Continuum Modeling and Analysis of Polymeric Sheet Reinforcement Subjected to Axial Pull
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Abstract
Stability of reinforced soil structures depends on the soil–reinforcement interaction. Continuum modeling of polymeric sheet reinforcement–backfill interaction when subjected to axial pull at one end of the reinforcement is presented. Analysis is carried out by FLAC, soil is modeled as linear elastic material and reinforcement as cable elements. Displacements of reinforcement are quantified for different pullout forces. Variations of displacement and tension along the reinforcement length are presented. Effect of axial stiffness of reinforcement, soil–reinforcement interface shear stiffness and length of reinforcement are quantified. For a given pullout load the displacement of reinforcement decreases with increase of interface shear stiffness, axial stiffness of reinforcement and length of reinforcement. Results of the present numerical model compare closely with other mechanical models and pullout tests.
Keywords
FLAC Axial force Reinforcement–backfill interaction Axial stiffness Interface shear stiffnessIntroduction
Reinforcement–soil interaction plays a key role in the design and analysis of reinforced soil structures such as walls, slopes, embankments and footings. Pullout resistance is quantified by laboratory and field pullout tests, mechanical and continuum analyses. A wide range of pullout tests are carried out based on different boundary conditions of the test apparatus on extensible reinforcement such as geogrids, geotextiles and inextensible reinforcement such as welded rods, steel strips, etc., for different types of backfill, granular soils with different particle sizes, shear strength, and unit weight. Pullout tests are conducted as displacement or load controlled tests [1, 2, 3, 4, 5, 6, 7]. Aiban and Ali [8] evaluated the interface friction characteristics of nonwoven geotextile for two different types of backfill. Khedkar and Mandal [9] conducted pullout tests on cellular reinforcement and results compared with the predictions based on finite element method. Suksiripattanapong et al. [10, 11] demonstrated the use of bearing reinforcement in reinforced earth wall and studied their pullout response. Lajevardi et al. [12, 13] conducted pullout tests on steel welded mesh reinforcement and results compared with the predictions based on analytical method mentioned in the French standard, NF P 94270.
Pullout test data is widely used to estimate the soil–reinforcement bond capacity for designs based on limit equilibrium analyses [14]. For reinforcements with planar surfaces, the mechanism of interaction in pullout tests is controlled by the friction mobilized between soil and reinforcement. Abramento and Whittle [15, 16] conducted pullout tests on inextensible and extensible reinforcements. Hayashi et al. [17] conducted laboratory pullout tests and measured the elongation at several points along the geosynthetic reinforcement. Pradhan et al. [18] investigated the effect of normal pressure during pullout tests on a saturated claygeosynthetic system.
Abramento and Whittle, Sobhi and Wu, and Long et al. [15, 19, 20] proposed mechanical models based on respectively shearlag analysis, rigidplastic and nonuniform shear stress distribution at the soil–reinforcement interface to predict the pullout capacity of planar polymeric reinforcement. Weerasekara and Wijewickreme [21] developed a new analytical model to predict the pullout response of geotextiles combining the nonlinear responses of the geotextile and soil–geotextile interface characteristics.
Pullout interaction mechanisms between soil and geogrid reinforcements are more complex than those between soil and strip or sheet reinforcement. Pullout resistance of geogrids includes two components, viz., the interface shear resistance that takes place along the longitudinal ribs and to a lesser extent along the transverse ribs and the passive resistance that develops against the front of transverse ribs (Koerner et al. [22]). Passive pullout resistance that develops against transverse ribs was determined by general shear failure mechanism by Peterson and Anderson [23], punching failure mechanism by Jewell et al. [24] and a modified punching failure mechanism by Chai [25]. General shear and punching shear failure mechanisms have been reported to provide upper and lower bounds of experimental pullout test results by Palmeira and Milligan [2] and Jewell [14]. Large scale pullout tests and individual rib pullout tests were conducted on individual longitudinal and transverse ribs to quantify the contributions of passive and interface shear mechanisms to the overall pullout resistance of geogrids [26].
Sugimoto and Alagiyawanna [27] conducted finite element analysis to model the pullout behaviour of geogrid and demonstrated that the interface properties play a significant role in the FEM simulations of geogrid pullout behavior. Alam and Lo [28] studied pullout of steel grid soil reinforcement using FLAC 2D. Alam et al. [29] investigated the pullout behavior of steel grid reinforcement embedded in silty sand and compared the results with numerical analysis.
Madhav et al. [30] presented a mechanical model to demonstrate the pullout response of geosynthetic reinforcement. Applied pullout force mobilizes soil reinforcement interface shear stress along the reinforcement quantified by the bilinear shear stress displacement response. Elongation of reinforcement was considered and the resulting finite difference equation was solved by Gauss–Siedel iteration method. Gurung and Iwao [31] compared satisfactorily the results of theoretical pullout tests and experimental results. Gurung [32] presented one dimensional expression for pullout of planar reinforcement that analyses small to large strain cases of inextensible to extensible reinforcements.
Problem Definition and Analysis
Geometric properties of sheet reinforcement such as cross sectional area and perimeter of the reinforcement are computed for 2 mm thick sheet for unit length perpendicular to the plane. Overburden pressure on sheet reinforcement is applied in the model by input of acceleration due to gravity of 10 m/sec^{2}. Model is rebuilt by several iterations and brought to equilibrium under the gravity stresses. Each grid point is surrounded by four material elements and the algebraic sum of the forces contributed by these surrounding elements at any specified grid point is defined as an unbalanced force. This unbalanced force should converge to zero when the model reaches equilibrium state. Other equilibrium criterion is stress ratio defined as maximum unbalanced force to the representative internal force. In the present work the equilibrium of the model is established based on the following criteria—number of iterations are restricted to one hundred thousand, the minimum stress ratio of 0.001 or the maximum unbalanced force of zero Newton. In most of the cases equilibrium state based on minimum stress ratio of 0.001 governs the solution.
Displacement of reinforcement, w and tension T at different nodes due to pullout force, T_{max} at node B is obtained from the numerical model and the obtained results are compared with mechanical model [30]. Large strain mode is adopted in the analysis to accurately predict the displacements and tensile forces at different nodes along the reinforcement.
Results and Discussion
Parametric study is carried for the following range of parameters: depth of reinforcement, D = 1–5 m, length of reinforcement, L = 3–7 m, shear stiffness of interface, k_{b} = 25–200 kN/m^{3}, axial stiffness of reinforcement, J = 0.1–5 MN/m. Tensile yield strength of sheet reinforcement = 200 kN/m and interface friction angle, φ_{r} = 30°.
Comparison of Results with Other Analytical and Field Test Results
Konami et al. [34] conducted field pullout tests on polymer strip reinforced soil wall. Three types of polymeric strip reinforcements, named PW3, PW5 and PW10 were used in the field pullout tests. These strips were made of fiber coated with polyethylene of 8.5, 9.0 and 9.0 cm wide respectively. Thickness of the strips was 2, 3 and 5 mm respectively. Details of secant rigidity at 2 and 5 % strains, nominal strength and elongation due to tension failure are mentioned in the paper by Konami et al. [34].
Estimated pullout test parameters for comparison with experimental results [30]
Test number  Shear stiffness, k_{b} in kN/m^{3}  Axial stiffness, J MN/m  Interface friction coefficient, μ 

1  272.9  5.97  0.30 
2  443.5  5.97  0.44 
3  483.0  3.88  0.45 
4  245.5  3.88  0.80 
7  483.5  5.97  0.33 
8  246.9  3.88  0.80 
Displacement of node B is quantified by FLAC for different pullout forces, T_{max} for the shear stiffness, k_{b}, axial stiffness, J and interface friction coefficient, μ mentioned in Table 1. The variation of displacement at node B for different pullout loads is presented in Figs. 16, 17, 18, 19, 20 and 21.
Variation of displacement of reinforcement at node B for different pullout loads predicted by mechanical model compare closely with the experimental results in Figs. 16, 17, 18, 19, 20 and 21. Displacements quantified by the present numerical model are marginally less for the same order of pullout force when compared with the predictions of mechanical model. Variation of pullout load with displacement is linear based on the predictions of numerical model and these results are comparable with the experimental predictions for smaller displacement/pullout force. Konami et al. [34] underestimate the displacements compared with the other results due to the assumption of full shear stress mobilization along the reinforcement–soil interface. Predictions of displacement of reinforcement for different axial pullout forces based on FLAC compare closely with other mechanical/theoretical and experimental results.
Conclusions
Response of polymeric sheet reinforcement subjected to axial force at its free end is studied using FLAC for a range of interface shear stiffnesses and axial stiffnesses of reinforcement and for different lengths of reinforcement.

Results of the axial pullout of sheet reinforcement based on FLAC compare closely with the results of mechanical model given by Madhav et al. [30]. Marginal difference between the two methods is due to the continuum and discrete approaches adopted.

Displacement of loaded end of reinforcement subjected to an axial pullout force of 24 kN/m increases from 27 to 113 mm with decrease of interface shear stiffness from 200 to 25 kN/m^{3} and elongation of reinforcement is nearly constant.

Inextensible reinforcement (J = 5 MN/m) subjected to an axial pullout force of 23 kN/m mobilizes a uniform shear stress and undergoes a rigid body displacement of about 25 mm.

Extensible reinforcement (J = 0.1 MN/m) subjected to an axial pullout force of 23 kN/m undergoes a larger displacement of 114 mm at loaded end of reinforcement while the displacements of the free of end of reinforcement is negligible. Shear stress mobilized at loaded end of reinforcement is large while it is negligible at the free end.

For the same order of axial pullout force, shorter length of reinforcement undergoes rigid body displacement and longer length of reinforcement elongates. Displacement at free end of reinforcement is small for longer length of reinforcement compared to that for shorter length of reinforcement.

Pullout force to be applied for a given displacement increases with increase of interface shear stiffness, axial stiffness of reinforcement and marginally with length of reinforcement.
Predicted pullout force versus displacement relation is comparable with the results of mechanical model by Konami et al. [34], Madhav et al. [30] and the field pullout tests by Konami et al. [34].
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