Pyol Kim ,Hyong-Sik Kim,Chol-U Pak,Chung-Hyok Paek,Gun-Hyang Ri,Hak-Bom Myong

Faculty of Geology,Kim Il Sung University,Pyongyang 999093,Democratic People’s Republic of Korea

Abstract This paper presents analytical solutions for one-dimensional nonlinear consolidation of saturated multi-layered soil under time-dependent loading.Analytical solutions are derived for one-dimensional nonlinear consolidation of multi-layered soil subjected to constant loading and ramp loading.The proposed solutions are verified through the comparison with the existing solutions for double-layered soil and singlelayered soil,which shows the proposed solutions are more general ones for one-dimensional nonlinear consolidation of saturated soils subjected to time-dependent loading.Based on the proposed solutions,nonlinear consolidation behavior of saturated multi-layered soil under time-dependent loading is investigated.

Keywords:Saturated soil;Multi-layered;Nonlinear consolidation;Analytical solution.

1.Introduction

In geotechnical and ocean engineering,a soil is often layered and has nonlinear characteristics during consolidation.Therefore,it is very significant to analyze the one-dimensional consolidation of saturated soil subjected to time-dependent loading by taking both the nonlinearity and the layered characteristics of the soil into account,which has drawn considerable attention and has become the focus of academic research.

Since Davis and Raymond [1]first proposed the nonlinear consolidation theory based on the assumptions that the decrease in permeability is proportional to the decrease in compressibility during the consolidation process and the initial effective stress is constant with depth,many attempts have been made to develop different one-dimensional consolidation models considering the nonlinear variations in permeability and compressibility [ 2,3 ].Poskitt [4],Mesri and Rokhasar[5],Lekha et al.[6],Zhuang et al.[7],Abbasi et al.[8]and Zheng et al.[9]investigated one-dimensional nonlinear consolidation on the basis of the relationship between void ratio e-log effective stress and void ratio e-log permeability.Based on Davis and Raymond’s theory,Xie et al.[10]and Conte and Troncone [11]derived analytical solutions to one-dimensional nonlinear consolidation of soil subjected to time-dependent loading.More recently,Li et al.[12]and Kim et al.[13]proposed analytical solutions for one-dimensional nonlinear consolidation of a saturated clay layer under ramp loading and cyclic loadings with the consideration of variable consolidation coefficient.All solutions mentioned above are for onedimensional nonlinear consolidation of single-layered soil.

In the past decades,numerous studies have been conducted on one-dimensional consolidation of multi-layered soil.Based on Terzaghi’s linear consolidation theory,Gray [14],Schiffman and Stein [15],Lee et al.[16],Xie and Pan [17],Huang and Griffiths [18]investigated one-dimensional consolidation of layered soil.Xie et al.[19]extended Davis and Raymond’s nonlinear consolidation theory to double-layered soil and time-dependent loading conditions.Chen et al.[20]and Hu et al.[21]studied one-dimensional nonlinear consolidation of multi-layered soil under time-dependent loading by using numerical method with the same assumptions as proposed by Davis and Raymond [1].Besides,many researchers have investigated multi-layered theories of structures by using numerical methods such as finite element solutions through Carrea Unified Formulation [22].

Fig.1.Schematic model of multi-layered soil subjected to time-dependent loading.

Compared with numerical solutions,analytical solutions can be more accurate and simple.In addition,analytical solutions can be used for verification of numerical methods.However,due to the complexity of the nonlinear consolidation equations for multi-layered soil,none of researchers have ever derived any analytical solutions for one-dimensional nonlinear consolidation of multi-layered soil under time-dependent loading.So,the purpose of this study is to derive analytical solutions for one-dimensional nonlinear consolidation of saturated multi-layered soil under time-dependent loading based on the assumptions proposed by Davis and Raymond [1].The proposed solutions are verified through the comparison with the existing solutions in literature.Based on the proposed solution,nonlinear consolidation behavior of multi-layered soil under time-dependent loading is investigated.

2.Governing equations

Fig.1 shows the schematic model for one-dimensional nonlinear consolidation of multi-layered soil subjected to time-dependent loading.In this model,hi,kvi0,mvi0and.civ.are the thickness,the initial coefficient of permeability,the initial coefficient of compressibility and the consolidation coefficient of theith soil layer (i=1,2,···,n),respectively.His the total thickness of soil which satisfiesis the uniformly distributed time-dependent loading applied on the top surface of the soil.

Based on the nonlinear consolidation theory proposed by Davis and Raymond [1],the governing equation for onedimensional nonlinear consolidation of multi-layered soil subjected to time-dependent loading can be written as follows:

whereui(z,t) andσ＇i(z,t) are the excess pore water pressure and the effective vertical stress in theith soil layer,respectively;zandtare the variables for space and time respectively.

According to the assumption that the decrease in permeability is proportional to the decrease in compressibility during the consolidation process of soil,the consolidation coefficientcviis given byin which,γwis the unit weight of water;the compression index and the initial void ratio in theith layer corresponding to the initial effective vertical stressr espectively.By the assumption that the distribution of initial effective stress is constant with depth,

Using Terzaghi’s principle of effective stress,σ＇i(z,t) can be written as:

By defining a new variable,

Eq.(1) can be simplified in terms ofωi(z,t) as follows:

where

The initial and boundary conditions for Eq.(4) in terms ofui(z,t) andωi(z,t) can be given by

3.Derivation of solutions

3.1. Excess pore water pressure

According to Eqs.(2) and (3),the excess pore water pressure can be expressed as follows:

In order to obtain analytical solution for excess pore water pressure,nondimensional parameters are defined as:

The general solutions of Eq.(4) satisfying relevant conditions (i.e.Eqs.(6) - (11)) can be assumed as follows [ 14,15 ]:

where

The coefficientsAmiandBmican be determined by the following recursive formula [16]:

where

in which

λmis the positive root of the eigen-equation as follows:

where

CmandDmcan be calculated by following equations:

3.2. Average degree of consolidation

Average degree of consolidation can be defined in terms of either settlement or effective stress.The former represents the settlement rate,while the latter denotes the dissipation rate of excess pore water pressure.

The average degree of consolidation defined in terms of settlement for each layer can be expressed as:

where ?iis the vertical strain in theith layer defined byεi=is the final vertical strain in theith layer defined byandefiare the void ratio and the final void ratio corresponding to the effective stressσ＇iand the final effective stressσ＇ifin theith layer respectively,in which the final effective stress can be given by=σ＇f=σ＇0+qufrom Eq.(2).

According to Eqs.(2) and (26),Eq.(27) can be written as follows:

whereNσis the loading parameter reflecting the ratio of final effective stress and initial effective stress defined byNσ=

The average degree of consolidation defined in terms of effective stress for each layer can be given by

Fig.2.Constant loading (a) and ramp loading (b).

The average degree of consolidation in terms of settlement for entire soil system can be expressed as follows:

The average degree of consolidation in terms of effective stress for entire soil system can be written as follows:

4.Solutions for special cases

Based on the above solutions,analytical solutions for onedimensional nonlinear consolidation of multi-layered soil under constant loading and ramp loading can be easily obtained.

4.1. Solutions for constant loading

For constant loading shown in Fig.2 (a),q(t)=quandR(t)=0.Then,the solutions for the excess pore water pressure and the average degree of consolidation of multi-layered soil under constant loading can be expressed as:

Fig.3.Comparison for double-layered soil under constant loading.

where

The average degrees of consolidation for entire soil systemUsandUpcan be given by Eqs.(30) and (31) respectively.

Fig.4.Comparison for double-layered soil under ramp loading.

4.2. Solutions for ramp loading

Ramp loading shown in Fig.2 (b) can be expressed as follows:

wheretcis construction time.

The solutions for ramp loading can be written as follows:

Fig.5.Comparison for single-layered soil under constant loading.

where

The average degrees of consolidation for entire soil systemUsandUpcan be also given by Eqs.(30) and (31) respectively.

Fig.6.Comparison for single-layered soil under ramp loading.

4.3. Verification

For verification,the proposed solutions are compared with the existing solutions for double-layered soil and singlelayered soil under constant loading and ramp loading respectively.

4.3.1.Double-layeredsoil

Whenn=2l,Ki=bi=1,(i=1,2,···,l),Kj=bj=2,(j=l+ 1,l+ 2,···,n)andci=cj=1,multi-layered soil can be reduced to double-layered soil withandc=h2/h1=1.The proposed solutions are compared with the solutions for double-layered soil under constant loading and ramp loading presented by Xie et al.[19].

Figs.3 and 4 show the comparison for double-layered soil under constant loading and ramp loading,respectively.It can be seen that the proposed solutions are consistent with the existing solutions,indicating the correctness of the analytical solutions in this paper.

Fig.7.The excess pore pressure isochrones for four-layered soil with single drainage condition under constant loading.

4.3.2.Single-layeredsoil

WhenKi=bi=ci=1,(i=1,2,···,n),multi-layered soil can be reduced to single-layered soil and the proposed solutions are compared with the solutions suggested by Davis and Raymond [1]for constant loading and Xie et al.[10]for ramp loading.

As shown in Figs.5 and 6,it can be found that the identical results are obtained from different solutions for constant loading (Fig.5) and ramp loading (Fig.6).

Through the verification mentioned above,it can be seen that the solutions developed in this paper are more general ones for one-dimensional nonlinear consolidation of saturated soils under time-dependent loading.

Fig.8.The excess pore pressure isochrones for four-layered soil with double drainage condition under constant loading.

Table1 The properties of four-layered soil.

Fig.9.The variation of average degrees of consolidation Us and U p for constant loading with different loading parameter Nσ.(a) Single drainage;(b) double drainage.

5.Nonlinear consolidation behavior of multi-layered soil

In order to investigate one-dimensional nonlinear consolidation behavior of multi-layered soil under time-dependent loading,the properties of four-layered soil proposed by Schiffman and Stein [15]are adopted as shown in Table1.All data in Table1 are converted to SI units.

5.1. The effect of loading parameter N σ

Based on the solutions for constant loading obtained above,the effect of loading parameterNσon the nonlinear consolidation behavior of four-layered soil under constant loading is investigated with the results of linear consolidation presented by Lee et al.[16].

Figs.7 and 8 illustrate the excess pore water pressure isochrones of one-dimensional consolidation of four-layered soil with single drainage condition and double drainage condition,respectively.It is obvious that both results from different solutions are in good agreement whenNσapproaches to one,whereas,the difference increases with the increase of the value ofNσ.It implies that the effects ofNσon nonlinear consolidation can be considered for the proposed solution,but not for the solution of linear consolidation.

Fig.10.The variation of excess pore pressures at different depths for ramp loading with different construction time factor T vc.(a) Single drainage;(b)double drainage.

The effect ofNσon nonlinear consolidation can be obviously found in Fig.9,which shows the variation of average degrees of consolidationUsandUpwith time factorTvfor single drainage condition (Fig.9 (a)) and double drainage condition (Fig.9 (b)).The time factorTvis defined byTv=cv1t/H2.It can be seen that the average degree of consolidation in terms of effective stressUpdecreases with the increase ofNσ,butNσhas no effect on the average degree of consolidation in terms of settlementUs,indicating that the bigger the loading parameterNσis,the slower the dissipation rate of excess pore water pressure is.In addition,it can be found that whenNσapproaches to one,both nonlinear and linear consolidation results are consistent,which shows that the solution for linear consolidation is a special case for the proposed solution for nonlinear consolidation.

Fig.11.The variation of average degrees of consolidation U s and U p for ramp loading with different construction time factor T vc.(a) Single drainage;(b) double drainage.

5.2. The effect of construction time factor T vc

Fig.10 presents the variation of excess pore water pressures at different depths with time factorTvunder ramp loading with different construction time factorTvc.The construction time factorTvcis given byTvc=cv1tc/H2.For both single drainage condition (Fig.10 (a)) and double drainage condition(Fig.10 (b)),it can be seen that it takes longer time to reach maximum values of excess pore water pressures at different depths and to dissipate when the construction time factorTvcincreases.

The effect of construction time can be also found in Fig.11,which shows the variation of average degrees of consolidationUsandUpfor ramp loading with different construction time factorTvc.It can be observed that the shorter the construction time is,the faster the consolidation develops,especially,the consolidation rate is the highest for constant loading (i.e.,Tvc=0).

Conclusion

In this paper,analytical solutions were derived for one-dimensional nonlinear consolidation of saturated multilayered soil under time-dependent loading.The proposed solutions were verified through the comparison with the existing solutions for double-layered soil and single-layered soil,which shows the proposed solutions are reliable and more general ones for one-dimensional nonlinear consolidation of saturated soils subjected to time-dependent loading.Based on the proposed solutions,nonlinear consolidation behavior of multi-layered soil under time-dependent loading was investigated.It is concluded that the bigger the loading parameterNσis,the slower the dissipation rate of excess pore water pressure is.And the shorter the construction time is,the faster the consolidation develops.

It should be noted that the present solutions are based on the assumptions that the distribution of initial effective stress is constant with depth and the consolidation coefficient of each layer does not vary with time under time-dependent loading.However,the initial effective stress increases with depth and the consolidation coefficients are not constant with time in practice.Nevertheless,the proposed solutions are more general for one-dimensional nonlinear consolidation of saturated soils subjected to time-dependent loading.Consequently,the present model can be used as a reference to check the efficiency of approximate numerical methods.

Declaration of Competing Interest

The Authors have no interests to declare.

Acknowledgment

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions.