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A simple mathematical model for soil nail and soil interaction analysis

Wan-Huan Zhou 1, Jian-Hua Yin *

The Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

Received 13 July 2006; received in revised form 19 June 2007; accepted 20 July 2007Available online 14 September 2007

Abstract

Soil nails have been widely used to stabilize slopes and earth retaining structures in many countries and regions, especially, in HongKong. The analysis of the interaction between a soil nail and the surrounding soil is of great interests to both design engineers andresearchers. In this paper, authors present a simple mathematical model for the interaction analysis of a soil nail and the surroundingsoil considering a few key factors which are soil dilation, bending of the soil nail, vertical pressure, and non-linear subgrade reactionstiffness. The lateral subgrade reaction between the soil and the soil nail is assumed to obey a hyperbolic relation. Reported test datain the literature are used to verify the present model. The contributions of the soil-nail bending on the pull-out resistance are evaluatedin two case studies.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Soil nail; Soil dilation; Interaction; Shear resistance; Slope, Model

1. Introduction

Soil nailing has been proved to be a versatile and cost-effective technique in the stabilization of slopes and earthretaining structures. Conventional limit equilibrium analy-sis and design methods depend on the concept of the nailsbeing ‘anchored’ in the passive zone (i.e. stable zone) anddeveloping tensile forces, in a passive way, which resistthe slope failure when the slope is moving (Fig. 1a). Therequired soil-nail length Lp in the passive zone dependson the ‘pull-out resistance’ Fa (Fig. 1b) which is a veryimportant parameter in the soil-nail design. The local anal-ysis on the soil nail (Fig. 1b) is useful in determining thepull-out strength of each soil nail so that the stability ofthe reinforced slope/wall can be evaluated in the globalanalysis.

Extensive tests [20,6,21,22] have demonstrated that soildilation has significant influence on the pull-out resistanceof soil inclusions in dilative soils. As suggested by Schlosser[20], the shear resistance on the interface between the inclu-sion and soil is essentially governed by the dilation behav-ior of the soil. The laboratory pull-out tests [22] showedthat the development of pull-out shear resistance of soilnails in the completely decomposed granite was mainlymobilized by the constrained dilatancy of the soil. Theoret-ically, Luo [15] developed an analytical model for calculat-ing the pull-out resistance of the soil-nail reinforcement indilative soils.

The soil nail is not only subjected to tensile forces, butalso shear forces and bending moments. All these loadsoriginate as reactions to the slope movement before andduring the slope failure. Many researchers have investi-gated the effects of the nail bending on the soil nailed struc-tures [14,11,12]. Yeung et al. [18] reported pull-out tests onFibre Reinforced Polymer (FRP) pipe soil nails in the field.An FRP nail was a pipe with outer diameter of 55 mm andinner diameter of 37 mm. During installation of a FRPpipe nail in a drill hole of normally 100 mm in Hong Kong,

0266-352X/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compgeo.2007.07.001

* Corresponding author. Tel.: +852 2766 6065; fax: +852 2334 6389.E-mail addresses: wanhuan.zhou@polyu.edu.hk(W.-H.Zhou),cejhyin@

polyu.edu.hk (J.-H. Yin).1 Tel.: +852 2766 6008; fax: +852 2334 6389.

www.elsevier.com/locate/compgeo

Available online at www.sciencedirect.com

Computers and Geotechnics 35 (2008) 479–488

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both inside and outside of the FRP pipe were grouted withcement grout. According to the study by Yeung et al. [18],one of the advantages of the FRP pipe nails is the signifi-cant increase of the bending stiffness and shear force forthis type of pipe nails, which lead to a large increase tothe slope stability. Bending and shear force of the FRP pipenails must be considered in the slope stability analysis whenusing this type of nails [18].

To date, the prevailing opinion is that soil-nails workpredominantly in tension, but they also mobilize stressesdue to shear and bending at the intersection of the slip sur-face with the soil nails [3]. Under service load conditions,the contribution of shear/bending is considered negligible.As failure conditions are approached, the contribution ofshear/bending action is more significant but still small.On the other hand, the bending of the soil nail is likelyto have influences on the normal pressure along the soil-nail length, and hence influence the shear resistance ofthe soil-nail interface. As shown in Fig. 1(b), the post-installation stress, the constrained soil dilation, and the nailbending exist together and contribute to the developmentof normal and shear stresses around the soil nail. However,the contribution of soil nail bending on the shear resistanceof the soil-nail interface has never been considered in for-mer researches. Furthermore, most researches evaluatedthe soil-nail resistance at the ultimate state other than ina progressive way, although the loads in the soil nails aremobilized gradually with the soil deformation.

In this paper, a rational simple mathematical model forthe complex load transfer mechanism at the interfacebetween soil nail and the surrounding soil is firstly devel-oped. The model aims to describe the evolution of shearstresses along the nail when considering the effects of soildilation and the nail bending together. The effects of bend-ing of nails are estimated in the process of the failure of asoil nail in a slope. The pull-out resistance of a soil nail isstudied in a progressive way. Numerical results of themathematical model are obtained and compared with thereported data in the literature for verification. The contri-

butions of the soil dilation and the soil-nail bending tothe pull-out resistance are evaluated in the case studies.

2. Development of a mathematic model

The soil nail in the passive zone is subjected to multipleforces. The normal stress r0n is composed of the post-instal-lation normal stress r0r, the normal stress r0di induced by soildilation, and the normal stress r0bm due to bending of thesoil nail, as shown in Fig. 2. The normal stress due to dila-tion is evenly distributed around the soil nail, while thenormal stress due to bending is only imposed on half ofthe nail. The post-installation stress may be unevendepending on the installation methods which we will dis-cuss later. In this study, we assume that the three normalstresses are induced independently, i.e. they do not influ-ence each other, but may exist together. In the following,the three types of normal stresses are normalized aroundthe perimeter of nail so that we have the relationr0n ¼ r0r þ r0di þ r0bm in analysis.

For the circular soil nail, the average post-installationstress r0r, can be calculated by the following:

pσ ′

(c)(b)(a)

02r02r 02r

rσ ′ diσ ′bmσ ′

Post-installationstress

Stress due to dilation

Stress due to bending

pσ ′rK

Fig. 2. Three normal stresses around the soil nail in the passive zone.

x

τ (x)

Potential slip surface

Passive zoneActive zone

Angle of the slip surface

sF

aF0M

)(n xσ′

)(n xσ′

θ

(a) (b)

pL

Fig. 1. Failure principle of a soil-nailed slope: (a) a slope reinforced with soil nails and (b) local mechanism of one soil nail in a slope.

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r0r ¼1þ Kr

2r0p ð1Þ

where r0p is the post-installation vertical effective normalstress on the soil-nail interface. The coefficient Kr is the ra-tio of horizontal normal pressure to vertical normal pres-sure around the soil nail, as shown in Fig. 3. For groutedsoil nails, because the stress has been released by drillingof the borehole, the coefficient Kr can be assumed as 1based on the axisymmetry of the hole. If not consideringthe effect of drilling process, Kr can be determined as thecoefficient of earth pressure at rest, Kr = K0. For normallyconsolidated soil, the widely accepted form of the coeffi-cient is proposed by Jaky [9], K0 = 1 � sin/ 0, where / 0 isthe internal friction angle of the soil. Therefore, the r0r va-lue is highly dependent on the installation method. Forgrouted soil nails, r0r can be very small and is approxi-mately constant with depth. For driven soil nails, r0r is re-lated to the overburden pressure or earth pressure.

The soil around a soil nail can be simplified as an axi-symmetric problem and is considered to be a hollow cylin-der with infinite thickness, after the soil nail is removed.The relationship between radial displacement and normalpressure at the circumference of the hole can be calculatedusing a linear elastic solution for a hollow cylinder withinternal pressure p1 at r = r0 and p2 = 0 at infinite distanceaccording to the following equation given by Jaeger andCook [8]:

p1 ¼2Gur0

r0

ð2Þ

where G is the soil shear modulus. ur0and r0 are the radial

displacement and radius of the soil nail, respectively. Assuggested by Luo et al. [15], the critical shear displacement(uc), i.e. the relative displacement in the interface shearzone to mobilize the peak interface shear resistance, andthe maximum dilation angle wmax can be used to calculatethe radial displacement uc

c0at the critical shear displace-

ment uc as follows:

ucc0¼Z uc

0

tan wdua

where ua is the shear displacement, w is the correspondingdilation angle. The tanw is assumed to be linear with theshear displacement as tanw = bua, where b is a constant.Thus

ucc0¼Z uc

0

buadua ¼1

2bu2

a

����uc

0

¼ 1

2uc tan wmax ð3Þ

Substituting Eq. (3) to Eq. (2) and assuming the normalpressure r0di due to soil dilation is equal to the internal pres-sure p1, the peak normal pressure r0di � peak due to soil dila-tion at the critical shear displacement is expressed by thefollowing:

r0di�peak ¼Guc

r0

tan wmax ð4Þ

The critical shear displacement uc is normally very smalland could be assumed to be constant along the soil nail.Experimental results 13,16] have shown that the critical rel-ative displacement is particularly small (in millimeters).The maximum dilation angle wmax can be decided by therelative density ID and the mean confining pressure p 0 fortriaxial strain problem [1]

wmax ¼ 3IR ¼ 3½IDð10� ln p0Þ � 1� ð5Þwhich is valid when the relative dilatancy index IR is in therange of 0 < IR < 4. In Eq. (5), p 0 has a unit of kN/m2. Thisempirical formula implies a limit value of dilation angle 12�for triaxial strain as suggested by Bolton [1]. In the prob-lem of soil-nail interface, the confining pressure p 0 repre-sents the effect of overburden pressure, i.e. it can becalculated as the average of effective vertical earth pressurer0v and horizontal earth pressure K0r0v. Therefore, the meanconfining pressure p0 ¼ ð1þ K0Þr0v=2 and it can be assumedconsistent along the soil nail.

The mean normal stress due to soil-nail bending can bedescribed as a hyperbolic relation with the lateral deflectionuy of the soil nail (positive is in the y direction), as shown inFig. 3

r0bm ¼2r0

2pr0

� absðuyÞ1=ks þ absðuyÞ=r0b

¼ ks � absðuyÞ=p1þ ks � absðuyÞ=r0b

ð6Þ

where ks is the initial soil modulus of subgrade reaction.For the case of punching shear failure around a nail, Jewelland Pedley [12] suggested the value of the lower safe limitof bearing stress r0b as

r0b ¼1þ Ka

2r0v tan

p4þ /0

2

� �exp

p2þ /0

� �tan /0

h ið7Þ

where Ka is the active earth pressure coefficient. For a givensoil, the modulus of subgrade reaction can be determinedby the limit bearing pressure r0b and the corresponding rel-ative displacement uf

ks ¼ r0b=uf ð8Þwhere the value of uf can be estimated in the range of 6–25 mm, or from inspection of a load-settlement curve if aload test was done [2].

bmσ ′

πσ b′

yu

πsk

0fu

Fig. 3. Development of the mean normal pressure due to bending of thesoil nail.

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The shear stress at the interface for a specific relativeshear displacement ua (positive in the x direction inFig. 1) is defined as

sðxÞ ¼ tan /0i � ðr0r þ r0di�peak þ r0bmÞ �minðua=uc; 1Þ ð9Þ

where /0i is the friction angle of the soil-nail interface.Substituting Eqs. (1,4) and (6) to Eq. (9), one can obtain

sðxÞ ¼ tan /0i � r0r þGuc tan wmax

r0

þ ks � absðuyÞ=p1þ ks � absðuyÞ=r0b

� �

�minðua=uc; 1Þ ð10Þ

Eq. (10) indicates that the initial development of shear stressat the soil-nail interface with pull-out displacement is mostfrom the post-installation stress and stress increased by re-strained dilation up to the point at which the critical sheardisplacement uc is reached. Beyond this point, the shearstress can be still increased by the bending of the nail. Thisdemonstrate the general understanding that at service load(or pre-failure) condition tensile reinforcement is dominantand that the contribution of bending of the nail would in-crease at large slippage displacement of the slope, so calledin the post-failure stage.

The equilibrium equation for the elastic axial tension ofthe soil nail is

d2ua

dx2� 2pr0

EtAsðxÞ ¼ 0 ð11Þ

where Et and A are the elastic modulus and section area ofthe soil nail. For grouted nails, Et is an equivalent elasticmodulus of the composite section Et = (EsAs + EgAg)/A,where Es and Eg are the elastic moduli of the steel barand the grout; and As and Ag are the cross-sectional areasof the steel bar and the grout, A = As + Ag.

In the analysis of bending, the soil nail is considered as abeam on the elastic foundation. The soil nail is undersimultaneous axial and transverse loading. If we cut outof this soil nail an infinitely small unloaded element

bounded by two verticals a distance dx apart (Fig. 4), theequilibrium of moments is analogue to that of a bar undervertical loading and a pair of equilibrating horizontal ten-sile forces N acting in the center of gravity of the end cross-sections of the bar. The reason is that the shear stress s(x)and tensile force increment dN are a pair of equilibratingforces which do not contribute any moment increment tothe element. Thus the equilibrium of moments of the ele-ment leads to the equation [7]

dMdxþ N

dydx� Qv ¼ 0 ð12Þ

Here we make the usual assumption that since d is generallysmall, we get Qn = dM/dx. If EI denotes the bending stiff-ness of the soil nail, we put the known differential equationof a beam in bending, M = �EI(d2uy/dx2), into Eq. (12),then differentiate with respect to x, and make the substitu-tion dQv=duy ¼ 2r0ksuy=ð1þ ks � absðuyÞ=r0bÞ, N = EtA Ædua/dx, and dN/dx = 2pr0s(x), we obtain the differentialequation as following:

EId4uy

dx4� EtA �

d2uy

dx2� dua

dx� 2pr0

duy

dxsðxÞ

þ 2r0ksuy

1þ ks � absðuyÞ=r0b¼ 0 ð13Þ

which is more complicated than the equation derived byHetenyi [7] because of the existence of the shear stresss(x). For grouted nails, EI is preserved by using an equiv-alent bending stiffness of the composite section of the nail.

It is assumed that the slope fails gradually along thepotential slip surface. As shown in Fig. 1, the slip surfacehas an inclination angle h. The soil-nail section at the slipsurface is subjected to an inclined displacement with thesame angle as the slip angle h, that is, a pull-out displace-ment ua0 and a shear displacement ua0tanh. The x-coordi-nate is selected to be zero at the slip surface and positivetoward the nail end in the passive zone. Therefore, theboundary conditions are

N+dN

M+dMM

N

y

oxdxx

by

y

uk

ukr

σ′⋅+ )(abs1

2

s

s0)(xτ

)(xτyuyy duu +

vv dQQ +

vQvQ nQ

(a) (b)

δ

Fig. 4. A soil-nail element with all forces.

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x ¼ 0 : ua ¼ ua0;d2uy

dx2¼ M0

EI; uy ¼ ua0 tan h ð14Þ

x ¼ Lp :dua

dx¼ 0;

d2uy

dx2¼ 0;

d3uy

dx3¼ 0 ð15Þ

Using dimensionless parameters defined as �ua ¼ ua=Lp,�uc ¼ uc=Lp, �x ¼ x=Lp, �uy ¼ uy=Lp, �s ¼ s=r0r, Eqs. (10,11)and (13,14,15) become

�s ¼ T 1 þ T 2 þT 3 � absð�uyÞ

1þ T 4 � absð�uyÞ

� ��minð�ua=�uc; 1Þ ð16Þ

d2�ua

d�x2¼ A1 � �s ð17Þ

d4�uy

d�x4þ B1

d2�uy

d�x2

d�ua

d�xþ B2

d�uy

d�x�sþ B3�uy

1þ T 4 � absð�uyÞ¼ 0 ð18Þ

�x ¼ 0 : �ua ¼ G1;d2�uy

d�x2¼ G2; �uy ¼ G3 ð19Þ

�x ¼ 1 :d�ua

d�x¼ 0;

d2�uy

d�x2¼ 0;

d3�uy

d�x3¼ 0 ð20Þ

where T 1 ¼ tan /0i, T 2 ¼ Guc tan /0i tan wmax

r0rr0, T 3 ¼ ks tan /0iLp

pr0r, T 4 ¼

ksLp

r0b

, A1 ¼ 2pr0Lpr0rEtA

, B1 ¼ �EtAL2

p

EI , B2 ¼ �2pr0L3

pr0rEI , B3 ¼

2r0ksL4p

EI ,

G1 ¼ ua0

Lp, G2 ¼ M0Lp

EI , and G3 ¼ ua0 tan hLp

. This boundary value

problem can be solved by a numerical method.

3. Verification and discussion

3.1. Case study 1 – comparisons with field test data by

Gassler [5]

An instrumented nailed wall failure was performed inGermany during 1975–1981 where bending moment distri-butions were measured in the nails [5,17]. The slope was a6m high cut inclined at 10� to the vertical into which fiverows of instrumented nails of 3 m and 3.5 m in length wereplaced. The nails were of the drilled and grouted type withthe nail diameter D = 55 mm and with a steel bar diameterd = 22 mm. A uniform surcharge was applied at the top ofthe slope.

In the field test, ‘‘failure’’ was observed under a peaksurcharge of 150 kPa. The surcharge was then reduced to110 kPa with an over all rupture surface (slip surface crossnails and soil mass). In this paper, the local analysis on theRow 4 nail under the surcharge of 110 kPa is conductedand the measured bending and shear forces are chosen toverify the present model.

The soil properties are taken from Pedley [17]: the fric-tion angle for the residual strength / 0 = 40.5�; the buckdensity c = 15.6 kN/m3; the depth of Row 4 nailz = 4.53 m; thus the vertical earth pressure r0v ¼ 181 kPa.The relative density and the shear modulus of the mediumdensity sand are assumed to be 50% and 8 MPa, respec-tively. The mean confining pressure is calculated

p0 ¼ ð1þ K0Þr0v=2 ¼ ð2� sin /0Þr0v=2 ¼ 122:2 kPa and theangle of dilation is obtained by Eq. (5), w = 4.8�. The bear-ing stress of the soil is obtained by Eq. (7) r0b ¼ 1665 kPa.The modulus of subgrade reaction is determined by Eq. (8)with uf of 25 mm, ks = 66.7 MN/m3, and the value is withinthe reasonable range for the medium dense sand accordingto the guide table listed in Bowles [2]. The friction angle ofthe soil-nail interface /0i is assumed equal to the frictionangle of the soil / 0. Since the critical shear displacementuc does not depend on the relative density of the soil [15],it is assumed to be 2 mm in calculations in this paper.For the grouted soil nail it is assumed the post-installationpressure at the soil-nail interface is relatively small,r0r ¼ 5 kPa.

The length of the soil nail in the passive zone isobserved from experimental results as 1.2 m. The elasticmoduli of the steel bar and cement grout are assumedto be 200 and 25 GPa, respectively. Thus the equivalentelastic modulus Et and the equivalent bending stiffnessEI of the grouted soil nail are estimated to be 53.0 GPaand 13.2 kN m2, respectively. The boundary conditionsat x = 0 are taken from experimental data [17]: 0 kNmfor bending moment and the slippage happened alongthe direction of 37� from the vertical, i.e. h = 53�. Theexperiment data showed that the bending moments inthe soil nails were quite small (<0.2 kNm), while the slip-page displacement of the soil was reported around 40 mm[10]. This implied that at the post-failure stage the soilnails had vertically moved with the surrounding soil.Therefore, the relative punching displacement uy in Eq.(6) should be re-examined by back analysis. It is finallyassigned as 0.71 mm in calculation to match the experi-mental data.

The calculated results of bending moment using thepresent model are shown together with the observed datafrom Pedley [17] in Fig. 5. The predicted curves agree wellwith the reported data. The calculated shear stresses on thenail surface are distributed unevenly along the soil nail, asshown in Fig. 6. Since this present model divides the nor-mal stresses around the soil nail into three parts: post-installation stress, stress due to soil dilation and stressdue to soil-nail bending, the contributions of the three nor-mal stresses to the pull-out resistance of the soil nail can beevaluated. In Figs. 7 and 8, we can see that the constrainedsoil dilation contribute most to the pull-out resistance ofthe soil nail, whereas the bending of the soil nail is of sec-ondary importance in the mechanism of soil nail pull-outforce. Although the bending behavior of the soil nailincreases the pull-out force at the large pull-out displace-ment, its contribution is still as low as around 10% of thetotal pull-out force.

3.2. Case study 2 – comparisons with large-scale direct shear

test data reported by Pedley [17]

Pedley [17] carried out a study of the soil-reinforcementinteraction with a series of large scale shear box tests. The

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direct shear test apparatus consisted of a rigid steel cubicbox of internal size 1 m · 1 m · 1 m, split at mid-height.The boundary conditions maintained symmetry about thecentral plane so that shearing took place along the horizon-tal central plane.

The soil used was quartz Leighton Buzzard sands. Thedilation angle was observed in the unreinforced direct

shear tests [17] wmax = 12.8�. The steel bars used were920 mm long with two diameters, that is, d = 15.9 mmand d = 25.4 mm. Each bar was placed in the soil in theshear box at inclination angles of 0�, 15� or 25� fromthe vertical direction, respectively. In calculation, onlyhalf length of the steel bar is analyzed because the presentmodel is for describing the nail in the passive zone. The

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

00 0.2 0.4 0.6 0.8 1 1.2

Soil nail in the passive zone, x (m)

Ben

ding

mom

ent,

M (

kNm

)

Calculated by the present model

Observed data (after Pedley 1990)

Fig. 5. Bending moment along the soil nail (Case study 1).

0

10

20

30

40

50

60

70

0.0 0.2 0.4 0.6 0.8 1.0 1.2Soil nail in the passive zone, x (m)

Shea

r st

ress

(kP

a)

Fig. 6. Shear stress along the soil nail (Case study 1).

0

3

6

9

12

15

0.8 3.1 5.4 7.7 10.1 12.4 14.7 17.0 19.4 21.7 24.0

Pull-out displacement (mm)

Pull-

out f

orce

(kN

)

Due to post-installation stress Due to soil dilationDue to soil nail bending

Fig. 7. Three components (due to post-installation stress, soil dilation, and soil nail bending) of the pull-out force and their variations with pull-outdisplacement (Case study 1).

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ultimate shear displacement was 60 mm, half of whichwas the displacement in the direction of slip surface,and according to the inclination angle, the steel bar wasaxially pulled 7.8 mm at the end of the test. The shearmodulus of the soil is assumed as 4 MPa by back analysis,considering the dimension effect of the reinforced steelbar. The critical shear displacement uc is assumed to be1 mm. The modulus of subgrade reaction value is deter-mined by back analysis from Eq. (8), ks = 220 MN/m3,with r0b of 2200 kPa and uf of 10 mm. Other parameters

used in the calculation are referred from Pedley [17] andsummarized in Table 1.

Tan et al. [23] used a series of failure modes to describethe behavior of soil-nail lateral interaction. Since the soil-nail interaction behaviour is mobilized gradually, it wouldbe more practical to analyze the interaction in a progressivemanner. The results of bending moment and shear forcealong the steel bar (d = 25.4 mm and h = 15�) at the sheardisplacement of 20 mm and 60 mm are shown in Figs. 9and 10, together with the experimental data from Pedley[17]. It should be noted that the shear force profiles inexperiments were obtained by differentiating the bendingmoment profiles by Pedley [17]. Correspondingly, the pres-ent model shows better predictions in bending momentsthan those in shear forces. It can be seen that the presentsimple model can be used to obtain results at differentshearing stages and the calculated results are in good agree-ments with the experimental data.

It should be noted that some researchers argue that thedirect shear tests could not represent the behaviour of thereinforcement in the slope [19]. One of the reasons is that

0

20

40

60

80

100

0.8 3.1 5.4 7.7 10.1 12.4 14.7 17.0 19.4 21.7 24.0Pull-out displacement (mm)

Con

trib

utio

ns to

the

pull-

out r

esis

tanc

e (%

)

Due to post-installation stress Due to soil dilationDue to soil nail bending

Fig. 8. Contributions of three normal stresses (due to post-installation stress, soil dilation, and soil nail bending) to the pull-out resistance (Case study 1).

Table 1Material parameters adopted in the verification study (Case study 2)

Parameters adopted Values

Initial pressure at the interface, r0r (kPa) 100Soil friction angle, / 0 (�) 46Soil dilation angle, wmax (�) 12.8Elastic modulus of the steel bar, Et (kPa) 205 · 106

Bending stiffness of the steel bar, EI (kN m2) 4.1885Bearing stress of the soil, r0b (kPa) 2200Modulus of subgrade reaction, ks (MN/m3) 220

0.0 0.1 0.2 0.3 0.4 0.5

0.0

-0.2

-0.4

-0.6

-0.8

-1.0 Shear disp. Present model 20 mm Measured (Pedley 1990) 20 mm Present model 60 mm Measured (Pedley 1990) 60 mm

Ben

ding

mom

ent,

M (

kNm

)

Steel bar in the axial direction, x (m)

Steel bar diameter : 25.4 mmInclination angle θ : 15˚

Fig. 9. Comparison of calculated bending moment with LDST data (Case study 2).

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the reinforcement in shear box fails in bending and shear-ing rather than tension. However, since the present modelconsiders the bending and tension behaviour altogether,it is suitable for analyzing both the local failure of a soil-

nailed wall and the direct shear tests with reinforcement.Figs. 11 and 12 show the contributions of the three normalstresses on the pull-out resistance of the steel bar. We cansee that the contributions of bending are significant in this

0.0 0.1 0.2 0.3 0.4 0.5

-12

-10

-8

-6

-4

-2

0

2

4

6

Steel bar diameter : 25.4 mmInclination angle θ : 15˚

Shea

r fo

rce,

Q (

kN)

Steel bar in the axial direction, x (m)

Shear disp. Present model 20 mm Measured (Pedley 1990) 20 mm Present model 60 mm Measured (Pedley 1990) 60 mm

Fig. 10. Comparison of shear force with LDST data (Case study 2).

0

5

10

15

20

25

30

35

0.25 1.00 1.75 2.50 3.26 4.01 4.76 5.51 6.26 7.01 7.76Pull-out displacement (mm)

Pull-

out f

orce

(kN

)

Due to post-installation stress Due to soil dilationDue to soil nail bending

Fig. 11. Three components (due to post-installation stress, soil dilation, and soil nail bending) of the pull-out force and their variations with pull-outdisplacement (Case study 2).

0

20

40

60

80

100

0.25 1.00 1.75 2.50 3.26 4.01 4.76 5.51 6.26 7.01 7.76Pull-out displacement (mm)

Con

trib

utio

ns to

the

pull-

out r

esis

tanc

e (%

)

Due to post-installation stress Due to soil dilation Due to soil nail bending

Fig. 12. Contributions of three normal stresses (due to post-installation stress, soil dilation, and soil nail bending) to the pull-out resistance (Case study 2).

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case, about 20% to 80% of the total pull-out resistance,during the shear tests.

It can be seen clearly that the contribution of bendingmoments in the shear box apparatus (Case study 2) is largerthan that in the large-scale field test (Case study 1). This isdue to the different failure mechanisms between the rein-forced wall and the direct shear tests with reinforcement.In the soil-nailed wall, the soil nail fails in tension with smallbending moments motivated, while in the direct shear tests,the steel bar fails in large shearing forces and bendingmoments. On the other hand, from the distribution of bend-ing moment, we can observe that the effect of bending on thenormal pressure along the soil-nail length is within a limitrange. After certain distance away from the failure point,the bending would have neglectable contribution to theshear resistance of the soil/nail interface. In the shear boxapparatus, as the length of nail is relatively small to generatea large axial force, the nail bending makes a larger contribu-tion to the shear resistance of the soil-nail interface. There-fore, in the comparisons between the results of two casestudies, we can find that the relative contributions of thethree normal stresses depend on the geometry characters,the bending stiffness of the reinforcement, the soil propertiesand the failure mechanism of the reinforced structure.

4. Summary and conclusions

A simple mathematical model for analyzing the interac-tion between a soil nail and the surrounding soil is devel-oped and presented in this paper. This model considersthe nail-soil interface dilation, non-linear subgrade reac-tion, and the bending of the nail. The progressive interac-tion can be simulated using the present model.

Two cases with measured data are simulated using thepresent model. The calculated results are compared withthe test data. It is found that the present model is suitablein modelling both the local failure of a soil-nailed wall anddirect shear tests with reinforcement. The analyses showthat the relative contributions of the bending behavior onthe pull-out resistance of the reinforcement differ in thelarge-scale soil-nailed wall and in the direct shear tests withreinforcement. It is concluded that for the reinforced slope/wall, the contribution of the bending of the soil nails to thepull-out resistance are of secondary importance as the ten-sion failure is dominant in the soil-nailed structures.

It shall be pointed out that the present model simplifiesthe soil lateral reaction on the nail using disconnectednon-linear springs. The actual case is that the soil surround-ing the nail is a continuum and can be better simulatedusing a finite element model based on continuum mechanicsand using proper constitutive models for the soil and theinterface. However, the present model is much simpler thanthe finite element model and can be easily used by engineers.In addition, the present model can clearly demonstrate theinfluences of a few key parameters, which are of great inter-ests to engineers. Nevertheless, two-dimensional and three-dimensional finite element models based on continuum

mechanics are being established by the authors to simulate:(a) the interaction of a soil nail and the surrounding soil and(b) soil nailed slopes. The finite element results will be com-pared with test data and the results obtained using the pres-ent simple model. The limitations and advantages of thepresent model will be further examined.

Acknowledgements

Financial supports from The Hong Kong PolytechnicUniversity and a grant from Research Grants Committee(RGC: PolyU 5174/04E) of the Hong Kong SpecialAdministrative Region Government of China are gratefullyacknowledged.

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