*Associate Professor, Civil Engineering Department, Faculty of Engineering, Minia University, Minia, egypt 

Elasto-plastic Finite Element One-Dimensional Consolidation Analysis

 

By 

Ahmed M. Hassan* 

 

Abstract  

In this study modified Cam clay plasticity model was utilized to evaluate the elasto-plastic behavior of clay during one-dimensional consolidation. A finite element one dimensional consolidation model was established and verified against Terzaghi's theory. Analysis was performed for two cases: elastic clay behavior and elasto-plastic behavior. The results indicated significant difference in consolidation settlement for both cases. The elasto-plastic analysis gave significant bigger settlements than elastic analysis. However, no significant change was observed regarding the time rate of consolidation.

Introduction

The conventional one-dimensional consolidation theory developed by Terzaghi treated soil as a linear elastic porous material in which change in volume is proportional to change in pressure. Other assumption considered by this theory included neglecting water and soil grains compressibility, application of Darcy’s law, and assuming soil homogeneity and constant coefficient of volume compressibility (mv) and coefficient of permeability (k). Duncan (1993) showed that taking mv and k as constants is a major shortcoming of the conventional consolidation theory. Many researches were conducted to overcome this shortcoming: e.g. Hassan and Abu Bakr (2006), Abbasi et al. (2007), Cai et al. (2007), Menéndeza et al. (2009), Zheng et al. (2010), and Tang et al. (2013).

Other main shortcoming of the conventional consolidation theory is considering the soil as a linear elastic porous material. The actual behavior of soil is highly non-linear. Modeling elasto-plastic behavior of clay during consolidation is made possible by numerical finite element method (FEM) which provides a wide range of soil constitutive models. Cam-clay models (Schofield and Wroth, 1968; Roscoe and Burland, 1968) have been acknowledged as efficient constitutive model for clay plasticity. Krishnamoorthy (2008) applied FEM analysis to model nonlinear behavior of soil. He used a tangent modulus of elasticity obtained for each time step to model the nonlinear behavior of soil in FEM consolidation analysis.

In the present paper clay plasticity was simulated by the modified Cam clay model (MCC). One-dimensional consolidation process was performed numerically using Abaqus finite element program (Abaqus/standard V 6.12). The model was first verified against Terzaghi's analytical solution assuming linear elastic behavior of soil.

Thecase

Fin

soil probonlydispDra

elempresFigu

e elastoplaste.

nite Eleme

The fincolumn of

blem. The y vertical dplacements. ainage is allo

The soments. The ssure variabure (2).

tic soil beh

ent Mode

nite elementf width 0.8 side bound

displacemenA vertica

owed at top

Figure (1

oil column four corne

bles. The o

havior using

el

model conmeter and

daries have nt. Bottom l stress, q

p boundary o

) One-dime

was modeer points ofother points

Figure

g MCC wa

nsidered in theight 8.0 symmetricboundary ris suddenl

only.

ensional con

eled with 8f this eleme have only

(2) CPE8P

s then com

the analysismeters was

cal boundarrestricts boly applied

nsolidation

8-noded Abent have boy displacem

element

mpared to th

s is shown is modeled ary conditionoth vertical

at the top

FE model

baqus CPE8oth displaceent variable

he linear el

in Figure (1as a plain stns which aand horizoof the mo

8P plain stement and es as show

astic

1). A train

allow ontal odel.

train pore

wn in

Mo

196ThedefoFigu

(slo

sloprelomea

domparaTheTheparathe effe

odified Ca

Modifie68) is a plase model is bormations wure (3) show

ope k) in the

The mape of the nooading/swellan effective

As shomain. In critameters: mee dry (softene shape of thameter) so tdry side. T

ective stress

am Clay M

ed Cam clasticity modebased on thwithout chaws the norm

e e-ln p' spa

Figure (3

aterial paramormal consoling, and eN

stress (1.0 wn in Figutical-state thean effectivning) regionhe yield curthat the ellipThe mean ees as follow

Model (M

y model (Rel capable ohe critical sanges in stremal consolid

ace. These tw

3) Normal c

meters λ, κ,olidation linN is the voikPa).

ure (4), theheory, the s

ve stress p', n and the wrve on the wptic arc on teffective strws:

MCC)

Roscoe and of describinstate theoryess orvolumdation line,

wo lines are

= −= −

consolidatio

, and eN arene and the cid ratio on

e critical-stastate of a sodeviator str

wet (hardeniwet side is mthe wet sideress can be

= 23

Burland, 19ng the stresy and henceme when th NCL (slop

e given resp

on and swel

e unique focritical-statethe normal

ate line (Coil specimenress (shear ing) region modified bye has a diffee calculated

968; Schofiss–strain bee it predictshe critical spe l) and th

pectively by

ling lines

r a particule line, κ is tl consolidat

SL) is plotn is charactstress) q anare separat

y the paramerent curvatud in terms o

ield and Wrehavior of ss unlimited state is reache swelling

y:

lar soil. λ isthe slope oftion line at

tted in the terized by tnd void ratited by the C

meter β (wetture than thaof the princ

roth, soils.

soil ched.

line

s the f the unit

p'-q three io, e. CSL. t cap at on cipal

She

The

The

 

Fin

Terzpermbothpore

Wh

ear stress, q

e slope of C

e yield curve

nite Eleme

The przaghi' analymeability coh analyticale pressure, u

ere cv is the

is expressed

F

SL, M is gi

e for the MC

ent Mode

reviously dytical resuloefficient, k and numeru with depth

e coefficient

d as:

Figure (4) M

ven by:

CC model i

+el Verifica

described filts for the k=1.1E-8m/srical analyseh, z is relate

t of consolid

= −

Modified cam

= 6 sin3 − sinis an ellipse

1 −ation

finite elemesame soil

s). The soil wes. Accordined to the rat

t

uc

z

uv

2

2

dation given

m clay mod

′ ′ in p'-q spac

= 0

ent model parameters

was modeleng to Terzate of change

t

u

n by:

el

ce given by

was verifis (E=1000 ed as an elasaghi's theorye of u with t

y: 

fied by agakPa, n =0

stic materiay, the changtime by:

ainst 0.30, al for ge of

Whby:

Wh

analshow

Figu

Mo

prevTab

ere mv is th

ere H is theThe deg

lytical (Terws good ag

ure (5) Veri

odeling Cl

The moviously expble (1).

he coefficie

e drainage hgree of conrzaghi) and reement bet

ification of

lay Plasti

odified Camlained. The

Table (1)

LogaritLogarit

wv

kc

ent of volum

height. nsolidation,

numerical tween analy

FEM nume

city Using

m clay modee MCC para

) MCC soil

Propehmic elastichmic elasticCritical stat

vwm

k, vm

me compres

2H

tcT v

v

U is plotte(FEM) me

ytical and nu

erical model

g MCC

el, MCC wameters use

parameters

erty c bulk moduc bulk modute ratio M

oe

e

1

/

ssibility. Th

ed versus thethods in Fiumerical re

l versus Ter

as utilized d to model

s used in the

ulus k

ulus l

he time fact

he time factigure (5). Tsults.

rzaghi's anal

to model clsoil plastic

e analysis

Value 0.08

0.45

0.98

tor, Tv is g

tor, Tv for The compar

lytical solut

lay plasticitcityare show

given

both rison

tion

ty as wn in

wet cap parameter b 0.5

Third stress invariant parameter K 1.0

 

The third stress invariant, K is the ratio of the flow stress in triaxial tension to the flow stress in triaxial compression. K = 1 gives a circle as in the original Cambridge formulation (Abaqus 6.12 Keywords Reference Manual).The Poisson's ratio was taken as 0.30 and coefficient of permeability was assumed 1.1x10-8 m/s. The soil model was assumed to be initially loaded by a surcharge of 10.0 kPa in addition to soil own weight. The initial yield curve size, a0=p'/2 (see Figure (4)) is assumed to vary with depth taking the value of 5.0 kPa at top and 30.0 kPa at bottom. The initial void ratio, e0 was assumed 2.2 at top of soil layer and 1.3 at the bottom.

The suddenly applied stress of 300 kPa was considered at the top of soil layer. The problem is run in two steps. The first step is a single increment with an arbitrary time step, with no drainage allowed across the top surface. This establishes a uniform pore pressure equal to the applied load throughout the body, with no stress carried by the soil skeleton. Then the actual consolidation is started in the second step of 200 increments. During consolidation analysis, the maximum pore pressure change permitted in any increment was set as 10.0 kPa. The initial time increment was 1x10-5 s, time period =5x1012 s (consolidation process ends after this time period), and minimum time increment allowed was taken as 1x10-5 s. It may be noted that the number of increments and time periods are estimated according Abaqus 6.12 Benchmarks Manual. These values were checked also by monitoring pore pressure and its dissipation during the two consolidation steps. Elastic Clay Behavior

In this step, consolidation process was performed assuming elastic behavior of soil. To model soil as an elastic material we need two parameters: Young's modulus and Poisson's ratio. Poisson's ratio was taken as 0.30 as in the previous analysis. For soils modeled using the modified Cam clay model, the Young's modulus, E is stress dependent. The Young's modulus depends on the mean effective stress p', initial void ratio, e0, logarithmic elastic bulk modulus k. The followingequation was used to determine the equivalent elasticity modulus used to model the elastic behavior of clay using same Cam parameters (Sam (2007):

= 3 1 − 2 1 +

Using values of e0 and p' used in the MCM at the top and the bottom of soil

column, values of E were determined as 2070 kPa and 480 kPa respectively. The consolidation process was perfumed applying the same two Abaqus steps described in the MCC model und using the same time increments and periods. For simplicity, permeability was assumed constant for both elastic and elasto-plastic cases.

An

for takesignanalanalthe obtadispgav

conelasdela

F

alysis of R

The oneboth elasti

en at the mnificantly bilysis resultelysis.Similasettlement

ained fromplacement oe a settleme

Howevesolidation.

stic and elaays the cons

Figure (6) V

Results

e-dimensionc and alast

middle of thigger consoed in a settar results wof a strip a

m the nonliobtained froent about 3.er, no sigFigure (7)

aso-plastic csolidation p

Vertical settl

nal consolidto-plastic behe clay layelidation settlement abo

were obtainea footing reinear FEMom the labo5 bigger tha

gnificant dishows degr

cases. It marocess.

lement versuelaso-pla

dation settleehavior of er. It is notttlements thout 3 timesed by Krishesting on s

M analysis oratory elasan that obtaifference wree of consay be noted

us time fact

astic behavio

ement is ploclay in Figted that ela

han elastic bs bigger thahnamoorthyoil mass. Hat various

stic consolidained by linewas noted solidation vd that elast

tor, comparior of clay.

otted versusgure (6). Thasto-plastic behavior. Than that obta

(2008) whHe compare

time intedation test. ear elastic a

regarding versus time o-plastic be

ison betwee

s time factohe values wbehavior g

he elasto-plained by elhen determied displacemervals with

The non-lianalysis.

time ratefactor for

ehavior slig

en elastic an

r, Tv were gives astic astic ning ment

the inear

e of both

ghtly

nd

 

Figu

 

werstreof tare

comdiffconsym

Res

ure (7) Deg

In this re tested. Pss in triaxia

the yield cunot materia

Two vampared keeference in solidation s

mmetrical yiChangin

sults of usin  

gree of conso

part of theParameter Kal compressurve on the al dependentalues of pareping all oconsolidatiosettlement wield curve) gng the valu

ng K=.78 ga

olidation veelaso-pla

e analysis, K is the ratio

ion whereaswet side. T

t; rather therameter β w

other paramon time rawas noted igave a slighue of paramve exactly s

ersus time fastic behavio

 

two paramo of the flows the wet cThese two pey are modewere applie

meters consate was obin Figure (9

ht bigger valameter K dsame results

factor, compor of clay.

eters charaw stress in cap parametparameters el dependented 0.50 andstant. As sserved. Ho9). The β=lues of cons

did not chas as that of

parison betw

cterizing Mtriaxial tenster, β determwere choset. d 1.0 and thshown in Fowever a s1.0 case (wsolidation senge consolK=0.98.

ween elastic

MCC (K ansion to the mines the shen because

he results wFigure (8),

slight effectwhich indicaettlement. lidation res

 

c and

nd β) flow hape they

were , no t on ate a

sults.

Fi

Fig

igure (8) De

gure (9) Co

egree of con

onsolidation

nsolidation vand

n settlement and

versus timed β = 1.0 ca

  

 

versus timed β = 1.0 ca

 

e factor, comases 

e factor, comases 

mparison be

mparison be

etween β=0.

etween β=0

50

 

.50

 

Conclusions

Elasoplastic clay behavior was modeled using a finite element one-dimensional model. The settlement behavior of clay was significantly influenced by elastplastic modeling. Significantly bigger settlements were obtained in the elastoplastic analysis compared to elastic analysis. The time rate of consolidation was not significantly affected. The MCC parameters K and β showed no significant effect on consolidation results.

References Abbasi, N.Rahimi, H., Javadi, A.A., and Fakher, A. (2007), "Finite difference approach for consolidation withvariable compressibility and permeability", Computers and Geotechnics 34 (2007) 41–52. Abaqus 6.12 Benchmarks Manual. Abaqus 6.12 Keywords Reference Manual. Cai, Y-Q, Geng, X-Y, and Xu, C-J (2007), "Solution of one-dimensional finite-strain consolidation of soilwith variable compressibility under cyclic loadings", Computers and Geotechnics 34 (2007) 31–40. Duncan, J.M. (1993), "Limitations of conventional analysis of consolidation settlement", Journal of GeotechnicalEngineering ASCE, 119(9):1333-1359. Hassan, A. M., and Abu Bakr, A. (2006) "Evaluation of Permeability and Compressibility Variation in Consolidation of Clays" 2nd International Conference on Problematic Soils, Petaling, Jaya Malaysia, pp: 155-162. Krishnamoorthy (2008), "Consolidation Analysis Using Finite Element Method", The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG), Goa, India. Menéndeza, C., Nieto, P.J.G., Ortega, F.A., and Bello, A., (2009), "Mathematical modelling and study of the consolidation of an elasticsaturated soil with an incompressible fluid by FEM", Mathematical and Computer Modelling 49 (2009) 2002_2018. Sam, H. (2007), "Applied Soil Mechanics", John Wiley and Sons Inc., Hoboken, New Jersey, USA. Schofield,  A.  N.  and  Worth,  C.  P.  (1968),  "Critical  State  Soil  Mechanics",  McGraw  Hill, 

London. 

Roscoe,  K.H.  and  Burland,  J.,  B.  (1968),  "On  the  Generalised  Behavior  of  Wet  Clays" 

Engineering Plasticity, Cambridge University Press: 535‐609. 

Tang, X., Cheng, B. N., and Shen, H., (2013), "Closed-form solution for consolidation of three-layer soil witha vertical drain system", Geotextiles and Geomembranes 36 (2013) 81-91.  Terzaghi, K. (1943) "Theoretical Soil Mechanics, John Wiley @ Sons, New York, USA. 

Terzaghi, K., Peck, R.B., 1948. Soil Mechanics in Engineering Practice.Wiley, New York. 

Zheng, J-J., Lu, Y-E, Yin, J-H, and Guo, J. (2010), "Radial consolidation with variable compressibility and permeability followingpile installation", Computers and Geotechnics 37 (2010) 408–412.