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- 1. “Consolidation Settlement with Sand Drains – Analytical and Numerical Approaches” Department of Civil Engineering, IIT Kanpur CE 632 By – Kundan Tripathi (10327365) Rajeev Verma (10572) Saurav Shekhar (10660) Shashank Kumar (10327670) Umed Paliwal (10327774) Dated: 5th April, 2014
- 2. Abstract & Objective Sand drains are often used in important construction projects in order to accelerate the process of consolidation settlement for the construction of some structures. Sand drains increase the rate of consolidation such that the settlement that would occur in years can be hastened to occur in months. When a surcharge is applied at ground surface, the pore water pressure in the clay will increase, and there will be drainage in the vertical and horizontal directions. The horizontal drainage is induced by the sand drains. Hence the process of dissipation of excess pore water pressure created by the loading (and hence the settlement) is accelerated. The objectives of this study are two-fold. Analytical and numerical approaches have been studied herein. The analytical part includes a review of the existing literature and presents useful extracts in regards to settlement, structure, installation and monitoring of sand drains. Popular subjects such as free strain and equal strain cases with and without smear have been glanced at. The numerical part is the result of finite element analysis of a drain unit cell using Plaxis 2d version 8.2. It addresses – 1. Reduction in time of consolidation by use of sand drains and also the changes in this reduction as the loading is changed. 2. Relationship between ultimate settlement and loading. 3. Relationship between ultimate settlement and drain diameter. *****
- 3. Table of Contents S.No Topic Page No. 1. Part 1 - Analytical Approach 4 1.1 Popular Theory 4 1.2 Recent Research 9 2. Part 2 – Numerical Approach 12 2.1 Objective 12 2.2 General Settings 12 2.3 Soil Properties 12 2.4 Boundary Conditions 15 2.5 Initial Conditions 15 2.6 Calculations 15 2.7 Observations 15 2.8 Results & Discussion 17 2.9 Conclusions 23 3. References 24 *****
- 4. Part 1. Settlement of Foundations built on Sand Drains - Analytical Approach" The consolidation settlement of soft clay subsoil creates a lot of problems in foundation and infrastructure engineering. Because of the very low clay permeability, the primary consolidation takes a long time to complete. To shorten this consolidation time, sand drains can be used. Sand drains are constructed by driving down casings or hollow mandrels into the soil. The holes are then filled with sand, after which the casings are pulled out. When a surcharge is applied at ground surface, the pore water pressure in the clay will increase, and there will be drainage in the vertical and horizontal directions. Hence the process of dissipation of excess pore water pressure created by the loading (and hence the settlement) is accelerated. The basic theory of sand drains was presented by Rendulic (1935) and Barron (1948) and later summarized by Richart (1959) and in the development of these theories; it is assumed that drainage takes place only in the radial direction, i.e., no dissipation of excess pore water pressure in the vertical direction. In the study of sand drains, two fundamental cases: 1. Free-strain case- When the surcharge applied at the ground surface is of a flexible nature, there will be equal distribution of surface load. This will result in an uneven settlement at the surface. 2. Equal-strain case.-When the surcharge applied at the ground surface is rigid, the surface settlement will be the same all over. However, this will result in an unequal distribution of stress. 1.1 Adapted from advanced soil mechanics from B.M Das 1.1.1Free-strain consolidation with no smear Figure 6.37bshows the general pattern of the layout of sand drains. For triangular spacing of the sand drains, the zone of influence of each drain is hexagonal in plan. This hexagon can be approximated as an equivalent circle of diameter de. Other notations used in this section are as follows: 1. re = radius of the equivalent circle = de/2. 2. rw= radius of the sand drain well. 3. rs= radial distance from the centerline of the drain well to the farthest point of the smear zone. Note that, in the no-smear case, rw= rs. The basic differential equation of Terzaghi’s consolidation theory for flow in the vertical direction is given in Eq. (1).
- 5. ……………….(1) For radial drainage, this equation can be written as Where u=excess pore water pressure r=radial distance measured from center of drain well Cvr =coefficient of consolidation in radial direction
- 6. For solution of above Eq., the following boundary conditions are used: 1. At time t=0, u=ui 2. At time t>0, u=0 at r=rw. 3. At r=re, du=dr. With the above boundary conditions, above Eq. yields the solution for excess pore water pressure at any time t and radial distance r: where J0=Bessel function of first kind of zero order J1=Bessel function of first kind of first order Y0=Bessel function of second kind of zero order Y1=Bessel function of second kind of first order Where kh is the coefficient of permeability in the horizontal direction. The average pore water pressure uav throughout the soil mass may now be obtained from as
- 7. 1.1.2 Equal-strain consolidation with no smear The problem of equal-strain consolidation with no smear (rw=rs) was solved by Barron (1948). The excess pore water pressure at any time t and radial distance r is given by Uav=average value of pore water pressure throughout clay layer. The average degree of consolidation due to radial drainage is For re/rw>5 the free-strain and equal-strain solutions give approximately the same results for the average degree of consolidation. Olson (1977) gave a solution for the average degree of consolidation Ur for time-dependent loading (ramp load) similar to that for vertical drainage. The surcharge increases from zero at time t=0 and q at time t=tc. For t ≥ tc, the surcharge is equal to q. For this case
- 8. 1.1.3 Effect of smear zone on radial consolidation Barron (1948) also extended the analysis of equal-strain consolidation by sand drains to account for the smear zone. The analysis is based on the assumption that the clay in the smear zone will have one boundary with zero excess pore water pressure and the other boundary with an excess pore water pressure that will be time dependent. Based on this assumption.
- 9. 1.2 Recent Research Recently several analytical and experimental studies have reported on sand drain consolidation of clayey soils, some of them are listed below: 1.2.1 TOYOAKI NAGOMI, AND MAOXIN LI, (2003) CONSOLIDATION OF CLAY WITH A SYSTEM OF VERTICAL AND HORIZONTAL DRAINS - Consolidation behavior with the drain system is formulated using the transfer matrix method. Special care is given to formulation of thin pervious layers for efficient computation. The developed formulation is verified using available numerical and field information. Parametric studies are conducted to study the consolidation characteristics of clay with the drain system. Based on the findings, a design method for an optimum system of horizontal and vertical drains is proposed and design charts are presented for such a design. The consolidation behavior of clay with a system of horizontal drains and vertical cylindrical drains is formulated using the transfer matrix approach. The developed formulation can handle the inhomogeneous profile in clay and multiple horizontal drains made of either thin sand layers or geotexstile sheets. The number of terms used in series is five terms in the r direction, and five to ten terms in the z direction depending on the behavior in the series expression. As Terzaghi’s consolidation solution, only one or two terms in the expansion in the z direction are sufficient to compute the consolidation behavior in the later stage of consolidation but the upper-side number of terms is required in the early stage of consolidation. The formulation is found to be very efficient and convenient for computation. 1.2.2. K.R.LEKHA, N.R.KRISHNASWAMY, AND P.BASAK, (1998) CONSOLIDATION OF CLAY BY SAND DRAIN UNDERTIME-DEPENDENT LOADING - The literature contains a nonlinear theory of sand drain consolidation under time-dependent loading that can take into account any effective stress/void ratio/permeability variations. A generalized governing equation, capable of yielding a large class of analytical solutions for these variations is derived in this paper. Closed-form solutions are presented for the variation of pore water pressure with a time factor and load increment ratio under time-dependent loading. The analytical formulation is validated by comparing the solution with the standard results available in the literature for instantaneous loading, constant permeability, and constant compressibility. Governing equation for equal strains and drain problems in time-dependent loading is given in its
- 10. most general form, which can conveniently account for any effective stress/void ratio/permeability variation. This equation is linear and requires evaluation of only one integral to yield the solution for a large class of problems in time-dependent loading with variable permeability and compressibility. The theory is an extension of the solution by Basakand Madhav(1970), for the case of instantaneous loading and variation of compressibility and permeability. The analytical formulation is validated by comparing the solution with the standard results available for instantaneous loading, constant permeability, and constant compressibility. The results are presented for the variation of pore water pressure with a time factor and load increment ratio. 1.2.3. IEW-ANN TAN, (1993) ULTIMATE SETTLEMENT BY HYPERBOLIC PLOT FOR CLAYS WITH VERTICAL DRAINS The rectangular hyperbola method (Tv/U versus Tv) is extended to the case of drains and surcharge by considering the hyperbolic plots for combined vertical and radial flow consolidation in clays of varying thickness and drain spacing ratio for typical soil properties of Cv of 1-5 m2 /yr. The results indicate that the hyperbolic plots are linear between U50% and U90%. For the lines radiating from the origin to U50% point , the slope is (1/0.5 = 2.0), and to the U90% point, the slope is (1/0.9 = 1.11). Thus, the ratio of the slopes of these radiating lines to the slope of the linear portion of the hyperbolic plots identifies the U50% and U90% for any settlement record using drains and surcharge. It is found that the estimate of ultimate settlement from the U50% and U90% is more accurate than the conventional inverse slope .approach of the hyperbolic method, especially for data between the 50% and 90% consolidation points. The hyperbolic method of settlement analysis can be extended to the practical case of vertical drains and surcharge. The use of the inverse of the slope of the first straight-line portion of the hyperbolic plot of settlement data tends to overestimate the amount of ultimate primary compression. To some extent, this overestimation compensates for the effects of secondary compression, but in field applications the amount of secondary compression is uncertain. However, when the hyperbolic method is used to obtain the 50% and 90% points of the settlement record, the ultimate compression obtained from these points agrees reasonably well with long-term compression data of the Skh-Edeby test fill. Therefore, this method can provide a useful and practical check on the progress of consolidation in field applications using vertical drains and surcharge, especially in the absence of reliable soil properties data.
- 11. 1.2.4. CHIN JIAN LEO, (2004) EQUAL STRAIN CONSOLIDATION BY VERTICAL DRAINS- Closed-form analytic solutions of equal strain consolidation by a vertical drain with smear and well resistance have been developed in the present paper. Solutions in this paper, however, have been derived for coupled radial and vertical drainage and covered a step-loading or a ramp- loading situation. Comparisons made with the corresponding analytic solutions of Hansbo and Barron showed that the differences between the solutions of the present paper and the solutions of Hansbo and/or Barron are generally quite small. In keeping with Barron (1948), consolidation is considered in the undisturbed soil mass only, not in the vertical drain or the smeared zone, and only radial drainage is assumed in smeared zone. 1.2.5 BUDDHIMA INDRARATNA, ALA AlIJORANY ANS CHOLACHAT RUJIKATKARNOM, ANALYTICAL AND NUMERICAL MODELLING OF CONSOLIDATION BY VERTICAL DRAIN BENEATH A CIRCULAR EMBANKMENT While analyzing the axisymmetric problems, it is tried that aspects of geometry, material properties, and loading characteristics are either maintained as constants or represented by continuous functions in the circumferential direction. In the case of radial consolidation beneath a circular embankment by vertical drains i.e., circular oil tanks or silos, the discrete system of vertical drains can be substituted by continuous concentric rings of equivalent drain walls. An equivalent value for the coefficient of permeability of the soil is obtained by matching the degree of consolidation of a unit cell model. A rigorous solution to the continuity equation of radial drainage towards cylindrical drain walls is presented and verified by comparing its results with the existing unit cell model. The proposed model is then adopted to analyze the consolidation process by vertical drains at the Skå-Edeby circular test embankment. The calculated values of settlement, lateral displacement, and excess pore-water pressure indicate good agreement with the field measurements. 1.2.6. TUNG WEN SHU and HUI_JYE LU, (2013) CONSOLIDATION FOR RADIAL DRAINAGE UNDER TIME-DEPENDENT LOADING It represents the details of consolidation for radial drainage under linear time-dependent loading with varying loading dependent coefficients of radial consolidation by using a visco elastic approach. By extending Barron’s solution for radial consolidation of small strain sustained constant load, the convolution integral with time as the variable was used to analyze the consolidation under time-dependent loading. Four different loading rates were applied in the consolidation tests on three types of remolded clay with various plasticity indices to study the behavior of radial consolidation. The findings indicate that the predicted consolidation settlements accounting for the loading rate-dependent Cr values more closely match the experimental results than the predictions using an assumed constant Cr.
- 12. PART 2. Numerical Approach toward study of Sand Drains 2.1 Objective: 1. Comparative analysis of consolidation times with and without sand drains. 2. Find variation of ultimate consolidation settlement with applied stress. 3. Find variation of consolidation time with sand drain diameter at constant applied stress. The finite element software Plaxis 8.2 was used for the modeling of vertical sand drains through a layer of saturated clay. The goal was to discover the qualitative relationship between the ultimate consolidation settlement and time taken vs. the load applied and the diameter of sand drains. The time required for consolidation with and without drains was compared. The geometry of the unit cell modelled is given in figures 2.1. and 2.2. 2.2 General Settings: The following general settings were used for the modeling- Model: Axisymmetry Elements: 15 noded x-acceleration: 0 y-acceleration: 0 Earth gravity : 9.8 m/s^2 Length: metre Force: kN Time: day 2.3 Soil Properties: Three material sets were created for this problem- Dense Sand, Stiff Clay and Soft Clay. Sand Drain: The sand used for making the drain was accorded the following properties- Material Set: Dense Sand Unsaturated unit weight = 17 kN/m3
- 13. Saturated unit weight = 20 kN/m3 Permeability Kx = ky = 1 m/day Cohesion = 1 kpa Internal angle of friction = 35 degrees Angle of dilatancy = 3 degrees Young’s Modulus = 40000 Kpa Poisson’s ratio = 0.30 Fig. 2.1 Sample geometry of unit cell without sand drain. Plaxis 8.2 Height of the unit cell = 15m Clay layer
- 14. Clay layer: Two different types of clay were studied- Clay 1 material set: Stiff Clay Unsaturated unit weight = 18 kN/m3 Saturated unit weight = 19 kN/m3 Permeability Kx = ky = 0.001 m/day Cohesion = 50 kPa Internal angle of friction = 0 degrees Angle of dilatancy = 0 degrees Young’s Modulus = 50000 Kpa Fig. 2.2 Sample geometry of the unit cell with sand drain. Plaxis 8.2 Height of clay layer = 15m Clay surrounding the sand drain in a unit cell Sand Drain
- 15. Poisson’s ratio = 0.35 Clay 2 material set: Soft clay Unsaturated unit weight = 15 kN/m3 Saturated unit weight = 17 kN/m3 Permeability Kx = ky = 0.01 m/day Cohesion = 15 kpa Internal angle of friction = 25 degrees Angle of dilatancy = 0 degrees Young’s Modulus = 10000 Kpa Poisson’s ratio = 0.25 2.4 Boundary Conditions: The two vertical sides of the unit cell and one bottom side were accorded the standard fixities boundary condition (no displacements) and closed consolidation boundary condition. In short, settlement and consolidation both were allowed only through the top surface. 2.5 Initial conditions: The effective stresses and ground water pore pressure were generated in the standard way (K0 procedure and phreatic surface respectively). 2.6 Calculations: Calculations were run till the excess pore pressure developed reached 1 kPa or below at all points in the unit cell. 2.7 Observations: The observations are summarized in the tables below- 2.7.1 U (settlement) vs. Δσ (Applied Stress) Taking diameter of sand drain = 0.4 m
- 16. Table 2.4 Clay 1 Δσ (kPa) Total Consolidation Settlement (m) Total Time with Sand drain, t1 (days) Total Time without Sand Drain, t2 (days) 100 200 300 400 500 600 700 800 900 1000 0.019 0.038 0.063 0.09 0.117 0.144 0.171 0.198 0.225 0.253 11.484 22.972 19.619 17.234 18.006 20.912 20.688 28.395 30.295 30.345 122.5 183.75 153.13 157.92 169.4 149.3 172.87 179.69 175.62 181.9 Table 2.5 Clay 2 Δσ (kPa) Total Consolidation Settlement (m) Total Time with Sand drain, t1 (days) Total Time without Sand Drain, t2 (days) 100 200 300 400 500 600 700 800 900 1000 0.077 0.169 0.261 0.353 0.445 0.537 0.629 0.721 0.814 0.906 22.968 45.938 45.938 45.938 45.938 45.938 45.938 45.938 61.251 61.251 61.25 91.876 91.876 91.876 91.876 91.876 91.876 91.876 91.876 91.876 2.7.2 U (Settlement) vs. Diameter (of drain) Keeping applied stress at 300 kPa Table 2.6 Clay 1 d (m) Total Consolidation Settlement (m) Total Time (days)
- 17. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.063 0.063 0.063 0.064 0.064 0.064 0.063 11.875 42.109 31.582 15.312 12.919 10.287 8.373 Table 2.7 Clay 2 d (m) Total Consolidation Settlement (m) Total Time (days) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.265 0.265 0.263 0.261 0.259 0.261 0.253 93.876 61.251 45.938 46.057 30.626 28.261 23.088 2.8 Results and Discussion: 2.8.1 Time vs. Time From Tables 2.4 and 2.5, it is clear that sand drains effectively reduce the time taken for consolidation of saturated clay for both stiff and soft clays. As expected, this reduction is much more pronounced in case of stiff clays where the time taken reduces by about 6 to 11 times. While in the case of soft clays, the reduction factor is 1.5 to 3 times. It is noticeable that as the applied stress increases and the final settlement (U) and time taken (t1 and t2) increase with it, the reduction factor is seen to decrease. Sand drains become less and less effective as the time of consolidation increases. All these results may be noticed in Tables 2.8 and 2.9.
- 18. Clay 1 Table 2.8 Time with sand drain, t1 (days) Time without sand drain, t2 (days) Reduction Ratio t2/t1 11.484 22.972 19.619 17.234 18.006 20.912 20.688 28.395 30.295 30.345 122.5 183.75 153.13 157.92 169.4 149.3 172.87 179.69 175.62 181.9 10.66701 7.998868 7.805189 9.163282 9.407975 7.139441 8.356052 6.328227 5.796996 5.994398 Clay 2 Table 2.9 Time with sand drain, t1 (days) Time without sand drain, t2 (days) Reduction Ratio t2/t1 22.968 45.938 45.938 45.938 45.938 45.938 45.938 45.938 61.251 61.251 61.25 91.876 91.876 91.876 91.876 91.876 91.876 91.876 91.876 91.876 2.666754 2 2 2 2 2 2 2 1.499992 1.499992 2.8.2 U vs Δσ For both stiff and soft clays, settlement steadily increases with applied load. This is in accordance with theory and intuition.
- 19. 0 0.05 0.1 0.15 0.2 0.25 0.3 100 200 300 400 500 600 700 800 900 1000 Total SeDlement (m) Applied Stress (kPa) Fig. 2.3 SeDlement vs Loading for Clay 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100 200 300 400 500 600 700 800 900 1000 Total SeDlement (m) Applied Stress (kPa) Fig. 2.4 SeDlement vs Loading for Clay 2
- 20. 2.8.3 Settlement vs. Diameter of Sand Drain As expected, the final settlement did not vary with the diameter of sand drain. And the time of consolidation steadily decreases with increase in the diameter of drains. This is also in accordance with theory. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 100 200 300 400 500 600 700 800 900 1000 RaOo Applied Stress (kPa) Fig. 2.5 RaOo of SeDlement SoP Clay to SOﬀ Clay
- 21. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Total SeDlement (m) Diameter of Sand Drain (m) Fig. 2.6 SeDlement vs. Drain Dia for Clay 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Total SeDlement (m) Diameter of Sand Drain (m) Fig. 2.6 SeDlement vs. Drain Dia for Clay 2
- 22. 0 10 20 30 40 50 60 70 80 90 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ConsolidaOon Time (days) Diameter of Sand Drain (m) Fig. 2.7 Total Time vs. Drain Diameter for Clay 1 0 10 20 30 40 50 60 70 80 90 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ConsolidaOon Time (days) Diameter of Sand Drain (m) Fig. 2.7 Total Time vs. Drain Diameter for Clay 2
- 23. 2.9 Conclusions 1. Sand drains effectively reduce the time taken for consolidation of saturated clay for both stiff and soft clays. 2. This reduction is much more pronounced in case of stiff clays where the time taken reduces by about 6 to 11 times. 3. Sand drains become less and less effective as the time of consolidation increases. 4. For both stiff and soft clays, settlement steadily increases with applied load. 5. The settlement of soft clay was found to be 3 to 5 times more than that for stiff clay. 6. The final settlement does not vary with the diameter of sand drain. 7. And the time of consolidation steadily decreases with increase in the diameter of drains. *****
- 24. References: 1. Leo, C. (2004). ”Equal Strain Consolidation by Vertical Drains.” J. Geotech. Geoenviron. Eng., 130(3), 316–327. 2. Xiao, D., Yang, H., and Xi, N. (2011) Effect of Smear on Radial Consolidation with Vertical Drains. Geo-Frontiers 2011: pp. 4339-4348. doi: 10.1061/41165(397)444 3. Hsu, T. and Liu, H. (2013). ”Consolidation for Radial Drainage under Time-Dependent Loading.” J. Geotech. Geoenviron. Eng., 139(12), 2096–2103. 4. Indraratna, B., Aljorany, A., and Rujikiatkamjorn, C. (2008). ”Analytical and Numerical Modeling of Consolidation by Vertical Drain beneath a Circular Embankment.” Int. J. Geomech., 8(3), 199–206. 5. Nogami, T. and Li, M. (2003). ”Consolidation of Clay with a System of Vertical and Horizontal Drains.” J. Geotech. Geoenviron. Eng.,129(9), 838–848. 6. Lekha, K., Krishnaswamy, N., and Basak, P. (1998). ”Consolidation of Clay by Sand Drain under Time-Dependent Loading.” J. Geotech. Geoenviron. Eng., 124(1), 91–94. 7. Tan, S. (1993). ”Ultimate Settlement by Hyperbolic Plot for Clays with Vertical Drains.” J. Geotech. Engrg., 119(5), 950–956. 8. Das, B. M. (2008). “Advanced Soil Mechanics”, 3rd Ed., Taylor and Francis, London and New York.

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