Characteriation of Anisotropic Aggregate Behavior Under Variable Confinement Conditions Erol Tutumluer 1, Member, Umit Seyhan 2, Student Member, And Navneet Garg 3, Member Abstract Compared to the standard constant confining pressure (CCP) tests, the variable confining pressure (VCP) type repeated load triaxial tests better simulate in the laboratory the effects of moving wheel loads since in the pavement structure the confining stress acting is cyclic in nature. In this paper, the important effects of VCP conditions in triaxial testing are considered in establishing anisotropic aggregate moduli. The anisotropic stiffnesses are next characteried using nonlinear models. Introduction Recent studies conducted at the University of Illinois have shown that unbound aggregate bases typically exhibit anisotropic behavior due to compaction and subsequent load induced stiffening in the vertical direction (Tutumluer and Thompson, 1997, Tutumluer, 1998). Anisotropic moduli were established from constant confining pressure (CCP) type repeated load triaxial test results with measured vertical and lateral specimen deformations. Both the important effects of load-induced directional stiffening and the nonlinear dilative behavior of granular materials under applied wheel loading were successfully modeled using an anisotropic approach (Tutumluer, 1997). This was a major advancement, compared to the isotropic assumption, towards correctly modeling of stress states and reducing/eliminating of the significant tensile stresses often predicted in unbound aggregate bases. To better characterie aggregate behavior, it is important to properly simulate in the laboratory the actual loading conditions. The pavement in the field is usually 1 Assistant Professor, 2 Graduate Research Assistant, 3 Post-Doctoral Research Associate, Department of Civil Engineering, University of Illinois, 205 N. Mathews, Urbana, IL 61801-2352. 1
loaded by moving wheel loads, which at any time impose varying magnitudes of vertical, horiontal, and shear stresses in the aggregate layer accompanied by the rotation of the principal stresses. This type of loading can not be ideally simulated in the laboratory by the CCP type repeated load triaxial tests, which have been commonly used in the U.S. since late 1960s and recognied as the standard procedure (AASHTO T294-94). The variable confining pressure (VCP) type repeated load triaxial tests, on the other hand, offer much wider loading possibilities by better simulating actual field conditions since in the pavement structure the confining stress acting on the material is cyclic in nature. The inherent differences between the CCP and VCP tests are such that in the VCP tests: (i) the confining pressure is also cycled in phase with the axial deviator stress and (ii) the axial specimen deformations are generally larger due to the lack of a constant all-around confinement on the specimen. This paper mainly focuses on the issues related to VCP testing of unbound aggregates. The important effects of VCP conditions in triaxial testing are considered in establishing anisotropic moduli and in modeling anisotropic stiffnesses. The VCP test data were also obtained from the same study that provided the CCP test results for the earlier anisotropic model development (Allen, 1973). The use of a consistent set of CCP and VCP test results is essential for studying the effects of variable confinement conditions and for comparing the predicted anisotropic behavior. Previous Work in VCP Testing Allen and Thompson (1974) investigated the effect of cyclic confining stress on the resilient modulus (M R ) and Poisson s ratio (ν) by performing a preliminary series of CCP and VCP tests on various granular materials (Allen, 1973). The peak values of confining stress from the VCP tests were used in the CCP tests. Accordingly, the CCP tests generally resulted in somewhat higher values for the two material properties than did the VCP tests. Brown and Hyde (1975) later showed that both the CCP and VCP tests yield the same M R -values provided that the confining stress used in the CCP tests is equal to the mean value of the confining stress used in the VCP tests. Nevertheless, the Poisson s ratios obtained from the CCP tests were still much higher than the ones from the VCP tests and were often in excess of 0.5 (dilation) especially at high stress ratios. Since the early work of Allen and Thompson (1974), there has been a complete lack of VCP type triaxial testing of granular materials in the U.S. In contrast, over the past two decades European researchers have pursued improved testing and characteriation of aggregates by the use of VCP tests. A number of nonlinear material models, such as the Boyce (1980) model, were proposed to describe the resilient behavior observed from VCP tests by separating stresses and strains into volumetric and shear components. In the proceedings of a recent EuroFlex symposium, Paute et al. (1996) summaried a set of recommendations for the proper use and testing procedures of VCP testing of granular materials on the basis of apparatuses that were used in different laboratories of four European countries. To characterie unbound aggregates for pavement purposes, VCP tests were indicated as suitable for 2
specimens having a diameter of about 150 mm (about 6 in.) which is suitable for aggregates with a maximum sie of 31.5 mm (1¼ in.). Whereas CCP tests were considered only as adequate for specimens having a diameter greater than 300 mm (approximately 6 in.) based on the fact that laboratory equipment needed for testing these large specimen sies had limited capabilities for applying VCP conditions. Anisotropic Moduli from CCP and VCP Tests A special type of anisotropy, known as cross-anisotropy, is commonly observed in pavement geomaterials due to stratification and applied wheel loading in the vertical direction. Resilient (elastic) pavement responses are primarily affected by this kind of directional dependency of stiffnesses. A total of five material properties are needed to define a cross-anisotropic material under conditions of axial symmetry as given by Zienkiewic and Taylor (1989): moduli in vertical and radial directions, M R and M R r ; shear modulus in vertical direction, G R ; Poisson s ratio for strain in the vertical direction due to a horiontal direct stress, ν ; and Poisson s ratio for strain in any horiontal direction due to a horiontal direct stress, ν r. Anisotropic resilient responses obtained from the CCP type repeated load triaxial tests due to the pulse deviator stress were previously defined from triaxial test data with measured axial and lateral deformations as follows (Tutumluer and Thompson, 1997): CCP: Vertical Resilient Modulus: MR = σd / ε1 (1) r CCP: Horiontal Resilient Modulus: MR = σ3 / ε3 (2) CCP & VCP: Resilient Shear Modulus: GR = σd / 2( ε1 ε3) (3) where M R r is defined under anisotropic elasticity as the horiontal resilient modulus resisting lateral specimen bulging, ε 1 and ε 3 are axial and radial resilient strains, and σ d (=σ 1 -σ 3 ) and σ 3 are the pulse deviator stress and constant confining pressure, respectively. Since for a cylindrical triaxial sample there is co-axiality between the material orientation and principal stress axes, the horiontal (or lateral) and vertical (axial) directions, as referred to in the above definitions, are used in the same context with the radial (r) and vertical () directions under axial symmetry. In the case of VCP tests, since confining stress (σ 3 = σ r ) is also pulsed simultaneously with the axial stress (σ 1 = σ ), it is necessary to use the axisymmetric stress-strain relations for the cross-anisotropic, linear elastic material (Zienkiewic and Taylor, 1989). Under the assumed condition of axial symmetry of the triaxial test, tangential stresses and strains are equal to radial stresses and strains. The following stress-strain relations can then be used to solve for the anisotropic resilient properties, M R and ν: 3
1 VCP: ε 1 = ( σ 2ν r ) M σ (4) R VCP: σr σ ε 3 = (1 ν r r ) ν (5) MR MR In using the above equations to obtain anisotropic resilient parameters as constants, the assumption is made that for any individual stress state, the material behaves linearly and elastically. Unlike CCP testing, the equations are not independent of the Poisson s ratios, ν r and ν, and the horiontal and shear moduli can only be obtained for the known values of Poisson s ratios. The shear modulus G R, however, can still be computed from Equation 3. Pickering (1970) studied the bounds of the elastic parameters in a crossanisotropic material. In addition to the requirement of each of the three moduli being greater than ero, the Poisson s ratios in horiontal and vertical directions were shown to be related to each other for a positive strain energy by (Pickering, 1970): r 2 1 < ν r <1 and MR 2ν > (6) M (1 νr ) R Figure 1 shows the variation of the horiontal to vertical modular ratio in the above equation with the positive values of Poisson s ratios, ν r and ν, ranging from 0 to 1. For the typical granular soils having greater horiontal compressibilities than vertical, no negative Poisson s ratios were noted in previous studies (Gaetas, 1982). Moreover, Tutumluer and Thompson (1997) reported modular ratios as low as 0.1 predicted in the granular base of a conventional pavement. For proper modeling of the anisotropic behavior, selection of lower values of Poisson s ratios is, therefore, recommended here to capture the considerably low modular ratios under the applied wheel loading and to ensure that the resulting strain energy is positive. Material Characteriation A new improved way of modeling granular materials using cross-anisotropic nonlinear elasticity was proposed recently at the University of Illinois to predict the dilative granular material behavior (Tutumluer and Thompson, 1997; Tutumluer, 1998). A comprehensive database of repeated load triaxial test results, which included vertical and lateral deformation measurements performed on a variety of aggregate types, was used to characterie the anisotropic moduli (horiontal, vertical, and shear) under axisymmetric conditions. Granular material response was shown to be reasonably described by using stress dependent models which express the moduli as nonlinear power functions of stress states. The characteriation models include two triaxial stress conditions, i.e., the bulk stress θ (= σ 1 +2σ 3 ) and the deviator stress σ d, and account for the effects of both 4
confinement and shear loading, respectively (Uan, 1992). The vertical and horiontal moduli are, therefore, modeled from the triaxial stress states using the equation: i M R 2i K = K 3i 1i( θ p K 0) ( σd p 0 ) (7) where i = or r for vertical or horiontal (radial) directions, respectively; K 1i to K 3i are model parameters, and p 0 = unit pressure (1 kpa or 1 psi). ν r 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.25 0.5 1.0 r M R / MR= 1.5 2.0 4.0 10.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ν Figure 1. Bounding Values of Poisson s Ratios for Positive Strain Energy Similarly, the nonlinear model for shear modulus can also be given by: GR K5 K = K 6 4( θ p0) ( σd p0) (8) where K 4 to K 6 are model parameters. Note that these model parameters are obtained from the multiple regression analyses of the triaxial test results. Alternatively, anisotropy of the stiffnesses expressed as vertical to horiontal and vertical to shear modular ratios can be obtained as functions of principal stress ratio σ 1 /σ 3 and confining stress σ 3 as follows: 1 B C B 1 C MR σ3 σ (1/A) 1 σ 2 1 = 1 r M p + (9) R 0 σ3 σ3 5
1 E-F E 1 F MR σ3 σ (1/D) 1 σ 2 1 = 1 G R p + (10) 0 σ3 σ3 where A = K 1r /K 1, B = K 2r -K 2, C = K 3r -K 3 +1, D = K 4 /K 1, E = K 5 -K 2, F = K 6 -K 3 +1. In the case of CCP tests, substituting M R = σd /ε 1 and M R r = -σ3 /ε 3 into the above stiffness ratio models, the principal strain ratio can be given by: CCP: B+ C 1 B C ε1 σ3 σ A 1 σ 2 1 = + 1 3 p ε 0 σ3 σ3 (11) where p 0 = unit pressure (1 kpa or 1 psi). The principal strain ratios derived from the above equation are therefore independent of the assumed Poisson s ratio values of ν and ν r. In the case of VCP tests, the principal strain ratios ε 1 /ε 3 can be computed using Equations 4 and 5 by substituting the vertical and horiontal moduli obtained from the anisotropic characteriation models given by Equation 7 with the known model parameters. It is important to note that the principal strain ratios computed this way will nonetheless depend upon the values of Poisson s ratios, which should consistently be taken as constants for the interpretation of VCP test results. Anisotropic Characteriation of CCP and VCP Tests The nonlinear anisotropic models summaried in the previous section were primarily based on laboratory tests conducted at the University of Illinois at Urbana- Champaign (Allen, 1973). A total of 9 individual repeated load triaxial tests were performed under both CCP and VCP conditions on gravels, partially crushed, and crushed aggregates at varying densities and saturation levels. In each test, increasing levels of deviator stresses were applied on 15-cm (6-in.) diameter by 30-cm (12-in.) height samples with confining pressures held constant at 14, 35, 55, 76, and 103 kpa (2, 5, 8, 11, and 15 psi). An average of 19 sets of vertical and lateral deformation measurements corresponding to various stress states was taken from each specimen. Typical variations of the vertical and horiontal moduli with strain levels are presented in Figures 2 and 3 for a high density, high quality partially saturated crushed stone (HD1) tested under both the CCP and VCP conditions by Allen (1973). The anisotropic moduli corresponding to the CCP tests were obtained using Equations 1 and 2 whereas the moduli for the VCP tests were computed from Equations 4 and 5 with the assumed Poisson s ratios ν r = 0.3 and ν = 0.1. Note that these constant Poisson s ratios, also used previously by Tutumluer and Thompson (1997) for predicting the anisotropic behavior of granular bases, were mainly selected for the purpose of providing consistent comparisons between the CCP and VCP test results. Allen (1973) recommended the use of a constant Poisson s ratio 6
based on his findings that the regression line for ν from the VCP tests was relatively flat over the range of σ 1 /σ 3 occurring in granular pavement layers and pavement response to load was least affected by changes in Poisson s ratio. Moreover, from the results of a recent sensitivity analysis by Tutumluer and Thompson (1997), a wide range and combination of admissible values considered for Poisson s ratios proved to have negligible effect on the computed stiffnesses. Vertical Resilient Modulus M R (MPa) 500 400 300 200 100 0 VCP CCP High Density Crushed Stone, HD1 VCP CCP σ 3 (kpa) 14 35 55 200 400 600 800 1000 1200 1400 1600 1800 Axial Strain ε 1 (µε) Figure 2. Variation of Vertical Moduli with Axial Strain for CCP and VCP Tests The vertical moduli typically increase with increasing axial strains in the CCP tests and for most of the confining stresses used in the VCP tests except for the high confinement levels at which the moduli stay almost constant (see Figure 2). Throughout the VCP tests, the effect of increasing deviator stresses (or principal stress ratios) on the measured vertical moduli is, therefore, not as dramatic as in the case of CCP tests. The horiontal moduli, on the other hand, tend to decrease with increasing radial strains for both the CCP and VCP tests (see Figure 3). The measured radial strains are, in general, smaller in the case of VCP tests due to the pulsing nature of confinement. The observed horiontal modulus reduction in the large strain one is in accordance with the dilation phenomenon often encountered in the granular bases under the wheel load (Uan et al., 1992 and Tutumluer, 1997). The experimental response of the HD1 granular material shown in Figures 2 and 3 was modeled by using the nonlinear anisotropic characteriation models. Table 1 summaries the model parameters for each modulus obtained from the multiple regression analysis of the individual test data. When compared to the VCP 76 103 7
case, the models developed from the CCP test results, in general, give a better statistical fit with high correlation coefficients (R 2 = 0.99, n = 19). Horiontal Resilient Modulus M R r (MPa) 600 500 400 300 200 100 0 High Density Crushed Stone, HD1 VCP CCP σ 3 (kpa) 14 35 55 VCP CCP 0 100 200 300 400 500 600 700 800 Radial Strain ε 3 (µε) 76 103 Figure 3. Variation of Horiontal Moduli with Axial Strain for CCP and VCP Tests Test Type Table 1. Anisotropic Stress Dependent Model Parameters for HD1 Vertical Modulus, M R Horiontal Modulus, M r R Shear Modulus, G R K 1 K 1r K 4 (MPa K / psi) 2 K 3 R 2 (MPa / psi) K R 2 (MPa / psi) R 2 2r K 3r K 5 K 6 40.8 / 287 3.416-2.808 143.8 0.99 0.99 / 754 0.834-0.167 0.99 CCP / 533.9 2,318 0.640-0.065 VCP / 353.3 7,137 0.539-0.143 0.90 48.0 / 678 2.545-2.077 137.8 0.87 / 2,508 0.264 0.153 Figure 4 shows variation of the vertical resilient moduli with increasing bulk stresses for the CCP and VCP tests. The VCP moduli are generally greater than the CCP values for bulk stresses up to about 400 kpa (57 psi), however, this trend changes after 400 kpa (57 psi) and the CCP tests consistently yield higher stiffnesses. This type of moduli variation at various confinement levels was successfully predicted by the nonlinear models (see Equation 7) using the model parameters presented in Table 1. While the CCP results represent a narrow, linearly increasing trend, the VCP moduli tend to be scattered and become almost unaffected 0.92 8
by the increasing stress state. Overall, the nonlinear model performed a much better curve fitting to the CCP data (see Table 1) and the predicted moduli followed more closely the experimental values. 450 Vertical Resilient Modulus, M R (MPa) 400 350 300 250 200 VCP CCP } From Equation 7 VCP CCP σ 3 (kpa) 14 150 35 100 55 High Density Crushed Stone, HD1 76 103 50 0 100 200 300 400 500 600 700 800 Bulk Stress θ (kpa) Figure 4. Variation of Vertical Moduli with Bulk Stress for CCP and VCP Tests Figure 5 shows variation of the principal strain ratios with increasing major to minor principal stress ratios for the CCP and VCP tests. Any value of strain ratio less than 2 corresponds to the case of resilient dilation. Overall, strain ratios predicted using the nonlinear anisotropic model match closely with the experimental ones. In the case of CCP tests, dilation begins to take place at a stress ratio of 3.5 and is adequately modeled by the nonlinear anisotropic model given by Equation 11. An almost constant degree of dilation was predicted for the CCP tests even at high principal stress ratios. Conversely, for the VCP tests, no dilative behavior is observed and the strain ratios stay completely on the compression side. Better strain ratio predictions are achieved at low to medium principal stress ratios for up to σ 1 /σ 3 = 6 after which the anisotropic model tends to slightly overestimate the strain ratios. The anisotropy of the stiffnesses, M R /M R r, in the cylindrical triaxial sample is next analyed for the CCP and VCP tests. Predictions obtained using the nonlinear anisotropic model (see Equation 9) for confining pressures σ 3 = 14, 55, and 103 kpa (3, 8, and 15 psi) are indicated in Figure 6 with the dashed and solid lines. For CCP tests, the anisotropic model is capable of matching all the experimental points, and also captures the trend of increasing degree of stiffness anisotropy with increasing principal stress ratios (Tutumluer, 1997). However, for VCP tests, the stiffness ratios are considerably higher than the CCP results especially at high stress ratios and 9
can not be adequately modeled using the nonlinear models. A distinct difference between the two tests is that the small dashed line representing the predictions for σ 3 = 14 kpa (3 psi) shifts from the bottom location in the CCP tests to the top location in the VCP tests in accordance with the shift observed in the experimental data. Overall, the trends in the anisotropic stiffness behavior can be characteried more accurately for the CCP tests using the nonlinear models than for the VCP tests. Strain Ratio, ε 1 / ε 3 14 12 10 8 6 4 2 Compression Dilation Experimental Anisotropic Model VCP CCP High Density Crushed Stone, HD1 0 0 1 2 3 4 5 6 7 8 9 10 Stress Ratio, σ 1 / σ 3 Figure 5. Variation of Strain Ratio with Stress Ratio for VCP and CCP Tests Conclusions 1. Compared to the standard constant confining pressure (CCP) tests, variable confining pressure (VCP) type repeated load triaxial tests are better suited to properly simulate in the laboratory the effects of moving wheel loads since in the pavement structure the confining stress acting on the material is variable in nature. 2. Directional dependency, i.e., anisotropy, of granular material stiffnesses due to applied wheel loading can be defined from both CCP and VCP type tests. The three resilient moduli, i.e., vertical, horiontal, and shear, were obtained for a high density crushed aggregate from the two tests with assumed constant Poisson s ratios. 3. Simple stress dependent models were developed to characterie the crossanisotropic granular material behavior from the consistent set of CCP and VCP test results with measured vertical and lateral deformations. 4. Dilative behavior was observed only from the CCP test results for the high density crushed aggregate. The trends in the anisotropic stiffness behavior could be characteried more accurately for the CCP tests using the nonlinear models than for the VCP tests. This is believed to be due to stress path independence of the CCP 10
tests. A volumetric-deviatoric type anisotropic model, such as the one proposed by Elhannani (1991), would be better suited for modeling of the stiffness anisotropy from VCP tests. Stiffness Anisotropy, M R / M R r 8 7 6 5 4 3 2 1 Anisot. Model σ 3 (kpa) 14 35 55 76 103 High Density Crushed Stone, HD1 VCP Experimental Experimental Anisot. Model σ 3 (kpa) 14 35 55 76 103 CCP 0 0 1 2 3 4 5 6 7 8 9 0 Stress Ratio, σ 1 / σ 3 1 2 3 4 5 6 7 8 9 Stress Ratio, σ 1 / σ 3 Figure 6. Variation of Stiffness Anisotropy with Stress Ratio for CCP and VCP Tests Acknowledgments / Disclaimer This paper was prepared from a study conducted in the Center of Excellence for Airport Pavement Research. Funding for the Center of Excellence is provided in part by the Federal Aviation Administration under Research Grant Number 95-C- 001. The Center of Excellence is maintained at the University of Illinois at Urbana- Champaign who works in partnership with Northwestern University and the Federal Aviation Administration. Ms. Patricia Watts is the FAA Program Manager for Air Transportation Centers of Excellence and Dr. Satish Agrawal is the FAA Technical Director for the Pavement Center. The contents of this paper reflect the views of the authors who are responsible for the facts and accuracy of the data presented within. The contents do not necessarily reflect the official views and policies of the Federal Aviation Administration. This paper does not constitute a standard, specification, or regulation. Appendix - References Allen, J.J. (1973). The Effects of Non-constant Lateral Pressures on the Resilient Response of Granular Materials. Ph.D. Thesis, Department of Civil Engineering, University of Illinois, Urbana, IL, May. 11
Allen, J.J. and Thompson, M.R. (1974). Resilient Response of Granular Materials Subjected to Time-Dependent Lateral Stresses. In Transportation Research Record 510, TRB, National Research Council, Washington D.C., pp. 1-13. AASHTO T294-94. Resilient Modulus of Unbound Granular Base/Subbase Materials and Subgrade Soils SHRP Protocol P46. Standard Specifications for Transportation Materials and Methods of Sampling and Testing, Seventeenth Edition, AASHTO, Washington D.C., 1995. Brown, S.F. and Hyde, A.F.L. (1975). Significance of Cyclic Confining Stress in Repeated- Load Triaxial Testing of Granular Material. In Transportation Research Record 537, TRB, National Research Council, Washington D.C., pp. 49-58. Elhannani, M. (1991). Modelisation et Simulation Numerique des Chaussees Souples. Ph.D. Thesis, University of Nantes, France. Gaetas, A.M. (1982). Stresses and Displacements in Cross-Anisotropic Soils. Journal of Geotechnical Engineering, ASCE, Vol. 108, No. GT4, pp. 532-553. Paute, J.-L., Dawson, A.R., and Galjaard, P.J. (1996) Recommendations for Repeated Load Triaxial Test Equipment and Procedure for Unbound Granular Materials. In Proceedings of the European Symposium Euroflex, Lisbon, Portugal, 1993, Flexible Pavements, Edited by A. Gomes Correia, A.A. Balkema Publishers, Rotterdam, The Netherlands, 1996, pp. 23-34. Pickering, D.J. (1970). Anisotropic Elastic Parameters for Soil. Geotechnique 20, No. 3, pp. 271-276. Tutumluer, E. and Thompson, M.R. (1997). Anisotropic Modeling of Granular Bases In Flexible Pavements. In Transportation Research Record 1577, TRB, National Research Council, Washington D.C., pp. 18-26. Tutumluer, E. (1997). Nonlinear Anisotropic Modeling of Dilative Granular In Proceedings of the Ninth International Conference on Computer Methods and Advances in Geomechanics, Wuhan, China, November 2-7, pp. 923-928. Tutumluer, E. (1998). "Anisotropic Behavior of Unbound Aggregate Bases State of the Art Summary. In Proceedings of the Sixth Annual Symposium of the International Center for Aggregate Research (ICAR), St. Louis, Missouri, April 19-21, pp. 11-33. Uan, J. (1992). Resilient Characteriation of Pavement Materials. International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 16, pp. 453-59. Zienkiewic, O.C. and Taylor, R.L. (1989). The Finite Element Method. Volume 1, Basic Formulation and Linear Problems, 4th Ed., McGraw Hill Book Co. (UK). 12