Research Article Consolidation by Prefabricated Vertical...

Research ArticleConsolidation by Prefabricated Vertical Drains witha Threshold Gradient

Xiao Guo1 Kang-He Xie1 and Yue-Bao Deng2

1 Institute of Geotechnical Engineering Zhejiang University Hangzhou 310058 China2Faculty of Architectural Civil Engineering and Environment Ningbo University Ningbo 315211 China

Correspondence should be addressed to Xiao Guo 52968791qqcom

Received 6 August 2014 Accepted 1 December 2014 Published 16 December 2014

Academic Editor Yuming Qin

Copyright copy 2014 Xiao Guo et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper shows the development of an approximate analytical solution of radial consolidation by prefabricated vertical drainswith a threshold gradient To understand the effect of the threshold gradient on consolidation a parametric analysis was performedusing the present solution The applicability of the present solution was demonstrated in two cases wherein the comparisons withHansborsquos results and observed data were conducted It was found that (1) the flow with the threshold gradient would not occurinstantaneously throughout the whole unit cell Rather it gradually occurs from the vertical drain to the outside (2) the movingboundary would never reach the outer radius of influence if 119877 + 1 lt 119899 whereas it will reach the outer radius of influence at sometime (3) the excess pore pressure will not be dissipated completely but it will maintain a long-term stable value at the end ofconsolidation (4) the larger the threshold gradient is the greater the long-term excess pore pressure will be and (5) the presentsolution could predict the consolidation behavior in soft clay better than previous methods

1 Introduction

Preloading with sand or prefabricated vertical drains hasbeen successfully applied to accelerate the consolidation ofcompressible fine-grained soil deposits for decades Manyauthors have developed the consolidation theory for thedesign of a vertical drain [1ndash10] Among these theories theanalytic solutions developed by Barron [1] and Hansbo [3]remain popular and widely used because of their simplicityand ease of use

All of the above-mentioned solutions assume that Darcyrsquoslaw is valid However several researchers have providedlaboratory evidence that Darcyrsquos law should be corrected forthe effect of the threshold gradient in some fine-grained soilMiller and Low [11] found that the threshold gradient is thehydraulic gradient where no flow occurs This seems to besupported by experimental observations reported by Byerlee[12] Hansbo [13 14] proposed a non-Darcian flow law whichimplied that the presence of a threshold gradient valuewhen the gradient is less than this value may not cease butmay dramatically decrease the flow rate On the other handthere are several studies that refute this conclusion [15ndash18]

For example having conducted permeability tests on fournatural clays Tavenas et al [18] concluded that Darcyrsquos lawwas valid in soft clays for hydraulic gradients ranging from 01to 50 However it is nearly impossible to verify the validity ofDarcyrsquos law with very small gradients

Although there are some inconsistencies among thesestudies it is helpful to examine and assess the impact on theconsolidation process due to the existence of the thresholdgradient which was demonstrated by previous experimentsIn this study the flow law of the threshold gradient is assumedto be

V = 0 |119894| le 1198940

119896 (|119894| minus 1198940) |119894| gt 1198940

(1)

in which 119896 is the coefficient of permeability 119894 is the hydraulicgradient and 119894

0is the threshold gradient When 119894

0= 0 (1)

degenerates to Darcyrsquos lawFollowing equation (1) Pascal et al [19] obtained a nu-

merical solution using the finite difference method to arriveat approximate analytical solutions for one-dimensional con-solidation Xie et al [20] presented a general approximate

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 410390 9 pageshttpdxdoiorg1011552014410390

2 Mathematical Problems in Engineering

analytical solution for one-dimensional consolidation undera time-dependent loading condition and further exploredthe movement of the seepage boundary in a similar wayto (1) Thus (1) provides an excellent approximation of theeffective flow rate Importantly it may be simpler to depictthe characteristics of the consolidation problem with a non-Darcy flow by using (1)

Various attempts have characterized the influence of thethreshold gradient for one-dimension consolidation in clayHowever limited research has focused on the simulation ofconsolidation behavior using vertical drains that consider thethreshold gradient Hansbo [3] made some approximationsand deduced a simple closed-form solution for radial consoli-dation using vertical drains assuming thatDarcyrsquos law is validFollowing the approach of Hansbo [3] a general analyticalsolution with a threshold gradient for radial consolidationby vertical drain is developed in this paper Afterwardsthe influence of the threshold gradient on the consolidationprocess is investigated

2 Mathematical Model

21 Basic Assumptions Figure 1 presents the schematic rep-resentation of an axisymmetric unit cell with the presence ofa vertical drain in the center The vertical drain completelypenetrates the soft soil layer and the top of the layer is freelydraining but the bottom is completely impermeable Thusthe length of the drain well 119897 is equal to the thickness ofthe soft layer 119867 119903

119908is the equivalent radius of the drainage

well 119903119890is the radius of the influence zone which is the

radius of the soil cylinder that is dewatered by a drain119903119905is the outer radius of the seepage zone A widespread

surcharge loading of 1199020is simulated by the instantaneous

application to the upper boundary and this was kept constantduring the consolidation process Additionally 119903 and 119911 inFigure 1 indicate the radial coordinates and the depth respec-tively

The assumptions of this paper some similar to thosepreviously presented by Barron [1] and Hansbo [3] are listedas follows

(1) Radial and vertical flow can be considered separately(2) The quantity of water flowing through the disturbed

soil zone into the drain well is equal to the quantity ofwater flowing out of the drain

(3) The radial flow in the well can be ignored(4) Uniform loading is instantaneously imposed and it is

kept constant during the entire consolidation process(5) All vertical loads are initially carried by the excess

pore water pressure(6) Theflow in clay obeys (1) while the flow in the vertical

drains obeys Darcyrsquos law(7) All compressive strains within the soil mass occur in

a vertical direction and the equal strain condition isvalid

(8) The smear effect is ignored

Seepage zone

Moving boundaryDrain kw

I INonseepage zone

rwrt

re

Plan viewof I-I

Cross-sectional

view

z

H

r

q0

Figure 1 The cylindrical unit cell for the vertical drain ground

22 Basic Equation According to assumption (1) the radialand vertical flow can be considered separately Thus asshown in Figure 2 to satisfy the continuity condition forincompressible flow in the radial direction (119903-direction inFigure 2) the rate of the soil elementrsquos volume changeconsidering that the radial flow must exclusively be equal tothe net flow rate in the radial direction yields

120597119881

120597119905= Δ119902 =

120597119902119903

120597119903119889119903 (2)

where 119902119903is the net rates of flow at the radial direction 119902

119903can

be expressed as

119902119903= V119903119903119889120579119889119911 (3)

in which V119903is the flow velocity at the radial directionThe rate

of volume change of the soil element considering the radialflow exclusively can be expressed as

120597119881

120597119905= minus

120597120576V

120597119905119903119889119903119889120579119889119911 (4)

in which 120576V is the volumetric strain of the soil element if onlythe radial flow is considered

Substituting (3) and (4) into (2) the separate generalcontinuity equation for the radial flow under the cylindricalcoordinates system can be presented as

120597120576V

120597119905+120597V119903

120597119903+V119903

119903= 0 (5)

In this paper the flow of pore water in the soil describedby (1) will be incorporated in this study By expressing thehydraulic gradient 119894 in terms of the radial excess porepressure 119906

119903 and the unit weight of water 120574

119908 V119903can bewritten

as0 119894 lt 119894

0

V119903=119896ℎ

120574119908

120597119906119903

120597119903minus 119896ℎ1198940

119894 ge 1198940

(6)

Mathematical Problems in Engineering 3

z

r

qr

dz

dr

d120579

Figure 2 Consolidating the soil element around a vertical drain ina cylindrical system

Substituting (6) into (5) the inquiry consequently be-comes a moving boundary problem as illustrated in Figure 1and it is governed by the equations that follow

119896ℎ

120574119908

1

119903

120597

120597119903(119903120597119906119903

120597119903) minus

119896ℎ

1199031198940= minus

120597120576V

120597119905119903119908le 119903 le 119903

119905

119906119903= 1199060

119903119905le 119903 le 119903

119890

(7)

in which 1199060is the initial excess pore pressure and is often

assumed to be 1199060= 1199020 119903119905is the outer radius of the seepage

zone and the radius of the moving boundary out of which theexcess pore pressure is equal to the value of 119906

0

The rate of volumetric strain change of the soil elementin the sweep zone depends on the outer radius of the seepagezone 119903

119905 and eventually relays the threshold gradient 119894

0 It can

be written as follows120597120576V

120597119905= minus119898V

120597119906119903119905

120597119905119903119908le 119903 le 119903

119905 (8)

where119898V is the coefficient of the volume compressibility and119906119903119905is the average pore pressure of the seepage zone at any

depth The average pore pressure of the seepage zone can beobtained by

119906119903119905=

1

120587 (1199032

119905minus 1199032119908)int

119903119905

119903119908

2120587119903119906119903119889119903 119903

119908le 119903 le 119903

119905 (9)

in which 119906119903is the excess pore water pressure in the radial

directionThe consolidation equation for the continuity condition

of the drain well can be derived using assumptions (3) and(4) Imagining an elementary column of vertical drain witha height of 119889119911 as illustrated in Figure 3 the vertical flowthrough the elementary column can be written as

1199021=1198961199081205871199032

119908

120574119908

120597119906119908

120597119911119889119905 (10)

dw = 2rw

q1

q2

q1 + dq1

dz

Figure 3 Schematic diagram of the flow rate continuation in avertical drain

The radial inflow entering into the elementary columnthrough its outside face in a unit time 119889119905 is

1199022= 2120587119903

119908(119896ℎ

120574119908

120597119906119903

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119908

minus 119896ℎ1198940)119889119911119889119905 (11)

According to assumption (3) the continuity equation in theelementary column of the vertical drain yields

1199021+1205971199021

120597119911119889119911 + 119902

2= 1199021 (12)

Substituting (10) and (11) into (12) the following expressionis obtained

120597119906119903

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119908

minus 1198940+119896119908

119896ℎ

119903119908

2

1205972119906119908

1205971199112= 0 (13)

Equations (7) (8) (9) and (13) constitute the governingequations of consolidation by vertical drains incorporatingthe threshold gradient under the equal strain condition Theboundary conditions and initial conditions are as follows

A (120597119906119903120597119903)|119903=119903119905

= 1205741199081198940

B 119906119908|119911=0

= 0C (120597119906

119908120597119911)|119911=119897= 0

D 119906119903|119903=119903119908

= 119906119908

E 119906119903119905|119905=0= 1199060

F 119906119903(119903119905 119905) = 119906

0

Obviously the governing equation describes a movingboundary problem which is addressed by a time-dependentparameter 119903

119905


23 Solution of System Taking the integral of (7) with respectto 119903 for both sides and introducing boundary condition Ayields

120597119906119903

120597119903=120574119908

2119896ℎ

(1199032

119905

119903minus 119903)

120597120576V

120597119905+ 1205741199081198940

119903119908le 119903 le 119903

119905 (14)

From (14) the following relationship can be obtained

120597119906119903

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119908

=120574119908

2119896ℎ

(1199032

119905

119903119908

minus 119903119908)120597120576V

120597119905+ 1205741199081198940 (15)

Substituting (15) into (13) yields

1205972119906119908

1205971199112= minus

120574119908

119896119908

(1199032

119905

1199032119908

minus 1)120597120576V

120597119905 (16)

Following Hansborsquos derivation procedure [3] in whichthe variation of 120597120576V120597119905 with depth 119911 is ignored (16) trans-forms into a second-order variable coefficient equationIntegrating (16) along the drain length at a given time andcombining boundary conditions B and C the excess porepressure in the drain can be derived as

119906119908=120574119908

119896119908

(1199032

119905

1199032119908

minus 1)(119897119911 minus1

21199112)120597120576V

120597119905 (17)

By integrating (14) again and combining it with continuityconditionE the excess pore pressure in the seepage zone canbe derived as

119906119903=120574119908

2119896ℎ

(1199032

119905ln 119903

119903119908

minus1199032minus 1199032

119908

2)120597120576V

120597119905+ 1205741199081198940(119903 minus 119903119908) + 119906119908

119903119908le 119903 le 119903

119905

(18)

Substituting the expression of 119906119903into (4) the average

excess pore pressure of the seepage zone can be expressed as

119906119903119905=1205741199081199032

119905

2119896ℎ

119865119886119905

120597120576V

120597119905+ 1205741199081199031199081198651198871199051198940+ 119906119908 (19)

where

119865119886119905=

1199032

119905

1199032

119905minus 1199032119908

ln119903119905

119903119908

+1

4

1199032

119908

1199032

119905

minus3

4

119865119887119905=

21199032

119905

3 (119903119905119903119908+ 1199032119908)minus1

3

(20)

Setting 119899119905= 119903119905119903119908 the coefficients 119865

119886119905and 119865

119887119905can also be

expressed as

119865119886119905=

1198992

119905

1198992

119905minus 1

ln 119899119905+

1

41198992

119905

minus3

4

119865119887119905=

21198992

119905

3 (119899119905+ 1)

minus1

3

(21)

Substituting the expression of 119906119908into (19) and using the

relationship of (8) the following differential equation can beobtained

120597119906119903119905

120597119905= minus

2119888ℎ

1199032

119905120583119905

(119906119903119905minus 1198651198871199051199031199081205741199081198940) (22)

in which

119888ℎ=

119896ℎ

(120574119908119898V)

120583119905= 119865119886119905+ 120573119911(1198992

119905minus 1

1198992

119905

)

120573119911=119896ℎ

119896119908

(119897

119903119908

)

2

(2119911

119897119905

minus1199112

1198972

119905

) 119866 =119896ℎ

119896119908

(119897

119889119908

)

2

(23)

in which 119866 denotes well resistanceA fully analytical solution for (22) is difficult since there

is a time-dependent parameter 119903119905in the expression Xie

et al [20] demonstrated an approximate approach for one-dimensional consolidation in consideration of the thresholdgradient certifying the reasonability of the solution througha comparative analysis with a numerical solution Thereforein this study the approximate approach is applied similarlyto Xie et alrsquos study [20] by assuming that 119903

119905is a constant for

solving (22)With the above assumption and according to the initial

conditionE the average excess pore pressure of the seepagezone can be obtained from (27) as

119906119903119905= (1199060minus 1198651198871199051199031199081205741199081198940) 119890minus8119879ℎ1205831015840

119905 + 1198651198871199051199031199081205741199081198940

119903119908le 119903 le 119903

119905

(24)

in which

1205831015840

119905=1198992

119905

1198992120583119905 119879

ℎ=119888ℎ

1198892119890

119905 (25)

The time factor 119879ℎis a dimensionless parameter for

119905 Furthermore if 1198940approaches 0 (24) will degenerate to

Hansborsquos solution [3]Combining (8) (22) and (24) there is

120597120576V

120597119905=

2119896ℎ

1205741199081199032

119905120583119905

(1199060minus 1198651198871199051199031199081205741199081198940) 119890minus8119879ℎ1205831015840

119905 119903119908le 119903 le 119903

119905 (26)

Substituting (24) and (26) into (18) then

119906119903= [ln 119903

119903119908


119908

21199032

119905

+ 120573119911(1198992

119905minus 1

1198992

119905

)]

times(1199060minus 1205721199051199031199081205741199081198940) 119890minus8119879ℎ1205831015840

119905

120583119905

+ (119903

119903119908

minus 1) 1199031199081205741199081198940

119903119908le 119903 le 119903

119905

(27)

119906119908=1

120583119905

1198992

119905minus 1

1198992

119905

120573119911(1199060minus 1198651198871199051199031199081205741199081198940) 119890minus8119879ℎ1205831015840

119905 (28)


The excess pore pressure 119906119903 should satisfy the boundary

condition F Thus for radius 119903 = 119903119905 the following equation

is obtained

[ln119903119905

119903119908

minus1199032

119905minus 1199032

119908

21199032

119905

+ 120573119911(1198992

119905minus 1

1198992

119905

)]


119905

120583119905

+ (119903119905

119903119908

minus 1) 1199031199081205741199081198940= 1199060

(29)

It can be rewritten as120582119905

120583119905

(119877 minus 119865119887119905) 119890minus8119879ℎ1205831015840

119905 + (119899119905minus 1) = 119877 (30)

119877 is a dimensionless parameter for 1198940

Consider

120582119905= ln 119899

119905+ (120573119911minus1

2)(

1198992

119905minus 1

1198992

119905

) 119877 =1199060

1199031199081205741199081198940

(31)

When 119905 approachesinfin it can be deduced from (30) that

119899119905= 119877 + 1 (32)

Parameters 119899119905and 119903

119905can be determined from (30) if

the time factor 119879ℎ is specified and vice versa Furthermore

the average excess pore pressure 119906119903119905 and the excess pore

pressure 119906119903 in seepage zone at any depth can be obtained

from (24) and (29) respectivelyIf 119899119905= 119877+1 ge 119899 themoving boundarywill reach the outer

radius of the influence zone Then the correspondent timecan be obtained from (30) by replacing 119899

119905with 119899 The average

excess pore pressure 119906119903119905 and the excess pore pressure 119906

119903 can

be calculated by substituting 119903119905= 119903119890into (24) and (27) On

the other hand if 119877 + 1 lt 119899 the moving boundary will neverreach the outer radius of the influence zone during the wholeconsolidation process

The average pore pressure of the whole zone includes thesweep zone and the nonsweep zone at a given depth 119911 and itcan be defined as follows

119906119903=

1

120587 (1199032119890minus 1199032119908)int

119903119890

119903119908

2120587119903119906119903119889119903 =

1198992minus 1198992

119905

1198992 minus 11199060+1198992

119905minus 1

1198992 minus 1119906119903119905

(33)

Generally the average degree of consolidation in terms ofstress is defined as the ratio of the average effective stress tothe average total stress Thus the degree of consolidation at agiven depth 119880

119903 can be calculated by integrating the average

excess pore pressure over the radius (from 119903119908to 119903119890)

119880119903=

1

120587 (1199032119890minus 1199032119908)int

119903119890

119903119908

1199060minus 119906119903

1199060

2120587119903119889119903 =1198992

119905minus 1

1198992 minus 1(1 minus

119906119903119905

1199060

)

(34)

The average degree of consolidation for the whole soilstratumbased on the dissipation of excess pore pressure119880

119903 is

then computed by integrating119880119903over the soil layer thickness

119880119903=1

119897int

119897

0

119880119903119889119911 =

1

119897int

119897

0

1198992

119905minus 1


119906119903119905

1199060

)119889119911 (35)

Note here that (35) can be computed through numericalintegration Therefore an analytical solution system for theradial consolidation with the threshold gradient is obtained

3 Parametric Analysis

This section conducts a parametric analysis of the consolida-tion results that is 119906

119903 119880119903 and 119880

119903 for the present solution

The dimensionless parameters 120585 = 119911119897 and 119879ℎrepresent the

depth 119911 and the time 119905 respectivelyThe parameter values areherein listed 119903

119890= 0525m 119903

119908= 0035m 119899 = 119903

119890119903119908= 15

119898V = 02MPaminus1 119897 = 20m 119896ℎ= 20 times 10

minus8ms and119896119908= 10 times 10

minus3msThe motions of the moving boundary predicted by (30)

along with the various values of the dimensionless thresholdgradients 119877 are presented in Figure 4 In Figure 4(a) therestriction of the outer boundary of influence is disregarded(ie 119899 rarr infin) while in Figure 4(b) 119899 is set as 25 It can bedrawn that (1) the flow gradually occurs from the verticaldrain to the outside influence zone (2) when 119905 rarr infin thevalue of 119899

119905approaches 119877 + 1 which confirms to a certain

extent the validity of (32) (3) if 119877 + 1 lt 119899 the movingboundary will never reach the outer radius of the influencezone during thewhole consolidation process on the contraryit will only reach the influence zone if119877+1 ⩾ 119899 (4) regardlessof the restriction of the outer boundary of the influence zonethe value of 119899

119905at 119905 = infin would increase along with 119877 (5) as

the value of 119877 increases the time for the moving boundaryto reach the outer boundary of the influence zone decreasesunder the condition that 119877 + 1 ⩾ 119899

The motions of the moving boundary predicted at differ-ent depths for value of119877 = 10 are presented in Figure 5 It canbe observed that (1) when 119905 rarr infin the value of 119899

119905at different

depths approaches the same value 119899119905= 119877 + 1 (2) the deeper

the depth of 119911 the smaller the radius of themoving boundaryat time 119905 and (3) with the increased value of 119866 the motion ofthe moving boundary decelerates

When considering the threshold gradient the averageexcess pore pressure is related to the motion of the movingboundary Figure 6(a) shows the average excess pore pressurevariations with varying depths and Figure 6(b) shows thecorresponding moving boundary In the two figures thedotted lines represent the results of 119877 = 10 while the solidlines represent the results of 119877 = 20 The comparison showsthat (1) at the end of consolidation the average pore pressurewill not completely dissipate but it will maintain a long-termstable value (2) the larger the value of119877 the smaller the long-term average pore pressure and (3) if119877+1 lt 119899 the radial flowwithin the range from 119877 + 1 to 119899 does not occur during theconsolidation process which results in a slow dissipation ratein the pore pressure

A further comparison of the average degree of consoli-dation (119880

119903) for various values of 119877 is presented in Figure 7

It can be seen from the figure that (1) in the early stage ofconsolidation the difference in the average degree of consol-idation with a different 119877 is minuscule With a consolidationtime that has elapsed the effect of 119877 on the degree ofconsolidation becomes obvious and (2) as 119877 increases the


Table 1 Parameters and values used in the case at SkaE-Edeby

119889119890(m) 119889

119908(m) 119899 = 119889

119890119889119908

119862ℎ(m2y) 119896

ℎ(my) 119902

119908(m3y) 119897 (m)

1575 018 875 045lowast 002 100 12

lowastrefers to the solution determined from back-calculation method by fitting the field data from Hansborsquos study [14]

001 01 1 10 1000

10

20

30

40

50

R = 1

R = 5

R = 10

R = 25

R = 50

120585 = 05G = 816

nt

Th

(a) Disregard the restriction of the outer boundary of influence

001 01 1 10 1000

10

20

30

40

50

R = 1

R = 5

R = 10

R = 25

R = 50

120585 = 05G = 816

nt

Th

nt = R + 1 t = infin

n = 25

R + 1 = 11

R + 1 = 6

R + 1 = 2

(b) Assume 119899 = 25

Figure 4 Comparison of the motion of the moving boundary for various 119877

001 01 1 10 1000

2

4

6

8

10

12

nt

Th

R = 10

120585 = 0 G = 816

120585 = 05 G = 816

120585 = 1 G = 816

120585 = 0 G = 3265

120585 = 05 G = 3265

120585 = 1 G = 3265

Figure 5 Comparison of the motion of moving boundary underdifferent depths and 119866 (119877 = 10)

development of consolidation accelerates and the averagedegree of consolidation 119880

119903 increases

4 Application of Theory

The field tests at SkaE-Edeby located about 25 km west ofStockholm are among the oldest and best-documented tests

on the behavior of vertical drains throughout the world[14] The details of this work and the results of the analysiswere reported in Hansborsquos papers [14] The study includedthree test areas with sand drains 018m in diameter (testareas IsimIII) The excess pore pressure between depths of25 and 75m in test areas II and III was selected for thisstudy The parameters used in the comparison are listed inTable 1 Except for the value of the consolidation coefficient119862ℎ the other values are sampled from Hansborsquos study [14]

Test area II and test area III were surcharged with anequivalent loading of 32 kPa and 46 kPa The radial averagedegree of consolidation 119880

ℎ at a depth of 5m as computed

by the solutions of Hansbo [3] and the present paper isplotted in Figure 8 respectively The measured data arealso plotted in this figure to aid comparative analysis Theconsolidation degree calculated by the present solution witha consideration of the threshold gradient becomes relatedto the magnitude of the initial stress which is not the casein Hansborsquos solution [3] with a Darcian flow It can be seenfrom the figure that the solutions both by Hansbo [3] (119862

ℎ=

040m2y) and in the present paper (119862ℎ= 045m2y and

1198940= 0365) confer with the measured data Relatively the

newly present solution considering a threshold gradient forthe flow gives a better agreement with the field measuredvalue

Another case involves a radial consolidation test that wasconducted by Kim et al [21] which was performed on a largeblock sample using a large consolidometer (diameter = 12mheight = 20m) The boundary for the consolidation test was


10

08

06

04

02

00

00 02 04 06 08 10

120585=zl

uru0

Th = 01

Th = 1

Th = 5

Th = infin

Solid line R = 25

Th = 01

Th = 1

Th = 5

Th = infin

Dash line R = 10

(a) The average EPP

10

08

06

04

02

00

4 6 8 10 12 14 16 18 20

n

120585=zl

nt

Th = 01

Th = 1

Th = 5

Th = infin

Solid line R = 25

Th = 01

Th = 1

Th = 5

Th = infin

Dash line R = 10

(b) The corresponding radius of moving boundary

Figure 6 Comparison of the average EPP and the corresponding radius of moving boundary at depth with 119877 = 10 and 119877 = 25

001 01 1 10 100100

80

60

40

20

0

Ur

Th

R = 1

R = 5

R = 10

R = 15

R = 25

R = 50

R = 100

Figure 7 Comparison of the average radial degree of consolidationwith times for different 119877

particularly processed to ensure a solely radial consolidationcondition Finally the block sample was consolidated undera vertical pressure of 300 kPa for 3 weeks The input param-eters applied in the analysis are presented in Table 2 Theparameters are summarized from Kim et al [21] except theconsolidation coefficient 119862

ℎ which is determined using a

back-calculation methodThe predictions from Hansborsquos solution [3] under differ-

ent values of 119862ℎand through the present solution are plotted

0 1 2 3 4100

80

60

40

20

0

Consolidation time (years)

Hansbo (1981)

Area II Area III

Area II R = 100

Area III R = 144

Deg

ree o

f con

solid

atio

nUh

()

Figure 8 Comparison of consolidation curves and observed data atSkaE-Edeby

in Figure 9 together with Kim et alrsquos [21] experimental dataIt can be seen from the figure that all three predictions usingHansborsquos solution (1981) deviate from themeasured data withelapsed time A better prediction of the consolidation processis obtained using the present solution with 119862

ℎ= 139 times

10minus10m2s and 119894

0= 2667 Figure 9 indicates that the present

solution considering the threshold gradient could be a betterfit with the experimental results


Table 2 Parameters and values of Kimrsquos radial consolidation test

119889119890(m) 119889

119908(m) 119899 = 119889

119890119889119908

119862ℎ(m2s) 119896

ℎ(ms) 119902

119908(m3y) 119897 (m)

060 005 12 139 times 10minus10lowast

36 times 10minus10 50 2

lowastrefers to the solution determined from back-calculation method by fitting the experimental results from Kim et alrsquos study [21]

0 200 400 600 800 1000100

80

60

40

20

0

Measured data

Elapsed time (hours)

Ch = 139E minus 10

Ch = 100E minus 10

Ch = 0694E minus 10

Ch = 139E minus 10 RR = 45

Deg

ree o

f con

solid

atio

n (

)

Figure 9 Comparison of theoretical results and observed data byKim et al [21]

5 Conclusions

Following Hansborsquos approximate approach [3] the solutionwas developed for an equal strain consolidation theory ofthe vertical drain in consideration of the threshold gradientParametric analysis for the present solution was carried outin order to understand the effect of the threshold gradienton consolidation The applicability of the present solutionwas demonstrated in two cases wherein the comparisonswith Hansborsquos results and observed data were conductedThe following conclusions were obtained (1) The flow withthe threshold gradient does not occur throughout the wholeunit cell instantaneously It gradually occurs from the verticaldrain to the outside (2) The moving boundary would neverreach the outer radius of influence if 119877 + 1 lt 119899 otherwise itwill reach the outer radius of influence at some time (3) Theexcess pore pressure will not completely dissipate but it willmaintain a long-term stable value at the end of consolidation(4)The larger the threshold gradient the larger the long-termexcess pore pressure (5) The present solution can predictthe consolidation behavior in soft clay better than previousmethods

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The support of this research by the National Natural ScienceFoundation of China (nos 51179170 and 51308309) is grate-fully acknowledged

References

[1] R A Barron ldquoConsolidation of fine-grained soils by drainwellsrdquo Transactions of the American Society of Civil Engineersvol 113 pp 718ndash742 1948

[2] H Yoshikuni and H Nakanodo ldquoConsolidation of soils byvertical drain wells with finite permeabilityrdquo Soils Found vol14 no 2 pp e35ndashe46 1974

[3] S Hansbo ldquoConsolidation of fine-grained soils by prefabricateddrainsrdquo in Proceedings of the 10th International Conference onSoil Mechanics and Foundation Engineering A A Balkema Edvol 3 pp 677ndash682 Stockholm Sweden 1981

[4] A Onoue ldquoConsolidation by vertical drains taking well resis-tance and smear into considerationrdquo Soils and Foundations vol28 no 4 pp 165ndash174 1988

[5] G X Zeng and K H Xie ldquoNew development of the verticaldrain theoriesrdquo in Proceedings of the 12th International Confer-ence on Soil Mechanics and Foundation Engineering vol 2 pp1435ndash1438 Rio de Janeiro Brazil 1989

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

[7] C Rujikiatkamjorn and B Indraratna ldquoDesign procedure forvertical drains considering a linear variation of lateral perme-ability within the smear zonerdquo Canadian Geotechnical Journalvol 46 no 3 pp 270ndash280 2009

[8] C Y Ong J C Chai and T Hino ldquoDegree of consolidationof clayey deposit with partially penetrating vertical drainsrdquoGeotextiles and Geomembranes vol 34 pp 19ndash27 2012

[9] S Artidteang D T Bergado J Saowapakpiboon N Teer-achaikulpanich and A Kumar ldquoEnhancement of efficiency ofprefabricated vertical drains using surcharge vacuum and heatpreloadingrdquoGeosynthetics International vol 18 no 1 pp 35ndash472011

[10] B Indraratna C Rujikiatkamjorn J Ameratunga and P BoyleldquoPerformance and prediction of vacuum combined surchargeconsolidation at Port of Brisbanerdquo Journal of Geotechnical andGeoenvironmental Engineering vol 137 no 11 pp 1009ndash10182011

[11] R J Miller and P F Low ldquoThreshold gradient for water flow inclay systemrdquo Proceedings of Soil Science Society of America vol27 no 6 pp 605ndash609 1963

[12] J Byerlee ldquoFriction overpressure and fault normal compres-sionrdquo Geophysical Research Letters vol 17 no 12 pp 2109ndash21121990

[13] S Hansbo Consolidation of Clay with Special Reference toInfluence of Vertical Sand Drains vol 18 Swedish GeotechnicalInstitute 1960


[14] S Hansbo ldquoAspects of vertical drain design Darcian or non-Darcian flowrdquo Geotechnique vol 47 no 5 pp 983ndash992 1997

[15] H W Olsen ldquoDeviation from Darcys law in saturated claysrdquoProceedings of Soil Science Society of America vol 29 no 2 pp135ndash140 1965

[16] J K Mitchell Fundamentals of Soil Behaviour John Wiley ampSons New York NY USA 1976

[17] G Imai and Y-X Tang ldquoConstitutive equation of one-dimensional consolidation derived from inter-connected testsrdquoSoils and Foundations vol 32 no 2 pp 83ndash96 1992

[18] F Tavenas P Leblond P Jean and S Leroueil ldquoThe per-meability of natural soft clays Part I methods of laboratorymeasurementrdquoCanadianGeotechnical Journal vol 20 no 4 pp629ndash644 1983

[19] F Pascal H Pascal and D W Murray ldquoConsolidation withthreshold gradientsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 5 no 3 pp 247ndash2611981

[20] K-H Xie K Wang Y-L Wang and C-X Li ldquoAnalyticalsolution for one-dimensional consolidation of clayey soils witha threshold gradientrdquo Computers and Geotechnics vol 37 no 4pp 487ndash493 2010

[21] R Kim S-J HongM-J Lee andW Lee ldquoTime dependent wellresistance factor of PVDrdquoMarine Georesources and Geotechnol-ogy vol 29 no 2 pp 131ndash144 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


analytical solution for one-dimensional consolidation undera time-dependent loading condition and further exploredthe movement of the seepage boundary in a similar wayto (1) Thus (1) provides an excellent approximation of theeffective flow rate Importantly it may be simpler to depictthe characteristics of the consolidation problem with a non-Darcy flow by using (1)

Various attempts have characterized the influence of thethreshold gradient for one-dimension consolidation in clayHowever limited research has focused on the simulation ofconsolidation behavior using vertical drains that consider thethreshold gradient Hansbo [3] made some approximationsand deduced a simple closed-form solution for radial consoli-dation using vertical drains assuming thatDarcyrsquos law is validFollowing the approach of Hansbo [3] a general analyticalsolution with a threshold gradient for radial consolidationby vertical drain is developed in this paper Afterwardsthe influence of the threshold gradient on the consolidationprocess is investigated

2 Mathematical Model

21 Basic Assumptions Figure 1 presents the schematic rep-resentation of an axisymmetric unit cell with the presence ofa vertical drain in the center The vertical drain completelypenetrates the soft soil layer and the top of the layer is freelydraining but the bottom is completely impermeable Thusthe length of the drain well 119897 is equal to the thickness ofthe soft layer 119867 119903

119908is the equivalent radius of the drainage

well 119903119890is the radius of the influence zone which is the

radius of the soil cylinder that is dewatered by a drain119903119905is the outer radius of the seepage zone A widespread

surcharge loading of 1199020is simulated by the instantaneous

application to the upper boundary and this was kept constantduring the consolidation process Additionally 119903 and 119911 inFigure 1 indicate the radial coordinates and the depth respec-tively

The assumptions of this paper some similar to thosepreviously presented by Barron [1] and Hansbo [3] are listedas follows

(1) Radial and vertical flow can be considered separately(2) The quantity of water flowing through the disturbed

soil zone into the drain well is equal to the quantity ofwater flowing out of the drain

(3) The radial flow in the well can be ignored(4) Uniform loading is instantaneously imposed and it is

kept constant during the entire consolidation process(5) All vertical loads are initially carried by the excess

pore water pressure(6) Theflow in clay obeys (1) while the flow in the vertical

drains obeys Darcyrsquos law(7) All compressive strains within the soil mass occur in

a vertical direction and the equal strain condition isvalid

(8) The smear effect is ignored

Seepage zone

Moving boundaryDrain kw

I INonseepage zone

rwrt

re

Plan viewof I-I

Cross-sectional

view

z

H

r

q0

Figure 1 The cylindrical unit cell for the vertical drain ground

22 Basic Equation According to assumption (1) the radialand vertical flow can be considered separately Thus asshown in Figure 2 to satisfy the continuity condition forincompressible flow in the radial direction (119903-direction inFigure 2) the rate of the soil elementrsquos volume changeconsidering that the radial flow must exclusively be equal tothe net flow rate in the radial direction yields

120597119881

120597119905= Δ119902 =

120597119902119903

120597119903119889119903 (2)

where 119902119903is the net rates of flow at the radial direction 119902

119903can

be expressed as

119902119903= V119903119903119889120579119889119911 (3)

in which V119903is the flow velocity at the radial directionThe rate

of volume change of the soil element considering the radialflow exclusively can be expressed as

120597119881

120597119905= minus

120597120576V

120597119905119903119889119903119889120579119889119911 (4)

in which 120576V is the volumetric strain of the soil element if onlythe radial flow is considered

Substituting (3) and (4) into (2) the separate generalcontinuity equation for the radial flow under the cylindricalcoordinates system can be presented as

120597120576V

120597119905+120597V119903

120597119903+V119903

119903= 0 (5)

In this paper the flow of pore water in the soil describedby (1) will be incorporated in this study By expressing thehydraulic gradient 119894 in terms of the radial excess porepressure 119906

119903 and the unit weight of water 120574

119908 V119903can bewritten

as0 119894 lt 119894

0

V119903=119896ℎ

120574119908

120597119906119903

120597119903minus 119896ℎ1198940

119894 ge 1198940

(6)


z

r

qr

dz

dr

d120579



119896ℎ

120574119908

1

119903

120597

120597119903(119903120597119906119903

120597119903) minus

119896ℎ

1199031198940= minus

120597120576V

120597119905119903119908le 119903 le 119903

119905

119906119903= 1199060

119903119905le 119903 le 119903

119890

(7)




0



0 It can


120597119905= minus119898V

120597119906119903119905

120597119905119903119908le 119903 le 119903

119905 (8)



119906119903119905=

1

120587 (1199032

119905minus 1199032119908)int

119903119905

119903119908

2120587119903119906119903119889119903 119903

119908le 119903 le 119903

119905 (9)




1199021=1198961199081205871199032

119908

120574119908

120597119906119908

120597119911119889119905 (10)

dw = 2rw

q1

q2

q1 + dq1

dz



1199022= 2120587119903

119908(119896ℎ

120574119908

120597119906119903

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119908

minus 119896ℎ1198940)119889119911119889119905 (11)


1199021+1205971199021

120597119911119889119911 + 119902

2= 1199021 (12)


120597119906119903

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119908

minus 1198940+119896119908

119896ℎ

119903119908

2

1205972119906119908

1205971199112= 0 (13)


A (120597119906119903120597119903)|119903=119903119905

= 1205741199081198940

B 119906119908|119911=0

= 0C (120597119906

119908120597119911)|119911=119897= 0

D 119906119903|119903=119903119908

= 119906119908

E 119906119903119905|119905=0= 1199060

F 119906119903(119903119905 119905) = 119906

0


119905



120597119906119903

120597119903=120574119908

2119896ℎ

(1199032

119905

119903minus 119903)

120597120576V

120597119905+ 1205741199081198940

119903119908le 119903 le 119903

119905 (14)


120597119906119903

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119908

=120574119908

2119896ℎ

(1199032

119905

119903119908

minus 119903119908)120597120576V

120597119905+ 1205741199081198940 (15)


1205972119906119908

1205971199112= minus

120574119908

119896119908

(1199032

119905

1199032119908

minus 1)120597120576V

120597119905 (16)


119906119908=120574119908

119896119908

(1199032

119905

1199032119908

minus 1)(119897119911 minus1

21199112)120597120576V

120597119905 (17)


119906119903=120574119908

2119896ℎ

(1199032

119905ln 119903

119903119908


119908

2)120597120576V

120597119905+ 1205741199081198940(119903 minus 119903119908) + 119906119908

119903119908le 119903 le 119903

119905

(18)



119906119903119905=1205741199081199032

119905

2119896ℎ

119865119886119905

120597120576V

120597119905+ 1205741199081199031199081198651198871199051198940+ 119906119908 (19)

where

119865119886119905=

1199032

119905

1199032

119905minus 1199032119908

ln119903119905

119903119908

+1

4

1199032

119908

1199032

119905

minus3

4

119865119887119905=

21199032

119905

3 (119903119905119903119908+ 1199032119908)minus1

3

(20)


119886119905and 119865

119887119905can also be

expressed as

119865119886119905=

1198992

119905

1198992

119905minus 1

ln 119899119905+

1

41198992

119905

minus3

4

119865119887119905=

21198992

119905

3 (119899119905+ 1)

minus1

3

(21)



120597119906119903119905

120597119905= minus

2119888ℎ

1199032

119905120583119905

(119906119903119905minus 1198651198871199051199031199081205741199081198940) (22)

in which

119888ℎ=

119896ℎ

(120574119908119898V)

120583119905= 119865119886119905+ 120573119911(1198992

119905minus 1

1198992

119905

)

120573119911=119896ℎ

119896119908

(119897

119903119908

)

2

(2119911

119897119905

minus1199112

1198972

119905

) 119866 =119896ℎ

119896119908

(119897

119889119908

)

2

(23)







119906119903119905= (1199060minus 1198651198871199051199031199081205741199081198940) 119890minus8119879ℎ1205831015840

119905 + 1198651198871199051199031199081205741199081198940

119903119908le 119903 le 119903

119905

(24)

in which

1205831015840

119905=1198992

119905

1198992120583119905 119879

ℎ=119888ℎ

1198892119890

119905 (25)




120597120576V

120597119905=

2119896ℎ

1205741199081199032

119905120583119905

(1199060minus 1198651198871199051199031199081205741199081198940) 119890minus8119879ℎ1205831015840

119905 119903119908le 119903 le 119903

119905 (26)


119906119903= [ln 119903

119903119908


119908

21199032

119905

+ 120573119911(1198992

119905minus 1

1198992

119905

)]


119905

120583119905

+ (119903

119903119908

minus 1) 1199031199081205741199081198940

119903119908le 119903 le 119903

119905

(27)

119906119908=1

120583119905

1198992

119905minus 1

1198992

119905

120573119911(1199060minus 1198651198871199051199031199081205741199081198940) 119890minus8119879ℎ1205831015840

119905 (28)




is obtained

[ln119903119905

119903119908

minus1199032

119905minus 1199032

119908

21199032

119905

+ 120573119911(1198992

119905minus 1

1198992

119905

)]


119905

120583119905

+ (119903119905

119903119908

minus 1) 1199031199081205741199081198940= 1199060

(29)


120583119905

(119877 minus 119865119887119905) 119890minus8119879ℎ1205831015840

119905 + (119899119905minus 1) = 119877 (30)


Consider

120582119905= ln 119899

119905+ (120573119911minus1

2)(

1198992

119905minus 1

1198992

119905

) 119877 =1199060

1199031199081205741199081198940

(31)


119899119905= 119877 + 1 (32)

Parameters 119899119905and 119903









119903 can




119906119903=

1

120587 (1199032119890minus 1199032119908)int

119903119890

119903119908

2120587119903119906119903119889119903 =

1198992minus 1198992

119905

1198992 minus 11199060+1198992

119905minus 1

1198992 minus 1119906119903119905

(33)




119880119903=

1

120587 (1199032119890minus 1199032119908)int

119903119890

119903119908

1199060minus 119906119903

1199060

2120587119903119889119903 =1198992

119905minus 1


119906119903119905

1199060

)

(34)


119903 is


119880119903=1

119897int

119897

0

119880119903119889119911 =

1

119897int

119897

0

1198992

119905minus 1


119906119903119905

1199060

)119889119911 (35)




119903 119880119903 and 119880




119890= 0525m 119903

119908= 0035m 119899 = 119903

119890119903119908= 15










119905at different









119889119890(m) 119889

119908(m) 119899 = 119889

119890119889119908

119862ℎ(m2y) 119896

ℎ(my) 119902

119908(m3y) 119897 (m)

1575 018 875 045lowast 002 100 12


001 01 1 10 1000

10

20

30

40

50

R = 1

R = 5

R = 10

R = 25

R = 50

120585 = 05G = 816

nt

Th


001 01 1 10 1000

10

20

30

40

50

R = 1

R = 5

R = 10

R = 25

R = 50

120585 = 05G = 816

nt

Th


n = 25

R + 1 = 11

R + 1 = 6

R + 1 = 2

(b) Assume 119899 = 25


001 01 1 10 1000

2

4

6

8

10

12

nt

Th

R = 10

120585 = 0 G = 816

120585 = 05 G = 816

120585 = 1 G = 816

120585 = 0 G = 3265

120585 = 05 G = 3265

120585 = 1 G = 3265



119903 increases







ℎ=






10

08

06

04

02

00

00 02 04 06 08 10

120585=zl

uru0

Th = 01

Th = 1

Th = 5

Th = infin

Solid line R = 25

Th = 01

Th = 1

Th = 5

Th = infin

Dash line R = 10

(a) The average EPP

10

08

06

04

02

00

4 6 8 10 12 14 16 18 20

n

120585=zl

nt

Th = 01

Th = 1

Th = 5

Th = infin

Solid line R = 25

Th = 01

Th = 1

Th = 5

Th = infin

Dash line R = 10



001 01 1 10 100100

80

60

40

20

0

Ur

Th

R = 1

R = 5

R = 10

R = 15

R = 25

R = 50

R = 100






0 1 2 3 4100

80

60

40

20

0


Hansbo (1981)

Area II Area III

Area II R = 100

Area III R = 144

Deg

ree o

f con

solid

atio

nUh

()



ℎ= 139 times






119889119890(m) 119889

119908(m) 119899 = 119889

119890119889119908

119862ℎ(m2s) 119896

ℎ(ms) 119902

119908(m3y) 119897 (m)




0 200 400 600 800 1000100

80

60

40

20

0

Measured data


Ch = 139E minus 10

Ch = 100E minus 10

Ch = 0694E minus 10

Ch = 139E minus 10 RR = 45

Deg

ree o

f con

solid

atio

n (

)


5 Conclusions




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










z

r

qr

dz

dr

d120579



119896ℎ

120574119908

1

119903

120597

120597119903(119903120597119906119903

120597119903) minus

119896ℎ

1199031198940= minus

120597120576V

120597119905119903119908le 119903 le 119903

119905

119906119903= 1199060

119903119905le 119903 le 119903

119890

(7)




0



0 It can


120597119905= minus119898V

120597119906119903119905

120597119905119903119908le 119903 le 119903

119905 (8)



119906119903119905=

1

120587 (1199032

119905minus 1199032119908)int

119903119905

119903119908

2120587119903119906119903119889119903 119903

119908le 119903 le 119903

119905 (9)




1199021=1198961199081205871199032

119908

120574119908

120597119906119908

120597119911119889119905 (10)

dw = 2rw

q1

q2

q1 + dq1

dz



1199022= 2120587119903

119908(119896ℎ

120574119908

120597119906119903

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119908

minus 119896ℎ1198940)119889119911119889119905 (11)


1199021+1205971199021

120597119911119889119911 + 119902

2= 1199021 (12)


120597119906119903

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119908

minus 1198940+119896119908

119896ℎ

119903119908

2

1205972119906119908

1205971199112= 0 (13)


A (120597119906119903120597119903)|119903=119903119905

= 1205741199081198940

B 119906119908|119911=0

= 0C (120597119906

119908120597119911)|119911=119897= 0

D 119906119903|119903=119903119908

= 119906119908

E 119906119903119905|119905=0= 1199060

F 119906119903(119903119905 119905) = 119906

0


119905



120597119906119903

120597119903=120574119908

2119896ℎ

(1199032

119905

119903minus 119903)

120597120576V

120597119905+ 1205741199081198940

119903119908le 119903 le 119903

119905 (14)


120597119906119903

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119908

=120574119908

2119896ℎ

(1199032

119905

119903119908

minus 119903119908)120597120576V

120597119905+ 1205741199081198940 (15)


1205972119906119908

1205971199112= minus

120574119908

119896119908

(1199032

119905

1199032119908

minus 1)120597120576V

120597119905 (16)


119906119908=120574119908

119896119908

(1199032

119905

1199032119908

minus 1)(119897119911 minus1

21199112)120597120576V

120597119905 (17)


119906119903=120574119908

2119896ℎ

(1199032

119905ln 119903

119903119908


119908

2)120597120576V

120597119905+ 1205741199081198940(119903 minus 119903119908) + 119906119908

119903119908le 119903 le 119903

119905

(18)



119906119903119905=1205741199081199032

119905

2119896ℎ

119865119886119905

120597120576V

120597119905+ 1205741199081199031199081198651198871199051198940+ 119906119908 (19)

where

119865119886119905=

1199032

119905

1199032

119905minus 1199032119908

ln119903119905

119903119908

+1

4

1199032

119908

1199032

119905

minus3

4

119865119887119905=

21199032

119905

3 (119903119905119903119908+ 1199032119908)minus1

3

(20)


119886119905and 119865

119887119905can also be

expressed as

119865119886119905=

1198992

119905

1198992

119905minus 1

ln 119899119905+

1

41198992

119905

minus3

4

119865119887119905=

21198992

119905

3 (119899119905+ 1)

minus1

3

(21)



120597119906119903119905

120597119905= minus

2119888ℎ

1199032

119905120583119905

(119906119903119905minus 1198651198871199051199031199081205741199081198940) (22)

in which

119888ℎ=

119896ℎ

(120574119908119898V)

120583119905= 119865119886119905+ 120573119911(1198992

119905minus 1

1198992

119905

)

120573119911=119896ℎ

119896119908

(119897

119903119908

)

2

(2119911

119897119905

minus1199112

1198972

119905

) 119866 =119896ℎ

119896119908

(119897

119889119908

)

2

(23)







119906119903119905= (1199060minus 1198651198871199051199031199081205741199081198940) 119890minus8119879ℎ1205831015840

119905 + 1198651198871199051199031199081205741199081198940

119903119908le 119903 le 119903

119905

(24)

in which

1205831015840

119905=1198992

119905

1198992120583119905 119879

ℎ=119888ℎ

1198892119890

119905 (25)




120597120576V

120597119905=

2119896ℎ

1205741199081199032

119905120583119905

(1199060minus 1198651198871199051199031199081205741199081198940) 119890minus8119879ℎ1205831015840

119905 119903119908le 119903 le 119903

119905 (26)


119906119903= [ln 119903

119903119908


119908

21199032

119905

+ 120573119911(1198992

119905minus 1

1198992

119905

)]


119905

120583119905

+ (119903

119903119908

minus 1) 1199031199081205741199081198940

119903119908le 119903 le 119903

119905

(27)

119906119908=1

120583119905

1198992

119905minus 1

1198992

119905

120573119911(1199060minus 1198651198871199051199031199081205741199081198940) 119890minus8119879ℎ1205831015840

119905 (28)




is obtained

[ln119903119905

119903119908

minus1199032

119905minus 1199032

119908

21199032

119905

+ 120573119911(1198992

119905minus 1

1198992

119905

)]


119905

120583119905

+ (119903119905

119903119908

minus 1) 1199031199081205741199081198940= 1199060

(29)


120583119905

(119877 minus 119865119887119905) 119890minus8119879ℎ1205831015840

119905 + (119899119905minus 1) = 119877 (30)


Consider

120582119905= ln 119899

119905+ (120573119911minus1

2)(

1198992

119905minus 1

1198992

119905

) 119877 =1199060

1199031199081205741199081198940

(31)


119899119905= 119877 + 1 (32)

Parameters 119899119905and 119903









119903 can




119906119903=

1

120587 (1199032119890minus 1199032119908)int

119903119890

119903119908

2120587119903119906119903119889119903 =

1198992minus 1198992

119905

1198992 minus 11199060+1198992

119905minus 1

1198992 minus 1119906119903119905

(33)




119880119903=

1

120587 (1199032119890minus 1199032119908)int

119903119890

119903119908

1199060minus 119906119903

1199060

2120587119903119889119903 =1198992

119905minus 1


119906119903119905

1199060

)

(34)


119903 is


119880119903=1

119897int

119897

0

119880119903119889119911 =

1

119897int

119897

0

1198992

119905minus 1


119906119903119905

1199060

)119889119911 (35)




119903 119880119903 and 119880




119890= 0525m 119903

119908= 0035m 119899 = 119903

119890119903119908= 15










119905at different









119889119890(m) 119889

119908(m) 119899 = 119889

119890119889119908

119862ℎ(m2y) 119896

ℎ(my) 119902

119908(m3y) 119897 (m)

1575 018 875 045lowast 002 100 12


001 01 1 10 1000

10

20

30

40

50

R = 1

R = 5

R = 10

R = 25

R = 50

120585 = 05G = 816

nt

Th


001 01 1 10 1000

10

20

30

40

50

R = 1

R = 5

R = 10

R = 25

R = 50

120585 = 05G = 816

nt

Th


n = 25

R + 1 = 11

R + 1 = 6

R + 1 = 2

(b) Assume 119899 = 25


001 01 1 10 1000

2

4

6

8

10

12

nt

Th

R = 10

120585 = 0 G = 816

120585 = 05 G = 816

120585 = 1 G = 816

120585 = 0 G = 3265

120585 = 05 G = 3265

120585 = 1 G = 3265



119903 increases







ℎ=






10

08

06

04

02

00

00 02 04 06 08 10

120585=zl

uru0

Th = 01

Th = 1

Th = 5

Th = infin

Solid line R = 25

Th = 01

Th = 1

Th = 5

Th = infin

Dash line R = 10

(a) The average EPP

10

08

06

04

02

00

4 6 8 10 12 14 16 18 20

n

120585=zl

nt

Th = 01

Th = 1

Th = 5

Th = infin

Solid line R = 25

Th = 01

Th = 1

Th = 5

Th = infin

Dash line R = 10



001 01 1 10 100100

80

60

40

20

0

Ur

Th

R = 1

R = 5

R = 10

R = 15

R = 25

R = 50

R = 100






0 1 2 3 4100

80

60

40

20

0


Hansbo (1981)

Area II Area III

Area II R = 100

Area III R = 144

Deg

ree o

f con

solid

atio

nUh

()



ℎ= 139 times






119889119890(m) 119889

119908(m) 119899 = 119889

119890119889119908

119862ℎ(m2s) 119896

ℎ(ms) 119902

119908(m3y) 119897 (m)




0 200 400 600 800 1000100

80

60

40

20

0

Measured data


Ch = 139E minus 10

Ch = 100E minus 10

Ch = 0694E minus 10

Ch = 139E minus 10 RR = 45

Deg

ree o

f con

solid

atio

n (

)


5 Conclusions




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597119906119903

120597119903=120574119908

2119896ℎ

(1199032

119905

119903minus 119903)

120597120576V

120597119905+ 1205741199081198940

119903119908le 119903 le 119903

119905 (14)


120597119906119903

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119908

=120574119908

2119896ℎ

(1199032

119905

119903119908

minus 119903119908)120597120576V

120597119905+ 1205741199081198940 (15)


1205972119906119908

1205971199112= minus

120574119908

119896119908

(1199032

119905

1199032119908

minus 1)120597120576V

120597119905 (16)


119906119908=120574119908

119896119908

(1199032

119905

1199032119908

minus 1)(119897119911 minus1

21199112)120597120576V

120597119905 (17)


119906119903=120574119908

2119896ℎ

(1199032

119905ln 119903

119903119908


119908

2)120597120576V

120597119905+ 1205741199081198940(119903 minus 119903119908) + 119906119908

119903119908le 119903 le 119903

119905

(18)



119906119903119905=1205741199081199032

119905

2119896ℎ

119865119886119905

120597120576V

120597119905+ 1205741199081199031199081198651198871199051198940+ 119906119908 (19)

where

119865119886119905=

1199032

119905

1199032

119905minus 1199032119908

ln119903119905

119903119908

+1

4

1199032

119908

1199032

119905

minus3

4

119865119887119905=

21199032

119905

3 (119903119905119903119908+ 1199032119908)minus1

3

(20)


119886119905and 119865

119887119905can also be

expressed as

119865119886119905=

1198992

119905

1198992

119905minus 1

ln 119899119905+

1

41198992

119905

minus3

4

119865119887119905=

21198992

119905

3 (119899119905+ 1)

minus1

3

(21)



120597119906119903119905

120597119905= minus

2119888ℎ

1199032

119905120583119905

(119906119903119905minus 1198651198871199051199031199081205741199081198940) (22)

in which

119888ℎ=

119896ℎ

(120574119908119898V)

120583119905= 119865119886119905+ 120573119911(1198992

119905minus 1

1198992

119905

)

120573119911=119896ℎ

119896119908

(119897

119903119908

)

2

(2119911

119897119905

minus1199112

1198972

119905

) 119866 =119896ℎ

119896119908

(119897

119889119908

)

2

(23)







119906119903119905= (1199060minus 1198651198871199051199031199081205741199081198940) 119890minus8119879ℎ1205831015840

119905 + 1198651198871199051199031199081205741199081198940

119903119908le 119903 le 119903

119905

(24)

in which

1205831015840

119905=1198992

119905

1198992120583119905 119879

ℎ=119888ℎ

1198892119890

119905 (25)




120597120576V

120597119905=

2119896ℎ

1205741199081199032

119905120583119905

(1199060minus 1198651198871199051199031199081205741199081198940) 119890minus8119879ℎ1205831015840

119905 119903119908le 119903 le 119903

119905 (26)


119906119903= [ln 119903

119903119908


119908

21199032

119905

+ 120573119911(1198992

119905minus 1

1198992

119905

)]


119905

120583119905

+ (119903

119903119908

minus 1) 1199031199081205741199081198940

119903119908le 119903 le 119903

119905

(27)

119906119908=1

120583119905

1198992

119905minus 1

1198992

119905

120573119911(1199060minus 1198651198871199051199031199081205741199081198940) 119890minus8119879ℎ1205831015840

119905 (28)




is obtained

[ln119903119905

119903119908

minus1199032

119905minus 1199032

119908

21199032

119905

+ 120573119911(1198992

119905minus 1

1198992

119905

)]


119905

120583119905

+ (119903119905

119903119908

minus 1) 1199031199081205741199081198940= 1199060

(29)


120583119905

(119877 minus 119865119887119905) 119890minus8119879ℎ1205831015840

119905 + (119899119905minus 1) = 119877 (30)


Consider

120582119905= ln 119899

119905+ (120573119911minus1

2)(

1198992

119905minus 1

1198992

119905

) 119877 =1199060

1199031199081205741199081198940

(31)


119899119905= 119877 + 1 (32)

Parameters 119899119905and 119903









119903 can




119906119903=

1

120587 (1199032119890minus 1199032119908)int

119903119890

119903119908

2120587119903119906119903119889119903 =

1198992minus 1198992

119905

1198992 minus 11199060+1198992

119905minus 1

1198992 minus 1119906119903119905

(33)




119880119903=

1

120587 (1199032119890minus 1199032119908)int

119903119890

119903119908

1199060minus 119906119903

1199060

2120587119903119889119903 =1198992

119905minus 1


119906119903119905

1199060

)

(34)


119903 is


119880119903=1

119897int

119897

0

119880119903119889119911 =

1

119897int

119897

0

1198992

119905minus 1


119906119903119905

1199060

)119889119911 (35)




119903 119880119903 and 119880




119890= 0525m 119903

119908= 0035m 119899 = 119903

119890119903119908= 15










119905at different









119889119890(m) 119889

119908(m) 119899 = 119889

119890119889119908

119862ℎ(m2y) 119896

ℎ(my) 119902

119908(m3y) 119897 (m)

1575 018 875 045lowast 002 100 12


001 01 1 10 1000

10

20

30

40

50

R = 1

R = 5

R = 10

R = 25

R = 50

120585 = 05G = 816

nt

Th


001 01 1 10 1000

10

20

30

40

50

R = 1

R = 5

R = 10

R = 25

R = 50

120585 = 05G = 816

nt

Th


n = 25

R + 1 = 11

R + 1 = 6

R + 1 = 2

(b) Assume 119899 = 25


001 01 1 10 1000

2

4

6

8

10

12

nt

Th

R = 10

120585 = 0 G = 816

120585 = 05 G = 816

120585 = 1 G = 816

120585 = 0 G = 3265

120585 = 05 G = 3265

120585 = 1 G = 3265



119903 increases







ℎ=






10

08

06

04

02

00

00 02 04 06 08 10

120585=zl

uru0

Th = 01

Th = 1

Th = 5

Th = infin

Solid line R = 25

Th = 01

Th = 1

Th = 5

Th = infin

Dash line R = 10

(a) The average EPP

10

08

06

04

02

00

4 6 8 10 12 14 16 18 20

n

120585=zl

nt

Th = 01

Th = 1

Th = 5

Th = infin

Solid line R = 25

Th = 01

Th = 1

Th = 5

Th = infin

Dash line R = 10



001 01 1 10 100100

80

60

40

20

0

Ur

Th

R = 1

R = 5

R = 10

R = 15

R = 25

R = 50

R = 100






0 1 2 3 4100

80

60

40

20

0


Hansbo (1981)

Area II Area III

Area II R = 100

Area III R = 144

Deg

ree o

f con

solid

atio

nUh

()



ℎ= 139 times






119889119890(m) 119889

119908(m) 119899 = 119889

119890119889119908

119862ℎ(m2s) 119896

ℎ(ms) 119902

119908(m3y) 119897 (m)




0 200 400 600 800 1000100

80

60

40

20

0

Measured data


Ch = 139E minus 10

Ch = 100E minus 10

Ch = 0694E minus 10

Ch = 139E minus 10 RR = 45

Deg

ree o

f con

solid

atio

n (

)


5 Conclusions




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












is obtained

[ln119903119905

119903119908

minus1199032

119905minus 1199032

119908

21199032

119905

+ 120573119911(1198992

119905minus 1

1198992

119905

)]


119905

120583119905

+ (119903119905

119903119908

minus 1) 1199031199081205741199081198940= 1199060

(29)


120583119905

(119877 minus 119865119887119905) 119890minus8119879ℎ1205831015840

119905 + (119899119905minus 1) = 119877 (30)


Consider

120582119905= ln 119899

119905+ (120573119911minus1

2)(

1198992

119905minus 1

1198992

119905

) 119877 =1199060

1199031199081205741199081198940

(31)


119899119905= 119877 + 1 (32)

Parameters 119899119905and 119903









119903 can




119906119903=

1

120587 (1199032119890minus 1199032119908)int

119903119890

119903119908

2120587119903119906119903119889119903 =

1198992minus 1198992

119905

1198992 minus 11199060+1198992

119905minus 1

1198992 minus 1119906119903119905

(33)




119880119903=

1

120587 (1199032119890minus 1199032119908)int

119903119890

119903119908

1199060minus 119906119903

1199060

2120587119903119889119903 =1198992

119905minus 1


119906119903119905

1199060

)

(34)


119903 is


119880119903=1

119897int

119897

0

119880119903119889119911 =

1

119897int

119897

0

1198992

119905minus 1


119906119903119905

1199060

)119889119911 (35)




119903 119880119903 and 119880




119890= 0525m 119903

119908= 0035m 119899 = 119903

119890119903119908= 15










119905at different









119889119890(m) 119889

119908(m) 119899 = 119889

119890119889119908

119862ℎ(m2y) 119896

ℎ(my) 119902

119908(m3y) 119897 (m)

1575 018 875 045lowast 002 100 12


001 01 1 10 1000

10

20

30

40

50

R = 1

R = 5

R = 10

R = 25

R = 50

120585 = 05G = 816

nt

Th


001 01 1 10 1000

10

20

30

40

50

R = 1

R = 5

R = 10

R = 25

R = 50

120585 = 05G = 816

nt

Th


n = 25

R + 1 = 11

R + 1 = 6

R + 1 = 2

(b) Assume 119899 = 25


001 01 1 10 1000

2

4

6

8

10

12

nt

Th

R = 10

120585 = 0 G = 816

120585 = 05 G = 816

120585 = 1 G = 816

120585 = 0 G = 3265

120585 = 05 G = 3265

120585 = 1 G = 3265



119903 increases







ℎ=






10

08

06

04

02

00

00 02 04 06 08 10

120585=zl

uru0

Th = 01

Th = 1

Th = 5

Th = infin

Solid line R = 25

Th = 01

Th = 1

Th = 5

Th = infin

Dash line R = 10

(a) The average EPP

10

08

06

04

02

00

4 6 8 10 12 14 16 18 20

n

120585=zl

nt

Th = 01

Th = 1

Th = 5

Th = infin

Solid line R = 25

Th = 01

Th = 1

Th = 5

Th = infin

Dash line R = 10



001 01 1 10 100100

80

60

40

20

0

Ur

Th

R = 1

R = 5

R = 10

R = 15

R = 25

R = 50

R = 100






0 1 2 3 4100

80

60

40

20

0


Hansbo (1981)

Area II Area III

Area II R = 100

Area III R = 144

Deg

ree o

f con

solid

atio

nUh

()



ℎ= 139 times






119889119890(m) 119889

119908(m) 119899 = 119889

119890119889119908

119862ℎ(m2s) 119896

ℎ(ms) 119902

119908(m3y) 119897 (m)




0 200 400 600 800 1000100

80

60

40

20

0

Measured data


Ch = 139E minus 10

Ch = 100E minus 10

Ch = 0694E minus 10

Ch = 139E minus 10 RR = 45

Deg

ree o

f con

solid

atio

n (

)


5 Conclusions




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119889119890(m) 119889

119908(m) 119899 = 119889

119890119889119908

119862ℎ(m2y) 119896

ℎ(my) 119902

119908(m3y) 119897 (m)

1575 018 875 045lowast 002 100 12


001 01 1 10 1000

10

20

30

40

50

R = 1

R = 5

R = 10

R = 25

R = 50

120585 = 05G = 816

nt

Th


001 01 1 10 1000

10

20

30

40

50

R = 1

R = 5

R = 10

R = 25

R = 50

120585 = 05G = 816

nt

Th


n = 25

R + 1 = 11

R + 1 = 6

R + 1 = 2

(b) Assume 119899 = 25


001 01 1 10 1000

2

4

6

8

10

12

nt

Th

R = 10

120585 = 0 G = 816

120585 = 05 G = 816

120585 = 1 G = 816

120585 = 0 G = 3265

120585 = 05 G = 3265

120585 = 1 G = 3265



119903 increases







ℎ=






10

08

06

04

02

00

00 02 04 06 08 10

120585=zl

uru0

Th = 01

Th = 1

Th = 5

Th = infin

Solid line R = 25

Th = 01

Th = 1

Th = 5

Th = infin

Dash line R = 10

(a) The average EPP

10

08

06

04

02

00

4 6 8 10 12 14 16 18 20

n

120585=zl

nt

Th = 01

Th = 1

Th = 5

Th = infin

Solid line R = 25

Th = 01

Th = 1

Th = 5

Th = infin

Dash line R = 10



001 01 1 10 100100

80

60

40

20

0

Ur

Th

R = 1

R = 5

R = 10

R = 15

R = 25

R = 50

R = 100






0 1 2 3 4100

80

60

40

20

0


Hansbo (1981)

Area II Area III

Area II R = 100

Area III R = 144

Deg

ree o

f con

solid

atio

nUh

()



ℎ= 139 times






119889119890(m) 119889

119908(m) 119899 = 119889

119890119889119908

119862ℎ(m2s) 119896

ℎ(ms) 119902

119908(m3y) 119897 (m)




0 200 400 600 800 1000100

80

60

40

20

0

Measured data


Ch = 139E minus 10

Ch = 100E minus 10

Ch = 0694E minus 10

Ch = 139E minus 10 RR = 45

Deg

ree o

f con

solid

atio

n (

)


5 Conclusions




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10

08

06

04

02

00

00 02 04 06 08 10

120585=zl

uru0

Th = 01

Th = 1

Th = 5

Th = infin

Solid line R = 25

Th = 01

Th = 1

Th = 5

Th = infin

Dash line R = 10

(a) The average EPP

10

08

06

04

02

00

4 6 8 10 12 14 16 18 20

n

120585=zl

nt

Th = 01

Th = 1

Th = 5

Th = infin

Solid line R = 25

Th = 01

Th = 1

Th = 5

Th = infin

Dash line R = 10



001 01 1 10 100100

80

60

40

20

0

Ur

Th

R = 1

R = 5

R = 10

R = 15

R = 25

R = 50

R = 100






0 1 2 3 4100

80

60

40

20

0


Hansbo (1981)

Area II Area III

Area II R = 100

Area III R = 144

Deg

ree o

f con

solid

atio

nUh

()



ℎ= 139 times






119889119890(m) 119889

119908(m) 119899 = 119889

119890119889119908

119862ℎ(m2s) 119896

ℎ(ms) 119902

119908(m3y) 119897 (m)




0 200 400 600 800 1000100

80

60

40

20

0

Measured data


Ch = 139E minus 10

Ch = 100E minus 10

Ch = 0694E minus 10

Ch = 139E minus 10 RR = 45

Deg

ree o

f con

solid

atio

n (

)


5 Conclusions




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119889119890(m) 119889

119908(m) 119899 = 119889

119890119889119908

119862ℎ(m2s) 119896

ℎ(ms) 119902

119908(m3y) 119897 (m)




0 200 400 600 800 1000100

80

60

40

20

0

Measured data


Ch = 139E minus 10

Ch = 100E minus 10

Ch = 0694E minus 10

Ch = 139E minus 10 RR = 45

Deg

ree o

f con

solid

atio

n (

)


5 Conclusions




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Consolidation by Prefabricated Vertical...

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