Equal Strain Consolidation by Vertical Drains

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Equal Strain Consolidation by Vertical DrainsChin Jian Leo1

Abstract: This paper shows the development of a series of closed-form solutions of equal strain consolidation in the presvertical drain with smear and well resistance. Using an approach that considers the effects of both the radial and vertical drainagcoupled fashion, solutions are obtained for the excess pore pressure and the degree of consolidation in the compressible soila step- or ramp-loading situation. The closed-form solutions in the present paper may be evaluated in an electronic spreadstandard personal computer.

DOI: 10.1061/~ASCE!1090-0241~2004!130:3~316!

CE Database subject headings: Vertical drain; Wick drains; Consolidation; Pore pressure; Soil improvement; Soft soils; Comapplications.

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Introduction

For a number of years now, vertical sand and prefabricated dhave been used successfully to accelerate the consolidatfine-grained compressible soils in combination with the effecpreloading. To facilitate the design and construction processperformance of a vertical drain is typically evaluated usinmathematical model—this could range from relatively simanalytic solutions to more complicated numerical models. Dethe presence of an increasing number of sophisticated nummodels, analytic solutions such as the ones developed by B~1948! and Hansbo~1981! remain popular and widely used. Itsuggested that these analytic solutions are often the solutiochoice because of their simplicity and ease of use, espewhere incomplete knowledge of the soil does not justify usingmore sophisticated methods.

Barron ~1948! developed solutions for two limiting casesconsolidation by vertical drains—free strain and equal strainsolidation. Barron’s approach was to decouple radial and vedrainage in the undisturbed soil mass. First, he found thetions for the excess pore water pressure by considering onlydrainage into the vertical drain and then obtained the comsolutions as a product of the contributions from the radialvertical drainage. It can be shown that the product solution ian exact solution if effects of well resistance and/or smeaincluded, since the interface boundary conditions are not rously complied with in these cases. Barron also did not giclosed-form solution for the degree of consolidation, suggeinstead that a numerical integration procedure such as Simprule could be used to obtain the degree of consolidation.

1Senior Lecturer, School of Engineering and Industrial DesWerrington South Campus, Univ. of Western Sydney, Locked Bag 1Penrith South DC NSW 1797, Australia. E-mail: [email protected]

Note. Discussion open until August 1, 2004. Separate discusmust be submitted for individual papers. To extend the closing daone month, a written request must be filed with the ASCE ManaEditor. The manuscript for this paper was submitted for review andsible publication on March 1, 2001; approved on May 30, 2003.paper is part of theJournal of Geotechnical and GeoenvironmentEngineering, Vol. 130, No. 3, March 1, 2004. ©ASCE, ISSN 1090-02
2004/3-316–327/$18.00.
316 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINE

J. Geotech. Geoenviron. Eng

Hansbo~1981! introduced some approximations and produa simple closed-form solution of the degree of consolidationgiven depth for equal strain consolidation by radial drainHansbo pointed out that the equal strain and free strain soluare in fact quite similar. The complete solution of Hansbonecessary, is also found as a product of vertical and radial dage solutions. While the solutions of Hansbo are attractive inmathematical simplicity, they are also not exact in the rigosense.

A series of rigorous coupled solutions considering verticalradial drainage simultaneously in a coupled sense, in the preof a vertical well with resistance, was given by YoshikuniNakanodo~1974!. Said to be equal strain solutions, thesemore correctly, free strain solutions. The solutions of Yoshiand Nakanodo appear even more unwieldy than Barron’stions, and require root finding of some Fourier-Bessel funcequations, which have to be evaluated based on some numprocedures.

The present paper solves the equal strain consolidationlem by considering vertical and radial drainage in a fully coufashion. A series of closed-form solutions is obtained for thecess pore pressure and degree of consolidation in the undissoil mass subjected to a step- or ramp-loading operation. Tsolutions are ‘‘exact’’ in the sense of the governing equationsboundary and initial conditions being satisfied rigorously inmathematical sense. Also, these solutions can be computedelectronic spreadsheet such asMS Excelwith the only provisionbeing that the appropriate Bessel mathematical functions nebe installed~if the Bessel functions are not available, it is necsary to run the setup to install theAnalysis ToolPak!.

This paper further attempts a comparison of the resultsdifferent analytic approaches—those of Barron~1948!, Hansbo~1981!, and the current solutions. Data from the StockhoArlanda airport soil improvement scheme are used to provbasis with which to make the comparison.

Development of Closed-Form Analytic Solutions

Fig. 1 shows the schematic representation of the problemconsideration where a vertical drain is surrounded by a smzone and undisturbed soil. At the center of the cylindrical cell
drainage well of radiusr w surrounded by a zone of remoulded
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soil bounded by an outer radiusr s , which is in turn surroundeby the undisturbed soil extending to a radius of influencer e . Thesoil has a layer thicknessh and it is freely draining at the top aimpermeable at the bottom. The cylindrical cell typically repsents one of many in a network of drainage wells in a consotion problem of practical interest~Fig. 2!. Although the cell unit isconsidered as somewhat of an idealization of the real probledoes however greatly facilitate the mathematical analysis wiunduly affecting the overall accuracy, given the incompknowledge of the soil properties in practice. Barron discussedidealization in his 1948 paper and it has also been discussmost current geotechnical textbooks.

The basic assumptions made in developing the solutionspresent paper were largely given previously by Barron in 1and are stated below for completeness:1. All vertical loads are initially carried by the excess po

water pressureu.2. Darcy’s law is obeyed.3. All compressive strains within the soil mass occur in a

tical direction. This arises from the fact that compressioconfined and no shear strain exists.

4. Horizontal sections remain horizontal during the consotion.

5. The zone of influence of each well is a circle.6. Load distribution is uniform over this area.7. The top of the layer is freely draining but the bottom

Fig. 1. Typical cylindrical cell representing a vertical drasurrounded by a smeared zone and undisturbed soil

completely impermeable.

JOURNAL OF GEOTECHNICAL AND G


It suffices, at this point, to mention that the solutions haveobtained by considering the dissipation of excess pore preand consolidation behavior in the three subdomains of thelem domain. These are the undisturbed soil, the smearemolded zone surrounding the vertical well, and the verticalitself ~Fig. 1!. In keeping with Barron~1948!, consolidation iconsidered in the undisturbed soil mass only, not in the vedrain or the smeared zone, and only radial drainage is assumthe smeared zone. The development of the solutions, inclthe limiting solutions, are elaborated in the Appendix.

Consideration of Undisturbed Soil

The consolidation of the undisturbed soil is related to the dpation of excess pore pressure resulting from the impositionsurcharge loading on the surface. This is described by thenuity equation, which is given by

kh

gwS ]2u~r ,z,t !

]r 2 11

r

]u~r ,z,t !

]r D1kv

gw

]2u~r ,z,t !

]z2 52]«z~z,t !

]t(1)

where u(r ,z,t)5excess pore-water pressure;t5elapsed timekh5horizontal soil permeability;kv5vertical soil permeabilitygw5unit weight of the fluid;r5radial distance from the cen

Fig. 2. Typical cylindrical cell within a network of vertical drain

Fig. 3. Step loading

EOENVIRONMENTAL ENGINEERING © ASCE / MARCH 2004 / 317

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of the well; and«z(z,t)5vertical strain. The undisturbed smass also has a coefficient of consolidation for radial~horizontal!flow, ch , and a coefficient of consolidation for vertical flocv . It may be noted that the strain is expressed as a funof thez spatial direction whereas thex,y directions are neglecteThis arises from the assumption that the strains occurin the vertical direction and shear strains are not presethe cylinder is laterally confined. Fig. 2 shows a typcell, which in practice would be constrained horizontally alits vertical sides due to the presence of neighboring cells in
network.
and

and

ay bee atin the

radial direction, giving



Complete Solutions for Step Loading

The solution of the excess pore water pressure in the undistsoil for a single loading step (q50,t,0;q5q0 ,t>0) ~Fig. 3! is aFourier sine series given by

u~r ,z,Th!5 (n50

`4u0

~2n11!pVne28Th /nn@c1nI 0~mnr !

1c2nK0~mnr !11#sinvnz (2)

whereu05 initial excess pore pressure and is often assumed to be

u05H 0, t,0

q0 , t>0(3)

Th5cht

4r e2 (4)

vn5~2n11!p

2h(5)

mn25

kv

khvn

2 (6)

Vn52c1n@mnr eI 1~mnr e!2mnr sI 1~mnr s!#22c2n@mnr eK1~mnr e!2mnr sK1~mnr s!#

mn2~r e

22r s2!

11 (7)

ius is

entirection,

I 1 ,K15modified Bessel functions of the first and second kindorder 1, respectively

nn52

mn2r e

2 Vn (8)

c1n5K1~mnr e!

Dn(9)

c2n5I 1~mnr e!

Dn(10)

Dn52anK1~mnr e!2bnI 1~mnr e! (11)

an5I 0~mnr s!2I 1~mnr s!Fn (12)

bn5K0~mnr s!1K1~mnr s!Fn (13)

I 0 ,K05modified Bessel functions of the first and second kindzero order, respectively

Fn5mnr sH kh

kslnS r s

r wD1

2

r w2

kh

kw

1

vn2J (14)

Degree of Consolidation for Step LoadingThe average excess pore pressure in the undisturbed soil mdefined in various ways. The average excess pore pressurparticular depth is obtained by averaging the pore pressure

a

u~z,Th!51

p~r e22r s

2!E

r s

r e

u~r ,z,Th!2prdr

5 (n50

`4u0

~2n11!pe28Th /nn sinvnz (15)

while the average excess pore pressure at a particular radobtained by averaging over the depth to give

u~r ,Th!51

h E0

h

u~r ,z,Th!dz

5 (n50

`8u0

~2n11!2p2Vn@c1nI 0~mnr !

1c2nK0~mnr !11#e28Th /nn (16)

Finally, the overall average excess pore pressure of theundisturbed soil mass is averaged over depth and radial diregiving

u~Th!51

h E0

h

(n50

`

u~z,Th!dz

51

h (n50

`4u0

~2n11!pe28Th /nnE

0

h

sinvnz dz

and leading to


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u~Th!5 (n50

`8u0

~2n11!2p2 e28Th /nn (17)

Various forms of degree of consolidation~for coupled radial anvertical drainage! may now be defined based on the averagecess pore-water pressure described in Eq.~15!, ~16!, or ~17!.These are as follows:

U~z,Th!5u02u~z,Th!

u0(18)

U~r ,Th!5u02u~r ,Th!

u0(19)

U~Th!5u02u~Th!

u0(20)

Now, using Eq.~20! the overall average degree of consolidaof the entire undisturbed soil mass considering coupled radiavertical drainage is therefore given by

U~Th!512 (n50

`8

~2n11!2p2 e28Th /nn (21)

which is a fairly simple expression, whereas the corresponsolution as given by Barron~1948! appears rather unwieldy. Tsolutions in the present paper, as previously noted, can beputed in an electronic spreadsheet and this is exemplified b

Fig. 4. Spreadsheet calculations of th

spreadsheet calculations presented in Fig. 4.



Ramp Loading

The loading sequence is often more realistically representedramp type loading~Fig. 5! where the load is applied uniformlytime, such that

q~ t !5H s0t, t,t0

q0 , t>t0(22)

Although Terzaghi had previously suggested a simple modtion to his step-loading solution to take into account ramp-loaconditions, the formal analytical solutions as shown belowquite easily obtained as well~see the Appendix!. The solution fothe excess pore pressure for the ramp-loading case is found

rall degree of consolidation in a typical case

Fig. 5. Ramp loading of the filling material

e ove


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u~r ,z,Th!5 (n50

`4s0

cvvn2~2n11!p

$e2@8~Th2Th0!/nn#H~Th2Th0!

2e28Th /nn%@c1nI 0~mnr !1c2nK0~gmnr !11#sinvnz

(23)

where

Th05cht0 /~4r e2! (24)

and

H~Th!5H 0 if Th,0

1 if Th>0(25)

is the Heaviside function. The radial-, depth-, and oveaveraged excess pore pressure are given as

u~z,Th!5 (n50

`4s0

cvvn2~2n11!p


2e28Th /nn%Vn sinvnz (26)

u~r ,Th!5 (n50

`8s0

cvvn2~2n11!2p2 $e2@8~Th2Th0!/nn#H~Th2Th0!

2e28Th /nn%@c1nI 0~mnr !1c2nK0~mnr !11# (27)

u~Th!5 (n50

`8s0


2e28Th /nn%Vn (28)

The overall average degree of consolidation for the entire uturbed soil mass is defined as

U~Th!5@q~ t !/q0#u02u~Th!

u0(29)

thus yielding

U~Th!5Th

Th01S 12

Th

Th0DH~Th2Th0!

2 (n50

`8s0 /u0


2e28Th /nn%Vn (30)

where it has been assumed thatu05q0 . This assumption is theretically correct if the soil is identically fully saturated evewhere in the undisturbed soil. In other situations of practicaterest, it may be more appropriate to determine the initial expore pressure caused by a surcharge loading from Skempore pressure relationship, which is given as

u~z,0!5u05As11~12A!s3 (31)

wheres1 ,s35 increments in major and minor principal stressand A5well-known Skempton pore pressure coefficient.Skempton’s relationship is adopted, some adjustments shoumade to ensure consistency in the effective stress. It is suggthat

s0* 5s0

u0

q0(32)

should now be used to substitute fors0 .



Comparison with Other Analytic Solutions

In this section, the solutions developed in the present paperapplied in some test cases and this is done for the purpomaking comparisons with the solutions of Barron~1948! andHansbo~1981!.

Radial Drainage, No Well ResistanceTable 1 shows some typical values from the solutions of Ba~1948!, Hansbo~1981!, and the present paper for consolidationradial drainage only (kv50), where effects of well resistance asmear were not included. The input data adopted in the cations are ch57.9 m2/year, kh50.0194 m/year, h510 m, r e

51.8 m, andr w50.2 m. The overall average consolidationthe entire mass of undisturbed soil was found by numericalgration, in the case of the solution presented by Barron. Aswas no well resistance, Hansbo’s solution for average constion at a particular depth becomes independent of the deptshould converge to the value of the overall average consolidfor the entire soil mass. The solutions of the present papercomputed using 20 Fourier terms on anExcelspreadsheet. Thisa test case of ideal and specialized flow conditions so allsolution methods were expected to give identical results.maximum absolute difference between any two given soluapproaches, for any nominated time level shown in Tableonly 0.009 ~at small Th); thus the solutions are consideredpractically identical.

Radial and Vertical Drainage with Smear and WellResistanceThe results from Hansbo and the present paper, for averagesolidation including well resistance for radial drainage only,shown in Table 2. Both the radial and vertical drainage wassidered in the analysis. The earlier input data were empl

Table 2. Comparison of Average and Overall Average Consolidafor Radial and Vertical Drainage: Effects of Well Smear and WResistance Included

Th

Average consolidation~at z5h)

Overall average consolidatio~entire soil!

Hansbo~1981! Present Paper Hansbo~1981! Present pape

0.2 0.053 0.032 0.163 0.1610.5 0.127 0.093 0.305 0.2921.0 0.239 0.211 0.471 0.4422.0 0.421 0.443 0.675 0.6365.0 0.745 0.831 0.909 0.892

10.0 0.935 0.978 0.987 0.986

Table 1. Comparison of Overall Average Consolidation~for EntireUndisturbed Soil Mass! Calculated from Solutions of BarroHansbo, and Present Paper: Radial Drainage without Effects oSmear and Well Resistance

Th Barron ~1948! Hansbo~1981! Present paper

0.01 0.052 0.053 0.0610.02 0.101 0.102 0.1100.05 0.234 0.237 0.2410.1 0.414 0.417 0.4180.2 0.657 0.660 0.6580.5 0.931 0.933 0.9311.0 0.995 0.995 0.995


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again in these solutions, in addition to adopting the valuekh /ks55, ch /cv55, r s50.45 m, and a finite well permeabilitykw5100kh . Two sets of average consolidation values aresented, one at depthz5h and the other for the entire mass ofundisturbed soil. The results in Table 2 show that in a nonvertical drain~where smear and well resistance are present! thedifferences remain small for practical purposes. Between thlution sets of Hansbo and the present paper, the maximumlute difference is 0.086 for the average consolidation atz5h and0.039 for the overall average consolidation.

Stockholm-Arlanda Airport Soil Improvement SchemeTheoretical solutions were also calculated for a soil improvemscheme using prefabricated vertical drain which was carried oSitek of the Stockholm-Arlanda airport. This work and the ansis results were reported in a paper by Eriksson et al.~2000!. Thefollowing data were used in the analysis:ch52.25 m2/year,kh /ks53, ch /cv53, h54.5 m, r e50.4725 m,r s50.075 m, andr w50.033 m. As prefabricated drains were used, no well rtance was considered in the calculations.

The selected site of interest was surcharged in threephases as shown in Fig. 6. The average degree of consolifor the entire undisturbed soil for the load increment of e

Fig. 6. Results of excess pore pressure observations~Eriksson et al.those of the present paper

phaseU1 ,U2 ,U3 was calculated based on the solutions of Barron



~1948!, Hansbo~1981!, and the present paper. The calculatusing Hansbo’s solutions are described in detail by Ericket al. ~2000!. As the solutions developed by Barron and Hanwere for step loading, a correction was made to the elapsedto account for the finite rate of the actual loading. Two solutfrom this paper, the first the single-step-loading solution of~21! and the second the ramp-loading solution of Eq.~30!, arealso shown. In the case of step loading, the elapsed timeadjusted by assuming that the step load is placed midway dthe period of the filling operation as suggested by Ericksson~2000!. No correction was made for the ramp-loading solu@Eq. ~30!# and the elapsed times for the ramp loading are tshown in parentheses in Tables 3~a, b, and c!.

The analysis shows that Eq.~21! for the step loading yieldesimilar results to the solutions from Hansbo and Barron. Eq.~30!for the ramp loading yielded values that were less similar at stimes of interest, but the results were similar when the nomintimes are large.

In calculating the excess pore pressure, Eriksson et asumed that no reduction in pore pressure had occurred durinplacement of the load step. Thus the initial excess pore prewas assumed to develop at the end of the load placement o

t SiteK compared with theoretical results from Hansbo solutions
2000! a
step, and the elapsed time was referenced from the time at the end


. 2004.130:316-327.

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of the load placement. This was justified on the basis thanamic disturbance effects occurred during filling operationsaddition, a radius of influencer e50.6675 m was used in this cabased partly on the arrangement of the piezometers used tosure the pore pressure.

The theoretical values of the average pore pressure presin Table 4 were computed from the solutions of Hansbo~1981!and Eqs.~16! and ~17! of this paper. The excess pore presswere averaged either over the entire undisturbed soil mass othe entire depth at the radius of influencer e . A sensitivity studyof the ratioch /cv was made by comparing the solutions ofpresent paper forcv50 ~no vertical drainage! andch /cv53, withthe solutions of Hansbo from Eriksson et al.’s paper whercv50. It may be observed in Table 4 that the differences inpressure values are generally small.

It is also evident that when the vertical drainage is inclu(ch /cv53), a greater divergence between the present and Hbo’s solutions was found to occur. However, the solutions wnot as sensitive to vertical drainage changes as might haveexpected, and in any case the differences were within the coerable ambiguity in the interpretation of measured pore predata, which is subjected to a number of possible interfereThis is more apparent when all the solutions were plotted agmeasured field results as shown in Fig. 6~where the excess popressure is shown expressed in terms of head!. It is observed thathere is quite a large variation in the field readings of the

Table 3. Degree of Consolidation~Entire Undisturbed Soil Mass!

Elapsedtime t1

~months!a

Step loading Ramp loading

Barron~1948!

Hansbo~1981!

Presentpaper

Presentpaper

~a! U1

1 ~2! 0.416 0.416 0.416 0.3892 ~3! 0.647 0.646 0.647 0.6324.5 ~5.5! 0.897 0.897 0.897 0.8937.5 ~8.5! 0.976 0.976 0.976 0.97510.5 ~11.5! 0.994 0.994 0.994 0.994

~b! U2

0.5 ~1! 0.246 0.245 0.246 0.2363 ~3.5! 0.785 0.784 0.785 0.7836 ~6.5! 0.951 0.950 0.951 0.9509 ~9.5! 0.989 0.988 0.989 0.989

~c! U3

1.25 ~1! 0.486 0.485 0.486 0.4484.25 ~3.5! 0.884 0.883 0.884 0.8767.25 ~6.5! 0.973 0.973 0.973 0.972aValues in parentheses indicate elapsed time for ramp loading.

Table 4. Average Excess Pore Pressureu ~kPa!

Elapsed timet1 ~months!a

Hansbo~1981!Depth average

(cv50) Depth average (cv50) Depth a

1 216 2133.5 316 3116.5 165 1639.5 86 85aNo reduction in excess pore pressure was assumed to occur du
placement of load step 1 has been completed.


-

pressure; therefore, in this context the variations in the valutheoretical solutions are quite insignificant.

Conclusion

Closed-form analytic solutions of equal strain consolidationvertical drain with smear and well resistance have been devein the present paper. Comparisons made with the correspoanalytic solutions of Hansbo and Barron showed that the dences between the solutions of the present paper and the soof Hansbo and/or Barron are generally quite small. Solutionthis paper, however, have been derived for coupled radiavertical drainage and covered a step-loading or a ramp-losituation. These solutions can be evaluated in an electspreadsheet.

Appendix. Development of Solutions

Consideration of the Undisturbed Soil

To solve the consolidation of the undisturbed soil mass as deby the partial differential equation~1!, it is useful to introduce thFourier sine series defined as

u~r ,z,t !5 (n50

`

Un~r ,t !sinvnz (33a)

]«z~z,t !

]t5 (

n50

`]«zn~ t !

]tsinvnz (33b)

wherevn5(2n11)p/(2h) and Un(r ,t), «zn(t) are the Fouriecoefficients of their respective Fourier series expansions. Oevidence of Eq.~33a!, it is apparent that the drainage boundconditions ~i.e., pore pressure! are satisfied rigorously in thezspatial direction.

If Eq. ~33! were to be substituted into Eq.~1! the followingequation applying to each of the Fourier terms results:

kh

gwS ]2Un~r ,t !

]r 2 11

r

]Un~r ,t !

]r D2kv

gwvn

2Un~r ,t !52]«zn~ t !

]t(34)

Now suppose that

Un~r ,t !5An~r !Bn~ t ! (35)

Then substitution into Eq.~34! yields

Present paper

(ch /cv53) Overall average (cv50) Overall average (ch /cv53)

201 1989 294 2834 154 136

81 68

lacement of each load step. Elapsed timet1 therefore commences only af

verage

209291472

ring p


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kh

gwS ]2An~r !

]r 2 11

r

]An~r !

]r D2kv

gwvn

2An~r !

521

B~ t !

]«zn~ t !

]t52ln (36)

whereln5separation constant. A solution of Eq.~36! is

An~r !5lnfn@c1nI 0~mnr !1c2nK0~mnr !11# (37)

where l 0 ,K05modified Bessel functions of the first and seckind of zero order, respectively; mn

25(kv /kh)vn2 ; fn

5gw /kvvn2 ; andc1n ,c2n5constants of integration the values

which are to be determined. Now consider again Eq.~36! where

1

B~ t !

]«zn~ t !

]t5ln (38)

Since the vertical strain is related to the effective stress in the

«z~z,t !5mv@q~r ,t !2u~r ,z,t !# (39)

in which mv5coefficient of volume compressibility; anq(r ,t)5applied surcharge, then a Fourier sine series expansEq. ~39! gives

(n50

`

«zn~ t !sinvnz5mv(n50

` F 4q~r ,t !

~2n11!p2Un~r ,t !Gsinvnz

(40)

Although the load distribution due to the fill material is assumuniform over the area, the effects manifested as applied surcare not necessarily uniform. These depend, instead, on theof arching and the load transfer mechanism being developthe fill material above the soil layer. In the context of this prlem, it is clear that the only way in which equal strain consoltion is ensured would be if the effects of arching were presethe fill material. Furthermore, because of the confining natuthe compression, the coefficient of volume compressibility,mn , isin fact the inverse of the constrained modulus of elasticity, v

mv5~11n8!~122n8!

~12n8!E8(41)

where n8, E85effective Poisson’s ratio and Young’s modulrespectively. It would be possible to measure the value ofmv in alaboratory ocdometer test. Then, following Eq.~40!, the Fourie
component of the strain rate gives
ingle

n.



t

]«zn~ t !

]t5mvH 4

~2n11!p

]q~r ,t !

]t2

]Un~r ,t !

]t J (42)

Given that the strains are equal on any horizontal section anintegral of Eq.~42! taken fromr 5r s to r 5r e shows that

]«zn~ t !

]t Er s

r e

2pr dr

5mvEr s

r eH 4

~2n11!p

]q~r ,t !

]t2

]Un~r ,t !

]t J 2pr dr

(43)

so that

]«zn~ t !

]t5mvH 4

~2n11!p

]q~ t !

]t2

]Un~ t !

]t J (44)

where

Un~ t !51

p~r e22r s

2!E

r s

r e

Un~r ,t !2pr dr (45)

q~ t !51

p~r e22r s

2!E

r s

r e

q~r ,t !2pr dr (46)

are radial-averaged quantities. It is also noted thatq(t) corre-sponds to uniform loading. Without loss of clarity, the overbaq(t) will be dropped from here onward for the sake of mathemcal convenience, i.e.,q(t) will denote the radial-averaged applsurcharge as well as the uniform loading, which may varytime.It immediately follows from Eqs.~35! and ~37! that the averagrate of change of the Fourier component of the excess poresure is

]Un~ t !

]t5

lnfn

p~r e22r s

2!

3Er s

r e

@c1nI 0~mnr !1c2nK0~mnr !11#2pr dr]Bn

]t

5lnfnVn

]Bn

]t(47)

where

Vn52c1n@mnr eI 1~mnr e!2mnr sI 1~mnr s!#22c2n@mnr eK1~mnr e!2mnr sK1~mnr s!#

mn2~r e

22r s2!

11 (48)

In the simplest loading case, which is considered first, a sstep loading ofq0 is applied att50 ~Fig. 3!; thus

]«zn~ t !

]t52mv

]Un~r ,t !

]t52

lnVn

chmn2

]Bn~ t !

]t(49)

where ch5kh /mvgw5horizontal coefficient of consolidatioCombining Eqs.~38! and ~49! gives

]Bn

]t52gnBn (50)

wheregn5chmn2/Vn . A solution of Eq.~50! is

Bn5an

lnfne2gnt5

an

lnfne28Th /nn (51)

wheren 5(2/m2r 2)V , the dimensionless timeT 5c t/4r 2, and
n n e n h h e

. 2004.130:316-327.

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an is an as yet undetermined constant of integration. The gesolution of the excess pore pressure therefore gives

u~r ,z,t !5 (n50

`

ane28Th /nn@c1nI 0~mnr !1c2nK0~mnr !11#sinvnz

(52)

It remains now to determine the constantsan , c1n , and c2n tocomplete the solution. The values of these constants depethe boundary and initial conditions in the undisturbed soil mawell as the conditions in the smeared remolded zone and theresistance in the vertical drain.

Consideration of Remolded Zone and Well Resistance

Remolded ZoneIn keeping with the assumptions of Barron~1948!, the presenpaper assumes that there is no consolidation in the remoldedand the drainage well, that is, consideration is given only todrainage in these zones. Continuity of flow through the remozoner w,r ,r s therefore gives

]2u8]r 2 1

1

r

]u8]r

50 (53)

whereu85excess pore pressure in the remolded zone. Invothe Fourier series expansion,

u85 (n50

`

Un8~r ,t !sinvnz (54)

and assuming thatUn8(r ,t)5An8(r )Bn(t), Eq. ~53! gives

]2An8

]r 2 11

r

]An8

]r50 (55)

yielding the solution

Un8~r ,t !5lnfn$c5n ln r 1c6n%Bn~ t ! (56)

wherec5n , c6n are once again the constants of integration. Itbe noted here that in order to ensure continuity of pore pre~and flow! at all times, the same time functionBn(t) must prevaiin each of the three regions. At the interface of the remoundisturbed soil, continuity of pore pressure and flow ensurethe following must hold:

Un~r s ,t !5Un8~r s ,t ! (57)

kh

]Un

]r Ur 5r s

5ks

]Un8

]rU

r 5r s

(58)

in which ks is the coefficient of permeability in the smeared zoSubstituting the appropriate relationships@Eqs. ~35!, ~37!, and~56!# into the continuity equations above, it is found that

c1nI 0~mnr s!1c2nK0~mnr s!115c5n ln r s1c6n (59)

c5n5mnr s

kh

ks@c1nI 1~mnr s!2c2nK1~mnr s!# (60)

where use has been made of the following Bessel function
tions:


I 08~mnr !5mnI 1~mnr ! and K08~mnr !52mnK1~mnr !

Well ResistanceFrom consideration of continuity and assuming that there iconsolidation as well as that flow exists in the vertical well inz spatial direction only, then~Barron 1948!

]2uw

]z2 12

r w

ks

kwS ]u8

]r Dr 5r w

50 (61)

where uw ,kw5pore pressure and coefficient of permeabilitythe well, respectively. Expandinguw as a finite Fourier seriegives

uw~z,t !5 (n50

`

Uwn~ t !sinvnz (62)

and substituting into Eq.~61! leads to

2vn2Uwn1

2

r w

ks

kwlnfn

c5n

r wBn~ t !50 (63)

so that

Uwn5lnfn

2

r w2

ks

kw

c5n

vn2 Bn~ t ! (64)

Sinceuw(z,t)5u8(r w ,z,t) it follows from Eqs.~56! and~64! that

lnfn~c5n ln r w1c6n!Bn~ t !5lnfn

2

r w2

ks

kw

c5n

vn2 Bn~ t !

giving

c6n5F 2

r w2

ks

kw

1

vn22 ln r wGc5n (65)

Backsubstituting Eq.~65! into Eq. ~59!, then rearranging Eq~59! and ~60! results in

anc1n1bnc2n1150 (66)

where

an5I 0~mnr s!2I 1~mnr s!Fn (67)

bn5K0~mnr s!1K1~mnr s!Fn (68)

Fn5mnr sH kh

kslnS r s

r wD1

2

r w2

kh

kw

1

vn2J (69)

Now, at the radius of influencer 5r e of the undisturbed soradial flow ceases, and therefore

]u

]r Ur 5r e

50

and it follows after taking the derivative of Eq.~52! that

I 1~mnr e!c1n2K1~mnr e!c2n50 (70)

Solving for c1n andc2n using Eqs.~66! and ~70! it is found that


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c1n5K1~mnr e!

Dn(71)

c2n5I 1~mnr e!

Dn(72)

where

Dn52anK1~mnr e!2bnI 1~mnr e! (73)

Excess Pore Pressure Solutions for a Step Loading

Finally, to determine the value of the constantan it is necessary trevert to the initial pore water pressure in the undisturbedFollowing Eq. ~52!, the radial-averaged excess pore pressut50 is given as

u~z,t50!51

p~r e22r s

2! (n50

`

anEr s

r e

@c1nI 0~mnr !

1c2nK0~mnr !11#2pr dr sinvnz

5 (n50

`

anVn sinvnz (74)

Suppose that the placement of a single uniform loadingq0 at t50 leads to an initial excess pore-water pressureu0 , then a Fourier series expansion gives

u05 (n50

`4u0

~2n11!psinvnz (75)

so that

an54u0

~2n11!pVn(76)

Hence, the final solution for the excess pore-water pressureundisturbed soil for a single loading step gives

u~r ,z,Th!5 (n50

`4u0

~2n11!pVne28Th /nn@c1nI 0~mnr !


Limiting Solutions

Case of no SmearSuppose there is no smear, but well resistance is presentr w→r s and

Fn52

r s

Akhkv

kw

1

vn

andan ,bn simplify to

an5I 0~mnr s!22

r s

Akhkv

kw

I 1~mnr s!

vn(78)

bn5K0~mnr s!12

r

Akhkv

k

K1~mnr s!

v(79)

s w n



while the solutions given in Eqs.~2! and ~15!–~21! remain with-out changes, except for Eqs.~78! and~79! replacing Eqs.~12! and~13!, respectively.

Case of Ideal Vertical DrainIn the case of an ideal vertical drain where there is no smeawell resistance, thenr w ,kw→` so that an5I 0(mnr s) andbn5K0(mnr s), while the rest of the solution remains intact.

Case of Radial ConsolidationIf only radial consolidation is considered in the undisturbedthen the continuity equation reduces to

kh

gwS ]2u~r ,z,t !

]r 2 11

r

]u~r ,z,t !

]r D52]«z~z,t !

]t(80)

and the solution gives

u~r ,z,Th!5 (n50

`4u0

~2n11!pnne28Th /nnF lnS r

r sD2

~r 22r s2!

2r e2

1kh

ks

~b22a2!

b2 ln a12kh

vn2r w

2 kw

~b22a2!

b2 Gsinvnz

(81)

whereb5r e /r w , a5r s /r w , and

nn5b2

b22a2 lnS b

aD23

41

a2

4b2 1kh

ks

~b22a2!

b2 ln a

12kh

vn2r w

2 kw

~b22a2!

b2 (82)

Based on the definitions of Eqs.~15!–~17!, the average excepore water pressure is found as

u~z,Th!5 (n50

`4u0

~2n11!pnne28Th /nnr w

2 Fb2 lnS b

aD23b2

41a2

1a4

4b2 1kh

ks

~b22a2!2

b2 ln a

12kh

vn2r w

2 kw

~b22a2!2

b2 Gsinvnz (83)

u~r ,Th!5 (n50

`8u0

~2n11!2p2nne28Th /nnF lnS r

r sD2

~r 22r s2!

2r e2

1kh

ks

~b22a2!

b2 ln a12kh

vn2r w

2 kw

~b22a2!

b2 G (84)

u~Th!5 (n50

`8u0

~2n11!2p2 e28Th /nn (85)

The degree of consolidation defined in accordance with~18!–~20! is found by substituting for the average excess pwater pressure in Eqs.~83!–~85!. Now, the overall average degr
of consolidation for the entire undisturbed soil mass is given by

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U~Th!512 (n50

`8

~2n11!2p2 e28Th /nn (86)

The last term of Eq.~82! clearly represents the effects of wresistance, and if this vanishes, for instance, in prefabricdrains where well resistance is negligible~Hansbo 1981!, Eq.~86!further reduces to

U~Th!512e28Th /n (87)

where

n5b2

b22a2 lnS b

aD23

41

a2

4b2 1kh

ks

~b22a2!

b2 ln a (88)

Now, if r e /r s is large,b2/(b22a2)'1, the solution in Eq.~87!approaches the solution of Hansbo~1981! identically.

Ramp Loading

The loading sequence is often more realistically simulatedramp type loading~Fig. 5! where the load is applied uniformlytime, such that

q~ t !5H s0t, t,t0

q0 , t>t0(89)

Consider first the loading sequence where loading is incremuniformly at a constant rate where]q(t)/]t5s0 ; then it followsfrom Eqs.~38! and ~44! that

]«zn

]t5mvH 4s0

~2n11!p2

]Un~ t !

]t J 5lnBn~ t ! (90)

so that

]Un~ t !

]t5lnfnVn

]Bn

]t52

ln

mnS Bn2

mv

ln

4s0

~2n11!p D (91)

The solution of Eq.~91! is

Bn51

lnfnH ane21/mvfnVn1mvfn

4s0

~2n11!pJ (92)

Now, at t50, u(z,0)50; therefore

an52mvfn

4s0

~2n11!p52

4s0

cvvn2~2n11!p

(93)

which gives the following solution for a uniformly incrementiloading:

u~r ,z,Th!5 (n50

`4s0

cvvn2~2n11!p

$12e28Th /nn%@c1nI 0~mnr !


Consideration of principles of superposition then leads to thelowing solutions for ramp loading shown in Fig. 5:

u~r ,z,Th!5 (n50

`4s0

cvvn2~2n11!p


28Th /nn
2e %@c1nI 0~mnr !1c2nK0~mnr !1I #sinvnz


whereTh0 and the Heaviside functionH(Th) were defined prevously in Eqs.~24! and ~25!.

Notation

The following symbols are used in this paper:A 5 Skempton’s pore pressure parameter;

An 5 substitution factor, defined by Eq.~37!;a 5 the ratior s /r w ;

an 5 constant of integration;Bn 5 substitution factor, defined by Eq.~51!;

b 5 ratio r e /r w ;c1n ,c2n ,c3n ,c4n ,c5n ,c6n

5 constants of integration;ch 5 coefficient of consolidation for radial

~horizontal! flow;cv 5 coefficient of consolidation for vertical flow;E8 5 effective Young’s modulus;

h 5 soil layer thickness;kh 5 coefficient of radial~horizontal! permeability in

undisturbed soil;ks 5 coefficient of permeability in remolded or

smeared zone;kv 5 coefficient of vertical permeability in

undisturbed soil;kw 5 coefficient of permeability in drain well;mv 5 coefficient of volume compressibility;q0 5 uniform step load;

q(r ,t) 5 applied surcharge at radiusr and timet;q(t) 5 q(t), uniform loading~which varies with

time!;r 5 radius, coordinate in cylindrical coordinate

system;r e 5 radius of zone of influence;r s 5 radius~outer! of smeared or remolded zone;r w 5 radius of drainage well;s0 5 slope of ramp loading;Th 5 cht/4r e

2, time factor;Th0 5 cht0/4r e

2, time factor;t 5 elapsed time;

Ui 5 overall average degree of consolidation forloading phasei;

Un 5 Fourier coefficient ofu;Un8 5 Fourier coefficient ofu8;

Uwn 5 Fourier coefficient ofuw ;U(r ,Th) 5 average degree of consolidation at given

radius;U(Th) 5 overall average degree of consolidation for

entire undisturbed soil mass;U(z,Th) 5 average degree of consolidation at given depth

u 5 excess pore pressure in undisturbed soil;u8 5 excess pore pressure in remolded zone;ui 5 initial excess pore pressure due to application

of step loadqi ;uw 5 excess pore pressure in drain well;u0 5 initial excess pore pressure due to application

of step loadq0 ;u(r ,z,Th) 5 excess pore pressure;u(r ,Th) 5 average excess pore pressure at given radius;

u(Th) 5 average excess pore pressure for entire
undisturbed soil mass;

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u(z,Th) 5 average excess pore pressure at given depth;x,y,z 5 Cartesian coordinates;

an 5 substitution factor defined by Eq.~12! or 78~limiting case!;

bn 5 substitution factor defined by Eq.~13! or 79~limiting case!;

gn 5 chmn2/Vn , a substitution factor;

gw 5 unit weight of water;Dn 5 substitution factor defined by Eq.~11!;

«z(z,t) 5 vertical strain;ln 5 separation constant;mn 5 substitution factor5A(kv /kn)vn;n8 5 effective Poisson’s ratio;nn 5 substitution factor52Vn /(mn

2r e2);

s ,s 5 major and minor principal stress increments;
1 3


Fn 5 substitution factor defined by Eq.~14!;fn 5 substitution factor5gw /(kvvn

2);Vn 5 substitution factor defined by Eq.~7!; andvn 5 (2n11)p/(2h).

References

Barron, R. A. ~1948!. ‘‘Consolidation of fine-grained soils by drawells.’’ Trans. Am. Soc. Civ. Eng.,113, 718–742.

Eriksson, U., Hansbo, S., and Torstensson, B. A.~2000!. ‘‘Soil improve-ment at Stockholm-Arlanda airport.’’Ground Improvement,4, 73–80

Hansbo, S.~1981!. ‘‘Consolidation of fine-grained soils by prefabricadrains.’’ Proc., 10th Int. Conf. on Soil Mechanics and FoundaEngineering, Balkema, Rotterdam, The Netherlands, 3, 677–682

Yoshikuni, H., and Nakanodo, H.~1974!. ‘‘Consolidation of soils by vertical drains with finite permeability.’’Jpn. Soc. Soil Mech. Foun
Eng.,14~2!, 35–46.

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Equal Strain Consolidation by Vertical Drains

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