Solution to 1-D consolidation of non-homogeneous soft clay *

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1 Xie et al. / J Zhejiang Uni SCI 25 6A(Suppl. I): Journal of Zhejiang Uniersity SCIENCE ISSN jzus@zju.edu.cn Solution to 1-D consolidation of non-homogeneous soft clay * XIE Kang-he ( 谢康和 ) 1, WEN Jie-bang ( 温介邦 ) 1, XIA Jian-zhong ( 夏建中 ) 2 ( 1 Institute of Geotechnical Engineering, Zhejiang Uniersity, Hangzhou 3127, China) ( 2 Department of Ciil Engineering, Zhejiang Uniersity of Science and Technology, Hangzhou 3112, China) zdkhxie@zju.edu.cn; 97crazyiori@163.com Receied Jan. 1, 25; reision accepted July 17, 25 Abstract: In this work, semi-analytical methods were used to sole the problem of 1-D consolidation of non-homogeneous soft clay with spatially arying coefficients of permeability and compressibility. The semi-analytical was programmed and then erified by comparison with the obtained analytical of a special case. Based on the results of some computations and comparisons with the 1-D homogeneous consolidation (by Terzaghi) and the 1-D non-linear consolidation theory (by Dais et al.) of soft clay, some diagrams were prepared and the releant consolidation behaior of non-homogeneous soils is discussed. It was shown that the result obtained differs greatly from Terzaghi s theory and that of the non-linear consolidation theory when the coefficients of permeability and compressibility ary greatly. Key words: Non-homogeneous soil, Time-dependent loading, One-dimensional consolidation, Layered system, Semi-analytical doi:1.1631/jzus.as29 Document code: A CLC number: TU432 INTRODUCTION It is well known that taking the coefficient of consolidation c as a constant is a major shortcoming of conentional consolidation theory (Duncan, 1993). To oercome this shortcoming, many studies were conducted on the 1-D nonlinear consolidation problem, in which the coefficient of permeability k and that of olume compressibility m both directly relate to c and ary with the effectie stress and oid ratio (Dais and Raymond, 1965; Xie and Leo, 1999; Li et al., 1999a; 1999b; Xie et al., 22). Meanwhile, the problem of 1-D consolidation of non-homogeneous soil was studied assuming that k and m ary with depth. Schiffman and Gibson (1964) first studied in detail the 1-D consolidation problem of non-homogeneous soil. By assuming k and m to be polynomial function or exponential function of depth and adopting differential method, they obtained a to * Project (No ) supported by the National Research Foundation for the Doctoral Program of Higher Education of China 1-D consolidation of single layer soft clay under instantaneous loading. Jiang et al.(25) presented an analytical to the problem of 1-D consolidation of single-layered soil, wherein k is constant and E s aries with depth, with this problem considered as a special case of the problem of consolidation of non-homogeneous soil aboe. The semi-analytical for the consolidation problem of single-layered or multi-layered soil with k and m (=1/E s ) arying with depth, but this may only be applicable to NC clay. Based on this, the consolidation behaior can be discussed. For either single-layered foundation or multi-layered foundation, this paper takes into consideration both single-drainage or double-drainage situations and time-dependent loading. MATHEMATICAL MODELING AND ANALYTI- CAL SOLUTIONS OF SPECIAL CASE The analysis scheme of consolidation problem of

2 3 Xie et al. / J Zhejiang Uni SCI 25 6A(Suppl. I):29-34 soft clay is shown in Fig.1 based on the assumption that k and m ary with depth. In this figure, H is the total thickness of soil layer. q(t) is the uniformly distributed load applied on the top surface of the soil as shown in Fig.2, in which, q, q u are the initial and ultimate load respectiely, and t c is the construction time. The top surface of the soil layer is perious but the bottom is perious or imperious. H h 1 h i h n Fig.1 Loading and boundary condition of non-homogeneous foundation q(t) q Perious Perious or imperious Schiffman and Gibson (1964) s eations goerning 1-D consolidation process of single-layered non-homogeneous soft clay were used to obtain the following goerning eations for the 1-D consolidation of layered non-homogeneous soil: u 1 dk u 1 u d qt ( ) + = z k ( z) d z z c ( z) t dt, 2 i i i i 2 i i t c (i=1,2,,n). (1) The conditions corresponding to Eq.(1) can Z Z 1 Z i 1 Z i Z n 1 Z n k 1, m 1 k i, m i k n, m n Fig.2 Loading-time relationship q u t be written as follows: t=: u i =q()=q, z=: u 1 =, z=h: u n / z= (for single-drainage situation), u n = (for double-drainage situation), z=z i : u i =u i+1, k i u i / z=k (i+1) u i+1 / z, i=1,2,n 1 in which k i =k i (z), m i =m i (z)=1/e si (z), c i =k i / (γ w m i )=k i E si /γ w, the coefficient of permeability, the coefficient of olume compressibility and the coefficient of consolidation in layer i respectiely; γ w is the unit weight of water; u i is the excess porewater pressure in layer i. The (Jiang et al., 25) to the 1-D consolidation problem of single-layered non-homogeneous soil can be obtained as follows by assuming that the coefficient of permeability k is constant that the modulus of compressibility E s aries linearly with depth, and that this may only be applicable to NC clay. 2 Dm u = x[ J1( λma) Y1( λmx) Y1( λma) J1( λmx)] m= 1 λm Em 2 t d λmt q λmc / H τ e q + e d τ, (2) dτ qt () 4 Us = ln(1 + α) 2 1 D 2 t d m λmt q λmc / H τ e q 2 + e d τ, (3) m= 1 λm E m dτ where U s is the aerage consolidation degree defined in terms of settlement or strain. qt () α 1 DF U = m m p 2 m= 1 λm Em e 2 λmt t dq e λmc / H τ d, (4) q + τ dτ where U p is the aerage consolidation degree defined in terms of porewater pressure or stress; a=2/α, b = α / α, x= αz/ H / α, Dm = J1( λma)[ Y( λma) Y( λmb)] Y( λ a)[ J ( λ a) J ( λ b)]. (5) 1 m m m

3 Xie et al. / J Zhejiang Uni SCI 25 6A(Suppl. I): Em = b [ J1( λmb) Y1( λma) Y1( λmb) J1( λma)] a [ J( λma) Y1( λma) Y( λma) J1( λma)], (for single-drainage situation) (6.1) Em = b [ J( λmb) Y1( λma) Y( λmb) J1( λma)] a [ J( λma) Y1( λma) Y( λma) J1( λma)]. (for double-drainage situation) (6.2) Fm = J1( λma)[ b Y2( λmb) a Y2( λma)] Y1( λma)[ b J2( λmb) a J2( λma)]. (7) The positie eigenalues λ m satisfy the following eigen-eation: J ( λ a) Y ( λ b) Y( λ a) J ( λ b) =, 1 m m 1 m m (for single-drainage situation) (8.1) J1( λma) Y1( λmb) Y1( λma) J1( λmb) =. (for double-drainage situation) (8.2) time-dependent loading can be used for calculation. This paper deeloped the computation program to calculate in two cases below: (1) k i and m i of each clay layer are the function of depth (z), namely, p m = m [1 + α ( z h ) / h] i, k [1 ( = k + β z i i i, i 1 i i i i )/ ] q h i,, i 1 hi in which, m i, k i are the coefficient of olume compressibility and the coefficient of permeability in the top of layer i respectiely; h,i 1 is the total thickness from layer 1 to layer i 1; h i is the thickness of layer i; α i, β i, p i, q i are parameters that can be determined according to the properties of each clay layer. When i=1, p 1 =1, β 1 =, the case becomes the one discussed in (Jiang et al., 25). (2) For soft clay, k and m are supposed to distribute not-uniformly because the initial effectie pressure aries with depth. m = m =.434C ci i i (1 + ei) σ i Cci / Cki i = i = 1i( σ 1i / σ i), k k k, (9) SEMI-ANALYTICAL SOLUTION UNDER NORMAL CONDITION Basic idea For general 1-D consolidation problem of single-layered or multi-layered non-homogeneous soil (Fig.1) under time-dependent loading (Fig.2), there seems to be no analytical s deeloped so far. Therefore, this paper adopts semi-analytical methods for soling the problem of 1-D consolidation of non-homogeneous soft clay. For some problems that hae been studied thoroughly in theory, the numerical method can be substituted by corresponding semi-analytical method. Thus, the antity of work can be greatly decreased. In the paper, by diiding the whole clay layer into many small clay layers until the coefficient of permeability k and the coefficient of olume compressibility m can be considered as constant in each small clay layer, the semi-analytical method is used to conert the consolidation problem of non-homogeneous clay of the whole clay stratum to the linear consolidation problem of many small clay layer. Thus, the analytical (Xie et al., 1995) for 1-D linear consolidation problem of layered soil under where e i, σ i are the initial oid ratio and the initial effectie pressure; C ci, C ki are the compression index and the permeability index; k 1i, σ are the coefficients of permeability and the effectie pressure cor- 1i responding to oid ratio e 1i at any point 1 on the e i log σ i cure and e logk i cure of the layer i respectiely. Validation and irtues of semi-analytical method The semi-analytical approach was alidated by comparing the semi-analytical in this paper against the analytical for single-layered non-homogeneous soil with linearly arying modulus of compressibility. The aerage degree of consolidation U s defined in terms of settlement under instantaneous loading and the aerage degree of consolidation U p defined in terms of the aerage excess pore in Fig.3 and Fig.4 respectiely, in which the parameter α=.9,.1, 1, 9 was taken into calculation. It can be seen that the semi-analytical agrees ite well with the analytical. In addition, two methods calculated results for the excess porewater pressure or settlement under other drainage and loading conditions were almost identical.

4 32 Xie et al. / J Zhejiang Uni SCI 25 6A(Suppl. I): U s (%) U p (%) α= E s =E s (1+αz/H) Analytical Semi-analytical Double-drainage One-step loading c =.1 Compared with the differential method proposed by Schiffman and Gibson (1964) and other numerical methods, the semi-analytical method can control the precision more easily and therefore speed up the calculation. The semi-analytical method can deal with eery case in which k and m ary with depth, for example, k and m are polynomial function, exponential function or other nonlinear function of z. Furthermore, the to single-layered problem can be easily deeloped to sole the multi-layered soil problem. CALCULATION ANALYSIS Single-drainage Instantaneous loading 1 1E E Fig.3 U s s cures with different α in comparison with other s (single-layered) Analytical Semi-analytical From left to right α=9, 1,.1,.9 1E E Fig.4 U p s cures with different α in comparison with other s (single-layered) The influence of the non-homogeneous characteristics of soil under instantaneous load on the consolidation process is analyzed and discussed below. k and E s are linear functions of z 1. Only E s aries with z linearly Fig.5 compares the obtained in a special case in which E s aries with z linearly, e.g., E s =E s (1+αz/H) with the corresponding of Terzaghi. In Fig.5 the aerage coefficient of consolidation C = C 1 H dz H of four kinds of non-homogeneous soil is eal to one another and is eal to the consolidation coefficient of Terzaghi s. Fig.5 shows that when the modulus of compressibility aries greatly, the excess porewater pressure of the soil is greatly different from Terzaghi s for homogeneous soil calculated by the aerage alue of the consolidation coefficient α=9 or α=.9 (the ratio of the compressibility modulus of top foundation to that of bottom foundation is 1 or.1). Furthermore, neglecting the non-homogeneous characteristic of the soil generally leads to oer-ealuating the dissipation rate of the porewater pressure z/h Terzaghi s α=9 α=.9 α=1 α=.5 Double-drainage u/q u Fig.5 Isochrones of excess porewater pressure with linear E s (t=1 d) 2. Only k aries with z linearly Fig.6 compares the obtained in the special case in which k aries linearly (e.g., k =k (1+βz/H)) with the corresponding of Terzaghi. When k aries greatly, the conclusion similar to that of Fig.5 can be deried. In addition, in Fig.5 and Fig.6, the coefficients of consolidation corresponding to α=9, β=9 respectiely both ary with depth linearly and the two ariations are identical. The other three cases α=.9, 1,.5 corresponding to β=.9, 1,.5 are also shown in these two figures respectiely. Comparison of the two

5 Xie et al. / J Zhejiang Uni SCI 25 6A(Suppl. I): figures shows that though the ariation of c is identical, the distribution of the porewater pressure is different when k, m ary differently. Furthermore, when the ariation extent of k and m is identical, the influence of k on the distribution of the porewater pressure is great. This conclusion can also be deried from the result of Terzaghi that the deiation of the porewater pressure of non-homogeneous foundation in Fig.6 is more significant than that in Fig.5. z/h Terzaghi s β=9 β=.9 β=1 β=.5 Double-drainage u/q u Fig.6 Isochrones of excess porewater pressure with linear k (t=1 d) Both k and E s are non-linear functions of z In Fig.7, the consolidation cure is gien based on the second case referred to in Section 3.1 and compared with the 1-D non-linear consolidation. To distinguish from the case that k and m (=1/E s ) ary non-linearly with both depth and time, the case when k and m ary non-linearly only with depth is still called consolidation of non-homogeneous soil. In this example, Nσ =( σ + ) / σ = 2.5. Fig.7 shows that the smaller the ratio C c /C k, the closer are the cures of U s. In two cases, different from non-linear consolidation, the rule of the ariation of non-homogeneous soil consolidation with C c /C k is consistent, that is, from beginning to end, with the increase beginning to end, with the increase of C c /C k, the rate of consolidation also increases. For the degree of consolidation U p defined in terms of the aerage porewater pressure, the same conclusion can be obtained Us (%) From left to right C c /C k =1.5, 1,.5 Non-homogeneous Non-linear Single-drainage From left to right C c /C k =.5,1, E Fig.7 U s 1E-3 s cures.1with different.1 C c /C1 k of non-homo- 1 geneous soil in comparison with nonlinear consolidation soil (single-layered) Based on non-linear and non-homogeneous assumptions, the problems of four-layer soil (Table 1) bellow can be studied respectiely and the corresponding cures of the degree of consolidation are gien in Fig.8. 2 Us (Up) (%) U s Non-homogeneous Non-linear Single-drainage 1 1E E Fig.8 U s (U p ) s cures of non-homogeneous soil in comparison with nonlinear consolidation soil (multi-layered) Fig.7 and Fig.8 show that for either single-layered soil or multi-layered soil, the rate of consolidation calculated based on non-homogeneous assumptions is greater than that calculated based on non-linearity assumptions. Furthermore, Fig.8 shows that the aerage degree of consolidation U s defined in terms of settlement is not eal to U p defined in terms U p Table 1 Geotechnical data on layered soil i C ci C ki k 1i (1 9 m/s) e 1i σ 1i (kpa) γ sati (kn/m 3 ) h i (m)

6 34 Xie et al. / J Zhejiang Uni SCI 25 6A(Suppl. I):29-34 of aerage porewater pressure. This can also be deried from the definition of U s and U p. Generally, U s is not eal to U p for non-homogeneous foundation (except for the case when m does not ary with depth). CONCLUSION This work resoles by semi-analytical method the problem of 1-D consolidation of single-layered and multi-layered soil with coefficient of compressibility m and coefficient of permeability k arying with depth. Through comparison and calculation analysis, some conclusions can be obtained as follows: (1) The 1-D consolidation problem of non-homogeneous soil is not dependent on the coefficient of olume compressibility m, although the coefficient of permeability k should be taken into consideration. When m, k ary greatly, there is great difference between the excess porewater pressure in non-homogeneous soil and the porewater pressure of Terzaghi s of homogeneous soil calculated by the aerage alue of the consolidation coefficient. (2) Except for the case when m does not ary with depth in single-layered soil, the degree of consolidation defined in terms of deformation is not eal to that defined in terms of the aerage porewater pressure for general non-homogeneous soil. (3) For the 1-D consolidation problem of soft clay, the rate of consolidation calculated based on the assumptions that k and m distribute non-uniformly because the initial effectie pressure ariation with depth is greater than that calculated based on non-lin- earity assumptions. Furthermore, the smaller the alue of C c /C k, the closer is the of non-homogeneous consolidation to the of non-linear consolidation. References Dais, E.H., Raymond, C.P., A non-linear theory of consolidation. Geotechnie, 15(2): Duncan, J.M., Limitations of conentional analysis of consolidation settlement. Journal of Geotechnical Engineering ASCE, 119(9): Jiang, W., Xie, K.H., Xia, J.Z., 25. Analytical to 1-D consolidation of soft clay considering the modulus of compressibility aries with depth linearly. Chinese Journal of Bulletin of Science and Technology, (in Chinese, to be published). Li, B.H., Xie, K.H., Ying, H.W., Zeng, G.X., 1999a. Semi-analytical of 1-D nonlinear consolidation considering the initial effectie stress distribution. Chinese Journal of Ciil Engineering, 32(6):47-52 (in Chinese). Li, B.H., Xie, K.H., Ying, H.W., Zeng, G.X., 1999b. Semi-analytical of one dimensional non-linear consolidation of soft clay under time-dependent loading. Chinese Journal of Geotechnical Engineering, 21(3): (in Chinese). Schiffman, R.L., Gibson, R.E., Consolidation of non-homogeneous clay layers. Journal of the Soil Mechanics and Foundations Diision, ASCE, 443:1-3. Xie, K.H., Pan, Q.Y., Theory of non dimensional consolidation of layered soils under ariable loading. Chinese Journal of Geotechnical Engineering, 17(5):8-85 (in Chinese). Xie, K.H., Wen, J.B., Hu, H.Y., 25. A study on one-dimensional consolidation of oer-consolidated saturated soil. Chinese Journal of bulletin of science and technology, (in Chinese, to be published). Xie, K.H., Xie, X.Y., Jiang, W., 22. A study on one-dimensional nonlinear consolidation of double-layered soil. Computers and Geotechnics, 29: