Ultimate bearing capacity analysis of strip footings on reinforced soil foundation - Research Paper Notes | EduRev

: Ultimate bearing capacity analysis of strip footings on reinforced soil foundation - Research Paper Notes | EduRev

 Page 1


HOSTED BY
Ultimate bearing capacity analysis of strip footings on reinforced
soil foundation
Qiming Chen
n
, Murad Abu-Farsakh
Louisiana Transportation Research Center, Louisiana State University, 4101 Gourrier Avenue, Baton Rouge, LA 70808, USA
Received 25 September 2013; received in revised form 21 August 2014; accepted 20 October 2014
Available online 2 January 2015
Abstract
Reinforcedsoilfoundations(RSFs)havebeenemployedinengineeringpracticetoincreasethesoilbearingcapacityandtoreducethepotential
footing settlement. The aim of this study is to develop analytical solutions for estimating the ultimate bearing capacity of strip footings on RSFs.
A general failure mode for RSFs was ?rst proposed based on previous studies conducted by the authors and test results from literature study.
A limit equilibrium stability analysis of RSFs was performed based on the proposed failure mechanism. New bearing capacity formulas, which
consider both the con?nement and the membrane effects of reinforcements on the increase in ultimate bearing capacity, were then developed for
strip footings on RSFs. Several special cases of RSFs were presented and discussed. The proposed model was veri?ed by the experimental data
reported in the published literature. The predicted ultimate bearing capacity was in good agreement with the results of model tests reported in the
literature. The study showed that the depth of the punching shear failure zone (D
P
) depends on the relative strength of the reinforced soil layer
and the underlying unreinforced soil layer, and is directly related to the reinforced ratio (R
r
).
& 2015 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.
Keywords: Reinforced soil foundation; Ultimate bearing capacity; Failure mechanism; Limit equilibrium; Reinforced ratio
1. Introduction
The use of reinforced soils to support shallow foundations
has recently received considerable attention. The bene?ts of
including reinforcements in the soil mass to increase the
bearing capacity and to reduce the settlement of the soil
foundation have been widely recognized. However, the devel-
opment of a rational design method and a theory for reinforced
soil foundations (RSFs) is lagging in comparison to RSF
applications. These restrictions, on the other hand, inhibit the
further development of reinforcement technology. Therefore, it
is essential to investigate the proper failure mechanisms for
reinforced soil applications. During the past forty years, many
experimental, numerical, and analytical studies have been
performed to investigate the behavior of reinforced soil foun-
dations (RSFs)for different soil types (e.g., Abu-Farsakh etal.,
2008,2013; Adams and Collin,1997; Binquet and Lee, 1975a,
1975b; Chakraborty and Kumar, 2014; Chen et al., 2007,
2009; Demir et al., 2013; Huang and Tatsuoka, 1990; Kurian
et al., 1997; Sharma et al., 2009).
The?rstexperimentalstudyreportedinliteraturewasconducted
by Binquet and Lee (1975a) to evaluate the bearing capacity of
sand reinforced by aluminum foil strips. Since then, several
experimental studies have been conducted to evaluate the bearing
capacity of footings on reinforced sandy soil (e.g., Abu-Farsakh
et al., 2013; Adams and Collin, 1997; Akinmusuru and
Akinbolade, 1981; Fragaszy and Lawton, 1984; Gabr et al.,
The Japanese Geotechnical Society
www.sciencedirect.com
journal homepage: www.elsevier.com/locate/sandf
Soils and Foundations
http://dx.doi.org/10.1016/j.sandf.2014.12.006
0038-0806/& 2015 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.
n
Corresponding author.
E-mail addresses: qchen1@lsu.edu (Q. Chen),
cefars@lsu.edu (M. Abu-Farsakh).
Peer review under responsibility of The Japanese Geotechnical Society.
Soils and Foundations 2015;55(1):74–85
Page 2


HOSTED BY
Ultimate bearing capacity analysis of strip footings on reinforced
soil foundation
Qiming Chen
n
, Murad Abu-Farsakh
Louisiana Transportation Research Center, Louisiana State University, 4101 Gourrier Avenue, Baton Rouge, LA 70808, USA
Received 25 September 2013; received in revised form 21 August 2014; accepted 20 October 2014
Available online 2 January 2015
Abstract
Reinforcedsoilfoundations(RSFs)havebeenemployedinengineeringpracticetoincreasethesoilbearingcapacityandtoreducethepotential
footing settlement. The aim of this study is to develop analytical solutions for estimating the ultimate bearing capacity of strip footings on RSFs.
A general failure mode for RSFs was ?rst proposed based on previous studies conducted by the authors and test results from literature study.
A limit equilibrium stability analysis of RSFs was performed based on the proposed failure mechanism. New bearing capacity formulas, which
consider both the con?nement and the membrane effects of reinforcements on the increase in ultimate bearing capacity, were then developed for
strip footings on RSFs. Several special cases of RSFs were presented and discussed. The proposed model was veri?ed by the experimental data
reported in the published literature. The predicted ultimate bearing capacity was in good agreement with the results of model tests reported in the
literature. The study showed that the depth of the punching shear failure zone (D
P
) depends on the relative strength of the reinforced soil layer
and the underlying unreinforced soil layer, and is directly related to the reinforced ratio (R
r
).
& 2015 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.
Keywords: Reinforced soil foundation; Ultimate bearing capacity; Failure mechanism; Limit equilibrium; Reinforced ratio
1. Introduction
The use of reinforced soils to support shallow foundations
has recently received considerable attention. The bene?ts of
including reinforcements in the soil mass to increase the
bearing capacity and to reduce the settlement of the soil
foundation have been widely recognized. However, the devel-
opment of a rational design method and a theory for reinforced
soil foundations (RSFs) is lagging in comparison to RSF
applications. These restrictions, on the other hand, inhibit the
further development of reinforcement technology. Therefore, it
is essential to investigate the proper failure mechanisms for
reinforced soil applications. During the past forty years, many
experimental, numerical, and analytical studies have been
performed to investigate the behavior of reinforced soil foun-
dations (RSFs)for different soil types (e.g., Abu-Farsakh etal.,
2008,2013; Adams and Collin,1997; Binquet and Lee, 1975a,
1975b; Chakraborty and Kumar, 2014; Chen et al., 2007,
2009; Demir et al., 2013; Huang and Tatsuoka, 1990; Kurian
et al., 1997; Sharma et al., 2009).
The?rstexperimentalstudyreportedinliteraturewasconducted
by Binquet and Lee (1975a) to evaluate the bearing capacity of
sand reinforced by aluminum foil strips. Since then, several
experimental studies have been conducted to evaluate the bearing
capacity of footings on reinforced sandy soil (e.g., Abu-Farsakh
et al., 2013; Adams and Collin, 1997; Akinmusuru and
Akinbolade, 1981; Fragaszy and Lawton, 1984; Gabr et al.,
The Japanese Geotechnical Society
www.sciencedirect.com
journal homepage: www.elsevier.com/locate/sandf
Soils and Foundations
http://dx.doi.org/10.1016/j.sandf.2014.12.006
0038-0806/& 2015 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.
n
Corresponding author.
E-mail addresses: qchen1@lsu.edu (Q. Chen),
cefars@lsu.edu (M. Abu-Farsakh).
Peer review under responsibility of The Japanese Geotechnical Society.
Soils and Foundations 2015;55(1):74–85
1998; Guido et al., 1986; Huang and Tatsuoka, 1990; Latha and
Somwanshi,2009;LavasanandGhazavi,2012;Omaretal.1993a,
1993b; Yetimoglu et al. 1994), clayey soil (e.g., Abu-Farsakh
et al., 2008; Chen et al., 2007; Chen and Abu-Farsakh, 2011; Das
et al., 1994; Ingold and Miller, 1982; Mandal and Sah, 1992;
Ramaswamy and Purushothaman (1992); Sakti and Das, 1987;
Shin et al., 1993), aggregate (e.g., Chen et al., 2009; DeMerchant
et al., 2002; James and Raymond, 2002), and pond ash (e.g., Bera
etal.,2005;Ghoshetal.,2005).Theaimofmanyoftheseresearch
efforts was to investigate the parameters and variables that would
contribute to the value of the bearing capacity ratio (BCR), which
is de?ned as the ratio of the bearing capacity of the RSF to that of
the unreinforced soil foundation. The results of the experimental
studies showed that the bearing capacity of soil was improved
when it was reinforced by reinforcements and that the amount of
improvement was highly dependent on the layout of the reinforce-
ments. Better improvements were obtained when the reinforce-
ments were placed within a certain depth (or in?uence depth)
beyond which no additional signi?cant improvement occurred. In
other words, the BCR value would approach a constant/limiting
value with an increasing number of reinforcement layers.
From the experimental studies reported in the literature, two
fundamental reinforcement mechanisms can be distinguished
as contributing to the increase in bearing capacity of reinforced
soil foundations (RSFs).
(1) Con?nement effect or lateral restraint effect: With the
applied load, the lateral forces are induced and the soil
List of symbols
B width of footing
u top layer spacing, i.e., spacing between top layer
of reinforcement and bottom of footing
h vertical spacing between reinforcement layers
l length of reinforcement
d total depth of reinforcement¼uþ(N1)h.
N number of reinforcement layers
N
p
number of reinforcement layers located in punch-
ing shear failure zone
N
T
number of reinforcement layers located above
point c
T tensile force in reinforcement
D
P
depth of punching shear failure zone
q
u(R)
ultimate bearing capacity of reinforced soil
foundation
q
u(R)1
ultimate bearing capacity of punching shear
failure zone
q
u(R)2
ultimate bearing capacity of underlying general
shear failure zone
P
p1
total passive earth pressure on vertical punching
failure surfaces aa
0
and bb
0
d mobilized friction angle along vertical punching
failure surfaces aa
0
and bb
0
C
a
adhesive force acting on vertical punching failure
surfaces aa
0
and bb
0
,¼c
a
D
P
c
a
unit adhesion of soil along vertical punching
failure surfaces aa
0
and bb
0
,
T
1
tensile force acting on vertical punching failure
surfaces aa
0
and bb
0
a angle of tensile force T
1
to horizontal
T
1x
horizontal component of tensile force T
1
T
1y
vertical component of tensile force T
1
? unit weight of soil
D
f
embedment depth of footing
K
pH
horizontal component of passive earth pressure
coef?cient
K
s
punching shear coef?cient
? friction angle of soil
P
p2
passive force acting on faces ac and bc
C cohesive force C acting on faces ac and bc
T
2L
,T
2
tensile force acting on faces ac and bc
P
pc
passive force due to cohesion c,
P
pq
passive force due to surcharge q
P
p?
, passive force due to weight of soil ?
c cohesion of soil
q surcharge load
P
pT
passive force due to tensile force of reinforcement
T
2L
? angle of tensile force T
2L
to horizontal
T
2Lx
,T
2x
horizontal component of tensile force T
2L
T
2Ly
vertical component of tensile force T
2L
T
2R
tensile force acting on face gd
? angle of tensile force T
2R
to horizontal
T
2RX
horizontal component of tensile force T
2R
T
2Ry
vertical component of tensile force T
2R
F resisting force along log spiral cd
r length of radial line of log spiral cd,¼r
0
e
?tan?
r
0
length of bc
? angle between line bc and radial line of log spiral
curve cd
X
TR
distance from center of footing to point where
tensile force T
2R
is applied
q
u(UR)
ultimate bearing capacity of unreinforced soil in
general shear failure zone
N
c
,N
q
, and N
?
bearing capacity factors
ß angle between s
1
direction and bedding plane
?
design
design friction angle of soil
?
peak
peak friction angle of soil
?
cv
residual design friction angle of soil
? percent of contribution of failure surfaces con-
trolled by soil’s peak friction angle
R
r
reinforced ratio
E
R
elastic modulus of reinforcement¼J/t
R
J tensile modulus of reinforcement
A
R
area of reinforcement per unit width¼Nt
R
1
t
R
thickness of reinforcement
E
s
modulus of elasticity of soil
A
s
area of reinforced soil per unit width¼d1
Q. Chen, M. Abu-Farsakh / Soils and Foundations 55 (2015) 74–85 75
Page 3


HOSTED BY
Ultimate bearing capacity analysis of strip footings on reinforced
soil foundation
Qiming Chen
n
, Murad Abu-Farsakh
Louisiana Transportation Research Center, Louisiana State University, 4101 Gourrier Avenue, Baton Rouge, LA 70808, USA
Received 25 September 2013; received in revised form 21 August 2014; accepted 20 October 2014
Available online 2 January 2015
Abstract
Reinforcedsoilfoundations(RSFs)havebeenemployedinengineeringpracticetoincreasethesoilbearingcapacityandtoreducethepotential
footing settlement. The aim of this study is to develop analytical solutions for estimating the ultimate bearing capacity of strip footings on RSFs.
A general failure mode for RSFs was ?rst proposed based on previous studies conducted by the authors and test results from literature study.
A limit equilibrium stability analysis of RSFs was performed based on the proposed failure mechanism. New bearing capacity formulas, which
consider both the con?nement and the membrane effects of reinforcements on the increase in ultimate bearing capacity, were then developed for
strip footings on RSFs. Several special cases of RSFs were presented and discussed. The proposed model was veri?ed by the experimental data
reported in the published literature. The predicted ultimate bearing capacity was in good agreement with the results of model tests reported in the
literature. The study showed that the depth of the punching shear failure zone (D
P
) depends on the relative strength of the reinforced soil layer
and the underlying unreinforced soil layer, and is directly related to the reinforced ratio (R
r
).
& 2015 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.
Keywords: Reinforced soil foundation; Ultimate bearing capacity; Failure mechanism; Limit equilibrium; Reinforced ratio
1. Introduction
The use of reinforced soils to support shallow foundations
has recently received considerable attention. The bene?ts of
including reinforcements in the soil mass to increase the
bearing capacity and to reduce the settlement of the soil
foundation have been widely recognized. However, the devel-
opment of a rational design method and a theory for reinforced
soil foundations (RSFs) is lagging in comparison to RSF
applications. These restrictions, on the other hand, inhibit the
further development of reinforcement technology. Therefore, it
is essential to investigate the proper failure mechanisms for
reinforced soil applications. During the past forty years, many
experimental, numerical, and analytical studies have been
performed to investigate the behavior of reinforced soil foun-
dations (RSFs)for different soil types (e.g., Abu-Farsakh etal.,
2008,2013; Adams and Collin,1997; Binquet and Lee, 1975a,
1975b; Chakraborty and Kumar, 2014; Chen et al., 2007,
2009; Demir et al., 2013; Huang and Tatsuoka, 1990; Kurian
et al., 1997; Sharma et al., 2009).
The?rstexperimentalstudyreportedinliteraturewasconducted
by Binquet and Lee (1975a) to evaluate the bearing capacity of
sand reinforced by aluminum foil strips. Since then, several
experimental studies have been conducted to evaluate the bearing
capacity of footings on reinforced sandy soil (e.g., Abu-Farsakh
et al., 2013; Adams and Collin, 1997; Akinmusuru and
Akinbolade, 1981; Fragaszy and Lawton, 1984; Gabr et al.,
The Japanese Geotechnical Society
www.sciencedirect.com
journal homepage: www.elsevier.com/locate/sandf
Soils and Foundations
http://dx.doi.org/10.1016/j.sandf.2014.12.006
0038-0806/& 2015 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.
n
Corresponding author.
E-mail addresses: qchen1@lsu.edu (Q. Chen),
cefars@lsu.edu (M. Abu-Farsakh).
Peer review under responsibility of The Japanese Geotechnical Society.
Soils and Foundations 2015;55(1):74–85
1998; Guido et al., 1986; Huang and Tatsuoka, 1990; Latha and
Somwanshi,2009;LavasanandGhazavi,2012;Omaretal.1993a,
1993b; Yetimoglu et al. 1994), clayey soil (e.g., Abu-Farsakh
et al., 2008; Chen et al., 2007; Chen and Abu-Farsakh, 2011; Das
et al., 1994; Ingold and Miller, 1982; Mandal and Sah, 1992;
Ramaswamy and Purushothaman (1992); Sakti and Das, 1987;
Shin et al., 1993), aggregate (e.g., Chen et al., 2009; DeMerchant
et al., 2002; James and Raymond, 2002), and pond ash (e.g., Bera
etal.,2005;Ghoshetal.,2005).Theaimofmanyoftheseresearch
efforts was to investigate the parameters and variables that would
contribute to the value of the bearing capacity ratio (BCR), which
is de?ned as the ratio of the bearing capacity of the RSF to that of
the unreinforced soil foundation. The results of the experimental
studies showed that the bearing capacity of soil was improved
when it was reinforced by reinforcements and that the amount of
improvement was highly dependent on the layout of the reinforce-
ments. Better improvements were obtained when the reinforce-
ments were placed within a certain depth (or in?uence depth)
beyond which no additional signi?cant improvement occurred. In
other words, the BCR value would approach a constant/limiting
value with an increasing number of reinforcement layers.
From the experimental studies reported in the literature, two
fundamental reinforcement mechanisms can be distinguished
as contributing to the increase in bearing capacity of reinforced
soil foundations (RSFs).
(1) Con?nement effect or lateral restraint effect: With the
applied load, the lateral forces are induced and the soil
List of symbols
B width of footing
u top layer spacing, i.e., spacing between top layer
of reinforcement and bottom of footing
h vertical spacing between reinforcement layers
l length of reinforcement
d total depth of reinforcement¼uþ(N1)h.
N number of reinforcement layers
N
p
number of reinforcement layers located in punch-
ing shear failure zone
N
T
number of reinforcement layers located above
point c
T tensile force in reinforcement
D
P
depth of punching shear failure zone
q
u(R)
ultimate bearing capacity of reinforced soil
foundation
q
u(R)1
ultimate bearing capacity of punching shear
failure zone
q
u(R)2
ultimate bearing capacity of underlying general
shear failure zone
P
p1
total passive earth pressure on vertical punching
failure surfaces aa
0
and bb
0
d mobilized friction angle along vertical punching
failure surfaces aa
0
and bb
0
C
a
adhesive force acting on vertical punching failure
surfaces aa
0
and bb
0
,¼c
a
D
P
c
a
unit adhesion of soil along vertical punching
failure surfaces aa
0
and bb
0
,
T
1
tensile force acting on vertical punching failure
surfaces aa
0
and bb
0
a angle of tensile force T
1
to horizontal
T
1x
horizontal component of tensile force T
1
T
1y
vertical component of tensile force T
1
? unit weight of soil
D
f
embedment depth of footing
K
pH
horizontal component of passive earth pressure
coef?cient
K
s
punching shear coef?cient
? friction angle of soil
P
p2
passive force acting on faces ac and bc
C cohesive force C acting on faces ac and bc
T
2L
,T
2
tensile force acting on faces ac and bc
P
pc
passive force due to cohesion c,
P
pq
passive force due to surcharge q
P
p?
, passive force due to weight of soil ?
c cohesion of soil
q surcharge load
P
pT
passive force due to tensile force of reinforcement
T
2L
? angle of tensile force T
2L
to horizontal
T
2Lx
,T
2x
horizontal component of tensile force T
2L
T
2Ly
vertical component of tensile force T
2L
T
2R
tensile force acting on face gd
? angle of tensile force T
2R
to horizontal
T
2RX
horizontal component of tensile force T
2R
T
2Ry
vertical component of tensile force T
2R
F resisting force along log spiral cd
r length of radial line of log spiral cd,¼r
0
e
?tan?
r
0
length of bc
? angle between line bc and radial line of log spiral
curve cd
X
TR
distance from center of footing to point where
tensile force T
2R
is applied
q
u(UR)
ultimate bearing capacity of unreinforced soil in
general shear failure zone
N
c
,N
q
, and N
?
bearing capacity factors
ß angle between s
1
direction and bedding plane
?
design
design friction angle of soil
?
peak
peak friction angle of soil
?
cv
residual design friction angle of soil
? percent of contribution of failure surfaces con-
trolled by soil’s peak friction angle
R
r
reinforced ratio
E
R
elastic modulus of reinforcement¼J/t
R
J tensile modulus of reinforcement
A
R
area of reinforcement per unit width¼Nt
R
1
t
R
thickness of reinforcement
E
s
modulus of elasticity of soil
A
s
area of reinforced soil per unit width¼d1
Q. Chen, M. Abu-Farsakh / Soils and Foundations 55 (2015) 74–85 75
particles are spread. Due to the relative displacement
between the soil and the reinforcements, frictional interac-
tion is induced at the soil-reinforcement interface. For
geogrid reinforcements, the interlocking can be developed
by the interaction of the soil and the geogrid. Conse-
quently,lateral deformationorpotentialtensilestrainofthe
reinforced soil is restrained. As a result, the vertical
deformation of the soil is reduced. Since soils are stress-
dependent, improved lateral con?nement increases the
compressive strength of the soil, and thus, improves the
bearing capacity of the reinforced soil.
(2) Membrane effect: With the applied load, the soil beneath
the footing moves downward; the reinforcement is then
deformed and tensioned. The deformed reinforcement
develops an upward force that supports part of the applied
load. A certain amount of settlement is generally required
to signi?cantly mobilize the tensioned membrane effect,
and the reinforcement should have enough length and
tensile strength to prevent it from failing by pull out and
rupture.
Several researchers have presented analytical models for esti-
mating the ultimate bearing capacity of RSFs based on either the
con?nement effect (e.g., Huang and Tatsuoka, 1990; Michalowski,
2004) or the membrane effect (e.g., Binquet and Lee, 1975b;
Kumar and Saran, 2003; Wayne et al., 1998). With these two
fundamentalreinforcingmechanisms, the“deepfooting” effect can
be formed under certain conditions. Here, the“deep footing” effect
means that the performance of footings on a reinforced soil
foundation is very similar to that of footings on an unreinforced
soil foundation with an additional embedment depth equal to the
depth of the reinforced zone. This effect was ?rst proposed by
Schlosser et al., (1983). Huang and Menq (1997) employed this
effect to derive the ultimate bearing capacity formulas for
reinforced sand. This effect is also considered in Wayne et al.’
(1998) solution. With an increase in the number of reinforcement
layers, the “deep footing” effect leads to an almost linear increase
in the bearing capacity ratio (BCR), and convergence cannot be
obtained(i.e.,theBCRvaluedoesnotapproachaconstant/limiting
value with an increasing number of reinforcement layers). This
poses a big problem, especially in granular soil with a high friction
angle. The authors made great efforts to develop analytical
solutions to overcome this problem (Chen et al., 2009; Sharma
et al., 2009). However, the developed analytical solutions were
only applied to square footings and took different forms for
different soil types. Also, only a single reinforcement mechanism
(either con?nement effect or membrane effect) was considered in
those analytical solutions.
Therefore, this study will focus on strip footings on RSFs by
considering the two reinforcement mechanisms (i.e., con?ne-
ment effect and membrane effect) together. A general failure
mechanism of RSFs is ?rstly proposed. The limit equilibrium
stability analysis for RSFs, based on the proposed failure
mechanism, is then performed as an attempt to develop a
rational uni?ed analytical model for evaluating the ultimate
bearing capacity of strip footings on RSFs. Finally, the
developed analytical solution is veri?ed through the results
of model tests reported in the literature.
2. Failure modes of reinforced soil foundations
As mentioned earlier, reinforcements can restrain the lateral
deformation or the potential tensile strain of the soil (con?ne-
ment effect). In addition, deformed reinforcements can develop
an upward force (membrane effect). These effects will result in
an increase in the bearing capacity of the RSF.
Three potential failure modes of RSFs are shown in Fig. 1.
The ?rst two failure modes, failure above the top layer of the
reinforcement (Binquet and Lee, 1975b)(Fig. 1a) and failure
between the reinforcement layers (Wayne et al., 1998)
(Fig. 1b), can be avoided by keeping the top layer spacing
(u) and the vertical spacing between the reinforcement layers
(h) within an acceptable/reasonable range.
Based on previous studies by Chen (2007), Chen et al. (2009),
Sharma et al. (2009), the third failure mechanism, i.e., a general
failuremechanismofRSFs,isidenti?edasapunchingshearfailure
followed by a general shear failure (Fig. 1c). The value of
punching shear failure depth D
p
depends on the relative strength
between the reinforced zone and the underlying unreinforced zone.
It can be zero (i.e., D
p
¼0) if the strength of the reinforced zone is
slightly larger than that of the underlying unreinforced zone, or if
the reinforcement depth ratio (d/B) is relatively large. It also can
punch all the way through the reinforced zone (i.e., D
p
¼d)ifthe
strength of the reinforced zone is much larger than that of the
underlying unreinforced zone and the reinforcement depth ratio (d/
B) is relatively small.
3. Limit equilibrium analysis of reinforcedsoil foundations
First, we will consider the strip footing case with two layers of
reinforcement. One layer is located in the punching shear failure
zone at a depth of u, and the other layer is located in the general
shearfailurezoneatadepthofuþh.Thefailuresurfaceinthesoil
for the strip footing at the ultimate load is shown in Fig. 1c. The
ultimate bearing capacity of the RSF includes the contribution of
punching shear failure zone a
0
abb
0
, q
u(R)1
, and an underlying
general shear failure zone, q
u(R)2
, i.e., q
u(R)
¼q
u(R)1
þq
u(R)2
.
Let’s consider the soil block a
0
abb
0
in the punching shear
failure zone, as shown in Fig. 2. The forces on the vertical
punching failure surfaces, aa
0
and bb
0
, include total passive
earth pressure P
p1
, inclined at an average angle d,and adhesive
force C
a
¼c
a
D
P
, acting upwards (Meyerhof and Hanna, 1978),
where c
a
istheunitadhesionofthesoilalongtwosidesand D
P
is the depth of the punching shear failure in the reinforced
zone. With the inclusion of the reinforcement, an upward force
will be induced by the tension effect of the reinforcement
along the failure surfaces. At the ultimate load, the reinforce-
ment will deform. The tensile force at the vertical failure
surface, T
1
, is assumed to have an angle of a in the horizontal
direction. This force can be decomposed into two components:
horizontal component T
1x
, which provides the con?nement
effect, and vertical component T
1y
, which provides the tensile
membrane effect.
Q. Chen, M. Abu-Farsakh / Soils and Foundations 55 (2015) 74–85 76
Page 4


HOSTED BY
Ultimate bearing capacity analysis of strip footings on reinforced
soil foundation
Qiming Chen
n
, Murad Abu-Farsakh
Louisiana Transportation Research Center, Louisiana State University, 4101 Gourrier Avenue, Baton Rouge, LA 70808, USA
Received 25 September 2013; received in revised form 21 August 2014; accepted 20 October 2014
Available online 2 January 2015
Abstract
Reinforcedsoilfoundations(RSFs)havebeenemployedinengineeringpracticetoincreasethesoilbearingcapacityandtoreducethepotential
footing settlement. The aim of this study is to develop analytical solutions for estimating the ultimate bearing capacity of strip footings on RSFs.
A general failure mode for RSFs was ?rst proposed based on previous studies conducted by the authors and test results from literature study.
A limit equilibrium stability analysis of RSFs was performed based on the proposed failure mechanism. New bearing capacity formulas, which
consider both the con?nement and the membrane effects of reinforcements on the increase in ultimate bearing capacity, were then developed for
strip footings on RSFs. Several special cases of RSFs were presented and discussed. The proposed model was veri?ed by the experimental data
reported in the published literature. The predicted ultimate bearing capacity was in good agreement with the results of model tests reported in the
literature. The study showed that the depth of the punching shear failure zone (D
P
) depends on the relative strength of the reinforced soil layer
and the underlying unreinforced soil layer, and is directly related to the reinforced ratio (R
r
).
& 2015 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.
Keywords: Reinforced soil foundation; Ultimate bearing capacity; Failure mechanism; Limit equilibrium; Reinforced ratio
1. Introduction
The use of reinforced soils to support shallow foundations
has recently received considerable attention. The bene?ts of
including reinforcements in the soil mass to increase the
bearing capacity and to reduce the settlement of the soil
foundation have been widely recognized. However, the devel-
opment of a rational design method and a theory for reinforced
soil foundations (RSFs) is lagging in comparison to RSF
applications. These restrictions, on the other hand, inhibit the
further development of reinforcement technology. Therefore, it
is essential to investigate the proper failure mechanisms for
reinforced soil applications. During the past forty years, many
experimental, numerical, and analytical studies have been
performed to investigate the behavior of reinforced soil foun-
dations (RSFs)for different soil types (e.g., Abu-Farsakh etal.,
2008,2013; Adams and Collin,1997; Binquet and Lee, 1975a,
1975b; Chakraborty and Kumar, 2014; Chen et al., 2007,
2009; Demir et al., 2013; Huang and Tatsuoka, 1990; Kurian
et al., 1997; Sharma et al., 2009).
The?rstexperimentalstudyreportedinliteraturewasconducted
by Binquet and Lee (1975a) to evaluate the bearing capacity of
sand reinforced by aluminum foil strips. Since then, several
experimental studies have been conducted to evaluate the bearing
capacity of footings on reinforced sandy soil (e.g., Abu-Farsakh
et al., 2013; Adams and Collin, 1997; Akinmusuru and
Akinbolade, 1981; Fragaszy and Lawton, 1984; Gabr et al.,
The Japanese Geotechnical Society
www.sciencedirect.com
journal homepage: www.elsevier.com/locate/sandf
Soils and Foundations
http://dx.doi.org/10.1016/j.sandf.2014.12.006
0038-0806/& 2015 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.
n
Corresponding author.
E-mail addresses: qchen1@lsu.edu (Q. Chen),
cefars@lsu.edu (M. Abu-Farsakh).
Peer review under responsibility of The Japanese Geotechnical Society.
Soils and Foundations 2015;55(1):74–85
1998; Guido et al., 1986; Huang and Tatsuoka, 1990; Latha and
Somwanshi,2009;LavasanandGhazavi,2012;Omaretal.1993a,
1993b; Yetimoglu et al. 1994), clayey soil (e.g., Abu-Farsakh
et al., 2008; Chen et al., 2007; Chen and Abu-Farsakh, 2011; Das
et al., 1994; Ingold and Miller, 1982; Mandal and Sah, 1992;
Ramaswamy and Purushothaman (1992); Sakti and Das, 1987;
Shin et al., 1993), aggregate (e.g., Chen et al., 2009; DeMerchant
et al., 2002; James and Raymond, 2002), and pond ash (e.g., Bera
etal.,2005;Ghoshetal.,2005).Theaimofmanyoftheseresearch
efforts was to investigate the parameters and variables that would
contribute to the value of the bearing capacity ratio (BCR), which
is de?ned as the ratio of the bearing capacity of the RSF to that of
the unreinforced soil foundation. The results of the experimental
studies showed that the bearing capacity of soil was improved
when it was reinforced by reinforcements and that the amount of
improvement was highly dependent on the layout of the reinforce-
ments. Better improvements were obtained when the reinforce-
ments were placed within a certain depth (or in?uence depth)
beyond which no additional signi?cant improvement occurred. In
other words, the BCR value would approach a constant/limiting
value with an increasing number of reinforcement layers.
From the experimental studies reported in the literature, two
fundamental reinforcement mechanisms can be distinguished
as contributing to the increase in bearing capacity of reinforced
soil foundations (RSFs).
(1) Con?nement effect or lateral restraint effect: With the
applied load, the lateral forces are induced and the soil
List of symbols
B width of footing
u top layer spacing, i.e., spacing between top layer
of reinforcement and bottom of footing
h vertical spacing between reinforcement layers
l length of reinforcement
d total depth of reinforcement¼uþ(N1)h.
N number of reinforcement layers
N
p
number of reinforcement layers located in punch-
ing shear failure zone
N
T
number of reinforcement layers located above
point c
T tensile force in reinforcement
D
P
depth of punching shear failure zone
q
u(R)
ultimate bearing capacity of reinforced soil
foundation
q
u(R)1
ultimate bearing capacity of punching shear
failure zone
q
u(R)2
ultimate bearing capacity of underlying general
shear failure zone
P
p1
total passive earth pressure on vertical punching
failure surfaces aa
0
and bb
0
d mobilized friction angle along vertical punching
failure surfaces aa
0
and bb
0
C
a
adhesive force acting on vertical punching failure
surfaces aa
0
and bb
0
,¼c
a
D
P
c
a
unit adhesion of soil along vertical punching
failure surfaces aa
0
and bb
0
,
T
1
tensile force acting on vertical punching failure
surfaces aa
0
and bb
0
a angle of tensile force T
1
to horizontal
T
1x
horizontal component of tensile force T
1
T
1y
vertical component of tensile force T
1
? unit weight of soil
D
f
embedment depth of footing
K
pH
horizontal component of passive earth pressure
coef?cient
K
s
punching shear coef?cient
? friction angle of soil
P
p2
passive force acting on faces ac and bc
C cohesive force C acting on faces ac and bc
T
2L
,T
2
tensile force acting on faces ac and bc
P
pc
passive force due to cohesion c,
P
pq
passive force due to surcharge q
P
p?
, passive force due to weight of soil ?
c cohesion of soil
q surcharge load
P
pT
passive force due to tensile force of reinforcement
T
2L
? angle of tensile force T
2L
to horizontal
T
2Lx
,T
2x
horizontal component of tensile force T
2L
T
2Ly
vertical component of tensile force T
2L
T
2R
tensile force acting on face gd
? angle of tensile force T
2R
to horizontal
T
2RX
horizontal component of tensile force T
2R
T
2Ry
vertical component of tensile force T
2R
F resisting force along log spiral cd
r length of radial line of log spiral cd,¼r
0
e
?tan?
r
0
length of bc
? angle between line bc and radial line of log spiral
curve cd
X
TR
distance from center of footing to point where
tensile force T
2R
is applied
q
u(UR)
ultimate bearing capacity of unreinforced soil in
general shear failure zone
N
c
,N
q
, and N
?
bearing capacity factors
ß angle between s
1
direction and bedding plane
?
design
design friction angle of soil
?
peak
peak friction angle of soil
?
cv
residual design friction angle of soil
? percent of contribution of failure surfaces con-
trolled by soil’s peak friction angle
R
r
reinforced ratio
E
R
elastic modulus of reinforcement¼J/t
R
J tensile modulus of reinforcement
A
R
area of reinforcement per unit width¼Nt
R
1
t
R
thickness of reinforcement
E
s
modulus of elasticity of soil
A
s
area of reinforced soil per unit width¼d1
Q. Chen, M. Abu-Farsakh / Soils and Foundations 55 (2015) 74–85 75
particles are spread. Due to the relative displacement
between the soil and the reinforcements, frictional interac-
tion is induced at the soil-reinforcement interface. For
geogrid reinforcements, the interlocking can be developed
by the interaction of the soil and the geogrid. Conse-
quently,lateral deformationorpotentialtensilestrainofthe
reinforced soil is restrained. As a result, the vertical
deformation of the soil is reduced. Since soils are stress-
dependent, improved lateral con?nement increases the
compressive strength of the soil, and thus, improves the
bearing capacity of the reinforced soil.
(2) Membrane effect: With the applied load, the soil beneath
the footing moves downward; the reinforcement is then
deformed and tensioned. The deformed reinforcement
develops an upward force that supports part of the applied
load. A certain amount of settlement is generally required
to signi?cantly mobilize the tensioned membrane effect,
and the reinforcement should have enough length and
tensile strength to prevent it from failing by pull out and
rupture.
Several researchers have presented analytical models for esti-
mating the ultimate bearing capacity of RSFs based on either the
con?nement effect (e.g., Huang and Tatsuoka, 1990; Michalowski,
2004) or the membrane effect (e.g., Binquet and Lee, 1975b;
Kumar and Saran, 2003; Wayne et al., 1998). With these two
fundamentalreinforcingmechanisms, the“deepfooting” effect can
be formed under certain conditions. Here, the“deep footing” effect
means that the performance of footings on a reinforced soil
foundation is very similar to that of footings on an unreinforced
soil foundation with an additional embedment depth equal to the
depth of the reinforced zone. This effect was ?rst proposed by
Schlosser et al., (1983). Huang and Menq (1997) employed this
effect to derive the ultimate bearing capacity formulas for
reinforced sand. This effect is also considered in Wayne et al.’
(1998) solution. With an increase in the number of reinforcement
layers, the “deep footing” effect leads to an almost linear increase
in the bearing capacity ratio (BCR), and convergence cannot be
obtained(i.e.,theBCRvaluedoesnotapproachaconstant/limiting
value with an increasing number of reinforcement layers). This
poses a big problem, especially in granular soil with a high friction
angle. The authors made great efforts to develop analytical
solutions to overcome this problem (Chen et al., 2009; Sharma
et al., 2009). However, the developed analytical solutions were
only applied to square footings and took different forms for
different soil types. Also, only a single reinforcement mechanism
(either con?nement effect or membrane effect) was considered in
those analytical solutions.
Therefore, this study will focus on strip footings on RSFs by
considering the two reinforcement mechanisms (i.e., con?ne-
ment effect and membrane effect) together. A general failure
mechanism of RSFs is ?rstly proposed. The limit equilibrium
stability analysis for RSFs, based on the proposed failure
mechanism, is then performed as an attempt to develop a
rational uni?ed analytical model for evaluating the ultimate
bearing capacity of strip footings on RSFs. Finally, the
developed analytical solution is veri?ed through the results
of model tests reported in the literature.
2. Failure modes of reinforced soil foundations
As mentioned earlier, reinforcements can restrain the lateral
deformation or the potential tensile strain of the soil (con?ne-
ment effect). In addition, deformed reinforcements can develop
an upward force (membrane effect). These effects will result in
an increase in the bearing capacity of the RSF.
Three potential failure modes of RSFs are shown in Fig. 1.
The ?rst two failure modes, failure above the top layer of the
reinforcement (Binquet and Lee, 1975b)(Fig. 1a) and failure
between the reinforcement layers (Wayne et al., 1998)
(Fig. 1b), can be avoided by keeping the top layer spacing
(u) and the vertical spacing between the reinforcement layers
(h) within an acceptable/reasonable range.
Based on previous studies by Chen (2007), Chen et al. (2009),
Sharma et al. (2009), the third failure mechanism, i.e., a general
failuremechanismofRSFs,isidenti?edasapunchingshearfailure
followed by a general shear failure (Fig. 1c). The value of
punching shear failure depth D
p
depends on the relative strength
between the reinforced zone and the underlying unreinforced zone.
It can be zero (i.e., D
p
¼0) if the strength of the reinforced zone is
slightly larger than that of the underlying unreinforced zone, or if
the reinforcement depth ratio (d/B) is relatively large. It also can
punch all the way through the reinforced zone (i.e., D
p
¼d)ifthe
strength of the reinforced zone is much larger than that of the
underlying unreinforced zone and the reinforcement depth ratio (d/
B) is relatively small.
3. Limit equilibrium analysis of reinforcedsoil foundations
First, we will consider the strip footing case with two layers of
reinforcement. One layer is located in the punching shear failure
zone at a depth of u, and the other layer is located in the general
shearfailurezoneatadepthofuþh.Thefailuresurfaceinthesoil
for the strip footing at the ultimate load is shown in Fig. 1c. The
ultimate bearing capacity of the RSF includes the contribution of
punching shear failure zone a
0
abb
0
, q
u(R)1
, and an underlying
general shear failure zone, q
u(R)2
, i.e., q
u(R)
¼q
u(R)1
þq
u(R)2
.
Let’s consider the soil block a
0
abb
0
in the punching shear
failure zone, as shown in Fig. 2. The forces on the vertical
punching failure surfaces, aa
0
and bb
0
, include total passive
earth pressure P
p1
, inclined at an average angle d,and adhesive
force C
a
¼c
a
D
P
, acting upwards (Meyerhof and Hanna, 1978),
where c
a
istheunitadhesionofthesoilalongtwosidesand D
P
is the depth of the punching shear failure in the reinforced
zone. With the inclusion of the reinforcement, an upward force
will be induced by the tension effect of the reinforcement
along the failure surfaces. At the ultimate load, the reinforce-
ment will deform. The tensile force at the vertical failure
surface, T
1
, is assumed to have an angle of a in the horizontal
direction. This force can be decomposed into two components:
horizontal component T
1x
, which provides the con?nement
effect, and vertical component T
1y
, which provides the tensile
membrane effect.
Q. Chen, M. Abu-Farsakh / Soils and Foundations 55 (2015) 74–85 76
By considering the force equilibrium of soil block a
0
abb
0
in
the vertical direction, the contribution of the punching shear
failure zone can be evaluated as
q
uðRÞ1
¼
2c
a
D
P
B
þ?D
P
2
1þ
2D
f
D
P

K
pH
tan d
B
þ
2T
1x
tan d
B
þ
2T
1y
B
?D
P
ð1Þ
Let
K
pH
tan d¼K
s
tan ? ð2Þ
then,
q
uðRÞ1
¼
2c
a
D
P
B
þ?D
2
P
1þ
2D
f
D
P

K
s
tan ?
B
þ
2T
1x
tan d
B
þ
2T
1y
B
?D
P
ð3Þ
where B is the footing width, ? is the unit weight of the soil, D
f
is the embedment depth of the footing, K
pH
is the horizontal
component of the passive earth pressure coef?cient, K
s
is the
punchingshearcoef?cient,whichdependsonthefrictionangle
of the soil in the reinforced zone and the ultimate bearing
capacity of the soil in both the reinforced zone and the
underlying unreinforced zone, ? is the friction angle of the
q
Ca
Pp1
B
Pp1
Ca
DP
q
u(R)1
q
T1
T1x
T1y
Fig. 2. Free body diagram of soil block a
0
abb
0
.
B
u>0.5B
q
u(R)
q q
h
h
d
l
h>0.5B
q
B
q
u(R)
q
h
u
d
l
d
q q
Ca
Pp1
B
q
u(R)
Pp1
Ca
DP
u
h
h
l
Fig. 1. Failure modes of reinforced soil foundation,
Q. Chen, M. Abu-Farsakh / Soils and Foundations 55 (2015) 74–85 77
Page 5


HOSTED BY
Ultimate bearing capacity analysis of strip footings on reinforced
soil foundation
Qiming Chen
n
, Murad Abu-Farsakh
Louisiana Transportation Research Center, Louisiana State University, 4101 Gourrier Avenue, Baton Rouge, LA 70808, USA
Received 25 September 2013; received in revised form 21 August 2014; accepted 20 October 2014
Available online 2 January 2015
Abstract
Reinforcedsoilfoundations(RSFs)havebeenemployedinengineeringpracticetoincreasethesoilbearingcapacityandtoreducethepotential
footing settlement. The aim of this study is to develop analytical solutions for estimating the ultimate bearing capacity of strip footings on RSFs.
A general failure mode for RSFs was ?rst proposed based on previous studies conducted by the authors and test results from literature study.
A limit equilibrium stability analysis of RSFs was performed based on the proposed failure mechanism. New bearing capacity formulas, which
consider both the con?nement and the membrane effects of reinforcements on the increase in ultimate bearing capacity, were then developed for
strip footings on RSFs. Several special cases of RSFs were presented and discussed. The proposed model was veri?ed by the experimental data
reported in the published literature. The predicted ultimate bearing capacity was in good agreement with the results of model tests reported in the
literature. The study showed that the depth of the punching shear failure zone (D
P
) depends on the relative strength of the reinforced soil layer
and the underlying unreinforced soil layer, and is directly related to the reinforced ratio (R
r
).
& 2015 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.
Keywords: Reinforced soil foundation; Ultimate bearing capacity; Failure mechanism; Limit equilibrium; Reinforced ratio
1. Introduction
The use of reinforced soils to support shallow foundations
has recently received considerable attention. The bene?ts of
including reinforcements in the soil mass to increase the
bearing capacity and to reduce the settlement of the soil
foundation have been widely recognized. However, the devel-
opment of a rational design method and a theory for reinforced
soil foundations (RSFs) is lagging in comparison to RSF
applications. These restrictions, on the other hand, inhibit the
further development of reinforcement technology. Therefore, it
is essential to investigate the proper failure mechanisms for
reinforced soil applications. During the past forty years, many
experimental, numerical, and analytical studies have been
performed to investigate the behavior of reinforced soil foun-
dations (RSFs)for different soil types (e.g., Abu-Farsakh etal.,
2008,2013; Adams and Collin,1997; Binquet and Lee, 1975a,
1975b; Chakraborty and Kumar, 2014; Chen et al., 2007,
2009; Demir et al., 2013; Huang and Tatsuoka, 1990; Kurian
et al., 1997; Sharma et al., 2009).
The?rstexperimentalstudyreportedinliteraturewasconducted
by Binquet and Lee (1975a) to evaluate the bearing capacity of
sand reinforced by aluminum foil strips. Since then, several
experimental studies have been conducted to evaluate the bearing
capacity of footings on reinforced sandy soil (e.g., Abu-Farsakh
et al., 2013; Adams and Collin, 1997; Akinmusuru and
Akinbolade, 1981; Fragaszy and Lawton, 1984; Gabr et al.,
The Japanese Geotechnical Society
www.sciencedirect.com
journal homepage: www.elsevier.com/locate/sandf
Soils and Foundations
http://dx.doi.org/10.1016/j.sandf.2014.12.006
0038-0806/& 2015 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.
n
Corresponding author.
E-mail addresses: qchen1@lsu.edu (Q. Chen),
cefars@lsu.edu (M. Abu-Farsakh).
Peer review under responsibility of The Japanese Geotechnical Society.
Soils and Foundations 2015;55(1):74–85
1998; Guido et al., 1986; Huang and Tatsuoka, 1990; Latha and
Somwanshi,2009;LavasanandGhazavi,2012;Omaretal.1993a,
1993b; Yetimoglu et al. 1994), clayey soil (e.g., Abu-Farsakh
et al., 2008; Chen et al., 2007; Chen and Abu-Farsakh, 2011; Das
et al., 1994; Ingold and Miller, 1982; Mandal and Sah, 1992;
Ramaswamy and Purushothaman (1992); Sakti and Das, 1987;
Shin et al., 1993), aggregate (e.g., Chen et al., 2009; DeMerchant
et al., 2002; James and Raymond, 2002), and pond ash (e.g., Bera
etal.,2005;Ghoshetal.,2005).Theaimofmanyoftheseresearch
efforts was to investigate the parameters and variables that would
contribute to the value of the bearing capacity ratio (BCR), which
is de?ned as the ratio of the bearing capacity of the RSF to that of
the unreinforced soil foundation. The results of the experimental
studies showed that the bearing capacity of soil was improved
when it was reinforced by reinforcements and that the amount of
improvement was highly dependent on the layout of the reinforce-
ments. Better improvements were obtained when the reinforce-
ments were placed within a certain depth (or in?uence depth)
beyond which no additional signi?cant improvement occurred. In
other words, the BCR value would approach a constant/limiting
value with an increasing number of reinforcement layers.
From the experimental studies reported in the literature, two
fundamental reinforcement mechanisms can be distinguished
as contributing to the increase in bearing capacity of reinforced
soil foundations (RSFs).
(1) Con?nement effect or lateral restraint effect: With the
applied load, the lateral forces are induced and the soil
List of symbols
B width of footing
u top layer spacing, i.e., spacing between top layer
of reinforcement and bottom of footing
h vertical spacing between reinforcement layers
l length of reinforcement
d total depth of reinforcement¼uþ(N1)h.
N number of reinforcement layers
N
p
number of reinforcement layers located in punch-
ing shear failure zone
N
T
number of reinforcement layers located above
point c
T tensile force in reinforcement
D
P
depth of punching shear failure zone
q
u(R)
ultimate bearing capacity of reinforced soil
foundation
q
u(R)1
ultimate bearing capacity of punching shear
failure zone
q
u(R)2
ultimate bearing capacity of underlying general
shear failure zone
P
p1
total passive earth pressure on vertical punching
failure surfaces aa
0
and bb
0
d mobilized friction angle along vertical punching
failure surfaces aa
0
and bb
0
C
a
adhesive force acting on vertical punching failure
surfaces aa
0
and bb
0
,¼c
a
D
P
c
a
unit adhesion of soil along vertical punching
failure surfaces aa
0
and bb
0
,
T
1
tensile force acting on vertical punching failure
surfaces aa
0
and bb
0
a angle of tensile force T
1
to horizontal
T
1x
horizontal component of tensile force T
1
T
1y
vertical component of tensile force T
1
? unit weight of soil
D
f
embedment depth of footing
K
pH
horizontal component of passive earth pressure
coef?cient
K
s
punching shear coef?cient
? friction angle of soil
P
p2
passive force acting on faces ac and bc
C cohesive force C acting on faces ac and bc
T
2L
,T
2
tensile force acting on faces ac and bc
P
pc
passive force due to cohesion c,
P
pq
passive force due to surcharge q
P
p?
, passive force due to weight of soil ?
c cohesion of soil
q surcharge load
P
pT
passive force due to tensile force of reinforcement
T
2L
? angle of tensile force T
2L
to horizontal
T
2Lx
,T
2x
horizontal component of tensile force T
2L
T
2Ly
vertical component of tensile force T
2L
T
2R
tensile force acting on face gd
? angle of tensile force T
2R
to horizontal
T
2RX
horizontal component of tensile force T
2R
T
2Ry
vertical component of tensile force T
2R
F resisting force along log spiral cd
r length of radial line of log spiral cd,¼r
0
e
?tan?
r
0
length of bc
? angle between line bc and radial line of log spiral
curve cd
X
TR
distance from center of footing to point where
tensile force T
2R
is applied
q
u(UR)
ultimate bearing capacity of unreinforced soil in
general shear failure zone
N
c
,N
q
, and N
?
bearing capacity factors
ß angle between s
1
direction and bedding plane
?
design
design friction angle of soil
?
peak
peak friction angle of soil
?
cv
residual design friction angle of soil
? percent of contribution of failure surfaces con-
trolled by soil’s peak friction angle
R
r
reinforced ratio
E
R
elastic modulus of reinforcement¼J/t
R
J tensile modulus of reinforcement
A
R
area of reinforcement per unit width¼Nt
R
1
t
R
thickness of reinforcement
E
s
modulus of elasticity of soil
A
s
area of reinforced soil per unit width¼d1
Q. Chen, M. Abu-Farsakh / Soils and Foundations 55 (2015) 74–85 75
particles are spread. Due to the relative displacement
between the soil and the reinforcements, frictional interac-
tion is induced at the soil-reinforcement interface. For
geogrid reinforcements, the interlocking can be developed
by the interaction of the soil and the geogrid. Conse-
quently,lateral deformationorpotentialtensilestrainofthe
reinforced soil is restrained. As a result, the vertical
deformation of the soil is reduced. Since soils are stress-
dependent, improved lateral con?nement increases the
compressive strength of the soil, and thus, improves the
bearing capacity of the reinforced soil.
(2) Membrane effect: With the applied load, the soil beneath
the footing moves downward; the reinforcement is then
deformed and tensioned. The deformed reinforcement
develops an upward force that supports part of the applied
load. A certain amount of settlement is generally required
to signi?cantly mobilize the tensioned membrane effect,
and the reinforcement should have enough length and
tensile strength to prevent it from failing by pull out and
rupture.
Several researchers have presented analytical models for esti-
mating the ultimate bearing capacity of RSFs based on either the
con?nement effect (e.g., Huang and Tatsuoka, 1990; Michalowski,
2004) or the membrane effect (e.g., Binquet and Lee, 1975b;
Kumar and Saran, 2003; Wayne et al., 1998). With these two
fundamentalreinforcingmechanisms, the“deepfooting” effect can
be formed under certain conditions. Here, the“deep footing” effect
means that the performance of footings on a reinforced soil
foundation is very similar to that of footings on an unreinforced
soil foundation with an additional embedment depth equal to the
depth of the reinforced zone. This effect was ?rst proposed by
Schlosser et al., (1983). Huang and Menq (1997) employed this
effect to derive the ultimate bearing capacity formulas for
reinforced sand. This effect is also considered in Wayne et al.’
(1998) solution. With an increase in the number of reinforcement
layers, the “deep footing” effect leads to an almost linear increase
in the bearing capacity ratio (BCR), and convergence cannot be
obtained(i.e.,theBCRvaluedoesnotapproachaconstant/limiting
value with an increasing number of reinforcement layers). This
poses a big problem, especially in granular soil with a high friction
angle. The authors made great efforts to develop analytical
solutions to overcome this problem (Chen et al., 2009; Sharma
et al., 2009). However, the developed analytical solutions were
only applied to square footings and took different forms for
different soil types. Also, only a single reinforcement mechanism
(either con?nement effect or membrane effect) was considered in
those analytical solutions.
Therefore, this study will focus on strip footings on RSFs by
considering the two reinforcement mechanisms (i.e., con?ne-
ment effect and membrane effect) together. A general failure
mechanism of RSFs is ?rstly proposed. The limit equilibrium
stability analysis for RSFs, based on the proposed failure
mechanism, is then performed as an attempt to develop a
rational uni?ed analytical model for evaluating the ultimate
bearing capacity of strip footings on RSFs. Finally, the
developed analytical solution is veri?ed through the results
of model tests reported in the literature.
2. Failure modes of reinforced soil foundations
As mentioned earlier, reinforcements can restrain the lateral
deformation or the potential tensile strain of the soil (con?ne-
ment effect). In addition, deformed reinforcements can develop
an upward force (membrane effect). These effects will result in
an increase in the bearing capacity of the RSF.
Three potential failure modes of RSFs are shown in Fig. 1.
The ?rst two failure modes, failure above the top layer of the
reinforcement (Binquet and Lee, 1975b)(Fig. 1a) and failure
between the reinforcement layers (Wayne et al., 1998)
(Fig. 1b), can be avoided by keeping the top layer spacing
(u) and the vertical spacing between the reinforcement layers
(h) within an acceptable/reasonable range.
Based on previous studies by Chen (2007), Chen et al. (2009),
Sharma et al. (2009), the third failure mechanism, i.e., a general
failuremechanismofRSFs,isidenti?edasapunchingshearfailure
followed by a general shear failure (Fig. 1c). The value of
punching shear failure depth D
p
depends on the relative strength
between the reinforced zone and the underlying unreinforced zone.
It can be zero (i.e., D
p
¼0) if the strength of the reinforced zone is
slightly larger than that of the underlying unreinforced zone, or if
the reinforcement depth ratio (d/B) is relatively large. It also can
punch all the way through the reinforced zone (i.e., D
p
¼d)ifthe
strength of the reinforced zone is much larger than that of the
underlying unreinforced zone and the reinforcement depth ratio (d/
B) is relatively small.
3. Limit equilibrium analysis of reinforcedsoil foundations
First, we will consider the strip footing case with two layers of
reinforcement. One layer is located in the punching shear failure
zone at a depth of u, and the other layer is located in the general
shearfailurezoneatadepthofuþh.Thefailuresurfaceinthesoil
for the strip footing at the ultimate load is shown in Fig. 1c. The
ultimate bearing capacity of the RSF includes the contribution of
punching shear failure zone a
0
abb
0
, q
u(R)1
, and an underlying
general shear failure zone, q
u(R)2
, i.e., q
u(R)
¼q
u(R)1
þq
u(R)2
.
Let’s consider the soil block a
0
abb
0
in the punching shear
failure zone, as shown in Fig. 2. The forces on the vertical
punching failure surfaces, aa
0
and bb
0
, include total passive
earth pressure P
p1
, inclined at an average angle d,and adhesive
force C
a
¼c
a
D
P
, acting upwards (Meyerhof and Hanna, 1978),
where c
a
istheunitadhesionofthesoilalongtwosidesand D
P
is the depth of the punching shear failure in the reinforced
zone. With the inclusion of the reinforcement, an upward force
will be induced by the tension effect of the reinforcement
along the failure surfaces. At the ultimate load, the reinforce-
ment will deform. The tensile force at the vertical failure
surface, T
1
, is assumed to have an angle of a in the horizontal
direction. This force can be decomposed into two components:
horizontal component T
1x
, which provides the con?nement
effect, and vertical component T
1y
, which provides the tensile
membrane effect.
Q. Chen, M. Abu-Farsakh / Soils and Foundations 55 (2015) 74–85 76
By considering the force equilibrium of soil block a
0
abb
0
in
the vertical direction, the contribution of the punching shear
failure zone can be evaluated as
q
uðRÞ1
¼
2c
a
D
P
B
þ?D
P
2
1þ
2D
f
D
P

K
pH
tan d
B
þ
2T
1x
tan d
B
þ
2T
1y
B
?D
P
ð1Þ
Let
K
pH
tan d¼K
s
tan ? ð2Þ
then,
q
uðRÞ1
¼
2c
a
D
P
B
þ?D
2
P
1þ
2D
f
D
P

K
s
tan ?
B
þ
2T
1x
tan d
B
þ
2T
1y
B
?D
P
ð3Þ
where B is the footing width, ? is the unit weight of the soil, D
f
is the embedment depth of the footing, K
pH
is the horizontal
component of the passive earth pressure coef?cient, K
s
is the
punchingshearcoef?cient,whichdependsonthefrictionangle
of the soil in the reinforced zone and the ultimate bearing
capacity of the soil in both the reinforced zone and the
underlying unreinforced zone, ? is the friction angle of the
q
Ca
Pp1
B
Pp1
Ca
DP
q
u(R)1
q
T1
T1x
T1y
Fig. 2. Free body diagram of soil block a
0
abb
0
.
B
u>0.5B
q
u(R)
q q
h
h
d
l
h>0.5B
q
B
q
u(R)
q
h
u
d
l
d
q q
Ca
Pp1
B
q
u(R)
Pp1
Ca
DP
u
h
h
l
Fig. 1. Failure modes of reinforced soil foundation,
Q. Chen, M. Abu-Farsakh / Soils and Foundations 55 (2015) 74–85 77
soil, and d is the mobilized friction angle along two sides, aa
0
and bb
0
. Since the strip footings studied in this paper sit on a
uniform soil layer, c
a
, d, and K
s
are equal to c, ?, and K
p
,
respectively. Punching shear coef?cient K
s
can then be easily
determined with the passive earth pressure coef?cient chart
proposed by Caquot and Kerisel (1948) and given in Fig. 3.
The contribution of the general shear failure zone to the
ultimate bearing capacity of the RSF can be obtained by
solving the equivalent regular case of a general shear failure
probleminsoil,asshowninFig.4a.Inthe?gure, q
0
isequal to
?(D
f
þD
P
).
Considering soil wedge abc (Fig. 4b), the forces acting on
faces ac and bc include passive force P
p2
, cohesive force C,
and tensile force T
2L
, which provide the bene?t of the tension
membrane effect of the reinforcement. Passive force P
p2
includes four components and can be written as follows:
P
p2
¼P
pc
þP
pq
þP
p?
þP
pT
ð4Þ
where P
pc
, P
pq
, P
p?
, and P
pT
are the passive forces due to
cohesion (c), surcharge (q), weight of the soil (?), and the
tensile force of the reinforcement (T), respectively. P
pT
can be
attributed to the con?nement effect of the reinforcement.
The derivation of C, P
pc
, P
pq
, and P
p?
can be found in many
foundation engineering books (e.g., Das, 1999). Therefore, the
discussion here will be focused on the derivation of P
pT
only.
Considering the free body diagram of soil wedge bcdg,
shown inFig. 4c,theforces perunitlengthofwedge bcdg,due
to the tensile force of reinforcement T, include P
pT
, the tensile
forcesofthereinforcement, T
2L
and T
2R
,andtheresistingforce
along log spiral cd, F. Tensile force T
2L
is assumed to have an
angle of ? in the horizontal direction and can be decomposed
into two components: the horizontal component, T
2Lx,
and the
component along line bc, T
2Ly
, as shown in Fig. 4c. Tensile
force T
2R
is assumed to have an angle of ? in the horizontal
direction and can also be decomposed into two components:
horizontal component T
2Rx
and vertical component T
2Ry
,as
shown in Fig. 4c.
Log spiral cd is described by the equation
r¼r
0
e
? tan ?
ð5Þ
where r
0
¼bc, and ? is the angle between line bc and the radial
line of log spiral curve cd. This means that the radial line at
any point makes an angle ? with the normal direction of the
log spiral. Resisting force F also makes an angle ? with the
normal direction of the log spiral. Taking the moment about
center point b of the log spiral curve, passive force P
pT
can be
given by the following relation (moment equilibrium):
P
pT
¼
4 T
2Lx
T
2Rx
ðÞ uþhD
P
ðÞþT
2Ry
X
TR
B=2
 
cos p=4þ?=2

1
B cos ?
ð6Þ
Considering the force equilibrium of soil wedge abc in the
vertical direction, as shown in Fig. 4b, bearing capacity q
u(R)2
can be given by the following equation:
q
uðRÞ2
¼q
uðURÞ
þ
2P
pT
sin p=4þ?=2

B1
þ
2T
2L
sin ?
B
ð7Þ
q
uðURÞ
¼cN
c
þ? D
f
þD
P

N
q
þ0:5?BN? ð8Þ
where q
u(UR)
is the bearing capacity of the unreinforced soil in
the general shear failure zone, c is the cohesion of the soil, and
N
c
,N
q
, and N
?
are the bearing capacity factors, which are
dependent on the friction angle of soil ?.
The distance, X
TR
, from the center of the footing to the point
where tensile force T
2R
is applied is greater than 2B when soil
friction angle ? is greater than 251. The measured strain
distribution along the reinforcement reported in the literature
(Chen, 2007; Huang and Tatsuoka, 1990; James and
Raymond, 2002) showed that the tensile force in the reinforce-
ment at this distance is negligible or in compression. There-
fore, tensile force T
2R
can betaken aszero andEq.(7)canthen
be simpli?ed as
q
uðRÞ2
¼q
uðURÞ
þ
4T
2Lx
uþhD
P
ðÞ
B
2
þ
T
2L
sin ?
B
¼q
uðURÞ
þ
4T
2x
uþhD
P
ðÞ
B
2
þ
T
2
sin ?
B
ð9Þ
where T
2
is equal to T
2L
and T
2x
is equal to T
2Lx
.
The ultimate bearing capacity of the strip footing on
reinforced soil with two layers of reinforcement can then be
given as follows:
q
uðRÞ
¼q
uðRÞ1
þq
uðRÞ2
¼q
uðURÞ
þ?q
P
þ?q
t
ð10Þ
Fig. 3. Coef?cients of punching shear resistance under vertical load (after
Meyerhof and Hanna, 1978), (a) equivalence of general shear failure zone, (b)
Free body diagram of soil wedge abc, (c) free body diagram of soil wedge
bcdg.
Q. Chen, M. Abu-Farsakh / Soils and Foundations 55 (2015) 74–85 78
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