A novel approach to estimating the bearingcapacity stability of geosynthetic reinforcedretaining walls constructed on yieldingfoundations
Graeme D. Skinner and R. Kerry Rowe
Abstract: Yielding foundation conditions have been shown to adversely affect the stability and behaviour of overlyinggeosynthetic reinforced soil walls. To avoid serious problems and maintain a cost-effective design, careful considerationmust be given to short-term stability. Previous research has shown that lengthening and stiffening the bottomreinforcement layer of the wall can increase the external stability, but the magnitude of this increase is not well under-stood. To provide insight regarding the potential benefit of lengthening and stiffening the bottom reinforcement layer, anumerical investigation is made of the plastic collapse mechanism due to bearing capacity failure of the foundation de-posit for the case of a 6 m high geosynthetic reinforced retaining wall on a 10 m thick soft to firm viscoplastic claystratum. The calculated behaviour of the wall is compared with that from typical and novel design considerations forboth a conventional reinforced wall and a wall where the bottom reinforcement layer has been extended and stiffened.A parametric study of the extended bottom reinforcement layer stiffness and interaction is reported, and the influenceon the external stability is discussed.
Résumé : Il a été démontré que les conditions de déformation à la limite élastique des fondations affectent défavora-blement la stabilité et le comportement des murs sus-jacents en sol armé de géotextile. Pour éviter de sérieux problè-mes et maintenir une conception économique, on doit considérer attentivement la stabilité à court terme. Des recherchesantérieures ont montré que l’allongement et le renforcement de l’armature de la couche d’assise du mur peut augmen-ter la stabilité externe, mais l’importance de cette augmentation n’est pas bien comprise. Pour fournir un éclairagequant au bénéfice de l’allongement et du renforcement de l’armature de la couche d’assise, on a étudié numériquementle mécanisme d’effondrement plastique dû à la rupture en capacité portante du dépôt de fondation pour le cas d’un murde soutènement armé de géotextile de 6 m de hauteur reposant sur une couche d’argile viscoplastique molle à ferme.Le comportement calculé du mur a été comparé à celui basé sur des considérations de conception nouvelle et typiquetant pour un mur armé conventionnel que pour un mur où la couche d’assise d’armature à été allongée et renforcée.On rapporte une étude paramétrique de la rigidité et de l’interaction de la couche d’assise d’armature allongée, et ondiscute de son influence sur la stabilité externe.
Mots clés : mur en sol armé, fondation molle déformante, conception de la capacité portante, analyse numérique.
[Traduit par la Rédaction] Skinner and Rowe 779
Introduction
Two of the major design considerations for a geosyntheticreinforced wall are stability (including an assessment of thepotential internal and external failure modes of the wall) anddeformation. Previous work (Rowe and Skinner 2001; Skin-ner 2002; Skinner and Rowe 2003) examined the effect ofconstructing a reinforced soil wall on a yielding foundationand design considerations for the internal stability and settle-
ment. The external stability and, specifically, the short-termbearing capacity of the wall has received limited attention todate.
When site and project conditions are such that the externalfactors of safety are less than the minimum desirable designvalues, methods of increasing the external stability are re-quired. Various methods of increasing the external stabilityof a reinforced retaining wall include, but are not limited to:avoidance, removal, treatment, preloading, and surcharging
Received 13 September 2003. Accepted 17 January 2005. Published on the NRC Research Press Web site at http://cgj.nrc.ca on8 June 2005.
G.D. Skinner.1 Golder Associates Limited, 940–6th Avenue SW, Suite 1000, Calgary, AB T2P 3T1, Canada.R.K. Rowe. GeoEngineering Centre at Queen’s–RMC, Department of Civil Engineering, Queen’s University at Kingston, Kingston,ON K7L 3N6, Canada.
of the clay deposit (with and without the use of verticaldrains); staged construction of the wall; and increased lengthand stiffness of the reinforcement layer at the bottom of thewall. Of specific interest to this paper is the case of a longerlayer of high-strength reinforcement at the base of the wall,which has been shown to increase the external stability andreduce the differential settlements of the wall in a number ofcases (Curtis et al. 1988; Bloomfield et al. 2001; Troung etal. 2001). However, the expected increase in stability andthe methods of predicting the associated external factors ofsafety are not well understood, especially in terms of the po-tential plastic collapse mechanism and the base reinforce-ment stiffness interaction.
The objective of this paper is to investigate the plastic col-lapse mechanism of a yielding foundation and the influenceof extending and stiffening the bottom reinforcement layer.A numerical analysis of a typical geosynthetic reinforcedsoil wall, constructed on a viscoplastic clay foundation, wasconducted to investigate the wall and foundation behaviour(Fig. 1). The behaviour of the wall calculated from the finite-element (FE) analysis will be compared with that expectedfor the current and proposed techniques for estimating exter-nal stability for both a conventional reinforced wall andwalls where the bottom reinforcement layer has been ex-tended. The effect of increasing the tensile strength and stiff-ness of the bottom reinforcement layer will be examined,and the resulting increase in external stability, effect on wallbehaviour, and implications for design will be discussed.
Bearing capacity design methods andconsiderations
Review of current bearing capacity design methodsTypical design practice for calculating the external stabil-
ity of a reinforced soil wall assumes that the reinforced sec-tion of the wall acts as a rigid block (CGS 1992; FHWA1996; NCMA 1996). The rigid soil block is assumed to actas a shallow rigid strip footing, and the potential plastic col-
lapse of the foundation deposit is analyzed using commonplasticity solutions assuming a semi-infinite soil deposit ofuniform strength (Terzaghi 1943; Hansen 1968). Rowe andSoderman (1987) synthesized the work of Davis and Booker(1973) and Matar and Salencon (1977) to account for both alinear increase in the undrained shear strength with depthand a limited foundation deposit depth for estimating the ul-timate undrained bearing capacity and depth of the failurezone below a rigid footing (Figs. 2 and 3). For all cases pre-sented in this study, the eccentricity was close to or less thanzero, and the inclination of the resulting load on the assumedrigid footing was accounted for in the bearing capacity cal-culation (along with the effects of an increasing undrainedshear strength with depth and the finite depth of the layer;Fig. 2).
Design considerations for a reinforced soil wallThe stability methods discussed above assume the base of
the rigid footing (B) to be equal to the width of the rein-forced soil block. From plasticity considerations, a rigid foot-ing on top of a fine-grained foundation layer has a maximumundrained resistance at the edges of the footing equal to (2 +π)suo, and the resistance increases toward the centreline ofthe footing. However, a reinforced soil wall does not have asymmetrical geometry, and if one were to treat the rein-forced block as a rigid footing (with an inclined load), oneedge of this assumed rigid footing would be within the back-fill and hence have a significant surcharge, whereas the outeredge (at the toe of the wall) would have little or no sur-charge. Therefore, the pressure distribution along the base ofthe rigid footing would not be symmetrical as assumed incurrent design methods. Because of the non-symmetricalpressure distribution, it may be argued that the width of theequivalent rigid footing needs to be greater than the width ofthe reinforced soil block (Fig. 4a) if one is to obtain a goodestimate of the bearing capacity. Thus, there would be an as-sociated equivalent rigid footing of the width (beq) at whichplastic collapse of the foundation deposit occurred under the
applied (inclined) wall loading, and the size of the equiva-lent rigid footing would depend on the geometry and charac-teristics of the wall and foundation.
The potential increase in bearing capacity of a reinforcedsoil wall due to extending and stiffening the bottom rein-forcement layer and the effective increase in the width of theequivalent rigid footing have not been investigated to date oraccounted for in the current design manuals. For these cases,the width of the equivalent rigid footing may be expected tobe a function of the extended length and, more significantly,the tensile stiffness of the layer (Fig. 4b).
Numerical model
A version of the FE program, AFENA (A Finite Element
Numerical Algorithm), originally developed by Carter andBalaam (1990) and modified (as noted below) to account forthe modelling of geosynthetic reinforced soil walls and theviscoplastic clay behaviour, was used to conduct the numeri-cal analyses reported herein. The soil retaining wall was ex-amined under two-dimensional (plane strain) conditions,consistent with normal design assumptions (CGS 1992;FHWA 1996; NCMA 1996). The FE mesh (Fig. 5) used4335 eight-noded isoparametric elements to model the soil,masonry, and concrete; 352 linear bar elements (with no sig-nificant compressive or bending strength) to model the rein-forcement; and 1434 interface elements between the variousmaterials. The initial geostatic stress conditions in the foun-dation were based on the unit weight and effective coeffi-cient of lateral earth pressure at rest (Ko) of the soil.
Fig. 2. Bearing capacity factor for nonhomogeneous soil, synthesized from results by Davis and Booker (1973) and Matar andSalencon (1977) (modified from Rowe and Soderman 1987).
Fig. 3. Effect of nonhomogeneity on depth of the failure zone beneath a rough rigid footing, based on the results obtained by Matarand Salencon (1977) (modified from Rowe and Soderman 1987).
The viscoplastic model adopted for the continuum ele-ments used for the clay foundation combined an ellipticalcap yield surface (Chen and Mizuno 1990) and a Drucker–Prager failure criterion with Perzyna’s (1963) overstressmodel and fully coupled Biot (1941) type consolidation(Rowe and Hinchberger 1998).
An elastoplastic stress–strain model with a Mohr–Coulomb failure criterion was adopted for the continuum el-ements used for the coarse-grained soils, masonry facing,and concrete key. The Young’s modulus, E, of the granularsoils was assumed to be nonlinear and to be given byJanbu’s (1963) equation, expressed in the form
[1]EP
KP
n
a a
=⎛
⎝⎜
⎞
⎠⎟σ3
where σ3 is the minor principal stress; Pa is the atmosphericpressure (e.g., 101.3 kPa); and the values of K and n wereselected on the basis of correlations (Duncan et al. 1980)with the assumed soil properties. To deal with the case oflow σ3 (i.e., σ3 = 1 kPa), a minimum stiffness was assumedequal to E = K(Pa)
n (with units of kPa for this case). Themasonry and concrete materials were assumed to be purelyelastic.
Rigid–plastic interface elements, as described by Roweand Soderman (1987), were used to model the behaviour be-tween the various materials. A Mohr–Coulomb failure crite-rion was used to model failure at interfaces.
The model adopted had been previously used to success-fully describe the behaviour of a full-scale reinforced soilwall (Rowe and Skinner 2001), the geosynthetic reinforcedSackville test embankment (Rowe and Hinchberger 1998),
Fig. 4. Definition of variables for bearing capacity analysis of a reinforced soil wall (a) without and (b) with an extended bottom rein-forcement layer.
and in the numerical analysis of reinforced embankmentsconstructed on viscoplastic foundation soils (Li and Rowe1999; Li 2000).
The wall construction was simulated layer by layer, fol-lowing the typical sequence used to build reinforced retain-ing walls (FHWA 1996) and using a numerical schemeselected to ensure the numerical stability of the solution andto minimize numerical errors, as described in detail by Skin-ner (2002). The wall was constructed over an assumed24 day period, and in this paper attention is focused on theshort-term stability and behaviour of the wall.
A limit equilibrium analysis was conducted to ensure ade-quate distance between the bottom rough rigid and right lat-eral smooth rigid boundaries and the primary zone ofinfluence of the wall, and they were modelled at distances of10 and 24 m, respectively. The distance to the left lateralsmooth rigid boundary was taken to be 10 times the initiallength of the reinforcement to minimize boundary effects, asdiscussed by Rowe and Skinner (2001). Finally, the top andbottom of the clay layer were assumed to be boundaries ofzero excess pore pressure seepage.
Model parameters and designconsiderations
The wall was designed on the basis of the current Na-tional Concrete Masonry Association (NCMA) workingstress design code (NCMA 1996) for segmental walls. Thisapproach specifically considers a segmental wall facing andis based on Coulomb’s active earth pressure theory. Al-though this investigation focuses on the external bearing ca-pacity stability of a geosynthetic reinforced soil wall and theeffects of varying the properties of the bottom reinforcementlayer, a check was made to ensure that the internal stabilityof the wall satisfied current design methods.
Design and description of “typical” wallThe wall was designed to a height of 6 m, with 10 layers
of 3.6 m long knitted polyester (PET) geogrid. The facingwas assumed to be constructed from 40 masonry facingblocks (e.g., Pisa II blocks from Unilock® (Unilock Ltd.,Georgetown, Ont.)), each having an infilled unit weight of21.8 kN/m3 and a natural setback of 20 mm due to interlock-ing shear keys. The wall was embedded 0.3 m (see Fig. 1).A prefabricated concrete key and a gravel layer were used atthe toe and base of the wall, respectively. The concrete keyacted only as a levelling surface for face alignment andserved no structural purpose. The thin (0.15 m) layer ofgravel at the base of the reinforced wall and around the keyacted as a level surface for construction of the wall and asthe top drainage boundary for the clay foundation below thewall. The water table was assumed to be located at the top ofthe clay foundation.
The allowable reinforcement tensile strength was assumedto be 20.4 kN/m (Stratagrid 200; see IFAI 1999) and ac-counted for the ultimate and creep-limited tensile strength ofthe geogrid and for additional strength reductions due toinstallation damage and durability, as indicated in Table 1.The geogrid secant tensile stiffness, J, was taken to be
Ultimate tensile strength (kN/m) ASTM standard D 4595 (ASTM 1998a) 39.7Creep tensile strength (kN/m) ASTM standard D 5262 (ASTM 1998b) 24.6Allowable tensile strength (kN/m) GRI standard and practice GG4(a) and GG4(b) (GRI 1991a, 1991b) 20.4Tensile stiffness, J (kN/m) ASTM standard D 4595 (ASTM 1998a) 400
Table 1. Geosynthetic material properties.
CharacteristicSandbackfill
Drainagegravel
Unit weight (kN/m3) 20 20Friction angle (°) 35 45Dilation angle (°) 6.0 12.5Poisson’s ratio, ν 0.30 0.35Coefficient of earth pressure at rest,
Ko
0.4 0.3
Janbu K and n 460 and 0.5 900 and 0.7
Table 2. Sand and drainage gravel material parameters.
400 kN/m, on the basis of ASTM standard D 4595 (ASTM1998a) and Bathurst (2000, unpublished)2. The polyestergeogrid was assumed to have limited susceptibility to creepdeformation (Koerner 1990; Rowe 2000).
The reinforced and retained backfill were assumed to bethe same cohesionless sand, and the drainage layer at thebase of the wall was assumed to be cohesionless gravel. Theunit weight, friction and dilation angles, and other assumedparameters for the sand and gravel are given in Table 2. Allparameters were taken from the range of typical values forthese materials (Craig 1992; Holtz and Kovacs 1981), withthe following exceptions. The dilation angle of each materialwas assumed to be given by Bolton’s (1986) equation, ψ′ =(φ′ – φ cv′ )/0.8, where the value of the constant volume fric-tion angle (φ cv′ ) was assumed to be 30° and 35° for the sandand gravel, respectively (Craig 1992). It should be noted thatthese dilation angles were assumed to be constant at alltimes in the analysis, and it has been shown (Skinner 2002)that this assumption would not affect the results presented inthis study. The Janbu (1963) nonlinear stiffness parameters,K and n, were selected for the assumed soil properties (seeDuncan et al. 1980).
The block–block and block–reinforcement interface pa-rameters were estimated from test protocol SRWU-1(NCMA 1996) for both the ultimate criterion and the ser-viceability criterion, as reported by Bathurst, Jarrett, and As-sociates Inc. (1996a, 1996b), and are given in Table 3. Thefriction angle for the interface between the facing blocks andthe backfill soil was taken as two-thirds the sand friction an-gle. The same assumption was made for the interface be-tween the gravel and both the facing blocks and the concretekey, based on the gravel material. The friction angle for theinterface between the backfill and the foundation was as-sumed to be equal to the normally consolidated (NC) fric-tion angle of the foundation, as it was the lesser value for thetwo soils. The angle of friction for the interface between thebackfill soil and reinforcement was taken as 90% of thebackfill friction angle on the basis of similar material param-eters (Krieger et al. 1994), rather than the conservative as-sumption of 70% indicated by NCMA (1996) for the casewhere no other data are available. The interface friction an-gles for various combinations of materials are summarizedin Table 3.
The minimum reinforcement length and wall embedmentdepth were taken as 0.6 times the height of the wall and theexposed height of the wall divided by 20, respectively, asspecified by the NCMA (1996) manual. Further, the maxi-mum reinforcement spacing was limited to two times thefacing block width, as recommended by the American Asso-ciation of State Highways and Transportation Officials(AASHTO 1996). The design coefficient for sliding for thegeogrid reinforcement was taken as 0.95 (NCMA 1996).
Hydraulic conductivity constant, Ck 0.5Ratio of horizontal to vertical hydraulic
conductivity, kh/kv
3
Viscoplastic fluidity constant (/h) 1.0×10–7
Viscoplastic strain rate exponent 30Vertical preconsolidation pressure at top of layer
(kPa)33
Change in vertical preconsolidation pressure withdepth (kPa/m)
4.5
Undrained shear strength at top of layer (kPa)* 18Change in undrained shear strength with depth
(kPa/m)*1.2
Note: NC, normally consolidated; OC, overconsolidated.*Based on the estimated corrected (Bjerrum 1973) shear vane results.
Table 4. General foundation material properties.
2 Unpublished test data for ASTM Standard D 4595 stress–strain data for Stratagrid 200. Obtained from Dr. R.J. Bathurst, Royal MilitaryCollege, Kingston, Ont., 2000.
The NCMA (1996) manual covers a wide range of poten-tial internal and facing failure modes and considers the char-acteristics of the interface between the various materials.The internal and facing stability of the wall were governedby the required minimum reinforcement length (3.6 m), theconnection strength between the reinforcement and the facingblocks for the second reinforcement layer from the base ofthe wall, and the maximum reinforcement spacing (0.6 m) forthe remaining layers. In this respect, a layer of reinforcementwas added at a height of 0.3 m to prevent reinforcement–fac-ing connection rupture from occurring in the layer above it(see Fig. 1).
Foundation descriptionThe 10 m thick foundation deposit was taken to be a
viscoplastic yielding clay, similar to that described by Li andRowe (1999), with properties as given in Table 4. The initialvoid ratios were taken to be 2.2–1.9 (from top to bottom ofthe deposit), and the average unit weight was 15.4 kN/m3.The vertical hydraulic conductivity was assumed to be afunction of the void ratio, given by the equation
[2] k ke eC
v voo
k
= −⎛
⎝⎜
⎞
⎠⎟exp
where the initial vertical hydraulic conductivity (kvo) andvoid ratio (eo) are given in Table 4; and the hydraulic con-ductivity change index (Ck) was taken to be 0.5 for bothcases (based on Mesri et al. 1994). Given that typically thehydraulic conductivity of clay is anisotropic (Tavenas et al.1983; Terzaghi et al. 1996), the ratio of horizontal to verticalhydraulic conductivity was assumed to be kh /kv = 3. Theviscoplastic characteristics of the clay were based on therate-dependent relationship between undrained shearstrength and strain rate, as presented by Kulhawy and Mayne(1990) and estimated by Li (2000). All other relevant soilproperties are summarized in Table 4. The initial vertical ef-fective stress and vertical preconsolidation pressure profilesare shown in Fig. 6. Given the assumed parameters and ac-counting for the relationship between plane strain and cor-rected (Bjerrum 1973) field vane strengths (Skinner 2002),the corrected undrained shear vane strength at the top of thefoundation stratum, suo, was calculated from the viscoplasticcap model (Rowe and Hinchberger 1998) to be 18 kPa andto increase with depth at a rate of 1.2 kPa/m, as shown inFig. 6. It should be noted that the higher than expected ratioof su /σp′ was due to the viscoplastic nature of the soil andthe applied strain rate (Rowe and Hinchberger 1998; Skinnerand Rowe 2003) and is not beyond a reasonable value for asoft viscoplastic clay.
Examination of conventional reinforced soilwall
The initial wall considered in the investigation had a uni-form reinforcement layer length of 3.6 m (including the baselayer). The external stability was governed by the short-termbearing capacity and plastic collapse of the soft to firm foun-dation deposit. The ultimate bearing capacity was estimatedby the method presented by Rowe and Soderman (1987),with an assumed rigid footing width equal to the width ofthe reinforced block (3.6 m). The factor of safety for the ul-timate bearing capacity was calculated as 0.66 and wastherefore below unity.
The initial FE analysis of the geosynthetic reinforced soilwall, with a uniform reinforcement length of 3.6 m, wasconducted to provide a reference point for the wall height atwhich collapse occurred and an initial comparison and vali-dation with respect to conventional stability methods. TheFE analysis predicted failure at a wall height of 4.8 m, corre-sponding to 80% of the target wall height of 6.0 m and anexternal bearing capacity factor of safety of 0.8, which didnot agree with the initial factor of safety prediction (0.66).
The deformations at the face and the base of the wall cor-responded to the general rotational movement around the topof the wall and tilting of the wall base associated with bear-ing capacity failure. The strains in all reinforcement layerswere below the allowable 5% limit (NCMA 1996) and werenot significantly affected by the external bearing capacityfailure of the foundation. It should be noted that all resultspresented in this study were taken for the specified heightsjust prior to failure of the foundation deposit and the onsetof indeterminate deformations.
Overconsolidated (OC) yielding and NC failure zones de-veloped beneath the wall as the wall height increased, andjust prior to failure the NC failure zone extended around the
Fig. 6. Initial effective stress, preconsolidation pressure, and un-drained shear strength of foundation (su = Bjerrum (1973) cor-rected shear vane strength).
base of the reinforced wall (Fig. 7). The FE model used inthis study is capable of accounting for both NC and OCyielding of a soil beyond the elastic range and before ulti-mate failure occurs (Atkinson and Bransby 1978). It shouldbe noted that there was a small amount of NC yielding andplastic failure (and an associated increase in the strain ratecontours) along the bottom clay boundary below the wall.This was a consequence of the initial soil parameters and thefact that the stress state was initially close to the yield sur-face and consolidation of the soil close to the seepageboundary occurred during construction. It was found thatthis behaviour was sufficiently deep that the base boundarydid not influence the general external stability of the wall orthe results of any part of this study (Skinner 2002). The ve-locity vectors and strain rate contours (major principal strain
rate) below the wall just prior to failure of the foundation ata height of 4.8 m are shown in Figs. 8 and 9, respectively.The velocity vectors agreed with the general bearing failuremovement of the foundation soil, and the strain rate contoursemphasize the general shape of the plastic failure zone.However, both showed soil movement and a plastic failurezone extending past the end of the reinforced soil block, thussignifying a larger equivalent rigid footing width than ini-tially assumed. The equivalent rigid footing width and thedepth of the plastic deformations were estimated from thecombined plasticity zones, velocity vectors, and strain ratecontour plots to be 5.5 and 3.3 m, respectively, as illustratedin Fig. 9. It should be noted that all equivalent rigid footingwidths and depths of the plastic failure zone were estimatedfrom the combined plasticity zones, velocity vectors, and
Fig. 7. Plasticity zones in wall and foundation for wall with a uniform reinforcement length of 3.6 m (just prior to failure at height =4.8 m). M/C, Mohr–Coulomb; N/C, normally consolidated; O/C, overconsolidated.
Fig. 8. Velocity vectors in wall and foundation for wall with a uniform reinforcement length of 3.6 m (just prior to failure at height =4.8 m).
strain rate contour plots for each case presented in thisstudy. The FE results showed that the equivalent rigid foot-ing width was larger than the typical design assumption, andthis difference moderately increased the bearing capacity ofthe wall in this case.
Back analysis of the bearing capacity factor of safety forthe collapse wall height of 4.8 m for the initially assumedwidth of the rigid footing (3.6 m) and the estimated equiva-lent rigid footing width (5.5 m) gave estimates of 0.87 and0.98, respectively. Thus, the factor of safety (0.87) for theinitial width of 3.6 m did not agree with the FE results,whereas the factor of safety (0.98) for an equivalent rigidfooting width of 5.5 m agreed well with the FE results andshowed the increase in bearing capacity associated with theincrease in bearing width. The depth of the plastic failurezone was estimated from Fig. 3 to be 3.5 m for the equiva-lent rigid footing width, which agreed well with the esti-mated depth of 3.3 m obtained from the FE analysis, incontrast to the estimated depth for the plastic failure zonefor the initial footing width (2.2 m, from Fig. 3). Therefore,the initial width of the rigid footing (3.6 m) assumed in cur-rent design methods appears to be gaining additional resis-tance from the foundation deposit, as a result of a largerrigid footing width, increased by a factor of 1.5, for anequivalent rigid footing width of 5.5 m in this case. Thus,current design methods that assume a bearing width equal tothe width of the reinforced soil section of the wall may bemoderately conservative.
Effect of extended and stiffened bottomreinforcement layer
Design considerations and description of analysesIn normal design, increasing the size of the reinforced soil
block can increase the bearing capacity of a wall. Addi-tionally, it has been shown above that for a reinforced soil
wall with uniform reinforcement length, the width of theequivalent rigid footing is larger than the width of the rein-forced soil block. Therefore, increasing the size of the rein-forced soil block for the initial case may be expected tofurther increase the width of the equivalent rigid footing, de-pending on the relative stiffness of the reinforced soil block.As an alternative to increasing the length of all the rein-forcement layers to increase the size of the reinforced soilblock, it is proposed that the bottom reinforcement layeralone be lengthened and stiffened to increase the overallrigid footing width.
To evaluate the effectiveness of increasing the length andstiffness of only the bottom reinforcement layer, a compari-son was made of the results for two walls. Both walls hadthe same properties as the initial wall described above, withthe following exceptions. In the first case (Wall-U7), thelength of all the reinforcement layers was increased to 7 m,but the tensile properties (ultimate strength, 39.7 kN/m; se-cant tensile stiffness, 400 kN/m) were the same as previ-ously described. In the second case (L7-J120), only thebottom layer was increased to 7 m, and the strength and ten-sile stiffness of this layer were increased to 1340 and12 000 kN/m, respectively (Paralink 1250S; see IFAI 1999).For this second case, the remaining reinforcement layersabove the bottom layer had the same properties as previ-ously described (length, 3.6 m; ultimate strength, 39.7 kN/m;secant tensile stiffness, 400 kN/m).
The bearing capacity of both walls was estimated from themethod presented by Rowe and Soderman (1987) and usingFigs. 2 and 3. The bearing capacity of Wall-U7 was initiallyestimated assuming the reinforced section of the wall actedas a rigid block, as specified in the design manual (NCMA1996). For case L7-J120, it was initially assumed that thebase of the wall was sufficiently stiffened and that the equiv-alent rigid footing width was at least equal to the length ofthe bottom reinforced layer (7.0 m) and it supported the en-
Fig. 9. Strain rate contours (10–4%/min) in foundation for wall with a uniform reinforcement length of 3.6 m (just prior to failure atheight = 4.8 m).
tire soil mass above the equivalent rigid footing width.Therefore, the walls had the same assumed footing width,applied loads, and bearing capacity factor of safety (0.82).
Results of analysesThe results of the FE analyses for Wall-U7 and L7-J120
indicated failure at wall heights of 5.25 and 5.4 m, respec-tively. This corresponds to 87.5% and 90% of the target wallheight (6.0 m), and again they did not agree with the initiallyestimated factors of safety. The walls showed the same gen-eral deformations associated with a bearing capacity failureat the face and base of the walls, and all reinforcementstrains were below the allowable 5% limit (NCMA 1996).
The OC yielding and NC failure zones in the foundation
were approximately the same for both walls, and again theNC failure zone extended around the base of the wall justprior to failure (Fig. 10). The associated velocity vectors andstrain rate contours were also similar for both cases andshow the extent of the foundation soil movement and col-lapse zones (Figs. 11 and 12, respectively). Although the FEresults did not show a significant increase in the size of theequivalent rigid footing, it can be seen from Figs. 11 and 12,respectively, that the soil movement and plastic failure zonesextend slightly past the end of the reinforcement layers, sig-nifying a larger rigid footing width than initially assumed.The equivalent rigid footing widths and depth of the plasticdeformations were estimated to be 7.7 and 4.3 m for Wall-U7 and 8.0 and 4.6 m for case L7-J120, as illustrated in
Fig. 10. Plasticity zones in wall and foundation just prior to failure. (a) Wall-U7 at height = 5.25 m (uniform reinforcement length of7.0 m). (b) Case L7-J120 at height = 5.4 m (bottom reinforcement layer extended to 7.0 m). M/C, Mohr–Coulomb; N/C, normallyconsolidated; O/C, overconsolidated.
Fig. 12. It was found that L7-J120 had a slightly largerequivalent rigid footing width, likely because of its greateroverall stiffness, compared with Wall-U7.
Back analysis of the bearing capacity factors of safety atthe failure heights of 5.25 and 5.40 m for each wall for theinitial rigid footing width (7 m in both cases) gave estimatesof 0.96 and 0.93, respectively, and for equivalent rigid foot-ing widths (7.7 m for Wall-U7 and 8.0 m for L7-J120) gaveestimates of 0.98 and 0.96, respectively. For three of the fourcases there was good agreement between the calculated val-ues from the plasticity methods and the FE results.
The back-calculated bearing capacity factors of safetywere all within 5% of the estimated FE wall behaviour for
both cases and are closer to the FE behaviour for the equiva-lent rigid footing widths, which shows excellent agreementbetween the two cases. This is with the exception of the fac-tor of safety (0.93) for case L7-J120 with the initial width of7.0 m where the factor of safety (0.96) for the equivalentrigid footing width of 8.0 m showed slightly better agree-ment with the plasticity solutions with the increased bearingwidth. It should be noted that although for these two cases(Wall-U7 and L7-J120) the back-calculated factors of safetyfor the larger equivalent rigid footing widths (7.7 and 8.0 m,respectively) were closer to the estimated FE results than theresults for the initial width (7.0 m) were, there was only amarginal difference between them. This smaller difference
Fig. 11. Velocity vectors just prior to failure. (a) Wall-U7 at height = 5.25 m (uniform reinforcement length of 7.0 m). (b) Case L7-J120 at height = 5.4 m (bottom reinforcement layer extended to 7.0 m).
was due to the fact that the soft to firm undrained shearstrength profile had a low increase in undrained shearstrength with depth (ρc) relative to the undrained shearstrength at the surface (suo). This translated into a low ρc /suoratio and low to moderate increases in the ultimate bearingcapacity with increased footing width, as shown in Fig. 2.This effect of a low ρc value is further reflected by the rela-tively small difference between the back-calculated factorsof safety discussed below.
The estimated depths of the plastic failure zones (fromFig. 3) based on the equivalent rigid footing widths were 4.2and 4.3 m for Wall-U7 and L7-J120, respectively, and theseestimates agreed well with the depths deduced from the FEanalyses (4.3 and 4.6 m for Wall-U7 and L7-J120, respec-tively). Further, the estimated depth of the plasticity zone(3.7 m, from Fig. 3) for the initial rigid footing width
(7.0 m) was significantly less than the predicted FE depthsas given above. Therefore, the proposed constructionmethod, which increases the external stability of a reinforcedsoil wall built on a soft to firm foundation by increasing thelength and stiffness of the bottom reinforcement layer alone,appears valid for a sufficiently reinforced wall base.
Parametric study of length and secanttensile stiffness of bottom reinforcementlayer
Design considerations and description of analysesIt has been shown above that (i) the proposed method of
increasing the bearing capacity of a geosynthetic reinforcedsoil wall by extending and sufficiently stiffening the bottom
Fig. 12. Strain rate contours (10–4%/min) just prior to failure. (a) Wall-U7 at height = 5.25 m (uniform reinforcement length of 7.0 m).(b) Case L7-J120 at height = 5.4 m (bottom reinforcement layer extended to 7.0 m).
reinforcement layer to increase the bearing width is sup-ported by the FE analysis; and (ii) the equivalent rigid foot-ing width considered for design may be greater than thelength of the bottom reinforcement layer.
To evaluate the effect of the tensile stiffness and length ofthe bottom reinforcement layer on the external bearing ca-pacity stability and behaviour of the wall, a parametric studyvarying these two parameters was conducted. All of thewalls investigated had the same properties as the initial walldescribed above, with the exception of the bottom reinforce-ment layer. For the parametric study, bottom layer lengths of3.6, 7.0, 11.0, and 14.0 m were considered. Reinforcementstrengths of 39.7–800.0 kN/m and tensile stiffnesses of 400–20 000 kN/m were examined (Tables 5 and 6). The remain-ing reinforcement layers above the bottom layer were 3.6 min length, with an ultimate strength of 39.7 kN/m and a se-cant tensile stiffness of 400 kN/m (as before). Unless other-wise stated, all reinforcement materials were assumed to bepolyester geogrids, with properties based on the assumedtensile stiffness, and the associated ultimate and allowablestrengths were based on a strain of 10% and a factor ofsafety of 2, respectively. It should be noted that case L7-J120 was the same as discussed in the previous section andhas been included here for comparison. The initial bearingcapacity for each case was estimated with the method pre-sented by Rowe and Soderman (1987); it was assumed thatthe base of each wall was sufficiently stiffened to have a
rigid footing width at least equal to the length of the bottomlayer (Table 5).
Results of analysesThe collapse heights and percentages of the 6 m target
height obtained from the FE analyses are summarized in Ta-ble 6. Only the cases in which the tensile stiffness of thebottom reinforcement layer was >2000 kN/m (see Tables 5and 6) resulted in good agreement between the collapseheights estimated from the plasticity solution and the FEanalyses. In the cases with lower tensile stiffness of the bot-tom layer (cases L7-J8, L11-J8, and L14-J8, with J =800 kN/m; and cases L7-J20, L11-J20, and L14-J20, withJ = 2000 kN/m), the general behaviour of the wall and foun-dation deposit and the collapse height of the wall were notsignificantly affected by the increased stiffness and length ofthe bottom layer. The plasticity zones, velocity vectors, andstrain rate contours just prior to failure for these six caseswere approximately the same as for the initial case examinedabove, where all reinforcement layers had a uniform lengthof 3.6 m (see Figs. 7, 8, and 9, respectively). The equivalentrigid footing widths and depths of the plastic failure zonesfor these six cases were estimated from the FE results andare summarized in Table 6. It can be seen that the equivalentrigid footing width was 5.5 m for each case (but see com-ment about L11-J20 below) and equal to the minimum widthestimated from the initial case examined above with a uni-
*Based on assumed strength at 10% stiffness unless otherwise noted.†Based on assumed general factor of safety for reinforcement rupture equal to 2, except as noted otherwise.‡Based on reported (IFAI 1999) material parameters (Paralink 1250S).
Table 5. Outline of cases investigated in parametric study.
form reinforcement length of 3.6 m, and the predicteddepths of the plastic failure zone agreed with the estimateddepth obtained from the plastic solution presented by Roweand Soderman (1987). It should be noted that case L14-J20did have a slightly higher collapse height and equivalentrigid footing width, but the general behaviour of the casewas approximately the same as that of cases L7-J20 andL11-J20. Therefore, there was no significant or notable in-crease in the external stability for any increase in the investi-gated bottom reinforcement layer length when the tensilestiffness of the layer was increased to 2000 kN/m or less.
The plasticity zones and velocity vectors for the remain-ing investigated cases, where there was an increase in exter-nal stability, generally showed the same plastic yield andfailure zones and soil movement trends as expected for abearing capacity failure and as observed for case L7-J120(see Figs. 10 and 11). It was found from the design calcula-tions and FE results that the load inclination tended to de-crease slightly as the width of the footing increased, asexpected. The equivalent rigid footing widths and depths ofthe plastic failure zone were estimated for each case fromthe plasticity zones, velocity vectors, and strain rate contourFE results just prior to failure and are summarized in Ta-ble 6. The bearing capacity factors of safety for each casewere back calculated for the wall failure heights deduced
from the FE analyses on the basis of the estimated equiva-lent footing widths and were found to agree well with eachother. Additionally, in most cases, the depths of the plasticfailure zones predicted from the FE analyses agreed wellwith the estimated depths obtained with the method pre-sented by Rowe and Soderman (1987). It should be notedthat for cases L14-J120 and L14-J200 the initial targetheight of 6 m was exceeded, so for estimating the collapseheight, loading beyond 6 m was achieved by adding a uni-formly distributed load to the top of the wall.
The results summarized in Table 6 indicate that when thebottom reinforcement layer was sufficiently stiffened (with athreshold of approximately J = 2000 kN/m for the cases in-vestigated), there was an increase in the external stability,and more specifically, an increase in the equivalent rigidfooting width corresponding to bearing capacity failure. Thesize of the equivalent rigid footing width was greatly af-fected by the length and tensile stiffness of the bottom rein-forcement layer. It was generally observed that increasingeither the tensile stiffness or the length of the bottom rein-forcement layer, or both, increased the equivalent rigid foot-ing width (provided the threshold stiffness was reached), asindicated in Fig. 13. The figure presents the increase inequivalent rigid footing width (normalized with respect tothe length of the reinforcement in the main section of the
Note: FE, finite element.*Based on bearing capacity plasticity solution presented by Rowe and Soderman (1987), a target wall height of 6 m, and a footing width equal to L
from Table 5.†Deduced from FE strain rate contours results just prior to failure.‡Estimated from Rowe and Soderman (1987), as shown in Fig. 3.§Calculated from bearing capacity plasticity solution presented by Rowe and Soderman (1987) for the collapse height of the wall and equivalent rigid
footing width, beq (m), deduced from FE results.
Table 6. Summary of FE results for parametric study.
wall, 3.6 m in this case) due to increases in the bottom rein-forcement layer tensile stiffness for the three bottom rein-forcement lengths investigated (normalized with respect tothe target wall height of 6 m).
ExampleFigure 13 can be used in a number of ways: either to esti-
mate the tensile stiffness required to achieve a specifiedequivalent rigid footing width or conversely to estimate theequivalent rigid footing width that can be achieved for aspecified reinforcement tensile stiffness. For example, ac-cording to the method of Rowe and Soderman (1987) (seeFigs. 2 and 3), a 2 m high reinforced retaining wall con-structed on a soft fine-grained soil with an undrained shearstrength at the surface of 14 kPa (suo) that is increasing at arate (ρc) of 1.5 kPa/m with depth would require a minimumequivalent rigid footing width of 3.6 m to achieve a bearingcapacity stability with a minimum factor of safety of 2. If itis assumed that the internal stability of the wall is satisfiedwith the minimum reinforcement length of 1.2 m, the ratioof the equivalent rigid footing width to the length of the re-inforcement layer above the bottom layer is calculated as 3,and this is the starting value along the vertical axis ofFig. 13 to estimate the required tensile stiffness (case 1). Inthis case, there is more than one combination of reinforce-ment tensile stiffness and base reinforcement length that canbe used to achieve the required equivalent rigid footingwidth, as shown for case 1. For example, the length of thebottom reinforcement layer could be taken as either 4.66 m(Lb /H = 2.33) or 3.66 m (Lb /H = 1.83), with reinforcementtensile stiffness values of 4750 and 7700 kN/m, respectively,to achieve the equivalent rigid footing width required.
Alternatively, if a reinforcement material with known ten-sile stiffness of 6000 kN/m is to be used for the bottom rein-forcement layer (case 2), one can start on the horizontal axisof Fig. 13 and estimate the equivalent rigid footing width
that can be achieved for a given length of the reinforcementlayers above the extended bottom layer. In this case, a bot-tom reinforcement layer length of 4.66 m can be used to ob-tain an equivalent rigid footing width of 4.1 m, greater thanthe minimum width of 3.6 m. The other two bottom rein-forcement layer lengths of 3.66 and 2.34 m can only achieveequivalent rigid footing widths of 3.1 and 2.6 m, respec-tively, and are not stiff enough, with a tensile stiffness of6000 kN/m, to achieve the required minimum footing width.
As previously noted, there is a lower threshold of rein-forcement tensile stiffness below which there is little effecton the bearing capacity of the wall. Correspondingly, there isan upper threshold of tensile stiffness for each extendedlength examined beyond which further increase has very lit-tle effect on stability and the equivalent rigid footing width.This upper threshold is likely a function of the reinforce-ment and backfill soil stiffness and interaction and a func-tion of the foundation soil and the development of the plasticfailure mechanism. Additional investigation of the upperthreshold is required to fully understand this aspect of theproblem. As well, increasing the bottom reinforcement layerlength to wall height ratio increases the equivalent rigidfooting width. It should be recognized that Fig. 13 is basedon results for one wall height and soil profile, and althoughthe results look very promising, further research is requiredto confirm the normalized behaviour for other geosyntheticreinforced soil wall cases.
Summary and conclusions
The foundation of a geosynthetic reinforced soil wallplays an integral role in the internal and external stabilityand deformation of a wall, particularly when the wall is con-structed on a viscoplastic yielding material. For estimatingbearing capacity, typical design methods consider the widthof the rigid footing to be equal to the width of the reinforced
Fig. 13. Design considerations for estimating width of equivalent rigid footing.
soil block. It has been shown that because of the non-symmetric geometry of a reinforced soil wall, the width ofthe equivalent rigid footing may be larger than the assumedreinforced soil block, and therefore typical design methodsare conservative. Further, it has been shown that typicalplasticity solutions (Rowe and Soderman 1987) agree withthe predicted FE results based on the equivalent rigid foot-ing widths deduced from the FE analyses and that the largerequivalent footing widths yield higher factors of safety.
A technique for increasing the external stability of a rein-forced wall by increasing the length and stiffness of the bot-tom reinforcement layer has been presented and examined. Ithas been shown that for the foundation soil examined, ex-tending and stiffening the bottom reinforcement layer can in-crease the external stability, and specifically the bearingcapacity, of a reinforced soil wall. Typical plasticity bearingcapacity solutions agree with the predicted FE results basedon the equivalent rigid footing widths deduced from the FEanalyses. However, it was also shown that the equivalentrigid footing width was significantly influenced by the ten-sile stiffness and length of the extended bottom reinforce-ment layer. Preliminary design considerations have beenpresented to account for the tensile stiffness and extendedlength of the bottom reinforcement layer when estimatingthe size of the equivalent rigid footing. It is suggested thatthe approach warrants more study, in particular to investigatewall height and increased undrained shear strength throughthe entire soil profile and in the plastic failure zone and toprovide experimental verification.
Acknowledgements
The research reported here was funded by the Natural Sci-ences and Engineering Research Council of Canada and theOntario Ministry of Training, Colleges and Universities (On-tario Graduate Scholarship Program). The constructive com-ments of the reviewers are much appreciated.
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List of symbols
B rigid footing width, width of reinforced soil blockbeq equivalent rigid footing widthCc compression indexCk hydraulic conductivity change indexCr recompression indexd depth of plastic failure zone from bearing capacity failureD thickness of foundation deposite void ratioE Young’s moduluseo initial void ratioH height of reinforced soil wallJ geosynthetic reinforcement tensile stiffnessK Janbu material constantkh horizontal hydraulic conductivityKo coefficient of lateral earth pressure at restkv vertical hydraulic conductivity
kvo initial vertical hydraulic conductivityL total length of reinforcement layer
Lb length of bottom reinforcement layern Janbu material constant
