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Analytical solution for the 1D consolidation of unsaturated multi-layered soil
Computers and Geotechnics 57 (2014) 17–23
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Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Analytical solution for the 1D consolidation of unsaturated multi-layered soil Zhendong Shan a, Daosheng Ling b,⇑, Haojiang Ding b a b
Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, PR China MOE Key Lab of Soft Soils and Geo-Environmental Engineering, Zhejiang University, Hangzhou 310058, PR China
a r t i c l e
i n f o
Article history: Received 15 April 2013 Received in revised form 10 November 2013 Accepted 30 November 2013
Keywords: Analytical solution Unsaturated soil Multi-layered soil Consolidation Arbitrary load
a b s t r a c t The 1D consolidation of unsaturated multi-layered soil is studied based on the theory proposed by Fredlund and Hasan, and an analytical solution for a typical boundary condition is obtained by assuming all material parameters remain constant during consolidation. In the derivation of the analytical solution, the eigenfunction and eigenvalue for the multi-layered problem are first derived through the transfer matrix method. Then, by using the method of undetermined coefficients and the orthogonal relation of the eigenfunction, the analytical solution is obtained. The present method is applicable to various types of boundary conditions. Finally, numerical examples are provided to investigate the consolidation behavior of unsaturated multi-layered soil. Ó 2014 Published by Elsevier Ltd.
1. Introduction Unsaturated soil is a three-phase system which contains solid particles, water, and air. The compressibility and seepage of unsaturated soil are more complex than those of saturated soil. Many different types of theories have been proposed to describe the behavior of unsaturated soil, such as Biot [1], Blight [2], Barden [3,4], and Fredlund and Hasan [5]. Fredlund and Hasan [5] is now widely accepted, in which two partial differential equations are used to describe the dissipation process of excess pore pressures in unsaturated soil. This theory was extended to the 3D case by Dakshanamurthy et al. [6]. As material properties are functions of the stress state in unsaturated soil, the consolidation problems are usually solved via integral transform method or numerical methods involving the discretisation of both spatial and temporal domains [7–10]. Assuming all the soil parameters remain constant during consolidation, Fredlund and Rahardjo [11] presented 1D consolidation equations in the form of linear equations. These simplified consolidation equations may not be practical (parameters remain constant during consolidation), but the solution for these equations can give a preliminary description of the dissipation laws of the
⇑ Corresponding author. Address: Anzong Building A421, Zijingang Campus, Zhejiang University, Hangzhou, Zhejiang Province 310058, PR China. Tel.: +86 571 88208756; fax: +86 571 88208793. E-mail addresses: [email protected] (Z. Shan), [email protected] (D. Ling), [email protected] (H. Ding). 0266-352X/$ - see front matter Ó 2014 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.compgeo.2013.11.009
air and water pressures, and enable the study of the evolution law of the volume change of unsaturated soil. Using the simplified consolidation equations, several analytical solutions for the 1D consolidation of unsaturated soil have been published. Qin et al. [12,13] adopted the Laplace transform method and gave analytical solutions for unsaturated single-layer soil subjected to step and exponentially loads, respectively. By using the Laplace transform and Bessel function, Qin et al. [14] presented a semi-analytical solution for unsaturated single-layer soil with the free drainage well. Using the separation of variables method, Shan et al. [15] obtained the exact solutions for unsaturated single-layer soil subjected to arbitrary loads with three types of boundary conditions. Shan et al. [16] obtained an analytical solution for unsaturated single-layer soil subjected to an arbitrary load with mixed boundary condition using the modal expansion method. These analytical solutions are useful to validate the numerical results of consolidation in unsaturated soil. The above mentioned analytical solutions are for unsaturated single-layer soil, but the soil stratum in real engineering site is not always layered one. This paper presents an analytical solution for unsaturated multi-layered soil with a typical boundary condition subjected to an arbitrary load. In Section 2, the consolidation equations for unsaturated multi-layered soil are outlined. The associated initial, boundary, and interface continuity conditions are discussed in Section 3. In Section 4, the derivation of the analytical solution are presented in detail. In Section 5, several illustrative examples are given to illustrate some interesting features of the consolidation in unsaturated multi-layered soil.
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Z. Shan et al. / Computers and Geotechnics 57 (2014) 17–23
2. Consolidation equations for an unsaturated multi-layered soil The 1D consolidation equations developed by Fredlund and Hasan [5] in the Cartesian coordinate system are used to study the consolidation of unsaturated soil. The consolidation equations for unsaturated multi-layered soil (Fig. 1) can be written as:
ð1Þ þms2
where
( u C
ðiÞ
¼
ðiÞ
uw
) ;
ðiÞ ua
¼
" K
ðiÞ
1
Cw
Cw
C w =C ðiÞ a
¼
;
C wðiÞ v 0
Q
ðiÞ
¼
0
# ;
C aðiÞ C w =C aðiÞ (v cw r q;t ðtÞ
) ð2Þ
aðiÞ
cr q;t ðtÞC w =C aðiÞ ðiÞ
ðiÞ
where superscript (i) denotes the ith layer; uw and ua are the water and air pressures of the ith layer; r is the total stress mentioned later; (),z and (),t denote the derivatives with respect to z and t, respectively. The parameters in Eq. (2) can be written as
w w C w ¼ mw 1k m2 =m2 ; C ðiÞ a ¼
ðiÞ w C wðiÞ v ¼ kw = cw m2 ;
w w Cw r ¼ m1k =m2
ma2 =ma1k ; ðiÞ ðiÞ 1 ma2 =ma1k n0 1 S0 = ua ma1k
C aðiÞ r ¼
C aðiÞ v ¼
ðiÞ ka
h i ; ðiÞ a 1 ma2 =ma1k nðiÞ a ðxa =RT Þgma1k u 0 1 S0 = ua m1k ðiÞ
ðiÞ
Z
hi hi1
hi1
ðiÞ ðiÞ ðiÞ ua ðz;tÞ uw ðz;tÞ uaðiÞ ðz;0Þ uw ðz;0Þ dz
) ð4Þ
where ms1k and ms2 are the coefficients of the soil volume changes, due to the net normal stress (r ua) and the matric suction a (ua uw), respectively, which can be expressed as ms1k ¼ mw 1k þ m1k a and ms2 ¼ mw þ m . Note that, when the water and air pressures 2 2 are totally dissipated, the compression of unsaturated multi-layered soil becomes rðtÞms1k H. 3. Initial, boundary, and interface continuity conditions To address the consolidation of an unsaturated multi-layered soil, associated initial and boundary conditions and interface continuity conditions should be imposed. The following initial conditions are used
uðiÞ ðz; 0Þ ¼ g ðiÞ ðzÞ; n
ði ¼ 1; 2; . . . ; nÞ;
ðiÞ ðiÞ g 1 ðzÞ; g 2 ðzÞ
oT
ð5Þ ðiÞ g 1 ðzÞ
ðiÞ g 2 ðzÞ
where g ðiÞ ðzÞ ¼ , with and being arbitrary functions. The following boundary conditions are considered
1 ðiÞ ðiÞ 1 ma2 =ma1k n0 1 S0 = ua ma1k
ðiÞ
hi1
i¼1
ðiÞ
ðiÞ ðiÞ K ðiÞ uðiÞ ;zz þ C u;t ¼ Q
ðiÞ
pressure. When ua is small or dissipates rapidly in the process of a can be considered to be constant [8], and let consolidation, u a ¼ uatm in this study. u The compression of the multi-layered soil can be written as [15,17]: ( Z hi hi1 n X
SðtÞ ¼ ms1k rðtÞ uaðiÞ ðz; tÞ rð0Þ uðiÞ a ðz;0Þ dz
ð1Þ ðnÞ ðnÞ uð1Þ w ð0;tÞ ¼ 0; ua ð0; tÞ ¼ 0; uw;z ðhn ; tÞ ¼ 0; ua;z ðhn ;tÞ ¼ 0;
ð3Þ
ðiÞ
where kw , ka , S0 , and n0 are different for different layers. However, w a a mw 1k , m2 , m1k , m2 , cw, xa, R, T, and uatm are parameters of water and air and do not change with depth. ðiÞ ðiÞ In Eq. (3), kw and ka are the water and air permeability coeffiðiÞ ðiÞ cients of the ith layer, respectively; S0 and n0 represent the saturation of water and porosity of the ith layer, respectively. mw 1k and ma1k are the coefficients of the water and air volume changes, respectively, due to the net normal stress (r ua) for a K0-load a condition; mw 2 and m2 are the coefficients of the water and air volume changes, respectively, due to the matric suction (ua uw)cw is the density of water. xa is the molecular mass of air, R is the universal gas constant, T is the absolute temperature, g is the gravity a is the absolute pore-air pressure which acceleration. In addition, u a ¼ ua þ u atm , where u atm is the atmospheric can be expressed as u
ð6Þ
which indicate that the top surface of the multi-layered soil is permeable to water and air, while the bottom surface is impermeable to water and air. The interface continuity conditions between layers are specified as
uðiÞ ðhi ; tÞ ¼ uðiþ1Þ ðhi ; tÞ;
wðiÞ ðhi ; tÞ ¼ wðiþ1Þ ðhi ; tÞ;
ði ¼ 1; 2; . . . ; n 1Þ;
ð7Þ
n oT ðiÞ ðiÞ ðiÞ ðiÞ where w(i)(hi, t) denotes flow, and wðiÞ ðz; tÞ ¼ kw uw;z ; ka ua;z . The first equation in Eq. (7) denotes the pressure continuity and the second equation in Eq. (7) denotes the flow continuity. 4. Derivation of the analytical solution The method of Shan et al. [16] can address the consolidation of unsaturated single-layer soil with various types of boundary conditions. By incorporating the transfer matrix method to the method of Shan et al. [16], the present method can be used to solve the consolidation of unsaturated multi-layered soil. The solution procedure is illustrated below. 4.1. Eigenvalue and eigenfunction To solve the nonhomogeneous partial differential equation (1) with initial conditions (5), boundary conditions (6), and interface continuity conditions (7), the following characteristic equation is first considered to derive the corresponding eigenvalue and eigenfunction: ðiÞ
ðiÞ K ðiÞ u;zz þ C ðiÞ u;t ¼ 0:
ð8Þ
The solution to Eq. (8) takes the form: Fig. 1. An unsaturated multi-layered soil subjected to an external load. (hi is the depth from the surface to the bottom of the ith layer, and h0 = 0; n is the number of layers; H is the total thickness; q(t) is the external load).
uðiÞ ¼ U ðiÞ ðzÞ expðx2 tÞ;
ð9Þ
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Z. Shan et al. / Computers and Geotechnics 57 (2014) 17–23
where x is the eigenvalue which is a non-negative real number, and
( U ðiÞ ðzÞ ¼
)
ðiÞ Uw ðzÞ ðiÞ U a ðzÞ
( W ðiÞ ðzÞ ¼
;
W ðiÞ w ðzÞ W ðiÞ a ðzÞ
)
1
¼ ½M ðiÞ U ;zðiÞ ðzÞ
M ðiÞ ¼
ðiÞ 1=kw
ð10Þ
1=ka
By substituting Eq. (9) into Eq. (8), boundary conditions (6), and interface continuity conditions (7), we obtain 2 ðiÞ ðiÞ K ðiÞ U ðiÞ ;zz ðzÞ x C U ðzÞ ¼ 0;
U ð1Þ ð0Þ ¼ 0;
ði ¼ 1; 2; . . . ; nÞ
ð12Þ
W ðnÞ ðhn Þ ¼ 0
U ðiÞ ðhi Þ ¼ U ðiþ1Þ ðhi Þ;
ð13Þ
W ðiÞ ðhi Þ ¼ W ðiþ1Þ ðhi Þ
ð14Þ
The solution of Eq. (12) under Eqs. (13) and (14) will be described when x = 0 and x > 0. When x = 0, Eq. (12) can be simplified as: ðiÞ
K ðiÞ U 0;zz ðzÞ ¼ 0;
ði ¼ 1; 2; :::; nÞ;
ð15Þ
where subscript 0 means x = 0. As the determinant of matrix K(i) is not equal to zero, the general solution of Eq. (15) is as follows: ðiÞ
ðiÞ
ðiÞ
U 0 ðzÞ ¼ U 0 ðhi1 Þ þ U 0;z ðhi1 Þðz hi1 Þ:
ð16Þ
Eqs. (14) and (16) can be rewritten as follows: ðiÞ
ði1Þ
X 0 ðhi1 Þ ¼ X 0 ðiÞ
ðiÞ
ðhi1 Þ
ð17Þ
ðiÞ
X 0 ðzÞ ¼ T 0 ðzÞX 0 ðhi1 Þ;
ð18Þ
where T i0 ðzÞ is usually called the transfer matrix, and
(
ðiÞ X0
¼
)
ðiÞ
U 0 ðzÞ ðiÞ
W 0 ðzÞ
"
;
ðiÞ T 0 ðzÞ
¼
I
ðz hi1 ÞM
0 I
ðiÞ
;
ðiÞ
ðiÞ
X 0 ðhi Þ ¼ T 0 ðhi ÞX 0 ðhi1 Þ:
ð19Þ
ð20Þ
Repeatedly making use of interface continuity conditions (17) and Eq. (20) and (18) can be rewritten as: ðiÞ X 0 ðzÞ
¼
ðiÞ ð1Þ V 0 ðzÞX 0 ð0Þ;
ð21Þ
ðiÞ
ðiÞ
V 0 ðzÞ ¼ T 0 ðzÞ
" ðpÞ
T 0 ðhp Þ ¼
p¼i1
ðiÞ
N 0 ðzÞ ¼ ðz hi1 ÞM ðiÞ þ
I 0
ðiÞ N 0 ðzÞ
#
I
i1 X ðhp hp1 ÞM ðiÞ :
ð22Þ
¼
ðnÞ X 0 ðhn Þ
ð23Þ
ð1Þ X 0 ð0Þ
Both and contains four variables, four of them are shown in Eq. (13), other four unknowns can be determined through Eq. (23). Then, using Eqs. (21) and (19), we obtain ðiÞ
U 0 ðzÞ ¼ 0;
ði ¼ 1; 2; . . . ; nÞ;
M ðiÞ
x2 LðiÞ
0
2
# ;
ðiÞ
L
ðiÞ
kw =C wðiÞ v 6 ðiÞ ðiÞ aðiÞ ¼ 4 k C =C a a v
ðiÞ
3
ðiÞ
7 5:
kw C w =C wðiÞ v ka =C aðiÞ v
ð26Þ
According to the Hamilton–Cayley law, exp[N(i)(z hi1)] can be expressed as
h i ðiÞ ðiÞ exp N ðiÞ ðz hi1 Þ ¼ a0 ðz hi1 ÞI þ a1 ðz hi1 ÞN ðiÞ 2 ðiÞ ðiÞ þ a2 ðz hi1 Þ N ðiÞ þ a3 ðz 3 hi1 Þ N ðiÞ
ð27Þ
ðiÞ
where ak ðz hi1 Þ(k = 0, 1, 2, 3) can be obtained by solving the following equations
3 X
akðiÞ ðz hi1 Þ rðiÞ j
k
h i ðiÞ ¼ exp r j ðz hi1 Þ ;
ðj ¼ 1; 2; 3; 4Þ;
ð28Þ
k¼0 ðiÞ
where r j (j = 1, 2, 3, 4) are distinct eigenvalue of matrix N(i), which satisfy the following characteristic equation in r(i)
jN ðiÞ rIj ¼ 0:
ð29Þ
It should be note that if Eq. (29) has multiple roots, then (28) has to take a different form [18]. Similarly, by using the transfer matrix method, Eq. (25) can be rewritten as:
ð30Þ
where
V ðiÞ ðzÞ ¼ T ðiÞ ðzÞ
1 Y
T ðpÞ ðhp Þ:
ð31Þ
p¼i1
Setting i = n and z = hn in Eq. (30), yields
X ðnÞ ðhn Þ ¼ V ðnÞ ðhn ÞX ð1Þ ð0Þ:
ð32Þ
According to the four boundary condition (13) at z = 0 and z = hn, we can solve for the four unknowns from Eq. (32). Substituting Eq. (13) into Eq. (32), we have
#
mðnÞ mðnÞ 33 ðhn Þ 34 ðhn Þ W ð1Þ ð0Þ ¼ 0: ðnÞ m43 ðhn Þ mðnÞ 44 ðhn Þ
ð33Þ
where mpq ðzÞ is the element of matrix V(i)(z). If, and only if, the determinant of the coefficient matrix is equal to zero, Eq. (33) has a nonzero solution, and thus we obtain the characteristic equation of x as follows: ðnÞ ðnÞ ðnÞ mðnÞ 33 ðhn Þm44 ðhn Þ m43 ðhn Þm34 ðhn Þ ¼ 0:
Setting i = n and z = hn in Eq. (21), yields ðnÞ ð1Þ V 0 ðhn ÞX 0 ð0Þ:
¼
0
ðiÞ
p¼1
ðnÞ X 0 ðhn Þ
N
"
where 1 Y
" ðiÞ
X ðiÞ ðzÞ ¼ V ðiÞ ðzÞX ð1Þ ð0Þ;
#
with I being a two-by-two identity matrix. Setting z = hi in Eq. (18), yields ðiÞ
n oT ðiÞ ðiÞ X ðiÞ ðzÞ ¼ U w ðzÞ; U aðiÞ ðzÞ; W w ðzÞ; W ðiÞ ; a ðzÞ
ð11Þ
:
ðiÞ
0
where T (z) is the transfer matrix, and
T ðiÞ ðzÞ ¼ exp½N ðiÞ ðz hi1 Þ
#
0
ð25Þ
(i)
where
"
X ðiÞ ðzÞ ¼ T ðiÞ ðzÞX ðiÞ ðhi1 Þ;
ð24Þ
which should be omitted. When x > 0, by using the state-space method, the solution to Eq. (12) can be written as:
ð34Þ
This is a transcendental equation which has infinite positive roots labeled xk (k = 1, 2, 3,. . .) from smallest to largest, respectively. Using Eq. (33), a nonzero solution of W(1)(0) can be obtained as follows: ð1Þ
W k ð0Þ ¼
n
oT
ðnÞ mðnÞ mk33 ðhn Þ ; k34 ðhn Þ;
ð35Þ
where the subscript k indicates eigenvalue xk. Considering Eqs. (30) and (35), the eigenfunction can be written as
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Z. Shan et al. / Computers and Geotechnics 57 (2014) 17–23
"
ðiÞ ðiÞ mk13 ðzÞ mk14 ðzÞ ðiÞ ðiÞ mk23 ðzÞ mk24 ðzÞ
ðiÞ
U k ðzÞ ¼
#(
)
mðnÞ k34 ðhn Þ ; ðnÞ mk33 ðhn Þ
ð36Þ
n Z X
h
hi
hi1
i¼1
iT 0 ðiÞ ðiÞ U ðiÞ q ðzÞ C U p ðzÞdz ¼ Gp
p–q p¼q
;
ð44Þ
which indicates that eigenfunction U pðiÞ ðzÞ is orthogonal to eigenfunction U qðiÞ ðzÞ with respect to the weight matrix C(i).
where i = 1, 2,. . ., n and k = 1, 2, 3,. . . 4.2. Orthogonality of the eigenfunction In this section, we demonstrate the orthogonality of the eigenfunction, which will be used in the next section. Let xp and xq be ðiÞ eigenvalue and U ðiÞ p ðzÞ and U q ðzÞ be the corresponding eigenfunction. Using Eq. (8), the following two equations are obtained:
4.3. Analytical solution
h
iT h iT ðiÞ U ðiÞ K ðiÞ U p;zz ðzÞ x2 U ðiÞ C ðiÞ U ðiÞ q q p ðzÞ ¼ 0
ð37aÞ
As the last step, Eq. (1) is to be solved. By using the method of undetermined coefficients and the principle of linear superposition, the solution of Eq. (1) subjected to initial condition (5), boundary conditions (6), and interface continuity conditions (7) can be written as:
h
iT h iT ðiÞ U ðiÞ K ðiÞ U q;zz ðzÞ x2 U ðiÞ C ðiÞ U ðiÞ p p q ðzÞ ¼ 0:
ð37bÞ
uðiÞ ðz; tÞ ¼
1 X ðiÞ U k ðzÞXk ðtÞ;
ð45Þ
k¼1
By subtracting Eq. (37b) from Eq. (37a), integrating Eq. (37a) with respect to z from 0 to H, then using the symmetry of matrix C, we obtain
Z
hi
h
hi1
x
iT h iT ðiÞ ðiÞ ðiÞ 2 U qðiÞ ðzÞ K ðiÞ U ðiÞ p;zz ðzÞ U p ðzÞ K U q;zz ðzÞ dz ðxp
2 qÞ
Z
h
hi
hi1
U qðiÞ ðzÞ
iT
C
ðiÞ
K ðiÞ
U pðiÞ ðzÞdz ð38Þ
By using the method of integration by parts and the symmetry of matrix K(i), Eq. (38) can be rewritten as 2 p
x x
2 q
Z
h
hi
hi1
iT
U qðiÞ ðzÞ
C
ðiÞ
U pðiÞ ðzÞdz
¼
n h X
i¼1
hi
h
hi1
ð39Þ
iT U qðiÞ ðzÞ C ðiÞ U pðiÞ ðzÞdz
hi iT h iT ðiÞ ðiÞ ðiÞ U qðiÞ ðzÞ K ðiÞ U p;z ðzÞ U ðiÞ ðzÞ K U ðzÞ p q;z (i)
According to Eqs. (2), (3), and (10), matrix K two matrixes, i.e.:
E¼
a
0
;
0 b
a¼
1
cw mw2
;
b¼
ð41Þ
C w RT : a mw u 2 xa
¼
i¼1
EW pðiÞ ðzÞ
h
U pðiÞ ðzÞ
iT
hi1
h
hi
hi1
iT ðiÞ U pðiÞ ðzÞ C ðiÞ U k ðzÞdz
iT
U pðiÞ ðzÞ Q ðiÞ dz
ð49Þ
Using the orthogonality of U ðiÞ p ðzÞ as shown in Eq. (44), Eq. (49) can be simplified as
Xp;t ðtÞ þ x2p Xp ðtÞ ¼ Sp ðtÞ
ð50Þ
Sp ðtÞ ¼
n 1 X Gp i¼1
Z
hi
h
hi1
iT U pðiÞ ðzÞ Q ðiÞ dz
ð51Þ
Similarly, using the orthogonal relation in Eq. (44), Eq. (47) can be simplified as
hi
EW qðiÞ ðzÞ
Xp ð0Þ ¼
hi1
h iT h iT h iT ðnÞ ð1Þ ð1Þ ¼ U qðnÞ ðhn Þ EW pðnÞ ðhn Þ U ðnÞ p ðhn Þ EW q ðhn Þ U q ð0Þ EW p ð0Þ h iT ð1Þ þ U ð1Þ p ð0Þ EW q ð0Þ n o ðnÞ ðnÞ ðnÞ ðnÞ ð1Þ ð1Þ ð1Þ ðhn ÞW wp ðhn Þ U wp ðhn ÞW wq ðhn Þ U ð1Þ ¼ U wq wq ð0ÞW wp ð0Þ þ U wp ð0ÞW wq ð0Þ a n o ðnÞ ðnÞ ðnÞ ð1Þ ð1Þ ð1Þ ð1Þ þ U ðnÞ aq ðhn ÞW ap ðhn Þ U ap ðhn ÞW aq ðhn Þ U aq ð0ÞW ap ð0Þ þ U ap ð0ÞW aq ð0Þ b
Z
i¼1
h
hi
hi1
iT
U qðiÞ ðzÞ
n X
where
hi n h iT h iT X ðiÞ ðiÞ ðiÞ U qðiÞ ðzÞ K ðiÞ U p;z ðzÞ U ðiÞ p ðzÞ K U q;z ðzÞ n h X
¼
n Z X
ð42Þ
Considering Eqs. (10) and (41), and the interface continuity conditions (14), the right hand of Eq. (40) can be rewritten as
i¼1
T
k¼1
ð40Þ
is shown in Eq. (11), and
ð48Þ
Pre-multiplying Eq. (48) by ½U pðiÞ ðzÞ , integrating Eq. (48) with respect to z from hi1 to hi, and then summing of Eq. (48) from the 1st layer to nth layer, we obtain
x2k Xk ðtÞ þ Xk;t ðtÞ
can be split into
1
where M
i¼1
K ðiÞ ¼ E½M ðiÞ ; E ¼ M ðiÞ K ðiÞ ¼ K ðiÞ M ðiÞ ; (i)
ð47Þ
Considering Eq. (12), Eq. (46) can be rewritten as
1 X
hi1
i¼1
ði ¼ 1; 2; . . . ; nÞ;
k¼1
k¼1
Summing from the 1st layer to the nth layer, Eq. (39) becomes: n Z X
1 X ðiÞ U k ðzÞXk ð0Þ ¼ g ðiÞ ðzÞ;
x2k Xk ðtÞ þ Xk;t ðtÞ C ðiÞ U kðiÞ ðzÞ ¼ Q ðiÞ
hi1
x2p x2q
ð46Þ
k¼1
1 X
h
hi iT h iT ðiÞ ðiÞ ðiÞ ¼ U qðiÞ ðzÞ K ðiÞ U ðiÞ p;z ðzÞ U p ðzÞ K U q;z ðzÞ
1 1 X X ðiÞ ðiÞ U k;zz ðzÞXk ðtÞ þ C ðiÞ U k ðzÞXk;t ðtÞ ¼ Q ðiÞ k¼1
¼0
ðiÞ
where U k ðzÞ are shown as Eq. (36). Therefore, if the scalar function Xk(t) are known, the air and water pressures for each layer can be obtained through Eq. (45), directly. Substituting Eq. (45) into Eqs. (1) and (5), we have
n 1 X Gp i¼1
Z
hi
hi1
h
iT ðiÞ ðiÞ U ðiÞ p ðzÞ C g ðzÞdz
ð52Þ
The solution of Eq. (50) with initial condition (52) can be written as 2
2
Xp ðtÞ ¼ exp t Xp ð0Þ þ exp t
Z
t
2
exp n Sp ðnÞdn
ð53Þ
0
ð43Þ
Substituting boundary conditions (13) into Eq. (43), we know Eq. (43) equals zero. Eq. (40) can thus be simplified as follows:
Substituting this expression and (36) into Eq. (45), we obtain the analytical solution of Eq. (1) with initial condition (5), boundary conditions (6), and interface continuity conditions (7).
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Z. Shan et al. / Computers and Geotechnics 57 (2014) 17–23
q;t ðtÞ ¼ q0 dðtÞ;
5. Examples The consolidation of an unsaturated three-layered soil with boundary condition (6) is considered in this section, as shown in Fig. 2. Three examples are used to validate the analytical solution and to illustrate the consolidation properties of unsaturated multi-layered soil. Unless otherwise specified, all examples in this section adopt the material parameters in Table 1. It is assumed that the total stress r(z, t) in unsaturated soil is zero before external excitation is applied, and the initial conditions are as follows:
ui ðz; 0Þ ¼ 0;
ði ¼ 1; 2; 3Þ:
ð53Þ
Note that both uw and ua are excess pore pressures. Assume the external excitation is a step load or an exponential load, as shown in Fig. 3, and
Step load :
qðtÞ ¼ q0 HðtÞ;
q0 ¼ 100 kPa;
ð54aÞ
ð54bÞ
Exponential load :
qðtÞ ¼ q0 ½1 expðbtÞ;
¼ 0:00005 q;t ðtÞ ¼ q0 b expðbtÞ;
b ð55aÞ ð55bÞ
where H(t) is an unit step function, d(t) is a Dirac delta function.
5.1. Validity of the analytical solution As no analytical or numerical solutions for the 1D consolidation of unsaturated multi-layered soil are published in the literature, a simplified case is chosen to validate the correctness of the present solution. Assume the unsaturated three-layered soil is subjected to the step load in Eq. (54) and different layers have the same permeability coefficients, porosity, and saturation of water (i.e., ðiÞ ðiÞ ðiÞ ðiÞ kw ¼ 1010 m=s, ka ¼ 109 m=s, n0 ¼ 0:45, S0 ¼ 0:80, i = 1, 2, 3, other parameters can be found in Table 1), which means the three-layered soil is the same as a single-layered soil with 10 m high. The consolidation of the three-layered soil is calculated by using the solution in this paper, and the consolidation of the single-layered soil with 10 m high is calculated by using the solution in Shan et al. [15]. The comparisons of the two results can be used to validate the correctness of the analytical solution. Fig. 4 shows the comparison of the numerical results for different methods. The results obtained with two methods are consistent, indicating the correctness of the analytical solution in this paper.
Fig. 2. An unsaturated three-layered soil subjected to an external load.
5.2. Step load Table 1 Material parameters. Layer
Parameters
1st
kw ¼ 1010 m=s;ka ¼ 109 m=s;h1 ¼ 3 m
2st 3rd Other parameters
ð1Þ
ð1Þ
ð1Þ ð1Þ n0 ¼ 0:45; S0 ¼ 0:80 ð2Þ ð2Þ 9 kw ¼ 10 m=s;ka ¼ 108 m=s;h2 ¼ 7 m ð2Þ ð2Þ n0 ¼ 0:50; S0 ¼ 0:60 ð3Þ ð3Þ kw ¼ 1010 m=s;ka ¼ 109 m=s;h3 ¼ 10 m ð3Þ ð3Þ n0 ¼ 0:40; S0 ¼ 0:70 1 ms1 ¼ 2:5 104 kPa ; ms2 =ms1 ¼ 0:4 s w w mw 1 =m1 ¼ 0:2; m2 =m1 ¼ 4 3 w = 10,000 N/m , uatm = 101 kPa, T = 293.16 K R = 8.31432 J/(mol K), a = 29 103 kg/mol
c
x
Assume an unsaturated three-layered soil with boundary condition (6) is subjected to the step load in Eq. (54). Fig. 5 shows the variation of the excess pore-air and pore-water pressures with time in different depth. Fig. 6 shows how the excess pore-air and pore-water pressures vary with depth at different time. The application of a step load to an unsaturated multi-layered soil results in the excess pore-air and pore-water pressures, while the excess pore pressures are different for different layers (Fig. 5). It is also can be seen in Fig. 6 that there are excess pore-air and pore-water pressure differences at the interface of two layers. In addition, during the dissipation of excess pore pressures, due to the influence of higher pressures in the first and third layers, the excess pore-air and pore-water pressures in the second layer increase firstly and then decrease.
Fig. 3. External excitation.
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Z. Shan et al. / Computers and Geotechnics 57 (2014) 17–23
Fig. 4. Comparison of the results for two different methods.
Fig. 5. Excess pore-air and pore-water pressures over time in different depth of the unsaturated soil subjected to a step load. (Note that z = 1.5 m, z = 4.5 m, and z = 8.5 m are the middle position of the first, second, and third layers, respectively).
Fig. 6. Excess pore-air and pore-water pressures over depth at different time.
Fig. 7. Excess pore-air and pore-water pressure over time in different depth of the unsaturated soil subjected to an exponential load. (Note that z = 1.5 m, z = 4.5 m, and z = 8.5 m are the middle position of the first, second, and third layers, respectively).
Z. Shan et al. / Computers and Geotechnics 57 (2014) 17–23
23
step load to an unsaturated multi-layered soil generates different excess pore-air and pore-water pressures in different layers, and the excess pore pressure differences can be observed at the interfaces of neighboring layers. Acknowledgements The authors express their gratitude for the grants provided by the National Basic Research Program of China (2014CB047005), the National Natural Science Foundation of China (51278451, 51308512), the Zhejiang Provincial Natural Science Foundation of China (LZ12E09001), and the Fundamental Research Funds of IEM (2013B04). Fig. 8. Compressions of unsaturated three-layered soil under step load and exponential load.
5.3. Exponential load Assume an unsaturated three-layered soil with boundary condition (6) is subjected to the exponential load in Eq. (55). Fig. 7 illustrates the variations of excess pore-air and porewater pressures with time in different depths subjected to the exponential load. The excess pore pressures first increase gradually with the external load increasing, and then decrease slowly after the increase of external load becomes slow. Furthermore, comparison of Figs. 5 and 7 shows that the maximums of excess pore-air and pore-water pressures generated by the exponential load is lower than those generated by the step load. Fig. 8 shows the compressions of unsaturated soil subjected to different external loads with time, where S0 ¼ ms1k q0 H. Once the step load is applied on the unsaturated soil, a sudden compression occurs. Instead, for the exponential load, the compression of unsaturated soil is gradually increasing as the dissipation of excess poreair and pore-water pressures. 6. Conclusions An analytical solution for the 1D consolidation of an unsaturated multi-layered soil with typical boundary condition was obtained based on the consolidation equations proposed by Fredlund and Hasan [5], in which the material parameters are assumed to be constants during consolidation. The method presented in this paper can be used to solve various types of boundary conditions. The analytical solution reveals the dissipation rules of the excess pore-air and pore-water pressures, and can be used to validate the accuracy of various numerical results. The consolidation behavior of unsaturated multi-layered soil was investigated through several numerical examples. It is concluded that the application of a
References [1] Biot MA. General theory of three-dimensional consolidation. J Appl Phys 1941;12:155–64. [2] Blight GE. Strength and consolidation characteristics of compacted soils. PhD Thesis. University of London, England; 1961. [3] Barden L. Consolidation of compacted and unsaturated clays. Géotechnique 1965;15:267–86. [4] Barden L. Consolidation of clays compacted dry and wet of optimum water content. Géotechnique 1974;24:605–25. [5] Fredlund DG, Hasan JU. One-dimensional consolidation theory: unsaturated soils. Can Geotech J 1979;17:521–31. [6] Dakshanamurthy VV, Fredlund DG, Rahardjo H. Coupled three-dimensional consolidation theory of unsaturated porous media. Fifth International Conference on Expansive Soils. Adelaide, South Australia; 1984. [7] Wong TT, Fredlund DG, Krahn J. A numerical study of coupled consolidation in unsaturated soils. Can Geotech J 1998;35:926–37. [8] Conte E. Consolidation analysis for unsaturated soils. Can Geotech J 2004;41:599–612. [9] Conte E. Plane strain and axially symmetric consolidation in unsaturated soils. Int J Geomech 2006;6:131–5. [10] Geng XY, Xu CJ, Cai YQ. Non-linear consolidation analysis of soil with variable compressibility and permeability under cyclic loadings. Int J Numer Anal Methods Geomech 2006;30:803–21. [11] Fredlund DG, Rahardjo H. Soil mechanics for unsaturated soil. New York: John Wiley and Sons; 1993. [12] Qin AF, Chen GJ, Tan YW, Sun DA. Analytical solution to one-dimensional consolidation in unsaturated soils. Appl Math Mech 2008;29:1329–40. [13] Qin AF, Sun DA, Tan YW. Analytical solution to one-dimensional consolidation in unsaturated soils under loading varying exponentially with time. Comput Geotech 2010;37:233–8. [14] Qin AF, Sun DA, Yang LP, Weng YF. A semi-analytical solution to consolidation of unsaturated soils with the free drainage well. Comput Geotech 2010;37:867–75. [15] Shan ZD, Ling DS, Ding HJ. Exact solutions for one dimensional consolidation of single layer unsaturated soil. Int J Numer Anal Methods Geomech 2012;36:708–22. [16] Shan ZD, Ling DS, Ding HJ. Analytical solution for 1D consolidation of unsaturated soil with mixed boundary condition. J Zhejiang Univ – Sci A 2013:61–70. [17] Ausilio E, Conte E. Settlement rate of foundations on unsaturated soils. Can Geotech J 1999;36:940–6. [18] Bellman R. Introduction to matrix analysis. New York: McGraw-Hill; 1970.
Contents lists available at ScienceDirect
Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Analytical solution for the 1D consolidation of unsaturated multi-layered soil Zhendong Shan a, Daosheng Ling b,⇑, Haojiang Ding b a b
Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, PR China MOE Key Lab of Soft Soils and Geo-Environmental Engineering, Zhejiang University, Hangzhou 310058, PR China
a r t i c l e
i n f o
Article history: Received 15 April 2013 Received in revised form 10 November 2013 Accepted 30 November 2013
Keywords: Analytical solution Unsaturated soil Multi-layered soil Consolidation Arbitrary load
a b s t r a c t The 1D consolidation of unsaturated multi-layered soil is studied based on the theory proposed by Fredlund and Hasan, and an analytical solution for a typical boundary condition is obtained by assuming all material parameters remain constant during consolidation. In the derivation of the analytical solution, the eigenfunction and eigenvalue for the multi-layered problem are first derived through the transfer matrix method. Then, by using the method of undetermined coefficients and the orthogonal relation of the eigenfunction, the analytical solution is obtained. The present method is applicable to various types of boundary conditions. Finally, numerical examples are provided to investigate the consolidation behavior of unsaturated multi-layered soil. Ó 2014 Published by Elsevier Ltd.
1. Introduction Unsaturated soil is a three-phase system which contains solid particles, water, and air. The compressibility and seepage of unsaturated soil are more complex than those of saturated soil. Many different types of theories have been proposed to describe the behavior of unsaturated soil, such as Biot [1], Blight [2], Barden [3,4], and Fredlund and Hasan [5]. Fredlund and Hasan [5] is now widely accepted, in which two partial differential equations are used to describe the dissipation process of excess pore pressures in unsaturated soil. This theory was extended to the 3D case by Dakshanamurthy et al. [6]. As material properties are functions of the stress state in unsaturated soil, the consolidation problems are usually solved via integral transform method or numerical methods involving the discretisation of both spatial and temporal domains [7–10]. Assuming all the soil parameters remain constant during consolidation, Fredlund and Rahardjo [11] presented 1D consolidation equations in the form of linear equations. These simplified consolidation equations may not be practical (parameters remain constant during consolidation), but the solution for these equations can give a preliminary description of the dissipation laws of the
⇑ Corresponding author. Address: Anzong Building A421, Zijingang Campus, Zhejiang University, Hangzhou, Zhejiang Province 310058, PR China. Tel.: +86 571 88208756; fax: +86 571 88208793. E-mail addresses: [email protected] (Z. Shan), [email protected] (D. Ling), [email protected] (H. Ding). 0266-352X/$ - see front matter Ó 2014 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.compgeo.2013.11.009
air and water pressures, and enable the study of the evolution law of the volume change of unsaturated soil. Using the simplified consolidation equations, several analytical solutions for the 1D consolidation of unsaturated soil have been published. Qin et al. [12,13] adopted the Laplace transform method and gave analytical solutions for unsaturated single-layer soil subjected to step and exponentially loads, respectively. By using the Laplace transform and Bessel function, Qin et al. [14] presented a semi-analytical solution for unsaturated single-layer soil with the free drainage well. Using the separation of variables method, Shan et al. [15] obtained the exact solutions for unsaturated single-layer soil subjected to arbitrary loads with three types of boundary conditions. Shan et al. [16] obtained an analytical solution for unsaturated single-layer soil subjected to an arbitrary load with mixed boundary condition using the modal expansion method. These analytical solutions are useful to validate the numerical results of consolidation in unsaturated soil. The above mentioned analytical solutions are for unsaturated single-layer soil, but the soil stratum in real engineering site is not always layered one. This paper presents an analytical solution for unsaturated multi-layered soil with a typical boundary condition subjected to an arbitrary load. In Section 2, the consolidation equations for unsaturated multi-layered soil are outlined. The associated initial, boundary, and interface continuity conditions are discussed in Section 3. In Section 4, the derivation of the analytical solution are presented in detail. In Section 5, several illustrative examples are given to illustrate some interesting features of the consolidation in unsaturated multi-layered soil.
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Z. Shan et al. / Computers and Geotechnics 57 (2014) 17–23
2. Consolidation equations for an unsaturated multi-layered soil The 1D consolidation equations developed by Fredlund and Hasan [5] in the Cartesian coordinate system are used to study the consolidation of unsaturated soil. The consolidation equations for unsaturated multi-layered soil (Fig. 1) can be written as:
ð1Þ þms2
where
( u C
ðiÞ
¼
ðiÞ
uw
) ;
ðiÞ ua
¼
" K
ðiÞ
1
Cw
Cw
C w =C ðiÞ a
¼
;
C wðiÞ v 0
Q
ðiÞ
¼
0
# ;
C aðiÞ C w =C aðiÞ (v cw r q;t ðtÞ
) ð2Þ
aðiÞ
cr q;t ðtÞC w =C aðiÞ ðiÞ
ðiÞ
where superscript (i) denotes the ith layer; uw and ua are the water and air pressures of the ith layer; r is the total stress mentioned later; (),z and (),t denote the derivatives with respect to z and t, respectively. The parameters in Eq. (2) can be written as
w w C w ¼ mw 1k m2 =m2 ; C ðiÞ a ¼
ðiÞ w C wðiÞ v ¼ kw = cw m2 ;
w w Cw r ¼ m1k =m2
ma2 =ma1k ; ðiÞ ðiÞ 1 ma2 =ma1k n0 1 S0 = ua ma1k
C aðiÞ r ¼
C aðiÞ v ¼
ðiÞ ka
h i ; ðiÞ a 1 ma2 =ma1k nðiÞ a ðxa =RT Þgma1k u 0 1 S0 = ua m1k ðiÞ
ðiÞ
Z
hi hi1
hi1
ðiÞ ðiÞ ðiÞ ua ðz;tÞ uw ðz;tÞ uaðiÞ ðz;0Þ uw ðz;0Þ dz
) ð4Þ
where ms1k and ms2 are the coefficients of the soil volume changes, due to the net normal stress (r ua) and the matric suction a (ua uw), respectively, which can be expressed as ms1k ¼ mw 1k þ m1k a and ms2 ¼ mw þ m . Note that, when the water and air pressures 2 2 are totally dissipated, the compression of unsaturated multi-layered soil becomes rðtÞms1k H. 3. Initial, boundary, and interface continuity conditions To address the consolidation of an unsaturated multi-layered soil, associated initial and boundary conditions and interface continuity conditions should be imposed. The following initial conditions are used
uðiÞ ðz; 0Þ ¼ g ðiÞ ðzÞ; n
ði ¼ 1; 2; . . . ; nÞ;
ðiÞ ðiÞ g 1 ðzÞ; g 2 ðzÞ
oT
ð5Þ ðiÞ g 1 ðzÞ
ðiÞ g 2 ðzÞ
where g ðiÞ ðzÞ ¼ , with and being arbitrary functions. The following boundary conditions are considered
1 ðiÞ ðiÞ 1 ma2 =ma1k n0 1 S0 = ua ma1k
ðiÞ
hi1
i¼1
ðiÞ
ðiÞ ðiÞ K ðiÞ uðiÞ ;zz þ C u;t ¼ Q
ðiÞ
pressure. When ua is small or dissipates rapidly in the process of a can be considered to be constant [8], and let consolidation, u a ¼ uatm in this study. u The compression of the multi-layered soil can be written as [15,17]: ( Z hi hi1 n X
SðtÞ ¼ ms1k rðtÞ uaðiÞ ðz; tÞ rð0Þ uðiÞ a ðz;0Þ dz
ð1Þ ðnÞ ðnÞ uð1Þ w ð0;tÞ ¼ 0; ua ð0; tÞ ¼ 0; uw;z ðhn ; tÞ ¼ 0; ua;z ðhn ;tÞ ¼ 0;
ð3Þ
ðiÞ
where kw , ka , S0 , and n0 are different for different layers. However, w a a mw 1k , m2 , m1k , m2 , cw, xa, R, T, and uatm are parameters of water and air and do not change with depth. ðiÞ ðiÞ In Eq. (3), kw and ka are the water and air permeability coeffiðiÞ ðiÞ cients of the ith layer, respectively; S0 and n0 represent the saturation of water and porosity of the ith layer, respectively. mw 1k and ma1k are the coefficients of the water and air volume changes, respectively, due to the net normal stress (r ua) for a K0-load a condition; mw 2 and m2 are the coefficients of the water and air volume changes, respectively, due to the matric suction (ua uw)cw is the density of water. xa is the molecular mass of air, R is the universal gas constant, T is the absolute temperature, g is the gravity a is the absolute pore-air pressure which acceleration. In addition, u a ¼ ua þ u atm , where u atm is the atmospheric can be expressed as u
ð6Þ
which indicate that the top surface of the multi-layered soil is permeable to water and air, while the bottom surface is impermeable to water and air. The interface continuity conditions between layers are specified as
uðiÞ ðhi ; tÞ ¼ uðiþ1Þ ðhi ; tÞ;
wðiÞ ðhi ; tÞ ¼ wðiþ1Þ ðhi ; tÞ;
ði ¼ 1; 2; . . . ; n 1Þ;
ð7Þ
n oT ðiÞ ðiÞ ðiÞ ðiÞ where w(i)(hi, t) denotes flow, and wðiÞ ðz; tÞ ¼ kw uw;z ; ka ua;z . The first equation in Eq. (7) denotes the pressure continuity and the second equation in Eq. (7) denotes the flow continuity. 4. Derivation of the analytical solution The method of Shan et al. [16] can address the consolidation of unsaturated single-layer soil with various types of boundary conditions. By incorporating the transfer matrix method to the method of Shan et al. [16], the present method can be used to solve the consolidation of unsaturated multi-layered soil. The solution procedure is illustrated below. 4.1. Eigenvalue and eigenfunction To solve the nonhomogeneous partial differential equation (1) with initial conditions (5), boundary conditions (6), and interface continuity conditions (7), the following characteristic equation is first considered to derive the corresponding eigenvalue and eigenfunction: ðiÞ
ðiÞ K ðiÞ u;zz þ C ðiÞ u;t ¼ 0:
ð8Þ
The solution to Eq. (8) takes the form: Fig. 1. An unsaturated multi-layered soil subjected to an external load. (hi is the depth from the surface to the bottom of the ith layer, and h0 = 0; n is the number of layers; H is the total thickness; q(t) is the external load).
uðiÞ ¼ U ðiÞ ðzÞ expðx2 tÞ;
ð9Þ
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Z. Shan et al. / Computers and Geotechnics 57 (2014) 17–23
where x is the eigenvalue which is a non-negative real number, and
( U ðiÞ ðzÞ ¼
)
ðiÞ Uw ðzÞ ðiÞ U a ðzÞ
( W ðiÞ ðzÞ ¼
;
W ðiÞ w ðzÞ W ðiÞ a ðzÞ
)
1
¼ ½M ðiÞ U ;zðiÞ ðzÞ
M ðiÞ ¼
ðiÞ 1=kw
ð10Þ
1=ka
By substituting Eq. (9) into Eq. (8), boundary conditions (6), and interface continuity conditions (7), we obtain 2 ðiÞ ðiÞ K ðiÞ U ðiÞ ;zz ðzÞ x C U ðzÞ ¼ 0;
U ð1Þ ð0Þ ¼ 0;
ði ¼ 1; 2; . . . ; nÞ
ð12Þ
W ðnÞ ðhn Þ ¼ 0
U ðiÞ ðhi Þ ¼ U ðiþ1Þ ðhi Þ;
ð13Þ
W ðiÞ ðhi Þ ¼ W ðiþ1Þ ðhi Þ
ð14Þ
The solution of Eq. (12) under Eqs. (13) and (14) will be described when x = 0 and x > 0. When x = 0, Eq. (12) can be simplified as: ðiÞ
K ðiÞ U 0;zz ðzÞ ¼ 0;
ði ¼ 1; 2; :::; nÞ;
ð15Þ
where subscript 0 means x = 0. As the determinant of matrix K(i) is not equal to zero, the general solution of Eq. (15) is as follows: ðiÞ
ðiÞ
ðiÞ
U 0 ðzÞ ¼ U 0 ðhi1 Þ þ U 0;z ðhi1 Þðz hi1 Þ:
ð16Þ
Eqs. (14) and (16) can be rewritten as follows: ðiÞ
ði1Þ
X 0 ðhi1 Þ ¼ X 0 ðiÞ
ðiÞ
ðhi1 Þ
ð17Þ
ðiÞ
X 0 ðzÞ ¼ T 0 ðzÞX 0 ðhi1 Þ;
ð18Þ
where T i0 ðzÞ is usually called the transfer matrix, and
(
ðiÞ X0
¼
)
ðiÞ
U 0 ðzÞ ðiÞ
W 0 ðzÞ
"
;
ðiÞ T 0 ðzÞ
¼
I
ðz hi1 ÞM
0 I
ðiÞ
;
ðiÞ
ðiÞ
X 0 ðhi Þ ¼ T 0 ðhi ÞX 0 ðhi1 Þ:
ð19Þ
ð20Þ
Repeatedly making use of interface continuity conditions (17) and Eq. (20) and (18) can be rewritten as: ðiÞ X 0 ðzÞ
¼
ðiÞ ð1Þ V 0 ðzÞX 0 ð0Þ;
ð21Þ
ðiÞ
ðiÞ
V 0 ðzÞ ¼ T 0 ðzÞ
" ðpÞ
T 0 ðhp Þ ¼
p¼i1
ðiÞ
N 0 ðzÞ ¼ ðz hi1 ÞM ðiÞ þ
I 0
ðiÞ N 0 ðzÞ
#
I
i1 X ðhp hp1 ÞM ðiÞ :
ð22Þ
¼
ðnÞ X 0 ðhn Þ
ð23Þ
ð1Þ X 0 ð0Þ
Both and contains four variables, four of them are shown in Eq. (13), other four unknowns can be determined through Eq. (23). Then, using Eqs. (21) and (19), we obtain ðiÞ
U 0 ðzÞ ¼ 0;
ði ¼ 1; 2; . . . ; nÞ;
M ðiÞ
x2 LðiÞ
0
2
# ;
ðiÞ
L
ðiÞ
kw =C wðiÞ v 6 ðiÞ ðiÞ aðiÞ ¼ 4 k C =C a a v
ðiÞ
3
ðiÞ
7 5:
kw C w =C wðiÞ v ka =C aðiÞ v
ð26Þ
According to the Hamilton–Cayley law, exp[N(i)(z hi1)] can be expressed as
h i ðiÞ ðiÞ exp N ðiÞ ðz hi1 Þ ¼ a0 ðz hi1 ÞI þ a1 ðz hi1 ÞN ðiÞ 2 ðiÞ ðiÞ þ a2 ðz hi1 Þ N ðiÞ þ a3 ðz 3 hi1 Þ N ðiÞ
ð27Þ
ðiÞ
where ak ðz hi1 Þ(k = 0, 1, 2, 3) can be obtained by solving the following equations
3 X
akðiÞ ðz hi1 Þ rðiÞ j
k
h i ðiÞ ¼ exp r j ðz hi1 Þ ;
ðj ¼ 1; 2; 3; 4Þ;
ð28Þ
k¼0 ðiÞ
where r j (j = 1, 2, 3, 4) are distinct eigenvalue of matrix N(i), which satisfy the following characteristic equation in r(i)
jN ðiÞ rIj ¼ 0:
ð29Þ
It should be note that if Eq. (29) has multiple roots, then (28) has to take a different form [18]. Similarly, by using the transfer matrix method, Eq. (25) can be rewritten as:
ð30Þ
where
V ðiÞ ðzÞ ¼ T ðiÞ ðzÞ
1 Y
T ðpÞ ðhp Þ:
ð31Þ
p¼i1
Setting i = n and z = hn in Eq. (30), yields
X ðnÞ ðhn Þ ¼ V ðnÞ ðhn ÞX ð1Þ ð0Þ:
ð32Þ
According to the four boundary condition (13) at z = 0 and z = hn, we can solve for the four unknowns from Eq. (32). Substituting Eq. (13) into Eq. (32), we have
#
mðnÞ mðnÞ 33 ðhn Þ 34 ðhn Þ W ð1Þ ð0Þ ¼ 0: ðnÞ m43 ðhn Þ mðnÞ 44 ðhn Þ
ð33Þ
where mpq ðzÞ is the element of matrix V(i)(z). If, and only if, the determinant of the coefficient matrix is equal to zero, Eq. (33) has a nonzero solution, and thus we obtain the characteristic equation of x as follows: ðnÞ ðnÞ ðnÞ mðnÞ 33 ðhn Þm44 ðhn Þ m43 ðhn Þm34 ðhn Þ ¼ 0:
Setting i = n and z = hn in Eq. (21), yields ðnÞ ð1Þ V 0 ðhn ÞX 0 ð0Þ:
¼
0
ðiÞ
p¼1
ðnÞ X 0 ðhn Þ
N
"
where 1 Y
" ðiÞ
X ðiÞ ðzÞ ¼ V ðiÞ ðzÞX ð1Þ ð0Þ;
#
with I being a two-by-two identity matrix. Setting z = hi in Eq. (18), yields ðiÞ
n oT ðiÞ ðiÞ X ðiÞ ðzÞ ¼ U w ðzÞ; U aðiÞ ðzÞ; W w ðzÞ; W ðiÞ ; a ðzÞ
ð11Þ
:
ðiÞ
0
where T (z) is the transfer matrix, and
T ðiÞ ðzÞ ¼ exp½N ðiÞ ðz hi1 Þ
#
0
ð25Þ
(i)
where
"
X ðiÞ ðzÞ ¼ T ðiÞ ðzÞX ðiÞ ðhi1 Þ;
ð24Þ
which should be omitted. When x > 0, by using the state-space method, the solution to Eq. (12) can be written as:
ð34Þ
This is a transcendental equation which has infinite positive roots labeled xk (k = 1, 2, 3,. . .) from smallest to largest, respectively. Using Eq. (33), a nonzero solution of W(1)(0) can be obtained as follows: ð1Þ
W k ð0Þ ¼
n
oT
ðnÞ mðnÞ mk33 ðhn Þ ; k34 ðhn Þ;
ð35Þ
where the subscript k indicates eigenvalue xk. Considering Eqs. (30) and (35), the eigenfunction can be written as
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Z. Shan et al. / Computers and Geotechnics 57 (2014) 17–23
"
ðiÞ ðiÞ mk13 ðzÞ mk14 ðzÞ ðiÞ ðiÞ mk23 ðzÞ mk24 ðzÞ
ðiÞ
U k ðzÞ ¼
#(
)
mðnÞ k34 ðhn Þ ; ðnÞ mk33 ðhn Þ
ð36Þ
n Z X
h
hi
hi1
i¼1
iT 0 ðiÞ ðiÞ U ðiÞ q ðzÞ C U p ðzÞdz ¼ Gp
p–q p¼q
;
ð44Þ
which indicates that eigenfunction U pðiÞ ðzÞ is orthogonal to eigenfunction U qðiÞ ðzÞ with respect to the weight matrix C(i).
where i = 1, 2,. . ., n and k = 1, 2, 3,. . . 4.2. Orthogonality of the eigenfunction In this section, we demonstrate the orthogonality of the eigenfunction, which will be used in the next section. Let xp and xq be ðiÞ eigenvalue and U ðiÞ p ðzÞ and U q ðzÞ be the corresponding eigenfunction. Using Eq. (8), the following two equations are obtained:
4.3. Analytical solution
h
iT h iT ðiÞ U ðiÞ K ðiÞ U p;zz ðzÞ x2 U ðiÞ C ðiÞ U ðiÞ q q p ðzÞ ¼ 0
ð37aÞ
As the last step, Eq. (1) is to be solved. By using the method of undetermined coefficients and the principle of linear superposition, the solution of Eq. (1) subjected to initial condition (5), boundary conditions (6), and interface continuity conditions (7) can be written as:
h
iT h iT ðiÞ U ðiÞ K ðiÞ U q;zz ðzÞ x2 U ðiÞ C ðiÞ U ðiÞ p p q ðzÞ ¼ 0:
ð37bÞ
uðiÞ ðz; tÞ ¼
1 X ðiÞ U k ðzÞXk ðtÞ;
ð45Þ
k¼1
By subtracting Eq. (37b) from Eq. (37a), integrating Eq. (37a) with respect to z from 0 to H, then using the symmetry of matrix C, we obtain
Z
hi
h
hi1
x
iT h iT ðiÞ ðiÞ ðiÞ 2 U qðiÞ ðzÞ K ðiÞ U ðiÞ p;zz ðzÞ U p ðzÞ K U q;zz ðzÞ dz ðxp
2 qÞ
Z
h
hi
hi1
U qðiÞ ðzÞ
iT
C
ðiÞ
K ðiÞ
U pðiÞ ðzÞdz ð38Þ
By using the method of integration by parts and the symmetry of matrix K(i), Eq. (38) can be rewritten as 2 p
x x
2 q
Z
h
hi
hi1
iT
U qðiÞ ðzÞ
C
ðiÞ
U pðiÞ ðzÞdz
¼
n h X
i¼1
hi
h
hi1
ð39Þ
iT U qðiÞ ðzÞ C ðiÞ U pðiÞ ðzÞdz
hi iT h iT ðiÞ ðiÞ ðiÞ U qðiÞ ðzÞ K ðiÞ U p;z ðzÞ U ðiÞ ðzÞ K U ðzÞ p q;z (i)
According to Eqs. (2), (3), and (10), matrix K two matrixes, i.e.:
E¼
a
0
;
0 b
a¼
1
cw mw2
;
b¼
ð41Þ
C w RT : a mw u 2 xa
¼
i¼1
EW pðiÞ ðzÞ
h
U pðiÞ ðzÞ
iT
hi1
h
hi
hi1
iT ðiÞ U pðiÞ ðzÞ C ðiÞ U k ðzÞdz
iT
U pðiÞ ðzÞ Q ðiÞ dz
ð49Þ
Using the orthogonality of U ðiÞ p ðzÞ as shown in Eq. (44), Eq. (49) can be simplified as
Xp;t ðtÞ þ x2p Xp ðtÞ ¼ Sp ðtÞ
ð50Þ
Sp ðtÞ ¼
n 1 X Gp i¼1
Z
hi
h
hi1
iT U pðiÞ ðzÞ Q ðiÞ dz
ð51Þ
Similarly, using the orthogonal relation in Eq. (44), Eq. (47) can be simplified as
hi
EW qðiÞ ðzÞ
Xp ð0Þ ¼
hi1
h iT h iT h iT ðnÞ ð1Þ ð1Þ ¼ U qðnÞ ðhn Þ EW pðnÞ ðhn Þ U ðnÞ p ðhn Þ EW q ðhn Þ U q ð0Þ EW p ð0Þ h iT ð1Þ þ U ð1Þ p ð0Þ EW q ð0Þ n o ðnÞ ðnÞ ðnÞ ðnÞ ð1Þ ð1Þ ð1Þ ðhn ÞW wp ðhn Þ U wp ðhn ÞW wq ðhn Þ U ð1Þ ¼ U wq wq ð0ÞW wp ð0Þ þ U wp ð0ÞW wq ð0Þ a n o ðnÞ ðnÞ ðnÞ ð1Þ ð1Þ ð1Þ ð1Þ þ U ðnÞ aq ðhn ÞW ap ðhn Þ U ap ðhn ÞW aq ðhn Þ U aq ð0ÞW ap ð0Þ þ U ap ð0ÞW aq ð0Þ b
Z
i¼1
h
hi
hi1
iT
U qðiÞ ðzÞ
n X
where
hi n h iT h iT X ðiÞ ðiÞ ðiÞ U qðiÞ ðzÞ K ðiÞ U p;z ðzÞ U ðiÞ p ðzÞ K U q;z ðzÞ n h X
¼
n Z X
ð42Þ
Considering Eqs. (10) and (41), and the interface continuity conditions (14), the right hand of Eq. (40) can be rewritten as
i¼1
T
k¼1
ð40Þ
is shown in Eq. (11), and
ð48Þ
Pre-multiplying Eq. (48) by ½U pðiÞ ðzÞ , integrating Eq. (48) with respect to z from hi1 to hi, and then summing of Eq. (48) from the 1st layer to nth layer, we obtain
x2k Xk ðtÞ þ Xk;t ðtÞ
can be split into
1
where M
i¼1
K ðiÞ ¼ E½M ðiÞ ; E ¼ M ðiÞ K ðiÞ ¼ K ðiÞ M ðiÞ ; (i)
ð47Þ
Considering Eq. (12), Eq. (46) can be rewritten as
1 X
hi1
i¼1
ði ¼ 1; 2; . . . ; nÞ;
k¼1
k¼1
Summing from the 1st layer to the nth layer, Eq. (39) becomes: n Z X
1 X ðiÞ U k ðzÞXk ð0Þ ¼ g ðiÞ ðzÞ;
x2k Xk ðtÞ þ Xk;t ðtÞ C ðiÞ U kðiÞ ðzÞ ¼ Q ðiÞ
hi1
x2p x2q
ð46Þ
k¼1
1 X
h
hi iT h iT ðiÞ ðiÞ ðiÞ ¼ U qðiÞ ðzÞ K ðiÞ U ðiÞ p;z ðzÞ U p ðzÞ K U q;z ðzÞ
1 1 X X ðiÞ ðiÞ U k;zz ðzÞXk ðtÞ þ C ðiÞ U k ðzÞXk;t ðtÞ ¼ Q ðiÞ k¼1
¼0
ðiÞ
where U k ðzÞ are shown as Eq. (36). Therefore, if the scalar function Xk(t) are known, the air and water pressures for each layer can be obtained through Eq. (45), directly. Substituting Eq. (45) into Eqs. (1) and (5), we have
n 1 X Gp i¼1
Z
hi
hi1
h
iT ðiÞ ðiÞ U ðiÞ p ðzÞ C g ðzÞdz
ð52Þ
The solution of Eq. (50) with initial condition (52) can be written as 2
2
Xp ðtÞ ¼ exp t Xp ð0Þ þ exp t
Z
t
2
exp n Sp ðnÞdn
ð53Þ
0
ð43Þ
Substituting boundary conditions (13) into Eq. (43), we know Eq. (43) equals zero. Eq. (40) can thus be simplified as follows:
Substituting this expression and (36) into Eq. (45), we obtain the analytical solution of Eq. (1) with initial condition (5), boundary conditions (6), and interface continuity conditions (7).
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Z. Shan et al. / Computers and Geotechnics 57 (2014) 17–23
q;t ðtÞ ¼ q0 dðtÞ;
5. Examples The consolidation of an unsaturated three-layered soil with boundary condition (6) is considered in this section, as shown in Fig. 2. Three examples are used to validate the analytical solution and to illustrate the consolidation properties of unsaturated multi-layered soil. Unless otherwise specified, all examples in this section adopt the material parameters in Table 1. It is assumed that the total stress r(z, t) in unsaturated soil is zero before external excitation is applied, and the initial conditions are as follows:
ui ðz; 0Þ ¼ 0;
ði ¼ 1; 2; 3Þ:
ð53Þ
Note that both uw and ua are excess pore pressures. Assume the external excitation is a step load or an exponential load, as shown in Fig. 3, and
Step load :
qðtÞ ¼ q0 HðtÞ;
q0 ¼ 100 kPa;
ð54aÞ
ð54bÞ
Exponential load :
qðtÞ ¼ q0 ½1 expðbtÞ;
¼ 0:00005 q;t ðtÞ ¼ q0 b expðbtÞ;
b ð55aÞ ð55bÞ
where H(t) is an unit step function, d(t) is a Dirac delta function.
5.1. Validity of the analytical solution As no analytical or numerical solutions for the 1D consolidation of unsaturated multi-layered soil are published in the literature, a simplified case is chosen to validate the correctness of the present solution. Assume the unsaturated three-layered soil is subjected to the step load in Eq. (54) and different layers have the same permeability coefficients, porosity, and saturation of water (i.e., ðiÞ ðiÞ ðiÞ ðiÞ kw ¼ 1010 m=s, ka ¼ 109 m=s, n0 ¼ 0:45, S0 ¼ 0:80, i = 1, 2, 3, other parameters can be found in Table 1), which means the three-layered soil is the same as a single-layered soil with 10 m high. The consolidation of the three-layered soil is calculated by using the solution in this paper, and the consolidation of the single-layered soil with 10 m high is calculated by using the solution in Shan et al. [15]. The comparisons of the two results can be used to validate the correctness of the analytical solution. Fig. 4 shows the comparison of the numerical results for different methods. The results obtained with two methods are consistent, indicating the correctness of the analytical solution in this paper.
Fig. 2. An unsaturated three-layered soil subjected to an external load.
5.2. Step load Table 1 Material parameters. Layer
Parameters
1st
kw ¼ 1010 m=s;ka ¼ 109 m=s;h1 ¼ 3 m
2st 3rd Other parameters
ð1Þ
ð1Þ
ð1Þ ð1Þ n0 ¼ 0:45; S0 ¼ 0:80 ð2Þ ð2Þ 9 kw ¼ 10 m=s;ka ¼ 108 m=s;h2 ¼ 7 m ð2Þ ð2Þ n0 ¼ 0:50; S0 ¼ 0:60 ð3Þ ð3Þ kw ¼ 1010 m=s;ka ¼ 109 m=s;h3 ¼ 10 m ð3Þ ð3Þ n0 ¼ 0:40; S0 ¼ 0:70 1 ms1 ¼ 2:5 104 kPa ; ms2 =ms1 ¼ 0:4 s w w mw 1 =m1 ¼ 0:2; m2 =m1 ¼ 4 3 w = 10,000 N/m , uatm = 101 kPa, T = 293.16 K R = 8.31432 J/(mol K), a = 29 103 kg/mol
c
x
Assume an unsaturated three-layered soil with boundary condition (6) is subjected to the step load in Eq. (54). Fig. 5 shows the variation of the excess pore-air and pore-water pressures with time in different depth. Fig. 6 shows how the excess pore-air and pore-water pressures vary with depth at different time. The application of a step load to an unsaturated multi-layered soil results in the excess pore-air and pore-water pressures, while the excess pore pressures are different for different layers (Fig. 5). It is also can be seen in Fig. 6 that there are excess pore-air and pore-water pressure differences at the interface of two layers. In addition, during the dissipation of excess pore pressures, due to the influence of higher pressures in the first and third layers, the excess pore-air and pore-water pressures in the second layer increase firstly and then decrease.
Fig. 3. External excitation.
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Z. Shan et al. / Computers and Geotechnics 57 (2014) 17–23
Fig. 4. Comparison of the results for two different methods.
Fig. 5. Excess pore-air and pore-water pressures over time in different depth of the unsaturated soil subjected to a step load. (Note that z = 1.5 m, z = 4.5 m, and z = 8.5 m are the middle position of the first, second, and third layers, respectively).
Fig. 6. Excess pore-air and pore-water pressures over depth at different time.
Fig. 7. Excess pore-air and pore-water pressure over time in different depth of the unsaturated soil subjected to an exponential load. (Note that z = 1.5 m, z = 4.5 m, and z = 8.5 m are the middle position of the first, second, and third layers, respectively).
Z. Shan et al. / Computers and Geotechnics 57 (2014) 17–23
23
step load to an unsaturated multi-layered soil generates different excess pore-air and pore-water pressures in different layers, and the excess pore pressure differences can be observed at the interfaces of neighboring layers. Acknowledgements The authors express their gratitude for the grants provided by the National Basic Research Program of China (2014CB047005), the National Natural Science Foundation of China (51278451, 51308512), the Zhejiang Provincial Natural Science Foundation of China (LZ12E09001), and the Fundamental Research Funds of IEM (2013B04). Fig. 8. Compressions of unsaturated three-layered soil under step load and exponential load.
5.3. Exponential load Assume an unsaturated three-layered soil with boundary condition (6) is subjected to the exponential load in Eq. (55). Fig. 7 illustrates the variations of excess pore-air and porewater pressures with time in different depths subjected to the exponential load. The excess pore pressures first increase gradually with the external load increasing, and then decrease slowly after the increase of external load becomes slow. Furthermore, comparison of Figs. 5 and 7 shows that the maximums of excess pore-air and pore-water pressures generated by the exponential load is lower than those generated by the step load. Fig. 8 shows the compressions of unsaturated soil subjected to different external loads with time, where S0 ¼ ms1k q0 H. Once the step load is applied on the unsaturated soil, a sudden compression occurs. Instead, for the exponential load, the compression of unsaturated soil is gradually increasing as the dissipation of excess poreair and pore-water pressures. 6. Conclusions An analytical solution for the 1D consolidation of an unsaturated multi-layered soil with typical boundary condition was obtained based on the consolidation equations proposed by Fredlund and Hasan [5], in which the material parameters are assumed to be constants during consolidation. The method presented in this paper can be used to solve various types of boundary conditions. The analytical solution reveals the dissipation rules of the excess pore-air and pore-water pressures, and can be used to validate the accuracy of various numerical results. The consolidation behavior of unsaturated multi-layered soil was investigated through several numerical examples. It is concluded that the application of a
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