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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 795962 14 pageshttpdxdoiorg1011552013795962

Research ArticleAnalytical Analysis and Field Test Investigation ofConsolidation for CCSG Pile Composite Foundation in Soft Clay

Jin Yu1 Yanyan Cai1 Zhibo Qi1 Yunfei Guan2 Shiyu Liu1 and Bingxiong Tu1

1 Geotechnical Engineering Institute Huaqiao University Xiamen 361021 China2Geotechnical Engineering Department Nanjing Hydraulic Research Institute Nanjing Jiangsu 210024 China

Correspondence should be addressed to Jin Yu bugyu0717163com

Received 6 June 2013 Revised 25 August 2013 Accepted 6 September 2013

Academic Editor Fayun Liang

Copyright copy 2013 Jin Yu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Low-grade concrete-cored sand-gravel (CCSG) pile composite foundation is a new kind of composite foundation for thick and softclay ground treatment An analytical solution was derived for calculating the consolidation process of this composite foundation byconsidering coefficients of horizontal permeability in smear zone the radial flow within the sand-gravel shell and the imperviousproperty of concrete-cored pile The results show that Terzaghirsquos one-dimensional consolidation solution and the consolidationanalytical solution of ordinary composite foundation were special cases of this solution Curves of the average consolidation degreeof the composite foundation under various nondimensional parameters were observed using the program based on the theoreticalformula Meanwhile a series of in situ measurements including the settlement of pile and soil the pore water pressure and thetotal stress under embankment load were obtained on the CCSG pile composite foundation on a section of Zhenjiang-Liyanghighway The analyzed results show that the new style composite foundation patent technology has many advantages such as smalldifferential postconstruction settlement (differential is not good small is) reliable quality high bearing capacity and stability Andthe consolidation of composite foundation is largely affected by the nondimensional parameters The analytical solution is finallyverified with the actual measurement data

1 Introduction

Composite foundation technology has been widely usedin the foundation treatment of soft soil Concrete-coredsand-gravel pile composite foundation (CCSG pile compos-ite foundation) which is composed by prefabricated low-grade concrete-cored pile and sand-gravel shell consistingof soil between piles cushion and the composite pile isa new type of multivariate composite foundation that hasbeen put forward in the recent years Based on the ideaof controlling and differentiating post-construction settle-ment this foundation uses low-grade concrete-cored pile asthe vertical reinforcement sand-gravel shell as the verticaldrainage body and cushion as the horizontal drainage bodyas shown in Figure 1 The outstanding advantages of thisnew style technology lie in using the concrete-cored sand-gravel shell as the vertical drainage body to speed up theconsolidation of soil between piles during construction andpreloading periods to control post-construction settlement

and differential post-construction settlement within allowedextent in applied periods and to make full use of the bearingcapacity of soil between piles A surcharge load larger than thestructure load was adopted for preloading to accelerate theprocess of compression In deep soft foundation treatmentthe hierarchical heap load and the drainage reinforcementadvantages of sand shell in preloading period help to speed upporewater pressure dissipation anddegree of consolidation insoft soil And the deep foundation compression consolidationeffect is remarkable

On the other hand the consolidation theories for com-posite foundations reinforced by columns are developedon the basis of those well-drained foundations The onlydifference between these theories is that the former oneconsiders the stress concentration between soil and columnbut the latter does notThemostwell-known theoretical studyon the radial consolidation of vertical drains was carried outby Barron [1] firstly A large number of studies have beenconducted after Recently the consolidation of composite

2 Journal of Applied Mathematics

Temporary surcharge fill

Permanent fill Sand cushionGeogrid

Concrete-cored pileSand-gravel shell

Figure 1 Schematic diagram of section

foundation theory based on the sand drain consolidationtheory has made great progress with the consideration of thestress concentration phenomenon well resistance and smeareffect Goughnour and Bayuk [2] thought that compositefoundation consolidation problem could be analyzed by thesand drain consolidation theory Tang and Onitsuka [3] andXie [4 5] have established the consolidation equation for dis-crete material-pile composite foundation based on the equalstrain assumption of Barron [1] and have also put forward theradial average degree of consolidation calculation formulaeand the analytical solution of shaft drainage consideringwell resistance and the smudge effect By incorporating theradial and vertical drainage in a coupling fashion Leo [6]presented a series of closed-form solutions for equal strainconsolidation of vertical drains subjected to instantaneousand ramp loading The smear effect and well resistance werestudied Furthermore this solution was extended by Lei et al[7] to consider a time and depth-dependent loading Zhuand Yin [8 9] presented an analytical solution for theconsolidation analysis of soil with a vertical drain underramp loading considering the smear effect Wang and Jiao[10] introduced the double porosity model into the analysisof vertical drain consolidation With this approach thevariation of horizontal soil permeability can be depicted by anarbitrary function which presents a relatively simple way toconsider the gradual variation of soil permeability within thesmear zone Walker and Indraratna [11] have shown that theoverlapping smear zone due to the reduction of drain spacingcould also influence the drain performance By incorporatingthe relationship of e-log1205901015840 and e-log119896

ℎ Indraratna et al [12]

found a new solution for the radial consolidation of verticaldrains Basu and Prezzi [13] Castro and Sagaseta [14] andXie et al [15 16] utilized a stress increment independentof time and depth for the simplicity and effectiveness insolving engineering problems in which the external load wasassumed to be applied instantly and the corresponding stressincrement resulted within the foundation was considered tobe uniformly distributed along the column depth A largenumber of laboratory studies [15ndash21] have shown that thecoefficient of permeability within the smear zone was highlyvariable To reflect the variability some researchers includedthe gradual decay of horizontal permeability of soil towardthe drain such as linear decay in their analyses of verticaldrain consolidation [22] Recently researchers have done lotsof work on composite foundation [15 16 23ndash29] A general

kk

EnEnks ks

usus

EsEs

Ec

khkh

Ew Ew

2rc

2rw

2rs2re

Huw uw

kw

khw

q

Figure 2 Computing model of single CCSG pile

theoretical solution has been put forward for the consolida-tion of a composite foundation and the consolidation theoryhas been presented for the composite foundation consideringradial and vertical flows within the column the variation ofsoil permeability within the disturbed soil zone the depth-varying stress induced by multistage loading and time-and depth-dependent stress increment along with differentdistribution patterns of soil permeability

The above results have important reference value to studythe reinforcement mechanism of composite foundation Butthe studies of consolidation characteristic of CCSG pilecomposite foundation which is new under complicatedconditions and multivariate have not been reported yet Theauthors have tried to study the consolidation calculationmethod of CCSG pile composite foundation in simple casesbut the influence factors of CCSG pile composite foundationconsolidation characteristic are not considered such as thechange of horizontal penetration parameter in influence arearadial flow in sand shell and the impervious character of lowgrade concrete core pile

This paper introduces a series of field tests on CCSG pilecomposite foundation and the consolidation analysis modelof CCSG pile composite foundation is established which isbased on equal strain hypothesis and considering the changeof horizontal penetration parameters in influence zone andradial flow within sand-gravel shell The consolidation gen-eral solution was obtained by the theoretical derivationThe analytical solution was finally validated with the dataobtained from field tests and has verified the correctness ofthe theoretical solution

2 General Solution for Consolidation ofCCSG Pile Composite Foundation

21 Calculation Diagram The idealized CCSG compositefoundation is shown in Figure 2 In this figure 119867 is thethickness of the soil 119903

119888 119903119908 119903119904 and 119903

119890are radius of the

concrete-cored pile the sand-gravel shell the smear zoneand the influence zone (consisting of strong smear zone

Journal of Applied Mathematics 3

and weak smear zone) respectively 119864119888 119864119908 119864119904 and 119864

119899are

modulus of compressibility of the concrete-cored pile thesand-gravel shell the strong smear zone and the weak smearzone respectively 119896

119904and 119896

ℎare the horizontal permeability

coefficient of the strong smear zone and the weak smearzone respectively 119896V is the vertical permeability coefficient ofsoil 119896

ℎ119908and 119896V119908 are the horizontal and vertical permeability

coefficient of sand-gravel shell respectively 119906119908and 119906

119904are

excess pore water pressures within the sand-gravel shell andin the soil 119902 is external load

22 Basic Assumptions In order to obtain a simplified ana-lytical solution the following assumptions were made for thecalculation

(1) The relative displacement between CCSG pile andsoil was ignored The column and the surroundingsoil were assumed to deform only vertically and hadequal strain at same depth Concrete-cored pile wassimplified as an impervious cylindrical pile with thecorresponding radius and the interaction betweenthe concrete-cored pile and the sand-gravel shell wasalso ignored

(2) Darcyrsquos law was obeyed(3) The soil within the scope of drainage influence zone

was divided into strong smear zone and weak smearzone in which the horizontal permeability coefficientchanged along radial direction as 119896

119903(119903)

(4) The radial flow was taken into account in sand shell(5) The load was applied instantly The additional stress

of composite foundation distributed uniformly alongthe depth

23 Consolidation Equations and Solving Conditions

231 Equilibrium Condition and Stress-Strain RelationshipIn order to investigate the consolidation properties of CCSGpile composite foundation the stress concentration effectshould be considered which concludes the stress of theconcrete-cored pile and the sand-gravel shell the excess porewater pressures of the sand-gravel shell and the compositemodulus of compression of the soil At any time both thecolumn and the surrounding soil share the total stress in acomposite foundation that is

1205871199032

119888120590119888+ 120587 (119903

2

119890minus 1199032

119908) 120590119904+ 120587 (119903

2

119908minus 1199032

119888) 120590119908= 1205871199032

1198901205900

120590119888

119864119888

=

(120590119904minus 119906119904)

119864

=

(120590119908minus 119906119908)

119864119908

= 120576V

(1)

where 120590119908 120590119888 and 120590

119904are the average total stresses within

the sand-gravel shell the concrete-cored pile and the soilrespectively and 120590

0is the additional stress of composite

foundation in any depth caused by the uniform load 120576V isthe vertical strain of the column and the surrounding soil119864 = ((119899

2minus1199042)(1198992minus1))119864

119899+((1199042minus1)(119899

2minus1))119864

119904is the composite

modulus of compression of the soil 119906119904and 119906

119908are the excess

porewater pressurewithin the soil andwithin the sand-gravelshell respectively which can be defined as

119906119904=

int

119903119890

119903119908

2120587119903119906119904119889119903

120587 (1199032

119890minus 1199032

119908)

(2)

119906119908=

int

119903119908

119903119888

2120587119903119906119908119889119903

120587 (1199032

119908minus 1199032

119888)

(3)

where 119903 is the radial distance away from the centre ofconcrete-cored pile

From (1) 120576V can be derived as

120576V =11989921205900minus (1198992minus 1) 119906

119904minus (1 minus 119886

2) 119906119908

119864 [120572 + (1198992minus 1 + 119884)]

=

11989921205900minus (1198992minus 1198862) 119906

119864 [120572 + (1198992minus 1 + 119884)]

(4)

Derivation (4) about time is as follow

120597120576V

120597119905

= minus

1198992minus 1198862

119864 [120572 + (1198992minus 1 + 119884)]

120597119906

120597119905

(5)

where 119906 is the total average excess pore water pressure in thesoil at any depth which can be defined as

119906 =

1

120587 (1199032

119890minus 1199032

119888)

(int

119903119908

119903119888

2120587119903119906119908119889119903+ int

119903119890

119903119908

2120587119903119906119904119889119903)

=

[(1 minus 1198862) 119906119908+ (1198992minus 1) 119906

119904]

(1198992minus 1198862)

(6)

119904 = 119903119904119903119908is the radius ratio of the drainage influence zone

to the column 119899 = 119903119890119903119908and 119886 = 119903

119888119903119908are the radius ratio

respectively 119884 = 119864119908119864 is the compression modulus ratio of

the sand-gravel shell to the surrounding soil119883 = 119864119888119864 is the

compression modulus ratio of the concrete-cored pile to thesurrounding soil 120572 = 1198862(119883 minus 119884) is an expression

232 Continuity Conditions of Seepage Theexpression of thehorizontal permeability coefficient of the soils in drainageinfluence zone that varies linearly with respect to the radialdistance away from the column can be assumed to be

119896119903(119903) = 119896

ℎ119891 (119903) (7)

where 119896ℎis the horizontal permeability coefficients of the

weak smear zone and 119891(119903) is the function depending onradial distance away from column The equation describesthe variation pattern of the soil permeability along horizontaldirection

The concrete-cored pile of CCSG pile is set to be impervi-ous pile and the sand-gravel shell is made of discrete materialpile The consolidation equations of the soil of composite


foundation and the sand-gravel shell are used which can bedefined as

120597

120597119903

[

119903 sdot 119896119903(119903)

120574119908

120597119906119904

120597119903

] = minus119903(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112) 119903

119908le 119903 le 119903

119890

(8)

120597

120597119903

(

119903 sdot 119896ℎ119908

120574119908

120597119906119908

120597119903

) = minus119903(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112) 119903

119888le 119903 le 119903

119908

(9)

where 120574119908is the unit weight of water

233 Solving Conditions Consider the following

A 119903 = 119903119890 120597119906119904120597119903 = 0

B 119903 = 119903119908 119906119904= 119906119908

C 119903 = 119903119888 120597119906119908120597119903 = 0 (considering the concrete-cored

pile as an impervious pile)D 119903 = 119903

119908 119896119903(119903119908)(120597119906119904120597119903) = 119896

ℎ119908(120597119906119908120597119903) (the radial

velocity of pile- soil interface is equal)

The vertical boundary conditions can be written as

E 119911 = 0 119906119908= 0 119906 = 0

F 119911 = 119867 120597119906119908120597119911 = 0 120597119906120597119911 = 0 (in single-drainage

condition)G 119911 = 119867 119906 = 0 119906 = 0 (in double-drainage condition)

Assuming that there is no deformation of the pile and thesoil at initial time and the external load is bore all by porewater so the initial condition can bewritten as 119905 = 0 119906(119911 0) =1205900

24 The Establishment of the Governing Equations Equation(10) can be obtained by integrating both sides of (8) about 119903and using solving conditionA

120597119906119904

120597119903

=

120574119908

2119896ℎ

(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112)[

1199032

119890

119903119891 (119903)

minus

119903

119891 (119903)

] (10)

Integrating both sides of (10) about 119903 again and using solvingconditionB the following can be obtained

119906119904(119903) = 119906

119908

1003816100381610038161003816119903=119903119908

+

120574119908

2119896ℎ

(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112)

times [1199032

1198901198600(119903) minus 119861

0(119903)]

(11)

1198600(119903) = int

119903

119903119908

119889120585

120585119891 (120585)

1198610(119903) = int

119903

119903119908

120585119889120585

119891 (120585)

(12)

Equation (11) is substituted into (2)

119906119904= 119906119908

1003816100381610038161003816119903=119903119908

+

1199032

119890120574119908119865119888

2119896ℎ

(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112) (13)

where 119865119888= 2(119860

11199032

119890minus 1198611)1199032

1198901199032

119908(1198992minus 1) 119860

1= int

119903119890

119903119908

1199031198600(119903) 119889119903

and 1198611= int

119903119890

119903119908

1199031198610(119903)119889119903

By integrating both sides of (9) about 119903 and using solvingconditionC the following is obtained

120597119906119908

120597119903

=

120574119908

2119896ℎ119908

(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112)(

1199032

119888

119903

minus 119903) (14)

Equation (14) is integrated about 119903 both sides to get

119906119908(119903) = 119906

119908

1003816100381610038161003816119903=119903119908

+

120574119908

2119896ℎ119908

(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112)

sdot [1199032

119908(

1

2

+ 1198862 ln 119903

119903119908

) minus

1

2

1199032]

(15)


119906119908= 119906119908

1003816100381610038161003816119903=119903119908

+

120574119908119877

8119896ℎ119908

(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112) (16)

where119877 = 1199032119908[(41198862(1minus119886

2))(ln 119903

119908minus1198862 ln 119886119903

119908)minus21198862minus41198862 ln 119903119908minus

(1 minus 1198864)(1 minus 119886

2) + 2]

Equation (13) minus (16) meanwhile combining theexpression of 119906 in (4) and (5) deduces

119906 minus 119906119908= minus

(1198992minus 1)

119864 (120572 + 1198992minus 1 + 119884)

(

1199032

119890119865119888120574119908

2119896ℎ

minus

120574119908119877

8119896ℎ119908

)

120597119906

120597119905

+

1199032

119890119865119888119896V

2119896ℎ

1205972119906

1205971199112

minus [

(1 minus 1198862)

(1198992minus 1198862)

1199032

119890119865119888119896V

2119896ℎ

+

(1198992minus 1)

(1198992minus 1198862)

119877119896V119908

8119896ℎ119908

]

1205972119906119908

1205971199112

(17)

Substituting (10) and (14) into solving condition Dmeanwhile combining (5) the equation can be deduced as

minus(1198992minus 1198862)

2

119864 (120572 + 1198992minus 1 + 119884)

120597119906

120597119905

= (1198862minus 1)

119896V119908

120574119908

1205972119906119908

1205971199112

minus (1198992minus 1)

119896V

120574119908

1205972119906119904

1205971199112

(18)

Substituting the expression of 119906 in (4) into (18) we getequation

1205972119906119908

1205971199112= 119860

120597119906

120597119905

minus 119861

1205972119906

1205971199112 (19)

where 119860 = (1198992minus 1198862)120574119908(1 minus 119886

2)(119896V119908 minus 119896V)119864(120572 + 119899

2minus 1 + 119884)

119861 = (1198992minus 1198862)119896V(1 minus 119886

2)(119896V119908 minus 119896V)

Equation (19) is substituted into (17) and we get

119906119908= 119906 + 119862

120597119906

120597119905

minus 119863

1205972119906

1205971199112 (20)

where 119862 = (120574119908[(1198992minus 1)119896V119908 + 119896V]119864(120572 + 119899

2minus 1 +

119884)(119896V119908 minus 119896V))[1199032

1198901198651198882119896ℎ+ 119877(119899

2minus 1)8119896

ℎ119908] 119863 = (119896V119908119896V(119896V119908 minus

119896V))[1199032

1198901198651198882119896ℎ+ (1198992minus 1)119877(1 minus 119886

2)8119896ℎ119908]


Deduce from (20) as

120597119906119908

1205971199112=

1205972119906

1205971199112+ 119862

1205973119906

1205971199051205971199112minus 119863

1205974119906

1205971199114 (21)

Substituting (19) into (21) is deduced as

119863

1205974119906

1205971199114minus 119862

1205973119906

1205971199051205971199112minus (119861 + 1)

1205972119906

1205971199112+ 119860

120597119906

120597119905

= 0 (22)

So far the governing equations as (20) and (22) areobtained

The solution can be obtained by using the method ofseparation of variables for (22) which can be expressed as

119906 (119911 119905) = 1205900

infin

sum

119898=1

2

119872

sin(119872119867

119911) 119890minus120573119898119905 (23)

119906119908(119911 119905) = 120590

0

infin

sum

119898=1

2

119872

[1 minus 119862120573119898+ 119863(

119872

119867

)

2

]

sdot sin(119872119867

119911) 119890minus120573119898119905

(24)

119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus120573119898119905 (25)

where119872 = ((2119898 + 1)2)120587 (119898 = 0 1 2 )

120573119898= 119864 (120572 + 119899

2minus 1 + 119884)119896V119908119896V(

119872

119867

)

2

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119877

(1 minus 1198862) 8119896ℎ119908

]

+ [(1198992minus 1) 119896V + 119896V119908]

times (120574119908

(1198992minus 1198862)

2

1 minus 1198862

sdot (

119867

119872

)

2

+ [(1198992minus 1) 119896V119908 + 119896V]

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119877

8119896ℎ119908

]

)

minus1

(26)

In order to verify the rationality of the assumptions andthemethods of the consolidation in this paper the consolida-tion solution can be degraded

kw

kh

ks

o

kr(r)

rw rs re r

Figure 3 Five variation patterns of horizontal permeability coeffi-cient in smear zone

When119883 = 1 and 119886 = 0 120573119898changes into

120573119898= 119864 (119899

2minus 1 + 119884)119896V119908119896V(

119872

119867

)

2

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119903

2

119908

8119896ℎ119908

]

+ [(1198992minus 1) 119896V + 119896V119908]

times (1205741199081198994(

119867

119872

)

2

+ [(1198992minus 1) 119896V119908 + 119896V]

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119903

2

119908

8119896ℎ119908

])

minus1

(27)

This is the consolidation solution of common compositefoundation provided by Lu et al [30] that considered theradial flow within the pile

When 119884 = 1 119896V = 119896V119908 and 119896ℎ = 119896ℎ119908 120573119898 and 119880 changeinto

120573119898=

119896V119864

120574119908

(

119872

119867

)

2

= 119888V(119872

119867

)

2

119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus1198722

119879V

(28)

where 119888V = 119896V119864120574119908 119879V = 119888V1199051198672

This is the one-dimensional consolidation solution ofTerzaghirsquos theory The rationality of the consolidation solu-tion in this paper can be reflected through the above answerto degradation

From (13) (23) and (25) it can be seen that the influenceof horizontal permeability coefficient in the influenced zoneto the consolidation solution is reflected mainly by theparameter 119865

119888 which is related to the changing pattern of

horizontal permeability coefficient Figure 3 displays a typical


changing pattern of horizontal permeability coefficient versus119903 The solutions of 119865

119888had been given by Zhang et al [24] as

119865119888=

1198992

1198992minus 1

119904 minus 1

120581119904 minus 1

ln (120581119904) minus (119904 minus 1)2

1198992(1 minus 120581)

+

2

1198992

(119904 minus 1) (120581119904 minus 1)

(1 minus 120581)2

ln 1120581

minus

2

1198994

119904 minus 1

1 minus 120581

times (

1199043minus 1

3

minus

1199042minus 1

3

) minus

1

1198994

(119904 minus 1) (120581119904 minus 1)

(1 minus 120581)2

times [

1199042minus 1

2

minus

(119904 minus 1) (120581119904 minus 1)

1 minus 120581

+

(120581119904 minus 1)2

(1 minus 120581)2ln 1120581

]

minus

1198992minus 1199042

1198994

(1 minus 119904)2

1 minus 120581

+ ln 119899119904

minus

3

4

+

411989921199042minus 1199044

41198994

(29)

3 Parametric Study and Discussion

In order to study the characters of consolidation of CCSGpilecomposite foundation120573

119898should be converted to dimension-

less which can be expressed as

120573119898119905 = 120591119898119879ℎ (30)

where 119879ℎis the horizontal time factor of the soil which can

be expressed as

119879ℎ=

119888ℎ119905

41199032

119890

119888ℎ=

119864119896ℎ

120574119908

(31)

In this case (25) changes into

119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus120591119898119879ℎ

(32)

where

120591119898= 1199032

119890(120572 + 119899

2minus 1 + 119884)

times

119896V119908

119896ℎ

(

119872

119867

)

2

[

1199032

119890119865119888

2

+

(1198992minus 1)119877119896

ℎ

(1 minus 1198862) 8119896ℎ119908

]

+ (1198992minus 1) +

119896V119908

119896V

00

02

04

06

08

1000001 0001 001 01 1 10

n = 4

n = 6

n = 8

n = 10

Th

U

Figure 4 Influence of 119899 on consolidation process

00

02

04

06

08

1000001 0001 001 01 1 10

Hdw = 20

Hdw = 50

Hdw = 100

Th

U

Figure 5 Influence of119867119889119908on consolidation process

times (

(1198992minus 1198862)

2

1 minus 1198862

sdot

119896ℎ

119896V

(

119867

119872

)

2

+ [(1198992minus 1)

119896V119908

119896V

+ 1]

sdot (

1199032

119890119865119888

2

+

119877 (1198992minus 1) 119896

ℎ

8119896ℎ119908

)

)

minus1

(33)

From the expression of 120591119898 the dimensionless parameters

influencing the consolidation character of CCSG pile com-posite foundation include 119899 119886 119904 119883 119884 119867119889

119908(119889119908= 2119903119908)

119896ℎ119896V 119896119904119896ℎ 119896ℎ119896V119908 119896ℎ119896ℎ119908 and 119896

ℎ119908119896V119908 The influences

of several dimensionless parameters on the consolidationbehaviour of CCSG pile composite foundation were inves-tigated and some numerical results from different solutionswere compared according to the above formula The specificcalculation results are shown in Figures 4ndash11 The calculatingparameters are shown in Table 1


Table 1 Parameters used for calculation (119868)

Figure no 119899 119886 119904 119867119889119908

119883 119884 119896119904119896ℎ

119896ℎ119896V 119896

ℎ119896V119908 119896

ℎ119908119896V119908

4 mdash 02 15 20 1000 10 06 2 0001 15 4 02 15 mdash 1000 10 06 2 0001 16 4 02 15 20 1000 10 02 2 0001 17 4 mdash 15 20 mdash mdash 06 2 0001 18 4 02 15 20 mdash mdash 02 2 0001 19 4 02 15 20 1000 10 02 2 mdash 110 4 02 15 20 1000 10 mdash 2 0001 111 4 02 15 20 1000 10 mdash 2 0001 1

00

02

04

06

08

1000001 0001 001 01 1 10

Y = 20 X = 1000Y = 20 X = 2000

Y = 10 X = 1000Y = 10 X = 2000

Th

U

Figure 6 Influence of 119883 and 119884 on consolidation process

00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

a = 0 X = 1 Y = 1

a = 0 X = 1 Y = 10

a = 02

a = 05


Figures 4 and 5 show the influence of 119899 and 119867119889119908on

consolidation behaviour of CCSG pile composite foundationIt can be seen that consolidation velocity is reduced withthe increase of 119899 and 119867119889

119908 In other words the bigger the

scope of influenced zone the slower the consolidation rateis Figure 4 also shows that the reducing rate of consolidationvelocity decreases with the increase of 119899

Figure 6 shows the influence of 119883 and 119884 on the consol-idation behaviour of CCSG pile composite foundation Asshown in the figure when the value of 119884 is constant the

00

02

04

06

08

1000001 0001 001 01 1 10

Th

khkhw = 00001

khkhw = 0001

khkhw = 001

khkhw = 1

U

Figure 8 Influence of 119896ℎ119896ℎ119908

on consolidation process

00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

kskh = 02

kskh = 04

kskh = 06

kskh = 08

Figure 9 Influence of 119896119904119896ℎon consolidation process

consolidation rate increases with the value of 119883 Similarlywith 119883 constant the larger the value of 119884 the faster theconsolidation rate is So the consolidation rate should beaccelerated with the increasing of119883 and 119884

Figure 7 shows the comparison between several solutionsincluding the solution for ordinary granular material pilecomposite foundation proposed by Lu et al [26] and Terza-ghirsquos solution for natural foundationWhen 119886 = 0119883 = 1119884 =1 119896V = 119896V119908 and 119896ℎ = 119896ℎ119908 the curve presents the changing ofconsolidation degree with time for natural foundationWhen


00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

s = 3s = 2

s = 15

s = 1


10

08

06

04

02

00

00 01 02 03 04 05 06 07 08 09 10

u120590

0

Th

zH = 02

zH = 03

zH = 04

zH = 06

zH = 08

Figure 11 Dissipation curves of average pore-pressure at differentdepth in foundation

119886 = 0 119883 = 1 and 119884 = 10 the curve shows the changing ofconsolidation degreewith time for ordinary granularmaterialpile composite foundation The curves of 119886 = 02 and 119886 = 05are the consolidation degree curves changing with time ofCCSG pile composite foundation Among these solutionsonly the solution presented in this study takes the charactersof CCSG pile composite foundation into account As shownin Figure 7 the consolidation rates given by the presentsolution are greater than those given by the other solutionsIt also can be seen that the consolidation rates given by thepresent solution and the Lu et al [26] solution are both greaterthan that given by Terzaghirsquos solution because the formertwo solutions both take the stress concentration from soil tocolumn into account These conclusions indicate that CCSGpile composite foundation performs best in the drainageconsolidation Because CCSG pile composite foundation haslarge diameter of the drainage channel and the concrete-corepile as vertical reinforcement the postloading settlement ofthe foundation finished quickly In addition the larger thevalue of 119886 the faster the consolidation rate of CCSG pilecomposite foundation is So the consolidation rate should be

accelerated with the increase of 119886 However when the valueof 119886 reaches 1 CCSG pile composite foundation will convertto pile foundation and has no drainage function

Figure 8 shows the influence of 119896ℎ119896ℎ119908

on consolidationbehaviour of CCSG pile composite foundation under threedifferent patterns of horizontal permeability coefficient Itcan be seen that when the horizontal permeability coefficientwithin soil is smaller than that within the sand-gravel shellthe consolidation rate is accelerated limited with the decreaseof 119896ℎ119896ℎ119908 In addition the curves of consolidation degree

are close to each other which indicate that the influence of119896ℎ119896ℎ119908

on consolidation behaviour of CCSG pile compositefoundation is weak As the concrete-cored pile is consideredas an impervious pile both the solutions of the soil and sand-gravel shell are obtained using the consolidation equations sotaking the radial flow within sand-gravel shell into account isnecessary

Figure 9 shows the influence of 119896119904119896ℎon consolidation

behaviour of CCSG pile composite foundation under thechange of horizontal permeability coefficient of the soil Thevalue of 119896

119904119896ℎcan reflect the intensity of disturbance to the

surrounding soil during column construction the bigger thevalue is the more the disturbance intensity is It can beseen from Figure 9 that the consolidation rate of a compositefoundation reduces with the decrease of 119896

119904119896ℎ In other

words the consolidation rate of a composite foundation isenhanced by reducing the disturbance intensity

Figure 10 shows the influence of 119904 on consolidationbehaviour of CCSG pile composite foundation which repre-sents the size of the disturbed zone It can be seen from thegraph that the consolidation rate of a composite foundationreduces with the increase of 119904 In another word the larger thedisturbance area the slower the consolidation rate is

Figure 11 represents the average pore-pressure dissipa-tion It can be seen from the graph that the deeper the depthof foundation the slower the pore-pressure dissipation speedis which means slow consolidation rate of deep foundationand is consistent with the actual situation

4 Case Application

41 Survey of Experimental Sections The construction sec-tion Liyang Second Bid of Zhenjiang Liyang Highway islocated in the Yangtze River valley The area of treatedembankment of thick and soft ground in section K63 + 046simK63 + 087 was 41m times 63m which was 2583m2 The CCSGpile composite foundation treatment was adopted with adiameter of 50 cm (the prefabricated low-grade concrete-cored pile was 20 cm times 20 cm and the outside diameter ofthe sand-gravel shell was 50 cm) and a length of 22m Thepiles were prefabricated in 3 parts and arranged in equilat-eral triangle with 21m spacing between section K63 + 046and K63 + 066 and 19m between section K63 + 066 andK63 + 087 One layer of hardcore bed with 50 cm thicknesswas placed as cushion together with one layer of geogridafter the construction of concrete-cored sand-gravel pileThe ground altitude was 31m the designed altitude at thecentre of roadbed superface was 85m and the surcharge


Table 2 Physical and mechanical parameters of soils

Soillayer

Name of soillayer

Soil layerthicknessm 120596 120574(kNm3) 119890

0119868119875

119868119871

120572(1-2)MPaminus1 119864

119904(1-2)MPa Direct shear (Quick shear)119891akkPa

ckPa 120593(∘)

AMiscellaneous

fill 20sim37

B Silty clay 16sim26 323 184 091 107 101 030 728 20 171 125C Muddy silty clay 66sim109 388 179 107 117 135 049 447 70 245 65D-1 Silty clay-silt 06sim25 352 183 096 116 127 045 537 180 165 105D-2 Silt-silty clay 07sim29 339 185 093 109 135 041 568 100 255 150D-3 Silty clay-silt 0sim44 266 193 075 117 076 045 394 250 137 165E Silty clay 96sim101 350 187 093 130 116 043 529 130 122 105F Angular pebbles 250

41

20 21

21 19

63

K63 + 046 K63 + 066 K63 + 087

K63 + 056 K63 + 076

Size units (m)

Monitoring sections

Figure 12 Test section layout drawing

preloading altitude was 111m The maximum dry densityof embankment filler was 184 gm3 and the slope of theembankment was 1 2 The subgrade was 35m wide and thegroundwater level was 1 sim 2m below surface Vibratingsunk-tube method was used for construction According toexploratory boring CPT vane shear test and geotechnicalparameter test the physical and mechanical parameters ofsoils in experimental sections are shown inTable 2The layoutof tested section is shown in Figure 12

The profile map of instruments at site monitored sectionis shown in Figure 13 The layout of instruments at each sec-tion was as follows

(1) Three ground settlement poles were placed in the leftmiddle and right position along the width of thesubgrade respectively

(2) One 30-meter long layered settlement pile wasembedded at subgrade centre Twelve settlement rings

were layered on the pile every two meters along thedepth Meanwhile nine pore-pressure detectors wereembedded every two meters along the depth

(3) One inclinometer pile was embedded at subgradeslope

(4) Ten soil pressure boxes were embedded in the piletop the sand top and the soil respectively at the tri-angular area consisting of three piles at the subgradecentre

(5) Six steel bar stress detectors including one at piletop one at pile bottom and four at pile body wereembedded in each pile of the two chosen piles

42 Comparative Analysis of the Engineering Example

421 Settlement of Pile Top and Soil Three settlement meterswere embedded in the right and left shoulders and the centreof section K63 + 056 and K63 + 076 respectively to measurethe settlement of soils Meanwhile another two meters wereembedded in the concrete-cored pile top in the centre ofsection K63 + 056 and K63 + 076 respectively to measure thesettlement of concrete-cored pile top Settlement meters witha size of 70 cmtimes 70 cm were used in the soil To prevent frominterfering with the settlement of the concrete-cored pile andto avoid the deflection of the settlement meter on the top ofthe pile meters with a size the same as the concrete-coredpile section which was 20 cm times 20 cm were used on the topof the pile and were welded to the top directly Settlementobservation points were set up on tops of the sand-gravel shelland concrete-cored pile Stable control point was establishedto measure the elevation changes with high precision waterlevel The measurement reading was obtained daily in theearly stage of construction and then in every two or threedays after the reading was stable for the settlement-timecurveThe settlement observation process lasted for 285 daysThe variation of settlement in soil in the centre of sectionK63 + 056 and K63 + 076 is shown in Figure 14

From Figure 14 by comparing the settlement of sectionK63 + 056 and K63 + 076 it can be seen that until the endthe settlement in the centre of the section K63 + 056 whichwas farther away from the abutment was 649 cm and theother one was 479 cmThe result indicates that the bigger the


Ground settlement Embankment

Pile (22m)

Inclinometerpipe (30m)

Pore-pressure detector

Layered settlement pipes (30m)

pole

(a) The layout section of monitoring section instruments

Soil pressure box

Pile

Steel bar stress detector

7m

7m

7m

4m

4m

21

m215

m

(b) The layout drawing of earth pressure boxesand steel bar stress detectors

Figure 13 Profile drawing of instruments

50 100 150 200 250 3002

4

6

8

60

40

20

0

Time (day)Hei

ght (

m)

Settl

emen

t (cm

)

(a) K63 + 056 Section centre

Hei

ght (

m)

50 100 150 200 250 3000

2

4

6

8

5040302010

Settl

emen

t (cm

)Time (day)

(b) K63 + 076 Section centre

Figure 14 Settlement of soil surface

pile spacing is the bigger the settlement ratio and settlementare which also reflects the influence of the pile spacing on thecomposite foundation settlement During the construction ofembankment (the first 110 days) the maximum settlementrates at section K63 + 056 and K63 + 076 were 06 cm and04 cm per day respectively During the period of preloading(the next 10 days) the maximum settlement rate at sectionK63 + 056was 15 cmper day andwas 11 cmper day at sectionK63 + 076 As shown in Figure 14 the settlement curvesbecame horizontal straight while the maximum lateral dis-placement was only 1mm per day by now indicating that theimmediate settlement of soil was larger After the applyingof dead load the settlement rate tended to reduce steadilyThe first-measured settlement rate was 5mm per month afterdead load had been applied for four months indicating thatthe surface settlement had become steady

In order to validate the theoretical formula derived inthis paper the construction data of the testing section andthe field measured data at section K63 + 076 were used forcalculating validation and contrastive analysis

Asaoka method was used to predict the final settlementof composite foundation according to the settlement datameasured from section K63 + 076 Then the field test curveof overall average consolidation degree of CCSG pile com-posite foundation changing with time was obtained Finally

the formula in this paper was amended using the improvedTerzaghi method to satisfy the conditions of step loading inpractical engineering

As shown in Figure 15 the manner of the theoreticalcurve of average consolidation degree changing with timeis similar to that of the curve obtained from field test Thehysteresis phenomenon is due to the existing settlementhysteresis after completion of loading From the figure itcan be seen that the consolidation rate given by the solutionin this paper is greater than that deduced from field testdata Considering the error between the observation andtest a good agreement can be affirmed which validates thetheoretical solution in this paper

422 Pore Water Pressure The steel-string type pore-pressure detector was used to monitor the pore-pressurewith a measuring range up to 200 kPa or 400 kPa Frequencymeter was used to sense the frequency In order to determinethe instrument sensitivity coefficient temperature coefficientand sealing performance the instruments were checked-out and calibrated before being embedded The embeddedway was stated in Section 41 Nine pore-pressure detectorswere embedded at different depth at section K63 + 056 andK63 + 076 respectively No test results were obtained fromsection K63 + 076 as the one detector was damaged during


10

08

06

04

02

00

8

6

4

2

0

0 100 200 300 400 500

Fill height curve

Test dataTheory solution

U

d

Figure 15 Variation of consolidation degree of the composite foun-dation

50 100 150 200 250 3000

10

20

30

40

Time (day)

Pore

wat

er p

ress

ure (

kpa)

2m6m10m

14m18m

Figure 16 Excess pore water pressure with depth

the abutment excavation The observation of pore-pressurewent on in parallel with the observation of soil pressure Theobservation results of pore water pressure at primary depthat section K63+056 were shown in Figure 16

As shown in Figure 16 at the first three months with afilling height of 42m the variation curves of pore-pressurewere smooth From day 100 to day 120 the filling heightrose from 42m to 79m and the pore-pressure at differentdepth went up by 9sim28 kPa meanwhile the steep and sharpvariation occurred on the curves After dead load on day 120the pore-pressure dissipated rapidly From the entire depthrange the pore-pressure did not decrease with the increaseof depth and the measured maximum pore-pressure wasat the depth of 10m below surface The drainage effect ofconcrete-cored sand gravel pile is very excellent which couldbe obtained by the phenomenon that pore-pressure rose anddissipated rapidly

Figure 17 shows the excess pore water pressure obtainedby the amended theoretical equation The figure is similar

05

10152025303540

0 50 100 150 200 250 300

Pore

wat

er p

ress

ure (

kpa)

Time (day)

2m6m10m

14m18m

Figure 17 Excess pore water pressure with depth obtained by thetheoretical equation

to Figure 16 which indicates the validity of the theoreticalequation in this paper

423 Soil Pressure under Embankment Load Ten soil pres-sure boxes were embedded in the triangular area consisting ofthree piles in the centre of section K63 + 056 and K63 + 076respectively including three on the top of the pile threeon the top of sand-gravel and four in the soil betweenpiles The pressure surface of the soil pressure boxes mustface to the measured soil Meanwhile the following issuesrequire attention including that the soil surface under soilpressure boxes must be strictly levelled the material ofbackfill soil should be the same with the surrounding soil(stone removed) and artificial compaction in layers mustbe used carefully The frequency was sensed by a frequencymeter No test result was obtained from section K63 + 076because of the damage of soil pressure boxes due to abutmentexcavation

Variations of the pile-soil stress ratio 119899119904and shared

load ratio 119873 are shown in Figure 18 During embankmentconstruction and preloading period 119899

119904and 119873 increased at

first and then decreased In the early filling period (the first24 days) when the cumulative filling height was 12m thepile-soil pressure ratio 119899

119904rose from 16 to 29 and the shared

load ratio 119873 rose from 009 to 016 slowly From day 25 today 100 the filling height grew from 12m to 42m the pile-soil pressure ratio rose from 29 to 168 and the shared loadratio rose from 016 to 092 On day 100 after the cumulativefilling height reaching 42m the pile-soil pressure ratio andthe shared load ratio reached the maximum values 168 and092 respectively From day 100 to day 120 the filling heightrose from 42m to 79m the pile-soil ratio reduced from 168to 132 and the shared load ratio reduced from 092 to 072From finishing preloading (day 100) to observation beingover (day 150) the pile-soil pressure ratio rose from 132 to15 and the shared load ratio rose from 072 to 082 indicatingthat the pile-soil pressure ratio and the shared load ratio stillrose gradually after dead load the load carried by the pile soiltransferred to the pile top and the effect of reinforcement onthe composite foundation strengthened continually


50 100 150 200 250 3000369

121518

Time (day)

ns

(a)

50 100 150 200 250 300Time (day)

0

02

04

06

08

1

N

(b)

Figure 18 Variation of pile-soil stress ratio 119899119904and shared load ratio119873

0

3

6

9

12

15

18

0 50 100 150 200 250 300Time (day)

zH = 04

zH = 02

zH = 08

ns

Figure 19 Variation of pile-soil stress ratio 119899119904obtained by theoreti-

cal equation

Figure 19 shows the variation of pile-soil stress ratio119899119904obtained by the amended equation which is similar to

Figure 18 It can be seen that the value of 119899119904varied in

different depth and increased when approaching to thesurface It means that the pile shared greater stress and thesoil undertakes smaller load at the smaller depth place

5 Conclusions

The general solution for consolidation of CCSG pile compos-ite foundation under equal strain hypothesis was obtained byconsidering the variation of horizontal penetration param-eter in influence zone radial flow within the sand-gravelshell and the impervious characteristic of concrete-coredpile in this paper Meanwhile this paper also introduced aseries of field tests on CCSG pile composite foundation toprove the correctness of the general solution including thesettlement of pile and soil the pore water pressure and thesoil pressure under embankment load It is concluded that thenew style composite foundation patent technology has manyadvantages such as small post-construction and differentialpost-construction settlement reliable quality high bearingcapacity high speed of settlement and stabilityThe followingmain conclusions are obtained

(1) The solution given by Terzaghi is a limited case ofthe present solution when 119886 = 0 119883 = 1 119884 = 1119896V = 119896V119908 and 119896ℎ = 119896

ℎ119908 the solution for ordinary

granular material pile composite foundation is also alimited case of the present solution when 119886 = 0 and119883 = 1The consolidation rate of CCSGpile compositefoundation is greater than that of natural foundationsand drained ground and ordinary granular materialpile composite foundation

(2) A parametric study shows that an increase in thevalues of 119886119883 and119884 and a decrease in the values of 119899119904 and 119867119889

119908will accelerate the consolidation rate of

CCSG pile composite foundation(3) The analytical solution in this paper was finally vali-

dated with the actual filed measurement data(4) The essence of the composite foundation soil is that

the reinforcement and subgrade soil mass undertakethe load from the upper structure To improve thebearing capacity of the foundation soil in thick andsoft ground area the area replacement ratio of pilescan be appropriately improved with the project costand the pile length or the bearing capacity of softsoil can be increased Meanwhile the existence of thegranular columns and the critical pile length in rigid-flexible piles made the increased pile length difficultto play a full role Compared with the rigid pile theconcrete-cored sand-gravel pile makes full use of theconcrete-cored sand gravel shell capacity of drainageand consolidation to increase the bearing capacity ofthe soft soil

(5) The total settlement of subgrade of CCSG pile com-posite foundation during embankment constructionand preloading periods is bigger than that of churningpile and other composite foundations The reasonis that the concrete-cored sand gravel shell provideslarge diameter vertical drainage channel and reducesthe drainage distance of the soil mass greatly so thesubsoil can produce larger consolidation settlementin a short time Excess pore-pressure can dissipate ina short time so the subgrade settlement is faster andbecomes stable quickly Four months after the pre-compression at the experimental section the settle-ment rate was 5mm per month which indicated that


the settlement becames stable the precompressiontime fits in with that of the common sand drain

Experimental results illustrate that adopting the concrete-cored sand-gravel pile composite foundation for thick andsoft ground treatment at bridge head and the controllingof post-construction settlement is feasible The test resultsalso verify the advance rationality and validity of CCSGpile composite foundation Because of the effect of soft soilconsolidation and composite foundation that take place at thesame time the soil and the composite foundation interact andthe relative displacement and interaction relationship amongconcrete-cored pile soil and sand-gravel shell are verycomplicated So the deformation and load characteristicsespecially the reinforcement mechanism under soft loadremain to be further studied

Acknowledgments

This research was supported by the National Nature Founda-tion of China (Grant nos 51374112 51308234 and 51208321)and Natural Science Foundation of Fujian Province (no2013J05079) all support is gratefully acknowledged

References

[1] R A Barron ldquoConsolidation of fine-grained soils by drainwellsrdquo Transactions of the American Society of Civil Engineersvol 113 no 2346 pp 718ndash742 1948

[2] R R Goughnour and A A Bayuk ldquoAnalysis of stone columnsoil matrix interaction under vertical loadrdquo in Proceedings ofInternational Conference on Soil Reinforcement pp 279ndash285Paris France 1979

[3] X W Tang and K Onitsuka ldquoConsolidation of ground withpartially penetrated vertical drainsrdquo Geotechnical Engineeringvol 29 no 2 pp 209ndash227 1998

[4] K H Xie ldquoPresent situation and development of consolidationtheory of composite groundrdquo Ground Improvement vol 4 no3 pp 1ndash14 1993

[5] KH Xie Sand drained ground analytical amp numerical solutionsof consolidation and optimum design [PhD dissertation] Zhe-jiang University Hangzhou China 1987

[6] C J Leo ldquoEqual strain consolidation by vertical drainsrdquo Journalof Geotechnical and Geoenvironmental Engineering vol 130 no3 pp 316ndash327 2004

[7] G H Lei C X Jiang and C J leo ldquoDiscussion of ldquoEqual strainconsolidation by vertical drainsrdquo by Chin Jian Leordquo Journal ofGeotechnical and Geoenvironmental Engineering vol 131 no 10pp 1315ndash1317 2005

[8] G Zhu and J H Yin ldquoConsolidation of soil with vertical andhorizontal drainage under ramp loadrdquoGeotechnique vol 51 no4 pp 361ndash367 2001

[9] G Zhu and J H Yin ldquoConsolidation analysis of soil with ver-tical and horizontal drainage under ramp loading consideringsmear effectsrdquo Geotextiles and Geomembranes vol 22 no 1-2pp 63ndash74 2004

[10] X S Wang and J J Jiao ldquoAnalysis of soil consolidationby vertical drains with double porosity modelrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 28 no 14 pp 1385ndash1400 2004

[11] R Walker and B Indraratna ldquoVertical drain consolidation withparabolic distribution of permeability in smear zonerdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 132 no 7pp 937ndash941 2006

[12] B Indraratna C Rujikiatkamjorn andL Sathananthan ldquoRadialconsolidation of clay using compressibility indices and varyinghorizontal permeabilityrdquo Canadian Geotechnical Journal vol42 no 5 pp 1330ndash1341 2005

[13] D Basu andM Prezzi ldquoEffect of the smear and transition zonesaround prefabricated vertical drains installed in a triangularpattern on the rate of soil consolidationrdquo International Journalof Geomechanics vol 7 no 1 pp 34ndash43 2007

[14] J Castro and C Sagaseta ldquoConsolidation around stonecolumns Influence of column deformationrdquo International Jour-nal for Numerical and Analytical Methods in Geomechanics vol33 no 7 pp 851ndash877 2009

[15] K H Xie M M Lu A F Hu and G H Chen ldquoA generaltheoretical solution for the consolidation of a composite foun-dationrdquo Computers and Geotechnics vol 36 no 1-2 pp 24ndash302009

[16] K H Xiel M M Lu and G B Liu ldquoEqual strain consolidationfor stone columns reinforced foundationrdquo International Journalfor Numerical and Analytical Methods in Geomechanics vol 33no 15 pp 1721ndash1735 2009

[17] J C Chai and N Miura ldquoInvestigation of factors affectingvertical drain behaviourrdquo Journal of Geotechnical and Geoenvi-ronmental Engineering vol 125 no 3 pp 216ndash226 1999

[18] J S Sharma and D Xiao ldquoCharacterization of a smear zonearound vertical drains by large-scale laboratory testsrdquoCanadianGeotechnical Journal vol 37 no 6 pp 1265ndash1271 2000

[19] B C Hawlader G Imai and B Muhunthan ldquoNumerical studyof the factors affecting the consolidation of clay with verticaldrainsrdquo Geotextiles and Geomembranes vol 20 no 4 pp 213ndash239 2002

[20] B Indraratna and I W Redana ldquoLaboratory determinationof smear zone due to vertical drain installationrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124 no 2pp 180ndash184 1998

[21] A Onoue N H Ting J T Germaine and R V WhitmanldquoPermeability of disturbed zone around vertical drainsrdquo inProceedings of the Geotechnical Engineering Congress pp 879ndash890 Boulder Colo USA 1991

[22] R Walker and B Indraratna ldquoVertical drain consolidation withoverlapping smear zonesrdquo Geotechnique vol 57 no 5 pp 463ndash467 2007

[23] X W Tang and K Onitsuka ldquoConsolidation by verticaldrains under time-dependent loadingrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 24 no9 pp 739ndash751 2000

[24] Y G Zhang K H Xie and ZWang ldquoConsolidation analysis ofcomposite ground improved by granular columns consideringvariation of permeability coefficient of soilrdquo in Ground Modifi-cation and Seismic Mitigation GSP 152 2006

[25] M M Lu K H Xie and B Guo ldquoConsolidation theory fora composite foundation considering radial and vertical flowswithin the column and the variation of soil permeability withinthe disturbed soil zonerdquo Canadian Geotechnical Journal vol 47no 2 pp 207ndash217 2010

[26] M M Lu K H Xie A F Hu and G H Chen ldquoReply to Com-ments on ldquo A general theoretical solution for the consolidationof a composite foundationrdquo [Computers and Geotechnics 36


(2009) 24ndash30]rdquo Computers and Geotechnics vol 37 no 4 pp582ndash583 2010

[27] M M Lu K H Xie C X Li and K Wang ldquoConsolidationsolution for composite foundation considering a time- anddepth-dependent stress increment alongwith three distributionpatterns of soil permeabilityrdquo Journal of Zhejiang UniversityScience A vol 12 no 4 pp 268ndash277 2011

[28] M M Lu K H Xie and S Y Wang ldquoConsolidation ofvertical drain with depth-varying stress induced by multi-stageloadingrdquo Computers and Geotechnics vol 38 no 8 pp 1096ndash1101 2011

[29] R C Wang K H Xie and S H Guan ldquoAnalytical solutionsfor consolidation of composite ground with granular columnsunder timedependent loadingrdquo Journal of Zhejiang Universityvol 36 no 1 pp 12ndash16 2002

[30] M M Lu K H Xie S Y Wang and C X Li ldquoAnalytical solu-tion for the consolidation of a composite foundation reinforcedby impervious column with an arbitrary stress incrementrdquoASCE International Journal of Geomechanics vol 13 no 1 pp33ndash40 2013

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Differential EquationsInternational Journal of

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Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Temporary surcharge fill

Permanent fill Sand cushionGeogrid

Concrete-cored pileSand-gravel shell

Figure 1 Schematic diagram of section

foundation theory based on the sand drain consolidationtheory has made great progress with the consideration of thestress concentration phenomenon well resistance and smeareffect Goughnour and Bayuk [2] thought that compositefoundation consolidation problem could be analyzed by thesand drain consolidation theory Tang and Onitsuka [3] andXie [4 5] have established the consolidation equation for dis-crete material-pile composite foundation based on the equalstrain assumption of Barron [1] and have also put forward theradial average degree of consolidation calculation formulaeand the analytical solution of shaft drainage consideringwell resistance and the smudge effect By incorporating theradial and vertical drainage in a coupling fashion Leo [6]presented a series of closed-form solutions for equal strainconsolidation of vertical drains subjected to instantaneousand ramp loading The smear effect and well resistance werestudied Furthermore this solution was extended by Lei et al[7] to consider a time and depth-dependent loading Zhuand Yin [8 9] presented an analytical solution for theconsolidation analysis of soil with a vertical drain underramp loading considering the smear effect Wang and Jiao[10] introduced the double porosity model into the analysisof vertical drain consolidation With this approach thevariation of horizontal soil permeability can be depicted by anarbitrary function which presents a relatively simple way toconsider the gradual variation of soil permeability within thesmear zone Walker and Indraratna [11] have shown that theoverlapping smear zone due to the reduction of drain spacingcould also influence the drain performance By incorporatingthe relationship of e-log1205901015840 and e-log119896

ℎ Indraratna et al [12]

found a new solution for the radial consolidation of verticaldrains Basu and Prezzi [13] Castro and Sagaseta [14] andXie et al [15 16] utilized a stress increment independentof time and depth for the simplicity and effectiveness insolving engineering problems in which the external load wasassumed to be applied instantly and the corresponding stressincrement resulted within the foundation was considered tobe uniformly distributed along the column depth A largenumber of laboratory studies [15ndash21] have shown that thecoefficient of permeability within the smear zone was highlyvariable To reflect the variability some researchers includedthe gradual decay of horizontal permeability of soil towardthe drain such as linear decay in their analyses of verticaldrain consolidation [22] Recently researchers have done lotsof work on composite foundation [15 16 23ndash29] A general

kk

EnEnks ks

usus

EsEs

Ec

khkh

Ew Ew

2rc

2rw

2rs2re

Huw uw

kw

khw

q

Figure 2 Computing model of single CCSG pile

theoretical solution has been put forward for the consolida-tion of a composite foundation and the consolidation theoryhas been presented for the composite foundation consideringradial and vertical flows within the column the variation ofsoil permeability within the disturbed soil zone the depth-varying stress induced by multistage loading and time-and depth-dependent stress increment along with differentdistribution patterns of soil permeability

The above results have important reference value to studythe reinforcement mechanism of composite foundation Butthe studies of consolidation characteristic of CCSG pilecomposite foundation which is new under complicatedconditions and multivariate have not been reported yet Theauthors have tried to study the consolidation calculationmethod of CCSG pile composite foundation in simple casesbut the influence factors of CCSG pile composite foundationconsolidation characteristic are not considered such as thechange of horizontal penetration parameter in influence arearadial flow in sand shell and the impervious character of lowgrade concrete core pile

This paper introduces a series of field tests on CCSG pilecomposite foundation and the consolidation analysis modelof CCSG pile composite foundation is established which isbased on equal strain hypothesis and considering the changeof horizontal penetration parameters in influence zone andradial flow within sand-gravel shell The consolidation gen-eral solution was obtained by the theoretical derivationThe analytical solution was finally validated with the dataobtained from field tests and has verified the correctness ofthe theoretical solution

2 General Solution for Consolidation ofCCSG Pile Composite Foundation

21 Calculation Diagram The idealized CCSG compositefoundation is shown in Figure 2 In this figure 119867 is thethickness of the soil 119903

119888 119903119908 119903119904 and 119903

119890are radius of the

concrete-cored pile the sand-gravel shell the smear zoneand the influence zone (consisting of strong smear zone



119899are


119904and 119896





119904are






119903(119903)





1205871199032

119888120590119888+ 120587 (119903

2

119890minus 1199032

119908) 120590119904+ 120587 (119903

2

119908minus 1199032

119888) 120590119908= 1205871199032

1198901205900

120590119888

119864119888

=

(120590119904minus 119906119904)

119864

=

(120590119908minus 119906119908)

119864119908

= 120576V

(1)

where 120590119908 120590119888 and 120590





2minus1199042)(1198992minus1))119864

119899+((1199042minus1)(119899

2minus1))119864





119906119904=

int

119903119890

119903119908

2120587119903119906119904119889119903

120587 (1199032

119890minus 1199032

119908)

(2)

119906119908=

int

119903119908

119903119888

2120587119903119906119908119889119903

120587 (1199032

119908minus 1199032

119888)

(3)



120576V =11989921205900minus (1198992minus 1) 119906

119904minus (1 minus 119886

2) 119906119908

119864 [120572 + (1198992minus 1 + 119884)]

=

11989921205900minus (1198992minus 1198862) 119906

119864 [120572 + (1198992minus 1 + 119884)]

(4)


120597120576V

120597119905

= minus

1198992minus 1198862

119864 [120572 + (1198992minus 1 + 119884)]

120597119906

120597119905

(5)


119906 =

1

120587 (1199032

119890minus 1199032

119888)

(int

119903119908

119903119888

2120587119903119906119908119889119903+ int

119903119890

119903119908

2120587119903119906119904119889119903)

=

[(1 minus 1198862) 119906119908+ (1198992minus 1) 119906

119904]

(1198992minus 1198862)

(6)


to the column 119899 = 119903119890119903119908and 119886 = 119903






119896119903(119903) = 119896

ℎ119891 (119903) (7)






120597

120597119903

[

119903 sdot 119896119903(119903)

120574119908

120597119906119904

120597119903

] = minus119903(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112) 119903

119908le 119903 le 119903

119890

(8)

120597

120597119903

(

119903 sdot 119896ℎ119908

120574119908

120597119906119908

120597119903

) = minus119903(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112) 119903

119888le 119903 le 119903

119908

(9)



A 119903 = 119903119890 120597119906119904120597119903 = 0

B 119903 = 119903119908 119906119904= 119906119908



119908 119896119903(119903119908)(120597119906119904120597119903) = 119896

ℎ119908(120597119906119908120597119903) (the radial



E 119911 = 0 119906119908= 0 119906 = 0





120597119906119904

120597119903

=

120574119908

2119896ℎ

(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112)[

1199032

119890

119903119891 (119903)

minus

119903

119891 (119903)

] (10)


119906119904(119903) = 119906

119908

1003816100381610038161003816119903=119903119908

+

120574119908

2119896ℎ

(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112)

times [1199032

1198901198600(119903) minus 119861

0(119903)]

(11)

1198600(119903) = int

119903

119903119908

119889120585

120585119891 (120585)

1198610(119903) = int

119903

119903119908

120585119889120585

119891 (120585)

(12)


119906119904= 119906119908

1003816100381610038161003816119903=119903119908

+

1199032

119890120574119908119865119888

2119896ℎ

(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112) (13)

where 119865119888= 2(119860

11199032

119890minus 1198611)1199032

1198901199032

119908(1198992minus 1) 119860

1= int

119903119890

119903119908

1199031198600(119903) 119889119903

and 1198611= int

119903119890

119903119908

1199031198610(119903)119889119903


120597119906119908

120597119903

=

120574119908

2119896ℎ119908

(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112)(

1199032

119888

119903

minus 119903) (14)


119906119908(119903) = 119906

119908

1003816100381610038161003816119903=119903119908

+

120574119908

2119896ℎ119908

(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112)

sdot [1199032

119908(

1

2

+ 1198862 ln 119903

119903119908

) minus

1

2

1199032]

(15)


119906119908= 119906119908

1003816100381610038161003816119903=119903119908

+

120574119908119877

8119896ℎ119908

(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112) (16)

where119877 = 1199032119908[(41198862(1minus119886

2))(ln 119903

119908minus1198862 ln 119886119903


(1 minus 1198864)(1 minus 119886

2) + 2]


119906 minus 119906119908= minus

(1198992minus 1)

119864 (120572 + 1198992minus 1 + 119884)

(

1199032

119890119865119888120574119908

2119896ℎ

minus

120574119908119877

8119896ℎ119908

)

120597119906

120597119905

+

1199032

119890119865119888119896V

2119896ℎ

1205972119906

1205971199112

minus [

(1 minus 1198862)

(1198992minus 1198862)

1199032

119890119865119888119896V

2119896ℎ

+

(1198992minus 1)

(1198992minus 1198862)

119877119896V119908

8119896ℎ119908

]

1205972119906119908

1205971199112

(17)


minus(1198992minus 1198862)

2

119864 (120572 + 1198992minus 1 + 119884)

120597119906

120597119905

= (1198862minus 1)

119896V119908

120574119908

1205972119906119908

1205971199112


119896V

120574119908

1205972119906119904

1205971199112

(18)


1205972119906119908

1205971199112= 119860

120597119906

120597119905

minus 119861

1205972119906

1205971199112 (19)


2)(119896V119908 minus 119896V)119864(120572 + 119899

2minus 1 + 119884)

119861 = (1198992minus 1198862)119896V(1 minus 119886

2)(119896V119908 minus 119896V)


119906119908= 119906 + 119862

120597119906

120597119905

minus 119863

1205972119906

1205971199112 (20)

where 119862 = (120574119908[(1198992minus 1)119896V119908 + 119896V]119864(120572 + 119899

2minus 1 +

119884)(119896V119908 minus 119896V))[1199032

1198901198651198882119896ℎ+ 119877(119899

2minus 1)8119896

ℎ119908] 119863 = (119896V119908119896V(119896V119908 minus

119896V))[1199032

1198901198651198882119896ℎ+ (1198992minus 1)119877(1 minus 119886

2)8119896ℎ119908]


Deduce from (20) as

120597119906119908

1205971199112=

1205972119906

1205971199112+ 119862

1205973119906

1205971199051205971199112minus 119863

1205974119906

1205971199114 (21)


119863

1205974119906

1205971199114minus 119862

1205973119906

1205971199051205971199112minus (119861 + 1)

1205972119906

1205971199112+ 119860

120597119906

120597119905

= 0 (22)



119906 (119911 119905) = 1205900

infin

sum

119898=1

2

119872

sin(119872119867

119911) 119890minus120573119898119905 (23)

119906119908(119911 119905) = 120590

0

infin

sum

119898=1

2

119872

[1 minus 119862120573119898+ 119863(

119872

119867

)

2

]

sdot sin(119872119867

119911) 119890minus120573119898119905

(24)

119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus120573119898119905 (25)

where119872 = ((2119898 + 1)2)120587 (119898 = 0 1 2 )

120573119898= 119864 (120572 + 119899

2minus 1 + 119884)119896V119908119896V(

119872

119867

)

2

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119877

(1 minus 1198862) 8119896ℎ119908

]

+ [(1198992minus 1) 119896V + 119896V119908]

times (120574119908

(1198992minus 1198862)

2

1 minus 1198862

sdot (

119867

119872

)

2

+ [(1198992minus 1) 119896V119908 + 119896V]

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119877

8119896ℎ119908

]

)

minus1

(26)


kw

kh

ks

o

kr(r)

rw rs re r



120573119898= 119864 (119899

2minus 1 + 119884)119896V119908119896V(

119872

119867

)

2

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119903

2

119908

8119896ℎ119908

]

+ [(1198992minus 1) 119896V + 119896V119908]

times (1205741199081198994(

119867

119872

)

2

+ [(1198992minus 1) 119896V119908 + 119896V]

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119903

2

119908

8119896ℎ119908

])

minus1

(27)



120573119898=

119896V119864

120574119908

(

119872

119867

)

2

= 119888V(119872

119867

)

2

119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus1198722

119879V

(28)

where 119888V = 119896V119864120574119908 119879V = 119888V1199051198672








119865119888=

1198992

1198992minus 1

119904 minus 1

120581119904 minus 1

ln (120581119904) minus (119904 minus 1)2

1198992(1 minus 120581)

+

2

1198992

(119904 minus 1) (120581119904 minus 1)

(1 minus 120581)2

ln 1120581

minus

2

1198994

119904 minus 1

1 minus 120581

times (

1199043minus 1

3

minus

1199042minus 1

3

) minus

1

1198994

(119904 minus 1) (120581119904 minus 1)

(1 minus 120581)2

times [

1199042minus 1

2

minus

(119904 minus 1) (120581119904 minus 1)

1 minus 120581

+

(120581119904 minus 1)2

(1 minus 120581)2ln 1120581

]

minus

1198992minus 1199042

1198994

(1 minus 119904)2

1 minus 120581

+ ln 119899119904

minus

3

4

+

411989921199042minus 1199044

41198994

(29)





120573119898119905 = 120591119898119879ℎ (30)


be expressed as

119879ℎ=

119888ℎ119905

41199032

119890

119888ℎ=

119864119896ℎ

120574119908

(31)


119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus120591119898119879ℎ

(32)

where

120591119898= 1199032

119890(120572 + 119899

2minus 1 + 119884)

times

119896V119908

119896ℎ

(

119872

119867

)

2

[

1199032

119890119865119888

2

+

(1198992minus 1)119877119896

ℎ

(1 minus 1198862) 8119896ℎ119908

]

+ (1198992minus 1) +

119896V119908

119896V

00

02

04

06

08

1000001 0001 001 01 1 10

n = 4

n = 6

n = 8

n = 10

Th

U


00

02

04

06

08

1000001 0001 001 01 1 10

Hdw = 20

Hdw = 50

Hdw = 100

Th

U


times (

(1198992minus 1198862)

2

1 minus 1198862

sdot

119896ℎ

119896V

(

119867

119872

)

2

+ [(1198992minus 1)

119896V119908

119896V

+ 1]

sdot (

1199032

119890119865119888

2

+

119877 (1198992minus 1) 119896

ℎ

8119896ℎ119908

)

)

minus1

(33)



119908(119889119908= 2119903119908)

119896ℎ119896V 119896119904119896ℎ 119896ℎ119896V119908 119896ℎ119896ℎ119908 and 119896





Figure no 119899 119886 119904 119867119889119908

119883 119884 119896119904119896ℎ

119896ℎ119896V 119896

ℎ119896V119908 119896

ℎ119908119896V119908


00

02

04

06

08

1000001 0001 001 01 1 10

Y = 20 X = 1000Y = 20 X = 2000

Y = 10 X = 1000Y = 10 X = 2000

Th

U


00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

a = 0 X = 1 Y = 1

a = 0 X = 1 Y = 10

a = 02

a = 05







00

02

04

06

08

1000001 0001 001 01 1 10

Th

khkhw = 00001

khkhw = 0001

khkhw = 001

khkhw = 1

U



00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

kskh = 02

kskh = 04

kskh = 06

kskh = 08





00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

s = 3s = 2

s = 15

s = 1


10

08

06

04

02

00

00 01 02 03 04 05 06 07 08 09 10

u120590

0

Th

zH = 02

zH = 03

zH = 04

zH = 06

zH = 08












119904119896ℎ In other




4 Case Application




Soillayer

Name of soillayer


0119868119875

119868119871

120572(1-2)MPaminus1 119864


ckPa 120593(∘)

AMiscellaneous

fill 20sim37


41

20 21

21 19

63

K63 + 046 K63 + 066 K63 + 087

K63 + 056 K63 + 076

Size units (m)

Monitoring sections















Pile (22m)




pole


Soil pressure box

Pile


7m

7m

7m

4m

4m

21

m215

m



50 100 150 200 250 3002

4

6

8

60

40

20

0

Time (day)Hei

ght (

m)

Settl

emen

t (cm

)


Hei

ght (

m)

50 100 150 200 250 3000

2

4

6

8

5040302010

Settl

emen

t (cm

)Time (day)










10

08

06

04

02

00

8

6

4

2

0

0 100 200 300 400 500

Fill height curve


U

d


50 100 150 200 250 3000

10

20

30

40

Time (day)

Pore

wat

er p

ress

ure (

kpa)

2m6m10m

14m18m





05

10152025303540

0 50 100 150 200 250 300

Pore

wat

er p

ress

ure (

kpa)

Time (day)

2m6m10m

14m18m











50 100 150 200 250 3000369

121518

Time (day)

ns

(a)

50 100 150 200 250 300Time (day)

0

02

04

06

08

1

N

(b)


0

3

6

9

12

15

18

0 50 100 150 200 250 300Time (day)

zH = 04

zH = 02

zH = 08

ns


cal equation




5 Conclusions














Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899are


119904and 119896





119904are






119903(119903)





1205871199032

119888120590119888+ 120587 (119903

2

119890minus 1199032

119908) 120590119904+ 120587 (119903

2

119908minus 1199032

119888) 120590119908= 1205871199032

1198901205900

120590119888

119864119888

=

(120590119904minus 119906119904)

119864

=

(120590119908minus 119906119908)

119864119908

= 120576V

(1)

where 120590119908 120590119888 and 120590





2minus1199042)(1198992minus1))119864

119899+((1199042minus1)(119899

2minus1))119864





119906119904=

int

119903119890

119903119908

2120587119903119906119904119889119903

120587 (1199032

119890minus 1199032

119908)

(2)

119906119908=

int

119903119908

119903119888

2120587119903119906119908119889119903

120587 (1199032

119908minus 1199032

119888)

(3)



120576V =11989921205900minus (1198992minus 1) 119906

119904minus (1 minus 119886

2) 119906119908

119864 [120572 + (1198992minus 1 + 119884)]

=

11989921205900minus (1198992minus 1198862) 119906

119864 [120572 + (1198992minus 1 + 119884)]

(4)


120597120576V

120597119905

= minus

1198992minus 1198862

119864 [120572 + (1198992minus 1 + 119884)]

120597119906

120597119905

(5)


119906 =

1

120587 (1199032

119890minus 1199032

119888)

(int

119903119908

119903119888

2120587119903119906119908119889119903+ int

119903119890

119903119908

2120587119903119906119904119889119903)

=

[(1 minus 1198862) 119906119908+ (1198992minus 1) 119906

119904]

(1198992minus 1198862)

(6)


to the column 119899 = 119903119890119903119908and 119886 = 119903






119896119903(119903) = 119896

ℎ119891 (119903) (7)






120597

120597119903

[

119903 sdot 119896119903(119903)

120574119908

120597119906119904

120597119903

] = minus119903(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112) 119903

119908le 119903 le 119903

119890

(8)

120597

120597119903

(

119903 sdot 119896ℎ119908

120574119908

120597119906119908

120597119903

) = minus119903(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112) 119903

119888le 119903 le 119903

119908

(9)



A 119903 = 119903119890 120597119906119904120597119903 = 0

B 119903 = 119903119908 119906119904= 119906119908



119908 119896119903(119903119908)(120597119906119904120597119903) = 119896

ℎ119908(120597119906119908120597119903) (the radial



E 119911 = 0 119906119908= 0 119906 = 0





120597119906119904

120597119903

=

120574119908

2119896ℎ

(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112)[

1199032

119890

119903119891 (119903)

minus

119903

119891 (119903)

] (10)


119906119904(119903) = 119906

119908

1003816100381610038161003816119903=119903119908

+

120574119908

2119896ℎ

(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112)

times [1199032

1198901198600(119903) minus 119861

0(119903)]

(11)

1198600(119903) = int

119903

119903119908

119889120585

120585119891 (120585)

1198610(119903) = int

119903

119903119908

120585119889120585

119891 (120585)

(12)


119906119904= 119906119908

1003816100381610038161003816119903=119903119908

+

1199032

119890120574119908119865119888

2119896ℎ

(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112) (13)

where 119865119888= 2(119860

11199032

119890minus 1198611)1199032

1198901199032

119908(1198992minus 1) 119860

1= int

119903119890

119903119908

1199031198600(119903) 119889119903

and 1198611= int

119903119890

119903119908

1199031198610(119903)119889119903


120597119906119908

120597119903

=

120574119908

2119896ℎ119908

(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112)(

1199032

119888

119903

minus 119903) (14)


119906119908(119903) = 119906

119908

1003816100381610038161003816119903=119903119908

+

120574119908

2119896ℎ119908

(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112)

sdot [1199032

119908(

1

2

+ 1198862 ln 119903

119903119908

) minus

1

2

1199032]

(15)


119906119908= 119906119908

1003816100381610038161003816119903=119903119908

+

120574119908119877

8119896ℎ119908

(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112) (16)

where119877 = 1199032119908[(41198862(1minus119886

2))(ln 119903

119908minus1198862 ln 119886119903


(1 minus 1198864)(1 minus 119886

2) + 2]


119906 minus 119906119908= minus

(1198992minus 1)

119864 (120572 + 1198992minus 1 + 119884)

(

1199032

119890119865119888120574119908

2119896ℎ

minus

120574119908119877

8119896ℎ119908

)

120597119906

120597119905

+

1199032

119890119865119888119896V

2119896ℎ

1205972119906

1205971199112

minus [

(1 minus 1198862)

(1198992minus 1198862)

1199032

119890119865119888119896V

2119896ℎ

+

(1198992minus 1)

(1198992minus 1198862)

119877119896V119908

8119896ℎ119908

]

1205972119906119908

1205971199112

(17)


minus(1198992minus 1198862)

2

119864 (120572 + 1198992minus 1 + 119884)

120597119906

120597119905

= (1198862minus 1)

119896V119908

120574119908

1205972119906119908

1205971199112


119896V

120574119908

1205972119906119904

1205971199112

(18)


1205972119906119908

1205971199112= 119860

120597119906

120597119905

minus 119861

1205972119906

1205971199112 (19)


2)(119896V119908 minus 119896V)119864(120572 + 119899

2minus 1 + 119884)

119861 = (1198992minus 1198862)119896V(1 minus 119886

2)(119896V119908 minus 119896V)


119906119908= 119906 + 119862

120597119906

120597119905

minus 119863

1205972119906

1205971199112 (20)

where 119862 = (120574119908[(1198992minus 1)119896V119908 + 119896V]119864(120572 + 119899

2minus 1 +

119884)(119896V119908 minus 119896V))[1199032

1198901198651198882119896ℎ+ 119877(119899

2minus 1)8119896

ℎ119908] 119863 = (119896V119908119896V(119896V119908 minus

119896V))[1199032

1198901198651198882119896ℎ+ (1198992minus 1)119877(1 minus 119886

2)8119896ℎ119908]


Deduce from (20) as

120597119906119908

1205971199112=

1205972119906

1205971199112+ 119862

1205973119906

1205971199051205971199112minus 119863

1205974119906

1205971199114 (21)


119863

1205974119906

1205971199114minus 119862

1205973119906

1205971199051205971199112minus (119861 + 1)

1205972119906

1205971199112+ 119860

120597119906

120597119905

= 0 (22)



119906 (119911 119905) = 1205900

infin

sum

119898=1

2

119872

sin(119872119867

119911) 119890minus120573119898119905 (23)

119906119908(119911 119905) = 120590

0

infin

sum

119898=1

2

119872

[1 minus 119862120573119898+ 119863(

119872

119867

)

2

]

sdot sin(119872119867

119911) 119890minus120573119898119905

(24)

119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus120573119898119905 (25)

where119872 = ((2119898 + 1)2)120587 (119898 = 0 1 2 )

120573119898= 119864 (120572 + 119899

2minus 1 + 119884)119896V119908119896V(

119872

119867

)

2

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119877

(1 minus 1198862) 8119896ℎ119908

]

+ [(1198992minus 1) 119896V + 119896V119908]

times (120574119908

(1198992minus 1198862)

2

1 minus 1198862

sdot (

119867

119872

)

2

+ [(1198992minus 1) 119896V119908 + 119896V]

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119877

8119896ℎ119908

]

)

minus1

(26)


kw

kh

ks

o

kr(r)

rw rs re r



120573119898= 119864 (119899

2minus 1 + 119884)119896V119908119896V(

119872

119867

)

2

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119903

2

119908

8119896ℎ119908

]

+ [(1198992minus 1) 119896V + 119896V119908]

times (1205741199081198994(

119867

119872

)

2

+ [(1198992minus 1) 119896V119908 + 119896V]

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119903

2

119908

8119896ℎ119908

])

minus1

(27)



120573119898=

119896V119864

120574119908

(

119872

119867

)

2

= 119888V(119872

119867

)

2

119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus1198722

119879V

(28)

where 119888V = 119896V119864120574119908 119879V = 119888V1199051198672








119865119888=

1198992

1198992minus 1

119904 minus 1

120581119904 minus 1

ln (120581119904) minus (119904 minus 1)2

1198992(1 minus 120581)

+

2

1198992

(119904 minus 1) (120581119904 minus 1)

(1 minus 120581)2

ln 1120581

minus

2

1198994

119904 minus 1

1 minus 120581

times (

1199043minus 1

3

minus

1199042minus 1

3

) minus

1

1198994

(119904 minus 1) (120581119904 minus 1)

(1 minus 120581)2

times [

1199042minus 1

2

minus

(119904 minus 1) (120581119904 minus 1)

1 minus 120581

+

(120581119904 minus 1)2

(1 minus 120581)2ln 1120581

]

minus

1198992minus 1199042

1198994

(1 minus 119904)2

1 minus 120581

+ ln 119899119904

minus

3

4

+

411989921199042minus 1199044

41198994

(29)





120573119898119905 = 120591119898119879ℎ (30)


be expressed as

119879ℎ=

119888ℎ119905

41199032

119890

119888ℎ=

119864119896ℎ

120574119908

(31)


119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus120591119898119879ℎ

(32)

where

120591119898= 1199032

119890(120572 + 119899

2minus 1 + 119884)

times

119896V119908

119896ℎ

(

119872

119867

)

2

[

1199032

119890119865119888

2

+

(1198992minus 1)119877119896

ℎ

(1 minus 1198862) 8119896ℎ119908

]

+ (1198992minus 1) +

119896V119908

119896V

00

02

04

06

08

1000001 0001 001 01 1 10

n = 4

n = 6

n = 8

n = 10

Th

U


00

02

04

06

08

1000001 0001 001 01 1 10

Hdw = 20

Hdw = 50

Hdw = 100

Th

U


times (

(1198992minus 1198862)

2

1 minus 1198862

sdot

119896ℎ

119896V

(

119867

119872

)

2

+ [(1198992minus 1)

119896V119908

119896V

+ 1]

sdot (

1199032

119890119865119888

2

+

119877 (1198992minus 1) 119896

ℎ

8119896ℎ119908

)

)

minus1

(33)



119908(119889119908= 2119903119908)

119896ℎ119896V 119896119904119896ℎ 119896ℎ119896V119908 119896ℎ119896ℎ119908 and 119896





Figure no 119899 119886 119904 119867119889119908

119883 119884 119896119904119896ℎ

119896ℎ119896V 119896

ℎ119896V119908 119896

ℎ119908119896V119908


00

02

04

06

08

1000001 0001 001 01 1 10

Y = 20 X = 1000Y = 20 X = 2000

Y = 10 X = 1000Y = 10 X = 2000

Th

U


00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

a = 0 X = 1 Y = 1

a = 0 X = 1 Y = 10

a = 02

a = 05







00

02

04

06

08

1000001 0001 001 01 1 10

Th

khkhw = 00001

khkhw = 0001

khkhw = 001

khkhw = 1

U



00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

kskh = 02

kskh = 04

kskh = 06

kskh = 08





00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

s = 3s = 2

s = 15

s = 1


10

08

06

04

02

00

00 01 02 03 04 05 06 07 08 09 10

u120590

0

Th

zH = 02

zH = 03

zH = 04

zH = 06

zH = 08












119904119896ℎ In other




4 Case Application




Soillayer

Name of soillayer


0119868119875

119868119871

120572(1-2)MPaminus1 119864


ckPa 120593(∘)

AMiscellaneous

fill 20sim37


41

20 21

21 19

63

K63 + 046 K63 + 066 K63 + 087

K63 + 056 K63 + 076

Size units (m)

Monitoring sections















Pile (22m)




pole


Soil pressure box

Pile


7m

7m

7m

4m

4m

21

m215

m



50 100 150 200 250 3002

4

6

8

60

40

20

0

Time (day)Hei

ght (

m)

Settl

emen

t (cm

)


Hei

ght (

m)

50 100 150 200 250 3000

2

4

6

8

5040302010

Settl

emen

t (cm

)Time (day)










10

08

06

04

02

00

8

6

4

2

0

0 100 200 300 400 500

Fill height curve


U

d


50 100 150 200 250 3000

10

20

30

40

Time (day)

Pore

wat

er p

ress

ure (

kpa)

2m6m10m

14m18m





05

10152025303540

0 50 100 150 200 250 300

Pore

wat

er p

ress

ure (

kpa)

Time (day)

2m6m10m

14m18m











50 100 150 200 250 3000369

121518

Time (day)

ns

(a)

50 100 150 200 250 300Time (day)

0

02

04

06

08

1

N

(b)


0

3

6

9

12

15

18

0 50 100 150 200 250 300Time (day)

zH = 04

zH = 02

zH = 08

ns


cal equation




5 Conclusions














Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597

120597119903

[

119903 sdot 119896119903(119903)

120574119908

120597119906119904

120597119903

] = minus119903(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112) 119903

119908le 119903 le 119903

119890

(8)

120597

120597119903

(

119903 sdot 119896ℎ119908

120574119908

120597119906119908

120597119903

) = minus119903(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112) 119903

119888le 119903 le 119903

119908

(9)



A 119903 = 119903119890 120597119906119904120597119903 = 0

B 119903 = 119903119908 119906119904= 119906119908



119908 119896119903(119903119908)(120597119906119904120597119903) = 119896

ℎ119908(120597119906119908120597119903) (the radial



E 119911 = 0 119906119908= 0 119906 = 0





120597119906119904

120597119903

=

120574119908

2119896ℎ

(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112)[

1199032

119890

119903119891 (119903)

minus

119903

119891 (119903)

] (10)


119906119904(119903) = 119906

119908

1003816100381610038161003816119903=119903119908

+

120574119908

2119896ℎ

(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112)

times [1199032

1198901198600(119903) minus 119861

0(119903)]

(11)

1198600(119903) = int

119903

119903119908

119889120585

120585119891 (120585)

1198610(119903) = int

119903

119903119908

120585119889120585

119891 (120585)

(12)


119906119904= 119906119908

1003816100381610038161003816119903=119903119908

+

1199032

119890120574119908119865119888

2119896ℎ

(

120597120576V

120597119905

+

119896V

120574119908

1205972119906119904

1205971199112) (13)

where 119865119888= 2(119860

11199032

119890minus 1198611)1199032

1198901199032

119908(1198992minus 1) 119860

1= int

119903119890

119903119908

1199031198600(119903) 119889119903

and 1198611= int

119903119890

119903119908

1199031198610(119903)119889119903


120597119906119908

120597119903

=

120574119908

2119896ℎ119908

(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112)(

1199032

119888

119903

minus 119903) (14)


119906119908(119903) = 119906

119908

1003816100381610038161003816119903=119903119908

+

120574119908

2119896ℎ119908

(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112)

sdot [1199032

119908(

1

2

+ 1198862 ln 119903

119903119908

) minus

1

2

1199032]

(15)


119906119908= 119906119908

1003816100381610038161003816119903=119903119908

+

120574119908119877

8119896ℎ119908

(

120597120576V

120597119905

+

119896V119908

120574119908

1205972119906119908

1205971199112) (16)

where119877 = 1199032119908[(41198862(1minus119886

2))(ln 119903

119908minus1198862 ln 119886119903


(1 minus 1198864)(1 minus 119886

2) + 2]


119906 minus 119906119908= minus

(1198992minus 1)

119864 (120572 + 1198992minus 1 + 119884)

(

1199032

119890119865119888120574119908

2119896ℎ

minus

120574119908119877

8119896ℎ119908

)

120597119906

120597119905

+

1199032

119890119865119888119896V

2119896ℎ

1205972119906

1205971199112

minus [

(1 minus 1198862)

(1198992minus 1198862)

1199032

119890119865119888119896V

2119896ℎ

+

(1198992minus 1)

(1198992minus 1198862)

119877119896V119908

8119896ℎ119908

]

1205972119906119908

1205971199112

(17)


minus(1198992minus 1198862)

2

119864 (120572 + 1198992minus 1 + 119884)

120597119906

120597119905

= (1198862minus 1)

119896V119908

120574119908

1205972119906119908

1205971199112


119896V

120574119908

1205972119906119904

1205971199112

(18)


1205972119906119908

1205971199112= 119860

120597119906

120597119905

minus 119861

1205972119906

1205971199112 (19)


2)(119896V119908 minus 119896V)119864(120572 + 119899

2minus 1 + 119884)

119861 = (1198992minus 1198862)119896V(1 minus 119886

2)(119896V119908 minus 119896V)


119906119908= 119906 + 119862

120597119906

120597119905

minus 119863

1205972119906

1205971199112 (20)

where 119862 = (120574119908[(1198992minus 1)119896V119908 + 119896V]119864(120572 + 119899

2minus 1 +

119884)(119896V119908 minus 119896V))[1199032

1198901198651198882119896ℎ+ 119877(119899

2minus 1)8119896

ℎ119908] 119863 = (119896V119908119896V(119896V119908 minus

119896V))[1199032

1198901198651198882119896ℎ+ (1198992minus 1)119877(1 minus 119886

2)8119896ℎ119908]


Deduce from (20) as

120597119906119908

1205971199112=

1205972119906

1205971199112+ 119862

1205973119906

1205971199051205971199112minus 119863

1205974119906

1205971199114 (21)


119863

1205974119906

1205971199114minus 119862

1205973119906

1205971199051205971199112minus (119861 + 1)

1205972119906

1205971199112+ 119860

120597119906

120597119905

= 0 (22)



119906 (119911 119905) = 1205900

infin

sum

119898=1

2

119872

sin(119872119867

119911) 119890minus120573119898119905 (23)

119906119908(119911 119905) = 120590

0

infin

sum

119898=1

2

119872

[1 minus 119862120573119898+ 119863(

119872

119867

)

2

]

sdot sin(119872119867

119911) 119890minus120573119898119905

(24)

119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus120573119898119905 (25)

where119872 = ((2119898 + 1)2)120587 (119898 = 0 1 2 )

120573119898= 119864 (120572 + 119899

2minus 1 + 119884)119896V119908119896V(

119872

119867

)

2

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119877

(1 minus 1198862) 8119896ℎ119908

]

+ [(1198992minus 1) 119896V + 119896V119908]

times (120574119908

(1198992minus 1198862)

2

1 minus 1198862

sdot (

119867

119872

)

2

+ [(1198992minus 1) 119896V119908 + 119896V]

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119877

8119896ℎ119908

]

)

minus1

(26)


kw

kh

ks

o

kr(r)

rw rs re r



120573119898= 119864 (119899

2minus 1 + 119884)119896V119908119896V(

119872

119867

)

2

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119903

2

119908

8119896ℎ119908

]

+ [(1198992minus 1) 119896V + 119896V119908]

times (1205741199081198994(

119867

119872

)

2

+ [(1198992minus 1) 119896V119908 + 119896V]

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119903

2

119908

8119896ℎ119908

])

minus1

(27)



120573119898=

119896V119864

120574119908

(

119872

119867

)

2

= 119888V(119872

119867

)

2

119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus1198722

119879V

(28)

where 119888V = 119896V119864120574119908 119879V = 119888V1199051198672








119865119888=

1198992

1198992minus 1

119904 minus 1

120581119904 minus 1

ln (120581119904) minus (119904 minus 1)2

1198992(1 minus 120581)

+

2

1198992

(119904 minus 1) (120581119904 minus 1)

(1 minus 120581)2

ln 1120581

minus

2

1198994

119904 minus 1

1 minus 120581

times (

1199043minus 1

3

minus

1199042minus 1

3

) minus

1

1198994

(119904 minus 1) (120581119904 minus 1)

(1 minus 120581)2

times [

1199042minus 1

2

minus

(119904 minus 1) (120581119904 minus 1)

1 minus 120581

+

(120581119904 minus 1)2

(1 minus 120581)2ln 1120581

]

minus

1198992minus 1199042

1198994

(1 minus 119904)2

1 minus 120581

+ ln 119899119904

minus

3

4

+

411989921199042minus 1199044

41198994

(29)





120573119898119905 = 120591119898119879ℎ (30)


be expressed as

119879ℎ=

119888ℎ119905

41199032

119890

119888ℎ=

119864119896ℎ

120574119908

(31)


119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus120591119898119879ℎ

(32)

where

120591119898= 1199032

119890(120572 + 119899

2minus 1 + 119884)

times

119896V119908

119896ℎ

(

119872

119867

)

2

[

1199032

119890119865119888

2

+

(1198992minus 1)119877119896

ℎ

(1 minus 1198862) 8119896ℎ119908

]

+ (1198992minus 1) +

119896V119908

119896V

00

02

04

06

08

1000001 0001 001 01 1 10

n = 4

n = 6

n = 8

n = 10

Th

U


00

02

04

06

08

1000001 0001 001 01 1 10

Hdw = 20

Hdw = 50

Hdw = 100

Th

U


times (

(1198992minus 1198862)

2

1 minus 1198862

sdot

119896ℎ

119896V

(

119867

119872

)

2

+ [(1198992minus 1)

119896V119908

119896V

+ 1]

sdot (

1199032

119890119865119888

2

+

119877 (1198992minus 1) 119896

ℎ

8119896ℎ119908

)

)

minus1

(33)



119908(119889119908= 2119903119908)

119896ℎ119896V 119896119904119896ℎ 119896ℎ119896V119908 119896ℎ119896ℎ119908 and 119896





Figure no 119899 119886 119904 119867119889119908

119883 119884 119896119904119896ℎ

119896ℎ119896V 119896

ℎ119896V119908 119896

ℎ119908119896V119908


00

02

04

06

08

1000001 0001 001 01 1 10

Y = 20 X = 1000Y = 20 X = 2000

Y = 10 X = 1000Y = 10 X = 2000

Th

U


00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

a = 0 X = 1 Y = 1

a = 0 X = 1 Y = 10

a = 02

a = 05







00

02

04

06

08

1000001 0001 001 01 1 10

Th

khkhw = 00001

khkhw = 0001

khkhw = 001

khkhw = 1

U



00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

kskh = 02

kskh = 04

kskh = 06

kskh = 08





00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

s = 3s = 2

s = 15

s = 1


10

08

06

04

02

00

00 01 02 03 04 05 06 07 08 09 10

u120590

0

Th

zH = 02

zH = 03

zH = 04

zH = 06

zH = 08












119904119896ℎ In other




4 Case Application




Soillayer

Name of soillayer


0119868119875

119868119871

120572(1-2)MPaminus1 119864


ckPa 120593(∘)

AMiscellaneous

fill 20sim37


41

20 21

21 19

63

K63 + 046 K63 + 066 K63 + 087

K63 + 056 K63 + 076

Size units (m)

Monitoring sections















Pile (22m)




pole


Soil pressure box

Pile


7m

7m

7m

4m

4m

21

m215

m



50 100 150 200 250 3002

4

6

8

60

40

20

0

Time (day)Hei

ght (

m)

Settl

emen

t (cm

)


Hei

ght (

m)

50 100 150 200 250 3000

2

4

6

8

5040302010

Settl

emen

t (cm

)Time (day)










10

08

06

04

02

00

8

6

4

2

0

0 100 200 300 400 500

Fill height curve


U

d


50 100 150 200 250 3000

10

20

30

40

Time (day)

Pore

wat

er p

ress

ure (

kpa)

2m6m10m

14m18m





05

10152025303540

0 50 100 150 200 250 300

Pore

wat

er p

ress

ure (

kpa)

Time (day)

2m6m10m

14m18m











50 100 150 200 250 3000369

121518

Time (day)

ns

(a)

50 100 150 200 250 300Time (day)

0

02

04

06

08

1

N

(b)


0

3

6

9

12

15

18

0 50 100 150 200 250 300Time (day)

zH = 04

zH = 02

zH = 08

ns


cal equation




5 Conclusions














Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Deduce from (20) as

120597119906119908

1205971199112=

1205972119906

1205971199112+ 119862

1205973119906

1205971199051205971199112minus 119863

1205974119906

1205971199114 (21)


119863

1205974119906

1205971199114minus 119862

1205973119906

1205971199051205971199112minus (119861 + 1)

1205972119906

1205971199112+ 119860

120597119906

120597119905

= 0 (22)



119906 (119911 119905) = 1205900

infin

sum

119898=1

2

119872

sin(119872119867

119911) 119890minus120573119898119905 (23)

119906119908(119911 119905) = 120590

0

infin

sum

119898=1

2

119872

[1 minus 119862120573119898+ 119863(

119872

119867

)

2

]

sdot sin(119872119867

119911) 119890minus120573119898119905

(24)

119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus120573119898119905 (25)

where119872 = ((2119898 + 1)2)120587 (119898 = 0 1 2 )

120573119898= 119864 (120572 + 119899

2minus 1 + 119884)119896V119908119896V(

119872

119867

)

2

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119877

(1 minus 1198862) 8119896ℎ119908

]

+ [(1198992minus 1) 119896V + 119896V119908]

times (120574119908

(1198992minus 1198862)

2

1 minus 1198862

sdot (

119867

119872

)

2

+ [(1198992minus 1) 119896V119908 + 119896V]

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119877

8119896ℎ119908

]

)

minus1

(26)


kw

kh

ks

o

kr(r)

rw rs re r



120573119898= 119864 (119899

2minus 1 + 119884)119896V119908119896V(

119872

119867

)

2

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119903

2

119908

8119896ℎ119908

]

+ [(1198992minus 1) 119896V + 119896V119908]

times (1205741199081198994(

119867

119872

)

2

+ [(1198992minus 1) 119896V119908 + 119896V]

times [

1199032

119890119865119888

2119896ℎ

+

(1198992minus 1) 119903

2

119908

8119896ℎ119908

])

minus1

(27)



120573119898=

119896V119864

120574119908

(

119872

119867

)

2

= 119888V(119872

119867

)

2

119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus1198722

119879V

(28)

where 119888V = 119896V119864120574119908 119879V = 119888V1199051198672








119865119888=

1198992

1198992minus 1

119904 minus 1

120581119904 minus 1

ln (120581119904) minus (119904 minus 1)2

1198992(1 minus 120581)

+

2

1198992

(119904 minus 1) (120581119904 minus 1)

(1 minus 120581)2

ln 1120581

minus

2

1198994

119904 minus 1

1 minus 120581

times (

1199043minus 1

3

minus

1199042minus 1

3

) minus

1

1198994

(119904 minus 1) (120581119904 minus 1)

(1 minus 120581)2

times [

1199042minus 1

2

minus

(119904 minus 1) (120581119904 minus 1)

1 minus 120581

+

(120581119904 minus 1)2

(1 minus 120581)2ln 1120581

]

minus

1198992minus 1199042

1198994

(1 minus 119904)2

1 minus 120581

+ ln 119899119904

minus

3

4

+

411989921199042minus 1199044

41198994

(29)





120573119898119905 = 120591119898119879ℎ (30)


be expressed as

119879ℎ=

119888ℎ119905

41199032

119890

119888ℎ=

119864119896ℎ

120574119908

(31)


119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus120591119898119879ℎ

(32)

where

120591119898= 1199032

119890(120572 + 119899

2minus 1 + 119884)

times

119896V119908

119896ℎ

(

119872

119867

)

2

[

1199032

119890119865119888

2

+

(1198992minus 1)119877119896

ℎ

(1 minus 1198862) 8119896ℎ119908

]

+ (1198992minus 1) +

119896V119908

119896V

00

02

04

06

08

1000001 0001 001 01 1 10

n = 4

n = 6

n = 8

n = 10

Th

U


00

02

04

06

08

1000001 0001 001 01 1 10

Hdw = 20

Hdw = 50

Hdw = 100

Th

U


times (

(1198992minus 1198862)

2

1 minus 1198862

sdot

119896ℎ

119896V

(

119867

119872

)

2

+ [(1198992minus 1)

119896V119908

119896V

+ 1]

sdot (

1199032

119890119865119888

2

+

119877 (1198992minus 1) 119896

ℎ

8119896ℎ119908

)

)

minus1

(33)



119908(119889119908= 2119903119908)

119896ℎ119896V 119896119904119896ℎ 119896ℎ119896V119908 119896ℎ119896ℎ119908 and 119896





Figure no 119899 119886 119904 119867119889119908

119883 119884 119896119904119896ℎ

119896ℎ119896V 119896

ℎ119896V119908 119896

ℎ119908119896V119908


00

02

04

06

08

1000001 0001 001 01 1 10

Y = 20 X = 1000Y = 20 X = 2000

Y = 10 X = 1000Y = 10 X = 2000

Th

U


00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

a = 0 X = 1 Y = 1

a = 0 X = 1 Y = 10

a = 02

a = 05







00

02

04

06

08

1000001 0001 001 01 1 10

Th

khkhw = 00001

khkhw = 0001

khkhw = 001

khkhw = 1

U



00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

kskh = 02

kskh = 04

kskh = 06

kskh = 08





00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

s = 3s = 2

s = 15

s = 1


10

08

06

04

02

00

00 01 02 03 04 05 06 07 08 09 10

u120590

0

Th

zH = 02

zH = 03

zH = 04

zH = 06

zH = 08












119904119896ℎ In other




4 Case Application




Soillayer

Name of soillayer


0119868119875

119868119871

120572(1-2)MPaminus1 119864


ckPa 120593(∘)

AMiscellaneous

fill 20sim37


41

20 21

21 19

63

K63 + 046 K63 + 066 K63 + 087

K63 + 056 K63 + 076

Size units (m)

Monitoring sections















Pile (22m)




pole


Soil pressure box

Pile


7m

7m

7m

4m

4m

21

m215

m



50 100 150 200 250 3002

4

6

8

60

40

20

0

Time (day)Hei

ght (

m)

Settl

emen

t (cm

)


Hei

ght (

m)

50 100 150 200 250 3000

2

4

6

8

5040302010

Settl

emen

t (cm

)Time (day)










10

08

06

04

02

00

8

6

4

2

0

0 100 200 300 400 500

Fill height curve


U

d


50 100 150 200 250 3000

10

20

30

40

Time (day)

Pore

wat

er p

ress

ure (

kpa)

2m6m10m

14m18m





05

10152025303540

0 50 100 150 200 250 300

Pore

wat

er p

ress

ure (

kpa)

Time (day)

2m6m10m

14m18m











50 100 150 200 250 3000369

121518

Time (day)

ns

(a)

50 100 150 200 250 300Time (day)

0

02

04

06

08

1

N

(b)


0

3

6

9

12

15

18

0 50 100 150 200 250 300Time (day)

zH = 04

zH = 02

zH = 08

ns


cal equation




5 Conclusions














Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119865119888=

1198992

1198992minus 1

119904 minus 1

120581119904 minus 1

ln (120581119904) minus (119904 minus 1)2

1198992(1 minus 120581)

+

2

1198992

(119904 minus 1) (120581119904 minus 1)

(1 minus 120581)2

ln 1120581

minus

2

1198994

119904 minus 1

1 minus 120581

times (

1199043minus 1

3

minus

1199042minus 1

3

) minus

1

1198994

(119904 minus 1) (120581119904 minus 1)

(1 minus 120581)2

times [

1199042minus 1

2

minus

(119904 minus 1) (120581119904 minus 1)

1 minus 120581

+

(120581119904 minus 1)2

(1 minus 120581)2ln 1120581

]

minus

1198992minus 1199042

1198994

(1 minus 119904)2

1 minus 120581

+ ln 119899119904

minus

3

4

+

411989921199042minus 1199044

41198994

(29)





120573119898119905 = 120591119898119879ℎ (30)


be expressed as

119879ℎ=

119888ℎ119905

41199032

119890

119888ℎ=

119864119896ℎ

120574119908

(31)


119880 = 1 minus

infin

sum

119898=1

2

1198722119890minus120591119898119879ℎ

(32)

where

120591119898= 1199032

119890(120572 + 119899

2minus 1 + 119884)

times

119896V119908

119896ℎ

(

119872

119867

)

2

[

1199032

119890119865119888

2

+

(1198992minus 1)119877119896

ℎ

(1 minus 1198862) 8119896ℎ119908

]

+ (1198992minus 1) +

119896V119908

119896V

00

02

04

06

08

1000001 0001 001 01 1 10

n = 4

n = 6

n = 8

n = 10

Th

U


00

02

04

06

08

1000001 0001 001 01 1 10

Hdw = 20

Hdw = 50

Hdw = 100

Th

U


times (

(1198992minus 1198862)

2

1 minus 1198862

sdot

119896ℎ

119896V

(

119867

119872

)

2

+ [(1198992minus 1)

119896V119908

119896V

+ 1]

sdot (

1199032

119890119865119888

2

+

119877 (1198992minus 1) 119896

ℎ

8119896ℎ119908

)

)

minus1

(33)



119908(119889119908= 2119903119908)

119896ℎ119896V 119896119904119896ℎ 119896ℎ119896V119908 119896ℎ119896ℎ119908 and 119896





Figure no 119899 119886 119904 119867119889119908

119883 119884 119896119904119896ℎ

119896ℎ119896V 119896

ℎ119896V119908 119896

ℎ119908119896V119908


00

02

04

06

08

1000001 0001 001 01 1 10

Y = 20 X = 1000Y = 20 X = 2000

Y = 10 X = 1000Y = 10 X = 2000

Th

U


00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

a = 0 X = 1 Y = 1

a = 0 X = 1 Y = 10

a = 02

a = 05







00

02

04

06

08

1000001 0001 001 01 1 10

Th

khkhw = 00001

khkhw = 0001

khkhw = 001

khkhw = 1

U



00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

kskh = 02

kskh = 04

kskh = 06

kskh = 08





00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

s = 3s = 2

s = 15

s = 1


10

08

06

04

02

00

00 01 02 03 04 05 06 07 08 09 10

u120590

0

Th

zH = 02

zH = 03

zH = 04

zH = 06

zH = 08












119904119896ℎ In other




4 Case Application




Soillayer

Name of soillayer


0119868119875

119868119871

120572(1-2)MPaminus1 119864


ckPa 120593(∘)

AMiscellaneous

fill 20sim37


41

20 21

21 19

63

K63 + 046 K63 + 066 K63 + 087

K63 + 056 K63 + 076

Size units (m)

Monitoring sections















Pile (22m)




pole


Soil pressure box

Pile


7m

7m

7m

4m

4m

21

m215

m



50 100 150 200 250 3002

4

6

8

60

40

20

0

Time (day)Hei

ght (

m)

Settl

emen

t (cm

)


Hei

ght (

m)

50 100 150 200 250 3000

2

4

6

8

5040302010

Settl

emen

t (cm

)Time (day)










10

08

06

04

02

00

8

6

4

2

0

0 100 200 300 400 500

Fill height curve


U

d


50 100 150 200 250 3000

10

20

30

40

Time (day)

Pore

wat

er p

ress

ure (

kpa)

2m6m10m

14m18m





05

10152025303540

0 50 100 150 200 250 300

Pore

wat

er p

ress

ure (

kpa)

Time (day)

2m6m10m

14m18m











50 100 150 200 250 3000369

121518

Time (day)

ns

(a)

50 100 150 200 250 300Time (day)

0

02

04

06

08

1

N

(b)


0

3

6

9

12

15

18

0 50 100 150 200 250 300Time (day)

zH = 04

zH = 02

zH = 08

ns


cal equation




5 Conclusions














Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Figure no 119899 119886 119904 119867119889119908

119883 119884 119896119904119896ℎ

119896ℎ119896V 119896

ℎ119896V119908 119896

ℎ119908119896V119908


00

02

04

06

08

1000001 0001 001 01 1 10

Y = 20 X = 1000Y = 20 X = 2000

Y = 10 X = 1000Y = 10 X = 2000

Th

U


00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

a = 0 X = 1 Y = 1

a = 0 X = 1 Y = 10

a = 02

a = 05







00

02

04

06

08

1000001 0001 001 01 1 10

Th

khkhw = 00001

khkhw = 0001

khkhw = 001

khkhw = 1

U



00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

kskh = 02

kskh = 04

kskh = 06

kskh = 08





00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

s = 3s = 2

s = 15

s = 1


10

08

06

04

02

00

00 01 02 03 04 05 06 07 08 09 10

u120590

0

Th

zH = 02

zH = 03

zH = 04

zH = 06

zH = 08












119904119896ℎ In other




4 Case Application




Soillayer

Name of soillayer


0119868119875

119868119871

120572(1-2)MPaminus1 119864


ckPa 120593(∘)

AMiscellaneous

fill 20sim37


41

20 21

21 19

63

K63 + 046 K63 + 066 K63 + 087

K63 + 056 K63 + 076

Size units (m)

Monitoring sections















Pile (22m)




pole


Soil pressure box

Pile


7m

7m

7m

4m

4m

21

m215

m



50 100 150 200 250 3002

4

6

8

60

40

20

0

Time (day)Hei

ght (

m)

Settl

emen

t (cm

)


Hei

ght (

m)

50 100 150 200 250 3000

2

4

6

8

5040302010

Settl

emen

t (cm

)Time (day)










10

08

06

04

02

00

8

6

4

2

0

0 100 200 300 400 500

Fill height curve


U

d


50 100 150 200 250 3000

10

20

30

40

Time (day)

Pore

wat

er p

ress

ure (

kpa)

2m6m10m

14m18m





05

10152025303540

0 50 100 150 200 250 300

Pore

wat

er p

ress

ure (

kpa)

Time (day)

2m6m10m

14m18m











50 100 150 200 250 3000369

121518

Time (day)

ns

(a)

50 100 150 200 250 300Time (day)

0

02

04

06

08

1

N

(b)


0

3

6

9

12

15

18

0 50 100 150 200 250 300Time (day)

zH = 04

zH = 02

zH = 08

ns


cal equation




5 Conclusions














Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00

02

04

06

08

1000001 0001 001 01 1 10

Th

U

s = 3s = 2

s = 15

s = 1


10

08

06

04

02

00

00 01 02 03 04 05 06 07 08 09 10

u120590

0

Th

zH = 02

zH = 03

zH = 04

zH = 06

zH = 08












119904119896ℎ In other




4 Case Application




Soillayer

Name of soillayer


0119868119875

119868119871

120572(1-2)MPaminus1 119864


ckPa 120593(∘)

AMiscellaneous

fill 20sim37


41

20 21

21 19

63

K63 + 046 K63 + 066 K63 + 087

K63 + 056 K63 + 076

Size units (m)

Monitoring sections















Pile (22m)




pole


Soil pressure box

Pile


7m

7m

7m

4m

4m

21

m215

m



50 100 150 200 250 3002

4

6

8

60

40

20

0

Time (day)Hei

ght (

m)

Settl

emen

t (cm

)


Hei

ght (

m)

50 100 150 200 250 3000

2

4

6

8

5040302010

Settl

emen

t (cm

)Time (day)










10

08

06

04

02

00

8

6

4

2

0

0 100 200 300 400 500

Fill height curve


U

d


50 100 150 200 250 3000

10

20

30

40

Time (day)

Pore

wat

er p

ress

ure (

kpa)

2m6m10m

14m18m





05

10152025303540

0 50 100 150 200 250 300

Pore

wat

er p

ress

ure (

kpa)

Time (day)

2m6m10m

14m18m











50 100 150 200 250 3000369

121518

Time (day)

ns

(a)

50 100 150 200 250 300Time (day)

0

02

04

06

08

1

N

(b)


0

3

6

9

12

15

18

0 50 100 150 200 250 300Time (day)

zH = 04

zH = 02

zH = 08

ns


cal equation




5 Conclusions














Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Soillayer

Name of soillayer


0119868119875

119868119871

120572(1-2)MPaminus1 119864


ckPa 120593(∘)

AMiscellaneous

fill 20sim37


41

20 21

21 19

63

K63 + 046 K63 + 066 K63 + 087

K63 + 056 K63 + 076

Size units (m)

Monitoring sections















Pile (22m)




pole


Soil pressure box

Pile


7m

7m

7m

4m

4m

21

m215

m



50 100 150 200 250 3002

4

6

8

60

40

20

0

Time (day)Hei

ght (

m)

Settl

emen

t (cm

)


Hei

ght (

m)

50 100 150 200 250 3000

2

4

6

8

5040302010

Settl

emen

t (cm

)Time (day)










10

08

06

04

02

00

8

6

4

2

0

0 100 200 300 400 500

Fill height curve


U

d


50 100 150 200 250 3000

10

20

30

40

Time (day)

Pore

wat

er p

ress

ure (

kpa)

2m6m10m

14m18m





05

10152025303540

0 50 100 150 200 250 300

Pore

wat

er p

ress

ure (

kpa)

Time (day)

2m6m10m

14m18m











50 100 150 200 250 3000369

121518

Time (day)

ns

(a)

50 100 150 200 250 300Time (day)

0

02

04

06

08

1

N

(b)


0

3

6

9

12

15

18

0 50 100 150 200 250 300Time (day)

zH = 04

zH = 02

zH = 08

ns


cal equation




5 Conclusions














Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Pile (22m)




pole


Soil pressure box

Pile


7m

7m

7m

4m

4m

21

m215

m



50 100 150 200 250 3002

4

6

8

60

40

20

0

Time (day)Hei

ght (

m)

Settl

emen

t (cm

)


Hei

ght (

m)

50 100 150 200 250 3000

2

4

6

8

5040302010

Settl

emen

t (cm

)Time (day)










10

08

06

04

02

00

8

6

4

2

0

0 100 200 300 400 500

Fill height curve


U

d


50 100 150 200 250 3000

10

20

30

40

Time (day)

Pore

wat

er p

ress

ure (

kpa)

2m6m10m

14m18m





05

10152025303540

0 50 100 150 200 250 300

Pore

wat

er p

ress

ure (

kpa)

Time (day)

2m6m10m

14m18m











50 100 150 200 250 3000369

121518

Time (day)

ns

(a)

50 100 150 200 250 300Time (day)

0

02

04

06

08

1

N

(b)


0

3

6

9

12

15

18

0 50 100 150 200 250 300Time (day)

zH = 04

zH = 02

zH = 08

ns


cal equation




5 Conclusions














Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10

08

06

04

02

00

8

6

4

2

0

0 100 200 300 400 500

Fill height curve


U

d


50 100 150 200 250 3000

10

20

30

40

Time (day)

Pore

wat

er p

ress

ure (

kpa)

2m6m10m

14m18m





05

10152025303540

0 50 100 150 200 250 300

Pore

wat

er p

ress

ure (

kpa)

Time (day)

2m6m10m

14m18m











50 100 150 200 250 3000369

121518

Time (day)

ns

(a)

50 100 150 200 250 300Time (day)

0

02

04

06

08

1

N

(b)


0

3

6

9

12

15

18

0 50 100 150 200 250 300Time (day)

zH = 04

zH = 02

zH = 08

ns


cal equation




5 Conclusions














Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










50 100 150 200 250 3000369

121518

Time (day)

ns

(a)

50 100 150 200 250 300Time (day)

0

02

04

06

08

1

N

(b)


0

3

6

9

12

15

18

0 50 100 150 200 250 300Time (day)

zH = 04

zH = 02

zH = 08

ns


cal equation




5 Conclusions














Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Analytical Analysis and Field Test ...downloads.hindawi.com/journals/jam/2013/795962.pdf · water pressures of the sand-gravel shell, and the composite modulus of