Ain Shams Engineering Journal (2014) 5, 1043–1057

Ain Shams University

Ain Shams Engineering Journal

www.elsevier.com/locate/asejwww.sciencedirect.com

CIVIL ENGINEERING

A layer equation method for 1-D consolidation

under time-dependent reloading

* Corresponding author. Address: P.O. Box 465, Port Said, Egypt.

Mobile: +20 1112584579.E-mail addresses: richard.herrmann@uni-siegen.de (R. Herrmann),

melgendy@elpla.com (M. El Gendy).

Peer review under responsibility of Ain Shams University.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.asej.2014.07.004

2090-4479 � 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University.

R. Herrmann a, M. El Gendy b,*

a Geotechnical Institute, Siegen University, Germanyb Faculty of Engineering, Port Said University, Egypt

Received 1 April 2013; revised 26 May 2014; accepted 10 July 2014Available online 12 August 2014

KEYWORDS

1-D consolidation settle-

ment;

Layered soil model;

Nonlinear consolidation

theory

Abstract Most of available meshless methods for time-dependent settlement problems depend on

deriving an algebraic equation for each layer, in which, the derived equation has an infinite number

of series functions with an infinite number of coefficients. In this paper, a Layer Equation Method

(LEM) is adapted for analyzing time-dependent settlement problems. The technique of LEM is a

semi-analytical solution that deals with a limited number of algebraic equations instead of solving

numerically the partial differential equations. Consequently, it leads to a significant reduction in

variables of the problem. This method can take into account any variable initial stress along the

clay layers. It is also applicable for stress coefficient technique, in which the vertical stress in any

node at coordinate (x, y, z) may be considered. An application for the method on reloading

time-dependent settlement of clay is presented in which a deep excavation is necessary for buildings

with basements. In this case, the soil stress reduces due to excavation, and the reloading of the soil

should be taken into account.� 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University.

1. Introduction

Most of the available meshless methods for time-dependentsettlement problems depend on deriving an algebraic equationfor each layer; some of these methods were introduced by Lee

et al. [1], Xie et al. [2–4], Zhuang et al. [5] and Morris [6]. In thesolution, the derived equation has an infinite number of series

functions with an infinite number of coefficients. Also the ori-ginal solution of the 1-D consolidation problem presented byTerzaghi [7] was a formula with infinite series. Infinite series

may be lead to oscillation. Furthermore, the methods assumeda uniformly initial applied stress on the clay layers or regularshapes of stress such as a triangular shape [8]. In a better case,

the stress was assumed as a continuous function in depth [1].The reason is that it is difficult to generate infinite coefficientsto represent the applied variable stress on the clay.

In this study, a Layer Equation Method (LEM) is proposedfor analyzing time-dependent settlement problems. LEMdepends on selecting a number of nodes in the clay layers alongthe z-axis. Consequently, a better representation for applied

Notation

Aj area around node j (m2)

Aij coefficient of basis functions for layer iBij coefficient of basis functions for layer iCj constants of basis functionsCC compression index (–)

Cci compressibility index of layer i (–)Cri recompression index of layer i (–)Cai coefficient of secondary consolidation for

layer i (–)Cari reloading coefficient of secondary consolidation

for layer i (–)

cvi coefficient of consolidation of layer i (year/m2)ei void ratio at time t of layer i (–)eoi initial void ratio of layer i (–)epi void ratio for layer i at the end of primary consol-

idation (–)fl,j stress coefficient of node l due to contact force at

node j on the surface (1/m2)

H total thickness of clay layers (m)hi thickness of layer i (m)hpi thickness of the compressible soil layer i at time tp

(m)kvi coefficient of permeability of layer i (m/year)kvoi initial coefficients of permeability in a layer

i (m/year)M number of steps-load increment (–)mvi coefficient of volume change of layer i (m2/kN)mvoi initial coefficients of volume change in a layer i

(m2/kN)m number of grid nodes (–)mi number of sub-layers in a layer i (–)

mvi coefficients of volume change in a layer i (m2/kN)mri coefficients of volume change for reloading in a

layer i (m2/kN)

N number of function terms (number of studiednodes)

Qj contact force at node j (kN)qc contact pressure at the surface due to construction

load (kN/m2)qj contact pressure at node j (kN/m2)r total number of studied nodes of clay layers (–)

ssi secondary consolidation settlement of a layer i (m)spi primary consolidation settlement of a layer i (m)Skt sum of primary consolidation settlements in all

layers at the required time t (m)Sku sum of finial consolidation settlements in all lay-

ers, when Dui = 0 (m)

t time for which excess pore water pressure is com-

puted (year)tc construction time (year)tp time at the end of primary consolidation (year)Tvi time factor for layer i, Tvi = cvit/H

2 (–)

u(z, t) excess pore water pressure at any vertical depth zand time t (kN/m2)

Up degree of consolidation at the required time t in

terms of stressUs degree of consolidation at the required time t in

terms of settlement

zi vertical coordinate of layer i (m)cw unit weight of the water (kN/m3)ni local depth ratio of layer i, ni = zi/hi (–)kj differential equation operator

li parameter of the coefficient of consolidation andthickness of layer i

rl initial vertical stress in a node depth l (kN/m2)

r0i effective stresses at time t of layer i (kN/m2)r0oi initial effective stresses at the middle of layer i

(kN/m2)

r0ci pre-consolidation pressure of a layer i (kN/m2)uj(z) a set of basis functions in the variable z onlywj(t) coefficients of basis functions in the variable t only

xj index of the exponential functions in matrix (Ev)Dqi load increment at variable interval of times

(kN/m2)Dti interval of times, Dt= tc/(M � 1) (year)

Dui average excess power water pressure at time t inlayer i (kN/m2)

Duoi initial average stress in a layer i (kN/m2)

Dzi depth increment in sub-layer i (m)Dr0i increment of vertical stress at time t in a layer

i (kN/m2)

Dr0ri reloading increment of vertical stress in a layeri (kN/m2)

Dr0ei loading increment of vertical stress in a layeri (kN/m2)

{C} vector of constants Cj, j = 1 to N[D] diagonal square matrix due to variable loading[Ev] diagonal square matrix of exponential functions

{R}n vector o obtained from boundary conditions{u} vector of the excess pore water pressure

uj, j= 1 to N

{u}o vector of initial excess pore water pressure{Dq}k vector of applied load at interval k[U] matrix of basis functions.

1044 R. Herrmann, M. El Gendy

stress on soil layers can be represented. The method is alsoideal for using a stress coefficient technique, which may be

extended to study the interaction of irregular loaded areason the surface or contact pressure due to foundation rigidity.LEM requires fewer equation terms, in which fewer termsare sufficient to give excellent results compared to the available

closed-form solution of time-dependent settlement problems.However, algebraic equations of clay layers are developedfrom an initial stress applied to a specified number of grid

nodes, which can represent the excess pore water pressure atany node on the layers.

LEM is used to investigate the behavior of the excess porewater pressure when the clay changes from an over-consoli-dated state to a normally-consolidated state during the consol-idation process because the stress applied to the clay layers

varies with time. These states were studied for a single claylayer by Xie et al. [9]. They had determined the moving depthof the interface between over-and normally-consolidated zones

cv1

kv1

mv1

Layer n

Layer i

Layer 1

zi

r

Pervious

H

h1

hi

hn

Pervious or impervious

z1

cvi

kvi

mvi

cvn

kvn

mvnr-1

12

43

Initial stress

Ground surface

Contact forces

k

l

Figure 2 Multi-layered system.

A layer equation method for 1-D consolidation under time-dependent reloading 1045

in a layer. The layer is considered to have an impervious base.The initial load applied to the layer in each interval incrementof time was uniform. Also, the initial vertical effective stress

due to weight of the entire layer itself was uniform. Besides,only two coefficients of consolidation were considered; onefor the normally-consolidated zone; and the other for the

over-consolidated zone. They had considered this case as adouble-layered soil. In fact, the initial applied stress generatedon the soil from the surface loading is not uniform throughout

the clay depth. It is greater near the surface than at the base. Inaddition, maybe the clay has a pervious top and base. It is alsoknown that the initial vertical effective stress increases withdepth. This means that at any sub-layer within the clay, the

state may change from over- to normally-consolidated, espe-cially for a thick clay layer. The analysis in this study takes intoaccount the nonlinear response of the excess pore water pres-

sure due to the change of compressibility and permeability ofthe soil during the consolidation process.

2. Mathematical modeling

2.1. Formulation of excess pore water pressure

2.1.1. Defining basic functions

To formulate the analysis, the loaded area on the surface isdivided into triangular elements as shown in Fig. 1. Then thecontact pressure is represented by a series of contact forcesQj on the element nodes. The soil under the loaded area may

consist of multi-layered system of clay with different soilparameters and is divided into n sub-layers with r nodes asshown in Fig. 2. Stress coefficients for the nodes under the

loaded area due to contact forces can be determined asdescribed by El Gendy [10].

According to El Gendy [10], for a set of grid nodes of m

contact forces Qj at the surface, the vertical stress rl in a nodedepth l under the surface node k is attributed to stresses causedby all of the contact forces on that node:

rl ¼Xmj¼1

fl;jQj ð1Þ

where fl, j is the stress coefficient of node l due to the contact

force at node j on the surface (1/m2). It depends only on thegeometry of the loaded area and the soil layer.

Each layer in Fig. 2 has different soil parameters and geom-

etries. kvi (m/year), cvi (year/m2), mvi (m

2/kN), zi (m) and hi (m)are the coefficient of permeability, the coefficient of consolida-tion, the coefficient of volume change, the depth and the thick-

ks

j

At node j is:qj = Contact pressure at node jQj = Contact force at node jAj = Area around node jQj = qj . Aj

Figure 1 Loaded area with mesh of elements on the surface.

ness of the ith soil layer, respectively. H (m) is the totalthickness of the clay layers.

At a time t= 0, Eq. (1), for the entire clay layers at a sec-

tion in the z-axis passing through point k, in matrix formbecomes:

frgo ¼ ½f�fQgo ð2Þ

where {r}o is the initial vertical stress vector at time t= 0; [f] isthe stress coefficient matrix; and {Q}o is the initial contactforce vector at time t= 0.

The solution depends on choosing a formula that representsthe excess pore water pressure along the z-axis and satisfies theboundary conditions. A partial differential equation such as

the consolidation equation can be solved and expressed in aseries of N terms as:

uðz; tÞ ¼XNj¼1

CjujðzÞwjðtÞ ð3Þ

where u(z, t) is the excess pore water pressure at any vertical

depth z and time t (kN/m2); uj(z) is a set of basic functionsin the variable z only; wj(t) are coefficients of basic functionsin the variable t only; Cj are constants of basic functions;

and N is the number of function terms (number of studiednodes); z is the vertical coordinate (m); and t is the time forwhich excess pore water pressure is computed (year).

Coefficients and constants of basic functions can be obtainedby selecting a set of N arbitrary nodes and their function valuesu(z, t). The basic functions are chosen to satisfy the boundary

condition. The boundary conditions for double drainage areu(0, t) = 0 and u(Ho, t) = 0, while those for single drainageare u(0, t) = 0 and ou(Ho, t)/oz= 0.To selectN arbitrary nodes,each layer is divided into mi sub-layers with depth increment

Dzi = hi/mi, which gives a total r nodes. For a pervious bottom

1046 R. Herrmann, M. El Gendy

boundary the excess pore water pressure at the bottom bound-ary is known and equal to zero. Therefore, studied nodes in thiscase are less than those of an impervious bottom boundary by a

node. The number of studied nodes will beN = r � 1 for a per-vious bottom boundary, while that for an impervious bottomboundary will beN = r. Suitable basic functions for excess pore

water pressure problems are as follows:

ujðzÞ ¼ Aij sinðlikjniÞ þ Bij cosðlikjniÞ ð4Þ

and corresponding coefficients are:

wjðtÞ ¼ expð�l2i k

2j TviÞ ð5Þ

where for a layer i, ni is the local depth ratio, ni = zi/hi; Aij and

Bij are coefficients of basic functions; kj is the differential equa-tion operator; and li is the parameter of the coefficient of con-solidation and thickness, li ¼ ðhi=h1Þ

ffiffiffiffiffiffiffiffiffiffiffiffifficv1=cvi

p. Tvi is the time

factor, Tvi ¼ cvit=h2i .

Now, the equation for excess pore water pressure ui(z, t) forlayer i in a multi-layered system may be expressed as follows:

uiðz; tÞ ¼XNj¼1

Cj Aij sinðlikjniÞ þ Bij cosðlikjniÞ� �

exp �l2i k

2j Tvi

� �

ð6Þ

To satisfy the condition of the governing differential equa-

tion for 1-D consolidation (Terzaghi’s equation) cvi@2ui@z2¼ @ui

@t,

the following equations should be satisfied:

l21k

2j Tv1 ¼ l2

2k2j Tv2 ¼ � � � ¼ l2

nk2j Tvn ¼ xjt ð7Þ

where xj ¼ cv1h21

k2j .

Eq. (6) may be written for N studied nodes in a matrix formas:

fug ¼ ½U�½Ev�tfCg ð8Þ

where {u} is the vector of the excess pore water pressure uj,j= 1 to N; {C} is the vector of constants Cj, j= 1 to N; [U]is the matrix of basic functions; and [Ev] is the diagonal square

matrix of the exponential functions.Details of the matrix [U] is listed in Appendix A, while the

diagonal square matrix [Ev] is given by

½Ev� ¼ diag expð�x1Þ expð�x2Þ expð�x3Þ � � � expð�xNÞ½ � ð9Þ

2.1.2. Determining coefficients Aij and Bij

Relations among coefficients Aij and Bij can be obtained usinginterface and boundary conditions. Equating the excess porewater pressures ui(zi, t) = ui+1(zi, t) at layer interfaces, leads to:

Aij sinðlikjniÞ þ Bij cosðlikjniÞ¼ Aðiþ1Þj sinðliþ1kjniþ1Þ þ Bðiþ1Þj cosðliþ1kjniþ1Þ ð10Þ

while equating the vertical velocity of flow

kvi@u@z

� �i¼ kvðiþ1Þ

@u@z

� �iþ1 at layer interfaces, leads to:

likvihiþ1lðiþ1Þkvðiþ1Þhi

Aij cosðlikjniÞ � Bij sinðlikjniÞ� �

¼ Aðiþ1Þj cosðliþ1kjniþ1Þ � Bðiþ1Þj sinðliþ1kjniþ1Þ ð11Þ

At the interface ni = 1 and ni+1 = 0, Eqs. (10) and (11) thenbecome:

Aij sinðlikjÞ þ Bij cosðlikjÞ ¼ Bðiþ1Þj ð12Þlikvihiþ1

lðiþ1Þkvðiþ1Þhi½Aij cosðlikjÞ � Bij sinðlikjÞ� ¼ Aðiþ1Þj ð13Þ

Satisfying free drainage at the top u1(0, t) = 0, requiresthat:

A1j ¼ 1 and B1j ¼ 0 ð14Þ

From Eqs. (12) and (13), coefficients Aij and Bij can beexpressed as:

fRgiþ1 ¼ ½h�ifRgi ð15Þ

where

fRgi ¼ Aij Bijf gT; i ¼ 1; 2; . . . ; n

½h�i ¼gi cosðlikjÞ �gi sinðlikjÞsinðlikjÞ cosðlikjÞ

ð16Þ

and:

gi ¼likvihiþ1

lðiþ1Þkvðiþ1Þhi¼ mvi

mvðiþ1Þ

ffiffiffiffiffiffiffiffiffiffifficvi

cvðiþ1Þ

rð17Þ

The vector {R}n is obtained from boundary conditions of

the two cases of single and double drainages. Applying bound-ary conditions at the base where u = 0 for double drainageand ou/oz = 0 for single drainage, thus gives:

½Sd�fRgn ¼ 0 ð18Þ

where the matrix [Sd] is given by:

½Sd� ¼ sinðlnkjÞ cosðlnkjÞ½ � for double drainage

½Sd� ¼ cosðlnkjÞ � sinðlnkjÞ½ � for single drainageð19Þ

From Eqs. (16) and (19), the following characteristic equationin the unknown Eigen values kj (differential equation opera-tors) can be obtained:

½Sd�½h�n�1fRgn�1 ¼ 0 ð20Þ

The operator kj is the positive root of the above character-

istic equation. Substituting the value of kj obtained from Eq.(20) into Eq. (15), gives coefficients Aij and Bij.

2.1.3. Determining constants Cj

Constants Cj can be found using the initial conditionuj(z,0) = uo(z). Consider a system of linear equations at a set

of N grid nodes at time t= 0 as follows:

fugo ¼ ½U�fCg ð21Þ

where {u}o is the vector of initial excess pore water pressureSubstituting Eq. (21) into Eq. (8), gives the following

matrix equation for excess pore water pressure:

fug ¼ ½U�½Ev�t½U��1fugo ð22Þ

The advantage of Eq. (22) is that raising the diagonal matrix[Ev] to any power of time t is carried out by raising its diagonalelements Evj to that power.

A layer equation method for 1-D consolidation under time-dependent reloading 1047

2.1.4. Degree of consolidation

Integrating Eq. (6) over the entire layer i, gives the average

excess pore water pressure Dui at any time factor in that layeras follows:

Dui ¼1

hi

Z hi

0

uiðz; tÞdz

¼ 1

li

XNj¼1

Cj

kj

Aij½1� cos likj

� �� þ Bij sinðlikjÞ

� �expð�xjtÞ

ð23Þ

The initial average stress Droi in a layer i is given by:

Droi ¼1

hi

Z hi

0

roiðzÞdz ð24Þ

where roi(z) is the initial stress in a layer i due to a foundationload (kN/m2).

The degree of consolidation Up and Us at the required timet can be obtained from either the stress:

Up ¼ 1�Pn

i¼1DuihiPni¼1Droihi

ð25Þ

or the settlement:

Us ¼ 1�Pn

i¼1mviDuihiPni¼1mviDroihi

ð26Þ

2.1.5. Determining excess pore water pressure due to variableloading on the surface

In practice, the total load on clay under a structure is appliedover a period of time. In this case, the total load of construc-

tion on the surface qc can be applied gradually over a time tc asshown in Fig. 3. The governing equation for 1-D consolida-tion, taking into account the variable loading with construc-tion time as indicated by Lee et al. [1], can be expressed as:

cvi@2ui@z2¼ @ui@t� dri

dtð27Þ

An analytical solution for Eq. (27) is difficult. To determinethe excess pore water pressure, the integral can be evaluated bya series of M steps-load increment at the surface Dq at interval

of times Dt (Fig. 3). The load increment at the surface Dq willlead to an increment of vertical stress dri at node i.

Loa

d q

[kN

/m2 ]

Time t [Year]

ΔqΔt

qc

tc t

qc= MΔqtc= (M-1)Δt

Figure 3 Applied load in a series of steps.

For determining the excess pore water pressure due to stepload increments, the excess pore water pressure induced by theprevious load is obtained, and at the same time the excess pore

water pressure induced by the additional load is determined.The results of this process may be expressed explicitly as:

fug1 ¼ ½U�½Ev�Dt½U��1fdugofug2 ¼ ½U�½Ev�Dt½U��1 fug1 þ fdugo

� �fug3 ¼ ½U�½Ev�Dt½U��1 fug2 þ fdugo

� �� � �fugtc ¼ ½U�½Ev�Dt½U��1 fugðM�2Þ þ fdugo

n o

fugt ¼ ½U�½Ev�tr ½U��1 fugtc þ fdugo� �

9>>>>>>>>>>=>>>>>>>>>>;

ð28Þ

In Eq. (28), the total load is applied by M steps of equal loadincrement. Therefore, the additional initial pore water pressurevectors in all steps are the same and equal to fdugo ¼ 1

Mfugo.

Now the vector of pore water pressure at time t may bewritten as:

fugt ¼ ½U�½Ev�trþðM�1ÞDt½U��1fdugo þ ½U�½Ev�trþðM�2ÞDt½U��1fdugoþ � � � þ ½U�½Ev�trþ2Dt½U��1fdugo þ ½U�½Ev�trþDt½U��1fdugoþ ½U�½Ev�tr ½U��1fdugo ð29Þ

where Dt is the time interval (year), Dt= tc/(M � 1); tc is the

construction time (year); tr = t � tc (year); {du}o is the vectorof additional initial pore water pressure; {u}tc is the vector ofpore water pressure at time tc, and {u}k is the vector of porewater pressure at interval k.

Replacing tr by t � tc in the above equation gives:

fugt ¼ ½U�½Ev�t½U��1fdugo þ ½U�½Ev�t�Dt½U��1fdugoþ ½U�½Ev�t�2Dt½U��1fdugo þ � � �

þ ½U�½Ev�t�ðM�1ÞDt½U��1fdugo ð30Þ

Replacing {du}o by 1Mfugo in the above equation and rewriting

the equation gives:

fugt ¼1

M½U�½Ev�t ½I� þ ½Ev��Dt þ ½Ev��2Dt þ ½Ev��3Dt þ � � �

h

þ½Ev��ðM�1ÞDti½U��1fugo ð31Þ

Eq. (31) is rewritten in matrix form as:

fugt ¼ ½U�½Ev�t½D�½U��1fugo ð32Þ

where [D] is a diagonal square matrix. The diagonal elementsof the matrix [D] are defined by:

Dj ¼1

M½1þ exp xjDt

� �þ exp 2xjDt

� �þ � � �

þ exp ðM� 1ÞxjDt� �

� ð33Þ

The summation of the above series when M =1 is givenby:

Dj ¼expðxjtcÞ � 1

xjtcð34Þ

In this case, the equation for excess pore water pressure ui(z, t)for layer i in a multi-layered system becomes:

uiðz;tÞ¼XNj¼1

DjCj½Aij sinðlikjniÞþBij cosðlikjniÞ�expðxjtÞ ð35Þ

Virgin compressionslop Cci

Recompression slope Cri

[-]

1048 R. Herrmann, M. El Gendy

and the average excess power water pressure Dui becomes:

Dui¼1

li

XNj¼1

DjCj

kj

Aij½1� cosðlikjÞ�þBij sinðlikjÞ� �

expð�xjtÞ ð36Þ

Voi

d ra

tio e

ei

eoi

Δσi-Δui

σ'iσ'ci σ'oi

Δe

Stress σ' [kN/m2] (Log scale)

Figure 4 Relationship between void ratio and stress (loading

case), r0oi P r0ci.

Voi

d ra

tio e

[-] Virgin compression

slop C ci

Recompression slope Cri

ei

eoi

Δσi-Δui

σ'iσ'oi σ 'ci

Δe

Stress σ' [kN/m2] (Log scale)

Figure 5 Relationship between void ratio and stress (reloading

case), r0i < r0ci.

2.2. Formulation of consolidation settlement

2.2.1. Nonlinear analysis

Many researchers, Zhung et al. [5], Zhuang and Xie [11], Conteand Troncone [12], Lekha et al. [13], Xie et al. [2,4,14], Chenet al. [15] and Abbasi et al. [16], had used the assumption of

nonlinear analysis proposed by Davis and Raymond [17] tointroduce a nonlinear analysis of 1-D consolidation with vari-able compressibility and permeability. They assumed that the

decrease in permeability is proportional to the decrease incompressibility. Therefore, if the coefficient of consolidationis considered as constant during the consolidation process(Eq. (37)), the only soil variable during the consolidation pro-

cess required for the nonlinear analysis will be the coefficient ofvolume change.

cvi ¼kvi

cwmvi

¼ kvoicwmvoi

ð37Þ

where cw is the unit weight of the water (kN/m3); mvoi is the ini-

tial coefficients of volume change in a layer i (m2/kN); and kvoiis the initial coefficients of permeability in a layer i (m/year).

During the consolidation process, void ratio-effective stressresponse is given by:

ei ¼ eoi � Cci logr0ir0oi

�ð38Þ

where ei (–) and eoi (–) are the void ratios at time t and the ini-tial void ratio of layer i corresponding to effective stresses r0i(kN/m2) and r0oi (kN/m2), respectively, and Cci (–) is the com-pressibility index of layer i.

From Eq. (38), the initial coefficient of volume change mvoi

in a layer i can be estimated as follows:

mvoi ¼�1

1þ eoi

@e

@r0

����r0i¼r0

oi

¼ Cci

ð1þ eoiÞr0oi lnð10Þð39Þ

As the analysis in this paper deals with either normally or

pre-consolidated clay, the compressibility index Cci (–) isreplaced by the recompression index Cri(–) in the case of pre-consolidated clay. Coefficient of volume change at any inter-mediate interval time during the consolidation process may

be determined from the previous time step as described inthe next sections.

2.2.2. Determining stresses in a soil layer

According to Terzaghi’s principle of effective stress, the effec-tive stress r0i in a layer i during the consolidation process canbe given by:

r0i ¼ r0oi þ Dri � Dui ð40Þ

where r0oi is the initial vertical effective stress caused by the

weight of the soil itself at the middle of layer i (kN/m2); Driis the increment of vertical stress at time t in a layer i due tothe applied load on the surface (kN/m2); and Dui is the averageexcess pore water pressure at time t in a layer i (kN/m2).

At the end of consolidation, Dui = 0 and the effective stressr0i in a layer i reaches its final value:

r0i ¼ r0oi þ Dri ð41Þ

As the load on the surface is applied gradually during theconstruction time, tc and the excess pore water pressure varies

during the consolidation process; the value of Dri � Dui defineswhen the clay layer is normally consolidated or pre-consolidated.

2.2.3. Normally consolidated clay (loading case)

According to Fig. 4, a layer i is considered as normally consol-idated clay during the consolidation process when the initial

vertical effective stress r0oi is greater than or equal to thepre-consolidation pressure r0ci.

For normally consolidated clay, the consolidation settle-ment spi of a layer i of thickness hi is given by:

spi ¼Ccihi1þ eoi

logr0oi þ Dri � Dui

r0oi

�ð42Þ

As the layer thickness is assumed to be small, the consolida-

tion settlement spi as a function in the coefficient of volumechange mvi may be also given by:

spi ¼ mviðDri � DuiÞhi ð43Þ

A layer equation method for 1-D consolidation under time-dependent reloading 1049

The coefficient of volume change at any time t during theconsolidation process can be obtained from Eqs. (42) and(43) as follows:

mvi ¼Cci

ð1þ eoiÞðDri � DuiÞlog

r0oi þ Dri � Duir0oi

�ð44Þ

2.2.4. Pre-consolidated clay (reloading case)

The reloading case for a layer i during the consolidation pro-cess is considered when the pre-consolidation pressure r0ci isgreater than the effective stress r0i (Fig. 5). For pre-consoli-

dated clay (reloading case), replacing the compression indexfor loading Cci by the compression index for reloading Cri,gives the settlement spi from Eq. (42) and the coefficient of vol-

ume change from Eq. (44).

2.2.5. Pre-consolidated clay (case of loading and reloading)

In the general case of loading and reloading (Fig. 6), the incre-

ment of vertical stress Dr0i in a layer i is expressed as:

Dr0i ¼ r0ri þ Dr0ei ð45Þ

where Dr0ri ¼ r0ci � r0oi is the reloading increment of verticalstress in a layer i (kN/m2) and Dr0ei ¼ r0oi þ Dri � Dui � r0ci isthe loading increment of vertical stress in a layer i (kN/m2).

The total consolidation settlement is divided into two partsaccording to Fig. 6. In the first part, the clay layer will consol-idate with reloading compression index Cri until the soil pres-

sure reaches pre-consolidation pressure r0ci. In the second partafter reaching the pressure r0ci, the clay layer will consolidatemore with loading compression index Cci until reaching the

final stress r0i.For pre-consolidated clay (the loading and reloading case),

the reloading pressure effect may be taken into consideration

by dividing the consolidation settlement in a layer i into twoterms such that:

spi ¼Crihi1þ eoi

logr0cir0oi

�þ Ccihi1þ eoi

logr0oi þ Dri � Dui

r0ci

�ð46Þ

For the loading and reloading case, the consolidation settle-ment spi as a function in coefficients of volume change mvi forloading and mri for reloading may be also given by:

spi ¼ mri r0ci � r0oi� �

hi þmvi r0oi þ Dri � Dui � r0ci� �

hi ð47Þ

σ'oi

Virgin compressionslope C ci

Recompression slope C ri

Stress σ' [kN/m2] (Log scale)

σ'ci σ'i

eoi

ei

Voi

d ra

tio e

[-]

σ'ci - σ'oi σ'i - σ'ci

σi - ui

e2

e1

Δ

Δ

Δ

Δ

Figure 6 Relationship between void ratio and stress (case of

loading and reloading), r0i > r0ci > r0oi.

The coefficients of volume change for loading and reload-ing at any time t during the consolidation process can beobtained from Eqs. (46) and (47) as follows:

mri ¼ Cri

ð1þeoiÞðr0ci�r0oiÞ logð

r0ci

r0oiÞ

mvi ¼ Cci

ð1þeoiÞ r0oiþDri�Dui�r0

cið Þ logr0oiþDri�Dui

r0ci

� �9>=>; ð48Þ

2.2.6. Degree of consolidation

The degree of consolidation Us of the entire soil layers can beexpressed as:

Us ¼Skt

Sku

ð49Þ

where Skt is the sum of primary consolidation settlements in all

layers at the required time t (m), and Sku is the sum of finalconsolidation settlements in all layers, when Dui = 0 (m).

3. Numerical results

A user-friendly computer program has been developed forsolving time-dependent problems of clay layers using the

method outlined in this paper. With the help of this program,an analysis of two examples was carried out first to verify andtest the proposed method. Then, a general time-dependentproblem was evaluated to show the behavior of excess pore

water pressure during the consolidation process taking intoaccount changing the state from over- to normally-consoli-dated clay.

3.1. Test Example 1: consolidation of a single soil layer

A closed form solution for 1-D consolidation of a single layer

in an infinite sin series is available in the reference Terzaghiand Peck [18]. To verify the proposed method (LEM) fortime-dependent settlement problems for deriving a closed form

solution, the excess pore water pressure and the degree of con-solidation for a single layer calculated analytically by the avail-able closed form solution are compared with those obtained bydescribed LEM in this paper.

According to Terzaghi and Peck [18], the excess pore waterpressure u(z, t) at any depth z and time t can be determinedfrom:

uðz; tÞ ¼ 2uoX1j¼1

1

Mj

sinðMjnÞ exp �M2j Tv

� �ð50Þ

while the degree of consolidation U(t) is determined from:

UðtÞ ¼ 1� 2X1j¼1

1

M2j

exp �M2j Tv

� �ð51Þ

where Mj ¼ p2ð2j� 1Þ; Tv ¼ cvt

H2d

is the time factor in which Hd isthe length of drainage pass, for double drainage Hd =H/2while for single drainage Hd = H; and uo is the initial excesspore water pressure which is constant with depth (kN/m2).

To apply LEM, the single layer is divided into three equalsub-layers, which gives a grid nodes of N = 3. Consequently,an equation in three terms for determining the excess pore

water pressure u(z, t) at any depth z and time t is obtained:

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0D

epth

rat

io z

/HExcess pore water pressure ratio u/q

Tv=0.05 (LEM)

TV=0.197 (LEM)

TV=0.848 (LEM)

Analytical

Figure 7 Excess pore water pressure ratio with depth ratio at

different Tv for double drainage layer.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Dep

th r

atio

z/H

Excess pore water pressure ratio u/q

Tv=0.05 (LEM)TV=0.197 (LEM)TV=0.848 (LEM)Analytical

Figure 8 Excess pore water pressure ratio with depth ratio at

different Tv for single drainage layer.

cv1=100 [m2/Year]

Layer 2

Layer 1

Pervious

H

h2=9 [m]

z1

1

3

2 uo=100 [kN/m2]

Ground surface Uniform stress q=100 [kN/m2]

cv2=25 [m2/Year]

h1=9 [m]

kv2/ kv1=0.25

Pervious

uo

uo

uoz2

u

Figure 10 Two clay layers with soil properties.

1050 R. Herrmann, M. El Gendy

uðz; tÞ ¼ uoX3j¼1

Cj sinðMjnÞ exp �M2j Tv

� �ð52Þ

Also an equation in three terms for determining the degree ofconsolidation U(t) is obtained:

UðtÞ ¼ 1�X3j¼1

Cj

Mj

exp �M2j Tv

� �ð53Þ

where C1 ¼ 2þffiffiffi3p� �

=3; C2 = 1/3; and C3 ¼ 2�ffiffiffi3p� �

=3

0

20

40

60

80

100

0.0 0.4 0.8 1.2 1.6 2.0

Deg

ree

of c

onso

lidat

ion

U [

%]

Time factor Tv [-]

LEMAnalytical

Figure 9 Degree of consolidation at different Tv.

Results of LEM are compared with those of the closed formsolution. Figs. 7 and 8 show excess pore water pressure ratiosand degrees of consolidation at different time factors. How-

ever, Eqs. (52) and (53) of LEM are derived in three terms;results obtained from LEM are in a good agreement with thoseof the analytical solution of Terzaghi for both cases of doubleand single drainage (see Fig. 9).

3.2. Test Example 2: consolidation of a double-layered soil

To illustrate the hand application of LEM, the excess pore

water pressure for a double-layered soil shown in Fig. 10 afterone year is determined and compared with that obtained byFDM. The two layers are equal in thickness, each of

h1 = h2 = 9 (m). The coefficient of consolidation for the firstlayer is cv1 = 100 (m2/year), while that for the second layeris cv2 = 25 (m2/year). The coefficient of permeability ratio

for the two layers is g = k2/k1 = 0.25. The initial excess porewater pressure is distributed uniformly on the soil layers andequal to uo = 100 (kN/m2). Pervious top and bottom bound-aries are assumed.

According to Eq. (6), equations of excess pore water pres-sure for the two layers when considering three nodes can beexpressed as:

u1ðz; tÞ ¼XN¼3j¼1

Cj½A1j sinðkjnÞ þ B1j cosðkjnÞ� exp �k2j Tv1

� �

u2ðz; tÞ ¼XN¼3j¼1

Cj½A2j sin lkjq� �

þ B2j cos lkjq� �

� expð�l2k2j Tv2Þ

ð54Þ

where ui(z, t) is the excess pore water pressure at any localdepth z and time t for layer i (kN/m2); n is the depth ratio in

local coordinate of layer 1, n = z/h1 with 0 6 n 6 1; q is thedepth ratio in local coordinate of layer 2, q = z/h2 with0 6 q 6 1.

Satisfying free drainage at the top u1(0, t) = 0, requires

that:

0

3

6

9

12

15

18

0 5 10 15 20 25

Dep

th z

[m

]

Excess pore water pressure u [kN/m2]

t=3 Years (LEM)

t=2 Years (LEM)

t=1 Year (LEM)

FDM

Figure 11 Excess pore water pressure with depth at different

times.

A layer equation method for 1-D consolidation under time-dependent reloading 1051

A1j

B1j

� �¼

1

0

� �ð55Þ

Eq. (54) at t= 0 for the first layer becomes:

u1ðz; tÞ ¼ C1 sinðk1nÞ þ C2 sinðk2nÞ þ C3 sinðk3nÞ ð56Þ

and for the second layer becomes:

u2ðz; tÞ ¼ C1½A21 sin lk1qð Þ þ B21 cos lk1qð Þ� þ C2½A22

� sin lk2qð Þ þ B22 cos lk2qð Þ� þ C3½A23

� sin lk2qð Þ þ B23 cosðlk2qÞ� ð57Þ

Coefficients A2j and B2j can be obtained using interface andboundary conditions. Equating the excess pore water pressuresu1(z, 0) = u2(z, 0) at layer interfaces, leads to:

sinðk1Þ ¼ B2j ð58Þ

while equating the vertical velocity of flow at layer interfaces,leads to:

g cosðk1Þ ¼ A2j ð59Þ

Applying the boundary condition for double drainage at

the bottom u2(h2, t) = 0, leads to:

A2j sinðlkjÞ þ B2j cosðlkjÞ ¼ 0 ð60Þ

Eliminating coefficients A2j and B2j from Eqs. (58)–(60),

gives the following characteristic equation in the unknownEigen values kj:

g cos kj

� �sinðlkjÞ þ sin kj

� �cosðlkjÞ ¼ 0 ð61Þ

Substituting the values of l ¼ ðh2=h1Þffiffiffiffiffiffiffiffiffiffiffiffifficv1=cv2

p¼ 2 and

g = (h2kv1)/lh1kv2 = 2 into Eq. (61).

The characteristic Eq. (61) becomes:

cos 2kj

� �¼ �2

3ð62Þ

The operator kj is the positive root of the above character-istic equation.

kj ¼ 1:1505; 1:991; 4:292 ðj ¼ 1; 2; 3Þ ð63Þ

Substituting the value of kj obtained from Eq. (63) into Eqs.(58) and (59), gives coefficients A2j and B2j.

A2j ¼ 0:816;�0:816;�0:816 ðj ¼ 1; 2; 3ÞB2j ¼ 0:913; 0:913;�0:913 ðj ¼ 1; 2; 3Þ

ð64Þ

Constant Cj can be obtained using the initial condition.

Consider a system of linear equations at the set of N = 3 gridpoints at time t= 0 as follows:

at node 1 where n = 0.5

uo ¼ C1 sinð1:150� 0:5Þ þ C2 sinð1:991� 0:5Þþ C3 sinð4:292� 0:5Þ ð65Þ

at node 2 where n = 1

uo ¼ C1 sinð1:150� 1Þ þ C2 sinð1:991� 1Þþ C3 sinð4:292� 1Þ ð66Þ

at node 3 where q = 0.5

uo¼C1½0:816sin 2�1:150�0:5ð Þþ0:913cos 2�1:150�0:5ð Þ�þC2½�0:816sin 2�1:991�0:5ð Þþ0:913cos 2�1:991�0:5ð Þ�þC3½�0:816sin 2�4:292�0:5ð Þ�0:913cos 2�4:292�0:5ð Þ� ð67Þ

Eqs. (65)–(67) in a matrix form are:

uo

uo

uo

8><>:

9>=>; ¼

0:544 0:839 0:839

0:913 0:913 �0:9131:119 �1:119 1:119

264

375

C1

C2

C3

8><>:

9>=>; ð68Þ

Through inverting the matrix of Eq. (68), the constants of

the basic functions Cj can be obtained as:

C1

C2

C3

8><>:

9>=>; ¼ uo

0:994

0:324

0:224

8><>:

9>=>; ð69Þ

According to Eq. (8), the excess pore water pressure fornodes 1, 2 and 3 of the two soil layers can be expressed as:

u1

u2

u3

8><>:

9>=>;¼ 100

0:544 0:839 0:839

0:913 0:913 �0:9131:119 �1:119 1:119

264

375

exp �1:1502�1:235� �

0 0

0 exp �1:9912�1:235� �

0

0 0 exp �4:2922�1:235� �

264

375

0:994

0:324

0:224

8><>:

9>=>;ð70Þ

where the time factor is given by Tv1 ¼ cv1t=h21 ¼ 1:235.

The values of excess pore water pressure at the three nodesusing LEM obtained from Eq. (70) are:

u1 u2 u3f gT ¼ 10:73 17:90 21:44f gT ð71Þ

The same example is analyzed using the FDM with 0.9 (m)depth increment. Results obtained from FDM after one yearare:

u1 u2 u3f gT ¼ 10:23 16:89 22:46f gT ð72Þ

Furthermore, the excess pore water pressure with depth at

different times obtained from LEM and FDM are comparedin Fig. 11. The comparison shows a good agreement betweenthe two results.

Cv1=4.419*10-8 [m2/Sec]kv1=2.783*10-11 [m/Sec]

Layer 2

Layer 1 1 h1=3.05 [m]

h2=6.10 [m]

Layer 3

Layer 4

2

12

11

10

9

8

7

6

5

4

3

h3=9.14 [m]

h4=6.10 [m]

Cv2=2.062*10-7 [m2/Sec] kv2=8.255*10-11 [m/Sec]

Cv3=5.892*10-8 [m2/Sec]kv3=1.175*10-11 [m/Sec]

Cv4=7.376*10-8 [m2/Sec]kv4=2.946*10-11 [m/Sec]

Pervious or impervious

Pervious

H=

24.3

9 [m

]

Figure 12 Four clay layers with studying nodes and soil

properties.

0.00.0 0.2 0.4 0.6 0.8 1.0

Excess pore water pressure ratio u/q

1052 R. Herrmann, M. El Gendy

3.3. Test Example 3: consolidation of a multi-layered soil

Many authors applied their developed methods for analyzing

the multi-layered system on the example presented by Schiffi-man and Stein [19]. The example was studied by Lee et al.[1] using a general analytical solution, by Chen et al. [15] usingfinite difference method and differential quadrature method,

by Walker [20] using spectral method and by Huang and Grif-fiths [21] using the finite-element method. The soil profile in theexample consists of four layers. The geometry and the soil

properties of the four layers are shown in Fig. 12. The appliedload on the layers was uniform.

To apply LEM, the four layers are divided into 2, 3, 4 and 3

sub-layers, respectively. The total grid nodes for the case ofdouble drainage is N = 11, while that for the case of singledrainage is N= 12. The case of double drainage that was stud-ied by Chen et al. [15] using differential quadrature method

required to divide the soil profile into four differential quadra-ture elements with equally spaced sub-layers. The number ofsampling nodes for the quadrature method of the four layers

was taken as 7, 13, 19 and 13, respectively. This leads to a totalsampling nodes of N = 52. The sub-layer thickness was 0.51

Table 1 Eigen values of the four layers.

Double drainage Single drainage

j kj j kj j kj j kj

1 0.6309 7 3.6397 1 0.2524 7 3.4559

2 0.9367 8 4.0673 2 0.7216 8 3.9392

3 1.4444 9 4.6724 3 1.2326 9 4.2328

4 2.2204 10 5.2145 4 1.9140 10 4.8953

5 2.6107 11 5.7231 5 2.3810 11 5.6192

6 3.0565 – – 6 2.7253 12 5.9248

(m). In LEM, studying nodes are arbitrarily. Sublayers inLEM are 1.53 (m), 2.29 (m), 2.03 (m) and 2.29 (m), respec-tively. Chen et al. [15] studied also the case of double drainage

by finite difference method with the same sub-layer thicknessof 0.51 (m). The results of the quadrature method agreed wellwith the analytical solution while those of the finite difference

had an obvious difference with the analytical solution.Table 1 listed the Eigen values kj of the four layers obtained

from the characteristic equation, Eq. (20). However, kj of LEMare computed by a similar manner to that of Lee et al. [1], theyare different from those of Lee et al. [1] for the same example.This is related to using a different depth ratio in LEM. InLEM, the number of kj is specified by the number of studying

nodes, while that in the general analytical solution of Lee et al.[1] is unlimited.

Figs. 13 and 14 show the excess pore water pressure ratio

with depth ratio at different time factors obtained fromLEM compared with that of the analytical solution presentedby Lee et al. [1] for both cases of single and double drainages.

Fig. 15 shows the degree of consolidation at different time fac-tors Tv = cv1t/H

2 for the double drainage case obtained fromLEM compared with that of the differential quadrature

method presented by Chen et al. [15]. It can be seen from thosefigures that results obtained by LEM are in a good agreementwith those of Lee et al. [1] and Chen et al. [15].

3.4. Application Example 4: nonlinear analysis of a squarefoundation on a finite thick clay layer

An application of LEM is carried out to study the behavior of

a foundation resting on a finite thick clay layer. A square foun-dation shape is considered with a side of B = 10 (m). Thefoundation is subjected to a uniform load of qo = 100 (kN/

m2) and rests on a clay layer of thickness H= 2B as shownin Fig. 16. The groundwater level lies at a depth equal to thefoundation depth Df.

The study examines effects of reloading contact pressure onthe nonlinear consolidation behavior. In the example, the totalcontact pressure on the raft is divided into two terms. The firstterm is reloading contact pressure qv = cDf and the second

term is loading contact pressure qe = qo � qv. Consequently,the total settlement is obtained from two parts. The first partis due to the reloading contact pressure qv, which is estimated

0.2

0.4

0.6

0.8

1.0

Dep

th r

atio

z/H

Tv=0.050 (LEM), t=20 Years

Tv=0.020 (LEM), t=8 Years

Tv=0.005 (LEM), t=2 Years

Analytical

Figure 13 Excess pore water pressure ratio with depth ratio at

different Tv for double drainage layer.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Dep

th r

atio

z/ H

Excess pore water pressure ratio u /q

Tv=0.200 (LEM), t=85 Years

Tv=0.050 (LEM), t=20 years

Tv=0.020 (LEM), t=8 Years

Tv=0.005 (LEM), t=2 Years

Analytical

Figure 14 Excess pore water pressure ratio with depth ratio at

different Tv for single drainage layer.

0

20

40

60

80

100

0.00 0.05 0.10 0.15 0.20 0.25

Deg

ree

of c

onso

lidat

ion

U [

%]

Time factor Tv [-]

LEM

Chen et al. (2005)

Figure 15 Degree of consolidation at different Tv for single

drainage layer.

A layer equation method for 1-D consolidation under time-dependent reloading 1053

from Cr; and the second part is due to the loading contact pres-

sure qe, which is estimated from Cc. Sequence of unloading,loading and reloading are not considered.

B=1

qo=1

GWPerviou

Clay propertieCv

-8

Cc

eo

γγ

=1×10= 0.15= 0.5= 18.5

' = 8.69

H1

GS

Pervious or impe

Figure 16 Section in clay layer wit

3.4.1. Clay properties

The compression indices for loading Cc and for reloading Cr

along with the initial void ratio eo are used to define the con-solidation characteristics of the clay. For the comparison pur-pose, the compression index Cc and the initial void ratio eo are

chosen to make the term Cc/(1+eo) as a constant and equal to0.1 for all calculations. The compression index for reloading Cr

is assumed such that a compression index ratio, rc = Cc/Cr,

gives values 1 (–), 5 (–), 10 (–) and 15 (–). The clay propertiesin this case are:

Coefficient of consolidation Cv = 1 · 10�8 (m2/s)

Compression index for loading Cc= 0.15 (–)

Initial void ratio eo = 0.5 (–)

Unit weight of the clay c = 18.5 (kN/m3)

0

0

s

s[m[-[-[k[k

rv

h

3.4.2. Variables

Referring to Fig. 16, the foundation depth Df is chosen to give

different values of reloading contact pressure qv. Values offoundation depth and their corresponding variables are shownin Table 2.

To carry out the analysis, the clay layer of thickness Hunder the foundation level is subdivided into 10 equal sub-layers with 11 nodes, each of thickness 2.0 (m), while the con-solidation time is divided into 10 equal intervals. The pre-con-

solidation pressure in a node i on the clay layer is given by:

r0ci ¼ r0oi þ Drvi ð73Þ

where Drvi is the increase of vertical stress due to the reloadingpressure at node i (kN/m2).

3.4.3. Analysis and results

To achieve the study, 592 computational analyses have beencarried out for the above variables and parameters. To get

general relations between the different variables, results are

Df

H

[m]

0 [kN/m2]

top

: 2/s]

] ]N/m3] N/m3]

ious base

dimension and soil properties.

0

20

40

60

80

100

0.0 0.2 0.4 0.6 0.8 1.0

Deg

ree

of c

onso

lidat

ion

U[%

]

Time factor Tv [-]

Up-Pervious

Us-Pervious

Up-Impervious

Us-Impervious

Figure 17 Comparison between Up and Us (rc = 1 and rq = 0.5).

0

20

40

60

80

100

0.0 0.2 0.4 0.6 0.8 1.0

Deg

ree

of c

onso

lidat

ion

U[%

]

Time factor Tv [-]

Up-Pervious

Us-Pervious

Up-Impervious

Us-Impervious

Figure 18 Comparison between Up and Us (rc = 10 and

rq = 0.5).

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Settl

emen

t rat

io rs

[-]

Time factor Tv [-]

rq=0.00

rq=0.25

rq=0.50

rq=0.75

rq=1.00

Figure 19 Effect of reloading pressure on the settlement ratio rs(pervious boundary-rc = 1).

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Settl

emen

t rat

io rs

[-]

Time factor Tv [-]

rq=0.00

rq=0.25

rq=0.50

rq=0.75

rq=1.00

Figure 20 Effect of reloading pressure on the settlement ratio rs(pervious boundary-rc = 10).

Table 2 Values of foundation depth and their corresponding variables.

Foundation depth Df (m) 0.00 1.35 2.70 4.05 5.41

Total clay layer H1 (m) 20.00 21.35 22.70 24.05 25.41

Reloading contact pressure qv (kN/m2) 0 25 50 75 100

1054 R. Herrmann, M. El Gendy

plotted in dimensionless ratios. The reloading effect isexpressed by a reloading pressure ratio rq(–), which is given by:

rq ¼qvqo

ð74Þ

where qo the contact pressure on the foundation (kN/m2); andqv = cDf is the reloading contact pressure (kN/m2).

Similarly, the settlement effect is expressed by a settlementratio rs (–), which is given by:

rs ¼Sv

So

ð75Þ

where So is the central settlement due to the applied load with-out reloading effect (rq = 0) (m); and Sv is the central settle-ment due to the applied load with reloading effect (rq > 0) (m).

Figs. 17 and 18 show comparisons between Up and Us forpervious and impervious boundaries at different values of Tv

when rc = 1, rc = 10 and rq = 0.5. Unlike the linear analysiswhere Up is equal to Us, the figures indicate that Up and Us

are not equal for either rc = 1 or rc = 10. The development

of settlement in the case of rc = 1 is quicker than the dissipa-tion of excess pore water pressure (i.e., Us > Up). This obser-vation is changed in the case of rc = 10. The greatest rate ofconsolidation occurs for a pervious boundary.

Figs. 19–22 show the variations in settlement ratio rs withthe reloading pressure ratio rq for rc = 1 and rc = 10 at differ-ent values of Tv.

It can be observed from these figures that the reloadingpressure has a considerable influence on the settlement. To fur-ther clarify, consider the following three values of reloading

0

20

40

60

80

100

0.0 0.2 0.4 0.6 0.8 1.0

Deg

ree

of c

onso

lidat

ion

Us

[%]

Time factor Tv [-]

rc=1

rc=5

rc=10

rc=15

Figure 23 Effect of index ratio on the degree of consolidation Us

(pervious boundary-rq = 0.5).

0

20

40

60

80

100

0.0 0.2 0.4 0.6 0.8 1.0

Deg

ree

of c

onso

lidat

ion

Us

[%]

Time factor Tv [-]

rc=1

rc=5

rc=10

rc=15

Figure 24 Effect of index ratio on the degree of consolidation Us

(impervious boundary-rq = 0.5).

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Settl

emen

t rat

io rs

[-]

Time factor Tv [-]

rq=0.00

rq=0.25

rq=0.50

rq=0.75

rq=1.00

Figure 22 Effect of reloading pressure on the settlement ratio rs(impervious boundary-rc = 10).

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Settl

emen

t rat

io rs

[-]

Time factor Tv [-]

rq=0.00

rq=0.25

rq=0.50

rq=0.75

rq=1.00

Figure 21 Effect of reloading pressure on the settlement ratio rs(impervious boundary-rc = 1).

A layer equation method for 1-D consolidation under time-dependent reloading 1055

pressures; rq = 0.75, rq = 0.25 and rq = 0.0 for a pervious

boundary at consolidation time of t= 40 years(Tv = 0.0315) and rc = 10, which may represent relativelylarge and small values of rq. Results of these three cases are

presented in values in Table 3. Referring to this table, as a sam-ple, rq = 0.75 meets a foundation depth of Df = 4.05 (m),which gives a reloading contact pressure of qv = 75 (kN/m2).

This means that the first term of settlement is determined fromreloading contact pressure of qv = 75 (kN/m2) and Compres-sion index for reloading Cr = 0.015 (–). In this case, the second

term of settlement is determined from loading contact pressureof qe = qo�qv = 25 (kN/m2) and Compression index for load-ing Cr = 0.15 (–). Similarly, the settlement terms for the othertwo cases can be determined.

Table 3 Values of results at consolidation time of 40 years.

Foundation depth Df (m) 0.00

Total clay layer H1 (m) 20.00

Reloading contact pressure qv = cDf (kN/m2) 0

Reloading pressure ratio rq = qv/qo (–) 0

Total settlement Sv (cm) 37.68

Settlement ratio rs = Sv/So (–) 1.00

From Table 3, it can be concluded that at a relatively smallvalue of rq = 25, the settlement reduces to 35 (%) of that with-

out reloading pressure (at rq = 0 and Df = 0), while at a rela-tively greater value of rq =75, the settlement reduces to 12 (%)of that without reloading pressure. These percentages become

51 (%) and 25 (%) when rc = 1.The development of settling begins quickly and becomes

constant until the end of consolidation process. Both two cases

of pervious and impervious boundaries give nearly the samesettlement magnitude at a specified rc.

The effect of the compression index ratio rc on the degree ofconsolidation Us at different values of Tv when rq = 0.5 for

pervious and impervious boundaries are shown in Figs. 23

1.35 2.70 4.05 5.41

21.35 22.70 24.05 25.41

25 50 75 100

0.25 0.50 0.75 1.0

13.22 4.66 1.24 0.66

0.35 0.12 0.03 0.02

0

20

40

60

80

100

0.0 0.2 0.4 0.6 0.8 1.0D

egre

e of

con

solid

atio

n U

s[%

]Time factor Tv [-]

Tvc=0.0

Tvc=0.03

Tvc=0.25

Figure 25 Effect of loading rate on the degree of consolidation

Us (pervious boundary rc = 10-rq = 0.5).

1056 R. Herrmann, M. El Gendy

and 24. The figures show that the rate of consolidation for apervious boundary is higher than that for an imperviousboundary. The rate of consolidation Us for all values of rcare nearly the same for either pervious or imperviousboundary.

Fig. 25 shows the effect of loading rate on the degree of

consolidation Us at different values of Tv and Tvc = Cvtc/H2

when rc = 10 and rq = 0.5 an for pervious boundary, whileFig. 26 shows this effect for an impervious boundary. As

expected, the quicker the loading the faster the consolidation.

3.4.4. Stability of the analysis

The solution of LEM is considered to be convergence when the

differential equation operator kj of a problem can be obtained.The stability of the solution is investigated by choosing differ-ent studying nodes with different element thicknesses of clay. It

is found that the solution for nonlinear analysis depends on thesub-layer thickness in which a thickness of 2 (m) gives a goodresults. For linear analysis the Test Example 3 is studied. Thesolution was always convergence even for a very small sub-

layer thickness of 0.05 (m).For case of a system having too many layers with extreme

differences in soil properties, the convergence of the solution

may be not occurred. It can overcome this problem by choos-ing sublayers lead to the parameter of the coefficient of

0

20

40

60

80

100

0.0 0.2 0.4 0.6 0.8 1.0

Deg

ree

of c

onso

lidat

ion

Us

[%]

Time factor Tv [-]

Tvc=0.0

Tvc=0.03

Tvc=0.25

Figure 26 Effect of loading rate on the degree of consolidation

Us (impervious boundary rc = 10-rq = 0.5).

consolidation and thickness, li ¼ ðhi=h1Þffiffiffiffiffiffiffiffiffiffiffiffifficv1=cvi

pfor all layers

nearly equal to 1.

4. Conclusions

A generalized procedure and method for the solution of thefundamental problem of time-dependent consolidation and

settlement of clay layers is presented. The proposed methodconsiders various analysis aspects of the consolidation prob-lem such as, among others: uniform and time-dependent load-

ing, cycling loading, nonlinear compressibility parameters,multi-layered soil, normal- and over-consolidated clay, andcreep compression. In reality, settlement due to consolidation

consists of two parts: loading and reloading settlements.In this paper, an analysis using the LEM method is devel-

oped to take into account the change of the soil state from over-

to normally-consolidated clay since both reloading and loadingcases can exist during the consolidation process. The initialapplied stress on the clay layer is considered non-uniform.Also, coefficients of compressibility and permeability are con-

sidered to decrease with time and are variable for each layer.The method provides a solution to differential equations forboundary condition problems in which closed form solutions

are not possible or difficult. Three tested examples are pre-sented to verify the method. A study for nonlinear analysis ofa square foundation on a finite thick clay layer is also presented.

The developed LEMmethod depends on selecting a numberof nodes in the soil layers. Consequently, a better representationfor applied stresses on the layers can be represented. It is alsoideal for using a stress coefficient technique, which may be

extended to study the interaction of irregularly loaded areason the surface, or contact pressure due to foundation rigidity.LEM requires fewer equation terms, in which a few terms are

sufficient to give good results. In this method only a limitednumber of Eigen values equal to the selected points are required.

Appendix A. Matrix [U] for double drainage boundaries is

given by:

½U� ¼

B21 B22 B23 � � � � � � � � � B2n

B31 B32 B33 � � � � � � � � � B3n

B41 B42 B43 � � � � � � � � � B4n

� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �

Bðn�1Þ1 Bðn�1Þ2 Bðn�1Þ3 � � � � � � � � � Bðn�1Þn

jBn1j jBn2j jBn3j � � � � � � � � � jBnnj

2666666666664

3777777777775ð76Þ

Matrix [U] for single drainage boundary is given by:

½U� ¼

B21 B22 B23 � � � � � � � � � B2n

B31 B32 B33 � � � � � � � � � B3n

B41 B42 B43 � � � � � � � � � B4n

� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �Bn1 Bn2 Bn3 � � � � � � � � � Bnn

Bðnþ1Þ1 Bðnþ1Þ2 Bðnþ1Þ3 � � � � � � � � � Bðnþ1Þn

2666666666664

3777777777775

ð77Þ

A layer equation method for 1-D consolidation under time-dependent reloading 1057

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Richard Herrmann is a professor of Geotech-

nical Engineering, in Department of Civil

Engineering, Head of Geotechnical Institute,

Siegen University, Germany. Author’s contact

e-mail id is ‘‘richard.herrmann@uni-siegen.

de’’.

Mohamed El Gendy is a professor of Geo-

technical and Foundation Engineering, in

Department of Civil Engineering and Vice

Dean of Faculty of Engineering in Port Said

University, Egypt. Author’s contact e-mail id

is ‘‘melgendy@elpla.com’’.