Uniersity of Wollongong Research Online Faculty of Engineering - Papers (Archie) Faculty of Engineering and Information Sciences Non-linear consolidation of soil ith ertical and horizontal drainage under time-dependent loading Xueyu Geng Zhejiang Uniersity, xgeng@uo.edu.au http://ro.uo.edu.au/engpapers/979 Publication Details Geng, X. (). Non-linear consolidation of soil ith ertical and horizontal drainage under time-dependent loading. Adanced Computer Theory and Engineering,. ICACTE. International Conference on (pp. -). Australia: IEEE. Research Online is the open access institutional repository for the Uniersity of Wollongong. For further information contact the UOW Library: research-pubs@uo.edu.au
International Conference on Adanced Computer Theory and Engineering Non-linear Consolidation of Soil ith Vertical and Horizontal Drainage under Time-dependent Loading Xueyu Geng College of Ciil Engineering and Architecture, Zhejiang Uniersity, Hangzhou, 317, China College of Computer Science and Technology, Zhejiang Uniersity, Hangzhou, 317, China gengxy@gmail.com Abstract This paper presents a nonlinear theory for sand drain consolidation of clayey soils under time dependent loading. The solution is obtained using the method of separation of ariables. Using the solutions obtained, some diagrams are prepared and the releant consolidation behaior of soil ith ertical and horizontal drainage under time-loadings is discussed. For nonlinear material properties, aerage degree of consolidation can be defined either in terms of settlement or in terms of effectie stress. And consolidation ill be sloer hen compared to the cases ith constant material properties. And the difference depends on the external loading pressure N, and the construction time T c. 1. Introduction Good quality geologic materials for construction are becoming scare. Since compressible soils are usually characterized by ery lo permeability, the time needed for the desired consolidation can be long. Due to these reasons and become of the enironmental restrictions on certain public orks, ground improement is becoming an essential part of infrastructure deelopment. Hence, a system of ertical drains combined ith acuum preloading is an effectie method to accelerate soil consolidation by promoting radial flo. For the classic ertical drain consolidation theory, a constant coefficient of olume compressibility, a constant coefficient of horizontal permeability and a constant coefficient of ertical permeability ere assumed for gien stress range[1-6]. Hoeer, for a relatiely large applied stress range, it is knon that both soil permeability and soil olume compressibility coefficients decrease as a result of physical reduction in oid ratio during the consolidation process [7-1]. And pre-consolidation pressure is essential for predicting the actual settlement. Otherise, loads in the construction of ciil engineering orks are mostly applied gradually ith time [11-1]. In many cases the loading process may endure oer a long period of time, so a significant part of the consolidation occurs accordingly. To the author s knoledge, hoeer, no comprehensie solution to consolidation ith both ertical and horizontal drainage including the non-line properties of soil and the changes of external loading is aailable in the literature. The content of this paper is specifically deeloped mathematical solutions ith the idea that ariable consolidation properties are particularly related to the aforementioned problems.. Mathematical models for ariable soil properties Here, take the e log relationship hich is used to determine the compressibility indices, and the e log k relationship hich is used to represent permeability ariation linear responses into account to present the non-linear relationship of the soil. Assuming that ee Cc lg( / ) (1) ee Ck lg( k / k) () ee Chlg( kh / kh) (3) here e is the oid ration, e is the initial oid ratio, is the ertical total stress, C c is the compression index, k is the ertical initial coefficient of permeability, k h is the horizontal initial coefficient of permeability, C k is the ertical hydraulic conductiity index, C h is the horizontal hydraulic conductiity index. Other assumptions are the same as Barron s. Based on equations (1)-(3): C / ( ) c C k k k () 97--7695-39-3/ $5. IEEE DOI 1.119/ICACTE..16
C / ( ) h C kh kh k (5) 1 e m m (6) 1e here m is the coefficient of compressibility. m =C c /(1+e )/ln1/ is the initial coefficient of compressibility. According to Terzaghi s Principle of effectie stress, can be expressed as: qt () u (7) in hich, u is the pore ater pressure, q(t) is the external loading. The analysis model is shon in Figure 1. Fig.1 Scheme of ground ith ertical drains The differential equation for the dissipation of excess pore ater pressure using the free strain assumption is 1 u 1 u u dq () ( rkh ) ( k ) m( ) r r r z z t t dt here r is the radial coordinate, z is the ertical coordinate, re is the radius of the equialent cylindrical block of soil, r is the radius of drain, is the unit eight of ater, H is the drainage path. Substituting Equations (), (5) and (7) into Equation () yields: (9) 1 u 1 u C 1 u dq [ rk ( ) ] [ k ( ) ] ( ) r r r z z e t dt C / C / c c k h k h (1 ) ln1 3. Boundary and initial conditions The folloing commonly encountered boundary conditions are studied in this paper: (,,) r t q() t, z zh (1) ( r, z, t) q( t), r rre If drainage is aailable at the bottom of the sol, the drainage path H is haled. The initial condition is: (,,) rz (11). Solutions for the goernmental equation Equation (9) is non-linear in and hence does not hae a general solution ith the boundary conditions mentioned aboe. Therefore, assuming the decrease in permeability is proportional to the decrease in compressibility during the consolidation of a soil and the distribution of initial effectie pressures is constant ith depth. Then, (1) 1 u 1 u C 1 u dq [ rk ( ) ] [ k ( ) ] ( ) r r r z z e t dt c h (1 ) ln1 By defining a ne parameter (,,) zrt ln( ) (13) qt () Equation (1) can be simplified as follos: 1 Ch [ r ] C Q( t) (1) r r r z t in hich, Cc m kh Ch k C (1 e) ln1 m m 1 dq Qt () qt () dt Using the method of separation of ariables, the consolidation equation (1) under the gien boundary and initial conditions in equations (11) and (1) can be soled. Letting Z z/ H, N r e, r C r L, R r/ r C H r Here, Z is a dimensionless ertical coordinate, N is the ratio of the equialent radius oer ell radius, and L is a nely defined dimensionless parameter. L is related to the ertical and horizontal consolidation coefficients (C and C h ), the radius of the ell (r ) and the ertical drainage distance (H). 1
The solution to equation (1) becomes A ( TR ) ( R)sin( Z) (15) mn, 1 mn m n n 1 in hich n. If C is zero or H is infinite, parameter L is zero, hich implies horizontal ater flo and horizontal consolidation only. If C h is zero, L is infinite, hich implies ertical ater flo and ertical consolidation only. If r is zero, though L is zero, but N is infinite, this is a ertical consolidation case. The expressions for A mn, R m and n in equation (1) are discussed as follos. R m in equation (15) can be expressed in terms of Bessel function [13] of the first kind ( J, J 1) and of the second kind ( Y, Y 1) as follos: Rm( r) Y1( Nm) J( mr) J1( Nm) Y( mr) (16) The quantity m is the mth positie root of the folloing equation: Y1( Nm) J( m) J1( Nm) Y( m) (17) A mn in equation (15) is gien by: qrzt (,,) q ( t ) (1) u Amncon( T ) Dmn here, Bmn m ml Dmn exp( T ) n 1 1L ; rr m( r) Bmn [ rr m( r)] ; rr m( r) my1( m) J1( Nm) J1( m) Y1( Nm) ; ( 1 1 LC ) ht T, ln r qu ln N, qu N 1, q u is the final external loading increment.. 6. Examples and discussions It has been reported that the parameter N is in the range of 5-, and the alue of parameter L is in the range of -.1. Fig shos that the excess pore ater pressure u and its rate of dissipation in such nonlinear consolidation is different from those in a linear consolidation. The bigger N is, the sloer dissipation of excess pore ater pressure is. Hoeer, the rate of settlement in a nonlinear consolidation subjecting to the assumptions made here is no different from that in a linear consolidation, hich is the same as Zhu s result. Because the nonlinear character of the soil considered in this paper, the decrease in permeability is proportional to the decrease in compressibility during the consolidation and the distribution of initial effectie pressures is constant ith depth, there should be no different from the situation in linear consolidation in theory. At the same time, it is proed from the theory that the results in this paper are correct. 1...6.. R = N/ L =.1. 1E-3.1.1 1 1 T (a) 1...6.. N = R = N/ L = N = 1 N = 1. 1E-3.1.1 T (c) 1 1 1...6... 1E-3.1 T.1 1 1 (b) 1...6.. R = N/ L = N = R = N/ L =.1 N = 1. 1E-3.1 T.1 1 1 (d) Fig. Dissipation of pore pressure at bottom under the constant loading For the same time factor T, N and L are the to main factors affect the degree of consolidation defined by the settlement. For the range of 5N, L.1, the maximum difference deiation is only 13.1%. This indicates that the relationship beteen the degree of consolidation ith the time factor T is approximately independent of the dimensionless parameters N and L [1].....6. results in this paper Zhus result 1. 1E-3.1.1 1 1 T (a)....6. L =. L = L =.1 1. 1E-3.1.1 1 1 T (c)....6. L =.1 N = N = N = 1 1. 1E-3.1.1 1 1 T (b)....6. N = L = L =.1 1. 1E-3.1.1 1 1 T (d) Fig. 3 Degree of consolidation defined in terms of settlement under the constant loading
In such nonlinear consolidation the degree of consolidation defined by the excess pore ater pressure is different from in a linear consolidation, hich as Figure shos. It can be seen that the difference ill disappear as the alue of N reproaches to 1. It also can be approed by mathematic method. Since the alue of N reflects the magnitude of load, it can be concluded that the discrepancy beteen linear and nonlinear consolidation is greatly related to the leel of the external load leel, and the smaller the load leel, the smaller the discrepancy. When N =1, the maximum difference deiation of U p ill almost reach 15% (as Figure shos). Therefore, the nonlinear character of the soil should be considered hen the external loading is big. In figure 5 and figure 6, cures of the aerage degree of consolidation U p and ersus time factor T are gien, respectiely. With the increasing of the load leel N, the difference of U p and beteen the current solution and the line s solution gets bigger. Moreoer, for the same parameters, U p is alays less than.....6. L =. L =.1 N = L =. N = L =.1 1. 1E-3.1.1 1 1 T Fig. 5 Degree of consolidation defined in terms of settlement for different N and L under the constant loading.. U....6. N = L =.1 U p N = 1 1. 1E-3.1.1 1 1 T Fig. Comparison beteen U p and under the constant loading 1 U p..6. N = 1 L =. L =.1 N = L =. N = L =.1 1. 1E-3.1.1 1 1 T Fig. 6 Degree of consolidation defined in terms of effectie stress for different N and L under the constant loading 7. Conclusion The deeloped method in this paper can handle the problem of non-linear consolidation ith ertical and horizontal drainage undergoing time-dependent loadings. 1) The equations presented gien an analytical solution to consolidation of soil ith ertical and horizontal drainage under time-dependent loading. ) Because of the nonlinear character of the soil, the aerage degree of consolidation can be defined either in terms of settlement ( ) or in terms of effectie stress (U p ). While the former shos the rate of settlement deelopment, the later indicates the rate of the increase of effectie pressure or the rate of the dissipation of excess pore ater pressure. And for the same parameters, U p is alays less than. 3
3) The aerage degree of consolidation defined in terms of settlement in non-linear theory has no different ith the aerage degree of consolidation in linear theory for the constant loading case. It means that the rate of settlement in a nonlinear consolidation ith ertical and horizontal drainage is no different from that in a linear consolidation hen the soil subjected to constant loading. ) Since the alue of N reflects the magnitude of load, it can be concluded that for the constant loading case the discrepancy beteen linear and nonlinear consolidation is greatly related to the leel of the external load leel, and the smaller the load leel, the smaller the discrepancy. 5) For the ramp loading case, the discrepancy beteen linear and nonlinear consolidation is related not only to the external load leel N, but also the construction time factor T c. References 1. Barron, R. A.: Consolidaiton of Fine-grained Soils by Drain Wells. Trans. ASCE 113, No. 36 (19) 71-7. Horne, M. R.: The Consolidation of a Stratified Soil ith Vertical and Horizontal Drainage. Int. J. Mech. Sci. 6(196) 17-197 3. Yoshikuni, H. and Nakanodo, H.: Consolidation of Soils by Vertical Drain Wells ith Finite Permeability. Soils Found. 1, No..(197) 35-6. Olson, R. E.: Consolidation under Time-dependent Loading. J. Geotech. Engng Di., ASCE. 13. No. GT1 (1977) 55-6 5. Basak, P. and Madha, M. R.: Analytical Solutions of Sand Drain Problem. Journal of Geotechnical engineering ASCE, 1(GT1) (197) 19-135 6. Tang, X. W. and Onitsuka, K.: Consolidation of Ground ith Partially Penetrated Vertical Drains. Geotechnical Engineering Journal, 9, No.. (199) 9 31 7. Dais, R. E. and Raymond, G. P.: A Non-linear of Consolidation. Geotechnique. 15, No.. (1965) 161-173. Xie, K. H., Xie, X. Y. and Jiang, W.: A Study on One-dimensional Nonlinear consolidation of double-layered Soil. 9 () 151-16 9. Geng, X. Y., Cai, Y. Q. and Xu, C. J.: Non-linear Consolidation Analysis of Soil ith Variable Compressibility and Permeability under Cyclic Loadings. Int. J. Numer. Anal. Meth. Geomech. 3 (6) 3-1. 1.Cai, Y. Q., Geng, X. Y. and Xu, C. J.: Solution of One-dimensional Finite-strain Consolidation of Soil ith Variable Compressibility under Cyclic Loadings. Computers and Geotechnics. 3 (7) 31-. 11.Lekha, K. R., Krishnasamy, N. R. and Basak, P.: Consolidation of Clay by Sand Drain under Timedependent Loading. J. Geo. Geoen. Engine. ASCE, 1, No. 1. (199) 91-9 1.Zhu, G. and Yin, J. H.: Consolidation of Soil ith Vertical and Horizontal Drainage under Ramp Load. Geotechnique 51. No.. (1) 361-367 13.Moshier, S. L. B, Methods and programs for mathematical functions. Chichster: Ellis Horood, 199