Quantifying the Influence of Geosynthetics on Pavement Performance (2017)

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D-1 APPENDIX D. ANALYTICAL MODEL FOR QUANTIFYING INFLUENCE OF GEOSYNTHETICS The repeated load triaxial test studies indicate that the placement of geosynthetics influences the cross-anisotropic properties (i.e., the vertical and horizontal modulus) and the permanent deformation properties of the UGM. An analytical model is proposed to predict the vertical and horizontal modulus and the permanent deformation of the geosynthetic-reinforced UGM when it is subjected to a triaxial load. Figure D-1a shows a schematic plot of a geosynthetic-reinforced UGM specimen in the triaxial load test. The geosynthetic-reinforced specimen is compressed in the axial direction and normally expands in the lateral direction due to the plastic and resilient deformation. The figure shows that the lateral movement of the UGM is restrained by the geosynthetic. The shear stress is generated due to the relative lateral displacement between the geosynthetic and aggregate, which results in the stretch of the embedded geosynthetic. Note that the lateral movement of the aggregate and geosynthetic cannot be identical. Figure D-1b shows the difference in lateral movement between the geosynthetic and aggregate during the test. A coefficient α was employed to account for the difference of radial displacement between geosynthetic and aggregate, as shown in Equation D-1. a rr g rr ε α ε = (D-1) where a r rε is the aggregate radial tensile strain at the interface between the geosynthetic and aggregate, and g r rε is the geogrid radial tensile strain. Note that the value of α is normally larger than 1, which illustrates that the aggregate has a larger lateral movement than the geosynthetic. The analytical solution to determine the coefficient α is shown in Equations D-2 and D-3 (1). 0 1 3 2 2 2 D DJ J D β β β σ   ⋅ − ⋅ =       (D-2) ( )( ) 1/222 1 1a gG M α νβ δ   − − =     (D-3) where ( )iJ x is the Bessel function of order i, D is the diameter of the aggregate specimen (i.e., D = 6 inches), and aG is the shear modulus of the aggregate. Equation D-2 is an implicit equation for the coefficient α . The stretch of the geosynthetic generates a reinforcement force T to confine the UGM specimen through the aggregate particle interlock and interface friction (2). Figure D-1c shows that the reinforcement force T is equivalent to a triangularly distributed additional confining stress 3σΔ , which only acts on a 6-inch geosynthetic-reinforced influence zone (3). This distribution takes into account the phenomenon that the geosynthetic reinforcement influence decreases with the distance between the aggregate and geosynthetic, and the geosynthetic reinforcement is negligible when the material is far away from the geosynthetic.

D-2 (a) Displacement Pattern of UGM Restraint by Geosynthetic (b) Difference in Radial Movement of Geosynthetic and Aggregate (c) Equivalence of Reinforcement Force to Additional Stress Δσ3 Figure D-1. Schematic Plot of Geosynthetic Reinforcement on UGM Specimen Aggregate Before test After test Geosynthetic Deformed Reinforcement Force T

D-3 Under an axisymmetric plane-stress condition, the reaction force T is determined by Equation D-4. ( ) ( )21 g g rr g g MT θθε ν ε ν = ⋅ + − (D-4) where M is the geosynthetic sheet stiffness, gν is the Poisson’s ratio of the geosynthetic, gr rε is the geosynthetic tensile strain in the radial direction, and gθ θε is the geosynthetic tensile strain in the circumferential direction. By assuming that the geosynthetic expands uniformly in both the radial and the circumferential directions, Equation D-4 is simplified as: ( )1 g rr g MT ε ν = ⋅ − (D-5) If the equivalent additional confining stress 3σΔ is triangularly distributed in the influential zone, the maximum additional confining stress 3 m a xσΔ can be calculated by Equation D-6. ( )3max 2 2 1 g rr g T M σ εδ ν δ Δ = = ⋅ − (D-6) where δ is the thickness of the influential zone (i.e., δ = 6 inches). Substituting Equation D-1 into Equation D-6 yields: ( )3max 2 1 a rr g M σ ε ν δα Δ = ⋅ − (D-7) In Equation D-7, the aggregate radial tensile strain a r rε is the summation of the radial elastic strain 3,a rε and the radial plastic strain 3,apε . The radial elastic strain 3,a rε is calculated by the generalized Hooke’s law, as shown in Equations D-8. ( ) ( )3 3max 33 3 3max13 1 3, a r H V HE E E σ σ ν σ σν σ ε +Δ +Δ = − − (D-8) where 3σ is the axial stress applied to the specimen, 1σ is the initial confining pressure, 13ν is the Poisson’s ratio to characterize the effect of axial strain on lateral strain, 33ν is the Poisson’s ratio to characterize the effect of lateral strain on lateral strain, HE is the horizontal modulus of the specimen, and VE is the vertical modulus of the specimen. Equation D-9 is used to calculate the axial plastic strain 1,a pε . ( ) ( )1, 0 2 1m na Np e J I K βρ ε ε α   −   = + (D-9)

D-4 where 2J = ( ) 21 3 3max13 σ σ σ − + Δ  , 1I = ( )1 3 3max2σ σ σ+ +Δ , and 0ε , ρ , β , m, and n are permanent deformation properties for the unreinforced specimen. The relationship between the radial plastic strain 3,a pε and the axial plastic strain 1,a pε is shown in Equation D-10. 3, 1, 1 1 sin 2 1 sin a a p p ψ ε ε ψ  + =   −  (D-10) where ψ is the dilation angle of the specimen. Assuming that the dilation angle ψ is 15 degrees, Equation D-10 is simplified as: 3, 1,0.85a ap pε ε= (D-11) Substituting Equations D-8, D-9, and D-11 into Equation D-7 yields: ( ) ( ) ( ) ( ) ( ) ( )3 3max 33 3 3max13 1 0 2 13max 2 0.851 m nN g H V H M e J I K E E E βρσ σ ν σ σν σ σ ε α ν δα −+ Δ + Δ Δ = ⋅ − − + + −      (D-12) In Equation D-12, the only unknown parameter is the maximum additional confining stress, 3maxσΔ . An iteration method is utilized to solve for this parameter. Since the thickness of the influence zone, δ , is a constant, the calculated maximum additional confining stress, 3maxσΔ , can be used to determine the distribution function of equivalent additional confining stress , ( )3 zσΔ , along the depth, z , of specimen. The determined equivalent additional confining stress distribution, ( )3 zσΔ , is then input into Equation D-13 to calculate the modified vertical modulus of the base course, ( )V ModifiedE z− , in the influence zone. ( ) ( ) 2 31 3 1 ( 1) k koct V Modified a a a I z E z k P P P σ τ −  + Δ = +   (D-13) where 1I is the first invariant of the stress tensor; octτ is the octahedral shear stress; aP is the atmospheric pressure; and 1k , 2k , and 3k are regression coefficients. The effective vertical modulus of the entire geosynthetic-reinforced UGM specimen, V EffectiveE − , is calculated using Equation D-14, which takes into account the variation of the location of the geosynthetic in the UGM specimen.

D-5 ( ) ( ) ( ) ( ) 0 2 0 2 0 2 2 2 2 2 2 V UGM V Modified l V UGM V Modified V Effective h l V UGM V Modified E h E z dz l h h E h l E z dz E l h E h l E z dz l h h δ δ δ δ δ δ δ δ δ δ − − + − − − + − − −  − +   < < −      − − +      = <      − − +      > −       (D-14) where V UGME − is the vertical modulus of the unreinforced base course, h is the thickness of the base course, and l is the distance between the geosynthetic layer and the bottom of the base course. The effective horizontal modulus of the geosynthetic-reinforced UGM specimen, H EffectiveE − , is calculated using Equation D-15. H Effective V EffectiveE n E− −= ⋅ (D-15) where n is the ratio of the horizontal modulus to the vertical modulus, which is determined from the repeated load test. Similarly, inputting the determined equivalent additional confining stress distribution, ( )3 zσΔ , into Equation D-9 can predict the permanent deformation of geosynthetic- reinforced UGMs at any given stress levels. Figure D-2 shows the comparison of the resilient moduli of geogrid-reinforced UGMs predicted by the proposed analytical models and those measured from the laboratory tests. The horizontal and vertical resilient moduli predicted by the analytical models match the measured values with R-squared values of 0.96 and 0.98, respectively. This indicates that the proposed analytical models can accurately predict both the horizontal and vertical moduli of the geogrid- reinforced UGMs.

D-6 (a) Predicted Horizontal Moduli vs. Measured Horizontal Moduli (b) Predicted Vertical Moduli vs. Measured Vertical Moduli Figure D-2. Comparison of Resilient Moduli Predicted by Analytical Models with Measured Values References 1. Lytton, R.L. (2015). Analytical Model for Quantifying Influence of Geosynthetics on Performance of Granular Material. Unpublished Work. Department of Civil Engineering, Texas A&M University, College Station, Texas. y = 1.0621x - 2.6616 R² = 0.9569 0 10 20 30 40 0 10 20 30 40 Pr ed ict ed H or izo nta l M od ulu s ( ks i) Measured Horizontal Modulus (ksi) y = 0.9589x - 0.9205 R² = 0.9766 0 30 60 90 0 30 60 90 Pr ed ict ed V ert ica l M od ulu s ( ks i) Measured Vertical Modulus (ksi)

D-7 2. Yang, X., and Han, J. (2013). Analytical Model for Resilient Modulus and Permanent Deformation of Geosynthetic-Reinforced Unbound Granular Material. Journal of Geotechnical and Geoenvironmental Engineering, Vol. 139, No. 9, pp. 1443–1453. 3. Schuettpelz, C., Fratta, D., and Edil, T.B. (2009). Evaluation of the Zone of Influence and Stiffness Improvement from Geogrid Reinforcement in Granular Materials. Transportation Research Record: Journal of the Transportation Research Board, No. 2116, pp. 76–84.