One-dimensional consolidation theories for layered soil and coupled and uncoupled solutions by the finite-element method

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1 Huang, J. & Griffiths, D. V. (1). Géotechnique 6, No. 9, [doi: 1.168/geot.8.P.8] TECHNICAL NOTE One-dimensional consolidation theories for layered soil and coupled and uncoupled solutions by the finite-element method J. HUANG and D. V. GRIFFITHS One-dimensional consolidation theories for layered soil have been re-examined. Coupled (settlement and excess pore pressure), uncoupled (excess pore pressure only) and the classical Terzaghi equation are solved by the finiteelement method. By accounting only for changes in the coefficient of consolidation (c v ), the classical Terzaghi approach is unable to satisfy the flow continuity conditions at the interface between layers. KEYWORDS: consolidation; compressibility; finite-element method; permeability On a réexaminé des théories de consolidations unidimensionnelles (1-D) pour sols stratifiés. On résout les équations couplées (tassement et excès de pression interstitielle), non couplées (excès de pression interstitielle seulement) et l équation classique de Terzaghi avec la méthode aux éléments finis (FE). En ne tenant compte que des changements dans le coefficient de consolidation (c v ), l équation de Terzaghi ne permet pas de satisfaire les conditions de continuité de débit à l interface entre les couches. INTRODUCTION Gray (1945) first discussed the nature of the consolidation of two contiguous layers of unlike compressible soils. The general analytical solution for the one-dimensional (1D) consolidation of a layered system has been developed by Schiffman & Stein (197). Desai & Saxena (1977) analysed the consolidation behaviour of layered anisotropic foundations. Abid & Pyrah (1988) presented some guidelines for using the finite-element (FE) method to predict 1D consolidation behaviour using both diffusion and coupled approaches. Lee et al. (199) developed a more efficient analytical solution technique, which showed that the effects of the permeability and coefficient of volume compressibility of soil on the consolidation of layered systems are different and cannot be embodied into a single coefficient of consolidation. The compressibility of the soil layer m v also plays an important role in the rate of consolidation. Xie & Pan (1995) further developed an analytical solution for a layered system under time-dependent loading. Pyrah (1996) showed that the 1D consolidation behaviour of layered soils consisting of two layers with the same value of the c v, but different k and m v were quite different. Zhu & Yin (1999) gave more analytical solutions for different loading cases. In the present work coupled, uncoupled and the Terzaghi 1D consolidation theories have been re-examined using the FE method and it is shown that the Terzaghi FE solutions do not satisfy the flow continuity conditions at the interfaces between soil layers. Numerical results show that applying the Terzaghi FE solution to a layered system can lead to incorrect results. It is also shown that the average degree of consolidation, as defined by settlement and excess pore pressure, are different for layered systems. Manuscript received 5 March 8; revised manuscript accepted 1 November 9. Published online ahead of print 14 April 1. Discussion on this paper closes on 1 February 11, for further details see p. ii. * Division of Engineering, Colorado School of Mines, Golden, CO 841 UNCOUPLED AND COUPLED 1D CONSOLIDATION THEORIES Consider a thin strip of soil within a layered saturated soil undergoing consolidation. At any time t by equilibrium p ¼ u þ ó 9 (1) where ó 9 and u are, respectively, the effective stress and the excess pore pressure at any given depth z and p is the total load on top of the soil layers. For the sake of simplicity, p is assumed to be constant. By taking a derivative of equation (1) with respect to time and 9 ¼ ¼ () Assuming small strains ó 9 ¼ (4) m where s is the settlement at any given depth z. From equations () and ¼ m The net flow rate from Darcy law is Q ª where ª w is the unit weight of water. The rate of volume change of k ª (6) ¼ (8) 79

2 71 HUANG AND GRIFFITHS Equations (5) and (8) are the coupled governing equations of 1D consolidation (Griffiths, 1994). [k c ]fugþ[m m ] ¼fg dt From equations (), (4) and (8) (17) where k c ] and [m m ] are the fluid conctivity and mass ¼ m v (9) matrices There are many ways of integrating this set of ordinary differential equations (e.g. Smith & Griffiths, 4). If linear Equation (9) is the uncoupled governing equation of 1D interpolations and fixed time steps are used, equation (17) consolidation with excess pore pressure u as the only can be written at two consecutive time steps and 1 as dependent variable (e.g. Schiffman & Arya, 1977; Verruijt, follows 1995). If m v and k=ª w are constant throughout the soil layer, the [k c ]fug þ [m m ] ¼fg (18) settlement variable s can be eliminated from equations (5) dt and (8) to give the classical Terzaghi consolidation u c ¼ [k c ]fug 1 þ [m m ] ¼fg (19) dt 1 (1) A third equation advances the solution from to 1 using where c v, the coefficient of consolidation, is defined as a weighted average of the gradients at the beginning and the c v ¼ k end of the time interval, thus (11) m v ª w fug 1 ¼fug þ t ð1 ŁÞ þ Ł! dt dt 1 and m v and k are the coefficient of volume compressibility () and the soil permeability and ª w is the unit weight of water. where < Ł < 1. Elimination of f=dtg and f=dtg 1 from equations (18) to () leads to the following recurrence equation after LAYERED 1D CONSOLIDATION THEORY Let u i be the excess pore pressure at any given depth z at any given time t. The governing equations of the layered system can be expressed as assembly between time steps and 1 ([M m ] þ Ł t[k c ])fug 1 ¼ ([M m ] (1 Ł) t[k c ])fug u i c ¼ i, i ¼ 1,,..., n (1) where c vi is the coefficient of consolidation of the ith layer. The boundary conditions are z ¼ : z ¼ H: ¼ (impermeable) or u 1 ¼ (drained) ¼ (impermeable) or u n ¼ (drained) (1) The interface flow continuity conditions are i k ¼ k iþ1 iþ1 i ¼ 1,,..., n 1 The initial conditions at t ¼, assumed in this case to be uniform with depth, are given by u i ¼ (15) The analytical solutions of equation (1), with the above conditions, have been developed by Schiffman & Stein (197), Lee et al. (199), Xie & Pan (1995) and Zhu & Yin (1999) among others. COUPLED AND UNCOUPLED FE SOLUTIONS Solution of the uncoupled equation If the FE method is used, equation (9) can be written in an uncoupled form as (Schiffman & Arya, 1977) ª ¼ m v (16) It is noted that equation (16) does not use the coefficient of consolidation c v. After solution by the Galerkin weighted resial process, equation (16) leads to the element matrix form The above solution gives the distribution of excess pore pressure. The corresponding settlement distribution can be obtained at every time step from s ¼ ð D z m v z ðp uþdz () Applying the above FE solution to Terzaghi equation (1) for layered systems will give the wrong excess pore pressure distribution. The solution is unable explicitly to model changes in the permeability k, and is therefore unable to enforce the interface flow continuity conditions given by equation (14). A numerical example is given in the Appendix to show that combining the permeability and coefficient of volume compressibility into a single coefficient of consolidation will give wrong excess pore pressure distributions. Solution of the coupled equations After solution by the Galerkin weighted resial method, equations (5) and (8) lead to the element matrix form [k m ]fsgþ[c]fug ¼ffg [c] T ds [k c ] fug ¼fg dt () where [k m ] and [k c ] are the solid stiffness and fluid conctivity matrices. The matrix [c] is the connectivity matrix. ffg is the total force applied. If f fg is the change in load between successive times, the incremental form of the first part of equation () is [k m ]f sgþ[c]f ug ¼f fg (4) where f sg and f ug are the resulting changes in displacement and excess pore pressure respectively. Linear interpolation in time using the Ł-method yields

3 CONSOLIDATION THEORIES FOR LAYERED SOIL AND COUPLED AND UNCOUPLED SOLUTIONS 711 f sg ¼ t ð1 ŁÞ ds þ Ł ds! NUMERICAL EXAMPLES (5) Two 1D consolidation FE programs using two-node rod dt dt 1 elements were developed in the same style as program 9. in the text written by Smith & Griffiths (4) (see Appendix and the second part of equation () can be written at the for link to downloadable version of this program). The two time levels to give expressions for the derivatives which coupled program was based on equations (5) and (8) and was can then be eliminated to give the following incremental called 8. 1_c. The uncoupled program was based on equation recurrence equations (e.g. Sandhu & Wilson, 1969; Griffiths, (16) and was called 8. 1_uc. Program 8. 1 from the same 1994) source was used for solving the Terzaghi equation (1). The results obtained by the uncoupled program were omitted since [k m ] [c] f sg f fg they are the same as the coupled ones. In all the following [c] T ¼ (6) Ł t[k c ] f ug t[k c ]fug analyses, the time interpolation parameter Ł ¼. 5. The example used by Schiffman & Stein (197) was reanalysed. The soil profile shown in Fig. 1 consists of four At each time step, all that remains is to update the dependent variables using compressible layers with double-drainage and properties shown in Table 1. An instantaneous applied load ( ) of unit magnitude was applied and maintained constant with time. fsg 1 ¼fsg þf sg (7) The thickness of the ith layer is given by h i. A time step of fug 1 ¼fug þf ug t ¼ 1 day was used. The calculated results for a set of excess pore pressure at If applying equation () to a layered system, the FE different times after loading are shown in Fig.. It can be solution not only enforces the excess pore pressure continuity, but also the interface flow continuity conditions (14), those presented by Schiffman & Stein (197). Also plotted seen that the coupled results are essentially the same as because the second line of equation () is derived from the in Fig. are the results which were obtained by Terzaghi s flow continuity condition. 1D equation (1) and the c vi values listed in Table 1. The results are quite different from the coupled ones except at the earliest time of t ¼ 74 days. Figure plots the average degree of consolidation as a function of time. It can be seen that the coupled approach predicts faster consolidation with respect both to settlement and excess pore pressure than the Terzaghi approach. AVERAGE DEGREE OF CONSOLIDATION OF A LAYERED SYSTEM The average degree of consolidation can be expressed in terms of either excess pore pressure or settlement. If the initial (uniform) excess pore pressure is given by and the maximum drainage path by D, the average degree of consolidation defined by excess pore pressure is U avp ¼ 1 1 D ð D and defined by settlement u dz (8) U avs ¼ s t (9) s u where s u is the long-term (ultimate) settlement and s t is the settlement at time t. For layered systems, equations (8) and (9) become U avp ¼ 1 1 D U avs ¼ u i dz () ( u i )dz h i ¼ 1 u i dz h i (1) CONCLUDING REMARKS The paper has examined the differences between coupled, uncoupled and Terzaghi 1D consolidation modelling by the FE method in layered systems. Results show that applying the Terzaghi FE equation to a layered system can lead to incorrect results because interface flow continuity conditions are violated. Coupled analyses show that the average degree of consolidation is different depending on whether it is defined by settlement or excess pore pressure for a layered system. D Layer 1 Layer Layer Layer 4 Drained Drained where h i is the thickness of ith layer. If is constant throughout the layers h i ¼ Dm v () u i dz ¼ m v Xn u i dz () U avp and U avs will be exactly the same (e.g. Xie & Pan, 1995). Fig. 1. Four-layer system Table 1. Geotechnical data of the four-layer system (Schiffman & Stein, 197) i layer h i :m k i : m/s m v i : kpa 1 c v i :m /s

4 71 HUANG AND GRIFFITHS 1 4 Excess pore pressure: u / t 9 days (Schiffman & Stein) t 9 days ( Schiffman & Stein) z / D ( Schiffman & Stein) (Terzaghi FE) t 9 days ( Terzaghi FE) ( Terzaghi FE) 9 1 Fig.. Excess pore pressure isochrones 1 Node 1 avp avs U or U Terzaghi U avp Coupled U avp Coupled U avs Element Time t: days Fig.. Coupled and Terzaghi results of the four-layer system When the uncoupled (excess pore pressure only) approach is used, equation () must be used to obtain the average degree of consolidation as defined by settlement, the only exception is when m v is constant throughout the soil layers. Node Element Node ACKNOWLEDGEMENT The authors wish to acknowledge the support of NSF grant CMS-4815 on Advanced probabilistic analysis of stability problems in geotechnical engineering. Fig. 4. A two-layer system APPENDIX Demonstration of contrasting global FE matrices obtained from equations (1) and (16) for a two-layered system with the same c v A two-layer system is modelled by two rod elements as shown in Fig. 4 with parameters given in Table. Note that the elements have the same c v but different k=ª w and m v. Table. Parameters of a two-layer system Element Length k=ª w m v c v

5 CONSOLIDATION THEORIES FOR LAYERED SOIL AND COUPLED AND UNCOUPLED SOLUTIONS 71 For the classical Terzaghi consolidation equation (equation (1)), the element fluid conctivity matrices are given as 1 1 [k c ] 1 ¼ [k c ] ¼ (4) 1 1 and the mass matrices as 1= 1=6 [m m ] 1 ¼ [m m ] ¼ 1=6 1= (5) After assembly, the global fluid conctivity and mass matrices are 1 1 [K c ] ¼ (6) 1 1 1= 1=6 [M m ] ¼ 4 1=6 = 1=6 5 (7) 1=6 1= For the uncoupled equation (16), the element fluid conctivity matrices are given as 1 1 [k c ] 1 ¼ (8) [k c ] ¼ (9) 1 1 and the element mass matrices as 1= 1=6 [m m ] 1 ¼ (4) 1=6 1= 1= 1=6 [m m ] ¼ (41) 1=6 1= After assembly, the global fluid conctivity and mass matrices are 1 1 [K c ] ¼ (4) 1 1 1= 1=6 [M m ] ¼ 4 1=6 11= 1=6 5 (4) 1=6 1= Clearly equations (1) and (16) lead to different FE formulations and will therefore deliver different excess pore pressure distributions. The formulation given by equation (16) is the correct one for layered soils and the authors program called p9_u.f95 can be downloaded from NOTATION c v coefficient of consolidation c v : 5 coefficient of consolidation determined by log time method c v : 9 coefficient of consolidation determined by root time method ffg total force applied D maximum drainage path h i thickness of ith layer i layer number k permeability [k c ] fluid conctivity matrices [m m ] mass matrices m v coefficient of volume compressibility n number of layers p total load on top of the soil layers s settlement s t settlement at time t s u long-term (ultimate) settlement T dimensionless time factor t time U average degree of consolidation U avp average degree of consolidation defined by excess pore pressure U avs average degree of consolidation defined by settlement u excess pore pressure initial (uniform) excess pore pressure z depth ª w unit weight of water Ł time interpolation parameter, < Ł < 1 REFERENCES Abid, M. M. & Pyrah, I. C. (1988). Guidelines for using the finite element method to predict one-dimensional consolidation behaviour. Comput. Geotech. 5, No., 1 6. Desai, C. S. & Saxena, S. K. (1977). Consolidation analysis of layered anisotropic foundations. Int. J. Numer. Analyt. Methods Geomech. 1, No. 1, 5. Gray, H. (1945). Simultaneous consolidation of contiguous layers of unlike compressive soils. ASCE Trans. 11, Griffiths, D. V. (1994). Coupled analyses in geomechanics. In Visco-plastic behaviour of geomaterials, chapter 5, (eds N. D. Cristescu and G. Gioda). Vienna, New York: Springer-Verlag. Lee, P. K. K., Xie, K. H. & Cheung, Y. K. (199). A study on onedimensional consolidation of layered systems. Int. J. Numer. Analyt. Methods Geomech. 16, No. 11, Pyrah, I. C. (1996). One-dimensional consolidation of layered soils. Géotechnique 46, No., Sandhu, R. S. & Wilson, E. L. (1969). Finite-element analysis of seepage in elastic media. J. Engng Mech., ASCE 95, No. EM, Schiffman, R. L. & Arya, S. K. (1977). One-dimensional consolidation. In Numerical methods in geotechnical engineering (eds C. S. Desai and J. T. Christian), pp New York: McGraw- Hill. Schiffman, R. L. & Stein, J. R. (197). One-dimensional consolidation of layered systems. J. Soil Mech. Found. Div., ASCE 96, Smith, I. M. & Griffiths, D. V. (4). Programming the finite element method, 4th edn. New York: Wiley. Verruijt, A. (1995). Computational geomechanics. Dordrecht: Kluwer. Xie, K. H. & Pan, Q. Y. (1995). One-dimensional consolidation of soil stratum of arbitrary layers under time-dependent loading. China J. Geotech. Engng 17, No. 5, Zhu, G. & Yin, J. H. (1999). Consolidation of double soil layers under depth-dependent ramp load. Géotechnique 49, No.,