Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 497161 14 pageshttpdxdoiorg1011552013497161
Research ArticleAxisymmetric Consolidation of Unsaturated Soils byDifferential Quadrature Method
Wan-Huan Zhou
Department of Civil and Environmental Engineering Faculty of Science and Technology University of Macau Macau
Correspondence should be addressed to Wan-Huan Zhou hannahzhouumacmo
Received 14 June 2013 Accepted 7 November 2013
Academic Editor Xi Frank Xu
Copyright copy 2013 Wan-Huan Zhou This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Axisymmetric consolidation in a sand drain foundation is a common problem in foundation engineering In unsaturated soils theexcess pore-water and pore-air pressures simultaneously change during the consolidation procedure and the solutions are not easyto obtain The present paper uses the differential quadrature method (DQM) for axisymmetric consolidation of unsaturated soilsin a sand drain foundation The radial seepage of sand drain foundation is considered based on the framework of Fredlundrsquos one-dimensional consolidation theory in unsaturated soils With the use of Darcyrsquos law and Fickrsquos law the polar governing equations ofexcess pore-air and pore-water pressures of axisymmetric consolidation are derived By using DQM the two governing equationsare transformed into two sets of ordinary differential equationsThen the solutions of excess pore-water and pore-air pressures canbe obtained by Rong-Kutta method The DQM solution can be used to deal with the case of nonuniform initial pore-air and pore-water distributions Finally case studies are presented to investigate the behavior of axisymmetric consolidation of unsaturatedsoils The convergence analysis and average degree of consolidation the settlements in radial and vertical direction and the effectsof different initial excess pore pressure distributions are presented and discussed in this paper
1 Introduction
As a common phenomenon of geotechnical engineeringconsolidation is a process of decreasing soil volume whensoil is subjected to increased stress Terzaghi [1] establisheda classical theory for the analysis of one-dimensional (1-D)consolidation in saturated soils which is still widely usedin engineering practice But in real cases soils related toengineering are usually in a state of unsaturation and aredefined by more characteristics than saturated soils such asthe coupled dissipation of pore-water and pore-air pressuresduring consolidation In the case of unsaturated soil theexcess pore-water pressure and pore-air pressure changesimultaneously during the consolidation process
For the consolidation of unsaturated soil Biot [2] deriveda general theory of consolidation for unsaturated soil whichhad occluded air bubbles The theory provides a couplingbetween the magnitude and progress of settlement Blight [3]presented a consolidation equation for the air phase wherethe air was in a dry rigid and unsaturated soil Based onBiotrsquostheory of consolidation Scott [4] derived a consolidation
equation for unsaturated soils with occluded air bubbles bychanging void ratio and degree of saturation SubsequentlyBarden [5] presented an analysis for 1-D consolidation ofcompacted unsaturated clay Fredlund and Hasan [6] pro-posed a 1-D consolidation theory for unsaturated soils Thisformulation is based on two continuity partial differentialequations one for the water phase and the other for the airphase which have to be solved simultaneously to give waterand air pressures at any time and elevation throughout thesoil According to Fredlundrsquos consolidation theory [7] withthe use of Darcyrsquos law and Fickrsquos law Qin et al [8] obtaineda semianalytical solution for consolidation of unsaturatedsoils with free drainage well They obtained the excess pore-air and pore-water pressures and the soil layer settlementin the Laplace transformed domains by utilizing the Besselfunction More recently Zhou et al [9] presented a simpleanalytical solution to 1-D consolidation for unsaturated soilwhich can be degenerated into Terzaghi consolidation solu-tion for fully saturated condition
As the consolidation equations of unsaturated soil arecoupled the analytical solutions cannot be easily obtained
2 Mathematical Problems in Engineering
Therefore numerical solutions such as the finite elementmethod (FEM) and the finite difference method (FDM) aretraditionally employed These two methods approximate thepartial derivatives of a function at a grid point only by using alimited number of function values in the vicinity of that gridpoint The accuracy and stability of these methods dependon the size of the grid spacing Compared with these twonumerical methods differential quadrature method (DQM)is a more efficient numerical method for the rapid solution oflinear and nonlinear partial differential equations involvingone dimension or multiple dimensions The fundamentalidea of DQM is that the derivatives of each node in acontinuous function can be expressed as a weighted linearsum of function values at all grid points Malik and Civan[10] have made a comprehensive comparison for linear andnonlinear convection-diffusion-reaction problems and haveshown that theDQM is superior to FEMandFDM innumeri-cal accuracy as well as computational efficiencyMany studieshave successfully employed DQM for solving engineeringproblems [11 12] Examples for the applications of DQM insolving complex problems in geotechnical engineering arepresented in these literatures [13ndash16] among others
In this paper DQM is introduced into the analysis ofaxisymmetric consolidation of unsaturated soils in a sanddrain foundation Following Fredlundrsquos one-dimensionalconsolidation theory of unsaturated soils with the use ofDarcyrsquos law and Fickrsquos law the polar governing equations ofexcess pore-air pressure and pore-water pressure of axisym-metric consolidation are obtained The behaviors of axisym-metric consolidation of unsaturated soils in the sand drainfoundation have been analyzedThe convergence analysis andaverage degree of consolidation the effects of settlement inradial and vertical directions the effects of different initialexcess pore-air and pore-water pressure distributions and theeffects of different boundary conditions are presented anddiscussed
2 Mathematical Model andGoverning Equations
21 Mathematical Model for Axisymmetric Consolidation ofUnsaturated Soils As shown in Figure 1 the unsaturated soillayer is described in a schematic diagram and thickness119867 Figures 1(a) and 1(b) are the sectional view and verticalview of the unsaturated soil layer respectively The radii ofsoil layer and sand drain are 119903
119890and 119903
119908 A surcharge 119902 is
applied on the top surface of the soil layer 119896119908and 119896
119886are the
permeability coefficients of water and air respectively Thecoordinated origin is selected at the center of the top surfaceand the 119911 coordinate is positive downward In this paper thevertical prismatic blocks of soil surrounding the sand drainsare simulated by cylindrical blocks of radius 119878 Square patternand triangular pattern are two common distributions of sanddrains For instance if square pattern is adopted as shown inFigure 1(c) then 119878 = 0564119903
119890 The basic assumptions are the
same as those of Fredlundrsquos one-dimensional consolidationtheory for unsaturated soils and the other assumptions arelisted as follows
(1) The seepage of water and air phases is considered onlyin radial direction
(2) The top and bottom surfaces are impermeable and theright boundary surface is impermeable or impeded
(3) The permeability coefficients of water and air phasesare assumed to be constant
(4) In the process of consolidation the strain is smallenough and the density of water is assumed to beconstant
22 Governing Equations The net flux of water through thesoil layer is computed from the volume of water entering andleaving the soil layer within a period of time with respect toDarcyrsquos law in the polar coordinate system
120597 (1198811199081198810)
120597119905=119896119908
120574119908
(1205972
119906119908
1205971199032+1
119903
120597119906119908
120597119903) (1)
where 1198810is the initial total volume of the soil 120597(119881
1199081198810)120597119905
is the net flux of water per unit volume of the soil 119906119908is the
excess pore-water pressure 120574119908is the water unit weight and 119905
is the time variableThe net flux of water per unit volume of the soil can
be obtained by differentiating the water phase constitutiverelation with respect to time
120597 (1198811199081198810)
120597119905= 119898119908
1119896
120597 (119902 minus 119906119886)
120597119905+ 119898119908
2
120597 (119906119886minus 119906119908)
120597119905 (2)
where 1198981199081119896
is the coefficient of water volume change withrespect to a change in the net normal stress (119902minus119906
119886)1198981199082is the
coefficient of water volume change with respect to a change inmatric suction (119906
119886minus119906119908) and119906
119886is the excess pore-air pressure
For constant loading 119889119902119889119905 = 0 substituting (1) into (2)the governing equation for the water phase can be written as
120597119906119908
120597119905= minus119862119908
120597119906119886
120597119905minus 119862119908
V (1205972
119906119908
1205971199032+1
119903
120597119906119908
120597119903) (3)
where 119862119908= (119898119908
1119896minus 119898119908
2)119898119908
2and 119862119908V = 119896
119908120574119908119898119908
2
The net flux of air through the soil layer is computedfrom the volume of air entering and leaving the soil layerwithin a period of time with respect to Fickrsquos law in the polarcoordinate system
120597 (1205881198811198861198810)
120597119905=119896119886
119892(1205972
119906119886
1205971199032+1
119903
120597119906119886
120597119903) (4)
where 120588 is the density of air phase 120597(1205881198811198861198810)120597119905 is the net
mass rate of air flow per unit volume of the soil and 119892 is thegravitational acceleration
The total volume change of an unsaturated soil can beassumed to be small during the consolidation process Thevolume of air119881
119886can be related to the volume-mass properties
of the soil 119881119886= (1 minus 119878
0)11989901198810 where 119899
0and 119878
1199030are the
initial porosity and initial degree of saturation before loading
Mathematical Problems in Engineering 3
re
q
0
H
r
rw
Z
ka kw
Impermeable
Impermeable
Perm
eabl
e
Sand
dra
in
Impe
rmea
ble
(impe
ded)
(a)
re
r
rw
ka k
w
Permeable
Sand drain
Impermeable(impeded)
(b)
re
rw
S
S
Square patternre = 0564S
(c)
Figure 1 (a) The sectional view of sand drain foundation (b) the vertical view of sand drain foundation and (c) the distribution of sanddrains (square pattern)
According to Boylersquos law assuming there is no initial excessair pressure in the soil before loading we have [17 18]
120597 (1198811198861198810)
120597119905=
119896119886119877119879
1198921199060
119886119872
(1205972
119906119886
1205971199032+1
119903
120597119906119886
120597119903)
minus119906atm1198990 (1 minus 1198780)
(1199060
119886)2
120597119906119886
120597119905
(5)
where 119877 is the universal air constant (8314 JmolK) 119879 =
29315K is the absolute temperature119872 = 29 kgkmol is themolecular mass of air 119906119886
0is absolute pore-air pressure (ie
119906119886
0= 119906119886
0+119906
atm) and 119906atm = 1013 kPa is atmospheric pressure
The derivative of the air phase constitutive relation withrespect to time is equal to the net flux of air per unit volumeof the soil
120597 (1198811198861198810)
120597119905= 119898119886
1119896
120597 (119902 minus 119906119886)
120597119905+ 119898119886
2
120597 (119906119886minus 119906119908)
120597119905 (6)
where1198981198861119896is the coefficient of air volume change with respect
to a change in the net normal stress (119902 minus 119906119886
) and 119898119886
2is the
coefficient of air volume change with respect to a change inmatric suction (119906119886 minus 119906119908)
Substituting (5) into (6) the governing equation for theair phase under constant loading 119889119902119889119905 = 0 can be written as
120597119906119886
120597119905= minus119862119886
120597119906119908
120597119905minus 119862119886
V (1205972
119906119886
1205971199032+1
119903
120597119906119886
120597119903) (7)
4 Mathematical Problems in Engineering
where 119862119886= 119898119886
2(119898119886
1119896minus119898119886
2minus 119906
atm1198990(1 minus 119878
1199030)(119906119886
0)2
) and 119862119908V =
119896119886119877119879119892119906
119886
0119872(119898119886
1119896minus 119898119886
2minus 119906
atm1198990(1 minus 119878
1199030)(119906119886
0)2
)Further two transformed governing equations of water
V (1minus119862119886119862119908)119860119886 =minus119862119886
V(1 minus 119862119886119862119908) and 119860119908 = minus119862119886119862119908
V (1 minus 119862119886119862119908)
23 Boundary and Initial Conditions As shown in Figure 1the wall of sand drain is permeable and lateral boundarysurface is impermeable or impeded the boundary conditionsfor radial consolidation are
1198961198861199030 (9) reflects that the lateral boundary surface is imperme-
able or impeded respectively 1198960119908and 1198960
119886are the coefficients
of permeability for water and air at lateral boundary respec-tively 119903
0is the thickness of lateral boundary
The initial condition can be written as
119906119886
(119903 0) = 119906119886
0 119906
119908
(119903 0) = 119906119908
0 (11)
where 119906119908
0and 119906
119886
0are the initial pore-water and pore-air
pressure distributions
3 Differential Quadrature Formulation
DQM is a numerical solution technique for initial andorboundary value problems proposed by Bellman et al [19]and Bellman and Casti [20] According to DQM a partialderivative of a function with respect to a variable can beapproximated by a weighted linear sum of the functionvalues at given discrete points Chen et al [13] employedDQM to solve one-dimensional consolidation problems inmultilayered soils
To show the mathematic detail of DQM consider afunction 120593 = 120593(119909) on the domain 0 le 119909 le 119886 and thedomain is dispersed as119873 pointsThen the general differentialquadrature approximation of the function at the 119894th discretepoint is given by
are the weighting coefficient of 119903th derivative 119903 lt119873
This paper adopts a method derived by Quan and Chang[21] which uses Lagrange polynomial to determine theweighting coefficients
119863(1)
119894119896=
prod119873
V=1V = 119894 (119909119894 minus 119909V)
(119909119894minus 119909119896)prod119873
V=1119896 = 119894 (119909119896 minus 119909V)
119894 119896 = 1 2 119873 (119896 = 119894)
119863(119903)
119894119894= minus
119873
sum
V=1V = 119894
119863(119903)
119894V 119894 = 1 2 119873 1 le 119903 le 119873 minus 1
119863(119903)
119894119896= 119903[119863
(119903minus1)
119894119894119863(1)
119894119896minus
119863(119903minus1)
119894119896
119909119894minus 119909119896
]
119894 119896 = 1 2 119873 (119896 = 119894) 2 le 119903 le 119873 minus 1
(13)
The soil layer is dispersed in the radial direction andthe number of discrete points is 119873 Then in order tosolve the equations conveniently the local coordinate 120576 isintroduced The relationship between local coordinate andintegral coordinate is
119903 = (05 minus 120576) 1199031+ (05 + 120576) 119903
119873 (minus05 le 120576 le 05) (14)
where 119903 is the integral coordinate of soil layer and 1199031and 119903119873
are the radius values of 1th and119873th point respectivelyThe differential of (14) can be expressed as
According to (19) and (20) (18) can be rewritten as
120597V119908
120597119879=119882119886
119866(AV119886 + 1
119903BV119886) + 119882
119908
119866(AV119908 + 1
119903BV119908)
120597V119886
120597119879=119860119886
119866(AV119886 + 1
119903BV119886) + 119860
119908
119866(AV119908 + 1
119903BV119908)
(23)
where V119908 = [119906119908
2 119906119908
3 119906
119908
119873minus1]1015840
V119886 = [119906119886
2 119906119886
3 119906
119886
119873minus1]1015840
A = A1 + A2 and B = B1 + B2
A1 =[[[
[
119863(2)
22sdot sdot sdot 119863
(2)
2(119873minus1)
d
119863(2)
(119873minus1)2sdot sdot sdot 119863
(2)
(119873minus1)(119873minus1)
]]]
]
B1 =[[[
[
119863(1)
22sdot sdot sdot 119863
(1)
2(119873minus1)
d
119863(1)
(119873minus1)2sdot sdot sdot 119863
(1)
(119873minus1)(119873minus1)
]]]
]
A2 = Mminus1 timesQ times
[[[
[
119863(2)
21119863(2)
2119873
119863(2)
(119873minus1)1119863(2)
(119873minus1)119873
]]]
]
B2 = Mminus1 timesQ times
[[[
[
119863(1)
21119863(1)
2119873
119863(1)
(119873minus1)1119863(1)
(119873minus1)119873
]]]
]
119866 =119896119908
minus120574119908119898119904
1119896
M = [1 0
119863(1)
1198731119863(1)
119873119873
]
Q = minus[0 sdot sdot sdot 0
119863(1)
1198732sdot sdot sdot 119863
(1)
119873(119873minus1)
]
(24)
Therefore the governing equations of water and air aretranslated into two sets of ordinary differential equations(ODEs) The solutions of ODEs can be obtained by usingRong-KuttamethodThenwe can apply the solutions into thefollowing formulas the average degree of consolidation theradial displacement and the vertical settlement
In unsaturated soils the average degree of consolidationcan be divided into two parts the average degree of consoli-dation with respect to water phase119880119908 and the average degreeof consolidation with respect to air phase 119880119886 To obtain theaverage degree of consolidation two formulations are givenby Fredlund and Rahardjo [7] Consider
According to the two stress-state variable approaches [67] volume strain is represented by the following constitutiveequation for soil layer
120597120576V
120597119905= (119898119908
1119896+ 119898119886
1119896)120597 (119902 minus 119906
119886
)
120597119905+ (119898119908
2+ 119898119886
2)120597 (119906119886
minus 119906119908
)
120597119905
(27)
By integrating (26) with respect to time 119905 from 0 to 119905 weget the expression of volumetric strain 120576V
120576V = (119898119908
1119896+ 119898119886
1119896) [(119902 minus 119906
119886
) minus (119902 minus 119906119886
0)]
+ (119898119908
2+ 119898119886
2) [(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)]
(28)
Volumetric strain consists of vertical strain and horizon-tal strain For the axisymmetric consolidation it is assumed
in this paper that one-third of the volume strain is contributedfrom vertical strain and two-thirds are fromhorizontal strainSo the radial displacement and vertical settlement can beobtained respectively Consider
119878ℎ=2
3(119898119908
1119896+ 119898119886
1119896) int
119903119890
0
[(119902 minus 119906119886
) minus (119902 minus 119906119886
0)] 119889119903
+2
3(119898119908
2+ 119898119886
2) int
119903119890
0
[(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)] 119889119903
119878V =1
3(119898119908
1119896+ 119898119886
1119896) int
119867
0
[(119902 minus 119906119886
) minus (119902 minus 119906119886
0)] 119889119911
+1
3(119898119908
2+ 119898119886
2) int
119867
0
[(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)] 119889119911
(29)
8 Mathematical Problems in Engineering
0
005
01
015
02
025
03
035
04
045
10minus3
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 10 mr = 12 m
uw
q
Figure 7 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
0
001
002
003
004
005
006
10minus3
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 10 mr = 12 m
uaq
Figure 8 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
4 An Example and Convergence Analysis
In this section the convergence analysis the average degree ofconsolidation and the effects of displacements in radial andvertical directions are discussed by using a simple example
As shown in Figure 1 a vertical loading 119902 = 100 kPa isapplied on the top surface of the soil layerThe lateral bound-ary is considered as impermeable Considering a uniforminitial excess pore-water and pore-air pressure distribution119906119886
0= 5 kPa 119906119908
0= 40 kPa Other parameters are 119899
0= 05 119878
1199030=
08 119896119908= 10minus10ms119898119908
1119896= minus015times10
minus4 kPaminus11198981199082= minus061times
10minus4 kPaminus1119898119886
1119896= 006 times 10
minus4 kPaminus11198981198862= 026 times 10
minus4 kPaminus1119903119908= 02m 119903
119890= 18m and 119906atm = 1013 kPa By applying the
DQM solution presented above the results of axisymmetric
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
1
08
06
04
02
0
uw
q
Figure 9 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
014
012
01
008
006
004
002
0
uaq
Figure 10 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
consolidation of unsaturated soils in a sand drain foundationwere obtained
Tables 1 and 2 list the solutions of excess pore-water andpore-air pressures at radius 119903 = 1m at different time factorsand different equally spaced discrete points respectively Theratio of permeability coefficient is 119896
119886119896119908= 01 From Tables
1 and 2 the accuracy of solutions increases when the numberof discrete point becomes big It is obvious that 9 equallyspaced grid points are sufficient to obtain the convergedresults of excess pore-water and pore-air pressuresThereforethe number of discrete points119873 = 9 is adopted in most casestudies
Figure 2 shows the curves of average degree of consoli-dation with respect to water and air phases when 119873 = 9The ratio of permeability coefficient is 119896
119886119896119908
= 1 FromFigure 2 it can be observed that the consolidation of air
Mathematical Problems in Engineering 9
Table 1 Different excess pore-water pressures 119906119908119902 under time factor 119879 with different discrete points119873 (119896
1 times 10minus6 0050 0050 0050 0050 0050 0050 0050
1 times 10minus4 0050 0050 0050 0050 0050 0050 0050
5 times 10minus2 0048 0048 0048 0048 0048 0048 0048
1 times 10minus1 0046 0045 0045 0045 0045 0045 0045
2 times 10minus1 0042 0041 0040 0040 0040 0040 0010
5 times 10minus1 0035 0033 0032 0032 0032 0031 0031
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 11 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
phase and water phase begins when 119879 = 10 times 10minus3 When
119879 = 10 times 100 and 119879 = 10 times 10
15 the consolidation ofair phase and water phase is almost finished respectivelyAt the early stages soil consolidation is mainly caused bythe dissipation of excess pore-air pressure But in the laterstages soil consolidation is mainly caused by the dissipationof excess pore-water pressure
Figure 3 shows the radial displacement development withtime and Figure 4 describes the settlement of top surface(119867 = 10m) at different radii with time factor 119879 The numberof discrete points is 119873 = 9 The ratio of permeabilitycoefficient is 119896
119886119896119908
= 01 From Figure 4 we can see thatthe soil settles earlier at the points with smaller radius thatis closer to the sand drain Figure 5 shows the settlement atdifferent heights of soil layer at radius 119903 = 1m with time
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 12 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
factor 119879 When the height of soil layer119867 is equal to 4m thesettlement is almost zero
5 Cases of Different Initial andBoundary Conditions
As the initial conditions do not need to be constant in thepresentDQMsolution it is easily used to analyze nonuniforminitial pore-water and pore-air distribution problems In thispart four different initial pore-water and pore-air distribu-tions and some different boundary conditions are consideredA vertical loading 119902 = 100 kPa is applied on the top surfaceof soil layer Other parameters are 119899
0= 05 119878
1199030= 08 119896
119908=
10minus10ms 119903
119908= 02m 119903
119890= 18m119898119908
1119896= minus015 times 10
minus4 kPaminus1
10 Mathematical Problems in Engineering
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 13 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 14 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
119898119908
2= minus061 times 10
minus4 kPaminus1 1198981198861119896
= 006 times 10minus4 kPaminus1 119898119886
2=
026 times 10minus4 kPaminus1 and 119906atm = 1013 kPa
51 Effect of Different Initial Pore-WaterAir Pressure Distribu-tions In this section in order to make the curves smooththe number of discrete points is chosen to be 119873 = 17 Thelateral boundary is considered as impermeable and the ratioof permeability coefficient with respect to air and water is119896119886119896119908= 01
Four different initial pore-waterair distributions aretaken into account In Figure 6(a) the case of uniform initialpore-air and pore-water pressures assumes that 119906119886
0= 5 kPa
and 1199061199080= 40 kPa Figure 6(b) presents the linear increasing
initial pore-air and pore-water distributions The lateral-skewed distributions initial pore-waterair pressures is shown
in Figures 6(c) and 6(d) and they are represented by thefollowing equations
119906119908
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
100105119887
119906119886
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
8 times 100105119887
(30)
where 119887 is the variable which can control the spread orskewness of the curves in Figures 6(c) and 6(d)
Under the condition shown in Figure 6(a) the dissipationcurves of the excess pore-water and pore-air pressure atdifferent distances from the sand drain are presented inFigures 7 and 8 respectively The excess pore-air and pore-water pressures dissipate faster at the points with smallerradius that is closer to the sand drain As the intervalbetween the two curves becomes smaller from 119903 = 06m to119903 = 12m it can be seen that the effect of the distance on thedissipation becomes less obvious
In Figure 6(b) the initial pore-air and pore-water pres-sures are assumed not to be constants and to be linearincreasing when the radius increases The initial pore-airpressure is assumed one-eighth of the initial pore-water pres-sure In other words from 119903 = 04m to 119903 = 16m the initialpore-water pressure will increase from 30 kPa to 110 kPa andthe initial pore-air pressure will increase from 375 kPa to1375 kPa Figures 9 and 10 show the dissipation of excesspore-water and pore-air pressures at different distances fromthe sand drain under Figure 6(b) distributions
Comparing Figures 7 and 9 it is found that the excesspore-water pressure dissipates completely at almost the sametime With the change of the initial pore-water pressuredistribution the time of dissipation with respect to wateris not significantly affected Comparing Figures 8 and 10it can be seen that the excess pore-air pressure dissipatescompletely at almost the same time The initial pore-airpressure distribution has little influence on the dissipationtime of air phase
Under the conditions of Figures 6(c) and 6(d) thedissipation curves of the excess pore-water and pore-airpressures for initial distributions where 119887 = 2 6 are shownin Figures 11 12 13 and 14 From these figures whatever theinitial pore-water and pore-air pressures are the excess pore-water and pore-air pressures dissipate completely at almostthe same timeThe plateau period is longer at the points withlarger radius
Here excess pore pressure isochrones for the above fourinitial conditions are presented in Figures 15 and 16 Figures15(a) and 16(a) show the excess pore-water and pore-airpressure isochrones when the initial excess pore pressure isuniform at all radii In Figures 15(b) and 16(b) the initialexcess pore pressure distributions are linearly increasingThepore pressure isochrones for lateral-skewed initial distribu-tions are shown in Figures 15(c) 16(c) 15(d) and 16(d)respectively From Figures 15(a) and 16(a) as the initial excesspore pressure is constant the peak of curves in Figure 15(a)approaches to 04 and the peak of curves in Figure 13(a)approaches to 005 Comparing Figures 15(a) and 15(b) as
Mathematical Problems in Engineering 11
0
005
01
015
02
025
03
035
04
02 04 06 08 1 12 14 16 18r (m)
02
1
2
5
05
T = 10
uw
q
(a)
02 04 06 08 1 12 14 16 18r (m)
1
1
2
5
09
08
07
06
05
05
04
03
02
02
01
01
0
T = 10
uw
q
(b)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
011
09
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(c)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
0109
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(d)
Figure 15Normalized pore-water pressure isochrones for different initial excess pore-water pressure distributions (119896119886119896119908= 01) (a) uniform
(b) linearly increasing (c) lateral skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
the initial excess pore-water pressure is linearly increasingthe skewness is more evident during early stages of dissipa-tion in Figure 15(b) But in the later stages the influence ofinitial excess pore-water pressure becomes less significant Soare the isochrones of the excess pore-air pressure in Figures16(a) and 16(b) Figures 15(c) 16(c) 15(d) and 16(d) showthe excess pore-water and pore-air pressure isochrones forthe lateral-skewed initial distribution when 119887 = 2 6 It isseen that some intersections appear in the isochrones duringthe early stages It means that a redistribution of excesspore pressures occurs toward the region of minimal initialexcess pore pressure during the early stages With the timeincreasing the isochrones of excess pore pressures becomeregular
52 Effect of Different Boundary Conditions In this partthe effect of different lateral boundary conditions is studiedUniform initial excess pore-waterair pressure distributions119906119886
0= 5 kPa and 119906119908
0= 40 kPa are adopted Four different ratios
119896119886119896119908are considered that is 01 1 10 and 100 with 119896
119908=
10minus10ms Two different drainage conditions are analyzed
lateral surface is impervious and lateral surface is impededFor the impeded surface 119886
119908and 119886
119886in (9) are chosen to be
05Figures 17 and 18 shows the dissipation curves for the
water and air pressures at radius 119903 = 12m under differentboundary conditions First the excess pore-water and pore-air pressures dissipate faster in impeded surface than that inimpermeable surface Second 119896
119908is the same in the cases
12 Mathematical Problems in Engineering
02 04 06 08 1 12 14 16 18r (m)
01
02
05
T = 2
006
005005
004
003
002
002
001
0
1
uaq
(a)
02 04 06 08 1 12 14 16 18r (m)
014
012
0101
02
05008
006
004
002
0
T = 2
005002
1
uaq
(b)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(c)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(d)
Figure 16 Normalized pore-air pressure isochrones for different initial excess pore-air pressure distributions (119896119886119896119908= 01) (a) uniform (b)
linearly increasing (c) lateral-skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
of four 119896119886119896119908values It is observed from the results that
the bigger 119896119886119896119908
is the faster the excess pore-water andpore-air pressures dissipate Third the dissipation curves ofthe excess pore-air and pore-water pressures have almostthe same shape under different values of 119896
119886119896119908 Comparing
Figure 17 with Figure 18 when the excess pore-air pressuredissipates almost completely the dissipation of excess pore-water pressure enters into a plateau period The bigger 119896
119886119896119908
is the earlier the excess pore-air pressure dissipates com-pletely Therefore it is shown that the value of 119896
119886affects the
dissipation of pore-water pressure during the consolidationprocess
6 Conclusion
This paper obtains a general solution to the axisymmet-ric consolidation of unsaturated soils by using differen-tial quadrature method (DQM) based on Fredlundrsquos one-dimensional consolidation theory for unsaturated soil Withthe use of DQM the two governing equations are trans-formed into two sets of ordinary differential equationsThe solutions to the transformed differential equations areobtained by Rong-Kutta method under different initial andboundary conditions A case study has been presented forthe analysis of unsaturated consolidation in a sand drain
Mathematical Problems in Engineering 13
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
045
04
035
03
025
02
015
01
005
0
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
uw
q
Figure 17 Variation of excess pore-water pressures 119906119908119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
007
006
005
004
003
002
001
0
uaq
Figure 18 Variation of excess pore-air pressures 119906119886
119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
foundation The convergence analysis the average degreeof consolidation for water and air phases the settlementin radial and vertical directions effects of different initialexcess pore-air and pore-water pressure distributions andeffects of different boundary conditions have been presentedFrom the analyses some conclusions can be drawn (a)through compiling the programs it is found that the DQsolution delivers accurate and stable results for unsaturatedsoil consolidation Due to the uniformmatrix structure of theDQ formulas it is easy to compile computer programs whenconsidering complicated conditions such as nonuniformpore-waterair distribution conditions (b) soil consolidationismainly caused by the dissipation of excess pore-air pressurein the early stages and by the dissipation of excess pore-water pressure in the later stages (c) the bigger the ratio
of the permeability coefficients for water and air is thefaster the excess pore-water and pore-air pressures dissipate(d) The excess pore-air and pore-water pressures dissipatefaster at the points closer to the sand drain (e) the initialdistribution has some effects on the early consolidationprocess of unsaturated soil and boundary condition has asignificant effect on the whole consolidation process
Notation
119862119908
V The consolidation coefficient of water119862119886
V The consolidation coefficient of air119863(119903)
119894119896 The weighting coefficient
120576V Volumetric strain119892 The gravitational acceleration119867 The thickness of soil layer119896119886 The permeability coefficients of air
1198960
119886 The coefficients of permeability for water
at lateral boundary119896119908 The permeability coefficients of water
1198960
119908 The coefficients of permeability for air at
lateral boundary119898119886
1119896 The coefficient of air volume change withrespect to a change in the net normal stress
119898119908
1119896 The coefficient of water volume changewith respect to a change in the net normalstress
119898119886
2 The coefficient of air volume change with
respect to a change in matric suction119898119908
2 The coefficient of water volume changewith respect to a change in matric suction
1198990 The initial porosity
119873 The discrete point119875V A vacuum pressure120588 The density of air phase119902 The vertical loading1199030 The thickness of lateral boundary
119903119890 The radius of soil layer
119903119908 The radius of sand drain
119903120572 The radius value of 120572th point
120574119908 The unit weight of water phase
119877 The universal air constant119878 The distance between two sand drains1198781199030 The initial degree of saturation
119878V Settlement119878ℎ The radial displacement
119905 Time119879 The time factor119906119886 The excess pore-air pressure119906atm Atmospheric pressure119906119886
0 The initial excess pore-air pressure
119906119908 The excess pore-water pressure119906119908
0 The initial excess pore-water pressure
119880119886 The average degree of consolidation with
respect to air119880119908 The average degree of consolidation with
respect to water119881119886 The air volume of soil layer
14 Mathematical Problems in Engineering
119881119908 The water volume of soil layer
1198810 Initial total volume of the soil
Acknowledgments
The author gratefully acknowledges the financial supportfrom the Macau Science and Technology Development Fund(Grant no FDCT0112013A1) and the University of MacauResearch Fund (Grants nos MYRG189(Y2-L3)-FST11-ZWHand MYRG067(Y2-L2)-FST12-ZWH)
References
[1] K Terzaghi Theoretical Soil Mechanics Wiley New York NYUSA 1943
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] G E Blight Strength and consolidation characteristics of com-pacted soils [PhD thesis] University of London London UK1961
[4] R F Scott Principles of Soil Mechanics Addison Wesley Pub-lishing Company Boston Mass USA 1963
[5] L Barden ldquoConsolidation of compacted and unsaturated clayrdquoGeotechnique vol 15 no 3 pp 267ndash286 1965
[6] D G Fredlund and J U Hasan ldquoOne-dimensional consolida-tion theory unsaturated soilsrdquo Canadian Geotechnical Journalvol 16 no 3 pp 521ndash531 1979
[7] D G Fredlund andH Rahardjo Soil Mechanics for UnsaturatedSoils John Wiley and Son 1993
[8] A Qin D Sun L Yang and Y Weng ldquoA semi-analyticalsolution to consolidation of unsaturated soils with the freedrainage wellrdquo Computers and Geotechnics vol 37 no 7-8 pp867ndash875 2010
[9] W H Zhou L S Zhao and X B Li ldquoA simple analyticalsolution to one-dimensional consolidation for unsaturated soilrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics 2013
[10] M Malik and F Civan ldquoA comparative study of differentialquadrature and cubature methods vis-a-vis some conventionaltechniques in context of convection-diffusion-reaction prob-lemsrdquo Chemical Engineering Science vol 50 no 3 pp 531ndash5471995
[11] P Malekzadeh M R Golbahar Haghighi and M M AtashildquoOut-of-plane free vibration of functionally graded circularcurved beams in thermal environmentrdquo Composite Structuresvol 92 no 2 pp 541ndash552 2010
[12] P Malekzadeh M R G Haghighi and M M Atashi ldquoFreevibration analysis of elastically supported functionally radedannular plates subjected to thermal environmentrdquo Meccanicavol 46 no 5 pp 893ndash913 2011
[13] R P Chen W H Zhou H Z Wang and Y M Chen ldquoOne-dimensional nonlinear consolidation of multi-layered soil bydifferential quadrature methodrdquo Computers and Geotechnicsvol 32 no 5 pp 358ndash369 2005
[14] R P Chen W H Zhou H Z Wang and Y M Chen ldquoStudyon one-dimensional nonlinear consolidation of multi-layeredsoil using differential quadrature methodrdquo Chinese Journal ofComputational Mechanics vol 22 no 3 pp 310ndash315 2005
[15] H Z Wang R P Chen W H Zhou and Y M ChenldquoComputation of 1-D nonlinear consolidation in double-layer
foundation by using differential quadrature methodrdquo Journal ofHydraulic Engineering vol 4 pp 8ndash14 2004
[16] W H Zhou and L S Zhao ldquoOne-dimensional consolidationof unsaturated soil subjected to time-dependent loading undervarious initial and boundary conditionsrdquo ASCE InternationalJournal of Geomechanics 2013
[17] E Conte ldquoConsolidation analysis for unsaturated soilsrdquo Cana-dian Geotechnical Journal vol 41 no 4 pp 599ndash612 2004
[18] E Conte ldquoPlane strain and axially symmetric consolidation inunsaturated soilsrdquo International Journal of Geomechanics vol 6no 2 pp 131ndash135 2006
[19] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 no 1 pp40ndash52 1972
[20] R Bellman and J Casti ldquoDifferential quadrature and long-termintegrationrdquo Journal of Mathematical Analysis and Applicationsvol 34 no 2 pp 235ndash238 1971
[21] J R Quan andC T Chang ldquoNew insights in solving distributedsystem equations by the quadrature method I analysisrdquo Com-puters and Chemical Engineering vol 13 no 7 pp 779ndash7881989
Therefore numerical solutions such as the finite elementmethod (FEM) and the finite difference method (FDM) aretraditionally employed These two methods approximate thepartial derivatives of a function at a grid point only by using alimited number of function values in the vicinity of that gridpoint The accuracy and stability of these methods dependon the size of the grid spacing Compared with these twonumerical methods differential quadrature method (DQM)is a more efficient numerical method for the rapid solution oflinear and nonlinear partial differential equations involvingone dimension or multiple dimensions The fundamentalidea of DQM is that the derivatives of each node in acontinuous function can be expressed as a weighted linearsum of function values at all grid points Malik and Civan[10] have made a comprehensive comparison for linear andnonlinear convection-diffusion-reaction problems and haveshown that theDQM is superior to FEMandFDM innumeri-cal accuracy as well as computational efficiencyMany studieshave successfully employed DQM for solving engineeringproblems [11 12] Examples for the applications of DQM insolving complex problems in geotechnical engineering arepresented in these literatures [13ndash16] among others
In this paper DQM is introduced into the analysis ofaxisymmetric consolidation of unsaturated soils in a sanddrain foundation Following Fredlundrsquos one-dimensionalconsolidation theory of unsaturated soils with the use ofDarcyrsquos law and Fickrsquos law the polar governing equations ofexcess pore-air pressure and pore-water pressure of axisym-metric consolidation are obtained The behaviors of axisym-metric consolidation of unsaturated soils in the sand drainfoundation have been analyzedThe convergence analysis andaverage degree of consolidation the effects of settlement inradial and vertical directions the effects of different initialexcess pore-air and pore-water pressure distributions and theeffects of different boundary conditions are presented anddiscussed
2 Mathematical Model andGoverning Equations
21 Mathematical Model for Axisymmetric Consolidation ofUnsaturated Soils As shown in Figure 1 the unsaturated soillayer is described in a schematic diagram and thickness119867 Figures 1(a) and 1(b) are the sectional view and verticalview of the unsaturated soil layer respectively The radii ofsoil layer and sand drain are 119903
119890and 119903
119908 A surcharge 119902 is
applied on the top surface of the soil layer 119896119908and 119896
119886are the
permeability coefficients of water and air respectively Thecoordinated origin is selected at the center of the top surfaceand the 119911 coordinate is positive downward In this paper thevertical prismatic blocks of soil surrounding the sand drainsare simulated by cylindrical blocks of radius 119878 Square patternand triangular pattern are two common distributions of sanddrains For instance if square pattern is adopted as shown inFigure 1(c) then 119878 = 0564119903
119890 The basic assumptions are the
same as those of Fredlundrsquos one-dimensional consolidationtheory for unsaturated soils and the other assumptions arelisted as follows
(1) The seepage of water and air phases is considered onlyin radial direction
(2) The top and bottom surfaces are impermeable and theright boundary surface is impermeable or impeded
(3) The permeability coefficients of water and air phasesare assumed to be constant
(4) In the process of consolidation the strain is smallenough and the density of water is assumed to beconstant
22 Governing Equations The net flux of water through thesoil layer is computed from the volume of water entering andleaving the soil layer within a period of time with respect toDarcyrsquos law in the polar coordinate system
120597 (1198811199081198810)
120597119905=119896119908
120574119908
(1205972
119906119908
1205971199032+1
119903
120597119906119908
120597119903) (1)
where 1198810is the initial total volume of the soil 120597(119881
1199081198810)120597119905
is the net flux of water per unit volume of the soil 119906119908is the
excess pore-water pressure 120574119908is the water unit weight and 119905
is the time variableThe net flux of water per unit volume of the soil can
be obtained by differentiating the water phase constitutiverelation with respect to time
120597 (1198811199081198810)
120597119905= 119898119908
1119896
120597 (119902 minus 119906119886)
120597119905+ 119898119908
2
120597 (119906119886minus 119906119908)
120597119905 (2)
where 1198981199081119896
is the coefficient of water volume change withrespect to a change in the net normal stress (119902minus119906
119886)1198981199082is the
coefficient of water volume change with respect to a change inmatric suction (119906
119886minus119906119908) and119906
119886is the excess pore-air pressure
For constant loading 119889119902119889119905 = 0 substituting (1) into (2)the governing equation for the water phase can be written as
120597119906119908
120597119905= minus119862119908
120597119906119886
120597119905minus 119862119908
V (1205972
119906119908
1205971199032+1
119903
120597119906119908
120597119903) (3)
where 119862119908= (119898119908
1119896minus 119898119908
2)119898119908
2and 119862119908V = 119896
119908120574119908119898119908
2
The net flux of air through the soil layer is computedfrom the volume of air entering and leaving the soil layerwithin a period of time with respect to Fickrsquos law in the polarcoordinate system
120597 (1205881198811198861198810)
120597119905=119896119886
119892(1205972
119906119886
1205971199032+1
119903
120597119906119886
120597119903) (4)
where 120588 is the density of air phase 120597(1205881198811198861198810)120597119905 is the net
mass rate of air flow per unit volume of the soil and 119892 is thegravitational acceleration
The total volume change of an unsaturated soil can beassumed to be small during the consolidation process Thevolume of air119881
119886can be related to the volume-mass properties
of the soil 119881119886= (1 minus 119878
0)11989901198810 where 119899
0and 119878
1199030are the
initial porosity and initial degree of saturation before loading
Mathematical Problems in Engineering 3
re
q
0
H
r
rw
Z
ka kw
Impermeable
Impermeable
Perm
eabl
e
Sand
dra
in
Impe
rmea
ble
(impe
ded)
(a)
re
r
rw
ka k
w
Permeable
Sand drain
Impermeable(impeded)
(b)
re
rw
S
S
Square patternre = 0564S
(c)
Figure 1 (a) The sectional view of sand drain foundation (b) the vertical view of sand drain foundation and (c) the distribution of sanddrains (square pattern)
According to Boylersquos law assuming there is no initial excessair pressure in the soil before loading we have [17 18]
120597 (1198811198861198810)
120597119905=
119896119886119877119879
1198921199060
119886119872
(1205972
119906119886
1205971199032+1
119903
120597119906119886
120597119903)
minus119906atm1198990 (1 minus 1198780)
(1199060
119886)2
120597119906119886
120597119905
(5)
where 119877 is the universal air constant (8314 JmolK) 119879 =
29315K is the absolute temperature119872 = 29 kgkmol is themolecular mass of air 119906119886
0is absolute pore-air pressure (ie
119906119886
0= 119906119886
0+119906
atm) and 119906atm = 1013 kPa is atmospheric pressure
The derivative of the air phase constitutive relation withrespect to time is equal to the net flux of air per unit volumeof the soil
120597 (1198811198861198810)
120597119905= 119898119886
1119896
120597 (119902 minus 119906119886)
120597119905+ 119898119886
2
120597 (119906119886minus 119906119908)
120597119905 (6)
where1198981198861119896is the coefficient of air volume change with respect
to a change in the net normal stress (119902 minus 119906119886
) and 119898119886
2is the
coefficient of air volume change with respect to a change inmatric suction (119906119886 minus 119906119908)
Substituting (5) into (6) the governing equation for theair phase under constant loading 119889119902119889119905 = 0 can be written as
120597119906119886
120597119905= minus119862119886
120597119906119908
120597119905minus 119862119886
V (1205972
119906119886
1205971199032+1
119903
120597119906119886
120597119903) (7)
4 Mathematical Problems in Engineering
where 119862119886= 119898119886
2(119898119886
1119896minus119898119886
2minus 119906
atm1198990(1 minus 119878
1199030)(119906119886
0)2
) and 119862119908V =
119896119886119877119879119892119906
119886
0119872(119898119886
1119896minus 119898119886
2minus 119906
atm1198990(1 minus 119878
1199030)(119906119886
0)2
)Further two transformed governing equations of water
V (1minus119862119886119862119908)119860119886 =minus119862119886
V(1 minus 119862119886119862119908) and 119860119908 = minus119862119886119862119908
V (1 minus 119862119886119862119908)
23 Boundary and Initial Conditions As shown in Figure 1the wall of sand drain is permeable and lateral boundarysurface is impermeable or impeded the boundary conditionsfor radial consolidation are
1198961198861199030 (9) reflects that the lateral boundary surface is imperme-
able or impeded respectively 1198960119908and 1198960
119886are the coefficients
of permeability for water and air at lateral boundary respec-tively 119903
0is the thickness of lateral boundary
The initial condition can be written as
119906119886
(119903 0) = 119906119886
0 119906
119908
(119903 0) = 119906119908
0 (11)
where 119906119908
0and 119906
119886
0are the initial pore-water and pore-air
pressure distributions
3 Differential Quadrature Formulation
DQM is a numerical solution technique for initial andorboundary value problems proposed by Bellman et al [19]and Bellman and Casti [20] According to DQM a partialderivative of a function with respect to a variable can beapproximated by a weighted linear sum of the functionvalues at given discrete points Chen et al [13] employedDQM to solve one-dimensional consolidation problems inmultilayered soils
To show the mathematic detail of DQM consider afunction 120593 = 120593(119909) on the domain 0 le 119909 le 119886 and thedomain is dispersed as119873 pointsThen the general differentialquadrature approximation of the function at the 119894th discretepoint is given by
are the weighting coefficient of 119903th derivative 119903 lt119873
This paper adopts a method derived by Quan and Chang[21] which uses Lagrange polynomial to determine theweighting coefficients
119863(1)
119894119896=
prod119873
V=1V = 119894 (119909119894 minus 119909V)
(119909119894minus 119909119896)prod119873
V=1119896 = 119894 (119909119896 minus 119909V)
119894 119896 = 1 2 119873 (119896 = 119894)
119863(119903)
119894119894= minus
119873
sum
V=1V = 119894
119863(119903)
119894V 119894 = 1 2 119873 1 le 119903 le 119873 minus 1
119863(119903)
119894119896= 119903[119863
(119903minus1)
119894119894119863(1)
119894119896minus
119863(119903minus1)
119894119896
119909119894minus 119909119896
]
119894 119896 = 1 2 119873 (119896 = 119894) 2 le 119903 le 119873 minus 1
(13)
The soil layer is dispersed in the radial direction andthe number of discrete points is 119873 Then in order tosolve the equations conveniently the local coordinate 120576 isintroduced The relationship between local coordinate andintegral coordinate is
119903 = (05 minus 120576) 1199031+ (05 + 120576) 119903
119873 (minus05 le 120576 le 05) (14)
where 119903 is the integral coordinate of soil layer and 1199031and 119903119873
are the radius values of 1th and119873th point respectivelyThe differential of (14) can be expressed as
According to (19) and (20) (18) can be rewritten as
120597V119908
120597119879=119882119886
119866(AV119886 + 1
119903BV119886) + 119882
119908
119866(AV119908 + 1
119903BV119908)
120597V119886
120597119879=119860119886
119866(AV119886 + 1
119903BV119886) + 119860
119908
119866(AV119908 + 1
119903BV119908)
(23)
where V119908 = [119906119908
2 119906119908
3 119906
119908
119873minus1]1015840
V119886 = [119906119886
2 119906119886
3 119906
119886
119873minus1]1015840
A = A1 + A2 and B = B1 + B2
A1 =[[[
[
119863(2)
22sdot sdot sdot 119863
(2)
2(119873minus1)
d
119863(2)
(119873minus1)2sdot sdot sdot 119863
(2)
(119873minus1)(119873minus1)
]]]
]
B1 =[[[
[
119863(1)
22sdot sdot sdot 119863
(1)
2(119873minus1)
d
119863(1)
(119873minus1)2sdot sdot sdot 119863
(1)
(119873minus1)(119873minus1)
]]]
]
A2 = Mminus1 timesQ times
[[[
[
119863(2)
21119863(2)
2119873
119863(2)
(119873minus1)1119863(2)
(119873minus1)119873
]]]
]
B2 = Mminus1 timesQ times
[[[
[
119863(1)
21119863(1)
2119873
119863(1)
(119873minus1)1119863(1)
(119873minus1)119873
]]]
]
119866 =119896119908
minus120574119908119898119904
1119896
M = [1 0
119863(1)
1198731119863(1)
119873119873
]
Q = minus[0 sdot sdot sdot 0
119863(1)
1198732sdot sdot sdot 119863
(1)
119873(119873minus1)
]
(24)
Therefore the governing equations of water and air aretranslated into two sets of ordinary differential equations(ODEs) The solutions of ODEs can be obtained by usingRong-KuttamethodThenwe can apply the solutions into thefollowing formulas the average degree of consolidation theradial displacement and the vertical settlement
In unsaturated soils the average degree of consolidationcan be divided into two parts the average degree of consoli-dation with respect to water phase119880119908 and the average degreeof consolidation with respect to air phase 119880119886 To obtain theaverage degree of consolidation two formulations are givenby Fredlund and Rahardjo [7] Consider
According to the two stress-state variable approaches [67] volume strain is represented by the following constitutiveequation for soil layer
120597120576V
120597119905= (119898119908
1119896+ 119898119886
1119896)120597 (119902 minus 119906
119886
)
120597119905+ (119898119908
2+ 119898119886
2)120597 (119906119886
minus 119906119908
)
120597119905
(27)
By integrating (26) with respect to time 119905 from 0 to 119905 weget the expression of volumetric strain 120576V
120576V = (119898119908
1119896+ 119898119886
1119896) [(119902 minus 119906
119886
) minus (119902 minus 119906119886
0)]
+ (119898119908
2+ 119898119886
2) [(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)]
(28)
Volumetric strain consists of vertical strain and horizon-tal strain For the axisymmetric consolidation it is assumed
in this paper that one-third of the volume strain is contributedfrom vertical strain and two-thirds are fromhorizontal strainSo the radial displacement and vertical settlement can beobtained respectively Consider
119878ℎ=2
3(119898119908
1119896+ 119898119886
1119896) int
119903119890
0
[(119902 minus 119906119886
) minus (119902 minus 119906119886
0)] 119889119903
+2
3(119898119908
2+ 119898119886
2) int
119903119890
0
[(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)] 119889119903
119878V =1
3(119898119908
1119896+ 119898119886
1119896) int
119867
0
[(119902 minus 119906119886
) minus (119902 minus 119906119886
0)] 119889119911
+1
3(119898119908
2+ 119898119886
2) int
119867
0
[(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)] 119889119911
(29)
8 Mathematical Problems in Engineering
0
005
01
015
02
025
03
035
04
045
10minus3
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 10 mr = 12 m
uw
q
Figure 7 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
0
001
002
003
004
005
006
10minus3
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 10 mr = 12 m
uaq
Figure 8 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
4 An Example and Convergence Analysis
In this section the convergence analysis the average degree ofconsolidation and the effects of displacements in radial andvertical directions are discussed by using a simple example
As shown in Figure 1 a vertical loading 119902 = 100 kPa isapplied on the top surface of the soil layerThe lateral bound-ary is considered as impermeable Considering a uniforminitial excess pore-water and pore-air pressure distribution119906119886
0= 5 kPa 119906119908
0= 40 kPa Other parameters are 119899
0= 05 119878
1199030=
08 119896119908= 10minus10ms119898119908
1119896= minus015times10
minus4 kPaminus11198981199082= minus061times
10minus4 kPaminus1119898119886
1119896= 006 times 10
minus4 kPaminus11198981198862= 026 times 10
minus4 kPaminus1119903119908= 02m 119903
119890= 18m and 119906atm = 1013 kPa By applying the
DQM solution presented above the results of axisymmetric
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
1
08
06
04
02
0
uw
q
Figure 9 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
014
012
01
008
006
004
002
0
uaq
Figure 10 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
consolidation of unsaturated soils in a sand drain foundationwere obtained
Tables 1 and 2 list the solutions of excess pore-water andpore-air pressures at radius 119903 = 1m at different time factorsand different equally spaced discrete points respectively Theratio of permeability coefficient is 119896
119886119896119908= 01 From Tables
1 and 2 the accuracy of solutions increases when the numberof discrete point becomes big It is obvious that 9 equallyspaced grid points are sufficient to obtain the convergedresults of excess pore-water and pore-air pressuresThereforethe number of discrete points119873 = 9 is adopted in most casestudies
Figure 2 shows the curves of average degree of consoli-dation with respect to water and air phases when 119873 = 9The ratio of permeability coefficient is 119896
119886119896119908
= 1 FromFigure 2 it can be observed that the consolidation of air
Mathematical Problems in Engineering 9
Table 1 Different excess pore-water pressures 119906119908119902 under time factor 119879 with different discrete points119873 (119896
1 times 10minus6 0050 0050 0050 0050 0050 0050 0050
1 times 10minus4 0050 0050 0050 0050 0050 0050 0050
5 times 10minus2 0048 0048 0048 0048 0048 0048 0048
1 times 10minus1 0046 0045 0045 0045 0045 0045 0045
2 times 10minus1 0042 0041 0040 0040 0040 0040 0010
5 times 10minus1 0035 0033 0032 0032 0032 0031 0031
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 11 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
phase and water phase begins when 119879 = 10 times 10minus3 When
119879 = 10 times 100 and 119879 = 10 times 10
15 the consolidation ofair phase and water phase is almost finished respectivelyAt the early stages soil consolidation is mainly caused bythe dissipation of excess pore-air pressure But in the laterstages soil consolidation is mainly caused by the dissipationof excess pore-water pressure
Figure 3 shows the radial displacement development withtime and Figure 4 describes the settlement of top surface(119867 = 10m) at different radii with time factor 119879 The numberof discrete points is 119873 = 9 The ratio of permeabilitycoefficient is 119896
119886119896119908
= 01 From Figure 4 we can see thatthe soil settles earlier at the points with smaller radius thatis closer to the sand drain Figure 5 shows the settlement atdifferent heights of soil layer at radius 119903 = 1m with time
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 12 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
factor 119879 When the height of soil layer119867 is equal to 4m thesettlement is almost zero
5 Cases of Different Initial andBoundary Conditions
As the initial conditions do not need to be constant in thepresentDQMsolution it is easily used to analyze nonuniforminitial pore-water and pore-air distribution problems In thispart four different initial pore-water and pore-air distribu-tions and some different boundary conditions are consideredA vertical loading 119902 = 100 kPa is applied on the top surfaceof soil layer Other parameters are 119899
0= 05 119878
1199030= 08 119896
119908=
10minus10ms 119903
119908= 02m 119903
119890= 18m119898119908
1119896= minus015 times 10
minus4 kPaminus1
10 Mathematical Problems in Engineering
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 13 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 14 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
119898119908
2= minus061 times 10
minus4 kPaminus1 1198981198861119896
= 006 times 10minus4 kPaminus1 119898119886
2=
026 times 10minus4 kPaminus1 and 119906atm = 1013 kPa
51 Effect of Different Initial Pore-WaterAir Pressure Distribu-tions In this section in order to make the curves smooththe number of discrete points is chosen to be 119873 = 17 Thelateral boundary is considered as impermeable and the ratioof permeability coefficient with respect to air and water is119896119886119896119908= 01
Four different initial pore-waterair distributions aretaken into account In Figure 6(a) the case of uniform initialpore-air and pore-water pressures assumes that 119906119886
0= 5 kPa
and 1199061199080= 40 kPa Figure 6(b) presents the linear increasing
initial pore-air and pore-water distributions The lateral-skewed distributions initial pore-waterair pressures is shown
in Figures 6(c) and 6(d) and they are represented by thefollowing equations
119906119908
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
100105119887
119906119886
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
8 times 100105119887
(30)
where 119887 is the variable which can control the spread orskewness of the curves in Figures 6(c) and 6(d)
Under the condition shown in Figure 6(a) the dissipationcurves of the excess pore-water and pore-air pressure atdifferent distances from the sand drain are presented inFigures 7 and 8 respectively The excess pore-air and pore-water pressures dissipate faster at the points with smallerradius that is closer to the sand drain As the intervalbetween the two curves becomes smaller from 119903 = 06m to119903 = 12m it can be seen that the effect of the distance on thedissipation becomes less obvious
In Figure 6(b) the initial pore-air and pore-water pres-sures are assumed not to be constants and to be linearincreasing when the radius increases The initial pore-airpressure is assumed one-eighth of the initial pore-water pres-sure In other words from 119903 = 04m to 119903 = 16m the initialpore-water pressure will increase from 30 kPa to 110 kPa andthe initial pore-air pressure will increase from 375 kPa to1375 kPa Figures 9 and 10 show the dissipation of excesspore-water and pore-air pressures at different distances fromthe sand drain under Figure 6(b) distributions
Comparing Figures 7 and 9 it is found that the excesspore-water pressure dissipates completely at almost the sametime With the change of the initial pore-water pressuredistribution the time of dissipation with respect to wateris not significantly affected Comparing Figures 8 and 10it can be seen that the excess pore-air pressure dissipatescompletely at almost the same time The initial pore-airpressure distribution has little influence on the dissipationtime of air phase
Under the conditions of Figures 6(c) and 6(d) thedissipation curves of the excess pore-water and pore-airpressures for initial distributions where 119887 = 2 6 are shownin Figures 11 12 13 and 14 From these figures whatever theinitial pore-water and pore-air pressures are the excess pore-water and pore-air pressures dissipate completely at almostthe same timeThe plateau period is longer at the points withlarger radius
Here excess pore pressure isochrones for the above fourinitial conditions are presented in Figures 15 and 16 Figures15(a) and 16(a) show the excess pore-water and pore-airpressure isochrones when the initial excess pore pressure isuniform at all radii In Figures 15(b) and 16(b) the initialexcess pore pressure distributions are linearly increasingThepore pressure isochrones for lateral-skewed initial distribu-tions are shown in Figures 15(c) 16(c) 15(d) and 16(d)respectively From Figures 15(a) and 16(a) as the initial excesspore pressure is constant the peak of curves in Figure 15(a)approaches to 04 and the peak of curves in Figure 13(a)approaches to 005 Comparing Figures 15(a) and 15(b) as
Mathematical Problems in Engineering 11
0
005
01
015
02
025
03
035
04
02 04 06 08 1 12 14 16 18r (m)
02
1
2
5
05
T = 10
uw
q
(a)
02 04 06 08 1 12 14 16 18r (m)
1
1
2
5
09
08
07
06
05
05
04
03
02
02
01
01
0
T = 10
uw
q
(b)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
011
09
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(c)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
0109
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(d)
Figure 15Normalized pore-water pressure isochrones for different initial excess pore-water pressure distributions (119896119886119896119908= 01) (a) uniform
(b) linearly increasing (c) lateral skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
the initial excess pore-water pressure is linearly increasingthe skewness is more evident during early stages of dissipa-tion in Figure 15(b) But in the later stages the influence ofinitial excess pore-water pressure becomes less significant Soare the isochrones of the excess pore-air pressure in Figures16(a) and 16(b) Figures 15(c) 16(c) 15(d) and 16(d) showthe excess pore-water and pore-air pressure isochrones forthe lateral-skewed initial distribution when 119887 = 2 6 It isseen that some intersections appear in the isochrones duringthe early stages It means that a redistribution of excesspore pressures occurs toward the region of minimal initialexcess pore pressure during the early stages With the timeincreasing the isochrones of excess pore pressures becomeregular
52 Effect of Different Boundary Conditions In this partthe effect of different lateral boundary conditions is studiedUniform initial excess pore-waterair pressure distributions119906119886
0= 5 kPa and 119906119908
0= 40 kPa are adopted Four different ratios
119896119886119896119908are considered that is 01 1 10 and 100 with 119896
119908=
10minus10ms Two different drainage conditions are analyzed
lateral surface is impervious and lateral surface is impededFor the impeded surface 119886
119908and 119886
119886in (9) are chosen to be
05Figures 17 and 18 shows the dissipation curves for the
water and air pressures at radius 119903 = 12m under differentboundary conditions First the excess pore-water and pore-air pressures dissipate faster in impeded surface than that inimpermeable surface Second 119896
119908is the same in the cases
12 Mathematical Problems in Engineering
02 04 06 08 1 12 14 16 18r (m)
01
02
05
T = 2
006
005005
004
003
002
002
001
0
1
uaq
(a)
02 04 06 08 1 12 14 16 18r (m)
014
012
0101
02
05008
006
004
002
0
T = 2
005002
1
uaq
(b)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(c)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(d)
Figure 16 Normalized pore-air pressure isochrones for different initial excess pore-air pressure distributions (119896119886119896119908= 01) (a) uniform (b)
linearly increasing (c) lateral-skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
of four 119896119886119896119908values It is observed from the results that
the bigger 119896119886119896119908
is the faster the excess pore-water andpore-air pressures dissipate Third the dissipation curves ofthe excess pore-air and pore-water pressures have almostthe same shape under different values of 119896
119886119896119908 Comparing
Figure 17 with Figure 18 when the excess pore-air pressuredissipates almost completely the dissipation of excess pore-water pressure enters into a plateau period The bigger 119896
119886119896119908
is the earlier the excess pore-air pressure dissipates com-pletely Therefore it is shown that the value of 119896
119886affects the
dissipation of pore-water pressure during the consolidationprocess
6 Conclusion
This paper obtains a general solution to the axisymmet-ric consolidation of unsaturated soils by using differen-tial quadrature method (DQM) based on Fredlundrsquos one-dimensional consolidation theory for unsaturated soil Withthe use of DQM the two governing equations are trans-formed into two sets of ordinary differential equationsThe solutions to the transformed differential equations areobtained by Rong-Kutta method under different initial andboundary conditions A case study has been presented forthe analysis of unsaturated consolidation in a sand drain
Mathematical Problems in Engineering 13
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
045
04
035
03
025
02
015
01
005
0
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
uw
q
Figure 17 Variation of excess pore-water pressures 119906119908119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
007
006
005
004
003
002
001
0
uaq
Figure 18 Variation of excess pore-air pressures 119906119886
119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
foundation The convergence analysis the average degreeof consolidation for water and air phases the settlementin radial and vertical directions effects of different initialexcess pore-air and pore-water pressure distributions andeffects of different boundary conditions have been presentedFrom the analyses some conclusions can be drawn (a)through compiling the programs it is found that the DQsolution delivers accurate and stable results for unsaturatedsoil consolidation Due to the uniformmatrix structure of theDQ formulas it is easy to compile computer programs whenconsidering complicated conditions such as nonuniformpore-waterair distribution conditions (b) soil consolidationismainly caused by the dissipation of excess pore-air pressurein the early stages and by the dissipation of excess pore-water pressure in the later stages (c) the bigger the ratio
of the permeability coefficients for water and air is thefaster the excess pore-water and pore-air pressures dissipate(d) The excess pore-air and pore-water pressures dissipatefaster at the points closer to the sand drain (e) the initialdistribution has some effects on the early consolidationprocess of unsaturated soil and boundary condition has asignificant effect on the whole consolidation process
Notation
119862119908
V The consolidation coefficient of water119862119886
V The consolidation coefficient of air119863(119903)
119894119896 The weighting coefficient
120576V Volumetric strain119892 The gravitational acceleration119867 The thickness of soil layer119896119886 The permeability coefficients of air
1198960
119886 The coefficients of permeability for water
at lateral boundary119896119908 The permeability coefficients of water
1198960
119908 The coefficients of permeability for air at
lateral boundary119898119886
1119896 The coefficient of air volume change withrespect to a change in the net normal stress
119898119908
1119896 The coefficient of water volume changewith respect to a change in the net normalstress
119898119886
2 The coefficient of air volume change with
respect to a change in matric suction119898119908
2 The coefficient of water volume changewith respect to a change in matric suction
1198990 The initial porosity
119873 The discrete point119875V A vacuum pressure120588 The density of air phase119902 The vertical loading1199030 The thickness of lateral boundary
119903119890 The radius of soil layer
119903119908 The radius of sand drain
119903120572 The radius value of 120572th point
120574119908 The unit weight of water phase
119877 The universal air constant119878 The distance between two sand drains1198781199030 The initial degree of saturation
119878V Settlement119878ℎ The radial displacement
119905 Time119879 The time factor119906119886 The excess pore-air pressure119906atm Atmospheric pressure119906119886
0 The initial excess pore-air pressure
119906119908 The excess pore-water pressure119906119908
0 The initial excess pore-water pressure
119880119886 The average degree of consolidation with
respect to air119880119908 The average degree of consolidation with
respect to water119881119886 The air volume of soil layer
14 Mathematical Problems in Engineering
119881119908 The water volume of soil layer
1198810 Initial total volume of the soil
Acknowledgments
The author gratefully acknowledges the financial supportfrom the Macau Science and Technology Development Fund(Grant no FDCT0112013A1) and the University of MacauResearch Fund (Grants nos MYRG189(Y2-L3)-FST11-ZWHand MYRG067(Y2-L2)-FST12-ZWH)
References
[1] K Terzaghi Theoretical Soil Mechanics Wiley New York NYUSA 1943
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] G E Blight Strength and consolidation characteristics of com-pacted soils [PhD thesis] University of London London UK1961
[4] R F Scott Principles of Soil Mechanics Addison Wesley Pub-lishing Company Boston Mass USA 1963
[5] L Barden ldquoConsolidation of compacted and unsaturated clayrdquoGeotechnique vol 15 no 3 pp 267ndash286 1965
[6] D G Fredlund and J U Hasan ldquoOne-dimensional consolida-tion theory unsaturated soilsrdquo Canadian Geotechnical Journalvol 16 no 3 pp 521ndash531 1979
[7] D G Fredlund andH Rahardjo Soil Mechanics for UnsaturatedSoils John Wiley and Son 1993
[8] A Qin D Sun L Yang and Y Weng ldquoA semi-analyticalsolution to consolidation of unsaturated soils with the freedrainage wellrdquo Computers and Geotechnics vol 37 no 7-8 pp867ndash875 2010
[9] W H Zhou L S Zhao and X B Li ldquoA simple analyticalsolution to one-dimensional consolidation for unsaturated soilrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics 2013
[10] M Malik and F Civan ldquoA comparative study of differentialquadrature and cubature methods vis-a-vis some conventionaltechniques in context of convection-diffusion-reaction prob-lemsrdquo Chemical Engineering Science vol 50 no 3 pp 531ndash5471995
[11] P Malekzadeh M R Golbahar Haghighi and M M AtashildquoOut-of-plane free vibration of functionally graded circularcurved beams in thermal environmentrdquo Composite Structuresvol 92 no 2 pp 541ndash552 2010
[12] P Malekzadeh M R G Haghighi and M M Atashi ldquoFreevibration analysis of elastically supported functionally radedannular plates subjected to thermal environmentrdquo Meccanicavol 46 no 5 pp 893ndash913 2011
[13] R P Chen W H Zhou H Z Wang and Y M Chen ldquoOne-dimensional nonlinear consolidation of multi-layered soil bydifferential quadrature methodrdquo Computers and Geotechnicsvol 32 no 5 pp 358ndash369 2005
[14] R P Chen W H Zhou H Z Wang and Y M Chen ldquoStudyon one-dimensional nonlinear consolidation of multi-layeredsoil using differential quadrature methodrdquo Chinese Journal ofComputational Mechanics vol 22 no 3 pp 310ndash315 2005
[15] H Z Wang R P Chen W H Zhou and Y M ChenldquoComputation of 1-D nonlinear consolidation in double-layer
foundation by using differential quadrature methodrdquo Journal ofHydraulic Engineering vol 4 pp 8ndash14 2004
[16] W H Zhou and L S Zhao ldquoOne-dimensional consolidationof unsaturated soil subjected to time-dependent loading undervarious initial and boundary conditionsrdquo ASCE InternationalJournal of Geomechanics 2013
[17] E Conte ldquoConsolidation analysis for unsaturated soilsrdquo Cana-dian Geotechnical Journal vol 41 no 4 pp 599ndash612 2004
[18] E Conte ldquoPlane strain and axially symmetric consolidation inunsaturated soilsrdquo International Journal of Geomechanics vol 6no 2 pp 131ndash135 2006
[19] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 no 1 pp40ndash52 1972
[20] R Bellman and J Casti ldquoDifferential quadrature and long-termintegrationrdquo Journal of Mathematical Analysis and Applicationsvol 34 no 2 pp 235ndash238 1971
[21] J R Quan andC T Chang ldquoNew insights in solving distributedsystem equations by the quadrature method I analysisrdquo Com-puters and Chemical Engineering vol 13 no 7 pp 779ndash7881989
Figure 1 (a) The sectional view of sand drain foundation (b) the vertical view of sand drain foundation and (c) the distribution of sanddrains (square pattern)
According to Boylersquos law assuming there is no initial excessair pressure in the soil before loading we have [17 18]
120597 (1198811198861198810)
120597119905=
119896119886119877119879
1198921199060
119886119872
(1205972
119906119886
1205971199032+1
119903
120597119906119886
120597119903)
minus119906atm1198990 (1 minus 1198780)
(1199060
119886)2
120597119906119886
120597119905
(5)
where 119877 is the universal air constant (8314 JmolK) 119879 =
29315K is the absolute temperature119872 = 29 kgkmol is themolecular mass of air 119906119886
0is absolute pore-air pressure (ie
119906119886
0= 119906119886
0+119906
atm) and 119906atm = 1013 kPa is atmospheric pressure
The derivative of the air phase constitutive relation withrespect to time is equal to the net flux of air per unit volumeof the soil
120597 (1198811198861198810)
120597119905= 119898119886
1119896
120597 (119902 minus 119906119886)
120597119905+ 119898119886
2
120597 (119906119886minus 119906119908)
120597119905 (6)
where1198981198861119896is the coefficient of air volume change with respect
to a change in the net normal stress (119902 minus 119906119886
) and 119898119886
2is the
coefficient of air volume change with respect to a change inmatric suction (119906119886 minus 119906119908)
Substituting (5) into (6) the governing equation for theair phase under constant loading 119889119902119889119905 = 0 can be written as
120597119906119886
120597119905= minus119862119886
120597119906119908
120597119905minus 119862119886
V (1205972
119906119886
1205971199032+1
119903
120597119906119886
120597119903) (7)
4 Mathematical Problems in Engineering
where 119862119886= 119898119886
2(119898119886
1119896minus119898119886
2minus 119906
atm1198990(1 minus 119878
1199030)(119906119886
0)2
) and 119862119908V =
119896119886119877119879119892119906
119886
0119872(119898119886
1119896minus 119898119886
2minus 119906
atm1198990(1 minus 119878
1199030)(119906119886
0)2
)Further two transformed governing equations of water
V (1minus119862119886119862119908)119860119886 =minus119862119886
V(1 minus 119862119886119862119908) and 119860119908 = minus119862119886119862119908
V (1 minus 119862119886119862119908)
23 Boundary and Initial Conditions As shown in Figure 1the wall of sand drain is permeable and lateral boundarysurface is impermeable or impeded the boundary conditionsfor radial consolidation are
1198961198861199030 (9) reflects that the lateral boundary surface is imperme-
able or impeded respectively 1198960119908and 1198960
119886are the coefficients
of permeability for water and air at lateral boundary respec-tively 119903
0is the thickness of lateral boundary
The initial condition can be written as
119906119886
(119903 0) = 119906119886
0 119906
119908
(119903 0) = 119906119908
0 (11)
where 119906119908
0and 119906
119886
0are the initial pore-water and pore-air
pressure distributions
3 Differential Quadrature Formulation
DQM is a numerical solution technique for initial andorboundary value problems proposed by Bellman et al [19]and Bellman and Casti [20] According to DQM a partialderivative of a function with respect to a variable can beapproximated by a weighted linear sum of the functionvalues at given discrete points Chen et al [13] employedDQM to solve one-dimensional consolidation problems inmultilayered soils
To show the mathematic detail of DQM consider afunction 120593 = 120593(119909) on the domain 0 le 119909 le 119886 and thedomain is dispersed as119873 pointsThen the general differentialquadrature approximation of the function at the 119894th discretepoint is given by
are the weighting coefficient of 119903th derivative 119903 lt119873
This paper adopts a method derived by Quan and Chang[21] which uses Lagrange polynomial to determine theweighting coefficients
119863(1)
119894119896=
prod119873
V=1V = 119894 (119909119894 minus 119909V)
(119909119894minus 119909119896)prod119873
V=1119896 = 119894 (119909119896 minus 119909V)
119894 119896 = 1 2 119873 (119896 = 119894)
119863(119903)
119894119894= minus
119873
sum
V=1V = 119894
119863(119903)
119894V 119894 = 1 2 119873 1 le 119903 le 119873 minus 1
119863(119903)
119894119896= 119903[119863
(119903minus1)
119894119894119863(1)
119894119896minus
119863(119903minus1)
119894119896
119909119894minus 119909119896
]
119894 119896 = 1 2 119873 (119896 = 119894) 2 le 119903 le 119873 minus 1
(13)
The soil layer is dispersed in the radial direction andthe number of discrete points is 119873 Then in order tosolve the equations conveniently the local coordinate 120576 isintroduced The relationship between local coordinate andintegral coordinate is
119903 = (05 minus 120576) 1199031+ (05 + 120576) 119903
119873 (minus05 le 120576 le 05) (14)
where 119903 is the integral coordinate of soil layer and 1199031and 119903119873
are the radius values of 1th and119873th point respectivelyThe differential of (14) can be expressed as
According to (19) and (20) (18) can be rewritten as
120597V119908
120597119879=119882119886
119866(AV119886 + 1
119903BV119886) + 119882
119908
119866(AV119908 + 1
119903BV119908)
120597V119886
120597119879=119860119886
119866(AV119886 + 1
119903BV119886) + 119860
119908
119866(AV119908 + 1
119903BV119908)
(23)
where V119908 = [119906119908
2 119906119908
3 119906
119908
119873minus1]1015840
V119886 = [119906119886
2 119906119886
3 119906
119886
119873minus1]1015840
A = A1 + A2 and B = B1 + B2
A1 =[[[
[
119863(2)
22sdot sdot sdot 119863
(2)
2(119873minus1)
d
119863(2)
(119873minus1)2sdot sdot sdot 119863
(2)
(119873minus1)(119873minus1)
]]]
]
B1 =[[[
[
119863(1)
22sdot sdot sdot 119863
(1)
2(119873minus1)
d
119863(1)
(119873minus1)2sdot sdot sdot 119863
(1)
(119873minus1)(119873minus1)
]]]
]
A2 = Mminus1 timesQ times
[[[
[
119863(2)
21119863(2)
2119873
119863(2)
(119873minus1)1119863(2)
(119873minus1)119873
]]]
]
B2 = Mminus1 timesQ times
[[[
[
119863(1)
21119863(1)
2119873
119863(1)
(119873minus1)1119863(1)
(119873minus1)119873
]]]
]
119866 =119896119908
minus120574119908119898119904
1119896
M = [1 0
119863(1)
1198731119863(1)
119873119873
]
Q = minus[0 sdot sdot sdot 0
119863(1)
1198732sdot sdot sdot 119863
(1)
119873(119873minus1)
]
(24)
Therefore the governing equations of water and air aretranslated into two sets of ordinary differential equations(ODEs) The solutions of ODEs can be obtained by usingRong-KuttamethodThenwe can apply the solutions into thefollowing formulas the average degree of consolidation theradial displacement and the vertical settlement
In unsaturated soils the average degree of consolidationcan be divided into two parts the average degree of consoli-dation with respect to water phase119880119908 and the average degreeof consolidation with respect to air phase 119880119886 To obtain theaverage degree of consolidation two formulations are givenby Fredlund and Rahardjo [7] Consider
According to the two stress-state variable approaches [67] volume strain is represented by the following constitutiveequation for soil layer
120597120576V
120597119905= (119898119908
1119896+ 119898119886
1119896)120597 (119902 minus 119906
119886
)
120597119905+ (119898119908
2+ 119898119886
2)120597 (119906119886
minus 119906119908
)
120597119905
(27)
By integrating (26) with respect to time 119905 from 0 to 119905 weget the expression of volumetric strain 120576V
120576V = (119898119908
1119896+ 119898119886
1119896) [(119902 minus 119906
119886
) minus (119902 minus 119906119886
0)]
+ (119898119908
2+ 119898119886
2) [(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)]
(28)
Volumetric strain consists of vertical strain and horizon-tal strain For the axisymmetric consolidation it is assumed
in this paper that one-third of the volume strain is contributedfrom vertical strain and two-thirds are fromhorizontal strainSo the radial displacement and vertical settlement can beobtained respectively Consider
119878ℎ=2
3(119898119908
1119896+ 119898119886
1119896) int
119903119890
0
[(119902 minus 119906119886
) minus (119902 minus 119906119886
0)] 119889119903
+2
3(119898119908
2+ 119898119886
2) int
119903119890
0
[(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)] 119889119903
119878V =1
3(119898119908
1119896+ 119898119886
1119896) int
119867
0
[(119902 minus 119906119886
) minus (119902 minus 119906119886
0)] 119889119911
+1
3(119898119908
2+ 119898119886
2) int
119867
0
[(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)] 119889119911
(29)
8 Mathematical Problems in Engineering
0
005
01
015
02
025
03
035
04
045
10minus3
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 10 mr = 12 m
uw
q
Figure 7 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
0
001
002
003
004
005
006
10minus3
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 10 mr = 12 m
uaq
Figure 8 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
4 An Example and Convergence Analysis
In this section the convergence analysis the average degree ofconsolidation and the effects of displacements in radial andvertical directions are discussed by using a simple example
As shown in Figure 1 a vertical loading 119902 = 100 kPa isapplied on the top surface of the soil layerThe lateral bound-ary is considered as impermeable Considering a uniforminitial excess pore-water and pore-air pressure distribution119906119886
0= 5 kPa 119906119908
0= 40 kPa Other parameters are 119899
0= 05 119878
1199030=
08 119896119908= 10minus10ms119898119908
1119896= minus015times10
minus4 kPaminus11198981199082= minus061times
10minus4 kPaminus1119898119886
1119896= 006 times 10
minus4 kPaminus11198981198862= 026 times 10
minus4 kPaminus1119903119908= 02m 119903
119890= 18m and 119906atm = 1013 kPa By applying the
DQM solution presented above the results of axisymmetric
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
1
08
06
04
02
0
uw
q
Figure 9 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
014
012
01
008
006
004
002
0
uaq
Figure 10 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
consolidation of unsaturated soils in a sand drain foundationwere obtained
Tables 1 and 2 list the solutions of excess pore-water andpore-air pressures at radius 119903 = 1m at different time factorsand different equally spaced discrete points respectively Theratio of permeability coefficient is 119896
119886119896119908= 01 From Tables
1 and 2 the accuracy of solutions increases when the numberof discrete point becomes big It is obvious that 9 equallyspaced grid points are sufficient to obtain the convergedresults of excess pore-water and pore-air pressuresThereforethe number of discrete points119873 = 9 is adopted in most casestudies
Figure 2 shows the curves of average degree of consoli-dation with respect to water and air phases when 119873 = 9The ratio of permeability coefficient is 119896
119886119896119908
= 1 FromFigure 2 it can be observed that the consolidation of air
Mathematical Problems in Engineering 9
Table 1 Different excess pore-water pressures 119906119908119902 under time factor 119879 with different discrete points119873 (119896
1 times 10minus6 0050 0050 0050 0050 0050 0050 0050
1 times 10minus4 0050 0050 0050 0050 0050 0050 0050
5 times 10minus2 0048 0048 0048 0048 0048 0048 0048
1 times 10minus1 0046 0045 0045 0045 0045 0045 0045
2 times 10minus1 0042 0041 0040 0040 0040 0040 0010
5 times 10minus1 0035 0033 0032 0032 0032 0031 0031
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 11 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
phase and water phase begins when 119879 = 10 times 10minus3 When
119879 = 10 times 100 and 119879 = 10 times 10
15 the consolidation ofair phase and water phase is almost finished respectivelyAt the early stages soil consolidation is mainly caused bythe dissipation of excess pore-air pressure But in the laterstages soil consolidation is mainly caused by the dissipationof excess pore-water pressure
Figure 3 shows the radial displacement development withtime and Figure 4 describes the settlement of top surface(119867 = 10m) at different radii with time factor 119879 The numberof discrete points is 119873 = 9 The ratio of permeabilitycoefficient is 119896
119886119896119908
= 01 From Figure 4 we can see thatthe soil settles earlier at the points with smaller radius thatis closer to the sand drain Figure 5 shows the settlement atdifferent heights of soil layer at radius 119903 = 1m with time
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 12 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
factor 119879 When the height of soil layer119867 is equal to 4m thesettlement is almost zero
5 Cases of Different Initial andBoundary Conditions
As the initial conditions do not need to be constant in thepresentDQMsolution it is easily used to analyze nonuniforminitial pore-water and pore-air distribution problems In thispart four different initial pore-water and pore-air distribu-tions and some different boundary conditions are consideredA vertical loading 119902 = 100 kPa is applied on the top surfaceof soil layer Other parameters are 119899
0= 05 119878
1199030= 08 119896
119908=
10minus10ms 119903
119908= 02m 119903
119890= 18m119898119908
1119896= minus015 times 10
minus4 kPaminus1
10 Mathematical Problems in Engineering
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 13 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 14 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
119898119908
2= minus061 times 10
minus4 kPaminus1 1198981198861119896
= 006 times 10minus4 kPaminus1 119898119886
2=
026 times 10minus4 kPaminus1 and 119906atm = 1013 kPa
51 Effect of Different Initial Pore-WaterAir Pressure Distribu-tions In this section in order to make the curves smooththe number of discrete points is chosen to be 119873 = 17 Thelateral boundary is considered as impermeable and the ratioof permeability coefficient with respect to air and water is119896119886119896119908= 01
Four different initial pore-waterair distributions aretaken into account In Figure 6(a) the case of uniform initialpore-air and pore-water pressures assumes that 119906119886
0= 5 kPa
and 1199061199080= 40 kPa Figure 6(b) presents the linear increasing
initial pore-air and pore-water distributions The lateral-skewed distributions initial pore-waterair pressures is shown
in Figures 6(c) and 6(d) and they are represented by thefollowing equations
119906119908
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
100105119887
119906119886
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
8 times 100105119887
(30)
where 119887 is the variable which can control the spread orskewness of the curves in Figures 6(c) and 6(d)
Under the condition shown in Figure 6(a) the dissipationcurves of the excess pore-water and pore-air pressure atdifferent distances from the sand drain are presented inFigures 7 and 8 respectively The excess pore-air and pore-water pressures dissipate faster at the points with smallerradius that is closer to the sand drain As the intervalbetween the two curves becomes smaller from 119903 = 06m to119903 = 12m it can be seen that the effect of the distance on thedissipation becomes less obvious
In Figure 6(b) the initial pore-air and pore-water pres-sures are assumed not to be constants and to be linearincreasing when the radius increases The initial pore-airpressure is assumed one-eighth of the initial pore-water pres-sure In other words from 119903 = 04m to 119903 = 16m the initialpore-water pressure will increase from 30 kPa to 110 kPa andthe initial pore-air pressure will increase from 375 kPa to1375 kPa Figures 9 and 10 show the dissipation of excesspore-water and pore-air pressures at different distances fromthe sand drain under Figure 6(b) distributions
Comparing Figures 7 and 9 it is found that the excesspore-water pressure dissipates completely at almost the sametime With the change of the initial pore-water pressuredistribution the time of dissipation with respect to wateris not significantly affected Comparing Figures 8 and 10it can be seen that the excess pore-air pressure dissipatescompletely at almost the same time The initial pore-airpressure distribution has little influence on the dissipationtime of air phase
Under the conditions of Figures 6(c) and 6(d) thedissipation curves of the excess pore-water and pore-airpressures for initial distributions where 119887 = 2 6 are shownin Figures 11 12 13 and 14 From these figures whatever theinitial pore-water and pore-air pressures are the excess pore-water and pore-air pressures dissipate completely at almostthe same timeThe plateau period is longer at the points withlarger radius
Here excess pore pressure isochrones for the above fourinitial conditions are presented in Figures 15 and 16 Figures15(a) and 16(a) show the excess pore-water and pore-airpressure isochrones when the initial excess pore pressure isuniform at all radii In Figures 15(b) and 16(b) the initialexcess pore pressure distributions are linearly increasingThepore pressure isochrones for lateral-skewed initial distribu-tions are shown in Figures 15(c) 16(c) 15(d) and 16(d)respectively From Figures 15(a) and 16(a) as the initial excesspore pressure is constant the peak of curves in Figure 15(a)approaches to 04 and the peak of curves in Figure 13(a)approaches to 005 Comparing Figures 15(a) and 15(b) as
Mathematical Problems in Engineering 11
0
005
01
015
02
025
03
035
04
02 04 06 08 1 12 14 16 18r (m)
02
1
2
5
05
T = 10
uw
q
(a)
02 04 06 08 1 12 14 16 18r (m)
1
1
2
5
09
08
07
06
05
05
04
03
02
02
01
01
0
T = 10
uw
q
(b)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
011
09
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(c)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
0109
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(d)
Figure 15Normalized pore-water pressure isochrones for different initial excess pore-water pressure distributions (119896119886119896119908= 01) (a) uniform
(b) linearly increasing (c) lateral skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
the initial excess pore-water pressure is linearly increasingthe skewness is more evident during early stages of dissipa-tion in Figure 15(b) But in the later stages the influence ofinitial excess pore-water pressure becomes less significant Soare the isochrones of the excess pore-air pressure in Figures16(a) and 16(b) Figures 15(c) 16(c) 15(d) and 16(d) showthe excess pore-water and pore-air pressure isochrones forthe lateral-skewed initial distribution when 119887 = 2 6 It isseen that some intersections appear in the isochrones duringthe early stages It means that a redistribution of excesspore pressures occurs toward the region of minimal initialexcess pore pressure during the early stages With the timeincreasing the isochrones of excess pore pressures becomeregular
52 Effect of Different Boundary Conditions In this partthe effect of different lateral boundary conditions is studiedUniform initial excess pore-waterair pressure distributions119906119886
0= 5 kPa and 119906119908
0= 40 kPa are adopted Four different ratios
119896119886119896119908are considered that is 01 1 10 and 100 with 119896
119908=
10minus10ms Two different drainage conditions are analyzed
lateral surface is impervious and lateral surface is impededFor the impeded surface 119886
119908and 119886
119886in (9) are chosen to be
05Figures 17 and 18 shows the dissipation curves for the
water and air pressures at radius 119903 = 12m under differentboundary conditions First the excess pore-water and pore-air pressures dissipate faster in impeded surface than that inimpermeable surface Second 119896
119908is the same in the cases
12 Mathematical Problems in Engineering
02 04 06 08 1 12 14 16 18r (m)
01
02
05
T = 2
006
005005
004
003
002
002
001
0
1
uaq
(a)
02 04 06 08 1 12 14 16 18r (m)
014
012
0101
02
05008
006
004
002
0
T = 2
005002
1
uaq
(b)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(c)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(d)
Figure 16 Normalized pore-air pressure isochrones for different initial excess pore-air pressure distributions (119896119886119896119908= 01) (a) uniform (b)
linearly increasing (c) lateral-skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
of four 119896119886119896119908values It is observed from the results that
the bigger 119896119886119896119908
is the faster the excess pore-water andpore-air pressures dissipate Third the dissipation curves ofthe excess pore-air and pore-water pressures have almostthe same shape under different values of 119896
119886119896119908 Comparing
Figure 17 with Figure 18 when the excess pore-air pressuredissipates almost completely the dissipation of excess pore-water pressure enters into a plateau period The bigger 119896
119886119896119908
is the earlier the excess pore-air pressure dissipates com-pletely Therefore it is shown that the value of 119896
119886affects the
dissipation of pore-water pressure during the consolidationprocess
6 Conclusion
This paper obtains a general solution to the axisymmet-ric consolidation of unsaturated soils by using differen-tial quadrature method (DQM) based on Fredlundrsquos one-dimensional consolidation theory for unsaturated soil Withthe use of DQM the two governing equations are trans-formed into two sets of ordinary differential equationsThe solutions to the transformed differential equations areobtained by Rong-Kutta method under different initial andboundary conditions A case study has been presented forthe analysis of unsaturated consolidation in a sand drain
Mathematical Problems in Engineering 13
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
045
04
035
03
025
02
015
01
005
0
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
uw
q
Figure 17 Variation of excess pore-water pressures 119906119908119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
007
006
005
004
003
002
001
0
uaq
Figure 18 Variation of excess pore-air pressures 119906119886
119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
foundation The convergence analysis the average degreeof consolidation for water and air phases the settlementin radial and vertical directions effects of different initialexcess pore-air and pore-water pressure distributions andeffects of different boundary conditions have been presentedFrom the analyses some conclusions can be drawn (a)through compiling the programs it is found that the DQsolution delivers accurate and stable results for unsaturatedsoil consolidation Due to the uniformmatrix structure of theDQ formulas it is easy to compile computer programs whenconsidering complicated conditions such as nonuniformpore-waterair distribution conditions (b) soil consolidationismainly caused by the dissipation of excess pore-air pressurein the early stages and by the dissipation of excess pore-water pressure in the later stages (c) the bigger the ratio
of the permeability coefficients for water and air is thefaster the excess pore-water and pore-air pressures dissipate(d) The excess pore-air and pore-water pressures dissipatefaster at the points closer to the sand drain (e) the initialdistribution has some effects on the early consolidationprocess of unsaturated soil and boundary condition has asignificant effect on the whole consolidation process
Notation
119862119908
V The consolidation coefficient of water119862119886
V The consolidation coefficient of air119863(119903)
119894119896 The weighting coefficient
120576V Volumetric strain119892 The gravitational acceleration119867 The thickness of soil layer119896119886 The permeability coefficients of air
1198960
119886 The coefficients of permeability for water
at lateral boundary119896119908 The permeability coefficients of water
1198960
119908 The coefficients of permeability for air at
lateral boundary119898119886
1119896 The coefficient of air volume change withrespect to a change in the net normal stress
119898119908
1119896 The coefficient of water volume changewith respect to a change in the net normalstress
119898119886
2 The coefficient of air volume change with
respect to a change in matric suction119898119908
2 The coefficient of water volume changewith respect to a change in matric suction
1198990 The initial porosity
119873 The discrete point119875V A vacuum pressure120588 The density of air phase119902 The vertical loading1199030 The thickness of lateral boundary
119903119890 The radius of soil layer
119903119908 The radius of sand drain
119903120572 The radius value of 120572th point
120574119908 The unit weight of water phase
119877 The universal air constant119878 The distance between two sand drains1198781199030 The initial degree of saturation
119878V Settlement119878ℎ The radial displacement
119905 Time119879 The time factor119906119886 The excess pore-air pressure119906atm Atmospheric pressure119906119886
0 The initial excess pore-air pressure
119906119908 The excess pore-water pressure119906119908
0 The initial excess pore-water pressure
119880119886 The average degree of consolidation with
respect to air119880119908 The average degree of consolidation with
respect to water119881119886 The air volume of soil layer
14 Mathematical Problems in Engineering
119881119908 The water volume of soil layer
1198810 Initial total volume of the soil
Acknowledgments
The author gratefully acknowledges the financial supportfrom the Macau Science and Technology Development Fund(Grant no FDCT0112013A1) and the University of MacauResearch Fund (Grants nos MYRG189(Y2-L3)-FST11-ZWHand MYRG067(Y2-L2)-FST12-ZWH)
References
[1] K Terzaghi Theoretical Soil Mechanics Wiley New York NYUSA 1943
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] G E Blight Strength and consolidation characteristics of com-pacted soils [PhD thesis] University of London London UK1961
[4] R F Scott Principles of Soil Mechanics Addison Wesley Pub-lishing Company Boston Mass USA 1963
[5] L Barden ldquoConsolidation of compacted and unsaturated clayrdquoGeotechnique vol 15 no 3 pp 267ndash286 1965
[6] D G Fredlund and J U Hasan ldquoOne-dimensional consolida-tion theory unsaturated soilsrdquo Canadian Geotechnical Journalvol 16 no 3 pp 521ndash531 1979
[7] D G Fredlund andH Rahardjo Soil Mechanics for UnsaturatedSoils John Wiley and Son 1993
[8] A Qin D Sun L Yang and Y Weng ldquoA semi-analyticalsolution to consolidation of unsaturated soils with the freedrainage wellrdquo Computers and Geotechnics vol 37 no 7-8 pp867ndash875 2010
[9] W H Zhou L S Zhao and X B Li ldquoA simple analyticalsolution to one-dimensional consolidation for unsaturated soilrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics 2013
[10] M Malik and F Civan ldquoA comparative study of differentialquadrature and cubature methods vis-a-vis some conventionaltechniques in context of convection-diffusion-reaction prob-lemsrdquo Chemical Engineering Science vol 50 no 3 pp 531ndash5471995
[11] P Malekzadeh M R Golbahar Haghighi and M M AtashildquoOut-of-plane free vibration of functionally graded circularcurved beams in thermal environmentrdquo Composite Structuresvol 92 no 2 pp 541ndash552 2010
[12] P Malekzadeh M R G Haghighi and M M Atashi ldquoFreevibration analysis of elastically supported functionally radedannular plates subjected to thermal environmentrdquo Meccanicavol 46 no 5 pp 893ndash913 2011
[13] R P Chen W H Zhou H Z Wang and Y M Chen ldquoOne-dimensional nonlinear consolidation of multi-layered soil bydifferential quadrature methodrdquo Computers and Geotechnicsvol 32 no 5 pp 358ndash369 2005
[14] R P Chen W H Zhou H Z Wang and Y M Chen ldquoStudyon one-dimensional nonlinear consolidation of multi-layeredsoil using differential quadrature methodrdquo Chinese Journal ofComputational Mechanics vol 22 no 3 pp 310ndash315 2005
[15] H Z Wang R P Chen W H Zhou and Y M ChenldquoComputation of 1-D nonlinear consolidation in double-layer
foundation by using differential quadrature methodrdquo Journal ofHydraulic Engineering vol 4 pp 8ndash14 2004
[16] W H Zhou and L S Zhao ldquoOne-dimensional consolidationof unsaturated soil subjected to time-dependent loading undervarious initial and boundary conditionsrdquo ASCE InternationalJournal of Geomechanics 2013
[17] E Conte ldquoConsolidation analysis for unsaturated soilsrdquo Cana-dian Geotechnical Journal vol 41 no 4 pp 599ndash612 2004
[18] E Conte ldquoPlane strain and axially symmetric consolidation inunsaturated soilsrdquo International Journal of Geomechanics vol 6no 2 pp 131ndash135 2006
[19] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 no 1 pp40ndash52 1972
[20] R Bellman and J Casti ldquoDifferential quadrature and long-termintegrationrdquo Journal of Mathematical Analysis and Applicationsvol 34 no 2 pp 235ndash238 1971
[21] J R Quan andC T Chang ldquoNew insights in solving distributedsystem equations by the quadrature method I analysisrdquo Com-puters and Chemical Engineering vol 13 no 7 pp 779ndash7881989
V (1minus119862119886119862119908)119860119886 =minus119862119886
V(1 minus 119862119886119862119908) and 119860119908 = minus119862119886119862119908
V (1 minus 119862119886119862119908)
23 Boundary and Initial Conditions As shown in Figure 1the wall of sand drain is permeable and lateral boundarysurface is impermeable or impeded the boundary conditionsfor radial consolidation are
1198961198861199030 (9) reflects that the lateral boundary surface is imperme-
able or impeded respectively 1198960119908and 1198960
119886are the coefficients
of permeability for water and air at lateral boundary respec-tively 119903
0is the thickness of lateral boundary
The initial condition can be written as
119906119886
(119903 0) = 119906119886
0 119906
119908
(119903 0) = 119906119908
0 (11)
where 119906119908
0and 119906
119886
0are the initial pore-water and pore-air
pressure distributions
3 Differential Quadrature Formulation
DQM is a numerical solution technique for initial andorboundary value problems proposed by Bellman et al [19]and Bellman and Casti [20] According to DQM a partialderivative of a function with respect to a variable can beapproximated by a weighted linear sum of the functionvalues at given discrete points Chen et al [13] employedDQM to solve one-dimensional consolidation problems inmultilayered soils
To show the mathematic detail of DQM consider afunction 120593 = 120593(119909) on the domain 0 le 119909 le 119886 and thedomain is dispersed as119873 pointsThen the general differentialquadrature approximation of the function at the 119894th discretepoint is given by
are the weighting coefficient of 119903th derivative 119903 lt119873
This paper adopts a method derived by Quan and Chang[21] which uses Lagrange polynomial to determine theweighting coefficients
119863(1)
119894119896=
prod119873
V=1V = 119894 (119909119894 minus 119909V)
(119909119894minus 119909119896)prod119873
V=1119896 = 119894 (119909119896 minus 119909V)
119894 119896 = 1 2 119873 (119896 = 119894)
119863(119903)
119894119894= minus
119873
sum
V=1V = 119894
119863(119903)
119894V 119894 = 1 2 119873 1 le 119903 le 119873 minus 1
119863(119903)
119894119896= 119903[119863
(119903minus1)
119894119894119863(1)
119894119896minus
119863(119903minus1)
119894119896
119909119894minus 119909119896
]
119894 119896 = 1 2 119873 (119896 = 119894) 2 le 119903 le 119873 minus 1
(13)
The soil layer is dispersed in the radial direction andthe number of discrete points is 119873 Then in order tosolve the equations conveniently the local coordinate 120576 isintroduced The relationship between local coordinate andintegral coordinate is
119903 = (05 minus 120576) 1199031+ (05 + 120576) 119903
119873 (minus05 le 120576 le 05) (14)
where 119903 is the integral coordinate of soil layer and 1199031and 119903119873
are the radius values of 1th and119873th point respectivelyThe differential of (14) can be expressed as
According to (19) and (20) (18) can be rewritten as
120597V119908
120597119879=119882119886
119866(AV119886 + 1
119903BV119886) + 119882
119908
119866(AV119908 + 1
119903BV119908)
120597V119886
120597119879=119860119886
119866(AV119886 + 1
119903BV119886) + 119860
119908
119866(AV119908 + 1
119903BV119908)
(23)
where V119908 = [119906119908
2 119906119908
3 119906
119908
119873minus1]1015840
V119886 = [119906119886
2 119906119886
3 119906
119886
119873minus1]1015840
A = A1 + A2 and B = B1 + B2
A1 =[[[
[
119863(2)
22sdot sdot sdot 119863
(2)
2(119873minus1)
d
119863(2)
(119873minus1)2sdot sdot sdot 119863
(2)
(119873minus1)(119873minus1)
]]]
]
B1 =[[[
[
119863(1)
22sdot sdot sdot 119863
(1)
2(119873minus1)
d
119863(1)
(119873minus1)2sdot sdot sdot 119863
(1)
(119873minus1)(119873minus1)
]]]
]
A2 = Mminus1 timesQ times
[[[
[
119863(2)
21119863(2)
2119873
119863(2)
(119873minus1)1119863(2)
(119873minus1)119873
]]]
]
B2 = Mminus1 timesQ times
[[[
[
119863(1)
21119863(1)
2119873
119863(1)
(119873minus1)1119863(1)
(119873minus1)119873
]]]
]
119866 =119896119908
minus120574119908119898119904
1119896
M = [1 0
119863(1)
1198731119863(1)
119873119873
]
Q = minus[0 sdot sdot sdot 0
119863(1)
1198732sdot sdot sdot 119863
(1)
119873(119873minus1)
]
(24)
Therefore the governing equations of water and air aretranslated into two sets of ordinary differential equations(ODEs) The solutions of ODEs can be obtained by usingRong-KuttamethodThenwe can apply the solutions into thefollowing formulas the average degree of consolidation theradial displacement and the vertical settlement
In unsaturated soils the average degree of consolidationcan be divided into two parts the average degree of consoli-dation with respect to water phase119880119908 and the average degreeof consolidation with respect to air phase 119880119886 To obtain theaverage degree of consolidation two formulations are givenby Fredlund and Rahardjo [7] Consider
According to the two stress-state variable approaches [67] volume strain is represented by the following constitutiveequation for soil layer
120597120576V
120597119905= (119898119908
1119896+ 119898119886
1119896)120597 (119902 minus 119906
119886
)
120597119905+ (119898119908
2+ 119898119886
2)120597 (119906119886
minus 119906119908
)
120597119905
(27)
By integrating (26) with respect to time 119905 from 0 to 119905 weget the expression of volumetric strain 120576V
120576V = (119898119908
1119896+ 119898119886
1119896) [(119902 minus 119906
119886
) minus (119902 minus 119906119886
0)]
+ (119898119908
2+ 119898119886
2) [(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)]
(28)
Volumetric strain consists of vertical strain and horizon-tal strain For the axisymmetric consolidation it is assumed
in this paper that one-third of the volume strain is contributedfrom vertical strain and two-thirds are fromhorizontal strainSo the radial displacement and vertical settlement can beobtained respectively Consider
119878ℎ=2
3(119898119908
1119896+ 119898119886
1119896) int
119903119890
0
[(119902 minus 119906119886
) minus (119902 minus 119906119886
0)] 119889119903
+2
3(119898119908
2+ 119898119886
2) int
119903119890
0
[(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)] 119889119903
119878V =1
3(119898119908
1119896+ 119898119886
1119896) int
119867
0
[(119902 minus 119906119886
) minus (119902 minus 119906119886
0)] 119889119911
+1
3(119898119908
2+ 119898119886
2) int
119867
0
[(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)] 119889119911
(29)
8 Mathematical Problems in Engineering
0
005
01
015
02
025
03
035
04
045
10minus3
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 10 mr = 12 m
uw
q
Figure 7 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
0
001
002
003
004
005
006
10minus3
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 10 mr = 12 m
uaq
Figure 8 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
4 An Example and Convergence Analysis
In this section the convergence analysis the average degree ofconsolidation and the effects of displacements in radial andvertical directions are discussed by using a simple example
As shown in Figure 1 a vertical loading 119902 = 100 kPa isapplied on the top surface of the soil layerThe lateral bound-ary is considered as impermeable Considering a uniforminitial excess pore-water and pore-air pressure distribution119906119886
0= 5 kPa 119906119908
0= 40 kPa Other parameters are 119899
0= 05 119878
1199030=
08 119896119908= 10minus10ms119898119908
1119896= minus015times10
minus4 kPaminus11198981199082= minus061times
10minus4 kPaminus1119898119886
1119896= 006 times 10
minus4 kPaminus11198981198862= 026 times 10
minus4 kPaminus1119903119908= 02m 119903
119890= 18m and 119906atm = 1013 kPa By applying the
DQM solution presented above the results of axisymmetric
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
1
08
06
04
02
0
uw
q
Figure 9 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
014
012
01
008
006
004
002
0
uaq
Figure 10 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
consolidation of unsaturated soils in a sand drain foundationwere obtained
Tables 1 and 2 list the solutions of excess pore-water andpore-air pressures at radius 119903 = 1m at different time factorsand different equally spaced discrete points respectively Theratio of permeability coefficient is 119896
119886119896119908= 01 From Tables
1 and 2 the accuracy of solutions increases when the numberof discrete point becomes big It is obvious that 9 equallyspaced grid points are sufficient to obtain the convergedresults of excess pore-water and pore-air pressuresThereforethe number of discrete points119873 = 9 is adopted in most casestudies
Figure 2 shows the curves of average degree of consoli-dation with respect to water and air phases when 119873 = 9The ratio of permeability coefficient is 119896
119886119896119908
= 1 FromFigure 2 it can be observed that the consolidation of air
Mathematical Problems in Engineering 9
Table 1 Different excess pore-water pressures 119906119908119902 under time factor 119879 with different discrete points119873 (119896
1 times 10minus6 0050 0050 0050 0050 0050 0050 0050
1 times 10minus4 0050 0050 0050 0050 0050 0050 0050
5 times 10minus2 0048 0048 0048 0048 0048 0048 0048
1 times 10minus1 0046 0045 0045 0045 0045 0045 0045
2 times 10minus1 0042 0041 0040 0040 0040 0040 0010
5 times 10minus1 0035 0033 0032 0032 0032 0031 0031
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 11 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
phase and water phase begins when 119879 = 10 times 10minus3 When
119879 = 10 times 100 and 119879 = 10 times 10
15 the consolidation ofair phase and water phase is almost finished respectivelyAt the early stages soil consolidation is mainly caused bythe dissipation of excess pore-air pressure But in the laterstages soil consolidation is mainly caused by the dissipationof excess pore-water pressure
Figure 3 shows the radial displacement development withtime and Figure 4 describes the settlement of top surface(119867 = 10m) at different radii with time factor 119879 The numberof discrete points is 119873 = 9 The ratio of permeabilitycoefficient is 119896
119886119896119908
= 01 From Figure 4 we can see thatthe soil settles earlier at the points with smaller radius thatis closer to the sand drain Figure 5 shows the settlement atdifferent heights of soil layer at radius 119903 = 1m with time
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 12 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
factor 119879 When the height of soil layer119867 is equal to 4m thesettlement is almost zero
5 Cases of Different Initial andBoundary Conditions
As the initial conditions do not need to be constant in thepresentDQMsolution it is easily used to analyze nonuniforminitial pore-water and pore-air distribution problems In thispart four different initial pore-water and pore-air distribu-tions and some different boundary conditions are consideredA vertical loading 119902 = 100 kPa is applied on the top surfaceof soil layer Other parameters are 119899
0= 05 119878
1199030= 08 119896
119908=
10minus10ms 119903
119908= 02m 119903
119890= 18m119898119908
1119896= minus015 times 10
minus4 kPaminus1
10 Mathematical Problems in Engineering
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 13 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 14 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
119898119908
2= minus061 times 10
minus4 kPaminus1 1198981198861119896
= 006 times 10minus4 kPaminus1 119898119886
2=
026 times 10minus4 kPaminus1 and 119906atm = 1013 kPa
51 Effect of Different Initial Pore-WaterAir Pressure Distribu-tions In this section in order to make the curves smooththe number of discrete points is chosen to be 119873 = 17 Thelateral boundary is considered as impermeable and the ratioof permeability coefficient with respect to air and water is119896119886119896119908= 01
Four different initial pore-waterair distributions aretaken into account In Figure 6(a) the case of uniform initialpore-air and pore-water pressures assumes that 119906119886
0= 5 kPa
and 1199061199080= 40 kPa Figure 6(b) presents the linear increasing
initial pore-air and pore-water distributions The lateral-skewed distributions initial pore-waterair pressures is shown
in Figures 6(c) and 6(d) and they are represented by thefollowing equations
119906119908
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
100105119887
119906119886
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
8 times 100105119887
(30)
where 119887 is the variable which can control the spread orskewness of the curves in Figures 6(c) and 6(d)
Under the condition shown in Figure 6(a) the dissipationcurves of the excess pore-water and pore-air pressure atdifferent distances from the sand drain are presented inFigures 7 and 8 respectively The excess pore-air and pore-water pressures dissipate faster at the points with smallerradius that is closer to the sand drain As the intervalbetween the two curves becomes smaller from 119903 = 06m to119903 = 12m it can be seen that the effect of the distance on thedissipation becomes less obvious
In Figure 6(b) the initial pore-air and pore-water pres-sures are assumed not to be constants and to be linearincreasing when the radius increases The initial pore-airpressure is assumed one-eighth of the initial pore-water pres-sure In other words from 119903 = 04m to 119903 = 16m the initialpore-water pressure will increase from 30 kPa to 110 kPa andthe initial pore-air pressure will increase from 375 kPa to1375 kPa Figures 9 and 10 show the dissipation of excesspore-water and pore-air pressures at different distances fromthe sand drain under Figure 6(b) distributions
Comparing Figures 7 and 9 it is found that the excesspore-water pressure dissipates completely at almost the sametime With the change of the initial pore-water pressuredistribution the time of dissipation with respect to wateris not significantly affected Comparing Figures 8 and 10it can be seen that the excess pore-air pressure dissipatescompletely at almost the same time The initial pore-airpressure distribution has little influence on the dissipationtime of air phase
Under the conditions of Figures 6(c) and 6(d) thedissipation curves of the excess pore-water and pore-airpressures for initial distributions where 119887 = 2 6 are shownin Figures 11 12 13 and 14 From these figures whatever theinitial pore-water and pore-air pressures are the excess pore-water and pore-air pressures dissipate completely at almostthe same timeThe plateau period is longer at the points withlarger radius
Here excess pore pressure isochrones for the above fourinitial conditions are presented in Figures 15 and 16 Figures15(a) and 16(a) show the excess pore-water and pore-airpressure isochrones when the initial excess pore pressure isuniform at all radii In Figures 15(b) and 16(b) the initialexcess pore pressure distributions are linearly increasingThepore pressure isochrones for lateral-skewed initial distribu-tions are shown in Figures 15(c) 16(c) 15(d) and 16(d)respectively From Figures 15(a) and 16(a) as the initial excesspore pressure is constant the peak of curves in Figure 15(a)approaches to 04 and the peak of curves in Figure 13(a)approaches to 005 Comparing Figures 15(a) and 15(b) as
Mathematical Problems in Engineering 11
0
005
01
015
02
025
03
035
04
02 04 06 08 1 12 14 16 18r (m)
02
1
2
5
05
T = 10
uw
q
(a)
02 04 06 08 1 12 14 16 18r (m)
1
1
2
5
09
08
07
06
05
05
04
03
02
02
01
01
0
T = 10
uw
q
(b)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
011
09
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(c)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
0109
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(d)
Figure 15Normalized pore-water pressure isochrones for different initial excess pore-water pressure distributions (119896119886119896119908= 01) (a) uniform
(b) linearly increasing (c) lateral skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
the initial excess pore-water pressure is linearly increasingthe skewness is more evident during early stages of dissipa-tion in Figure 15(b) But in the later stages the influence ofinitial excess pore-water pressure becomes less significant Soare the isochrones of the excess pore-air pressure in Figures16(a) and 16(b) Figures 15(c) 16(c) 15(d) and 16(d) showthe excess pore-water and pore-air pressure isochrones forthe lateral-skewed initial distribution when 119887 = 2 6 It isseen that some intersections appear in the isochrones duringthe early stages It means that a redistribution of excesspore pressures occurs toward the region of minimal initialexcess pore pressure during the early stages With the timeincreasing the isochrones of excess pore pressures becomeregular
52 Effect of Different Boundary Conditions In this partthe effect of different lateral boundary conditions is studiedUniform initial excess pore-waterair pressure distributions119906119886
0= 5 kPa and 119906119908
0= 40 kPa are adopted Four different ratios
119896119886119896119908are considered that is 01 1 10 and 100 with 119896
119908=
10minus10ms Two different drainage conditions are analyzed
lateral surface is impervious and lateral surface is impededFor the impeded surface 119886
119908and 119886
119886in (9) are chosen to be
05Figures 17 and 18 shows the dissipation curves for the
water and air pressures at radius 119903 = 12m under differentboundary conditions First the excess pore-water and pore-air pressures dissipate faster in impeded surface than that inimpermeable surface Second 119896
119908is the same in the cases
12 Mathematical Problems in Engineering
02 04 06 08 1 12 14 16 18r (m)
01
02
05
T = 2
006
005005
004
003
002
002
001
0
1
uaq
(a)
02 04 06 08 1 12 14 16 18r (m)
014
012
0101
02
05008
006
004
002
0
T = 2
005002
1
uaq
(b)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(c)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(d)
Figure 16 Normalized pore-air pressure isochrones for different initial excess pore-air pressure distributions (119896119886119896119908= 01) (a) uniform (b)
linearly increasing (c) lateral-skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
of four 119896119886119896119908values It is observed from the results that
the bigger 119896119886119896119908
is the faster the excess pore-water andpore-air pressures dissipate Third the dissipation curves ofthe excess pore-air and pore-water pressures have almostthe same shape under different values of 119896
119886119896119908 Comparing
Figure 17 with Figure 18 when the excess pore-air pressuredissipates almost completely the dissipation of excess pore-water pressure enters into a plateau period The bigger 119896
119886119896119908
is the earlier the excess pore-air pressure dissipates com-pletely Therefore it is shown that the value of 119896
119886affects the
dissipation of pore-water pressure during the consolidationprocess
6 Conclusion
This paper obtains a general solution to the axisymmet-ric consolidation of unsaturated soils by using differen-tial quadrature method (DQM) based on Fredlundrsquos one-dimensional consolidation theory for unsaturated soil Withthe use of DQM the two governing equations are trans-formed into two sets of ordinary differential equationsThe solutions to the transformed differential equations areobtained by Rong-Kutta method under different initial andboundary conditions A case study has been presented forthe analysis of unsaturated consolidation in a sand drain
Mathematical Problems in Engineering 13
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
045
04
035
03
025
02
015
01
005
0
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
uw
q
Figure 17 Variation of excess pore-water pressures 119906119908119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
007
006
005
004
003
002
001
0
uaq
Figure 18 Variation of excess pore-air pressures 119906119886
119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
foundation The convergence analysis the average degreeof consolidation for water and air phases the settlementin radial and vertical directions effects of different initialexcess pore-air and pore-water pressure distributions andeffects of different boundary conditions have been presentedFrom the analyses some conclusions can be drawn (a)through compiling the programs it is found that the DQsolution delivers accurate and stable results for unsaturatedsoil consolidation Due to the uniformmatrix structure of theDQ formulas it is easy to compile computer programs whenconsidering complicated conditions such as nonuniformpore-waterair distribution conditions (b) soil consolidationismainly caused by the dissipation of excess pore-air pressurein the early stages and by the dissipation of excess pore-water pressure in the later stages (c) the bigger the ratio
of the permeability coefficients for water and air is thefaster the excess pore-water and pore-air pressures dissipate(d) The excess pore-air and pore-water pressures dissipatefaster at the points closer to the sand drain (e) the initialdistribution has some effects on the early consolidationprocess of unsaturated soil and boundary condition has asignificant effect on the whole consolidation process
Notation
119862119908
V The consolidation coefficient of water119862119886
V The consolidation coefficient of air119863(119903)
119894119896 The weighting coefficient
120576V Volumetric strain119892 The gravitational acceleration119867 The thickness of soil layer119896119886 The permeability coefficients of air
1198960
119886 The coefficients of permeability for water
at lateral boundary119896119908 The permeability coefficients of water
1198960
119908 The coefficients of permeability for air at
lateral boundary119898119886
1119896 The coefficient of air volume change withrespect to a change in the net normal stress
119898119908
1119896 The coefficient of water volume changewith respect to a change in the net normalstress
119898119886
2 The coefficient of air volume change with
respect to a change in matric suction119898119908
2 The coefficient of water volume changewith respect to a change in matric suction
1198990 The initial porosity
119873 The discrete point119875V A vacuum pressure120588 The density of air phase119902 The vertical loading1199030 The thickness of lateral boundary
119903119890 The radius of soil layer
119903119908 The radius of sand drain
119903120572 The radius value of 120572th point
120574119908 The unit weight of water phase
119877 The universal air constant119878 The distance between two sand drains1198781199030 The initial degree of saturation
119878V Settlement119878ℎ The radial displacement
119905 Time119879 The time factor119906119886 The excess pore-air pressure119906atm Atmospheric pressure119906119886
0 The initial excess pore-air pressure
119906119908 The excess pore-water pressure119906119908
0 The initial excess pore-water pressure
119880119886 The average degree of consolidation with
respect to air119880119908 The average degree of consolidation with
respect to water119881119886 The air volume of soil layer
14 Mathematical Problems in Engineering
119881119908 The water volume of soil layer
1198810 Initial total volume of the soil
Acknowledgments
The author gratefully acknowledges the financial supportfrom the Macau Science and Technology Development Fund(Grant no FDCT0112013A1) and the University of MacauResearch Fund (Grants nos MYRG189(Y2-L3)-FST11-ZWHand MYRG067(Y2-L2)-FST12-ZWH)
References
[1] K Terzaghi Theoretical Soil Mechanics Wiley New York NYUSA 1943
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] G E Blight Strength and consolidation characteristics of com-pacted soils [PhD thesis] University of London London UK1961
[4] R F Scott Principles of Soil Mechanics Addison Wesley Pub-lishing Company Boston Mass USA 1963
[5] L Barden ldquoConsolidation of compacted and unsaturated clayrdquoGeotechnique vol 15 no 3 pp 267ndash286 1965
[6] D G Fredlund and J U Hasan ldquoOne-dimensional consolida-tion theory unsaturated soilsrdquo Canadian Geotechnical Journalvol 16 no 3 pp 521ndash531 1979
[7] D G Fredlund andH Rahardjo Soil Mechanics for UnsaturatedSoils John Wiley and Son 1993
[8] A Qin D Sun L Yang and Y Weng ldquoA semi-analyticalsolution to consolidation of unsaturated soils with the freedrainage wellrdquo Computers and Geotechnics vol 37 no 7-8 pp867ndash875 2010
[9] W H Zhou L S Zhao and X B Li ldquoA simple analyticalsolution to one-dimensional consolidation for unsaturated soilrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics 2013
[10] M Malik and F Civan ldquoA comparative study of differentialquadrature and cubature methods vis-a-vis some conventionaltechniques in context of convection-diffusion-reaction prob-lemsrdquo Chemical Engineering Science vol 50 no 3 pp 531ndash5471995
[11] P Malekzadeh M R Golbahar Haghighi and M M AtashildquoOut-of-plane free vibration of functionally graded circularcurved beams in thermal environmentrdquo Composite Structuresvol 92 no 2 pp 541ndash552 2010
[12] P Malekzadeh M R G Haghighi and M M Atashi ldquoFreevibration analysis of elastically supported functionally radedannular plates subjected to thermal environmentrdquo Meccanicavol 46 no 5 pp 893ndash913 2011
[13] R P Chen W H Zhou H Z Wang and Y M Chen ldquoOne-dimensional nonlinear consolidation of multi-layered soil bydifferential quadrature methodrdquo Computers and Geotechnicsvol 32 no 5 pp 358ndash369 2005
[14] R P Chen W H Zhou H Z Wang and Y M Chen ldquoStudyon one-dimensional nonlinear consolidation of multi-layeredsoil using differential quadrature methodrdquo Chinese Journal ofComputational Mechanics vol 22 no 3 pp 310ndash315 2005
[15] H Z Wang R P Chen W H Zhou and Y M ChenldquoComputation of 1-D nonlinear consolidation in double-layer
foundation by using differential quadrature methodrdquo Journal ofHydraulic Engineering vol 4 pp 8ndash14 2004
[16] W H Zhou and L S Zhao ldquoOne-dimensional consolidationof unsaturated soil subjected to time-dependent loading undervarious initial and boundary conditionsrdquo ASCE InternationalJournal of Geomechanics 2013
[17] E Conte ldquoConsolidation analysis for unsaturated soilsrdquo Cana-dian Geotechnical Journal vol 41 no 4 pp 599ndash612 2004
[18] E Conte ldquoPlane strain and axially symmetric consolidation inunsaturated soilsrdquo International Journal of Geomechanics vol 6no 2 pp 131ndash135 2006
[19] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 no 1 pp40ndash52 1972
[20] R Bellman and J Casti ldquoDifferential quadrature and long-termintegrationrdquo Journal of Mathematical Analysis and Applicationsvol 34 no 2 pp 235ndash238 1971
[21] J R Quan andC T Chang ldquoNew insights in solving distributedsystem equations by the quadrature method I analysisrdquo Com-puters and Chemical Engineering vol 13 no 7 pp 779ndash7881989
According to (19) and (20) (18) can be rewritten as
120597V119908
120597119879=119882119886
119866(AV119886 + 1
119903BV119886) + 119882
119908
119866(AV119908 + 1
119903BV119908)
120597V119886
120597119879=119860119886
119866(AV119886 + 1
119903BV119886) + 119860
119908
119866(AV119908 + 1
119903BV119908)
(23)
where V119908 = [119906119908
2 119906119908
3 119906
119908
119873minus1]1015840
V119886 = [119906119886
2 119906119886
3 119906
119886
119873minus1]1015840
A = A1 + A2 and B = B1 + B2
A1 =[[[
[
119863(2)
22sdot sdot sdot 119863
(2)
2(119873minus1)
d
119863(2)
(119873minus1)2sdot sdot sdot 119863
(2)
(119873minus1)(119873minus1)
]]]
]
B1 =[[[
[
119863(1)
22sdot sdot sdot 119863
(1)
2(119873minus1)
d
119863(1)
(119873minus1)2sdot sdot sdot 119863
(1)
(119873minus1)(119873minus1)
]]]
]
A2 = Mminus1 timesQ times
[[[
[
119863(2)
21119863(2)
2119873
119863(2)
(119873minus1)1119863(2)
(119873minus1)119873
]]]
]
B2 = Mminus1 timesQ times
[[[
[
119863(1)
21119863(1)
2119873
119863(1)
(119873minus1)1119863(1)
(119873minus1)119873
]]]
]
119866 =119896119908
minus120574119908119898119904
1119896
M = [1 0
119863(1)
1198731119863(1)
119873119873
]
Q = minus[0 sdot sdot sdot 0
119863(1)
1198732sdot sdot sdot 119863
(1)
119873(119873minus1)
]
(24)
Therefore the governing equations of water and air aretranslated into two sets of ordinary differential equations(ODEs) The solutions of ODEs can be obtained by usingRong-KuttamethodThenwe can apply the solutions into thefollowing formulas the average degree of consolidation theradial displacement and the vertical settlement
In unsaturated soils the average degree of consolidationcan be divided into two parts the average degree of consoli-dation with respect to water phase119880119908 and the average degreeof consolidation with respect to air phase 119880119886 To obtain theaverage degree of consolidation two formulations are givenby Fredlund and Rahardjo [7] Consider
According to the two stress-state variable approaches [67] volume strain is represented by the following constitutiveequation for soil layer
120597120576V
120597119905= (119898119908
1119896+ 119898119886
1119896)120597 (119902 minus 119906
119886
)
120597119905+ (119898119908
2+ 119898119886
2)120597 (119906119886
minus 119906119908
)
120597119905
(27)
By integrating (26) with respect to time 119905 from 0 to 119905 weget the expression of volumetric strain 120576V
120576V = (119898119908
1119896+ 119898119886
1119896) [(119902 minus 119906
119886
) minus (119902 minus 119906119886
0)]
+ (119898119908
2+ 119898119886
2) [(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)]
(28)
Volumetric strain consists of vertical strain and horizon-tal strain For the axisymmetric consolidation it is assumed
in this paper that one-third of the volume strain is contributedfrom vertical strain and two-thirds are fromhorizontal strainSo the radial displacement and vertical settlement can beobtained respectively Consider
119878ℎ=2
3(119898119908
1119896+ 119898119886
1119896) int
119903119890
0
[(119902 minus 119906119886
) minus (119902 minus 119906119886
0)] 119889119903
+2
3(119898119908
2+ 119898119886
2) int
119903119890
0
[(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)] 119889119903
119878V =1
3(119898119908
1119896+ 119898119886
1119896) int
119867
0
[(119902 minus 119906119886
) minus (119902 minus 119906119886
0)] 119889119911
+1
3(119898119908
2+ 119898119886
2) int
119867
0
[(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)] 119889119911
(29)
8 Mathematical Problems in Engineering
0
005
01
015
02
025
03
035
04
045
10minus3
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 10 mr = 12 m
uw
q
Figure 7 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
0
001
002
003
004
005
006
10minus3
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 10 mr = 12 m
uaq
Figure 8 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
4 An Example and Convergence Analysis
In this section the convergence analysis the average degree ofconsolidation and the effects of displacements in radial andvertical directions are discussed by using a simple example
As shown in Figure 1 a vertical loading 119902 = 100 kPa isapplied on the top surface of the soil layerThe lateral bound-ary is considered as impermeable Considering a uniforminitial excess pore-water and pore-air pressure distribution119906119886
0= 5 kPa 119906119908
0= 40 kPa Other parameters are 119899
0= 05 119878
1199030=
08 119896119908= 10minus10ms119898119908
1119896= minus015times10
minus4 kPaminus11198981199082= minus061times
10minus4 kPaminus1119898119886
1119896= 006 times 10
minus4 kPaminus11198981198862= 026 times 10
minus4 kPaminus1119903119908= 02m 119903
119890= 18m and 119906atm = 1013 kPa By applying the
DQM solution presented above the results of axisymmetric
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
1
08
06
04
02
0
uw
q
Figure 9 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
014
012
01
008
006
004
002
0
uaq
Figure 10 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
consolidation of unsaturated soils in a sand drain foundationwere obtained
Tables 1 and 2 list the solutions of excess pore-water andpore-air pressures at radius 119903 = 1m at different time factorsand different equally spaced discrete points respectively Theratio of permeability coefficient is 119896
119886119896119908= 01 From Tables
1 and 2 the accuracy of solutions increases when the numberof discrete point becomes big It is obvious that 9 equallyspaced grid points are sufficient to obtain the convergedresults of excess pore-water and pore-air pressuresThereforethe number of discrete points119873 = 9 is adopted in most casestudies
Figure 2 shows the curves of average degree of consoli-dation with respect to water and air phases when 119873 = 9The ratio of permeability coefficient is 119896
119886119896119908
= 1 FromFigure 2 it can be observed that the consolidation of air
Mathematical Problems in Engineering 9
Table 1 Different excess pore-water pressures 119906119908119902 under time factor 119879 with different discrete points119873 (119896
1 times 10minus6 0050 0050 0050 0050 0050 0050 0050
1 times 10minus4 0050 0050 0050 0050 0050 0050 0050
5 times 10minus2 0048 0048 0048 0048 0048 0048 0048
1 times 10minus1 0046 0045 0045 0045 0045 0045 0045
2 times 10minus1 0042 0041 0040 0040 0040 0040 0010
5 times 10minus1 0035 0033 0032 0032 0032 0031 0031
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 11 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
phase and water phase begins when 119879 = 10 times 10minus3 When
119879 = 10 times 100 and 119879 = 10 times 10
15 the consolidation ofair phase and water phase is almost finished respectivelyAt the early stages soil consolidation is mainly caused bythe dissipation of excess pore-air pressure But in the laterstages soil consolidation is mainly caused by the dissipationof excess pore-water pressure
Figure 3 shows the radial displacement development withtime and Figure 4 describes the settlement of top surface(119867 = 10m) at different radii with time factor 119879 The numberof discrete points is 119873 = 9 The ratio of permeabilitycoefficient is 119896
119886119896119908
= 01 From Figure 4 we can see thatthe soil settles earlier at the points with smaller radius thatis closer to the sand drain Figure 5 shows the settlement atdifferent heights of soil layer at radius 119903 = 1m with time
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 12 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
factor 119879 When the height of soil layer119867 is equal to 4m thesettlement is almost zero
5 Cases of Different Initial andBoundary Conditions
As the initial conditions do not need to be constant in thepresentDQMsolution it is easily used to analyze nonuniforminitial pore-water and pore-air distribution problems In thispart four different initial pore-water and pore-air distribu-tions and some different boundary conditions are consideredA vertical loading 119902 = 100 kPa is applied on the top surfaceof soil layer Other parameters are 119899
0= 05 119878
1199030= 08 119896
119908=
10minus10ms 119903
119908= 02m 119903
119890= 18m119898119908
1119896= minus015 times 10
minus4 kPaminus1
10 Mathematical Problems in Engineering
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 13 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 14 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
119898119908
2= minus061 times 10
minus4 kPaminus1 1198981198861119896
= 006 times 10minus4 kPaminus1 119898119886
2=
026 times 10minus4 kPaminus1 and 119906atm = 1013 kPa
51 Effect of Different Initial Pore-WaterAir Pressure Distribu-tions In this section in order to make the curves smooththe number of discrete points is chosen to be 119873 = 17 Thelateral boundary is considered as impermeable and the ratioof permeability coefficient with respect to air and water is119896119886119896119908= 01
Four different initial pore-waterair distributions aretaken into account In Figure 6(a) the case of uniform initialpore-air and pore-water pressures assumes that 119906119886
0= 5 kPa
and 1199061199080= 40 kPa Figure 6(b) presents the linear increasing
initial pore-air and pore-water distributions The lateral-skewed distributions initial pore-waterair pressures is shown
in Figures 6(c) and 6(d) and they are represented by thefollowing equations
119906119908
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
100105119887
119906119886
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
8 times 100105119887
(30)
where 119887 is the variable which can control the spread orskewness of the curves in Figures 6(c) and 6(d)
Under the condition shown in Figure 6(a) the dissipationcurves of the excess pore-water and pore-air pressure atdifferent distances from the sand drain are presented inFigures 7 and 8 respectively The excess pore-air and pore-water pressures dissipate faster at the points with smallerradius that is closer to the sand drain As the intervalbetween the two curves becomes smaller from 119903 = 06m to119903 = 12m it can be seen that the effect of the distance on thedissipation becomes less obvious
In Figure 6(b) the initial pore-air and pore-water pres-sures are assumed not to be constants and to be linearincreasing when the radius increases The initial pore-airpressure is assumed one-eighth of the initial pore-water pres-sure In other words from 119903 = 04m to 119903 = 16m the initialpore-water pressure will increase from 30 kPa to 110 kPa andthe initial pore-air pressure will increase from 375 kPa to1375 kPa Figures 9 and 10 show the dissipation of excesspore-water and pore-air pressures at different distances fromthe sand drain under Figure 6(b) distributions
Comparing Figures 7 and 9 it is found that the excesspore-water pressure dissipates completely at almost the sametime With the change of the initial pore-water pressuredistribution the time of dissipation with respect to wateris not significantly affected Comparing Figures 8 and 10it can be seen that the excess pore-air pressure dissipatescompletely at almost the same time The initial pore-airpressure distribution has little influence on the dissipationtime of air phase
Under the conditions of Figures 6(c) and 6(d) thedissipation curves of the excess pore-water and pore-airpressures for initial distributions where 119887 = 2 6 are shownin Figures 11 12 13 and 14 From these figures whatever theinitial pore-water and pore-air pressures are the excess pore-water and pore-air pressures dissipate completely at almostthe same timeThe plateau period is longer at the points withlarger radius
Here excess pore pressure isochrones for the above fourinitial conditions are presented in Figures 15 and 16 Figures15(a) and 16(a) show the excess pore-water and pore-airpressure isochrones when the initial excess pore pressure isuniform at all radii In Figures 15(b) and 16(b) the initialexcess pore pressure distributions are linearly increasingThepore pressure isochrones for lateral-skewed initial distribu-tions are shown in Figures 15(c) 16(c) 15(d) and 16(d)respectively From Figures 15(a) and 16(a) as the initial excesspore pressure is constant the peak of curves in Figure 15(a)approaches to 04 and the peak of curves in Figure 13(a)approaches to 005 Comparing Figures 15(a) and 15(b) as
Mathematical Problems in Engineering 11
0
005
01
015
02
025
03
035
04
02 04 06 08 1 12 14 16 18r (m)
02
1
2
5
05
T = 10
uw
q
(a)
02 04 06 08 1 12 14 16 18r (m)
1
1
2
5
09
08
07
06
05
05
04
03
02
02
01
01
0
T = 10
uw
q
(b)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
011
09
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(c)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
0109
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(d)
Figure 15Normalized pore-water pressure isochrones for different initial excess pore-water pressure distributions (119896119886119896119908= 01) (a) uniform
(b) linearly increasing (c) lateral skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
the initial excess pore-water pressure is linearly increasingthe skewness is more evident during early stages of dissipa-tion in Figure 15(b) But in the later stages the influence ofinitial excess pore-water pressure becomes less significant Soare the isochrones of the excess pore-air pressure in Figures16(a) and 16(b) Figures 15(c) 16(c) 15(d) and 16(d) showthe excess pore-water and pore-air pressure isochrones forthe lateral-skewed initial distribution when 119887 = 2 6 It isseen that some intersections appear in the isochrones duringthe early stages It means that a redistribution of excesspore pressures occurs toward the region of minimal initialexcess pore pressure during the early stages With the timeincreasing the isochrones of excess pore pressures becomeregular
52 Effect of Different Boundary Conditions In this partthe effect of different lateral boundary conditions is studiedUniform initial excess pore-waterair pressure distributions119906119886
0= 5 kPa and 119906119908
0= 40 kPa are adopted Four different ratios
119896119886119896119908are considered that is 01 1 10 and 100 with 119896
119908=
10minus10ms Two different drainage conditions are analyzed
lateral surface is impervious and lateral surface is impededFor the impeded surface 119886
119908and 119886
119886in (9) are chosen to be
05Figures 17 and 18 shows the dissipation curves for the
water and air pressures at radius 119903 = 12m under differentboundary conditions First the excess pore-water and pore-air pressures dissipate faster in impeded surface than that inimpermeable surface Second 119896
119908is the same in the cases
12 Mathematical Problems in Engineering
02 04 06 08 1 12 14 16 18r (m)
01
02
05
T = 2
006
005005
004
003
002
002
001
0
1
uaq
(a)
02 04 06 08 1 12 14 16 18r (m)
014
012
0101
02
05008
006
004
002
0
T = 2
005002
1
uaq
(b)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(c)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(d)
Figure 16 Normalized pore-air pressure isochrones for different initial excess pore-air pressure distributions (119896119886119896119908= 01) (a) uniform (b)
linearly increasing (c) lateral-skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
of four 119896119886119896119908values It is observed from the results that
the bigger 119896119886119896119908
is the faster the excess pore-water andpore-air pressures dissipate Third the dissipation curves ofthe excess pore-air and pore-water pressures have almostthe same shape under different values of 119896
119886119896119908 Comparing
Figure 17 with Figure 18 when the excess pore-air pressuredissipates almost completely the dissipation of excess pore-water pressure enters into a plateau period The bigger 119896
119886119896119908
is the earlier the excess pore-air pressure dissipates com-pletely Therefore it is shown that the value of 119896
119886affects the
dissipation of pore-water pressure during the consolidationprocess
6 Conclusion
This paper obtains a general solution to the axisymmet-ric consolidation of unsaturated soils by using differen-tial quadrature method (DQM) based on Fredlundrsquos one-dimensional consolidation theory for unsaturated soil Withthe use of DQM the two governing equations are trans-formed into two sets of ordinary differential equationsThe solutions to the transformed differential equations areobtained by Rong-Kutta method under different initial andboundary conditions A case study has been presented forthe analysis of unsaturated consolidation in a sand drain
Mathematical Problems in Engineering 13
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
045
04
035
03
025
02
015
01
005
0
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
uw
q
Figure 17 Variation of excess pore-water pressures 119906119908119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
007
006
005
004
003
002
001
0
uaq
Figure 18 Variation of excess pore-air pressures 119906119886
119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
foundation The convergence analysis the average degreeof consolidation for water and air phases the settlementin radial and vertical directions effects of different initialexcess pore-air and pore-water pressure distributions andeffects of different boundary conditions have been presentedFrom the analyses some conclusions can be drawn (a)through compiling the programs it is found that the DQsolution delivers accurate and stable results for unsaturatedsoil consolidation Due to the uniformmatrix structure of theDQ formulas it is easy to compile computer programs whenconsidering complicated conditions such as nonuniformpore-waterair distribution conditions (b) soil consolidationismainly caused by the dissipation of excess pore-air pressurein the early stages and by the dissipation of excess pore-water pressure in the later stages (c) the bigger the ratio
of the permeability coefficients for water and air is thefaster the excess pore-water and pore-air pressures dissipate(d) The excess pore-air and pore-water pressures dissipatefaster at the points closer to the sand drain (e) the initialdistribution has some effects on the early consolidationprocess of unsaturated soil and boundary condition has asignificant effect on the whole consolidation process
Notation
119862119908
V The consolidation coefficient of water119862119886
V The consolidation coefficient of air119863(119903)
119894119896 The weighting coefficient
120576V Volumetric strain119892 The gravitational acceleration119867 The thickness of soil layer119896119886 The permeability coefficients of air
1198960
119886 The coefficients of permeability for water
at lateral boundary119896119908 The permeability coefficients of water
1198960
119908 The coefficients of permeability for air at
lateral boundary119898119886
1119896 The coefficient of air volume change withrespect to a change in the net normal stress
119898119908
1119896 The coefficient of water volume changewith respect to a change in the net normalstress
119898119886
2 The coefficient of air volume change with
respect to a change in matric suction119898119908
2 The coefficient of water volume changewith respect to a change in matric suction
1198990 The initial porosity
119873 The discrete point119875V A vacuum pressure120588 The density of air phase119902 The vertical loading1199030 The thickness of lateral boundary
119903119890 The radius of soil layer
119903119908 The radius of sand drain
119903120572 The radius value of 120572th point
120574119908 The unit weight of water phase
119877 The universal air constant119878 The distance between two sand drains1198781199030 The initial degree of saturation
119878V Settlement119878ℎ The radial displacement
119905 Time119879 The time factor119906119886 The excess pore-air pressure119906atm Atmospheric pressure119906119886
0 The initial excess pore-air pressure
119906119908 The excess pore-water pressure119906119908
0 The initial excess pore-water pressure
119880119886 The average degree of consolidation with
respect to air119880119908 The average degree of consolidation with
respect to water119881119886 The air volume of soil layer
14 Mathematical Problems in Engineering
119881119908 The water volume of soil layer
1198810 Initial total volume of the soil
Acknowledgments
The author gratefully acknowledges the financial supportfrom the Macau Science and Technology Development Fund(Grant no FDCT0112013A1) and the University of MacauResearch Fund (Grants nos MYRG189(Y2-L3)-FST11-ZWHand MYRG067(Y2-L2)-FST12-ZWH)
References
[1] K Terzaghi Theoretical Soil Mechanics Wiley New York NYUSA 1943
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] G E Blight Strength and consolidation characteristics of com-pacted soils [PhD thesis] University of London London UK1961
[4] R F Scott Principles of Soil Mechanics Addison Wesley Pub-lishing Company Boston Mass USA 1963
[5] L Barden ldquoConsolidation of compacted and unsaturated clayrdquoGeotechnique vol 15 no 3 pp 267ndash286 1965
[6] D G Fredlund and J U Hasan ldquoOne-dimensional consolida-tion theory unsaturated soilsrdquo Canadian Geotechnical Journalvol 16 no 3 pp 521ndash531 1979
[7] D G Fredlund andH Rahardjo Soil Mechanics for UnsaturatedSoils John Wiley and Son 1993
[8] A Qin D Sun L Yang and Y Weng ldquoA semi-analyticalsolution to consolidation of unsaturated soils with the freedrainage wellrdquo Computers and Geotechnics vol 37 no 7-8 pp867ndash875 2010
[9] W H Zhou L S Zhao and X B Li ldquoA simple analyticalsolution to one-dimensional consolidation for unsaturated soilrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics 2013
[10] M Malik and F Civan ldquoA comparative study of differentialquadrature and cubature methods vis-a-vis some conventionaltechniques in context of convection-diffusion-reaction prob-lemsrdquo Chemical Engineering Science vol 50 no 3 pp 531ndash5471995
[11] P Malekzadeh M R Golbahar Haghighi and M M AtashildquoOut-of-plane free vibration of functionally graded circularcurved beams in thermal environmentrdquo Composite Structuresvol 92 no 2 pp 541ndash552 2010
[12] P Malekzadeh M R G Haghighi and M M Atashi ldquoFreevibration analysis of elastically supported functionally radedannular plates subjected to thermal environmentrdquo Meccanicavol 46 no 5 pp 893ndash913 2011
[13] R P Chen W H Zhou H Z Wang and Y M Chen ldquoOne-dimensional nonlinear consolidation of multi-layered soil bydifferential quadrature methodrdquo Computers and Geotechnicsvol 32 no 5 pp 358ndash369 2005
[14] R P Chen W H Zhou H Z Wang and Y M Chen ldquoStudyon one-dimensional nonlinear consolidation of multi-layeredsoil using differential quadrature methodrdquo Chinese Journal ofComputational Mechanics vol 22 no 3 pp 310ndash315 2005
[15] H Z Wang R P Chen W H Zhou and Y M ChenldquoComputation of 1-D nonlinear consolidation in double-layer
foundation by using differential quadrature methodrdquo Journal ofHydraulic Engineering vol 4 pp 8ndash14 2004
[16] W H Zhou and L S Zhao ldquoOne-dimensional consolidationof unsaturated soil subjected to time-dependent loading undervarious initial and boundary conditionsrdquo ASCE InternationalJournal of Geomechanics 2013
[17] E Conte ldquoConsolidation analysis for unsaturated soilsrdquo Cana-dian Geotechnical Journal vol 41 no 4 pp 599ndash612 2004
[18] E Conte ldquoPlane strain and axially symmetric consolidation inunsaturated soilsrdquo International Journal of Geomechanics vol 6no 2 pp 131ndash135 2006
[19] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 no 1 pp40ndash52 1972
[20] R Bellman and J Casti ldquoDifferential quadrature and long-termintegrationrdquo Journal of Mathematical Analysis and Applicationsvol 34 no 2 pp 235ndash238 1971
[21] J R Quan andC T Chang ldquoNew insights in solving distributedsystem equations by the quadrature method I analysisrdquo Com-puters and Chemical Engineering vol 13 no 7 pp 779ndash7881989
According to (19) and (20) (18) can be rewritten as
120597V119908
120597119879=119882119886
119866(AV119886 + 1
119903BV119886) + 119882
119908
119866(AV119908 + 1
119903BV119908)
120597V119886
120597119879=119860119886
119866(AV119886 + 1
119903BV119886) + 119860
119908
119866(AV119908 + 1
119903BV119908)
(23)
where V119908 = [119906119908
2 119906119908
3 119906
119908
119873minus1]1015840
V119886 = [119906119886
2 119906119886
3 119906
119886
119873minus1]1015840
A = A1 + A2 and B = B1 + B2
A1 =[[[
[
119863(2)
22sdot sdot sdot 119863
(2)
2(119873minus1)
d
119863(2)
(119873minus1)2sdot sdot sdot 119863
(2)
(119873minus1)(119873minus1)
]]]
]
B1 =[[[
[
119863(1)
22sdot sdot sdot 119863
(1)
2(119873minus1)
d
119863(1)
(119873minus1)2sdot sdot sdot 119863
(1)
(119873minus1)(119873minus1)
]]]
]
A2 = Mminus1 timesQ times
[[[
[
119863(2)
21119863(2)
2119873
119863(2)
(119873minus1)1119863(2)
(119873minus1)119873
]]]
]
B2 = Mminus1 timesQ times
[[[
[
119863(1)
21119863(1)
2119873
119863(1)
(119873minus1)1119863(1)
(119873minus1)119873
]]]
]
119866 =119896119908
minus120574119908119898119904
1119896
M = [1 0
119863(1)
1198731119863(1)
119873119873
]
Q = minus[0 sdot sdot sdot 0
119863(1)
1198732sdot sdot sdot 119863
(1)
119873(119873minus1)
]
(24)
Therefore the governing equations of water and air aretranslated into two sets of ordinary differential equations(ODEs) The solutions of ODEs can be obtained by usingRong-KuttamethodThenwe can apply the solutions into thefollowing formulas the average degree of consolidation theradial displacement and the vertical settlement
In unsaturated soils the average degree of consolidationcan be divided into two parts the average degree of consoli-dation with respect to water phase119880119908 and the average degreeof consolidation with respect to air phase 119880119886 To obtain theaverage degree of consolidation two formulations are givenby Fredlund and Rahardjo [7] Consider
According to the two stress-state variable approaches [67] volume strain is represented by the following constitutiveequation for soil layer
120597120576V
120597119905= (119898119908
1119896+ 119898119886
1119896)120597 (119902 minus 119906
119886
)
120597119905+ (119898119908
2+ 119898119886
2)120597 (119906119886
minus 119906119908
)
120597119905
(27)
By integrating (26) with respect to time 119905 from 0 to 119905 weget the expression of volumetric strain 120576V
120576V = (119898119908
1119896+ 119898119886
1119896) [(119902 minus 119906
119886
) minus (119902 minus 119906119886
0)]
+ (119898119908
2+ 119898119886
2) [(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)]
(28)
Volumetric strain consists of vertical strain and horizon-tal strain For the axisymmetric consolidation it is assumed
in this paper that one-third of the volume strain is contributedfrom vertical strain and two-thirds are fromhorizontal strainSo the radial displacement and vertical settlement can beobtained respectively Consider
119878ℎ=2
3(119898119908
1119896+ 119898119886
1119896) int
119903119890
0
[(119902 minus 119906119886
) minus (119902 minus 119906119886
0)] 119889119903
+2
3(119898119908
2+ 119898119886
2) int
119903119890
0
[(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)] 119889119903
119878V =1
3(119898119908
1119896+ 119898119886
1119896) int
119867
0
[(119902 minus 119906119886
) minus (119902 minus 119906119886
0)] 119889119911
+1
3(119898119908
2+ 119898119886
2) int
119867
0
[(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)] 119889119911
(29)
8 Mathematical Problems in Engineering
0
005
01
015
02
025
03
035
04
045
10minus3
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 10 mr = 12 m
uw
q
Figure 7 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
0
001
002
003
004
005
006
10minus3
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 10 mr = 12 m
uaq
Figure 8 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
4 An Example and Convergence Analysis
In this section the convergence analysis the average degree ofconsolidation and the effects of displacements in radial andvertical directions are discussed by using a simple example
As shown in Figure 1 a vertical loading 119902 = 100 kPa isapplied on the top surface of the soil layerThe lateral bound-ary is considered as impermeable Considering a uniforminitial excess pore-water and pore-air pressure distribution119906119886
0= 5 kPa 119906119908
0= 40 kPa Other parameters are 119899
0= 05 119878
1199030=
08 119896119908= 10minus10ms119898119908
1119896= minus015times10
minus4 kPaminus11198981199082= minus061times
10minus4 kPaminus1119898119886
1119896= 006 times 10
minus4 kPaminus11198981198862= 026 times 10
minus4 kPaminus1119903119908= 02m 119903
119890= 18m and 119906atm = 1013 kPa By applying the
DQM solution presented above the results of axisymmetric
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
1
08
06
04
02
0
uw
q
Figure 9 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
014
012
01
008
006
004
002
0
uaq
Figure 10 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
consolidation of unsaturated soils in a sand drain foundationwere obtained
Tables 1 and 2 list the solutions of excess pore-water andpore-air pressures at radius 119903 = 1m at different time factorsand different equally spaced discrete points respectively Theratio of permeability coefficient is 119896
119886119896119908= 01 From Tables
1 and 2 the accuracy of solutions increases when the numberof discrete point becomes big It is obvious that 9 equallyspaced grid points are sufficient to obtain the convergedresults of excess pore-water and pore-air pressuresThereforethe number of discrete points119873 = 9 is adopted in most casestudies
Figure 2 shows the curves of average degree of consoli-dation with respect to water and air phases when 119873 = 9The ratio of permeability coefficient is 119896
119886119896119908
= 1 FromFigure 2 it can be observed that the consolidation of air
Mathematical Problems in Engineering 9
Table 1 Different excess pore-water pressures 119906119908119902 under time factor 119879 with different discrete points119873 (119896
1 times 10minus6 0050 0050 0050 0050 0050 0050 0050
1 times 10minus4 0050 0050 0050 0050 0050 0050 0050
5 times 10minus2 0048 0048 0048 0048 0048 0048 0048
1 times 10minus1 0046 0045 0045 0045 0045 0045 0045
2 times 10minus1 0042 0041 0040 0040 0040 0040 0010
5 times 10minus1 0035 0033 0032 0032 0032 0031 0031
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 11 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
phase and water phase begins when 119879 = 10 times 10minus3 When
119879 = 10 times 100 and 119879 = 10 times 10
15 the consolidation ofair phase and water phase is almost finished respectivelyAt the early stages soil consolidation is mainly caused bythe dissipation of excess pore-air pressure But in the laterstages soil consolidation is mainly caused by the dissipationof excess pore-water pressure
Figure 3 shows the radial displacement development withtime and Figure 4 describes the settlement of top surface(119867 = 10m) at different radii with time factor 119879 The numberof discrete points is 119873 = 9 The ratio of permeabilitycoefficient is 119896
119886119896119908
= 01 From Figure 4 we can see thatthe soil settles earlier at the points with smaller radius thatis closer to the sand drain Figure 5 shows the settlement atdifferent heights of soil layer at radius 119903 = 1m with time
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 12 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
factor 119879 When the height of soil layer119867 is equal to 4m thesettlement is almost zero
5 Cases of Different Initial andBoundary Conditions
As the initial conditions do not need to be constant in thepresentDQMsolution it is easily used to analyze nonuniforminitial pore-water and pore-air distribution problems In thispart four different initial pore-water and pore-air distribu-tions and some different boundary conditions are consideredA vertical loading 119902 = 100 kPa is applied on the top surfaceof soil layer Other parameters are 119899
0= 05 119878
1199030= 08 119896
119908=
10minus10ms 119903
119908= 02m 119903
119890= 18m119898119908
1119896= minus015 times 10
minus4 kPaminus1
10 Mathematical Problems in Engineering
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 13 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 14 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
119898119908
2= minus061 times 10
minus4 kPaminus1 1198981198861119896
= 006 times 10minus4 kPaminus1 119898119886
2=
026 times 10minus4 kPaminus1 and 119906atm = 1013 kPa
51 Effect of Different Initial Pore-WaterAir Pressure Distribu-tions In this section in order to make the curves smooththe number of discrete points is chosen to be 119873 = 17 Thelateral boundary is considered as impermeable and the ratioof permeability coefficient with respect to air and water is119896119886119896119908= 01
Four different initial pore-waterair distributions aretaken into account In Figure 6(a) the case of uniform initialpore-air and pore-water pressures assumes that 119906119886
0= 5 kPa
and 1199061199080= 40 kPa Figure 6(b) presents the linear increasing
initial pore-air and pore-water distributions The lateral-skewed distributions initial pore-waterair pressures is shown
in Figures 6(c) and 6(d) and they are represented by thefollowing equations
119906119908
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
100105119887
119906119886
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
8 times 100105119887
(30)
where 119887 is the variable which can control the spread orskewness of the curves in Figures 6(c) and 6(d)
Under the condition shown in Figure 6(a) the dissipationcurves of the excess pore-water and pore-air pressure atdifferent distances from the sand drain are presented inFigures 7 and 8 respectively The excess pore-air and pore-water pressures dissipate faster at the points with smallerradius that is closer to the sand drain As the intervalbetween the two curves becomes smaller from 119903 = 06m to119903 = 12m it can be seen that the effect of the distance on thedissipation becomes less obvious
In Figure 6(b) the initial pore-air and pore-water pres-sures are assumed not to be constants and to be linearincreasing when the radius increases The initial pore-airpressure is assumed one-eighth of the initial pore-water pres-sure In other words from 119903 = 04m to 119903 = 16m the initialpore-water pressure will increase from 30 kPa to 110 kPa andthe initial pore-air pressure will increase from 375 kPa to1375 kPa Figures 9 and 10 show the dissipation of excesspore-water and pore-air pressures at different distances fromthe sand drain under Figure 6(b) distributions
Comparing Figures 7 and 9 it is found that the excesspore-water pressure dissipates completely at almost the sametime With the change of the initial pore-water pressuredistribution the time of dissipation with respect to wateris not significantly affected Comparing Figures 8 and 10it can be seen that the excess pore-air pressure dissipatescompletely at almost the same time The initial pore-airpressure distribution has little influence on the dissipationtime of air phase
Under the conditions of Figures 6(c) and 6(d) thedissipation curves of the excess pore-water and pore-airpressures for initial distributions where 119887 = 2 6 are shownin Figures 11 12 13 and 14 From these figures whatever theinitial pore-water and pore-air pressures are the excess pore-water and pore-air pressures dissipate completely at almostthe same timeThe plateau period is longer at the points withlarger radius
Here excess pore pressure isochrones for the above fourinitial conditions are presented in Figures 15 and 16 Figures15(a) and 16(a) show the excess pore-water and pore-airpressure isochrones when the initial excess pore pressure isuniform at all radii In Figures 15(b) and 16(b) the initialexcess pore pressure distributions are linearly increasingThepore pressure isochrones for lateral-skewed initial distribu-tions are shown in Figures 15(c) 16(c) 15(d) and 16(d)respectively From Figures 15(a) and 16(a) as the initial excesspore pressure is constant the peak of curves in Figure 15(a)approaches to 04 and the peak of curves in Figure 13(a)approaches to 005 Comparing Figures 15(a) and 15(b) as
Mathematical Problems in Engineering 11
0
005
01
015
02
025
03
035
04
02 04 06 08 1 12 14 16 18r (m)
02
1
2
5
05
T = 10
uw
q
(a)
02 04 06 08 1 12 14 16 18r (m)
1
1
2
5
09
08
07
06
05
05
04
03
02
02
01
01
0
T = 10
uw
q
(b)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
011
09
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(c)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
0109
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(d)
Figure 15Normalized pore-water pressure isochrones for different initial excess pore-water pressure distributions (119896119886119896119908= 01) (a) uniform
(b) linearly increasing (c) lateral skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
the initial excess pore-water pressure is linearly increasingthe skewness is more evident during early stages of dissipa-tion in Figure 15(b) But in the later stages the influence ofinitial excess pore-water pressure becomes less significant Soare the isochrones of the excess pore-air pressure in Figures16(a) and 16(b) Figures 15(c) 16(c) 15(d) and 16(d) showthe excess pore-water and pore-air pressure isochrones forthe lateral-skewed initial distribution when 119887 = 2 6 It isseen that some intersections appear in the isochrones duringthe early stages It means that a redistribution of excesspore pressures occurs toward the region of minimal initialexcess pore pressure during the early stages With the timeincreasing the isochrones of excess pore pressures becomeregular
52 Effect of Different Boundary Conditions In this partthe effect of different lateral boundary conditions is studiedUniform initial excess pore-waterair pressure distributions119906119886
0= 5 kPa and 119906119908
0= 40 kPa are adopted Four different ratios
119896119886119896119908are considered that is 01 1 10 and 100 with 119896
119908=
10minus10ms Two different drainage conditions are analyzed
lateral surface is impervious and lateral surface is impededFor the impeded surface 119886
119908and 119886
119886in (9) are chosen to be
05Figures 17 and 18 shows the dissipation curves for the
water and air pressures at radius 119903 = 12m under differentboundary conditions First the excess pore-water and pore-air pressures dissipate faster in impeded surface than that inimpermeable surface Second 119896
119908is the same in the cases
12 Mathematical Problems in Engineering
02 04 06 08 1 12 14 16 18r (m)
01
02
05
T = 2
006
005005
004
003
002
002
001
0
1
uaq
(a)
02 04 06 08 1 12 14 16 18r (m)
014
012
0101
02
05008
006
004
002
0
T = 2
005002
1
uaq
(b)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(c)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(d)
Figure 16 Normalized pore-air pressure isochrones for different initial excess pore-air pressure distributions (119896119886119896119908= 01) (a) uniform (b)
linearly increasing (c) lateral-skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
of four 119896119886119896119908values It is observed from the results that
the bigger 119896119886119896119908
is the faster the excess pore-water andpore-air pressures dissipate Third the dissipation curves ofthe excess pore-air and pore-water pressures have almostthe same shape under different values of 119896
119886119896119908 Comparing
Figure 17 with Figure 18 when the excess pore-air pressuredissipates almost completely the dissipation of excess pore-water pressure enters into a plateau period The bigger 119896
119886119896119908
is the earlier the excess pore-air pressure dissipates com-pletely Therefore it is shown that the value of 119896
119886affects the
dissipation of pore-water pressure during the consolidationprocess
6 Conclusion
This paper obtains a general solution to the axisymmet-ric consolidation of unsaturated soils by using differen-tial quadrature method (DQM) based on Fredlundrsquos one-dimensional consolidation theory for unsaturated soil Withthe use of DQM the two governing equations are trans-formed into two sets of ordinary differential equationsThe solutions to the transformed differential equations areobtained by Rong-Kutta method under different initial andboundary conditions A case study has been presented forthe analysis of unsaturated consolidation in a sand drain
Mathematical Problems in Engineering 13
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
045
04
035
03
025
02
015
01
005
0
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
uw
q
Figure 17 Variation of excess pore-water pressures 119906119908119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
007
006
005
004
003
002
001
0
uaq
Figure 18 Variation of excess pore-air pressures 119906119886
119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
foundation The convergence analysis the average degreeof consolidation for water and air phases the settlementin radial and vertical directions effects of different initialexcess pore-air and pore-water pressure distributions andeffects of different boundary conditions have been presentedFrom the analyses some conclusions can be drawn (a)through compiling the programs it is found that the DQsolution delivers accurate and stable results for unsaturatedsoil consolidation Due to the uniformmatrix structure of theDQ formulas it is easy to compile computer programs whenconsidering complicated conditions such as nonuniformpore-waterair distribution conditions (b) soil consolidationismainly caused by the dissipation of excess pore-air pressurein the early stages and by the dissipation of excess pore-water pressure in the later stages (c) the bigger the ratio
of the permeability coefficients for water and air is thefaster the excess pore-water and pore-air pressures dissipate(d) The excess pore-air and pore-water pressures dissipatefaster at the points closer to the sand drain (e) the initialdistribution has some effects on the early consolidationprocess of unsaturated soil and boundary condition has asignificant effect on the whole consolidation process
Notation
119862119908
V The consolidation coefficient of water119862119886
V The consolidation coefficient of air119863(119903)
119894119896 The weighting coefficient
120576V Volumetric strain119892 The gravitational acceleration119867 The thickness of soil layer119896119886 The permeability coefficients of air
1198960
119886 The coefficients of permeability for water
at lateral boundary119896119908 The permeability coefficients of water
1198960
119908 The coefficients of permeability for air at
lateral boundary119898119886
1119896 The coefficient of air volume change withrespect to a change in the net normal stress
119898119908
1119896 The coefficient of water volume changewith respect to a change in the net normalstress
119898119886
2 The coefficient of air volume change with
respect to a change in matric suction119898119908
2 The coefficient of water volume changewith respect to a change in matric suction
1198990 The initial porosity
119873 The discrete point119875V A vacuum pressure120588 The density of air phase119902 The vertical loading1199030 The thickness of lateral boundary
119903119890 The radius of soil layer
119903119908 The radius of sand drain
119903120572 The radius value of 120572th point
120574119908 The unit weight of water phase
119877 The universal air constant119878 The distance between two sand drains1198781199030 The initial degree of saturation
119878V Settlement119878ℎ The radial displacement
119905 Time119879 The time factor119906119886 The excess pore-air pressure119906atm Atmospheric pressure119906119886
0 The initial excess pore-air pressure
119906119908 The excess pore-water pressure119906119908
0 The initial excess pore-water pressure
119880119886 The average degree of consolidation with
respect to air119880119908 The average degree of consolidation with
respect to water119881119886 The air volume of soil layer
14 Mathematical Problems in Engineering
119881119908 The water volume of soil layer
1198810 Initial total volume of the soil
Acknowledgments
The author gratefully acknowledges the financial supportfrom the Macau Science and Technology Development Fund(Grant no FDCT0112013A1) and the University of MacauResearch Fund (Grants nos MYRG189(Y2-L3)-FST11-ZWHand MYRG067(Y2-L2)-FST12-ZWH)
References
[1] K Terzaghi Theoretical Soil Mechanics Wiley New York NYUSA 1943
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] G E Blight Strength and consolidation characteristics of com-pacted soils [PhD thesis] University of London London UK1961
[4] R F Scott Principles of Soil Mechanics Addison Wesley Pub-lishing Company Boston Mass USA 1963
[5] L Barden ldquoConsolidation of compacted and unsaturated clayrdquoGeotechnique vol 15 no 3 pp 267ndash286 1965
[6] D G Fredlund and J U Hasan ldquoOne-dimensional consolida-tion theory unsaturated soilsrdquo Canadian Geotechnical Journalvol 16 no 3 pp 521ndash531 1979
[7] D G Fredlund andH Rahardjo Soil Mechanics for UnsaturatedSoils John Wiley and Son 1993
[8] A Qin D Sun L Yang and Y Weng ldquoA semi-analyticalsolution to consolidation of unsaturated soils with the freedrainage wellrdquo Computers and Geotechnics vol 37 no 7-8 pp867ndash875 2010
[9] W H Zhou L S Zhao and X B Li ldquoA simple analyticalsolution to one-dimensional consolidation for unsaturated soilrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics 2013
[10] M Malik and F Civan ldquoA comparative study of differentialquadrature and cubature methods vis-a-vis some conventionaltechniques in context of convection-diffusion-reaction prob-lemsrdquo Chemical Engineering Science vol 50 no 3 pp 531ndash5471995
[11] P Malekzadeh M R Golbahar Haghighi and M M AtashildquoOut-of-plane free vibration of functionally graded circularcurved beams in thermal environmentrdquo Composite Structuresvol 92 no 2 pp 541ndash552 2010
[12] P Malekzadeh M R G Haghighi and M M Atashi ldquoFreevibration analysis of elastically supported functionally radedannular plates subjected to thermal environmentrdquo Meccanicavol 46 no 5 pp 893ndash913 2011
[13] R P Chen W H Zhou H Z Wang and Y M Chen ldquoOne-dimensional nonlinear consolidation of multi-layered soil bydifferential quadrature methodrdquo Computers and Geotechnicsvol 32 no 5 pp 358ndash369 2005
[14] R P Chen W H Zhou H Z Wang and Y M Chen ldquoStudyon one-dimensional nonlinear consolidation of multi-layeredsoil using differential quadrature methodrdquo Chinese Journal ofComputational Mechanics vol 22 no 3 pp 310ndash315 2005
[15] H Z Wang R P Chen W H Zhou and Y M ChenldquoComputation of 1-D nonlinear consolidation in double-layer
foundation by using differential quadrature methodrdquo Journal ofHydraulic Engineering vol 4 pp 8ndash14 2004
[16] W H Zhou and L S Zhao ldquoOne-dimensional consolidationof unsaturated soil subjected to time-dependent loading undervarious initial and boundary conditionsrdquo ASCE InternationalJournal of Geomechanics 2013
[17] E Conte ldquoConsolidation analysis for unsaturated soilsrdquo Cana-dian Geotechnical Journal vol 41 no 4 pp 599ndash612 2004
[18] E Conte ldquoPlane strain and axially symmetric consolidation inunsaturated soilsrdquo International Journal of Geomechanics vol 6no 2 pp 131ndash135 2006
[19] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 no 1 pp40ndash52 1972
[20] R Bellman and J Casti ldquoDifferential quadrature and long-termintegrationrdquo Journal of Mathematical Analysis and Applicationsvol 34 no 2 pp 235ndash238 1971
[21] J R Quan andC T Chang ldquoNew insights in solving distributedsystem equations by the quadrature method I analysisrdquo Com-puters and Chemical Engineering vol 13 no 7 pp 779ndash7881989
According to the two stress-state variable approaches [67] volume strain is represented by the following constitutiveequation for soil layer
120597120576V
120597119905= (119898119908
1119896+ 119898119886
1119896)120597 (119902 minus 119906
119886
)
120597119905+ (119898119908
2+ 119898119886
2)120597 (119906119886
minus 119906119908
)
120597119905
(27)
By integrating (26) with respect to time 119905 from 0 to 119905 weget the expression of volumetric strain 120576V
120576V = (119898119908
1119896+ 119898119886
1119896) [(119902 minus 119906
119886
) minus (119902 minus 119906119886
0)]
+ (119898119908
2+ 119898119886
2) [(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)]
(28)
Volumetric strain consists of vertical strain and horizon-tal strain For the axisymmetric consolidation it is assumed
in this paper that one-third of the volume strain is contributedfrom vertical strain and two-thirds are fromhorizontal strainSo the radial displacement and vertical settlement can beobtained respectively Consider
119878ℎ=2
3(119898119908
1119896+ 119898119886
1119896) int
119903119890
0
[(119902 minus 119906119886
) minus (119902 minus 119906119886
0)] 119889119903
+2
3(119898119908
2+ 119898119886
2) int
119903119890
0
[(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)] 119889119903
119878V =1
3(119898119908
1119896+ 119898119886
1119896) int
119867
0
[(119902 minus 119906119886
) minus (119902 minus 119906119886
0)] 119889119911
+1
3(119898119908
2+ 119898119886
2) int
119867
0
[(119906119886
minus 119906119908
) minus (119906119886
0minus 119906119908
0)] 119889119911
(29)
8 Mathematical Problems in Engineering
0
005
01
015
02
025
03
035
04
045
10minus3
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 10 mr = 12 m
uw
q
Figure 7 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
0
001
002
003
004
005
006
10minus3
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 10 mr = 12 m
uaq
Figure 8 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
4 An Example and Convergence Analysis
In this section the convergence analysis the average degree ofconsolidation and the effects of displacements in radial andvertical directions are discussed by using a simple example
As shown in Figure 1 a vertical loading 119902 = 100 kPa isapplied on the top surface of the soil layerThe lateral bound-ary is considered as impermeable Considering a uniforminitial excess pore-water and pore-air pressure distribution119906119886
0= 5 kPa 119906119908
0= 40 kPa Other parameters are 119899
0= 05 119878
1199030=
08 119896119908= 10minus10ms119898119908
1119896= minus015times10
minus4 kPaminus11198981199082= minus061times
10minus4 kPaminus1119898119886
1119896= 006 times 10
minus4 kPaminus11198981198862= 026 times 10
minus4 kPaminus1119903119908= 02m 119903
119890= 18m and 119906atm = 1013 kPa By applying the
DQM solution presented above the results of axisymmetric
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
1
08
06
04
02
0
uw
q
Figure 9 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
014
012
01
008
006
004
002
0
uaq
Figure 10 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
consolidation of unsaturated soils in a sand drain foundationwere obtained
Tables 1 and 2 list the solutions of excess pore-water andpore-air pressures at radius 119903 = 1m at different time factorsand different equally spaced discrete points respectively Theratio of permeability coefficient is 119896
119886119896119908= 01 From Tables
1 and 2 the accuracy of solutions increases when the numberof discrete point becomes big It is obvious that 9 equallyspaced grid points are sufficient to obtain the convergedresults of excess pore-water and pore-air pressuresThereforethe number of discrete points119873 = 9 is adopted in most casestudies
Figure 2 shows the curves of average degree of consoli-dation with respect to water and air phases when 119873 = 9The ratio of permeability coefficient is 119896
119886119896119908
= 1 FromFigure 2 it can be observed that the consolidation of air
Mathematical Problems in Engineering 9
Table 1 Different excess pore-water pressures 119906119908119902 under time factor 119879 with different discrete points119873 (119896
1 times 10minus6 0050 0050 0050 0050 0050 0050 0050
1 times 10minus4 0050 0050 0050 0050 0050 0050 0050
5 times 10minus2 0048 0048 0048 0048 0048 0048 0048
1 times 10minus1 0046 0045 0045 0045 0045 0045 0045
2 times 10minus1 0042 0041 0040 0040 0040 0040 0010
5 times 10minus1 0035 0033 0032 0032 0032 0031 0031
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 11 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
phase and water phase begins when 119879 = 10 times 10minus3 When
119879 = 10 times 100 and 119879 = 10 times 10
15 the consolidation ofair phase and water phase is almost finished respectivelyAt the early stages soil consolidation is mainly caused bythe dissipation of excess pore-air pressure But in the laterstages soil consolidation is mainly caused by the dissipationof excess pore-water pressure
Figure 3 shows the radial displacement development withtime and Figure 4 describes the settlement of top surface(119867 = 10m) at different radii with time factor 119879 The numberof discrete points is 119873 = 9 The ratio of permeabilitycoefficient is 119896
119886119896119908
= 01 From Figure 4 we can see thatthe soil settles earlier at the points with smaller radius thatis closer to the sand drain Figure 5 shows the settlement atdifferent heights of soil layer at radius 119903 = 1m with time
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 12 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
factor 119879 When the height of soil layer119867 is equal to 4m thesettlement is almost zero
5 Cases of Different Initial andBoundary Conditions
As the initial conditions do not need to be constant in thepresentDQMsolution it is easily used to analyze nonuniforminitial pore-water and pore-air distribution problems In thispart four different initial pore-water and pore-air distribu-tions and some different boundary conditions are consideredA vertical loading 119902 = 100 kPa is applied on the top surfaceof soil layer Other parameters are 119899
0= 05 119878
1199030= 08 119896
119908=
10minus10ms 119903
119908= 02m 119903
119890= 18m119898119908
1119896= minus015 times 10
minus4 kPaminus1
10 Mathematical Problems in Engineering
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 13 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 14 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
119898119908
2= minus061 times 10
minus4 kPaminus1 1198981198861119896
= 006 times 10minus4 kPaminus1 119898119886
2=
026 times 10minus4 kPaminus1 and 119906atm = 1013 kPa
51 Effect of Different Initial Pore-WaterAir Pressure Distribu-tions In this section in order to make the curves smooththe number of discrete points is chosen to be 119873 = 17 Thelateral boundary is considered as impermeable and the ratioof permeability coefficient with respect to air and water is119896119886119896119908= 01
Four different initial pore-waterair distributions aretaken into account In Figure 6(a) the case of uniform initialpore-air and pore-water pressures assumes that 119906119886
0= 5 kPa
and 1199061199080= 40 kPa Figure 6(b) presents the linear increasing
initial pore-air and pore-water distributions The lateral-skewed distributions initial pore-waterair pressures is shown
in Figures 6(c) and 6(d) and they are represented by thefollowing equations
119906119908
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
100105119887
119906119886
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
8 times 100105119887
(30)
where 119887 is the variable which can control the spread orskewness of the curves in Figures 6(c) and 6(d)
Under the condition shown in Figure 6(a) the dissipationcurves of the excess pore-water and pore-air pressure atdifferent distances from the sand drain are presented inFigures 7 and 8 respectively The excess pore-air and pore-water pressures dissipate faster at the points with smallerradius that is closer to the sand drain As the intervalbetween the two curves becomes smaller from 119903 = 06m to119903 = 12m it can be seen that the effect of the distance on thedissipation becomes less obvious
In Figure 6(b) the initial pore-air and pore-water pres-sures are assumed not to be constants and to be linearincreasing when the radius increases The initial pore-airpressure is assumed one-eighth of the initial pore-water pres-sure In other words from 119903 = 04m to 119903 = 16m the initialpore-water pressure will increase from 30 kPa to 110 kPa andthe initial pore-air pressure will increase from 375 kPa to1375 kPa Figures 9 and 10 show the dissipation of excesspore-water and pore-air pressures at different distances fromthe sand drain under Figure 6(b) distributions
Comparing Figures 7 and 9 it is found that the excesspore-water pressure dissipates completely at almost the sametime With the change of the initial pore-water pressuredistribution the time of dissipation with respect to wateris not significantly affected Comparing Figures 8 and 10it can be seen that the excess pore-air pressure dissipatescompletely at almost the same time The initial pore-airpressure distribution has little influence on the dissipationtime of air phase
Under the conditions of Figures 6(c) and 6(d) thedissipation curves of the excess pore-water and pore-airpressures for initial distributions where 119887 = 2 6 are shownin Figures 11 12 13 and 14 From these figures whatever theinitial pore-water and pore-air pressures are the excess pore-water and pore-air pressures dissipate completely at almostthe same timeThe plateau period is longer at the points withlarger radius
Here excess pore pressure isochrones for the above fourinitial conditions are presented in Figures 15 and 16 Figures15(a) and 16(a) show the excess pore-water and pore-airpressure isochrones when the initial excess pore pressure isuniform at all radii In Figures 15(b) and 16(b) the initialexcess pore pressure distributions are linearly increasingThepore pressure isochrones for lateral-skewed initial distribu-tions are shown in Figures 15(c) 16(c) 15(d) and 16(d)respectively From Figures 15(a) and 16(a) as the initial excesspore pressure is constant the peak of curves in Figure 15(a)approaches to 04 and the peak of curves in Figure 13(a)approaches to 005 Comparing Figures 15(a) and 15(b) as
Mathematical Problems in Engineering 11
0
005
01
015
02
025
03
035
04
02 04 06 08 1 12 14 16 18r (m)
02
1
2
5
05
T = 10
uw
q
(a)
02 04 06 08 1 12 14 16 18r (m)
1
1
2
5
09
08
07
06
05
05
04
03
02
02
01
01
0
T = 10
uw
q
(b)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
011
09
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(c)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
0109
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(d)
Figure 15Normalized pore-water pressure isochrones for different initial excess pore-water pressure distributions (119896119886119896119908= 01) (a) uniform
(b) linearly increasing (c) lateral skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
the initial excess pore-water pressure is linearly increasingthe skewness is more evident during early stages of dissipa-tion in Figure 15(b) But in the later stages the influence ofinitial excess pore-water pressure becomes less significant Soare the isochrones of the excess pore-air pressure in Figures16(a) and 16(b) Figures 15(c) 16(c) 15(d) and 16(d) showthe excess pore-water and pore-air pressure isochrones forthe lateral-skewed initial distribution when 119887 = 2 6 It isseen that some intersections appear in the isochrones duringthe early stages It means that a redistribution of excesspore pressures occurs toward the region of minimal initialexcess pore pressure during the early stages With the timeincreasing the isochrones of excess pore pressures becomeregular
52 Effect of Different Boundary Conditions In this partthe effect of different lateral boundary conditions is studiedUniform initial excess pore-waterair pressure distributions119906119886
0= 5 kPa and 119906119908
0= 40 kPa are adopted Four different ratios
119896119886119896119908are considered that is 01 1 10 and 100 with 119896
119908=
10minus10ms Two different drainage conditions are analyzed
lateral surface is impervious and lateral surface is impededFor the impeded surface 119886
119908and 119886
119886in (9) are chosen to be
05Figures 17 and 18 shows the dissipation curves for the
water and air pressures at radius 119903 = 12m under differentboundary conditions First the excess pore-water and pore-air pressures dissipate faster in impeded surface than that inimpermeable surface Second 119896
119908is the same in the cases
12 Mathematical Problems in Engineering
02 04 06 08 1 12 14 16 18r (m)
01
02
05
T = 2
006
005005
004
003
002
002
001
0
1
uaq
(a)
02 04 06 08 1 12 14 16 18r (m)
014
012
0101
02
05008
006
004
002
0
T = 2
005002
1
uaq
(b)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(c)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(d)
Figure 16 Normalized pore-air pressure isochrones for different initial excess pore-air pressure distributions (119896119886119896119908= 01) (a) uniform (b)
linearly increasing (c) lateral-skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
of four 119896119886119896119908values It is observed from the results that
the bigger 119896119886119896119908
is the faster the excess pore-water andpore-air pressures dissipate Third the dissipation curves ofthe excess pore-air and pore-water pressures have almostthe same shape under different values of 119896
119886119896119908 Comparing
Figure 17 with Figure 18 when the excess pore-air pressuredissipates almost completely the dissipation of excess pore-water pressure enters into a plateau period The bigger 119896
119886119896119908
is the earlier the excess pore-air pressure dissipates com-pletely Therefore it is shown that the value of 119896
119886affects the
dissipation of pore-water pressure during the consolidationprocess
6 Conclusion
This paper obtains a general solution to the axisymmet-ric consolidation of unsaturated soils by using differen-tial quadrature method (DQM) based on Fredlundrsquos one-dimensional consolidation theory for unsaturated soil Withthe use of DQM the two governing equations are trans-formed into two sets of ordinary differential equationsThe solutions to the transformed differential equations areobtained by Rong-Kutta method under different initial andboundary conditions A case study has been presented forthe analysis of unsaturated consolidation in a sand drain
Mathematical Problems in Engineering 13
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
045
04
035
03
025
02
015
01
005
0
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
uw
q
Figure 17 Variation of excess pore-water pressures 119906119908119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
007
006
005
004
003
002
001
0
uaq
Figure 18 Variation of excess pore-air pressures 119906119886
119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
foundation The convergence analysis the average degreeof consolidation for water and air phases the settlementin radial and vertical directions effects of different initialexcess pore-air and pore-water pressure distributions andeffects of different boundary conditions have been presentedFrom the analyses some conclusions can be drawn (a)through compiling the programs it is found that the DQsolution delivers accurate and stable results for unsaturatedsoil consolidation Due to the uniformmatrix structure of theDQ formulas it is easy to compile computer programs whenconsidering complicated conditions such as nonuniformpore-waterair distribution conditions (b) soil consolidationismainly caused by the dissipation of excess pore-air pressurein the early stages and by the dissipation of excess pore-water pressure in the later stages (c) the bigger the ratio
of the permeability coefficients for water and air is thefaster the excess pore-water and pore-air pressures dissipate(d) The excess pore-air and pore-water pressures dissipatefaster at the points closer to the sand drain (e) the initialdistribution has some effects on the early consolidationprocess of unsaturated soil and boundary condition has asignificant effect on the whole consolidation process
Notation
119862119908
V The consolidation coefficient of water119862119886
V The consolidation coefficient of air119863(119903)
119894119896 The weighting coefficient
120576V Volumetric strain119892 The gravitational acceleration119867 The thickness of soil layer119896119886 The permeability coefficients of air
1198960
119886 The coefficients of permeability for water
at lateral boundary119896119908 The permeability coefficients of water
1198960
119908 The coefficients of permeability for air at
lateral boundary119898119886
1119896 The coefficient of air volume change withrespect to a change in the net normal stress
119898119908
1119896 The coefficient of water volume changewith respect to a change in the net normalstress
119898119886
2 The coefficient of air volume change with
respect to a change in matric suction119898119908
2 The coefficient of water volume changewith respect to a change in matric suction
1198990 The initial porosity
119873 The discrete point119875V A vacuum pressure120588 The density of air phase119902 The vertical loading1199030 The thickness of lateral boundary
119903119890 The radius of soil layer
119903119908 The radius of sand drain
119903120572 The radius value of 120572th point
120574119908 The unit weight of water phase
119877 The universal air constant119878 The distance between two sand drains1198781199030 The initial degree of saturation
119878V Settlement119878ℎ The radial displacement
119905 Time119879 The time factor119906119886 The excess pore-air pressure119906atm Atmospheric pressure119906119886
0 The initial excess pore-air pressure
119906119908 The excess pore-water pressure119906119908
0 The initial excess pore-water pressure
119880119886 The average degree of consolidation with
respect to air119880119908 The average degree of consolidation with
respect to water119881119886 The air volume of soil layer
14 Mathematical Problems in Engineering
119881119908 The water volume of soil layer
1198810 Initial total volume of the soil
Acknowledgments
The author gratefully acknowledges the financial supportfrom the Macau Science and Technology Development Fund(Grant no FDCT0112013A1) and the University of MacauResearch Fund (Grants nos MYRG189(Y2-L3)-FST11-ZWHand MYRG067(Y2-L2)-FST12-ZWH)
References
[1] K Terzaghi Theoretical Soil Mechanics Wiley New York NYUSA 1943
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] G E Blight Strength and consolidation characteristics of com-pacted soils [PhD thesis] University of London London UK1961
[4] R F Scott Principles of Soil Mechanics Addison Wesley Pub-lishing Company Boston Mass USA 1963
[5] L Barden ldquoConsolidation of compacted and unsaturated clayrdquoGeotechnique vol 15 no 3 pp 267ndash286 1965
[6] D G Fredlund and J U Hasan ldquoOne-dimensional consolida-tion theory unsaturated soilsrdquo Canadian Geotechnical Journalvol 16 no 3 pp 521ndash531 1979
[7] D G Fredlund andH Rahardjo Soil Mechanics for UnsaturatedSoils John Wiley and Son 1993
[8] A Qin D Sun L Yang and Y Weng ldquoA semi-analyticalsolution to consolidation of unsaturated soils with the freedrainage wellrdquo Computers and Geotechnics vol 37 no 7-8 pp867ndash875 2010
[9] W H Zhou L S Zhao and X B Li ldquoA simple analyticalsolution to one-dimensional consolidation for unsaturated soilrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics 2013
[10] M Malik and F Civan ldquoA comparative study of differentialquadrature and cubature methods vis-a-vis some conventionaltechniques in context of convection-diffusion-reaction prob-lemsrdquo Chemical Engineering Science vol 50 no 3 pp 531ndash5471995
[11] P Malekzadeh M R Golbahar Haghighi and M M AtashildquoOut-of-plane free vibration of functionally graded circularcurved beams in thermal environmentrdquo Composite Structuresvol 92 no 2 pp 541ndash552 2010
[12] P Malekzadeh M R G Haghighi and M M Atashi ldquoFreevibration analysis of elastically supported functionally radedannular plates subjected to thermal environmentrdquo Meccanicavol 46 no 5 pp 893ndash913 2011
[13] R P Chen W H Zhou H Z Wang and Y M Chen ldquoOne-dimensional nonlinear consolidation of multi-layered soil bydifferential quadrature methodrdquo Computers and Geotechnicsvol 32 no 5 pp 358ndash369 2005
[14] R P Chen W H Zhou H Z Wang and Y M Chen ldquoStudyon one-dimensional nonlinear consolidation of multi-layeredsoil using differential quadrature methodrdquo Chinese Journal ofComputational Mechanics vol 22 no 3 pp 310ndash315 2005
[15] H Z Wang R P Chen W H Zhou and Y M ChenldquoComputation of 1-D nonlinear consolidation in double-layer
foundation by using differential quadrature methodrdquo Journal ofHydraulic Engineering vol 4 pp 8ndash14 2004
[16] W H Zhou and L S Zhao ldquoOne-dimensional consolidationof unsaturated soil subjected to time-dependent loading undervarious initial and boundary conditionsrdquo ASCE InternationalJournal of Geomechanics 2013
[17] E Conte ldquoConsolidation analysis for unsaturated soilsrdquo Cana-dian Geotechnical Journal vol 41 no 4 pp 599ndash612 2004
[18] E Conte ldquoPlane strain and axially symmetric consolidation inunsaturated soilsrdquo International Journal of Geomechanics vol 6no 2 pp 131ndash135 2006
[19] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 no 1 pp40ndash52 1972
[20] R Bellman and J Casti ldquoDifferential quadrature and long-termintegrationrdquo Journal of Mathematical Analysis and Applicationsvol 34 no 2 pp 235ndash238 1971
[21] J R Quan andC T Chang ldquoNew insights in solving distributedsystem equations by the quadrature method I analysisrdquo Com-puters and Chemical Engineering vol 13 no 7 pp 779ndash7881989
Figure 7 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
0
001
002
003
004
005
006
10minus3
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 10 mr = 12 m
uaq
Figure 8 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(a) distribution (119896
119886119896119908= 01)
4 An Example and Convergence Analysis
In this section the convergence analysis the average degree ofconsolidation and the effects of displacements in radial andvertical directions are discussed by using a simple example
As shown in Figure 1 a vertical loading 119902 = 100 kPa isapplied on the top surface of the soil layerThe lateral bound-ary is considered as impermeable Considering a uniforminitial excess pore-water and pore-air pressure distribution119906119886
0= 5 kPa 119906119908
0= 40 kPa Other parameters are 119899
0= 05 119878
1199030=
08 119896119908= 10minus10ms119898119908
1119896= minus015times10
minus4 kPaminus11198981199082= minus061times
10minus4 kPaminus1119898119886
1119896= 006 times 10
minus4 kPaminus11198981198862= 026 times 10
minus4 kPaminus1119903119908= 02m 119903
119890= 18m and 119906atm = 1013 kPa By applying the
DQM solution presented above the results of axisymmetric
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
1
08
06
04
02
0
uw
q
Figure 9 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 04 mr = 08 m
r = 12 mr = 16 m
014
012
01
008
006
004
002
0
uaq
Figure 10 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(b) distribution (119896
119886119896119908= 01)
consolidation of unsaturated soils in a sand drain foundationwere obtained
Tables 1 and 2 list the solutions of excess pore-water andpore-air pressures at radius 119903 = 1m at different time factorsand different equally spaced discrete points respectively Theratio of permeability coefficient is 119896
119886119896119908= 01 From Tables
1 and 2 the accuracy of solutions increases when the numberof discrete point becomes big It is obvious that 9 equallyspaced grid points are sufficient to obtain the convergedresults of excess pore-water and pore-air pressuresThereforethe number of discrete points119873 = 9 is adopted in most casestudies
Figure 2 shows the curves of average degree of consoli-dation with respect to water and air phases when 119873 = 9The ratio of permeability coefficient is 119896
119886119896119908
= 1 FromFigure 2 it can be observed that the consolidation of air
Mathematical Problems in Engineering 9
Table 1 Different excess pore-water pressures 119906119908119902 under time factor 119879 with different discrete points119873 (119896
1 times 10minus6 0050 0050 0050 0050 0050 0050 0050
1 times 10minus4 0050 0050 0050 0050 0050 0050 0050
5 times 10minus2 0048 0048 0048 0048 0048 0048 0048
1 times 10minus1 0046 0045 0045 0045 0045 0045 0045
2 times 10minus1 0042 0041 0040 0040 0040 0040 0010
5 times 10minus1 0035 0033 0032 0032 0032 0031 0031
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 11 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
phase and water phase begins when 119879 = 10 times 10minus3 When
119879 = 10 times 100 and 119879 = 10 times 10
15 the consolidation ofair phase and water phase is almost finished respectivelyAt the early stages soil consolidation is mainly caused bythe dissipation of excess pore-air pressure But in the laterstages soil consolidation is mainly caused by the dissipationof excess pore-water pressure
Figure 3 shows the radial displacement development withtime and Figure 4 describes the settlement of top surface(119867 = 10m) at different radii with time factor 119879 The numberof discrete points is 119873 = 9 The ratio of permeabilitycoefficient is 119896
119886119896119908
= 01 From Figure 4 we can see thatthe soil settles earlier at the points with smaller radius thatis closer to the sand drain Figure 5 shows the settlement atdifferent heights of soil layer at radius 119903 = 1m with time
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 12 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
factor 119879 When the height of soil layer119867 is equal to 4m thesettlement is almost zero
5 Cases of Different Initial andBoundary Conditions
As the initial conditions do not need to be constant in thepresentDQMsolution it is easily used to analyze nonuniforminitial pore-water and pore-air distribution problems In thispart four different initial pore-water and pore-air distribu-tions and some different boundary conditions are consideredA vertical loading 119902 = 100 kPa is applied on the top surfaceof soil layer Other parameters are 119899
0= 05 119878
1199030= 08 119896
119908=
10minus10ms 119903
119908= 02m 119903
119890= 18m119898119908
1119896= minus015 times 10
minus4 kPaminus1
10 Mathematical Problems in Engineering
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 13 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 14 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
119898119908
2= minus061 times 10
minus4 kPaminus1 1198981198861119896
= 006 times 10minus4 kPaminus1 119898119886
2=
026 times 10minus4 kPaminus1 and 119906atm = 1013 kPa
51 Effect of Different Initial Pore-WaterAir Pressure Distribu-tions In this section in order to make the curves smooththe number of discrete points is chosen to be 119873 = 17 Thelateral boundary is considered as impermeable and the ratioof permeability coefficient with respect to air and water is119896119886119896119908= 01
Four different initial pore-waterair distributions aretaken into account In Figure 6(a) the case of uniform initialpore-air and pore-water pressures assumes that 119906119886
0= 5 kPa
and 1199061199080= 40 kPa Figure 6(b) presents the linear increasing
initial pore-air and pore-water distributions The lateral-skewed distributions initial pore-waterair pressures is shown
in Figures 6(c) and 6(d) and they are represented by thefollowing equations
119906119908
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
100105119887
119906119886
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
8 times 100105119887
(30)
where 119887 is the variable which can control the spread orskewness of the curves in Figures 6(c) and 6(d)
Under the condition shown in Figure 6(a) the dissipationcurves of the excess pore-water and pore-air pressure atdifferent distances from the sand drain are presented inFigures 7 and 8 respectively The excess pore-air and pore-water pressures dissipate faster at the points with smallerradius that is closer to the sand drain As the intervalbetween the two curves becomes smaller from 119903 = 06m to119903 = 12m it can be seen that the effect of the distance on thedissipation becomes less obvious
In Figure 6(b) the initial pore-air and pore-water pres-sures are assumed not to be constants and to be linearincreasing when the radius increases The initial pore-airpressure is assumed one-eighth of the initial pore-water pres-sure In other words from 119903 = 04m to 119903 = 16m the initialpore-water pressure will increase from 30 kPa to 110 kPa andthe initial pore-air pressure will increase from 375 kPa to1375 kPa Figures 9 and 10 show the dissipation of excesspore-water and pore-air pressures at different distances fromthe sand drain under Figure 6(b) distributions
Comparing Figures 7 and 9 it is found that the excesspore-water pressure dissipates completely at almost the sametime With the change of the initial pore-water pressuredistribution the time of dissipation with respect to wateris not significantly affected Comparing Figures 8 and 10it can be seen that the excess pore-air pressure dissipatescompletely at almost the same time The initial pore-airpressure distribution has little influence on the dissipationtime of air phase
Under the conditions of Figures 6(c) and 6(d) thedissipation curves of the excess pore-water and pore-airpressures for initial distributions where 119887 = 2 6 are shownin Figures 11 12 13 and 14 From these figures whatever theinitial pore-water and pore-air pressures are the excess pore-water and pore-air pressures dissipate completely at almostthe same timeThe plateau period is longer at the points withlarger radius
Here excess pore pressure isochrones for the above fourinitial conditions are presented in Figures 15 and 16 Figures15(a) and 16(a) show the excess pore-water and pore-airpressure isochrones when the initial excess pore pressure isuniform at all radii In Figures 15(b) and 16(b) the initialexcess pore pressure distributions are linearly increasingThepore pressure isochrones for lateral-skewed initial distribu-tions are shown in Figures 15(c) 16(c) 15(d) and 16(d)respectively From Figures 15(a) and 16(a) as the initial excesspore pressure is constant the peak of curves in Figure 15(a)approaches to 04 and the peak of curves in Figure 13(a)approaches to 005 Comparing Figures 15(a) and 15(b) as
Mathematical Problems in Engineering 11
0
005
01
015
02
025
03
035
04
02 04 06 08 1 12 14 16 18r (m)
02
1
2
5
05
T = 10
uw
q
(a)
02 04 06 08 1 12 14 16 18r (m)
1
1
2
5
09
08
07
06
05
05
04
03
02
02
01
01
0
T = 10
uw
q
(b)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
011
09
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(c)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
0109
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(d)
Figure 15Normalized pore-water pressure isochrones for different initial excess pore-water pressure distributions (119896119886119896119908= 01) (a) uniform
(b) linearly increasing (c) lateral skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
the initial excess pore-water pressure is linearly increasingthe skewness is more evident during early stages of dissipa-tion in Figure 15(b) But in the later stages the influence ofinitial excess pore-water pressure becomes less significant Soare the isochrones of the excess pore-air pressure in Figures16(a) and 16(b) Figures 15(c) 16(c) 15(d) and 16(d) showthe excess pore-water and pore-air pressure isochrones forthe lateral-skewed initial distribution when 119887 = 2 6 It isseen that some intersections appear in the isochrones duringthe early stages It means that a redistribution of excesspore pressures occurs toward the region of minimal initialexcess pore pressure during the early stages With the timeincreasing the isochrones of excess pore pressures becomeregular
52 Effect of Different Boundary Conditions In this partthe effect of different lateral boundary conditions is studiedUniform initial excess pore-waterair pressure distributions119906119886
0= 5 kPa and 119906119908
0= 40 kPa are adopted Four different ratios
119896119886119896119908are considered that is 01 1 10 and 100 with 119896
119908=
10minus10ms Two different drainage conditions are analyzed
lateral surface is impervious and lateral surface is impededFor the impeded surface 119886
119908and 119886
119886in (9) are chosen to be
05Figures 17 and 18 shows the dissipation curves for the
water and air pressures at radius 119903 = 12m under differentboundary conditions First the excess pore-water and pore-air pressures dissipate faster in impeded surface than that inimpermeable surface Second 119896
119908is the same in the cases
12 Mathematical Problems in Engineering
02 04 06 08 1 12 14 16 18r (m)
01
02
05
T = 2
006
005005
004
003
002
002
001
0
1
uaq
(a)
02 04 06 08 1 12 14 16 18r (m)
014
012
0101
02
05008
006
004
002
0
T = 2
005002
1
uaq
(b)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(c)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(d)
Figure 16 Normalized pore-air pressure isochrones for different initial excess pore-air pressure distributions (119896119886119896119908= 01) (a) uniform (b)
linearly increasing (c) lateral-skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
of four 119896119886119896119908values It is observed from the results that
the bigger 119896119886119896119908
is the faster the excess pore-water andpore-air pressures dissipate Third the dissipation curves ofthe excess pore-air and pore-water pressures have almostthe same shape under different values of 119896
119886119896119908 Comparing
Figure 17 with Figure 18 when the excess pore-air pressuredissipates almost completely the dissipation of excess pore-water pressure enters into a plateau period The bigger 119896
119886119896119908
is the earlier the excess pore-air pressure dissipates com-pletely Therefore it is shown that the value of 119896
119886affects the
dissipation of pore-water pressure during the consolidationprocess
6 Conclusion
This paper obtains a general solution to the axisymmet-ric consolidation of unsaturated soils by using differen-tial quadrature method (DQM) based on Fredlundrsquos one-dimensional consolidation theory for unsaturated soil Withthe use of DQM the two governing equations are trans-formed into two sets of ordinary differential equationsThe solutions to the transformed differential equations areobtained by Rong-Kutta method under different initial andboundary conditions A case study has been presented forthe analysis of unsaturated consolidation in a sand drain
Mathematical Problems in Engineering 13
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
045
04
035
03
025
02
015
01
005
0
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
uw
q
Figure 17 Variation of excess pore-water pressures 119906119908119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
007
006
005
004
003
002
001
0
uaq
Figure 18 Variation of excess pore-air pressures 119906119886
119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
foundation The convergence analysis the average degreeof consolidation for water and air phases the settlementin radial and vertical directions effects of different initialexcess pore-air and pore-water pressure distributions andeffects of different boundary conditions have been presentedFrom the analyses some conclusions can be drawn (a)through compiling the programs it is found that the DQsolution delivers accurate and stable results for unsaturatedsoil consolidation Due to the uniformmatrix structure of theDQ formulas it is easy to compile computer programs whenconsidering complicated conditions such as nonuniformpore-waterair distribution conditions (b) soil consolidationismainly caused by the dissipation of excess pore-air pressurein the early stages and by the dissipation of excess pore-water pressure in the later stages (c) the bigger the ratio
of the permeability coefficients for water and air is thefaster the excess pore-water and pore-air pressures dissipate(d) The excess pore-air and pore-water pressures dissipatefaster at the points closer to the sand drain (e) the initialdistribution has some effects on the early consolidationprocess of unsaturated soil and boundary condition has asignificant effect on the whole consolidation process
Notation
119862119908
V The consolidation coefficient of water119862119886
V The consolidation coefficient of air119863(119903)
119894119896 The weighting coefficient
120576V Volumetric strain119892 The gravitational acceleration119867 The thickness of soil layer119896119886 The permeability coefficients of air
1198960
119886 The coefficients of permeability for water
at lateral boundary119896119908 The permeability coefficients of water
1198960
119908 The coefficients of permeability for air at
lateral boundary119898119886
1119896 The coefficient of air volume change withrespect to a change in the net normal stress
119898119908
1119896 The coefficient of water volume changewith respect to a change in the net normalstress
119898119886
2 The coefficient of air volume change with
respect to a change in matric suction119898119908
2 The coefficient of water volume changewith respect to a change in matric suction
1198990 The initial porosity
119873 The discrete point119875V A vacuum pressure120588 The density of air phase119902 The vertical loading1199030 The thickness of lateral boundary
119903119890 The radius of soil layer
119903119908 The radius of sand drain
119903120572 The radius value of 120572th point
120574119908 The unit weight of water phase
119877 The universal air constant119878 The distance between two sand drains1198781199030 The initial degree of saturation
119878V Settlement119878ℎ The radial displacement
119905 Time119879 The time factor119906119886 The excess pore-air pressure119906atm Atmospheric pressure119906119886
0 The initial excess pore-air pressure
119906119908 The excess pore-water pressure119906119908
0 The initial excess pore-water pressure
119880119886 The average degree of consolidation with
respect to air119880119908 The average degree of consolidation with
respect to water119881119886 The air volume of soil layer
14 Mathematical Problems in Engineering
119881119908 The water volume of soil layer
1198810 Initial total volume of the soil
Acknowledgments
The author gratefully acknowledges the financial supportfrom the Macau Science and Technology Development Fund(Grant no FDCT0112013A1) and the University of MacauResearch Fund (Grants nos MYRG189(Y2-L3)-FST11-ZWHand MYRG067(Y2-L2)-FST12-ZWH)
References
[1] K Terzaghi Theoretical Soil Mechanics Wiley New York NYUSA 1943
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] G E Blight Strength and consolidation characteristics of com-pacted soils [PhD thesis] University of London London UK1961
[4] R F Scott Principles of Soil Mechanics Addison Wesley Pub-lishing Company Boston Mass USA 1963
[5] L Barden ldquoConsolidation of compacted and unsaturated clayrdquoGeotechnique vol 15 no 3 pp 267ndash286 1965
[6] D G Fredlund and J U Hasan ldquoOne-dimensional consolida-tion theory unsaturated soilsrdquo Canadian Geotechnical Journalvol 16 no 3 pp 521ndash531 1979
[7] D G Fredlund andH Rahardjo Soil Mechanics for UnsaturatedSoils John Wiley and Son 1993
[8] A Qin D Sun L Yang and Y Weng ldquoA semi-analyticalsolution to consolidation of unsaturated soils with the freedrainage wellrdquo Computers and Geotechnics vol 37 no 7-8 pp867ndash875 2010
[9] W H Zhou L S Zhao and X B Li ldquoA simple analyticalsolution to one-dimensional consolidation for unsaturated soilrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics 2013
[10] M Malik and F Civan ldquoA comparative study of differentialquadrature and cubature methods vis-a-vis some conventionaltechniques in context of convection-diffusion-reaction prob-lemsrdquo Chemical Engineering Science vol 50 no 3 pp 531ndash5471995
[11] P Malekzadeh M R Golbahar Haghighi and M M AtashildquoOut-of-plane free vibration of functionally graded circularcurved beams in thermal environmentrdquo Composite Structuresvol 92 no 2 pp 541ndash552 2010
[12] P Malekzadeh M R G Haghighi and M M Atashi ldquoFreevibration analysis of elastically supported functionally radedannular plates subjected to thermal environmentrdquo Meccanicavol 46 no 5 pp 893ndash913 2011
[13] R P Chen W H Zhou H Z Wang and Y M Chen ldquoOne-dimensional nonlinear consolidation of multi-layered soil bydifferential quadrature methodrdquo Computers and Geotechnicsvol 32 no 5 pp 358ndash369 2005
[14] R P Chen W H Zhou H Z Wang and Y M Chen ldquoStudyon one-dimensional nonlinear consolidation of multi-layeredsoil using differential quadrature methodrdquo Chinese Journal ofComputational Mechanics vol 22 no 3 pp 310ndash315 2005
[15] H Z Wang R P Chen W H Zhou and Y M ChenldquoComputation of 1-D nonlinear consolidation in double-layer
foundation by using differential quadrature methodrdquo Journal ofHydraulic Engineering vol 4 pp 8ndash14 2004
[16] W H Zhou and L S Zhao ldquoOne-dimensional consolidationof unsaturated soil subjected to time-dependent loading undervarious initial and boundary conditionsrdquo ASCE InternationalJournal of Geomechanics 2013
[17] E Conte ldquoConsolidation analysis for unsaturated soilsrdquo Cana-dian Geotechnical Journal vol 41 no 4 pp 599ndash612 2004
[18] E Conte ldquoPlane strain and axially symmetric consolidation inunsaturated soilsrdquo International Journal of Geomechanics vol 6no 2 pp 131ndash135 2006
[19] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 no 1 pp40ndash52 1972
[20] R Bellman and J Casti ldquoDifferential quadrature and long-termintegrationrdquo Journal of Mathematical Analysis and Applicationsvol 34 no 2 pp 235ndash238 1971
[21] J R Quan andC T Chang ldquoNew insights in solving distributedsystem equations by the quadrature method I analysisrdquo Com-puters and Chemical Engineering vol 13 no 7 pp 779ndash7881989
1 times 10minus6 0050 0050 0050 0050 0050 0050 0050
1 times 10minus4 0050 0050 0050 0050 0050 0050 0050
5 times 10minus2 0048 0048 0048 0048 0048 0048 0048
1 times 10minus1 0046 0045 0045 0045 0045 0045 0045
2 times 10minus1 0042 0041 0040 0040 0040 0040 0010
5 times 10minus1 0035 0033 0032 0032 0032 0031 0031
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 11 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
phase and water phase begins when 119879 = 10 times 10minus3 When
119879 = 10 times 100 and 119879 = 10 times 10
15 the consolidation ofair phase and water phase is almost finished respectivelyAt the early stages soil consolidation is mainly caused bythe dissipation of excess pore-air pressure But in the laterstages soil consolidation is mainly caused by the dissipationof excess pore-water pressure
Figure 3 shows the radial displacement development withtime and Figure 4 describes the settlement of top surface(119867 = 10m) at different radii with time factor 119879 The numberof discrete points is 119873 = 9 The ratio of permeabilitycoefficient is 119896
119886119896119908
= 01 From Figure 4 we can see thatthe soil settles earlier at the points with smaller radius thatis closer to the sand drain Figure 5 shows the settlement atdifferent heights of soil layer at radius 119903 = 1m with time
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 12 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(c) distribution (119896
119886119896119908= 01)
factor 119879 When the height of soil layer119867 is equal to 4m thesettlement is almost zero
5 Cases of Different Initial andBoundary Conditions
As the initial conditions do not need to be constant in thepresentDQMsolution it is easily used to analyze nonuniforminitial pore-water and pore-air distribution problems In thispart four different initial pore-water and pore-air distribu-tions and some different boundary conditions are consideredA vertical loading 119902 = 100 kPa is applied on the top surfaceof soil layer Other parameters are 119899
0= 05 119878
1199030= 08 119896
119908=
10minus10ms 119903
119908= 02m 119903
119890= 18m119898119908
1119896= minus015 times 10
minus4 kPaminus1
10 Mathematical Problems in Engineering
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
1
09
08
07
06
05
04
03
02
01
0
uw
q
Figure 13 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 14 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
119898119908
2= minus061 times 10
minus4 kPaminus1 1198981198861119896
= 006 times 10minus4 kPaminus1 119898119886
2=
026 times 10minus4 kPaminus1 and 119906atm = 1013 kPa
51 Effect of Different Initial Pore-WaterAir Pressure Distribu-tions In this section in order to make the curves smooththe number of discrete points is chosen to be 119873 = 17 Thelateral boundary is considered as impermeable and the ratioof permeability coefficient with respect to air and water is119896119886119896119908= 01
Four different initial pore-waterair distributions aretaken into account In Figure 6(a) the case of uniform initialpore-air and pore-water pressures assumes that 119906119886
0= 5 kPa
and 1199061199080= 40 kPa Figure 6(b) presents the linear increasing
initial pore-air and pore-water distributions The lateral-skewed distributions initial pore-waterair pressures is shown
in Figures 6(c) and 6(d) and they are represented by thefollowing equations
119906119908
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
100105119887
119906119886
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
8 times 100105119887
(30)
where 119887 is the variable which can control the spread orskewness of the curves in Figures 6(c) and 6(d)
Under the condition shown in Figure 6(a) the dissipationcurves of the excess pore-water and pore-air pressure atdifferent distances from the sand drain are presented inFigures 7 and 8 respectively The excess pore-air and pore-water pressures dissipate faster at the points with smallerradius that is closer to the sand drain As the intervalbetween the two curves becomes smaller from 119903 = 06m to119903 = 12m it can be seen that the effect of the distance on thedissipation becomes less obvious
In Figure 6(b) the initial pore-air and pore-water pres-sures are assumed not to be constants and to be linearincreasing when the radius increases The initial pore-airpressure is assumed one-eighth of the initial pore-water pres-sure In other words from 119903 = 04m to 119903 = 16m the initialpore-water pressure will increase from 30 kPa to 110 kPa andthe initial pore-air pressure will increase from 375 kPa to1375 kPa Figures 9 and 10 show the dissipation of excesspore-water and pore-air pressures at different distances fromthe sand drain under Figure 6(b) distributions
Comparing Figures 7 and 9 it is found that the excesspore-water pressure dissipates completely at almost the sametime With the change of the initial pore-water pressuredistribution the time of dissipation with respect to wateris not significantly affected Comparing Figures 8 and 10it can be seen that the excess pore-air pressure dissipatescompletely at almost the same time The initial pore-airpressure distribution has little influence on the dissipationtime of air phase
Under the conditions of Figures 6(c) and 6(d) thedissipation curves of the excess pore-water and pore-airpressures for initial distributions where 119887 = 2 6 are shownin Figures 11 12 13 and 14 From these figures whatever theinitial pore-water and pore-air pressures are the excess pore-water and pore-air pressures dissipate completely at almostthe same timeThe plateau period is longer at the points withlarger radius
Here excess pore pressure isochrones for the above fourinitial conditions are presented in Figures 15 and 16 Figures15(a) and 16(a) show the excess pore-water and pore-airpressure isochrones when the initial excess pore pressure isuniform at all radii In Figures 15(b) and 16(b) the initialexcess pore pressure distributions are linearly increasingThepore pressure isochrones for lateral-skewed initial distribu-tions are shown in Figures 15(c) 16(c) 15(d) and 16(d)respectively From Figures 15(a) and 16(a) as the initial excesspore pressure is constant the peak of curves in Figure 15(a)approaches to 04 and the peak of curves in Figure 13(a)approaches to 005 Comparing Figures 15(a) and 15(b) as
Mathematical Problems in Engineering 11
0
005
01
015
02
025
03
035
04
02 04 06 08 1 12 14 16 18r (m)
02
1
2
5
05
T = 10
uw
q
(a)
02 04 06 08 1 12 14 16 18r (m)
1
1
2
5
09
08
07
06
05
05
04
03
02
02
01
01
0
T = 10
uw
q
(b)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
011
09
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(c)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
0109
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(d)
Figure 15Normalized pore-water pressure isochrones for different initial excess pore-water pressure distributions (119896119886119896119908= 01) (a) uniform
(b) linearly increasing (c) lateral skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
the initial excess pore-water pressure is linearly increasingthe skewness is more evident during early stages of dissipa-tion in Figure 15(b) But in the later stages the influence ofinitial excess pore-water pressure becomes less significant Soare the isochrones of the excess pore-air pressure in Figures16(a) and 16(b) Figures 15(c) 16(c) 15(d) and 16(d) showthe excess pore-water and pore-air pressure isochrones forthe lateral-skewed initial distribution when 119887 = 2 6 It isseen that some intersections appear in the isochrones duringthe early stages It means that a redistribution of excesspore pressures occurs toward the region of minimal initialexcess pore pressure during the early stages With the timeincreasing the isochrones of excess pore pressures becomeregular
52 Effect of Different Boundary Conditions In this partthe effect of different lateral boundary conditions is studiedUniform initial excess pore-waterair pressure distributions119906119886
0= 5 kPa and 119906119908
0= 40 kPa are adopted Four different ratios
119896119886119896119908are considered that is 01 1 10 and 100 with 119896
119908=
10minus10ms Two different drainage conditions are analyzed
lateral surface is impervious and lateral surface is impededFor the impeded surface 119886
119908and 119886
119886in (9) are chosen to be
05Figures 17 and 18 shows the dissipation curves for the
water and air pressures at radius 119903 = 12m under differentboundary conditions First the excess pore-water and pore-air pressures dissipate faster in impeded surface than that inimpermeable surface Second 119896
119908is the same in the cases
12 Mathematical Problems in Engineering
02 04 06 08 1 12 14 16 18r (m)
01
02
05
T = 2
006
005005
004
003
002
002
001
0
1
uaq
(a)
02 04 06 08 1 12 14 16 18r (m)
014
012
0101
02
05008
006
004
002
0
T = 2
005002
1
uaq
(b)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(c)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(d)
Figure 16 Normalized pore-air pressure isochrones for different initial excess pore-air pressure distributions (119896119886119896119908= 01) (a) uniform (b)
linearly increasing (c) lateral-skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
of four 119896119886119896119908values It is observed from the results that
the bigger 119896119886119896119908
is the faster the excess pore-water andpore-air pressures dissipate Third the dissipation curves ofthe excess pore-air and pore-water pressures have almostthe same shape under different values of 119896
119886119896119908 Comparing
Figure 17 with Figure 18 when the excess pore-air pressuredissipates almost completely the dissipation of excess pore-water pressure enters into a plateau period The bigger 119896
119886119896119908
is the earlier the excess pore-air pressure dissipates com-pletely Therefore it is shown that the value of 119896
119886affects the
dissipation of pore-water pressure during the consolidationprocess
6 Conclusion
This paper obtains a general solution to the axisymmet-ric consolidation of unsaturated soils by using differen-tial quadrature method (DQM) based on Fredlundrsquos one-dimensional consolidation theory for unsaturated soil Withthe use of DQM the two governing equations are trans-formed into two sets of ordinary differential equationsThe solutions to the transformed differential equations areobtained by Rong-Kutta method under different initial andboundary conditions A case study has been presented forthe analysis of unsaturated consolidation in a sand drain
Mathematical Problems in Engineering 13
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
045
04
035
03
025
02
015
01
005
0
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
uw
q
Figure 17 Variation of excess pore-water pressures 119906119908119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
007
006
005
004
003
002
001
0
uaq
Figure 18 Variation of excess pore-air pressures 119906119886
119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
foundation The convergence analysis the average degreeof consolidation for water and air phases the settlementin radial and vertical directions effects of different initialexcess pore-air and pore-water pressure distributions andeffects of different boundary conditions have been presentedFrom the analyses some conclusions can be drawn (a)through compiling the programs it is found that the DQsolution delivers accurate and stable results for unsaturatedsoil consolidation Due to the uniformmatrix structure of theDQ formulas it is easy to compile computer programs whenconsidering complicated conditions such as nonuniformpore-waterair distribution conditions (b) soil consolidationismainly caused by the dissipation of excess pore-air pressurein the early stages and by the dissipation of excess pore-water pressure in the later stages (c) the bigger the ratio
of the permeability coefficients for water and air is thefaster the excess pore-water and pore-air pressures dissipate(d) The excess pore-air and pore-water pressures dissipatefaster at the points closer to the sand drain (e) the initialdistribution has some effects on the early consolidationprocess of unsaturated soil and boundary condition has asignificant effect on the whole consolidation process
Notation
119862119908
V The consolidation coefficient of water119862119886
V The consolidation coefficient of air119863(119903)
119894119896 The weighting coefficient
120576V Volumetric strain119892 The gravitational acceleration119867 The thickness of soil layer119896119886 The permeability coefficients of air
1198960
119886 The coefficients of permeability for water
at lateral boundary119896119908 The permeability coefficients of water
1198960
119908 The coefficients of permeability for air at
lateral boundary119898119886
1119896 The coefficient of air volume change withrespect to a change in the net normal stress
119898119908
1119896 The coefficient of water volume changewith respect to a change in the net normalstress
119898119886
2 The coefficient of air volume change with
respect to a change in matric suction119898119908
2 The coefficient of water volume changewith respect to a change in matric suction
1198990 The initial porosity
119873 The discrete point119875V A vacuum pressure120588 The density of air phase119902 The vertical loading1199030 The thickness of lateral boundary
119903119890 The radius of soil layer
119903119908 The radius of sand drain
119903120572 The radius value of 120572th point
120574119908 The unit weight of water phase
119877 The universal air constant119878 The distance between two sand drains1198781199030 The initial degree of saturation
119878V Settlement119878ℎ The radial displacement
119905 Time119879 The time factor119906119886 The excess pore-air pressure119906atm Atmospheric pressure119906119886
0 The initial excess pore-air pressure
119906119908 The excess pore-water pressure119906119908
0 The initial excess pore-water pressure
119880119886 The average degree of consolidation with
respect to air119880119908 The average degree of consolidation with
respect to water119881119886 The air volume of soil layer
14 Mathematical Problems in Engineering
119881119908 The water volume of soil layer
1198810 Initial total volume of the soil
Acknowledgments
The author gratefully acknowledges the financial supportfrom the Macau Science and Technology Development Fund(Grant no FDCT0112013A1) and the University of MacauResearch Fund (Grants nos MYRG189(Y2-L3)-FST11-ZWHand MYRG067(Y2-L2)-FST12-ZWH)
References
[1] K Terzaghi Theoretical Soil Mechanics Wiley New York NYUSA 1943
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] G E Blight Strength and consolidation characteristics of com-pacted soils [PhD thesis] University of London London UK1961
[4] R F Scott Principles of Soil Mechanics Addison Wesley Pub-lishing Company Boston Mass USA 1963
[5] L Barden ldquoConsolidation of compacted and unsaturated clayrdquoGeotechnique vol 15 no 3 pp 267ndash286 1965
[6] D G Fredlund and J U Hasan ldquoOne-dimensional consolida-tion theory unsaturated soilsrdquo Canadian Geotechnical Journalvol 16 no 3 pp 521ndash531 1979
[7] D G Fredlund andH Rahardjo Soil Mechanics for UnsaturatedSoils John Wiley and Son 1993
[8] A Qin D Sun L Yang and Y Weng ldquoA semi-analyticalsolution to consolidation of unsaturated soils with the freedrainage wellrdquo Computers and Geotechnics vol 37 no 7-8 pp867ndash875 2010
[9] W H Zhou L S Zhao and X B Li ldquoA simple analyticalsolution to one-dimensional consolidation for unsaturated soilrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics 2013
[10] M Malik and F Civan ldquoA comparative study of differentialquadrature and cubature methods vis-a-vis some conventionaltechniques in context of convection-diffusion-reaction prob-lemsrdquo Chemical Engineering Science vol 50 no 3 pp 531ndash5471995
[11] P Malekzadeh M R Golbahar Haghighi and M M AtashildquoOut-of-plane free vibration of functionally graded circularcurved beams in thermal environmentrdquo Composite Structuresvol 92 no 2 pp 541ndash552 2010
[12] P Malekzadeh M R G Haghighi and M M Atashi ldquoFreevibration analysis of elastically supported functionally radedannular plates subjected to thermal environmentrdquo Meccanicavol 46 no 5 pp 893ndash913 2011
[13] R P Chen W H Zhou H Z Wang and Y M Chen ldquoOne-dimensional nonlinear consolidation of multi-layered soil bydifferential quadrature methodrdquo Computers and Geotechnicsvol 32 no 5 pp 358ndash369 2005
[14] R P Chen W H Zhou H Z Wang and Y M Chen ldquoStudyon one-dimensional nonlinear consolidation of multi-layeredsoil using differential quadrature methodrdquo Chinese Journal ofComputational Mechanics vol 22 no 3 pp 310ndash315 2005
[15] H Z Wang R P Chen W H Zhou and Y M ChenldquoComputation of 1-D nonlinear consolidation in double-layer
foundation by using differential quadrature methodrdquo Journal ofHydraulic Engineering vol 4 pp 8ndash14 2004
[16] W H Zhou and L S Zhao ldquoOne-dimensional consolidationof unsaturated soil subjected to time-dependent loading undervarious initial and boundary conditionsrdquo ASCE InternationalJournal of Geomechanics 2013
[17] E Conte ldquoConsolidation analysis for unsaturated soilsrdquo Cana-dian Geotechnical Journal vol 41 no 4 pp 599ndash612 2004
[18] E Conte ldquoPlane strain and axially symmetric consolidation inunsaturated soilsrdquo International Journal of Geomechanics vol 6no 2 pp 131ndash135 2006
[19] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 no 1 pp40ndash52 1972
[20] R Bellman and J Casti ldquoDifferential quadrature and long-termintegrationrdquo Journal of Mathematical Analysis and Applicationsvol 34 no 2 pp 235ndash238 1971
[21] J R Quan andC T Chang ldquoNew insights in solving distributedsystem equations by the quadrature method I analysisrdquo Com-puters and Chemical Engineering vol 13 no 7 pp 779ndash7881989
Figure 13 Variation of pore-water pressures 119906119908119902 at different radiiwith time factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
r = 06 mr = 08 m
r = 12 mr = 14 m
014
012
01
008
006
004
002
0
uaq
Figure 14 Variation of pore-air pressures 119906119886119902 at different radii withtime factor 119879 under Figure 6(d) distribution (119896
119886119896119908= 01)
119898119908
2= minus061 times 10
minus4 kPaminus1 1198981198861119896
= 006 times 10minus4 kPaminus1 119898119886
2=
026 times 10minus4 kPaminus1 and 119906atm = 1013 kPa
51 Effect of Different Initial Pore-WaterAir Pressure Distribu-tions In this section in order to make the curves smooththe number of discrete points is chosen to be 119873 = 17 Thelateral boundary is considered as impermeable and the ratioof permeability coefficient with respect to air and water is119896119886119896119908= 01
Four different initial pore-waterair distributions aretaken into account In Figure 6(a) the case of uniform initialpore-air and pore-water pressures assumes that 119906119886
0= 5 kPa
and 1199061199080= 40 kPa Figure 6(b) presents the linear increasing
initial pore-air and pore-water distributions The lateral-skewed distributions initial pore-waterair pressures is shown
in Figures 6(c) and 6(d) and they are represented by thefollowing equations
119906119908
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
100105119887
119906119886
119902=(119903119903119890)025119887
(2 minus 119903119903119890)119887
8 times 100105119887
(30)
where 119887 is the variable which can control the spread orskewness of the curves in Figures 6(c) and 6(d)
Under the condition shown in Figure 6(a) the dissipationcurves of the excess pore-water and pore-air pressure atdifferent distances from the sand drain are presented inFigures 7 and 8 respectively The excess pore-air and pore-water pressures dissipate faster at the points with smallerradius that is closer to the sand drain As the intervalbetween the two curves becomes smaller from 119903 = 06m to119903 = 12m it can be seen that the effect of the distance on thedissipation becomes less obvious
In Figure 6(b) the initial pore-air and pore-water pres-sures are assumed not to be constants and to be linearincreasing when the radius increases The initial pore-airpressure is assumed one-eighth of the initial pore-water pres-sure In other words from 119903 = 04m to 119903 = 16m the initialpore-water pressure will increase from 30 kPa to 110 kPa andthe initial pore-air pressure will increase from 375 kPa to1375 kPa Figures 9 and 10 show the dissipation of excesspore-water and pore-air pressures at different distances fromthe sand drain under Figure 6(b) distributions
Comparing Figures 7 and 9 it is found that the excesspore-water pressure dissipates completely at almost the sametime With the change of the initial pore-water pressuredistribution the time of dissipation with respect to wateris not significantly affected Comparing Figures 8 and 10it can be seen that the excess pore-air pressure dissipatescompletely at almost the same time The initial pore-airpressure distribution has little influence on the dissipationtime of air phase
Under the conditions of Figures 6(c) and 6(d) thedissipation curves of the excess pore-water and pore-airpressures for initial distributions where 119887 = 2 6 are shownin Figures 11 12 13 and 14 From these figures whatever theinitial pore-water and pore-air pressures are the excess pore-water and pore-air pressures dissipate completely at almostthe same timeThe plateau period is longer at the points withlarger radius
Here excess pore pressure isochrones for the above fourinitial conditions are presented in Figures 15 and 16 Figures15(a) and 16(a) show the excess pore-water and pore-airpressure isochrones when the initial excess pore pressure isuniform at all radii In Figures 15(b) and 16(b) the initialexcess pore pressure distributions are linearly increasingThepore pressure isochrones for lateral-skewed initial distribu-tions are shown in Figures 15(c) 16(c) 15(d) and 16(d)respectively From Figures 15(a) and 16(a) as the initial excesspore pressure is constant the peak of curves in Figure 15(a)approaches to 04 and the peak of curves in Figure 13(a)approaches to 005 Comparing Figures 15(a) and 15(b) as
Mathematical Problems in Engineering 11
0
005
01
015
02
025
03
035
04
02 04 06 08 1 12 14 16 18r (m)
02
1
2
5
05
T = 10
uw
q
(a)
02 04 06 08 1 12 14 16 18r (m)
1
1
2
5
09
08
07
06
05
05
04
03
02
02
01
01
0
T = 10
uw
q
(b)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
011
09
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(c)
02
02 04 06 08 1 12 14 16 18r (m)
1
2
5
05
0109
08
07
06
05
04
03
02
01
0
T = 10
uw
q
(d)
Figure 15Normalized pore-water pressure isochrones for different initial excess pore-water pressure distributions (119896119886119896119908= 01) (a) uniform
(b) linearly increasing (c) lateral skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
the initial excess pore-water pressure is linearly increasingthe skewness is more evident during early stages of dissipa-tion in Figure 15(b) But in the later stages the influence ofinitial excess pore-water pressure becomes less significant Soare the isochrones of the excess pore-air pressure in Figures16(a) and 16(b) Figures 15(c) 16(c) 15(d) and 16(d) showthe excess pore-water and pore-air pressure isochrones forthe lateral-skewed initial distribution when 119887 = 2 6 It isseen that some intersections appear in the isochrones duringthe early stages It means that a redistribution of excesspore pressures occurs toward the region of minimal initialexcess pore pressure during the early stages With the timeincreasing the isochrones of excess pore pressures becomeregular
52 Effect of Different Boundary Conditions In this partthe effect of different lateral boundary conditions is studiedUniform initial excess pore-waterair pressure distributions119906119886
0= 5 kPa and 119906119908
0= 40 kPa are adopted Four different ratios
119896119886119896119908are considered that is 01 1 10 and 100 with 119896
119908=
10minus10ms Two different drainage conditions are analyzed
lateral surface is impervious and lateral surface is impededFor the impeded surface 119886
119908and 119886
119886in (9) are chosen to be
05Figures 17 and 18 shows the dissipation curves for the
water and air pressures at radius 119903 = 12m under differentboundary conditions First the excess pore-water and pore-air pressures dissipate faster in impeded surface than that inimpermeable surface Second 119896
119908is the same in the cases
12 Mathematical Problems in Engineering
02 04 06 08 1 12 14 16 18r (m)
01
02
05
T = 2
006
005005
004
003
002
002
001
0
1
uaq
(a)
02 04 06 08 1 12 14 16 18r (m)
014
012
0101
02
05008
006
004
002
0
T = 2
005002
1
uaq
(b)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(c)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(d)
Figure 16 Normalized pore-air pressure isochrones for different initial excess pore-air pressure distributions (119896119886119896119908= 01) (a) uniform (b)
linearly increasing (c) lateral-skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
of four 119896119886119896119908values It is observed from the results that
the bigger 119896119886119896119908
is the faster the excess pore-water andpore-air pressures dissipate Third the dissipation curves ofthe excess pore-air and pore-water pressures have almostthe same shape under different values of 119896
119886119896119908 Comparing
Figure 17 with Figure 18 when the excess pore-air pressuredissipates almost completely the dissipation of excess pore-water pressure enters into a plateau period The bigger 119896
119886119896119908
is the earlier the excess pore-air pressure dissipates com-pletely Therefore it is shown that the value of 119896
119886affects the
dissipation of pore-water pressure during the consolidationprocess
6 Conclusion
This paper obtains a general solution to the axisymmet-ric consolidation of unsaturated soils by using differen-tial quadrature method (DQM) based on Fredlundrsquos one-dimensional consolidation theory for unsaturated soil Withthe use of DQM the two governing equations are trans-formed into two sets of ordinary differential equationsThe solutions to the transformed differential equations areobtained by Rong-Kutta method under different initial andboundary conditions A case study has been presented forthe analysis of unsaturated consolidation in a sand drain
Mathematical Problems in Engineering 13
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
045
04
035
03
025
02
015
01
005
0
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
uw
q
Figure 17 Variation of excess pore-water pressures 119906119908119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
007
006
005
004
003
002
001
0
uaq
Figure 18 Variation of excess pore-air pressures 119906119886
119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
foundation The convergence analysis the average degreeof consolidation for water and air phases the settlementin radial and vertical directions effects of different initialexcess pore-air and pore-water pressure distributions andeffects of different boundary conditions have been presentedFrom the analyses some conclusions can be drawn (a)through compiling the programs it is found that the DQsolution delivers accurate and stable results for unsaturatedsoil consolidation Due to the uniformmatrix structure of theDQ formulas it is easy to compile computer programs whenconsidering complicated conditions such as nonuniformpore-waterair distribution conditions (b) soil consolidationismainly caused by the dissipation of excess pore-air pressurein the early stages and by the dissipation of excess pore-water pressure in the later stages (c) the bigger the ratio
of the permeability coefficients for water and air is thefaster the excess pore-water and pore-air pressures dissipate(d) The excess pore-air and pore-water pressures dissipatefaster at the points closer to the sand drain (e) the initialdistribution has some effects on the early consolidationprocess of unsaturated soil and boundary condition has asignificant effect on the whole consolidation process
Notation
119862119908
V The consolidation coefficient of water119862119886
V The consolidation coefficient of air119863(119903)
119894119896 The weighting coefficient
120576V Volumetric strain119892 The gravitational acceleration119867 The thickness of soil layer119896119886 The permeability coefficients of air
1198960
119886 The coefficients of permeability for water
at lateral boundary119896119908 The permeability coefficients of water
1198960
119908 The coefficients of permeability for air at
lateral boundary119898119886
1119896 The coefficient of air volume change withrespect to a change in the net normal stress
119898119908
1119896 The coefficient of water volume changewith respect to a change in the net normalstress
119898119886
2 The coefficient of air volume change with
respect to a change in matric suction119898119908
2 The coefficient of water volume changewith respect to a change in matric suction
1198990 The initial porosity
119873 The discrete point119875V A vacuum pressure120588 The density of air phase119902 The vertical loading1199030 The thickness of lateral boundary
119903119890 The radius of soil layer
119903119908 The radius of sand drain
119903120572 The radius value of 120572th point
120574119908 The unit weight of water phase
119877 The universal air constant119878 The distance between two sand drains1198781199030 The initial degree of saturation
119878V Settlement119878ℎ The radial displacement
119905 Time119879 The time factor119906119886 The excess pore-air pressure119906atm Atmospheric pressure119906119886
0 The initial excess pore-air pressure
119906119908 The excess pore-water pressure119906119908
0 The initial excess pore-water pressure
119880119886 The average degree of consolidation with
respect to air119880119908 The average degree of consolidation with
respect to water119881119886 The air volume of soil layer
14 Mathematical Problems in Engineering
119881119908 The water volume of soil layer
1198810 Initial total volume of the soil
Acknowledgments
The author gratefully acknowledges the financial supportfrom the Macau Science and Technology Development Fund(Grant no FDCT0112013A1) and the University of MacauResearch Fund (Grants nos MYRG189(Y2-L3)-FST11-ZWHand MYRG067(Y2-L2)-FST12-ZWH)
References
[1] K Terzaghi Theoretical Soil Mechanics Wiley New York NYUSA 1943
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] G E Blight Strength and consolidation characteristics of com-pacted soils [PhD thesis] University of London London UK1961
[4] R F Scott Principles of Soil Mechanics Addison Wesley Pub-lishing Company Boston Mass USA 1963
[5] L Barden ldquoConsolidation of compacted and unsaturated clayrdquoGeotechnique vol 15 no 3 pp 267ndash286 1965
[6] D G Fredlund and J U Hasan ldquoOne-dimensional consolida-tion theory unsaturated soilsrdquo Canadian Geotechnical Journalvol 16 no 3 pp 521ndash531 1979
[7] D G Fredlund andH Rahardjo Soil Mechanics for UnsaturatedSoils John Wiley and Son 1993
[8] A Qin D Sun L Yang and Y Weng ldquoA semi-analyticalsolution to consolidation of unsaturated soils with the freedrainage wellrdquo Computers and Geotechnics vol 37 no 7-8 pp867ndash875 2010
[9] W H Zhou L S Zhao and X B Li ldquoA simple analyticalsolution to one-dimensional consolidation for unsaturated soilrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics 2013
[10] M Malik and F Civan ldquoA comparative study of differentialquadrature and cubature methods vis-a-vis some conventionaltechniques in context of convection-diffusion-reaction prob-lemsrdquo Chemical Engineering Science vol 50 no 3 pp 531ndash5471995
[11] P Malekzadeh M R Golbahar Haghighi and M M AtashildquoOut-of-plane free vibration of functionally graded circularcurved beams in thermal environmentrdquo Composite Structuresvol 92 no 2 pp 541ndash552 2010
[12] P Malekzadeh M R G Haghighi and M M Atashi ldquoFreevibration analysis of elastically supported functionally radedannular plates subjected to thermal environmentrdquo Meccanicavol 46 no 5 pp 893ndash913 2011
[13] R P Chen W H Zhou H Z Wang and Y M Chen ldquoOne-dimensional nonlinear consolidation of multi-layered soil bydifferential quadrature methodrdquo Computers and Geotechnicsvol 32 no 5 pp 358ndash369 2005
[14] R P Chen W H Zhou H Z Wang and Y M Chen ldquoStudyon one-dimensional nonlinear consolidation of multi-layeredsoil using differential quadrature methodrdquo Chinese Journal ofComputational Mechanics vol 22 no 3 pp 310ndash315 2005
[15] H Z Wang R P Chen W H Zhou and Y M ChenldquoComputation of 1-D nonlinear consolidation in double-layer
foundation by using differential quadrature methodrdquo Journal ofHydraulic Engineering vol 4 pp 8ndash14 2004
[16] W H Zhou and L S Zhao ldquoOne-dimensional consolidationof unsaturated soil subjected to time-dependent loading undervarious initial and boundary conditionsrdquo ASCE InternationalJournal of Geomechanics 2013
[17] E Conte ldquoConsolidation analysis for unsaturated soilsrdquo Cana-dian Geotechnical Journal vol 41 no 4 pp 599ndash612 2004
[18] E Conte ldquoPlane strain and axially symmetric consolidation inunsaturated soilsrdquo International Journal of Geomechanics vol 6no 2 pp 131ndash135 2006
[19] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 no 1 pp40ndash52 1972
[20] R Bellman and J Casti ldquoDifferential quadrature and long-termintegrationrdquo Journal of Mathematical Analysis and Applicationsvol 34 no 2 pp 235ndash238 1971
[21] J R Quan andC T Chang ldquoNew insights in solving distributedsystem equations by the quadrature method I analysisrdquo Com-puters and Chemical Engineering vol 13 no 7 pp 779ndash7881989
Figure 15Normalized pore-water pressure isochrones for different initial excess pore-water pressure distributions (119896119886119896119908= 01) (a) uniform
(b) linearly increasing (c) lateral skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
the initial excess pore-water pressure is linearly increasingthe skewness is more evident during early stages of dissipa-tion in Figure 15(b) But in the later stages the influence ofinitial excess pore-water pressure becomes less significant Soare the isochrones of the excess pore-air pressure in Figures16(a) and 16(b) Figures 15(c) 16(c) 15(d) and 16(d) showthe excess pore-water and pore-air pressure isochrones forthe lateral-skewed initial distribution when 119887 = 2 6 It isseen that some intersections appear in the isochrones duringthe early stages It means that a redistribution of excesspore pressures occurs toward the region of minimal initialexcess pore pressure during the early stages With the timeincreasing the isochrones of excess pore pressures becomeregular
52 Effect of Different Boundary Conditions In this partthe effect of different lateral boundary conditions is studiedUniform initial excess pore-waterair pressure distributions119906119886
0= 5 kPa and 119906119908
0= 40 kPa are adopted Four different ratios
119896119886119896119908are considered that is 01 1 10 and 100 with 119896
119908=
10minus10ms Two different drainage conditions are analyzed
lateral surface is impervious and lateral surface is impededFor the impeded surface 119886
119908and 119886
119886in (9) are chosen to be
05Figures 17 and 18 shows the dissipation curves for the
water and air pressures at radius 119903 = 12m under differentboundary conditions First the excess pore-water and pore-air pressures dissipate faster in impeded surface than that inimpermeable surface Second 119896
119908is the same in the cases
12 Mathematical Problems in Engineering
02 04 06 08 1 12 14 16 18r (m)
01
02
05
T = 2
006
005005
004
003
002
002
001
0
1
uaq
(a)
02 04 06 08 1 12 14 16 18r (m)
014
012
0101
02
05008
006
004
002
0
T = 2
005002
1
uaq
(b)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(c)
02 04 06 08 1 12 14 16 18r (m)
01
02
05
012
01
008
006
004
002
0
T = 2
005
002
1
uaq
(d)
Figure 16 Normalized pore-air pressure isochrones for different initial excess pore-air pressure distributions (119896119886119896119908= 01) (a) uniform (b)
linearly increasing (c) lateral-skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
of four 119896119886119896119908values It is observed from the results that
the bigger 119896119886119896119908
is the faster the excess pore-water andpore-air pressures dissipate Third the dissipation curves ofthe excess pore-air and pore-water pressures have almostthe same shape under different values of 119896
119886119896119908 Comparing
Figure 17 with Figure 18 when the excess pore-air pressuredissipates almost completely the dissipation of excess pore-water pressure enters into a plateau period The bigger 119896
119886119896119908
is the earlier the excess pore-air pressure dissipates com-pletely Therefore it is shown that the value of 119896
119886affects the
dissipation of pore-water pressure during the consolidationprocess
6 Conclusion
This paper obtains a general solution to the axisymmet-ric consolidation of unsaturated soils by using differen-tial quadrature method (DQM) based on Fredlundrsquos one-dimensional consolidation theory for unsaturated soil Withthe use of DQM the two governing equations are trans-formed into two sets of ordinary differential equationsThe solutions to the transformed differential equations areobtained by Rong-Kutta method under different initial andboundary conditions A case study has been presented forthe analysis of unsaturated consolidation in a sand drain
Mathematical Problems in Engineering 13
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
045
04
035
03
025
02
015
01
005
0
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
uw
q
Figure 17 Variation of excess pore-water pressures 119906119908119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
007
006
005
004
003
002
001
0
uaq
Figure 18 Variation of excess pore-air pressures 119906119886
119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
foundation The convergence analysis the average degreeof consolidation for water and air phases the settlementin radial and vertical directions effects of different initialexcess pore-air and pore-water pressure distributions andeffects of different boundary conditions have been presentedFrom the analyses some conclusions can be drawn (a)through compiling the programs it is found that the DQsolution delivers accurate and stable results for unsaturatedsoil consolidation Due to the uniformmatrix structure of theDQ formulas it is easy to compile computer programs whenconsidering complicated conditions such as nonuniformpore-waterair distribution conditions (b) soil consolidationismainly caused by the dissipation of excess pore-air pressurein the early stages and by the dissipation of excess pore-water pressure in the later stages (c) the bigger the ratio
of the permeability coefficients for water and air is thefaster the excess pore-water and pore-air pressures dissipate(d) The excess pore-air and pore-water pressures dissipatefaster at the points closer to the sand drain (e) the initialdistribution has some effects on the early consolidationprocess of unsaturated soil and boundary condition has asignificant effect on the whole consolidation process
Notation
119862119908
V The consolidation coefficient of water119862119886
V The consolidation coefficient of air119863(119903)
119894119896 The weighting coefficient
120576V Volumetric strain119892 The gravitational acceleration119867 The thickness of soil layer119896119886 The permeability coefficients of air
1198960
119886 The coefficients of permeability for water
at lateral boundary119896119908 The permeability coefficients of water
1198960
119908 The coefficients of permeability for air at
lateral boundary119898119886
1119896 The coefficient of air volume change withrespect to a change in the net normal stress
119898119908
1119896 The coefficient of water volume changewith respect to a change in the net normalstress
119898119886
2 The coefficient of air volume change with
respect to a change in matric suction119898119908
2 The coefficient of water volume changewith respect to a change in matric suction
1198990 The initial porosity
119873 The discrete point119875V A vacuum pressure120588 The density of air phase119902 The vertical loading1199030 The thickness of lateral boundary
119903119890 The radius of soil layer
119903119908 The radius of sand drain
119903120572 The radius value of 120572th point
120574119908 The unit weight of water phase
119877 The universal air constant119878 The distance between two sand drains1198781199030 The initial degree of saturation
119878V Settlement119878ℎ The radial displacement
119905 Time119879 The time factor119906119886 The excess pore-air pressure119906atm Atmospheric pressure119906119886
0 The initial excess pore-air pressure
119906119908 The excess pore-water pressure119906119908
0 The initial excess pore-water pressure
119880119886 The average degree of consolidation with
respect to air119880119908 The average degree of consolidation with
respect to water119881119886 The air volume of soil layer
14 Mathematical Problems in Engineering
119881119908 The water volume of soil layer
1198810 Initial total volume of the soil
Acknowledgments
The author gratefully acknowledges the financial supportfrom the Macau Science and Technology Development Fund(Grant no FDCT0112013A1) and the University of MacauResearch Fund (Grants nos MYRG189(Y2-L3)-FST11-ZWHand MYRG067(Y2-L2)-FST12-ZWH)
References
[1] K Terzaghi Theoretical Soil Mechanics Wiley New York NYUSA 1943
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] G E Blight Strength and consolidation characteristics of com-pacted soils [PhD thesis] University of London London UK1961
[4] R F Scott Principles of Soil Mechanics Addison Wesley Pub-lishing Company Boston Mass USA 1963
[5] L Barden ldquoConsolidation of compacted and unsaturated clayrdquoGeotechnique vol 15 no 3 pp 267ndash286 1965
[6] D G Fredlund and J U Hasan ldquoOne-dimensional consolida-tion theory unsaturated soilsrdquo Canadian Geotechnical Journalvol 16 no 3 pp 521ndash531 1979
[7] D G Fredlund andH Rahardjo Soil Mechanics for UnsaturatedSoils John Wiley and Son 1993
[8] A Qin D Sun L Yang and Y Weng ldquoA semi-analyticalsolution to consolidation of unsaturated soils with the freedrainage wellrdquo Computers and Geotechnics vol 37 no 7-8 pp867ndash875 2010
[9] W H Zhou L S Zhao and X B Li ldquoA simple analyticalsolution to one-dimensional consolidation for unsaturated soilrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics 2013
[10] M Malik and F Civan ldquoA comparative study of differentialquadrature and cubature methods vis-a-vis some conventionaltechniques in context of convection-diffusion-reaction prob-lemsrdquo Chemical Engineering Science vol 50 no 3 pp 531ndash5471995
[11] P Malekzadeh M R Golbahar Haghighi and M M AtashildquoOut-of-plane free vibration of functionally graded circularcurved beams in thermal environmentrdquo Composite Structuresvol 92 no 2 pp 541ndash552 2010
[12] P Malekzadeh M R G Haghighi and M M Atashi ldquoFreevibration analysis of elastically supported functionally radedannular plates subjected to thermal environmentrdquo Meccanicavol 46 no 5 pp 893ndash913 2011
[13] R P Chen W H Zhou H Z Wang and Y M Chen ldquoOne-dimensional nonlinear consolidation of multi-layered soil bydifferential quadrature methodrdquo Computers and Geotechnicsvol 32 no 5 pp 358ndash369 2005
[14] R P Chen W H Zhou H Z Wang and Y M Chen ldquoStudyon one-dimensional nonlinear consolidation of multi-layeredsoil using differential quadrature methodrdquo Chinese Journal ofComputational Mechanics vol 22 no 3 pp 310ndash315 2005
[15] H Z Wang R P Chen W H Zhou and Y M ChenldquoComputation of 1-D nonlinear consolidation in double-layer
foundation by using differential quadrature methodrdquo Journal ofHydraulic Engineering vol 4 pp 8ndash14 2004
[16] W H Zhou and L S Zhao ldquoOne-dimensional consolidationof unsaturated soil subjected to time-dependent loading undervarious initial and boundary conditionsrdquo ASCE InternationalJournal of Geomechanics 2013
[17] E Conte ldquoConsolidation analysis for unsaturated soilsrdquo Cana-dian Geotechnical Journal vol 41 no 4 pp 599ndash612 2004
[18] E Conte ldquoPlane strain and axially symmetric consolidation inunsaturated soilsrdquo International Journal of Geomechanics vol 6no 2 pp 131ndash135 2006
[19] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 no 1 pp40ndash52 1972
[20] R Bellman and J Casti ldquoDifferential quadrature and long-termintegrationrdquo Journal of Mathematical Analysis and Applicationsvol 34 no 2 pp 235ndash238 1971
[21] J R Quan andC T Chang ldquoNew insights in solving distributedsystem equations by the quadrature method I analysisrdquo Com-puters and Chemical Engineering vol 13 no 7 pp 779ndash7881989
Figure 16 Normalized pore-air pressure isochrones for different initial excess pore-air pressure distributions (119896119886119896119908= 01) (a) uniform (b)
linearly increasing (c) lateral-skewed with 119887 = 2 and (d) lateral skewed with 119887 = 6
of four 119896119886119896119908values It is observed from the results that
the bigger 119896119886119896119908
is the faster the excess pore-water andpore-air pressures dissipate Third the dissipation curves ofthe excess pore-air and pore-water pressures have almostthe same shape under different values of 119896
119886119896119908 Comparing
Figure 17 with Figure 18 when the excess pore-air pressuredissipates almost completely the dissipation of excess pore-water pressure enters into a plateau period The bigger 119896
119886119896119908
is the earlier the excess pore-air pressure dissipates com-pletely Therefore it is shown that the value of 119896
119886affects the
dissipation of pore-water pressure during the consolidationprocess
6 Conclusion
This paper obtains a general solution to the axisymmet-ric consolidation of unsaturated soils by using differen-tial quadrature method (DQM) based on Fredlundrsquos one-dimensional consolidation theory for unsaturated soil Withthe use of DQM the two governing equations are trans-formed into two sets of ordinary differential equationsThe solutions to the transformed differential equations areobtained by Rong-Kutta method under different initial andboundary conditions A case study has been presented forthe analysis of unsaturated consolidation in a sand drain
Mathematical Problems in Engineering 13
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
045
04
035
03
025
02
015
01
005
0
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
uw
q
Figure 17 Variation of excess pore-water pressures 119906119908119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
007
006
005
004
003
002
001
0
uaq
Figure 18 Variation of excess pore-air pressures 119906119886
119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
foundation The convergence analysis the average degreeof consolidation for water and air phases the settlementin radial and vertical directions effects of different initialexcess pore-air and pore-water pressure distributions andeffects of different boundary conditions have been presentedFrom the analyses some conclusions can be drawn (a)through compiling the programs it is found that the DQsolution delivers accurate and stable results for unsaturatedsoil consolidation Due to the uniformmatrix structure of theDQ formulas it is easy to compile computer programs whenconsidering complicated conditions such as nonuniformpore-waterair distribution conditions (b) soil consolidationismainly caused by the dissipation of excess pore-air pressurein the early stages and by the dissipation of excess pore-water pressure in the later stages (c) the bigger the ratio
of the permeability coefficients for water and air is thefaster the excess pore-water and pore-air pressures dissipate(d) The excess pore-air and pore-water pressures dissipatefaster at the points closer to the sand drain (e) the initialdistribution has some effects on the early consolidationprocess of unsaturated soil and boundary condition has asignificant effect on the whole consolidation process
Notation
119862119908
V The consolidation coefficient of water119862119886
V The consolidation coefficient of air119863(119903)
119894119896 The weighting coefficient
120576V Volumetric strain119892 The gravitational acceleration119867 The thickness of soil layer119896119886 The permeability coefficients of air
1198960
119886 The coefficients of permeability for water
at lateral boundary119896119908 The permeability coefficients of water
1198960
119908 The coefficients of permeability for air at
lateral boundary119898119886
1119896 The coefficient of air volume change withrespect to a change in the net normal stress
119898119908
1119896 The coefficient of water volume changewith respect to a change in the net normalstress
119898119886
2 The coefficient of air volume change with
respect to a change in matric suction119898119908
2 The coefficient of water volume changewith respect to a change in matric suction
1198990 The initial porosity
119873 The discrete point119875V A vacuum pressure120588 The density of air phase119902 The vertical loading1199030 The thickness of lateral boundary
119903119890 The radius of soil layer
119903119908 The radius of sand drain
119903120572 The radius value of 120572th point
120574119908 The unit weight of water phase
119877 The universal air constant119878 The distance between two sand drains1198781199030 The initial degree of saturation
119878V Settlement119878ℎ The radial displacement
119905 Time119879 The time factor119906119886 The excess pore-air pressure119906atm Atmospheric pressure119906119886
0 The initial excess pore-air pressure
119906119908 The excess pore-water pressure119906119908
0 The initial excess pore-water pressure
119880119886 The average degree of consolidation with
respect to air119880119908 The average degree of consolidation with
respect to water119881119886 The air volume of soil layer
14 Mathematical Problems in Engineering
119881119908 The water volume of soil layer
1198810 Initial total volume of the soil
Acknowledgments
The author gratefully acknowledges the financial supportfrom the Macau Science and Technology Development Fund(Grant no FDCT0112013A1) and the University of MacauResearch Fund (Grants nos MYRG189(Y2-L3)-FST11-ZWHand MYRG067(Y2-L2)-FST12-ZWH)
References
[1] K Terzaghi Theoretical Soil Mechanics Wiley New York NYUSA 1943
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] G E Blight Strength and consolidation characteristics of com-pacted soils [PhD thesis] University of London London UK1961
[4] R F Scott Principles of Soil Mechanics Addison Wesley Pub-lishing Company Boston Mass USA 1963
[5] L Barden ldquoConsolidation of compacted and unsaturated clayrdquoGeotechnique vol 15 no 3 pp 267ndash286 1965
[6] D G Fredlund and J U Hasan ldquoOne-dimensional consolida-tion theory unsaturated soilsrdquo Canadian Geotechnical Journalvol 16 no 3 pp 521ndash531 1979
[7] D G Fredlund andH Rahardjo Soil Mechanics for UnsaturatedSoils John Wiley and Son 1993
[8] A Qin D Sun L Yang and Y Weng ldquoA semi-analyticalsolution to consolidation of unsaturated soils with the freedrainage wellrdquo Computers and Geotechnics vol 37 no 7-8 pp867ndash875 2010
[9] W H Zhou L S Zhao and X B Li ldquoA simple analyticalsolution to one-dimensional consolidation for unsaturated soilrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics 2013
[10] M Malik and F Civan ldquoA comparative study of differentialquadrature and cubature methods vis-a-vis some conventionaltechniques in context of convection-diffusion-reaction prob-lemsrdquo Chemical Engineering Science vol 50 no 3 pp 531ndash5471995
[11] P Malekzadeh M R Golbahar Haghighi and M M AtashildquoOut-of-plane free vibration of functionally graded circularcurved beams in thermal environmentrdquo Composite Structuresvol 92 no 2 pp 541ndash552 2010
[12] P Malekzadeh M R G Haghighi and M M Atashi ldquoFreevibration analysis of elastically supported functionally radedannular plates subjected to thermal environmentrdquo Meccanicavol 46 no 5 pp 893ndash913 2011
[13] R P Chen W H Zhou H Z Wang and Y M Chen ldquoOne-dimensional nonlinear consolidation of multi-layered soil bydifferential quadrature methodrdquo Computers and Geotechnicsvol 32 no 5 pp 358ndash369 2005
[14] R P Chen W H Zhou H Z Wang and Y M Chen ldquoStudyon one-dimensional nonlinear consolidation of multi-layeredsoil using differential quadrature methodrdquo Chinese Journal ofComputational Mechanics vol 22 no 3 pp 310ndash315 2005
[15] H Z Wang R P Chen W H Zhou and Y M ChenldquoComputation of 1-D nonlinear consolidation in double-layer
foundation by using differential quadrature methodrdquo Journal ofHydraulic Engineering vol 4 pp 8ndash14 2004
[16] W H Zhou and L S Zhao ldquoOne-dimensional consolidationof unsaturated soil subjected to time-dependent loading undervarious initial and boundary conditionsrdquo ASCE InternationalJournal of Geomechanics 2013
[17] E Conte ldquoConsolidation analysis for unsaturated soilsrdquo Cana-dian Geotechnical Journal vol 41 no 4 pp 599ndash612 2004
[18] E Conte ldquoPlane strain and axially symmetric consolidation inunsaturated soilsrdquo International Journal of Geomechanics vol 6no 2 pp 131ndash135 2006
[19] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 no 1 pp40ndash52 1972
[20] R Bellman and J Casti ldquoDifferential quadrature and long-termintegrationrdquo Journal of Mathematical Analysis and Applicationsvol 34 no 2 pp 235ndash238 1971
[21] J R Quan andC T Chang ldquoNew insights in solving distributedsystem equations by the quadrature method I analysisrdquo Com-puters and Chemical Engineering vol 13 no 7 pp 779ndash7881989
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
uw
q
Figure 17 Variation of excess pore-water pressures 119906119908119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
10minus3
10minus4
10minus5
10minus6
10minus2
10minus1
100
101
102
T
Lateral surface is impermeableLateral surface is impeded
100 10 1 kakw = 01
007
006
005
004
003
002
001
0
uaq
Figure 18 Variation of excess pore-air pressures 119906119886
119902 underdifferent 119896
119886119896119908with time factor 119879 (119903 = 12m)
foundation The convergence analysis the average degreeof consolidation for water and air phases the settlementin radial and vertical directions effects of different initialexcess pore-air and pore-water pressure distributions andeffects of different boundary conditions have been presentedFrom the analyses some conclusions can be drawn (a)through compiling the programs it is found that the DQsolution delivers accurate and stable results for unsaturatedsoil consolidation Due to the uniformmatrix structure of theDQ formulas it is easy to compile computer programs whenconsidering complicated conditions such as nonuniformpore-waterair distribution conditions (b) soil consolidationismainly caused by the dissipation of excess pore-air pressurein the early stages and by the dissipation of excess pore-water pressure in the later stages (c) the bigger the ratio
of the permeability coefficients for water and air is thefaster the excess pore-water and pore-air pressures dissipate(d) The excess pore-air and pore-water pressures dissipatefaster at the points closer to the sand drain (e) the initialdistribution has some effects on the early consolidationprocess of unsaturated soil and boundary condition has asignificant effect on the whole consolidation process
Notation
119862119908
V The consolidation coefficient of water119862119886
V The consolidation coefficient of air119863(119903)
119894119896 The weighting coefficient
120576V Volumetric strain119892 The gravitational acceleration119867 The thickness of soil layer119896119886 The permeability coefficients of air
1198960
119886 The coefficients of permeability for water
at lateral boundary119896119908 The permeability coefficients of water
1198960
119908 The coefficients of permeability for air at
lateral boundary119898119886
1119896 The coefficient of air volume change withrespect to a change in the net normal stress
119898119908
1119896 The coefficient of water volume changewith respect to a change in the net normalstress
119898119886
2 The coefficient of air volume change with
respect to a change in matric suction119898119908
2 The coefficient of water volume changewith respect to a change in matric suction
1198990 The initial porosity
119873 The discrete point119875V A vacuum pressure120588 The density of air phase119902 The vertical loading1199030 The thickness of lateral boundary
119903119890 The radius of soil layer
119903119908 The radius of sand drain
119903120572 The radius value of 120572th point
120574119908 The unit weight of water phase
119877 The universal air constant119878 The distance between two sand drains1198781199030 The initial degree of saturation
119878V Settlement119878ℎ The radial displacement
119905 Time119879 The time factor119906119886 The excess pore-air pressure119906atm Atmospheric pressure119906119886
0 The initial excess pore-air pressure
119906119908 The excess pore-water pressure119906119908
0 The initial excess pore-water pressure
119880119886 The average degree of consolidation with
respect to air119880119908 The average degree of consolidation with
respect to water119881119886 The air volume of soil layer
14 Mathematical Problems in Engineering
119881119908 The water volume of soil layer
1198810 Initial total volume of the soil
Acknowledgments
The author gratefully acknowledges the financial supportfrom the Macau Science and Technology Development Fund(Grant no FDCT0112013A1) and the University of MacauResearch Fund (Grants nos MYRG189(Y2-L3)-FST11-ZWHand MYRG067(Y2-L2)-FST12-ZWH)
References
[1] K Terzaghi Theoretical Soil Mechanics Wiley New York NYUSA 1943
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] G E Blight Strength and consolidation characteristics of com-pacted soils [PhD thesis] University of London London UK1961
[4] R F Scott Principles of Soil Mechanics Addison Wesley Pub-lishing Company Boston Mass USA 1963
[5] L Barden ldquoConsolidation of compacted and unsaturated clayrdquoGeotechnique vol 15 no 3 pp 267ndash286 1965
[6] D G Fredlund and J U Hasan ldquoOne-dimensional consolida-tion theory unsaturated soilsrdquo Canadian Geotechnical Journalvol 16 no 3 pp 521ndash531 1979
[7] D G Fredlund andH Rahardjo Soil Mechanics for UnsaturatedSoils John Wiley and Son 1993
[8] A Qin D Sun L Yang and Y Weng ldquoA semi-analyticalsolution to consolidation of unsaturated soils with the freedrainage wellrdquo Computers and Geotechnics vol 37 no 7-8 pp867ndash875 2010
[9] W H Zhou L S Zhao and X B Li ldquoA simple analyticalsolution to one-dimensional consolidation for unsaturated soilrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics 2013
[10] M Malik and F Civan ldquoA comparative study of differentialquadrature and cubature methods vis-a-vis some conventionaltechniques in context of convection-diffusion-reaction prob-lemsrdquo Chemical Engineering Science vol 50 no 3 pp 531ndash5471995
[11] P Malekzadeh M R Golbahar Haghighi and M M AtashildquoOut-of-plane free vibration of functionally graded circularcurved beams in thermal environmentrdquo Composite Structuresvol 92 no 2 pp 541ndash552 2010
[12] P Malekzadeh M R G Haghighi and M M Atashi ldquoFreevibration analysis of elastically supported functionally radedannular plates subjected to thermal environmentrdquo Meccanicavol 46 no 5 pp 893ndash913 2011
[13] R P Chen W H Zhou H Z Wang and Y M Chen ldquoOne-dimensional nonlinear consolidation of multi-layered soil bydifferential quadrature methodrdquo Computers and Geotechnicsvol 32 no 5 pp 358ndash369 2005
[14] R P Chen W H Zhou H Z Wang and Y M Chen ldquoStudyon one-dimensional nonlinear consolidation of multi-layeredsoil using differential quadrature methodrdquo Chinese Journal ofComputational Mechanics vol 22 no 3 pp 310ndash315 2005
[15] H Z Wang R P Chen W H Zhou and Y M ChenldquoComputation of 1-D nonlinear consolidation in double-layer
foundation by using differential quadrature methodrdquo Journal ofHydraulic Engineering vol 4 pp 8ndash14 2004
[16] W H Zhou and L S Zhao ldquoOne-dimensional consolidationof unsaturated soil subjected to time-dependent loading undervarious initial and boundary conditionsrdquo ASCE InternationalJournal of Geomechanics 2013
[17] E Conte ldquoConsolidation analysis for unsaturated soilsrdquo Cana-dian Geotechnical Journal vol 41 no 4 pp 599ndash612 2004
[18] E Conte ldquoPlane strain and axially symmetric consolidation inunsaturated soilsrdquo International Journal of Geomechanics vol 6no 2 pp 131ndash135 2006
[19] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 no 1 pp40ndash52 1972
[20] R Bellman and J Casti ldquoDifferential quadrature and long-termintegrationrdquo Journal of Mathematical Analysis and Applicationsvol 34 no 2 pp 235ndash238 1971
[21] J R Quan andC T Chang ldquoNew insights in solving distributedsystem equations by the quadrature method I analysisrdquo Com-puters and Chemical Engineering vol 13 no 7 pp 779ndash7881989
The author gratefully acknowledges the financial supportfrom the Macau Science and Technology Development Fund(Grant no FDCT0112013A1) and the University of MacauResearch Fund (Grants nos MYRG189(Y2-L3)-FST11-ZWHand MYRG067(Y2-L2)-FST12-ZWH)
References
[1] K Terzaghi Theoretical Soil Mechanics Wiley New York NYUSA 1943
[2] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941
[3] G E Blight Strength and consolidation characteristics of com-pacted soils [PhD thesis] University of London London UK1961
[4] R F Scott Principles of Soil Mechanics Addison Wesley Pub-lishing Company Boston Mass USA 1963
[5] L Barden ldquoConsolidation of compacted and unsaturated clayrdquoGeotechnique vol 15 no 3 pp 267ndash286 1965
[6] D G Fredlund and J U Hasan ldquoOne-dimensional consolida-tion theory unsaturated soilsrdquo Canadian Geotechnical Journalvol 16 no 3 pp 521ndash531 1979
[7] D G Fredlund andH Rahardjo Soil Mechanics for UnsaturatedSoils John Wiley and Son 1993
[8] A Qin D Sun L Yang and Y Weng ldquoA semi-analyticalsolution to consolidation of unsaturated soils with the freedrainage wellrdquo Computers and Geotechnics vol 37 no 7-8 pp867ndash875 2010
[9] W H Zhou L S Zhao and X B Li ldquoA simple analyticalsolution to one-dimensional consolidation for unsaturated soilrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics 2013
[10] M Malik and F Civan ldquoA comparative study of differentialquadrature and cubature methods vis-a-vis some conventionaltechniques in context of convection-diffusion-reaction prob-lemsrdquo Chemical Engineering Science vol 50 no 3 pp 531ndash5471995
[11] P Malekzadeh M R Golbahar Haghighi and M M AtashildquoOut-of-plane free vibration of functionally graded circularcurved beams in thermal environmentrdquo Composite Structuresvol 92 no 2 pp 541ndash552 2010
[12] P Malekzadeh M R G Haghighi and M M Atashi ldquoFreevibration analysis of elastically supported functionally radedannular plates subjected to thermal environmentrdquo Meccanicavol 46 no 5 pp 893ndash913 2011
[13] R P Chen W H Zhou H Z Wang and Y M Chen ldquoOne-dimensional nonlinear consolidation of multi-layered soil bydifferential quadrature methodrdquo Computers and Geotechnicsvol 32 no 5 pp 358ndash369 2005
[14] R P Chen W H Zhou H Z Wang and Y M Chen ldquoStudyon one-dimensional nonlinear consolidation of multi-layeredsoil using differential quadrature methodrdquo Chinese Journal ofComputational Mechanics vol 22 no 3 pp 310ndash315 2005
[15] H Z Wang R P Chen W H Zhou and Y M ChenldquoComputation of 1-D nonlinear consolidation in double-layer
foundation by using differential quadrature methodrdquo Journal ofHydraulic Engineering vol 4 pp 8ndash14 2004
[16] W H Zhou and L S Zhao ldquoOne-dimensional consolidationof unsaturated soil subjected to time-dependent loading undervarious initial and boundary conditionsrdquo ASCE InternationalJournal of Geomechanics 2013
[17] E Conte ldquoConsolidation analysis for unsaturated soilsrdquo Cana-dian Geotechnical Journal vol 41 no 4 pp 599ndash612 2004
[18] E Conte ldquoPlane strain and axially symmetric consolidation inunsaturated soilsrdquo International Journal of Geomechanics vol 6no 2 pp 131ndash135 2006
[19] R Bellman BG Kashef and J Casti ldquoDifferential quadrature atechnique for the rapid solution of nonlinear partial differentialequationsrdquo Journal of Computational Physics vol 10 no 1 pp40ndash52 1972
[20] R Bellman and J Casti ldquoDifferential quadrature and long-termintegrationrdquo Journal of Mathematical Analysis and Applicationsvol 34 no 2 pp 235ndash238 1971
[21] J R Quan andC T Chang ldquoNew insights in solving distributedsystem equations by the quadrature method I analysisrdquo Com-puters and Chemical Engineering vol 13 no 7 pp 779ndash7881989
