dacem

aringerivee colufiguratationsymmresultlimitaby com

apacit

role in the prediction of the settlement. Since the stresses induced sis in which column material and initial soil are separately con-

Dow

nloa

ded

from

asc

elib

rary

.org

by

SCM

S Sc

hool

of E

ngin

eerin

g &

Tec

hnol

ogy

on 0

5/27

/15.

Cop

yrig

ht A

SCE.

For

per

sona

l use

onl

y; al

l rig

hts r

eser

ved.by the loaded foundation within the reinforced soil mass decreasewith depth, the length of columns should not be greater than thethreshold value beyond which the settlement of the soft soil be-comes negligible.

Brauns 1978 studied the bearing capacity of soft soils rein-forced by an isolated floating column, assumed to be a rigid pile,with purely frictional material. He concluded that the predicted

sidered for equilibrium and strength verifications. Meanwhile theformalism of limit analysis can be investigated using the homog-enization approach as studied recently by Jellali et al. 2005 and2007. The three-dimensional modeling used in this investigationtakes into account the effect of unit weight of the reinforced soilconstituents. A comparison with Brauns and Broms methods ispresented to verify the reliability of the results obtained from limitanalysis approaches.

Problem Statement

Consider a rigid foundation, with an arbitrary area denoted by S,resting on a soil reinforced by floating columns, as shown in Fig.1. All columns have circular cross sections with different diam-eters, and are located under the foundation in random arrange-ment see Fig. 2. A vertical surcharge is applied uniformlyaround the foundation. The native soil and the column materialare assumed to be homogeneous and isotropic. A perfect adhesionis assumed to exist along the soil-column interfaces. The geom-etry of the columns is defined by length Hc and total cross

1Professor, Dept. of Civil Engineering, Ecole Nationale dIngnieursde Tunis, BP 37, Le Belvdre 1002 Tunis, Tunisia. E-mail: mounir.bouassida@enit.rnu.tn

2Institut Suprieur des Technologies de lEnvironnement, delUrbanisme Et du Btiment, 02 Rue de l Artisanat Charguia II, Tunis,Tunisia. E-mail: belgacem.jellali@enit.rnu.tn

3Assistant Professor, Dept. of Civil Engineering, California StateUniv., Sacramento, Riverside Hall 4032, 6000 J St., Sacramento, CA95819-6029 corresponding author. E-mail: porbaha@gmail.com

Note. This manuscript was submitted on May 18, 2007; approved onNovember 11, 2008; published online on May 15, 2009. Discussion pe-riod open until November 1, 2009; separate discussions must be submit-ted for individual papers. This paper is part of the International Journalof Geomechanics, Vol. 9, No. 3, June 1, 2009. ASCE, ISSN 1532-3641/2009/3-89101/$25.00.

INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE / MAY/JUNE 2009 / 89Limit Analysis of Rigid FounMounir Bouassida1; Belga

Abstract: This study applies the limit analysis to estimate the befloating columns. The lower bounds of the bearing capacity are dteristics of the soil and the inclusion, the area replacement ratio, thThe established results are valid for a variety of reinforcing condistributed column arrangements under the foundation. The computhe limit analysis for reinforcement of purely cohesive soils. Forfactor is calculated by the Prandtls mechanism failure. Previousremain valid when considering floating columns. Nevertheless, acases. The reliability of the results of limit analysis is illustrated

DOI: 10.1061/ASCE1532-364120099:389

CE Database subject headings: Foundations; Load bearing c

Introduction

When deep foundation is unavoidable and the bearing layer isreasonably close to the ground surface, the rational approach is totransfer the load of the foundation to a bearing layer using avertical load carrying system such as piles, piers, or columns.

Estimation of bearing capacity for reinforcement of soils byend-bearing columns has been the subject of interest for a numberof investigators Hughes and Withers 1974; Barksdale and Ba-chus 1983; Barksdale 1987. Such a reinforcement scheme couldbe adopted if the rigidity of the weak soil layer is relatively lowcompared to the rigidity of the bearing layer. However, in a situ-ation where the soft soil deposit is thick, the practical approach isto install floating columns due to the absence of rigid stratumclose to the ground surface. In this reinforcement scheme, frompractical consideration, the length of columns plays an importantInt. J. Geomech. 200tions on Floating ColumnsJellali2; and Ali Porbaha3

capacity of a rigid foundation on a soil reinforced by a group ofd in terms of a dimensionless factor that depends on the charac-mns length, and a uniform surcharge surrounding the foundation.tions in terms of arbitrary foundation shape and any randomlyof the bearing capacity is extended to the kinematic approach of

etrical column arrangement the upper bound of bearing capacitys established for the reinforcement case by end-bearing columnstion on the length of columns is required for some reinforcement

parison with two other published methods.

y; Limit analysis; Soil stabilization; Floating structures; Columns.

maximum length of the column was not always compatible withpractical use. In addition, Broms 1982 presented the calculationof bearing capacity of soft soils mixed with floating lime andlime-cement columns.

Using the framework of limit analysis, Drucker and Prager1952, Chen 1975, and Salenon 1990 the bearing capacity offoundations on soils reinforced by columns resting on a rigidstratum was investigated Bouassida et al. 1995. Furthermore, forthe end-bearing group of reinforcing columns, the prediction ofbearing capacity estimation by limit analysis was verified usingsmall-scale models of soft soil improved by sand columnsBouassida 1996, and by soil cement columns Bouassida andPorbaha 2004. The objective of this study is to establish thelower and upper bounds of the bearing capacity of a rigid foun-dation resting on soils reinforced by a group of floating columnsbased on the static and kinematic direct approaches of limit analy-9.9:89-101.

QRigid foundation Q

Dow

nloa

ded

from

asc

elib

rary

.org

by

SCM

S Sc

hool

of E

ngin

eerin

g &

Tec

hnol

ogy

on 0

5/27

/15.

Cop

yrig

ht A

SCE.

For

per

sona

l use

onl

y; al

l rig

hts r

eser

ved.section Sc. The degree of reinforcement is characterized by areareplacement ratio defined as

=ScS

1

The bearing stratum is encountered at depth z=HHc. Thestrength of the reinforced soil is governed by Coulombs failurecriterion. When compressive stresses are counted positively, thiscriterion is written as

11 sin c 31 + sin c 2Cc cos c 0 2

where 1 and 3 denote the major and minor principal stresses,respectively; C and =cohesion and the friction angle of the na-tive soil; and Cc and c represent the cohesion and the frictionangle of the column material, respectively.

Consider the three-dimensional axes coordinates x ,y ,z andthe equilibrium equations are as follows:

ijxj

c = 0 3

where iji , j=x ,y ,z denote the components of the Cauchy stresstensor in xjx ,y ,z, frame; and and c denote unit weights ofnative soil and column material, respectively such as c. Theloading parameter Q that results from the distribution of verticalstress zz beneath the rigid foundation is calculated as follows:

Q =Szzds 4

In the framework of limit analysis, the goal is to compute theultimate bearing capacity which is defined as the exact value ofthe yielding load, denoted as Q*, when failure is reached withinthe reinforced soil mass. When performing the approach by the

Rigid stratum

Cc

c

cccHH

P P

HHH

X

C

C

Fig. 1. Modeling of reinforced soil

Circular columns (Ac)

Rigid foundation (A)

Fig. 2. Geometry of foundation and columns distribution in staticapproach90 / INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE / MAY/JUN

Int. J. Geomech. 200inside of the limit analysis, the statically admissible stress fieldscomplying with failure strength criteria are constructed. As a re-sult a lower bound Q of failure load Q* is derived. From Eq. 4Q should be maximized in compression to obtain the best lowerbound of bearing capacity.

The bearing capacity is characterized by the dimensionlessfactor called the bearing capacity factor BCF denoted byQ /CS. Therefore, using the approach by inside of limit analysisthe result can be written as: Q* /CS Q /CS.

Lower Bound Estimation of Bearing Capacity

Since the reinforced soil mass is subjected to a vertical load ap-plied by the rigid foundation, the shear stresses induced on thecontact surface i.e., between foundation and the reinforced soilmay be neglected. This simple approximation implies that x ,y ,zwill remain the principal directions with respective stresses x,y, z.

Consider the continuous piecewise stress field with four zonesshown in Fig. 3. The corresponding components of Cauchy stresstensors in x ,y ,z system coordinates are as follows

x = y = z + A1while vertical stresses are written

In zone 1

z = z + p 5

In zones 2, 3, and 4

z = cz + Am 6

m=2, 3, 4where A1, A2, A3, and A4=constant values. The continuity of

stress vector between zones 3 and 4 requires that

A3 = A4 c Hc 7

From Eqs. 1 and 46, the lower bound of BCF is calculatedas

QS

= 1 A2 + A3 8

From Eq. 8 since: 01, constants A2 and A3 should bemaximized positively to provide the best value of the Q solution

PP

HH

C(2)

(4)

CcccHH

C

C

c

c

(1)

(1)

(2)

(3)

C

C

Fig. 3. Stress field with four zonesE 2009

9.9:89-101.

to the problem. The constructed stress field complies with theCoulombs criterion in zones 1 and 2, respectively, by satisfy-

Kpc Kp 15b

Dow

nloa

ded

from

asc

elib

rary

.org

by

SCM

S Sc

hool

of E

ngin

eerin

g &

Tec

hnol

ogy

on 0

5/27

/15.

Cop

yrig

ht A

SCE.

For

per

sona

l use

onl

y; al

l rig

hts r

eser

ved.ing Eq. 9 and Ineq. 10, as follows:

A1 = pKp + Kp 1z + 2CKp 9

A1Ka + Ka 1z 2CKa A2 A1Kp + Kp 1z + 2CKp10

Substituting Eq. 9 in Ineq. 10 one obtains

A2 = pKp2 + Kp

2 1z + 2C1 + KpKp 11

In zones 4 and 3 the equilibrium condition satisfies the Cou-lombs criterion with the following conditions respectively:

p A4 pKp2 + Kp

2 1Hc + 2C1 + KpKp 12

pKpKac + KpKac cz + 2CKacKp kKac A3 pKpKpc + KpKpc cz + 2CKpcKp + kKpc

13

where

Kp and Kpc = tan24 + c2 = 1Kacin Ineqs. 14a and 14b are the coefficients of passive earthpressure of the native soil and the column material; k=cohesionfactor such that Cc=kC.

Substituting Eq. 7 in Ineqs. 12 two new inequations ofconstant A3 are obtained. Combining the new inequations withIneqs. 12, solutions for constant A3 are determined by discuss-ing the sign of the term KpKpcc which appears in Ineq. 13.From the calculations detailed in Appendix I a maximum lengthof columns is derived after the verification of strength criterion ofcolumns material, that is

Hc2C1 + KpKp KacKp + kKac + pKp2 KpKac

c Kp2 = Hmax

14a

where Hmax denotes the maximum length of columns when thefollowing condition is satisfied:

c Kp2 0 14b

The maximum length of the columns given by Ineq 14a doesnot depend on the area replacement ratio because the verifica-tions of strength criterion and continuity of the stress vector aredone locally.

To calculate the lower bound of BCF, two cases should bedistinguished through determination of constant A3 from Ineq.13, as described in the following two reinforcement cases.

First Reinforcement CaseThe first reinforcement case corresponds to

Kpcc

Kp15a

Ineqs. 14a and 15a are simultaneously satisfied whenINTERNATION

Int. J. Geomech. 200Ineq. 15b implies c which is the case of soft soils rein-forced by granular material or soil-cement columns, and com-pacted loose sand by vibrocompaction.

Consider, for instance, the reinforcement of a purely cohesivesoil =0 by a cohesive-frictional material Cc0;c0,without uniform surcharge p=0. Then, Eq. 14a gives

Hmax =2Cc

2 Kac + kKac 16

Ineq. 16 is only valid if c, which is a common case inpractice.

To illustrate a numerical example, consider the following data:C=15 kPa; =16.5 kN /m3; c=37; k=0; and c=18 kN /m3.From condition 14a it follows that 2.5 which are typicalcharacteristics of soft soils. The verification of condition 14afrom Eq. 16 leads to: Hc35 m. Fig. 4 illustrates a linear varia-tion of the maximum length of columns as a function of thecohesion factor using Eq. 16. The difference in unit weights ofthe column material and native soil c - plays an importantrole in predicting Hmax. The value of Hmax decreases significantlywhen c increases.

Nevertheless, if the cohesion of column material increases forinstance from 0 to 10 kPa, the predicted maximum length of thecolumns is not significantly affected, even when the friction angleof the columns increases see Fig. 5. The predicted maximumlength of the columns is in accordance with those practiced in thefield, indicating the usefulness of the limit analysis approach.

The derivation presented in Appendix I demonstrates that thelower bound of BCF is determined when A3 is as follows:

A3 = pKpKpc + 2CKpcKp + kKpc 17

The determination of A1, A2, A3, and A4 from Eqs. 10, 11, 17,and 7, respectively, satisfies the Coulombs strength criterionand the constructed stress field in each point of the reinforcedsoil. Substituting Eqs. 11 and 17 in Eq. 8, the lower bound ofBCF is calculated as

QCS

= 21 1 + KpKp + k + KpKpcKpc

+pC

1 Kp2 + KpKpc 18

From Eq. 18 the lower bound of BCF does not depend on unitweights. Eq. 18 demonstrates that the lower bound of BCF ap-proaches zero for loose sands C=0 without vertical surchargep=0. In this case, the reinforced soil only generates potentialresistance for positive vertical surcharge p0.

Consider a purely frictional column material Cc=0, thelower bound of BCF is calculated as

QpS

= 1 Kp2 + KpKpc 19AL JOURNAL OF GEOMECHANICS ASCE / MAY/JUNE 2009 / 91

9.9:89-101.

Dow

nloa

ded

from

asc

elib

rary

.org

by

SCM

S Sc

hool

of E

ngin

eerin

g &

Tec

hnol

ogy

on 0

5/27

/15.

Cop

yrig

ht A

SCE.

For

per

sona

l use

onl

y; al

l rig

hts r

eser

ved.To illustrate by numerical example, consider the following data:=0.15, =30, c=45, =18 kN /m3, c=21 kN /m3, and C=0 which comply with condition 16a but not with condition14b. It follows that the maximum length of columns from Eq.14a does not intervene. In other words, the length of the col-umns is arbitrary in the case of columnar reinforcement of purelyfrictional media.

Fig. 4. Variation of maximum

Fig. 5. Limit analysis estimation of m92 / INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE / MAY/JUN

Int. J. Geomech. 200Second Reinforcement Case

The second reinforcement case corresponds to

Kpcc

Kp20

n length with cohesion factor

m column length versus friction angleaximucolumE 2009

9.9:89-101.

t on p

Dow

nloa

ded

from

asc

elib

rary

.org

by

SCM

S Sc

hool

of E

ngin

eerin

g &

Tec

hnol

ogy

on 0

5/27

/15.

Cop

yrig

ht A

SCE.

For

per

sona

l use

onl

y; al

l rig

hts r

eser

ved.The detailed derivation, presented in Appendix I, demonstratesthat Q is maximized by setting

A3 = pKpKpc + KpKpc cHc + 2CKpcKp + kKpc 21Substituting Eqs. 21 and 11 into Eq. 8, the lower bound ofBCF is

QCS

= 21 1 + KpKp + k + KpKpcKpc

+pC

1 Kp2 + KpKpc c KpKpc

HcC

22

Consider the reinforcement of a purely cohesive soil without ver-tical surcharge Kp=1; p=0. Then, condition 20 requires: 1Kpc c /. Since the c / ratio ranges between 1 and 2,then it follows c19.5. Then, the resulting lower bound ofBCF is simplified as

QCS

= 41 + 2Kpc + kKpc c KpcHcC

23

Calculation of BCF from Eq. 23 is feasible, if the length ofcolumns remains smaller than the maximum value given by Eq.16. Nevertheless, considering the practical values of c and ,this case is generally devoted to the reinforcement of purely co-hesive media CcC and Kac=1. Eq. 16 can be written

Hmax =2C1 + kc

24

The lower bound of BCF is simplified as

Fig. 6. Influence of unit weighINTERNATION

Int. J. Geomech. 200QCS

= 4 + 2k 1 +pC

c HcC

25

In the case =c from Eq. 25 the lower bound of BCF is inde-pendent of unit weights of reinforced soil. In the case c thelower bound of BCF decreases insignificantly Fig. 6.

The maximum length of the column given by Eq. 14a wasderived because unit weights, for initial soil and column material,were assumed to be different. As an interpretation the maximumlength of the column resulted as a required condition on verticalstress related to the weight of the soil. In order to demonstrate thephysical meaning of this result, consider a reinforcement schemerelevant to substitution of the native soil with unit weight bya column material with unit weight c. Consequently, inaddition to the loading parameter Q, the presence of a column inthe native soil plays the role of a second loading parameter, whichdecreases the lower bound value of BCF. Even if this decrease inlower bound of BCF is not significant, the maximum length of thecolumn is the main finding of the lower bound approach by theinside of limit analysis. Such a result was attained because differ-ent unit weights for initial soil and column material were takeninto account.

Comparison between Two Reinforcing Alternatives

Consider the reinforcement of a purely cohesive soil as soft claywith two alternative column materials: either purely cohesive orpurely frictional material. The lower bound BCF predicted bylimit analysis using Eqs. 18 and 25 predicts the reinforcementusing purely cohesive materials is preferable from k=7 as shownin Fig. 7 compared to a column constituted with cohesive fric-tional material. For k=20, the lower bound of BCF is 1.42 times

redicted lower bound of BCFAL JOURNAL OF GEOMECHANICS ASCE / MAY/JUNE 2009 / 93

9.9:89-101.

Pp Bz(a)

Dow

nloa

ded

from

asc

elib

rary

.org

by

SCM

S Sc

hool

of E

ngin

eerin

g &

Tec

hnol

ogy

on 0

5/27

/15.

Cop

yrig

ht A

SCE.

For

per

sona

l use

onl

y; al

l rig

hts r

eser

ved.which corresponds to the reinforcement by a purely frictionalcolumn material. This observation demonstrates better efficiencyof soil improvement using deep mixing techniques. Evidently,such an observation needs to be verified by experimental work.

Lower bounds given by Eqs. 19 and 25 are identical tothose established for soils reinforced by end-bearing columnsBouassida and Hadhri 1998. Meanwhile, when designing float-ing columns the maximum length of the column given by Eq.14a should be considered.

The lower bound of BCF suggested here is based on the use ofa direct approach where the ambient soil and columns materialwere treated separately. Using such a framework by looking for asophisticated lower bound estimate using inclined stress legproposed earlier by Chen 1975 and Davis and Selvadurai 2002will be difficult to handle. However, such sophisticated stressfields have been considered recently by using homogenizationframework Jellali et al. 2007 which is not an aim of the presentwork.

In the next section the kinematic approach of limit analysis isdiscussed to establish the upper bounds of BCF in the case of softsoils reinforced by purely cohesive columns material. The situa-tion considered here is for a group of floating cylindrical columnslocated symmetrically under a rigid rectangular foundation.

Upper Bound Estimation of Bearing Capacityof Soils Reinforced with Purely Cohesive Material

A foundation model on soft soil reinforced by purely cohesivecolumns is shown in Figs. 8a and b. The Prandtls failuremechanism is used to derive an upper bound estimation of BCFfor this model foundation.

Hypotheses and Boundary ConditionsThe materials of the improved ground i.e., soft clay and thecolumns are assumed to be a homogeneous isotropic continuum.The strength is governed by Trescas criterion for purely cohesivematerials when =c=0 in Ineq. 2. Such a hypothesis is wellrepresentative for soft clays, and it is also recommended for softclays mixed with cement particularly in the undrained condition

Fig. 7. Prediction of BCF using two alternatives for soft soil rein-forcement94 / INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE / MAY/JUN

Int. J. Geomech. 200Broms 1982. Indeed, when the percentage of the binder is lowsay less than 12% the frictional strength component of the ce-mented clay is insignificant.

Fig. 8a shows the rectangular rigid foundation with area S=BL on reinforced ground by floating columns in symmetricallocations. The geometry of the columns is characterized by theirlength Hc and radius Rc. The total number of columns N locatedsymmetrically under the rigid foundation is

N = Naxis + Nlat 26

where Naxis=number of columns located along the axis x=0; andNlat=number of columns located laterally

x = B2 B1with respect to the axis of the rigid foundation.

A vertical surcharge p is also acting at the top of the reinforcedground. The boundary conditions are as follows Figs. 8a andb:

x,y S v = Uez U 0 27

x,y S T = pez p 0 28

where U=constant velocity displacement counted positivelydownward induced by the rigid foundation at the top of the re-inforced ground.

The work of external forces is calculated from Eq. 51 seeAppendix II taking into account the conditions in Eqs. 27 and28. In addition, assuming a smooth contact between the softclay and side walls of the rigid box

x

B0

H

Hc N

(3)(2)(2)

N

MMPP

O

(3) (1)

B1

L

By

x

B1

2Rc

(b)

Fig. 8. a Model test and Prandtls mechanism failure; b geometryand columns location under rigid foundationE 2009

9.9:89-101.

B L 2,N=3(a)

Dow

nloa

ded

from

asc

elib

rary

.org

by

SCM

S Sc

hool

of E

ngin

eerin

g &

Tec

hnol

ogy

on 0

5/27

/15.

Cop

yrig

ht A

SCE.

For

per

sona

l use

onl

y; al

l rig

hts r

eser

ved.x = 2 and y = 2and the condition of null horizontal displacements on lateral bor-ders, the contribution of external forces will be zero on this partof the model ground. For practical purposes, it will be assumedthat =c because treatment of the soft clay by a small percent-age in weight lower than 15% of the binder, such as lime orcement, will not have a significant effect on the unit weight of theimproved soil Bouassida and Porbaha 2004. This assumption isalso adopted because the exact value of BCF does not depend onunit weights for purely cohesive reinforcing material Bouassidaand Hadhri 1996. Therefore, the external work is written

Wextv = QU pUS 29The kinematic approach of limit analysis is performed by con-structing the kinematically admissible velocity fields that lead toan upper bound estimate Q+ of Q*. In terms of the BCF, it iswritten that: Q+ /CS Q* /CS. From the kinematic theorembriefly presented in Appendix II the best upper bound Q+ /CSis obtained after a minimization procedure, as detailed bySalenon 1990.

Upper Bound Approach Using Prandtls FailureMechanismFig. 8a shows the Prandtls failure mechanism for the improvedground. Block OMM is virtually constrained to a rigid motionat velocity U to satisfy Eq. 27. This mechanism induces defor-mations in zones OMN and OMN and tangent discontinui-ties of velocity along surfaces OM, ON, NP, and theirsymmetric corresponding sections, i.e., OM, ON, andNP. Arcs ON and ON should be circular to fulfill therequirement for Trescas material and to ensure a finite maximumresisting power, as presented in Eq. 52a in Appendix II.

The Prandtls failure mechanism is kinematically admissible,if the following conditions are fulfilled see Fig. 8a:

HcB

2 cos 30a

Bsin 2

B B0

230b

where B0 denotes the width of the rigid box containing the rein-forced soil model. Using the hodograph construction Fig. 9a,discontinuities of velocity are determined as discussed in Appen-dix III.

As shown in Fig. 9b, the ON and ON surfaces that in-tercept the column axes are assumed to be planer and orthogonalto the MA radius. Therefore, the area of ON and ON sur-faces denoted by c, with inclination with respect to horizontal,is calculated as

c = Nlat Rc2cos

= Scos

30c

where angle is defined by see Fig. 9b

tan =a1 cos

1 a12 cos2 31a

withINTERNATION

Int. J. Geomech. 200a1 =2B1B

31b

Calculation of the maximum resisting work from Eq. 50 in Ap-pendix III gives

Pv = P1v + P2v + P3v 32

The upper bound estimation of BCF is obtained by applying thekinematic theorem in Ineq. 48 and taking Eqs. 29 and 32 intoconsideration as follows:

Q*CS

Q+CS

= k 1tan + NaxisN 1 + 4F1 cos2 + NlatN cos cos + F2 cos2 + G + pC + cotg g + + tan 33

where expressions of F1 and F2 are given in Appendix III. Thefunction G is then defined as

2,O

U

1

0

Assumed planar sections on (ON) Arch

N

B/2

2Rc

O

M

B1

O

A

(b)

Fig. 9. a Hodograph construction for Prandtls mechanism failure;b details of surfaces of discontinuities of velocity intercepting col-umnsAL JOURNAL OF GEOMECHANICS ASCE / MAY/JUNE 2009 / 95

9.9:89-101.

of upper bound BCF prediction when compared to Prandtls

ft Clay

Dow

nloa

ded

from

asc

elib

rary

.org

by

SCM

S Sc

hool

of E

ngin

eerin

g &

Tec

hnol

ogy

on 0

5/27

/15.

Cop

yrig

ht A

SCE.

For

per

sona

l use

onl

y; al

l rig

hts r

eser

ved.G = 18 sin 2 sin + 8 sin cos2 BL 34The upper bound values of BCF are determined by minimizingthe right side of Ineq. 33 with respect to angle to fulfill theconditions in Eqs. 30a and 30b. Fig. 10 shows the variation ofthe upper bound of BCF as a function of the cohesion ratio andthe area replacement ratio use of high values of k.

Effect of Column LengthThe upper bound of BCF derived from Ineq. 33 does not dependon length of columns i.e., is equal to that of end-bearing columntype. If condition 31a is not verified, the upper bound of BCFof floating columns is lower than that obtained in the case ofend-bearing columns.

Consider three foundation models, as described in Figs. 8aand b, whose mechanical characteristics are summarized in Table1. No surcharge load exists at the top of the reinforced soil p=0.

Geometrical parameters are: B=9 cm; B0=50 cm; H=17.5 cm; L=20 cm. Calculation of minimized upper bound BCFvalues from Ineq. 33 are presented in Table 1 with predictedupper bounds BCF derived by a mechanism failure of five blocksdetailed in Bouassida and Porbaha 2004. Table 1 also includesestimation of lower bounds BCF from Eq. 25 by assuming equalunit weights for soft clay and columns material.

Comparing predictions in Table 1 it can be noted that themechanism failure of five blocks provides a better estimation, ofabout 10% as the mean value, of predicted upper bounds by thePrandtls mechanism failure. Because the five block mechanismincluded two kinematic parameters it gives a light improvement

Fig. 10. Comparison of different limit analysis approaches to esti-mate BCF

Table 1. BCF Predicted by Limit Analysis for Model Foundations of SoModel I

Mechanical characteristics =0.149; N=9k=21.5

Prandtl mechanism 12.97Optimized angle 39.97Lower bound 10.11Five-block mechanism 11.8896 / INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE / MAY/JUN

Int. J. Geomech. 200mechanism investigated here including only one kinematic pa-rameter that is angle . Related to optimized values of angle from Table 1 it can be noted that for representative values ofcohesion ratio k, we have /4. Further, efficiency of the fiveblock mechanism has been confirmed by Kasama et al. 2006 inthe case of the deep mixing columns technique after numericallimit analysis computation. Indeed, the three numerically pre-dicted edge modes of failure are close to the five block mecha-nism proposed here and the analytical solution of Bouassida andPorbaha 2004 is very close to a numerical upper bound solutionobtained by Kasama et al. 2006 and prior experimental recordson scaled test models for end-bearing columns Porbaha andBouassida 2004. Kasama et al. 2006 concluded that the agree-ment between analytical and numerical methods was obtained forall ranges of cohesion and area improvement ratios.

For practical purposes, the Prandtls mechanism will be suit-able with the columnar reinforcement scheme in which both co-hesion and substitution factors are limited in prefixed ranges, e.g.,40% and k40. This permits avoid once of high dispropo-tioned rigidities between reinforced soil mass and lower soft claylayer. In such a case 40% or k40 the Prandtls failuremechanism will not be realistic and the composite ground will becloser to a floating pile foundation model. In this case the failuremechanism to be considered is that of the transferred load at thebase of reinforced soil mass as suggested by Broms 1982.

Combining Ineq. 30a and Eq. 24 one obtains

Hmin =B2 Hc

2C1 + kc

= Hmax 35

The minimum length of the columns Hmin is solely related to thegeometry of the foundation, whereas the maximum length Hmax ofthe columns is related to the mechanical parameters of the rein-forced soil. The maximum length of the column is the depth tonot exceed when installing columns in a soft clay layer in whichthe stratum is located at depth HHc. The minimum length ofthe column is the smallest depth along which column installationshould be done. For practical purposes the minimum length of thecolumn might be adequate to determine the bearing capacity ofthe reinforced soil.

Table 1 compares the lower and upper bounds of limit analysisin terms of relative differences. As shown, the exact theoreticalvalue of BCF is determined with an accuracy of about 12.417.5%. Such an encouraging result should be confirmed throughexperimental investigations. Meanwhile, due to very few experi-ments carried out, especially for a rigid foundation resting onreinforced floating columnar foundations, a comparison is onlyfeasible between analytical and numerical results.

s Reinforced by Floating Cement Columns

Model II Model III

=0.192; N=11k=19.7

=0.244; N=14k=19.2

14.12 15.9839.5 40.7111.18 12.8912.58 14.68E 2009

9.9:89-101.

Hmin =21 + k

39Occ

Dow

nloa

ded

from

asc

elib

rary

.org

by

SCM

S Sc

hool

of E

ngin

eerin

g &

Tec

hnol

ogy

on 0

5/27

/15.

Cop

yrig

ht A

SCE.

For

per

sona

l use

onl

y; al

l rig

hts r

eser

ved.Comparison with Existing Methods

In this section the ultimate bearing capacity of a soft clay rein-forced by floating columns by limit analysis is compared withmethods discussed by Brauns 1978 and Broms 1982. Brauns1978 presented the model of an isolated column, whereasBroms 1982 discussed a method assuming that the columns actas piles with different unit weights for the soft soil and the rein-forcing material. These methods are briefly discussed here.

Brauns Approach 1978Consider the reinforcement of a purely cohesive soft soil by anisolated floating column in which the reinforcing material is as-sumed to be purely frictional. The column is assumed to be a rigidpile. Such a hypothesis appears to be unrealistic when the rein-forcing element is a column, particularly for the installation pro-cess of soil improvement techniques such as stone columns orvibrocompated columns where the contact soil-column is perfecttotal adhesion. From the limit equilibrium of vertical forces, thevertical stress within the column is as shown in Fig. 11

vz = c + c 2CRc z 36where c=ultimate vertical stress which is assumed to be actinguniformly at the top of the column with radius Rc. By setting:vz=0, from Eq. 36 the minimum length of the column i.e.,the depth along which the reinforcement is needed is obtained asfollows:

Hmin =c

2CRc

c

37

Depending on the strength characteristics of the column materialto be either purely frictional Cc=0, or purely cohesive c=0,the ultimate vertical stress acting at the top of the column is: c=2CKpc, or c=2C1+k. These values are substituted in Eq. 37

Hmin =2Kpc

2Rc

c

C

38

z

cc22RR

Column

mmiinnHH

Rigid Stratum

c

c

Soft clay

CC

Fig. 11. Equilibrium of floating column assumed as rigid pile byBrauns 1978INTERNATION

Int. J. Geomech. 2002Rc

c

C

Consider the reinforcement by floating columns of a soft claywith the following properties: C=15.7 kPa; =16 kN /m3:1. If the reinforcement is by mixing the soft soil with 12% of

cement, the properties of the column material Bouassida andPorbaha 2004 are: c=21 kN /m3; Cc=292 kPa; and c=0;Based on the limit analysis approach using Eq. 24 themaximum length is estimated to be Hmax=153 m! Using theBrauns method with Rc=0.5 m using Eq. 39, the mini-mum length is predicted to be Hmin=13.6 m. For this case itappears that a reasonable prediction is attained by theBrauns approach compared to the limit analysis. Meanwhileit can also be said that the limit analysis estimation of maxi-mum length of columns well applies if the stratum depth ishigh more than 60 m; and

2. Consider the reinforcement of the same soft clay by a purelyfrictional column material having characteristics: c=20 kN /m3; c=42.

From Eq. 16 the maximum length predicted by limit analysis isHmax=14.2 m. The prediction obtained by Brauns method usingEq. 39 leads to: Hmin=3.7 m! This result is mainly due to thehypothesis assuming the column to be a rigid pile. Such an as-sumption is applicable only for soft clay reinforced by a treatmentmethod with a binder such as cement or lime. In this situation thecohesion of the column material is greater than 20 times that ofthe soft clay k20.

Broms Approach 1982The bearing capacity of a soft clay reinforced by floating columnsunder a rigid foundation was investigated by Broms 1982. Themode of failure proposed for floating piles is adopted in thismethod. Such a mechanism is apparently more justified when areaimprovement ratio is high, like for the deep mixing treatment, andcemented columns have very high cohesion by 100 times thecohesion of soft clay. In this reinforcement scheme the Prandtlsmechanism considered for scaled test models is rather not prefer-able.

The ultimate bearing capacity of a rigid foundation with lengthL and width B, is the sum of: 1 the soft clay resistance at the tipof the columns that is usually estimated by 6CBL9CBL; and2 the lateral skin developed within the soft soil on the lateralparallelepipedic surface surrounding the edges of the foundationalong the length of columns 2CHcB+L.

Such a hypothesis, when the reinforcing element is a column,appears to be unrealistic, particularly for the installation processof soil improvement techniques such as stone columns or vi-brocompated columns where the contact soil-column is perfecttotal adhesion. The minimum ultimate bearing capacity is

QCS = 6 + 2Hc 1B + 1L 40From Eq. 40, it can be noted that the BCF predicted by theBroms approach does not depend on the properties of columnmaterial and improvement area ratio. These are the limitation ofthe Broms approach. By setting p=0 in Eq. 25 and equaling itwith Eq. 40 one obtains the cohesion ratio from which the BCFwill be definitely underestimated by Broms approach regardingthe lower bound prediction of limit analysisAL JOURNAL OF GEOMECHANICS ASCE / MAY/JUNE 2009 / 97

9.9:89-101.

k = 1 +1 1 + Hc + Hc + c Hc 41 c Kp 42a

Dow

nloa

ded

from

asc

elib

rary

.org

by

SCM

S Sc

hool

of E

ngin

eerin

g &

Tec

hnol

ogy

on 0

5/27

/15.

Cop

yrig

ht A

SCE.

For

per

sona

l use

onl

y; al

l rig

hts r

eser

ved. L B 2C

Consider a case example with the following data: =0.16, Hc=10 m, H=20 m; =16 kN /m3; c=17.5 kN /m3; C=15 kPa; B=25 m; and L=35 m. Substituting these values in Eq. 41 willlead to k=12. Then, the applicability of the Broms approachseems acceptable only for restricted margin of values of k. In fact,the relative strength of the soil cement columns to that of the softsoil commonly used ranges from k=15 to 40. This result impliesthat the bearing capacity prediction using the Broms approach isunderestimated when k10 Bouassida and Porbaha 2004. Thisis contradictory to the consideration of the mechanism used byBroms which failure is assumed to take place only along par-allepepidic surface in soft soil surrounding the reinforced soilmass, especially when cohesion and area replacement ratios arehigh. Consequently, it is concluded that Prandtls failure mecha-nism suggested in this paper is more efficient than Broms mecha-nism for estimating the bearing capacity of floating columnarreinforced soil.

Conclusions

This study presents the estimation of a bearing capacity for foun-dations resting on soils reinforced by floating columns. Using thelimit analysis the lower bounds of the bearing capacity factor arederived for all reinforcement schemes. The bearing capacity re-sults are independent of the distribution of the columns and theshape of the foundation.

Predicted lower bounds of BCF for a cohesive-frictional soilreinforced by floating columns with cohesive-frictional materialrequires that the unit weight of the column material be greaterthan that of the native soil. It was found that the lower bounds ofBCF decrease insignificantly with unit weight when c19.5.

Regarding reinforcement by end-bearing columns, the lowerbound values of bearing capacity remain unchanged for floatingcolumns. Nevertheless, a maximum length of columns is requireddepending on the characteristics of the reinforced soil.

When dealing with soft soil reinforced with granular columnsand soil cement columns the prediction of the maximum length ofcolumns is determined from the limit analysis approach andBrauns method, respectively.

In this study new bounds of bearing capacity and columnlength are presented for practical purposes. Estimation of bearingcapacity by the limit analysis demonstrates a more efficient ap-proach compared with the existing methods, as evidenced by thepresented design charts.

Acknowledgments

The writers appreciate the comments by the anonymous reviewersthat enhanced the quality of this presentation. Financial supportby a U.S. Fulbright program for the first writer is gratefully ac-knowledged.

Appendix I: Calculation of BCFs Lower Bounds

Consider the reinforcement case KpKpcc0. Since Kpc1,from condition 15a we have98 / INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE / MAY/JUN

Int. J. Geomech. 200From condition 42a it can also be written

KpKac c 0 42b

Taking into account conditions 15a and 42b, Ineqs. 13 arevalid by setting z=0, as follows:

pKpKac + 2CKacKp kKac A3 pKpKpc + 2CKpcKp+ kKpc 43

From Ineq. 43 there is always a solution for constant A3. Sub-stituting Eq. 7 into Ineqs. 43, a new bounding is obtained

pKpKac + 2CKacKp kKac + c Hc A4 pKpKpc+ 2CKpcKp + kKpc + c Hc 44

Solutions for constant A4 are derived if Ineqs. 12 and 44 aresimultaneously fulfilled from which Eq. 14a is therefore ob-tained. From Ineqs. 43 the maximum value of constant A3 isgiven by Eq. 17.

Consider the reinforcement case: KpKpcc0. SinceKac1, from condition 20 it can be written

KpKac c 0 45

Taking into account condition 20, Ineq. 13 are always valid bysetting: z=Hc in the right inequation, and z=0 in the left one, itfollows that

pKpKac + 2CKacKp kKac A3 pKpKpc + KpKpc cHc+ 2CKpcKp + kKpc 46

Substituting condition 7 in Ineq. 46, it follows that

pKpKac + 2CKacKp kKac + c Hc A4 pKpKpc+ KpKpc 1Hc + 2CKpcKp + kKpc 47

Possible solutions only exist for constant A4, if boundings Eqs.12 and 47 are simultaneously verified. Such a system leads toEq. 14a. Therefore, Ineq. 46 being satisfied, the constant A3will be determined from Eq. 21.

Appendix II: Kinematic Theorem of Limit Analysis

An upper bound of the extreme load is obtained by applying thekinematic theorem that states

Wextv Pv 48

where Wextv denotes the work of external forces in a givenvelocity field v, which is expressed as Salenon 1990

Wextv =

f . vd +

T . vdS 49

where =domain of the improved ground; =its boundary;dS=elementary surface; f=vector of volumetric forces; and T=stress vector. Pv in Eq. 48 =maximum resisting work in-volved, in the constructed velocity field v, which is defined asSalenon 1990E 2009

9.9:89-101.

Pv = d= d + n,vd 50 dr = dr = U ; otherwise dij = 0 56

Dow

nloa

ded

from

asc

elib

rary

.org

by

SCM

S Sc

hool

of E

ngin

eerin

g &

Tec

hnol

ogy

on 0

5/27

/15.

Cop

yrig

ht A

SCE.

For

per

sona

l use

onl

y; al

l rig

hts r

eser

ved.

where v=velocity jump across the velocity discontinuity sur-faces in the domain ; and d= =strain rate tensor defined by

dij =12 vixj + v jxi 51

Starting from yield criterion expression, functions represent themaximum resisting energy in the kinematically admissible veloc-ity field constructed Salenon 1990.

Maximum Resisting PowerFor Trescas material functions introduced in Eq. 50 are de-fined by

n,v = Cv if v . n = 0 52a

d= = Cd1 + d2 + d3 if d1 + d2 + d3 = 0 52bdi denotes the eigenvalue i=13 of the strain rate tensor definedby Eq. 51. vij denotes the discontinuity of velocity field onsurface separating two blocks or zones numbered i and j.The condition 52a traduces that only tangent discontinuities ofvelocity are allowed on within Trescas material.

Appendix III: Upper Bound of BCF by KinematicApproach

For this mechanism the velocity field in zones 2 and 2 isdefined by

v = U cos e 53a

From the hodograph construction Fig. 9b the velocity field inblocks 3 and 3 is written

v = U cos2 ex + U cos sin ez 53b

Modulus of discontinuities of velocities and the surfaces wherethey occur are given, respectively, by

v12 = v12 =U

2 sin 54a

OM = OM =B

2 cos 54b

v02 = v02 =U

2 sin 55a

ON = PN =B

2 sin 55b

ON = ON =

4B

2 cos 55c

Term of Deformation in Zones 2 and 2Using Eqs. 53a and 51 the strain rate components are writtenasINTERNATION

Int. J. Geomech. 2004r sin

The eignenvalues of tensor d verifying conditions 52b are writ-ten as

d1 = d2 =U

4r sin ; d3 = 0 57

The first term of maximum resisting power in Eq. 51 corre-sponding to deformation in zones 2 and 2 that is calculatedfrom Eqs. 52b and 57 is written as

P1v =2&2

d= d 58

In the case where the soil is unreinforced in zones 2 and 2i.e., k=1 the calculation of maximum resisting work leads to

P1v =

2CSU

sin 259a

The volume of zones 2 and 2 being

V2&2 =BS

8 cos2 59b

From Eqs. 59a and 59b we get

P1v = CU2 cot

BV2&2 59c

From Eqs. 58 and 59a59c the maximum resisting powerdue to deformation in zones 2 and 2 of the reinforced soil is

P1v = CU2 cot

BV2&2 V2&2

c + kV2&2c

60a

where V2&2c

represents the columns volume in zones 2 and2 is given by

V2&2c

=

SBN

F1Naxis + F2Nlat 60b

where N, Naxis, and Nlat are defined by Eq. 28. We also have

F1 =23

.

RcB

. tan 61a

F2 =1

4 cos . cos a1 . sin 61b

Then

P1v = CUS 2 sin 2 + 4k 1F1NaxisN + F2NlatN Cot g62

Maximum Resisting Power due to Discontinuitiesof VelocitiesMaximum resisting power due to discontinuities of velocitiescomprises two terms; we haveAL JOURNAL OF GEOMECHANICS ASCE / MAY/JUNE 2009 / 99

9.9:89-101.

n;v = P v + P v 63d= strain rate tensor;

e ; e ; e unit vectors;

Dow

nloa

ded

from

asc

elib

rary

.org

by

SCM

S Sc

hool

of E

ngin

eerin

g &

Tec

hnol

ogy

on 0

5/27

/15.

Cop

yrig

ht A

SCE.

For

per

sona

l use

onl

y; al

l rig

hts r

eser

ved.

2 3

This second contribution of discontinuities of velocity is locatedwithin the reinforced soil model, i.e., OM, ON, NP, andtheir symmetric parts. From Eqs. 52a, 54a, 54b, and 55a55c calculation gives

P2v = CUS2 + tan + cotg g + k 1

tan + cotg gNaxisN + cos cos NlatN 64

The third contribution of maximum resisting power is due to dis-continuities of velocity occurring on lateral surfaces of blocks 1,3, and zone 2 located on sides y= L /2.

Respective discontinuities of velocity on these surfaces areidentical to those given by Eqs. 54a for blocks 3 and 3 andzones 2 and 2, while for block 1 we have v01=U. Modu-lus of surfaces where these discontinuities of velocity occur are asfollows: Surfaces OMM

S1 =B2 tan

265a

Surfaces OMN and OMN

S2 =B2

4 cos2 65b

Surfaces PMN and PMN

S3 =B2

4 sin 265c

Using Eqs 52a, 53a, 53b, and 65a65c the third contri-bution of maximum resisting power is

P3v = CUS tan 2 + 18 sin sin 2 + 8 sin cos2 BL66

The upper bound of BCF expression given by Ineq. 33 is ob-tained after derivation of the total maximum resisting power asthe sum of terms given by Eqs. 62, 64, and 66, and aftersubstituting Eq. 29 in Ineq. 48. It can be noted that Eq. 66 isindependent of all column parameters number, locations, andcharacteristics.

For the case of models with Naxis=0, the calculation of maxi-mum resisting work Eq. 50 gives

Pv = CUS + 1 + tan + cotg + k 1

tan + cos

cos + a1 cos

2 1tan

tan 67

Notation

The following symbols are used in this paper:Ai i=14 constants;

B width of foundation;C cohesion of native soil;

Cc cohesion of column material;100 / INTERNATIONAL JOURNAL OF GEOMECHANICS ASCE / MAY/JUN

Int. J. Geomech. 200x y zF1; F2 parametric functions

H stratum depth;Hmaxc maximum column length;Hminc minimum column length;

Ka coefficient of active earth pressure of nativesoil;

Kac coefficient of active earth pressure of columnmaterial;

Kp coefficient of passive earth pressure of nativesoil;

Kpc coefficient of passive earth pressure ofcolumn material;

k cohesion factor;L length of foundation;N total number of columns;

Naxis number of columns located at foundationsaxis;

Nlat number of columns located at symmetricsides;

p uniform surcharge;Q load exerted by foundation;

Q lower bound of Q;Q+ upper bound of Q;Q* extreme load;Rc column radius;S total area of rigid foundation;

Sc total area of columns;T stress vector;v velocity field vector;

vij discontinuity of velocity vector betweenadjacent zones or blocks i and j;

x ,y ,z three-dimensional variables; unit weight of native soil;c unit weight of column material; area replacement ratio; set of surfaces where discontinuities of

velocity occur;c crossed columns area by surface ;c vertical stress acting at head of column;ij components of Cauchy stress tensor;1 major principal stress;3 minor principal stress; friction angle of native soil; andc friction angle of column material.

References

Barksdale, R. D. 1987. Applications of the state of the art of stonecolumnsLiquefaction, local bearing and example calculations.Technical Rep. No. REMR GT-7, U.S. Corps of Engineers, Washing-ton, D.C.

Barksdale, R. D., and Bachus, R. C. 1983. Design and construction ofstone columns. Vol., 1 Rep. No., 1, FHWA/RD 83/026, Federal High-way Administration, Washington, D.C.

Bouassida, M. 1996. Etude exprimentale du renforcement de la vasede Tunis par colonnes de sableApplication pour la validation de larsistance en compression thorique dune cellule composite con-fine. Revue Franaise de Gotechnique, 75, 312.

Bouassida, M., de Buhan, P., and Dormieux, L. 1995. Bearing capacityof a foundation resting on a soil reinforced by a group of columns.E 2009

9.9:89-101.

Geotechnique, 451, 2534.Bouassida, M., and Hadhri, T. 1996. Closure to Extreme load of soils

reinforced by columns: The case of an isolated column. Soils Found.,361, 118119.

Bouassida, M., and Hadhri, T. 1998. Capacit portante dune fondationpose sur un sol renforc par un groupe de colonnes. Revue Ma-rocaine de Gnie Civil, 78, 216.

Bouassida, M., and Porbaha, A. 2004. Ultimate bearing capacity ofsoft clays reinforced by a group of columnsApplication to a deepmixing technique. Soils Found., 443, 91101.

Brauns, J. 1978. Die anfangstraglast von schottersulen im bindigenuntergrund. Die Bautechnik, 558, 263271.

Broms, B. G. 1982, Lime columns in theory and practice. Proc. Int.Conf. of Soil Mechanics, Mexico, 149165.

Chen, W. F. 1975. Limit analysis and soil plasticity, Elsevier, Amster-dam, The Netherlands.

Davis, R. O., and Selvadurai, A. P. S. 2002. Plasticity and geomechan-ics, Cambridge University Press, London.

Drucker, D. C., and Prager, W. 1952. Soil mechanics and plasticityanalysis or limit design. Q. Appl. Math., 10, 157165.

Hughes, J. M., and Withers, N. J. 1974. Reinforcing of soft cohesivesoils with stone columns. Ground Eng.

Jellali, B., Bouassida, M., and de Buhan, P. 2005. Homogenisationmethod for bearing capacity estimation. Int. J. Numer. Analyt. Meth.Geomech., 2910, 9891004.

Jellali, B., Bouassida, M., and de Buhan, P. 2007. A homogenizationapproach to estimate the ultimate bearing capacity of a stone columnreinforced foundation. Int. J. Geotechnical Engineering, 1, 6169.

Kasama, K., Zen, K., and Whittle, A. J. 2006. Effects of spatial vari-ability of cement-treated soil on undrained bearing capacity. Numeri-cal modeling of construction processes in geotechnical engineeringfor urban environment, T. Triantafyllidis, ed., Ruhr University Press,Bochum, Germany, 305313.

Porbaha, A., and Bouassida, M. 2004. Bearing capacity of foundationsresting on soft ground improved by soil cement columns. Proc., Int.Conf. on Geotechnical Engineering, A. Basma and M. Omar, eds.,University of Sharjah, United Arab Emirates, 173180.

Salenon, J. 1990. An introduction to the yield design theory and itsapplications to soil mechanics. Eur. J. Mech. A/Solids, 95, 477500.

Dow

nloa

ded

from

asc

elib

rary

.org

by

SCM

S Sc

hool

of E

ngin

eerin

g &

Tec

hnol

ogy

on 0

5/27

/15.

Cop

yrig

ht A

SCE.

For

per

sona

l use

onl

y; al

l rig

hts r

eser

ved.INTERNATION

Int. J. Geomech. 200AL JOURNAL OF GEOMECHANICS ASCE / MAY/JUNE 2009 / 101

9.9:89-101.