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Appl. Math. Mech. -Engl. Ed., 2008, 29(12):1617–1624DOI 10.1007/s10483-008-1209-9cShanghai University and Springer-Verlag 2008

Applied Mathematicsand Mechanics(English Edition)

A new analytical solution to axisymmetric Biot’s consolidation

of a finite soil layer ∗

AI Zhi-yong ( ), WANG Quan-sheng ( )

(Department of Geotechnical Engineering and Key Laboratory of Geotechnical and Underground

Engineering of Ministry of Education, Tongji University, Shanghai 200092, P. R. China)

(Communicated by GUO Xing-ming)

Abstract A new analytical method is presented to study the axisymmetric Biot’sconsolidation of a finite soil layer. Starting from the governing equations of axisymmetricBiot’s consolidation, and based on the property of Laplace transform, the relation of basic variables for a point of a finite soil layer is established between the ground surface(z = 0) and the depth z in the Laplace and Hankel transform domains. Combined withthe boundary conditions of the finite soil layer, the analytical solution of any point inthe transform domain can be obtained. The actual solution in the physical domain canbe obtained by inverse Laplace and Hankel transforms. A numerical analysis for theaxisymmetric consolidation of a finite soil layer is carried out.

Key words axisymmetric Biot’s consolidation, finite soil layer, Laplace transform,Hankel transform

Chinese Library Classification TU43, O3432000 Mathematics Subject Classification 74F10

Introduction

Biot[1] first developed a theory of three-dimensional consolidation which can consider thecoupling between the solid and the fluid in saturated soil. Many research on how to solve theBiot’s consolidation equations have been made afterwards. One of the most successful analyticaltechnique for solving Biot’s consolidation is the displacement function methods proposed byMcNamee and Gibson[2−3], as well as Schiffman and Fungaroli[4]. Numerical techniques such asthe finite element method[5], the boundary element method[6], and the finite layer method[7−9]

have been used to solve more complicated consolidation problems. In this study, a new analyticalmethod is proposed to solve axisymmetric Biot’s consolidation of a finite soil layer, which

can avoid introducing the displacement functions[2−4], and obtain the analytical solution intransform domain directly by using the technique of Hankel transform and Laplace transform.

∗ Received Jun. 15, 2008 / Revised Oct. 27, 2008Project supported by the National Natural Science Foundation of China (No. 50578121)Corresponding author AI Zhi-yong, E-mail: [email protected]

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1618 AI Zhi-yong and WANG Quan-sheng

1 Governing equations

The governing equations of axisymmetric Biot’s consolidation are[1]

∇2ur − 1

r2ur +

1

1 − 2ν

∂e

∂r −

1

G

∂σ

∂r = 0, (1a)

∇2uz + 11 − 2ν

∂e∂z

− 1G

∂σ∂z

= 0, (1b)

∂e

∂t =

k

γ w∇2σ, (1c)

where ur and uz are displacements in the r and z coordinate directions, respectively; e =∂ur∂r

+ urr

+ ∂uz∂z

is the dilatation; ∇2 = ∂ 2

∂r2 + 1

r∂ ∂r

+ ∂ 2

∂z2 is Laplacian operator in the cylindrical

coordinate system; G and ν are the shear modulus and Poisson’s ratio of the soil, respectively;k is the coefficient of the permeability; γ w is the unit weight of water; and σ is the excess porewater pressure (positive under compression). According to Darcy’s law, the flux Q in the zdirection is defined as Q = k

γ w

∂σ∂z

.The constitutive equations of axisymmetric Biot’s consolidation are

σr + σ = λe + 2Gεr, (2a)σθ + σ = λe + 2Gεθ, (2b)

σz + σ = λe + 2Gεz, (2c)

σrz = Gγ rz = E

2(1 + ν )γ rz , (2d)

where σr, σθ and σz are the normal total stresses acting on the plane normal to the r , θ andz coordinate directions, respectively; and tensile stress is taken as positive; σrz is the shearstress; εr = ∂ur

∂r and εz = ∂uz

∂z denote the normal strain components in the r and z coordinate

directions, respectively; γ rz = ∂ur∂z

+ ∂uz∂r

denotes the shear strain component; E is Young’smodulus.

2 The analytical solution of a finite homogeneous saturated soil layer

The Laplace transform and its inversion[10] are defined as

f (r,z,s) =

∞

0

f (r,z,t)e−stdt, f (r,z,t) = 1

2πi

γ +i∞γ −i∞

f (r,z,s)estds, (3)

in which, s denotes the Laplace transform parameter corresponding to the variable t.The mth order Hankel transform and its inversion[11] are defined as

f (ξ , z , s) =

∞

0

f (r,z,s)J m(ξr)rdr, f (r,z,s) =

∞

0

f (ξ , z , s)J m(ξr)ξ dξ, (4)

where J m(ξr) is the mth order Bessel function.

Applying the Laplace transform with respect to variable t to Eq. (1a) yields

∇2ur −

1

r2ur +

1

1 − 2ν

∂ e∂r −

1

G

∂ σ∂r

= 0, (5)

where ur is the corresponding variable of ur in the Laplace transform domain with respect tot, etc.

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A new analytical solution to axisymmetric Biot’s consolidation of a finite soil layer 1619

Taking the first order Hankel transform of Eq. (5), we have

d2ur

dz2 −

2(1 − ν )

(1 − 2ν )ξ 2ur −

1

(1 − 2ν )ξ

duz

dz +

1

Gξσ = 0, (6)

where ur is the corresponding variable of

ur in the Hankel transform domain, etc.

The Laplace transform and its inversion[10]

of z domain are defined as

f (ξ,p,s) =

∞

0

f (ξ , z , s)e− pzdz, f (ξ , z , s) = 1

2πi

γ +i∞γ −i∞

f (ξ,p,s)e pzdp, (7)

in which, p denotes the Laplace transform parameter corresponding to the variable z .Taking the Laplace transform of Eq. (6) with respect to variable z yields

p2 − 2(1 − ν )

(1 − 2ν )ξ 2 ur −

1

(1 − 2ν ) pξ uz +

1

Gξ σ = pur(0) + ur(0) −

1

(1 − 2ν )ξuz(0), (8)

where ur is the corresponding variable of ur in the Laplace transform domain of z , etc. ur(0)denotes differentiating with respect to z and has its value at z = 0. ur(0) and uz(0) are thevalues of ur and uz at z = 0, respectively.

Application of the Laplace transform with respect to t of Eq. (2d) yields

σrz = G

∂ ur

∂z +

∂ uz

∂r

. (9)

Taking the first order Hankel transform with respect to r of Eq. (9) yields

σrz = G

dur

∂z − ξuz

. (10)

Let z = 0, we have

ur(0) = ξuz(0) + 1

Gσrz(0). (11)

Substituting Eq. (11) into Eq. (8) yields p2 −

M

G ξ 2

ur − M − G

G pξ uz +

1

Gξ σ = pur(0) −

M − 2G

G ξuz(0) −

1

Gσrz(0), (12)

where M = E (1−ν )(1+ν )(1−2ν ) .

Based on Eq. (1b), the following equation can be obtained similarly:M

G p2

− ξ 2 uz +

M − G

G pξ ur −

1

G p σ = ξur(0) +

M

G puz(0) +

1

Gσz(0). (13)

Taking the Laplace transform with respect to t of Eq. (1c), and assuming that the initialdilatation e is zero everywhere, we have

s∂ ur

∂r +

ur

r +

∂ uz

∂z

=

k

γ w

∂ 2

∂r2 +

∂ 2

∂z2

σ. (14)

Taking the zero order Hankel transform with respect to r of Eq. (14) yields

d2σ

dz2 − ξ 2σ −

sγ wk

ξur − sγ w

k

duz

dz = 0. (15)

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Taking the Laplace transform with respect to z of Eq. (15) yields

p2 − ξ 2

σ − sγ w

k p uz −

sγ wk

ξ ur = pσ(0) + σ(0) − sγ w

k uz(0). (16)

Taking the Laplace transform with respect to t and the zero order Hankel transform withrespect to r of the flux Q yields

Q = kγ w

∂σ∂z

. (17)

Let z = 0, we have

σ(0) = γ w

k Q (0) . (18)

Substituting Eq. (18) into Eq. (16) yields

p2 − ξ 2

σ − sγ w

k p uz −

sγ wk

ξ ur = pσ(0) − sγ w

k uz(0) +

γ wk

Q (0) . (19)

Eqs. (12), (13) and (19) are three simultaneous equations in three variables of ur, uz and σ.Solving these equations and applying the inversion of the Laplace transform with respect to pleads to

ur(z) = Φ11ur(0) + Φ12uz(0) + Φ13σ(0) + Φ14σrz(0) + Φ15σz(0) + Φ16Q(0), (20)

uz(z) = Φ21ur(0) + Φ22uz(0) + Φ23σ(0) + Φ24σrz(0) + Φ25σz(0) + Φ26Q(0), (21)

σ(z) = Φ31ur(0) + Φ32uz(0) + Φ33σ(0) + Φ34σrz(0) + Φ35σz(0) + Φ36Q(0). (22)

From the Eqs. (2c), (10) and (17), and combined the Eqs. (20), (21) and (22), the followingequations can be obtained:

σrz(z) = Φ41ur(0) + Φ42uz(0) + Φ43σ(0) + Φ44σrz(0) + Φ45σz(0) + Φ46Q(0), (23)

σz(z) = Φ51ur(0) + Φ52uz(0) + Φ53σ(0) + Φ54σrz(0) + Φ55σz(0) + Φ56Q(0), (24)

Q(z) = Φ61ur(0) + Φ62uz(0) + Φ63σ(0) + Φ64σrz(0) + Φ65σz(0) + Φ66Q(0), (25)

in which, Φkj(k = 1, 2, · · ·6, j = 1, 2, · · · 6) are functions of ξ and s.Equations (20)–(25) can be expressed in form of matrix as follows:

B (ξ , z , s) = Φ(ξ , z , s)B (ξ, 0, s), (26)

in which, B (ξ , z , s) = [ur, uz, σ , σrz , σz, Q]Tz=z, Φ(ξ , z , s) is the transfer matrix of 6th order,which establishes the relationship between the ground surface (z = 0) and an arbitrary depthz in Laplace and Hankel transform domain. The elements of Φkj are listed in Appendix.

a a

q

h

z

r

Fig. 1 A finite homoge-neous saturatedsoil layer

Based on Eq. (26) and combined with boundary conditions,we can obtain the solutions for displacements, stresses, excesspore water pressure and flux of any point in the transform do-main.

As shown in Fig. 1, we consider a finite homogeneous satu-rated soil layer subjected to a uniform circular loading q , which

occupies the area 0 ≤ r ≤ a and may be supposed to havebeen applied at time t = 0+, and then held constant. A morecomplex loading case can be analyzed in the same way.

Taking the Laplace transform of the load with respect to tleads to q (r, 0, s) =

∞

0

q (r, 0, t)e−stdt = q

s. (27)

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Taking the zero order Hankel transform of the above equation yields

q (ξ, 0, s) =

∞

0

q

sJ 0(ξr)rdr =

qaJ 1(ξa)

ξs . (28)

Supposing the surface of the soil is permeable, then

σz(ξ, 0, s) = −q (ξ, 0, s), σrz(ξ, 0, s) = 0, σ(ξ, 0, s) = 0. (29)

If the base of a finite homogeneous saturated soil layer is fixed, there are two possibledrainage conditions as follows:

When the base is permeable, then

ur(ξ,h ,s) = 0, uz(ξ,h ,s) = 0, σ(ξ,h ,s) = 0. (30)

When the base is impermeable, then

ur(ξ,h ,s) = 0, uz(ξ,h ,s) = 0, Q(ξ,h ,s) = 0. (31)

All components of B (ξ, 0, s) and B (ξ,h ,s) can be derived from Eq. (26) combined with the

boundary conditions. So the solution for any point in the transform domain in the layer canbe expressed as

B (ξ , z , s) = Φ(ξ, z − h, s) ·B (ξ,h ,s). (32)

Equation (32) is the solution of axisymmetric consolidation of a finite homogeneous saturatedsoil layer in the transform domain.

3 Numerical results

In order to obtain the actual solutions in the physical domain, we take the inversion of Laplace transform and Hankel transform to the variables with respect to s and ξ in the trans-fom domain, respectively. In this study, the inverse of Laplace transform is obtained by thenumerical scheme proposed by Talbot[10], because the feasibility and efficiency of this scheme

for the consolidation problems had been successfully demonstrated by Booker and Small

[7–9]

.The inverse of Hankel transform is obtained by the technique which can be found in the pa-per published by Ai et al.[12]. For the convenience of analysis, the dimensionless parametersc = 2Gk/γ w and τ = ct/a2 are introduced, and all the results are obtained by considering thesurface of the finite homogeneous saturated soil layer is permeable and the base of the soil isfixed and impermeable.

To verify the reasonableness and accuracy of the proposed solutions from this study, thenumerical results for the axisymmetric consolidation of a finite homogeneous saturated soillayer are presented to validate against the existing results suggested by Booker and Small[9].The parameters and results of calculation are shown in Fig. 2. The comparisons in Fig. 2 showthat the results obtained using the solution from this study are in good agreement with thoseby Booker and Small[9].

In this study, we consider the influence of Poisson’s ratio ν on the vertical displacement and

excess pore water pressure due to axisymmetric consolidation. The parameters and results areschematically in Fig. 2 and Fig. 3. As shown in Fig. 2 and Fig. 3, the reduction of Poisson’sratio ν ( ν from 0.49 to 0.0) increases the vertical displacement uz of the soil, and decreasesthe dissipation of excess pore water pressure.

The distribution of excess pore water pressures within a finite homogeneous saturated soillayer along the centerline (r/a = 0) at different time factor τ due to axisymmetric consolidationis considered in this study. The parameters and results are shown schematically in Fig. 4. It

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is shown that the excess pore water pressure decreases with the increase of time and increaseswith the increase of depth z .

Meantime, the influence of thickness of the finite soil layer on excess pore water pressuredue to axisymmetric consolidation is considered in this study. The parameters and results areshown schematically in Fig. 5. It is shown that the excess pore pressure decreases with theincrease of thickness h, and there exists a peak value when thickness h increases.

0.1

0.2

0.3

0.4

0.50.001 0.01 0.1 1 10

τ

r /a = 0 z /a = 0h/a = 1

The authors

Booker and Small [9]

0.49

0.4

0.3

0.2

0.1

0.0

G u z

a q

Fig. 2 Influence of Poisson’s ratio ν on surfacedisplacement of a finite soil layer

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

τ =0.1

r /a=0

h/a=1

0

0.2

0.4

0.6

0.8

1.0

z / a

σ /q

0.49 0.4 0.3 0.2 0.1 0.0

Fig. 3 Influence of Poisson’s ratio ν on excesspore water pressure of a finite soil

0

0.2

0.4

0.6

0.8

1.00 0.2 0.4 0.6 0.8

σ /q

z / a

v=0.3

τ =0.2τ =0.1

τ =0.055

τ =0.01

h/a=1

r /a=0

Fig. 4 Variation of excess pore water pres-sure with time of a finite soil layer

0 0.1 0.2 0.3 0.4 0.5

σ /q

z / h h/a=5

h/a=2

h/a=1

v=0.3

r /a=0

τ =ct /a2=0.1h/a=10

0

0.2

0.4

0.6

0.8

1.0

Fig. 5 Influence of thickness h on excess porewater pressure of a finite soil layer

4 Conclusions

In this paper, a new method is developed for solving axisymmetric Biot’s consolidation of afinite soil layer. Starting from the governing equations of axisymmetric Biot’s consolidation, andbased on the property of Laplace transform, the relation of displacements, stresses, excess porewater pressure, and flux for a point of a finite soil layer is established at between the groundsurface (z = 0) and the depth z in the Laplace and Hankel transform domain. Combined

with the boundary conditions of the finite soil layer, the analytical solutions for displacements,stresses, excess pore water pressure and flux of any point in the transform domain can beobtained. The actual solutions in the physical domain can be obtained by inverse Laplaceand Hankel transforms. The numerical analysis for the axisymmetric consolidation of a finitehomogeneous saturated soil layer is carried out. The results of calculation show that the methodproposed in this study is accurate and efficient to solve the consolidation problem. Meanwhile,the influence of Poisson’s ratio ν on the progress of settlement and the excess pore water

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pressure, the distribution of excess pore water pressures with time, and the influence of soilthickness on excess pore pressure are examined in this paper. This method presented in thispaper can be extended to the Biot’s consolidation of a multilayered soil by using the transfermatrix concept[12−13]. Additionally, the analytical technique presented in this paper can beused to solve more complicated Biot’s consolidation problems, such as the Biot’s consolidationwith anisotropic permeability and the Biot’s consolidation with compressible constituents, in

those problems it is difficult to obtain the displacement functions, the relative research resultswill be reported by the authors in the next future.

References

[1] Biot M A. General theory of three-dimensional consolidation[J]. Journal of Applied Physics , 1941,12(2):155–164.

[2] McNamee J, Gibson R E. Displacement functions and linear transforms applied to diffusionthrough porous elastic media[J]. Quarterly Journal of Mechanics and Applied Mathematics , 1960,13(1): 98–111.

[3] McNamee J, Gibson R E. Plane strain and axially symmetric problem of the consolidation of a semi-infinite clay stratum[J]. Quarterly Journal of Mechanics and Applied Mathematics , 1960,13(2):210–227.

[4] Schiffman R L, Fungaroli A A. Consolidation due to tangential loads[C]. In: Proceedings of the

6th International Conference on Soil Mechanics and Foundation Engineering , Vol 1. Montreal,Canada, 1965, 188–192.

[5] Christian J T, Boehmer J W. Plane strain consolidation by finite elements[J]. Journal of the Soil

Mechanics and Foundations Division, ASCE , 1970, 96(4):1435–1457.

[6] Cheng A H D, Liggett J A. Boundary integral equation method for linear porous-elasticity withapplications to soil consolidation[J]. International Journal for Numerical Methods in Engineering ,1984, 20(2):255–278.

[7] Booker J R, Small J C. Finite layer analysis of consolidation I[J]. International Journal for Nu-

merical and Analytical Methods in Geomechanics , 1982, 6(2):151–171.

[8] Booker J R, Small J C. Finite layer analysis of consolidation II[J]. International Journal for

Numerical and Analytical Methods in Geomechanics , 1982, 6(2):173–194.

[9] Booker J R, Small J C. A method of computing the consolidation behavior of layered soils us-ing direct numerical inversion of Laplace transforms[J]. International Journal for Numerical and

Analytical Methods in Geomechanics , 1987, 11(4):363–380.

[10] Talbot A. The accurate numerical inversion of Laplace transforms[J]. Journal of Institute of Math-

ematics and Its Application , 1979, 23(1):97–120.

[11] Sneddon I N. The use of integral transform[M]. New York: McGraw-Hill, 1972.

[12] Ai Z Y, Yue Z Q, Tham L G, Yang M. Extended Sneddon and Muki solutions for multilayeredelastic materials[J]. International Journal of Engineering Science , 2002, 40(13):1453–1483.

[13] Ai Z Y, Cheng Z Y, Han J. State space solution to three-dimensional consolidation of multi-layeredsoils[J]. International Journal of Engineering Science , 2008, 46(5):486–498.

Appendix A

Φ11 = 2Gξ2C

Ms (chξz − chqz ) + (chξz + ξzshξz ) = Φ44,

Φ12 = 2Gξ2C

Ms (shξz −

ξ

q shqz ) + ξzchξz = −Φ54,

Φ13 = Cξ

Ms(chξz − chqz ) = −

ξC Φ26

M = −

C Φ31

2GMs = −

ξC Φ35

Ms =

C Φ46

2GM = −

Φ53

2Gξ = −

Φ62

2Gξs = −

Φ64

s ,

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Φ14 = Cξ

Ms(shξz −

ξ

q shqz ) +

1

2Gξ(shξz + ξzchξz ),

Φ15 = Cξ

Ms(chξz − chqz ) +

1

2Gξξzshξz =

Φ42

4G2ξ2 = −

Φ51

4G2ξ2 = −Φ24,

Φ16 = 1

s(shξz −

ξ

q shqz ) = −

Φ32

2Gξs = −

Φ34

s = −

Φ56

2Gξ,

Φ21 = 2GξC Ms (qshqz − ξshξz )− ξzchξz = −Φ45,

Φ22 = 2Gξ2C

Ms (chqz − chξz ) + (chξz − ξzshξz ) = Φ55,

Φ23 = C

Ms(qshqz − ξshξz ) = −

Φ43

2Gξ =

Φ61

2Gξs =

Φ65

s ,

Φ25 = C

Ms(qshqz − ξshξz ) +

1

2Gξ(shξz − ξzchξz ),

Φ33 = chqz = Φ66,

Φ36 = M

qC shqz =

M 2Φ63

q 2C 2 ,

Φ41 = 2G[2Gξ2C

Ms (ξshξz − qshqz ) + ξ(shξz + ξzchξz )],

Φ52 = 2G[ 2Gξ3C Ms

( ξq shqz − shξz ) + ξ(shξz − ξzchξz )],

where C = 2Gk(1−ν)γ w(1−2ν)

, q =p ξ2 + s/C .

s-866

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