analytical solution for one-dimensional consolidation of...

Research ArticleAnalytical Solution for One-DimensionalConsolidation of Soil with Exponentially Time-GrowingDrainage Boundary under a Ramp Load

Ming Sun12 Meng-fan Zong2 Shao-junMa23 Wen-bing Wu 123 and Rong-zhu Liang 2

1School of Civil and Environmental Engineering Hunan University of Science and Engineering Yongzhou Hunan 425199 China2Engineering ResearchCentre of Rock-Soil Drillingamp Excavation and Protection Ministry of Education Faculty of Engineering ChinaUniversity of Geosciences Wuhan Hubei 430074 China3Zhejiang Province Institute of Architect Ural Design and Research Hangzhou Zhejiang 310006 China

Correspondence should be addressed to Wen-bingWu zjuwwb1126163com and Rong-zhu Liang liangcug163com

Received 25 April 2018 Revised 22 August 2018 Accepted 4 September 2018 Published 3 October 2018

Academic Editor Enrico Conte

Copyright copy 2018 Ming Sun et alThis is an open access article distributed under theCreative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

By introducing the exponentially time-growing drainage boundary this paper investigated the one-dimensional consolidationproblem of soil under a ramp load Firstly the one-dimensional consolidation equations of soil are established when there is aramp load acting on the soil surfaceThen the analytical solution of excess pore water pressure and consolidation degree is derivedby means of the method of separation of variables and the integral transform technique The rationality of this solution is alsoverified by comparing it with other existing analytical solutions Finally the consolidation behavior of soil is studied in detail fordifferent interface parameters or loading scheme The results show that the exponentially time-growing drainage boundary canreflect the phenomenon that the excess pore water pressure at the drainage boundaries dissipates smoothly rather than abruptlyfrom its initial value to the value of zero By adjusting the values of interface parameters 119887 and c the presented solution can bedegraded to Schiffmanrsquos solution which can compensate for the shortcoming that Terzaghirsquos drainage boundary can only considerthe two extreme cases of fully pervious and impervious boundaries The significant advantage of the exponentially time-growingdrainage boundary is that it can be applied to describe the asymmetric drainage characteristics of the top and bottom drainagesurfaces of the actual soil layer by choosing the appropriate interface parameters 119887 and c

1 Introduction

The consolidation problem of soil has always been one ofthe key problems of soil mechanics for it has close relationwith the deformation strength stability and seepage of soilmechanics Since Terzaghi [1] published his consolidationtheory in 1925 noteworthy process has been achieved inconsolidation theory In order to accurately describe the soilproperties a considerable amount of scholars has made agreat effort to build the governing equation of consolidationproblem for soil according to different soil constitutivemodels such as Terzaghi-Rendulic diffusion equation [2]three-dimensional Biotrsquos consolidation equation [3]Maxwellmodel [4] Kelvin model [5 6] and Merchant model [7]Considering the significant influence of initial condition

on the consolidation process of soil a large number ofinvestigators have also proposed various solutions for one-dimensional consolidation problem of soil under differentloads such as linear load [8] rectangular load [9] randomload [10] cyclic loading [11ndash13] and ramp loading [13ndash15]

From the aforementioned literature review it can befound that the drainage boundaries are normally treated aspervious or impervious that is Terzaghirsquos drainage boundaryNevertheless the drainage at the boundaries of consolidatingsoil is impeded for most engineering applications [16] Withregard to this aspect Gray [16] first proposed an impededdrainage boundary which was later pursued by Schiffmanand Stein [17] to investigate the consolidation problem ofsoil with variable permeability and compressibility FollowingGrayrsquoswork solutions of one-dimensional consolidation have

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 9385615 10 pageshttpsdoiorg10115520189385615

2 Mathematical Problems in Engineering

been presented for two- [18] and multilayered soils [19 20]with impeded drainage boundaries However Lee et al [21]pointed out that the impeded drainage boundary conditionscan be regarded as special cases of permeable yet incom-pressible top or bottom soil layer in multilayered systemRecently Mei et al [22] have put forward the exponentiallytime-growing drainage boundary condition (ie continuousdrainage boundary as Meirsquos definition) including permeableand impermeable boundary conditions as two extremities toconsistently describe the all drainage boundary conditionsof saturated soil Soon afterwards studies on consolidationtheory with exponentially time-growing drainage boundarycondition have been conducted for saturated soils with asingle layer [23] and multilayers [24] and unsaturated soilwith a single layer [25] Since the exponentially time-growingdrainage boundary can allow the excess pore water pressureto dissipate smoothly rather than abruptly from its initialvalue given by the initial conditions to the value of zero itmay have board application prospect in engineering

The objective of this article is to develop a generalanalytical solution for the one-dimensional consolidationof soil with exponentially time-growing drainage boundary(abbreviated as ETGD boundary for convenience) under aramp load Moreover the detailed solution is obtained for theexcess pore water pressure and overall average consolidationdegree in terms of excess pore water pressure Then acomparison is made between the present solution and Schiff-manrsquos solution [8] to verify the present solution Comparedwith Contersquos solution [13] and Olsonrsquos solution [14] it isalso worth noting that the proposed solution draws closerto engineering reality for it can account for the variationof drainage boundaries with time during the consolidationprocess In addition the influence of the interface parametersof the ETGD boundary on the consolidation behavior of soilis discussed in detail based on the present solution

2 Problem Description

The thickness coefficient of permeability and coefficient ofcompressibility of soil are denoted as 119867 = 2ℎ 119896 and 119898Vrespectively ℎ is half the thickness of the soil As shown inFigure 1 a ramp load 119901(119905) is applied on the top surface of soillayer and the ultimate load and the loading time of the rampload are represented as 1199010 and 1199050 respectively For a singlestage load with constant loading rate it can be expressed as

119901 (119905) = 119898119905 0 le 119905 lt 11990501199010 1199050 le 119905 (1)

where 119898 = 11990101199050 indicates the loading rate For a constantultimate load the smaller 1199050 means the larger loading rate

According to Terzaghirsquos consolidation theory the govern-ing equation of one-dimensional consolidation problem ofsoil can be expressed as

120597119906120597119905 = 119862V

12059721199061205971199112 + d119901 (119905)

d119905 (0 le 119911 le 2ℎ) (2)

where 119906 represents the excess pore water pressure of soil 119911is the downward vertical coordinate originated from the top

Time t

0

0tp

m =

0t

0p

Exte

rnal

load

P

0

loading stage constant load stage

Figure 1 Relation of loading and time

boundary 119905 is the loading time 119862V = 119896120574119908119898V indicates theconsolidation coefficient of soil layer and 120574119908 denotes the unitweight of water respectively

Then this work introduces the ETGD boundary asfollows [20]

119906 (0 119905) = 119901 (119905) 119890minus119887119905119906 (2ℎ 119905) = 119901 (119905) 119890minus119888119905 (3)

where 119887 and 119888 are interface parameters which reflect thedrainage properties of the top and bottom drainage surfacesof soil layer respectivelyThe two parameters can be obtainedby experimental simulation or engineering measurementinversion

The initial boundary is given as

119906 (119911 0) = 119901 (0) (4)

3 Analytical Solution for This Model

Letting 119906 = V + (2ℎ minus 119911)119901(119905)119890minus1198871199052ℎ + 119911119901(119905)119890minus1198881199052ℎ andsubstituting it into (2) yield

120597V120597119905 = 119862V

1205972V1205971199112 + 119891 (119911 119905) (5)

where

119891 (119911 119905) = 119887119901 (119905) (2ℎ minus 119911) 119890minus1198871199052ℎ + 119888119901 (119905) 119911119890minus119888119905

2ℎ+ (1 minus (2ℎ minus 119911) 119890minus119887119905 + 119911119890minus119888119905

2ℎ ) d119901 (119905)d119905

(6)

Combining with the expression form of ramp load 119901(119905)119891(119911 119905) can be rewritten as

119891 (119911 119905) = 1198911 (119911 119905) 0 le 119905 lt 11990501198912 (119911 119905) 1199050 le 119905 (7)

Mathematical Problems in Engineering 3

where

1198911 (119911 119905) = 119898 (2ℎ minus 119911) (119887119905 minus 1) 119890minus1198871199052ℎ + 119898119911 (119888119905 minus 1) 119890minus119888119905

2ℎ+ 119898

(8)

1198912 (119911 119905) = 1198871199010 (2ℎ minus 119911) 119890minus1198871199052ℎ + 1198881199010119911119890minus1198881199052ℎ (9)

Then the boundary conditions and initial condition canbe rewritten as

V (0 119905) = 0V (2ℎ 119905) = 0 (10)

V (119911 0) = 0 (11)

The general solution of (5) can be set as

V (119911 119905) = infinsum119899=1

V119899 (119905) sin 1198991205871199112ℎ (12)

Substituting (12) into (5) givesinfinsum119899=1

V1015840119899 (119905) sin 1198991205871199112ℎ + 119862V

infinsum119899=1

(1198991205872ℎ )2 V119899 (119905) sin 119899120587119911

2ℎ= infinsum119899=1

119891119899 (119905) sin 1198991205871199112ℎ

(13)

where 119891119899(119905) can be expressed according to Fourier series as

119891119899 (119905) = 1ℎ int2ℎ0

119891 (119911 119905) sin 1198991205871199112ℎ d119911

= 1198911198991 (119905) 0 le 119905 lt 11990501198911198992 (119905) 1199050 le 119905

(14)

where

1198911198991 (119905) = 2119898119899120587 [(119887119905 minus 1) 119890minus119887119905 minus (minus1)119899 (119888119905 minus 1) 119890minus119888119905]+ 2119898

119899120587 [1 minus (minus1)119899](15)

1198911198992 (119905) = 21199010119899120587 [119887119890minus119887119905 minus (minus1)119899 119888119890minus119888119905] (16)

In order to make (13) always true the following equationcan be derived

V1015840119899 (119905) + 119862V (1198991205872ℎ )2 V119899 (119905) minus 119891119899 (119905) = 0 (17)

For a single stage load with constant loading rate it canbe determined that 119906(119911 0) = 119901(0) = 0 Moreover it can alsoobtain that V119899(0) = 0 because V(119911 0) = 0

Utilizing the Laplace transform technique to solve (17)yields

V119899 (119905) = int1199050119891119899 (120591) 119890minus119863(119905minus120591)d120591 (18)

where

119863 = 119862V (1198991205872ℎ )2 (19)

31 Solution of Loading Stage For loading stage (ie 0 le 119905 lt1199050) one can derive

V119899 (119905) = 2119898119899120587 119890minus119863119905 int119905

01198911198991 (120591) 119890120591d120591 (20)

Substituting (20) into (12) gives

V (119911 119905)= infinsum119899=1

2119898 [(119887119905 minus 1) 119890minus119887119905 + 119890minus119863119905]119899120587 (119863 minus 119887) sin 119899120587119911

2ℎ

minus infinsum119899=1

2119898 (minus1)119899 [(119888119905 minus 1) 119890minus119888119905 + 119890minus119863119905]119899120587 (119863 minus 119888) sin 119899120587119911

2ℎ


2119898119887 (119890minus119887119905 minus 119890minus119863119905)119899120587 (119863 minus 119887)2 sin 119899120587119911

2ℎ

+ infinsum119899=1

2119898119888 (minus1)119899 (119890minus119888119905 minus 119890minus119863119905)119899120587 (119863 minus 119888)2 sin 119899120587119911

2ℎ

+ infinsum119899=1

2119898 (1 minus (minus1)119899) (1 minus 119890minus119863119905)119899120587119863 sin 119899120587119911

2ℎ

(21)

Furthermore the excess porewater pressure of soil duringthe loading stage can be derived as

119906 = V + (2ℎ minus 119911)119898119905119890minus1198871199052ℎ + 119911119898119905119890minus119888119905



2ℎ



2ℎ



2ℎ

+ infinsum119899=1


2ℎ

+ infinsum119899=1


2ℎ+ (2ℎ minus 119911)119898119905119890minus119887119905

2ℎ + 119911119898119905119890minus1198881199052ℎ

(22)


32 Solution of Constant Load Stage For constant load stage(ie 1199050 le 119905) one can obtain

V119899 (119905) = 2119898119899120587 119890minus119863119905 [int1199050

01198911198991 (120591) 119890120591d120591 + int119905

1199050

1198911198992 (120591) 119890120591d120591]

= infinsum119899=1

21199010119887119890minus119887119905119899120587 (119863 minus 119887) sin1198991205871199112ℎ


(minus1)119899 21199010119888119890minus119888119905119899120587 (119863 minus 119888) sin 1198991205871199112ℎ

+ infinsum119899=1

21199010119860119899119890minus119863119905119899120587 sin 1198991205871199112ℎ

(23)

where

119860119899 = minus119890(119863minus119887)1199050 + 11199050 (119863 minus 119887) minus (minus1)119899 [minus119890(119863minus119888)1199050 + 1]

1199050 (119863 minus 119888)minus 119887 (119890(119863minus119887)1199050 minus 1)

1199050 (119863 minus 119887)2 + 119888 (minus1)119899 (119890(119863minus119888)1199050 minus 1)1199050 (119863 minus 119888)2

+ (1 minus (minus1)119899) (1198901198631199050 minus 1)1199050119863

(24)

Then the excess pore water pressure of soil during theconstant load stage can be obtained as

119906 = V + 1199010 (2ℎ minus 119911) 119890minus1198871199052ℎ + 1199010119911119890minus1198881199052ℎ

= infinsum119899=1

21199010119887119890minus119887119905119899120587 (119863 minus 119887) sin1198991205871199112ℎ



+ infinsum119899=1

21199010119860119899119890minus119863119905119899120587 sin 1198991205871199112ℎ + 1199010 (2ℎ minus 119911) 119890minus119887119905

2ℎ+ 1199010119911119890minus1198881199052ℎ

(25)

4 Verification and Parametric Study

41 Degeneration of Solution In order to verify the effective-ness and accuracy of the presented solution it is needed todegenerate the proposed solution and compare it with exist-ing solutions Therefore this section conducts the verificationfor both loading stage and constant load stage

411 Degeneration of Solution at Loading Stage When theinterface parameters 119887 and 119888 approach infinity (22) can bedegenerated as

119906 (119911 119905) = infinsum119899=1


2ℎ

= 161199010119879V01205873infinsum119899=135

11198993 (1 minus 119890minus11989921205872119879V4) sin 119899120587119911

2ℎ(26)

412 Degeneration of Solution at Constant Load Stage Whenthe interface parameters 119887 and 119888 tend to infinity (24) can bereduced as

119860119899 = (1 minus (minus1)119899) (1198901198631199050 minus 1)1199050119863 (27)

Then (25) can be degenerated as

119906 = infinsum119899=1

21199010119860119899119890minus119863119905119899120587 sin 1198991205871199112ℎ = 161199010119879V01205873

sdot infinsum119899=135

11198993 (1 minus 119890minus11989921205872119879V0 4) 119890minus11989921205872(119879Vminus119879V0 )4 sin 119899120587119911

2ℎ(28)

Apparently (26) and (28) are consistent with the corre-sponding equations proposed by Schiffman utilizing Terza-ghirsquos drainage boundary [8]

Then a relevant computer program is also developed andapplied to compare the presented solution with Schiffmanrsquossolution [8] For the degeneration of the ETGD boundary theinterface parameters both 119887 and 119888 are set as sufficiently smallvalues (ie 119887 = 119888 = 0001dminus1) and large enough values (ie119887 = 119888 = 1000dminus1) which can be regarded as the imperviousboundary and pervious boundary respectively The unitweight elastic modulus Poissonrsquos ratio and permeabilitycoefficient of soil are 181kN sdot mminus3 10MPa 03 and 74 times10minus3m sdot dminus1 respectively Lastly the time factor which isexpressed as 119879V = 119862V1199051198672 is also introduced for convenientcomparison and 119879v0 is the corresponding time factor withrespect to the final loading time

As shown in Figure 2 when the interface parameters 119887and c and the time factor 119879V have smaller values 1199061199010 tendsto 09 because the drainage velocity is small and the drainagetime is very short For the curve obtained by Schiffmanrsquossolution there are mutations at 119911 = 0 and 119911 = 2ℎ whichare different with the actual situations When the interfaceparameters 119887 and c and the time factor 119879V have larger valuesboth the values of 1199061199010 obtained by the presented solutionand Schiffmanrsquos solution are consistent and in symmetri-cal distribution along depth The values of 1199061199010 tend tozero because the drainage velocity is large enough and thedrainage time is long enough The reason for that the valuesof 1199061199010 are not zero is that the consolidation process is atloading stage and the increment of applied loading should befirst borne by the excess porewater pressureThe results showthat the solution obtained by Terzaghirsquos drainage boundarycan not be reduced to the solution degenerated from thepresented solution and the curves obtained by the solutionbased on Terzaghirsquos drainage boundary are not smooth andhave mutations at the top and bottom drainage surfaces Inother words whatever the value of time factor is the excesspore water pressure based on Terzaghirsquos drainage boundaryat 119911 = 0 and 119911 = 2ℎ is always zero Actually the pore water


Schiffman [8] Tv0 =0001 Tv=00009Schiffman [8] Tv0 =5 Tv =4

b=c=0001d-1 Tv0 =0001 Tv=00009b=c=1000d-1 Tv0 =5 Tv =4

0

02

04

06

08

1

up 0

04 08 12 16 20zh

Figure 2 Comparison of presented solution with Schiffmanrsquossolution

pressure can not be completely dissipated when the drainagetime is zero or very small and the exponentially time-growing drainage boundary can reflect this phenomenonthat the excess pore water pressure at the drainage boundarydissipates smoothly rather than abruptly from its initial valuegiven by the initial conditions to the value of zero

Moreover further comparison of the proposed solutionwith Schiffmanrsquos solution is conducted by analyzing theinfluence of 119879v0 on excess pore water pressure for bothloading stage and constant load stage The smaller 119879v0 meansthe shorter loading time from loading stage to constantload stage and also implies the faster loading rate As canbe observed in Figure 3 at the loading stage (119879v lt 119879v0)the excess pore water pressure increases with the increasingloading rate for the external load increases with the increaseof loading rate at a given time However at the constantload stage (119879v gt 119879v0) as shown in Figure 4 the excesspore water pressure at a given time decreases as the loadingrate increases for there is a longer time to dissipate theexcess pore water pressure if the loading process is completedwithin a short period of time In addition it can also beseen from the diagrams that when the interface parameters119887 and 119888 equal to 1000dminus1 the pore pressure curve obtainedby the ETGD boundary coincides with the pore pressurecurve obtained by Schiffmanrsquos solution This indicates thatthe solution obtained by the ETGD boundary can be reducedto Schiffmanrsquos solution when 119887 and 119888 value are large enoughwhich further verifies the correctness of the present solution

42 Influence of Number of Term in Series on CalculatedResults Figure 5 indicates the influence of number of termsin series on calculated results For the parameters of 119887 =119888 = 1dminus1 119879V0 = 1 119911 = ℎ it can be seen that the excess

b=c=1000d-1 Tv =05

0507Tv0 =09

Schiffman [8]

2

16

12

08

04

0

zh

02 04 06 080uq0

Figure 3 Influence of 119879v0 on excess pore water pressure at loadingstage

b=c=1000d-1 Tv =05Schiffman [8]

Tv0 =01 Tv0 =03 Tv0 =05

02 04 06 080uq0

2

16

12

08

04

0

zh

Figure 4 Influence of 119879v0 on excess pore water pressure at constantloading stage

pore water pressure of loading stage is larger than that ofconstant load stage The excess pore water pressure increasesas the time factor increases during the loading stage Butfor the constant load stage the excess pore water pressuredecreases with the increase of time factor because the porewater pressure dissipates gradually When the time factor hassmaller value it needs more number of terms in series tomeet the convergence of solution However if the time factoris large enough the fluctuation of the curves is small andthe convergence of solution can be satisfied by only takingtwo or three terms Therefore unless otherwise specified the


b=c=1d-1 Tv0 =1

0 4 8 12 16 20

02

03

04

05

06

Number of terms in series

up 0

Tv =1

Tv =08

Tv =05

Tv =13

Tv =15

Figure 5 Influence of number of terms in series on calculatedresults

0 01 02 03 04 05 06 072

16

12

08

04

0

up0

zh

Schiffman [8] Tv=01Schiffman [8] Tv=03Schiffman [8] Tv=09

b=c=1d-1 Tv =01

b=c=1d-1 Tv =03

b=c=1d-1 Tv =09

Tv0 =1

Figure 6 The excess pore water pressure distribution curves atloading stage while b=c

number of terms in series is set as five terms for efficientcalculation in the following analysis

43 Influence of Interface Parameters 119887 and 119888 on the ExcessPoreWater Pressure of Soil Figure 6 illustrates the excess porewater pressure distribution curves at loading stage while b=cWhen 119887 is equal to c the top and bottom drainage surfacesof soil exhibit the same permeability which results in the factthat the excess pore water pressure curves are in symmetricaldistribution about 119911 = ℎ With the gradual increase of timefactor 119879V the excess pore water pressures at the drainagesurfaces and interior of soil gradually dissipate instead ofthat the excess pore water pressures obtained by Schiffmanrsquos

0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =01Tv =03

Tv =05Tv =09

b=3d-1c=2d-1

T0=1

Figure 7 The excess pore water pressure distribution curves atloading stage while b =c

solution are always zero at the drainage surfaces It canbe also observed from Figure 6 that the excess pore waterpressure obtained by the presented solution will increasefirst and then decrease at the drainage surfaces The reasonfor the abovementioned phenomena may be that at thebeginning the excess pore water pressure at the drainagesurfaces increases with the increase of the applied loadingafter that the rate of dissipation becomes greater than the rateof loading increment It is also worth noting that the solutionbased on Terzaghirsquos drainage boundary can not reflect theabove-mentioned phenomena

As depicted in Figure 7 when 119887 is not equal to c forexample 119887 = 3dminus1 119888 = 2dminus1 the upper part of the excesspore water pressure distribution curves at loading stage isobviously steeper than the lower part which means thatthe drainage velocity of the upper part of the soil is largerthan that of the lower part of soil Therefore it can alsobe understood that the greater the value of the interfaceparameter the larger the drainage velocity of the correspond-ing haft part of the soil Terzaghirsquos drainage boundary cannot consider the difference of the drainage capacity betweenthe upper and lower parts of soil layer for the excess porewater pressure distribution curves at loading stage based onTerzaghirsquos drainage boundary are always symmetrical about119911 = ℎ [8] It can also be seen from Figure 7 that thesignificant advantage of the ETGD boundary is that it canbe applied to approximately model the asymmetric drainagecharacteristics of the top and bottom drainage surfaces ofactual soil layer by adjusting the interface parameters 119887 andc

As demonstrated in Figure 8 when 119887 is equal to cthe top and bottom drainage surfaces of soil also exhibitthe same permeability for the excess pore water pressurecurves at constant load stage also distribute symmetricallyabout 119911 = ℎ It is also found that the excess pore water


0 01 02 03 04 05 06 072

16

12

08

04

0


b=c=1d-1 Tv =1b=c=1d-1 Tv =13b=c=1d-1 Tv =2

Tv0 =1

up0

zh

Figure 8 The excess pore water pressure distribution curves atconstant load stage while b=c

0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =1Tv =13

Tv =15Tv =2

b=1d-1c=3d-1

T0=1

Figure 9 The excess pore water pressure distribution curves atconstant load stage while b =c

pressure at the drainage surfaces and interior of soil graduallydissipates as the time factor 119879V increases but the excess porewater pressures at constant load stage obtained by Schiffmanrsquossolution are also zero at the drainage surfaces

As shown in Figure 9 when 119887 is not equal to c for example119887 = 1dminus1 119888 = 3dminus1 the lower part of the excess porewater pressure distribution curves at constant load stage isobviously steeper than the upper part which implies that thedrainage velocity of the lower part of the soil is larger thanthat of the upper part of the soil Meanwhile the solution

0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =09Tv =09Tv =09

Tv =12Tv =12Tv =12

C=1m2d 5m2

d 5m2d10m2

d 10m2d1m2

d

T0

=1b=c=1d-1

Figure 10 Influence of consolidation coefficient on excess porewater pressure distribution curves

based on Terzaghirsquos single-drainage condition can not beextended to the soil layer with the upper and bottom surfaceshaving different drainage capacity for the excess pore waterpressure distribution curves at constant load stage based onTerzaghirsquos drainage boundary are always symmetrical about119911 = ℎ44 Influence of Consolidation Coefficient on the Excess PoreWater Pressure of Soil Figure 10 shows the influence ofconsolidation coefficient on the excess pore water pressureof soil For the loading stage for example 119879V0 = 1 119879V =09 the excess pore water pressure of soil decreases as theconsolidation coefficient decreases For the constant loadstage for example 119879V0 = 1 119879V = 12 the excess porewater pressure of soil also decreases with the decrease ofconsolidation coefficient The reason for these phenomena isthat for the same time factor 119879V the smaller consolidationcoefficient means the longer consolidation time and thelonger consolidation time will lead tomore sufficient dissipa-tion of the excess pore water pressure It is also worth notingthat the smaller consolidation coefficient means the weakerpermeability of soil which results in smaller drainage velocityof soil Therefore the results show that the consolidationtime has more significant influence on the excess porewater pressure than consolidation coefficient As can be seenfrom Schiffmanrsquos results [8] the change of consolidationcoefficient for constant time factor has very weak impact onthe excess pore water pressure which means that Terzaghirsquosdrainage boundary can not completely reflect the actualmechanism of consolidation process of soil influenced bytime It can also be observed from Figure 10 that the degreeof the influence of consolidation coefficient on excess porewater pressure decreases with the decrease of consolidationcoefficient


0 02 04 06 08 12

16

12

08

04

0

up0

zh b=c=1d-1

Tv0 =05Tv=05Tv0 =1Tv=05Tv0 =15Tv=05

Tv0 =05Tv=15Tv0 =1Tv=15Tv0 =15Tv=15

Figure 11 Influence of loading rate on excess pore water pressuredistribution curves

45 Influence of Loading Rate on the Excess Pore WaterPressure of Soil Figure 11 indicates the influence of loadingrate on excess pore water pressure when the other parametersof soil remain constant As can be observed in Figure 11the excess pore water pressure increases with the increase ofloading rate because the excess porewater pressure can not besufficiently dissipated within the shorter consolidation timewhen the loading rate is too fast For the constant load stagethe excess pore water pressure under the same time factordecreases as the loading rate increases for there is a longertime to dissipate the excess pore water pressure if the loadingprocess is completed within a short period of time

5 Analysis of Consolidation Degree

According to Soil Mechanics the overall average consolida-tion degree in terms of excess pore water pressure can beexpressed as

119880119875 = int2ℎ0

(119901 (119905) minus 119906)d1199112ℎ1199010 (29)

Then by substituting (22) into (29) the overall averageconsolidation degree in terms of excess pore water pressureat loading stage can be obtained as

119880119875 = int2ℎ0

(119901 (119905) minus 119906) d1199112ℎ1199010 = 119901 (119905)

1199010 minus int2ℎ0

119906d1199112ℎ1199010

= 119879V119879V0minus infinsum119899=1

2 (1 minus (minus1)119899) [(119887119905 minus 1) 119890minus119887119905 + 119890minus119863119905](119899120587)2 (119863 minus 119887) 1199050

001 01 1 101

08

06

04

02

0

Up

Tv

Schiffman [8] Tv0 =01

Tv0 =01 b=c=1d-1

Tv0 =01 b=c=10d-1b=1d-1

c=10d-1T0=01

Figure 12 Influence of interface parameters 119887 and 119888 on theconsolidation degree of soil



+ infinsum119899=1

2119887 (1 minus (minus1)119899) (119890minus119887119905 minus 119890minus119863119905)(119899120587)2 (119863 minus 119887)2 1199050

+ infinsum119899=1



4 (1 minus (minus1)119899) (1 minus 119890minus119863119905)(119899120587)21198631199050 minus 119905119890minus119887119905

21199050 minus 119905119890minus11988811990521199050

(30)

Furthermore by substituting (25) into (29) the overallaverage consolidation degree defined by excess pore waterpressure at constant load stage can be derived as

119880119901 = int2ℎ0

(1199010 minus 119906)d1199112ℎ1199010 = 1 minus int2ℎ

0119906d119911

2ℎ1199010= 1 minus infinsum

119899=1

2 (1 minus (minus1)119899) 119887119890minus119887119905(119899120587)2 (119863 minus 119887)




2 (1 minus (minus1)119899)119860119899119890minus119863119905(119899120587)2 minus 119890minus119887119905

2 minus 119890minus1198881199052

(31)

Figure 12 depicts the influence of interface parameters 119887and 119888 on the consolidation degree of soil The consolidation


degree obtained by the presented solution will approach thatcalculated by Schiffmanrsquos solution if the interface parameters119887 and 119888 are large enough for the boundary surfaces withincreasing interface parameters will tend to be perviousboundary In contrast if the interface parameters 119887 and 119888are becoming smaller the consolidation degrees obtainedby the proposed solution will gradually deviate from thatcalculated by Schiffmanrsquos solution It can also be found thatthe difference between the consolidation degree obtainedby the proposed solution and that calculated by Schiffmanrsquossolution increases with the decrease of time factor It is alsoworth noting from Figure 12 that the early consolidationprocess of Terzaghirsquos drainage boundary is faster than thatof the ETGD boundary but the late consolidation process ofthe ETGD boundary is faster than that of Terzaghirsquos drainageboundary Furthermore if the interface parameters 119887 and 119888are large enough the consolidation curves obtained by theETGDboundarywill essentially coincidewith those obtainedby Terzaghirsquos drainage boundary at the late consolidationprocess The results show that for practical engineeringeven though the early consolidation process of the soil withthe boundary surfaces designed according to the ETGDboundary is slow it can make more sufficient dissipation ofthe excess pore water pressure of the whole soil layer and itwill not lead to the slowdrainage of the lower soil layer or eventhe discharge of water after the consolidation of the uppersoil layer At the same time the ultimate consolidation timebased on the ETGDboundary is almost the same as that basedon Terzaghirsquos drainage boundary When utilizing the ETGDboundary to design the actual drainage boundaries there isno need to deal with the drainage surfaces as fully perviousboundaries It means that the overall economy of the designbased on the ETGD boundary is clearly superior to that basedon Terzaghirsquos drainage boundary

As shown in Figure 13 the smaller initial time factor 119879V0implies the faster loading process of ramp load It can be seenthat the faster the loading process of ramp load the fasterthe consolidation process When the loading rate has largervalue the early consolidation process is faster but the lateconsolidation process is slower In contrast when the loadingrate has slower value the early consolidation process is slowerbut the late consolidation process is faster For the ETGDboundary when the consolidation degree is equal to 80the time factor is about 11 13 and 88 when 119879V0 is equal to01 1 and 10 respectively For Terzaghirsquos drainage boundarywhen consolidation degree is equal to 80 the time factoris about 063 12 and 84 when 119879V0 is equal to 01 1 and 10respectivelyThe results imply that the ultimate consolidationtime based on the ETGD boundary is almost the same as thatbased on Terzaghirsquos drainage boundary when the loading rateis slow enough

6 Conclusions

In this article an analytical solution for the one-dimensionalconsolidation of soil layer under a ramp load is derivedby adopting the ETGD boundary The proposed analyticalsolution is also verified by comparing it with Schiffmanrsquossolution [8] Then selective results are presented to illustrate

Up

Tv

001 01 1 101

08

06

04

02

0

Schiffman [8]Tv0 =01Schiffman [8] Tv0 =1


Tv0 =01 b=c=1d-1

Tv0 =1 b=c=1d-1

Tv0 =10 b=c=1d-1

Figure 13 Influence of loading rate of ramp load on the consolida-tion degree of soil

the consolidation behavior of soil with ETGD boundarySchiffmanrsquos solution can be regarded as a special case of thepresent solution and the excess pore water pressure andconsolidation degree calculated by the presented solutionwill approach those calculated by Schiffmanrsquos solution if theinterface parameters 119887 and 119888 are large enough The importantadvantage of the ETGD boundary is that it can be utilizedto describe the asymmetric drainage characteristics of thetop and bottom drainage surfaces of the actual soil layerby choosing the appropriate interface parameters 119887 and 119888Significantly the present solution is just for homogeneoussoil and the consolidation problem of multilayered soil willbe investigated by the authorsrsquo team in near future

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grants nos 51578164 5167854751878634 51878185 and 41807262) and the China Post-doctoral Science Foundation Funded Project (Grants no2016M600711 and no 2017T100664) The Research Fundsprovided by MOE Engineering Research Center of Rock-Soil Drilling amp Excavation and Protection (Grant no 201402)and the Fundamental Research Funds for the Central


Universities-Cradle Plan for 2015 (Grant no CUGL150411)are also acknowledged

References

[1] K Terzaghi Erdbaumechanik and bodenphysikalischer grund-lage Lpz Deuticke 1925

[2] L Rendulic Porenziffer und porenwasserdruck in Tonen vol 7Bauingenieur 1936

[3] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941

[4] T K Tan ldquoSecondary time effects and consolidation of claysrdquoScience in China Series A-Mathematics vol 1 no 11 pp 1060ndash1075 1958

[5] R E Gibson and K Y Lo A theory of consolidation forsoils exhibiting secondary compression Norwegian GeotechnicalInstitute Oslo Norway 1961

[6] K Y Lo ldquoSecondary compression of claysrdquo Journal of the SoilMechanics and Foundation Division ASCE vol 87 no 4 pp 61ndash88 1961

[7] K H Xie and X W Liu ldquoA study on one dimensionalconsolidation of soils exhibiting rheological characteristicsrdquo inInternational Symposium on Compression and Consolidation ofClayey Soils pp 385ndash388 Rotterdam The Netherlands 1995

[8] R L Schiffman ldquoConsolidation of soil under time-dependentloading andvarying permeabilityrdquo inProceedings of theHighwayResearch Board vol 37 pp 584ndash617 1958

[9] N EWilson andMM Elgohary ldquoConsolidation of SoilsUnderCyclic Loadingrdquo Canadian Geotechnical Journal vol 11 no 3pp 420ndash423 1974

[10] E E Alonso and R J Krizek ldquoRandomness of settlement rateunder stochastic loadrdquo Journal of the Geotechnical EngineeringDivisio pp 1211ndash1226 1974

[11] M Favaretti andM Soranzo ldquoA simplified consolidation theoryin cyclic loading conditionsrdquo in International Symposium onCompression and Consolidation of Clayey Soils A A BalkemaEd pp 405ndash409 Rotterdam The Netherlands 1995

[12] J Z Chen X R Zhu K H Xie Q Pan and G X Zeng ldquoOnedimensional consolidation of soft clay under trapezoidal cyclicloadingrdquo in Proceedings of 2nd International Conference on SoSoil Engineering pp 211ndash216 Hohai University Press Nanjing1996

[13] E Conte and A Troncone ldquoOne-dimensional consolidationunder general time-dependent loadingrdquoCanadianGeotechnicalJournal vol 43 no 11 pp 1107ndash1116 2006

[14] R E Olson ldquoConsolidation under time-dependent loadingrdquoJournal of the Geotechnical Engineering Division vol 103 no 1pp 55ndash60 1977

[15] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007

[16] H Gray ldquoSimultaneous consolidation of contiguous layers ofunlike compressive soilsrdquo Transactions of the American Societyof Civil Engineers vol 110 pp 1327ndash1356 1945

[17] R L Schiffmann and J R Stein ldquoOne-dimensional consol-idation of layered systemsrdquo Journal of the Soil Mechanics ampFoundations Division vol 96 no 4 pp 1499ndash1504 1970

[18] K-H Xie X-Y Xie and X Gao ldquoTheory of one-dimensionalconsolidation of two-layered soil with partially drained bound-ariesrdquo Computers amp Geosciences vol 24 no 4 pp 265ndash2781999

[19] Y-Q Cai X Liang and S-M Wu ldquoOne-dimensional consol-idation of layered soils with impeded boundaries under time-dependent loadingsrdquo Applied Mathematics and Mechanics-English Edition vol 25 no 8 pp 937ndash944 2004

[20] R P Chen W H Zhou H Z Wang and Y M Chen ldquoOne-dimensional nonlinear consolidation of multi-layered soil bydifferential quadrature methodrdquo Computers amp Geosciences vol32 no 5 pp 358ndash369 2005

[21] P K K Lee K H Xie and Y K Cheung ldquoA study on one-dimensional consolidation of layered systemsrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 16 no 11 pp 815ndash831 1992

[22] G-X Mei J Xia and L Mei ldquoTerzaghirsquos one-dimensional con-solidation equation and its solution based on asymmetric con-tinuous drainage boundaryrdquo Yantu Gongcheng XuebaoChineseJournal of Geotechnical Engineering vol 33 no 1 pp 28ndash31 2011

[23] G-X Mei and Q-M Chen ldquoSolution of Terzaghi one-dimensional consolidation equation with general boundaryconditionsrdquo Journal of Central South University vol 20 no 8pp 2239ndash2244 2013

[24] J-C Liu and G H Lei ldquoOne-dimensional consolidation oflayered soils with exponentially time-growing drainage bound-ariesrdquo Computers amp Geosciences vol 54 pp 202ndash209 2013

[25] L Wang D Sun and A Qin ldquoSemi-analytical solution to one-dimensional consolidation for unsaturated soils with exponen-tially time-growing drainage boundary conditionsrdquo Interna-tional Journal of Geomechanics vol 18 no 2 2018

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been presented for two- [18] and multilayered soils [19 20]with impeded drainage boundaries However Lee et al [21]pointed out that the impeded drainage boundary conditionscan be regarded as special cases of permeable yet incom-pressible top or bottom soil layer in multilayered systemRecently Mei et al [22] have put forward the exponentiallytime-growing drainage boundary condition (ie continuousdrainage boundary as Meirsquos definition) including permeableand impermeable boundary conditions as two extremities toconsistently describe the all drainage boundary conditionsof saturated soil Soon afterwards studies on consolidationtheory with exponentially time-growing drainage boundarycondition have been conducted for saturated soils with asingle layer [23] and multilayers [24] and unsaturated soilwith a single layer [25] Since the exponentially time-growingdrainage boundary can allow the excess pore water pressureto dissipate smoothly rather than abruptly from its initialvalue given by the initial conditions to the value of zero itmay have board application prospect in engineering

The objective of this article is to develop a generalanalytical solution for the one-dimensional consolidationof soil with exponentially time-growing drainage boundary(abbreviated as ETGD boundary for convenience) under aramp load Moreover the detailed solution is obtained for theexcess pore water pressure and overall average consolidationdegree in terms of excess pore water pressure Then acomparison is made between the present solution and Schiff-manrsquos solution [8] to verify the present solution Comparedwith Contersquos solution [13] and Olsonrsquos solution [14] it isalso worth noting that the proposed solution draws closerto engineering reality for it can account for the variationof drainage boundaries with time during the consolidationprocess In addition the influence of the interface parametersof the ETGD boundary on the consolidation behavior of soilis discussed in detail based on the present solution

2 Problem Description

The thickness coefficient of permeability and coefficient ofcompressibility of soil are denoted as 119867 = 2ℎ 119896 and 119898Vrespectively ℎ is half the thickness of the soil As shown inFigure 1 a ramp load 119901(119905) is applied on the top surface of soillayer and the ultimate load and the loading time of the rampload are represented as 1199010 and 1199050 respectively For a singlestage load with constant loading rate it can be expressed as

119901 (119905) = 119898119905 0 le 119905 lt 11990501199010 1199050 le 119905 (1)

where 119898 = 11990101199050 indicates the loading rate For a constantultimate load the smaller 1199050 means the larger loading rate

According to Terzaghirsquos consolidation theory the govern-ing equation of one-dimensional consolidation problem ofsoil can be expressed as

120597119906120597119905 = 119862V

12059721199061205971199112 + d119901 (119905)

d119905 (0 le 119911 le 2ℎ) (2)

where 119906 represents the excess pore water pressure of soil 119911is the downward vertical coordinate originated from the top

Time t

0

0tp

m =

0t

0p

Exte

rnal

load

P

0

loading stage constant load stage

Figure 1 Relation of loading and time

boundary 119905 is the loading time 119862V = 119896120574119908119898V indicates theconsolidation coefficient of soil layer and 120574119908 denotes the unitweight of water respectively

Then this work introduces the ETGD boundary asfollows [20]

119906 (0 119905) = 119901 (119905) 119890minus119887119905119906 (2ℎ 119905) = 119901 (119905) 119890minus119888119905 (3)

where 119887 and 119888 are interface parameters which reflect thedrainage properties of the top and bottom drainage surfacesof soil layer respectivelyThe two parameters can be obtainedby experimental simulation or engineering measurementinversion

The initial boundary is given as

119906 (119911 0) = 119901 (0) (4)

3 Analytical Solution for This Model

Letting 119906 = V + (2ℎ minus 119911)119901(119905)119890minus1198871199052ℎ + 119911119901(119905)119890minus1198881199052ℎ andsubstituting it into (2) yield

120597V120597119905 = 119862V

1205972V1205971199112 + 119891 (119911 119905) (5)

where

119891 (119911 119905) = 119887119901 (119905) (2ℎ minus 119911) 119890minus1198871199052ℎ + 119888119901 (119905) 119911119890minus119888119905

2ℎ+ (1 minus (2ℎ minus 119911) 119890minus119887119905 + 119911119890minus119888119905

2ℎ ) d119901 (119905)d119905

(6)

Combining with the expression form of ramp load 119901(119905)119891(119911 119905) can be rewritten as

119891 (119911 119905) = 1198911 (119911 119905) 0 le 119905 lt 11990501198912 (119911 119905) 1199050 le 119905 (7)


where


2ℎ+ 119898

(8)



V (0 119905) = 0V (2ℎ 119905) = 0 (10)

V (119911 0) = 0 (11)


V (119911 119905) = infinsum119899=1

V119899 (119905) sin 1198991205871199112ℎ (12)


V1015840119899 (119905) sin 1198991205871199112ℎ + 119862V

infinsum119899=1

(1198991205872ℎ )2 V119899 (119905) sin 119899120587119911


119891119899 (119905) sin 1198991205871199112ℎ

(13)


119891119899 (119905) = 1ℎ int2ℎ0

119891 (119911 119905) sin 1198991205871199112ℎ d119911

= 1198911198991 (119905) 0 le 119905 lt 11990501198911198992 (119905) 1199050 le 119905

(14)

where


119899120587 [1 minus (minus1)119899](15)



V1015840119899 (119905) + 119862V (1198991205872ℎ )2 V119899 (119905) minus 119891119899 (119905) = 0 (17)




where

119863 = 119862V (1198991205872ℎ )2 (19)


V119899 (119905) = 2119898119899120587 119890minus119863119905 int119905

01198911198991 (120591) 119890120591d120591 (20)


V (119911 119905)= infinsum119899=1


2ℎ



2ℎ



2ℎ

+ infinsum119899=1


2ℎ

+ infinsum119899=1


2ℎ

(21)





2ℎ



2ℎ



2ℎ

+ infinsum119899=1


2ℎ

+ infinsum119899=1


2ℎ+ (2ℎ minus 119911)119898119905119890minus119887119905

2ℎ + 119911119898119905119890minus1198881199052ℎ

(22)



V119899 (119905) = 2119898119899120587 119890minus119863119905 [int1199050

01198911198991 (120591) 119890120591d120591 + int119905

1199050

1198911198992 (120591) 119890120591d120591]

= infinsum119899=1

21199010119887119890minus119887119905119899120587 (119863 minus 119887) sin1198991205871199112ℎ



+ infinsum119899=1

21199010119860119899119890minus119863119905119899120587 sin 1198991205871199112ℎ

(23)

where





(24)



= infinsum119899=1

21199010119887119890minus119887119905119899120587 (119863 minus 119887) sin1198991205871199112ℎ



+ infinsum119899=1


2ℎ+ 1199010119911119890minus1198881199052ℎ

(25)




119906 (119911 119905) = infinsum119899=1


2ℎ

= 161199010119879V01205873infinsum119899=135

11198993 (1 minus 119890minus11989921205872119879V4) sin 119899120587119911

2ℎ(26)




119906 = infinsum119899=1

21199010119860119899119890minus119863119905119899120587 sin 1198991205871199112ℎ = 161199010119879V01205873



2ℎ(28)






b=c=0001d-1 Tv0 =0001 Tv=00009b=c=1000d-1 Tv0 =5 Tv =4

0

02

04

06

08

1

up 0

04 08 12 16 20zh





b=c=1000d-1 Tv =05

0507Tv0 =09

Schiffman [8]

2

16

12

08

04

0

zh

02 04 06 080uq0



Tv0 =01 Tv0 =03 Tv0 =05

02 04 06 080uq0

2

16

12

08

04

0

zh




b=c=1d-1 Tv0 =1

0 4 8 12 16 20

02

03

04

05

06


up 0

Tv =1

Tv =08

Tv =05

Tv =13

Tv =15


0 01 02 03 04 05 06 072

16

12

08

04

0

up0

zh


b=c=1d-1 Tv =01

b=c=1d-1 Tv =03

b=c=1d-1 Tv =09

Tv0 =1




0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =01Tv =03

Tv =05Tv =09

b=3d-1c=2d-1

T0=1






0 01 02 03 04 05 06 072

16

12

08

04

0



Tv0 =1

up0

zh


0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =1Tv =13

Tv =15Tv =2

b=1d-1c=3d-1

T0=1




0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =09Tv =09Tv =09

Tv =12Tv =12Tv =12

C=1m2d 5m2

d 5m2d10m2

d 10m2d1m2

d

T0

=1b=c=1d-1




0 02 04 06 08 12

16

12

08

04

0

up0

zh b=c=1d-1

Tv0 =05Tv=05Tv0 =1Tv=05Tv0 =15Tv=05

Tv0 =05Tv=15Tv0 =1Tv=15Tv0 =15Tv=15





119880119875 = int2ℎ0

(119901 (119905) minus 119906)d1199112ℎ1199010 (29)


119880119875 = int2ℎ0

(119901 (119905) minus 119906) d1199112ℎ1199010 = 119901 (119905)


119906d1199112ℎ1199010



001 01 1 101

08

06

04

02

0

Up

Tv


Tv0 =01 b=c=1d-1

Tv0 =01 b=c=10d-1b=1d-1

c=10d-1T0=01




+ infinsum119899=1


+ infinsum119899=1




21199050 minus 119905119890minus11988811990521199050

(30)


119880119901 = int2ℎ0


0119906d119911


119899=1






2 minus 119890minus1198881199052

(31)





6 Conclusions


Up

Tv

001 01 1 101

08

06

04

02

0



Tv0 =01 b=c=1d-1

Tv0 =1 b=c=1d-1

Tv0 =10 b=c=1d-1



Data Availability




Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018









where


2ℎ+ 119898

(8)



V (0 119905) = 0V (2ℎ 119905) = 0 (10)

V (119911 0) = 0 (11)


V (119911 119905) = infinsum119899=1

V119899 (119905) sin 1198991205871199112ℎ (12)


V1015840119899 (119905) sin 1198991205871199112ℎ + 119862V

infinsum119899=1

(1198991205872ℎ )2 V119899 (119905) sin 119899120587119911


119891119899 (119905) sin 1198991205871199112ℎ

(13)


119891119899 (119905) = 1ℎ int2ℎ0

119891 (119911 119905) sin 1198991205871199112ℎ d119911

= 1198911198991 (119905) 0 le 119905 lt 11990501198911198992 (119905) 1199050 le 119905

(14)

where


119899120587 [1 minus (minus1)119899](15)



V1015840119899 (119905) + 119862V (1198991205872ℎ )2 V119899 (119905) minus 119891119899 (119905) = 0 (17)




where

119863 = 119862V (1198991205872ℎ )2 (19)


V119899 (119905) = 2119898119899120587 119890minus119863119905 int119905

01198911198991 (120591) 119890120591d120591 (20)


V (119911 119905)= infinsum119899=1


2ℎ



2ℎ



2ℎ

+ infinsum119899=1


2ℎ

+ infinsum119899=1


2ℎ

(21)





2ℎ



2ℎ



2ℎ

+ infinsum119899=1


2ℎ

+ infinsum119899=1


2ℎ+ (2ℎ minus 119911)119898119905119890minus119887119905

2ℎ + 119911119898119905119890minus1198881199052ℎ

(22)



V119899 (119905) = 2119898119899120587 119890minus119863119905 [int1199050

01198911198991 (120591) 119890120591d120591 + int119905

1199050

1198911198992 (120591) 119890120591d120591]

= infinsum119899=1

21199010119887119890minus119887119905119899120587 (119863 minus 119887) sin1198991205871199112ℎ



+ infinsum119899=1

21199010119860119899119890minus119863119905119899120587 sin 1198991205871199112ℎ

(23)

where





(24)



= infinsum119899=1

21199010119887119890minus119887119905119899120587 (119863 minus 119887) sin1198991205871199112ℎ



+ infinsum119899=1


2ℎ+ 1199010119911119890minus1198881199052ℎ

(25)




119906 (119911 119905) = infinsum119899=1


2ℎ

= 161199010119879V01205873infinsum119899=135

11198993 (1 minus 119890minus11989921205872119879V4) sin 119899120587119911

2ℎ(26)




119906 = infinsum119899=1

21199010119860119899119890minus119863119905119899120587 sin 1198991205871199112ℎ = 161199010119879V01205873



2ℎ(28)






b=c=0001d-1 Tv0 =0001 Tv=00009b=c=1000d-1 Tv0 =5 Tv =4

0

02

04

06

08

1

up 0

04 08 12 16 20zh





b=c=1000d-1 Tv =05

0507Tv0 =09

Schiffman [8]

2

16

12

08

04

0

zh

02 04 06 080uq0



Tv0 =01 Tv0 =03 Tv0 =05

02 04 06 080uq0

2

16

12

08

04

0

zh




b=c=1d-1 Tv0 =1

0 4 8 12 16 20

02

03

04

05

06


up 0

Tv =1

Tv =08

Tv =05

Tv =13

Tv =15


0 01 02 03 04 05 06 072

16

12

08

04

0

up0

zh


b=c=1d-1 Tv =01

b=c=1d-1 Tv =03

b=c=1d-1 Tv =09

Tv0 =1




0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =01Tv =03

Tv =05Tv =09

b=3d-1c=2d-1

T0=1






0 01 02 03 04 05 06 072

16

12

08

04

0



Tv0 =1

up0

zh


0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =1Tv =13

Tv =15Tv =2

b=1d-1c=3d-1

T0=1




0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =09Tv =09Tv =09

Tv =12Tv =12Tv =12

C=1m2d 5m2

d 5m2d10m2

d 10m2d1m2

d

T0

=1b=c=1d-1




0 02 04 06 08 12

16

12

08

04

0

up0

zh b=c=1d-1

Tv0 =05Tv=05Tv0 =1Tv=05Tv0 =15Tv=05

Tv0 =05Tv=15Tv0 =1Tv=15Tv0 =15Tv=15





119880119875 = int2ℎ0

(119901 (119905) minus 119906)d1199112ℎ1199010 (29)


119880119875 = int2ℎ0

(119901 (119905) minus 119906) d1199112ℎ1199010 = 119901 (119905)


119906d1199112ℎ1199010



001 01 1 101

08

06

04

02

0

Up

Tv


Tv0 =01 b=c=1d-1

Tv0 =01 b=c=10d-1b=1d-1

c=10d-1T0=01




+ infinsum119899=1


+ infinsum119899=1




21199050 minus 119905119890minus11988811990521199050

(30)


119880119901 = int2ℎ0


0119906d119911


119899=1






2 minus 119890minus1198881199052

(31)





6 Conclusions


Up

Tv

001 01 1 101

08

06

04

02

0



Tv0 =01 b=c=1d-1

Tv0 =1 b=c=1d-1

Tv0 =10 b=c=1d-1



Data Availability




Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018










V119899 (119905) = 2119898119899120587 119890minus119863119905 [int1199050

01198911198991 (120591) 119890120591d120591 + int119905

1199050

1198911198992 (120591) 119890120591d120591]

= infinsum119899=1

21199010119887119890minus119887119905119899120587 (119863 minus 119887) sin1198991205871199112ℎ



+ infinsum119899=1

21199010119860119899119890minus119863119905119899120587 sin 1198991205871199112ℎ

(23)

where





(24)



= infinsum119899=1

21199010119887119890minus119887119905119899120587 (119863 minus 119887) sin1198991205871199112ℎ



+ infinsum119899=1


2ℎ+ 1199010119911119890minus1198881199052ℎ

(25)




119906 (119911 119905) = infinsum119899=1


2ℎ

= 161199010119879V01205873infinsum119899=135

11198993 (1 minus 119890minus11989921205872119879V4) sin 119899120587119911

2ℎ(26)




119906 = infinsum119899=1

21199010119860119899119890minus119863119905119899120587 sin 1198991205871199112ℎ = 161199010119879V01205873



2ℎ(28)






b=c=0001d-1 Tv0 =0001 Tv=00009b=c=1000d-1 Tv0 =5 Tv =4

0

02

04

06

08

1

up 0

04 08 12 16 20zh





b=c=1000d-1 Tv =05

0507Tv0 =09

Schiffman [8]

2

16

12

08

04

0

zh

02 04 06 080uq0



Tv0 =01 Tv0 =03 Tv0 =05

02 04 06 080uq0

2

16

12

08

04

0

zh




b=c=1d-1 Tv0 =1

0 4 8 12 16 20

02

03

04

05

06


up 0

Tv =1

Tv =08

Tv =05

Tv =13

Tv =15


0 01 02 03 04 05 06 072

16

12

08

04

0

up0

zh


b=c=1d-1 Tv =01

b=c=1d-1 Tv =03

b=c=1d-1 Tv =09

Tv0 =1




0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =01Tv =03

Tv =05Tv =09

b=3d-1c=2d-1

T0=1






0 01 02 03 04 05 06 072

16

12

08

04

0



Tv0 =1

up0

zh


0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =1Tv =13

Tv =15Tv =2

b=1d-1c=3d-1

T0=1




0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =09Tv =09Tv =09

Tv =12Tv =12Tv =12

C=1m2d 5m2

d 5m2d10m2

d 10m2d1m2

d

T0

=1b=c=1d-1




0 02 04 06 08 12

16

12

08

04

0

up0

zh b=c=1d-1

Tv0 =05Tv=05Tv0 =1Tv=05Tv0 =15Tv=05

Tv0 =05Tv=15Tv0 =1Tv=15Tv0 =15Tv=15





119880119875 = int2ℎ0

(119901 (119905) minus 119906)d1199112ℎ1199010 (29)


119880119875 = int2ℎ0

(119901 (119905) minus 119906) d1199112ℎ1199010 = 119901 (119905)


119906d1199112ℎ1199010



001 01 1 101

08

06

04

02

0

Up

Tv


Tv0 =01 b=c=1d-1

Tv0 =01 b=c=10d-1b=1d-1

c=10d-1T0=01




+ infinsum119899=1


+ infinsum119899=1




21199050 minus 119905119890minus11988811990521199050

(30)


119880119901 = int2ℎ0


0119906d119911


119899=1






2 minus 119890minus1198881199052

(31)





6 Conclusions


Up

Tv

001 01 1 101

08

06

04

02

0



Tv0 =01 b=c=1d-1

Tv0 =1 b=c=1d-1

Tv0 =10 b=c=1d-1



Data Availability




Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018










b=c=0001d-1 Tv0 =0001 Tv=00009b=c=1000d-1 Tv0 =5 Tv =4

0

02

04

06

08

1

up 0

04 08 12 16 20zh





b=c=1000d-1 Tv =05

0507Tv0 =09

Schiffman [8]

2

16

12

08

04

0

zh

02 04 06 080uq0



Tv0 =01 Tv0 =03 Tv0 =05

02 04 06 080uq0

2

16

12

08

04

0

zh




b=c=1d-1 Tv0 =1

0 4 8 12 16 20

02

03

04

05

06


up 0

Tv =1

Tv =08

Tv =05

Tv =13

Tv =15


0 01 02 03 04 05 06 072

16

12

08

04

0

up0

zh


b=c=1d-1 Tv =01

b=c=1d-1 Tv =03

b=c=1d-1 Tv =09

Tv0 =1




0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =01Tv =03

Tv =05Tv =09

b=3d-1c=2d-1

T0=1






0 01 02 03 04 05 06 072

16

12

08

04

0



Tv0 =1

up0

zh


0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =1Tv =13

Tv =15Tv =2

b=1d-1c=3d-1

T0=1




0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =09Tv =09Tv =09

Tv =12Tv =12Tv =12

C=1m2d 5m2

d 5m2d10m2

d 10m2d1m2

d

T0

=1b=c=1d-1




0 02 04 06 08 12

16

12

08

04

0

up0

zh b=c=1d-1

Tv0 =05Tv=05Tv0 =1Tv=05Tv0 =15Tv=05

Tv0 =05Tv=15Tv0 =1Tv=15Tv0 =15Tv=15





119880119875 = int2ℎ0

(119901 (119905) minus 119906)d1199112ℎ1199010 (29)


119880119875 = int2ℎ0

(119901 (119905) minus 119906) d1199112ℎ1199010 = 119901 (119905)


119906d1199112ℎ1199010



001 01 1 101

08

06

04

02

0

Up

Tv


Tv0 =01 b=c=1d-1

Tv0 =01 b=c=10d-1b=1d-1

c=10d-1T0=01




+ infinsum119899=1


+ infinsum119899=1




21199050 minus 119905119890minus11988811990521199050

(30)


119880119901 = int2ℎ0


0119906d119911


119899=1






2 minus 119890minus1198881199052

(31)





6 Conclusions


Up

Tv

001 01 1 101

08

06

04

02

0



Tv0 =01 b=c=1d-1

Tv0 =1 b=c=1d-1

Tv0 =10 b=c=1d-1



Data Availability




Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018









b=c=1d-1 Tv0 =1

0 4 8 12 16 20

02

03

04

05

06


up 0

Tv =1

Tv =08

Tv =05

Tv =13

Tv =15


0 01 02 03 04 05 06 072

16

12

08

04

0

up0

zh


b=c=1d-1 Tv =01

b=c=1d-1 Tv =03

b=c=1d-1 Tv =09

Tv0 =1




0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =01Tv =03

Tv =05Tv =09

b=3d-1c=2d-1

T0=1






0 01 02 03 04 05 06 072

16

12

08

04

0



Tv0 =1

up0

zh


0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =1Tv =13

Tv =15Tv =2

b=1d-1c=3d-1

T0=1




0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =09Tv =09Tv =09

Tv =12Tv =12Tv =12

C=1m2d 5m2

d 5m2d10m2

d 10m2d1m2

d

T0

=1b=c=1d-1




0 02 04 06 08 12

16

12

08

04

0

up0

zh b=c=1d-1

Tv0 =05Tv=05Tv0 =1Tv=05Tv0 =15Tv=05

Tv0 =05Tv=15Tv0 =1Tv=15Tv0 =15Tv=15





119880119875 = int2ℎ0

(119901 (119905) minus 119906)d1199112ℎ1199010 (29)


119880119875 = int2ℎ0

(119901 (119905) minus 119906) d1199112ℎ1199010 = 119901 (119905)


119906d1199112ℎ1199010



001 01 1 101

08

06

04

02

0

Up

Tv


Tv0 =01 b=c=1d-1

Tv0 =01 b=c=10d-1b=1d-1

c=10d-1T0=01




+ infinsum119899=1


+ infinsum119899=1




21199050 minus 119905119890minus11988811990521199050

(30)


119880119901 = int2ℎ0


0119906d119911


119899=1






2 minus 119890minus1198881199052

(31)





6 Conclusions


Up

Tv

001 01 1 101

08

06

04

02

0



Tv0 =01 b=c=1d-1

Tv0 =1 b=c=1d-1

Tv0 =10 b=c=1d-1



Data Availability




Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018









0 01 02 03 04 05 06 072

16

12

08

04

0



Tv0 =1

up0

zh


0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =1Tv =13

Tv =15Tv =2

b=1d-1c=3d-1

T0=1




0 01 02 03 04 05 062

16

12

08

04

0

up0

zh

Tv =09Tv =09Tv =09

Tv =12Tv =12Tv =12

C=1m2d 5m2

d 5m2d10m2

d 10m2d1m2

d

T0

=1b=c=1d-1




0 02 04 06 08 12

16

12

08

04

0

up0

zh b=c=1d-1

Tv0 =05Tv=05Tv0 =1Tv=05Tv0 =15Tv=05

Tv0 =05Tv=15Tv0 =1Tv=15Tv0 =15Tv=15





119880119875 = int2ℎ0

(119901 (119905) minus 119906)d1199112ℎ1199010 (29)


119880119875 = int2ℎ0

(119901 (119905) minus 119906) d1199112ℎ1199010 = 119901 (119905)


119906d1199112ℎ1199010



001 01 1 101

08

06

04

02

0

Up

Tv


Tv0 =01 b=c=1d-1

Tv0 =01 b=c=10d-1b=1d-1

c=10d-1T0=01




+ infinsum119899=1


+ infinsum119899=1




21199050 minus 119905119890minus11988811990521199050

(30)


119880119901 = int2ℎ0


0119906d119911


119899=1






2 minus 119890minus1198881199052

(31)





6 Conclusions


Up

Tv

001 01 1 101

08

06

04

02

0



Tv0 =01 b=c=1d-1

Tv0 =1 b=c=1d-1

Tv0 =10 b=c=1d-1



Data Availability




Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018









0 02 04 06 08 12

16

12

08

04

0

up0

zh b=c=1d-1

Tv0 =05Tv=05Tv0 =1Tv=05Tv0 =15Tv=05

Tv0 =05Tv=15Tv0 =1Tv=15Tv0 =15Tv=15





119880119875 = int2ℎ0

(119901 (119905) minus 119906)d1199112ℎ1199010 (29)


119880119875 = int2ℎ0

(119901 (119905) minus 119906) d1199112ℎ1199010 = 119901 (119905)


119906d1199112ℎ1199010



001 01 1 101

08

06

04

02

0

Up

Tv


Tv0 =01 b=c=1d-1

Tv0 =01 b=c=10d-1b=1d-1

c=10d-1T0=01




+ infinsum119899=1


+ infinsum119899=1




21199050 minus 119905119890minus11988811990521199050

(30)


119880119901 = int2ℎ0


0119906d119911


119899=1






2 minus 119890minus1198881199052

(31)





6 Conclusions


Up

Tv

001 01 1 101

08

06

04

02

0



Tv0 =01 b=c=1d-1

Tv0 =1 b=c=1d-1

Tv0 =10 b=c=1d-1



Data Availability




Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018











6 Conclusions


Up

Tv

001 01 1 101

08

06

04

02

0



Tv0 =01 b=c=1d-1

Tv0 =1 b=c=1d-1

Tv0 =10 b=c=1d-1



Data Availability




Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018










References

































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








analytical solution for one-dimensional consolidation of...

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