.Whenapropermodelismade,PLAXIScanstartthecalculation.Inthecalculationprocesstheexclusivephasesmustdefinethen,theoutputs canbeobserved in eachphase. In this study, a comparisonbetween thetensile forceof reinforcementwhich inducedbyhorizontal slicemethodandcasestudybyPLAXIS,isdoneandtheresultsareshown.ThecharacteristicsofthecaseinTehranwhichischosentostudyandcomparearesimilartothesupposedmodelandisshownintheTable3.Theviewofmodelalsoisshowninthefigure5.Figs.7and8illustrate comparison between the lateral earth pressure distribution and themaximumvaluesbasedonEqs.(13)and(14)andtheresultsobtainedfromPLAXISandalsoHSMmethod.

Table3.CharacteristicsoftheCaseinTehran

degree 30 FrictionAngle )j( 2kg/cm 0.2 Cohesion

m 11 Heightofwall -9 Numberofreinforcement

FIG.5.RepresentativemodelofcasestudyinTehran

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Fig.6.ComparisionamongProposedformulaandHSMandPLAXIScalculationfor50kN/m

linesurchargealongheightofwall

Fig.7.ComparisionamongProposedformulaandHSMandPLAXIcalculationfor100kN/mlinesurchargealongheightofwall

8. Conclusion Thispaperproposesanewcomparisonamongthreeapproaches.ThefirstbasedonHSMtocalculateactiveearthpressurefromalinesurcharge,thesecondisrelatedontheformulafromTaheriandGhanbari(2012)andthelastbasedoncasestudywhichiscalculatedbyPLAXIS.Theresultsshowedthatanincreaseinthethesurchargeleadtoincreasesthelateralpressure.PLAXIScalculationshowsthehighervaluesthantheothermethods. Comparisons show that the results from the proposedmethod are at the

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lowerboundsforthevaluesobtainedbyPLAXISandalsosomecodessuchas,AASHTO(2007)

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