Download Paper - IGS - International Geosynthetics Society

Technical Paper by M.R. Madhav, N. Gurung and Y. Iwao 

ATHEORETICAL MODEL FOR THE PULL-OUT 

RESPONSE OF GEOSYNTHETIC REINFORCEMENT 

ABSTRACT: Soil-reinforcement pull-out tests are essential for evaluating the strength, integrity, 

and effectiveness of the soil-reinforcement system. In this paper, a new pull-out test 

model that calculates the soil-geosynthetic reinforcement interface shear stress for highly 

extensible geosynthetic reinforcement is proposed. Based on a new bilinear interface shear 

model, the geosynthetic pull-out test results are calculated with regard to the variation of the 

mobilised geosynthetic tension with distance, geosynthetic pre-yield and post-yield behaviour, 

and the effective and extended length of the geosynthetic reinforcement The resulting 

nonlinear equation for the soil-geosynthetic interface shear stress pull-out mechanism is 

nondimensionalised, expressed in a finite difference form, and solved numerically using the 

Gauss-Siedel technique. A parametric study is carried out for a range of relative stiffness values 

and interface shear stresses. The normalised load-displacement relationship and the 

variation of the pull-out force and reinforcement displacements, with distance along the reinforcement, 

are presented. The values calculated using the proposed model are compared 

with experimental pull-out test results for a needle-punched, nonwoven geotextile, polyester 

fibres coated with polyethylene, and nylon reinforcements. 

KEYWORDS: Analytical model, Finite difference method, Geosynthetic, Geotextile, 

Geomembrane, Numerical solution, Parametric study, Pull-Out test. 

AUTHORS: M.R. Madhav, Visiting Professor, Institute of Lowland Technology, Saga 

University, Saga 840, Japan and Professor of Civil Engineering, Indian Institute of 

Technology, Kanpur 208016, India, Telephone: 91/512-59-7144, Telefax: 

91/512-59-7395/0260/0007, E-mail: madhav@iitk.ernet.in; N. Gurung, Ph.D. Student, Civil 

Engineering, Saga University, Saga 840, Japan, Telephone: 81/952-28-8689, Telefax: 

81/952-28-8699, E-mail: 97ts53@edu.cc.saga-u.ac.jp; and Y. Iwao, Professor of Civil 

Engineering, Saga University, Saga 840, Japan, Telephone: 81/952-28-8687, Telefax: 

81/952-28-8699; E-mail: iwaoy@cc.saga-u.ac.jp. 

PUBLICATION: Geosynthetics International is published by the Industrial Fabrics 

Association International, 1801 County Road B West, Roseville, Minnesota 55113-4061, 

USA, Telephone: 1/612-222-2508, Telefax: 1/612-631-9334. Geosynthetics International is 

registered under ISSN 1072-6349. 

DATES: Original manuscript received 20 December 1997, revised version received 16 

March 1998 and accepted 21 March 1998. Discussion open until 1 March 1999. 

REFERENCE: Madhav, M.R., Gurung, N. and Iwao, Y., 1998, “A Theoretical Model for 

the Pull-Out Response of Geosynthetic Reinforcement”, Geosynthetics International, Vol. 

5, No. 4, pp. 399-424. 

GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NO. 4 

399

MADHAV, GURUNG AND IWAO D Theoretical Model for Pull-Out Response of Geosynthetics 

1 INTRODUCTION 

The use of soil reinforcement to improve the strength and stability, and to mitigate 

total and differential settlements, of foundations and soil structures has become common 

practice in geotechnical engineering. Earth structures, such as airport pavements, 

embankments, landfills, earth slopes, and retaining walls are built with soil reinforcement 

for improved safety against sliding or bearing failure and to improve settlement 

response. Geosynthetics are now often used in weak or soft ground, reclaimed lowlands, 

and in geo-environmental engineering applications, e.g. landfills and reservoirs. The 

use of geosynthetics has unique advantages over other soil strengthening techniques, 

due to their low mass per unit area, strength, and stiffness characteristics. However, the 

use of geosynthetics requires a proper understanding of soil-geosynthetic interaction 

mechanisms. Field and laboratory pull-out tests are widely used to interpret soil-reinforcement 

interaction mechanisms. Yang (1972) and Schlosser and Long (1973) proposed 

anisotropic cohesion and enhanced confining pressure concepts, respectively, to 

explain the increased strength of reinforced soil. Hausmann (1976) reported the absence 

of anisotropic cohesion at low confining stresses, but postulated that reinforcement 

pull-out failure occurs by slippage or loss of soil-reinforcement interface shear 

stress. McGown et al. (1978) reported the different load-deformation responses (Figure 

1) for extensible and inextensible reinforcement materials. To understand the behaviour 

of geotextile-reinforced clays, various laboratory tests such as triaxial, direct shear, 

pull-out, and physical model tests were performed by Ingold (1981, 1983a, 1983b) and 

Ingold and Miller (1983). The basic design criteria for reinforced earth structures requires 

an analysis of the external and internal stability of the structure (Mitchell and 

Villet 1987; Christopher et al. 1989). 

Soil and inextensible reinforcement 

Soil and extensible reinforcement 

Axial load 

Unreinforced soil 

Axial strain 

Figure 1. Axial load-strain relationship for unreinforced soil and soil reinforced with 

extensible and inextensible reinforcement (McGown et al. 1978) 

400 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NO. 4


Jewell and Wroth (1987) established that reinforcement oriented in the direction of 

tensile strain strengthens the soil. Schlosser and de Buhan (1991) simplified the soil-reinforcement 

interaction mechanisms as either direct shear or pull-out by neglecting the 

bending resistance of the soil reinforcement. The pull-out resistance of geogrids was 

obtained in terms of friction and bearing components. Peterson and Anderson (1980) 

proposed a general bearing failure mechanism for welded wire mesh reinforcement, 

while Jewell et al. (1985) considered a punching shear failure mechanism for the passive 

bearing component of geogrid reinforcement. The two theories of general bearing 

and punching shear failure gave upper- and lower-bound estimates of the passive bearing 

contribution in pull-out test results (Palmeira and Milligan 1989; Jewell 1990). Jewell 

(1993) suggested the use of the critical state soil friction angle instead of the peak 

soil friction angle to calculate the bearing resistance for the case of extensible reinforcement, 

due to different degrees of bearing resistance mobilisation along the length 

of the reinforcement. Extensive pull-out tests have been reported by Berg and Swan 

(1991) (geogrid), Bergado et al. (1992, 1993, 1995) (steel grid) and Bergado and Chai 

(1994) (geogrid). Hayashi et al. (1994) conducted laboratory pull-out tests and measured 

the elongation at several points along the geosynthetic reinforcement. Inextensible 

steel grid reinforcement, as well as extensible geogrid reinforcement-soil systems 

were studied in detail by Alfaro (1996). Pradhan et al. (1997) investigated the effect of 

normal pressure during pull-out tests on a saturated clay-geosynthetic system. Based 

on field geosynthetic pull-out tests, Konami et al. (1997) proposed an elastic model for 

polymer strips. For the analysis of the pull-out behaviour of planar geosynthetic reinforcement, 

a model based on shear-lag theory was proposed by Abramento and Whittle 

(1995a). Segrestin and Bastick (1997) compared the pull-out capacity of extensible 

geosynthetic and inextensible steel grid reinforcement. A soil-geosynthetic reinforcement 

interface model based on rigid-plastic shear stress mobilisation has been reported 

by Sobhi and Wu (1996) for extensible reinforcement (geotextile). Long et al. (1997) 

used curve-fitting by parabolas to describe the nonuniform shear distribution at the 

soil-reinforcement interface. 

The model proposed in the current study considers highly extensible reinforcement, 

i.e. geosynthetics, and incorporates a bilinear shear stress-displacement relationship for 

the soil-reinforcement interface shear stress response during pull-out tests. Thus, the 

model considers prefailure deformations of the soil-reinforcement interface caused by 

shear stress. The nonlinear governing equation relating the applied pull-out force, displacement, 

and distance along the reinforcement from the applied pull-out force are 

normalised and solved numerically to obtain the pull-out force-displacement relationships 

and the variation of pull-out force and displacement of any point along the reinforcement. 

The numerical results of the current study are compared with the following 

published data: field and model pull-out test results on polyester fibres coated with 

polyethylene (Konami et al. 1997); laboratory and model pull-out test results using 

needle-punched, nonwoven geotextiles (Sobhi and Wu 1996); and laboratory pull-out 

test results for two nylon 6/6 sheet specimens (Abramento and Whittle 1995b). 


401


2 PULL-OUT MODEL 

2.1 Formulation of the Model 

Figure 2a is a schematic of a pull-out test on a geosynthetic sheet reinforcement of 

length, L, elastic modulus, E r , and thickness, t r . The tensile strength, T, of the reinforcement 

should be significantly high when compared to the pull-out force to prohibit reinforcement 

breakage. The pull-out force per unit width, T 0 , should be applied ormeasured 

slightly away from the face to avoid reinforcement end effects (Jewell and Wroth 1987). 

The applied pull-out force, T 0 , mobilises the soil-reinforcement interface shear stress, 

τ, acting along the reinforcement length (Figure 2b). The interface shear stress, τ, is 

governed by the interface shear stress response shown in Figure 2c (typical direct shear 

test results). The interface shear stress increases with displacement, w, and asymptotically 

reaches the maximum soil-reinforcement interface shear stress value, τ max . 

In the current study, the strain softening response that was observed in physical tests 

reported in the literature was not considered. For simplicity, the τ versus w curve was 

simplified using a bilinear response curve as follows: 

τ = k s1 w for 0 ≤ w ≤ w max 

(1) 

and 

τ = k s1 w max + k s2 (w − w max ) for w > w max 

(2) 

(a) 

T 

q s 

Soil (γ, φ) 

L 

Soil (γ, φ ) 

(b) 

σ n =q s + γ D 

φ s/r 

D 

T 0 

Reinforcement (E r , t r ) 

τ 

x 

(c) 

1 

k s2 

(d) 

T 

τ 

T + ΔT 

τ max 

w max w 

τ 

1 

k s1 

Δx 

τ 

Δw 

Δw = ε Δx 

Figure 2. Schematic of the forces acting on an extensible geosynthetic reinforcement layer 

during pull-out: (a) soil-reinforcement system; (b) forces acting on the reinforcement; (c) 

soil-reinforcement shear stress-displacement curve; (d) forces on a differential 

reinforcement segment, Δx, during pull-out. 



where k s1 and k s2 equal the slope of the bilinear τ versus w curve in the pre- and post-yield 

ranges, respectively (Figure 2c), or the soil-reinforcement interface shear stiffnesses. 

The value of k s2 is typically equal to zero. However, for numerical simplicity, it was assumed 

to be 1/100 to 1/1000 the value of k s1 ,andw max = τ max / k s1 = displacement corresponding 

to the maximum interface shear stress. 

The maximum soil-reinforcement interface shear stress, τ max , is limited to: 

τ max = σ n tan φ s∕r = (q s + γ D) tanφ s∕r 

(3) 

where: σ n =(q s + γ D) = normal stress acting on the soil-reinforcement interface; q s = 

surcharge stress; γ = bulk unit weight of the fill; D = depth of the reinforcement; and 

tanφ s/r = soil-reinforcement interface friction angle. 

In the current study, the reinforcement was considered to be highly extensible as is 

the case for some geotextiles and geomembranes. Therefore, considering the extended 

length of a small differential element of length Δx and of unit width (Sobhi and Wu 

1996) (Figure 2d), the equilibrium of horizontal forces is satisfied by: 

(T + ΔT) − T + 2τ(Δx + Δw) = 0 

(4) 

where: T and (T + ΔT) = pull-out forces, i.e. tensile forces, in the reinforcement at the 

left and right ends; τ = mobilised soil-reinforcement interface shear stress; and Δw = 

elongation of the element. 

The elongation, Δw, is related to the strain, ε, by: 

Δw = ε Δx 

(5) 

while ε is related to the tensile force, T, by: 

ε = T ∕ E r t r 

(6) 

Equation 4 can be expressed as: 

dT 

+ 2 τ (1 + ε) = 0 

dx 

(7) 

Note that for the positive x-axis (to the right), ε =-dw/dx, Equations 6 and 7 can be 

combined to give: 

E r t 

d 2 w 

r + 2τ dw 

dx2 dx − 1 = 0 

Equation 8 is the basic equation governing the response of an extensible reinforcement. 

The interface shear stressgiven byEquations 1and 2are coupled with Equation 8 to give: 

(8) 

E r t r 

d 2 w 

dx 2 + 2k s1 

dw 

dx − 1 w = 0 for 0 ≤ w ≤ w max 

E r t r 

d 2 w 

dx 2 + 2[k s1 w max + k s2 (w − w max )] dw 

dx − 1 = 0 for w > w max 

(9) 

(10) 


403


The boundary conditions are: 

at x = 0, T = T 0 or ε = T 0 

E r t r 

=− dw 

dx 

at x = L, T = 0 or ε = 0 

(11) 

(12) 

If k s1 = ∞ and k s2 = 0, Equation 10 reduces to the simple and elegant expression reported 

by Sobhi and Wu (1996). Equations 9 and 10 can be nondimensionalised and 

simplified to: 

d 2 W 

+ α β dW 

dX2 dX 

− 1 W = 0 for 0 ≤ W ≤ 1 

(13) 

and 

d 2 W 

dX + α [1 + R 2 k (W − 1)] β dW − 1 = 0 for W > 1 

dX 

(14) 

where: W = w / w max ; X = x / L; R k = k s2 / k s1 = stiffness ratio; and: 

α = 2 k s1 L 2 

and β = w max 

E r t r L 

or 

τ max 

k s1 L 

where: α = normalised interface stiffness (or relative stiffness parameter); and β = relative 

displacement parameter. 

The boundary conditions in nondimensional form become: 

ε =− dw 

dx = T 0 

E r t r 

or 

dW 

dX =−α T* at X = 0 (15) 

ε = dW 

dX = 0 at X = 1 (16) 

where: T * = T 0 / T max ;andT max =2τ max L = maximum pull-out force. 

2.2 Numerical Solution 

Equations 13 and 14 are nonlinear differential equations that cannot be solved analytically. 

Discretising the reinforcement into n elements each of length ∆L = L/n or ∆X = 

1/n (Figure 3), and expressing the derivatives in a finite difference form, Equations 13 

and 14 can be rewritten for the i th node as: 

W i−1 −2W i +W i+1 

+ αβ W i+1−W i−1 

−1W 

(ΔX) 2 

i = 0 for 0 ≤ W i ≤ 1 (17) 

2ΔX 

and 



W i−1 −2W i +W i+1 

+ αβ W i+1+W i−1 

−1 [1 + R 

(ΔX) 2 

k (W i −1)] = 0 for W i > 1 

2ΔX 

The normalised displacement at node n = i, W i , is obtained iteratively from: 

(18) 

W i = W i−1 + W i+1 

if W i < 1.0 

2 − α C 1 ∕ n 2 

W i = W i−1 + W i+1 + α (1 − R k ) C 1 ∕ n 2 

2 − α C 1 R k ∕ n 2 if W i > 1.0 

where 

(19) 

(20) 

C 1 = β n W i+1 − W i−1 

− 1 

2 

To solve for displacements at nodes i =1andi = n + 1, two fictitious nodes i =2i to 

the left of node i =1andi = n + 2 to the right of node i = n + 1 were assumed. The displacements 

at these nodes can easily be derived from the boundary conditions (Equations 

15 and 16) as: 

W 2′ = W 2 + (2 α T)∕n 

(21) 

W n = W n+2 

Knowing W 2i and W n+2 , the normalised displacements at node i =1andi = n +1are 

once again obtained from Equations 19 and 20. From the known displacements along 

the reinforcement length, the strain, ε i , and normalised pull-out force, T i , at node i,are 

calculated as follows: 

ε i = n W i−1 − W i+1 

2 

 

T i = n W i−1 − W i+1 

2α 

 

(22) 

(23) 

(24) 

2.3 Estimation of the Parameter α 

The first approximate value of α may be estimated by neglecting the term βdW/dX 

→ 0 at the lower pull-out force, but the value of α at a higher pull-out force may slightly 

X =0 X =1 

n = 

2i 

1 2 

i --- 1 i i +1 n n +1 n +2 

Figure 3. Discretisation of the geosynthetic reinforcement that is used in the current study 

to model reinforcement pull-out. 


405


change as per the slope value k s1 in the τ versus w response. For the initial estimation, 

Equation 13 can be approximated as: 

d 2 W 

dX − α W = 0 for 0 ≤ W ≤ 1 

2 

The closed form solution of Equation 25 is: 

W = A′ cosh α 

X + B′ sinh α 

X for 0 ≤ W ≤ 1 

(25) 

(26) 

Differentiating Equation 26 gives: 

dW 

dX = 

A′ sinh α 

α 

X + B′ cosh α 

X 

Solving for the following boundary conditions (Equations 15 and 16), i.e.: 

− dW 

dX = α T* and B′ =−α 

T 

* 

at X = 0 

− dW = 0 

dX 

and A′ =−B′ cothα 

at X = 1 

The solution is: 

W = α 

T 

* 

cothα 

(27) 

cosh α 

X − sinh α 

X for 0 ≤ W ≤ 1 (28) 

Equation 28 may be redefined as: 

 

W = w∕w max = α T 

T maxcoth 

 

α 

cosh α 

X − sinh α 

X 

or 

At the pull-out end, i.e. at X =0: 

w ⎧ α T 

w = 

max 

⎪ 

w 

T = w max 

2 k s1 w max L ⎪ ⎧ ⎩ 

T 

⎩ max 

⎪ ⎫ ⎭ coth 

α 

tanh 

⎪⎫ ⎭ = 

α 

α 

L 

α 

Er t r tanh 

 

α 

(29) 

(30) 

Equation 30 is an expression for the initial slope of the displacement versus pull-out 

force in terms of the reinforcement characteristics, E r , t r ,andL, and the interface shear 

stiffness, k s1 . Typically, E r , t r ,andL are known; therefore, k s1 can easily be estimated 

from the initial slope of the pull-out force-displacement curve. Equation 30 can be nondimensionalised 

to give: 

wE r t r 

LT = 1 

 

(31) 

α tanh = f (α) 

α 



1.6 

f (α), α -1/2 

1.2 

0.8 

0.4 

α -1/2 

f (α) 

0 

1 10 100 

Normalised stiffness parameter, α 

Figure 4. Plot of α -1/2 and f(α) versus α. 

Equation 31 is proposed to estimate the initial values of α from the pull-out force-displacement 

test results. The variation of the function in Equation 31 with α is shown in 

Figure 4, and can be used to estimate the value of α and, subsequently, the value of k s1 . 

However, the value of k s1 estimated from the initial portion of the τ versus w curve corresponds 

to the initial tangent value of the interface shear stiffness. If a more accurate value 

of k s1 , covering a larger range of stresses, is required, it is preferable to estimate the 

secant value of k s1 from the latter portion of the τ versus w curve (e.g. T 0.3 T max ). For 

large values of α, the function f(α) is almost equal to α -1/2 . 

3 RESULTS AND DISCUSSION 

The solutions for the normalised displacements, strains, and normalised pull-out 

forces along the reinforcement length are obtained by solving Equations 19 to 24 numerically. 

The accuracy of the solution is verified by varying n: the number of elements 

into which the reinforcement is discretised. The difference in the displacement values 

for n = 20 and 40 was negligible; therefore, n = 20 was adopted for all further calculations. 

Parametric studies were performed for the following ranges of parameters: T * = 

0to1.0,α = 2 to 200, and β = 0.00001 to 0.2. The product αβ =(T max / E r t r )=(2σ n tanφ s/r 

L) /( E r t r ) is a function of the maximum pull-out force and the reinforcement stiffness 

only and is independent of the unit shear stiffness, k s1 . The test results reported by Sobhi 

and Wu (1996) were derived from the above formulation with a finite value of αβ but 

with either α → 0orβ → 0andR k = 0. The normalised pull-out force versus normalised 

displacement relationship for α =50andβ = 0.01 was obtained for different values of 

R k (0.01, 0.001, and 0.0001). For any given value of αβ , the values of R k 0.001 did 

not affect the resulting solution in terms of the normalised pull-out force versus normalised 

displacement response (Figure 5); therefore, a value of R k = 0.001 was used in the 

current parametric study. 


407


Normalised pull-out force, T* 

1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

R k = 0.0001 

R k = 0.001 

R k =0.01 

α = 50, β =0.01 

0 5 10 15 

Normalised displacement of the pull-out end, W 0 

Figure 5. Effect of R k on the pull-out force versus the normalised displacement for α =50 

and β = 0.01. 


1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

2 

5 

10 

20 

100 

0 10 20 


50 

α = 200 

β =0.01 

Figure 6. Normalised pull-out force versus the normalised displacement for β =0.01and 

different α values. 

The variation of the normalised displacement, W 0 , at the pull-out end (X = 0) with 

the normalised pull-out force, T * , is presented in Figure 6, for β = 0.01 and for different 

values of the relative stiffness parameter, α. The displacements increase linearly with 

the pull-out force for values of T * < 0.5. The rate of increase in W 0 is greater at higher 

pull-out forces because the elements near the pull-out force end attain the maximum 

interface shear stress values and slip without mobilising any additional interface shear 

stress. Theoretically, as T * → 1.0, the displacement, W 0 , should approach infinity but 

is finite because of the nonzero values of R k that correspond to the ratios of slopes of 



the bilinear unit interface shear stress relationship in the post- and pre-peak regions. The 

normalised displacements at any given pull-out force, T * , increase with increasing values 

of α. For a given soil-reinforcement interface, values of α increase either if the 

length of the reinforcement, L, is larger, or if the stiffness of the reinforcement, E r ,is 

smaller. In either case, it was expected that the displacements at all pull-out force levels 

would increase with increasing α values because of the increased specimen length, or 

because the reinforcement is highly extensible. The normalised displacements at T * = 

0.6 for β = 0.01 are 0.946, 1.427, 2.249, 3.830, 7.995, 13.390, and 20.750 for α =2,5, 

10, 20, 50, 100, and 200, respectively. Results, similar to those shown in Figure 6, are 

obtained for β = 0.1 (Figure 7). For the case of β = 0.1, the normalised displacements 

at the pull-out end, W 0 , are smaller because β = 0.1 implies that there is a higher maximum 

interface shear stress, τ max . The normalised displacements at T * =0.6forβ =0.1 

are 0.913, 1.280, 1.800, 2.534, 3.590, 3.880, and 4.180, for α = 2, 5, 10, 20, 50, 100, 

and 200, respectively. 

The influence of the relative displacement parameter, β, on the normalised pull-out 

end displacement versus the normalised pull-out force response is shown in Figure 8 

for α = 5. By varying the β value (0.2, 0.1, 0.05, 0.01, and 0.001) and maintaining a 

constant α value, it is implied that the maximum interface shear stress, τ max , varies, but 

the interface shear stiffness, k s1 , is constant. As the β value increases, greater maximum 

pull-out forces, T max , and greater maximum displacement values, w max , are required to 

reach the yield or maximum interface shear stress. Consequently, the curves for greater 

β values have an extended linear portion and exhibit relatively smaller pull-out end displacements 

for all pull-out force values. For example, at T * = 0.5, the displacements are 

0.987, 1.102, and 1.153 for β = 0.2, 0.05, and 0.001, respectively. The differences in 

the displacement values increase with increasing pull-out force. The results are not affected 

by the value of β for β 0.001. 

1.0 


0.8 

0.6 

0.4 

0.2 

0.0 

2 5 

0 2 4 


10 

20 

50 

100 

α = 200 

β =0.01 

Figure 7. Normalised pull-out force versus the normalised displacement for β = 0.1 and 

different α values. 


409



1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

β =0.2 

β =0.1 

β =0.0 

β = 0.001 

β =0.01 

α =5 

0 1 2 3 


Figure 8. Normalised pull-out force versus the normalised displacement for α =5. 


1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

α =5, β =0.02 

α =2, β =0.05 

α = 10, β =0.01 

αβ=0.1 

0 1 2 3 


Figure 9. 

Normalised pull-out force versus the normalised displacement for αβ=0.1. 

The displacement response for different values of α (10, 5, and 2) and β (0.01, 0.02, 

and 0.05), but for a constant αβ value of 0.1, and different pull-out forces is shown in 

Figure 9. A significant decrease in the displacement at any given pull-out force occurs 

with decreasing α values. The displacement values are very sensitive to the value of β. 

Low β values ( (w max / L)=(τ max / k s1 L)) imply a low shear stress, significantly long reinforcement, 

or a significantly high interface stiffness response (i.e. high k s1 value); thus, 

the response curves are highly nonlinear for low values of β. Therefore, Figure 9 shows 

the importance of considering both the stiffness and the maximum shear stress of the 

interface. Displacement dampening may be the assumed mechanism causing a reduc- 



tion in α values. Also, higher β values indicate a reduced stiffness of the interface response. 

The variation of displacement with distance along the length of the 

reinforcement for various pull-out force values is shown in Figure 10 for a constant value 

of αβ = 1, but with different values of α and β: α =50andβ = 0.02; and α =5and 

β = 0.2. The displacements are rapidly dampened with increasing normalised distance 

values for higher α and lower β values, which implies that only a part of the reinforcement 

actively mobilises the pull-out resistance. In contrast, if α is smaller or β is larger, 

the full length of the reinforcement participates in mobilising pull-out resistance. These 

results emphasise the need to consider the interface stiffness parameter k s1 . 

The importance of considering the interface stiffness is realised by considering the 

normalised pull-out force versus the normalised distance along the length of the reinforcement 

for αβ products comprising different values of α and β. The curves for the 

two pairs of α and β values (α =5,β =0.1andα = 50, β = 0.01), but with the same 

product αβ = 0.5, demonstrate the effect of k s1 or α on the pull-out interaction (Figure 

11). The stiffer the reinforcement, the shorter the length of resistance at all stress levels. 

From Figure 11, the extended length of the reinforcement can be determined for a given 

applied pull-out force. For any given maximum interface shear stress, τ max ,andreinforcement 

characteristics (E r , t r ,andL), the product αβ is unique. However, it can be 

concluded that the interface shear stiffness, k s1 , has a significant influence on the pullout 

response for highly extensible reinforcement. The normalised pull-out force versus 

the normalised distance along the length of the reinforcement curves are highly nonlinear, 

particularly for higher relative stiffness parameter values, α . 

Normalised displacement, W 

10 

α = 50, β =0.02 

α =5, β =0.2 

T* = 0.95 

0.6 

0.2 

αβ=1 

0 

0.0 0.2 0.4 0.6 0.8 1.0 

Normalised distance along the reinforcement, X = x / L 

Figure 10. Displacement versus the normalised distance along the length of the 

reinforcement for αβ=1. 


411


Normalised pull-out force, T * 

1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

α = 50, β =0.01 

α =5, β =0.1 

αβ=0.5 

0.0 0.2 0.4 0.6 0.8 1.0 

Normalised distance along the reinforcement, X = x / L 

Figure 11. Pull-out force versus the normalised distance along the length of the 

reinforcement for αβ=0.5. 

4 VALIDATION OF THE PROPOSED MODEL 

4.1 Konami et al. (1997) 

Konami et al. (1997) report results from field pull-out tests on three types of polymer 

strip reinforcements, PW-3, PW-5, and PW-10, that were 85, 90, and 95 mm wide and 

2, 3, and 5 mm thick, respectively. The polymer was made of polyester fibres coated with 

polyethylene. The reinforcement strips were tested at various depths in a fill compacted 

to a unit weight of 20.6 kN/m 3 and 18.6 kN/m 3 to represent natural and dry conditions, 

respectively. The reinforcement lengths were 3.5 and 5.5 m. The parameter E r t r =3.43 

× 10 3 ,5.39× 10 3 , and 1.02 × 10 3 kN/m at 2% strain, and 2.30 × 10 3 ,3.92× 10 3 , 

and 8.71 × 10 3 kN/m at 5% strain, for PW-3, PW-5, and PW-10, respectively. The interface 

friction angle, φ s/r , between the fill and the reinforcement was 38.2 _ based on direct 

shear tests. The maximum pull-out stress could not exceed the mobilised maximum interface 

shear stress or the breaking strength of the material itself (which ever is less). 

From the published data, using the concept of mobilised friction varying with the mean 

stress level (Jewell and Wroth 1987) or depth (Bolton and Powerie 1988), the value of 

the mobilised interface coefficient of friction versus normal stress was plotted (Figure 

12). At approximately 14% elongation, the reinforcement broke due to the tensile stress. 

From the data reported by Konami et al. (1997), the normal stress, σ n , maximum interface 

shear stress, τ max = σ n φ s/r , and maximum pull-out force, T max =2τ avg L for various 

depths were calculated and are summarised in Table 1. The values of the interface shear 

stiffness, k s1 , were not available; therefore, these values were estimated using Equation 

31 or Figure 4 by the method presented in Section 2.3. Knowing the value of the product 

αβ and estimating the value of α, the value of β was calculated. The estimated k s1 value, 

the adopted E r t r value, and the calculated values of α and β are given in Table 2. 



Coefficient of friction, μ s/r 

1.4 

0.7 

0.0 

0 20 40 60 80 100 

Normal stress (kN/m 2 ) 

Figure 12. Soil-reinforcement interface coefficient of friction versus normal stress (data 

interpreted from Konami et al. 1997). 

Table 1. Pull-out test parameters interpreted from the data reported by Konami et al. (1997). 

Test number 

D 

(m) 

σ n 

(kN/m 2 ) 

Estimated μ s/r 

(_) 

τ max 

(kN/m 2 ) 

L 

(m) 

T max 

(kN/m) 

1 4.8 86.4 0.30 25.92 3.5 181.440 

2 3.2 57.6 0.44 25.34 3.5 177.408 

3 1.6 28.8 0.45 12.96 3.5 90.720 

4 0.8 14.4 0.80 11.52 3.5 80.640 

6 4.0 72.0 0.36 25.92 5.5 285.120 

7 2.4 43.2 0.33 14.26 5.5 156.816 

8 0.8 14.4 0.80 11.52 5.5 126.720 

Notes: D = depth of reinforcement; σ n = normal stress; μ s/r = interface coefficient of friction; τ max =maximum 

interface shear stress; L = length of reinforcement; and T max = maximum pull-out force. 

Table 2. Estimated pull-out test parameters adopted in the current study for comparison 

with the data reported by Konami et al. (1997). 

Test number 

D 

(m) 

Specimen 

number 

E r t r 

(kN/m) 

αβ= T max / E r t r 

Estimated k s1 

(kN/m 3 ) 

1 4.8 PW-5 59.7 0.030 272.9 1.12 0.027136 

2 3.2 PW-5 59.7 0.030 443.5 1.82 0.016328 

3 1.6 PW-3 38.8 0.023 483.0 3.05 0.007666 

4 0.8 PW-3 38.8 0.021 245.5 1.55 0.013409 

6 4.0 PW-10 102 0.028 413.1 2.45 0.011409 

7 2.4 PW-5 59.7 0.026 483.5 4.90 0.005361 

8 0.8 PW-3 38.8 0.033 246.9 3.85 0.008483 

Notes: D = depth of reinforcement; E r = Young’s modulus of reinforcement; t r = thickness of reinforcement; 

α = relative stiffness parameter; β = relative displacement parameter; T max = maximum pull-out force; and k s1 

= initial slope of shear stress-displacement curve in Figure 2c = soil-reinforcement interface shear stiffness. 

α 

β 


413


The estimated value of k s1 corresponds to the initial tangent value of the shear-stress 

versus displacement curves for the pull-out tests. By choosing appropriate combinations 

of α and β, it is possible to predict the pull-out force versus displacement relations. 

For a comparative illustration, upper- and lower-bound responses for different α and 

β values, but for the same αβ value, are compared with the predicted and field test results 

reported by Konami et al. (1997). The first approximate values of α and β, based 

on the lower pull-out force, were estimated (Table 2). At higher pull-out forces, the α 

and β values may change as the slope, k s1 , of the stress-displacement response changes. 

The second refined values of α and β in Figures 13 (1.12 and 0.0181, respectively) and 

14 (1.82 and 0.0163, respectively) predicted displacement values that are closer to the 

field test results reported by Konami et al. (1997). The calculated displacement was underpredicted 

for α and β values that were based on the initial tangent value of k s1 , while 

the calculated displacement based on an appropriate choice of α and β values appears 

to closely fit the field test results (Figures 13 to 19). 

Figures 13 to 16 compare the field test results with the predicted, or calculated results 

(based on a simple approach assuming mobilisation of the full interface shear stress 

over an effective reinforcement length), and the model proposed in the current study. 

The Konami et al. (1997) predicted results for field tests Nos. 6, 7, and 8 were not available 

(Figures 17, 18, and 19). The deviations present in Figures 17, 18, and 19 could 

be minimised by choosing a smooth τ versus w curve instead of a bilinear curve. 

4.2 Sobhi and Wu (1996) 

Pull-out test results for 300 mm long and 3.18 mm thick needle-punched, nonwoven 

geotextile specimens (Sobhi and Wu 1996) are compared with the results of the model 

proposed in the current study (Figures 20 to 23). For a normal stress of 40 kPa, the 

maximum pull-out force was 6.96 kN/m assuming μ s/r = 0.29. Using a geotextile stiff- 

Pull-out force, T (kN/m) 0 (kN/m) 

1.8 

1.6 

1.4 

1.2 

1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

Predicted (Konami et al. 1997) 

Field test No. 1 (Konami et al. 1997) 

Proposed model, α = 1.12, β = 0.0271 (Table 2) 

Proposed model, α = 1.12, β = 0.0181 

0 40 4 808 120 160 

Displacement (mm) 

Figure 13. Comparison of the predicted reinforcement pull-out displacements with the 

pull-out test data (predicted and field test No. 1) reported by Konami et al. (1997). 



Pull-out force (kN/m) 

180 1.8 

160 1.6 

140 1.4 

120 1.2 

100 1.0 

0.8 80 

0.6 60 

0.4 40 

0.2 20 




Proposed model, α = 1.12, β = 0.0074 

0.00 

0 40 4 80 8 120 160 





0.9 90 

0.8 80 

0.7 70 

0.6 60 

0.5 50 

0.4 40 

0.3 30 

0.2 20 

0.1 10 




0.00 

0 50 5 100 




ness E r t r = 31.8 kN/m, the product αβ was estimated to be 0.212. Table 3 summarises 

the known and estimated parameter values. The estimated values of α and β using a 

low pull-out force were 25.0 and 0.0087 in the first test, respectively, and α =7.00and 

β = 0.0031 in the second test, which is closer to the results reported by Sobhi and Wu 

(1996). The response at higher pull-out forces may be improved by refining the values 

of α and β from the initial slope, k s1 ,oftheτ - w curve. Deviations in the values calcu- 


415



140 1.4 

120 1.2 

100 1.0 

0.8 80 

0.6 60 

0.4 40 

0.2 20 




0.00 

0 50 

5 

100 

10 

150 

15 

200 

20 





280 2.8 

240 2.4 

160 2.0 

200 1.6 

120 1.2 

0.8 80 

0.4 40 



0.00 

0 50 5 100 150 200 




lated using the proposed model may be reduced by adopting different values of α and 

β as per the real slope, k s1 ,oftheτ - w curve. Sobhi and Wu (1996) obtained closedform 

solutions for the pull-out force assuming full shear resistance mobilisation and a 

rigid, plastic response. 



1.6 160 


1.2 120 

0.8 80 

0.4 40 



0.00 

0 50 100 150 200 



pull-out field test data (field test No. 6) reported by Konami et al. (1997). 

120 1.2 


0.8 80 

0.4 40 



0.00 

0 50 5 100 150 200 



pull-out field test data (field test No. 8) reported by Konami et al. (1997). 

Table 3. Pull-out test parameters adopted for comparison with results reported by Sobhi 

and Wu (1997). 

σ n 

(kPa) 

μ s/r 

(_) 

E r t r 

(kN/m) 

L 

(mm) 

αβ= T max /E r t r 

Estimated, k s1 

(kN/m 3 ) 

40 0.29 31.8 300 0.218868 1237 7 0.0031 

40 0.29 31.8 300 0.218868 4417 25 0.0087 

α 

β 


417


5 



4 

3 

2 

1 

0 

Experimental, x = 0 mm (Sobhi and Wu 1996) 

Predicted (Sobhi and Wu 1996) 



0 10 20 30 



pull-out test data (laboratory and predicted) reported by Sobhi and Wu (1996) at the applied 

pull-out force end of the reinforcement, x =0. 

5 


4 

3 

2 

1 

0 





0 5 10 15 20 




pull-out force end of the reinforcement, x = 75 mm. 

4.3 Abramento and Whittle (1995b) 

Laboratory pull-out tests on nylon 6/6 sheet specimens reported by Abramento and 

Whittle (1995b) were analysed using the model proposed in the current study. The characteristic 

length, width, and thickness of the test specimens, designated as ASPR 57 and 

59, were 420.0, 133.4, and 0.508 mm, respectively. The tests were conducted at a confining 

pressure, σ 1 = 24.5 kPa, and at displacement rates of 35 and 3.5 μm/s. The sand- 



5 



4 

3 

2 

1 

0 





0 4 8 12 





5 

Pull-out 

Pull-out 

force 

force 

(kN/m) 

(kN/m) 

4 

3 

2 

1 

0 





0 2 4 6 8 10 





nylon interface friction angle values varied from φ s/r =20to25 _ . The estimated value 

of T max =2L σ 1 tanφ s/r = 9.596 kN/m resulted in the product αβ = T max / E r t r = 0.0084. 

The laboratory pull-out test results for specimens ASPR 57 and 59 are plotted and 

compared to the proposed model pull-out test results for α =3.25andβ = 0.00258, and 

α =1.39andβ = 0.00604 (Figure 24). Table 4 summarises the adopted parameters. 

Thus, it may be concluded that the proposed model reasonably predicts the laboratory 

test results. The information on k s1 , α,andβ can improve the accuracy of pull-out forcedisplacement 

prediction. 


419


1.2 

Pull-out force (kN) 

1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

ASPR 59 (Abramento and Whittle 1995b) 


ASPR 57 (Abramento and Whittle 1995b) 


0 1 2 3 4 

Displacement at reinforcement pull-out end (mm) 

Figure 24. Comparison of the predicted pull-out displacements with the measured 

laboratory pull-out test displacements reported by Abramento and Whittle (1995b) for 

nylon 6/6 sheet specimens (ASPR 59 and 57). 

Table 4. 

(1995). 

Pull-out test parameters adopted for comparison with Abramento and Whittle 

Nylon6 

ASPR 

σ n 

(kPa) 

μ s/r 

(_) 

E r t r 

(kN/m) 

L 

(mm) 

αβ= T max / E r t r 

Estimated, k s1 

(kN/m 3 ) 

57 24.5 20 to 25_ 975.36 420 0.0084 8985 3.25 0.00258 

59 24.5 20 to 25_ 975.36 420 0.0084 3843 1.39 0.00604 

α 

β 

5 CONCLUSIONS 

A new model for pull-out of highly extensible reinforcement has been proposed. The 

resulting nonlinear governing equation was nondimensionalised in order to perform a 

parametric analysis. The nondimensionalised equation was expressed in a finite difference 

form and solved using the Gauss-Siedel iteration method. The accuracy of the numerical 

solution was checked by varying the number of elements. The results showed 

that the discretisation of the reinforcement length into 20 elements gives accurate values 

within a reasonable amount of computational time. 

Two new nondimensional interaction terms, the relative stiffness parameter, α, and 

the relative displacement parameter, β, for soil-reinforcement interaction were 

introduced. A bilinear form of the soil-geosynthetic interface shear stress mobilisation 

with displacement was taken into consideration. A method for estimating the first 

approximate value of the normalised soil-reinforcement interface stiffness, or relative 

stiffness parameter, α , based on α values within the low pull-out force-displacement 

range is presented. A refinement of the stiffness parameter at higher pull-out loads requires 

the value of k s1 , which may be deduced from the slope of the τ - w curve. Normal- 



ised pull-out force versus displacement responses are presented for various ranges of 

relative stiffness parameters α and β . The predicted displacements are compared with 

the following field and laboratory pull-out test results: field pull-out tests on polymer 

strips (Konami et. al 1997), laboratory pull-out tests on geotextile specimens (Sobhi and 

Wu 1997, and laboratory pull-out tests on nylon 6/6 sheet specimens (Abramento and 

Whittle 1995b). The model presented in the current paper gives fair predictions of the 

pull-out behaviour of extensible reinforcement. A better estimation of the soil-reinforcement 

interaction parameters such as τ, μ s/r , k s1 , α,andβ could improve the predictability 

of the test results even further. The effective length of the reinforcement can be 

easily predicted from the pull-out force versus displacement plots; thus, obviating the 

need to estimate the effective length of the reinforcement by using simplifying assumptions. 

REFERENCES 

Abramento, M. and Whittle, J.A., 1995a, “Analysis of Pullout Tests for Planar Reinforcements 

in Soil”, Journal of Geotechnical Engineering, Vol. 121, No. 6, pp. 

476-485. 

Abramento, M. and Whittle, J.A., 1995b, “Experimental Evaluation of Pullout Analyses 

for Planar Reinforcements”, Journal of Geotechnical Engineering, Vol. 121, No. 

6, pp. 486-492. 

Alfaro, M.C., 1996, “Reinforced Soil Wall-Embankment System on Soft Foundation using 

Inextensible and Extensible Grid Reinforcements”, Ph.D Thesis, Saga University, 

Saga, Japan, 225 p. 

Berg, R.R. and Swan, R.H., 1991, “Investigation into Geogrid Pullout Mechanism”, 

Performance of Reinforced Soil Structures, McGown, A., Yeo, K., and Andrawes, 

K.Z., Editors, Thomas Telford, 1991, Proceedings of the International Reinforced 

Soil Conference held in Glasgow, Scotland, September 1990, pp. 353-357. 

Bergado, D.T., Lo, K.H., Chai, J.C., Alfaro, M.C. and Anderson, L.R., 1992, “Pullout 

Tests using Steel Grid Reinforcement with Low-quality Backfill”, Journal of Geotechnical 

Engineering, Vol. 118, No. 7, pp. 1047-1063. 

Bergado, D.T., Macatol, K.C., Amin, N.U., Chai, J.C. and Alfaro, M.C., 1993, “Interaction 

of Lateritic Soil and Steel Grid Reinforcement”, Canadian Geotechnical Journal, 

Vol. 30, No. 2, pp. 376-384. 

Bergado, D.T. and Chai, J.C., 1994, “Pullout Force-Displacement Relationship of Extensible 

Grid Reinforcements”, Geotextile and Geomembranes, Vol. 13, No. 5, pp. 

295-316. 

Bergado, D.T., Miura, N. and Alfaro, M.C., 1995, “Evaluations of Inextensible and Extensible 

Grid Reinforcements in Infrastructure Projects on Soft Bangkok Clay” Research 

of Lowland Technology No.4, March 1995, ILT, Saga University, Saga, Japan, 

pp. 102-112. 

Bolton, M.D., and Powerie, W., 1988, “Behaviour of Diaphragm Walls in Clay prior 

to Collapse”, Geotechnique, Vol. 38, No. 2, pp. 167-189. 


421


Christopher, B.R., Gill, S.A., Giroud, J.P., Juran, I., Schlosser F., Mitchell, J.K. and 

Dunnicliff, J., 1989, “Reinforced Soil Structures: Volume I. Design and Construction 

Guidelines”, Report No. FHWA-RD-89-043, Washington, DC, USA, November 

1989, 287 p. 

Hausmann, M.R., 1976, “Strength of Reinforced Soil”, Proceedings of the Eight Australian 

Road Research Conference, Vol. 8, Section 13, pp. 1-8. 

Hayashi, S., Makiuchi, K., Ochiai, H., Fukuoka, M. and Hirai, T., 1994, “Testing Methods 

for Soil-Geosynthetic Friction at Geosynthetic Interfaces”, Proceedings of the 

Fifth International Conference on Geotextiles, Geomembranes and Related Products, 

Vol. 1, Singapore, September 1994, pp. 411-414 

Ingold, T.S., 1981, “A Laboratory Simulation of Clay Walls”, Geotechnique, Vol. 31, 

No. 3, pp. 399-412. 

Ingold, T.S., 1983a, “Reinforced Clay Subjected to Undrained Triaxial Loading”, Journal 

of Geotechnical Engineering, Vol. 109, No. 5, pp. 738-744. 

Ingold, T.S., 1983b, “A Laboratory Investigation of Grid Reinforcement in Clay”, 

ASTM Geotechnical Testing Journal, Vol. 6, No. 3, pp. 112-119 

Ingold, T.S. and Miller, K.S., 1983, “Drained Axisymmetric Loading of Reinforced 

Clay”, Journal of Geotechnical Engineering, Vol. 109, No. 7, pp. 883-898. 

Jewell, R.A., Milligan G.W.E., Sarsby, R.W. and DuBois, D., 1985, “Interaction Between 

Reinforcement and Geogrids”, Polymer Grid Reinforcement, Thomas Telford, 

1985, Proceedings of a conference held in London, UK, March 1984, pp. 18-30. 

Jewell, R.A., and Wroth, C.P., 1987, “Direct Shear Tests on Reinforced Sand”, Geotechnique, 

Vol. 37, No. 1, pp. 53-68. 

Jewell, R.A., 1990, “Reinforcement Bond Capacity”, Geotechnique, Vol. 40, No. 3, pp. 

513-518. 

Jewell, R.A., 1993, “Links Between the Testing, Modelling and Design of Reinforced 

Soil”, Keynote Lecture, Earth Reinforcement Practice, Ochiai, H., Hayashi, S. and 

Otani, J., Editors, Balkema, 1993, Proceedings of the International Symposium on 

Earth Reinforcement Practice, Vol. 2, Kyushu University, Fukuoka, Japan, November 

1992, pp. 755-772. 

Konami, T., Imaizumi, S. and Takahashi, S., 1997, “Elastic Considerations of Field 

Pull-out Tests of Polymer Strip”, Earth Reinforcement, Ochiai, H., Yaufuku, N. and 

Omine, K., Editors, Balkema, 1997, Proceedings of the International Symposium on 

Earth Reinforcement, Vol. 1, Fukuoka, Kyushu, Japan, November 1996, pp. 57-62. 

Long, P.V., Bergado, D.T. and Balasubramanian, A.S., 1997 “Interaction Between Soil 

and Geotextile Reinforcement”, Ground Improvement, Ground Reinforcement and 

Ground Treatment: Developments 1987-1997, Schaefer, V.R., Editor, ASCE Geotechnical 

Special Publication No. 69, proceedings of the symposium held in Logan, 

Utah, USA, July 1997, pp. 560-578. 

McGown, A., Andrawes, K.Z. and Al-Hasani, M.M., 1978, “Effect of Inclusion Properties 

on the Behaviour of Sand”, Geotechnique, Vol. 28, No. 3, pp. 327-346. 



Mitchell, J.K. and Villet, W.C.B., 1987, “Reinforcement of Slopes and Embankments”, 

National Cooperative Highway Research Program Report No. 290, Transportation 

Research Board, National Research Council, Washington, DC, USA, 323 p. 

Palmeira, E.M. and Milligan, G.W.E., 1989, “Scale and Other Factors Affecting the Results 

of Pullout Tests of Grids Buried in Sand”, Geotechnique, Vol. 39, No. 3, pp. 

511-524. 

Peterson, L.M. and Anderson, L.R., 1980, “Pullout Resistance of Welded Wire Mats 

Embedded in Soil”, Research Report submitted to Hilfiker Company, Department of 

Civil and Environmental Engineering, Utah State University, Utah, USA, 106 p. 

Pradhan, T.B.S., Shiwakoti, D.R. and Imai, G., 1997, “Effect of Normal Pressure and 

Width of Geosynthetic Horizontal Drain in Pullout Behaviour using Saturated Clay”, 

Earth Reinforcement, Ochiai, H., Yaufuku, N. and Omine, K., Editors, Balkema, 

1997, Proceedings of the International Symposium on Earth Reinforcement, Vol. 1, 

Fukuoka, Kyushu, Japan, November 1996, pp. 133-138. 

Schlosser, F. and Long, N.T., 1973, “Recent Results in French Research on Reinforced 

Soil”, Journal of Construction Engineering, Vol. 100, No. 3, pp. 223-237. 

Schlosser, F. and de Buhan, P., 1991, “Theory and Design Related to the Performance 

of Reinforced Soil Walls”, Performance of Reinforced Soil Structures, McGown, A., 

Yeo, K., and Andrawes, K.Z., Editors, Thomas Telford, 1991, Proceedings of the International 

Reinforced Soil Conference held in Glasgow, Scotland, September 1990, 

pp. 1-14. 

Segrestin, P. and Bastick, M., 1997, “Comparative Study and Measurement of the Pullout 

Capacity of Extensible and In-extensible Reinforcements”, Earth Reinforcement, 

Ochiai, H., Yaufuku, N. and Omine, K., Editors, Balkema, 1997, Proceedings of the 

International Symposium on Earth Reinforcement, Vol. 1, Fukuoka, Kyushu, Japan, 

November 1996, pp. 139-144. 

Sobhi, S. and Wu, J.T.H., 1996, “Interface Pullout Formula for Extensible Sheet Reinforcement”, 

Geosynthetics International, Vol. 3, No. 5, pp. 565-581. 

Tzong, W.H. and Cheng-Kuang, S., 1987, “Soil-Geotextile Interaction Mechanism in 

Pull-out Test”, Proceedings of Geosynthetics ’87, IFAI, Vol. 1, New Orleans, Louisiana, 

USA, February 1987, pp. 250-259. 

Yang, Z.Z., 1972, “Strength and Deformation Characteristics of Reinforced Sand”, 

Ph.D Thesis, University of California, Los Angeles, California, USA. 

NOTATIONS 

Basic SI units are given in parentheses. 

D = depth of reinforcement (m) 

E r = Young’s modulus of reinforcement (Pa) 

i = node number (dimensionless) 

k s1 = initial slope of shear stress-displacement curve in Figure 2c 

= soil-reinforcement interface shear stiffness (N/m 3 ) 


423


k s2 = post-yield slope of shear stress-displacement curve in Figure 2c 

= soil-reinforcement interface shear stiffness (N/m 3 ) 

L = length of reinforcement (m) 

n = number of discretisation elements (dimensionless) 

q s = surcharge stress (Pa) 

R k = k s2 / k s1 = stiffness ratio (dimensionless) 

W = w / w max = normalised displacement of reinforcement (dimensionless) 

W 0 = normalised displacement of reinforcement pull-out end (dimensionless) 

W i = normalised displacement of i th element (dimensionless) 

w = displacement of reinforcement (m) 

w max = τ / k s1 = displacement of reinforcement at yield, i.e. at maximum interface 

shear stress (m) 

t r = thickness of reinforcement (m) 

T = tensile force at any point, x, along reinforcement (N/m) 

T max = maximum pull-out force in reinforcement (N/m) 

T 0 = reinforcement pull-out force per unit width (N/m) 

T * = T / T max = normalised tensile force in reinforcement (dimensionless) 

X = x / L = normalised co-ordinate along length of reinforcement 

(dimensionless) 

x = co-ordinate along length of reinforcement (m) 

α = normalised soil-reinforcement interface stiffness, or relative stiffness 

parameter (dimensionless) 

β = relative displacement parameter (dimensionless) 

ε = axial strain (dimensionless) 

γ = soil bulk unit weight (N/m 3 ) 

μ s/r = tanφ s/r = soil-reinforcement interface coefficient of friction 

(dimensionless) 

σ n = normal stress (Pa) 

τ = soil-reinforcement interface shear stress (Pa) 

τ avg = average soil-reinforcement interface shear stress value (Pa) 

τ max = limiting or maximum soil-reinforcement interface shear stress (Pa) 

φ = soil friction angle (_) 

φ s/r = soil-reinforcement interface friction angle (_)

Download Paper - IGS - International Geosynthetics Society

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?