a new working stress method for prediction of reinforcement loads in

A new working stress method for prediction ofreinforcement loads in geosynthetic walls

T.M. Allen, Richard J. Bathurst, Robert D. Holtz, D. Walters, and Wei F. Lee

Abstract: Proper estimation of soil reinforcement loads and strains is key to accurate internal stability design of rein-forced soil structures. Current design methodologies use limit equilibrium concepts to estimate reinforcement loads forinternal stability design of geosynthetic and steel reinforced soil walls. For geosynthetic walls, however, it appears thatthese methods are excessively conservative based on the performance of geosynthetic walls to date. This paper presentsa new method, called the K-stiffness method, that is shown to give more accurate estimates of reinforcement loads,thereby reducing reinforcement quantities and improving the economy of geosynthetic walls. The paper is focused onthe new method as it applies to geosynthetic walls constructed with granular (noncohesive, relatively low silt content)backfill soils. A database of 11 full-scale geosynthetic walls was used to develop the new design methodology based onworking stress principles. The method considers the stiffness of the various wall components and their influence on rein-forcement loads. Results of simple statistical analyses show that the current American Association of State Highway andTransportation Officials (AASHTO) Simplified Method results in an average ratio of measured to predicted loads (bias) of0.45, with a coefficient of variation (COV) of 91%, whereas the proposed method results in an average bias of 0.99 and aCOV of 36%. A principle objective of the method is to design the wall reinforcement so that the soil within the wallbackfill is prevented from reaching a state of failure, consistent with the notion of working stress conditions. This conceptrepresents a new approach for internal stability design of geosynthetic-reinforced soil walls because prevention of soil fail-ure as a limit state is considered in addition to the current practice of preventing reinforcement rupture.

Key words: geosynthetics, reinforcement, walls, loads, strains, design, K-stiffness method.

Résumé : L’évaluation appropriée des contraintes et des déformations du renforcement est primordiale pour la concep-tion précise de la stabilité interne des structures du sol renforcé. Les concepts actuels utilisent la méthode d’équilibrelimite pour évaluer les charges de renforcement pour la stabilité interne des murs de soutènement renforcés par les ma-tériaux géosynthétiques et métalliques. Cependant, au moins pour les murs géosynthétiques, il s’avère que ces métho-des sont excessivement conservatrices, d’après la performance des murs géosynthétiques construits jusqu’à présent. Cetarticle présente une nouvelle méthode, appelée « K-stiffness, » qui donne une évaluation plus précise des charges durenforcement, réduisant la quantité de renforcement et améliorant le coût des murs geosynthetiques. Cet article estconcentré sur une nouvelle méthode qui est limitée aux murs géosynthétiques construits avec les sols granulaires (noncohérent et relativement faible en particules fines). Une base de données, de 11 murs géosynthétiques à l’échelle réelle,a été utilisée pour développer la nouvelle méthodologie de la conception, basée sur les principes des contraintes admis-sibles. Cette méthode considère la rigidité des divers composants de mur et de leur influence sur des charges de renfor-cement. Les résultats de simples analyses statistiques montrent que la méthode simplifiée actuelle de l’AASHTOrésulte en un ratio moyen des chargements mesurés sur estimés de 0,45 avec un coefficient de variation (CDV) de91 %, tandis que la méthode proposée a comme conséquence un ratio moyen de 0,99 et un CDV de 36 %. Un objectifprincipal de cette méthode est de concevoir le renforcement de mur de sorte que le sol dans le remblai soit empêchéd’atteindre la rupture, conformément à la notion des contraintes admissibles. Le concept représente une nouvelle approchepour le calcule de la stabilité interne des murs de sol renforcés par geosynthetique, puisque la prévention de la rupture dusol à l’état limite est considérée en plus de la pratique actuelle de la prévention de la rupture de renforcement.

Mots clés : géosynthétiques, renforcement, murs, chargements, contraintes, conception, méthode de « K-stiffness ».

Allen et al. 994

Can. Geotech. J. 40: 976–994 (2003) doi: 10.1139/T03-051 © 2003 NRC Canada

976

Received 20 December 2001. Accepted 13 June 2003. Published on the NRC Research Press Web site at http://cgj.nrc.ca on24 September 2003.

T.M. Allen. State Materials Laboratory, Washington State Department of Transportation, Olympia, WA 98504-7365, U.S.A.R.J. Bathurst.1 GeoEngineering Centre at Queen’s–RMC, Civil Engineering Department, Royal Military College of Canada,Kingston, ON K7K 7B4, Canada.R.D. Holtz. Department of Civil Engineering, University of Washington, More Hall: FX 10, Box 352700, Seattle, WA 98195-2700, U.S.A.D. Walters. GeoEngineering Centre at Queen’s–RMC, Department of Civil Engineering, Queen’s University, Kingston, ONK7L 3N6, Canada.W.F. Lee. Research and Development Section, Taiwan Construction Research Institute, HsinTien City, Taiwan.

1Corresponding author (e-mail: [email protected]).

I:\cgj\CGJ40-05\T03-051.vpSeptember 19, 2003 8:42:52 AM

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Introduction

Accurate prediction of loads and their distribution in rein-forcement layers is necessary to produce cost-effective, in-ternally stable, reinforced soil wall designs. The predictedreinforcement loads affect the strength and spacing requiredfor the reinforcement and the reinforcement length requiredto resist pullout.

The three primary methods identified in the most recentdesign specifications in North America for estimating loadsin geosynthetic reinforced soil walls are the American Asso-ciation of State Highway and Transportation Officials(AASHTO) Coherent Gravity Method (AASHTO 1996), theFederal Highway Administration (FHWA) Structure Stiff-ness Method (Christopher et al. 1990), and the AASHTOSimplified Method (AASHTO 2002; Elias et al. 2001) orvariants (Simac et al. 1993; Bathurst et al. 1993a). Allen etal. (2002) provided an assessment of the predictive accuracyof these methods for geosynthetic walls, and Allen et al.(2001) evaluated the predictive accuracy of these methodsfor steel reinforced structures. These approaches haveworked reasonably well for typical steel reinforced soil walls(Allen et al. 2001), but have worked poorly for predictingloads in geosynthetic reinforced structures (Bell et al. 1983;Rowe and Ho 1993; Allen et al. 2002).

All of these methods are semiempirical in nature, usinglimit equilibrium concepts to develop the design model butworking stress observations to adjust the models to fit whathas been observed in full-scale structures. In the develop-ment of these methods, it was assumed that reinforcementloads can be equated directly to the soil state of stress andthat limit equilibrium concepts are applicable. There hasbeen little hesitation to adjust the load predictions to matchthe empirical data for steel reinforced soil walls because thereinforcement loads in steel reinforced structures have beenmeasured to be equal to or greater than the loads calculatedby integration of active or at-rest lateral earth pressures overthe tributary area of the reinforcement.

For geosynthetic reinforced walls, however, the measuredstrains converted to loads using reinforcement stiffnessvalues have shown that reinforcement loads were less thanthose predicted by integrating the active earth pressure overthe tributary area. To maintain the assumption that the rein-forcement loads should directly reflect the soil state ofstress, the data that appeared to support lower reinforcementload levels have in effect been ignored, and the design rein-forcement load has been maintained at active earth pressurelevels for these methods.

Uncertainties regarding the effect of time (creep or stressrelaxation), temperature, and soil confinement on the deter-mination of the geosynthetic stiffness values have hinderedacceptance of the lower reinforcement loads inferred fromstrain measurements. Bathurst (1990), Bathurst andBenjamin (1990), and Fannin and Hermann (1991) sug-gested that in-isolation isochronous creep stiffness valuesshould be used to convert measured geosynthetic strains toloads for geogrid reinforcement products rather than thestiffness value from an index tensile test such as AmericanSociety for Testing and Materials (ASTM) test methodD4595-96 (ASTM 1996). Walters et al. (2002) investigatedthis issue in detail, confirming that for geogrids and wovengeotextiles the in-isolation isochronous creep stiffness, with

consideration of the time it takes to construct the wall and toapply any surcharges, provides a reasonably accurate valueto convert measured reinforcement strains to loads and thatthe short-term wide-width tensile test stiffness is much toohigh for this purpose.

The proper estimation of the geosynthetic stiffness valueneeded to convert measured strain to load is a source of un-certainty in determining actual load levels in geosyntheticreinforcement layers. This uncertainty is compounded by theneed to correctly interpret measured strains. Through properstrain gauge calibration and redundancy with respect tonumber of monitoring points and strain measurement type,however, reasonable estimates of in-soil reinforcement strainare possible. Bathurst et al. (2002) addressed and quantifiedthese sources of uncertainty and their effect on the “mea-sured” reinforcement loads in instrumented full-scale struc-tures. Even considering these uncertainties, Allen andBathurst (2002a) showed that current design approachesgreatly overestimate geosynthetic reinforcement loads, evenwhen using less conservative (for design) plane strain soilstrength parameters.

Past performance of geosynthetic reinforced soil walls hasalso provided strong evidence that current design methodol-ogies for internal stability are very conservative, particularlythe prediction of reinforcement loads. Allen et al. (2002)showed that a number of well-documented geosyntheticwalls that have demonstrated good long-term performancefor up to 25 years were designed with significantly lowerglobal resistance to demand ratios than would be required bycurrent practice. Furthermore, Allen and Bathurst (2002b)demonstrated that the measured long-term creep rates infull-scale geosynthetic structures corroborate reinforcementload levels that are much lower than previously thought.

In this paper a new working stress methodology, calledthe K-stiffness method, is proposed. The method has beencalibrated against measurements of strain and load in moni-tored full-scale field walls reported in the references at theend of this paper. The design methodology, in addition tobeing relatively simple to apply, is developed to providea seamless transition between geosynthetic and steel rein-forced soil walls.

The scope of this paper and the proposed new designmethodology are limited to walls with granular (non-cohesive) backfills.

Summary of case histories evaluated

Key properties and parameters for each of the case histo-ries referenced in this paper are summarized in Table 1. Ad-ditional details for each of these case histories, includingfacing type, reinforcement geometry, reinforcement type,soil properties, and construction history, are provided by Al-len et al. (2002).

A total of 11 geosynthetic wall cases from Table 1 wereanalysed (the same wall with and without a surcharge wasconsidered to be one case). These wall cases included arange of wall facing geometry and materials, surcharge con-ditions, and granular backfill. Wall reinforcement productsincluded geotextiles and geogrids, different polymers (poly-propylene (PP), high-density polyethylene (HDPE), andpolyester (PET)), strip and continuous reinforcements, a

© 2003 NRC Canada

Allen et al. 977



© 2003 NRC Canada

978 Can. Geotech. J. Vol. 40, 2003

Wal

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range of tensile strengths from 12 to 200 kN/m, and rein-forcement stiffness values from 90 to 7400 kN/m. Rein-forcement vertical spacing varied from 0.3 to 1.6 m. Wallfacing batter angles varied from 0° (vertical) to 27°, al-though most of the walls had facing batter angles of 5° orless. Wall heights varied from 4.0 to 12.6 m, with surchargeheights of up to 5.3 m of soil. Facing types included geosyn-thetic wrapped-face, welded wire, precast concrete panels,and modular concrete blocks (segmental retaining wallunits). Estimated peak plane strain soil friction angles variedfrom 42 to 57°. Many of the conditions that are likely to beencountered in the field are included in the database of casehistories described previously.

Note that the peak plane strain friction angle and the mea-sured peak triaxial or direct shear friction angle are includedin Table 1 for each case history. Plane strain conditions typi-cally exist in reinforced soil walls. Peak plane strain frictionangles for granular soils are larger than values from triaxialcompression or direct shear testing and hence are less con-servative for design. Furthermore, recent work indicates thatthe peak plane strain soil friction angle in calculations givesa better estimate of reinforcement loads, at least for geosyn-thetic walls (Rowe and Ho 1993; Zornberg et al. 1998a,1998b; Lee et al. 1999; Allen and Bathurst 2002a).

Peak friction angles reported in the source referencesfrom triaxial compression tests (φtx, in degrees) were cor-rected to peak plane strain friction angles using the equationby Lade and Lee (1976):

[1] φps = 1.5φtx – 17

Based on interpretation of data presented by Bolton(1986) and Jewell and Wroth (1987) for dense sands, valuesof φps were calculated from peak direct shear friction angles,φds, reported in the source references using the following re-lationship:

[2] φps = tan–1(1.2 tan φds)

The isochronous reinforcement stiffness at 2% strain (J2%)was estimated from project-specific in-isolation creep testingresults where available. If these data were not available,in-isolation creep testing for the same product reported inthe literature or supplied by the manufacturer was used. Thestiffness values were corrected for temperature as required.A detailed description of the determination of isochronousstiffness values for each case study in Table 1 can be foundin the paper by Allen and Bathurst (2002a).

Analysis of reinforcement loads

Allen and Bathurst (2002a) investigated current NorthAmerican methods for the prediction of reinforcement loadsand concluded that the AASHTO Simplified Method gaveresults similar to those of the other methods, yet had the ad-vantage of being simpler to use and more broadly applica-ble. Therefore, the Simplified Method is used here as thebaseline of comparison for predicted reinforcement loads us-ing the new working stress method.

Figure 1 shows how well the Simplified Method predictsreinforcement loads in geosynthetic walls built in the field.Triaxial or direct shear friction angles were used in the origi-nal development of the Simplified Method. This method was

only adjusted to fit the empirical data for steel reinforcedwalls, however. To provide a common basis of comparisonwith the proposed K-stiffness method which uses planestrain soil friction angles and to eliminate any conservatismin the prediction of loads resulting from conservative soilparameter selection, peak plane strain soil friction anglesestimated from triaxial or direct shear strength tests and pro-ject-specific measured unit soil weights were used to esti-mate loads using the Simplified Method.

Loads predicted for geosynthetic walls using the Sim-plified Method are generally very conservative relative to themeasured loads (Fig. 1). The only exceptions are walls GW7(section N) and GW19. The reinforcement loads in wallsGW7 (section N) and GW19 were underpredicted by theSimplified Method. These walls had unusual features, how-ever, that may have contributed to their different behaviourwith respect to reinforcement loads. For example, the face ofwall GW7 is heavily battered and could be classified as a re-inforced slope. Wall GW19 is the only wall reinforced withPET straps rather than continuous sheets. The global wallstiffness for this wall is also very high, indicating that rein-forcement stiffness may significantly affect the amount ofload carried by the reinforcement. In practice, the SimplifiedMethod as described in AASHTO (2002) would not be usedto design heavily battered or polymer strap walls. For thesake of direct comparison of the Simplified Method to theproposed K-stiffness method, however, the data for wallsGW7 and GW19 have been included in Fig. 1. Even whendata for walls GW7 and GW19 are omitted from Fig. 1,there is a large amount of scatter in the predicted loads, andthe correlation of predicted to measured values using thecurrent design methodology remains poor.

Development of a new approach to predictmaximum reinforcement loads

GeneralThe following key factors will influence the magnitude of

maximum reinforcement load, Tmax: (i) height of the walland any surcharge loads, (ii) global and local stiffness ofthe soil reinforcement, (iii) resistance to lateral movementcaused by the stiffness of the facing and restraint at the walltoe, (iv) face batter, (v) shear strength and stress–strain be-haviour (e.g., modulus) of the soil, (vi) unit weight of thesoil, and (vii) vertical spacing of the reinforcement. Thesefactors are introduced analytically in the following generalexpression for the maximum load per running unit length ofwall in reinforcement layer i:

[3] T S Di imax max= v h tσ Φ

where Siv is the tributary area (equivalent to the vertical

spacing of the reinforcement in the vicinity of each layerwhen analyses are carried out per unit length of wall); σh isthe lateral earth pressure acting over the tributary area; Dtmaxis the load distribution factor that modifies the reinforcementload based on layer location; and Φ is the influence factorthat is the product of factors that account for the effects oflocal and global reinforcement stiffness, facing stiffness, andface batter.

© 2003 NRC Canada

Allen et al. 979



The lateral earth pressure is calculated as the averagevalue acting over the height of the wall according to conven-tional earth pressure theory, hence

[4] σ γh = +12

K H S( )

where K is the lateral earth pressure coefficient, γ is the unitweight of the soil, H is the height of the wall, and S is theequivalent height of uniform surcharge pressure q (i.e., S =q/γ). The coefficient of lateral earth pressure K is calculatedusing the Jaky equation (Holtz and Kovacs 1981):

[5] K = K0 = 1 – sin φps

where φps is the peak plane strain friction angle of the soil.The use of K = K0 in this proposed method does not implythat at-rest conditions exist within the reinforced backfill. K0is simply used as a familiar index parameter to characterizesoil behaviour. This point is discussed in more detail later inthe paper.

Substitution of eq. [4] into eq. [3] leads to

[6] T K H S S Di imax max( )= +1

2γ v t Φ

Equation [6] contains an expression for reinforcementloads that is similar to the conventional expression used incurrent limit equilibrium methods of analysis but representsthe average load applied to the reinforcement layers ratherthan a load that increases linearly as a function of the verti-cal overburden stress. The empirical reinforcement load dis-tribution parameter Dtmax is used to distribute the load as afunction of depth, accounting for the reinforcement proper-ties, load redistribution among layers, and foundation condi-tions. It is expressed here as a function of normalized depthbelow the top of the wall (z + S)/(H + S), including the effect

of the soil surcharge S, and varies over the range 0 ≤ Dtmax ≤1. The modifier Φ is an empirically determined parameterthat captures the effect the major wall components have onreinforcement load development. These parameters are usedto improve the correlation between predicted and measuredreinforcement loads at working stress conditions based onexamination of a large number of case studies. For brevity,the influence factor Φ in eq. [6] is used to represent theproduct of four factors as follows:

[7] Φ = Φg × Φlocal × Φfs × Φfb

Parameter Φg is a global stiffness factor that accounts forthe influence of the stiffness and spacing of the reinforce-ment layers over the entire wall height and has the generalform

[8] Φgglobal

a

=

α

βS

p

where Sglobal is the global reinforcement stiffness, and α andβ are constant coefficients. The nondimensionality of the ex-pression is preserved by dividing the global reinforcementstiffness by the atmospheric pressure pa = 101 kPa. Theglobal reinforcement stiffness value for a wall is calculatedas (Christopher et al. 1990)

[9] SJHn

J

H

ii

n

globalave= = =

∑1

where Jave is the average tensile stiffness of all n reinforce-ment layers over the wall height, and Ji is the tensile stiff-

© 2003 NRC Canada


Fig. 1. Predicted versus measured values of Tmax in reinforcement layers for geosynthetic walls using the AASHTO Simplified Methodand peak plane strain soil friction angles.



ness of an individual reinforcement layer expressed in unitsof force per unit length of wall.

Parameter Φlocal is a local stiffness factor that accounts forthe relative stiffness of the reinforcement layer with respectto the average stiffness of all reinforcement layers and is ex-pressed as

[10] Φlocallocal

glocal

=

SS

a

The coefficient term is taken as a = 1 for geosynthetic re-inforced soil walls. To accommodate a seamless transition tosteel reinforced soil walls, a similar back-analysis by Allenand Bathurst (2003) gave a = 0 for steel systems. Slocal is thelocal reinforcement stiffness for reinforcement layer i calcu-lated as

[11] SJS

i

localv

=

and is used to quantify the local combined influence of theindividual layer stiffness and spacing on reinforcement load,where J is the tensile stiffness of the reinforcement, and Sv isthe tributary area for the reinforcement layer.

Parameters Φfs (facing stiffness factor) and Φfb (facing bat-ter factor) in eq. [7] are factors that account for the influenceof the facing stiffness and facing batter, respectively, and areconstant values for a given wall.

Equations [3] and [7] show that the maximum load in areinforcement layer is the product of seven terms that havesome uncertainty associated with their value and (or) requireback-analyses to determine the magnitude of coefficientterms. In addition, some terms are highly nonlinear. It is as-sumed a priori that parameters K and γ and factors Dtmax, Φg,Φlocal, Φfs, and Φfb are for practical purposes uncorrelated.This assumption allows the influence of each term on pre-dicted reinforcement loads to be examined separately whilekeeping other parameters at baseline values. Baseline valuesfor coefficient terms in expressions for Φg, Φlocal, Φfs, and Φfbare identified in the following sections. For example, theconstant in eq. [10] is taken as a = 1, corresponding to thecase of geosynthetic reinforced soil walls unless noted other-wise.

The accuracy of the K-stiffness method using differentvalues for the influence factors identified previously is eval-uated in the following sections in two ways: (i) direct com-parison of predicted and measured reinforcement load valuesfor the walls summarized in Table 1; and (ii) comparison ofthe mean and coefficient of variation (COV) of the bias, de-fined as the ratio of the reinforcement loads estimated fromstrain measurements to the predicted reinforcement loads forall case studies (Table 2). Values of the mean of the bias ofreinforcement loads close to but slightly less than unity aredesirable while maintaining a minimum value for the coeffi-cient of variation (COV).

Load distribution factor DtmaxCurrent design methodologies assume a triangular distri-

bution of Tmax with depth below the wall top for geosyn-thetic walls (Fig. 2a) and a modified triangular distribution

© 2003 NRC Canada

Allen et al. 981

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ch:

K=

K0

(a=

1in

eq.

[10]

;d

=0.

25in

eq.

[18]

)K

=K

ah

K=

0.3

for

all

soil

s

Soi

l( γ

=19

.5kN

/m3

for

all

wal

ls)

a=

0( Φ

loca

l=

1)a

=0.

5

Fac

ing

stif

fnes

sfa

ctor

, Φfs

=1,

for

all

wal

ls

Φfb

=1

(d=

0in

eq.

[18]

)

Φfb

<1

(d=

1in

eq.

[18]

)

S v(0

.6m

for

all

wal

ls)

Mea

nT

max

560.

650.

450.

200.

991.

160.

900.

961.

101.

020.

790.

901.

621.

15C

OV

Tm

ax(%

)56

9591

5936

6439

3561

4452

3486

58M

ean

Tm

xmx

120.

580.

390.

230.

991.

070.

830.

981.

051.

010.

740.

911.

421.

34C

OV

Tm

xmx

(%)

1299

8653

1740

3018

3524

3722

6083

Not

e:K

ahus

ing

Cou

lom

ban

alys

isw

ithfu

llw

all–

soil

fric

tion

( δ=

φ)an

dw

all

batte

rω

.C

OV

,co

effi

cien

tof

vari

atio

n(=

(sta

ndar

dde

viat

ion

ofbi

asva

lues

)/(m

ean

ofbi

asva

lues

)×

100%

;N

,nu

mbe

rof

data

poin

ts;

Tm

ax,

max

imum

load

inre

info

rcem

ent

laye

r;T

mxm

x,m

axim

umre

info

rcem

ent

load

inth

ew

all.

Tab

le2.

Sum

mar

yof

the

rati

oof

mea

sure

dto

pred

icte

dre

info

rcem

ent

load

Tm

axor

Tm

xmx

(bia

s)fo

rge

osyn

thet

icre

info

rced

soil

wal

ls.



with depth for steel reinforced soil walls (Fig. 2b), where Kvaries as a function of depth. Other types of reinforced soilstructure design such as anchored walls have used trape-zoidal distributions for Tmax versus depth (Fig. 2d). Twoother distributions that have been proposed for geosyntheticreinforced soil walls are also illustrated (Figs. 2c, 2e).

Lee (2000) carried out a numerical investigation of geo-synthetic reinforced soil wall performance. The numericalmodel was calibrated against some of the case studies refer-enced in this paper, and the numerical study extended to awider range of wall geometry, reinforcement stiffness, andsoil properties. Lee found that a trapezoidal distribution ofmaximum reinforcement layer loads similar to that shown inFig. 2d was applicable to all of the cases evaluated.

Allen and Bathurst (2002a) adopted a similar approachand proposed the trapezoidal envelope in Fig. 3a for thegeosynthetic wall case studies in Table 1. In Fig. 3a the ratioof maximum reinforcement load in a layer Tmax to the maxi-mum reinforcement load for all layers Tmxmx is plottedagainst the depth of the layer plus surcharge height normal-ized by the total wall height plus average surcharge height(z + S)/(H + S). The coordinates for the distribution inFig. 3a are approximate only and have been selected to cap-ture the majority of the data while simplifying the envelopegeometry.

Numerical simulation results of geosynthetic reinforcedsoil walls reported by Lee (2000) and Rowe and Ho (1993)also predicted that load in the reinforcement layers near thebottom of the wall is less than the reinforcement loadswithin the middle third of the wall height. Rowe and Ho pro-vided a summary of physical data from reduced-scale andfull-scale walls which confirm this observation for wallswith a pinned toe. Bathurst and Hatami (1998) demonstratedthe same effect through numerical parametric analyses of anidealized full-height panel reinforced soil wall with a toe

that was free to rotate. They also showed, however, that at-tenuation of reinforcement loads at the base of the wall didnot occur if the toe was free to slide. Based on the observa-tions made here, it is clear that the stiffness of the founda-tion and the degree of fixity of the wall facing toe influencethe distribution of reinforcement loads at the base of a geo-synthetic reinforced soil wall. Most walls have a fixed toecondition due to wall embedment, however, and hence theattenuation of reinforcement load in proximity to the foun-dation predicted by a trapezoidal distribution is reasonablefor walls constructed on stiff competent foundations.

It is shown later in the paper that local reinforcement stiff-ness, Φlocal , can have a significant effect on the magnitudeand distribution of Tmax. Figure 3b also illustrates this effect.With one notable exception (wall GW20 (PP section) withan unusually large vertical reinforcement spacing) the distri-bution of normalized Tmax values was improved when thedata were normalized by the local stiffness factor in additionto Tmxmx. Based on Fig. 3b, truncating the trapezoidal distri-bution at a normalized depth of 0.8 appears to be appropriateand also lends support to the argument that increasing foun-dation stiffness and wall toe fixity tends to reduce reinforce-ment loads near the bottom of the wall.

Overall, the global stiffness of the reinforcement does ap-pear to influence the Dtmax distribution, causing the distribu-tion to tend toward a more triangular shape as the globalstiffness value increases (Allen and Bathurst 2003). Due tothe significantly greater stiffness of wall GW19 (PET strapreinforcement), the Dtmax distribution is modified for poly-mer strap walls as illustrated in Fig. 3c.

Global reinforcement stiffness factor, ΦgThe stiffness of the various internal components of the

wall will directly affect the distribution of loads to each ofthe wall components at working stress conditions. This is

© 2003 NRC Canada


Fig. 2. Typical distributions of Tmax with depth below the wall top for reinforced soil walls: (a) triangular distribution used in theTieback Wedge Method (Bell et al. 1975), (b) modified triangular distribution used in the Coherent Gravity Method (AASHTO 1996)and Simplified Method (AASHTO 2002), (c) rectangular distribution for geosynthetic walls (Broms 1978), (d) trapezoidal distributionfor anchored walls (Sabatini et al. 1999), and (e) distribution for geogrid reinforced soil walls (Collin 1986).



true of any composite material in which the components ofthe system have different stiffness values and in which thecomponents are perfectly bonded together (e.g., steel or fibrereinforced concrete). To account for the effect of stiffness,the relationship between reinforcement load and reinforce-ment stiffness must be quantified. The influence of rein-forcement stiffness on reinforcement loads can be assessedfrom both a global (i.e., influence of all reinforcement layersin the wall section; eq. [9]) and a local perspective (i.e., in-dividual reinforcement layer; eq. [11]). Christopher (1993)showed that maximum reinforcement loads increased withincreasing magnitude of the global reinforcement stiffnessvalue, Sglobal. Equation [3] is rewritten as follows to enableback-calculation of global stiffness factor values, Φg(mea-sured), from measured maximum reinforcement load (Tmxmx)values:

[12] ΦΦ Φ Φg

mxmx

v h t local fs fb

measuredmeasured

( )( )

max

= TS Diσ

Data for Φg(measured) versus Sglobal/pa are plotted inFig. 4 for all of the geosynthetic wall case histories in Ta-ble 1. From the data in Fig. 3 the maximum reinforcementload in the wall corresponds to the case with Dtmax = 1.Hence, Dtmax was set equal to 1 in eq. [12]. The constant forthe local stiffness factor Φlocal in these calculations (eq. [10])has been taken as a = 1 for geosynthetic reinforced soilwalls and a = 0 for steel reinforced soil walls as noted ear-lier in the paper. Values for Φfs (facing stiffness factor) andΦfb (facing batter factor) are presented later in the paper. Su-perimposed on Fig. 4 is a regressed approximation to thetrend in data for the geosynthetic case histories in Table 1and for steel reinforced wall case studies reported by Allen

et al. (2001) and Allen and Bathurst (2003) using a powerfunction. The data for steel reinforced soil walls were usedin the regression analysis to extend the predicted relation-ship to reinforcement stiffness values beyond those availablefor the geosynthetic reinforced soil walls in Table 1. Hence,the regression equation is applicable to both the geosyntheticand steel data sets. The power curve fit to the physical data

© 2003 NRC Canada

Allen et al. 983

Fig. 3. Distribution of Dtmax as a function of normalized depth plus average surcharge height versus (a) Tmax normalized by Tmxmx,(b) Tmax normalized by Tmxmx × Φlocal, and (c) Tmax normalized by Tmxmx × Φlocal for wall GW19 and showing the distribution proposedby Allen and Bathurst (2003) for polymer strap walls.

Fig. 4. Measured Φg (eq. [12]) versus normalized global rein-forcement stiffness values (Sglobal/pa).



is reasonably accurate, although there is some scatter for thesteel data, which may be due to factors unique to steel rein-forced soil walls (Allen and Bathurst 2003). In the paramet-ric analyses to follow, coefficient terms α = 0.25 and β =0.25 are used in the power function plotted in Fig. 4.

Local stiffness factor, ΦlocalLocal deviations from overall trends in reinforcement

loads can be expected when the reinforcement stiffness and(or) spacing of the reinforcement change from average val-ues over the height of the wall (i.e., Slocal/Sglobal ≠ 1). Thiseffect is captured by a local stiffness factor Φlocal expressedby eq. [10]. Figure 5 shows that the best predictions formaximum load in the geosynthetic reinforcement layers forthree different case studies using the working stress methodcorrespond to local stiffness factor calculations with a = 1.A value of a = 1 has been selected as a preliminary estimatein the working stress method for structures built with geo-synthetic reinforcement layers.

A parametric investigation using the same values for vari-ables in the denominator of eq. [12] was conducted for thelocal stiffness factor. Equation [3] was rewritten as followsto back-calculate values of local stiffness factor, Φlocal(mea-sured), from measured maximum reinforcement load (Tmxmx)values:

[13] ΦΦ Φ Φlocal

mxmx

v h t g fs fb

measuredmeasured

( )( )

max

= TS Diσ

Values of Φlocal(measured) versus Slocal/Sglobal are plotted inFig. 6 for the geosynthetic wall case histories in Table 1. Asshown in Fig. 6, an exponent value a = 1 provides the bestfit for geosynthetic walls using eq. [10].

A similar parametric investigation by Allen and Bathurst(2003) showed that for steel reinforced soil walls, a value ofa = 0 for the constant coefficient in the local stiffness factorequation is more accurate. An explanation for the differencein values is that steel reinforcement is much stiffer than thesoil, and hence local variations in reinforcement stiffnessmay have little effect on redistribution of reinforcementloads. Table 2 shows the influence of assigning values of a =0 (Φlocal = 1, i.e., ignoring any possible local reinforcementstiffness effect) and a = 0.5 in stiffness method calculationson predicted reinforcement loads in geosynthetic walls (seealso Figs. 5a, 5b, and 5c, in particular comparing curves fora = 0 and a = 1 with the same facing stiffness factor Φfs).The COV values in Table 2 are nearly twice as large if thelocal stiffness effect is not considered (a = 0), indicating astrong relationship between Tmax and the corresponding localstiffness value.

Facing stiffness factor, ΦfsPrevious research has indicated that the stiffness of the

facing and the lateral restraint of the wall facing at the walltoe can have a significant influence on the loads carriedby the soil reinforcement, at least for geosynthetic walls.Tatsuoka (1993) provided an overview of facing stiffness ef-fects on soil wall reinforcement loads. He categorized fac-ings based on their stiffness characteristics as follows: typesA and B, very flexible wrapped geosynthetic, gabion, orsteel skin facings; type C, articulated (incremental) concrete

panels; type D, full-height precast concrete panels; and typeE, concrete gravity structures. Facing rigidity was defined interms of local, axial, shear, and bending rigidity, and overallmass as a gravity structure. Tatsuoka concluded that soil re-inforcement strains tend to decrease as facing rigidity in-creases due to the increase in soil confinement caused bya very stiff facing, thereby reducing reinforcement loads.Loads carried axially by the facing to the toe may also con-tribute to the increased stability that occurs in stiffer facings.If the wall facing is massive enough to behave as a gravitystructure, the loads in the reinforcement may be reduced tovery low values.

Rowe and Ho (1993) concluded that both the facing andfoundation stiffness affect the overall stiffness of the systemand influence the portion of horizontal load carried by thereinforcement and the footing. In fact, for stiff facings (e.g.,full-height concrete panel walls), force equilibrium cannotbe satisfied without considering the toe forces transferred tothe bottom of a facing with a restrained toe (Rowe and Ho1993; Bathurst et al. 1989). Bathurst (1993) investigated theissue of facing stiffness – toe restraint for two full-scale lab-oratory test walls with full-height propped and incrementalaluminium panel facings. For these two walls, Bathurstfound that 25% of the total lateral load at collapse due tosurcharging was carried by the wall toe. In more recentwork, using 3.6 m high modular block-faced systems,Bathurst et al. (2000) found that the wall toe carried approx-imately 40% of the lateral load when the wall was loaded tonear collapse. This more recent work appears to indicate,however, that as wall lateral deformations develop, the rein-forcement layers carry a greater proportion of the total lat-eral load while the facing approaches a limiting capacitybeyond which additional surcharge load increments are car-ried by the reinforcement layers. Allen and Bathurst (2002a)also indicated that propping a full-height panel wall duringbackfilling caused the wall facing to behave as a very stiffcolumn, even after final equilibrium was reached followingprop release. After surcharge loading of the wall, however,reinforcement loads approached values recorded for an oth-erwise identical wall with a more flexible incremental panelfacing that was not externally supported during wall con-struction.

The numerical parametric studies by Lee (2000) men-tioned previously were also used to investigate the effect offacing stiffness on reinforcement loads, using two categoriesof facings: (i) flexible-faced walls (e.g., wrapped-face walls),and (ii) stiff-faced walls (e.g., propped precast concretepanel walls and modular block-faced walls). Lee found thatthe reinforcement loads in the stiff-faced walls were approx-imately 50% of the reinforcement loads in the flexible-facedwalls.

The empirical data available from the full-scale wall casehistories with stiff facings and those with flexible facingswere compared to determine the effect of facing stiffness onreinforcement loads. A parametric investigation similar tothat described in the previous sections was carried out forthe facing stiffness factor Φfs. Equation [3] was rewritten asfollows to back-calculate values of local stiffness factor,Φfs(measured), from maximum reinforcement load (Tmxmx)values estimated from strain measurements:

© 2003 NRC Canada




[14] ΦΦ Φ Φfs

mxmx

v h t g local fb

measuredmeasured

( )( )

max

= TS Diσ

The facing for each of the field walls can be treated asa conventional, uniformly loaded, cantilevered beam. Thestiffness of this “equivalent beam” is a function of its elastic

© 2003 NRC Canada

Allen et al. 985

Fig. 5. Influence of magnitude of facing stiffness factor and local stiffness factor on magnitude and distribution of reinforcement loadTmax: (a) GW16 (wrapped-face wall) with soil surcharge, (b) GW9 (modular block wall) with soil surcharge, and (c) GW5 (incrementalprecast concrete panel wall).



modulus, E, and moment of inertia, I. The maximum elasticbeam deflection can be expressed as (Popov 1978)

[15] ywH

ELbmax

.= 15 4

3

where b is the thickness of the facing column, L is the unitlength of the facing (e.g., L = 1 m), H is the height of thefacing column, and w is the distributed load. The elasticbeam model is admittedly crude given that the wall toe maynot be completely fixed, the facing column often containsjoints (i.e., the beam is not continuous), and the facing col-umn is attached to the reinforcement at intervals. The objec-tive here is not to predict wall deflections, however, butrather to introduce a normalized facing column stiffness pa-rameter Ff that captures the trend in relative facing columnstiffness for a range of facing types and geometry. Ff is de-fined here as

[16] FH

ELbhH

Pfeff

a= 15 4

3

.

where heff is the equivalent height of an unjointed facing col-umn that is 100% efficient in transmitting moment throughthe height of the facing column. The ratio heff/H is used toestimate the efficiency of a jointed facing system to transmitmoment throughout the facing column. The nondimen-sionality of the expression is preserved by the use of pa =101 kPa.

For modular block wall systems, heff = H, since these sys-tems have greater width (typically ≥300 mm) than otherconcrete facing systems. In addition, the blocks are in com-pression, partially due to self-weight and partially due todown-drag forces on the back of the facing, and can transmitmoment through the height of the column (Bathurst et al.2000, 2001). Incremental concrete panel systems are gener-

ally thinner (approximately 100–140 mm) and tend to be-have as beams that are pinned at the panel joints. Therefore,heff is assumed equal to the panel height for this type of fac-ing. The influence of facing type on the stiffness of flexiblewall facings is more challenging. The approach taken here isto consider the column of soil confined by the flexible fac-ing system to be the facing column. For welded wire walls,the length of the horizontal leg of the welded wire facingpanel is taken as the width of the facing, b. For wrapped-face geosynthetic walls, b is taken as the approximate widthof the facing wrap. The elastic modulus of the column inboth cases was taken as 35 000 kPa, which corresponds tothe soil modulus value used by Lee (2000) in numericalmodelling work that included some of the case studies inthis paper.

The value of Ff for the walls in this study varies overthree orders of magnitude and thus allows the structures tobe differentiated based on relative facing stiffness. Values ofΦfs (measured) versus Ff are plotted in Fig. 7 for the geo-synthetic wall case histories in Table 1. Figure 7 shows thatthe measured data can be correlated to the facing stiffnessparameter (eq. [16]) using a power function expressed hereas

[17] Φfs = η(Ff)κ

From regression analysis, η and κ are 0.5 and 0.14, re-spectively. This equation results in Φfs being equal to ap-proximately 1.0 for flexible-faced walls and as low as 0.35for the relatively stiff modular block systems. Incrementalconcrete panel faced systems fall between these two values.

It should be noted that the regression analysis in Fig. 7 tocalculate the constant coefficient values η and κ did not in-clude the data point for wall GW19 which falls well abovethe rest of the data. Recall that wall GW19 was constructedwith a very stiff reinforcement and had the highest global

© 2003 NRC Canada


Fig. 6. Measured Φlocal (eq. [13]) versus Slocal/Sglobal forgeosynthetic reinforced soil walls.

Fig. 7. Measured Φfs (eq. [14]) versus facing stiffness parameter,Ff, for geosynthetic reinforced soil walls.



stiffness value of all the walls in the database (i.e., one totwo orders of magnitude higher). An explanation for this re-sult is that the influence of the facing stiffness on reinforce-ment loads is low for very high global stiffness values. Inother words, regardless of the facing type, for very high re-inforcement global stiffness values, the reinforcement car-ries most of the load acting at the back of the facing, and thestiffness of the facing type plays a lesser role to resist earthpressures.

Figures 5b and 5c demonstrate the improvement in pre-dicted maximum reinforcement loads for modular block andincremental concrete panel walls using Φfs = 0.35 and 0.5,respectively, compared with a value of Φfs = 1.0 representingflexible face structures. Table 2 also shows the result of as-suming Φfs = 1.0 for all case studies, in effect treating allwall facings as flexible. Doing so resulted in substantial in-creases in COV values for the ratio of measured to predictedloads, demonstrating the importance of considering the fac-ing stiffness when estimating Tmax for typical geosyntheticwalls.

For preliminary design purposes, eq. [17] results in thefollowing facing stiffness factor values for typical geo-synthetic walls: (i) Φfs = 0.35 for modular block and proppedconcrete panel faced walls (stiff facings), (ii) Φfs = 0.5 forincremental precast concrete facings, and (iii) Φfs = 1.0 forall other types of wall facings (flexible facings, e.g., wrapped-face, welded wire, or gabion faced).

More data are required to quantify the relationship be-tween facing stiffness and reinforcement global stiffness andtheir combined influence on the magnitude of reinforcementloads in geosynthetic reinforced walls that are constructedwith very stiff reinforcement products and for walls withgreater facing flexibility than those available at the time ofthis investigation.

Facing batter factor, ΦfbIn current practice, wall face batter (i.e., inclination from

the vertical) is taken into account explicitly using Coulombearth pressure theory. Although calculations using the newworking stress method described up to this point in the paperimproved reinforcement load predictions, significant dis-crepancies remained for the battered walls in case studyGW7. As demonstrated by Allen and Bathurst (2002a) andas shown later, the Coulomb earth pressure coefficient tendsto reduce reinforcement loads excessively for heavily bat-tered walls. The influence of reduced confining pressure inthe vicinity of the wall face cannot be captured explicitly bylimit equilibrium methods.

The influence of wall facing batter on maximum rein-forcement loads is adjusted in the proposed working stressmethod using an empirical facing batter factor expressed as

[18] Φfbabh

avh

d

=

K

K

where Kabh is the horizontal component of active earth pres-sure coefficient accounting for wall face batter, Kavh is thehorizontal component of active earth pressure coefficient(assuming the wall is vertical), and d is a constant coeffi-cient. The form of the equation shows that as the wall facebatter angle ω → 0 (i.e., wall facing batter approaches the

vertical) the facing batter factor Φfb → 1. With the exceptionof case study GW7, the structures in Table 1 correspond tovalues of Φfb that are greater than 0.85. Due to the lack ofwall cases with a significant wall batter, back-analyses ofthe type carried out in previous sections were not possible.Rather, a trial and error approach was used to determine areasonable value for the constant d in eq. [18].

Table 2 provides a comparison of the spread (representedby COV) in the ratios of measured to predicted reinforce-ment loads for the proposed method by ignoring the facingbatter effect (d = 0), using d = 0.25, or assigning a constantcoefficient d = 1 (which corresponds to the full effect of fac-ing batter in the Coulomb equation) in eq. [18] for all cases.A value of d = 0.25 gives the best fit based on the availableTmax data and is recommended as the default value in theproposed K-stiffness method. Note that if the full facing bat-ter effect is allowed (i.e., d = 1), the COV value for Tmax ismore than twice the value obtained using d = 0.25.

Influence of soil strength on reinforcement loadsFor working stress conditions in reinforced soil walls, the

soil property that most likely affects the distribution of loadto the reinforcement layers in the wall is the soil modulus.This is due to the relatively low strain levels and the fact thatlimit equilibrium conditions have not been reached. The soilmodulus is difficult to determine, however, and is strain andstress level dependent. Furthermore, for most of the case his-tories reported herein, a measured soil modulus was notavailable. The peak soil friction angle is routinely available,familiar to designers, and relatively easy to measure or esti-mate. In general, as peak friction angle for a granular soil in-creases the soil modulus also increases (Duncan et al. 1980).Hence, the peak friction angle can be interpreted as an indi-cator of relative soil modulus value among soil types.

In the development of the proposed stiffness method, val-ues of K = K0 (eq. [5]) and K = Kah were examined to inves-tigate the relative accuracy of predicted values of Tmax andTmxmx. K0 was used because it is simple to calculate and itsvalue is independent of wall face batter. Kah (the horizontalcomponent of active earth pressure) was determined usingthe Coulomb method, assuming full interface friction for allwalls (i.e., δ = φ, where δ is the wall interface friction angleand φ is the friction angle of the soil) and continuous ornearly continuous reinforcement layers. For these structuresthe reinforcement–facing connections will restrict downwardmovement of the backfill soil against the face, effectively re-sulting in an interface friction angle at the back of the wallface equal to the backfill soil friction angle. In wall GW19,the reinforcement comprised discrete straps, and hence fullmobilization of soil shear strength behind the wall facingpanels may not be expected. For this wall, an interface fric-tion angle of two thirds the soil backfill friction angle wasused to calculate the Coulomb Kah value, which is typicalpractice for concrete–soil interface friction angles. The datain Table 2 show that there was less spread in the ratios ofmeasured to predicted reinforcement loads using K = K0rather than Kah. Some of this spread using Kah is likely dueto an inaccurate accounting of the wall face batter effect onthe calculation of Tmax. Interestingly, setting K equal to aconstant value of 0.3 for all soil friction angles yielded onlya modest increase in the COV value for the bias in all Tmax

© 2003 NRC Canada

Allen et al. 987



values compared with the COV value using K = K0 calcu-lated from the measured peak friction angle. The improve-ment in prediction accuracy of the proposed method usingeq. [5] was more pronounced when the COV values arecompared for the bias in Tmxmx values.

A parametric investigation, similar to that for the globalreinforcement stiffness factor, was carried out to back-calculate K in the K-stiffness method from measured maxi-mum reinforcement load (Tmxmx) values normalized asfollows:

[19] KT

S H S Di( )

( )[ . ( )] max

measuredmeasuredmxmx

v t g l

=+05γ Φ Φ ocal fb fsΦ Φ

The values of factors in the denominator are calculatedusing baseline values described earlier in the paper. Theback-calculated values of K determined from the measuredvalues of Tmxmx versus K from eq. [5] are plotted in Fig. 8for the geosynthetic wall case histories in Table 1. Figure 8indicates that the load in the reinforcement is influenced bythe soil response to load, if K0 is used as the soil indexparameter. There is scatter in the data, however, which isconsistent with the fact that the use of K = K0 can only ap-proximate the real parameter of interest, the soil modulus. Inaddition, this scatter may indicate that other factors relatedto soil properties and construction may be in play (e.g., com-paction effort). Until these other factors are identified, theuse of K = K0 is considered to be a reasonable approach toapproximate the role of soil strength and modulus in the de-velopment of reinforcement load.

Effect of soil unit weight on soil reinforcement loadsThe soil unit weight recorded for each case study was

within 16% of the mean value for all of the walls (γmean =19.5 kN/m3) and ranged from 16.4 to 21.1 kN/m3 (Allen etal. 2002). This variation was considered to be small com-pared with the uncertainty associated with other parametervalues in this investigation, including estimated reinforce-ment loads. The fundamental expression for reinforcementloads (eq. [6]) using the stiffness method, however, showsthat loads (and hence strains) should vary linearly with soilunit weight. To investigate the influence of soil unit weighton predicted reinforcement loads, calculations for Tmaxand Tmxmx were carried out using a constant value γ =19.5 kN/m3. Table 2 shows that there was only a minor dif-ference in the accuracy of predicted reinforcement loads us-ing a default constant unit weight of 19.5 kN/m3 in thecalculations rather than project-specific values. This minordifference is consistent with the small variation in the mag-nitude of measured soil unit weight values. A practical im-plication of this result is that the selection of soil unit weightis not a critical factor for design accuracy using theK-stiffness method.

Effect of reinforcement layer spacing on soilreinforcement loads

The vertical distance between reinforcement layers in thecase studies for this investigation varied from 0.3 to 1.6 m(Allen et al. 2002). Note that this is not necessarily the sameas the vertical zone in the wall that contributes to load in agiven reinforcement layer (i.e., the tributary area). Sv is rep-

resentative of the tributary area when loads are calculated onthe basis of load per unit of wall length and the spacing be-tween layers is uniform. When the spacing is not uniform,this parameter is representative of the average distance be-tween the layers that are adjacent to the layer in question. Atthe top of a wall, Sv includes the full distance between thetop layer and the top of the wall, plus the distance to themidpoint between the top layer and the next layer below.

The magnitude of reinforcement loads (and strains) can beexpected to vary linearly with Sv, as assumed in this stiffnessmethod and conventional design methods. Calculations wereredone using a default value of Sv = 0.6 m, which is a typicalreinforcement spacing value for the walls in Table 1. Table 2shows that the value of Sv has a significant effect on the ac-curacy of predicted reinforcement loads for all of the casehistories.

A parametric investigation, similar to that described forthe global reinforcement stiffness factor, was carried out toevaluate the accuracy of the assumption that maximum rein-forcement loads in a wall vary linearly with Sv. Equation [3]is rewritten as follows to back-calculate values of Sv frommeasured maximum reinforcement load (Tmxmx) values:

[20] STD

vmxmx

h t g local fs fb

back-calcmeasured

( )( )

max

=σ Φ Φ Φ Φ

Again, the values of factors in the denominator are calcu-lated using baseline values described previously. Values ofSv, back-calculated from the measured values of Tmxmx, ver-sus Sv determined directly from the spacing of the reinforce-ment in the wall are plotted in Fig. 9 for the geosyntheticwall case histories in Table 1. The regression suggests alinear relationship between reinforcement load and Sv, as as-sumed in the proposed method and current design methods,is correct. This correlation appears to hold reasonably well,even for large values of Sv. From a practical point of view,the multiplier on Sv that results from the linear regression inFig. 9 can be ignored without influencing design accuracy.

Overall performance of the K-stiffnessmethod

The accuracy of the proposed K-stiffness method (eq. [6])for geosynthetic walls is illustrated in Fig. 10 for all of thefull-scale field wall case histories in Table 1. The improve-ment in predicted loads versus loads estimated from mea-sured strains using the proposed stiffness method comparedto the AASHTO Simplified Method is apparent whenFig. 10 is compared to Fig. 1. The same conclusion isreached by examination of values for the mean and spread(COV) of the bias values for the two methods shown in Ta-ble 2.

Based on Table 2, there is twice as much variation in theprediction of reinforcement load Tmax, considering all rein-forcement layers, than for the prediction of the maximum re-inforcement load, Tmxmx (compare the COV value for Tmaxwith the COV value for Tmxmx). Therefore, better predictionof the distribution of Tmax versus depth could greatly im-prove the prediction accuracy of the K-stiffness method.

The strain level in the reinforcement also appears to havea significant effect on the prediction accuracy of the

© 2003 NRC Canada




K-stiffness method. Figure 11 shows predicted maximum re-inforcement strains using the K-stiffness method and Tmxmx,plotted against the maximum measured reinforcement strainin the backfill in each geosynthetic wall at the end of wallconstruction (i.e., data do not include any locally high con-nection strains). The predicted strains were calculated by di-viding Tmxmx at the end of construction by the reinforcementlayer stiffness value, J. It appears that once reinforcementstrains exceed approximately 3–4% for the available casehistories, the K-stiffness method consistently underpredictsthe measured strain. Note that all of the strains greater thanabout 3% were measured in full-scale laboratory test wallsthat were surcharged to loads well in excess of workingstress conditions (see Bathurst 1993; and Bathurst et al.1993b, 2000, 2001 for details regarding these full-scale lab-oratory test walls).

Bathurst (1993) and Bathurst et al. (1993b, 2000, 2001)noted that in their full-scale laboratory tests, soil failure oc-curred before reinforcement rupture. Evidence of soil failureincluded a sudden and large outward movement of the wallface, soil settlement directly behind the wall face, and a con-current increase in reinforcement strains. None of the full-scale field walls recorded reinforcement strains that wereconsistent with soil failure, with the exception of wallGW10, which exhibited signs of soil failure, but only afterthe strain gauges mounted directly on the reinforcementceased to function (at about 3% strain). Wall GW20 ap-peared to be stable at the end of surcharge completion at areinforcement strain of just over 3%. After several months ofreinforcement creep, however, the reinforcement strains ex-ceeded 4%. Based on observations made by Carrubba et al.(1999) and the pattern observed in the creep data (Allen andBathurst 2002b), it can be concluded that the wall was ap-proaching soil failure.

Based on correlating physical observations of soil failurewithin the reinforced wall backfill made by Bathurst (1993)and Bathurst et al. (1993b, 2000, 2001) with the measuredmaximum strain in the wall backfill reinforcement, itappears that backfill soil failure matches strain values inFig. 11 where the K-stiffness method begins to consistentlyunderpredict the reinforcement strain. Soil failure does notappear to be the only cause of this underprediction, however.The crudeness of the facing stiffness factor to capture the ef-fect of facing stiffness on reinforcement strain and load de-velopment appears to also contribute to this underprediction.Note that in Fig. 11, at high reinforcement strains, the datapoints for walls with stiff facings are consistently belowthose for walls with flexible facings. In fact, while theK-stiffness method tends to begin to underpredict loads atjust over 3% strain for the flexible-faced walls, this under-prediction begins at approximately 1.5–2% strain for thestiff-faced walls. This may be an indicator that the facingstiffness correction factor used in the K-stiffness method isnot a constant as proposed, but instead increases toward 1.0(i.e., less effect of facing stiffness) as strain increases. Athigher strains, the facing appears to have reduced reserve ca-pacity to carry additional load, consistent with the observa-tions made by Bathurst et al. (2000). Although the facinghas not failed at reinforcement strains greater than 1.5–2%,the reinforcement takes on additional load to maintain facingcolumn equilibrium. Once the reinforcement strain exceeds3–4%, the soil begins to fail for both flexible- and stiff-facedwalls.

Plane strain shear strength data for the same backfill usedin wall GW16 indicated that peak soil strength occurred atstrains of about 2–3% (Boyle 1995). Since the backfill usedfor this wall was at the upper end of the range of soil shearstrength for the case histories considered herein, these plane

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Allen et al. 989

Fig. 8. Back-calculated K (eq. [19]) versus K calculated fromeq. [5].

Fig. 9. Back-calculated values of Sv from measured Tmxmx

(eq. [20]) versus Sv determined directly from spacing of rein-forcement in geosynthetic reinforced soil walls.



strain peak strains are likely at the lower end of the typicalrange for granular soils. Plane strain shear strength data re-ported by Lee (2000) for Royal Military College of Canada(RMC) soil specimens at confining pressures comparable tosurcharge pressures required to initiate large wall deforma-tions indicate that peak soil strains were also about 2–3%. Inboth cases (Boyle 1995; Lee 2000), the plane strain testresults indicated that the peak soil strain increases with in-creasing confining stress.

Based on data from the RMC full-scale walls, geosyn-thetic reinforcement tensile strains large enough to causesigns of soil failure are typically numerically greater thanthe 2–3% peak strain values required to cause the soil to be-gin to fail in a plane strain shear test. That is, reinforcementstrain levels of approximately 3–4% or more appear to cor-respond to soil peak shear strains of 2–3% based on labora-tory plane strain testing of RMC granular soils.

Prevention of reinforcement strains that are great enoughto allow failure of the soil should be an objective of anyworking stress design method. Soil failure is defined as con-tiguous or near-contiguous zones of soil with shear strains inexcess of the strain at peak strength. Contiguous shear zoneshave been observed in test walls taken to collapse underuniform surcharge loading (Bathurst 1990; Bathurst et al.1993b; Allen and Bathurst 2002b). Based on analysis oflong-term creep strains measured in walls taken from work-ing stress conditions to near collapse, Allen and Bathurst(2002b) found that once a wall goes beyond working stressconditions, the load levels in the reinforcement begin to in-crease as internal soil shear surfaces continue to develop andthe soil approaches a residual strength. This, in turn, leads tohigher creep rates and an acceleration of strain developmentin the wall. Nevertheless, this condition does not necessarilyresult in reinforcement rupture and wall collapse if the rein-forcement has sufficient strength. It does mean, however,that wall deformations may become excessive. Once the soil

has failed, for all practical purposes the wall has failed andan internal strength limit state for the soil has been achieved.

The key to prevent reaching the soil failure limit state is toestimate how much strain can be allowed in the reinforcedwall system (i.e., the soil reinforcement) without causing thesoil to reach a soil failure condition. It appears that prevent-ing the reinforcement strain from exceeding a 3–3.5%design value will be adequate for the high shear strengthgranular backfill soils in this study and likely conservativefor weaker backfill soils. Since the maximum reinforcementstrain to prevent soil failure was derived from high shearstrength soils, the 3–3.5% strain value represents what is ef-fectively a lower bound value and is therefore recommendedfor design based on available data. The relationship betweenreinforcement strain and the soil shear strain at peak strengthneeds to be investigated for a wider range of soils, however.

Conclusions

A new approach, called the K-stiffness method, is pro-posed to predict reinforcement loads and strains for internalstability design of geosynthetic reinforced soil walls. Thenew working stress methodology described in this paper hasbeen developed and calibrated using a database of reinforcedsoil wall reinforcement strain and load data. The methodol-ogy considers the stiffness of the various wall componentsand their influence on reinforcement loads. The objective ofthe method is to design the wall reinforcement so that thesoil within the wall backfill is prevented from reaching astate of failure consistent with the notion of working stressconditions. This soil failure limit state is not considered inthe reinforced soil wall internal stability design methods cur-rently available, yet, based on the research results presentedherein, is likely to be a controlling limit state for geosyn-thetic structures. This new approach is largely empiricallybased, using back-analysis and curve fitting of measured

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Fig. 10. Predicted versus measured values of Tmax in reinforcement layers for geosynthetic walls using the K-stiffness method.



data from full-scale reinforced soil walls. The database usedcaptures the typical range of wall types and geosynthetic re-inforcement materials. This gives confidence that the newmethod is applicable to most typical geosynthetic reinforcedsoil walls constructed with granular backfill soils.

Acknowledgements

The writers would like to acknowledge the financial sup-port of the following State Departments of Transportation:Washington, Alaska, Arizona, California, Colorado, Idaho,Minnesota, New York, North Dakota, Oregon, and Wyo-ming. The writers are also grateful for the financial supportof the National Concrete Masonry Association, the Rein-forced Earth Company, the Natural Sciences and Engi-neering Research Council of Canada (NSERC), the

Academic Research Program at RMC, and grants from theDepartment of National Defence (Canada). Lastly, the writ-ers would like to acknowledge the contributions of Dr. BarryChristopher, Dr. Stanley Boyle, and Mr. Ryan Berg for re-viewing the background materials to this paper, assistingwith the gathering of data, and for many technical discus-sions that have helped the writers to develop the new designmethod.

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Allen et al. 991

Fig. 11. Predicted versus measured reinforcement strain (based on Tmxmx) using the K-stiffness method for full-scale production (field)and full-scale laboratory geosynthetic walls. The inset figure corresponds to a reinforcement strain range from 0% to 5%.



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List of symbols

a constant coefficient (dimensionless)b thickness of facing column (m)d constant coefficient (dimensionless)

Dtmax reinforcement load distribution factor (dimen-sionless)

E elastic modulus (N/m2)Ff facing column stiffness parameter (dimensionless)

heff equivalent height of facing column (m)H height of the wall or facing column (m)i counter (1, 2, 3, … n)I moment of inertia (m4)J tensile stiffness of the reinforcement (N/m)

J2% tensile stiffness of the reinforcement at 2% strain(N/m)

Jave average tensile stiffness for all the reinforcement lay-ers (N/m)

Ji tensile stiffness of an individual reinforcement layer(N/m)

K coefficient of lateral earth pressure (dimensionless)Kabh horizontal component of active earth pressure coeffi-

cient accounting for wall face batter (dimensionless)Kah horizontal component of active earth pressure coeffi-

cient (dimensionless)Kavh horizontal component of active earth pressure coeffi-

cient for vertical wall (dimensionless)Kback-calc back-calculated coefficient of lateral earth pressure

K0 coefficient of lateral earth pressure at rest (dimen-sionless)

L unit length of wall facing (m)n total number of reinforcement layers in wall sectionN number of data pointspa atmospheric pressure (101 kPa)q surcharge pressure (Pa)R coefficient of correlationS equivalent height of uniform surcharge pressure (m)

Sglobal global reinforcement stiffness value (N/m2)Slocal local reinforcement stiffness value (N/m2)

Sv, S iv tributary area for reinforcement layer i (assumed

equivalent to the vertical spacing of the reinforce-ment when analyses are carried out per unit length ofwall) (m)

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Allen et al. 993



Sv back-calc back-calculated tributary area for reinforcement layer(m)

T imax, Tmax maximum measured reinforcement load in layer i

(N/m)Tmxmx maximum reinforcement load from all layers in the

wall (N/m)w uniformly distributed load (N/m)

ymax maximum deflection of uniformly loaded cantileverbeam (m)

z depth below the top of the wall (m)α constant coefficient (dimensionless)β constant coefficient (dimensionless)δ wall interface friction angle (°)φ friction angle of the soil (°)

φds peak soil friction angle from direct shear tests (°)

φps peak plane strain friction angle of the soil (°)φtx peak soil friction angle from triaxial tests (°)

γ unit weight of soil (N/m3)γmean mean unit weight of soil (N/m3)

η constant coefficient (dimensionless)κ constant coefficient (dimensionless)

σh lateral earth pressure acting over the tributary area(Pa)

ω wall facing batter from vertical (°)Φ influence factor = Φg × Φlocal × Φfs × Φfb (dimension-

less)Φfb facing batter factor (dimensionless)Φfs facing stiffness factor (dimensionless)Φg global stiffness factor (dimensionless)

Φlocal local stiffness factor (dimensionless)

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a new working stress method for prediction of reinforcement loads in

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