3D Analysis of Emb on Soft Soils Incorporating Vertical Drains by FEM_Borges

7/31/2019 3D Analysis of Emb on Soft Soils Incorporating Vertical Drains by FEM_Borges

1/12

Three-dimensional analysis of embankments on soft

soils incorporating vertical drains by finite element method

Jose Leitao Borges *

Department of Civil Engineering, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

Received 22 October 2003; received in revised form 1 November 2004; accepted 3 November 2004

Available online 15 December 2004

Abstract

Three-dimensional behaviour of an embankment on soft soils incorporating vertical drains is analysed by a numerical model

based on the finite element method. The model, which incorporates the Biots consolidation theory (coupled formulation of the flow

and equilibrium equations) and constitutive relations simulated by the pqh critical state model, is applied on both the embank-

ment with vertical drains (three-dimensional analysis) and the same problem but without vertical drains (two-dimensional analysis).

Special emphasis is given to the analysis, during and after the construction period, of the excess pore pressures, settlements, hori-

zontal displacements and stress levels.

2004 Elsevier Ltd. All rights reserved.

Keywords: Vertical drains; Embankment; Soft soils; Finite element method; Three-dimensional analysis; Consolidation

1. Introduction

The study of embankments on soft soils is one of the

permanent problems of the soil mechanics and has been

analysed by a large number of authors. At the present

time, in spite of all experience obtained over the last dec-

ades, the execution of this kind of constructions still col-

locates diverse and delicate problems that are

determined by the weak geotechnical characteristics of

the foundation soils: (i) low strength significantly limits

the load (embankment height) that is possible to apply

with adequate safety for short term stability; (ii) high

deformability and low permeability determine large set-

tlements that develop slowly as pore water flows and ex-

cess pore pressure dissipates (consolidation).To design embankments on soft soils it is essential to

take into account the multiple constructive techniques

that allow to solve those problems. The constructive

solutions usually based on both foundation soil prop-

erties improvement and construction procedures or fillproperties alteration provide one or more of the fol-

lowing effects: increase of global stability, consolidation

acceleration and decrease of long term settlements [13].

The most used technique when the main purpose is to

accelerate the consolidation is the use of vertical drains

in the foundation soils (Fig. 1), which usually determines

drastic decreases of hydrodynamic consolidation time.

In the paper, the geotechnical behaviour of an

embankment on soft soils incorporating vertical drains

(geosynthetic band drains) is analysed during and after

the construction period by a numerical model developed

by Borges [4] for plane strain and axysymmetric analyses

(initial version) and three-dimensional analysis(improvement included in the program in 2001). Special

emphasis is given to the three-dimensional behaviour of

this kind of constructions by the comparison of the re-

sults of the embankment incorporating the drains

(three-dimensional analysis) with the results of the plane

strain analysis of the same problem without vertical

drains.

0266-352X/$ - see front matter 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compgeo.2004.11.001

* Tel.: +351225081928; fax: +351225081940/1440.

E-mail address: [email protected].

www.elsevier.com/locate/compgeo

Computers and Geotechnics 31 (2004) 665676


2/12

Basically, for the present applications, the model uses

the following theoretical hypotheses: (a) coupled formu-

lation of the flow and equilibrium equations considering

soil constitutive relations (elasto-plastic models) formu-

lated in effective stresses (Biots consolidation theory)

[47]; this formulation is applied to all phases of the

problem, both during the embankment construction

and in the post-construction period; (b) utilisation of

the pqh critical state model [46,8], an associated plas-

tic flow model, to simulate constitutive behaviour of the

foundation and embankment soils.Fig. 2(a) shows, in the principal effective stress space,

the yield and critical state surfaces of the pqh critical

state model. In the pq coordinate system (where p is

the effective mean stress and q the deviatoric stress),

the yielding function is a ellipse (Fig. 2(b)) and, depend-

ing on the stress path, the pqh model simulates hard-

ening behaviour (as shown in Figs. 2(b) and (c) for stress

path 12, where ev is the volumetric strain and ed the

shear strain) or softening behaviour (stress path 34,

Figs. 2(b) and (d)).

In this model, the parameter that defines the slope

of the critical state line, M, is not constant as in the

Cam-Clay and Modified Cam-Clay models, but de-

pends on the angular stress invariant, h, and friction

angle of the soil defined in effective terms, / 0, asfollows:

M 3sin/0

ffiffiffi

3p

cos h sin/0 sin h : 1

This defines the MohrCoulomb criteria (whose sur-

face in the principal effective stress space is shown inFig.

2(a)) when Mis introduced in the equation of the criticalstate line

q M p: 2

Fig. 2. Yield and critical state surfaces of the pqh critical state model in (a) principal effective stress space; (b) pq coordinate system; (c) stress path

12 (hardening behaviour); (d) stress path 34 (softening behaviour).

EMBANKMENT

HARD STRATUM

DRAINAGE LAYER(0.5 to1m)

VERTICALDRAINS

SOFTSOILS

Fig. 1. Vertical drains acceleration of hydrodynamic consolidation.

666 J.L. Borges / Computers and Geotechnics 31 (2004) 665676


3/12

This is an important characteristic of the pqh mod-

el because, as shown by triaxial tests [33], the critical

state of soil depends on h. (DruckerPrager is the crite-

ria of the Cam-Clay and Modified Cam-Clay models

and does not depend on h).

About embankments on soft ground, in order to ver-

ify accuracy of the finite element program in this kind of

works, Borges [4] compared numerical and field results

of two embankments on soft soils, one constructed up

to failure [9] and the other observed until the end of

the consolidation [1012]. The accuracy was considered

adequate in both cases, as numerical and field results are

similar, namely in terms of settlements and pore pres-

sures. Only some quantitative differences were observed

in the horizontal displacements, despite an overall qual-

itative similarity too.

In general terms, the verification of the program

was made by comparing numerical results not only

with field results, as said above (which is the adequate

way for complex problems without theoretical solu-

tions), but also with theoretical results of several

closed problems, namely one-dimensional Terzaghi

consolidation, circular load on an elastic foundation

[34], consolidation of a semi-infinite elastic medium

under an uniform surface loading over a circular area

[35], drained and undrained triaxial tests for soils sim-

ulated with critical state models. Comparisons with

numerical results presented by other authors, as Lewis

and Schrefler [5] with CRISP (critical state program),

were also verified.

It should be remembered that the first consistent and

general theory of multi-dimensional consolidation tak-

ing into account interdependence between soil effectivestrains and pore water flow (coupled formulation of

the flow and equilibrium equations) was presented by

Biot [13,14]. This theory, which was initially developed

considering soil with isotropic and linear elastic behav-

iour, was posteriorly improved for more complex behav-

iours, namely anisotropy and viscosity [1517].

After the presentation of Biots consolidation theory,

several authors have applied the finite element method

on consolidation problems using mathematical formula-

tions in which some of the initial hypotheses of Biot are

reformulated (extensions of the Biots theory). Small

et al. [18] considered the plasticity using MohrCou-

lomb model and Desai and Siriwardane [19] and Runes-son [20] applied critical state models. The variation of

the permeability in the course of the consolidation was

considered by Lewis et al. [21]. Norris [22] extended

the study to the use of constitutive models with kinemat-

ical hardening.

Concerning the use of vertical drains in embankments

on soft soils, Zeng et al. [23] conducted coupled consol-

idation analyses to assess the effectiveness of a method

that calculates an equivalent horizontal permeability

for plane strain analysis in which the vertical drains

are represented as sheets. Hird and Kwok [24] per-

formed a parametric study of a test embankment where

the effect of the vertical drains was considered by

increasing the subsoil permeability by an estimated fac-

tor. Poran et al. [25] defined an equivalent vertical per-

meability for the subsoils by comparing an

axisymmetric analysis of a cylindrical unit cell, with

drainage occurring both horizontally, inwards, and ver-

tically, upwards, with a plane strain unit cell with drain-

age towards the upper boundary only. This equivalent

vertical permeability was used in a plane strain analysis

of a trial embankment.

Because embankments on soft soils incorporating

vertical drains behave three-dimensionally (in terms of

stress and water flow, as shown below in Section 3),

one of the aims of the paper is to achieve a more com-

plex phenomenological interpretation of the 3D gheo-

technical behaviour of this kind of works, by

performing a three-dimensional numerical analysis with

the program developed by the author.

For three-dimensional applications, the program uses

two types of the 20-noded brick element. Fig. 3(a)

shows the type used in the foundation soils (element

with 60 displacement degrees of freedom, at the corners

and at middle of the sides, and with 8 more excess pore

pressure degrees of freedom, at the corners), where con-

solidation analysis is considered. In the fill, it is the 20-

noded brick element with only 60 displacement degrees

of freedom (at the corners and at middle of the sides)

that is used.

Similarly, for two-dimensional analyses, two types

of the six-noded triangular element are considered

(Fig. 4): (i) with 12 displacement degrees of freedom,at the vertices and at middle of the edges (for fill ele-

ments) and (ii) with 3 more excess pore pressure de-

grees of freedom at the vertices (for foundation

elements).

2. Description of the problem

The problem concerns the construction of a 2 m

height symmetric embankment, with a 10.6 m crest

width, 2/3 (V/H) inclined slopes and very large longitu-

dinal length. The foundation is a 5 m thick saturated

clay layer lying on a rigid and impermeable soil, whichconstitutes the lower boundary. The clay is lightly over-

consolidated to 1.8 m depth and normally consolidated

from 1.8 to 5 m. It will be analysed the embankment

with and without vertical band-shaped drains (geosyn-

thetic prefabricated drains) with a 200 5 mm2 section

and installed in a square grid with drain spacing of 2

m. The grid limit is 1.7 m beyond the toe to take up

any lateral spread of excess pore pressures. It is not in-

tended that the band drain reproduces any commercial

product.

J.L. Borges / Computers and Geotechnics 31 (2004) 665676 667


4/12

Fig. 5 shows the finite element mesh used in the three-

dimensional analysis of the embankment incorporating

the vertical drains.

The displacement boundary conditions were definedtaking into account that the soft clay lays on a hard

stratum (y = 0 plane, where displacements are set as

zero in the three directions, x, y and z). One the other

hand, symmetry conditions imply: (i) zero displacement

in x-direction for nodes on the x = 0 plane; (ii) zero

displacement in z-direction for nodes on the z = 0

plane, vertical plane containing one row of drains;

(iii) zero displacement in z-direction for nodes on the

z = 1 m plane, vertical plane equidistant from two rows

of drains in x-direction. Assuming that the horizontal

displacement can be defined as zero at nodes that are

enough distant from the embankment, the plane of

Fig. 3. 3-D finite element used in the program, 20-noded brick element: (a) with 60 displacement degrees of freedom and 8 excess pore pressure

degrees of freedom; (b) with 60 displacement degrees of freedom.

- displacement unknown- excess pore pressure

unknown

Fig. 4. 2-D finite element used in the program, 6-noded triangular

element: (a) with 12 displacement degrees of freedom and 3 excess pore

pressure degrees of freedom; (b) with 12 displacement degrees of

freedom.

Fig. 5. 3D finite element mesh for the problem with vertical drains.



5/12

x = 23.9 m was considered as the lateral boundary with

zero displacement in x-direction.

With regard to drainage boundary conditions, excess

pore pressure was set as zero on the ground level (upper

drainage surface), i.e., on the y = 5 m plane, and on the

drainage surfaces defined by the drains considered as

sheets, namely on the following planes: x = 0, x = 2,

x = 4, x = 6, x = 8 and x = 10 m, with y-coordinate

varying from 0 to 5 m and z-coordinate from 0 to 0.1

m (which means that centres of the drains are on the

z = 0 boundary plane and each drain was installed with

its larger dimension, 0.20 m, in z-direction).

The embankment construction was simulated activat-

ing the elements that form the fill layers. Four 0.5 m

height layers were considered and, in order to assess

the drainage effect even during the construction period,

a discontinuous sequence of construction was defined as

indicated in Fig. 6. The first three layers were con-

structed in 3.5 days each, and the fourth in 7 days.

The pause periods, which took place after each layer

construction, were respectively 3.5, 3.5 and 38.4 days.

As said in Section 1, the coupled analysis was performed

in all phases of the problem, both during the embank-

ment construction and in the post-construction period.

The constitutive relations of both the embankment

and foundation soils were simulated using the pqh

critical state model [46,8] with the parameters indicated

in Table 1 (k, slope of normal consolidation line and

critical state line; k, slope of swelling and recompression

line; C, specific volume of soil on the critical state line at

mean normal stress equal to 1 kPa; N, specific volume of

normally consolidated soil at mean normal stress equal

to 1 kPa). Table 1 also shows other geotechnical proper-ties: c, unit weight; m 0, Poissons ratio for drained load-ing; c0 and / 0, cohesion and angle of friction definedin effective terms; kx and ky, coefficients of permeability

in x and y directions. Table 2 indicates the variation

with depth of the at rest earth pressure coefficient, K0,

and over-consolidation ratio, OCR, in the foundation.

The embankment soil was considered with 0.43 for K0and 1 for OCR. All these parameters were defined tak-

ing into account typical experimental values for this

kind of soils [4,26].

Fig. 7 shows the 2D finite element mesh for the

embankment without the drains, problem that can be

considered as a plane strain problem, given the very

large longitudinal length of the embankment; y axis is

the symmetry line and, with exception of the boundary

conditions for excess pore pressure (set as zero only on

the upper drainage surface, i.e., at nodes with y = 5

m), all the other characteristics of the problem, whencompared with the three-dimensional problem, are

maintained.

Given the non-linearity of the constitutive model

used in the soils (pqh critical state model) and the

boundary conditions, in both problems (with and with-

out drains) mesh sensitivity in terms of variation of

numerical results was analysed by considering several

meshes and time increments (different geometry and

time discretizations). The meshes (Figs. 5 and 7) and

the increments used in the paper were assessed adequate

by analysing, at each calculation of the coupled analysis,

the global equilibrium of the problem (comparing exter-

nal forces with stresses at all Gauss points of the ele-ments). The smooth geometric variation of the stress

results in the medium (presented in the following

5 10 15 20 25 30 35 40 45 50 55 60 65 time (days)0

0.5

11.5

2

0

embankment

height

(m)

Fig. 6. Embankment construction sequence.

Table 1

Geotechnical properties of the foundation and embankment soils

c (kN/m3) m0 c0 (kPa) /0 () kx (m/s) ky (m/s) pqh critical state model

k k C N

Foundation 17 0.25 0 30 109 109 0.22 0.02 3.26 3.40

Embankment 20 0.30 0 35 0.03 0.005 1.80 1.817

Table 2

At rest earth pressure coefficient, K0, and over-consolidation ratio,

OCR, in the foundation

Depth (m) K0 OCR

01 0.7 2.43

11.8 0.70.5 2.431

1.85 0.5 1

Fig. 7. 2D finite element mesh for the problem without vertical drains.



6/12

section) corroborated that the numerical convergence of

this non-linear problem was achieved adequately.

In the problem with the drains the total number of

increments (i.e., total number of finite element calcula-

tions) was 359 (209 during the construction and 150 dur-

ing the post-construction period). In the embankment

without the drains the same number was used during

construction, but a higher number (177) was considered

during the post-construction period (since the consolida-

tion time is longer).

3. Analysis of the results

When a load is applied on a saturated soil mass, the

distribution of the excess pore pressure has, usually, gra-

dients that determine a field of relative velocity among

different zones of the soil. Initial conditions of a tran-

sient flow process are determined and transferences of

load from the water (pore pressure) to the soil skeleton

(effective stress) take place. Therefore, until a steady

state is reached, the soil mass behaviour is determined

by the variation of the fields of stress (pore pressure

and effective and total stress), strain and displacement

(consolidation).

Figs. 8 and 9 show results of excess pore pressures for

the two analyses of the problem, with and without ver-

tical drains, at different phases, during and after the con-

struction period. For the 3D analysis, Fig. 8 shows

results both on the vertical plane that contains one

row of drain centres, z = 0 plane (on the right side),

and on the vertical plane equidistant from two rows of

drains, z = 1 m plane (on the left side).

Based on these results, and considering the founda-

tion divided into four typical zones as illustrated in

Fig. 10, one can say that, for the problem without verti-

cal drains, during construction period: (a) maximum

values happen in zone A and are approximately similar

to the vertical stress due to the embankment weight; (b)

in zone B, excess pore pressures decrease from zone A to

zone D, where their values are not significant. After con-

struction, when the problem is, above all, characterised

by the transient water flow, one can see that isovalue

curves have a very regular shape, normal to the flow

lines.

With regard to the results of the problem incorporat-

ing the vertical drains (Fig. 8), the shape of the isovalue

curves clearly shows the three-dimensional condition of

the problem, with drainage occurring both horizontally

and vertically towards the several drainage surfaces

(band drains and upper drainage surface). Maximum

values also occur in zone A and also with similar values

to the vertical stress determined by the embankment

weight. However, the most important fact (see below

the analysis of the settlements) concerns the significantly

effect of consolidation (mainly in zone A) due to the ver-

Fig. 8. Excess pore pressure (u) for the embankment with vertical drains. (a) 1 m height embankment (time = 14 days); u max = 20.15 kPa. (b) 2 m

height embankment (end of construction); umax = 38.16 kPa. (c) 64 days after construction; umax = 29.29 kPa.



7/12


8/12

expressed by generalised settlements and horizontal dis-

placements that can be outwards too, as shown in Fig.

16 for the embankment without the vertical drains. As

explained by Borges [4], these outward horizontal dis-

placements, in consonance with experimental results ob-

served in real works [912,2831], are associated with

shear strains during the consolidation process which

are properly simulated only by elastoplastic models with

closed yielding surfaces, which is the case of the pqh

critical state model used in this study.

Three more important effects about the use of the ver-tical drains, obtained from the analysis of the results, are

pointed out below.

The first effect, as expected, is the very expressive de-

crease of the consolidation time (reduction from approx-

imately 5000 to 500 days, as shown in Fig. 15).

The second effect is the reduction of the maximum va-

lue of the long term settlements (about 16%, from 48.7

to 40.9 cm, as illustrated in Fig. 14). This effect is asso-

ciated with a certain improvement of the foundation soil

properties (decrease of voids ratio) by consolidation

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.00 2.00 4.00 6.00 8.00 10.00

Distance from x=0 plane (m)

Settlem

ent(m)

h = 0.5 m (t = 7 days)

h = 1 m (t = 14 days)

h= 1.5 m (t= 17.5 days)

h =1.5 m (t = 55.9 days)

h = 2 m (end of construction)

h = 2 m (end of cosolidation)

h embankment height

t time after beginning

of construction

Fig. 13. Settlements at the embankment base for the embankment with vertical drains.

-0.600

-0.500

-0.400

-0.300

-0.200

-0.100

0.000

0.100

0.000 2.000 4.000 6.000 8.000 10.000

Distance from x=0 plane (m)

Settlement(m)

Embankment without

vertical drains (end of

construction)

Embankment without


consolidation)

Embankment with


construction)

Embankment with


consolidation)

Fig. 14. Settlements at the embankment base for the embankment with and without vertical drains.

Settlement(m)

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0 1000 2000 3000 4000 5000 6000

Time after the end of construction (days)

Embankment without

vertical drains

Embankment with

vertical drains

Fig. 15. Settlement in time at the middle point under the embankment

on the ground level (point with x = 0, y = 5 and, for three-dimensional

case, z = 0) for the embankment with and without vertical drains.



9/12

during the construction period. This is a kind of a soil

hardening effect that influences the decrease of long

term settlements (as well as the reduction of long term

horizontal displacements, as shown in Fig. 16). This ef-fect only happens because shear stress increases during

the load periods and can reach higher values in the prob-

lem with the vertical drains (the consolidation effect dur-

ing the pause periods increases undrained strength of the

soil). If the problem was one-dimensional (as in the

oedometer tests), this effect would not take place be-

cause in the load periods there is no variation of effective

stress (in undrained conditions) and it is indifferent

whether there is pause periods or not, if total load is

the same.

The third effect is the uniformity of settlements along

z-direction for the embankment with band drains, de-

spite its three-dimensional behaviour in terms of stressesand water flow. This point is justified by the existence of

arch effect inside the fill, which is expressed by the

application of a non-uniform vertical load on the foun-

dation surface (see Fig. 17). The vertical load, at a deter-

mined phase, is smaller on zones that tend, by

consolidation, to settle more than the others, i.e., near

the vertical drains. This effect, as shown by field results

[1012], tends to be equilibrated in a non-uniform distri-

bution of the vertical load that approximately unifor-

mizes the corresponding settlements.

Finally, Figs. 18 and 19 show values of the stress lev-

els (which vary from 0 to 1, being 1 the critical state le-

vel) at different phases of the problem; Fig. 20 presents

the effective principal stresses for the embankment with-out vertical drains at the ends of construction and

consolidation.

The definition of stress level, SL, is given in Fig. 21

for an arbitrary stress state represented by the point A

in the pq plane, where p is the effective mean stress

and q the deviatoric stress.

For the embankment without vertical drains (Fig.

18), one can point out that: (i) during the construction

period, the main characteristic of the problem is

Fig. 16. Horizontal displacements at the end of construction and at the end of consolidation along vertical line under the toe (points with x = 8.3 m

and, for three-dimensional case, z = 0) for the embankment with and without vertical drains.

Fig. 17. Increment of vertical effective stress (kPa) at the end of

construction for the embankment with vertical drains.



10/12

expressed by the increase of the stress levels in the foun-

dation (and therefore by the decrease of the problem

safety [27,32]), especially in Zone B (Fig. 10); as shown

in Fig. 20(a), this effect is associated with an effective

stress path basically characterised by the rotation of

the principal stresses directions (i.e., low variations of

effective mean stress and significant increases of devia-

toric stress); (ii) during the post-construction period,

the effective stress path (associated with the dissipationof the excess pore pressures) is characterized by expres-

sive increases of effective mean stress and low variations

of deviatoric stress (for in Fig. 20 the magnitude of the

principal stresses increases without their directions sig-

nificantly changing), which implies a generalised reduc-

tion of the stress levels, as illustrated in Fig. 18.

With regard to the results of stress levels of the

embankment with vertical drains (Fig. 19), the main

difference in relation to the embankment without

drains concerns the very significant reduction of the

stress levels at all phases of the problem (and particu-

Fig. 18. Stress levels for the embankment without vertical drains. (a) 1m height embankment (time = 14 days). (b) 2 m height embankment

(end of construction). (c) End of consolidation.

Fig. 19. Stress levels for the embankment with vertical drains. (a) 1 m height embankment (time = 14 days). (b) 2 m height embankment (end of

construction). (c) End of consolidation.

Fig. 20. Effective principal stresses for the embankment without

vertical drains. (a) End of construction. (b) End of consolidation.



11/12

larly at the end of the construction period) due to the

consolidation acceleration determined by use of the

vertical drains.

4. Conclusions

In the paper, a numerical model based on the finite

element method was used to analyse the structural

behaviour of an embankment on soft soils incorporating

vertical drains. The model, which incorporates the Biots

consolidation theory and constitutive relations simu-

lated by the pqh critical state model, was applied on

both the embankment with vertical drains (three-dimen-

sional analysis) and the same problem without verticaldrains (two-dimensional analysis). The analysis of the

results (excess pore pressures, settlements, horizontal

displacements and stress levels) allows to point out the

following conclusions on the effects of the use of vertical

drains in embankments on soft soils:

1. The effect on the total time of consolidation is very

expressive (reduction of about 10 times, from approx-

imately 5000 to 500 days).

2. This fact is obviously associated with the faster dissi-

pation of the excess pore pressures (and consequent

decrease of the stress levels) at all phases of the prob-

lem, during and after the construction period.3. The increase of the maximum settlement value at the

end of the construction is significant (about 64%,

from 8.4 to 13.8 cm), which implies a certain

improvement of the foundation soil properties

(decrease of voids ratio) by consolidation during the

construction period.

4. This effect, which is a kind of a hardening effect of

the soil, influences the decrease of the long term settle-

ments (about 16%, from 48.7 to 40.9 cm), as well as

the reduction of long term horizontal displacements.

5. In spite of its three-dimensional behaviour in terms of

stresses and water flow, settlements of the embank-

ment with vertical drains are approximately uniform

along longitudinal direction. This is justified by the

existence of arch effect inside the fill, which is

expressed by the application of a non-uniform verti-

cal load on the foundation surface.

References

[1] Terzaghi K, Peck RB. Soil mechanics in engineering practi-

ce. New York: Wiley; 1948.

[2] Johnson SJ. Treatment of soft foundations for highway embank-

ments. Synthesis of Highway Practice 29, USA: National Coop-

erative Highway Program; 1975. p. 25.

[3] Holtz RD. Treatment of problem foundations for highway

embankments. Synthesis of highway practice 147, Washington

(DC): National Cooperative Highway Research Program; 1989. p.

72.[4] Borges JL. Geosynthetic-reinforced embankments on soft soils.

Analysis and design. Ph.D. thesis in civil engineering, Faculty of

Engineering, University of Porto, Portugal; 1995 [in Portuguese].

[5] Lewis RW, Schrefler BA. The finite element method in the

deformation and consolidation of porous media. New

York: Wiley; 1987.

[6] Britto AM, Gunn MJ. Critical soil mechanics via finite ele-

ments. England: Ellis Horwood Limited; 1987.

[7] Borges JL, Cardoso AS. Numerical simulation of the consolida-

tion processes in embankments on soft soils. R Geotecnia

2000;89:5775. [in Portuguese].

[8] Borges JL, Cardoso AS. Numerical simulation of the pqh

critical state model in embankments on soft soils. R Geotecnia

1998;84:3963 (in Portuguese).

[9] Quaresma MG. Behaviour and modelling of an embankment over

soft soils reinforced by geotextile. Ph.D. thesis, Universite Joseph

Fourier, Grenoble I, 1992 [in French].

[10] Yeo KC. Simplified foundation data to predictors. Proceedings of

the prediction symposium on a reinforced embankment on soft

ground. London: Kings College; 1986.

[11] Basset RH. The instrumentation of the trial embankment and of

the Tensar SR2 grid. Proceedings of the prediction symposium on

a reinforced embankment on soft ground. London: Kings Col-

lege; 1986.

[12] Basset RH. Presentation of instrumentation data. Proceedings of

the prediction symposium on a reinforced embankment on soft

ground. London: Kings College; 1986.

[13] Biot MA. The consolidation problems of clayey materials under a

load. Annales de la Societe Scientifique de Bruxelles 1935;55(Ser-

ies B):1103. [in French].

[14] Biot MA. General theory of three-dimensional consolidation. JAppl Phys 1941;12:15564.

[15] Biot MA. Theory of elasticity and consolidation for a porous

anisotropic solid. J Appl Phys 1955;26:1825.

[16] Biot MA. Theory of deformation of a porous viscoelastic

anisotropic solid. J Appl Phys 1956;27:45967.

[17] Biot MA. General solutions of the equations of elasticity and

consolidation for a porous material. Transactions. J Appl Mech

1956;78:916.

[18] Small JC, Booker JR, Davis EH. Elastoplastic consolidation of

soil. Int J Solids Struct 1976;12:43148.

[19] Desai CS, Siriwardane THJ. Subsidence due to consolida-

tion including nonlinear behaviour. In: Saxena SK, editor.

p

A

criticalstate line

SL=tg

tg

q

Fig. 21. Definition of stress level, SL, in the pq plane.



12/12

Evaluation and prediction of subsidence. New York: ASCE;

1979. p. 50015.

[20] Runesson K. On nonlinear consolidation of soft clays. Thesis,

Dept. of Struct. Mech., Chalmers University of Technology,

Goteborg; 1978.

[21] Lewis RW, Roberts GK, Zienkiewicz OC. A non-linear flow and

deformation analysis of consolidation problems. In: Desai CS,

editor. Numer Meth Geomech, 2. ASCE; 1976. p. 110618.

[22] Norris VA. The elasto-plastic analysis of soil consolidation withspecial reference to kinematic hardening. Ph.D. thesis, University

College of Swansea; 1980.

[23] Zeng GX, Xie KH, Shi ZY. Consolidation analysis of sand-

drained ground by F.E.M. In: Proceedings of the eighth

regional conference on soil mechanics (vol. 1). Kyoto; 1987.

p. 13942.

[24] Hird CC, Kwok CK. Predictions for the Stanstead Abbotts trial

embankment. Predictions symposium on a reinforced embank-

ment. London: Kings College; 1986.

[25] Poran CJ, Kaliakin VN, Hermenn LR, Romstadd KM, Lee DF,

Shen CK. Prediction of trial embankment behaviour Hertford-

shire county council Stanstead Abbotts. Prediction symposium

on a reinforced embankment. London: Kings College; 1986.

[26] Lambe TW, Whitman RV. Soil mechanics. New York: Wiley;

1969.[27] Borges JL, Cardoso AS. Structural behaviour and parametric

study of reinforced embankments on soft clays. Comput Geotech

2001;28/3:20933.

[28] Soderman KL. The Behaviour of Geotextile Reinforced Embank-

ments. Ph.D. thesis, Ontario: University of Western Ontario;

1986.

[29] Litwinowicz A, Wijeyakulasuriya CV, Brandon AN. Perfor-

mance of a reinforced embankment on sensitive soft clay

foundation. In: Proceedings of the fifth international conference

on geotextiles, Geomembranes and Related Products (vol. 1).

Singapore; 1994. p. 116.

[30] Loke KH, Ganeshan V, Werner G, Bergado DT. Compositebehaviour of geotextile reinforcement embankment on soft clay.

In: Proceedings of the fifth international conference on geotex-

tiles, geomembranes and related products (vol. 1). Singapore;

1994; vol. 1 p. 258.

[31] Matichard Y, Tanays E, Carbon M. Geotextile reinforced

embankment to cross a peat-bog. Proceedings of the fifth

international conference on geotextiles, geomembranes and

related products (vol. 1). Singapore; 1994. p. 3740.

[32] Borges JL, Cardoso AS. Overall stability of geosynthetic-rein-

forced embankments on soft soils. Geotext Geomembranes

2002;20/6:395421.

[33] Lade PV, Duncan JM. Cubical triaxial tests on cohesionless soils,

1973, J Soil Mech. Found Div. ASCE, vol. 99, SM10.

[34] Poulos HG. Stresses and displacements in an elastic layer

underlain by a rough rigid base. Geotechnique 1967;17:378410.

[35] Booker JR, Randolph MF. Consolidation of a cross-anisotropic

soil medium. Quart J Mech Appl Math 1984;37:47995.


Documents

3D Analysis of Emb on Soft Soils Incorporating Vertical Drains by FEM_Borges