simplifiedmethodforcalculatingtheactiveearthpressureon...

Research ArticleSimplified Method for Calculating the Active Earth Pressure onRetaining Walls of Narrow Backfill Width Based onDEM Analysis

Minghui Yang1 and Bo Deng 2

1Associate Professor College of Civil Engineering Hunan University Changsha Hunan 410082 China2PhD Candidate College of Civil Engineering Hunan University Changsha Hunan 410082 China

Correspondence should be addressed to Bo Deng parl_d126com

Received 22 December 2018 Revised 27 February 2019 Accepted 10 March 2019 Published 4 April 2019

Academic Editor Victor Yepes

Copyright copy 2019 Minghui Yang and Bo Deng +is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Spaces for backfills are often constrained and narrowed when retaining walls must be built close to existing stable walls in urban areasor near rock faces in mountainous areas +e discrete element method (DEM) using Particle Flow Code (PFC-2D) software wasemployed to simulate the behavior of cohesionless soil with narrow width behind a rigid retaining wall when the wall translationmoved away from the soils+e simulations focused on the failure model of the soil when the movement of the wall reaches the valuewhere active earth pressure occurs and the shape of the sliding surface was captured +en based on the limit equilibrium methodwith the obtained slip surfaces in PFC-2D a simplified analytical method is presented to obtain a solution of the active earth pressureacting on rigid retaining with narrow backfill width +e point of application of the active earth pressure is also obtained +ecalculated values agree well with those from physical tests in the previous literature Furthermore the effects of the width of thebackfill internal friction angle of soil and wall-soil friction angle on the distribution of active earth pressure are discussed

1 Introduction

In engineering practice for simplification traditionalmethods such as those based on the Rankine theory andCoulomb theory are commonly used to estimate the activelateral earth pressure However both of the theories assumethe backfills can extend to a sufficient distance in which thefailure plane can fully develop so that these approachescannot take account of the influence of the backfill widthbehind the wall [1 2]+is is not a realistic case in fact moreand more cases with narrow backfill width are considered inrecent years eg rock formations are close to the wall inmountain area [3ndash6] mechanically stabilized earth (MSE)walls built in front of previously stabilized walls in order towiden the existed highways for traffic [7ndash9] and the pitsupporting structure built near the existing building inurban agglomeration areas all as shown in Figure 1 In thesecases the narrow backfill has an obvious influence on the fulldevelopment of a failure wedge when predicted by the

Rankine theory or Coulombrsquos theory It indicates the lim-itation of those methods in the utilization

Furthermore the discrepancy of the active earth pressureof narrow cohesionless backfill against rigid retaining wallsis calculated by traditional theories and the field data havebeen verified by centrifugemodel tests Frydman and Keissar[10] carried out a series of centrifugal model tests on rigidretaining walls with sand backfill to observe the changes inearth pressures behind the wall from at-rest conditions toactive conditions It is observed that the coefficient of activelateral earth pressure decreases with the depth and is smallerthan the value calculated with the Rankine theory Cen-trifugal tests [11] were carried out to investigate on thearching effects on unyielding retaining walls with narrowbackfill width while the lateral earth pressure acting on aretaining wall with narrow backfills is clearly smaller thanthe estimation based on the Rankine theory or Coulombrsquostheory Afterwards centrifugal model tests on reinforced soilwalls adjacent to a stable face conducted by Woodruff [12]

HindawiAdvances in Civil EngineeringVolume 2019 Article ID 1507825 12 pageshttpsdoiorg10115520191507825

indicate that the slip line of the backfill is bilinear rather thanthe linear failure plane with an inclination angle of (45deg +φ2) from the horizontal assumed by the Rankine theorywhere φ is the internal friction angle of the soil All thesesuggest that traditional methods that assume the Rankinefailure plane to evaluate the active thrusts are not applicablefor narrow walls and an appropriate approach must be used

As a consequence some novel approaches should beprovided for solving this problem In this regard numericalmethods such as the finite element method (FEM) thediscrete element method (DEM) and finite element limitanalysis (FELA) were used to reveal the failure mechanic ofthe narrow backfill behind the retaining wall under morecomplex geometrical and geotechnical conditions in practice[4 8 13ndash15] Taking the various geometries of the limitedbackfill space Fan and Fang [4] found that the coefficients ofthe active earth pressure are considerably less than theCoulomb solution based on FEM while the result of activeearth pressure is generally located at locations higher thanthe lower one-third of the wall height In addition the in-stability mechanism of the backfill under constrained spaceswas revealed and the reduction factors of the active earthpressure had been also proposed based on FEM [8 14] whenthe soil was reinforced Li et al [13] built a series of discreteelement models with different soil widths to simulate thetransition of the resultant lateral force and found that theresultant lateral force exerted on the wall decreases with the

wall movement and eventually reaches a constant valueChen et al [15] explored the active earth pressure acting onthe retaining wall of a narrow backfill under the translationmode using a finite element limit analysis +e results showthat due to the boundary conditions reflective shear bandsoccur in the backfill when it is failing

However the numerical method is seldom used becauseof its complexity hence many formulas deduced fromanalytical studies were proposed to satisfy the requirementof the practice engineers Spangler and Handy [16] pio-neered to develop an equation based on Janssenrsquos archingtheory [17] for calculating the lateral pressure caused bycohesiveness soil acting on the wall of a silo Li and Aubertin[18] modified the Marston-based arching solution in orderto consider the nonuniform distribution of the verticalstresses exerted on the walls retaining stope fills and thepore pressure effects were further considered [18] +esestudies found that the reason for the reduction of the lateralearth pressure acting on a retaining wall with narrowbackfills is mostly due to soil arching effects Additionallytaking the advantage of simplicity of the formulation an-alytical solutions also have been derived for retaining wallswith narrow backfills under the assumption that failuresurfaces are planar and segmental [5 19] All these methodsare useful for assessing the earth pressure for retaining wallswith narrow backfills though with various degrees ofapproximation

Retaining wall

Backfill

Rock face

Rankine active wedge

45deg + φ2

(a)

Retaining wall

Existing road Extended roadhelliphellip

Stable face


Backfill

45deg + φ2

(b)

The foundation of the existing building

Deep excavation

Backfill

Retainingstructure

45deg + φ2


(c)

Figure 1 A schematic for narrow width soil behind a retaining wall (a) near a rock (b) near an existing road (c) near an existing building

2 Advances in Civil Engineering

It is well worth pointing out that in theoretical analyses areasonable assumption for the slip surface is fundamental tothe study of active earth pressure In this regardmany types ofthe planar slip surface have been proposed so far such as lines(Coulomb and Rankine theory) segment lines [5 19] circles[20] catenary lines [21] and curves composed of a line and alogarithmic spiral [22 23] Nevertheless little attention inprevious researches has been paid to the change of the slipsurface with which the narrow soil behind a retaining wallvaries with the width of the backfill To address this problemthe DEM was involved to analyze the exact slip surface shapeof the soil behind a retaining wall with different aspect ratiosof the limited backfill space Additionally a simple solutionfor evaluating the active earth pressure exerted by narrowbackfill based on the exact slip surface in terms of the limitequilibrium method was presented Meanwhile the distri-bution as well as the location of the resultant of active earthpressures was also discussed in the paper

2 Numerical Modeling by Using DEMand Verification

21 PFC-2D Over the last few decades the discrete elementmethod (DEM) has been widely used by many researchers[24ndash29] especially in the study of particulate mechanics formacroscopic granular materials +e prime advantage of thismethod is its simplicity to simulate the interaction betweengranular particles and other boundaries which means nodescription of the deformation behavior of granular materialsitself is required Furthermore each distinct element isallowed to translate rotate and even arbitrarily separate tosimulate the large deformation problems of granular soilscaused by the movements of retaining wall +erefore theParticle Flow Code PFC-2D3D based on the theory of DEMcan be used to analyze the behavior of granular soils It is wellknown that there are limitations when using the PFC-2Dmodel when compared with the PFC-3D model eg smallernumber of particles and increasing arching effects in thesimulation Nevertheless at present it is still very usual to usethe PFC-2Dmodel to simulate behavior of granular soils egearth pressure [13 30 31] and these studies can obtain goodsimulation results Herein a numerical model establishedusing the PFC-2D code program [32] was developed to in-vestigate the behavior of narrow width soil behind a retainingwall which was moved away from the soil gradually untilreached the active state and formulated the final slip surface

Unlike the analytical approach based on the continuousmedium theory the soil sample is assumed to be assembledby plenty of discrete rigid particles in PFC-2D with whichonly the equilibrium equation of the element should betaken into account Hence the unbalanced forces or mo-ments acting on the every particle will prompt it to moveaccording to Newtonrsquos second law of motion while themovement of each particle is affected by the resistance forcesfrom other particles in contact as well PFC-2D repeatedlyutilizes explicit time-stepping successive cycles until theequilibrium equations for every single particle are satisfied+e forces and displacements of the particles are based onNewtonrsquos second law of motion and the contact forces based

on the force-displacement law Additionally the macro-mechanical properties of granular soils and their contact willbe determined by some micromechanical parameters whichare discussed subsequently in this study

22 General Model for Soil behind Retaining Wall +e nu-merical model used for the failure mode of narrow co-hesionless soil behind a retaining wall investigation is 9m inheight with upper 85m-high section and the lower 05m-high section as shown in Figure 2 +e lower section wasfixed as the cushion layer for reducing boundary effects +ewall on the right is fixed to model the stationary wall and theleft side wall moves away from the material and the ho-mogeneous soil was modeled as disc-shaped particles withradius ranging from 003m to 005m Notice that the se-lected particles were chosen here just for saving computingtime To ensure the uniformity of the numerical specimensthe nine horizontal layers were prepared in this study Foreach layer a set of particles is first generated inside a boxwithout contacting and then allowing them to movedownward and come into contact with each other undergravity (g 981ms2) and the standard of the equilibriumstate was the maximum contact force ratio reaching 0001Noted that all the wall elements are fixed and the frictioncoefficient of the particle-wall interface is set to 0 during thesample preparation stage

After the last layer reaching the equilibrium state thefriction coefficient of the particles was then increased to amaximum value so as to compensate the lack of angularity forcircular particles +e friction coefficient (μw) between theparticles and the walls were kept constant during the wholeprocess +us the general calculated model was generated

23 Material Properties All the tiny particles the rigidsegments and the contact between particles should beendowed some micromechanical parameters to reflect theknown macromechanical properties In this study a linearcontact stiffness model governed by the value of normal andtangential stiffness was used to describe the relationshipbetween the force and displacement of each particle as wellas the rigid segments+e parameters of stiffness and contactbond strength of the particles can be typically calculated bythe following relationships [26 32]

ks 12kn

kn t middot Ec(1)

where ks is the tangential stiffness of the particles kn is thenormal stiffness of the particles t is the thickness of particlesalong the plane of paper with the default value 10m Ec isthe Youngrsquos modulus of the particleparticle contact

Furthermore in PFC-2D some physical experiments atthe macrolevel eg biaxial test simulation was also usuallyused to calibrate the micromechanical parameters of theparticle mentioned before ie ks kn and μs and this ap-proach was adopted in the present paper Note that here thephysical experiments are used for inversion of laboratory tests(eg triaxial test or direct shear test) Since the computational

Advances in Civil Engineering 3

time is highly depending on the particle numbers in DEMsimulation the ldquoupscalingrdquo technique of modeling particleswas used here which has been presented by many scholars[33ndash35] As such the biaxial sample with a width of 42m anda height of 84m generated in the same manner as mentionedabove herein was used to calibrate the micromechanicalparameters of soils and the dimensions of the sample weredetermined by the data of a laboratory test in the previousliterature [10] as a veried case discussed later

e dimensions (width and height) of the biaxial samplewere not realistic for a laboratory test however they werechosen to allow the generation of a sucient number ofparticles (number of particles 3422) for the biaxial testsimulatione deviatoric stress versus the axial strain in thebiaxial test with three dierent conning pressures (10 kPa20 kPa and 40 kPa) is shown in Figure 3e q-pprime plot can becurved by the three Mohrrsquos stress circle with dierentconning pressures erefore the friction angle of theassembly dened as φ can be determined by the inclinationangle of the q-pprime line

q

pprime sinφ 0588 (2)

where φ 36deg which was much smaller than the frictionangle of the particleparticle contact e micromechanicalparameters governing the response of the materials at themacrolevel are presented in Table 1

24 Estimation of the Active Earth Pressure along the WallAs shown in Figure 2 the translational wall moves outwardfrom the soil with a slight linear velocity (00001ms) which

can satisfy the quasi-static conditions and is close to thevelocity adopted by Jiang et al [30] It was paused whendisplacement of the wall reached the 01 percent of the heightof the wall At this moment the active earth pressure wasassumed to occur [14] However the stress of the soil inDEM is discontinuous Only the average contact stressbetween the particle and wall in a certain zone can beattained us the mean lateral stress acting on the wall in acertain zone was identied as the active earth pressure [32]Furthermore in total twenty measurement circles with aradius of 04m were arranged at the contact point closed tothe removable wall from top to bottom as shown in Figure 2to measure the mean lateral stress at the nal stage of thesimulation Note that the above results were obtained after

Soil particles

Zoom-in

Fixed wall

Rigid wall

Cushionlayer

bprimeaprime Measurement circle

aprime translation movementbprime rotation about the base

05m

b

Zoom-in

R

H

Figure 2 DEM model for soil behind a retaining wall

Confining pressure

40kPa

180

160

Axi

al d

evia

toric

stre

ss (k

Pa) 140

120

100

80

60

40

20

0

20kPa

10kPa

0000Axial strain

0005 0010 0015 0020

Figure 3 Deviatoric stress versus axial strain from PFC-2D biaxialtest simulations


the translational wall stopped moving and the backfillreached the equilibrium state under gravity

25 Validation Frydman and Keissar [10] conducted aseries of centrifuge tests to investigate the earth pressureacting on retaining walls near rock faces in active conditionswhich were used as a classical verified example by manyresearchers [4 8] In the tests the uniform fine sand with theparticle size in the range of 01sim03mm average unit weight(c) of 152 kNm3 the relative density (Dr) of 70 theinternal friction angle (φ) of 36deg and interface friction angle(δ) of 25deg was used as backfill materials +e micro-mechanical parameters are listed in Table 1 +e retainingwall was made of wooden board with a height of 195mmwhich can simulate the similar stress level (437 g) for a full-scale wall with a height of 85m Some load cells were thenarranged at the wall face near the top and bottom of the wallto capture the lateral pressure acting on the wall when thewall rotated around its base

Figure 4 shows the comparison among the computednormalized active earth pressures (σxcz) along the normal-ized depth from the DEM analyses where z is the depth belowthe wall top and b 1m is the soil width the theoretic solutionbased on arching equation [16] and the measured data fromthe centrifuge model wall test Obviously DEM simulation isin a good agreement with measurement which means thatDEM can simulate the distribution of the active earth pressureled by the movement of coarse-grain materials very well

3 Analysis of the Failure Slip SurfaceBased on DEM

+e typical model of the granular soil behind retaining wallsconsidered in this study is shown in Figure 2 +e height ofthe wall was kept as a constant value (H 85m) whereas theinternal frictional angle of the soil (φ) varied from 20deg to 45degFurthermore the movement of the wall was considered ashorizontal translation only and the other associated pa-rameters used in the analysis are shown in Table 1

In order to study the effect of the aspect ratio (β bH) ofbackfill on the failure model of the soil behind a rigidretaining wall different widths (b) of the backfill wereconsidered Figure 5 shows the displacement field of the soilwith different widths and a fixed internal friction angle value(φ 30deg) when the translation movement of the wall reaches

01 percent of the height of the wall According to theRankine and Coulomb theory the minimum width of thebackfill for active failure wedge that can be fully developed iseasily calculated as 50m and 5889m respectively +ere-fore the widths of backfills are arranged in a range from20m to 80m which are equivalent to the aspect ratio of thebackfill denoted as β varies from 0235 to 0941

+e displacement field of the soil in the dark blue zone isalmost close to zero as illustrated in Figure 5 and this areacan be identified as a stationary region when the wall movesthe similar way has been used by Nadukuru and Micha-lowski [29] +erefore the slip surface of the soil which isinterpreted as the boundary between the static and movingsoil can be approximately obtained with sufficient accuracywhile red lines were used to simulate the slipping of the soilwith an inclination angle denoted as θ

As can be seen in Figure 5 all the slipping lines arepassing through the heel of the retaining wall to the topsurface of the soil when the value of β meets the minimumdistance requirement according to the Rankine theory (ieβgt 0588) However with narrow backfill widths partialfailure surfaces were developed without passing throughthe heel of the retaining wall to the top surface of the soilMore particularly with precise measurements it is notedthat the value of θ is kept as constant (θ 60deg) regardless ofthe width of the backfill which indicates that θ is almostindependent of β

For granular soil the inclination angle of the slippingsurface of the soil behind a retaining wall is only related to theinternal friction angle based on the Rankine theory+e shapeof the active failure wedge was also investigated with differentφ which is on a scale from 20deg to 35deg which covers theproperties for most of the granular soil as shown in Figure 6According to the one-to-one relationship between φ and μs(friction coefficient of soil particles in PFC-2D) the targetedvalue of φ can be achieved by adjusting the value of μs

Table 1 Micromechanical properties for the DEM analysis

Parameter ValueSoilUnit weight c 152 KNmNormal stiffness of particles kn 75MNmShear stiffness of particles ks 625MNmFriction coefficient μs 15Particle radius R 003ndash005mRetaining wallNormal stiffness of particles kn 2000MNmShear stiffness of particles ks 2000MNmFriction coefficient μw 10

Measured data-top cellMeasured data-bottom cell

Rankine theoryArching equationPFC result

40

35

30

25

20zb

15

10

05

0000 01

σxγz02 03 04 05 06 07

Figure 4 Comparison of the distribution of normalized activeearth pressures (σxcz) along normalized depth (zb) among thePFC-2D results experimental data and theoretical solutions


As can be seen from Figure 6 partial failure planarsurfaces passing through the heel of the retaining wall can bestill obtained with dierent values of φ whereas slightchanges of the inclination angle of the slipping surface isobserved e relationship between the values of θ and φ isapproximately linear and can be expressed as follows

θ 45deg +φ2 (3)

Exact same relations can be obtained from the Rankinetheory which indicates that limited width of the backllchanges the length rather than the inclination angle of theactive failure wedge In other words the two-dimensionalsection of the active failure wedge can be simplied as atrapezoid shape if the width of the backll is less than theminimum distance with which the active failure wedge canbe fully developederefore a newmethod to calculate the

23221E ndash 01

22000E ndash 01

20000E ndash 01

18000E ndash 01

16000E ndash 01

14000E ndash 01

12000E ndash 01

10000E ndash 01

80000E ndash 02

60000E ndash 02

40000E ndash 02

Unit m

θ

b = 80m

θ

b = 70m

θ

b = 60m

θ

b = 30m

θ

b = 40m

θ

b = 50m

θ

b = 20m

20000E ndash 02

20538E ndash 14

Figure 5 e slip surface of soil behind a retaining wall with widths varying from 20m to 80m (φ 30deg)

φ = 20degθ = 55deg

θ


θ


θ


θ


θ


θ


θ θ


5521 24 27 30 33 36

565758596061626364

θ

φ

θ = 45deg + (φ2)

Figure 6 e slip surface of soil behind a retaining wall with φ varying from 20deg to 35deg (d 3m)


active earth pressure more reasonable in such case isrequired

4 Simplified Analytical Approaches for ActiveEarth Pressure

+e estimation of active earth pressures acting on theretaining wall is extremely important for geotechnical de-sign and analytical approaches are imperative for practicaluse Due to the stress redistribution caused by soil archingphenomenon verified by many models and centrifuge teststhe classic Coulomb and Rankine active earth pressuretheories usually underestimate the lateral pressure acting onretaining walls especially when the backfill widths arenarrow As a result Spangler and Handy [16] developed ananalytical solution to simply calculate the lateral pressureacting on the wall of a silo accounting for the possible soilarching in the backfill By choosing an arbitrary horizontalslice element from the backfill with a depth z confined bytwo rigid surfaces (Figure 7(a)) the vertical frictional forcesat the interface between the backfill and the rigid surfaces

will reduce the overall vertical stress together with the lateralpressure acting on the wall +e force equilibrium of thehorizontal element in the vertical direction can be written asfollows

bdσz

dz+ 2 tan δσx minus cb 0 (4)

where σz and σx are the vertical stress at depth z re-spectively Here δ is the interface frictional angle betweenthe backfill and the rigid wall c is the unit weight of thebackfill dz is the thickness of the horizontal element andthe earth pressure coefficient K is defined as the ratio of thehorizontal stress σx over the vertical stress σz(K σxσz)which depends on the state of movement of the wall andmaterial properties In particular for active pressurecondition using the trigonometry of the Mohrrsquos circlewhich represents the stress state of a soil element adjacentto the retaining wall (Figures 7(a) and 7(b)) under activeconditions the value of K can be derived as followingequation [10]

K sin2 φ + 1( 1113857minus

sin2 φ + 1( 11138572 minus 1minus sin2 φ( 1113857 middot 4 tan2 δ minus sin2 φ + 1( 1113857

1113969

4 tan2 δ minus sin2 φ + 1( 1113857 (5)

+e solution for the lateral active pressure at a givendepth z can be obtained from the following equation

σx cb

2 tan δ1minus exp minus2K

z

bmiddot tan δ1113874 11138751113876 1113877 (6)

However arching equation pays no attention to theeffect of the slip surface shape to the distribution of thelateral earth pressure acting on the retaining wall +is isbecause the soil slice element is identified as perfectlysymmetrical with respect to the central line and the researchconducted herein aims to address this problem

As mentioned before the sliding surface of the narrowedsoil behind the retaining wall was limited to a partial planarsurface with an inclination angle dependent on the char-acteristic of the soil Hence the active earth pressure actingon the retaining wall with different depths accounting for thenarrow width of the soil can be derived +e calculationmodel is schematically shown in Figure 8 where the trap-ezoid sliding soil block denoted as ABCD is divided intotwo parts+e active earth pressure along the wall herein canbe obtained based on the conditions of equilibrium of forcesacting on each wedge AECD or EBD

For an arbitrary horizontal thin-layer element withinwedge AECD all the external force is illustrated inFigure 8(b) and the equilibrium equation in the z-directioncan be expressed as follows (0le zleHminus b middot tan θ)

cb middot dz + σzbminus σz + dσz( 1113857bminus 2σx tan δ middot dz 0 (7)

Similar to the derivation process of arching equation(Spangler and Handy 1984) defining σx K middot σz where K

is the coefficient of lateral pressure on the wall is de-termined by equation (5) So equation (7) can be improvedas follows

dσz

dz cminus 2K middot σz tan

δb (8)

+e differential equation can be solved accounting forthe boundary condition at the top point of the soil (z 0)where the vertical stress σz equals to zero

σz b cminus c middot eminus2Kmiddotzmiddottan δb( 1113857

2K middot tan δ (9)

And the lateral stress σx is thus given by the followingequation

σx b cminus c middot eminus2Kmiddotzmiddottan δb( 1113857

2 tan δ (10)

Similarly according to the force analysis for an arbitraryhorizontal thin-layer element within wedge EBD the lateralthrust acting on the lower part of the wall is also deducedbased on the equilibrium conditions of both directions x

and z directions Under such a balanced force system theapplied forces acting on the wedge EBD can be simplified astwo concentrated loads ie normal force N and tangentialforce T along the sliding surface Meanwhile N and T satisfya specific relationship ie T N tanφ

+erefore for 1113936 Fx 0 the equilibrium equation in thex-direction can be expressed as follows (0lexle b)

σx +T

tan θ R (11)


Similarly for 1113936 Fz 0 the equilibrium equation in the x-direction can be expressed as follows (Hminus b middot tan θle zleH)where the second-order small quantities are ignored

dσz

dz c +

σz minusRminusT tan θminusKσz middot tan δ middot tan θHminus z

(12)

+en equation (11) can be substituted in equation (12) toobtain the following equation

dσz

dz c +

σz middot b0

Hminus z (13)

where b0 1minusK(tan δ middot tan θ + tan θ + tanφ middot tan2 θtan θminus tanφ)

+e lateral stress σx along the wall can be obtained byintegrating equation (13) to obtain the following equation

σx K middotzminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭ (14)

where C0 is an integral constant which can be obtained bycombining equations (10) and (14) with the same value ofz(z h Hminus b middot tan θ) accounting for the continuousdistribution conditions of the lateral stress along the wall

Furthermore by integrating equations (10) and (14) interms of z along the wall the resultant of active earthpressure acting on the rigid retaining wall can be obtained asfollows

Eh 1113946h

0

b cminus c middot eminus2Kmiddotzmiddottan δb( 1113857

2 tan δdz

+ 1113946H

hK middot

zminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭dz

(15)

where Eh is the resultant of active earth pressure+e total moment caused by active earth pressure to the

heel of the retaining wall can be illustrated as follows

dz

bV + dV

V = σz times b

W

z

τxz

σzτzx

σx

x

σx σx

τ τ

(a)

(σx τxz)

(σz τzx)

σ

τ

o φ δ

(b)

Figure 7 Force equilibrium analysis for slice element (a) horizontal element of the backfill between two rigid faces and (b) stress conditionsof a soil element adjacent to the back of retaining wall

Ax

z

z

b

dz

H

dz

h

C

E

B

E

D

D

dw

σz

τ τ

σz + dσz

σxσx

dw NTτ

σz + dσ

σz

σx

(a) (b)

Figure 8 Schematics of calculation model (a) block above the sliding surface and (b) analysis for thin-layer element


M inth

0

b cminus c middot eminus2Kmiddotzmiddot tan δb( )2 tan δ

middot (Hminus z)dz

+ intH

hK middot

zminusHb0 + 1

middot cminusC0

(Hminus z)2[ ]

b0+1( )2

middot (Hminus z)dz

(16)

e distance from the location of the resultant of activeearth pressures to the heel of the wall denoted as h0 can beeasily derived as the ratio of M to Eh

h0 M

Eh (17)

5 Comparison with Other Studies

e proposed new formulations for calculating the activeearth pressure acting on the retaining wall with narrowbackll are applied to some previous model tests which arewell documented in the literature

Case 1 e rst laboratory test was conducted by Tsagareli[1] where the distribution of the active earth pressuresacting on the translating rigid retaining wall with the heightof 4m and a width of 12m for narrow backll was mea-sured e other values of the parameters designed in thetests are given as follows c17658 kNm3 δ 10deg andφ 37deg Figure 9 shows the comparison among the computedearth pressure along the retaining wall from the proposedequation other theoretic solutions and the measured datafrom themodel test As can be seen in Figure 9 the estimatedvalues of earth pressure from the approach in this studyshow a good agreement with the measured data and providea better prediction in comparison with other analyticalsolutions in the latter part of the curves where the values ofearth pressure decrease with the increase of depth ephenomena have also been observed in other experiments[36 37] and analytical solutions [38] ese results show theconventional approaches are overly conservative

Case 2 A comparison is presented between the predictionsof active earth pressure and the measured data due to a sandbackll behind a rigid wall in a laboratory test which wasreported by Fang and Ishibashi [36] In the test the activeearth pressure distributions were observed with three dif-ferent wall movement modes (1) rotation about top (2)rotation about heel and (3) translation e rigid wall usedin the test had a height of 10m and a width of 102m whilethe parameters of the backll are designed as c 154 kNm3δ 17deg and φ 34deg e active earth pressure versus thedepth relationships from the measured and the predictedwith the proposed and other previous approaches when therigid wall translated are presented in Figure 10 e esti-mated active earth pressure based on the proposed methodin this study is closer to the measured data than thoseobtained from other approaches especially in the lower partof the wall It is noted that the reduction of active earthpressure was not observed which is partially due to enough

width of the backll preventing the formation of soil archingduring the test

6 Parameter Analysis

In addition the eects of some prime parameters such as thewidth of the backll internal friction angle of soil and wall-soil friction angle on the distribution of active earth pressureare further analyzed in which the basic parameters are givenas follows H 85m b 30m c 180 kNm3 δ 20deg andφ 30deg

Figure 11 shows the lateral active earth pressure dis-tribution along the height of a translating rigid wall for thevarious widths of backll (the value of b is in the range from10m to 50m in which the value of aspect ratio of thebackll is from 0117 to 0588) As shown in Figure 11 themagnitude of the active earth pressure decreases with thedecrease of the backll widths and the attenuation becomesclearer when b value is smaller It can be explained that thewidth of the narrow backll plays an important role in thedevelopment of arching in the backll and the arching eectcan be strengthened with the narrower distance between theretaining wall and the existed rigid wall by undertakingmoreself-weight of the soil

In order to study the eect of the soil internal frictionangle (φ) and wall-soil friction angle (δ) on the lateral activeearth pressure against the rigid various values of φ varyingfrom 24deg to 35deg and δ varying from 6deg to 20deg were employede change of active earth pressure along the height of thewall is shown in Figures 12 and 13 As seen in Figures 12 and13 a clear similar trend is observed with dierent values of φand δ is indicates that those two parameters inpounduence themagnitude of the active earth pressure signicantly but notthe shape of its distribution In particular the values of earthpressure increase with the increasing φ value whereas

σx (kPa)

Dep

th (m

)

40

35

30

25

20

15

10

05

000 5 10 15 20 25

Coulomb theoryArching equation

Rakine theory

Proposed methodExperimental data [1]

Figure 9 Comparison of the distribution of active earth pressuresalong depth between the theoretical solution and experimental data[1]


decrease with the value of δ owing to the reduction of thewall-soil interaction supporting the self-weight of the soil

It is widely accepted that the height where the lateralresultant active force is applied normalized by wall height (h0H) maintains one-third according to the Rankine theory andCoulomb theory because of the triangular distribution of theactive earth pressure along the retaining wall However thecalculations for active earth pressure based on the Rankinetheory and Coulomb theory ignore the eect of the width ofthe backll e point of application of the resultant activeearth pressure is also investigated in this study

As can be seen in Table 2 b varies from 20 to 40 byusing the present approach Only slight changes of h0H can

be observed and the approximate h0H value is regarded as038 which is closer to the measured data from experimentsthan the computed value based on the Rankine theory andCoulomb theory

Figure 14 shows the variation of the location (h0H) ofthe resultant of active earth pressures with the dierentinternal frictional angles of soil and wall-soil frictional anglee locations of the resultant of active earth pressures de-crease with the increase of φ values while increases with theincrease of δ values For a φ value varying from 22deg to 38deg adecrease of the h0H value from 0408 to 0368 can beachieved Whereas the h0H value will increases from 0344to 0380 when the φ value varies from 6deg to 20deg In generalthe h0H value is basically in a range of 034 to 040 and thisis signicantly higher than one-third of the wall height

σx (kPa)

Dep

th (m

)

10

08

06

04

02

00 05 10 15 20 25 30 35 40 45


Rakine theory



b = 1mb = 2mb = 3m

b = 4mb = 5m

8

7

6

5

4

3

2

1

0

Dep

th (m

)

3 6 9 12 15 18 21 24 27 30σx (kPa)

Figure 11 Variation of the active earth pressures along the lengthof the retaining wall at various ll widths

φ = 24degφ = 26degφ = 28deg


8

7

6

5

4

3

2

1

0

Dep

th (m

)

5 10 15 20 25 30 35σx (kPa)

Figure 12 Variation of the active earth pressures along the lengthof the retaining wall at various internal friction angles of soil (φ)

0 5 10 15 20 25 30 35 40 45

δ = φ5δ = φ3

δ = φ2δ = φ15

σx (kPa)

8

7

6

5

4

3

2

1

0

Dep

th (m

)

Figure 13 Variation of the active earth pressures along the lengthof the retaining wall at various wall-soil friction angle of soil (δ)


7 Conclusions

e estimation of active earth pressures acting on theretaining wall is extremely important for geotechnical designand its rationality depends on the accuracy of the simulatedfailure model of the soils Compared to the conventionalapproach such as the Rankine or Coulomb theory which issuitable only for the retaining walls with narrow width of thebacklls this study presented a new simplied method usinglimit equilibrium analysis for determining the nonlineardistribution of the active earth pressure on rigid retainingwalls with limited backllsis new approach is based on thehorizontal thin-layer element method and the exact slipsurface shape that was determined by the DEM simulation(PFC-2D) A good agreement can be obtained between theproposed method and experimental results In addition theeect of the aspect ratio of the backll width and the frictionangle of the backll on the distribution of active earthpressures along the wall was also discussedemain featuresof this study are summarized below

(1) e two-dimensional section shape of the activefailure wedge of the backll with a limited width canbe conrmed as a trapezoid composed of an upperrectangle part and a lower triangle part e in-clination angle of the triangle failure wedge is relatedto the value of the internal friction angle of the soils

(2) e lateral active earth pressure decreases signi-cantly with the decrease of the width of the backllwhich indicates that the soil arching plays a moreimportant role with narrower width of the backll

(3) e point of active earth pressures is noticeablyhigher than one-third of the wall height described inthe Rankine and Coulomb theory

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conpoundicts of interest

Acknowledgments

e authors would like to acknowledge the nancial supportfrom the National Science Foundation of China undercontract no 51278184

References

[1] Z V Tsagareli ldquoExperimental investigation of the pressure ofa loose medium on retaining walls with a vertical back faceand horizontal backll surfacerdquo Soil Mechanics and Foun-dation Engineering vol 2 no 4 pp 197ndash200 1965

[2] X L Yang and J H Yin ldquoEstimation of seismic passive earthpressures with nonlinear failure criterionrdquo EngineeringStructures vol 28 no 3 pp 342ndash348 2006

[3] S G Lee and S R Hencher ldquoe repeated failure of a cut-slope despite continuous reassessment and remedial worksrdquoEngineering Geology vol 107 no 1-2 pp 16ndash41 2009

[4] C C Fan and Y S Fang ldquoNumerical solution of active earthpressures on rigid retaining walls built near rock facesrdquoComputers and Geotechnics vol 37 no 7-8 pp 1023ndash10292010

[5] J J Chen M G Li and J H Wang ldquoActive earth pressureagainst rigid retaining walls subjected to conned co-hesionless soilrdquo International Journal of Geomechanicsvol 17 no 6 article 06016041 2016

[6] M H Yang and X Tang ldquoRigid retaining walls with narrowcohesionless backlls under various wall movement modesrdquoInternational Journal of Geomechanics vol 17 no 11 article04017098 2017

[7] K T Kniss K H Yang S G Wright and J G ZornbergldquoEarth pressures and design consideration of narrow MSEwallrdquo in Proceedings of the Texas Section ASCE Spring ASCEReston VA USA 2007

[8] K H Yang and C N Liu ldquoFinite-element analysis of earthpressures for narrow retaining wallsrdquo Journal of Geo-Engineering vol 2 no 2 pp 43ndash52 2007

[9] K H Yang J Ching and J G Zornberg ldquoReliability-baseddesign for external stability of narrow mechanically stabilizedearth walls calibration from centrifuge testsrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 137no 3 pp 239ndash253 2011

[10] S Frydman and I Keissar ldquoEarth pressure on retaining wallsnear rock facesrdquo Journal of Geotechnical Engineering vol 113no 6 pp 586ndash599 1987

[11] W A Take and A J Valsangkar ldquoEarth pressures on un-yielding retaining walls of narrow backll widthrdquo CanadianGeotechnical Journal vol 38 no 6 pp 1220ndash1230 2001

Table 2 Variation of the location of the resultant (hH) of activeearth pressures with various ll widths

Methodsb

Average20m 25m 30m 35m 40m

Coulomb theory 033 033 033 033 033 033Experimental study[1] 043 042 043 041 041 042

Presented approach 0391 0380 0392 0371 0372 0381

δ (deg

)

φ (deg

)

Variation of h0Hwith various φ

Variation of h0Hwith various δ

22

24

26

28

30

32

34

36

38

035 036 037 038 039 040 0410346

8

10

12

14

16

18

20

Figure 14 Variation of the location of the resultant (h0H) of activeearth pressures with dierent internal frictional angles of soil (φ)and wall-soil frictional angle (δ)


[12] R Woodruff ldquoCentrifuge modeling of MSE-shoring com-posite wallsrdquo Master thesis Department of Civil Engineering+e University of Colorado Boulder CO USA 2003

[13] M G Li J J Chen and J H Wang ldquoArching effect on lateralpressure of confined granular material numerical and the-oretical analysisrdquo Granular Matter vol 19 no 2 2017

[14] D Leshchinsky Y Hu and J Han ldquoLimited reinforced spacein segmental retaining wallsrdquo Geotextiles and Geomembranesvol 22 no 6 pp 543ndash553 2004

[15] F Q Chen Y J Lin and D Y Li ldquoSolution to active earthpressure of narrow cohesionless backfill against rigid retainingwalls under translation moderdquo Soils and Foundations 2018 inPress

[16] M G Spangler and R L Handy Soil Engineering Harper andRow New York NY USA 1984

[17] H A Janssen ldquoVersuche uber getreidedruck in silozellenrdquoZeitschrift des Vereines Deutscher Ingenieure vol 39pp 1045ndash1049 1895

[18] L Li and M Aubertin ldquoAn analytical solution for the non-linear distribution of effective and total stresses in verticalbackfilled stopesrdquo Geomechanics and Geoengineering vol 5no 4 pp 237ndash245 2010

[19] V Greco ldquoActive thrust on retaining walls of narrow backfillwidthrdquo Computers and Geotechnics vol 50 pp 66ndash78 2013

[20] W F Chen and N Snitbhan ldquoOn slip surface and slopestability analysisrdquo Soils and Foundations vol 15 no 3pp 41ndash49 1975

[21] R L Handy ldquo+e arch in soil archingrdquo Journal of Geo-technical Engineering vol 111 no 3 pp 302ndash318 1985

[22] J Kumar ldquoSeismic passive earth pressure coefficients forsandsrdquo Canadian Geotechnical Journal vol 38 no 4pp 876ndash881 2001

[23] K S Subba Rao and D Choudhury ldquoSeismic passive earthpressures in soilsrdquo International Journal of GeotechnicalEngineering vol 131 no 1 pp 131ndash135 2005

[24] P A Cundall and O D L Strack ldquoA discrete numericalmodel for granular assembliesrdquo Geotechnique vol 29 no 1pp 47ndash65 1979

[25] C +ornton ldquoNumerical simulations of deviatoric sheardeformation of granular mediardquo Geotechnique vol 50 no 1pp 43ndash53 2000

[26] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of RockMechanics andMiningSciences vol 41 no 8 pp 1329ndash1364 2004

[27] L Cui C OrsquoSullivan and S OrsquoNeill ldquoAn analysis of thetriaxial apparatus using a mixed boundary three-dimensionaldiscrete element modelrdquo Geotechnique vol 57 no 10pp 831ndash844 2007

[28] J Han A Bhandari and F Wang ldquoDEM analysis of stressesand deformations of geogrid-reinforced embankments overpilesrdquo International Journal of Geomechanics vol 12 no 4pp 340ndash350 2012

[29] S S Nadukuru and R L Michalowski ldquoArching in distri-bution of active load on retaining wallsrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 138 no 5pp 575ndash584 2012

[30] M Jiang J He J Wang F Liu and W Zhang ldquoDistinctsimulation of earth pressure against a rigid retaining wallconsidering inter-particle rolling resistance in sandy backfillrdquoGranular Matter vol 16 no 5 pp 797ndash814 2014

[31] Y Gao and Y H Wang ldquoExperimental and DEM exami-nations of K0 in Sand under different loading conditionsrdquoJournal of Geotechnical and Geoenvironmental Engineeringvol 140 no 5 article 04014012 2014

[32] Itasca Consulting Group PFC2D =eory and BackgroundItasca Consulting Group Inc Minneapolis MN USA 2004

[33] Y HWang and S C Leung ldquoA particulate-scale investigationof cemented sand behaviorrdquo Canadian Geotechnical Journalvol 45 no 1 pp 29ndash44 2008

[34] V D H Tran M A Meguid and L E Chouinard ldquoA finite-discrete element framework for the 3D modeling of geogrid-soil interaction under pullout loading conditionsrdquo Geotextilesand Geomembranes vol 37 pp 1ndash9 2013

[35] Z Wang F Jacobs and M Ziegler ldquoVisualization of loadtransfer behaviour between geogrid and sand using PFC2DrdquoGeotextiles and Geomembranes vol 42 no 2 pp 83ndash90 2014

[36] Y S Fang and I Ishibashi ldquoStatic earth pressures with variouswall movementsrdquo Journal of Geotechnical Engineeringvol 112 no 3 pp 317ndash333 1986

[37] Z C Tang ldquoA rigid retaining wall centrifuge model test ofcohesive soilrdquo Journal of Chongqing Jiao tong Universityvol 25 no 2 pp 48ndash56 1988 in Chinese

[38] K H Paik and R Salgado ldquoEstimation of active earth pressureagainst rigid retaining walls considering arching effectsrdquoGeotechnique vol 53 no 7 pp 643ndash653 2003


International Journal of

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Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Journal of

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Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

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Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation


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wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

indicate that the slip line of the backfill is bilinear rather thanthe linear failure plane with an inclination angle of (45deg +φ2) from the horizontal assumed by the Rankine theorywhere φ is the internal friction angle of the soil All thesesuggest that traditional methods that assume the Rankinefailure plane to evaluate the active thrusts are not applicablefor narrow walls and an appropriate approach must be used

As a consequence some novel approaches should beprovided for solving this problem In this regard numericalmethods such as the finite element method (FEM) thediscrete element method (DEM) and finite element limitanalysis (FELA) were used to reveal the failure mechanic ofthe narrow backfill behind the retaining wall under morecomplex geometrical and geotechnical conditions in practice[4 8 13ndash15] Taking the various geometries of the limitedbackfill space Fan and Fang [4] found that the coefficients ofthe active earth pressure are considerably less than theCoulomb solution based on FEM while the result of activeearth pressure is generally located at locations higher thanthe lower one-third of the wall height In addition the in-stability mechanism of the backfill under constrained spaceswas revealed and the reduction factors of the active earthpressure had been also proposed based on FEM [8 14] whenthe soil was reinforced Li et al [13] built a series of discreteelement models with different soil widths to simulate thetransition of the resultant lateral force and found that theresultant lateral force exerted on the wall decreases with the

wall movement and eventually reaches a constant valueChen et al [15] explored the active earth pressure acting onthe retaining wall of a narrow backfill under the translationmode using a finite element limit analysis +e results showthat due to the boundary conditions reflective shear bandsoccur in the backfill when it is failing

However the numerical method is seldom used becauseof its complexity hence many formulas deduced fromanalytical studies were proposed to satisfy the requirementof the practice engineers Spangler and Handy [16] pio-neered to develop an equation based on Janssenrsquos archingtheory [17] for calculating the lateral pressure caused bycohesiveness soil acting on the wall of a silo Li and Aubertin[18] modified the Marston-based arching solution in orderto consider the nonuniform distribution of the verticalstresses exerted on the walls retaining stope fills and thepore pressure effects were further considered [18] +esestudies found that the reason for the reduction of the lateralearth pressure acting on a retaining wall with narrowbackfills is mostly due to soil arching effects Additionallytaking the advantage of simplicity of the formulation an-alytical solutions also have been derived for retaining wallswith narrow backfills under the assumption that failuresurfaces are planar and segmental [5 19] All these methodsare useful for assessing the earth pressure for retaining wallswith narrow backfills though with various degrees ofapproximation

Retaining wall

Backfill

Rock face


45deg + φ2

(a)

Retaining wall

Existing road Extended roadhelliphellip

Stable face


Backfill

45deg + φ2

(b)

The foundation of the existing building

Deep excavation

Backfill

Retainingstructure

45deg + φ2


(c)

Figure 1 A schematic for narrow width soil behind a retaining wall (a) near a rock (b) near an existing road (c) near an existing building










ks 12kn

kn t middot Ec(1)






q





Soil particles

Zoom-in

Fixed wall

Rigid wall

Cushionlayer



05m

b

Zoom-in

R

H


Confining pressure

40kPa

180

160

Axi

al d

evia

toric

stre

ss (k

Pa) 140

120

100

80

60

40

20

0

20kPa

10kPa

0000Axial strain

0005 0010 0015 0020

















40

35

30

25

20zb

15

10

05

0000 01

σxγz02 03 04 05 06 07




θ 45deg +φ2 (3)


23221E ndash 01

22000E ndash 01

20000E ndash 01

18000E ndash 01

16000E ndash 01

14000E ndash 01

12000E ndash 01

10000E ndash 01

80000E ndash 02

60000E ndash 02

40000E ndash 02

Unit m

θ

b = 80m

θ

b = 70m

θ

b = 60m

θ

b = 30m

θ

b = 40m

θ

b = 50m

θ

b = 20m

20000E ndash 02

20538E ndash 14



θ


θ


θ


θ


θ


θ


θ θ


5521 24 27 30 33 36

565758596061626364

θ

φ

θ = 45deg + (φ2)







bdσz



K sin2 φ + 1( 1113857minus


1113969

4 tan2 δ minus sin2 φ + 1( 1113857 (5)


σx cb


z

bmiddot tan δ1113874 11138751113876 1113877 (6)







dσz


δb (8)






2 tan δ (10)




σx +T

tan θ R (11)



dσz

dz c +


(12)


dσz

dz c +

σz middot b0

Hminus z (13)



σx K middotzminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭ (14)



Eh 1113946h

0


2 tan δdz

+ 1113946H

hK middot

zminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭dz

(15)



dz

bV + dV

V = σz times b

W

z

τxz

σzτzx

σx

x

σx σx

τ τ

(a)

(σx τxz)

(σz τzx)

σ

τ

o φ δ

(b)


Ax

z

z

b

dz

H

dz

h

C

E

B

E

D

D

dw

σz

τ τ

σz + dσz

σxσx

dw NTτ

σz + dσ

σz

σx

(a) (b)



M inth

0


middot (Hminus z)dz

+ intH

hK middot

zminusHb0 + 1

middot cminusC0

(Hminus z)2[ ]

b0+1( )2

middot (Hminus z)dz

(16)


h0 M

Eh (17)










σx (kPa)

Dep

th (m

)

40

35

30

25

20

15

10

05

000 5 10 15 20 25


Rakine theory









σx (kPa)

Dep

th (m

)

10

08

06

04

02

00 05 10 15 20 25 30 35 40 45


Rakine theory



b = 1mb = 2mb = 3m

b = 4mb = 5m

8

7

6

5

4

3

2

1

0

Dep

th (m

)

3 6 9 12 15 18 21 24 27 30σx (kPa)




8

7

6

5

4

3

2

1

0

Dep

th (m

)

5 10 15 20 25 30 35σx (kPa)


0 5 10 15 20 25 30 35 40 45

δ = φ5δ = φ3

δ = φ2δ = φ15

σx (kPa)

8

7

6

5

4

3

2

1

0

Dep

th (m

)



7 Conclusions





Data Availability




Acknowledgments


References













Methodsb




δ (deg

)

φ (deg

)



22

24

26

28

30

32

34

36

38

035 036 037 038 039 040 0410346

8

10

12

14

16

18

20

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










ks 12kn

kn t middot Ec(1)






q





Soil particles

Zoom-in

Fixed wall

Rigid wall

Cushionlayer



05m

b

Zoom-in

R

H


Confining pressure

40kPa

180

160

Axi

al d

evia

toric

stre

ss (k

Pa) 140

120

100

80

60

40

20

0

20kPa

10kPa

0000Axial strain

0005 0010 0015 0020

















40

35

30

25

20zb

15

10

05

0000 01

σxγz02 03 04 05 06 07




θ 45deg +φ2 (3)


23221E ndash 01

22000E ndash 01

20000E ndash 01

18000E ndash 01

16000E ndash 01

14000E ndash 01

12000E ndash 01

10000E ndash 01

80000E ndash 02

60000E ndash 02

40000E ndash 02

Unit m

θ

b = 80m

θ

b = 70m

θ

b = 60m

θ

b = 30m

θ

b = 40m

θ

b = 50m

θ

b = 20m

20000E ndash 02

20538E ndash 14



θ


θ


θ


θ


θ


θ


θ θ


5521 24 27 30 33 36

565758596061626364

θ

φ

θ = 45deg + (φ2)







bdσz



K sin2 φ + 1( 1113857minus


1113969

4 tan2 δ minus sin2 φ + 1( 1113857 (5)


σx cb


z

bmiddot tan δ1113874 11138751113876 1113877 (6)







dσz


δb (8)






2 tan δ (10)




σx +T

tan θ R (11)



dσz

dz c +


(12)


dσz

dz c +

σz middot b0

Hminus z (13)



σx K middotzminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭ (14)



Eh 1113946h

0


2 tan δdz

+ 1113946H

hK middot

zminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭dz

(15)



dz

bV + dV

V = σz times b

W

z

τxz

σzτzx

σx

x

σx σx

τ τ

(a)

(σx τxz)

(σz τzx)

σ

τ

o φ δ

(b)


Ax

z

z

b

dz

H

dz

h

C

E

B

E

D

D

dw

σz

τ τ

σz + dσz

σxσx

dw NTτ

σz + dσ

σz

σx

(a) (b)



M inth

0


middot (Hminus z)dz

+ intH

hK middot

zminusHb0 + 1

middot cminusC0

(Hminus z)2[ ]

b0+1( )2

middot (Hminus z)dz

(16)


h0 M

Eh (17)










σx (kPa)

Dep

th (m

)

40

35

30

25

20

15

10

05

000 5 10 15 20 25


Rakine theory









σx (kPa)

Dep

th (m

)

10

08

06

04

02

00 05 10 15 20 25 30 35 40 45


Rakine theory



b = 1mb = 2mb = 3m

b = 4mb = 5m

8

7

6

5

4

3

2

1

0

Dep

th (m

)

3 6 9 12 15 18 21 24 27 30σx (kPa)




8

7

6

5

4

3

2

1

0

Dep

th (m

)

5 10 15 20 25 30 35σx (kPa)


0 5 10 15 20 25 30 35 40 45

δ = φ5δ = φ3

δ = φ2δ = φ15

σx (kPa)

8

7

6

5

4

3

2

1

0

Dep

th (m

)



7 Conclusions





Data Availability




Acknowledgments


References













Methodsb




δ (deg

)

φ (deg

)



22

24

26

28

30

32

34

36

38

035 036 037 038 039 040 0410346

8

10

12

14

16

18

20

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




q





Soil particles

Zoom-in

Fixed wall

Rigid wall

Cushionlayer



05m

b

Zoom-in

R

H


Confining pressure

40kPa

180

160

Axi

al d

evia

toric

stre

ss (k

Pa) 140

120

100

80

60

40

20

0

20kPa

10kPa

0000Axial strain

0005 0010 0015 0020

















40

35

30

25

20zb

15

10

05

0000 01

σxγz02 03 04 05 06 07




θ 45deg +φ2 (3)


23221E ndash 01

22000E ndash 01

20000E ndash 01

18000E ndash 01

16000E ndash 01

14000E ndash 01

12000E ndash 01

10000E ndash 01

80000E ndash 02

60000E ndash 02

40000E ndash 02

Unit m

θ

b = 80m

θ

b = 70m

θ

b = 60m

θ

b = 30m

θ

b = 40m

θ

b = 50m

θ

b = 20m

20000E ndash 02

20538E ndash 14



θ


θ


θ


θ


θ


θ


θ θ


5521 24 27 30 33 36

565758596061626364

θ

φ

θ = 45deg + (φ2)







bdσz



K sin2 φ + 1( 1113857minus


1113969

4 tan2 δ minus sin2 φ + 1( 1113857 (5)


σx cb


z

bmiddot tan δ1113874 11138751113876 1113877 (6)







dσz


δb (8)






2 tan δ (10)




σx +T

tan θ R (11)



dσz

dz c +


(12)


dσz

dz c +

σz middot b0

Hminus z (13)



σx K middotzminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭ (14)



Eh 1113946h

0


2 tan δdz

+ 1113946H

hK middot

zminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭dz

(15)



dz

bV + dV

V = σz times b

W

z

τxz

σzτzx

σx

x

σx σx

τ τ

(a)

(σx τxz)

(σz τzx)

σ

τ

o φ δ

(b)


Ax

z

z

b

dz

H

dz

h

C

E

B

E

D

D

dw

σz

τ τ

σz + dσz

σxσx

dw NTτ

σz + dσ

σz

σx

(a) (b)



M inth

0


middot (Hminus z)dz

+ intH

hK middot

zminusHb0 + 1

middot cminusC0

(Hminus z)2[ ]

b0+1( )2

middot (Hminus z)dz

(16)


h0 M

Eh (17)










σx (kPa)

Dep

th (m

)

40

35

30

25

20

15

10

05

000 5 10 15 20 25


Rakine theory









σx (kPa)

Dep

th (m

)

10

08

06

04

02

00 05 10 15 20 25 30 35 40 45


Rakine theory



b = 1mb = 2mb = 3m

b = 4mb = 5m

8

7

6

5

4

3

2

1

0

Dep

th (m

)

3 6 9 12 15 18 21 24 27 30σx (kPa)




8

7

6

5

4

3

2

1

0

Dep

th (m

)

5 10 15 20 25 30 35σx (kPa)


0 5 10 15 20 25 30 35 40 45

δ = φ5δ = φ3

δ = φ2δ = φ15

σx (kPa)

8

7

6

5

4

3

2

1

0

Dep

th (m

)



7 Conclusions





Data Availability




Acknowledgments


References













Methodsb




δ (deg

)

φ (deg

)



22

24

26

28

30

32

34

36

38

035 036 037 038 039 040 0410346

8

10

12

14

16

18

20

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia
















40

35

30

25

20zb

15

10

05

0000 01

σxγz02 03 04 05 06 07




θ 45deg +φ2 (3)


23221E ndash 01

22000E ndash 01

20000E ndash 01

18000E ndash 01

16000E ndash 01

14000E ndash 01

12000E ndash 01

10000E ndash 01

80000E ndash 02

60000E ndash 02

40000E ndash 02

Unit m

θ

b = 80m

θ

b = 70m

θ

b = 60m

θ

b = 30m

θ

b = 40m

θ

b = 50m

θ

b = 20m

20000E ndash 02

20538E ndash 14



θ


θ


θ


θ


θ


θ


θ θ


5521 24 27 30 33 36

565758596061626364

θ

φ

θ = 45deg + (φ2)







bdσz



K sin2 φ + 1( 1113857minus


1113969

4 tan2 δ minus sin2 φ + 1( 1113857 (5)


σx cb


z

bmiddot tan δ1113874 11138751113876 1113877 (6)







dσz


δb (8)






2 tan δ (10)




σx +T

tan θ R (11)



dσz

dz c +


(12)


dσz

dz c +

σz middot b0

Hminus z (13)



σx K middotzminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭ (14)



Eh 1113946h

0


2 tan δdz

+ 1113946H

hK middot

zminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭dz

(15)



dz

bV + dV

V = σz times b

W

z

τxz

σzτzx

σx

x

σx σx

τ τ

(a)

(σx τxz)

(σz τzx)

σ

τ

o φ δ

(b)


Ax

z

z

b

dz

H

dz

h

C

E

B

E

D

D

dw

σz

τ τ

σz + dσz

σxσx

dw NTτ

σz + dσ

σz

σx

(a) (b)



M inth

0


middot (Hminus z)dz

+ intH

hK middot

zminusHb0 + 1

middot cminusC0

(Hminus z)2[ ]

b0+1( )2

middot (Hminus z)dz

(16)


h0 M

Eh (17)










σx (kPa)

Dep

th (m

)

40

35

30

25

20

15

10

05

000 5 10 15 20 25


Rakine theory









σx (kPa)

Dep

th (m

)

10

08

06

04

02

00 05 10 15 20 25 30 35 40 45


Rakine theory



b = 1mb = 2mb = 3m

b = 4mb = 5m

8

7

6

5

4

3

2

1

0

Dep

th (m

)

3 6 9 12 15 18 21 24 27 30σx (kPa)




8

7

6

5

4

3

2

1

0

Dep

th (m

)

5 10 15 20 25 30 35σx (kPa)


0 5 10 15 20 25 30 35 40 45

δ = φ5δ = φ3

δ = φ2δ = φ15

σx (kPa)

8

7

6

5

4

3

2

1

0

Dep

th (m

)



7 Conclusions





Data Availability




Acknowledgments


References













Methodsb




δ (deg

)

φ (deg

)



22

24

26

28

30

32

34

36

38

035 036 037 038 039 040 0410346

8

10

12

14

16

18

20

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



θ 45deg +φ2 (3)


23221E ndash 01

22000E ndash 01

20000E ndash 01

18000E ndash 01

16000E ndash 01

14000E ndash 01

12000E ndash 01

10000E ndash 01

80000E ndash 02

60000E ndash 02

40000E ndash 02

Unit m

θ

b = 80m

θ

b = 70m

θ

b = 60m

θ

b = 30m

θ

b = 40m

θ

b = 50m

θ

b = 20m

20000E ndash 02

20538E ndash 14



θ


θ


θ


θ


θ


θ


θ θ


5521 24 27 30 33 36

565758596061626364

θ

φ

θ = 45deg + (φ2)







bdσz



K sin2 φ + 1( 1113857minus


1113969

4 tan2 δ minus sin2 φ + 1( 1113857 (5)


σx cb


z

bmiddot tan δ1113874 11138751113876 1113877 (6)







dσz


δb (8)






2 tan δ (10)




σx +T

tan θ R (11)



dσz

dz c +


(12)


dσz

dz c +

σz middot b0

Hminus z (13)



σx K middotzminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭ (14)



Eh 1113946h

0


2 tan δdz

+ 1113946H

hK middot

zminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭dz

(15)



dz

bV + dV

V = σz times b

W

z

τxz

σzτzx

σx

x

σx σx

τ τ

(a)

(σx τxz)

(σz τzx)

σ

τ

o φ δ

(b)


Ax

z

z

b

dz

H

dz

h

C

E

B

E

D

D

dw

σz

τ τ

σz + dσz

σxσx

dw NTτ

σz + dσ

σz

σx

(a) (b)



M inth

0


middot (Hminus z)dz

+ intH

hK middot

zminusHb0 + 1

middot cminusC0

(Hminus z)2[ ]

b0+1( )2

middot (Hminus z)dz

(16)


h0 M

Eh (17)










σx (kPa)

Dep

th (m

)

40

35

30

25

20

15

10

05

000 5 10 15 20 25


Rakine theory









σx (kPa)

Dep

th (m

)

10

08

06

04

02

00 05 10 15 20 25 30 35 40 45


Rakine theory



b = 1mb = 2mb = 3m

b = 4mb = 5m

8

7

6

5

4

3

2

1

0

Dep

th (m

)

3 6 9 12 15 18 21 24 27 30σx (kPa)




8

7

6

5

4

3

2

1

0

Dep

th (m

)

5 10 15 20 25 30 35σx (kPa)


0 5 10 15 20 25 30 35 40 45

δ = φ5δ = φ3

δ = φ2δ = φ15

σx (kPa)

8

7

6

5

4

3

2

1

0

Dep

th (m

)



7 Conclusions





Data Availability




Acknowledgments


References













Methodsb




δ (deg

)

φ (deg

)



22

24

26

28

30

32

34

36

38

035 036 037 038 039 040 0410346

8

10

12

14

16

18

20

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






bdσz



K sin2 φ + 1( 1113857minus


1113969

4 tan2 δ minus sin2 φ + 1( 1113857 (5)


σx cb


z

bmiddot tan δ1113874 11138751113876 1113877 (6)







dσz


δb (8)






2 tan δ (10)




σx +T

tan θ R (11)



dσz

dz c +


(12)


dσz

dz c +

σz middot b0

Hminus z (13)



σx K middotzminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭ (14)



Eh 1113946h

0


2 tan δdz

+ 1113946H

hK middot

zminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭dz

(15)



dz

bV + dV

V = σz times b

W

z

τxz

σzτzx

σx

x

σx σx

τ τ

(a)

(σx τxz)

(σz τzx)

σ

τ

o φ δ

(b)


Ax

z

z

b

dz

H

dz

h

C

E

B

E

D

D

dw

σz

τ τ

σz + dσz

σxσx

dw NTτ

σz + dσ

σz

σx

(a) (b)



M inth

0


middot (Hminus z)dz

+ intH

hK middot

zminusHb0 + 1

middot cminusC0

(Hminus z)2[ ]

b0+1( )2

middot (Hminus z)dz

(16)


h0 M

Eh (17)










σx (kPa)

Dep

th (m

)

40

35

30

25

20

15

10

05

000 5 10 15 20 25


Rakine theory









σx (kPa)

Dep

th (m

)

10

08

06

04

02

00 05 10 15 20 25 30 35 40 45


Rakine theory



b = 1mb = 2mb = 3m

b = 4mb = 5m

8

7

6

5

4

3

2

1

0

Dep

th (m

)

3 6 9 12 15 18 21 24 27 30σx (kPa)




8

7

6

5

4

3

2

1

0

Dep

th (m

)

5 10 15 20 25 30 35σx (kPa)


0 5 10 15 20 25 30 35 40 45

δ = φ5δ = φ3

δ = φ2δ = φ15

σx (kPa)

8

7

6

5

4

3

2

1

0

Dep

th (m

)



7 Conclusions





Data Availability




Acknowledgments


References













Methodsb




δ (deg

)

φ (deg

)



22

24

26

28

30

32

34

36

38

035 036 037 038 039 040 0410346

8

10

12

14

16

18

20

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



dσz

dz c +


(12)


dσz

dz c +

σz middot b0

Hminus z (13)



σx K middotzminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭ (14)



Eh 1113946h

0


2 tan δdz

+ 1113946H

hK middot

zminusH

b0 + 1middot cminus

C0

(Hminus z)21113890 1113891

b0+1( )2⎧⎨

⎩

⎫⎬

⎭dz

(15)



dz

bV + dV

V = σz times b

W

z

τxz

σzτzx

σx

x

σx σx

τ τ

(a)

(σx τxz)

(σz τzx)

σ

τ

o φ δ

(b)


Ax

z

z

b

dz

H

dz

h

C

E

B

E

D

D

dw

σz

τ τ

σz + dσz

σxσx

dw NTτ

σz + dσ

σz

σx

(a) (b)



M inth

0


middot (Hminus z)dz

+ intH

hK middot

zminusHb0 + 1

middot cminusC0

(Hminus z)2[ ]

b0+1( )2

middot (Hminus z)dz

(16)


h0 M

Eh (17)










σx (kPa)

Dep

th (m

)

40

35

30

25

20

15

10

05

000 5 10 15 20 25


Rakine theory









σx (kPa)

Dep

th (m

)

10

08

06

04

02

00 05 10 15 20 25 30 35 40 45


Rakine theory



b = 1mb = 2mb = 3m

b = 4mb = 5m

8

7

6

5

4

3

2

1

0

Dep

th (m

)

3 6 9 12 15 18 21 24 27 30σx (kPa)




8

7

6

5

4

3

2

1

0

Dep

th (m

)

5 10 15 20 25 30 35σx (kPa)


0 5 10 15 20 25 30 35 40 45

δ = φ5δ = φ3

δ = φ2δ = φ15

σx (kPa)

8

7

6

5

4

3

2

1

0

Dep

th (m

)



7 Conclusions





Data Availability




Acknowledgments


References













Methodsb




δ (deg

)

φ (deg

)



22

24

26

28

30

32

34

36

38

035 036 037 038 039 040 0410346

8

10

12

14

16

18

20

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


M inth

0


middot (Hminus z)dz

+ intH

hK middot

zminusHb0 + 1

middot cminusC0

(Hminus z)2[ ]

b0+1( )2

middot (Hminus z)dz

(16)


h0 M

Eh (17)










σx (kPa)

Dep

th (m

)

40

35

30

25

20

15

10

05

000 5 10 15 20 25


Rakine theory









σx (kPa)

Dep

th (m

)

10

08

06

04

02

00 05 10 15 20 25 30 35 40 45


Rakine theory



b = 1mb = 2mb = 3m

b = 4mb = 5m

8

7

6

5

4

3

2

1

0

Dep

th (m

)

3 6 9 12 15 18 21 24 27 30σx (kPa)




8

7

6

5

4

3

2

1

0

Dep

th (m

)

5 10 15 20 25 30 35σx (kPa)


0 5 10 15 20 25 30 35 40 45

δ = φ5δ = φ3

δ = φ2δ = φ15

σx (kPa)

8

7

6

5

4

3

2

1

0

Dep

th (m

)



7 Conclusions





Data Availability




Acknowledgments


References













Methodsb




δ (deg

)

φ (deg

)



22

24

26

28

30

32

34

36

38

035 036 037 038 039 040 0410346

8

10

12

14

16

18

20

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







σx (kPa)

Dep

th (m

)

10

08

06

04

02

00 05 10 15 20 25 30 35 40 45


Rakine theory



b = 1mb = 2mb = 3m

b = 4mb = 5m

8

7

6

5

4

3

2

1

0

Dep

th (m

)

3 6 9 12 15 18 21 24 27 30σx (kPa)




8

7

6

5

4

3

2

1

0

Dep

th (m

)

5 10 15 20 25 30 35σx (kPa)


0 5 10 15 20 25 30 35 40 45

δ = φ5δ = φ3

δ = φ2δ = φ15

σx (kPa)

8

7

6

5

4

3

2

1

0

Dep

th (m

)



7 Conclusions





Data Availability




Acknowledgments


References













Methodsb




δ (deg

)

φ (deg

)



22

24

26

28

30

32

34

36

38

035 036 037 038 039 040 0410346

8

10

12

14

16

18

20

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


7 Conclusions





Data Availability




Acknowledgments


References













Methodsb




δ (deg

)

φ (deg

)



22

24

26

28

30

32

34

36

38

035 036 037 038 039 040 0410346

8

10

12

14

16

18

20

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


simplifiedmethodforcalculatingtheactiveearthpressureon...

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