Geotextiles and Geomembranes 26 (2008) 1–13 Experimental and theoretical investigation of strength of soil reinforced with multi-layer horizontal–vertical orthogonal elements M.X. Zhang a, , H. Zhou a , A.A. Javadi b , Z.W. Wang a a Department of Civil Engineering, Shanghai University, 149 Yanchang Road, Shanghai 200072, China b Department of Engineering, School of Engineering and Computer Science, University of Exeter, Exeter, Devon EX4 4QF, UK Received 24 November 2006; received in revised form 28 May 2007; accepted 4 June 2007 Available online 7 August 2007 Abstract In conventional reinforced soil structures, the reinforcements are often laid horizontally in the soil. In this paper, a new concept of soil reinforced with horizontal–vertical (H–V) orthogonal reinforcing elements is proposed. In the proposed method of soil reinforcement, H–V orthogonal elements instead of conventional horizontal inclusions are placed in the soil. A fundamental difference between the H–V orthogonal reinforcing elements presented in this paper and other forms of inclusions is that the soil enclosed within the H–V orthogonal reinforcing elements will provide passive resistances against shearing that will increase the strength and stability of the reinforced soil. A comprehensive set of triaxial tests were carried out on sand reinforced with multi-layer H–V orthogonal elements and vertical ones. The behavior of sand reinforced with different H–V orthogonal elements was studied in terms of stress–strain relationship and shear strength. Based on experimental results, a strength model of the soil reinforced with H–V orthogonal elements was developed by means of the theory of limit equilibrium. The results of proposed strength model are compared with those obtained from the triaxial tests. It is shown that the results of prediction are in good agreement with those of the triaxial tests. r 2007 Elsevier Ltd. All rights reserved. Keywords: Horizontal–vertical (H–V) orthogonal element; Vertical reinforcement; Reinforced soil; Triaxial test; Strength model; Limit equilibrium 1. Introduction Reinforced soils have been widely used in geotechnical engineering. Numerous papers have examined the reinfor- cement of soil (e.g. Fleming et al., 2006; Iizuka et al., 2004; Katarzyna, 2006; Latha and Murthy, 2006; Park and Tan, 2005; Patra et al., 2005; Varuso et al., 2005; Yang, 1972; Yetimoglu et al., 2005). Current researches mainly focus on soil reinforced with conventional horizontal inclusions (Haeri et al., 2000; Ingold, 1983; Michalowski, 2004; Moraci and Recalcati, 2006). Schlosser and Long (1972) conducted a more detailed study on reinforced sand using triaxial tests and proposed pseudo-cohesion concept and strength relationship. Broms (1977) tested a dry fine sand reinforced with geotextile in a triaxial apparatus and proposed an equation for calculating the ultimate load in a reinforced soil. Rajagopal et al. (1999) carried out a large number of triaxial compression tests on granular soil encased in single and multiple geocells to study the influence of geocell confinement on the strength and stiffness behavior of granular soils. Model tests and centrifuge modeling test were performed on model geosynthetic retaining walls to examine the reinforcing effect and the failure mechanism. Smith and Brigilson (1979) used model tests to study the strength and bearing capacity of earth retaining walls reinforced with inclined reinforcements. Lawton et al. (1993) carried out a series of CBR, triaxial and permeability tests on soil reinforced with multioriented geosynthetic inclusions to compare the effectiveness of the multioriented elements with fibers. Irsyam and Hryciw (1991) performed theoretical analysis of stress transfer between sand and ribbed reinforcement and laboratory investigations to evaluate the individual contributions of friction and passive resistance to overall pullout resistance. The effects of reinforcement form on strength improvement of geosynthetic-reinforced sand were studied through triaxial compression tests, in which ARTICLE IN PRESS www.elsevier.com/locate/geotexmem 0266-1144/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.geotexmem.2007.06.001 Corresponding author. Tel.: +86 21 56331972; fax: +86 21 56331971. E-mail address: [email protected] (M.X. Zhang).
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ARTICLE IN PRESS
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Geotextiles and Geomembranes 26 (2008) 1–13
www.elsevier.com/locate/geotexmem
Experimental and theoretical investigation of strength of soil reinforcedwith multi-layer horizontal–vertical orthogonal elements
M.X. Zhanga,�, H. Zhoua, A.A. Javadib, Z.W. Wanga
aDepartment of Civil Engineering, Shanghai University, 149 Yanchang Road, Shanghai 200072, ChinabDepartment of Engineering, School of Engineering and Computer Science, University of Exeter, Exeter, Devon EX4 4QF, UK
Received 24 November 2006; received in revised form 28 May 2007; accepted 4 June 2007
Available online 7 August 2007
Abstract
In conventional reinforced soil structures, the reinforcements are often laid horizontally in the soil. In this paper, a new concept of soil
reinforced with horizontal–vertical (H–V) orthogonal reinforcing elements is proposed. In the proposed method of soil reinforcement,
H–V orthogonal elements instead of conventional horizontal inclusions are placed in the soil. A fundamental difference between the H–V
orthogonal reinforcing elements presented in this paper and other forms of inclusions is that the soil enclosed within the H–V orthogonal
reinforcing elements will provide passive resistances against shearing that will increase the strength and stability of the reinforced soil.
A comprehensive set of triaxial tests were carried out on sand reinforced with multi-layer H–V orthogonal elements and vertical ones.
The behavior of sand reinforced with different H–V orthogonal elements was studied in terms of stress–strain relationship and shear
strength. Based on experimental results, a strength model of the soil reinforced with H–V orthogonal elements was developed by means
of the theory of limit equilibrium. The results of proposed strength model are compared with those obtained from the triaxial tests. It is
shown that the results of prediction are in good agreement with those of the triaxial tests.
Reinforced soils have been widely used in geotechnicalengineering. Numerous papers have examined the reinfor-cement of soil (e.g. Fleming et al., 2006; Iizuka et al., 2004;Katarzyna, 2006; Latha and Murthy, 2006; Park and Tan,2005; Patra et al., 2005; Varuso et al., 2005; Yang, 1972;Yetimoglu et al., 2005). Current researches mainly focus onsoil reinforced with conventional horizontal inclusions(Haeri et al., 2000; Ingold, 1983; Michalowski, 2004;Moraci and Recalcati, 2006). Schlosser and Long (1972)conducted a more detailed study on reinforced sand usingtriaxial tests and proposed pseudo-cohesion concept andstrength relationship. Broms (1977) tested a dry fine sandreinforced with geotextile in a triaxial apparatus andproposed an equation for calculating the ultimate load in areinforced soil. Rajagopal et al. (1999) carried out a large
e front matter r 2007 Elsevier Ltd. All rights reserved.
otexmem.2007.06.001
ing author. Tel.: +8621 56331972; fax: +86 21 56331971.
number of triaxial compression tests on granular soilencased in single and multiple geocells to study theinfluence of geocell confinement on the strength andstiffness behavior of granular soils. Model tests andcentrifuge modeling test were performed on modelgeosynthetic retaining walls to examine the reinforcingeffect and the failure mechanism. Smith and Brigilson(1979) used model tests to study the strength and bearingcapacity of earth retaining walls reinforced with inclinedreinforcements. Lawton et al. (1993) carried out a series ofCBR, triaxial and permeability tests on soil reinforced withmultioriented geosynthetic inclusions to compare theeffectiveness of the multioriented elements with fibers.Irsyam and Hryciw (1991) performed theoretical analysisof stress transfer between sand and ribbed reinforcementand laboratory investigations to evaluate the individualcontributions of friction and passive resistance to overallpullout resistance. The effects of reinforcement form onstrength improvement of geosynthetic-reinforced sand werestudied through triaxial compression tests, in which
A area of specimen section (m2)Cu coefficient of uniformity (dimensionless)Cc coefficient of curvature (dimensionless)c cohesion of soil (kPa)cr apparent cohesion of reinforced sand (kPa)fu frictional coefficient between reinforcement and
soil (dimensionless)H height of vertical reinforcement (m)Hi height of vertical reinforcements at each
layer (m)DH spacing between reinforcements (m)h height of specimen (m)Kp coefficient of passive earth pressure (dimension-
less)M secant modulus of the membrane of the hoop at
axial strain of ea (kPa)n the number of reinforcing layers (dimension-
less)R resultant of the shear and normal forces on the
failure surface (kN)DRi additional resultant of the shear and normal
forces on the failure surface for specimen ateach vertical reinforcement (kN)
RT tensile strength of reinforcement per unit width(kN/m)
T resultant of tensile forces acted on all reinforce-ments pulled out (kN)
th thickness of horizontal reinforcements (m)tv thickness of vertical reinforcement (m)r radius of specimen (m)
r0 (initial) radius of vertical hoop reinforce-ment (m)
V volume of the upper and lower parts of thespecimen on either side of the failure surface (m3)
z vertical distance between bottom of specimenand calculated section (m)
Greek letters
a angle of failure surface to the horizontal plane(deg)
s1 major principal stress (kPa)s3 minor principal stress (kPa)Ds3 additional confining pressure (kPa)st tensile stresses within vertical reinforcement (kPa)[st] tensile strength of vertical reinforcement (kPa)ea axial strain of specimen at failure (dimension-
less)ec circumferential strain (dimensionless)o the total area of horizontal inclusions pulled
out (m2)j angle of internal friction (deg)mv vertical reinforcing ratio parameter (dimension-
less)Z correction for the coefficient of friction between
reinforcements and soil (dimensionless)l correction coefficient for tensile strength of
vertical reinforcement (dimensionless)x correction coefficient for tensile strength of
horizontal reinforcement (dimensionless)z a variable defined as (dimensionless): z ¼ 1�
(8/3p)ZfuKp(r/DH)
M.X. Zhang et al. / Geotextiles and Geomembranes 26 (2008) 1–132
samples of sand reinforced with geosynthetics in horizontallayers, geocells, and randomly distributed discrete fibers(Latha and Murthy, 2007).
Meanwhile, the behavior on fiber-reinforced soils waswidely studied (e.g. Kumar et al., 2006; Michalowski andCermak, 2003; Prabakar and Sridhar, 2002; Yetimoglu andSalbas, 2003). Gray and Ohashi (1983) considered areinforcement embedded perpendicularly or at an inclina-tion to the shear zone in a shear box to study the behaviorof a dry sand reinforced with different types of fibers.Arenicz and Choudhury (1988) carried out a series oflaboratory investigations to study the effects of differenttypes of random reinforcements on soil strength. Thecontributions related to new arrangements of reinforce-ment have played an active role in the development ofreinforced soil technology. Besides the study on conven-tional reinforced soil where the reinforcements are puthorizontally, some new configurations of inclusions weredeveloped. For example, Zhang et al. (2006) proposed anew concept of soil reinforced with three-dimensional (3D)elements and carried out a series of triaxial tests to study
the behavior of sand reinforced with a single-layer 3Dinclusion.In this paper, a new concept of soil reinforcement with
specific types of horizontal–vertical (H–V) orthogonalreinforcing elements, as one specific example of 3Dinclusions, was proposed. A fundamental differencebetween the H–V orthogonal reinforcing elements pre-sented in this paper and other forms of inclusions as well asfiber-reinforced soil is that in the presented H–V orthogo-nal reinforced soil, the soil enclosed within the H–Vorthogonal reinforcing elements will provide passiveresistances against shearing that will increase the strengthand stability of the reinforced soil. Typical soil structuresreinforced with H–V orthogonal reinforcing inclusions forin situ applications are shown in Fig. 1a. Most ofreinforced soil structures were built under (or close to)plane strain conditions. However, triaxial test underaxi-symmetric conditions is one of the important methodsto investigate the effects of H–V orthogonal reinforcinginclusions on the mechanical behavior of reinforced sand.So, 52 series of triaxial tests were carried out on sand
M.X. Zhang et al. / Geotextiles and Geomembranes 26 (2008) 1–13 3
reinforced with two-layer H–V orthogonal inclusionsand vertical elements. The results of experiments arepresented and discussed. Comparison is made betweenshear strength of the soil reinforced with horizontalreinforcements and with H–V orthogonal inclusions. Basedon experimental results, the interaction of H–V orthogonalreinforcing elements with soil is analyzed. Using the limitequilibrium theory, a strength model is developed for soilreinforced with multi-layer H–V orthogonal inclusions.The results of prediction by proposed model are comparedwith those obtained from the triaxial tests. It is shown thatthe results of analytical solution are in good agreementwith results of the triaxial tests.
0
10
20
30
40
50
60
70
80
90
100
0.01 0.1 1 10
Particle size (mm)
Per
cent
finer
(%
)
Fig. 2. Grain size distribution curve (Zhang et al., 2006).
2. Types of soil reinforced with H–V orthogonal inclusions
In soils reinforced with H–V orthogonal reinforcinginclusions, besides conventional horizontal reinforcement,vertical reinforcing elements are also laid in the soil. Themain configurations of H–V reinforcements can be dividedinto three categories:
(1) Vertical reinforcements are laid upon conventionalhorizontal reinforcements in soil; these are typicallyrectangular or hexagonal in shape, as shown in Fig. 1b.
(2) Vertical inclusions are laid in soil without anyhorizontal reinforcements; these are grid and ring shapereinforcements. Geocell is one special example of this typeof H–V reinforcement. These vertical inclusions can beconnected to each other by a series of rigid bars, as shownin Fig. 1c.
(3) Axial denti-reinforcements are laid in soil, as shownin Fig. 1d; these are typically rectangular or hexagonal inshape. The influence of denti-reinforcements is differentfrom that of conventional ribbed inclusions. Besidesfrictional resistances, the former will provide passiveresistance against shearing that will increase the strength
and stability of the reinforced soil, but the latter mainlyenlarges frictional resistance.
3. Experimental procedure
3.1. Test materials
Uniform, clean, quartz beach sand from shores wasused. The particle size distribution curve for the sand isshown in Fig. 2. The sand has a relatively uniform grain-size distribution with coefficients of uniformity (Cu) andcurvature (Cc) of 2.30 and 1.01, respectively. All thespecimens of sand were prepared at a unit weight of16.79 kN/m3, a void ratio of 0.586, an angle of internalfriction of 31.91 and a specific gravity of 2.64, in drycondition within a split cylinder mold. All samples werecarefully prepared to maintain a relative density of 79.8%.Galvanized iron sheet with a thickness of 0.12mm was usedto reinforce the sand specimen in the triaxial tests. Thefriction coefficient between sand and reinforcements,obtained from direct shear tests, is 0.47.
3.2. Test procedure
A set of triaxial compression tests was performed toinvestigate the effects of H–V orthogonal reinforcinginclusions, and vertical reinforcing elements on themechanical behavior of reinforced sand. The results wereused to compare the behavior of unreinforced sand, sandreinforced with horizontal inclusions and with H–Vorthogonal ones. In order to evaluate the effects of H–Vorthogonal reinforcements, a vertical reinforcing ratioparameter, mv is introduced, which is defined by the ratioof area of all vertical reinforcements at the radius r0 to thelateral surface area of cylindrical specimen. It is expressedby percent as follows:
mv ¼r0Pni¼1
Hi
rh� 100%, (1)
where n is the number of reinforcing layers; Hi is the heightof vertical reinforcements at each layer; r0 is the radius of
ARTICLE IN PRESSM.X. Zhang et al. / Geotextiles and Geomembranes 26 (2008) 1–134
vertical reinforcements; h is the height of specimen; r is theradius of specimen.
The values of the vertical reinforcement ratio parametermv for different configurations of H–V reinforcementshown in Fig. 3, are summarized in Table 1 for 13 casesof triaxial experiments. The shear strength parameters ofthe reinforced sands are also presented in the table. Thetriaxial tests were conducted at four different confiningpressures of 50, 100, 150 and 200 kPa.
A standard medium-sized triaxial shear apparatuswas used for testing specimens of unreinforced sand andsand reinforced with H–V orthogonal reinforcing elements.
Fig. 3. Configurations of H–V orthogonal reinforcements (unit: mm): (a) v
inclusions.
Table 1
Experimental cases and strength parameters of H–V reinforced sand
Case Reinforcing type Height of vertical
reinforcementsa, H (cm)
1 Unreinforced
2 Horizontally
reinforced
0.0, 0.0
3 Vertically reinforced 0.5, 0.5
4 1.0, 1.0
5 2.0, 2.0
6 2.0, 4.0
7 4.0, 4.0
8 H–V reinforced
(single-sided)
0.5, 0.5
9 1.0, 1.0
10 H–V reinforced
(double-sided)
0.5, 0.5
11 1.0, 1.0
12 2 (single-sided), 2 (double-sided)
13 2.0, 2.0
aThe two numerals in this column denote heights of vertical reinforcements
The specimens had a diameter of 61.8mm and a heightof 135mm. All dry specimens were subjected to triaxialcompression with a strain rate of 0.5% per minute. Mostof the tests were continued up to an axial strain levelof 8%.A standard procedure was adopted for preparing dry
cohesionless samples and testing with triaxial apparatus asrecommended by Bishop and Henkel (1969) and Head(1982). The samples were compacted in six layers throughtamping with a tamper consisting of a circular diskattached to a steel rod. The H–V orthogonal reinforce-ments with axial symmetry were composed of ring-shaped
on the first and second layer within the specimen, respectively.
ARTICLE IN PRESSM.X. Zhang et al. / Geotextiles and Geomembranes 26 (2008) 1–13 5
vertical elements with different heights and a horizontalone, whose diameter was slightly less than that of thespecimen. The vertical elements were fixed upon horizontalones. After compacting and leveling each layer of sand,H–V orthogonal inclusion was placed in the specimenaccording to the configurations shown in Fig. 3.
4. Test results and discussions
4.1. Stress–strain curves and failure pattern
Typical stress–strain curves for sand reinforced withH–V orthogonal inclusions are presented in Fig. 4. Thesefigures indicate that the maximum deviator stress increaseswith increasing the vertical reinforcing ratio. The peakstrength for most specimens occurs at an axial strain ofabout 1.5–3%.
Typical photographs of failed specimens are shown inFig. 5. A close examination of the failed specimens revealsthat reinforced specimens failed by bulging between twoadjacent layers of reinforcement. Moreover, inspection ofthe inclusions after failure shows that the vertical elementsappear to break at the joints, whereas H–V elements show
0 21 3 4 5 6 7 8 9 10 11 12
Dev
iato
rstr
ess
(kP
a)
Axial strain (%)
0 21 3 4 5 6 7 8 9 10
Axial strain (%)
0
100
200
300
400
500
600
700
800
900
0
100
200
300
400
500
600
700
800
900
Dev
iato
r st
ress
(kP
a)
H.reinforced �v=4.79%(S)
�v=9.59%(S) �v=9.59%(D)
�v=19.18%(D) �v=28.77%(S,D)
�v=38.36%(D)
Unreinforced �v=4.79%
�v=9.59% �v=19.18%
�v=28.77% �v=38.36%
Fig. 4. Deviator stress–axial strain curves for sand reinforced with vertical and
(s3 ¼ 200 kPa) (Note: S-single-sided and D-double-sided).
frictional failure for horizontal elements and breakage ofvertical ones.
4.2. Strength behavior
Typical p–q (mean stress versus deviator stress) diagramsfor the specimens of sand reinforced with vertical elementsand with H–V orthogonal inclusions are shown in Fig. 6,where p ¼ 1/3(s1+2s3) and q ¼ s1�s3. The shear strengthparameters (apparent cohesion and angle of internalfriction) for different reinforcement configurations arepresented in Table 1. Linear shear strength envelopesfor sand reinforced with vertical elements and withH–V orthogonal inclusions are shown in Fig. 7. Theexperimental results indicate that:(1) For sand reinforced with vertical elements, the
reinforced soil specimens have exhibited a significantincrease in angle of internal friction while the differencein apparent cohesion is marginal. As compared withunreinforced sand, the increase in angle of internal frictionfor the sand reinforced with vertical elements is about7.7–19.01.
0
200
400
600
800
1000
1200
1400
0 1 2 3 4 5 6 7 8 9 10
Axial strain (%)
0 1 2 3 4 5 6 7 8 9 10
Axial strain (%)
Dev
iato
r st
ress
(kP
a)
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Dev
iato
r st
ress
(kP
a)
H.reinforced �v=4.79%(S)
�v=9.59%(S) �v=9.59%(D)
�v=19.18%(D) �v=28.77%(S,D)
�v=38.36%(D)
Unreinforced �v=4.79%
�v=9.59% �v=19.18%
�v=28.77% �v=38.36%
H–V reinforcing elements with different reinforcement ratios and confining
Fig. 6. p–q diagrams for sand reinforced with: (a) vertical elements
and (b) H–V elements (two-layer) for different reinforcement ratios.
Note: S-single-sided and D-double-sided.
0
100
200
300
400
500
600
700
0 100 200 300 400 500 600
Normal stress (kPa)
Shea
r st
ress
(kP
a)
Unreinforced
�v=4.79%
�v=9.59%
�v=19.18%
�v=28.77%
�v=38.36%
0
100
200
300
400
500
600
700
800
0 100 200 300 400 500 600
Normal stress (kPa)
Shea
r st
ress
(kP
a)
Unreinforced
Horizontally reinforced
�v=4.79%(S)
�v=9.59%(S)
�v=9.59%(D)
�v=19.18%(D)
�v=28.77%(S,D)
�v=38.36%(D)
Fig. 7. Linear shear strength envelopes for sand reinforced with: (a)
vertical elements and (b) H–V elements (two-layer) for different
reinforcement ratios. Note: S-single-sided and D-double-sided.
M.X. Zhang et al. / Geotextiles and Geomembranes 26 (2008) 1–136
(2) For sand reinforced with H–V elements, the angle ofinternal friction has increased remarkably while displayinga slight increase in apparent cohesion. As compared withhorizontally reinforced sand, the angle of internal frictionof the sand reinforced with H–V elements increased about1.6–13.971.
(3) It can be seen that the angle of internal friction forsand reinforced with double-sided H–V elements(2� 0.5 cm high) was greater than that for sand reinforcedwith single-sided reinforcements (1 cm high). From thisobservation, it can be concluded that with the same heightof vertical reinforcements (the same vertical reinforcing
ARTICLE IN PRESSM.X. Zhang et al. / Geotextiles and Geomembranes 26 (2008) 1–13 7
ratio), the shear strength of sand reinforced with double-sided H–V elements is significantly greater than that withsingle-sided reinforcement.
5. Strength model of soil-reinforced with H–V orthogonal
inclusions
5.1. Strength model for soil
From the Mohr–Coulomb failure criterion for soils, therelationship between the major and minor principal stressesin a soil at failure can be expressed in the following form:
s1 ¼ s3Kp þ 2cffiffiffiffiffiffiKp
p, (2)
where s1 is the major principal stress (axial stress); s3 is theminor principal stress (confining pressure); Kp is thecoefficient of passive earth pressure given by Kp ¼
tan2(451+j/2); c is the cohesion; j is the angle of internalfriction.
Fig. 8a shows the free body diagram of a part of asoil specimen (above the failure surface) sheared in atriaxial test under a confining pressure s3 and axial stress
Fig. 8. Free body and force diagrams for soil specimens: (a) unreinforced; reinf
and at (c) failure by pullout of reinforcement.
s1. If the cross sectional area of the specimen is A, the areaof the failure surface (which is elliptical in shape) will beA/cos a, where a is the angle between the failure surface andthe horizontal (see Fig. 8a). The resultant R of the shearand normal forces on the failure surface can then becalculated as
R cos ða� jÞ ¼ s1A� cA tan a
or
R ¼ðs3Kp þ 2c
ffiffiffiffiffiffiKp
p� c tan aÞA
cosða� jÞ, (3)
where A ¼ pr2 and a ¼ 451+j/2.
5.2. Strength model for soil reinforced with horizontal
elements
5.2.1. Failure by breakage of reinforcement
Fig. 8b shows the free body diagram and the forcediagram for a soil specimen, reinforced with horizontalinclusions, which has been sheared in a triaxial test.According to the pseudo-cohesion concept, reinforcing a
orced with horizontal inclusions at (b) failure by breakage of reinforcement
ARTICLE IN PRESSM.X. Zhang et al. / Geotextiles and Geomembranes 26 (2008) 1–138
soil with conventional horizontal reinforcements results inan additional confining pressure of Ds3 or an apparentcohesion of cr. Consequently, the relationship between theultimate vertical stress and confining pressure applied onthe specimen can be written as
s1 ¼ ðs3 þ Ds3ÞKp þ 2cffiffiffiffiffiffiKp
p. (4a)
On the other hand, the above equation can also beexpressed to the Rankine equation for a cohesive-frictionalsoil, that is,
s1 ¼ s3Kp þ 2ðcþ crÞffiffiffiffiffiffiKp
p. (4b)
The apparent cohesion is obtained as
cr ¼Ds32
ffiffiffiffiffiffiKp
p. (5a)
However, it is difficult to measure the additionalconfining pressure Ds3. The apparent cohesion cr atfailure can be evaluated based on the Mohr–Coulombfailure criterion (Schlosser and Long, 1974; Ling,2003) as
cr ¼RT
ffiffiffiffiffiffiKp
p2DH
, (5b)
where RT is the tensile strength of reinforcement per unitwidth and DH is the spacing between reinforcing layer,DH ¼ h/(n+1).
Substituting Eq. (5b) into Eq. (4b) results in thefollowing relationship:
s1 ¼ s3Kp þ 2cffiffiffiffiffiffiKp
pþ Kp
RT
DH. (6)
It should be noted that in a unit cell, non-uniformstresses and strains may prevail inside the specimen.Moreover, it is unlikely for all the reinforcements to beruptured simultaneously. As the total tensile strength ofreinforcements has been considered in Eq. (6), therefore, acorrection coefficient x has to be applied to the tensilestrength of reinforcements computed from Eq. (6). Basedon experimental results, the correction coefficient x canadopt a value in the range of 0.2–0.4. Hence, Eq. (6) can bewritten as
s1 ¼ s3Kp þ 2cffiffiffiffiffiffiKp
pþ xKp
RT
DH. (7)
5.2.2. Failure by pullout of reinforcement (frictional failure)
Fig. 8c shows a triaxial soil specimen failed due topullout of reinforcement. The force polygon is alsoshown in the figure. When frictional failure occurs, it isassumed that the horizontal reinforcements are pulled outfrom the upper and lower parts of the specimen on eitherside of the failure surface (shaded parts in Fig. 8c). Thevolume of the both parts of the cylindrical specimen can bedetermined as
V ¼ 2
Z p=2
�ðp=2ÞdyZ r
0
rdrZ r cos y tan a
0
dz ¼ 43r3 tan a. (8)
The total volume of horizontal inclusions pulled outfrom both the parts can be obtained as
VR ¼ VAnth
Ah¼ V
th
h=n� V
th
DH, (9)
where th is the thickness of horizontal inclusions.Then, the total area of horizontal inclusions pulled out is
o ¼VR
th¼
V
DH¼
4
3
r3
DHtan a. (10)
The resultant of the tensile forces acted on all reinforce-ments pulled out is
T ¼ 2s1of uZ ¼8
3ps1f uZ
Ar
DHtan a, (11)
where fu is the frictional coefficient between reinforcementand soil and Z is the correction for the coefficient of frictionbetween reinforcements and soil.From the force polygon (see in Fig. 8c), the following
equation can be given:
s1A tanða� jÞ ¼ s3A tan aþ 2cAþ8
3ps1f uZ
Ar
DHtan a.
(12a)
Eq. (12a) can then be written in the following form:
s1 ¼ s3Kp þ 2cffiffiffiffiffiffiKp
pþ
8
3ps1f uZKp
r
DH(12b)
or
s1 ¼ s3Kp
zþ 2c
ffiffiffiffiffiffiKp
pz
, (13)
where z ¼ 1� 83p Zf uKp
rDH
.The resultant of the shear and normal forces on the
failure surface for the specimen reinforced with horizontalinclusions can be expressed as
R cosða� jÞ ¼ s1A� cA tan a. (14)
Substituting Eq. (7) into Eq. (14), the force limitequilibrium condition for the specimen of soil reinforcedwith horizontal inclusions for failure by breakage ofreinforcement can be expressed as
R ¼ðs3Kp þ 2c
ffiffiffiffiffiffiKp
pþ xKpðRT=DHÞ � c tan aÞAcosða� jÞ
. (15)
Similarly, substituting Eq. (13) into Eq. (14), the forcelimit equilibrium condition for the specimen of soilreinforced with horizontal inclusions for failure by pulloutof reinforcement can be obtained as
R ¼ðs3Kp þ 2c
ffiffiffiffiffiffiKp
pÞð1=zÞA� cA tan a
cosða� jÞ. (16)
5.3. Strength model for soil reinforced with vertical elements
The effect of vertical reinforcements is assumed as anadditional confining pressure Ds3, which will increase the
ARTICLE IN PRESS
Fig. 10. Free-body diagram: (a) 3D view of section ABDE and
(b) additional confining pressure.
M.X. Zhang et al. / Geotextiles and Geomembranes 26 (2008) 1–13 9
strength of the soil. The relationship between theadditional confining pressure Ds3 and ultimate axialstress s1 was established based on the theory of limitequilibrium. Considering the vertical reinforcing effect,the additional confining pressure Ds3 will result in theadditional resultant (DRi) of the shear and normalforces on the failure surface at each reinforcing layerfor specimen reinforced with vertical reinforcement(see Fig. 9).
Considering the forces acting on the free body ABDE(see Fig. 10), DRi can be deduced from application of thetheory of limit equilibrium:
where n is the number of layers of vertical reinforcements(in the experiments presented in this paper, n ¼ 2); H is theheight of vertical reinforcements; b ¼ arcsin((z�r tan a)/r tan a); z is the vertical distance between bottom ofspecimen and calculated section.
Fig. 9. Forces on vertically reinforced specimen: (a) a specimen with stress
Considering the forces acting on the upper part of thespecimen above the failure surface (see Fig. 9), the verticalequilibrium equation can be written as
s1A ¼ RþXn
i¼1
DRi
!cos ða� jÞ þ cA tan a. (19)
es and forces acting on it and (b) free body diagram of section ABDE
ARTICLE IN PRESSM.X. Zhang et al. / Geotextiles and Geomembranes 26 (2008) 1–1310
5.3.1. Failure by breakage of vertical reinforcement
Fig. 11 shows the free body diagram for a verticalreinforcing element subjected to radial stresses (Ds3)and tensile stresses (st). Summing the components offorces that act on the element in the vertical directionyields
2sttvH ¼Z p
0
Ds3H sin y r0 dy; (20)
where tv is the thickness of vertical reinforcements.Integrating the term on the right-hand side of Eq. (20),
the following equation will be deduced:
2sttv ¼ 2Ds3 r0 (21)
or
Ds3 ¼sttvr0
. (22)
At failure by breakage of vertical reinforcement,tensile stress (st) will reach the tensile strength of thevertical reinforcement. Thus, Eq. (22) can then be re-written as
Ds3 ¼st½ �tvr0
, (23)
where [st] is the tensile strength of vertical reinforcements.Similarly, it is unlikely for all the vertical reinforcements
to be ruptured simultaneously, i.e., the total tensile strengthof reinforcements will not be fully mobilized at failure.Consequently, a correction coefficient of l (0olo1) has tobe applied to the tensile strength of vertical reinforcementscomputed from Eq. (23). The preceding equation can bewritten as
If the soil fails before the stress within vertical hoopreinforcement reaches its tensile strength, the additionalconfining pressure Ds3 on the soil due to the membranestresses will develop on lateral sides of the verticalreinforcements (see Fig. 11). The additional confiningpressure Ds3 was calculated from the rubber membranetheory (Rajagopal et al., 1999), which was originallydeveloped to correct for the effects of stiff rubbermembrane. The additional confining pressure due to themembrane stresses can be expressed as (Henkel andGilbert, 1952)
where r0 is the initial radius of vertical hoop reinforcement.ea and ec are the axial strain and the circumferentialstrain for the specimen at failure, respectively. M is thesecant modulus of the membrane of the hoop at axialstrain of ea.Summing Eq. (27) into Eq. (18) leads to:
To validate the developed analytical strength model, thetheoretical values obtained using Eq. (30) were compared
Proposed Test
model result
35 40 45
)
σ3=50kPa
σ3=100kPa
σ3=150kPa
σ3=200kPa
Proposed Test
model results
35 40 45
(%)
σ3=50kPa
σ3=100kPa
σ3=150kPa
σ3=200kPa
d the experimental results for H–V reinforced sand: (a) l ¼ 0.45, Z ¼ 0.60
)), Z ¼ 0.48.
ARTICLE IN PRESS
Table 2
The soil and reinforcement parameters of soil and inclusions
Parameter Value
Height of specimen, h (cm) 13.5
Radius of specimen, r (cm) 3.09
Radius of horizontal reinforcements, r1 (cm) 3.0
Radius of vertical reinforcements, r0 (cm) 2.0
Height of vertical reinforcements, H (cm) 0.5, 1.0, 2.0, 3.0, 4.0
Spacing between reinforcements, DH (cm) 4.5 (for vertical
reinforcements)
Thickness of horizontal reinforcements, th (cm) 0.012
Thickness of vertical reinforcements, tv (cm) 0.012
Tensile strength of vertical reinforcement [st](MPa)
200
Friction coefficient between reinforcement and
soil, fu
0.47
Cohesion of sand, c (kPa) 0.0
Angle of internal friction of sand, j (deg) 31.9
M.X. Zhang et al. / Geotextiles and Geomembranes 26 (2008) 1–1312
with the experimental results of specimens of granular soilreinforced with H–V orthogonal elements, as shown inFig. 12. The parameters were presented in Table 2.
It is shown that the results calculated from strengthmodels presented in this paper are in good agreement withthose of triaxial tests. However, in case of higher verticalreinforcement and at high confining pressure, it is foundthat when l and Z are constant and height of the verticalreinforcement is more than 2 cm, there is a deviator. Themain reason is that the assumption of uniformity distribu-tion of the stress within specimen and between soil andreinforcement is not valid at above condition. Moreover,the stress distribution and failure mechanism may becomevery complex. If Z ¼ 0.48 and l can be varied when thevertical reinforcing ratio parameter is more than 19.18%and confining pressure is more than 100 kPa, e.g.,
l ¼ 0:45� 1:0þ 0:85�s3 � 100
100
�
þ0:75�mv � 19:18%
19:18%
�, ð33Þ
the agreement between predictions and measurements willbe improved significantly.
7. Conclusions
In this paper, a new concept of soil reinforced with H–Vorthogonal reinforcements has been proposed. A compre-hensive set of triaxial tests were carried out on samples ofdry sand reinforced with H–V orthogonal and verticalelements. A strength model for soil reinforced with multi-layer horizontal–vertical orthogonal elements has beenproposed. The following conclusions are drawn from theresults:
(1) The experimental results in this paper show that theinclusion of H–V orthogonal reinforcing elements (espe-cially the double-sided H–V elements) leads to an increase
in the angle of internal friction of the soil as well as a slightincrease in the apparent cohesion.(2) The strength of sand reinforced with H–V orthogonal
elements increases with increasing height of the verticalreinforcements.(3) For sand reinforced with H–V orthogonal elements
with the same vertical height, double-sided H–V orthogo-nal elements will result in greater increase in strength thansingle-sided ones.(4) It is shown that the results of analytical predictions
by the proposed strength model for soil reinforced withH–V orthogonal elements are in good agreement withresults of triaxial tests.
Acknowledgment
The financial assistance from the National NaturalScience Foundation of China under Grant No. 50678100is herein much acknowledged.
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