Effect of subgrade soil stiffness on the design of geosynthetic tube

Geotextiles and Geomembranes 29 (2011) 277e284

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Effect of subgrade soil stiffness on the design of geosynthetic tube Wei Guo a, Jian Chu a, *, Shuwang Yan b a

School of Civil and Environmental Engineering, Nanyang Technological University, Centre for Infrastructure Systems, Blk N1, #01A-10, 50 Nanyang Avenue, Singapore 639798, Singapore b Geotechnical Research Institute, School of Civil Engineering, Tianjin University, 92 Weijin Road, Nankai District, Tianjin, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 September 2010 Received in revised form 16 December 2010 Accepted 17 December 2010 Available online 10 February 2011

Water or soil filled geotextile or geosynthetic tubes have been used for coastal or river protection projects in recent years. How to design and analyze geosynthetic tube is still an important research topic. Although a number of solutions for geosynthetic tube have been proposed in the past, most of these solutions assume that the geosynthetic tube is resting on a rigid foundation. In this paper, a twodimensional analysis of geosynthetic tube resting on deformable foundation soil is presented. The deformable foundation is assumed to be an elastic Winkler type represented by the modulus of subgrade reaction, Kf. The study shows that the smaller the modulus, the smaller the height of the geosynthetic tube above the ground surface and the higher the tensile force in the geotextile or geosynthetic given the other conditions the same. When the foundation soil has a modulus higher than 1000 kPa/m which is representative of soft clay, the foundation soil can be assumed to be rigid in the analysis. The results obtained from the method proposed in this paper are compared with those from the solutions of Leshchinsky et al. and Plaut and Suherman for verification. The differences between the solutions are also discussed. Ó 2011 Published by Elsevier Ltd.

Keywords: Geotextile Geosynthetic Geosynthetic tube Numerical analysis

1. Introduction There has been an increasing use of geosynthetic tubes in river or coastal protection projects in recent years (Leshchinsky et al., 1996; Pilarczyk, 2003; Kim et al., 2004; Oh and Shin, 2006; Shin and Oh, 2007; Yan and Chu, 2010) or waste sludge dewatering projects (Mori et al., 2002; Koerner and Koerner, 2006; Muthukumaran and Ilamparuthi, 2006). Geosynthetic tubes have also been used for other constructions such as for small dams or spillways, flood control, water diversion, groundwater recharging, and dewatering of high water content, contaminated waste or lagoon solid (Perry, 1993; Tam, 1997; Alvarez et al., 2007; Sehgal, 1996; Plaut and Suherman, 1998). Normally geosynthetic tubes are formed by sewing or gluing geotextile or geosynthetic sheets together and then filled with water, clay slurry, or sand. The geosynthetic tubes are sometimes stacked together to form a dike or other types of geotechnical structures (Yan and Chu, 2010). In this paper, geosynthetic tube refers to a cross-section geometry that is more or less circular like a sausage. It is assumed in the following discussion that the geosynthetic tube is either watertight and inflated with water or air, or permeable but the fills

* Corresponding author. Tel.: þ65 67904563; fax: þ65 67910676. E-mail address: [email protected] (J. Chu). 0266-1144/$ e see front matter Ó 2011 Published by Elsevier Ltd. doi:10.1016/j.geotexmem.2011.01.014

in the geosynthetic tubes has dissipated fully so the volume or geometry of the geosynthetic tube does not change anymore. Therefore, the solutions developed in this paper are only applicable to watertight geosynthetic tubes or those filled with sand where consolidation is fast and does not affect much the final geometry of the geosynthetic tubes. Analytical solutions for water filled impervious geosynthetic tubes resting on rigid foundation have been derived by Liu and Silvester (1977), Silvester (1986), Leshchinsky et al. (1996), Kazimirowicz (1994), Plaut and Suherman (1998), Antman and Schagerl (2005), Malík (2007) and Ghavanloo and Daneshmand (2009). Most of these solutions are based on the assumptions that the tube is long enough to be simplified into a plane strain problem and the tubes are resting on a rigid base. The only exception is the solution given by Plaut and Suherman (1998) in which the geosynthetic tube is assumed to be resting on tensionless Winker foundation. It uses non-dimensional parameters in which H and Hf are normalized by the length of crosssection, L; the pumping pressure, p, by gL; the tensile force, T, by gL2; and the modulus of subgrade reaction, Kf, by g. The tube was modeled as an inextensible membrane and filled with an incompressible fluid. The geometry of geosynthetic tube was resolved by partial differential equations with non-dimensional parameters. In Plaut and Suherman’s solution, the perimeter of the cross-section of the tube and the pumping pressure

W. Guo et al. / Geotextiles and Geomembranes 29 (2011) 277e284

An analytical solution for the geometry of the cross-section of an impervious geosynthetic tube filled with water at the equilibrium state can be derived based on the differential formulas. In proposing this solution, the following assumptions need to be made. i. The geosynthetic tube is sufficiently long to be assumed into a plane strain problem. ii. The geosynthetic sheet is thin, flexible so that its weight and extension can be neglected. iii. The friction between the geosynthetic tube and the fill material, or that between the geosynthetic tube and the rigid foundation are neglected. iv. The foundation soil can be assumed to be an elastic Winkler foundation with a Winkler modulus of subgrade reaction equal to Kf . The lateral reaction of subgrade is not considered. Some of the above hypotheses are also made in the existing theoretical solutions such as Liu and Silvester (1977), Szyszkowski and Glockner (1987), Kazimirowicz (1994), Leshchinsky et al. (1996), Plaut and Suherman (1998), Cantré (2002), Cantré and Saathoff (in press) . Furthermore, it is assumed the geosynthetic tube is filled with water or slurry with a unit weight of g. A simplification of the cross-section of the geosynthetic tube is shown in Fig. 1. The coordinates are set up with y in the horizontal direction and x in the vertical direction. The origin is taken as the top of cross-section. The cross-section of the tube can be considered symmetric with respect to x-axis. The height and width of the cross-section are represented by H and B, respectively, in Fig. 1. Hf denotes the height of the geosynthetic tube below ground surface or the amount of settlement. The contact width of the tube with the ground is denoted as b. As shown in Fig. 1, the angle between the tangent direction along the cross-section of geosynthetic tube and the y axis is q. The force analysis of a section of the geosynthetic tube along the cross-section is given in Fig. 2. The pumping pressure used to fill in water or slurry is denoted as p0. The hydraulic pressure acting internally on the tube at a given depth is p0 þ gx. The circumferential tension force in the tube per unit width along the crosssection is T. Using Fig. 2, the force equilibrium equations on the normal and tangential directions can be written as:

θ

H

2. Analytical solutions for geosynthetic tube resting on deformable foundation

y

0

H

(expressed as pressure head) are assumed to be known. The height and width of the geosynthetic tube are the results of the solutions. However, in practice, the height of geosynthetic tube above ground surface and the pumping water pressure are often the targeted design parameters and the perimeter of the crosssection of the tube and the tensile force of geotextile material required are calculated. In this paper, a numerical solution in the form of a computer program for geosynthetic tube resting on deformable foundations soil is presented. A modified Winker foundation model is used in the solution. As the height of a dike is normally specified as a design requirement, the height of the geosynthetic tube and pumping pressure are taken as the two input parameters in the analysis. The tensile force and the geometry of the cross-section of the tube are calculated as solutions. The proposed method can be used to analyze impervious geosynthetic tubes filled with water or permeable geosynthetic tubes filled with sand. The solutions are not applicable to geosynthetic tube filled with low permeability clay.

f

278

b B Fig. 1. Cross-section of geosynthetic tube resting on deformable foundation.

dT ¼ aKf ðx  HÞsin q ds

(1)

i dq 1h ¼  aKf ðx  HÞjcos qj þ P0 þ gx ds T

(2)

where a ¼ 0 for x < H, and a ¼ 1.0 for x > H. Using Fig. 3, two geometrical equations relating the angle between the tangent direction along cross-section and the x and y coordinates can also be given as:

dx ¼ sin q ds

(3)

dy ¼ cos q ds

(4)

As the problem is symmetric as assumed, the tensile force direction on the top and bottom of the cross-section of the geosynthetic tube must be horizontal as shown in Fig. 3. The hydraulic pressure acting in horizontal direction on the geosynthetic tube is also shown in Fig. 3. The reaction of the Winkler foundation is vertically. Therefore, the forces acting along the y coordinate only involves the hydraulic pressure and the tensile force in the geosynthetic tube. The equilibrium between these two forces yields the following equation:

  1  2 Tx¼HþHf þ Tx¼0 ¼ p0 H þ Hf þ g H þ Hf 2

(5)

Integrating Eq. (1) by combining (3)e(5) gives the expression of tensile force along the cross-section of the geosynthetic tube as shown in Eqs. (6a) and (6b). It can be interpreted from these equations that the tensile force in the tube at a depth above the

T

p0 + γ x θ

T+dT

α K f (x − H )

Fig. 2. Calculation unit of geosynthetic tube.

W. Guo et al. / Geotextiles and Geomembranes 29 (2011) 277e284

Table 1 Comparison of present with Kazimirowicz’s (1994) solutions. (For L ¼ 3.6 m, g ¼ 14 kN/m3).

y

p0

T x=0

p0 (kPa)

Source

H (m)

θ 17.5

H

Kazimirowicz Present Kazimirowicz Present Kazimirowicz Present Kazimirowicz Present

10.4 4.6 3.0

x

T x=H+H f p0 + γ ( H + H f )

279

b/2

b (m)

T (kN/m)

Values

%

Values

(%)

Values

1.00 0.99 0.90 0.90 0.80 0.81 0.70 0.76

1.50

0.46 0.46 0.64 0.65 0.84 0.82 0.96 0.89

1.09

11.80 11.90 6.80 6.77 4.00 4.11 2.70 3.03

0.33 1.75 8.29

1.25 2.14 3.65

% 0.85 0.49 2.83 15.48

p0, are taken as inputs. The tensile force, T, can be calculated using Eq. (6). The following two initial boundary conditions can be established to resolve this equation:

B/2 Fig. 3. Force equilibrium of a geosynthetic tube resting on deformable foundation.

(1) When x ¼ 0, then y ¼ 0, dy/dx ¼ N, q ¼ 0. (2) When x ¼ H þ Hf, then y ¼ 0, dy/dx ¼ N, and q ¼ p. ground surface, Tx¼0eH, is constant because the friction is neglected and the tensile force below the ground surface, Tx¼Hw(HþHf), is not constant, but increase with the depth x. It can also been seen from Eq. (6a), that the tensile force T will become the largest when the x coordinate equals to the total height, (H þ Hf):

Tx¼0wH

 1  2 1 1  ¼ p0 H þ Hf þ g H þ Hf  Kf Hf2 2 4 4

3.1. Solutions for geosynthetic tube resting on rigid foundation

(6b)

For a geosynthetic tube resting on rigid foundation, the modulus of subgrade reaction is infinite, i.e., Kf ¼ N, and thus the geosynthetic tube has no settlement or Hf ¼ 0. Based on this condition, it can be derived from Eq. (1) that dT/ds ¼ 0. It implies that the tensile force along the cross-section of the geosynthetic tube is constant. The final governing equations of Eqs. (6) and (7) can then be re-written as:

Tx¼HwðHþHf Þ ¼ Kf

Combining Eqs. (2)e(4), a nonlinear differential equation relating geometry of the cross-section of geosynthetic tube can be derived as Eq. (7) which has to be resolved numerically as it contains an elliptical differential equation that has no closed form solution:

y00 ¼

 1=2 3=2 i 1h aKf ðx  HÞjy0 j 1 þ y02 ðP0 þ gxÞ 1 þ y02 T

3. Numerical solutions for geosynthetic tube

(6a)



  1 2 1  x  Hx þ p0 H þ Hf 2 2 2 1   1  þ g H þ Hf  Kf Hf2  2H 2 4 4

Combining these conditions and Eq. (6) to Eq. (7), the relationship between the x and y coordinates that depicts the geometry of the cross-section of the tube can then be solved.

(7)

In Eq. (7), the unit weight of filling slurry, g, the height of geosynthetic tube above ground surface, H, and the pumping pressure,

T ¼

1 1 p H þ gH 2 2 0 4

(8)

3=2  1 y00 ¼  ðP0 þ gxÞ 1 þ y02 T

(9)

The same equations have also been derived by Kazimirowicz (1994), Leshchinsky et al. (1996), and Plaut and Suherman (1998).

Table 2 Comparison of present with Leshchinsky’s et al.’s (1996) solutions. (For L ¼ 9 m, g ¼ 12 kN/m3). p0 (kPa)

0 4.8 6.9 34.5 52.4 103.5 122.8 593.4

Source

Leshchinsky Present Leshchinsky Present Leshchinsky Present Leshchinsky Present Leshchinsky Present Leshchinsky Present Leshchinsky Present Leshchinsky Present

H (m)

B (m)

A (m)

T (kN/m)

Values

(%)

Values

%

Values

%

Values

%

0.90 0.90 1.80 1.79 2.00 1.90 2.50 2.40 2.60 2.50 2.70 2.70 2.70 2.70 2.90 2.80

0.00

4.09 4.08 3.60 3.57 3.64 3.49 3.21 3.13 3.13 3.06 3.00 2.98 2.96 2.96 2.96 2.89

0.24

3.36 3.23 5.56 5.35 5.76 5.56 6.45 6.25 6.51 6.33 6.57 6.40 6.57 6.41 6.66 6.44

3.87

2.50 2.40 14.60 13.90 18.10 17.60 61.70 59.80 87.50 86.00 162.00 159.80 189.70 187.60 875.00 862.70

4.00

0.56 5.00 4.00 3.85 0.00 0.00 3.45

0.83 4.12 2.49 2.24 0.67 0.00 2.36

3.78 3.47 3.10 2.76 2.59 2.44 3.30

4.79 2.76 3.08 1.71 1.36 1.11 1.41

280

W. Guo et al. / Geotextiles and Geomembranes 29 (2011) 277e284

Table 3 Comparison of present with Plaut and Suherman’s (1998) solution. Kf (kPa/m)

10 25 50 100 200

Source

Plaut Present Plaut Present Plaut Present Plaut Present Plaut Present

p0 (kPa)

H (m)

Hf (m)

Tmax (m)

Values

%

Values

(%)

Values

0.0565 0.0558 0.0514 0.0504 0.0498 0.0499 0.0491 0.0488 0.0487 0.0485

1.33

0.1935 0.1943 0.1986 0.1996 0.2002 0.2001 0.2009 0.2012 0.2013 0.2015

0.39

0.0264 0.0287 0.0104 0.0114 0.0051 0.0059 0.0025 0.0028 0.0013 0.0015

1.95 0.18 0.65 0.41

0.50 0.04 0.16 0.10

By combining with the initial conditions, Eqs. (8) and (9) can be used to calculate the geometry of the cross-section of the geosynthetic tube. To verify computer program developed based on the above equations, the solutions given by the proposed method are compared with those by Kazimirowicz (1994) and Leshchinsky et al. (1996) in Tables 1 and 2, respectively. It can be seen that very good agreements have been achieved for both cases.

9.62 15.69 12.00 15.38

%

Values

%

0.02005 0.02043 0.01696 0.01698 e 0.01591 e 0.01533 e 0.01506

1.90

0.01656 0.01657 0.01562 0.01561 e 0.01527 e 0.01512 e 0.01487

0.06

0.12

0.06

The above numerical procedure was verified by comparing the solutions with those given by Plaut and Suherman (1998). As reviewed in Section 1, Plaut and Suherman’s method uses nondimensional parameters in which H and Hf are normalized by the

a

Pumping pressure, p0 = 10 kPa 3.0 2.5

Vertical axis (m)

2.0 1.5 Kf= 1.0

Kf= 200 kPa/m Kf= 100 kPa/m

0.5 0.0 -0.5 -2.0

b

-1.5

-1.0

-0.5 0.0 0.5 Horizontal axis (m)

1.0

1.5

2.0

Pumping pressure, p0=50 kPa 3.0 2.5

Vertical axis (m)

(1) Input the initial parameters: g, p0, Kf, and H; (2) Assume the trail height below the deformable foundation is Hft, then the trial tensile force on the top point of tube, Tt, can be calculated using Eq. (6); (3) Input g, Hft, p0, H, Kf, and Tt into Eq. (7) to calculate the x and y coordinates along the cross-section of the geosynthetic tube; (4) If y s 0 or dy/dx s N, for x ¼ H þ Hf, then change the value of Hft and repeat step (2) to (3) until y ¼ 0 and dy/dx ¼ N. The final value of Hft is the amount of settlement of the geosynthetic tube, Hf, and Tt is the tensile force on the top point of tube, T. Using Hf, the x and y coordinates along the cross-section of the geosynthetic tube, the perimeter, the width, the contact width, and the tensile force can be obtained; (5) If the length of the geosynthetic tube, L, is taken as an input parameter, instead of height, H, the iteration can be done as follows: assume H ¼ L/p as the input parameter to calculate the trial length, Lt. If Lt s L, then modify H and repeat the calculation until a good agreement between Lt and L is achieved.

8.71

Tmin (m)

Values

3.3. Verification of the proposed solutions

3.2. Solutions for geosynthetic tube resting on deformable foundation In this section, a numerical solution for the general case of geosynthetic tube resting on deformable foundation soil is presented. A computer program was written using COMPAQ VISUAL FORTRAN in which the RungeeKutta method was used to solve Eqs. (6) and (7). The following iteration procedure has been developed to calculate the cross-sectional shape of the geosynthetic tube resting on deformable foundation. The unit weight of the slurry, g, the pumping pressure, p0, the modulus of subgrade reaction, Kf, and the height of geosynthetic tube above ground surface, H, are taken as given designing parameters. The tensile force, T, and the x and y coordinates that define the geometry of the cross-section can then be calculated by combining Eqs. (6) and (7) and initial conditions. The width of the geosynthetic tube, the contact width of the geosynthetic tube with ground, and the cross-section perimeter of the geosynthetic tube can then be calculated. The calculation adopted in this numerical procedure for the design of geosynthetic tube is explained as follows.

%

2.0 1.5 Kf= 1.0

Kf= 200 kPa/m Kf= 100 kPa/m

0.5 0.0 -0.5 -2.0

-1.5

-1.0

-0.5 0.0 0.5 1.0 Horizontal axis (m)

1.5

2.0

Fig. 4. Changes in the cross-section of geosynthetic tube with pumping pressure, p0, and modulus of subgrade reaction, Kf (for g ¼ 12 kN/m3, L ¼ 9 m, Kf ¼ 100, 200 and f kPa/m, respectively). (a) Pumping pressure, p0 ¼ 10 kPa. (b) Pumping pressure, p0 ¼ 50 kPa.

W. Guo et al. / Geotextiles and Geomembranes 29 (2011) 277e284

length of cross-section, L; the pumping pressure, p, by gL; tensile force, T, by gL2; and modulus of subgrad reaction, Kf, by g. The calculations were made by taking the length of the cross-section of the tube, L, to be 1 m and the unit weight of the slurry to be 1.0 kN/m3. The numerical results obtained from both methods are shown in Table 3. It can be seen that a good agreement between the two solutions is achieved. 4. Parametric studies Using the proposed method, some parametric studies for a geosynthetic tube resting on deformable foundation can be conducted to assess the influence of some of the control parameters. The effect of the modulus of subgrade reaction on the deformation of geosynthetic tube is studied. The parameter study was carried out for an impervious geosynthetic tube with a perimeter of 9 m filled with slurry of a unit weight of 12 kN/m3. Using a pumping pressure of 10 and 50 kPa, respectively, the shape of the crosssection of the geosynthetic tube is calculated as shown in Fig. 4a and b. The solutions are given for modulus of subgrade reaction, Kf, of 100, 200, and f kPa/m, respectively. It can be seen from Fig. 4

a

Relationship between percentage error and modulus of subgrade reaction 25 p0 = 10 kPa p0 = 20 kPa

20

Δ H/H (%)

p0 = 50 kPa 15

281

Table 4 Range of values of modulus of subgrade reaction Kf (with reference to Bowles, 1988). (Use values as guide and for comparison when using approximate equations). Kf, kPa/m

Soil Loose sand

4800e16,000

Stiff to hard clayey soil qu 200< qu qu Soft to stiff soil Very soft soil

200 kPa 400 kPa >800 kPa

12,000e24,000 24,000e48,000 >48,000 1000e12,000 <1000

that the height of the geosynthetic tube above the ground surface is influenced by the modulus of foundation. For a given pumping pressure, the weaker the foundation, the smaller the height of the geosynthetic tube above ground surface and the greater the settlement of the geosynthetic tube. This is consistent with the intuitive expectation. Based on the solutions shown in Fig. 4, the differences between solutions with and without considering the modulus of subgrade reaction can be evaluated. The percentages of differences in the height of the tube versus the modulus of subgrade reaction Kf are plotted in Fig. 5a. It can be seen that when the modulus of subgrade reaction is less than 1000 kPa/m, the height of the geosynthetic tube is much influenced by the modulus of subgrade reaction. The differences become negligible when the modulus of subgrade reaction is greater than 1000 kPa/m. The settlement of the geosynthetic tube (or the height below the ground surface), Hf, is also plotted versus Kf in Fig. 5b. For the pumping pressures used, the settlement of the geosynthetic tube Hf varies drastically with the modulus of subgrade reaction.

a

Change in the cross-section of geosynthetic tube with pumping pressure, p0 2.5

p0= 0 kPa

10

p0= 1 kPa

2.0

5

0

0

500

1000

1500

2000

Vertical axis (m)

p0= 5 kPa

H

p0= 50 kPa p0= 100 kPa

0.5 0.0

Hf -0.5 -2.5

Relationship between the settlement Hf and modulus of subgrade reaction

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

Horizontal axis (m)

0.8

b3

p0 = 10 kPa

Relationships between height and pumping pressure

p0 = 20 kPa 0.6

p0= 20 kPa

1.0

Modulus of subground reaction, Kf , kPa/m

b

p0= 10 kPa

1.5

H Hf

p0 = 50 kPa

Height (m)

Hf (m)

2

0.4

1

0.2

0.0

0

500

1000

1500

2000

Modulus of subground reaction, Kf (kPa/m) Fig. 5. Effect of modulus of subgrade reaction on the settlement of geosynthetic tube. (For g ¼ 12 kN/m3, L ¼ 9 m, p0 ¼ 10, 20, 50 kPa, respectively). Relationship between (a) percentage error and modulus of subgrade reaction; and (b) the settlement Hf and modulus of subgrade reaction.

0

0

20

40

60

80

100

120

Pumping pressure (kPa) Fig. 6. Effect of pumping pressure on the cross-section of geosynthetic tube resting on deformable foundation (For g ¼ 12 kN/m3, L ¼ 9 m, Kf ¼ 100 kPa/m). (a) Change in the cross-section of geosynthetic tube with pumping pressure, p0. (b) Relationships between height and pumping pressure.

Height of a point on the cross section (m)

282

W. Guo et al. / Geotextiles and Geomembranes 29 (2011) 277e284

2.5

p0= 0 kPa p0= 1 kPa

2.0

p0= 5 kPa p0= 10kPa

1.5

p0= 20kPa p0= 50kPa

1.0

p0=100kPa 0.5 0.0 -0.5

0

20

40

60

80

100

120

140

160

180

Tensile force of the point on the cross section (kN/m) Fig. 7. Relationship between the height of a point on the cross-section and the tensile force of the geosynthetic tube (For g ¼ 12 kN/m3, L ¼ 9 m, Kf ¼ 100 kPa/m, p0 ¼ 0, 1, 5, 10, 20, 50, 100 kPa, respectively).

However, when the modulus of subgrade reaction is greater than 1000 kPa/m, the influence of modulus becomes insignificant. When the settlement of the geosynthetic tube is small, Eqs. (8) and (9) can be used as a good approximation for Eqs. (6) and (7). Table 4 gives

a

Modulus of subgrade reaction versus maximum tensile force of geosynthetic tube 100

a

Maximum tensile force of geosynthetic tube (kN/m)

p0 = 10 kPa, Kf = 100, 200, 500 and kPa/m separately 2.5 2.0 1.5

Vertical axis (m)

the modulus of subgrade reaction of different soil types. As seen from this table, almost the sand and clay have a larger modulus of subgrade than 1000 kPa/m. In other words, all these kinds of soil can be considered as rigid in the calculation. However, if the geosynthetic tube used to build dikes in soil reclamation projects, the modulus of subgrade is so low that the subgrade cannot be considered as rigid. The effect of pumping pressure on the geometry of the geosynthetic tube is also evaluated. Using the same initial parameters as for Fig. 4 with g ¼ 12 kN/m3, L ¼ 9 m, Kf ¼ 100 kPa/m, the crosssection of the geosynthetic tube at p0 of 0, 1, 5, 10, 20, 50 and 100 kPa are calculated and shown in Fig. 6a. The height of the geosynthetic tube versus pumping pressure is also plotted in Fig. 6b. It can be seen that the higher the pumping pressure, the higher the geosynthetic tube above the ground surface, and the greater the settlement below the ground surface. Fig. 6b shows that the influence of pumping pressure p0 on H and Hf is greater when p0 is smaller than 30 kPa in this case. When p0 is greater than 30 kPa, the cross-section of the geosynthetic tube has reached almost a circular shape and then the pumping pressure will have little

Kf= 100 kPa/m 1.0

Kf= 200 kPa/m Kf= 500 kPa/m

0.5

Kf=

0.0

p0=20 kPa

80

p0=50 kPa

60

40

20

-0.5 -1.0 -2.0

p0=10 kPa

0

500

1000

1500

2000

Modulus of subground reaction, Kf (kPa/m) -1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Horizontal axis (m)

b

Modulus of subgrade reaction versus perimeter of cross section 12

b

Perimeter of cross section of geosynthetic tube (m)

Kf = 100 kPa/m and p0 = 10, 30, and 50 kPa separately

2.5

Vertical axis (m)

2.0 1.5

p0= 10 kPa p0= 30 kPa

1.0

p0= 50 kPa

0.5 0.0 -0.5 -2.0

p0=10 kPa p0=20 kPa

8

6

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Horizontal axis (m) Fig. 8. Effect of modulus of subgrade reaction and pumping pressure on the crosssection of geosynthetic tube (for g ¼ 12 kN/m3 and H ¼ 2 m). (a) p0 ¼ 10 kPa, Kf ¼ 100, 200, 500 and N kPa/m, separately. (b) Kf ¼ 100 kPa/m and p0 ¼ 10, 30, and 50 kPa separately.

p0=50 kPa

10

0

500

1000

1500

2000

Modulus of subground reaction, Kf (kPa/m) Fig. 9. Effect of modulus of subgrade reaction on the tensile force and perimeter (For g ¼ 12 kN/m3, H ¼ 2 m, p0 ¼ 10, 20, 50 kPa, respectively). Modulus of (a) subgrade reaction versus maximum tensile force of geosynthetic tube; and (b) subgrade reaction versus perimeter of cross-section.

W. Guo et al. / Geotextiles and Geomembranes 29 (2011) 277e284

Maxium tensile force in geosynthetic tube (kN/m)

a

Perimeter of cross section of geosynthetic tube (m)

b

Pumping pressure versus maximum tensile force 200

150

100

Kf = 100 kPa/m

50

Kf = 200 kPa/m Kf = 500 kPa/m Kf = 1000 kPa/m

0

0

20

40 60 80 Pumping pressure, p0 (kPa)

100

120

Pumping pressure versus perimeter of cross section 12 Kf = 100 kPa/m Kf = 200 kPa/m Kf = 500 kPa/m Kf = 1000 kPa/m

10

8

283

that given the height of the tube being the same, the higher the pumping pressure, the smaller the perimeter as the tube is more circular. This is a different reflection of the fact shown in Fig. 6a where it is seen that given the perimeter the same, the higher the pumping pressure, the higher the tube. Based on the data shown in Fig. 8 and calculations for other modulus of subgrade reaction and pumping pressures, the influences of pumping pressure and modulus of the subgrade reaction of soil Kf on the maximum tensile force and the perimeter of the crosssection of a geosynthetic tube with a design height of 2 m are shown in Fig. 9a and b. It can be seen from Fig. 9a that the tensile force increases with pumping pressure and modulus of the foundation. However, the effect of Kf becomes insignificant when Kf > 1000 kPa/m. Fig. 9b shows the perimeter length required to achieve a geosynthetic tube of 2 m height above the ground surface increases when the soil is getting softer or in other words when the modulus of subgrade reaction becomes smaller. The variations of the maximum tensile force and perimeter of the cross-section of the tube with pumping pressure are also shown in Fig. 10a and b. As expected, the tensile force in the geosynthetic tube is affected considerably by the pumping pressure. As can be seen from Fig. 10a, the maximum tensile force increases almost linearly with increasing pumping pressure at different Kf values. Fig. 10b shows the perimeter for a geosynthetic tube of 2 m height when resting on a soil with different Kf reduces quickly as the pumping pressures increases. However, the influence becomes insignificant when the pumping pressure is greater than 50 kPa. This is because when the pumping pressure is greater than 30 kPa, the shape of the geosynthetic tube reaches almost a perfect circle and then the pumping pressure will not affect much the cross-section of the tube anymore. 5. Conclusions

6

0

20

40 60 80 Pumping pressure, p0 (kPa)

100

120

Fig. 10. Effect of pumping pressure on the tensile force and perimeter (For g ¼ 12 kN/ m3, H ¼ 2 m, Kf ¼ 100, 200, 500, 1000 kPa/m, respectively). (a) Pumping pressure versus maximum tensile force. (b) Pumping pressure versus perimeter of cross-section.

effect on the cross-section of the tube. However, a higher p0 will increase the tensile force heavily as shown in Fig. 7. It is also seen in Fig. 7 that the tensile forces along the cross-section of the geosynthetic tube in the section above the ground surface are constant. However, the tensile forces along the section below the ground surface varies. The lower the point on the geosynthetic tube, the greater the tensile force will be induced. The tensile force attains the highest value at the bottom of the geosynthetic tube. It should be pointed out that the above results regarding tensile force in geotextile is only valid when the shear friction between geotextile and soil can be ignored as is assumed in the Winkler foundation model. However, in practice, the friction between geotextile and soil will play a role and often lead to a reduction in the tensile force in the geotextile along the portion in contact with the ground will reduce because of the contact friction (Saathoff et al., 2007). For a real project, the design target is normally the height of geosynthetic tube above ground surface. Therefore, parametric studies were carried out to investigate the effect of modulus of subgrade reaction and pumping pressure. For a geosynthetic tube with a design height of 2 m under a pumping pressure of 10 kPa with slurry of a unit weight 12 kPa/m, the variation in the crosssection area and settlement is shown in Fig. 8a. The influence of the pumping pressure on the cross-sections of a geosynthetic tube with a fixed height of 2 m resting on soft foundation with a modulus of 100 kPa/m is shown in Fig. 8b. The solutions given in Fig. 8b are for pumping pressures of 10, 30, 50 kPa, respectively. It can be seen

A numerical method for conducting 2D analysis for geosynthetic tube resting on deformable foundation is presented. In the analysis, the geosynthetic tube is assumed to be watertight, weightless and inextensible. The deformable foundation is assumed to be elastic Winkler Foundation. The frictions along the soil geotextile interfaces are neglected. The method can be used to calculate the geometry and tensile force in the geotextile for an impervious geosynthetic tube inflated under a certain pressure, or the geometry and tensile force for a permeable geosynthetic tube with the fills in the geosynthetic tubes dissipated fully so the volume or geometry of the geosynthetic tube does not change anymore. Using the solutions, the effect of the modulus of subgrade reaction on the design of a geosynthetic tube has been discussed. When the modulus of subgrade reaction is less than 1000 kPa/m (that is, for soft clay), the geosynthetic tube has greater settlement and higher tensile strength. For a given design height, the perimeter required is also greater for smaller modulus of subgrade reaction. However, when the modulus of subgrade reaction is greater than 1000 kPa/m, its influence becomes insignificant and the foundation can be assumed to be rigid without affecting the design. The influence of the pumping pressure on the geometry of a geosynthetic tube was discussed. For the selected tube and foundation properties, the shape of the geosynthetic tube is much affected by the pumping pressure. However, the effect becomes insignificant when the pumping pressure is greater than 50 kPa. Factors affecting the tensile force along the cross-section of the geosynthetic tube were also discussed. For the cases studied where the interface friction between geotextile and soil is neglected, the tensile force increases almost linearly with pumping pressure. The parametric study shows that the tensile force in the section of the tube above the ground surface is constant. However, the tensile force in the section of the tube below the ground surface varies. The

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