- Email: [email protected]
Consolidation analysis of soil with vertical and horizontal drainage under ramp loading considering smear effects
ARTICLE IN PRESS
Geotextiles and Geomembranes 22 (2004) 63–74
Consolidation analysis of soil with vertical and horizontal drainage under ramp loading considering smear effects Guofu Zhu, Jian-Hua Yin* Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Received 12 February 2002; received in revised form 16 April 2003
Abstract The installation process of a vertical drain with a mandrel normally causes a smear zone with reduced permeability around the drain. The smear zone affects the consolidation of the soil. This paper presents a new analytical solution for the consolidation analysis of soil with a vertical drain under ramp loading considering the smear effects. The solution is described in detail in this paper. A new normalised time factor, T; is suggested. It is shown in the paper that the average degree of consolidation exhibits very good normalised behaviour using this new factor T: Results are presented in tables and charts for practical applications. The influence of time-dependent loading is incorporated in the solution charts. r 2003 Elsevier Ltd. All rights reserved. Keywords: Vertical drain; Consolidation; Theoretical analysis; Permeability; Time dependence; Smear zone; Soil
1. Introduction Vertical drains are widely used to facilitate the consolidation of soil. The installation process of a vertical drain, for example, a prefabricated band drain, involves penetration, displacement of the soil by the mandrel, and consequently disturbance of the soil around the drain. Two principal disturbing mechanisms exist: first, displacement of the soil to create space for the mandrel; and second the *Corresponding author. Tel. +852-2766-6065; fax: +852-2334-6389. E-mail address: [email protected] (J.-H. Yin). 0266-1144/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0266-1144(03)00052-9
ARTICLE IN PRESS 64
G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
dragging of soil by the sides of the mandrel. Substantial reduction in the soil permeability occurs around the drain due to this disturbance. A fully coupled finite element consolidation analysis (Zhu and Yin, 2000) has shown that, the vertical effective stress is much higher within about 5 times the equivalent radius of the vertical drain than in the rest part even for an ideally installed vertical drain. This in turn will cause a zone of reduced permeability due to the effective stress increase. The term ‘smear’ is used to describe all the effects mentioned above. The extent and permeability of the smear zone depends on many factors, such as the installation procedure, size and shape of the mandrel and soil fabrics. Field and laboratory observations (Bergado et al., 1991; Madhav et al., 1993; Indraratna and Redana, 1998; Hird and Moseley, 2000) have shown a continuous variation of soil permeability with radial distance away from the centre of the drain. However, it has always been hard to quantify the smear effects (Hansbo, 1997) although some attempts (Madhav et al., 1993; Chai et al., 1997; Hawlader et al., 2002) have been made to model a gradual variation of permeability with radius. The detrimental effect of smear on the efficiency of a vertical drain can be grossly taken into account by assuming that there will be a uniform smear zone around the drain for application purposes. Barron (1948) assumed two types of vertical strain that might occur in the clay layer: (a) ‘free vertical strain’ resulting from a uniform distribution of surface load and (b) ‘equal vertical strain’ resulting from imposing the same vertical deformation on the surface for soil. The soil in the smeared zone was treated as an incompressible material. Onoue (1988) extended Yoshikuni and Nakanodo’s (1974) rigorous solution to include the effects of smear. Simplified solutions were also obtained by some other researchers (Hansbo, 1981; Zeng and Xie, 1989; Xie et al., 1994). All the solutions mentioned above are based on the assumption that external loads are applied suddenly. A foundation construction loading or surcharge loading process takes some time, which in the case of vertical drains, may have a considerable influence on the consolidation behaviour, especially during the early stage of consolidation (Terzaghi, 1943; Olson, 1977; Zhu and Yin, 1998, 1999, 2001a,b; Li and Rowe, 2001). Olson (1977) obtained a solution using the equal strain assumption for the case of vertical drain without smear zone under a ramp load, that is, the vertical total stress increase varies linearly with time up to a maximum value. Olson’s solution (1977) was based on a formula derived by Carrillo (1942), ð1 UÞ ¼ ð1 Ur Þð1 Uv Þ; where U; Ur ; and Uv are the overall, the radial and the vertical average degree of consolidation. Theoretically speaking, this formula (Carrillo, 1942) is only valid for homogeneous equations, that is, for instantaneously applied loading. For non-homogeneous equations (loading is gradually applied), Carrillo’s (1942) formula can only be regarded as an approximate relationship. Zhu and Yin (2001a,b) presented a rigorous solution to the consolidation problem of vertical drain with horizontal and vertical drainage under ramp loading. However, to the authors’ knowledge, no rigorous solution to the consolidation problem of vertical drain with smear zone under ramp loading is available in the literature. In this paper, the authors present a mathematical solution to the above problem. The time is normalised using the related eigenvalues of the consolidation problem. It is
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
65
shown that the average degree of consolidation exhibits very good normalised behaviour. Eigenvalues and average degree of consolidation of the solution are presented in tables or charts for practical use. The influence of time-dependent loading is incorporated in the solution charts.
2. Basic equations and solutions To obtain the governing equation for consolidation of soil with vertical drains, it is assumed that (a) the soil is fully saturated, (b) water and soil particles are incompressible, (c) Darcy’s law is valid, (d) strains are small, (e) all compressive strains within the soil mass occur in the vertical direction and (f) the coefficient of compressibility is constant (Terzaghi, 1943; Barron, 1948). As in Barron (1948), the problem is simplified to an axisymmetric one as shown in Fig. 1. The differential equation for the dissipation of excess porewater pressure using a free strain assumption is 8 2 @ u 1 @u @2 u @s > > > c þ þ cv 2 þ ; rw orors ; rs < 2 @r r @r @z @t @u ð1Þ ¼ 2 2 @t > @ u 1 @u @ u @s > > þ þ cv 2 þ ; rs orore ; : cr @r2 r @r @z @t where u is the excess porewater pressure; s is the vertical total stress increase; t is the time; r is the radial co-ordinate; z is the vertical co-ordinate; rw is the radius of drain; rs is the radius of the smear zone; re is the radius of the equivalent cylindrical block of soil; cr ¼ kr =gw mv is the horizontal (or radial) consolidation coefficient in undisturbed zone; crs ¼ ks =gw mv is the horizontal (or radial) consolidation coefficient in smear zone; cv ¼ kv =gw mv is the vertical consolidation coefficient; gw is the unit weight of water; mv is the compressibility; kr is the horizontal hydraulic conductivity in undisturbed zone; ks is the horizontal hydraulic conductivity in smear zone; and kv is the vertical hydraulic conductivity. Eq. (1) is similar to the equation derived by Barron (1948) except for the non-homogenous term @s=@t and a compressible smear zone (Hansbo, 1981; Zhu and Yin, 2001a). Smear zone
rs
u=0 H 0
rw z
r re
Fig. 1. Co-ordinates and boundary conditions for axisymmetric consolidation.
ARTICLE IN PRESS 66
G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
σ
σo
tc
t
Fig. 2. Variation of the vertical total stress.
The vertical total stress increase is assumed to vary linearly with time and remain unchanged after time tc (see Fig. 2), that is, t sðr; z; tÞ ¼ s0 min 1; ; ð2Þ tc where s0 is a constant (final vertical total stress increment). The ‘‘min’’ means taking the minimum value between 1 and t=tc : At the interface of the smeared zone and undisturbed zone, the excess pore pressure is the same and the rate of flow out of the undisturbed zone must be equal to that into the smeared zone, so that 8 > < ujr¼rs ¼ ujr¼rþs ; ð3Þ @u @u > ¼ kr : : ks @r @r r¼rþs r¼r s The following commonly encountered boundary conditions are studied in this paper: 8 @u > > ¼ 0; > uðr; 0; tÞ ¼ 0; < @z z¼H ð4Þ > @u > > ¼ 0: : uðrw ; z; tÞ ¼ 0; @r r¼re Eq. (4) means that water is freely drained at the top of the soil and along well ðr ¼ rw ), but impermeable at r ¼ re and at the bottom. In the following solution, the initial excess porewater pressure is assumed to be zero. If drainage is available at the bottom of the soil, the drainage path H is halved. Using the method of separation of variables, the consolidation equation (1) under the given loading condition in Eq. (2) and the boundary conditions in Eqs. (3) and (4) can be solved. Letting rffiffiffiffiffi z re cv r2w rs cr Z¼ ; N¼ ; L¼ ; s ¼ ; Z ¼ ; ð5Þ H rw cr H 2 rw crs
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
the solution to Eq. (1) becomes N X Amn ðTÞRm ðrÞ sinðln ZÞ: u¼
67
ð6Þ
m;n¼1
In Eq. (5), Z is a dimensionless vertical co-ordinate, N is the ratio of the equivalent radius over well radius, s is the ratio of the smeared zone radius over well radius, Z expresses the difference in the coefficients of consolidation between the undisturbed zone and the smeared zone, and L is related to vertical and horizontal consolidation coefficients (cv and cr ), radius of well ðrw Þ and vertical drainage distance H: If cv is zero or H is infinite, parameter L is zero, which means horizontal water flow and horizontal consolidation only. If cr is zero, L is infinite, which implies vertical water flow and vertical consolidation only. If rw is zero, N is infinite although L is zero. This case is a vertical consolidation case. The expressions for Amn ; Rm and ln in Eq. (6) are discussed as follows: Rm in Eq. (6) can be expressed as a function of Bessel functions of the first kind ðJ0 ; J1 Þ and of the second kind ðY0 ; Y1 Þ as in Eq. (7). 8 r > m > > V Zm ; rw orors ; m < 0 rw ð7aÞ Rm ðrÞ ¼ > r > > W0m mm ; rs orore ; : rw Vnm ðxÞ ¼
Y0 ðZmm ÞJn ðxÞ J0 ðZmm ÞYn ðxÞ ; Y0 ðZmm ÞJ0 ðZsmm Þ J0 ðZmm ÞY0 ðZsmm Þ
ð7bÞ
Y1 ðNmm ÞJn ðxÞ J1 ðNmm ÞYn ðxÞ ; Y1 ðNmm ÞJ0 ðsmm Þ J1 ðNmm ÞY0 ðsmm Þ
ð7cÞ
Wnm ðxÞ ¼
where Jn ; Yn denote, respectively, Bessel functions of the first and second kind of order n: The quantity mm is the mth positive root of the following Eq. (8): ½Y0 ðZmm ÞJ1 ðZsmm Þ J0 ðZmm ÞY1 ðZsmm Þ½Y1 ðNmm ÞJ0 ðsmm Þ J1 ðNmm ÞY0 ðsmm Þ Z½Y0 ðZmm ÞJ0 ðZsmm Þ J0 ðZmm ÞY0 ðZsmm Þ½Y1 ðNmm ÞJ1 ðsmm Þ J1 ðNmm ÞY1 ðsmm Þ ¼ 0:
ð8Þ The first eigenvalue m1 of Eq. (8) is listed in Table 1 for different N; s and Z values. Amn in Eq. (6) is " !# 8 2 2 2 2 2B s ðm þ l LÞ m þ l L > m 0 1 1 n > > T ; TpTc ; 1 exp m2 > > > ln ðm2m þ l2n LÞTc m1 þ l21 L > > > " !# > < 2B s ðm2 þ l2 LÞ m2m þ l2n L m 0 1 1 ð9Þ Tc 1 exp 2 Amn ðTÞ ¼ > ln ðm2m þ l2n LÞTc m1 þ l21 L > > > " # > > > > m2m þ l2n L > > ðT Tc Þ ; TXTc ; : exp 2 m1 þ l21 L
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
68
Table 1 Eigenvalue m1 for different N; s; and Z N s ¼ 1:2 1.4
1.6
2
2.8
4.5
2
s ¼ 1:2 1.4
1.6
2
2.8
4.5
2
5 6 7 8 10 12 14 16 18 20 23 26 30 35 40 45 50
Z ¼1 0.282 0.218 0.177 0.148 0.11 0.0874 0.072 0.0611 0.0529 0.0465 0.0393 0.0339 0.0287 0.0239 0.0204 0.0178 0.0158
5 6 7 8 10 12 14 16 18 20 23 26 30 35 40 45 50
Z2 ¼ 2 0.262 0.204 0.166 0.139 0.105 0.0835 0.069 0.0586 0.0509 0.0448 0.0379 0.0328 0.0277 0.0232 0.0199 0.0173 0.0154
0.247 0.194 0.159 0.134 0.101 0.0805 0.0667 0.0568 0.0493 0.0435 0.0369 0.032 0.027 0.0226 0.0194 0.017 0.015
0.237 0.186 0.153 0.129 0.0977 0.0782 0.0649 0.0553 0.0481 0.0425 0.0361 0.0313 0.0265 0.0222 0.019 0.0166 0.0148
0.223 0.176 0.145 0.122 0.0931 0.0748 0.0622 0.0531 0.0463 0.0409 0.0348 0.0302 0.0256 0.0215 0.0184 0.0161 0.0143
5 6 7 8 10 12 14 16 18 20
Z2 ¼ 3:5 0.238 0.213 0.187 0.169 0.153 0.139 0.129 0.118 0.098 0.0903 0.0784 0.0726 0.0651 0.0605 0.0555 0.0518 0.0482 0.0451 0.0426 0.0399
0.197 0.157 0.13 0.111 0.0849 0.0685 0.0573 0.0491 0.0429 0.038
0.178 0.142 0.118 0.101 0.0778 0.063 0.0529 0.0454 0.0398 0.0353
0.282 0.218 0.177 0.148 0.11 0.0874 0.072 0.0611 0.0529 0.0465 0.0393 0.0339 0.0287 0.0239 0.0204 0.0178 0.0158
0.282 0.218 0.177 0.148 0.11 0.0874 0.072 0.0611 0.0529 0.0465 0.0393 0.0339 0.0287 0.0239 0.0204 0.0178 0.0158
0.282 0.218 0.177 0.148 0.11 0.0874 0.072 0.0611 0.0529 0.0465 0.0393 0.0339 0.0287 0.0239 0.0204 0.0178 0.0158
0.282 0.218 0.177 0.148 0.11 0.0874 0.072 0.0611 0.0529 0.0465 0.0393 0.0339 0.0287 0.0239 0.0204 0.0178 0.0158
Z ¼ 1:4 0.274 0.267 0.212 0.207 0.172 0.169 0.144 0.142 0.108 0.106 0.0858 0.0844 0.0708 0.0698 0.0601 0.0592 0.052 0.0514 0.0458 0.0452 0.0387 0.0383 0.0335 0.0331 0.0283 0.028 0.0236 0.0234 0.0202 0.02 0.0176 0.0175 0.0156 0.0155
0.261 0.204 0.166 0.139 0.105 0.0834 0.0689 0.0586 0.0508 0.0448 0.0379 0.0328 0.0277 0.0232 0.0198 0.0173 0.0154
0.253 0.198 0.161 0.136 0.102 0.0816 0.0676 0.0575 0.0499 0.044 0.0373 0.0323 0.0273 0.0228 0.0196 0.0171 0.0152
0.244 0.191 0.156 0.131 0.0992 0.0793 0.0657 0.056 0.0486 0.0429 0.0364 0.0316 0.0267 0.0224 0.0192 0.0168 0.0149
0.239 0.185 0.151 0.127 0.0956 0.0764 0.0634 0.0541 0.0471 0.0416 0.0353 0.0306 0.026 0.0218 0.0187 0.0163 0.0145
0.208 0.164 0.135 0.115 0.0874 0.0704 0.0587 0.0503 0.0438 0.0388 0.0331 0.0288 0.0244 0.0205 0.0177 0.0155 0.0138
0.2 0.155 0.127 0.107 0.0816 0.0657 0.0549 0.047 0.0411 0.0364 0.0311 0.0271 0.0231 0.0194 0.0167 0.0147 0.0131
Z2 ¼ 2:5 0.253 0.234 0.198 0.184 0.161 0.151 0.136 0.128 0.102 0.097 0.0817 0.0776 0.0676 0.0645 0.0575 0.055 0.0499 0.0478 0.044 0.0422 0.0373 0.0359 0.0323 0.0311 0.0273 0.0263 0.0228 0.0221 0.0196 0.0189 0.0171 0.0166 0.0152 0.0147
0.221 0.175 0.144 0.122 0.0929 0.0746 0.0621 0.053 0.0462 0.0408 0.0347 0.0301 0.0256 0.0214 0.0184 0.0161 0.0143
0.204 0.162 0.134 0.114 0.0871 0.0702 0.0586 0.0501 0.0438 0.0388 0.033 0.0287 0.0244 0.0205 0.0176 0.0155 0.0137
0.187 0.149 0.123 0.105 0.0802 0.0649 0.0543 0.0466 0.0408 0.0362 0.0309 0.0269 0.0229 0.0193 0.0166 0.0146 0.013
0.179 0.139 0.114 0.0963 0.0736 0.0595 0.0499 0.0428 0.0375 0.0333 0.0285 0.0249 0.0212 0.0179 0.0155 0.0136 0.0121
0.16 0.127 0.106 0.0905 0.07 0.0569 0.0478 0.0412 0.0361 0.0322
0.151 0.118 0.0965 0.0819 0.0629 0.0511 0.0429 0.037 0.0325 0.029
Z2 ¼ 5 0.219 0.173 0.143 0.121 0.0923 0.0742 0.0618 0.0528 0.046 0.0407
0.172 0.138 0.115 0.0984 0.076 0.0617 0.0518 0.0446 0.039 0.0347
0.152 0.122 0.102 0.0877 0.0681 0.0555 0.0467 0.0403 0.0354 0.0315
0.135 0.108 0.0899 0.0771 0.06 0.049 0.0414 0.0357 0.0314 0.028
0.126 0.0985 0.0809 0.0688 0.0531 0.0432 0.0364 0.0315 0.0277 0.0247
0.282 0.218 0.177 0.148 0.11 0.0874 0.072 0.0611 0.0529 0.0465 0.0393 0.0339 0.0287 0.0239 0.0204 0.0178 0.0158
0.189 0.151 0.126 0.107 0.0824 0.0667 0.0558 0.0479 0.0418 0.0371
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
69
Table 1 (continued) N s ¼ 1:2 1.4
1.6
2
2.8
4.5
s ¼ 1:2 1.4
1.6
2
2.8
4.5
23 26 30 35 40 45 50
0.0361 0.0313 0.0265 0.0222 0.0191 0.0167 0.0148
0.0324 0.0282 0.024 0.0202 0.0174 0.0152 0.0135
0.0302 0.0263 0.0224 0.0189 0.0163 0.0143 0.0127
0.0276 0.0241 0.0206 0.0174 0.015 0.0132 0.0118
0.0249 0.0217 0.0186 0.0158 0.0137 0.012 0.0107
0.0346 0.03 0.0255 0.0214 0.0184 0.0161 0.0143
0.0317 0.0276 0.0235 0.0197 0.017 0.0149 0.0133
0.0297 0.0259 0.0221 0.0186 0.0161 0.0141 0.0126
0.027 0.0236 0.0202 0.0171 0.0148 0.013 0.0116
0.0241 0.0211 0.0181 0.0153 0.0133 0.0117 0.0105
0.0213 0.0187 0.016 0.0136 0.0118 0.0104 0.00934
5 6 7 8 10 12 14 16 18 20 23 26 30 35 40 45 50
Z2 ¼ 7 0.199 0.159 0.132 0.112 0.0861 0.0695 0.058 0.0497 0.0434 0.0384 0.0328 0.0285 0.0242 0.0204 0.0175 0.0154 0.0137
0.13 0.105 0.0885 0.0761 0.0594 0.0487 0.0411 0.0356 0.0313 0.0279 0.024 0.021 0.018 0.0153 0.0133 0.0117 0.0104
Z2 ¼ 10 0.114 0.107 0.178 0.0918 0.0833 0.143 0.0768 0.0685 0.119 0.066 0.0583 0.102 0.0515 0.0451 0.0787 0.0423 0.0368 0.0638 0.0358 0.0311 0.0535 0.031 0.0269 0.0459 0.0273 0.0237 0.0402 0.0244 0.0212 0.0357 0.021 0.0183 0.0305 0.0185 0.0161 0.0266 0.0159 0.0138 0.0227 0.0135 0.0118 0.0191 0.0117 0.0102 0.0165 0.0103 0.00905 0.0145 0.00925 0.00811 0.0129
0.145 0.117 0.0986 0.0848 0.066 0.0539 0.0455 0.0393 0.0345 0.0307 0.0264 0.0231 0.0197 0.0167 0.0144 0.0127 0.0113
0.128 0.104 0.0875 0.0754 0.059 0.0483 0.0408 0.0353 0.0311 0.0278 0.0239 0.0209 0.0179 0.0152 0.0132 0.0116 0.0104
0.11 0.0896 0.0754 0.0651 0.051 0.0419 0.0355 0.0308 0.0272 0.0243 0.0209 0.0184 0.0158 0.0134 0.0117 0.0103 0.00922
0.0961 0.0773 0.0648 0.0558 0.0437 0.0359 0.0305 0.0264 0.0234 0.0209 0.0181 0.0159 0.0137 0.0116 0.0101 0.00895 0.00802
0.0894 0.0697 0.0574 0.0489 0.0379 0.031 0.0262 0.0227 0.0201 0.018 0.0155 0.0136 0.0118 0.01 0.00873 0.00773 0.00693
where
0.034 0.0295 0.0251 0.021 0.0181 0.0158 0.0141
0.167 0.135 0.112 0.0963 0.0745 0.0606 0.0509 0.0438 0.0384 0.0341 0.0292 0.0255 0.0217 0.0183 0.0158 0.0139 0.0124
R re
0.15 0.121 0.101 0.0869 0.0676 0.0551 0.0465 0.0401 0.0352 0.0313 0.0269 0.0235 0.0201 0.017 0.0147 0.0129 0.0115
r
rRm dr
rw
rR2m dr
Bm ¼ R rwe
ð10Þ
;
ln ¼ ðn 12Þp; m21 cv p2 p2 cr t 2 T ¼ cr 2 þ t ¼ m1 þ L 2 ; 2 rw rw 4H 4 2 2 2 m cv p p cr t c Tc ¼ cr 21 þ tc ¼ m21 þ L 2 : rw 4H 2 4 rw
ð11Þ ð12Þ ð13Þ
T is a dimensionless time factor and Tc is the time factor corresponding to construction time tc : Both T and Tc are dependent on N; s and Z through eigenvalue m1 : It will be shown that the average degree of consolidation has very good normalised behaviour if the time is normalised using Eq. (12). The average degree of consolidation UðTÞ is defined as RH R re SðtÞ rw r dr 0 mv ðs uÞ dz ¼ R re UðTÞ ¼ ; ð14Þ RH Sf r r dr 0 mv st¼N dz w
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
70
T 0.001 0
0.01
0.1
1
10
10 20
cr c rs
30
U (%)
40
1
50
2 60 70 80
4
N=20 s=2 L=1E-4 Tc=0.3
8
90 100 Fig. 3. Normalised time vs. degree of consolidation for different cr =crs :
where Sf and SðtÞ are average final settlement and settlement at t; respectively. Using this definition, the average degree of consolidation UðTÞ can be obtained from Eq. (6), and is presented in Eq. (15). 8 2 > 32Cm ðm21 þ p4 LÞ > > T PN P > 2 2 m¼1 n¼1;3;5;y; 2 2 2 > > Tc n p ðmm þ n 4p LÞTc > > > " ! # > > 2 n 2 p2 > > > 1 exp mm þ 4 L T > TpTc ; > 2 > > m21 þ p4 L > > > > 2 < PN P 32Cm ðm21 þ p4 LÞ ð15Þ UðTÞ ¼ 1 m¼1 n¼1;3;5;y; 2 2 > n2 p2 ðm2m þ n 4p LÞTc > > > " !# > > 2 n 2 p2 > m þ L > m 4 > > 1 exp Tc 2 > > m21 þ p4 L > > > " # > > 2 2 > > m2m þ n 4p L > > TXTc ; > : exp m2 þ p2 L ðT Tc Þ 1
4
Fig. 4. (a) Degree of consolidation for s ¼ 2:5 and L ¼ 0: (b) Degree of consolidation for s ¼ 2:5 and L ¼ 1 105 : (c) Degree of consolidation for s ¼ 2:5 and L ¼ 3 104 : (d) Degree of consolidation for s ¼ 2:5 and L ¼ 1 102 : (e) Degree of consolidation for s ¼ 4:5 and L ¼ 0: (f) Degree of consolidation for s ¼ 4:5 and L ¼ 1 105 : (g) Degree of consolidation for s ¼ 4:5 and L ¼ 3 104 : (h) Degree of consolidation for s ¼ 4:5 and L ¼ 1 102 :
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
(a)
(b)
(c)
(d)
(e)
(f )
(g)
(h)
71
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
72
where Cm ¼
ðN 2
½V1m ðZmm Þ2 : ½V1m ðZmm Þ2 þ ð1 ccrsr Þs2 ½V1m ðsZmm Þ2 g
1ÞZ2 m2m fN 2 ½W0m ðNmm Þ2
ð16Þ Eq. (15) can be used to calculate the average degree of consolidation of soils with vertical drains considering smear effects. It is clear from Eq. (15) that the vertical drainage becomes insignificant if L is considerably smaller than m21 such as in the case of a thick soil layer. If the vertical total stress s is composed of several ramp loads, the superposition method may be used to calculate the excess porewater pressure and the average degree of consolidation. It should be noted that Eq. (15) is also valid if the final total stress s0 is linear with depth for double drainage boundary conditions.
3. Influence of N; s; Z and L on average degree of consolidation The average degree of consolidation depends on N; s; Z and L in addition to Tc and T: Clearly, these parameters have very large influences on the average degree of consolidation. It is not easy to prepare solution charts to include all the parameters. However, if the time factor T is defined by Eq. (12), the average degree of consolidation shows very little sensitivity to N; s; Z and L: Fig. 3 shows a typical pffiffiffiffiffiffiffiffiffiffiffi result of the effect of Z ð¼ cr =crs Þ on the relations between normalised time factor T and the average degree of consolidation U for N ¼ 20; s ¼ 2; L ¼ 104 ; and Tc ¼ 0:3: Obviously, the difference is very small for different Z-values. Similar results are also obtained for N; s; and L: This indicates that the relationship of average degree of consolidation U vs. time factor T is approximately independent of the dimensionless parameters N; s; Z and L: This is because the influences of N; s; Z and L are included in the normalised time factor T in Eq. (12) since m1 is related to N; s; Z in Table 1. The normalised time factor T makes that the relationship of U vs. T is primarily dependent on the construction time factor Tc only. In this way, the total number of charts of U vs. T can be largely reduced. For practical applications, the average degree of consolidation is prepared for cr =crs ¼ 2:5; s ¼ 2:5 and for cr =crs ¼ 2:5; s ¼ 4:5 as shown in Fig. 4. If s is in the range of 1.2–2.5, the solution charts for cr =crs ¼ 2:5; s ¼ 2:5 can be used. If s is in the range of 2.5–4.5, the solution charts for cr =crs ¼ 2:5; s ¼ 4:5 can be adopted. The maximum error in average degree of consolidation induced in the substitution of cr =crs by 2.5 and s by 2.5 or 4.5 is less than 3% for the whole range calculated (N ¼ 5–80, s ¼ 1:2–4.5, cr =crs ¼ 1–10, and L ¼ 0–0.01). 4. Example of application As an example of calculation, let us consider the case where H ¼ 2:5 m; rw ¼ 25 mm; rs ¼ 100 mm; re ¼ 0:5 m; cv ¼ crs ¼ 1:5 m2 =year; cr ¼ 3:0 m2 =year;
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
73
subjected to a single ramp loading with a construction time of tc ¼ 0:15 year: Using Eq. (5), N ¼ 20; s ¼ 4; L ¼ 5 105 ; cr =crs ¼ 2 and m1 ¼ 0:0371 from Table 1. The construction time factor Tc is calculated to be 1.08 using Eq. (13). The time factor T for real consolidation time t ¼ 0:3 year is calculated using Eq. (12) to be T ¼ 2:16: Using Figs. 4(f) and (g) and interpolation, the average degrees of consolidation at time t ¼ tc and at time t ¼ 0:3 year since the start of construction are 45.5% and 82.5%, respectively. For a thick soil layer with H ¼ 10 m while the other parameters are the same as the above example, N ¼ 20; s ¼ 4; L ¼ 3:125 106 ; cr =crs ¼ 2 and m1 ¼ 0:0371: The construction time factor Tc is calculated to be 1.0 using Eq. (13). The time factor T for real consolidation time t ¼ 0:3 year is calculated using Eq. (12) to be T ¼ 2:0: Using Figs. 4(f) and (g) and interpolation, the average degrees of consolidation at time t ¼ tc and at time t ¼ 0:3 year since the start of construction are 39.0% and 78.0%, respectively. The influence of vertical drainage on the average degree of consolidation in this case is small.
5. Summary and conclusion An analytical solution is obtained for the consolidation analysis of soil with a vertical drain under ramp loading considering smear effects. The solution is described in detail. A normalised time factor T is suggested. Results are presented in figures and tables for practical applications. It is found in the paper that the average degree of consolidation U vs. the time factor T can be considered to be approximately independent of parameters N; s; Z and L; showing good normalisation behaviour.
Acknowledgements Financial supports from the Hong Kong Polytechnic University and from the RGC grant of the University Grant Council of the Hong Kong SAR Government of China are gratefully acknowledged.
References Barron, R.A., 1948. Consolidation of fine-grained soils by drain wells. Transactions of the ASCE 113 (2346), 718–742. Bergado, D.T., Asakami, H., Alfaro, M.C., Balasubramaniam, A.S., 1991. Smear effects of vertical drains on soft Bangkok clay. Journal of Geotechnical Engineering, ASCE 117 (10), 1509–1530. Carrillo, N., 1942. Simple two- and three-dimensional cases in the theory of consolidation of soils. Journal of Mathematical Physics 21 (1), 1–5. Chai, J.C., Miura, N., Sakajo, S., 1997. A theoretical study of smear effect around vertical drain. Proceedings of the 14th ICSMFE, Hamburg, Vol. 3, pp. 1581–1584.
ARTICLE IN PRESS 74
G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
Hansbo, S., 1981. Consolidation of fine-grained soils by prefabricated drains. Proceedings of the Tenth International Conference on Soil Mechanics and Foundation Engineering, Stochholm, Sweden, Vol. 3, pp. 667–682. Hansbo, S., 1997. Aspect of vertical drain design: Darcian or non-Darcian flow. Geotechnique 47 (5), 983–992. Hawlader, B.C., Imai, G., Muhunthan, B., 2002. Numerical study of the factors affecting the consolidation of clay with vertical drains. Geotextiles and Geomembranes 20 (4), 213–239. Hird, C.C., Moseley, V.J., 2000. Model study of seepage in smear zones around vertical drains in layered soil. Geotechnique 50 (1), 89–97. Indraratna, B., Redana, I.W., 1998. Laboratory determination of smear zone due to vertical drain installation. Journal of Geotechnical and Geoenvironmental Engineering 124 (2), 180–184. Li, A.L., Rowe, R.K., 2001. Combined effects of reinforcement and prefabricated vertical drains on embankment performance. Canadian Geotechnical Journal 38, 1266–1282. Madhav, R., Park, Y.M., Miura, N., 1993. Modeling and study of smear zones around band shaped drains. Soils and Foundations 33 (4), 135–147. Olson, R.E., 1977. Consolidation under time-dependent loading. Journal of the Geotechnical Engineering Division ASCE 103 (GT1), 55–60. Onoue, A., 1988. Consolidation by vertical drains taking well resistance and smear into consideration. Soils and Foundations 28 (4), 65–174. Terzaghi, K., 1943. Theoretical Soil Mechanics. Wiley, New York. Xie, K.H., Lee, P.K.K., Cheung, Y.K., 1994. Consolidation of a two-layer system with ideal drains. Proceedings of the Eighth International Conference on Computer Methods and Advances in Geomechanics, Morgantown, West Virginia, USA, Vol. 1, pp. 789–794. Yoshikuni, H., Nakanodo, H., 1974. Consolidation of soils by vertical drain wells with finite permeability. Soils and Foundations 14 (2), 35–46. Zeng, G.X., Xie, K.H., 1989. New development of the vertical drain theories. Proceedings of the 12th International Conference on Soil Mechanics and Foundation Engineering, Rotterdam, The Netherlands, Vol. 2, pp. 1435–1438. Zhu, G.F., Yin, J.-H., 1998. Consolidation of soil under depth-dependent ramp load. Canadian Geotechnical Journal 35 (3), 344–350. Zhu, G.F., Yin, J.-H., 1999. Consolidation of double soil layers under depth-dependent ramp load. Geotechnique 49 (3), 415–421. Zhu, G.F., Yin, J.-H., 2000. Finite element analysis of consolidation of soils with vertical drain. International Journal for Numerical Analysis and Methods in Geomechanics 24 (4), 337–366. Zhu, G.F., Yin, J.-H., 2001a. Consolidation of soil with vertical and horizontal drainage. Geotechnique 51 (4), 361–367. Zhu, G.F., Yin, J.-H., 2001b. Design charts for vertical drains considering construction time. Canadian Geotechnical Journal 38 (5), 1142–1148.
Geotextiles and Geomembranes 22 (2004) 63–74
Consolidation analysis of soil with vertical and horizontal drainage under ramp loading considering smear effects Guofu Zhu, Jian-Hua Yin* Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Received 12 February 2002; received in revised form 16 April 2003
Abstract The installation process of a vertical drain with a mandrel normally causes a smear zone with reduced permeability around the drain. The smear zone affects the consolidation of the soil. This paper presents a new analytical solution for the consolidation analysis of soil with a vertical drain under ramp loading considering the smear effects. The solution is described in detail in this paper. A new normalised time factor, T; is suggested. It is shown in the paper that the average degree of consolidation exhibits very good normalised behaviour using this new factor T: Results are presented in tables and charts for practical applications. The influence of time-dependent loading is incorporated in the solution charts. r 2003 Elsevier Ltd. All rights reserved. Keywords: Vertical drain; Consolidation; Theoretical analysis; Permeability; Time dependence; Smear zone; Soil
1. Introduction Vertical drains are widely used to facilitate the consolidation of soil. The installation process of a vertical drain, for example, a prefabricated band drain, involves penetration, displacement of the soil by the mandrel, and consequently disturbance of the soil around the drain. Two principal disturbing mechanisms exist: first, displacement of the soil to create space for the mandrel; and second the *Corresponding author. Tel. +852-2766-6065; fax: +852-2334-6389. E-mail address: [email protected] (J.-H. Yin). 0266-1144/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0266-1144(03)00052-9
ARTICLE IN PRESS 64
G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
dragging of soil by the sides of the mandrel. Substantial reduction in the soil permeability occurs around the drain due to this disturbance. A fully coupled finite element consolidation analysis (Zhu and Yin, 2000) has shown that, the vertical effective stress is much higher within about 5 times the equivalent radius of the vertical drain than in the rest part even for an ideally installed vertical drain. This in turn will cause a zone of reduced permeability due to the effective stress increase. The term ‘smear’ is used to describe all the effects mentioned above. The extent and permeability of the smear zone depends on many factors, such as the installation procedure, size and shape of the mandrel and soil fabrics. Field and laboratory observations (Bergado et al., 1991; Madhav et al., 1993; Indraratna and Redana, 1998; Hird and Moseley, 2000) have shown a continuous variation of soil permeability with radial distance away from the centre of the drain. However, it has always been hard to quantify the smear effects (Hansbo, 1997) although some attempts (Madhav et al., 1993; Chai et al., 1997; Hawlader et al., 2002) have been made to model a gradual variation of permeability with radius. The detrimental effect of smear on the efficiency of a vertical drain can be grossly taken into account by assuming that there will be a uniform smear zone around the drain for application purposes. Barron (1948) assumed two types of vertical strain that might occur in the clay layer: (a) ‘free vertical strain’ resulting from a uniform distribution of surface load and (b) ‘equal vertical strain’ resulting from imposing the same vertical deformation on the surface for soil. The soil in the smeared zone was treated as an incompressible material. Onoue (1988) extended Yoshikuni and Nakanodo’s (1974) rigorous solution to include the effects of smear. Simplified solutions were also obtained by some other researchers (Hansbo, 1981; Zeng and Xie, 1989; Xie et al., 1994). All the solutions mentioned above are based on the assumption that external loads are applied suddenly. A foundation construction loading or surcharge loading process takes some time, which in the case of vertical drains, may have a considerable influence on the consolidation behaviour, especially during the early stage of consolidation (Terzaghi, 1943; Olson, 1977; Zhu and Yin, 1998, 1999, 2001a,b; Li and Rowe, 2001). Olson (1977) obtained a solution using the equal strain assumption for the case of vertical drain without smear zone under a ramp load, that is, the vertical total stress increase varies linearly with time up to a maximum value. Olson’s solution (1977) was based on a formula derived by Carrillo (1942), ð1 UÞ ¼ ð1 Ur Þð1 Uv Þ; where U; Ur ; and Uv are the overall, the radial and the vertical average degree of consolidation. Theoretically speaking, this formula (Carrillo, 1942) is only valid for homogeneous equations, that is, for instantaneously applied loading. For non-homogeneous equations (loading is gradually applied), Carrillo’s (1942) formula can only be regarded as an approximate relationship. Zhu and Yin (2001a,b) presented a rigorous solution to the consolidation problem of vertical drain with horizontal and vertical drainage under ramp loading. However, to the authors’ knowledge, no rigorous solution to the consolidation problem of vertical drain with smear zone under ramp loading is available in the literature. In this paper, the authors present a mathematical solution to the above problem. The time is normalised using the related eigenvalues of the consolidation problem. It is
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
65
shown that the average degree of consolidation exhibits very good normalised behaviour. Eigenvalues and average degree of consolidation of the solution are presented in tables or charts for practical use. The influence of time-dependent loading is incorporated in the solution charts.
2. Basic equations and solutions To obtain the governing equation for consolidation of soil with vertical drains, it is assumed that (a) the soil is fully saturated, (b) water and soil particles are incompressible, (c) Darcy’s law is valid, (d) strains are small, (e) all compressive strains within the soil mass occur in the vertical direction and (f) the coefficient of compressibility is constant (Terzaghi, 1943; Barron, 1948). As in Barron (1948), the problem is simplified to an axisymmetric one as shown in Fig. 1. The differential equation for the dissipation of excess porewater pressure using a free strain assumption is 8 2 @ u 1 @u @2 u @s > > > c þ þ cv 2 þ ; rw orors ; rs < 2 @r r @r @z @t @u ð1Þ ¼ 2 2 @t > @ u 1 @u @ u @s > > þ þ cv 2 þ ; rs orore ; : cr @r2 r @r @z @t where u is the excess porewater pressure; s is the vertical total stress increase; t is the time; r is the radial co-ordinate; z is the vertical co-ordinate; rw is the radius of drain; rs is the radius of the smear zone; re is the radius of the equivalent cylindrical block of soil; cr ¼ kr =gw mv is the horizontal (or radial) consolidation coefficient in undisturbed zone; crs ¼ ks =gw mv is the horizontal (or radial) consolidation coefficient in smear zone; cv ¼ kv =gw mv is the vertical consolidation coefficient; gw is the unit weight of water; mv is the compressibility; kr is the horizontal hydraulic conductivity in undisturbed zone; ks is the horizontal hydraulic conductivity in smear zone; and kv is the vertical hydraulic conductivity. Eq. (1) is similar to the equation derived by Barron (1948) except for the non-homogenous term @s=@t and a compressible smear zone (Hansbo, 1981; Zhu and Yin, 2001a). Smear zone
rs
u=0 H 0
rw z
r re
Fig. 1. Co-ordinates and boundary conditions for axisymmetric consolidation.
ARTICLE IN PRESS 66
G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
σ
σo
tc
t
Fig. 2. Variation of the vertical total stress.
The vertical total stress increase is assumed to vary linearly with time and remain unchanged after time tc (see Fig. 2), that is, t sðr; z; tÞ ¼ s0 min 1; ; ð2Þ tc where s0 is a constant (final vertical total stress increment). The ‘‘min’’ means taking the minimum value between 1 and t=tc : At the interface of the smeared zone and undisturbed zone, the excess pore pressure is the same and the rate of flow out of the undisturbed zone must be equal to that into the smeared zone, so that 8 > < ujr¼rs ¼ ujr¼rþs ; ð3Þ @u @u > ¼ kr : : ks @r @r r¼rþs r¼r s The following commonly encountered boundary conditions are studied in this paper: 8 @u > > ¼ 0; > uðr; 0; tÞ ¼ 0; < @z z¼H ð4Þ > @u > > ¼ 0: : uðrw ; z; tÞ ¼ 0; @r r¼re Eq. (4) means that water is freely drained at the top of the soil and along well ðr ¼ rw ), but impermeable at r ¼ re and at the bottom. In the following solution, the initial excess porewater pressure is assumed to be zero. If drainage is available at the bottom of the soil, the drainage path H is halved. Using the method of separation of variables, the consolidation equation (1) under the given loading condition in Eq. (2) and the boundary conditions in Eqs. (3) and (4) can be solved. Letting rffiffiffiffiffi z re cv r2w rs cr Z¼ ; N¼ ; L¼ ; s ¼ ; Z ¼ ; ð5Þ H rw cr H 2 rw crs
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
the solution to Eq. (1) becomes N X Amn ðTÞRm ðrÞ sinðln ZÞ: u¼
67
ð6Þ
m;n¼1
In Eq. (5), Z is a dimensionless vertical co-ordinate, N is the ratio of the equivalent radius over well radius, s is the ratio of the smeared zone radius over well radius, Z expresses the difference in the coefficients of consolidation between the undisturbed zone and the smeared zone, and L is related to vertical and horizontal consolidation coefficients (cv and cr ), radius of well ðrw Þ and vertical drainage distance H: If cv is zero or H is infinite, parameter L is zero, which means horizontal water flow and horizontal consolidation only. If cr is zero, L is infinite, which implies vertical water flow and vertical consolidation only. If rw is zero, N is infinite although L is zero. This case is a vertical consolidation case. The expressions for Amn ; Rm and ln in Eq. (6) are discussed as follows: Rm in Eq. (6) can be expressed as a function of Bessel functions of the first kind ðJ0 ; J1 Þ and of the second kind ðY0 ; Y1 Þ as in Eq. (7). 8 r > m > > V Zm ; rw orors ; m < 0 rw ð7aÞ Rm ðrÞ ¼ > r > > W0m mm ; rs orore ; : rw Vnm ðxÞ ¼
Y0 ðZmm ÞJn ðxÞ J0 ðZmm ÞYn ðxÞ ; Y0 ðZmm ÞJ0 ðZsmm Þ J0 ðZmm ÞY0 ðZsmm Þ
ð7bÞ
Y1 ðNmm ÞJn ðxÞ J1 ðNmm ÞYn ðxÞ ; Y1 ðNmm ÞJ0 ðsmm Þ J1 ðNmm ÞY0 ðsmm Þ
ð7cÞ
Wnm ðxÞ ¼
where Jn ; Yn denote, respectively, Bessel functions of the first and second kind of order n: The quantity mm is the mth positive root of the following Eq. (8): ½Y0 ðZmm ÞJ1 ðZsmm Þ J0 ðZmm ÞY1 ðZsmm Þ½Y1 ðNmm ÞJ0 ðsmm Þ J1 ðNmm ÞY0 ðsmm Þ Z½Y0 ðZmm ÞJ0 ðZsmm Þ J0 ðZmm ÞY0 ðZsmm Þ½Y1 ðNmm ÞJ1 ðsmm Þ J1 ðNmm ÞY1 ðsmm Þ ¼ 0:
ð8Þ The first eigenvalue m1 of Eq. (8) is listed in Table 1 for different N; s and Z values. Amn in Eq. (6) is " !# 8 2 2 2 2 2B s ðm þ l LÞ m þ l L > m 0 1 1 n > > T ; TpTc ; 1 exp m2 > > > ln ðm2m þ l2n LÞTc m1 þ l21 L > > > " !# > < 2B s ðm2 þ l2 LÞ m2m þ l2n L m 0 1 1 ð9Þ Tc 1 exp 2 Amn ðTÞ ¼ > ln ðm2m þ l2n LÞTc m1 þ l21 L > > > " # > > > > m2m þ l2n L > > ðT Tc Þ ; TXTc ; : exp 2 m1 þ l21 L
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
68
Table 1 Eigenvalue m1 for different N; s; and Z N s ¼ 1:2 1.4
1.6
2
2.8
4.5
2
s ¼ 1:2 1.4
1.6
2
2.8
4.5
2
5 6 7 8 10 12 14 16 18 20 23 26 30 35 40 45 50
Z ¼1 0.282 0.218 0.177 0.148 0.11 0.0874 0.072 0.0611 0.0529 0.0465 0.0393 0.0339 0.0287 0.0239 0.0204 0.0178 0.0158
5 6 7 8 10 12 14 16 18 20 23 26 30 35 40 45 50
Z2 ¼ 2 0.262 0.204 0.166 0.139 0.105 0.0835 0.069 0.0586 0.0509 0.0448 0.0379 0.0328 0.0277 0.0232 0.0199 0.0173 0.0154
0.247 0.194 0.159 0.134 0.101 0.0805 0.0667 0.0568 0.0493 0.0435 0.0369 0.032 0.027 0.0226 0.0194 0.017 0.015
0.237 0.186 0.153 0.129 0.0977 0.0782 0.0649 0.0553 0.0481 0.0425 0.0361 0.0313 0.0265 0.0222 0.019 0.0166 0.0148
0.223 0.176 0.145 0.122 0.0931 0.0748 0.0622 0.0531 0.0463 0.0409 0.0348 0.0302 0.0256 0.0215 0.0184 0.0161 0.0143
5 6 7 8 10 12 14 16 18 20
Z2 ¼ 3:5 0.238 0.213 0.187 0.169 0.153 0.139 0.129 0.118 0.098 0.0903 0.0784 0.0726 0.0651 0.0605 0.0555 0.0518 0.0482 0.0451 0.0426 0.0399
0.197 0.157 0.13 0.111 0.0849 0.0685 0.0573 0.0491 0.0429 0.038
0.178 0.142 0.118 0.101 0.0778 0.063 0.0529 0.0454 0.0398 0.0353
0.282 0.218 0.177 0.148 0.11 0.0874 0.072 0.0611 0.0529 0.0465 0.0393 0.0339 0.0287 0.0239 0.0204 0.0178 0.0158
0.282 0.218 0.177 0.148 0.11 0.0874 0.072 0.0611 0.0529 0.0465 0.0393 0.0339 0.0287 0.0239 0.0204 0.0178 0.0158
0.282 0.218 0.177 0.148 0.11 0.0874 0.072 0.0611 0.0529 0.0465 0.0393 0.0339 0.0287 0.0239 0.0204 0.0178 0.0158
0.282 0.218 0.177 0.148 0.11 0.0874 0.072 0.0611 0.0529 0.0465 0.0393 0.0339 0.0287 0.0239 0.0204 0.0178 0.0158
Z ¼ 1:4 0.274 0.267 0.212 0.207 0.172 0.169 0.144 0.142 0.108 0.106 0.0858 0.0844 0.0708 0.0698 0.0601 0.0592 0.052 0.0514 0.0458 0.0452 0.0387 0.0383 0.0335 0.0331 0.0283 0.028 0.0236 0.0234 0.0202 0.02 0.0176 0.0175 0.0156 0.0155
0.261 0.204 0.166 0.139 0.105 0.0834 0.0689 0.0586 0.0508 0.0448 0.0379 0.0328 0.0277 0.0232 0.0198 0.0173 0.0154
0.253 0.198 0.161 0.136 0.102 0.0816 0.0676 0.0575 0.0499 0.044 0.0373 0.0323 0.0273 0.0228 0.0196 0.0171 0.0152
0.244 0.191 0.156 0.131 0.0992 0.0793 0.0657 0.056 0.0486 0.0429 0.0364 0.0316 0.0267 0.0224 0.0192 0.0168 0.0149
0.239 0.185 0.151 0.127 0.0956 0.0764 0.0634 0.0541 0.0471 0.0416 0.0353 0.0306 0.026 0.0218 0.0187 0.0163 0.0145
0.208 0.164 0.135 0.115 0.0874 0.0704 0.0587 0.0503 0.0438 0.0388 0.0331 0.0288 0.0244 0.0205 0.0177 0.0155 0.0138
0.2 0.155 0.127 0.107 0.0816 0.0657 0.0549 0.047 0.0411 0.0364 0.0311 0.0271 0.0231 0.0194 0.0167 0.0147 0.0131
Z2 ¼ 2:5 0.253 0.234 0.198 0.184 0.161 0.151 0.136 0.128 0.102 0.097 0.0817 0.0776 0.0676 0.0645 0.0575 0.055 0.0499 0.0478 0.044 0.0422 0.0373 0.0359 0.0323 0.0311 0.0273 0.0263 0.0228 0.0221 0.0196 0.0189 0.0171 0.0166 0.0152 0.0147
0.221 0.175 0.144 0.122 0.0929 0.0746 0.0621 0.053 0.0462 0.0408 0.0347 0.0301 0.0256 0.0214 0.0184 0.0161 0.0143
0.204 0.162 0.134 0.114 0.0871 0.0702 0.0586 0.0501 0.0438 0.0388 0.033 0.0287 0.0244 0.0205 0.0176 0.0155 0.0137
0.187 0.149 0.123 0.105 0.0802 0.0649 0.0543 0.0466 0.0408 0.0362 0.0309 0.0269 0.0229 0.0193 0.0166 0.0146 0.013
0.179 0.139 0.114 0.0963 0.0736 0.0595 0.0499 0.0428 0.0375 0.0333 0.0285 0.0249 0.0212 0.0179 0.0155 0.0136 0.0121
0.16 0.127 0.106 0.0905 0.07 0.0569 0.0478 0.0412 0.0361 0.0322
0.151 0.118 0.0965 0.0819 0.0629 0.0511 0.0429 0.037 0.0325 0.029
Z2 ¼ 5 0.219 0.173 0.143 0.121 0.0923 0.0742 0.0618 0.0528 0.046 0.0407
0.172 0.138 0.115 0.0984 0.076 0.0617 0.0518 0.0446 0.039 0.0347
0.152 0.122 0.102 0.0877 0.0681 0.0555 0.0467 0.0403 0.0354 0.0315
0.135 0.108 0.0899 0.0771 0.06 0.049 0.0414 0.0357 0.0314 0.028
0.126 0.0985 0.0809 0.0688 0.0531 0.0432 0.0364 0.0315 0.0277 0.0247
0.282 0.218 0.177 0.148 0.11 0.0874 0.072 0.0611 0.0529 0.0465 0.0393 0.0339 0.0287 0.0239 0.0204 0.0178 0.0158
0.189 0.151 0.126 0.107 0.0824 0.0667 0.0558 0.0479 0.0418 0.0371
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
69
Table 1 (continued) N s ¼ 1:2 1.4
1.6
2
2.8
4.5
s ¼ 1:2 1.4
1.6
2
2.8
4.5
23 26 30 35 40 45 50
0.0361 0.0313 0.0265 0.0222 0.0191 0.0167 0.0148
0.0324 0.0282 0.024 0.0202 0.0174 0.0152 0.0135
0.0302 0.0263 0.0224 0.0189 0.0163 0.0143 0.0127
0.0276 0.0241 0.0206 0.0174 0.015 0.0132 0.0118
0.0249 0.0217 0.0186 0.0158 0.0137 0.012 0.0107
0.0346 0.03 0.0255 0.0214 0.0184 0.0161 0.0143
0.0317 0.0276 0.0235 0.0197 0.017 0.0149 0.0133
0.0297 0.0259 0.0221 0.0186 0.0161 0.0141 0.0126
0.027 0.0236 0.0202 0.0171 0.0148 0.013 0.0116
0.0241 0.0211 0.0181 0.0153 0.0133 0.0117 0.0105
0.0213 0.0187 0.016 0.0136 0.0118 0.0104 0.00934
5 6 7 8 10 12 14 16 18 20 23 26 30 35 40 45 50
Z2 ¼ 7 0.199 0.159 0.132 0.112 0.0861 0.0695 0.058 0.0497 0.0434 0.0384 0.0328 0.0285 0.0242 0.0204 0.0175 0.0154 0.0137
0.13 0.105 0.0885 0.0761 0.0594 0.0487 0.0411 0.0356 0.0313 0.0279 0.024 0.021 0.018 0.0153 0.0133 0.0117 0.0104
Z2 ¼ 10 0.114 0.107 0.178 0.0918 0.0833 0.143 0.0768 0.0685 0.119 0.066 0.0583 0.102 0.0515 0.0451 0.0787 0.0423 0.0368 0.0638 0.0358 0.0311 0.0535 0.031 0.0269 0.0459 0.0273 0.0237 0.0402 0.0244 0.0212 0.0357 0.021 0.0183 0.0305 0.0185 0.0161 0.0266 0.0159 0.0138 0.0227 0.0135 0.0118 0.0191 0.0117 0.0102 0.0165 0.0103 0.00905 0.0145 0.00925 0.00811 0.0129
0.145 0.117 0.0986 0.0848 0.066 0.0539 0.0455 0.0393 0.0345 0.0307 0.0264 0.0231 0.0197 0.0167 0.0144 0.0127 0.0113
0.128 0.104 0.0875 0.0754 0.059 0.0483 0.0408 0.0353 0.0311 0.0278 0.0239 0.0209 0.0179 0.0152 0.0132 0.0116 0.0104
0.11 0.0896 0.0754 0.0651 0.051 0.0419 0.0355 0.0308 0.0272 0.0243 0.0209 0.0184 0.0158 0.0134 0.0117 0.0103 0.00922
0.0961 0.0773 0.0648 0.0558 0.0437 0.0359 0.0305 0.0264 0.0234 0.0209 0.0181 0.0159 0.0137 0.0116 0.0101 0.00895 0.00802
0.0894 0.0697 0.0574 0.0489 0.0379 0.031 0.0262 0.0227 0.0201 0.018 0.0155 0.0136 0.0118 0.01 0.00873 0.00773 0.00693
where
0.034 0.0295 0.0251 0.021 0.0181 0.0158 0.0141
0.167 0.135 0.112 0.0963 0.0745 0.0606 0.0509 0.0438 0.0384 0.0341 0.0292 0.0255 0.0217 0.0183 0.0158 0.0139 0.0124
R re
0.15 0.121 0.101 0.0869 0.0676 0.0551 0.0465 0.0401 0.0352 0.0313 0.0269 0.0235 0.0201 0.017 0.0147 0.0129 0.0115
r
rRm dr
rw
rR2m dr
Bm ¼ R rwe
ð10Þ
;
ln ¼ ðn 12Þp; m21 cv p2 p2 cr t 2 T ¼ cr 2 þ t ¼ m1 þ L 2 ; 2 rw rw 4H 4 2 2 2 m cv p p cr t c Tc ¼ cr 21 þ tc ¼ m21 þ L 2 : rw 4H 2 4 rw
ð11Þ ð12Þ ð13Þ
T is a dimensionless time factor and Tc is the time factor corresponding to construction time tc : Both T and Tc are dependent on N; s and Z through eigenvalue m1 : It will be shown that the average degree of consolidation has very good normalised behaviour if the time is normalised using Eq. (12). The average degree of consolidation UðTÞ is defined as RH R re SðtÞ rw r dr 0 mv ðs uÞ dz ¼ R re UðTÞ ¼ ; ð14Þ RH Sf r r dr 0 mv st¼N dz w
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
70
T 0.001 0
0.01
0.1
1
10
10 20
cr c rs
30
U (%)
40
1
50
2 60 70 80
4
N=20 s=2 L=1E-4 Tc=0.3
8
90 100 Fig. 3. Normalised time vs. degree of consolidation for different cr =crs :
where Sf and SðtÞ are average final settlement and settlement at t; respectively. Using this definition, the average degree of consolidation UðTÞ can be obtained from Eq. (6), and is presented in Eq. (15). 8 2 > 32Cm ðm21 þ p4 LÞ > > T PN P > 2 2 m¼1 n¼1;3;5;y; 2 2 2 > > Tc n p ðmm þ n 4p LÞTc > > > " ! # > > 2 n 2 p2 > > > 1 exp mm þ 4 L T > TpTc ; > 2 > > m21 þ p4 L > > > > 2 < PN P 32Cm ðm21 þ p4 LÞ ð15Þ UðTÞ ¼ 1 m¼1 n¼1;3;5;y; 2 2 > n2 p2 ðm2m þ n 4p LÞTc > > > " !# > > 2 n 2 p2 > m þ L > m 4 > > 1 exp Tc 2 > > m21 þ p4 L > > > " # > > 2 2 > > m2m þ n 4p L > > TXTc ; > : exp m2 þ p2 L ðT Tc Þ 1
4
Fig. 4. (a) Degree of consolidation for s ¼ 2:5 and L ¼ 0: (b) Degree of consolidation for s ¼ 2:5 and L ¼ 1 105 : (c) Degree of consolidation for s ¼ 2:5 and L ¼ 3 104 : (d) Degree of consolidation for s ¼ 2:5 and L ¼ 1 102 : (e) Degree of consolidation for s ¼ 4:5 and L ¼ 0: (f) Degree of consolidation for s ¼ 4:5 and L ¼ 1 105 : (g) Degree of consolidation for s ¼ 4:5 and L ¼ 3 104 : (h) Degree of consolidation for s ¼ 4:5 and L ¼ 1 102 :
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
(a)
(b)
(c)
(d)
(e)
(f )
(g)
(h)
71
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
72
where Cm ¼
ðN 2
½V1m ðZmm Þ2 : ½V1m ðZmm Þ2 þ ð1 ccrsr Þs2 ½V1m ðsZmm Þ2 g
1ÞZ2 m2m fN 2 ½W0m ðNmm Þ2
ð16Þ Eq. (15) can be used to calculate the average degree of consolidation of soils with vertical drains considering smear effects. It is clear from Eq. (15) that the vertical drainage becomes insignificant if L is considerably smaller than m21 such as in the case of a thick soil layer. If the vertical total stress s is composed of several ramp loads, the superposition method may be used to calculate the excess porewater pressure and the average degree of consolidation. It should be noted that Eq. (15) is also valid if the final total stress s0 is linear with depth for double drainage boundary conditions.
3. Influence of N; s; Z and L on average degree of consolidation The average degree of consolidation depends on N; s; Z and L in addition to Tc and T: Clearly, these parameters have very large influences on the average degree of consolidation. It is not easy to prepare solution charts to include all the parameters. However, if the time factor T is defined by Eq. (12), the average degree of consolidation shows very little sensitivity to N; s; Z and L: Fig. 3 shows a typical pffiffiffiffiffiffiffiffiffiffiffi result of the effect of Z ð¼ cr =crs Þ on the relations between normalised time factor T and the average degree of consolidation U for N ¼ 20; s ¼ 2; L ¼ 104 ; and Tc ¼ 0:3: Obviously, the difference is very small for different Z-values. Similar results are also obtained for N; s; and L: This indicates that the relationship of average degree of consolidation U vs. time factor T is approximately independent of the dimensionless parameters N; s; Z and L: This is because the influences of N; s; Z and L are included in the normalised time factor T in Eq. (12) since m1 is related to N; s; Z in Table 1. The normalised time factor T makes that the relationship of U vs. T is primarily dependent on the construction time factor Tc only. In this way, the total number of charts of U vs. T can be largely reduced. For practical applications, the average degree of consolidation is prepared for cr =crs ¼ 2:5; s ¼ 2:5 and for cr =crs ¼ 2:5; s ¼ 4:5 as shown in Fig. 4. If s is in the range of 1.2–2.5, the solution charts for cr =crs ¼ 2:5; s ¼ 2:5 can be used. If s is in the range of 2.5–4.5, the solution charts for cr =crs ¼ 2:5; s ¼ 4:5 can be adopted. The maximum error in average degree of consolidation induced in the substitution of cr =crs by 2.5 and s by 2.5 or 4.5 is less than 3% for the whole range calculated (N ¼ 5–80, s ¼ 1:2–4.5, cr =crs ¼ 1–10, and L ¼ 0–0.01). 4. Example of application As an example of calculation, let us consider the case where H ¼ 2:5 m; rw ¼ 25 mm; rs ¼ 100 mm; re ¼ 0:5 m; cv ¼ crs ¼ 1:5 m2 =year; cr ¼ 3:0 m2 =year;
ARTICLE IN PRESS G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
73
subjected to a single ramp loading with a construction time of tc ¼ 0:15 year: Using Eq. (5), N ¼ 20; s ¼ 4; L ¼ 5 105 ; cr =crs ¼ 2 and m1 ¼ 0:0371 from Table 1. The construction time factor Tc is calculated to be 1.08 using Eq. (13). The time factor T for real consolidation time t ¼ 0:3 year is calculated using Eq. (12) to be T ¼ 2:16: Using Figs. 4(f) and (g) and interpolation, the average degrees of consolidation at time t ¼ tc and at time t ¼ 0:3 year since the start of construction are 45.5% and 82.5%, respectively. For a thick soil layer with H ¼ 10 m while the other parameters are the same as the above example, N ¼ 20; s ¼ 4; L ¼ 3:125 106 ; cr =crs ¼ 2 and m1 ¼ 0:0371: The construction time factor Tc is calculated to be 1.0 using Eq. (13). The time factor T for real consolidation time t ¼ 0:3 year is calculated using Eq. (12) to be T ¼ 2:0: Using Figs. 4(f) and (g) and interpolation, the average degrees of consolidation at time t ¼ tc and at time t ¼ 0:3 year since the start of construction are 39.0% and 78.0%, respectively. The influence of vertical drainage on the average degree of consolidation in this case is small.
5. Summary and conclusion An analytical solution is obtained for the consolidation analysis of soil with a vertical drain under ramp loading considering smear effects. The solution is described in detail. A normalised time factor T is suggested. Results are presented in figures and tables for practical applications. It is found in the paper that the average degree of consolidation U vs. the time factor T can be considered to be approximately independent of parameters N; s; Z and L; showing good normalisation behaviour.
Acknowledgements Financial supports from the Hong Kong Polytechnic University and from the RGC grant of the University Grant Council of the Hong Kong SAR Government of China are gratefully acknowledged.
References Barron, R.A., 1948. Consolidation of fine-grained soils by drain wells. Transactions of the ASCE 113 (2346), 718–742. Bergado, D.T., Asakami, H., Alfaro, M.C., Balasubramaniam, A.S., 1991. Smear effects of vertical drains on soft Bangkok clay. Journal of Geotechnical Engineering, ASCE 117 (10), 1509–1530. Carrillo, N., 1942. Simple two- and three-dimensional cases in the theory of consolidation of soils. Journal of Mathematical Physics 21 (1), 1–5. Chai, J.C., Miura, N., Sakajo, S., 1997. A theoretical study of smear effect around vertical drain. Proceedings of the 14th ICSMFE, Hamburg, Vol. 3, pp. 1581–1584.
ARTICLE IN PRESS 74
G. Zhu, J.-H. Yin / Geotextiles and Geomembranes 22 (2004) 63–74
Hansbo, S., 1981. Consolidation of fine-grained soils by prefabricated drains. Proceedings of the Tenth International Conference on Soil Mechanics and Foundation Engineering, Stochholm, Sweden, Vol. 3, pp. 667–682. Hansbo, S., 1997. Aspect of vertical drain design: Darcian or non-Darcian flow. Geotechnique 47 (5), 983–992. Hawlader, B.C., Imai, G., Muhunthan, B., 2002. Numerical study of the factors affecting the consolidation of clay with vertical drains. Geotextiles and Geomembranes 20 (4), 213–239. Hird, C.C., Moseley, V.J., 2000. Model study of seepage in smear zones around vertical drains in layered soil. Geotechnique 50 (1), 89–97. Indraratna, B., Redana, I.W., 1998. Laboratory determination of smear zone due to vertical drain installation. Journal of Geotechnical and Geoenvironmental Engineering 124 (2), 180–184. Li, A.L., Rowe, R.K., 2001. Combined effects of reinforcement and prefabricated vertical drains on embankment performance. Canadian Geotechnical Journal 38, 1266–1282. Madhav, R., Park, Y.M., Miura, N., 1993. Modeling and study of smear zones around band shaped drains. Soils and Foundations 33 (4), 135–147. Olson, R.E., 1977. Consolidation under time-dependent loading. Journal of the Geotechnical Engineering Division ASCE 103 (GT1), 55–60. Onoue, A., 1988. Consolidation by vertical drains taking well resistance and smear into consideration. Soils and Foundations 28 (4), 65–174. Terzaghi, K., 1943. Theoretical Soil Mechanics. Wiley, New York. Xie, K.H., Lee, P.K.K., Cheung, Y.K., 1994. Consolidation of a two-layer system with ideal drains. Proceedings of the Eighth International Conference on Computer Methods and Advances in Geomechanics, Morgantown, West Virginia, USA, Vol. 1, pp. 789–794. Yoshikuni, H., Nakanodo, H., 1974. Consolidation of soils by vertical drain wells with finite permeability. Soils and Foundations 14 (2), 35–46. Zeng, G.X., Xie, K.H., 1989. New development of the vertical drain theories. Proceedings of the 12th International Conference on Soil Mechanics and Foundation Engineering, Rotterdam, The Netherlands, Vol. 2, pp. 1435–1438. Zhu, G.F., Yin, J.-H., 1998. Consolidation of soil under depth-dependent ramp load. Canadian Geotechnical Journal 35 (3), 344–350. Zhu, G.F., Yin, J.-H., 1999. Consolidation of double soil layers under depth-dependent ramp load. Geotechnique 49 (3), 415–421. Zhu, G.F., Yin, J.-H., 2000. Finite element analysis of consolidation of soils with vertical drain. International Journal for Numerical Analysis and Methods in Geomechanics 24 (4), 337–366. Zhu, G.F., Yin, J.-H., 2001a. Consolidation of soil with vertical and horizontal drainage. Geotechnique 51 (4), 361–367. Zhu, G.F., Yin, J.-H., 2001b. Design charts for vertical drains considering construction time. Canadian Geotechnical Journal 38 (5), 1142–1148.