Two Dimensional Consolidations for Clay Soil of Non-Homogeneous and Anisotropic Permeability

Transcription

1 Two Diensional Consolidations for Clay Soil of Non-Hoogeneous and Anisotropic Pereability Ressol R. Shakir, Muhaed Majeed Thiqar University, College of Engineering, Thiqar, Iraq University of Technology, Building and Construction Departent, Baghdad, Iraq Abstract: In this research a coputer progra was odified to solve two-diensional consolidation probles. The odified coputer progra results showed a very good agreeent copared with the analytical solution of two-diensional consolidation. The nonhoogeneity in one and two directions with the an-isotropic effect of pereability was included in the progra. It was found that the increasing of anisotropic ratio of pereability increases the average degree of consolidation while the increase in non-hoogeneous pereability ratio decreases the average degree of consolidation according to the drainage face taken there. This effect should be taken in to account when dealing with the analysis of consolidation of copressible soil and average degree of consolidation. Keywords: Finite Eleent, Consolidation, pereability. Introduction The conventional Terzaghi theory [] is derived based on assuptions such as the soil fluid flow is in one direction only and also that the soil is hoogenous and isotropic. These assuptions are ost often-violated in real life situations. They are functioned to siulate a special case of subjecting soil layer to unifor load such as fill load. The hoogeneous and isotropic soil eans that every soil paraeters are constant with any direction and so the pereability of soil is assued constant along the depth of soil layer. Two and three-diensional consolidation behavior that governed by two and three diffusion equations were presented by any investigators to overcoe the defect of assuing fluid flow in one direction [, 3, 4]. Rendulic- Terzaghi equation is considered as extending to Terzaghi's equation. For siple geoetry and one diensional consolidation, analytical solution can be used, otherwise nuerical analysis such finite difference and finite eleent ethod are utilized. Using two diensional consolidation analyses is ore realistic than one diensional consolidation. A considerable aount of research is achieved to include the effect of non hoogeneous and isotropic soil paraeters into account when analyzing consolidation behavior. Analytical solution for consolidation in two layer soil colun was provided by Zhu and Yin (005)[5]. A finite difference approach was used to solve the consolidation settleent for a layer with variable copressibility, pereability and coefficient of consolidation [6]. Xie et al. presented a fully explicit analytical solution for one diensional consolidation of two layered soils with partially drained boundaries. Airi and Esaelly [8] investigated the effect of variable pereability and copressibility on the consolidation behavior. In the present work, the pereability is considered variable in one or two different direction and is included in the finite eleent code of two diensional consolidation settleent proble. The study contribute in investigating the effect of an isotropic pereability and non hoogeneous pereability on the average degree of consolidation. Geverning Equations The consolidation governing equation for excess pore water pressure u is c vx u u u + cvy = () x y t Where c vx, c vy are the coefficient of consolidation in x, and y k direction and Cv = γ w v It is practically accepted to assue C v as a constant through the consolidation progressing since k and v both decrease through the process. Where K is the pereability and v is the coefficient of volue change. According to Griffiths and Sith [9], the discritization of equation using Galerkin ethod leads to the following equation du []{} u + [ ] = {} 0 k () Where [ K ] and [ ] are the fluid conductivity and ass atrices respectively. The above first order tie dependent equation can be solved by finite difference as follow using eleent assebly ethod du [ K]{} U 0 + [ M ] = {} 0 (3) 0 0 du [ K]{} U + [ M ] = {} 0 (4) Paper ID: J03370 of 6

2 A third equation advances the solution fro 0 to using a weighted average of the gradient at the beginning and end of tie interval, thus, du du {} U = {} U + Δt ( -θ) + (5) ([ M ] + [ ]){ } ([ ] ( ) [ K ]){ U } 0 (6) 0 θ 0 θ Δt K U = M - θ Δt The above equation gives the distribution of excess pore water pressure [0] 3. Verification of the Coputer Progra To check the accuracy of the results of finite eleent progra, a proble which represents the dissipation of excess pore water pressure fro a square of soil [9]. The square of soil is previous fro four directions, and because of syetry, it can be used as a part of square of soil (Figure ). The load is unifor and the coefficient of consolidation is constant with tie and equal in agnitude. The pressure at the center of the esh is calculated at tie interval 0., and then the results are plotted in Figure, where the results are copared with series solution values obtained by []. The crude finite eleent idealization gives excellent results. The solution of the finite eleent progra is copared with analytical solution giving good agreeent. Figure : Coparisons of finite eleent results with series results, After Sith and Griffiths Variation of Pereability The finite eleent progra was used to study the effect of the variable pereability on the average degree of consolidation for the two-diensional consolidation proble. The variation of pereability can be discussed as follows:. The pereability in x direction is different fro that in y direction (anisotropic pereability).. The pereability is variable with depth in one direction (non- hoogeneous pereability in one direction) 3. The pereability is variable with depth in each direction (non-hoogeneous pereability in two direction) The proble will depend on the type of variation of pereability (if it is stratified, linear or nonlinear variation) and if the type of variation is the sae in two directions or different. The progra can be odified to include all these types of variation of pereability. In this study the variation of pereability is used as linear variation with depth. The sketch for the pereability variation with depth has been introduced to siplify obtaining the average degree of consolidation fro these variables (Figure 3). The variables considered in this study are as follows: Figure : Grid of the verification proble. k x is the pereability in x direction. k y is the pereability in y direction 3. k x /k y is the anisotropic pereability ratio (-0) 4. k x /k x is the non-hoogeneous pereability ratio in the x direction (-8) 5. k y /k y is the non-hoogeneous pereability ratio in the y direction (-8) 6. H h /H v is the ratio of drainage path in x direction to drainage path in y direction. 7. X is the distance fro the centerline of the esh to the right side in x direction. To study the effect of the factor (X/H h ) and the anisotropic pereability ratio (k x /k y ) on the average degree of consolidation, two cases are studied. For the first case the Paper ID: J03370 of 6

3 pereability of soil is considered isotropic and hoogeneous and the excess pore pressures are recorded at different location fro the centerline of the esh, X/H h. The second case considered the pereability of soil as anisotropic paraeter and the excess pore pressures are recorded at two locations fro the centerline of the esh X/H h = 0 and X/H h = 0.4. Figure (4) shows the effects of factor (X/H h ) on the average degree of consolidation for values (0, 0., 0.4, 0.6, 0.8). It shows that the average degree of consolidation increases with the increasing of (X/H h ). The increase of (X/H h ) eans the region that the average degree of consolidation is calculated close to the drainage face. Figure (5) shows the effect of anisotropic ratio of pereability on the average degree of consolidation for different ratio of X/H h (0, and 0.4), and different ratio of anisotropic ratio of pereability (, 4, 6, 0). The average degree of consolidation increases at anisotropic ratio of pereability greater than one. Also increase of X/H h will increase the average degree of consolidation. K NY =k y -(NY(k y -k y )/NYE) K NX =k x -(NX(k x -k x )/NXE) Figure 3: Linear variation of pereability 5. Effect of Non Hoogeneous Pereability Ratio The results of the finite eleent progra for all cases studied are presented as average degree of consolidation, U av. It can be coputed at any location in the esh by the following forula (U av = -u e /u o ) where u e is the excess pore water pressure and u o is the initial pore water pressure. For the case of non-hoogeneous pereability ratio in y direction k y /k y =8 and anisotropic pereability ratio k x /k y >, when the anisotropic pereability ratio increases fro 4 to 0, keeping constant value of non-hoogeneous pereability ratio, the average degree of consolidation increases fro 0.7 to 0.9 (Table ). The increase in U av is due to the increase in anisotropic pereability ratio. Effect of the factor (X/H h ) at which the average degree of consolidation was calculated are also presented in Table. Two cases are discussed the first is for U av coputed at X/H h = 0 and the second is for U av coputed at X/H h = 0.4. The U av calculated at X/H h =0.4 is greater than U av calculated at X/H h =0 because this region is close to the right face of the esh which is idealized as a drainage face in the presented proble. Figure 4: average degree of consolidation at different location X/H h for hoogeneous and isotropic soil. Table : The effect of the anisotropic ratio of pereability and distance ratio X/H h on the average degree of consolidation H v /H h K x /K y X/H h Tv K y /K y U av For the case of non-hoogeneous pereability in two directions together with the anisotropic pereability the U av increases due to anisotropic of pereability of soil. With high pereability ratio the U av decreases because the high pereability is concentrated close to the ipervious face. The high ratio of non-hoogeneous pereability decreases the average degree of consolidation (Table ) because the high pereability in x direction and y direction are close to the ipervious side of the esh. Table : show the effect of the non-hoogeneity in two direction and the distance ratio X/H h on the average degree of consolidation H v /H h K x /K y X/H h Tv K y /K y, K x /K x U av , Table also shows the result of average degree of consolidation for the case of two diensional consolidation for soil with anisotropic pereability and non hoogeneous pereability in x and y direction. The average degree of consolidation is calculated at two location represented by X/H h. It is shown that the average degree of consolidation increases as the location closes to the face of drainage. Paper ID: J of 6

4 non-hoogeneous pereability in x-direction eans increase in pereability in the region that is far fro the drainage face of the soil while the region close to the drainage face still has the sae value of pereability. Figure 9a and Figure0a show the U av versus tie factor, Tv, for the sae previous case of soil with non-hoogeneous pereability in y direction and hoogenous pereability in x direction but for anisotropic pereability k x /k y = 4 and k x /k y = 0. The results of U av are approxiately the sae. The anisotropic pereability has no effect on the U av. Figure 5: average degree of consolidation at X/H h =0 for different anisotropic pereability ratio Figure 7: Grid of the studied proble Figure 6: average degree of consolidation at X/H h =0.4for different anisotropic pereability ratio 6. Effect of Aniostropic and Hoogeneous Pereability The study also considered rectangular esh of diensions H v and H h = H v (Figure 7). Three cases are studied, the first case is for soil with isotropic pereability and nonhoogeneous pereability in y or/and in x direction where the U av is calculated at X/H h = 0.4. The effect of anisotropic pereability on the U av are studied in the other two cases by using anisotropic pereability ratio of k x /k y = 4 and 0. The non-hoogeneous and anisotropic pereability reflects the nature of assued soils which approxiately represent real soil. Figure 8a shows the result of U av for case of soil with isotropic and non-hoogeneous pereability in y direction and hoogeneous pereability in x direction. Figure 8b shows the results of U av for the case of soil with isotropic and non-hoogeneous pereability in x and y direction. Fro the two Figures, it is found the effect of nonhoogeneous pereability ratio in x-direction has slightly effect on U av. It is attributed to that the increase of ratio of Figure 8a: Average degree of consolidation, U av, versus tie factor, Tv, for soil with isotropic pereability, k x /k y =, hoogeneous pereability in x direction and nonhoogeneous pereability in x direction This ay be attributed to that the high pereability is close to the ipervious face. Figure 9b and Figure 0b show the result of U av against Tv for soil with non-hoogeneous pereability in both x and y direction and with sae ratio of anisotropic pereability. It is obvious that the U av increases as non-hoogeneous pereability increases fro k x /k x = to k x /k x =8. Paper ID: J of 6

5 Figure 8b: Average degree of consolidation, U av, versus tie factor, Tv, for soil with isotropic pereability, k x /k y = 0, non-hoogeneous pereability in y direction and nonhoogeneous pereability in x direction Figure 9b: Average degree of consolidation, U av, versus tie factor, Tv, for soil with anisotropic pereability, k x /k y = 4, non-hoogeneous pereability in y direction and nonhoogeneous pereability in x direction Figure 9a: Average degree of consolidation, U av, versus tie factor, Tv, for soil with anisotropic pereability, k x /k y = 4, non-hoogeneous pereability in y direction and hoogeneous pereability in x direction Figure 0a: Average degree of consolidation, U av, versus tie factor, Tv, for soil with anisotropic pereability, k x /k y =0, non-hoogeneous pereability in y direction and hoogeneous pereability in x direction Paper ID: J of 6

6 Figure 0b: Average degree of consolidation, U av, versus tie factor, Tv, for soil with anisotropic pereability k x /k y =0, non-hoogeneous pereability in y direction and nonhoogeneous pereability in x direction 7. Conclusion In this study a coputer progra which depends on the uncoupled solution using finite eleent ethod has been used to solve proble of two- diensional consolidation settleent. This coputer progra is a good tool to study the non-hoogeneous and anisotropic pereability effects on the average degree of consolidation. According to the study, the following conclusions are drawn:. It is concluded that the average degree of consolidation decreases with the decrease of pereability with depth "non-hoogeneous pereability in y direction.. The soil layer of low pereability which located close to the drainage face reduces the average degree of consolidation. 3. The average degree of consolidation increases when the anisotropic pereability ratio increases ore than one 4. Including real variation of pereability in the soil to copute the average degree of consolidation has to be depended on in design. References [] Terzaghi, K.,(943) Theoretical Soil Mechanics, Wiley Publications, New York. [] Rendulic L.,(936), Porenziffer and Porenwasserdruck in Tonen entioned by N. Foldin and B. Brows, In Soft clay in engineering, 98, Elsevier Scientific publishing Copany. [3] Carillo, (94), Siple Two and Three Diensional Cases in the Theory of Consolidation of Soils, J. Math. And Phys., No., P. -5. [4] Poskitt, T.,J.(970) Settelent Charts for An Isotropic Soil, Geotechnique, Vol.0, No.3, pp [5] Zhu, G. and Yin, J-H, 005. Solution charts for the consolidation of double soil layers. Canadian Geotechnical Journal, 4: [6] Abbasi, N., Rahii, H., and Javadi, A. A. And Fakher, A. (007) Finite difference approach for consolidation with variable copressibility and pereability. Coputer and Geotechnics. 34(), (007), pp [7] Kang-He Xie, Xin-Yu Xie, Xiang Gao, Theory of one diensional consolidation of two-layered soil with partially drained boundaries, Coputers and Geotechnics, Volue 4, Issue 4, June 999, Pages 65-78, ISSN X. [8] Airi S. N and Eaaely A. (04) A realistic theory of soils consolidation" Journal of science and geotechnical engineering" Vol 4, no, [9] Sith, I. M. and Griffiths, D.V. (998) Prograing the Finite Eleent Method third edition, Jon Wiley & Sons, England. [0] J Huang, DV Griffiths (00) "One-diensional consolidation theories for layered soil and coupled and uncoupled solutions by the finite-eleent ethod" Geotechnique, Volue 60. Issue 9 Pages [] Carslaw, H.S., and Jaeger, J.C.(959) Conduction of Heat in Solids, nd edn. Clarendon Press, Oxford Paper ID: J of 6