Preliminary research on the theory and application of ... · railway construction. The embankment...

ORIGINAL ARTICLE

Preliminary research on the theory and application of unsaturatedRed-layers embankment settlement based on rheologyand consolidation theory

Junxin Liu1,2 • Wei Liu2 • Peng Liu1 • Chunhe Yang2 • Qiang Xie3 •

Yutian Liu1

Received: 17 June 2015 / Accepted: 24 November 2015 / Published online: 16 March 2016

� Springer-Verlag Berlin Heidelberg 2016

Abstract The settlement of an embankment body is

always a significant technological issue in highway and

railway construction. The embankment body is usually

constructed using unsaturated soil, whose deformation

consists of three parts, instantaneous deformation, primary-

consolidation deformation and secondary-consolidation

deformation. The primary consolidation and the secondary

compression deformations are not two separate processes,

but they influence each other, that are the deformation of

the embankment body is the coupling-effect result of

unsaturated-rheology and consolidation. Based on the

results of uniaxial compression creep test of crushed Red-

layers mudstone soils and the principle of the single stress

state variable in unsaturated soils, the theory of unsaturated

embankment settlement of Red-layers is put forward.

Physical centrifugal tests and numerical simulation of

centrifugal loading process are adapted not only to study

the relationships of the settlement of the embankment body

filled with different compaction coefficients versus

embankment filling height, as well as the settlement after

construction versus time, but also to predict the settlement

after construction. Based on these studies, a project case,

the settlement characteristics of a section of the railway

line for passenger traffic from Chongqing to Suining have

been studied by numerical simulation considering actual

filling process. These show that the physical centrifugal

tests and site monitoring data largely coincide with the

corresponding numerical simulation results, showing the

high validation of the proposed coupling-theory.

Keywords Creep � Unsaturated soils � Single stress state

variable � Settlement � Site monitoring

Introduction

Settlement of embankment bodies is a main research issue

for highway and railway roadbed practices. Analysis of the

settlement characteristics of the embankment body is the

basis for the correct understanding and evaluation of the

subgrade stability, the improvements for the roadbed

design, as well as for selecting effective ways for the

construction and management technologies. In high

embankments, gravity stress and stress level are high, and

the settlement produced by compression of filling soils is

more obvious, and moreover excessive settlement will

cause harm to the roadbed itself of highway, the railway

and their pavement structures (Tang et al. 2012; Wu et al.

2014). So, more attentions have been paid to the research

of the embankment settlement calculation. Generally,

because the properties of the unsaturated soil itself are

complicated and unsaturated soil mechanics theory is not

also mature enough, the settlement deformation of unsat-

urated embankment still lacks reasonable theoretical cal-

culation methods, especially for passenger dedicated

railway and highway due to more demanding requirements

of the settlement control. On the other hand, there are still

no reliable technologies and project experiences of similar

projects at home or abroad. Moreover, systematic

& Wei Liu

[email protected]

1 School of Civil Engineering and Architecture, Southwest

University of Science and Technology,

Mianyang 621010, Sichuan, China

2 State Key Laboratory of Coal Mine Disaster Dynamics and

Control, Chongqing University, Chongqing 400044, China

3 School of Geosciences and Environmental Engineering,

Southwest Jiaotong University, Chengdu 610031, China

123

Environ Earth Sci (2016) 75:503

DOI 10.1007/s12665-016-5313-2

http://crossmark.crossref.org/dialog/?doi=10.1007/s12665-016-5313-2&domain=pdf

http://crossmark.crossref.org/dialog/?doi=10.1007/s12665-016-5313-2&domain=pdf

observations for the application are lacking (Wang et al.

2000). Currently, for the instantaneous compression

deformation and consolidation deformation at different

times, the calculation method of filling body settlement is

mainly the normative method, numerical method and

experience method (Li 2006).

Unsaturated soils are typical three-phase media, con-

sisting of a deformable soil skeleton body composed of

mineral particles and the two fluid phases, water and air.

These two fluids filling the entire void space, are separated

by interfaces, and play an important role in the mechanical

behavior of the soil skeleton. Deformations of soil particles

are very small compared to those of the soil skeleton, thus

it is generally considered that the particles are incom-

pressible. Therefore, deformations of the unsaturated soils

are the results of the change of pore fluid and the com-

pression of pore gas, rearrangement of soil particles, dis-

tance reduction among particles, as well as dislocation of

the soil skeleton (Fredlund and Rahardjo 1993).

An embankment body is mainly filled with unsaturated

soils, the deformations of which consist of three parts, the

instantaneous deformation, the deformation of primary

consolidation, and the deformation of secondary com-

pression. The deformation due to the primary consolidation

is induced mainly due to the dissipation of compressive

pore gas and fluid pressures. When comparing this to the

saturated soil, this process takes longer. The deformation

produced by the secondary compression is characterized by

creep process due to the squirm of the bound water

membrane around particles’ surfaces and the rearrange-

ment of the particles’ structure. However, the primary and

the secondary compression are not two completely inde-

pendent processes and they significantly interactively

influence each other (Li 2006; Fatahi et al. 2013), and

consequently, the deformations of the embankment body,

filled with unsaturated soil, are the coupling results of the

unsaturated rheology and consolidation.

Consolidation of unsaturated soil is a common issue in

engineering practice and theoretical research, which

includes the dissipation laws of pore gas and water pres-

sures changing along with the time under loading. A lot of

different consolidation equations have been proposed to

describe the consolidation behaviors of unsaturated soil.

Biot (1941) and Scott (1963) proposed consolidation

equations for unsaturated soil with occluded air bubbles.

Blight (1961) derived a consolidation equation for the air

phase of a dry, rigid, and unsaturated soil. Barden (1965,

1974) presented an analysis method for the consolidation

of compacted and unsaturated clay. Assuming that the air

and water phases are continuous, Fredlund and Hasan

(1979) proposed a one-dimensional consolidation theory,

which is now widely adopted. In this theory, two partial

differential equations are employed to describe the

dissipation processes of excess pore pressures in unsatu-

rated soil. This theory was later extended to the 2D and 3D

case by Dakshanamurthy and Fredlund (1980) and Dak-

shanamurthy et al. (1984). In the past three decades, with

the rapid development of computational science and tech-

nology, the numerical simulation methods are widely used

in the field of geotechnical engineering. To estimate the

physical behaviors of the consolidation of unsaturated soil,

some researchers (Lloret and Alonso 1980; Wong et al.

1998; Conte 2004; Qin et al. 2010a; Zhou and Tu 2012)

conduct numerical modelling methods for plane strain and

axisymmeric problems and obtained remarkable results. In

addition, analytical methods have also been comparatively

progressive. Most recent analytical developments were

based on the concept of one-dimensional continuity equa-

tions given by Fredlund and Hasan (1979). Using the

techniques of eigenfunction expansion and Laplace trans-

formation and et al. researchers (Qin et al. 2008, 2010b;

Shan et al. 2012; Zhou et al. 2014; Ho et al. 2014)

developed some analytical methods of one-dimensional

consolidation of unsaturated soils subjected to different

types of loading and all kinds of given condition boundary

and gave a good agreement with the numerical predictions

and physical model results. However when the consolida-

tion concept is expanded to a 2D problem, in which both

horizontal and vertical drainage are considered, these

numerical solutions are overestimated (Ho and Fatahi

2015), so Ho and Fatahi (2015) introduced an exact ana-

lytical solution predicting variations in excess pore-air and

pore-water pressures and settlement considering the two-

dimensional (2D) plane strain consolidation of an unsatu-

rated soil stratum subjected to different time-dependent

loadings using the continuity equations proposed by Dak-

shanamurthy and Fredlund (1980) and the mathematical

development also adopts eigenfunction expansions and

Laplace transformation methods along with homogeneous

drainage boundary conditions and uniform initial

conditions.

Recently, to investigate the solid–water–air coupling

phenomenon in detail, many researchers have formulated

conservation equations for a three-phase system consisting

of a solid and two immiscible fluids, namely liquid and gas

(Schrefler and zhan 1993; Khalili and Valliappan 1996;

Khalili et al. 2005; Gray and Schrefler 2001, 2007; Borja

and Koliji 2009). These researchers also derived expres-

sions for the effective stress tensor in multi-phase porous

media exhibiting two porosity scales, micro- and macro-

porosity, during the course of loading. Unsaturated soils are

different from saturated soils, where both water and air are

present in their voids, so their mechanical behaviors are

much more complex than those of saturated soils. Unsat-

urated soils have commonly been viewed as a three-phase

system (Lambe and Whitman 1979). More recently, the

503 of 21 Environ Earth Sci (2016) 75:503

123

contractile skin (i.e. the air–water interface) has been

introduced as a fourth and independent phase (Fredlund

and Morgenstern 1977). Crucial for a proper description of

the material behavior of soils is the definition of the stress

state variables. Historically, three types of stress variable

have been proposed, namely the single stress state variable

describing the mechanical behavior of a unsaturated soil,

which is equivalent to the concept of single-valued effec-

tive stress in saturated soil (Bishop 1959; Aitchison 1961;

Jennings and Burland 1962; Hutter et al. 1999), the two

stress state variables, namely the net stress (r� ua) and the

matrix suction (ua � uw) (Coleman 1962; Bishop and

Blight 1963; Matyas and Radhakrishna 1968; Fredlund and

Morgenstern 1977), and more recently the two modified

stress state variables (Kohgo et al. 1993, 1995; Wheeler

and Karube 1995; Gallipoli et al. 2003). Advantages of the

approach of the single stress state variable over the two

stress state variables are that complete characterization of

the stress/strain behaviors of soil requires expressing of the

behavior in terms of single stress state variable rather than

two independent stresses. Also, the laboratory testing

required for determining soil properties in terms of effec-

tive stress could be performed in any geotechnical engi-

neering laboratory. It is substantially less cumbersome and

time-consuming than that required to determine soil prop-

erties in terms of two independent stress variables. But the

theory of the single stress state variable is not capable of

completely describing irrecoverable (plastic) deformations,

such as volume expansion and collapse phenomena of

unsaturated soils upon wetting within a linearly elastic

theoretical framework (Jennings and Burland 1962; Bur-

land 1965; Aitchison 1965; Matyas and Radhakrishna

1968; Brackley 1971; Gudehus 1995; Wheeler and Karube

1995). Expressed as a function of the externally applied

stresses and the internal fluid pressures, the effective stress

converts a multi-phase, multi porous media to a mechani-

cally equivalent, single-phase and single-stress state con-

tinuum. It enters elastic as well as elastic–plastic

constitutive equations of the solid phase, linking a change

in stress to straining or any other relevant quantity of the

soil skeleton. So difficulties of describing irrecoverable

volumetric deformations such as dilation and/or collapse

can be overcome in a critical state framework with con-

ventional plasticity theory of the single stress state variable

(Biot 1941; Rice and Cleary 1976; de Boer and Ehlers

1990; Coussy 1995; Loret and Khalili 2000; Laloui et al.

2003; Khalili et al. 2004).

In the long period of studies, researchers have proposed

many suitable rheological constitutive models for

geotechnical materials. From the formal point of view,

these macro-rheological models can be divided into three

types, the general rheological theory model, the empirical

and semi-empirical model and the visco-elastic–plastic

model. In general rheological theory model, the component

model and genetic theory are widely used (Feda 1992; Sun

1999; Chen and Bai 2003; Liingaard et al. 2004; Xia et al.

2009). But because of the influences of the loading mode,

loading path, loading rate, and other factors, the rheologi-

cal behavior of the soil is nonlinear, and these models

cannot describe the nonlinear behavior. At the same time,

the identification of the models and the determination of

the parameters can also give a lot of difficulties (Zhu et al.

2006). The empirical and semi-empirical model generally

uses the linear relationship between the strain (or strain

rate) and the time of the strain (or strain rate) in the semi

logarithmic (or double logarithmic) coordinate system

(Taylor 1942; Singh and Mitchell 1968; Mesri and God-

lewski 1977; Kavazanjian and Mitchell 1977; Tavenas

et al. 1978; Tian et al. 1994), and subsequently many

researchers also have found that the long term relationship

between the strain and logarithm of time may not be linear

(Berre and Iversen 1972; Yin 1999), and the following

question is that the determination of the parameters of non-

linear creep function is a challenging task. To solve the

problem of obtaining the parameters of the nonlinear creep

function, Le et al. (2015) introduced a numerical opti-

mization procedure to obtain non-linear elastic visco-

plastic model parameters and what’s more, the model

parameters can be obtained simultaneously. The empirical

model established by laboratory experiments is lack of

rigorous theoretical basis, which can only reflect the creep

phenomenon of simple loading or specific stress paths. But

the advantage of empirical creep model is that only a small

number of parameters can achieve good fitting effect, so it

has certain application value in engineering practice. The

key question is how to take the complex factors into

account, namely, stress conditions, loading rate, drainage

conditions and so on (Zhu et al. 2006). The visco-elastic–

plastic model is developed from the classical elastic–plastic

theory, and the characteristic of the model is to use the

concept of creep potential function to facilitate the

numerical analysis (Prager 1949; Adachi and Okano 1974;

Desai and zhang 1987; Satake 1989; Yin and Graham

1994a, b). Micro-rheological models, which are established

from the point of view of microstructure, mainly have three

types, cavity channel network model (De Jong 1968),

micro-slip-surface model (Shi et al. 1997) and discrete

particle model (Kuhn and Mitchell 1993). Although these

models have many disadvantages, such as lots of parame-

ters and difficulty to identify, they reveal the microscopic

rheological characteristics of soil, which is helpful to

understand the rheological properties of soil.

Regardless of the understanding of the soil rheology, the

time-dependent characteristics of the deformation of the

soil are determined by the common action of the creep and

the consolidation and the investigation of its mechanism

Environ Earth Sci (2016) 75:503 of 21 503

123

and model has never stopped (Fatahi et al. 2013). The

classical theory of soil mechanics cannot consider the

consolidation effect and study the creep effect simultane-

ously, especially the creep effect at the complex stress

states, and moreover the rheological theory from other

visco-elastic material is not considered as a result of the

dissipation of excess pore water pressure. Therefore, it

cannot be used to analyze the creep effect in the process of

soil consolidation and only the consolidation theory or the

creep theory to describe the time-dependent characteristics

of soil deformation are insufficient (Fatahi et al. 2013).

The coupling model of rheology and consolidation is

based on the consideration of the dissipation of pore water

pressure and the rheological deformation of soil skeleton,

and the early researches are to establish a combined model

of the component rheological model and the consolidation

model in series or in parallel (Taylor and Merchant 1940;

Chen 1958; Folque 1961), and subsequently with the needs

of the engineering of soft soil, all kinds of new models

continue to emerge, mainly including three types, namely,

the visco-elastic and visco-plastic consolidation model

established by the application of the Biot’s consolidation

theory to visco-elastic and visco-plastic soil (Zhan et al.

1993; Zhao and Shi 1996), the visco-elastic–plastic con-

solidation model according to the experimental results of

the soil (Yin and Graham 1989, 1994a, b, 1996; Tang et al.

2000), the empirical formula including the consolidation

and creep effect according to the experimental results

(Wang et al. 2000),the coupling model according to the soil

constitutive model, Biot’s consolidation theory and the

creep model considering the drainage conditions (Chen and

Bai 2001, 2003, 2006; Zhu et al. 2006), and the creep

model of soil under the consolidation state established by

the multi-functional characteristics of the disturbed state

surface model (DSC) and the triaxial test results (Desai

et al. 1995; Desai and Sane 2007), etc. Aforementioned

research results of the coupling model of rheology and

consolidation are mainly focused on saturated soil and very

few with regard to unsaturated soil, but actually, the stra-

tums above groundwater table are basically composed of

unsaturated soils, whose mechanical properties are differ-

ent from saturated soil and the related long-term settlement

of unsaturated soil, especially the embankment has been

the focus of attention, which is involved in the coupling

effect of unsaturated rheology and consolidation.

Red-layers mudstone is mainly distributed throughout

Southwest, Southeast, Southern and Northwest areas of

China. Because of its wide distribution, it is often

encountered in the construction and engineering practices.

Red-layers mudstone, as a kind of soft rock, has the

characteristics of easy weathering, water softening, and

disintegration, and belongs to filler Class C (Code for

Design of High Speed Railway 2009). So Red-layers

mudstone as fillers for embankment of highway and rail-

way and earth rock dam, must be crushed at first, and then

be filled (Wang 2000; Liu et al. 2013; Nahazanan et al.

2013). In this article, based on results of uniaxial com-

pression creep tests of crushed Red-layers mudstone soils

and the principle of the single stress state variable in

unsaturated soils, the theory of unsaturated embankment

settlement is proposed. In addition, according to the results

of physical centrifuge tests and site monitoring data of

embankments of crushed Red-layers mudstone soils, the

theoretical model has been verified with corresponding

numerical simulations considering actual loading process

(Fig. 1).

Creep constitutive model of crushed Red-layersmudstone soils

To indicate the creep deformation properties of Red-layers

mudstone filler, moderately weathered Red-layers mud-

stones were crushed to fine particles with diameters below

Fig. 1 The method keeping the sample moisture


123

of 2 mm. The soil materials, produced by crushed Red-

layers mudstone, were prepared with the optimal moisture

content of 12.81 %. The samples were made by dynamic

compaction with six superimposed layers, and with sizes of

61.8 mm 9 124 mm. The compaction coefficients, defined

as the ratio of dry density to the maximum density, are

0.87, 0.90, 0.93 and 0.95, respectively. And in this exper-

iment series, the maximum dry density is 1.855 g/cm3. To

insure the uniformity of the samples, the density error of

each sample was controlled to be less than 0.02 g/cm3. The

uniaxial compression creep tests were carried out using a

modified high pressure consolidation apparatus, whose

leverage ratio is 8.53, and the axial deformation was

measured by dial gauge. To maintain the moisture content

of sample during testing, two impermeable membranes

were placed on the sample’s lateral surface at first, and then

rubber bands were used to fix both ends of the sample. A

moist towel was used to wrap on the outside of the sample,

and water was sprayed frequently to keep the towel moist

during the whole process of each test.

Generally, the stress level of creep test is determined

according to the uniaxial compressive strength. Under

different compaction coefficients, the corresponding values

of the uniaxial compressive strength (UCS) were listed in

Table 1. The stress levels of the uniaxial compression

creep tests were respectively set as 0.25, 0.35, 0.45, 0.55,

0.65, 0.75 and 0.85 times of the uniaxial compression

strength. The duration of each loading test was more than

30 days. The test results are in Table 1.

The adjacently immediately preceding loading before

failure was taken as the long-term strength and the corre-

sponding uniaxial compressive long-term strength of dif-

ferent compaction samples can be calculated from Fig. 2,

as shown in Table 2.

According to the data from Fig. 2, the creep curve under

corresponding loads can be obtained through the Eq. (1) of

the Boltzmann superposition principle (Ferry 1980), and

the results are shown in Fig. 3.

eðr3ðtÞÞ ¼ eðr1ðtÞÞ þ eðr2ðtÞÞ ð1Þ

where, r is applied stress, in which r3 ¼ r1 þ r2; e is thecorresponding strain of stress.

Regardless of the instantaneous deformation under

loading, the first five-level loadings of the creep curves

were investigated with the generalized Kelvin creep model,

shown in Fig. 4, whose fitting formulas are shown in

Table 1 Uniaxial compressive strengthof different compaction samples

Compaction coefficient 0.87 0.90 0.93 0.95

Actual water content/% 12.95 12.90 12.88 12.66

UCS, Rc/kPa 222.60 279.45 385.54 484.32

02468

10121416182022 σ1=207.95kPa

σ1=187.16kPa

σ1=166.36kPa

σ1=145.57kPa

σ1=103.98kPa

stra

in/%

time/d

σ1=62.39kPa

(a) Compaction coefficient: 0.87

0

2

4

6

8

10

12

14

16

18

20 σ1=250.35kPa

σ1=229.49kPa

σ1=187.76kPa

σ1=146.04kPa

σ1=104.31kPaσ1=62.59kPa

stra

in/%

time/d(b) Compaction coefficient: 0.90

2

4

6

8

10

12

14

16

18

20 σ1=354.67kPa

σ1=312.94kPa

σ1=271.21kPa

σ1=229.49kPa

σ1=187.76kPa

σ1=146.04kPaσ1=104.31kPa

stra

in/%

time/d(c) Compaction coefficient: 0.93

0 50 100 150 200 250

0 50 100 150 200 250 300

0 50 100 150 200 250 300

-50 0 50 100 150 200 250 300 3500369

1215182124273033

σ1=457.50kPa

σ1=395.11kPa

σ1=353.52kPaσ1=311.93kPa

σ1=270.34kPaσ1=228.75kPa

σ1=187.16kPaσ1=103.98kPa

stra

in/%

time/d(d) Compaction coefficient: 0.95

Fig. 2 Uniaxial compression creep curves of different compaction

samples subjected to multi-stage loadings


123

Eq. (2) and fitting parameters shown in Table 3. The cor-

relation coefficients were all above 0.98.

eij ¼sij

2G1

1� e�G1

b1t

h iþ sij

2G2

1� e�G2

b2t

h ið2Þ

where eij are the components of the deviatoric strain tensor,

sij are the components of the deviatoric stress tensor, G1,

G2, b1 and b2 are parameters of the Kelvin bodies.

Description of the single stress state variableof unsaturated soil

Since the application of the Terzaghi effective stress

principle in saturated soils, the principle of the single

stress state variable that can well describe the mechanical

behaviors of unsaturated soils or not has been under

exploration. Unsaturated soil behaviors are much more

complex than the behavior of saturated soil. Therefore,

how to establish the unsaturated formula of the single

stress state variable has undergone a long development

process.

(a) Formula of Bishop effective stress (Bishop 1959).

Based on the effective stress principle for saturated

soils and the characteristics of saturation and unsat-

urated soil condition, Bishop (1959) suggested a

tentative expression for effective stress with a

parameter v, which has gained widespread reference.

The formula is:

r0 ¼ r� ua þ vðua � uwÞ ð3Þ

where r0, r, ua and uw are the effective stress, total

stress, pore air pressure and pore water pressure of

the unsaturated soil, respectively; v is a parameter

related to the degree of saturation of unsaturated soil,

0� v� 1.

(b) Formula of Aitchison (1961).

Aitchison (1961) proposed the following effective

stress equation at the Conference on Pore Pressure

and Suction in Soils, London, in 1960:

r0 ¼ rþ up00 ð4Þ

where p00 is pore-water pressure deficiency; u is a

parameter with values ranging from 0 to 1.

(c) Formula of Jennings (1961).

Jennings (1961) also proposed an effective stress

equation at the Conference on Pore Pressure and

Suction in Soils, London, in 1960:

r0 ¼ rþ bp00 ð5Þ

where p00 is negative pore-water pressure taken as a

positive value; b is a statistical factor of the same

type as the contact area, the value for which should

be measured experimentally.

Equations (3–5) are equivalent when the pore-air

pressure used in all four equations is the same (i.e.

v ¼ u ¼ b). Only Bishop’s form (i.e. Eq. 3) refer-

ences the total and pore-water pressures to the pore-

air pressure. The other equations simply use gauge

pressures which are referenced to the external air

pressure.

(d) Formula of Richards (1966).

Richards (1966) incorporated a solute suction com-

ponent into the effective stress equation, and

obtained:

r0 ¼ r� ua þ vmðhm þ uaÞ þ vsðhs þ uaÞ ð6Þ

where vm is the effective stress parameter for matric

suction; hm is matric suction; vs is the effective stressparameter for solute suction; hs is solute suction.

Aitchison (1973) presented an effective stress

equation in a slightly modified form of Richards’s

equation (1966):

r0 ¼ rþ vmp00m þ vsp

00s

where p00m is matric suction, p00m ¼ ua � uw; p00s is

solute suction; vm and vs are soil parameters which

are normally within the range of 0–1, and which

depend upon the stress path.

(e) Formula of Sparks (1963).

Sparks (1963) proposed the following formula:

r0 ¼ r� 11ua � 12uw þ 13Tc ð7Þ

where Tc is surface tension; 11, 13 and 13 are

experimental parameters.

(f) Formula of Lambe (1960).

Lambe(1960) proposed the following formula:

r0 ¼ r� aaua � awuw � ðR� AÞ ð8Þ

Table 2 Uniaxial compressive

long-term strength of different

compaction samples

Compaction coefficient 0.87 0.90 0.93 0.95

Actual water content at failure/% 12.36 13.48 12.82 12.55

The long-term strength Rr/kPa 187.16 229.49 312.94 395.11

Rr/Rc 0.84 0.82 0.81 0.82


123

where aa is a contact area coefficient between soil

particles and air, aa ¼ ð1� vÞ; aw is a contact area

coefficient between soil particles and water;

aw ¼ ðv� aÞ, in which a is an inter-granular contact

coefficient with values ranging from 0.01 to 0.03,

approximately equal to 0; R is unit inter-granular

repulsion, and A is unit inter-granular suction.

(g) Formula of Chen et al. (1994).

Through the investigation of the stress state and

deformation of unsaturated soil, Chen et al. (1994)

derived a theoretical formula of effective stress for

anisotropic porous media and unsaturated soil based

on elasticity theory, where there are a variety of

immiscible fluids.

r0 ¼ r� ua þKn

Ksnðua � uwÞ ð9Þ

where Kn and Ksn are the bulk modulus of soil

skeleton when corresponding porosities are n and sn,

which can be obtained by experiments on samples

when there is no fluid or when the fluid exists but

with zero-value of pore pressure. By compared to

Bishop formula (Bishop 1959), relationship of v ¼Kn

Ksn can be obtained.

(h) Formula of Loret and Khalili (2000).

According to the test data of more than a dozen kinds

of soil samples from 14 investigators, Loret and

Khalili (2000) obtained the formula:

x ¼ ðua � uwÞðua � uwÞb

� ��0:55

ðua � uwÞ[ ðua � uwÞb ð10Þ

(i) Formula of Shen (1996).

Shen (1996) proposed a generalized suction form:

r0 ¼ ðr� uaÞ þ s0 ¼ ðr� uaÞ þ cstg/s ð11Þ

where s0 is generalized suction, including cement

force between particles, bite force and suction; csand /s are the indexes of the inter-granular resis-

tance against sliding.

(j) Formula of Liu (1999).

Liu (1999) proposed the effective stress formula of

unsaturated soil based on detailed investigations of

existing states of unsaturated soil matrix, water and

gas and the influence of soil structure:

2

4

6

8

10

12

14

16

18

stra

in/%

time/d

the loading:62.39kPa the loading:103.98kPa the loading:145.57kPa the loading:166.36kPa the loading:187.16kPa the loading:207.95kPa

(a) Compaction coefficient: 0.87

2468

101214161820

stra

in/%

time/d

the loading:62.59kPa the loading:104.31kPa the loading:146.04kPa the loading:187.76kPa the loading:229.49kPa the loading:250.35kPa

(b) Compaction coefficient: 0.90

4

6

8

10

12

14

16

18

20

stra

in/%

time/d

the loading:104.31kPa the loading:146.04kPa the loading:187.76kPa the loading:229.49kPa the loading:271.21kPa the loading:312.94kPa the loading:354.67kPa

(c) Compaction coefficient: 0.93

0 5 10 15 20 25 30

0 5 10 15 20 25 30

0 5 10 15 20 25 30

(d) Compaction coefficient: 0.95

Fig. 3 Uniaxial compression creep curves of different com-

paction samples subjected to corresponding loadings

σ

1G 2G

1β 2β

σ

Fig. 4 The generalized Kelvin creep model


123

r0 ¼ ðr� uaÞ � vðua � uwÞ ð12Þ

where v is fundamentally different from the conven-

tional concept of Bishop expression, and it has two

important properties: the one is that a minus sign should

be put before v, and the other is that its absolute value

may be less than 1, equal to 1 or much larger than 1,

depending on the soil properties or conditions. So this

effective stress formula can be applied to all soils

including collapsible soil and expansive soil. It has not

onlyclarified the transmissionand sharingofunsaturated

soil stress and the mechanism of influences on defor-

mation and strength, but also laid the theoretical foun-

dation for the unsaturated principle of the single stress

state variable and enlarged the scopes of application.

(k) Formula of Hutter (1999).

Based on the assumptions that (1) the mixture is

made up of density preserving constituents, (2) water

is a perfect fluid, and (3) no effective stress is

introduced for the fluid, Hutter et al. (1999) derived

the following formula using mixture theory:

r0 ¼ r� ua þ Swðua � uwÞ ð13Þ

where Sw is the saturation of water. The formula has

been verified by Dangla (1999). If Sw is substituted

for v in the Bishop’s formula, the same formula can

be obtained. Because parameters of this formula are

explicit and the formula achieves a smooth transition

from theory mechanics of saturated soil to unsatu-

rated soil, it has been widely used. In this way, the

unsaturated fluid–solid coupling calculation is car-

ried out in this paper.

Unsaturated fluid–solid coupling model basedon the unsaturated principle of the single stressstate variable

The solid grains forming the matrix of the soil are assumed

to be incompressible. The following features of the fluids–

solid interaction are captured using the built-in logic:

(a) Changes in effective stress cause volumetric strain to

occur (the effective stress increment for two-phase

flow is the Terzaghi effective stress increment, with

pore pressure increment replaced by mean, satura-

tion weighted, fluid pressure increments (Dangla

1999);

(b) The volumetric strain is produced by effective stress

according to Terzaghi theory;

(c) Volumetric deformation causes changes in fluid

pressures;

(d) Bishop effective stress is used in the detection of

yield in constitutive models involving plasticity

(Schrefler and Zhan 1993; Hutter et al. 1999).

So, according to Biot consolidation theory, fluid–solid

coupling of unsaturated soil must embody fluid balance

laws, momentum balance laws and compatibility laws;

simultaneously, it also should embody the conduction law,

the capillary law, the fluid constitutive laws and the con-

stitutive laws of mechanics (Itasca 2005).

(a) Transport laws.

Wetting (such as water) and non-wetting (such as

gas) fluids transports are described by Darcy’s law:

Table 3 Fitting parameters of creep curves subjected to the first five levels loadings

Compaction coefficient Parameter name 1st level 2nd level 3rd level 4th level 5th level Average value

0.87 G1/kPa 2.39e6 3.22e6 2.35e6 2.56e6 2.72e6 2.629e6

b1/kPa s 2.76e10 2.77e10 3.10e10 3.18e10 3.76e10 2.951e10

G2/kPa 9.20e5 1.08e6 1.06e6 1.00e6 7.96e5 1.014e6

b2/kPa s 2.72e11 2.63e11 3.93e11 3.88e11 3.04e11 3.290e11

0.90 G1/kPa 3.43e6 4.18e6 3.45e6 3.35e6 3.85e6 3.603e6

b1/kPa s 3.97e10 4.54e10 4.40e10 4.53e10 6.47e10 4.360e10

G2/kPa 3.88e6 4.51e6 3.07e6 2.17e6 1.85e6 3.408e6

b2/kPa s 1.49e12 1.33e12 9.09e11 9.62e11 9.54e11 1.171e12

0.93 G1/kPa 4.21e6 4.88e6 4.72e6 5.13e6 5.25e6 4.732e6

b1/kPa s 3.13e10 2.72e10 4.74e10 5.81e10 7.55e10 4.102e10

G2/kPa 1.43e06 1.58e06 1.56e06 1.56e06 1.55e06 1.532e06

b2/kPa s 4.10e11 4.34e11 5.48e11 6.09e11 5.74e11 5.001e11

0.95 G1/kPa 5.00e06 5.74e06 4.90e06 4.79e06 5.15e06 5.106e06

b1/kPa s 8.27e10 8.33e10 9.28e10 7.96e10 1.09e11 8.461e10

G2/kPa 3.75e06 3.53e06 2.48e06 2.35e06 2.26e06 3.028e06

b2/kPa s 1.16e12 1.50e12 1.50e12 1.33e12 1.16e12 1.370e12


123

qwi ¼ �kijjwr

o

oxjðPw � qwgkxkÞ ð14Þ

qgi ¼ �kwij

lwlg

jgro

oxjðPg � qggkxkÞ ð15Þ

where qwi and qgi are the specific discharge vector of

wetting fluid and non-wetting fluid, respectively; kijis saturated mobility coefficient (which is a tensor);

jr is relative permeability for the fluid (which is a

function of saturation Sw, see, e.g. van Genuchten

1980); l is dynamic viscosity; P is pore pressure; qis fluid density and g is gravity. Note that the

mobility coefficient is defined as the ratio of intrinsic

permeability to dynamic viscosity (Itasca 2005).

(b) Relative permeability laws.

Relative permeabilities are related to saturation Swby empirical laws of the van Genuchten form (van

Genuchten 1980):

jwr ¼ Sbe ½1� ð1� S1=ae Þa�2 ð16Þ

jgr ¼ ð1� SeÞc½1� ð1� S1=ae Þ2a ð17Þ

In these laws, a, b and c are constant parameters; in

this paper, b and c equal 0.5 and Se is the effective

saturation. The effective saturation is defined as:

Se ¼Sw � Swr1� Swr

ð18Þ

where Swr is residual wetting fluid saturation (the

residual saturation, which remains in spite of high

capillary pressures, is referred to as ‘‘connate’’ in the

case of water).

(c) Capillary pressure law.

The capillary pressure law relates the difference in

fluid pore pressures to saturation:

PcðSwÞ ¼ ðPg � PwÞ ð19Þ

where Pc is capillary pressure, Pg is non-wetting fluid

pore pressure and Pw is wetting fluid pore pressure.

This empirical law is of the van Genuchten form

(1980) according to the Leverett scaling law (Niko-

laevskij 1990):

PcðSwÞ ¼ P0½S�1=ae � 1�1�a ð20Þ

where P0 is a constant parameter of the material and

Se is the effective saturation. Generally, P0 is larger

for finer material, and its dependency on material

properties may be assessed using the Leverett scal-

ing law (Nikolaevskij 1990). This law, derived using

dimensional analysis, has the form:

P0 ¼rffiffiffiffiffiffiffiffij=n

p ð21Þ

where r is surface tension, a property of the matrix,

j is intrinsic permeability, and n is porosity. The

‘‘alpha coefficient’’ is sometimes introduced in the

literature in place of P0. The relationship between

those two scaling parameters is.

P0 ¼qwga

ð22Þ

where qw is wetting fluid density and g is gravity.

(d) Saturation.

The two fluids completely fill the pore space, and we

have:

Sg þ Sw ¼ 1 ð23Þ

where Sg and Sw are non-wetting and wetting fluid

saturation respectively.

(e) Fluid balance laws.

For slightly compressible fluids, the balance rela-

tions are:

onwot

¼ � oqwioxi

þ qwv ð24Þ

ongot

¼ � oqgi

otþ qgv ð25Þ

where nw and ng are the variation of wetting and

non-wetting fluids content, respectively (variation of

fluid volume per unit volume of porous material); qwvand qgv are the volumetric source intensity wetting

and non-wetting fluid, respectively; other parameters

are the same as above.

(f) Fluid constitutive laws.

The constitutive laws for the fluids are:

swoPw

ot¼ Kw

n

onwot

� nosw

ot� sw

oeot

� �ð26Þ

sgoPg

ot¼ Kg

n

ongot

� nosg

ot� sg

oeot

� �ð27Þ

where Kw and Kg are the bulk moduli of the wetting

and non-wetting fluid, and e is volumetric strain,

other parameters are the same as above.

Finally, by substituting Eq. (24) in (26), Eq. (25) in

(27), and making some rearrangement of terms, the

following formulas can be obtained:

nsw

Kw

oPw

otþ osw

ot

� �¼ � oqwi

otþ sw

oeot

� �ð28Þ

nsg

Kg

oPg

otþ osg

ot

� �¼ � oq

gi

otþ sg

oeot

� �ð29Þ

(g) Balance of momentum.

The balance equation can be expressed as:


123

orijoxj

þ qgi ¼ qd _uidt

ðð30Þ

where q is bulk density, with expression

q ¼ qd þ nðswqw þ sgqgÞ, in which qg and qw are

respectively non-wetting and wetting fluid densities;

qd is the matrix dry density of soil; _ui are velocity

components; t is time; xj are components of coor-

dinate vector; gi are components of gravitational

acceleration and rij are components of the stress

tensor.

(h) Mechanical constitutive laws.

The incremental constitutive response for the porous

solid has the form:

Dr0ij ¼ Hðrij;Deij; jÞ ð31Þ

where Dr0ij are the changes of effective stress com-

ponents, Dr0ij ¼ Drij þ �D�Pdij, in which �D�P ¼swDPw þ sgDPg (Dangla 1999); H is the functional

form of the constitutive law, and in this paper a creep

model is adopted; j is a history parameter of stress

and Deij are the changes of strain components

(Schrefler and zhan 1993; Hutter et al. 1999; Dangla

1999).

(i) Compatibility equation.

According to the assumption of small strain and the

convention that the strain is positive in compres-

sion, the relation between strain rate and velocity

gradient is given as usual. It is listed here for

completeness:

_eij ¼ � 1

2

o _uioxj

þ o _ujoxi

� �ð32Þ

where _eij are strain-rate components.

Establishment of settlement theory for Red-layers

mudstone embankment

One-dimensional models of consolidation and creep in

saturated soil proposed by Chen (1958) and Taylor and

Merchant (1940) are a direct application of component

model theory, or in series or in parallel of the Terzaghi

elastic model, but is actually based on a new creep con-

stitutive model instead of Terzaghi elastic model (Fredlund

and Rahardjo 1993; Bishop 1959).

The unsaturated principle of the single stress state

variable has played important foundations for the transition

from unsaturated soil mechanics to saturated soil

mechanics. So, the same as with rheology and consolida-

tion theory of saturated soil, only the constitutive equations

of soil are modified to establish unsaturated rheology and

consolidation coupling theory.

From the ‘‘Creep constitutive model of crushed Red-

layers mudstone soils’’ section, the generalized Kelvin

model with two Kelvin bodies can describe well the

decelerating creep of crushed Red-layers mudstone soils.

So based on the assumption that the instantaneous

deformation of soil can be obtained according to the

Duncan–Chang nonlinear elastic model (1970), and the

deformations at constant pressure obey the Kelvin creep

equation, the creep constitutive model of the soil can be

established, as shown in Fig. 5. The equations for this are

as follows:

p0sij þ p1 _sij þ p2sij ¼ 2q0eij þ 2q1 _eij þ 2q2eijrm ¼ 3Kem

�ð33Þ

where sij, _sij and sij are respectively components of the

deviatoric stress tensor, the first deviatoric stress tensor

with respect to time and the second deviatoric stress tensor

with respect to time; eij, _eij, eij are respectively components

of the deviatoric strain tensor, the first deviatoric strain

tensor derivative with respect to time and the second

deviatoric strain tensor derivative with respect to time; rmis the spherical tensor of stress; em is the spherical tensor of

strain; G and K are shear and bulk modulus, respectively;

p0 ¼ 2GG1þ2GG2þ2G1G2

Gand p1 ¼ 2Gb1þ2Gb2þ2G1b2þ2G2b1

G; p2 ¼

2b1b2G

; q0 ¼ 4G1G2; q1 ¼ 4G2b1 þ 4G1b2; q3 ¼ 4b1b2. K

and G are functions of r3, which agree with the nonlinear

Duncan–Chang elastic model, and have the expressions

K ¼ KbPar3Pa

� �m, G ¼ 3EK

9K�E; E ¼ 1� Rf 1�sinusð Þ r1�r3ð Þ

2cs cosusþ2r3 sinus

h i2

KPa r3Pa

� �n, kb, k, M, N are fitting parameters; k1ðG1Þ,

g1ðb1Þ, k2ðG2Þ and g2ðb2Þ are elastic and viscous constants

of the first and second Kelvin sub-models related with

stress path, which can be obtained through experiments

according to specific conditions.

The Eq. (33) is solved by a finite difference scheme as

follows:

From Fig. 5, equations for the first Kelvin sub-model

are:

_u1 ¼Fd1

2gð34Þ

Fd1 ¼ �F � G1�u1 ð35Þ

( )F σ( )k G

1 1( )k G

1 1( )η β

the first Kelvin sub-model

0u 1u 2u

1dF 2dF2 2( )η β

2 2( )k G ( )F σthe second Kelvin sub-model

the nonlinear Duncan-Chang elastic model

Fig. 5 The theoretical model for unsaturated rheology and consol-

idation coupling


123

where _u1 is the velocity of displacement; F is the applied

force for the whole model; Fd1 is the applied force of

Newton viscous body Kelvin sub-model; k1 (G1) is elastic

constant; g1 (b1) is viscous constant; Dt is time-step; �F and

�u1 equal the mean values of F and u1 over the time-step.

Combining Eqs. (34) and (35) in finite-difference form

(the superscripts N and o denote new and old values,

respectively), we obtain:

uN1 ¼ uo1 þ FN þ Fo � k1 uN1 þ uo1� �� Dt

g1ð36Þ

Equation (34) is simplified as:

uN1 ¼ 1

A1

B1uo1 þ FN þ Fo

� � Dt2g1

�ð37Þ

where A1 ¼ 1þ G1Dt2g1

, B1 ¼ 1� G1Dt2g1

.

The same for the second Kelvin sub-model is:

uN2 ¼ 1

A2

B2uo2 þ FN þ Fo

� � Dt2g2

�ð38Þ

where u2 is displacement of the second Kelvin sub-model,

A2 ¼ 1þ G2Dt2g2

, B2 ¼ 1� G2Dt2g2

.

The equation for the Hooke sub-model is:

uN0 ¼ uo0 þFN � Fo

k0ð39Þ

where u0 is displacement for Hooke sub-model; k0 is elastic

constant for Hooke sub-model.

Finally, the first and the second Kelvin and Hooke dis-

placement increments are combined to calculate the

applied displacement increment:

Du ¼ uN0 � uO0 þ uN1 � uO1 þ uN2 � uO2 ð40Þ

By combining Eqs. (37–40), we obtain:

FN ¼ 1

XuN � uO� �

þ YFO � uO1 � B1

A1

� 1

� �

� uO2 � B2

A2

� 1

� �� ð41Þ

where X ¼ 1k0þ Dt

2A1g1þ Dt

2A2g2, Y ¼ 1

k0� Dt

2A1g1� Dt

2A2g2.

Equation (39) is rewritten as the deviatoric stress versus

strain tensor form, and we obtain:

sNij ¼2

XeNij � eoij

�þ Y

2soij � eoij;1 �

B1

A1

� 1

� �� eoij;2 �

B2

A2

� 1

� � �

rNkk ¼ rokk þ 3K _ekkDt

9=;

ð42Þ

where sij are components of the deviatoric stress tensor,

sij ¼ rij � rmdij, rm is spherical tensor of stress, rm ¼ rkk3;

rkk are principal stress components; eij;1 are components of

the deviatoric strain tensor for the first Kelvin sub-model;

eij;2 are components of the deviatoric strain tensor for the

second Kelvin sub-model; eij ¼ eij � emdij, em is spherical

tensor of strain, em ¼ ekk3, ekk are principal strain compo-

nents, _ekk are principal strain-rate components.

Verification of the settlement theoryof unsaturated embankments

The settlement of an embankment is an unsaturated cou-

pling consolidation process based on the rheology, which is

a long process. The consolidation time of a centrifugal

model is 1/m2 of the prototype model (m, scale ratio of the

model), thus, the consolidation time of the model can be

shortened with centrifugal tests (Taylor 1995). In this

section, without considering the influence of embankment

filler properties on embankment shortcomings, the defor-

mation characteristics under different compaction coeffi-

cient of crushed Red-layers mudstone soils were

investigated with centrifugal model tests, and then the

parameters of embankment fill can be determined. In

addition, based on centrifuge tests and site monitoring data,

unsaturated embankment settlement theory of the Red-

layers was validated through corresponding numerical

simulation of centrifugal loading process.

Analysis of the embankment settlement of crushedRed-layers mudstone soils with centrifugal modeltests and verification of the theory of settlementcalculation

Embankment centrifugal model design

In the centrifugal mdel test, the length, width and height of

the model box L 9 W 9 H are 600 mm 9 400 mm 9

400 mm. According to the requirements of the ‘‘Interim

Provisions’’ for the new railway design speed of 200 km/

h, the compaction coefficients of the centrifugal model

tests are 0.87, 0.90, 0.93 and 0.95, with the corresponding

number as T-1, T-2, T-3 and T-4.

The height of the prototype embankment is 15 m, and

the slope of the embankment consists of two different

level slopes, of which the two slopes are 1:1.5 and

1:1.75, respectively, as shown in Fig. 6. The scale rate of

the model is 1:100, so the height of the centrifugal

model is 15 cm and the section shape is the same as the

prototype.

Operation modes of the four test groups were the same:

the centrifugal acceleration of 0–100 g is divided into five

steps, namely the centrifugal machine operates for 5 min at

20, 40, 60, 80 g, and steady operation for 120 min at

100 g. Data is recorded every 5 min record (Fig. 7).


123

Analysis of the results of centrifugal model tests

(a) The relationship between settlement during con-

struction and height of embankment.

Figure 8 shows the settlement change of the

embankment top surface versus the centrifugal

acceleration variation for the four groups of tests,

which can also be interpreted as the height change

curve of the embankment.

From Fig. 8, the settlement tendency of T-4 group is

slightly different from those of T-1, T-2 and T-3.

When the embankment height is less than 6 m

(centrifugal acceleration less than 40 g), the settle-

ment tendency of the T-4 group is almost the same as

for T-1, T-2 and T-3; however, when the embank-

ment height is higher than 6 m (centrifugal accel-

eration greater than 40 g), the settlements of T-4

tend to change slowly.

The relationship between the settlement and

embankment height of T-1 to T-3 can be described

by the following formula:

s ¼ ah2 þ bh ð43Þ

where s is settlement, h is embankment height, a and

b are experimental parameters. Values of parameters

a are 1.7E-4, 1.7E-4 and 1.4E-4; the values of

parameters b are 1.92e-3, 1.63e-3 and 1.61e-3 for

T-1, T-2 and T-3, respectively.

From Fig. 8, the settlements increase with the

embankment height increasing during construction.

At 100 g (fill height of 15 m), the settlement values

are 0.066, 0.063, 0.057 and 0.031 m, corresponding

to compaction coefficients of 0.87, 0.90, 0.93 and

0.95, respectively. So, the ratios of settlement to

height during construction are 4.4, 4.2, 3.8 and

2.06 %, respectively.

150

80

40

Embankment

The rigid foundation

40

Centrifugal model tank boundary

Eddy current sensor

1:1.5

1:1.75

Fig. 6 Embankment centrifugal

model design

Fig. 7 Embankment centrifugal model

0 2 4 6 8 10 12 14 160.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

height of embankment/m

defo

rmat

ion/

m

Fig. 8 Relationship of deformation of the embankment versus height


123

(b) The relationship between the total settlement of

embankment and time.

In the centrifugal model tests, the settlement of the

embankment is composed of two parts with the

increment of centrifugal acceleration: one part is the

increment with the height of embankment, which

leads to lateral displacement of the slope, and is also

associated with strength and rheology; the other part

is compaction deformation, which is associated with

the compaction coefficient. When this coefficient

reaches a certain level, the compaction deformation

tends to remain steady.

Relations between the total settlement of embank-

ment and time are shown in Fig. 9 during steady

operation at 100 g (the equivalent of a fill height to

15 m step by step). Total settlements increase with

increasing time and total settlement rates gradually

slow down. Total settlements of a high embankment

of 15 m in 28 months after construction are 0.099,

0.084, 0.074 and 0.053 m. The ratios between the

total settlement and embankment height are 6.6, 5.6,

4.9 and 3.5 %, corresponding to compaction coef-

ficients of 0.87, 0.90, 0.93 and 0.95, respectively.

(c) The relationship between the settlement of embank-

ment after construction and time.

During steady state of the operation at 100 g,

settlement of embankment after construction versus

time is shown in Fig. 10. Settlements after construc-

tion increase with time and gradually become steady.

This indicates that the higher the compaction coeffi-

cient is, the smaller the settlement will be. Post-

construction settlements 28 months after construction

are 0.033, 0.022, 0.021 and 0.021 m and ratios

between post-construction settlement and embank-

ment height are 2.2, 1.5, 1.4 and 1.4 % corresponding

to 0.87, 0.90, 0.93 and 0.95, respectively.

(d) Embankment settlement calculation.The relationship

of the embankment settlement versus time can be

described by the following formula:

S1 ¼t

at þ bð44Þ

where S1 is settlement after construction (units: m);

t is time (units: months); a and b are fitting param-

eters. Fitting parameters for the four group tests are

shown in Table 7.

We obtain the value when time tends to infinite:

S1 ¼ lim S1 ¼1

að45Þ

where S1 is the final settlement, which is the

reciprocal of the parameter a.The final settlements are shown in Table 7 with the

Eq. (45). The final settlements are 0.0361, 0.0287,

0.0275 and 0.0251 m, and the ratios between final

settlement and embankment height are 2.4, 1.913,

1.833 and 1.673 % corresponding to compaction

coefficients of 0.87, 0.90, 0.93 and 0.95, respec-

tively.

With respect to the empirical formula of the final

settlement of the embankment body after construc-

tion, the following form is adopted in Germany and

Japan (Wang 2000):

S1 ¼ h2

3000ð46Þ

where S1 is final settlement after construction

(units: m); h is height of embankment (units: m);

the final settlement of the embankment with a

height of 15 m calculated by Eq. (44) is 0.075 m,

which is larger than the ones the centrifugal model

tested.

0 5 10 15 20 25 300.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

tota

l set

tlem

ent/m

time t/month

T-1 T-2 T-3 T-4

Fig. 9 Total settlement of embankment versus time

0 5 10 15 20 25 300.005

0.010

0.015

0.020

0.025

0.030

0.035

settl

emen

t afte

r con

stru

ctio

n/m

time td/month

T-1 T-2 T-3 T-4

Fig. 10 Settlement of embankment after construction versus time


123

(e) Relationship between final settlement and com-

paction coefficient.

According to the final settlements as shown in

Table 7, one can obtain the relation between final

settlement S1 and compaction coefficient as shown

in Fig. 11. When the compaction coefficient is larger

than 0.90, the settlements only change slowly, and

the final settlements are 2.75 and 2.51 cm corre-

sponding to compaction coefficients of 0.93 and

0.95. According to the necessary requirements of the

‘‘Interim Provisions’’ for the new railway design

speed of 200 km/h, the subgrade settlement after

construction is not permitted to be more than 15 cm,

settlement rate not to be more than 4 cm per year,

and compaction coefficient should not be less than

0.93 considering the effect of construction factors

and construction season.

Centrifugal test verification for the settlement

calculation theory

The loading in the centrifugal test is an unsaturated rhe-

ology and consolidation coupling process based on the

rheological theory. Thus, to verify the rationality of

unsaturated embankment settlement theory, the loading

process of the centrifugal model test is simulated by finite

element method. The acceleration changes are shown in

Fig. 12. The model sizes are set as same as those of the

centrifugal model, and the finite element model is shown as

Fig. 13. The unsaturated soil parameters, the rheological

parameters, the average values of the first five levels

loading and other mechanical parameters of different

compaction coefficient are shown in Tables 3, 4 and 5.

Relations between central deformations of centrifugal

model test and numerical simulation at the top surface of

the embankment for different compaction coefficients and

filling heights, as well as the settlements after construction

versus time are shown in Figs. 14, 15, 16 and 17. The

numerical results are basically in agreement with the data

from centrifugal model tests. However due to seepage in

unsaturated fluid–solid calculation, which causes pore

water transfers to lower parts of the embankment, the

gravitational stress and effective stress will change, and the

results based on the rheological theory tend to be larger

than those from unsaturated rheology and consolidation

coupling theory. With the Eq. (43), the data of deforma-

tions and filling height are fitted, and Fitting parameters are

shown in Table 6. Due to large measurement error data

during the experiment, fitting parameters from compaction

coefficient of 0.95 vary greatly, and the remaining results

are very similar, so Eq. (43) can well describe the rela-

tionship of embankment settlement and filling height dur-

ing construction. When the filling height reaches 15 m

(acceleration reaches 100 g), ratios of deformation for

centrifugal model test and filling height are basically the

same as the ones of the numerical calculation. Using

Eq. (44), relations between the settlement after

0.86 0.88 0.90 0.92 0.94 0.96 0.980.024

0.026

0.028

0.030

0.032

0.034

0.036

0.038S ∞

/m

compaction coefficient

Fig. 11 Relationship curves of the final settlement versus com-

paction coefficient

0.0 2.0E3 4.0E3 6.0E3 8.0E3 1.0E4 1.2E4 1.4E40g

20g

40g

60g

80g

100g

acce

lera

tion/

g

time/s

Fig. 12 Acceleration versus time

x/m

Y/m

0 0.1 0.2 0.3 0.4 0.5

0

0.05

0.1

0.15

Fig. 13 The finite element model


123

Table 4 Unsaturated soil parameters for crushed Red-layers mudstone soils of different compaction coefficients

Compaction

coefficient

qd/gcm-3

K

/kPa

G

/kPa

Swr/%

n Cs

/kPa

/s

/�kws/m-1s

Van Genuchten model

p0/kPa a

0.87 1.614 4.98e4 2.85e4 3.33 0.42 10.88 24.82 6.79e-7 42.82 0.356

0.90 1.670 5.53e4 3.48e4 4.04 0.40 13.38 27.03 2.81e-7 47.79 0.319

0.93 1.725 6.09e4 4.19e4 4.60 0.38 16.09 29.83 1.33e-7 53.53 0.288

0.95 1.762 8.84e4 6.63e4 4.88 0.37 18.15 30.29 5.09e-8 59.38 0.241

Table 5 Mechanical

parameters for crushed Red-

layers mudstone soils of

different compaction

coefficients

Compaction

coefficient

qd/gcm-3

Swr/%

Rf k N kb M

0.87 1.614 5.1 0.85 354.5 0.2476 78.92 0.1219

0.90 1.670 5.5 0.85 369.65 0.1746 81.78 0.2431

0.93 1.725 6.1 0.85 502.94 0.1885 99.25 0.2927

0.95 1.762 6.4 0.85 729.41 0.3144 124.16 0.2932

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

defo

rmat

ion/

m

filling height/m

The numerical calculation (based on the rheology and consolidation theory )

The numerical calculation (based on the rheological theory) the results of centrifugal model tests

(1) Relation curves between deformation and filling height

-2 0 2 4 6 8 10 12 14 16

0 5 10 15 20 25 300.00

0.01

0.02

0.03

0.04

settl

emen

t/m

time/month

The numerical calculation (based on the rheology and consolidation coupling theory)


(2) Settlement after construction vs. time

Fig. 14 The result of centrifugal model tests and numerical calcu-

lation for compaction coefficients of 0.87

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

defo

rmat

ion/

m

filling height/m



(1) Deformation vs. filling height

-2 0 2 4 6 8 10 12 14 16

0 5 10 15 20 25 30

0.00

0.01

0.02

0.03

0.04

settl

emen

t/m

time/month

The numerical calculation (based on the rheology and consolidation coupling theory )


(2) Settlement after construction vs. time

Fig. 15 The results of centrifugal model tests and numerical

calculation for compaction coefficients of 0.90


123

construction for different compaction coefficients and

times are fitted. Fitting parameters are shown in Table 7.

Fitting parameters show a large difference between the

centrifugal model tests and numerical simulations. Final

settlement for the compaction coefficient of 0.87 from

centrifugal model test is 0.0361 m; however, both numer-

ical results are 0.107 and 0.104 m, which may be due to

human factors affecting filling compaction of the

embankment model during the filling process.

Verification of the theory of settlement calculation

with engineering examples

The railway from Suining to Chongqing is the first express

railway of 200 km/h constructed in southwest China. The

railway passes through a Red-layers mudstone area in the

Suining section. Because of the lack of high-quality fill, the

limitations of project investment, transmission and

construction periods, Red-layers mudstone (belonging to

filler of Class C and easily collapsing) of poor engineering

properties has to be adopted as the filler for the embank-

ment on the Suining section. In addition, the Red-layers

mudstone has to be crushed into soil, with a maximum

particle size of less than 5 cm, and the filling compaction

coefficient is controlled to be about 0.95 (the largest dry

density is 1.855 g/cm3), and the moisture content to be

11.5 %. According to the strict requirements of the ‘‘In-

terim Provisions’’ for the new railway design speed of

200 km/h,the embankment settlement after construction

should be less than 15 cm, and the settlement rate should

not be more than 4 cm per year. To verify whether the

settlement can meet the requirements of the passenger

dedicated line if the compaction coefficient for the Red-

layers mudstone filler reaches 0.95, site monitoring works

have been carried out at a construction site of Suining

section.

0.00

0.01

0.02

0.03

0.04

0.05

filling height/m

defo

rmat

ion/

m The numerical calculation (based on the rheology and

consolidation coupling theory) The numerical calculation (based on the rheological theory) the results of centrifugal model tests

(1) Deformation vs. filling height

-2 0 2 4 6 8 10 12 14 16

0 5 10 15 20 25 30

0.00

0.01

0.02

0.03

0.04 The numerical calculation (based on the rheology and consolidation coupling theory )


settl

emen

t/m

time/month(2) Settlement after construction vs. time

Fig. 16 The results of centrifugal model tests and numerical

calculations for compaction coefficient of 0.93

0.00

0.01

0.02

0.03

0.04 The numerical calculation (based on the rheology and

consolidation coupling theory) The numerical calculation (based on the rheological theory) the results of centrifugal model tests

defo

rmat

on/m

filling height/m(1) Relation curves between deformation and filling height

-2 0 2 4 6 8 10 12 14 16

0 5 10 15 20 25 30

0.00

0.01

0.02



settl

emen

t/m

time/month(2) Settlement after construction vs. time

Fig. 17 The result of centrifugal model tests and numerical calcu-

lation for compaction coefficients of 0.95


123

The compacting process of the embankment is as

follows:

1. Sand cushion was compacted and the embankment

began to be filled; settlement pipes were installed at

the bottom of embankment body (Fig. 18) on 5th

Feb. 2004.

2. Filling height of embankment reached 2.5 m on 14th

Feb. 2004.

3. Filling height of embankment reached 7 m on 27th

Feb. 2004.

4. Filling height reached 7.3 m on 4th March, 2004.

5. Filling height reached 8.5 m on 9th March, 2004.

6. Filling height reached 9.0 m on 2nd April, 2004.

7. Filling height reached 9.5 m on 9th April, 2004.

8. Filling height reached 10 m on 6th May, 2004; at

this time, the construction of the embankment body

Table 6 Settlement results for

embankments with different

compaction coefficients during

construction

Name Result of centrifugal

model test

Numerical

calculation�Numerical

calculation`

0.87

a/10-4 1.70 1.50 1.60

b/10-3 1.92 1.85 2.06

s/H/(%) 4.40 4.03 4.45

0.90

a/10-4 1.70 1.20 1.40

b/10-3 1.63 1.73 1.80

s/H/(%) 4.20 3.52 3.84

0.93

a/10-4 1.40 1.10 1.20

b/10-3 1.61 1.68 1.81

s/H/(%) 3.53 3.23 3.52

0.95

a/10-4 -0.8 0.8 0.9

b/10-3 3.33 1.06 1.29

s/H/(%) 2.07 2.16 2.61

s/H is the ratio of the central settlement of the embankment top surface at the acceleration of 100 g versus

filling height; the numerical calculation� is numerical results with unsaturated rheology and consolidation

coupling theory; The numerical calculation` is numerical results with rheological theory

Table 7 Settlement results for

embankments with different

compaction coefficients after

construction

Name Result of centrifugal

model test

Numerical


calculation`

0.87

a 27.66 9.37 9.62

b 110.64 473.46 438.39

S?/m 0.0361 0.107 0.104

0.90

a 34.85 13.17 11.89

b 239.65 626.91 597.73

S?/m 0.0287 0.076 0.084

0.93

a 36.4 15.91 14.91

b 288.36 607.57 569.75

S?/m 0.0275 0.063 0.067

0.95

a 39.85 17.89 15.69

b 325.0 966.23 874.41

S?/m 0.0251 0.056 0.064


123

was completed and the settlement pipes were

installed at the top of the embankment body

(Fig. 18);

9. It was 40 days after construction on 15th June,

2004 (filling height of embankment was 10 m with

graded broken stone of 0.6 m, and the filling height

was converted to be 10.8 m according to the

density);

10. It was 59 days after construction on 4th July, 2004

(rail had been laid and filling height was converted to

be 11.1 m according to the requirements of code for

design on subgrade of railway, China).

Relationship between filling height and constructing

time (Fig. 19) was fitted with a hyperbola:

H ¼ t=ð0:0763t þ 1:765Þ ð47Þ

The process of loading on filling of embankment body

was assumed to follow Eq. (47), and the settlement after

filling to 10 m high was as the settlement after construc-

tion. This was regarded as the starting time of settlement

monitoring and numerical simulation after construction.

Numerical simulation was carried out of the filling of the

embankment, calculation parameters are shown in

Tables 3, 4 and 5.

Relationship between settlement after construction and

time is shown in Fig. 20. Equation 44 was adopted to fit

curves. Fitting results are shown in Table 8. The site

monitoring results agree well with numerical simulating

results. Post-construction settlement is about 0.042 m, the

ratio of which to the height of embankment body is about

4.2 %.

(1) Bottom of embankment (2) Top of embankment

Fig. 18 Layout of settlement pipes

0 20 40 60 80 100 120 140 1600

2

4

6

8

10

12

fill h

eigh

t H/m

constructive time t/d

post-construction

2 0.980.0763 1.765

tH Rt

= =+

Fig. 19 Filling heights versus construction time

0 5 10 15 20 25 30 35 40 45 500.00

0.01

0.02

0.03

0.04

0.05

settl

emen

t afte

r con

stru

ctio

n/m

time/m

The measured settlement of embankment The numerical calculation (based on the rheology and

consolidation coupling theoy) The numerical calculation (based on the rheological theory)

Fig. 20 Embankment settlements after construction

Table 8 Fitting results parameters a and b

Name Site

monitoring

Numerical


calculation`

a 23.273 24.622 23.59

b 58.613 95.945 95.945

S?/m 0.043 0.041 0.042


123

Conclusions

In this paper, based on the uniaxial compression creep tests

of crushed Red-layers mudstone soils under different

compaction coefficients, the embankment settlement cal-

culating theory of Red-layers unsaturated soil with the

principle of the single stress state variable in unsaturated

soil is proposed. This indicates that the instantaneous

deformation of soil at any moment can be described by the

Duncan nonlinear stress–strain relationship, and moreover

the deformation under constant stress conforms to the creep

equations with two Kelvin bodies, considering coupling-

effect of the unsaturated rheology and consolidation. In

addition, physical centrifugal tests and corresponding

numerical simulation of centrifugal loading process were

adopted to study the relationships of the settlement of the

embankment body during construction versus filling height

and the settlement after construction versus time. Based on

these studies, through the comparative analysis of the site

monitoring data and numerical simulation results of the

actual filling process at a constructing site at the Suining

section of the Suining-Chongqing Railway, the settlement

calculating theory of unsaturated Red-layers filling

embankment is adopted to investigate embankment settle-

ment, which reflects well the deformation laws of settle-

ment during and after the construction of the embankment,

and moreover, reveals the good suitability of the proposed

settlement calculating theory.

Acknowledgments The authors first acknowledge Professor Jaak J.

K. Daemen for his help on editing and modifying the manuscript. The

authors also acknowledge the financial support from the International

Cooperation Project of Sichuan (No. 2014HH007), the visiting

scholar funded project of the State Key Laboratory of Coal Mine

Disaster Dynamics and Control(Chongqing University) (No.

2011DA105287—FW201401) and the National Natural Science

Foundation of China (No. 41472285, No. 51304256; No. 51404241).

References

Adachi T, Okano M (1974) A constitutive equation for normally

consolidated clay. Soils Found 14(4):55–73

Aitchison GD (1961) Relationship of Moisture and Effective Stress

Functions in Unsaturated Soils. In: Proceedings of the confer-

ence on pore pressure and suction in soils, Butterworths,

London, England, pp 47–52

Aitchison GD (1965) Soil properties, shear strength, and consolida-

tion. In: Proceedings of the 6th international conference on soil

mechanics and foundation engineering, Montreal, Canada, 3,

pp 318–321

Aitchison GD (1973) The quantitative description of the stress-

deformation behavior of expansive soils-preface to set of papers.

In: Proceeding of the 3rd international conference expansive

soils, Haifa, Israel, 2, pp 79–82

Barden L (1965) Consolidation of compacted and unsaturated clays.

Geotechnique 15(3):267–286

Barden L (1974) Consolidation of clays compacted dry and wet of

optimum water content. Geotechnique 24(4):605–625

Berre T, Iversen K (1972) Oedometer tests with different specimen

heights on a clay exhibiting large secondary compression.

Geotechnique 22:53–70

Biot MA (1941) General theory of three-dimensional consolidation.

J Appl Phys 12(2):155–164

Bishop AW (1959) The principle of effective stress. Teknisk Ukeblad

106(39):859–863

Bishop AW, Blight GE (1963) Some aspects of effective stress in

saturated and partly saturated soils. Geotechnique 13(3):177–197

Blight GE (1961) Strength and consolidation characteristics of

compacted soils. PhD Thesis, University of London, England

Borja RI, Koliji A (2009) On the effective stress in unsaturated porous

continua with double porosity. J Mech Phys Solid

57(8):1182–1193

Brackley IJA (1971) Partial collapse in unsaturated expansive clay.

In: Proceedings of 5th regional conference on soil mechanics and

foundation engineering. [s. n.], South Africa, pp 23–30

Burland JB (1965) Some aspects of the mechanical behaviour of

partially saturated soils. Moisture equilibria and moisture

changes beneath covered areas. Butterworths, Sydney, Australia,

pp 270–278

Chen ZJ (1958) One-dimensional problems of consolidation and

secondary time effects. Chin Civil Eng J 5(1):1–10 (in Chinesewith English abstract)

Chen XP, Bai SW (2001) The numerical analysis of visco-elastic-

plastic Biot’s consolidation for soft clay foundation. Chin J

Geotech Eng 23(4):481–484 (in Chinese with English abstract)Chen XP, Bai SW (2003) Research on creep-consoildation charac-

teristics and calculating model of soft soil. Chin J Rock Mech

Eng 22(5):728–734 (in Chinese with English abstract)Chen ZH, Xie DY, Wang YS (1994) Effective stress in unsaturated

soil. Chin J Geotech Eng 3:62–69 (in Chinese with Englishabstract)

Chen XP, Zhu HH, Zhou QJ (2006) Study on modified generalized

kelvin creep consolidation model. Chin J Rock Mech Eng

25(2):3428–3434 (in Chinese with English abstract)Code for Design of High Speed Railway (2009) TB10621-2009. (in

Chinese)Coleman JD (1962) Stress strain relations for partly saturated soil.

Geotechnique 12(4):348–350

Conte E (2004) Consolidation analysis for unsaturated soils. Can

Geotech J 41:599–612

Coussy O (1995) Mechanics of Porous Continua. Wiley, New York

Dakshanamurthy V, Fredlund DG (1980) Moisture and air flow in an

unsaturated soil. In: Proceedings of the fourth international

conference on expansive soils, 1, pp 514–532

Dakshanamurthy V, Fredlund, DG, Rahardjo H (1984) Coupled three-

dimensional consolidation theory of unsaturated porous media.

The 5th international conference on expansive soils. Adelaide,

South Australia, pp 99–103

Dangla P (1999) Approches Energetique Et Numerique Des Milieux

Poreux Non Satures. Memoire D’Habilitation A Diriger Des

Recherches. Ecole Nationale Des Ponts Et Chaussees. Ecole

Normale Superieure De Cachan. Universite De Marne-La-Valee

De Boer R, Ehlers W (1990) The development of the concept of

effective stresses. Acta Mech 83:77–92

De Jong GDJ (1968) Consolidation models consisting of an assembly

of viscous elements or a cavity channel network. Geotechnique

18(2):195–228

Desai CS, Sane S (2007) Unified constitutive models: simplified for

practical applications. Constitutive modelling development,

implementation, evaluation, and application, Hong Kong,

pp 39–64


123

Desai CS, Zhang D (1987) Viscoplastic model for geological

materials with generalized flow rule. Int J Num Analyt Methods

Geomech 11(6):603–620

Desai CS, Samtani NC, Vulliet L (1995) Constitutive modeling and

analysis of creeping slopes. J Geotech Eng Div ASCE

121(1):43–56

Duncan JM, Chang CY (1970) Nonlinear analysis of stress and strain

in soils. J Soil Mech Found Div, ASCE 96:1629–1653

Fatahi B, Le TM, Le MQ, Khabbaz H (2013) Soil creep effects on

ground lateral deformation and pore water pressure under

embankments. Geomech Geoeng 8(2):107–124

FEDA J (1992) Creep of soils and related phenomena. North-Holland,

Amsterdam, The Netherlands

Ferry JD (1980) Vescoelastic properties of polymer, 3rd edn. Wiley,

New York

Folque J (1961) Rheology properties of compacted unsaturated soils.

In: Proceedings of 5th international conference on soil mechan-

ics and foundation engineering. [s.n.], Paris, 1, pp 113–116

Fredlund DG, Hasan JU (1979) One-dimensional consolidation

theory: unsaturated soils. Can Geotech J 16(3):521–531

Fredlund DG, Morgenstern NR (1977) Stress state variables for

unsaturated soils. J Geotech Eng Div ASCE 103:447–466

Fredlund DG, Rahardjo H (1993) Soil mechanics for unsaturated

soils. Wiley, New York

Gallipoli D, Gens A, Sharma R, Vaunat J (2003) An elasto–plastic

model for unsaturated soil incorporating the effects of suction

and degree of saturation on mechanical behavior. Geotechnique

53(1):123–135

Gray WG, Schrefler BA (2001) Thermodynamic approach to effective

stress in partially saturated porous media. Eur J Mech A Solid

20(4):521–538

Gray WG, Schrefler BA (2007) Analysis of the solid phase stress

tensor in multiphase porous media. Int J Num Analyt Methods

Geomech 31(4):541–581

Gudehus G (1995) A comprehensive concept for non-saturated

granular bodies. In: Alonso EE, Delage P (eds) Unsaturated

soils. In: Proceedings of the 1st international conference on

unsaturated soils, Paris, France, 2, pp 725–737, Rotterdam:

Balkema

Ho L, Fatahi B (2015) Analytical solution for the two-dimensional

plane strain consolidation of an unsaturated soil stratum

subjected to time-dependent loading. Comput Geotech 67:1–16

Ho L, Fatahi B, Khabbaz H (2014) Analytical solution for one-

dimensional consolidation of unsaturated soils using eigenfunc-

tion expansion method. Int J Numer Anal Methods

38(10):1058–1077

Hutter K, Laloui L, Vulliet L (1999) Thermodynamically Based

Mixture models of saturated and unsaturated soils. Mech Cohes

Frict Mat 4(4):295–338

Itasca Consulting Group, Inc (2005) Fast lagrangian analysis of

continua in two-dimensions, version 5.0 user’s manual. Itasca

Consulting Group, Inc

Jennings JE (1961) A revised effective stress law for use in the

prediction of the behavior of unsaturated soils. In: Proceedings

of the conference on pore pressure and suction in soils,

Butterworths, London, England, pp 26–30

Jennings JE, Burland JB (1962) Limitations to the use of effective

stresses in partly saturated soils. Geotechnique 12(2):125–144

Kato S, Matsuoka H, Sun DA (1995) A constitutive model for

unsaturated soil based on extended SMP. In: Alonso EE, Delage

P (eds) Unsaturated soils. Proceedings of the 1st international

conference on unsaturated soils, Paris, France, 2, pp 739–744,

Rotterdam: Balkema

Kavazanjian E, mitchell JK (1977) A general stress-strain-time

formulation for soils. In; Proceedings of 9th international

conference on soil mechanics and foundation engineering,

Japanese Society of Soil Mechanics and Foundation Engineer-

ing, Tokyo, pp 113–120

Khalili N, Valliappan S (1996) Unified theory of flow and deforma-

tion in double porous media. Eur J Mech A-Solid 15(2):321–336

Khalili N, Geiser F, Blight GE (2004) Effective stress in unsaturated

soils, a review with new evidence. Int J Geomech ASCE

4(2):115–126

Khalili N, Witt R, Laloui L, Vulliet L, Koliji A (2005) Effective stress

in double porous media with two immiscible fluids. Geophys Res

Lett 32(15):L15309

Kohgo Y, Nakano M, Miyazaki T (1993) Theoretical aspects of

constitutive modelling for unsaturated soils. Soils Found

33(4):49–63

Kuhn MR, Mitchell JK (1993) New perspective on soil creep.

J Geotech Eng Div ASCE 119(3):507–524

Laloui L, Klubertanz G, Vulliet L (2003) Solid-liquid-air coupling in

multiphase porous media. Int J Num Analyt Methods Geomech

27(3):183–206

Lambe TW (1960) A mechanistic picture of shear strengtgh in clay. In:

Proceeding of ASCE research conference on shear strength of

cohesive soils, University of Colorado, Boulder, Colo, pp 555–580

Lambe TW, Whitman RV (1979) Soil mechanics. Wiley, New York

Le TM, Fatahi B, Khabbaz H (2015) Numerical optimisation to obtain

elastic visco-plastic model parameters for soft clay. Int J Plasti

65:1–21

Li GX (2006) Advanced soil mechanics. Tsinghua University Press,

Beijing (in Chinese)Liingaard M, Augustesen A, Lade PV (2004) Characterization of

models for time-dependent behavior of soils. Int J Geomech

4(3):157–177

Liu FY (1999) Discussion on developments of unsaturated soil

mechanics fundamental test equipment and new effective stress

principle. PhD Thesis, Xian University of Technology, Xi’an,

China (in Chinese with English abstract)Liu JX, Hu QJ, Qiu EX et al (2013) The engineering characteristics

and embankment stability of Red-layers in Southwest China.

Science Press, Beijing (in Chinese)Lloret A, Alonso EE (1980) Consolidation of unsaturated soils

including swelling and collapse behaviour. Geotechnique

30:449–477

Loret B, Khalili N (2000) A three phase model for unsaturated soils.

Int J Num Analyt Methods Geomech 24:893–927

Matyas EL, Radhakrishna HS (1968) Volume change characteristics

of partially saturated soils. Geotechnique 18:432–448

Mesri G, Godlewski PM (1977) Time and stress compressibility

interrelationship. J Geotech Eng Div ASCE 103(5):417–430

Nahazanan H, Clarke SD, Asadi A, Yusoff ZM, Huat BK (2013)

Effect of inundation on shear strength characteristics of

mudstone backfill. Eng Geol 158:48–56

Nikolaevskij VN (1990) Mechanics of porous and fractured media.

In: Hsieh RKT (ed) Series in theoretical and applied mechanics,

vol 8. World Scientific Pub Co, Singapore

Prager W (1949) Recent developments in the mathematical theory of

plasticity. J Appl Phys 20(3):235–241

Qin AF, Chen GJ, Tan YW, Sun DA (2008) Analytical solution to

one-dimensional consolidation in unsaturated soils. Appl Math

Mech 29:1329–1340

Qin AF, Sun DA, Yang LP, Weng YF (2010a) A semi-analytical

solution to consolidationof unsaturated soils with the free

drainage well. Comput Geotech 37:867–875

Qin AF, Sun DA, Tan YW (2010b) Analytical solution to one-

dimensional consolidation in unsaturated soils under loading

varying exponentially with time. Comput Geotech 37:233–238

Rice JR, Cleary MP (1976) Some basic stress diffusion solutions of

fluid saturated elastic porous media with compressible con-

stituents. Rev Geophys 14(2):227–241


123

Richards BG (1966) The significance of moisture flow and equilibrian

unsaturated soils in relation to the design of engineering

structures built on shallow foundations in Australia. Symposium

on permeability and capillarity, American Society for Testing

Materials, Atlantic City, NJ

Satake M (1989) Mechanics of granular materials. In: Proceedings of

12th International conference on soil mechanics and foundation

engineering, Taylor and Francis, Rio De Janeiro, pp 62–79

Schrefler BA, Zhan ZX (1993) A fully coupled model for water flow

and airflow in deformable porous media. Water Resour Res

1(29):155–167

Scott RF (1963) Principles of soil mechanics. Addison-Wesley

Publishing Company, Inc., Reading

Shan ZD, Ling DS, Ding HJ (2012) Exact solutions for one-

dimensional consolidationof single-layer unsaturated soil. Int J

Numer Anal Methods Geomech 36:708–722

Shen ZJ (1996) Generalized suction and unified deformation theory

for unsaturated soils. Chin J Geotech Eng 18(2):1–9 (in Chinesewith English abstract)

Shi B, Wang BJ, Ning WW (1997) Micromechanical model on creep

of anisotropic clay. Chin J Geotech Eng 19(3):7–13 (in Chinesewith English abstract)

Singh A, Mitchell JK (1968) General stress-strain-time function for

soils. J Soil Mech Found Engrg Div ASCE 94(SM1):21–46

Sparks ADW (1963) Various aspects of soil moisture. In: Proceeding

of the 3rd regional conference for Africa on soil mechanics and

foundation engineering, Salisbury, Rhodesia, pp 1211–214

Sun J (1999) Rheological properties of rock and soil materials and its

engineering application. China Building Industry Press, Beijing

(in Chinese)Tang YF, Feng ZL, Lin H, Chen X (2000) Consolidation nonlinear

rheology coupling analysis on soft soil engineering. Build Struct

30(11):47–50 (in Chinese with English abstract)Tang YQ, Ren XW, Chen B, Song SP, Wang JX, Yang P (2012)

Study on land subsidence under different plot ratios through

centrifuge model test in soft-soil territory. Environ Earth Sci

66:1809–1816

Tavenas F, Leroueil S, Rochelle PL, Roy M (1978) Creep behavior of

an undisturbed lightly overconsolidated clay. Can Geotech J

15(3):402–423

Taylor DW (1942) Research on consolidation of clays. Department of

Civil Engineering, MIT, Massachusetts

Taylor RN (1995) Geotechnical centrifuge technology. Blackie

Academic & Professional, Glasgow

Taylor DW, Merchant W (1940) Theory of clay consolidation

accounting for secondary compression. J Math Phys 19:167–185

Tian WM, Szilva AJ, Veyera GE, Sadd MH (1994) Drained creep of

undisturbed cohesive marine sediments. Can Geotech J

31(6):841–855

Van Genuchten M (1980) A closed form equation for predicting the

hydraulic conductivity of unsaturated soils. Int Soil Sc Soc Am J

44:892–898

Wang QC (2000) Speed railway civil engineering. Southwest

Jiaotong University Press, Chengdu (in Chinese)Wang CM, Xiao SF, Xia YB, Liu CM (2000) On consolidation creep

behavior of marine soft soil. J Changchun Univ Sci Techno

30(1):57–60 (in Chinese with English abstract)Wheeler SJ, Karube D (1995) Constitutive modeling. In: Alonso EE,

Delage P (eds) Unsaturated soils. Proceedings of the 1st

international conference on unsaturated soils, Paris, France, 2,

pp 1323–1356, Rotterdam: Balkema

Wong TT, Fredlund DG, Krahn J (1998) A numerical study of coupled

consolidation in unsaturated soils. Can Geotech J 35:926–937

Wu QB, Niu FJ, Ma W, Liu YZ (2014) The effect of permafrost

changes on embankment stability along the Qinghai-Xizang

Railway. Environ Earth Sci 71:3321–3328

Xia CC, Xu CB, Wang XD, Zhang CS (2009) Method for parameters

determination with unified rheological mechanical model. Chin J

Rock Mech Eng 28(2):425–432 (in Chinese with Englishabstract)

Yin JH (1999) Non-linear creep of soils in oedometer tests.

Geotechnique 49:699–707

Yin JH, Graham J (1989) Viscous-elastic-plastic modelling of one-

dimensional time-dependent behavior of clays. Can Geotech J

26(2):199–209

Yin JH, Graham J (1994a) Equivalent times and onedimensional

elastic visco-plastic modeling of time-dependent stress-strain

behaviour of clays. Can Geotech J 31(1):45–52

Yin JH, Graham J (1994b) Equivalent times and elastic visco-plastic

modelling of time-dependent stress-strain behavior of clays. Can

Geotech J 31(1):42–52

Yin JH, Graham J (1996) Elastic visco-plastic modelling of one-

dimensional consolidation. Geotechnique 46(3):515–527

Zhan ML, Qian JH, Chen XL (1993) Test on rheological behavior of

soft soil and rheological model. Chin J Geotech Eng 15(3):54–62

(in Chinese with English abstract)Zhao WB, Shi JY (1996) The consolidation and rheology of soft soil.

Hehai University Press, Nanjing (in Chinese)Zhou WH, Tu S (2012) Unsaturated consolidation in a sand drain

foundation by differential quadrature method. Pro Earth Planet

Sci 5:52–57

Zhou WH, Zhao LS, Li XB (2014) A simple analytical solution to

one-dimensional consolidation for unsaturated soils. Int J Numer

Anal Methods Geomech 38:794–810

Zhu HH, Chen XP, Chen XJ et al (2006) Study on creep

characteristics and model of soft soil considering drainage

condition. Rock Soil Mech 27(5):694–698 (in Chinese withEnglish abstract)


123

Preliminary research on the theory and application of ... · railway construction. The embankment...

Documents

Transcript of Preliminary research on the theory and application of ... · railway construction. The embankment...