j 2Ben .nthng and tension Ifa torque Tis aaPP 'ed in comb.

combin t.the limitingforce p . ination

'tionofforcesarrelated irres ect'espective of the cr

dli 'd1985'I

'e condition (Calladine I

The apparent error of mefr h ''al

'c sought to distinguish

'ps —reinforced earinf db

biM in

ysis for the role oforcementbendin s

'

rib dbelee ow for ciric, perfectl 'irion). The a(Tresca crite '. in

literature andan itsconsction 4.

e

(-')'8-'hich

can e representelo toth alle owable comb

Fi l Thisanalogous toth limi

'tressesfore 'tin

subject to comb'a single clem ent of material

Fignres lb and cined

tens'-'j;

'(—:,)''re

k = v,J2 for a Trescaerek v esca,ork= (),JV3

contrast to equation 2combination of axial andbe din

oesdepenen g

sectional shape of the bar.on the cross-

A lower bound or safe sor safe solution is

combinations f'sofforcesirrross sectio al h d these lie withinn s ape,an

th er lommg pomts (+P 0),whereM is ic

t Ei 2(Callquadrilateral definin

e combination ofloe comb 'ads is adoptedsteel codes, such as BS

For a bar of rectangularthe limiting mbco inationmoment canb de efined by equation 4.

This paper addresses athat 4

rsyp p

d ig hi h as published in ther issue of Gro

E gi

Soil nailing desi,.l..fb..~"'"'well

and MJ PedleUniversity of Oxf0Di

When a pure axialial force P is areinforcement bar

ili d''se overt epport a limiting fullea to su

orce:yIntroduction

Therere appears to be aer o e a fundamental ecs, and a conce

error

reinforced soil desi giled in the French and

'ratureons'eupon these h b

evelo e t'la

'g esignwhichin soil nailin d

what can actuaee moved far

w yoccurmsb .Ift, thi oui

'1 dd esigners. Ase totheUnib '

'terature by Bridleri e (1989),it ish b id tif'd

Thi drie and

apparent erraws atte ntion to the

rror concerninb

ibl rol fbr 't endin st

o eofbendin s'

an source of'il

reinforced bimproveme

t'nvestigated.Th'

ybars' 'st e

e conclusionan

'at endinanalysis is th ban

'stiffness isonl

is the wrong focus for de'onpaerP Per discusses the

remforceory mtroduced wi oce with the use of

micspiralsasthe' '

urt e limiting failurs raightforwart'l hod ofdesi

described.esi'gn software, is then

g)

GROUNDD ENGINEERING . MARCH 1990

pp ——Ao,'v

pure torsion T is aWhena urort limit

torque T the magnitud

nthectional sh pn e cross-se

e yield coeither the Tr ises

criterion).resca or Von Mises

Hg. l.Iimi tincombinations of

~ ~

(a)combinedtoorsion and axialforceinreinforcement,and (b) and (c)shear and axialstress.

Ib)

L D'«)

Vo«Mis«s: s« =

Tr«s«o: K«s

2

3

2~ I Bars undewh

er combined,

Muar1on(4): square cross.section

fe for any cross.secuon

4—+ —

I =1

which is shown in Figure 2.No such simple form exists for a bar of

circular cross-section but the limitingcapacity slightly exceeds that given byequation 4. In this work, equation 4 hasbeen used to represent the limitingcombination of forces for a circular bar asit has a simple form while only beingslightly conservative.

2.2Models for shear force in asoil nail

The interaction between a deforming orrupturing soil and the reinforcement barsit contains is highly staticallyindeterminate. However, the likely rangefor the relationship between the maximumshear force and moment in thereinforcement bars appears to berelatively well defined, as describedbelow.

An elastic analysis proposed bySchlosser (1983)has been widelypublished and included in the USATransportation Research Board report onreinforced soil, Mitchell and Villet (1987).It has now appeared in the draft of amanual of practice for soil nailing, Eliasand Juran (1988).

An elastic solution is used to determinethe distance 1,between the two pointsexperiencing the maximuim momentM on either side of the centerline of apotential rupture surface through the soil,Figure 3:

PP

Fig.2. limitingcombinations ofbending moment,Mand axialforce, Pinreinforcement.

2.3Limiting combinations ofshear force and moment

The force which may be supported by abar under combined axial force and

The relation between the maximumshear force in the bar P, and the maximummoment M is then given as:

0.32(21gyr) 1,

A comparable plasticity model forshear interaction between soil andreinforcement is suggested in Figure 4.The requirement for symmetry is zeromoment on the centerline of the potentialrupture surface through the soil, whichdetermines the loading geometry as:

V'3

ls 2which leads to the following relationbetween the maximum shear force andthe moment:

p 4M~1,

A simpler limiting plasticity model issuggested in Figure S and assumes rigidbody shear displacement with encastrereinforcement, and gives:

p 2Mmox

1,

moment is limited by envelopes of thetype shown in Figure 2.As stated earlier,structural codes typically use the safeinner quadilateral in Figure 2, whichapplies to any cross-sectional shape. Thelarger envelope defined by equation 4 hasbeen adopted in this paper as it is themaximum possible benefit from bendingstiffness which is being investigated.

In practice the magnitude of themobilised axial force P in thereinforcement would be limited by bondcapacity. Thus the approach would be todeterminathe axial force P acting at thepoint of maximum moment and then to useequation 4 to determine the maximumallowable moment M = M

sr „,(r)'he

maximum shear force P„whichdepends on the distance l„would then be

p C mar n 1

where the constant C in the range 2 5depends on the elastic or plastic model ofinteraction used, equations 6, 8 or 9.

The improvement in soil shearingresistance stemming from reinforcementbending stiffness depends directly on themagnitude of the shear force P,. In orderto maximise the improvement in the soil,therefore, as small a value as possible ofthe shear width 1,would be desirable inequation (11).The likely magnitude of thiscritical parameter is investigated below.

where 8is the Young's modulus, 1thesecond moment of area and d thediameter of the reinforcement bar. Theparameter K, is the modulus of subgradereaction which is discussed further inSection 2.4.

Fig.3.Elasticanalysis ofsoil-nail interactionenvisaged byScMosser(1983).

GROUND ENGINEERING MARCH 1990

2.4Minimum shear widthElastic analysis

Substituting the second moment of areafor a circular bar into equation 5 providesan expression for the shear width I, fromthe elastic analysis for circular bars:

I, ~w'M12 4 VK>

Typical parameters for soil nailing withsteel bars are a bar diameter d = 25mmand a Young's modulus E= 210x 106

kN/m, Bruce and Jewell (1987).Unfortunately neither the magnitude nor

an adequate method of measurement forthe modulus of subgrade reaction K, hasbeen described in the texts on the elasticanalysis for soil-nail interaction. Schlosser(1983)suggests an equation to evaluate K,which is expressed in terms of a MenardPressuremeter modulus E,a parameterB„setequal to 0.6m, and a coefficient ofstructure of the soil a which is notevaluated at all. Mitchell and Villet (1987)give no guidance on the magnitude ofK,.

Juran et al (1988)have quoted (withoutreference) a single value K, = 50000kN/m forasiltysandina12mdeepcut. Thisis the number subsequently used byBridle (1989).

Values of the shear width IJd from theelastic analysis, equation (12),are given inTable I forK, = 50 000 kN/m and forarange of bar diameters.

d Igd

10 27 Assumptions:20 22 E= 210x10 kN/m25 21 K, = 50000kN/m30 2050 18

Tale I.Range ofshear widths'romthe elastic analysis for cucuiarreinforcement bars, equation l2.

To investigate how significant aninfluence on Igd the poorly definedparameter K,might have, Table 2 showsthe effect of increasing or decreasing K,by an order of magnitude on the behaviourfor a typical soil nail d = 25mm.

The conclusion to be drawn from theelastic analysis is that the shear width IJdis likely to be of the order 15to 30 for soilnailing with steel bars of 15mm to 30mm

32 diameter.GROUND ENGINEERING MARCH 1990

Plastic analysis

There is no existing analysis based onplasticity to determine the shear widthIgd. The analysis below is derived fromthe limiting stress distribution shown inFigure 4.

The lateral bearing stress on thereinforcement is given by:

2Ps 8Mm~13

dis dis

The limiting bearing pressure o'~ forsoil against reinforcement is analogous tothe problem of soil bearing againstreinforcement to mobilise bond stress,such as happens with grid reinforcement.A theoretical relation between the bearingstress a'~ and the stress normal to thereinforcement a'„, and the angle of frictionfor the soil P', was derived by Jewell et al(1984).Recent detailed experimentalresearch on such interaction hasconfirmed the relation to be satisfactory,Palmeira and Milligan (1987).Theequation is: e'z (x

exp —+ P'an P

For soil nailing in slopes the stress a'„ isthe one acting in the soil parallel to theslope face. For normally consolidatedgranular soils this stress is approximatelyequal to the mean stress (o'„s' (o'> +a'3)/2.) As failure is approached in a steepslope it may be assumed that o'> ——a'„anda'3 = o'q ——K,o'„to give:

(I +Kg15 Gv

2

A higher estimate for the lateral stresswouldbegivenbya', = a'„.

The required shear width Igd for areinforcement bar supporting no axialforce, P = 0, and loaded to the plastic limit

M = M~, may now be determined fromequations 13, 14 and 15.Recalling that

M~ = aQr/6 for circular bar:

I, 4 a„

K, lgdkN/m3

5000 38 Assumptions:50000 21 E = 210x 10

kN/m'00000

12 d = 25mm

Table 2.Sensitivity ofthe shear widthI,Id to the parameter ff for circularreinforcement bars.

Fig. 4.Plastic analysis ofsoil-nailinteraction.

Results from this analysis aresummarised in Table 3for the typicalrange of reinforcement yield stress andsoil shearing resistance, and forreinforcement at a depth of approximatelySm.

oy Oy AsslunptloIIS:200x 10'00x 10'ircular barkN/m'N/m o'„from eqn 15,

25'1 44 o'„= 100kN/mr35'2 31 K, = (1—sin41)/(1+sing)45'3 19

Table 3.Range ofshear widths'equiredto allow plastic bending ofsoil

nails.The results show that plastic bending in

a reinforcement bar can be developedover a smaller shear width in stronger soil,higher P'alues. This is because of thehigher bearing stress which can bedeveloped, equations 14and 16.

The conclusion to be drawn for theplastic analysis is that the shear width 1, /dfor soil nailing is likely to be in the range15to 30.

Conclusion

The agreement between the predictedmagnitude for the minimum shear width

1, /d for soil nailing from the entirelyseparate and independent elastic andplastic analyses is remarkable. This lendsconsiderable support to the expectedrange calculated for this ratio of 15 to 30.

Fig. S.Simple rigid plastic analysis ofsoil-nailinteraction.

S+4S

3 Improvement in shearingresistance

The improvement to shearing resistancethat a circular bar can provide inreinforced soil is now investigated at thetwo extremes ofpure axial force (P = Pp'

=0)andpurebending(P= 0,M= Mp), and for limiting combinations ofaxial and shear force loading betweenthese two extremes.

Because it is the improvement due to thereinforcement shear force which is beinginvestigated the most favourable case isexamined with the reinforcementperpendicular to the potential shearsurface in the soil, Figure 6.For thisorientation, the improvement in theshearing resistance ESdue to areinforcement bar is:

1? AS = P. +Ptany'here

P, and P are the limitingcombination of shear force and axial forcein the reinforcement.

It is convenient to normalise the resultsagainst the improvement when there is thefull axial force inthe bar(P = Pp, P, = 0),and to plot improvement on an axis ofDSIPp tan p.

3.1Mobilised shearing force

The results in Figures? and 8showlimiting combinations of shear force P,and axial force P, both normalised by Pp.

Fig. 6.Forceresultants actingon the soll wfthreinforcementperpendicular toa potentialrupture surface.

M =P,+P tang'= P. tan

4'he

likely combination of forces for soilnailing is given by the curve for lgd = 20.

The derivation ofFigures? and Smaybe described as follows. As discussed inSection 2.3,equation 10determines themagnitude ofM~Mp available in areinforcement bar carrying an axial forcein the range 1>PI'Pp>0. Equation 11determines the corresponding maximumshear force P, that can be mobilised in thebar.

To be explicit, equation 11can bemanipulated by dividing both sides withMp'.

P. C l Mm~

and noting that for circular bar:

M 2d18

Pp 3)rwhich allows both sides ofequation 18tobe multiplied to give:

20P, 2 l (MP 3 lgdiM

Remember that the constant C in the range2 to 5 stems from the uncertainty as to theexact interaction between the nail and thesoil, Section 2.2.The results in Figure 7

are for the lower limit C = 2 given by thelimiting plasticity analysis, and the resultsin Figure Sare for the higher limit C = 5given by the elastic analysis.

Both figures show that only a relativelysmall magnitude of shear force P,compared to the plastic axial force Pp canbe mobilised in soQ nails for shear widthslgd in the expected range 15to 30.

3.2Improvement in soilshearing resistance

The improvement in the soil shearingresistance due to a reinforcement bar is,normalising equation l7:

QS P, PPp tan 4) Pp tan p P„

The calculated improvement shown inFigures9and l0are for two soils, p = 45to represe)2t good quality granular soil,and 4) = 25'o represent poorer qualitysoil, and are for the values PJPp given bythe elastic analysis, Figure S.

The finding is that even the most

0.5-~- 2MP =—

I,0.5-

SMP e—I,

OA-

hOA-

02-

Sgosdoh(22)Sohlossw(199$ )Lsl'd e 5Lsht 10Lsld a 20Lsld 00Lsld

OA-

0.0.04

Soosooh(22)Sohlossor(1999Lsld 5Lsld 10Laid e 20Lsld 00Lshl

0.1 0.1-

00 02 OA os OA

PIP

Fig.?.limiting combinations ofshear and axial forceassuming the lower limit for the constant C = 2in Equation(20).

00 OA o.o OA

GROUND ENGINEERING MARCH 1990

PIP,

Fig.S.limiting combinations ofahear and axial forceassuming the upper limit for the constant C= Sin Equation(20). 33

45a

1.2

~. 0.0-

0.2

00 0.0

PIP

0.4

optimistic value for the reinforcementshear force P, in the bar gives acontribution to soil shearing resistance(stemming from bending stiffness) whichis small compared with that for thereinforcement acting in tension only, P =PP. This can be seen by comparing the leftand right sides in Figures 9and 10.

Bridle (1989)sought to examine anextreme case of a perfectly smooth bar inwhich no axial force could be generatedbydeformationinthesoil(P = 0).Theanalysis shows that in this case theimprovement in the soil due only toreinforcement bending stiffness amountstonomorethan5%0 for/ = 45,to 10% forP = 25', of that which could be mobilisedfrom axial force only (P = PP). Thesevalues can be found on the left axis ofFigures 9and 10,and assume a typicalcase le = 20.

4 Error in the literatureWhy, then, has there been such atrumpeting of the benefits of bendingstiffness for soil nailing'

The apparent error in the literatureappears to stem from Schlosser (1983),work which has been reproducedrepeatedly. More serious issue is that theanalysis is starting to be incorporated intomanuals for soil nailing, such as Mitchelland Villet (1987)or Elias and Juran (1988).

The error appears to be in theassumption made by Schlosser (1983)thatthe shearing force in reinforcementsustaining combined moment and axialforce may be calculated from an equation:

22

The equation appears to have beendeduced by incorrect analogy withequation 3.While the analogy is correctfor combined axial force and torsion,equation 2, it is entirely incorrect forcombined axial force and bending, seeequation 4.

The papers and draft manualsdiscussed above also assume that a soilnail can support a maximum moment M= MP irrespective of the magnitude of theaxial force P. This is in clear violation ofequation (4), for example.

34 The consequence of the erroneous

GROUND ENGINEERING MARCH 1990

0.4

Fig.9.Improvementinsheanngresistance forgranular soilwith/ = 4S'andCombinedloadingin thereinforcementbar.

The degree of improvement which can beachieved in soil by reinforcement axialforce and shear force (developed throughbending stiffness) has been determinedfor the range of soils which might bereinforced by soil nailing. This shows thatreinforcement bending stiffness is not asignificant parameter for soil nailing sincethe contribution to increased shearingresistance is relatively small comparedwith that due only to axial force.

The widespread misapprehension onthe role of reinforcement bendingstiffness appears to stem from equation(22) which is incorrect, but which

Fig. 10. 1.4

lmprovementinp'

25'heanng

resistance forpoorer soil with P=2S'andcombinedloadingin thereinforcementbar. 0.4-

~ 0.~-

00 02

stress analysis is to radically overestimatethe magnitudee of shear force P, whichcan be mobilised in a soil nail. The value ofP, calculated from equation (22) is shownas the upper line in Figures 7and 8.Equation (22) can be seen to overestimateby a factor of 10 to 20 the likely magnitudeof shear force in a typical soil nail within= 20.

It is these unrealistically high values ofP, that have apparently caused previousauthors to conclude that reinforcementbending stiffness is a significantmechanism in soil nailing. This isillustrated in Figures 9and 10,whereequation (22) suggests that smoothreinforcement (P = 0)would be able togenerate 50%, for P = 45', to 105%,for P =25', of the improvement due to axial forceonly(P = P ).

5 Conclusions and comments

0.0 0.4 0.~

PIP2

nevertheless has been widely adoptedand publicised. The consequence is thatthe magnitude of shear force P, that maybe mobilised in a soil nail has beenoverestimated by a factor of the order 10to 20.

The stress equilibrium envisaged inequation 22 assumes a single and uniquestate of stress with a constant normal andshear stress across the bar, representedby a unique Mohr's circle of stress. Thisstress state requires a complementaryshear stress to act on the outer freesurface of the reinforcement bar forequilibrium. Presumably it is hoped thatthe soil will provide this shear stresswhich acts in an opposite sense on eitherside of the bar. For a Tresca material theshear stress could be as great as r = II/2,which for steel bar would amount to ashear stress r = 100 '0'kNlm .

This paper has presented an analysisfor reinforcement bending stiffness andhas shown that it is not a significant factorsoilnailing design. Figures 9and10inthis paper provide a sound basis fordetermining the relative benefits of shearand axial reinforcement forces. Thefigures are for reinforcement in the mostbeneficial orientation for mobilisingimprovement due to bending stiffness.

An entirely separate approach to thepossible role of bending stiffness wouldbe to examine the likely deformat ton orincremental strain field in reinforced soil,as reported by Jewell and Wroth (1987),and applied to the analysis of reinforcedsoil walls by Jewell and Milligan (1989).Such studies clearly show, for typicalreinforcement orientations, that the tensileincremental strain in the soil in thedirection of the reinforcement is thedominant deformation, and that axial forcein the reinforcement is the dominant force

36

A report on theofte

sixth informal meetingo teachers of geotechnical subjectsheld at Sheffield University,September 1988.

mobilised. Thus not onl is threlativel

yist ereonlya've y poor enhancement in soil

strength due to reinforcement bendinstiffness, as sho

n g

paper but town by the analyses in this

he shear stresses required tomohilise this benefit ar unlikbe mohilised in the reinforcement.

only a proportion of the design axialcapacity of the reinforcement can b

because of limitations of bonden can e

strength with the soil, or becaus fngth. In such cases the

reinforcement bendin stiffng s 'ess can offerp ovement. For example,

'eaxial forcewaslimitedtoPIP =

inforcement was perpendicularto the potential slip surface Figur 9

w that the reinforcement bendingstiffness could provide an extra 5Ys to 10'Yo

improvement for the typical valen g on the mobilised value of

P'clmowledgements

geotechnical education

The authors are grateful to George

DA PONNIAH , BSc,MEng, PhD,

Milligan for his detailed

CEng, MICE, Department of Civil

paper and helpful support for the ideas

', niversity of Edinburgh.

expressed. Thishis meeting was held at Sheffield

ReferencesUniversityon12and13Se temband was attend

pte er 1988,ed hy about 60 delegate

om departm

ruce, D, and Jevvell, RA (1986/7). Soil nailin:Universities and Polytechnic .I

li tio and p ctice Gm

cs. twaso a series of meetings bringing

o 'together teachers of geotechnical

. tructural Use ofSteetw subjects to discuss topics of relevance in

graduates courses ofBrit'

, Vend Juran, 1(1988 .Man 'i 'on.

the con tex1of the ce 'raft, USA DTp, FHA

angest'

DTp, FHACo

at there should be a reappraisal f hot e

p orcement, Thomas Telford.reams o he meeting was giventhe them 'T

J ll RA d Milli

essentials ofan 'gan, GWE (1989).Deformati

i s o geotechnical education'. Th

ays o ing divi in oe

ec and Fndn Engng, Rio de Janeiro, VoL 3,spe ers in the

first three sessions, the fourth

d, F and Elias, V(1981). general diS'ussion.being left for

e st session was devoted to a

enforcement, Bd oka, Ja~ ~rsonal view'n the teachn f'o soil

y ofe orJo an of

entofeartft lo d transom~rtation Research m soil mechazucs and foundat

engmeermg, held m 1987 Dubhn.m soil 'on

ttestsof Professor Burlandbe ane results ofpull-ou

111

'gue 39.No 3, 611-824.

egan bymp ising that the views expressed

ement et le calcul des ouvrages de 'point that the student

ols et Fondations f84, arid tliat soilso'ec 'cs is taught as an

whether it be to undergraduatpo graduates or to practitionest

a uates,

'ersonal views were inQuenced byhis interest and enthusiasm for the

sub'or

which he gave two reasons. The firste ject,

was the continuing and ever varying

the eolchallenge offered by the complexiti feso

and theg ogy and the material properti

second was that he consideredes

as much as asoil mechanics to be a craftscience. He went on to say that soilmechanics teaching sh uld heo concernedwith a training of the mind and a feel for

ofthe subject rather than just th

fact. He thought that this could bee conveying

achieved in lectures in three ways:

for the subject.To transmit the lecturer' s ent usiasm

To focus on key topics and areas ofparticular difficulty to students.CI To pass on the lecturer's own ct'caland research expertise.

s own practical

He identified the greatest problem

thefaced by students as a difficul

'tyinfixing

e boundaries between reality,

trcateempiricism and theory, and th hatt ey

rasoQ mechanics as a'bl k art'c

Bther than a science and aft. Pra cr . ofessor

urland believed that some of thedifficulties encountered b dystu entscouldbe overcome by the use of a soilmechanics triangle made up of the

'applied mechanics', these forming theempiricism'pexesof the triangle with 'e

occupying the centre. In using the soilmechanics triangle he emphasised the

aintaining a balance whenimportance of main'

the various aspects, of identifyinthe methodolo ogy and rigour associatedwith each activity, and the sicontribu 'ion made by empiricism'smor wellwinnowed'xperience.

The 'ground profile'ould be theoutcome of a site investigation giving anengineering desniption of the soil basedfor example, on mineralogy andclassification, ground water table andgeological history. On the engin

'p ion of the soil he suggested thengineering

use of a methodology developed at the

thUniversity of Witwatersrand h

at provided in the UK Code of Practice(BS5930:1981).Hefurtherstressedtheimportance of 'seeing'nd 'feelin '

materialProfessor Burland emphasised the

maimportance of treating soil as art'cul

terial, and demonstrated h hisap i ate

with a base friction'odel. Thidemonst rated the movements and

o e. 's