AN INTERFACE PULLOUT FORMULA FOR EXTENSIBLE SHEET REINFORCEMENT

Technical Paper by S. Sobhi and J.T.H. Wu AN INTERFACE PULLOUT FORMULA FOR EXTENSIBLE SHEET REINFORCEMENT ABSTRACT: Pullout tests have been widely used to evaluate soil-reinforcement interface properties for the design and analysis of reinforced soil structures. However, there are difficulties interpreting pullout test results particularly when extensible materials, such as geotextiles, are used as reinforcement. In this study, an analytical model (an interface pullout formula ) is presented for predicting and interpreting pullout test results in a unified and consistent manner. The model is based on three postulates that were deduced from the measured behavior of laboratory pullout tests and numerical results from finite element analyses. A number of applications of the interface pullout formula for predicting and interpreting the results of pullout tests are presented, including: how to predict the active length at a given applied pullout force; how to predict the pullout failure force for reinforcement of a given length; how to determine the coefficient of friction from results of a pullout test; and, how to predict the displacement at any point along the reinforcement for a given applied pullout force. Results using the interface pullout formula are shown to be in good agreement with the results of an instrumented pullout test and finite element analyses. KEYWORDS: Pullout test, Geotextile, Geosynthetic, Analytical model, Soil-Geosynthetic interface, Reinforced soil. AUTHORS: S. Sobhi, Engineer, Colorado Department of Transportation, Transportation Safety and Traffic Engineering Branch, Denver, Colorado 80222, USA, Telephone: 1/303-757-9348, Telefax: 1/303-757-9219; and, J.T.H. Wu, Professor, Department of Civil Engineering, University of Colorado at Denver, Denver, Colorado 80217-3364, USA, Telephone: 1/303-556-8585, Telefax: 1/303-556-2368. PUBLICATION: Geosynthetics International is published by the Industrial Fabrics Association International, 345 Cedar St., Suite 800, St. Paul, Minnesota 55101-1088, USA, Telephone: 1/612-222-2508, Telefax: 1/612-222-8215. Geosynthetics International is registered under ISSN 1072-6349. DATES: Original manuscript received 9 February 1996, revised version received 13 September 1996 and accepted 18 September 1996. Discussion open until 1 July 1997. REFERENCE: Sobhi, S. and Wu, J.T.H., 1996, An Interface Pullout Formula for Extensible Sheet Reinforcement, Geosynthetics International, Vol. 3, No. 5, pp. 565-582. GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5 565

1 INTRODUCTION Over the past two decades, reinforcing a soil mass with extensible inclusions, such as geotextiles, has gained increasing popularity in earth structure construction. Design of reinforced soil structures requires that the tensile inclusion (tensile reinforcement) have adequate interface bond strength with the confining soil. Interface bonding is required to effectively transfer the tensile stress induced in the soil to the reinforcement, and to prevent pullout failure of the reinforced soil structure. The interface bond, or shear strength between soil and reinforcement has commonly been evaluated by two test methods: the pullout test, and the direct shear test. These two tests are fundamentally different in terms of geometric configurations, stress paths, and boundary conditions. Published data have demonstrated that the interface shear strengths determined by the two tests are often very different (Collios et al. 1980; Ingold 1983; Richards and Scott 1985; Rowe et al. 1985; Koerner 1986; Juran et al. 1988). It is generally recognized that the direct shear test is more appropriate for evaluating a sliding failure occurring between the reinforcement and the soil above it, and the pullout test gives a better representation of a pullout failure at the far end (i.e. the free end) of an embedded reinforcement. There are difficulties interpreting the test results of both the direct shear and the pullout test. These difficulties are primarily the result of boundary conditions and scaling that can affect the test results significantly (Palmeira and Milligan 1989). This study is focused on the pullout test. The purpose of this study is to develop an analytical model for predicting and interpreting results of a pullout test in a unified and consistent manner. This paper: presents the results of finite element analyses that were conducted to identify factors affecting pullout tests; describes three postulates that the interface pullout formula is based on; and, presents the derivation of the analytical model (the interface pullout formula ). A number of important applications of the interface pullout formula are also presented. These applications include: how to predict the active reinforcement length at a given applied pullout force; how to predict the pullout failure force for reinforcement of a given length; how to determine the coefficient of friction from the results of a pullout test; and, how to predict the displacement at any point for a given applied pullout force. In addition, preliminary verification of the interface pullout formula by comparison with the results of an instrumented pullout test and finite element analyses is presented. 2 FINITE ELEMENT SIMULATIONS 2.1 Introduction Tzong and Cheng-Kuang (1987) performed a series of large-scale laboratory pullout tests on a needle-punched nonwoven geotextile. The test box was 1450 mm high, 1220 mm long, and 610 mm wide. The geotextile specimen was embedded in an Ottawa sand under a constant surcharge and subjected to incrementally increasing pullout forces until a failure condition developed. The soil was prepared to a unit weight of 16.8 kn/m 3 (70% relative density) and the surcharge pressure was 30 kn/m 2. Geotextile properties 566 GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5

are shown in Table 1. The tests were instrumented to monitor pullout behavior, including internal displacements along the length of the geotextile at each load increment. Using material parameters obtained from element tests of the soil, the reinforcement, and the soil-reinforcement interface, Wu and Helwany (1987) conducted finite element analyses of the pullout tests and obtained excellent agreement with measured data. Figure 1 shows the cumulative displacements along the length of a geotextile specimen versus applied pullout force obtained from one of the experimental tests and from the finite element analysis. The finite element code used for the analyses was CANDE (Katona et al. 1976), which incorporates a unique interface model based on the constraint equations developed by Chan and Tuba (1971). Wu (1992) pointed out that such an interface model has many advantages over the commonly used stiffness method originally developed by Goodman et al. (1968) and later modified by a number of researchers (e.g. Clough and Duncan 1969; Ghaboussi et al. 1973). Table 1. Standard properties of the geotextile as provided by the manufacturer. Mass/unit area (g/m 2 ) Thickness (mm) (ASTM D 1777) Grab tensile strength (kn/m) (ASTM D 1682) Machine direction Cross-machine direction Grab tensile elongation (%) (ASTM D 1682) Machine direction Cross-machine direction 272 3.18 1.16 1.00 85 95 (a) (b) 50 40 30 20 10 0 50 40 30 20 Front end 75 mm from front 150 mm from front 225 mm from front Front end 75 mm from front 150 mm from front 225 mm from front 10 0 Applied pullout force, F (kn/m) Figure 1. Cumulative displacement versus applied pullout force: (a) pullout test; (b) finite element analysis. Note: Specimen length = 300 mm. GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5 567

In this study, the CANDE code was employed to investigate the effects of various factors on pullout behavior. The factors that significantly affected pullout behavior were the soil-reinforcement interface friction coefficient, reinforcement stiffness, reinforcement length, and overburden pressure. Some of the analytical results are presented in Sections 2.2, 2.3 and 2.4. 2.2 Reinforcement Length Geotextile specimen lengths of 300 and 530 mm were analyzed. Figure 2 shows the cumulative displacements along the length of the geotextile at different pullout forces. It should be noted that the displacement at a 6.96 kn/m pullout force is not shown for the 300 mm geotextile specimen because pullout failure occurred at 5.87 kn/m. Figure 2 reveals two important points concerning the reinforcement length. Firstly, at a given pullout force, movement occurs only within a certain length of the reinforcement specimen. This length, referred to as the active length, is practically independent of the total length of the reinforcement. Secondly, the variation of displacements along the active length is not affected by the total length of the reinforcement. 2.3 Reinforcement Stiffness Analyses were conducted using different values of reinforcement stiffness. Figure 3 depicts the cumulative displacements along the reinforcement at two different applied pullout forces using two different reinforcement stiffness values. It is seen that the reinforcement stiffness affects both the active length and the magnitude of the displacements. The effect on reinforcement displacement is particularly significant. L = 530 mm F =5.22kN/m F =6.96kN/m L = 300 mm F =5.22kN/m Distance from the front edge (mm) Figure 2. Finite element analysis of the effect of reinforcement length on pullout behavior. Note: L = specimen length; F = applied pullout force. 568 GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5

E =1.0 10 4 kpa F =1.74kN/m E =3.5 10 3 kpa F =1.74kN/m Distance from the front edge (mm) Figure 3. Finite element analysis of the effect of reinforcement stiffness on pullout behavior. Note: E = reinforcement stiffness. Overburden pressure, σ n =40kPa F =1.74kN/m Overburden pressure, σ n =23kPa F =1.74kN/m Figure 4. Distance from the front edge (mm) Finite element analysis of the effect of overburden pressure on pullout behavior. 2.4 Overburden Pressure Figure 4 shows the cumulative displacements along the reinforcement at two different pullout forces and under two different overburden pressures. As expected, the larger GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5 569

the overburden pressure, the smaller the displacements. It should be noted that a higher overburden pressure also results in a shorter active length. 3 POSTULATES OF THE ANALYTICAL MODEL Three postulates were prescribed in the development of the analytical model. They are: the postulate of stationary confining soil; the postulate of interface shear stress mobilization; and, the postulate of cumulative deformation. 3.1 Postulate of Stationary Confining Soil In a pullout test, the differential movement, du, at a point along the reinforcement can be considered to be the sum of two components: du s, and du r. The component du s is due to shear straining of the soil at the soil-reinforcement interface, while the component du r is due to tensile elongation of the reinforcement. The postulate of stationary confining soil states that the confining soil remains stationary at all times during a pullout test; i.e. du =du r, and du s = 0. In other words, before slippage occurs at the soil-reinforcement interface, the displacement undergone by the soil and reinforcement in a bonded manner is negligible. 3.2 Postulate of Interface Shear Stress Mobilization Data from instrumented pullout tests for extensible sheet reinforcement have consistently indicated that the displacements along the length of a reinforcement do not vary linearly; thus, the strain along the length of the reinforcement is not uniform. By assuming a linear relationship between the displacement and interface shear stress, many researchers (Juran and Christopher 1989; Yuan and Chua 1991; Chan et al. 1993) have concluded that the interface shear stress along an extensible reinforcement is not uniform prior to failure, and that this shear stress is less than the ultimate (limiting) interface shear stress, τ u = P u /(2wL), where P u is the ultimate pullout force, w is the reinforcement width, and L is the reinforcement length. The postulate of interface shear stress mobilization states that the shear stress induced at the soil-reinforcement interface along the active length is uniform and is equal to the limiting interface shear stress. The limiting interface shear stress can be computed as: τ u = σ n f,whereσ n is the overburden pressure, and f is the coefficient of interface friction between the soil and reinforcement. Depending on the extensibility and length of the reinforcement, the value of the limiting interface shear stress may be very different from that determined by the formula, τ u = P u /(2wL). Using this postulate, the limiting interface shear stress becomes a characteristic parameter of a given type of reinforcement and soil under a specified overburden pressure. Its value is independent of the length of the reinforcement. Results of the finite element analyses conducted by Wu and Helwany (1987) support the validity of this postulate. As shown in Figure 5, the top portion of the curves do not deviate significantly from a horizontal line. This implies that the shear stress can be assumed uniform, even before pullout failure occurs. Also, the somewhat abrupt drop of the curves can be idealized as a vertical line, suggesting that the pullout force will not 570 GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5

(a) (b) Interface shear stress (kpa) (c) Interface shear stress (kpa) Idealization Distance from the front edge (mm) Idealization Interface shear stress (kpa) Interface shear stress (kpa) (d) Idealization Distance from the front edge (mm) Idealization Distance from the front edge (mm) Distance from the front edge (mm) Figure 5. Finite element analysis interface shear stress results along the length of the reinforcement, and idealization: (a) F = 1.74kN/m;(b) F = 2.61kN/m;(c) F = 3.48kN/m; (d) F = 5.00 kn/m (Wu and Helwany 1987). be transferred from one point to the next unless a large enough pullout force is applied to overcome the limiting interface frictional resistance at that point. In other words, the shear stress at the soil-reinforcement interface at a given pullout force is only induced within a certain length of the reinforcement (the active length). The interface shear stress is of a constant magnitude, τ u = σ n f. 3.3 The Postulate of Cumulative Deformation Extensible reinforcement may undergo considerable elongation in a pullout test. The postulate of cumulative deformation states that the frictional resistance developed GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5 571

within the elongated length of the reinforcement is significant and must be accounted for in the analytical model. For a differential segment, dx, of an extensible pullout test specimen, the tensile force per unit width at one end of dx is T 1, and the force per unit width transferred to the other end is T 2 (Figure 6). The difference between the two forces is T 1 - T 2 = τ u (dx + εdx), rather than T 1 - T 2 = τ u dx,whereε is the strain developed in the differential segment, dx. 4 PULLOUT TEST CONDITIONS ASSUMED IN THE ANALYTICAL MODEL It has been recognized that the test conditions, including the top boundary conditions (for applying the vertical pressure), frictional characteristics of the front wall, proximity of the front wall, and scale of the test box, can all have strong influences on the results of pullout tests. In developing the analytical model, the pullout test is assumed to be conducted under the following conditions: S The reinforcement specimen is sufficiently strong that rupture failure does not occur during the pullout test. S The reinforcement specimen remains confined within the soil throughout the test. If a clamp is employed to exert pullout forces and is embedded in the soil, the frictional resistance on the clamp should be accounted for in data interpretation. S The normal stress (overburden pressure) on the reinforcement is uniform. A uniform normal stress can be achieved using a flexible pneumatic bag, as opposed to applying a force on a rigid plate. S The reinforcement specimen is wide enough so that the Poisson s effect (i.e. necking) is negligible. S The interior of all side walls of the test apparatus is properly lubricated. The frictional force between the side wall and the soil should be estimated and accounted for in the interpretation of the results. S The confining soil is uniform. dx εdx τ u T 2 T 1 τ u Figure 6. Forces and deformation in a differential segment of reinforcement, dx. 572 GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5

S The size of the test box is sufficiently large that the relative dimensions of the reinforcement and soil do not significantly affect the pullout test results. 5 DERIVATION OF THE INTERFACE PULLOUT FORMULA Considering a differential segment of a reinforcement specimen, dx, that undergoes an elongation, εdx (Figure 7), the static equilibrium equation can be written as: T 2σ n f(dx + εdx) (T dt) = 0 (1) where: T = tensile force per unit width at any given point along the reinforcement; σ n = normal stress (overburden pressure) on the reinforcement; and f = friction coefficient between the soil and reinforcement. Therefore: 1 2σ n f dt dx = 1 + ε (2) Let 1/(2σ n f)=k 1, thus: k 1 dt dx ε 1 = 0 (3) (a) dx F Positive x values Initial length, L x =0 (front edge) (b) dx εdx τ u T--- dt T τ u Figure 7. An illustration of the conventions used to derive the interface pullout formula, Equation 8: (a) initial position of reinforcement and coordinate system; (b) forces on a differential segment of reinforcement during deformation. GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5 573

Assuming the reinforcement is linearly elastic in tension: ε(x) = T(x) Et (4) where: E = inherent confined elastic modulus of reinforcement; and, t = initial thickness of the reinforcement. The product E t is the slope of the load/width versus strain curve obtained under pressure confinement (i.e. obtained under an overburden pressure, σ n ), and free from soil-reinforcement frictional resistance. It should be noted that the assumption of linear elasticity is not necessary for reinforcement that exhibits significant nonlinear behavior. For these reinforcements, E (or Et) can be formulated as a nonlinear function of the stress level (Ling et al. 1992). Wu (1991) has devised a test method for determining the inherent stiffness and strength of geosynthetics under pressure-confinement conditions. Ballegeer and Wu (1993) have presented the values of inherent confined stiffness for a number of different geosynthetics. Substituting Equation 4 into Equation 3 yields: k 1 dt dx T Et 1 = 0 (5) Let 1/(Et) =k 2, thus: k 1 dt dx k 2T 1 = 0 (6) Solving Equation 6 with the boundary condition, T = F at x =0(Note:F is the applied pullout force per unit width of reinforcement), the tensile force, T, induced at a point in the reinforcement test specimen is: T = 1 k 2 + F + 1 k 2 e mx (7) where: m = k 2 /k 1 ; k 1 =1/(2σ n f); and, k 2 =1/(Et). Alternatively, Equation 7 can be expressed as: T = (F + Et)e 2σ nf Et x Et (8) where T is the tensile force per unit width at any point, x, in the reinforcement specimen. Equation 8 is called the interface pullout formula. 574 GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5

6 APPLICATIONS OF THE INTERFACE PULLOUT FORMULA In this section, applications of the interface pullout formula for predicting and interpreting pullout test results are presented. For each case, the reader can refer to Figure 7 for clarification. 6.1 Active Length of Reinforcement In order to determine the distance, x, along the reinforcment at which the tensile force, T, in the reinforcement equals zero for a given applied pullout force, F, per unit width, Equation 7 can be used with T = 0. Hence: 1 k 2 = F + 1 k 2 e mx (9) Solving for x leads to: 1 k 2 x = m 1 ln F + k 1 2 = Et 2σ n f ln Et F + Et (10) 6.2 Pullout Force to Induce Pullout Failure In order to determine the applied pullout force, F, per unit width such that T =0at x =-L (L is the total length of reinforcement), substitute x =-L and T = 0 into Equation 7 to obtain the following: 1 k 2 = F + 1 k 2 e ml (11) thus: F = 1 k 2 e ml 1 = Et e 2σ nf Et L 1 (12) 6.3 Coefficient of Friction, f The calculation of the friction coefficient, f, for a given applied pullout force, F,per unit width at failure and a given total length, L, of the reinforcement specimen can be based upon Equation 11 (corresponding to failure condition) as follows: e ml = Fk 2 + 1 GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5 575

or: m = ln(fk 2 + 1) L that is to say: 2σ n f Et = ln(fk 2 + 1) L (13) therefore: f = Et ln(fk 2 + 1) 2σ n L = Et ln F Et + 1 2σ n L (14) It should be noted that if the active length and the corresponding pullout force prior to failure are known, they can be used in lieu of the total length and pullout force at failure to determine the friction coefficient by following the same procedure outlined above. 6.4 Reinforcement Displacement In order to determine the displacement at any point along the reinforcement for a given pullout force, F (before or at failure) let x 0 = the coordinate of the active length, as determined by Equation 9, and x 1 = the coordinate of the point at which the displacement is to be determined. Since ε = T/(Et) =du/dx, the displacement, u, can be determined by: u = T Et dx (15) Substituting T from Equation 7: 1 u = 1 1 Et x + (F + 1 )e k 2 k mx dx 2 x 0 1 = 1 2σnf Et x Et + (F + Et)e x 0 Et x dx (16) 576 GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5

7 VERIFICATION OF THE INTERFACE PULLOUT FORMULA Preliminary verification of the interface pullout formula was performed by comparing the predicted behavior of pullout test results with those obtained from an instrumented laboratory test and from finite element analyses. 7.1 Validation by Instrumented Pullout Test Results Two large-scale instrumented pullout tests performed by Tzong and Cheng-Kuang (1987) satisfied all of the test conditions prescribed in Section 4. The width of the geotextile test specimens was 450 mm. The gage lengths used were 300 and 450 mm. The soil conditions are described in Section 2.1. Pullout forces were applied at one end of the geotextile specimen in equal increments of 0.48 kn/m until pullout failure occurred. At each load increment, the displacement at selected points along the length of the geotextile specimen was measured. Equation 8, the interface pullout formula, was used to predict the behavior of the large-scale pullout tests. Equation 14 was first used to calculate the coefficient of friction, f. A coefficient of friction value of f = 0.29 was obtained for both tests despite the difference in the reinforcement length. This is very significant as it demonstrates that the interface pullout formula can indeed provide a consistent interpretation of pullout test results. Equation 15 was then used to predict the displacements at selected points along the length of the geotextile specimen. The predicted displacements were in good agreement with the measured displacements. Figure 8 shows a comparison of the displacements obtained from the interface pullout formula and from the experimental test for a 300 mm long geotextile specimen. The agreement was excellent except near the free end of the geotextile specimen where shear failure had occurred. 7.2 Verification by Finite Element Analyses Results Since other experimental data that satisfy the prescribed test conditions and include the internal response of the reinforcement during a pullout test are not readily available, further verification of the interface pullout formula was performed by comparison with the finite element analyses presented in Section 2. Figure 9 shows a comparison of the displacements determined using the interface pullout formula and the result of finite element analyses for different reinforcement lengths, different values of reinforcement stiffness, and different overburden pressures. Good agreement between the interface pullout formula and the finite element analyses results was obtained. 8 SUMMARY An interface pullout formula, Equation 8, was developed based on three postulates: the postulate of stationary confining soil; the postulate of interface shear stress mobilization; and, the postulate of cumulative deformation. Equation 8 describes the relationship between the forces and displacements of an extensible sheet reinforcement in a pullout test under prescribed test conditions. GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5 577

(a) (b) (c) Applied pullout force, F (kn/m) (d) Applied pullout force, F (kn/m) Applied pullout force, F (kn/m) Applied pullout force, F (kn/m) Figure 8. Comparison of displacements using the interface pullout formula and measured values from pullout tests on a 300 mm long geotextile specimen: (a) x = 0 (front edge); (b) x = 75 mm; (c) x = 150 mm; (d) x = 225 mm (data from Tzong and Cheng-Kuang 1987). A number of important applications of the interface pullout formula for predicting and interpreting pullout test results have been presented, including: how to predict the active length at a given applied pullout force; how to predict the applied pullout force required to induce failure for a given length of reinforcement; how to determine the coefficient of friction; and, how to predict the displacement at any point for a given applied pullout force. Comparisons of the interface pullout formula with the results of an instrumented pullout test and finite element analyses show that the interface pullout formula is capable of predicting pullout test results with very good accuracy. More importantly, the interface pullout formula provides a unified and consistent method for interpretation of pullout test results. 578 GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5

(a) Interface pullout formula L = 300 and 530 mm L = 300 and 530 mm F =5.22kN/m L = 300 and 530 mm F =6.96kN/m Finite element analyses L = 300 mm L = 530 mm F =5.22kN/m L = 530 mm F =6.96kN/m (b) Interface pullout formula E =1.0 10 4 kpa F =1.74kN/m E =3.5 10 3 kpa Finite element analyses E =1.0 10 4 kpa F =1.74kN/m E =3.5 10 3 kpa (c) Interface pullout formula σ n =23kPa F =1.74kN/m σ n =40kPa σ n =23kPa Finite element analyses σ n =23kPa F =1.74kN/m σ n =40kPa σ n =23kPa Distance from the front edge (mm) Figure 9. Comparison of displacements using the interface pullout formula and results of finite element analyses for different: (a) reinforcement lengths; (b) reinforcement stiffnesses; (c) overburden pressures. GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5 579

REFERENCES ASTM D 1682, Standard Test Methods for Breaking Load and Elongation of Textile Fabric, American Society for Testing and Materials, West Conshohocken, Pennsylvania, USA. ASTM D 1777, Standard Test Method for Measuring Thickness of Textile Materials, American Society for Testing and Materials, West Conshohocken, Pennsylvania, USA. Ballegeer, J.P. and Wu, J.T.H., 1993, Intrinsic Load-Deformation Properties of Geotextiles, Geosynthetic Soil Reinforcement Testing Procedures, Cheng, S.C.J., Editor, ASTM Special Technical Publication 1190, Proceedings of a symposium held in San Antonio, Texas, USA, January 1993, pp. 16-31. Chan, D.H., Yi, C.T. and Scott, J.D., 1993, An Interpretation of the Pullout Test, Proceedings of Geosynthetics 93, IFAI, Vol. 2, Vancouver, British Columbia, Canada, March 1993, pp. 593-605. Chan, S.K. and Tuba, I.S., 1971, A Finite Element Method for Contact Problems of Solid Bodies-Part I. Theory and Validation, International Journal of Mechanical Sciences, Vol. 13, No., pp. 615-625. Clough, G.W. and Duncan, J.M., 1969, Finite Element Analyses of Fort Allen and Old River Locks, Report No. S-69-6, U.S. Army Waterway Experiment Station, Vicksburg, Mississippi, USA. Collios, A., Delmas, P., Gourc, J.P. and Giroud, J.P., 1980, Experiments on Soil Reinforcement with Geotextiles, The Use of Geotextiles for Soil Improvement, ASCE National Convention, Portland, Oregon, USA, April 1980, pp. 53-73. Ghaboussi, J., Wilson, E.L. and Isenburg, J., 1973, Finite Element for Rock Joints and Interfaces,Journal of the Soil Mechanics and Foundation Division, ASCE, Vol. 99, SM. 10, pp. 833-848. Goodman, R.E., Taylor, R.L. and Brekke, J.L., 1968, A Model of the Mechanics of Jointed Rock, Journal of the Soil Mechanics and Foundation Division, ASCE, Vol. 94, SM. 3, pp. 637-659. Ingold, T.S., 1983, Laboratory Pull-Out Testing of Grid Reinforcements in Sand, Geotechnical Testing Journal, Vol. 6, No. 3, pp. 101-111. Juran, I. and Christopher, B.R., 1989, Laboratory Model Study on Geosynthetic Reinforced Soil Retaining Walls, Journal of Geotechnical Engineering,ASCE,No.115, No. 7, pp. 905-926. Juran, I., Knochenmus, G., Acar, Y.B. and Arman, A., 1988, Pull-Out Response of Geotextiles and Geogrids (Synthesis of Available Experimental Data), Geosynthetics for Soil Improvement, Holtz, R.D., Editor, Geotechnical Special Publication No. 18, ASCE, proceedings of a symposium held in Nashville, Tennessee, USA, May 1988, pp. 92-111. Katona, M.G., Odello, R.S. and Allgood, J.R., 1976, CANDE - A Modern Approach for the Structural Design and Analysis of Buried Culverts, Report No. FHWA- RD-77-5, Federal Highway Administration, Washington, DC, USA. 580 GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5

Koerner, R.M., 1986, Direct Shear/Pull-Out Tests on Geogrids, Report No. 1, Department of Civil Engineering, Drexel University, Philadelphia, Pennsylavania, USA. Ling, H.I., Wu, J.T.H. and Tatsuoka, F., 1992, Short-Term Strength and Deformation Characteristics of Geotextiles Under Typical Operational Conditions, Geotextiles and Geomembranes, Vol. 11, No. 2, pp. 185-219. Palmeira, E.M. and Milligan, G.W.E., 1989, Scale and Other Factors Affecting the Results of Pull-Out Tests of Grids Buried in Sand, Geotechnique, Vol. 39, No.3, pp. 511-524. Richards, E.A. and Scott, J.D., 1985, Soil Geotextile Frictional Properties, Proceedings of the Second Canadian Symposium on Geotextiles and Geomembranes, Edmonton, Alberta, Canada, September 1985, pp. 13-24. Rowe, R.K., Ho, S.K. and Fisher, D.G., 1985, Determination of Soil-Geotextile Interface Strength Properties. Proceedings of the Second Canadian Symposium on Geotextiles and Geomembranes, Edmonton, Alberta, Canada, September 1985, pp. 25-34. Tzong, W.H. and Cheng-Kuang, S., 1987, Soil-Geotextile Interaction Mechanism in Pullout Test, Proceedings of Geosynthetics 87, IFAI, Vol. 1, New Orleans, Louisiana, USA, February 1987, pp. 250-259. Wu, J.T.H., 1991, Measuring Inherent Load-Extension Properties of Geotextiles for Design of Reinforced Structures, Geotechnical Testing Journal, Vol. 14, No. 2, pp. 157-165. Wu, J.T.H., 1992, Discussion on Embankments, Earth Reinforcement Practice, Ochiai, H., Hayashi, S. and Otani, J., Editors, Balkema, Proceedings of the International Symposium on Earth Reinforcement Practice, Vol. 2, Fukuoka, Kyushu, Japan, November 1992, pp. 928-929. Wu, J.T.H. and Helwany, M.B., 1987, Numerical Simulation of Soil-Geotextile Interface in Pullout Test, Geosynthetic Research Report 87-03, Department of Civil Engineering, University of Colorado at Denver, Colorado, USA, 120 p. Yuan, Z. and Chua, K.M., 1991, Analytical Model for Pullout of Soil Reinforcement, Transportation Research Record 1330, pp. 64-71. NOTATIONS Basic SI units are given in parentheses. dt = differential tensile force over the length of a differential segment, dx (N/m) du = differential movement at a point along the reinforcement (m) du r = differential movement at a point along the reinforcement due to tensile elongation of the reinforcement (m) du s = differential movement at a point along the reinforcement due to shear straining of the soil at the soil-reinforcement interface (m) GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5 581

dx = length of a differential segment of reinforcement (m) E = inherent Young s modulus of reinforcement, obtained under a given confining pressure and free from soil-reinforcement frictional resistance (Pa) f = coefficient of friction between soil and reinforcement (dimensionless) F = applied pullout force per unit width of reinforcement (N/m) k 1 = 1/(2σ n f)(m 2 /N) k 2 = 1/(Et) (m/n) L = length of reinforcement (m) m = k 2 /k 1 (m -1 ) P u = ultimate pullout force (N) T = induced tensile force at any point, x, along the specimen (N/m) T 1 = tensile force per unit width at one end of a differential segment (N/m) T 2 = tensile force per unit width transferred to the opposite end of a differential segment (N/m) t = initial thickness of reinforcement (m) u = displacement (m) w = width of reinforcement (m) x = coordinate along the length of reinforcement (x = 0 represents the front edge of reinforcement) (m) x 0 = coordinate of the active length of reinforcement (m) x 1 = coordinate of a point within the active length where the displacement of that point is to be determined (m) ε = axial strain developed in a differential segment, dx, of reinforcement (dimensionless) σ n = overburden pressure (normal stress) in a pullout test (Pa) τ u = limiting interface shear stress (N/m 2 ) 582 GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 5