-
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
Int. J. Geomech. 2014.14:45-58.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Gal
gotia
s U
nive
rsity
on
02/2
0/15
. Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
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
Int. J. Geomech. 2014.14:45-58.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Gal
gotia
s U
nive
rsity
on
02/2
0/15
. Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
-
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
INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE / JANUARY/FEBRUARY
2014 / 47
Int. J. Geomech. 2014.14:45-58.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Gal
gotia
s U
nive
rsity
on
02/2
0/15
. Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
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
48 / INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE /
JANUARY/FEBRUARY 2014
Int. J. Geomech. 2014.14:45-58.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Gal
gotia
s U
nive
rsity
on
02/2
0/15
. Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
-
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
INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE / JANUARY/FEBRUARY
2014 / 49
Int. J. Geomech. 2014.14:45-58.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Gal
gotia
s U
nive
rsity
on
02/2
0/15
. Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
-
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
50 / INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE /
JANUARY/FEBRUARY 2014
Int. J. Geomech. 2014.14:45-58.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Gal
gotia
s U
nive
rsity
on
02/2
0/15
. Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
-
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
INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE / JANUARY/FEBRUARY
2014 / 51
Int. J. Geomech. 2014.14:45-58.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Gal
gotia
s U
nive
rsity
on
02/2
0/15
. Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
-
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
52 / INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE /
JANUARY/FEBRUARY 2014
Int. J. Geomech. 2014.14:45-58.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Gal
gotia
s U
nive
rsity
on
02/2
0/15
. Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
-
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
INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE / JANUARY/FEBRUARY
2014 / 53
Int. J. Geomech. 2014.14:45-58.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Gal
gotia
s U
nive
rsity
on
02/2
0/15
. Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
-
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.
54 / INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE /
JANUARY/FEBRUARY 2014
Int. J. Geomech. 2014.14:45-58.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Gal
gotia
s U
nive
rsity
on
02/2
0/15
. Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
-
(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
INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE / JANUARY/FEBRUARY
2014 / 55
Int. J. Geomech. 2014.14:45-58.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Gal
gotia
s U
nive
rsity
on
02/2
0/15
. Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
-
(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
56 / INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE /
JANUARY/FEBRUARY 2014
Int. J. Geomech. 2014.14:45-58.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Gal
gotia
s U
nive
rsity
on
02/2
0/15
. Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
-
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.
References
Adams, M. T., and Collin, J. G. (1997). Large model spread
footing loadtests on geosynthetic reinforced soil foundations. J.
Geotech. Geo-environ. Eng., 10.1061/(ASCE)1090-0241(1997)123:1(66),
6672.
Anthoine, A. (1989). Mixed modelling of reinforced soils within
theframework of the yield design theory. Comput. Geotech.,
7(12),6782.
Asaoka, A., Kodaka, T., and Pokhaerl, G. (1994). Stability
analysis ofreinforced soil structures using rigid plastic finite
element method. SoilsFound., 34(1), 107118.
ASTM. (2011). Standard specification for steel sheet,
zinc-coated (gal-vanized) or zinc-iron alloy-coated (galvannealed)
by the hot-dip pro-cess. A653/A653M, West Conshohocken, Pa.
Binquet, J., andLee,K. L. (1975). Bearing capacity tests on
reinforced earthslabs. J. Geotech. Engrg. Div., 101(12),
12411255.
Blatz, J. A., and Bathurst, R. J. (2003). Limit equilibrium
analysis ofreinforced and unreinforced embankments loaded by a
strip footing.Can. Geotech. J., 40(6), 10841092.
Bolton, M. D., and Lau, C. K. (1993). Vertical bearing capacity
factors forcircular and strip footings on Mohr-Coulomb soil. Can.
Geotech. J.,30(6), 10241033.
Bottero, A., Negre, R., Pastor, J., and Turgeman, S. (1980).
Finite elementmethod and limit analysis theory for soil mechanics
problem. Comput.Methods Appl. Mech. Eng., 22(1), 131149.
Das, B. M., and Omar, M. T. (1994). The effects of foundation
width onmodel tests for the bearing capacity of sandwith geogrid
reinforcement.Geotech. Geol. Eng., 12(2), 133141.
Das, B. M., Shin, E. C., and Omar, M. T. (1994). The bearing
capacity ofsurface strip foundation on geogrid-reinforced sand and
clayAcomparative study. Geotech. Geol. Eng., 12(1), 114.
Dash, S. K., Rajagopal, K., and Krishnaswamy, N. R. (2004).
Performanceof different geosynthetic reinforcement materials in
sand foundations.Geosynth. Int., 11(1), 3542.
Deb, K., Sivakugan, N., Chandra, S., and Basudhar, P. K. (2007).
Nu-merical analysis of multilayer geosynthetic-reinforced granular
bed oversoft fill. Geotech. Geol. Eng., 25(6), 639646.
Fragaszy, R., and Lawton, E. (1984). Bearing capacity of
reinforced sandsubgrades. J. Geotech. Engrg.,
10.1061/(ASCE)0733-9410(1984)110:10(1500), 15001507.
Gourvenec, S., Randolph, M., and Kingsnorth, O. (2006).
Undrainedbearing capacity of square and rectangular footings. Int.
J. Geomech.,10.1061/(ASCE)1532-3641(2006)6:3(147), 147157.
Griffiths, D.V. (1982). Computation of bearing capacity factors
usingfiniteelements. Geotechnique, 32(3), 195202.
Griffiths, D. V., Fenton, G. A., and Manoharan, N. (2006).
Undrainedbearing capacity of two-strip footings on spatially random
soil. Int. J.Geomech., 10.1061/(ASCE)1532-3641(2006)6:6(421),
421427.
Guido, V. A., Chang, D. K., and Sweeney, M. A. (1986).
Comparison ofgeogrid and geotextile reinforced earth slabs. Can.
Geotech. J., 23(4),435440.
Jewell, R. A. (1990). Reinforcement bond capacity.Geotechnique,
40(3),513518.
Jewell, R. A., Milligan, G. W. E., Sarsby, R. W., and Dubois, D.
(1984).Interaction between soil and geogrids. Proc., Symp. on
Polymer GridReinforcement, Thomas Telford, London, 1830.
INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE / JANUARY/FEBRUARY
2014 / 57
Int. J. Geomech. 2014.14:45-58.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Gal
gotia
s U
nive
rsity
on
02/2
0/15
. Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
http://dx.doi.org/10.1061/(ASCE)1090-0241(1997)123:1(66)http://dx.doi.org/10.1061/(ASCE)1090-0241(1997)123:1(66)http://dx.doi.org/10.1016/0266-352X(89)90007-4http://dx.doi.org/10.1016/0266-352X(89)90007-4http://dx.doi.org/10.3208/sandf1972.34.107http://dx.doi.org/10.3208/sandf1972.34.107http://dx.doi.org/10.1139/t03-053http://dx.doi.org/10.1139/t93-099http://dx.doi.org/10.1139/t93-099http://dx.doi.org/10.1016/0045-7825(80)90055-9http://dx.doi.org/10.1016/0045-7825(80)90055-9http://dx.doi.org/10.1007/BF00429771http://dx.doi.org/10.1007/BF00425933http://dx.doi.org/10.1680/gein.2004.11.1.35http://dx.doi.org/10.1007/s10706-007-9136-5http://dx.doi.org/10.1061/(ASCE)0733-9410(1984)110:10(1500)http://dx.doi.org/10.1061/(ASCE)0733-9410(1984)110:10(1500)http://dx.doi.org/10.1061/(ASCE)1532-3641(2006)6:3(147)http://dx.doi.org/10.1061/(ASCE)1532-3641(2006)6:3(147)http://dx.doi.org/10.1680/geot.1982.32.3.195http://dx.doi.org/10.1061/(ASCE)1532-3641(2006)6:6(421)http://dx.doi.org/10.1061/(ASCE)1532-3641(2006)6:6(421)http://dx.doi.org/10.1139/t86-073http://dx.doi.org/10.1139/t86-073http://dx.doi.org/10.1680/geot.1990.40.3.513http://dx.doi.org/10.1680/geot.1990.40.3.513
-
Khing, K., Das, B. M., Puri, V. K., Cook, E. E., and Yen, S. C.
(1993). Thebearing capacity of a strip foundation on
geogrid-reinforced sand.Geotext. Geomembr., 12(4), 351361.
Kumar, J. (2009). ThevariationofNg with footing roughnessusing
themethodof characteristics. Int. J.Numer.Anal.MethodsGeomech.,
33(2), 275284.
Kumar, J., and Khatri, V. N. (2008). Effect of footing roughness
on lowerbound Ng values. Int. J. Geomech.,
10.1061/(ASCE)1532-3641(2008)8:3(176), 176187.
MATLAB 7.9 [Computer software]. Natick, MA, MathWorks.Meyerhof,
G. G. (1963). Some recent research on the bearing capacity of
foundations. Can. Geotech. J., 1(1), 1626.Michalowski, R. L.
(2004). Limit loads on reinforced foundation soils. J.
Geotech.Geoenviron. Eng.,
10.1061/(ASCE)1090-0241(2004)130:4(381),381390.
Michalowski, R. L., and Zhao, A. (1995). Continuum versus
structuralapproach to stability of reinforced soil. J. Geotech.
Engrg., 10.1061/(ASCE)0733-9410(1995)121:2(152), 152162.
Omar, M. T., Das, B. M., Puri, V. K., and Yen, S. C. (1993).
Ultimatebearing capacity of shallow foundations on sand with
geogrid re-inforcement. Can. Geotech. J., 30(3), 545549.
Otani, J., Ochiai, H., and Yamamoto, K. (1998). Bearing capacity
analysisof reinforced foundations on cohesive soil.Geotext.
Geomembr., 16(4),195206.
Rodriguez-Gutierrez, J. A., and Aristizabal-Ochoa, J. D.
(2012a). Rigidspread footings resting on soil subjected to axial
load and biaxial
bending. I: Simplified analytical method. Int. J. Geomech.,
10.1061/(ASCE)GM.1943-5622.0000218, 109119.
Rodriguez-Gutierrez, J. A., and Aristizabal-Ochoa, J. D.
(2012b). Rigidspread footings resting on soil subjected to axial
load and biaxialbending. II: Design aids. Int. J. Geomech.,
10.1061/(ASCE)GM.1943-5622.0000210, 120131.
Shin, E. C., Das, B.M., Puri, V.K., Yen, S.-C., andCook, E. E.
(1993). Bearingcapacity of strip foundation on geogrid-reinforced
clay. J. ASTM GeotechTest., 16(4), 534541.
Sloan, S. W. (1988). Lower bound limit analysis using finite
elements andlinear programming. Int. J. Numer. Anal. Methods
Geomech., 12(1),6177.
Sugimoto, M., and Alagiyawanna, A. M. N. (2003). Pullout
behavior ofgeogrid by test and numerical analysis. J. Geotech.
Geoenviron. Eng.,10.1061/(ASCE)1090-0241(2003)129:4(361),
361371.
Terzaghi, K. (1943). Theoretical soil mechanics, Wiley, New
York.Ukritchon, B., Whittle, A. W., and Klangvijit, C. (2003).
Calculation of
bearing capacity factor Ng using numerical limit analysis. J.
Geotech.Geoenviron. Eng., 10.1061/(ASCE)1090-0241(2003)129:6(468),
468474.
Yamamoto, N., Randolph, M. F., and Einav, I. (2008). Simple
formulasfor the response of shallow foundations on compressible
sands. Int.J. Geomech., 10.1061/(ASCE)1532-3641(2008)8:4(230),
230239.
Yu, H. S., and Sloan, S. W. (1997). Finite element limit
analysis ofreinforced soils. Comput. Struct., 63(3), 567577.
58 / INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE /
JANUARY/FEBRUARY 2014
Int. J. Geomech. 2014.14:45-58.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Gal
gotia
s U
nive
rsity
on
02/2
0/15
. Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
http://dx.doi.org/10.1016/0266-1144(93)90009-Dhttp://dx.doi.org/10.1002/nag.716http://dx.doi.org/10.1061/(ASCE)1532-3641(2008)8:3(176)http://dx.doi.org/10.1061/(ASCE)1532-3641(2008)8:3(176)http://dx.doi.org/10.1139/t63-003http://dx.doi.org/10.1061/(ASCE)1090-0241(2004)130:4(381)http://dx.doi.org/10.1061/(ASCE)1090-0241(2004)130:4(381)http://dx.doi.org/10.1061/(ASCE)1090-0241(2004)130:4(381)http://dx.doi.org/10.1061/(ASCE)0733-9410(1995)121:2(152)http://dx.doi.org/10.1061/(ASCE)0733-9410(1995)121:2(152)http://dx.doi.org/10.1139/t93-046http://dx.doi.org/10.1016/S0266-1144(98)00005-3http://dx.doi.org/10.1016/S0266-1144(98)00005-3http://dx.doi.org/10.1061/(ASCE)GM.1943-5622.0000218http://dx.doi.org/10.1061/(ASCE)GM.1943-5622.0000218http://dx.doi.org/10.1061/(ASCE)GM.1943-5622.0000210http://dx.doi.org/10.1061/(ASCE)GM.1943-5622.0000210http://dx.doi.org/10.1520/GTJ10293Jhttp://dx.doi.org/10.1520/GTJ10293Jhttp://dx.doi.org/10.1002/nag.1610120105http://dx.doi.org/10.1002/nag.1610120105http://dx.doi.org/10.1061/(ASCE)1090-0241(2003)129:4(361)http://dx.doi.org/10.1061/(ASCE)1090-0241(2003)129:4(361)http://dx.doi.org/10.1061/(ASCE)1090-0241(2003)129:6(468)http://dx.doi.org/10.1061/(ASCE)1090-0241(2003)129:6(468)http://dx.doi.org/10.1061/(ASCE)1090-0241(2003)129:6(468)http://dx.doi.org/10.1061/(ASCE)1532-3641(2008)8:4(230)http://dx.doi.org/10.1061/(ASCE)1532-3641(2008)8:4(230)http://dx.doi.org/10.1016/S0045-7949(96)00353-7