1

Consolidation Settlement of Sand Drains – Analytical and Numerical Approaches

CE 632

INSTRUCTOR: DR. RAJESH SATHIYAMOORTHY

2Objectives

1. Analytical Approach

To review the available literature and theory pertaining to consolidation of clay installed with sand drains.

To investigate recent improvements in the popular theory.

2. Numerical Approach

To model a drain unit cell in the finite element program Plaxis 2d in order to study –

i. Improvement in consolidation time by use of vertical sand drains.

ii. Variation of ultimate settlement with loading

iii. Variation of consolidation time with diameter of sand drains.

3

PART 1 ANALYTICAL APPROACH

4Sand Drains

Definition

Fundamental approach

1). Free Strain

2). Equal Strain

Assumptions involved

Smear Consideration

Recent Improvements

5Nogami & 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

Care is given to formulation of thin pervious layers for efficient computation

Can handle the inhomogeneous profile in clay and multiple horizontal drains made of either thin sand layers or geotextile sheets

Developed formulation is verified using available numerical and field information

6Lekha et. al. (1998)

Consolidation Of Clay By Sand Drain Under Time-dependent Loading

Non Linear theory of sand drain consolidation

Took account of effective stress/void ratio/ permeability variations

Closed form solutions are provided for the variation of pore water pressure with a time factor and load increment ratio

Verified with standard results for instantaneous loading, constant permeability, and constant compressibility

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Indraratna et. al. (2008)

Analytical And Numerical Modelling Of Consolidation By Vertical Drain Beneath A Circular Embankment

Consolidation by vertical drains below a circular loaded area where the system of vertical drains in the field was transformed by a series of equivalent concentric cylindrical 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.

8Hsu et. al. (2013)

Consolidation For Radial Drainage Under Time-dependent Loading

Radial drainage under linear time-dependent loading with varying loading dependent coefficients of radial consolidation by using a visco-elastic approach.

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.

9References

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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PART 2NUMERICAL APPROACH

11Geometry of the Problem

12

13Material Sets

Material Set: Dense Sand• Unsaturated unit weight = 17

kN/m3

• 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

• Three material sets were created for this problem –

• Dense Sand for Sand Drain

• Soft Clay for Clay layer

• Stiff Clay for Clay layer

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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

• Poisson’s ratio = 0.35

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

15Observations• U vs. Δσ (Taking diameter of sand drain = 0.4 m)

Table 2.1 for Stiff Clay

Δσ (kPa) Total Consolidation Settlement (m)

Total Time with Sand drain, t1

(days)

Total Time without Sand Drain, t2

(days)

100 0.019 11.484 122.5200 0.038 22.972 183.75300 0.063 19.619 153.13400 0.09 17.234 157.92500 0.117 18.006 169.4600 0.144 20.912 149.3700 0.171 20.688 172.87800 0.198 28.395 179.69900 0.225 30.295 175.621000 0.253 30.345 181.9

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Δσ (kPa) Total Consolidation Settlement (m)

Total Time with Sand drain, t1

(days)

Total Time without Sand Drain, t2

(days)

100 0.077 22.968 61.25

200 0.169 45.938 91.876

300 0.261 45.938 91.876

400 0.353 45.938 91.876

500 0.445 45.938 91.876

600 0.537 45.938 91.876

700 0.629 45.938 91.876

800 0.721 45.938 91.876

900 0.814 61.251 91.876

1000 0.906 61.251 91.876

U vs. Δσ ; Table 2.2 for Soft Clay

17U vs. Diameter

Table 2.3 for Stiff Clayd (m) Total Consolidation

Settlement (m)Total Time (days)

0.1 0.063 91.875

0.2 0.063 42.109

0.3 0.063 31.582

0.4 0.064 15.312

0.5 0.064 12.919

0.6 0.064 10.287

0.7 0.063 8.373

18U vs. Diameter

Table 2.4 for Soft Clayd (m) Total Consolidation

Settlement (m)Total Time (days)

0.1 0.265 93.876

0.2 0.265 61.251

0.3 0.263 45.938

0.4 0.261 46.057

0.5 0.259 30.626

0.6 0.261 28.261

0.7 0.253 23.088

19Results & Discussion

• Time vs. Time

• From Tables 2.1 and 2.2, 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.

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Time with sand drain, t1 (days)

Time without sand drain, t2 (days)

Reduction Ratio t2/t1

11.484 122.5 10.667

22.972 183.75 7.998

19.619 153.13 7.805

17.234 157.92 9.163

18.006 169.4 9.407

20.912 149.3 7.139

20.688 172.87 8.356

28.395 179.69 6.328

30.295 175.62 5.796

30.345 181.9 5.994

Table 2.5 for stiff clay

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Time with sand drain, t1 (days)

Time without sand drain, t2 (days)

Reduction Ratio t2/t1

22.968 61.25 2.66

45.938 91.876 2

45.938 91.876 2

45.938 91.876 2

45.938 91.876 2

45.938 91.876 2

45.938 91.876 2

45.938 91.876 2

61.251 91.876 1.49

61.251 91.876 1.49

Table 2.6 for soft clay

22• U vs Δσ

For both stiff and soft clays, settlement steadily increases with applied load.

100 200 300 400 500 600 700 800 900 10000

0.05

0.1

0.15

0.2

0.25

0.3

Fig. 2.3 Settlement vs Loading for Stiff Clay

Applied Stress (kPa)

Tot

al S

ettle

men

t (m

)

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100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 2.4 Settlement vs Loading for Soft Clay

Applied Stress (kPa)

Tot

al S

ettle

men

t (m

)

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100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Fig. 2.5 Ratio of Settlement Soft Clay to Stiff Clay

Applied Stress (kPa)

Rat

io

25• 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.

0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Fig. 2.6 Settlement vs. Drain Dia for Stiff Clay

Diameter of Sand Drain (m)

Tot

al S

ettle

men

t (m

)

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0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Fig. 2.6 Settlement vs. Drain Dia for Soft Clay

Diameter of Sand Drain (m)

Tot

al S

ettle

men

t (m

)

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0.1 0.2 0.3 0.4 0.5 0.6 0.70

10

20

30

40

50

60

70

80

90

100

Fig. 2.7 Total Time vs. Drain Diameter for Stiff Clay

Diameter of Sand Drain (m)

Con

solid

atio

n T

ime

(day

s)

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0.1 0.2 0.3 0.4 0.5 0.6 0.70

10

20

30

40

50

60

70

80

90

100

Fig. 2.8 Total Time vs. Drain Diameter for Soft Clay

Diameter of Sand Drain (m)

Con

solid

atio

n T

ime

(day

s)

29Conclusions

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. For both stiff and soft clays, settlement steadily increases with applied load.

4. The settlement of soft clay was found to be 3 to 5 times more than that for stiff clay.

5. The final settlement does not vary with the diameter of sand drain.

6. And the time of consolidation steadily decreases with increase in the diameter of drains.