Bearing Capacity of Strip Foundations in Reinforced Soils

Bearing Capacity of Strip Foundations in Reinforced SoilsDebarghya Chakraborty1 and Jyant Kumar2

Abstract: Amethod is proposed to determine the ultimate bearing capacity of a strip footing placed over granular and cohesive-frictional soilsthat are reinforced with horizontal layers of reinforcements. The reinforcement sheet is assumed to resist axial tension but not bendingmoment.The analysis was performed by using the lower bound theorem of the limit analysis in combination with finite elements. A single layer anda group of two layers of reinforcements were considered. The efficiency factors hg and hc that need to be multiplied with the respectivebearing capacity factor Ng and Nc to account for the inclusion of the reinforcements were established. The results were obtained for differentvalues of the soil internal friction angle (f). The critical positions of the reinforcements, which would result in a maximum increase in thebearing capacity, were established. The required tensile strength of the reinforcement to avoid its breakage during the loading of the foundationwas also computed. The results from the analysis were compared with those available in the literature. DOI: 10.1061/(ASCE)GM.1943-5622.0000275. 2014 American Society of Civil Engineers.

Author keywords: Bearing capacity; Failure; Limit analysis; Plasticity; Reinforced soil.

Introduction

It is understood that an inclusion of any formof reinforcement in the soilmass below a footing not only increases its bearing capacity but alsoreduces its settlement. Various forms of reinforcement layers, such asgalvanized steel strips, geotextiles, and geogrids, are often used in theconstruction of foundations. Among available important experimentalstudies, Binquet and Lee (1975) and Fragaszy and Lawton (1984)conducted model tests by using metal strips to examine the response offootings loaded over a reinforced soil bed. Binquet and Lee (1975)noted that the bearing capacities of shallow foundations, with the use ofgalvanized steel strips, could be increased by two to four times com-pared with unreinforced soils. Binquet and Lee (1975) also identifiedthree different types of failuremechanisms, namely (1) the shear failureat the interface of reinforcement strips and adjoining soil mass, (2) theshear failure within the soil mass above the top layer of the re-inforcement, and (3) the breakage (tensile failure) of the reinforcementstrips. By conducting laboratory model tests on square footings, Guidoet al. (1986) determined the bearing capacity of foundations reinforcedwith geogrids and geotextiles. Through laboratory model tests, Khinget al. (1993) examined the bearing capacity of a strip foundation placedover sand reinforced with geogrids. Omar et al. (1993), Shin et al.(1993), Das et al. (1994), and Das and Omar (1994) also conductedlaboratory tests using multiple layers of geogrids. Adams and Collin(1997) carriedout full-scalemodel tests tofind the effect of geosyntheticreinforcement on the bearing capacity of foundations.Dash et al. (2004)compared the performance of different types of geosynthetics for stripfoundations. Compared with the existing experimental studies, notmany theoretical studies have been reported in literature for examiningthe effect of soil reinforcements on the bearing capacity of foundations.

With the use of the elastoplastic FEM, solutions have been obtained bydifferent researchers for determining the bearing capacity of founda-tions without any reinforcements for various soils and loading con-ditions (Griffiths et al. 2006; Gourvenec et al. 2006; Yamamoto et al.2008). With the use of the rigid plastic FEM, Asaoka et al. (1994) andOtani et al. (1998) determined the stability of reinforced soil structures.Considering reinforced soil mass as a homogeneous but an anisotropicmaterial, Yu and Sloan (1997) used finite-element formulations of thelower and upper bound limit analysis for a reinforced soil mass. An-alytical techniques, which are generally based on the limit equilibriummethod, are also quite popular for solving the different bearing capacityproblems without any reinforcement (Terzaghi 1943; Meyerhof 1963;Rodriguez-Gutierrez and Aristizabal-Ochoa 2012a, b). Blatz andBathurst (2003) used the limit equilibriummethodbyassuming a failuremechanism to incorporate the effect of the reinforcement on the bearingcapacity of foundations. Michalowski (2004) used the upper boundtheorem of the limit analysis, but by assuming a geometry of thecollapse mechanism, for calculating the bearing capacity of reinforcedfoundations. Deb et al. (2007) used Fast Langrangian Analysis ofContinua (FLAC) to examine the performance of a multilayered geo-synthetic reinforced granular bed. In the present paper, an analysis wasproposed by using the lower bound theorem of the limit analysis incombination with finite elements to determine the bearing capacity ofa strip footing that is placed over granular and cohesive-frictional soilembeddedwith layers of horizontal reinforcements. It was assumed thatthe reinforcements can resist axial tension but not the bendingmoment.The computational results were obtained for a single and two layers ofreinforcements for different values of soil friction angle (f). The criticaldepths of the reinforcements were determined. The tensile strength ofreinforcements that is required to avoid the possibility of any tensilefailure (breakage) of the reinforcement was also computed for differentcases. The results obtained from the analysis were comparedwith thoseavailable in the literature. It is expected that the research studywould bebeneficial from a design point of view.

Problem Definition

It is required to determine the ultimate bearing capacity of a roughstrip footing placed over a soil medium that is reinforced with (1)a single layer and (2) a group of two layers of horizontal

1Research Scholar, Civil Engineering Dept., Indian Institute of Science,Bangalore 560012, India. E-mail: [email protected]

2Professor, Civil EngineeringDept., Indian Institute of Science, Bangalore560012, India (corresponding author). E-mail: [email protected]

Note. This manuscript was submitted on May 23, 2012; approved onDecember 4, 2012; published online on December 6, 2012. Discussionperiod open until July 1, 2014; separate discussions must be submitted forindividual papers. This paper is part of the International Journal ofGeomechanics, Vol. 14, No. 1, February 1, 2014. ASCE, ISSN 1532-3641/2014/1-4558/$25.00.

INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE / JANUARY/FEBRUARY 2014 / 45

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http://dx.doi.org/10.1061/(ASCE)GM.1943-5622.0000275http://dx.doi.org/10.1061/(ASCE)GM.1943-5622.0000275mailto:[email protected]:[email protected]

reinforcements. The soil mass is assumed to follow an associatedflow rule and Mohr-Coulombs failure criterion. The analysis isbased on the approximation that the reinforcement sheet has a re-sistance against the axial tension but not against the bending mo-ment. Such an assumption is generally applicable for flexiblereinforcement, such as geotextiles; on the other hand, other forms ofrelatively rigid reinforcements, such as galvanized steel strips andgeogrids, also offer some resistance to the bending moment apartfrom the axial tension. The improvement in the bearing capacity thatwould be estimated with this assumption will, therefore, remain onthe conservative side compared with the use of rigid reinforcements.

It was also assumed that the reinforcement will not fail (break)structurally in axial tension but rather a shear failure would occuralong the interface between the reinforcement and adjoining soilmass. It is also intended to determine the axial strength of thereinforcements that would be needed to avoid any tensile failure.

Analysis

To perform the analysis for the present reinforced earth problem, itwas assumed that the soil mass follows a two-dimensional contin-uum model and the reinforcement layer acts as a one-dimensional

Fig. 1. Chosen domain and the stress boundary conditions for (a) single layer of reinforcement; (b) group of two layers of reinforcement

46 / INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE / JANUARY/FEBRUARY 2014

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flexible structural element. Such a modeling is usually referred to aseither mixed modeling (Anthoine 1989) or a structural approach(Michalowski and Zhao 1995). It was also assumed that the re-inforcement layer (1) is perfectly flexible such that no bendingmoment can develop at any point along its length, (2) always remainshorizontal, and (3) has a substantial axial strength such that notension failure of the reinforcement can take place. To assure that nobending moment (M) develops at any point along the length of thereinforcement, the magnitude of the soil overburden pressure on thetop surface of the reinforcement layer should become exactly equalto the soil reaction pressure exerted on the bottom surface of thereinforcement. In other words, themagnitudes of the normal stressesat any point above and below the surface of the layer of the re-inforcement need to be exactly the same (continuous). However, thereinforcement layer can resist axial tension; therefore, a disconti-nuity in the shear stress must prevail along the place of the re-inforcement sheet, which is assumed to have infinitesimal thickness.This is demonstrated in Fig. 1(a) by means of a free body diagram ofan element of the reinforcement having length dx. Note that (1) thevalues of the normal stresses above and below the layers of thereinforcements become exactly the same, that is, suy 5s

ly; (2) a dis-

continuity exists in the magnitudes of the shear stress; and (3) anincrease in the tensile force per unit width over the length of thereinforcement having the length (dx) becomes equal to Dft5 tuxydx1 2tlxydx; the superscripts u and l refer to the upper andlower surface of the reinforcement layer. It should be pointed out that attwo extreme ends of the reinforcement, the magnitude of the tensileforce becomes equal to zero; in the analysis, there is no need to specifyany additional boundary constraint to satisfy this condition.

The analysis was carried out based on the lower bound theoremof the limit analysis in combination with finite elements and linearprogramming. The formulation proposed by Sloan (1988) for anyplane-strain problem was used. Nodal stresses (sx, sy, txy) are keptas basic unknown variables. Three-noded triangular elements areused to discretize the stress field. Statically admissible stressdiscontinuities are permitted everywhere along the interfacesbetween adjacent elements; that is, along any stress discontinuity,

the magnitudes of normal and shear stresses always remain con-tinuous. The element equilibrium conditions are satisfied every-where in the domain. The stress boundary conditions are satisfiedalong different known boundaries. At all the nodes, it is assuredthat the yield condition is not violated. The linear optimizationtechnique is adopted to solve the problem. To ensure that the finite-element formulation leads to a linear programming problem,following Bottero et al. (1980), the original Mohr-Coulomb yieldsurface is linearized by a regular polygon of p sides inscribed to theparent yield surface. The value of p in the current study is takenequal to 24.

Provision to Incorporate the Inclusion ofthe Reinforcement

In the analysis, to incorporate the inclusion of the reinforcementbetween different element interfaces lying above and below thereinforcement sheet, a shear stress discontinuity is permitted; con-versely, the normal stress continuity is retained. It should be men-tioned that the reinforcement layer itself was not modeled with anytype of explicit element. Rather, a shear stress continuity condition isrelaxed on the edges of the elements lying above and below the layerof the reinforcement; the presence of the reinforcement layer makesthe shear stress discontinuous on either surface of the reinforcementsheet. It was assumed that the angle of internal friction (d) betweenthe reinforcement material and adjoining granular soil mass issimply equal to f. Because of this condition, it is specified in theanalysis that the absolute magnitude of the shear stress along all theelement edges above and below the reinforcement layer remainsalways smaller than (c2sn tanf), where sn is the magnitude of thetensile normal stress at any point along the layer of the reinforcement.At present, no attempt has been made to consider different values offriction angles along the reinforcement-soil interface.

The normal stress (sn) and shear stress (tnt) acting on a planeinclined at an angle v to the horizontal axis (measured positivecounterclockwise) are given by

Fig. 2. Mesh used in analysis, along with a zoomed view around the footing, for f5 30


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d.

sn sx sin2v sy cos2v2 txy sin 2v (1a)

tnt 20:5sx sin 2v 0:5sy sin 2v txy cos 2v (1b)

The continuity of the normal stresses condition along the edges ofthe elements, lying above and below the horizontal reinforcementlayer, would generate two equality constraints on 12 nodal stresses.Using only Eq. (1a), these equality constraints can be summarizedby the following matrix equation:

hAdcstat

i212

sdc

121 fbdcstatg21 (2)

where

Adcstat

212

G 2G 0 0

0 0 G 2G

(3a)

G13 sin2 v cos2 v 2sin 2v

(3b)

bdcstat

T f 0 0 g12 (3c)

sdc

T1x12

nsqx,1 s

qy,1 t

qxy,1 s

rx,2 s

ry,2 t

rxy,2 s

qx,3 s

qy,3 t

qxy,3 s

rx,4 s

ry,4 t

rxy,4

o(3d)

wherev5 anglemade by the discontinuity edgewith the horizontal;for the present case with a horizontal reinforcement, v5 0. Thesuperscripts q and r in Eq. (3d) indicate two adjacent elements oneither side of the layer of the reinforcement.

As in everywhere within the soil domain, the continuity of nor-mal and shear stresses along any discontinuity line would generatefour equality constraints on 12 nodal stresses (associated with thefour nodes). Using Eq. (1), these equality constraints can be sum-marized by the following matrix equation:

Adcstat

412

sdc

121

bdcstat

41 (4)

where

Adcstat

412

K 2K 0 0

0 0 K 2K

(5a)

K23

sin2 v cos2 v 2sin 2v

20:5 sin 2v 0:5 sin 2v cos 2v

(5b)

Table 1. Comparison of the Obtained Ng Values for a Rough Footing(d5f) without Any Reinforcement

f

(degrees)Presentworka

Kumar andKhatri (2008)a

Ukritchon et al.(2003)a

Kumar(2009)b

30 13.65 13.65 13.20 14.6835 31.43 31.90 29.30 34.3140 76.81 77.88 69.90 85.1045 205.46 204.53 165.00 232.65aLower bound limit analysis with finite elements and linear programming.bMethod of characteristics.

Table 2. Comparison of the Obtained Nc Values without AnyReinforcement

f

(degrees)Presentworka

Meyerhof(1963)b

Bolton andLau (1993)c

Griffiths(1982)d

0 5.09 5.14 5.1010 8.22 8.34 8.34 8.3020 14.47 14.83 14.84 14.8030 29.16 30.13 30.14 30.10aLower bound limit analysis with finite elements and linear programming.bLimit equilibrium method.cMethod of characteristics.dElastoplastic finite elements.

Fig. 3. Variation of the efficiency factors with d1=B for a single layerof reinforcement in (a) sand; (b) c-f soil


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Tab

le3.

Variatio

nof

d 1cr=B,d

2cr=B,T

1=gB2,andT2=gB2a

ndMaxim

umValuesof

hgforSand

f(degrees)

Layersof

reinforcem

ent

Single

Two

d 1cr=B

hg2max

(maxim

umefficiency

factor)

T1-max=gB2(

maxim

umtensionin

Layer

1)d 1

cr=B

d 2cr=B

hg2max

(maxim

umefficiency

factor)

T1-max=gB2(

maxim

umtensionin

Layer

1)T2-max=gB2(

maxim

umtensionin

Layer

2)

300.29

1.55

3.87

0.29

0.29

2.23

4.19

5.31

350.43

1.68

11.36

0.36

0.38

2.53

14.64

15.27

400.50

1.80

30.91

0.43

0.50

2.97

43.89

44.83

450.57

1.95

97.39

0.50

0.64

3.63

148.67

136.77

Tab

le4.

Variatio

nof

d 1cr=B,d

2cr=B,T

1=cB

,T2=cB

and

maxim

umvalues

ofhcforc-fsoil

f(degrees)

Layersof

reinforcem

ent

Single

Two

d 1cr=B

hc2

max

(Maxim

umefficiency

factor)

T1-max=cB

(Maxim

umtensionin

Layer

1)d 1

cr=B

d 2cr=B

hc2

max

(Maxim

umefficiency

factor)

T1-max=cB

(Maxim

umtensionin

Layer

1)T2-max=cB

(Maxim

umtensionin

Layer

2)

00.22

1.13

3.64

0.22

0.22

1.21

3.93

1.69

100.36

1.20

5.06

0.29

0.29

1.33

5.90

2.72

200.50

1.31

9.46

0.36

0.43

1.54

9.54

5.30

300.64

1.47

17.86

0.50

0.64

1.84

24.72

16.05


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bdcstat

T f 0 0 0 0 g14 (5c)In the previous expressions, the terms Adcstat and fbdcstatg are

known; the term fsdcg will be an unknown.

Finite-Element Mesh and Boundary Conditions

By considering the symmetry of the domain about the vertical line(AE) passing through the center line of the foundation, a rectan-gular domain in x-y coordinate axes, as shown in Figs. 1(b and c), isused. The center of the footing surface is taken as the origin of thecoordinate axes; the y-axis is considered positive in the upwarddirection.

The chosen domain and the associated stress boundary con-ditions are shown in Fig. 1. Along the stress free ground surface(HG), the values of the shear and normal stresses become equal tozero. Along the center line (AE) of the footing, txy 5 0. The footingbase is assumed to be fully rough. Therefore, along the soil-footingbase (AH), it was specified that

txy# c2sy tanf; the normalstress is taken as positive when tensile. The depth (D) of the domainbelow the footing base and the horizontal extent (Le) of the domainfrom the edge of the footing are found by using a number of trialssuch that (1) the plastic zones generated from the analysis do notextend up to the domain boundaries EF and FG; and (2) the mag-nitude of the collapse load remains almost constant even if a larger

size of the domain is being selected. The value of D is varied from6:5B for f5 0 to 17:6B for f5 45 (where B 5 width of thefooting). Similarly, the value of Le is varied from 6:0B for f5 0 to18:1B forf5 45. The domain is discretized into a number of three-noded triangular elements. The mesh is generated in a manner suchthat the sizes of the elements gradually reduce approaching the edge(Point H in Fig. 1) of the footing. Typical chosen finite-elementmesh for f5 30 is shown in Fig. 2; in this figure, the parametersE, N, Ni, and Dc refer to the total number of elements, nodes, nodesalong the footing base (AH), and discontinuities, respectively. Thechosen domain and the stress boundary conditions for a singlelayer and two layers of reinforcement are shown in Figs. 1(b and c),respectively.

The chosen meshes in all the cases are sufficiently fine. It wasassured that in all the cases for an unreinforced soil mass, thecomputational results from the present analysis were almost thesame as reported in the literature based on the similar computationaltechnique.

Final Form of the Formulation

After the satisfaction of (1) the stress-boundary conditions alongdifferent boundaries; (2) equilibrium equations; (3) the differentstress discontinuity requirements along different element edges,including that along the interface of the reinforcement soil; and(4) linearized yield criterion, the basic expression for finding the

Fig. 4. Variation of hg with d2=B for different d1=B for two layers of reinforcement in sand with (a) f5 30; (b) f5 35; (c) f5 40; (d) f5 45


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magnitude of the total collapse load is then derived from the nu-merical integration of the normal stresses along the footing-soil in-terface. The magnitude of the collapse load (objective function) isthen maximized subject to a number of equality and nonequalitylinear constraints on the nodal stresses.

After constituting the global matrices and vectors, the linearprogramming problem is defined in a standard canonical form:maximize the objective function

2fcgTfsg (6a)

Subjected to equality constraints

A1fsg fb1g (6b)

Inequality constraints

A2fsg# fb2g (6c)

where fsg is a vector of nodal stresses given by

fsgT sx,1 sy,1 txy,1 sx,2 sy,2 txy,2 . . . . . . . . . sx,N sy,N txy,N

For solving the problem, a computer program is developed inMATLAB 7.9. The optimization is carried out by using LINPROG,a libraryprogram inMATLAB7.9, to dealwith the linear programming.Kumar and Khatri (2008) provided a description of the computationalprocedure.

Results

The improvement in the bearing capacity with the inclusion of thereinforcement was expressed in terms of the efficiency factor, hc, and

hg, because of the components of soil cohesion and unit weight,respectively, where the efficiency factor is the ratio of the respectivebearingcapacity componentof the soilmasswith reinforcement to thatwithout any reinforcement. Computations were carried out for twodifferent cases: (1) a single layer of reinforcement and (2) a group oftwo layers of reinforcement. For determining hc, the value of g wastaken equal to zero.On theother hand, for computinghg, the value of cis kept equal to zero. In doing so, the principle of superposition wasassumed to be valid. The results are presented herein.

Fig. 5.Variation of hc with d2=B for different d1=B for two layers of reinforcement in c-f soil with (a) f5 0; (b) f5 10; (c) f5 20; (d) f5 30


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Validation of the Results

For unreinforced soil mass, the magnitude of the collapse load (Qu)per unit length of the footing caused by the components of soilcohesion and soil unit weight was determined with the use of thestandard bearing capacity expression

qu Qu=B cNc 0:5gBNg (7)

whereNc andNg 5 bearing capacity factors from the components ofsoil cohesion and unit weight, respectively. The computed values ofNc and Ng for a rough foundation without any reinforcement arepresented in Tables 1 and 2. In these two tables, the results obtainedfrom the analysis were also compared with those available in theliterature. Table 1 provides the variation of Ng with f. On the otherhand, the variation of Nc with f and m is presented in Table 2. Thevalues of the bearing capacity factors increase continuously withincreases in the value off. The values ofNg for different values off,obtained from the present analysis, were compared with the lowerbound limit analysis with finite elements provided by Ukritchonet al. (2003) and Kumar and Khatri (2008) by using linear pro-gramming and the results ofKumar (2009) using themethod of stresscharacteristics. It can be seen that for different values of the soilfriction angle, the present results compare quite favorably with theexisting computational results of Ukritchon et al. (2003) and Kumarand Khatri (2008). The values of Ng obtained from the presentanalysis are marginally lower than that from the method of the stresscharacteristics. In the case of Nc, the comparison of the present

results for different values of fwas made with the limit equilibriumanalysis of Meyerhof (1963), the elastoplastic finite-element anal-ysis of Griffiths (1982), and the method of the stress characteristicsapproaches of Bolton and Lau (1993). The present results comparequite favorably with most of the reported results. The differencebetween the values of Nc reported by the different researchers wasfound to be generally quite marginal.

Variation of hg and hc with d1/ B for a Single Layerof Reinforcement

The computations were performed for four different values off (30,35, 40, and 45) for computinghg for different values offwith c5 0and four different values off (0, 10, 20, and 30) for determining hcfor different values of f with g5 0. The variation of the efficiencyfactor with changes in d1=B corresponding to a single layer of re-inforcement is illustrated in Figs. 3(a and b), which provide thevariation of hg and hc with d1=B for different values of f. It can beinvariably seen from these two figures that there always existsa certain critical depth (d1cr) of the reinforcement layer corre-sponding to which the values of hg and hc always become themaximum. Tables 3 and 4 present the values of the critical depthsalongwith the correspondingmaximumvalues of the efficiency factorfor granular soil and cohesive-frictional soil, respectively. The criticaldepth of the reinforcement varies between 0:29 and 0:57B in the caseof hg for a granular soil mass and 0:22 and 0:64B in the case of hc fora cohesive-frictional soil mass. The value of d1cr increases with anincrease inf. The maximum values of hg and hc, associated with thecritical depths of reinforcements, were found to vary between 1.55and 1.95 in the case of hg and 1.13 and 1.47 in the case of hc.The maximum value of the efficiency factor always continuouslyincreases with an increase in f.

Variation of hg and hc with d2/ B for Two Layersof Reinforcement

For two layers of reinforcement, to determine the maximum valueof hg and hc, a number of independent computations need to beperformed for various combinations of the values of d1=B and d2=B,where d1 refers to the depth of the upper layer of the horizontalreinforcements from the footing base, and d2 refers to the spacingbetween the lower and upper layers of the reinforcements as shownin Fig. 1(c). In the first step, a certain value of d1 is chosen, andcomputations are then performed for several values of d2. In thesecond step, a new value of d1 is chosen, and computations are againrepeated for several values of d2. This procedure is continuouslyrepeated to determine the value of efficiency factors for severalcombinations of d1 and d2. This procedure is then used to find finallythe maximum value of the efficiency factor and correspondingcritical values of d1 and d2. The computations were performed bychoosing a depth increment interval for the values of d1 and d2, equalto 0:072B; because of the requirement of large computational timebecause of a number of program runs, at present, the computationscould not be carried out for a depth interval of the reinforcementsmaller than 0:072B. For two layers of reinforcement, Figs. 4 and 5show the variation of the efficiency factor with d1=B and d2=B fortwo different cases. Fig. 4 presents the variation of hg with d1=B andd2=B for four different values of f (30, 35, 40, and 45). Fig. 5illustrates the variation of hc with d1=B and d2=B for four differentvalues of f (0, 10, 20, and 30). In all cases, there exists certaincritical values of both d1 and d2, namely,d1cr and d2cr, correspondingto the maximum efficiency factor. Tables 3 and 4 provide the valuesof d1cr and d2cr alongwith correspondingmaximumvalues of hg andhc for granular soil and cohesive-frictional soil, respectively. The

Fig. 6. Variation of the BCR with (a) d1=B for a single layer of re-inforcement; (b) d2=B for d1=B5 0:29 for two layers of reinforcement


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maximum efficiency factor for two layers of reinforcementsbecomes significantly higher than that for a single layer of re-inforcement. The difference between the two cases becomes quitehigh, especially when reinforcements are embedded in sand withgreater friction angles. For f5 45, it was found that for co-hesionless soil media, with the use of two layers of reinforcements(d1cr 5 0:50B; d2cr 5 0:64B), the ultimate bearing capacity can beincreased up to 3.63 times of that of unreinforced earth. The values ofd2cr generally remain marginally greater than d1cr. In general, thevalues of d1cr for two layers of reinforcement become a little smallerthan the values of d1cr for a single layer of reinforcement. It was alsoseen that the values of d2cr for two layers of reinforcement, in manycases, become very close to the value of d1cr for a single layer ofreinforcement.

Bearing Capacity Ratio for Single and Two Layersof Reinforcements

The present results can be used for a general cohesive-frictional soilto determine the bearing capacity ratio (BCR), which is the ratio ofthe bearing capacity of foundations with reinforcements to thatwithout reinforcements. The expression for computing the BCR isgiven here. For a general c-f soil

BCR chcNc 0:5gBhgNgcNc 0:5gBNg (8)

Figs. 6(a and b) provide for the variation of BCR with d1=B fora single layer of reinforcement and d2=B for two layers of

reinforcement with d1=B5 0:29. This figure corresponds tof5 30and for different values of c varying from20 to 100 kPa; the values ofg and B are kept equal to 17 kN=m3 and 1 m, respectively. Thecritical value of d1=B remains closer to 0.6with the single layer of thereinforcement and the maximum value of BCR increases from 1.40to 1.45 with an increase in c from 20 to 100 kPa. For two layers ofreinforcement, the critical value of d2=B lies closer to 0.6 withd1=B5 0:29. The maximum value of the BCR in this case increasesfrom 1.74 to 1.78, with an increase in c from 20 to 100 kPa.

Tension in the Reinforcement Layer

Starting from the free end of the reinforcement, by numerically in-tegrating the mobilized shear stress along the lower and upper surfacesof the reinforcement sheet, the magnitude of the axial tension in thereinforcement layer is determined. The tension (T) in the reinforcement,per unit lengthof the strip footing, is expressed in termsofdimensionlessquantities T=gB2 and T=cB for sand and c-f soil, respectively,where g 5 unit weight of soil mass and c5 cohesion of the soil mass.For a single layer and a group of two layers of horizontal reinforcement,the maximum values of T=gB2 and T=cB corresponding to dcr inthe case of a single layer and d1cr and d2cr for a group of two layers arepresented in Tables 3 and 4, respectively; the term Tmax refers to themaximum value of T . The computational results were obtained fordifferent values of f. The value of Tmax in the reinforcement increasescontinuously with an increase in f.

FromTable 3, note that forf5 45, the value of Tmax=gB2 (forthe upper sheet) is 148.67 with two layers of reinforcement.

Fig. 7. Variation of the tension with x=B for a single layer ofreinforcement in (a) sand; (b) c-f soil Fig. 8. Variation of the tension with x=B for two layers of re-

inforcement in (a) sand; (b) c-f soil


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Considering g5 17 kN=m3 and B5 1 m, the value of Tmax willbecome 2,527:39 kN=m, which is quite high, and it should bementioned here that hardly any of the geosynthetic sheet will exhibitsuch a high tensile strength; in fact, for dense sand (f5 45),generally there would be no need to use any kind of reinforcement insoils. However, for such cases, galvanized steel strips/sheet could bean option. A galvanized steel sheet usually has an ultimate tensilestrength in the range of 400550 MPa (ASTM 2011). The requiredthickness of galvanized steel sheets (not strips) in dense sand withf5 45 would be 4.606.32 mm. From Table 3, it can also be seenthat for sand with f5 30, the value of Tmax=gB2 (for the lowersheet) is 5.31. The value of Tmax will become 90:27 kN=m forg5 17 kN=m3 and B5 1:0m. There are quite a few number ofgeogrid and geotextile materials on the market that have ultimatetensile strengths greater than this value.

Required Length of the Reinforcement

For a single layer and a group of two layers of reinforcement, thevariation of T=gB2 and T=cB (for sand and c-f soil, re-spectively) with an increase in x=B for different values of f ispresented in Figs. 7 and 8, respectively; in these figures, the pa-rameter x refers to the horizontal distance from the center line of thefooting. For the sake of better presentation, the results are presentedwith respect tox=B instead of x=B. Themagnitude of T becomesmaximum along the center line of the footing (x5 0). In the case ofpurely granular soil for the value of x approximately smaller than B,the magnitude of T decreases quite considerably, with an increase inx and the value of T remains only marginally greater than 0. Forx.B, the value of T hardly reduces further with an increase in x.This is because of the fact that beyond a certain distance, no shearstress gets mobilized over either the top or the bottom surface of thereinforcement layer and the additional length of the reinforcementwill have no beneficial effect in further increasing the ultimatebearing capacity. At the free end of the reinforcement, in all cases,the magnitude of T always becomes equal to zero. By using Figs. 7and 8, the required optimum length (Lopt) along with the requiredtensile strength of the reinforcement layer can, therefore, be de-termined. For sand, the optimum length of the reinforcement liesapproximately closer to 2B for the value off varying between 30 and45; the length of the reinforcement is 2x. For c-f soil, the optimumlength of the reinforcement is only a bit higher.

Comparisonof thePresentResultswithThoseReportedfor Reinforced Soil

For a single layer and for two layers of reinforcement, the variationsof the efficiency factor, hg and hc, with d1=B obtained from thepresent analysis were compared with that reported by Michalowski

Fig. 9. Comparison of the present results with those obtained byMichalowski (2004) with one layer of reinforcement for the (a) variationof hg ; (b) variation of hc; (c) with two layers of reinforcement for thevariation of hg

Table 5. Comparison of the Obtained hg Values with the ExperimentalWork of Khing et al. (1993) and Das et al. (1994)

Reference

Layers of reinforcement

Single Two

f

(degrees) d1=B hg

f

(degrees) d1=B d2=B hg

Present work 40 0.36 1.60 40 0.36 0.36 2.66Khing et al. (1993) 40.3 0.375 2.06 40.3 0.375 0.375 2.98Das et al. (1994) 41 0.33 1.55 41 0.33 0.33 2.41

Note: In the present work, the computations were performed by choosinga depth increment interval, for the values of d1 and d2, equal to 0:072B.Therefore, the nearest d1=B and d2=B values are chosen.

Table 6. A Comparison of the Obtained hc Values with the ExperimentalWork of Shin et al. (1993) and Das et al. (1994)

Reference

Layers of reinforcement

Single Two

f

(degrees) d1=B hc

f

(degrees) d1=B d2=B hc

Present work 0 0.36 1.09 0 0.36 0.36 1.15Shin et al. (1993) 0 0.4 1.11 0 0.4 0.333 1.23Das et al. (1994) 0 0.4 1.09 0 0.4 0.333 1.21

Note: In the present work, the computations were performed by choosinga depth increment interval, for the values of d1 and d2, equal to 0:072B.Therefore, the nearest d1=B and d2=B values are chosen.


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(2004) based on upper bound limit analysis; it needs to bementionedthat the limit analysis approach ofMichalowski (2004) is based on theuse of a number of triangular rigid blockswith a commonvertex at theedge of the footing. Michalowski (2004) obtained the solutions fortwo different cases: strong reinforcement andweak reinforcement. Inthe case of strong reinforcement, the reinforcement is considered tobe structurally strong so that no tensile failure of the reinforcementtakes place. On the other hand, in the case of weak reinforcement, thetensile strength of the reinforcement was taken into account whileperforming the analysis. To make a comparison of the present so-lution with that given by Michalowski (2004), the results for strong

reinforcements were used. For a single layer of reinforcement, thecomparison of the results from two different approaches is presentedin Figs. 9(a and b) with reference to hg and hc. Similarly, for twolayers of reinforcement, the comparison of the results from twodifferent approaches is presented in Fig. 9(c) with reference to hg.The two approaches compare favorably with each other. However,the analysis of Michalowski (2004) provides generally higher valuesof efficiency factors and dcr compared with the present results.

For single and two layers of reinforcements, the results from thepresent analysis for granular soil were also compared with the ex-perimental results reported by Khing et al. (1993) and Das et al.

Fig. 10. Failure pattern for sand with f5 30: (a) without reinforcement; (b) with a single layer of the reinforcement; (c) with a group of two layersof reinforcement


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(1994). Khing et al. (1993) andDas et al. (1994) tested a strip footingplaced over unreinforced and reinforced sand by using a number ofhorizontal layers of geogrid reinforcements below the footing. Thecomparison is presented in Table 5. It can be seen that the presentanalysis compares well with the experimental results of Khing et al.(1993) and Das et al. (1994).

Similarly, for single and two layers of reinforcement, the resultsfrom the present analysis for clay (f5 0) were also compared withthe experimental results reported by Shin et al. (1993) and Das et al.(1994). Shin et al. (1993) and Das et al. (1994) tested a strip footingplaced over unreinforced and reinforced clay by using a number ofhorizontal layers of geogrid reinforcements below the footing. The

comparison is presented in Table 6. It can be seen that the presentanalysis compares well with the experimental results of Shin et al.(1993) and Das et al. (1994).

Failure Patterns

The state of stress, with respect to shear failure, at the centroid ofeach element is defined in terms of a ratio a=s, where a5 sx 2sy21 2txy2, and s5 2c cosf2 sx1sy sinf2. For a point atshear failure, a=s5 1; conversely, for nonyielding points, the valueof a=s remains invariably smaller than 1. The failure patterns wereplotted for a rough footing placed on sand with f5 30 and c-f soil

Fig. 11. Failure pattern for c-f soil with c5 20 kPa and f5 30: (a) without reinforcement; (b) with a single layer of the reinforcement; (c) witha group of two layers of reinforcement


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with c5 20 kPa andf5 30with no reinforcement, a single layer ofreinforcement, and two layers of reinforcement. The correspondingfailure patterns are illustrated in Figs. 10 and 11; the horizontal linesin thesefigures indicate the position of the reinforcement. The area ofthe plastic zone, below and around the footing base, reduces with aninclusion of the reinforcements; the area of the plastic zone fora group of two layers of reinforcement becomes smaller comparedwith the single layer. In the presence of the reinforcement, the outerboundary of the plastic zone tends to become tangential to the layerof the reinforcement both in the case of a single layer and a group oftwo layers. In c-f soil, the size of the plastic zone becomes greatercompared with cohesionless soil for the same value of f.

Remarks

The analysis presented in this paper is based on the assumption thatthe reinforcement layer has a tensile strength greater than thatneeded on the basis of the results provided in Tables 3 and 4. In otherwords, the results presented herewould be applicable for a casewhenno tensile (structural) failure of the reinforcement would take place.However, if the tensile strength of the reinforcement is smaller thanthat required, the ultimate bearing capacity of the footing will besmaller than that determined from the present calculations.

In this analysis, it was assumed that the interface friction angle (d)between the reinforcement sheet and adjoining soil mass simplybecomes equal to f. It is known that the value of d significantlyaffects the BCR (Jewell et al. 1984; Jewell 1990; Sugimoto andAlagiyawanna 2003). Depending on the nature of the reinforcementmaterial, the analysis can be easily extended for any prescribed valueof the soil-reinforcement interface friction angle. If the BCR isdefined with respect to a certain magnitude of the footing dis-placement, the stiffness of the reinforcement sheet will also affect thevalue of the BCR. Because the current study only refers to at failure,this aspect cannot be taken into consideration in the analysis.

Conclusions

Based on the lower bound finite-element limit analysis, a methodwas proposed to determine the bearing capacity of a strip foundationthat is placed over a soil mass reinforced with a single and a group oftwo layers of horizontal reinforcement sheets. The analysis is basedon the assumption that the reinforcement sheet can resist only axialtension but not the bendingmoment. The analysis clearly shows thatthe inclusion of the reinforcement causes a significant increase in thebearing capacity. The effect of the reinforcement on the failure loadbecomes maximum corresponding to certain critical depths of thereinforcements.

In sand, for a single layer of reinforcement, the critical depth wasfound to vary between 0:29 and 0:57B, and the associated maximumincrease in the bearing capacity varies approximately between 55and 95%. On the other hand, for a group of two layers of re-inforcement, the d1cr lies between 0:29 and 0:50B, and d2cr liesbetween 0:29 and 0:64B. The associated maximum increase in thebearing capacity for a group of two layers varies between 123 and263%.

Similarly, for the cohesion component, with a single layer ofreinforcement, the critical depth was found to vary between0:22 and 0:64B, and the associated maximum increase in the co-hesion component of the bearing capacity varies approximatelybetween 13 and 47%. On the other hand, for the cohesion com-ponent, with a group of two layers of reinforcement, d1cr liesbetween lies between 0:22 and 0:50B, and d2cr lies between

0:22 and 0:64B. The associated maximum increase in the bearingcapacity caused by the cohesion component for a group of two layersvaries between 21 and 84%. It was noted that the maximum increasein the bearing capacity increases continuously with an increase inf.

The required tensile strength of the reinforcement, to avoidbreakage, increases continuously with an increase in f. Comparedwith a single layer of the reinforcement sheet, a group of two layersof reinforcement, associated with the critical depths of the re-inforcement layers, needs to be designed for greater axial tensilestrength. The results presented in this study are expected to bebeneficial in designing reinforced earth foundations.

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Bearing Capacity of Strip Foundations in Reinforced Soils

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