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PAGE 1 1 NONLINEAR EQUIVALENT LINEAR DESIGN METHOD TO DETERMINE UNBOUND AGGREGATE BASE LAYER MODULUS By ADITYA AYITHI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUI REMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011 PAGE 2 2 2011 Aditya Ayithi PAGE 3 3 To my parents PAGE 4 4 ACKNOWLEDGMENTS First of all, I would like to express my sincere gratitude to my advisor, Dr. Dennis R. Hiltunen whose valuabl e guidance, support and encouragement during my doctoral research made this possible. I have benefited tremendously from his experience and research methodology. He has provided me not only his knowledge but also emotional support as a great human being. I would like to thank my c oadvisor Dr. Reynaldo Roque, for his guidance and encouragement in my research and emotional support during my stay at University of Florida I would like to thank my other committee members Dr. Mang Tia and Dr. Bhavani V Sankar f or their valuable comments and suggestions. Special acknowledgements are extended to Florida Department of Transportation, project panel Dr. David Harahota, John shoucair and FDOT State Materials Office staff members Daniel Pitocchi, Dwayne Kirkland, Melis sa Barrs, Mike Davis, Gregg Sapp and Timothy Blanton for their technical support and encouragement. I would like to thank our lab engineer George A. Lopp whose technical support made my lab testing possible. I would like to thank my parents for encouragi ng my studies. I would like to thank my brother s sister in laws niece and nephews for their encouragement to achieve my goals I would like to thank my mentor and friend Dr. Raman G.V. for his encouragement and support in pursuing doctoral studies. I woul d like to thank my college friends Ravi Chandra (banda) Madhukar, Raghu, Ravi Tanniru, Hari (jandu) and CSS for their support Last but not least, I would like to thank my graduate friends CS Koh, Weiato Li, Tran and Patrick for their help and support. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...............................................................................................................4 LIST OF TABLES ...........................................................................................................................9 LIST OF FIGURES .......................................................................................................................12 ABSTRACT ...................................................................................................................................29 CHAPTER 1 INTRODUCTION ..................................................................................................................31 1.1 Problem Statement .......................................................................................................31 1.2 Hypothesis ....................................................................................................................33 1.3 Objectives ....................................................................................................................34 1.4 Scope of Research ........................................................................................................34 1.5 Organization of Dissertation ........................................................................................36 2 LITERATURE REVIEW .......................................................................................................39 2.1 Modulus of Particulat e Material ..................................................................................39 2.2 Resilient Modulus ........................................................................................................40 2.3 Modulus Characterization for Complete Range of Strains ..........................................41 2.3.1 SmallStrain Modulus (Gmax or Emax) ...............................................................42 2.3.2 Shear Modulus (G) at Different Strain Levels: ................................................44 2.3.3 Analytical Models to Estimate Shear Modulus at Different Strain Levels ......46 2.4 Influence of Suction on Soil Modulus .........................................................................47 2.5 Importance of Small Strain Modulus Nonlinearity for Design and Analysis ..............51 2.6 Pavement Response Models ........................................................................................52 2.6.1 MEPDG Nonlinear Response Model ...............................................................52 2.6.2 Single Effective Moduli Determination Based on Nonlinear Versus Linear Analysis Responses Comparison ...........................................................................53 2.6.3 PLAXIS HSsmall Model ........................................................................................54 2.7 Closing Remarks ..........................................................................................................55 3 EXPERIMENTS .....................................................................................................................68 3.1 Fixed Free Resonant Column Testing .........................................................................68 3.1.1 Background and Testing Mechanism ...............................................................68 3.1.2 Shear Modulus (G) ...........................................................................................69 3.1.3 .................................................................................................70 3.1.4 Equipment Setup...............................................................................................71 3.1.5 Calibrat ion of the Drive System: ......................................................................71 3.1.6 Limitations (We Experienced) of This Testing Equipment: ............................73 3.1.7 Equipment Credibility Verific ation ..................................................................73 PAGE 6 6 3.2 Free Free Resonant Column Testing (Free Free RC) ..................................................74 3.2.1 Back Ground and Mechanism ..........................................................................74 3.2.2 Equipment Setup: .............................................................................................76 4 MATERI A LS AND SPECIMEN PREPA RATION ..............................................................85 4.1 Materials ......................................................................................................................85 4.1.1 Sources ..............................................................................................................85 4.1.2 Characterization ................................................................................................85 4.2 Specimen Preparation and I nstallation .........................................................................86 4.2.1. Dry Specimens ..................................................................................................86 4.2.1.1 Dry specimen preparation ..........................................................................86 4.2.1.2 Dry specimen installation ..........................................................................87 4.2.2 Wet (Partially Saturated) Specimens ................................................................87 4.2.2.1 Wet specimen preparation .........................................................................87 4.2.2.2 Wet specimen installation .........................................................................88 5 EXPERIMENTAL RESULTS AND ANALYSIS .................................................................93 5.1 Fixed Free Resonant Column Torsional Shear Testing ...............................................93 5.1.1 Dry Specimen Testing Results .........................................................................93 5.1.2 Unsaturated (Wet) Specimen Testing Results ..................................................96 5.1.2.1 Equipment limitations ................................................................................97 5.1.2.2 Confinement effect on unsaturated specimens ..........................................98 5.1.2.3 Results and analysis ...................................................................................98 5.1.2.4 Additional effective stress or additional confinement pressure provided due to suction, at 105% strain magnitude .................................102 5.2 Free Free Resonant Column Testing .........................................................................103 5.2.1 Results and Discussion ...................................................................................103 5.2.2 A Method to Estimate Approximate Modulus at Given Conditions of Water Content, Confining Pressure and Strain Magnitude: ...........................104 5.3 Laboratory Testing R esults Closing Remarks ...........................................................105 6 NONLINEAR FINITE ELEMENT MODELING OF BASE LAYER ................................118 6.1 PLAXIS: Hardening Soil Small Model (Plaxis HSsmall Model) .............................118 6.1.1 Parameters of HSsmall Model ........................................................................119 6.1.2 Compatibility of Plaxis HSsmall Model Modulus Reduction Model to Our Laboratory Determined Base Soils Modulus Reduction Behavior .................120 6.2 Flexible Pavement Nonlinear Response Model .........................................................121 6.2.1 Ax isymmetric Model ......................................................................................121 6.2.2 Pavement Cross Sections ................................................................................122 6.2.3 Loading Conditions ........................................................................................122 6.2.4 Input Parameters for Surface and Subgrade Layers of Flexible Pavement ....122 6.3 Initial Plaxis HSsmall Pavement Model Runs and Recalibration .............................122 6.3.1 Footing Model Analysis and Verification ......................................................123 6.3.2 Model Recalibration .......................................................................................123 PAGE 7 7 6.4 Recalibrated Nonlinear Input Parameters of Base Soils for Plaxis ............................124 6.4.1 Recalibrated Parameters of Newberry Limerock and Georgia Granite ..........124 6.4.2 Input Parameters for Miami Limerock ...........................................................125 6.5 Demonstration of Response Model Nonlinear Behavior ...........................................125 6.5.1 Demo nstration of Input Parameters Nonlinearity .........................................126 6.5.2 Demonstration of Pavement Models Nonlinearity ........................................126 7 BASE LAYER NONLINEAR MODELING ANALYSIS AND RESULTS ......................139 7.1 Nonlinear Equivalent Single Effective Base Modulus Determination ......................139 7.1.1 Methodology ...................................................................................................139 7.1.2 An Example of Surface Deflection Basin Matching Between Nonlinear Modulus Base Case and Single Effective Elastic Modulus Base Case ..........140 7.1.3 Demonstration of Nonlinear Reduction of Single Effective Modulus with Increase in Load ..............................................................................................141 7.2 Single Effective Moduli Data for Base Layer ............................................................142 7.2.1 Nonlinear Equivalent Single Effective Design Moduli Values for Base Layer ..142 7.2.1.1 Influence of moisture content on base layer single effective de sign modulus ....................................................................................................142 7.2.1.2 Influence of subgrade modulus on single effective base design modulus ....................................................................................................143 7.2.1.3 Influence of structur e type on single effective base design modulus ......144 7.2.1.4 Comparison of single effective design moduli for different materials ....144 7.2.1.5 Comparison of equivalent single effective design moduli with mepdg moisture effect model design moduli .......................................................146 7.3 Evaluation of Applicability of Single Effective Modulus in place of Nonlinear Modulus .....................................................................................................................147 7.3.1 Comparison of Nonlinear and Equivalent Linear Analysis Responses ..........147 7.3.2 Analysis of Nonlinear an d Linear Responses in the Perspective of Rutting Performance Criteria .......................................................................................151 8 SUBGRADE LAYER NONLINEAR MODELING ANALYSIS AND RESULTS ...........174 8.1 Nonlinear Equivalent Single Effective Modulus Determination for Base and Subgrade Layers .........................................................................................................175 8.1.1 Methodology ...................................................................................................175 8.1.2 Material Parameters and Structural Inputs .....................................................176 8.1.3 An Example of Surface Deflection Basin Matching Between Nonlinear Base and Subgrade Case and Equivalent Linear Base and Subgrade Case ....176 8.2 Comparison of Nonlinear and Equivalent Linear Analysis Results ..........................177 8.2.1 Nonlinear Equivalent Single Effective Design M oduli Values for Base and Subgrade ..................................................................................................177 8.2.2 Comparison of Nonlinear and Equivalent Linear Analyses Pavement Responses .......................................................................................................178 8.2.3 Analysis of Nonlinear and Linear responses in the Perspective of Cracking and Rutting Performance Criteria ...................................................180 PAGE 8 8 9 CLOSURE ............................................................................................................................190 9.1 Summary of Findings .................................................................................................190 9.1.1 Laboratory Testing on Unbound Aggregate Base Soils .................................191 9.1.2 Development of Nonlinear Respons e Model and Base Layer Nonlinear Modeling and Analysis ...................................................................................193 9.1.3 Nonlinear Modeling and Analysis of Base and Subgrade Layers ..................196 9.2 Conclusions ................................................................................................................198 9.3 Recommendations ......................................................................................................199 APPENDIX A FIXED FREE RESONANT COLUMN TESTING DATA FOR DIFFERENT BASE SOILS ...................................................................................................................................201 B NONLINEAR EQUIVALENT LINEAR EFFECTIVE BASE MODULI DATA FOR DIFFERENT TYPES OF BASE SOILS ..............................................................................205 C COMPARISON OF NONLINEAR AND EQUIVA LENT LINEAR RESPONSES OBTAINED FROM NONLINEAR BASE ANALYSIS .....................................................212 D COMPARISON OF NONLINEAR AND EQUIVALENT LINEAR RESPONSES OBTAINED F0R NONLINEAR BASE AND NONLINEAR SUBGRADE ANALYSIS .280 LIST OF REFERENCES .............................................................................................................308 BIOGRAPHICAL SKETCH .......................................................................................................310 PAGE 9 9 LIST OF TABLES Table page 41 List of materials used for testing and their sources ............................................................89 42 Basic material parameters ..................................................................................................89 43 Selected void ratios for dry specimen testing ....................................................................89 51 nG and AG values for dry Newberry lime rock and Georgia granite. ...............................107 52 List of unsaturated specimen tested water contents .........................................................107 61 Input parameters of Plaxis HSsmall model .....................................................................127 62 Differe nt types of pavement structures considered for analysis ......................................127 63 Material input parameters for surface asphalt concrete and subgrade layers ..................128 64 Input parameters used in HSsmall model for footing settlement predictions (Lehane et al. 2008) .......................................................................................................................128 65 Comparison of Actual and Modified calibration parameters ...........................................129 66 Recalibrated HSsmall input parameters for Newberry limerock .....................................129 67 Recalibrated HSsmall input parameters for Georgia granite ...........................................130 68 Recalibrated HSsmall input parameters for Miami limerock ..........................................130 71 Equivalent elastic moduli values obtained for different paveme nt structures, with different subgrade moduli for Newberry limerock base ..................................................154 72 Effective equivalent elastic moduli values obtained for different pavement structures, with different subgrade moduli for Georgia granite base ................................................155 73 Effective equivalent elastic moduli values obtained for different pavement structures and with different subgrade moduli for Miami limerock base ........................................156 74 List of structures analyzed for pavements various responses comparison .....................156 75 Asphalt Institutes rutting criteria analysis f or structure 1 (200mm AC surface, 450 mm base) ..........................................................................................................................157 76 Asphalt Institutes rutting criteria analysis for structure 4 (100 mm AC surface, 300 mm base) ..........................................................................................................................158 81 Different material properties used for nonlinear base and nonlinear subgrade analysis .182 PAGE 10 10 82 HS Small model nonlinear material inputs for Ottawa sand subgrade la yer ...................182 83 Equivalent elastic moduli values obtained for structure 1 with Newberry limerock base and Ottawa sand subgrade .......................................................................................183 84 Equivalent elastic moduli values obtained for structure 4 with Newberry limerock base and Ottawa sand subgrade .......................................................................................183 85 Equivalent elastic moduli values obtained for structure 1 with Georgia granite base and Ottawa sand subgrade ...............................................................................................184 86 Equivalent elastic moduli values obtained for structure 4 with Georgia Granite base and Ottawa sand subgrade ...............................................................................................184 87 List of structures analyzed for pavements various responses comparison .....................184 88 Asphalt Institutes rutting criteria analysis for structure 1 with nonlinear base and nonlinear subgrade ...........................................................................................................185 B 1 For structure 1, nonlinear equivalent linear effective moduli data for Newberry limerock base layer. .........................................................................................................205 B 2 For structure 2, nonlinear equivalent linear effective moduli data for Newberry limerock base layer. .........................................................................................................206 B 3 For structure 3, nonlinear equivalent linear effective moduli dat a for Newberry limerock base layer. .........................................................................................................206 B 4 For structure 4, nonlinear equivalent linear effective moduli data for Newberry limerock base layer. .........................................................................................................207 B 5 For structure 5, nonlinear equivalent linear effective moduli data for Newberry limerock base layer. .........................................................................................................207 B 6 For structure 6, nonlinear equivalent linear effective moduli data for Newberry limerock base layer. .........................................................................................................208 B 7 For structure 1, nonlinear equivalent linear effective moduli data for Georgia granite base layer. ........................................................................................................................209 B 8 For structure 2, nonlinear equivalent linear effective moduli data for Georgia granite base layer. ........................................................................................................................209 B 9 For structure 3, nonlinear equivalent linear effective moduli data for Georgia granite base layer. ........................................................................................................................210 B 10 For structure 4, nonlinear equivalent linear effective moduli data for Georgia granite base layer. ........................................................................................................................210 PAGE 11 11 B 11 For structure 1, nonlinear equivalent linear effective moduli data for Maimi limerock base layer. ........................................................................................................................211 B 12 For structure 4, nonlinear equivalent linear effecti ve moduli data for Miami limerock base layer. ........................................................................................................................211 PAGE 12 12 LIST OF FIGURES Figure page 21 Characteristic modulus strain behavior of soils with typical str ain ranges for laboratory tests and structures ..........................................................................................56 22 G/Gmax ...........................................................56 23 G/Gmax versu cc=0.59 ............................................................................................57 24 max = 1.27 cm) .......................................................58 25 Variation of normalized shear modulus with shear strain .................................................58 26 G ..................................................................................................................59 27 Variation of Gmax 3 ....................................................................59 28 G/Gmax ....................................................................60 29 Comparison of c alculated and measured values of normalized shear modulus .................60 210 Data points defining G/Gmax ................................61 211 max containing 0, 20, 40 and 60% gravel size particles ............................................................62 212 Mean curves defining G/Gmax ationships for gravelly soils at various confining pressures along with standard deviation boundaries for reduced data set .........63 213 Shear modulus degradation curve proposed by Santos and Corre ia ..................................63 214 Shear wave velocity versus degree of satu ration for different materials ........................64 215 FFRC test results of specimen exp osed to constant moisture ............................................65 216 The FFRC test results of specimens exposed to laboratory ambient .................................65 217 The FFRC test resul ts of each material during first cycle of oven drying .........................66 218 The FFRC test results of each material during first cycle of wetting ................................66 219 Load displacement curves at Bothkennar footing test, fitted using the proposed model ..................................................................................................................................67 220 Comparison of measured foundation load settlement response at Shenton Par k to predict ion using Plaxis HSs mall model .............................................................................67 PAGE 13 13 31 Fixed Free RC testing specimen i dealized testing specimen and Differential soil element ...............................................................................................................................77 32 Shear Strain in soil specimen .............................................................................................77 33 Fixed Free RC equipment used for our testing ..................................................................78 34 Sectional view of Fixed Free R C testing equipment .........................................................79 35 Controlling units of FixedFree RC equipment .................................................................80 36 Real time test execution window, showing resona nt frequency reaching loop .................81 37 Calibration specimen with added mass ..............................................................................82 38 Shear modulus versus % shear strain curves at di fferent confining pressures for Ottawa sand. .......................................................................................................................83 39 Gmax c relationship curve for Ottawa sand. .....................................................83 310 Free Free R esonant Column test equipment setup ............................................................84 311 Free Free resonant column testing specimen .....................................................................84 41 Representative samples ......................................................................................................90 42 Grain size distribution ........................................................................................................90 43 Dry specim en compaction and installation .......................................................................91 44 Wet specimen compaction .................................................................................................92 51 eOMC ..................................................................................................................................108 52 OMC ...108 53 G/Gmax OMC for Newberry limerock ..................109 54 G/Gmax OMC for Georgia granite ........................109 55 Comparison of G/Gmax f Newberry limerock and Georgia granite with Seed and Idriss maximum and minimum limits ..........................................110 56 Gmax versus confining pressure curve Newberry limerock at different void ratios .........110 57 Gmax versus confining pressure curve Georgia granite at different void ratios ................111 58 Gmax empirical equations for Newberry limerock and Georgia granite ...........................111 PAGE 14 14 59 Shear moduli (G) versus % shear st water contents ..................................................................................................................112 510 water contents ..................................................................................................................112 511 G/Gmax versus shear strain curves for Newberry limerock for both dry & wet specimens .........................................................................................................................113 512 G/Gmax versus % shear strain curves for G eorgia granite, for both dry and wet specimens .........................................................................................................................113 513 G/GOMC versus water content (%) at different strain levels for Newberry limerock .......114 514 G/GOMC versus water content (%) at different strain levels for Georgia granite .............114 515 Additional confinement provided due to suction at 105% strain magnitude, at different water contents ....................................................................................................115 516 Additional confinement provided due to suction at 105% strain magnitude at different degree of saturation ...........................................................................................115 517 Youngs Modulus versus % water content results obtained from Free Free RC test on Newberry limerock ..........................................................................................................116 518 Youngs Modulus versus % water content results obtained from Fre e Free RC test on Georgia Granite ................................................................................................................116 519 Comparison of Newberry limerock Free Free RC test data with Fixed Free RC very smallstrain data and T oros (2008) data ...........................................................................117 520 Comparison of Georgia granite Free Free RC test data with Fixed Free RC very smallstrain data and T oros (2008) data ...........................................................................117 61 Characteristic modulus strain behavior of soil with typical strain ranges for laboratory tests and structures. .........................................................................................131 62 Results from modified Hardin Drnevich relationship compared to test data by Santos and Corre ia .......................................................................................................................131 63 Comparison of Newbery limerock and Georgia granite actual testing data with HSsmall models Santos and Correia modified hyperbolic curve ..................................132 64 Typical Plaxis HSsmall model cross section model used for nonlinear analysis. ...........133 65 Typical HSsmall model finite element mesh of a pavement cross section ......................134 66 Cross sections considered for nonlinear base pavement analysis ....................................135 PAGE 15 15 67 Finite element mesh used for model recalibration an alysis .............................................136 68 Comparison of actual settlements with settlements predicted using regular parameters via Plaxis HSsmall footing model. ................................................................136 69 Comparison of actual settlements with settlements predicted using regular parameters and calibrated settlements via Plaxis HSsmall footing model. .....................137 610 Approximate Miami limerock Gmax versus water content plot. .......................................137 611 Load versus settlement curve to demonstrate material input parameters nonlinearity. ...138 612 Pavement surface deflection basin for different load demonstrating pavement model nonlinearity. .....................................................................................................................138 71 Critical locations for pavement response analysis ...........................................................159 72 Surface deflection basin comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer. ................................................................................................159 73 Nonlinear variation of single effective modulus with increase in load, for Newberry limerock base at 13% and 10% moisture contents. ..........................................................160 74 Nonlinear variation of effective modulus with increase in load, for Georg ia granite base at 5.5% and 3.5% moisture contents. .......................................................................160 75 Newberry li m e rock effective base modulus versus subgrade elastic modulus relationship with decrease in moisture content, for differ ent structures. .........................161 76 Ge or gia granite effective base modulus versus subgrade elastic modulus relationship with decrease in moisture content, for different structures ..............................................162 77 Miami lime rock effective base modulus versus subgrade elastic modulus relationship with decrease in moisture content, for different structures ..............................................163 78 Newberry lim e rock effective base modulus versus subgrade elastic modulus relationship for different struct ures, at constant moisture content. ..................................164 79 Georgia granite effective base modulus versus subgrade elastic modulus relationship for different structres, at constant moisture content .........................................................165 710 Miami limerock effective base modulus versus subgrade elastic modulus relat ionship for different struct ures, at constant moisture content.......................................................166 711 Compar i son of effe c tive moduli of all three materials at different base water contents and with subgrade modulus of 50 MPa and 125 MPa .....................................................167 712 Compar i son of normalized effe c tive moduli of all three materials obtained with subgrade modulus of 50 MPa and 125 MPa ....................................................................168 PAGE 16 16 713 Comparison of Newberry limerock effective base moduli va l ues with MEPDG mositure model .................................................................................................................169 714 Comparison of Georgia granite effective base moduli vlaues with MEPD G modulus mositure model .................................................................................................................170 715 Comparison of Miami limerock effective base moduli vlaues with MEPDG mositure model ................................................................................................................................171 716 Compa xx) obtaine d for structure 1 and str ucture 4 wit h different base water contents ........................................172 717 Com pa rison of vertical compressive s yy) obta ine d for structure 1 and str u cture4 wit h different base water contents ..................................173 81 FWD analysis of structure 4 ............................................................................................186 82 Comparison of nonlinear and linear surface deflections for structure 1 with 10% w.c. base and Ottawa sand subgrade. ......................................................................................187 83 Com pa rison of horizontal tensile st xx) obtaine d for structure 1 with Ottawa sand subgrade and different base water contents ......................188 84 Compa rison of vertical compressive strain at top of subgrade lay yy) obtained for structure 1 with Ottawa sand subgrade and different base water contents ................189 A 1 e=0.5 ................................................................................................................................201 A 2 e=0.55 ..............................................................................................................................201 A 3 G/Gmax versus % shear str .......................202 A 4 G/Gmax ....................202 A 5 e=0.25 ..............................................................................................................................203 A 6 e=0.29 ..............................................................................................................................203 A 7 G/Gmax ............................204 A 8 G/Gmax 9 for Georgia granite ............................204 C 1 Surface deflection comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer. ........................................................................................................212 PAGE 17 17 C 2 Surface deflection comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer. ........................................................................................................213 C 3 Surface deflection comparison for nonlinear and linear cases, f or structure 1 with 8% w.c. base layer. .................................................................................................................213 C 4 Surface deflection comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer ........................................................................................................214 C 5 Surface deflection comparison for nonlinear and linear cases, for structure 4 with 13% w.c. base layer. ........................................................................................................214 C 6 Surface deflection comparison for nonlinear a nd linear cases, for structure 4 with 10% w .c. base layer. ........................................................................................................215 C 7 Surface deflection comparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer. .................................................................................................................215 C 8 Surface deflection comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. .......................................................................................................216 C 9 xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer. ........................................................................................................217 C 10 xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer. ........................................................................................................217 C 11 xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer. ..........................................................................................................218 C 12 xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. .......................................................................................................218 C 13 xx at top of AC layer comparison f or nonlinear and linear cases, for structure 4 with 13% w.c. base layer. ........................................................................................................219 C 14 xx at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer. ........................................................................................................219 C 15 xx at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer. ..........................................................................................................220 C 16 x x at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. .......................................................................................................220 C 17 xx at top of AC layer comparison for nonlinear and linear cases, for struc ture 1 with 13% w.c. base layer. ........................................................................................................221 PAGE 18 18 C 18 xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer. ........................................................................................................221 C 19 xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer. ..........................................................................................................222 C 20 xx at top of AC layer comparison for nonl inear and linear cases, for structure1 with 5.5% w.c. base layer. .......................................................................................................222 C 21 xx at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 13% w.c. base layer. ........................................................................................................223 C 22 xx at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer. ........................................................................................................223 C 23 xx at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer ...........................................................................................................224 C 24 xx at top of AC layer comparison for nonlinear and linear cases, for structure 4 w ith 5.5% w.c. base layer. .......................................................................................................224 C 25 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer. ................................................................................................225 C 26 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer. ................................................................................................225 C 27 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer. ..................................................................................................226 C 28 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. ...............................................................................................226 C 29 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 13% w.c. base layer. ................................................................................................227 C 30 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer. ................................................................................................227 C 31 xx at bottom of AC layer comparison for nonlinear and linear cases, f or structure 4 with 8% w.c. base layer. ..................................................................................................228 C 32 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. ...............................................................................................228 C 33 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer. ................................................................................................229 PAGE 19 19 C 34 xx at bottom of AC layer c omparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer. ................................................................................................229 C 35 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer. ..................................................................................................230 C 36 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. ...............................................................................................230 C 37 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 13% w.c. base layer. ................................................................................................231 C 38 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure4 with 10% w.c. base layer. ................................................................................................231 C 39 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer. ..................................................................................................232 C 40 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. ...............................................................................................232 C 41 yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer. ..............................................................................233 C 42 yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer. ..............................................................................233 C 43 yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer .................................................................................234 C 44 yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. .............................................................................234 C 45 yy at top of base layer comparison for nonlinear and linear cases, for structure 4 with 13% w.c. base layer .................................................................................................235 C 46 yy vertical stress at top of base layer comparison fo r nonlinear and linear cases, for structure 4 with 10% w.c. base layer. ..............................................................................235 C 47 yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer. ................................................................................236 C 48 yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. .............................................................................236 C 49 yy at top of base layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer. ........................................................................................................237 PAGE 20 20 C 50 yy at top of base layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer. ........................................................................................................237 C 51 yy at top of base layer comparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer. ..........................................................................................................238 C 52 yy at top of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. .......................................................................................................238 C 53 yy at top of base layer comparison for nonlinear and linear cases, for structure 4 with 13% w.c. base layer. ........................................................................................................239 C 54 yy at top of base layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer. ........................................................................................................239 C 55 yy at top of base layer comparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer. ..........................................................................................................240 C 56 yy at top of base layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. .......................................................................................................240 C 57 yy at bottom of base layer co mparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer. ................................................................................................241 C 58 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c base layer. ................................................................................................241 C 59 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer. ..................................................................................................242 C 60 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. ...............................................................................................242 C 61 yy at bottom of base layer comparison for nonline ar and linear cases, for structure4 with 13% w.c. base layer. ................................................................................................243 C 62 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer. ................................................................................................244 C 63 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer. ..................................................................................................244 C 64 y y at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. ...............................................................................................244 C 65 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer. ................................................................................................245 PAGE 21 21 C 66 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer .................................................................................................246 C 67 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer. ..................................................................................................246 C 68 yy at bottom of base l ayer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. ...............................................................................................247 C 69 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 13% w.c. base layer. ................................................................................................247 C 70 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer. ................................................................................................248 C 71 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer. ..................................................................................................248 C 72 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer ................................................................................................248 C 73 yy at top of subgrade comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer. ........................................................................................................249 C 74 yy at top of subgrade comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer .........................................................................................................250 C 75 yy a t top of subgrade comparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer. ..........................................................................................................250 C 76 yy at top of subgrade comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. .......................................................................................................251 C 77 yy at top of subgrade comparison for nonlinear and linear cases, for structure 4 with 13% w.c. base layer. ........................................................................................................251 C 78 yy at top of subgrade comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer. ........................................................................................................252 C 79 yy at top of subgrade comparison for nonline ar and linear cases, for structure4 with 8% w.c. base layer. ..........................................................................................................252 C 80 yy at top of subgrade comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. .......................................................................................................252 C 81 yy at top of subgrade layer comparison for nonlinear and linear cases for structure 1 with 13% w.c. base layer. ................................................................................................253 PAGE 22 22 C 82 yy at top of subgrade layer comparison for nonlinear and linear cases for structure 1 with 10% w.c. base layer. ................................................................................................254 C 83 yy at top of subgrade layer comparison for nonlinear and linear cases for structure 1 with 8% w.c. base layer. ..................................................................................................254 C 84 yy at top of subgrade layer comparison for nonlinear and linear cases for structure 1 with 5.5% w.c. base layer. ...............................................................................................255 C 85 yy at top of subgrade layer comparison for nonlinear and linear cases for structure 4 with 13% w.c. base layer. ................................................................................................255 C 86 yy at top of subgrade lay er comparison for nonlinear and linear cases for structure 4 with 10% w.c. base layer. ................................................................................................256 C 87 yy at top of subgrade layer comparison for nonlinear and linear cases for structure 4 with 8% w.c. base layer. ..................................................................................................256 C 88 yy at top of subgrade layer comparison for nonlinear and linear cases for structure 4 with 5.5% w.c. base layer. ...............................................................................................257 C 89 Surface deflection comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. .......................................................................................................258 C 90 Surface deflection comparison for nonlinear and linear cases, for structure1 with 3.5% w.c. base layer. .......................................................................................................258 C 91 Surface deflection comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. .......................................................................................................259 C 92 Surface deflection comparison for nonlinear and linear cases, for structure 4 with 3.5% w.c. base layer. .......................................................................................................259 C 93 xx at top of AC layer compar ison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. .......................................................................................................260 C 94 xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer ........................................................................................................260 C 95 xx at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. .......................................................................................................261 C 96 xx at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 3.5% w.c. base layer. .......................................................................................................261 C 97 xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. .......................................................................................................262 PAGE 23 23 C 98 xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer. .......................................................................................................262 C 99 xx at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. .......................................................................................................263 C 100 xx at top of AC layer compa rison for nonlinear and linear cases, for structure 4 with 3.5% w.c. base layer. .......................................................................................................263 C xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. ...............................................................................................264 C xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer. ...............................................................................................264 C xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer ................................................................................................265 C xx at bottom of AC layer comparison for nonlinear a nd linear cases, for structure 4 with 3.5% w.c. base layer. ...............................................................................................265 C xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. ...............................................................................................266 C xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer. ...............................................................................................266 C xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. ...............................................................................................267 C xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 3.5% w.c. base layer. ...............................................................................................267 C yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. .............................................................................268 C yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer. .............................................................................268 C yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. .............................................................................269 C yy vertical stress at top of base layer compar ison for nonlinear and linear cases, for structure 4 with 3.5% w.c. base layer. .............................................................................269 C yy at top of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. ba se layer. .......................................................................................................270 PAGE 24 24 C yy at top of base layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer. .......................................................................................................270 C yy at top of base layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. .......................................................................................................271 C yy at top of base layer comparison for nonlinear and l inear cases, for structure 4 with 3.5% w.c. base layer. .......................................................................................................271 C yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. ...............................................................................................272 C yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer ................................................................................................272 C yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. ...............................................................................................273 C yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 3.5% w.c. base layer. ...............................................................................................273 C yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. ...............................................................................................274 C yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer. ...............................................................................................274 C yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. ...............................................................................................275 C yy at bottom of base layer comparison for nonlinear and linear cases, for structur e 4 with 3.5% w.c. base layer. ...............................................................................................275 C yy at top of subgrade comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. .......................................................................................................276 C yy at top of subgrade layer comparison for nonlinear and linear cases for structure 1 with 3.5% w.c. base layer. ...............................................................................................276 C yy at top of subgrade layer comp arison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. ...............................................................................................277 C yy at top of subgrade layer comparison for nonlinear and linear cases for structure 4 ith 3.5% w.c base layer. ..................................................................................................277 C yy at top of subgrade layer comparison for nonlinear and linear cases for structure 1 with 5.5% w.c. base layer. ...............................................................................................278 PAGE 25 25 C yy at top of subgrade layer comparison for nonlinear and linear cases for structure 1 with 3.5% w.c. base layer. ...............................................................................................278 C yy at top of subgrade layer comparison for nonlinear and linear cases for structure 4 with 5.5% w.c. base layer. ...............................................................................................279 C yy at top of subgrade layer comparison for nonlinear and linear cases for structure 4 with 3.5% w.c. base layer ...............................................................................................279 D 1 S urface deflection comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade ..............................................280 D 2 Surface deflection comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. .............................................281 D 3 Surface deflection comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. .............................................281 D 4 xx at top of AC layer comparison for nonlinear and linear cases, for structure1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade. .............................................282 D 5 xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. ba se layer and nonlinear Ottawa sand subgrade. .............................................282 D 6 xx at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer and nonlinear Ottawa sand subgr ade. .............................................283 D 7 xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade. .............................................283 D 8 xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. .............................................284 D 9 x x at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. .............................................284 D 10 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade. .....................................285 D 11 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure1 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. .....................................285 D 12 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. .....................................286 D 13 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade. .....................................286 PAGE 26 26 D 14 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. .....................................287 D 15 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. .....................................287 D 16 yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade. ...................288 D 17 yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. ...................288 D 18 yy vertical stress at top of base layer co mparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. ...................289 D 19 yy at top of base layer comparison for nonlinear and linear cas es, for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade. .............................................289 D 20 yy at top of base layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. bas e layer and nonlinear Ottawa sand subgrade. .............................................290 D 21 yy at top of base layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. .............................................290 D 22 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade. .....................................291 D 23 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. .....................................291 D 24 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. .....................................292 D 25 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade. .....................................292 D 26 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. .....................................293 D 27 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 wi th 10% w.c. base layer and nonlinear Ottawa sand subgrade. .....................................293 D 28 yy at top of subgrade comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer and nonlinear Ot tawa sand subgrade. .............................................294 D 29 yy at top of subgrade comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. .............................................294 PAGE 27 27 D 30 yy at top of subgrade comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. .............................................295 D 31 yy at top of subgrade layer comparison for nonlinear and linear cases for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade ......................................295 D 32 yy at top of subgr ade layer comparison for nonlinear and linear cases for structure 1 with 10% w.c. base layer and nonlinear Ottawa sand subgrade ......................................296 D 33 yy at top of subgrade layer comparison for nonlinear and linear cases for structure4 with 10% w.c. base layer and nonlinear Ottawa sand subgrade ......................................296 D 34 Surface deflection comparison for nonlinear and linear cases, for structure 1 with 5.5% w .c. base layer and nonlinear Ottawa sand subgrade. ............................................297 D 35 Surface deflection comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand s ubgrade. ............................................297 D 36 xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade. ............................................298 D 37 xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade. ............................................298 D 38 xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade. ............................................299 D 39 xx at top of AC layer comparis on for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade. ............................................299 D 40 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade. ....................................300 D 41 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade. ....................................300 D 42 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade. ....................................301 D 43 xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade. ....................................301 D 44 yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade. ..................302 D 45 yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade. ..................302 PAGE 28 28 D 46 yy at top of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade. ............................................303 D 47 yy at top of base layer comp arison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base laye D 48 yy at bottom of base layer comparison for nonlinear and linear c ases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade. ....................................304 D 49 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w .c. base layer and nonlinear Ottawa sand subgrade. ....................................304 D 50 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Otta wa sand subgrade. ....................................305 D 51 yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade. ....................................305 D 52 yy at top of subgrade comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade. ............................................306 D 53 yy at top of subgrade comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade. ............................................306 D 54 yy at top of sub grade layer comparison for nonlinear and linear cases for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade .....................................307 D 55 yy at top of subgrade layer comparison for nonline ar and linear cases for structure1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade. ....................................307 PAGE 29 29 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NONLINEAR EQUIVALENT LINEAR DESIGN METHOD TO DETERMINE UNBOUND AGGREGATE BASE LAYER MODULUS By Aditya Ayithi August 2011 Chair: Dennis R Hiltunen Co Chair: Reynaldo Roque Major: Civil Engine ering MechanisticEmpirical Pavement Design Guidelines (MEPDG) recommends use of material modulus in lieu of structural number for pavement base layer thickness design. Soil modulus is nonlinear with respect to effective confinement stress, loading strain and moisture (suction); and modulus nonlinearity need to be considered for an efficient base layer design and analysis. For practical design purposes, single effective modulus value of base layer need to be known and this modulus value should be able to a pproximately account for nonlinearities of the whole base layer. However, MEPDG does not describe a standard procedure for determining this single modulus value. This research study focuses on laborator y characterization of base soil modulus nonlinearity, developing a nonlinear response model for nonlinear pavement analysis and developing a practical linear design methodology to determine nonlinear equivalent single effective design modulus value for whole base layer First, Fixed Free and FreeFree resonan t column tests are conducted on two gravelly base soils used in the State of Florida, to characterize shear modulus (G) nonlinearity in the strain range of 105% to 101%, including small level strains, under different loading confinements and moisture contents. Moisture s uction effect on nonlinear modulus is evaluated. It is found that unsaturated gravelly soils modulus is linear at strains lower than 105% and nonlinear thereafter. PAGE 30 30 Compared to dry soils, presence of moisture in unsaturated soils makes it more nonlinear with respect to strain. Suction effect can increase G in the strain range of 105% to 101%, but very significantly at strain levels less than 103%. At any given moisture content, additional confinement due to suction does not decrease with increase in strain. Empirical equations are developed to calculate very small strain modulus (Gmax) of dry soils. A procedure to calculate approximate G value at given water content, confinement and strain magnitude is developed. Second, using laboratory nonlinear modulus characterization data as material parameter inputs, a stress and strain dependent nonlinear response model for base layer analysis is developed via Plaxis HSsmall model. Nonlinearity of the response model is verified with respect to stres s and strain variation. Considering maximum surface deflection of nonlinear analysis as matching factor between nonlinear and linear design methods, a practical design methodology to determine equivalent single effective elastic modulus for base layer is p roposed. Effect of moisture, subgrade modulus and layer thickness on base single effective modulus is analyzed. Stress and strain responses for both nonlinear and equivalent linear analysis are compared at critical locations. Influence of base nonlinearity on pavement performance is evaluated. It is found that use of single effective modulus in place of nonlinear modulus over estimates rutting performance, when structure thickness and base water content decreases together. Last, effect of subgrade modulus nonlinearity on pavement performance is also analyzed for limited cases. Equivalent single effective modulus for all layers is back calculated via Falling Weight Deflectometer (FWD) analysis of nonlinear analysis surface deflection. It is found that, subgra de nonlinearity may significantly influence rutting performance and elastic based rutting performance criterion overestimates rutting performance. PAGE 31 31 CHAPTER 1 INTRODUCTION 1.1 Problem Statement MechanisticEmpirical Pavement Design Guidelines ( MEPDG ( 2004) ) for flexible pavement structures recommend use of modulus in place of layer coefficient, for unbound aggregate base layer thickness design. Resilient Modulus (MR two primary input parameters required for thickness design. MR represents modulus of a material subjected to repeated traffic loading and can be determined in lab via a standard testing pro tocol (AASHTO T307) Soil is a nonlinear material and its modulus nonlinearity is dependent primarily on effective confinement stress, loading strain, moisture content (suction) and some other parameters. MEPDG proposes three different levels of MR input for pavement design analysis response model. Level 1 MR input takes material modulus nonlinearity into account, whereas level 2 and level 3 MR input s assume material is elastic and assigns a single effective elastic modulus value for the whole layer. However; nonlinear design analysis response model based on level 1 nonlinear MR input has not been calibrated for practical applicati ons yet, because of various complicated nonlinear modeling issues involved in it. Thus, it seems that single effective elastic modulus approach using either level 2 or level 3 MR input would be most commonly used analysis in near future. Therefore determin ing single elastic modulus value is very critical for pavement response model analysis. However, since soil is a nonlinear material, single elastic modulus approach should not discard the importance of modulus nonlinearity. Hence, the single elastic modulus value should be able to reflect the nonlinear behavior of base layer up to certain extent under real loading conditions. In parallel, if we can built a database of either nonlinear modulus parameters or PAGE 32 32 single effective elastic moduli value s that can als o reflect nonlinear behavior approximately, for different types of base materials, it may not be required to conduct expensive MR laboratory testing at initial design stage and the whole design process would become more economical. MEPDG is primarily based on MR and its determination via AASHTO protocol. By following AASHTO T307 testing procedure, material modulus can be characterized only from intermediate (103%) to larger strains (101%), due to high deviatoric stresses applied in this test procedure and external measurement of loads and deformations. Recent research studies in geotechnical engineering revealed that it is necessary to consider modulus nonlinearity at small level strains ( 3%) along with nonlinearity at intermediate to larger strains, to predetermine accurate pavement responses. Since moduli values at small level strains cannot be determined via AASHTO protocol due to procedural limitations, small strain modulus nonlinearity cannot be characterized. Hence, accurate pavement responses can not be calculated using moduli values obtained via AASHTO testing protocol. Therefore there is a requirement to seek for a different testing procedure, which can characterize material modulus nonlinearity at small level strains also along with intermediate to larger strains. Moreover, AASHTO T307 gives us a set of MR values corresponding to different stress levels and strain magnitudes, but not a single elastic modulus value which is required for MEPDG level 2 and level 3 material parameter inputs. Till now no proper methodology is defined either in MEPDG or in literature about how to determine the single elastic modulus value, required for level 2 and level 3 MR input for response model? Therefore, there is a requirement to develop a proper methodology to determine single effective design modulus value, which can also approximately reflect the modulus nonlinearity of whole base layer. PAGE 33 33 Suction in partially saturated soils provides additional confinement, which in turn increases soil modulus Research stu dies related to suction effect on soil modulus at different strain levels have revealed that suction effect can increase modulus at small level strains very significantly compared to intermediate and large strains. Since it is important to consider modulus nonlinearity at smalllevel strain s it is important to consider suction effect on modulus at small level strains too. MEPDG incorporates moisture effect (which includes suction effect also) on modulus via Integrated Enhanced Climatic Model (ECIM). ECIM i s developed based on MR testing data base for different types of soils. Since, these MR values are determined via AASHTO protocol, which can measure modulus at intermediate to large strains only; ECIM does not consider suction effect at small level strains Therefore it is necessary to develop a pavement response model which can consider suction effect at different strain levels including small level strains also. In view of above discussed issues regarding testing protocol adequacies, moisture (suction eff ect) model inabilities and no properly defined methodology to calculate single effective design modulus for base layer, there is a necessity to develop i) A lab testing method for modulus determination, which can accurately characterize modulus nonlinearit y at different strain levels including small level strains ii) A design methodology to determine single effective modulus value for MEPDG level 2 and level 3 modulus inputs, which can also reflect modulus nonlinearity approximately and iii) Response model that can incorporate suction effect on modulus at small level strains, along with intermediate and larger strains. 1.2 Hypothesis A design methodology to determine single effective elastic design modulus of unbound aggregate base soils that can be used as MEPDG level 2 and level 3 base material parameter input needs to be developed. Single effective elastic modulus can be determined such that it can PAGE 34 34 approximate base soil modulus nonlinearity with respect to effective stress confinement, loading strain including small level strains and moisture content. 1.3 Objectives The p rimary objective of this research work is to develop a design methodology for characterizing base layer design modulus for use in MEPDG that can address the issues discussed above. Detaile d objectives would be: Conduct a lab oratory testing program on selected base materials, following a suitable testing methodology that can effectively account for the effects of confinement, strain magnitude and moisture on nonlinear modulus, including smal l strain nonlinear modulus; and can characterize its nonlinear behavior To evaluate and quantify suction effect on nonlinear small strain modulus To develop an appropriate pavement response model that can utilize the laboratory testing results and account for the above discussed modulus nonlinearity, under performance conditions. Based on deflections, stress and strain responses obtained from pavement response model analysis develop a practical d esign methodology to calculate single effective modulus for whole base layer that can approximate know n nonlinearities, for use as MEPDG level 2 and level 3 material inputs. Demonstrate by various techniques that the overall approach, as well as the laboratory test results and pavement response model results are cr edible and ap p e a r to agree with other known properties and behavior. 1.4 Scope of Research In the State of Florida, limerock type aggregates are commonly used for base layer construction. It is proposed to select one limerock material and one non limerock material for our testing. The two materials selected for our testing are i) Newberry limerock and ii) Georgia granite Material collection and test specimen preparation are performed following standard procedures. PAGE 35 35 Fixed Free Resonant Column (FixedFree RC) test method and Free Free Resonant Column (F ree F ree RC) test method are selected for material testing program. Fixed Free RC tests can determine shear modulus (G) of gr a vel type soils at di fferent strain magnitudes including small level strains. In Fixed Free RC tes ting, compacted specimens can be subjected to required stress confinement. Suction effect can be evaluated by conducting tests on specimens dried to different water contents. Therefore, via FixedFree RC testing, modulus nonlinearity of gravell y base soils can be characterized with respect to i) effective stress confinement, ii) loading strain magnitude including small level strains and iii) moisture or suction effect. These tests are conducted on standard size modified proctor compacted specime ns F ree F ree RC test can determine shear modulus at very small level strain (Gmax) at different water contents under no confinement, to evaluate suction effect on Gmax. Credibility of these testing results is established by comparing them with literature data. A nonlinear pavement response model is developed via PLAXIS HSsmall; a nonlinear finite element model based system that can utilize laboratory testing results. Using laboratory test results as material parameter inputs above response model can characterize material modulus nonlinearity properly with respect to stress confinement, strain magnitude including small level strains and moisture or suction effect. Flexible pavement structures with different layer thicknesses or cross sections are analyzed under performance loading conditions, using the nonlinear response model. A methodology to determine single effective elastic design modulus value for whole base layer that can be utilized as MEPDG level2 and level 3 material parameter input is developed. Single effective base modulus determination methodology is developed such that various pavement responses such as surface deflections, stresses and strains obtained by nonlinear base PAGE 36 36 modulus analysis and equivalent linear single elastic base modulus analys is are approximately same. Effect of base layer nonlinearity on pavement performance is evaluated by comparing the responses obtained from nonlinear base modulus analysis and equivalent linear single base modulus analysis. Out of professional interest, eff ect of subgrade nonlinearity on pavement performance is also briefly evaluated for limited cases. 1.5 Organization of Dissertation An over view of remaining chapters of this dissertation is presented here. Chapter 2 gives an over view of Resilient Modulus (MR), MR testing method and its adequacies, importance of modulus characterization including smallstrain modulus nonlinearity in soils and possible laboratory testing methods, modulus nonlinearity influencing factors and it s importance in geotechnical str uctural designs. Different models proposed by various researchers to calculate very small strain modulus and modulus at different strain levels are discussed. Suction effect on smallstrain modulus and importance of considering suction effect in determining design modulus is also discussed. Recent research efforts about importance of considering smallstrain modulus nonlinearity for geotechnical structural analysis and design are also discussed. Different types of materials selected for testing and their pr operties are presented in C hapter 3. A detailed explanation about specimen preparation methods, for both wet and dry materials, is also given in chapter three. In C hapter 4, Fixed Free RC and Free Free RC test methods used in this research work are presen ted. Detailed explanations about equipment development background and types of parameters that can be measured with these equipments are discussed. Equipment setup, calibration, verification and limitations are also explained. Chapter 5 mainly analyze and discuss various testing results obtained from both FixedFree RC and Free Free RC tests. Specimen testing conditions like different moisture contents PAGE 37 37 tested void ratios and pressure confinements are mentioned. E mpirical equation s are proposed to calculate very small strain modulus of dry soils. Suction effect on small strain modulus is evaluated and an indirect approach to calculate addition confinement due to suction is explained. Based on these testing results, a method to determine approximate modulus a t any given water content, confinement and strain magnitude is explained. Chapter 6 talks about PLAXIS HSsmall model and required model material input parameters. Development of a nonlinear response model for base layer analysis via HSsmall model is disc ussed and how to obtain input parameters from lab testing results is explained. Chapter 7 mainly talks about base layer nonlinear analysis. It presents a methodology to determine single effective design modulus for base layer thickness design. Different pavement cross sections are analyzed using H S small nonlinear response model. Single effective elastic base moduli values for different pavement structures at different base water contents and subgrade moduli are reported and analyzed. Effect of base nonlinea rity on pavement performance is evaluated. Importance of subgrade nonlinearity in pavement design and analysis is discussed in Chapter 8. A methodology to determine equivalent single elastic modulus for both base and subgrade layers is explained and effect of subgrade nonlinearity on pavement performance is analyzed briefly for limited number of structures. Based on the results presented in C hapter s 5,6,7 and 8, total findings are summarized in C hapter 9. Based on this entire research work information discussed in C hapters 2 through 8, conclusions and recommendations are presented Further test result s details are presented in the appendices. Fixed Free RC test results on dry specimens at different void ratios are given in Appendix A. Nonlinear equivalent si ngle PAGE 38 38 effective base moduli database is given in Appendix B. Comparison plots of surface deflection, stress and strain responses obtained from nonlinear base analysis and equivalent linear base analysis, for different pavement structures is presented in Appendix C. Comparison plots of pavement responses obtained from base and subgrade nonlinearity analysis and equivalent linear analysis are presented in Appendix D PAGE 39 39 CHAPTER 2 LITERATURE REVIEW The center point of proposed research work is determining effecti ve base modulus for practical design purposes, while considering modulus nonlinearity. It is proposed to achieve this through laboratory testing and nonlinear modeling. Hence this literature review mainly talks about soil modulus and its influencing factor s including suction, modulus at different strain levels including small level strains and its determination in laboratory, various analytical models developed to determine modulus at different strain levels, importance of modulus nonlinearity in geotechnic al design calculations and pavement response models. 2.1 Modulus of Particulate Material Material modulus represents its resistance to deformation under loading. Modulus or stiffness is the relationship between change of stress and change of strain, and is defined by ratio of stress over strain. Being a particulate material, soil modulus is nonlinear and primarily c Idriss (1970), Hardin and Drnevich (1972), Yasuda and Mastumoto (1993)). In unsaturated soils, modulus is dependent on moisture content and its suction effect also (Wu et al. (1984), Quin et al. (1993), Cho and Santamarina (2001)). In a pavement base layer, strain magnitudes are not constant throughout the layer. They are highest near or under the wheel load and diminishes to zero as moving away from wheel load, in a nonlinear manner. Soil modulus is maximum (Gmax or Emax) at strain levels lower than 104% and decreases nonlinearly with increase in strain (Figure 2 1), generally in an S shaped pattern. As shown in Figure 2 1, Atkinson and Sallfors (1981) presented a typical modulus reduction curve with increase in strain for soils, along with different strain levels involved in various types of geotechnical structures. Modulus nonlinearity for different strain levels that can be observed PAGE 40 40 under real loading conditions should be considered for accurate geotechnical design calculations (Atkinson (1991) ) Elhakim and Mayne (2008), Lehane et al. (2008)). 2.2 Resi lient Modulus According to MEPDG (2004), r esilient modulus (MRtwo basic material parameter inputs required for pavement layer thickness design. Behavior of a soil subjected to repeated traffic loading in a pavement layer is characterized by MR. MR is an elastic modulus based on recov erable strain under repeated loads and is defined as : r d RM (2 1) d= deviator ic stress and r= recoverable axial strain. Since its inception in AASHTO 1986 guidelines, lot of research has been done and well documented regar ding MR. MR of soils can be determined in lab via standard testing procedure (AASHTO T307). For design calculations, MR calculated using the basic constitutive equation : MR=k1k 2 (2 2) in which i s sum of three principal stresses, parameters k1 and k2 are material dependent and their values are determined from lab testing data. In standard MR testing procedure, testing specimen is subjected to different predetermined combinations of confinement str ess and deviatoric stress and resulting strain magnitudes are measured using strain gauges placed outside the triaxial testing chamber. S train magnitudes that can be measured in AASHTO T307 testing are typically in the range of 103% and higher. Hence t he moduli value s obtained from lab testing are good enough for the limited strain range that can be measured in lab. MEPDG level 2 and level 3 design procedure s for unbound aggregate base layer thickness design use single modulus value for the whole layer wh ich should be obtained from MR lab PAGE 41 41 testing. MR lab testing produces a set of moduli data corresponding to specific stress and strain levels, but not a single modulus value. MEPDG does not define a procedure about how to determine single modulus value required for design, from laboratory data? To consider modulus nonlinearity with respect to stress and strain in pavement design procedure s; first of all we should be able to measure modulus for a wide range of strain levels at any given confinement stress O ne of the main challenges in measuring the modulus of soil at small level strains is choosing the right equipment and testing method for accurate strain measurements. A good testing method should be reliable, simple and quicker. Conventional lab testing eq uipments can measure modulus at strains as small as 103% with reasonable accuracy, where the strain gauges are placed outside the testing chamber. Moduli corresponding to strain range 105% 103 % can be measured only by using local strain gauges attach ed directly to the sample (Jardine et al. ( 1984), Atkinson (1991) ). But these local strain gauges must operate satisfactorily in water or oil for a long time and also can not measure very small strains (i.e. 5% ), which correspond to fundamental material property Gmax AASHTO T307 testing protocol requires to attac h the strain gauges outside of the testing chamber. Thus standard MR test cannot measure small level strains and accuracy of strain measurements less than 103% becomes questionable. A lso MR lab testing procedure is very tedious, time taking and skill req uired one, which makes the testing very expensive. 2.3 Modulus Characterization for Complete Range of Strains Past research efforts have proved that nonlinearity of soil s for a wide range of strains including small level strains need to be considered for a ccurate settlement s /deformation s predictions in soils (Jardine et al. (1982), Atkinson (1991), Elhakim and Mayne (2008), Lehane et al. (2008)). It is now well acknowledged that strain/deformation predictions for geotechnical structures (e.g.: footings, ret aining walls and braced excavations etc.) are not at acceptable level PAGE 42 42 of accuracy and one of the primary reasons for this inaccuracy is choosing a single linear elastic modulus value which corresponds to high level strain, to represent the soil behavior. Q uite a few numbers of analytical models were developed to calculate G at different strain levels, using Gmax as the benchmark. Determination of accurate Gmax representing in situ soil condition is very critical. Research studies conducted about determinati on of G at different strain levels including Gmax for different soil types and analytical models are developed to calculate G under known conditions are discussed in following sections. 2.3.1 Small Strain Modulus (Gmax or Emax) Since 1960, lot of research investigations were conducted and documented on Gmax of sands, silts and clay type soils and its influencing factors. Hardin and Richart (1963) measured longitudinal and shear wave velocities of Ottawa sand, crushed quartz sand and crushed quartz silt at smallstrain level (103 rad) using free free and fixed free resonant column methods. They found that Vs for sands varied with approximately power of confining pressure. At very small strain, Vs of sands is function of mean effective confining pressure ( ) and void ratio (e); and Vs of Ottawa sand can be expressed as: for <2000 psf (2 3) for >2000 psf (2 4) where the unit of Vs is fps. Hardin and Drnevich (1972) conducted resonant column tests on different types of soils to analyze th e effects of confining pressure, strain amplitude, void ratio, number of loading cycles degree of saturation and thixotrop y. They found that shear modulus decrease s and damping ratio increases ver y rapidly with increas e in strain amplitude. The rate of decrease or increase in Gmax depends on many parameters and a single relationship between modulus or damping and strain PAGE 43 43 amplitude is not sufficient. At small strain amplitudes, modulus varies with 0. 5 power of effective mean principal stress, but at larger strains modulus depends primarily on strength of soil and the variation is more nearly with 1.0 power. Modulus and damping ratio decreases with increase in void ratio and the effect is accounted by : (2 5) w here eis void ratio. Shear modulus decreases for cohesive soils and increases for cohesionless soils with increase in number of loading cycles. Degree of saturation has no effect on modulus of cohesionless soils In cohesi ve soils, modulus increases rapidly with decrease in degree of saturation. Hardin and Drnevich (1972), Krizek (1974) and Kuribayashi (1974) have clearly shown that modul i values for sands are strongly influenced by three main factors: Confining pressure, str ain magnitude and void ratio. Seed et al. (1986) investigated shear moduli and damping characteristics of soils via cyclic undrained triaxial tests and derived an empirical equation to determine Gmax as: (2 6) They found that values of (K2)max for relatively dense well graded gravels are likely to range from 80 180, compared with the range of about 5580 for sands and gradation does not have much influence on shear moduli of gravels. They developed a normalized shear modulus versu s shear strain curve for gravels similar to the curve for sands, and curve for gravels is little bit flatter compared to sand s curve ( Figure 2 2). Based on in situ shear wave velocity measurements, shear moduli of gravelly soils are between 1.25 and 2.5 t imes greater than that of sands. PAGE 44 44 Hardin and Richart (1963) proposed following empirical equation to calculate Gmax, for known void ratio (e) and effective confinement ( c ): (2 7) where AG and nG are material constants, F(e) is function of void ratio and c is effective mean principal stress. Kokusho (1987).pointed out that as the grain size increase from clays to sands to gravel s ; rate of reductio n in strain dependent modulus becomes high at a smaller strain level. Different formulations were developed between 1960 and 2000, to calculate Gmax and most of them generally use a power law to describe the effect of confining pressure on Gmax. Power law exponent n=0.5 tends to fit for sands and n=0.50.75 tends to fit for gravels. According to Pestana and Salavati (2006) the best formulation to calculate Gmax is: (2 8) Where Gbmaterial constant and can be in the range of 400 800. 2.3.2 Shear Modulus (G) at Different Strain Levels: Yasuda and Matsumoto (1993) conducted cyclic torsional simple shear tests and cyclic triaxial tests to investigate dynamic deformation characteristics of sand and rock fill materials. They found that G ca n be expressed as a function of shear strain, void ratio and confining stress. They also found that the absolute value of G for the rock fill materials is substantially higher than sand under the same relative density. The alteration of G of rock fill mate rial occurred at smaller strains than for sands. Yasuda et al. (1996) conducted large scale cyclic triaxial tests on undisturbed and reconstituted specimens obtained from river bed gravel foundation of an embankment dam, to determine G in the strain range of 106 to 103. Undisturbed specimens of size 300 mm diameter PAGE 45 45 and 550 mm height were sampled by using freezing sampling method. From their testing investigation, they found that Gmax of undisturbed specimens was 1.5 to 2 times greater than reconstituted s pecimens. They reported that this difference is due to cementation and fabric produced by the effects of the geological time of sedimentation in undisturbed specimens. G/Gmax undisturbed and reconstituted specimens (Figure 2 undisturbed specimens can be determined from in situ Gmax. Li n et al. (2000) investigated characteri stics of shear modulus (G) and damping r atio (D) of gravelly cobble deposits. They conducted both resonant column and cyclic triaxial tests to obtain G for a strain range of 104% to 1%. They observed decrease of G with increase in until 0.1% and increase in G thereafter (Figure 2 4 and 2 5) and attributed this behavior to gap gradation of the soils tested. Gravel cobble deposits contain 80% gravel and remaining is filling material. When they a rtificially increas ed the proportion of filling material, G decreased with increase in as mentioned in literature. G incr eases as the max particle size (Dmax) increase and also with increase in confining pressure (Figure 2 6). There is no significant effect of either Dmax or c on G/Gmax versus relationship. Based on their testing data, they developed an empirical equation to calculate Gmax of deposit at different confining pressures as shown in Figure (2 7). Menq (2003) conducted free free resonant column tests on sandy and gravelly soils and measured G in the strain range of 105% to 101%. He observed Gmax at strains 4% for gravels and at strains 3% for sands. Normalized modulus versus strain curves for gravels are more flatter than for sand and follow the same trend as reported by Seed et al. (1986) (Figure 28) Gravelly soils are commonly used in base layer construction. Many researchers investigated smallstrain mod ulus of sands, silts and clays; but little efforts were made to PAGE 46 46 determine moduli of gravel type soils. Being a large size aggregate, testing on gravel requires 4 to 6 diameter specimens and large s ize equipment, which makes testing complicated and also one of the reason for scarce literature of moduli of gravels. 2.3.3 Analytical Models to Estimate Shear Modulus at Different Strain Levels General hyperbolic form of stress strain relation to estimate written as (2 9) w here Hardin and Drnevich (1972), based on their testing data base for different types of soils, modified above hyperbolic expression as: (2 10) w here a& b are soil constants. They proposed a = 0.5 and b= 0.16 for clean dry sands. They compared measured values with calculated values and shown in Figure 2 9. Hardin and Kalinski (2005) based on their testing data, found that G/Gmax relationship vary with confinement stress level, whereas G/Gmax versus r (normalized strain) relationship is independent of stress level Thus, r modulus reduction relationship for gravels similar to sands as proposed by Hardin and Drnevich (1972). Rollins et al. ( 1998) conducted an experimental program on gravels and reviewed results of fifteen testing programs (including their results) from literature and concluded that normalized PAGE 47 47 she ar modulus curve is dependent on confining pressure and independent of sample disturbance, relative density and gradation. Based on 15 different testing results data, they proposed a best fit hyperbolic curve model as given below and shown in Figure 210. (2 11) From their testing results, they observed an increase in G with increase in gravel content (Figure 2 11). They also observed that G/Gmax with increase in confining pressure (Figure 212). Accord ing to their review, for gravels, at shear strain s lower than 104%, shear modulus and damping remain constant and shear modulus is at its maximum, i.e. Gmax. Santos and Correia (2001) proposed a modified HardinDrnevich hyperbolic relationship to calculat e shear modulus for different strain levels. The unique straindependent shear modulus degradation curve for soils is: (2 12) where Gmax maximum shear modulus 0.7shear strain at G=0.7Gmax to threshold shear 0.7) makes it possible to define almost a unique strain dependent stiffness degradation curve for sand and clays. Their proposed stiffness degradation curve with higher and lower limits is shown in Figure 213. 2.4 Influence of Suction on Soil Modulus Wu et al. (1983), conducted resonant column tests on fine grained cohesionless soils to investigate capillary effects on Gmax. For their testing, they used 3.6 cm diameter and 8 cm height specimens compacted at required different water contents and void ratios. Specimens were tested PAGE 48 48 at 24.8, 49 and 98 kPa confinement pressure. For a given void ratio, starting form dry, Gmax increases with increase in degree of saturation and reaches a peak value and start decreasing thereafter with further increase in water content. Degree of saturation, where Gmax is at its peak is called as optimum degree of saturation. Capillary effects were greatest for soils with smallest effective grain diameter (D10) and at lowest confining pressure ( c ). Additional confinement provided due to capill ary suction is almost equal to 1.6 m of overburden of completely saturated soils. They found that the max imum capillary suction occurs between 5% and 20% water content. Qian et al. (1993) conducted a n experimental investigation of capillary effects on low strain shear modulus (at strains lower than 103% ) of partially saturated sands. They ran Hall type resonant column tests on fourteen sands with different gradations (four natural sands and remaining man m ade form these four natural sands) to study the effect of void ratio (e), confining pressure ( c ), grain shape and grain size distribution. Specimens were prepared at different water content s and void ratios same as procedure followed by Wu et al. ( 1983) They found that capillary effects increase small strain shear modulus significantly and more pronounced for soils with low void ratio. Optimum degree of saturation increases with increase in void ratio and is not affected by confinement. Capillary influence decreases with confinement. The content of 400 minus sieve size fraction can affect both m odulus and optimum degree of saturation. Picornell and Nazarian (1998) conducted bender element tests on specimens of coarse sand, fine sand, silt and clay, prepared by separating a local soil into size fractions, to investigate the effect of soil suction on low strain shear modulus (Gmax) of soils. Predetermined soil suction was applied on specimens at different water contents using pressure plate apparatus They PAGE 49 49 observed that as the soil particle size decreases, there is a progressive increase of satura ted and residual water contents. As the water content decreases from saturated water content to residual water content, Gmax increase for all soils, for sands by a factor of 1.8 and for silts by a factor of 2.5 and for clays by a factor of 10. M uch larger effect in clay specimens was attributed to presence of flatly particles that deform under forces imposed by the menisci resulting in increase in number of contact points where menisci can develop and act. Cho and Santamarina (2001) conducted microscale par ticle level studies to investigate capillary effects on low strain stiffness at different water contents. They con ducted bender element tests on 1) glass beads, 2) mixture of Kaolinite and glass beads 3) Granite powder and 4) sandboil sand. They also conducted microscale experimental study on menisci failure and recovery. Shear wave velocit ies (Vs) measured from bender element tests for the above four materials at different degree s of saturation, are shown in Figure 2 14. Some of the conclusions of their study are: Contribution of capillarity to inter particle forces involves both matric suction and surface tension. Equivalent effective stress due to capillary forces increase with decreasing water content, decreasing particle size and increasing coordinat ion. Remolding is not an appropriate specimen preparation method (as followed by Wu et al. ( 1983) and Quin et al. ( 1992) ) to study the behavior of low water content soils, since drying influences particle contact forces. Observed strains for menisci failur e are in the range of = 0.01 to 1 and higher than threshold strain for sands, so partial saturation is a stabilizing force for the soils skeleton. This strain at menisci failure decreases with decrease in water content. On the other hand, small menisci may fail before reachin g the strain at peak strength of soils. Thus, capillary forces at low water contents cause an increase in low strain stiffness of soils, but may not contribute to the pea k strength. Toros (2008) investigated suction effects on sm all strain Youngs modulus (Emax) of Florida base course gravelly soils. He conducted FreeFree Resonant Column (FFRC) tests on 6 PAGE 50 50 inch 12 inch cylindrical specimens of Miami limerock, Newberry limerock, Ocala limerock, Loxahatchee Shell rock and Georgia Granite; which are typically used base layer materials in the State of Florida. Samples were compacted at optimum moisture content and expos ed to four different ambiences 1 ) laboratory 2) outside 3) constant moisture 4) wetting and drying. For drying, specimens were placed in a oven and for wetting, oven dried specimens were soaked in a partially water filled tank. He investigated the time effect on Emax at constant moisture, water content effect on Emax at laboratory ambience, outside ambience and oven drying and water soaked wetting ambience. Under constant moisture condition, Emax increases with time initially and remain more or less same, after that (Figure 2 15). He hypothesized that the behavior observed could also be due to increased suction or negative pore water pressure that occurs as the water in the material redistributes following compaction into more preferential positions within the inter particle void spaces. This increased suction effectively adds confining stress to the particulate material and thereby increases the r esistance to deformation (stiffness). In both laboratory and oven drying methods, Emax increased significantly with decrease in water content (Figure 2 1 6 and 217). This stiffening with drying was explained by increase in suction, which effectively increase confinement and hence modulus. Regardless of the drying method, the Miami limerock changed the most and this behavior is partially explained by the fact that this material is coarsest, well graded, and at low void ratio. Drying and wetting responses do not follow the same relationship and there is a hysteretic phenomenon whereby a different modulus is measured while drying to certain moisture content compared to while wetting to the same moisture content (Figure 2 18) This hysteretic PAGE 51 51 phenomenon is well known in unsaturated soil mechanics where the suction values reached at common moisture content are different between drying and wetting. MEPDG incorporates suction effect on MR of unbound aggregate via Enhanced Integrated Climatic Model (ECIM). ECIM is b ased on MR lab testing database, and incorporates suction effect at MR lab testing strain levels only (i.e. higher than 103%) and does not consider suction effect for complete range of strains including very small level strains Moreover this model is dev eloped based on testing data obtained from limited selected base soils only 2.5 Importance of Small Strain Modulus Nonlinearity for Design and Analysis Recent research studies in geotechnical engineering have helped in increasing the awareness of consider ing modulus nonlinearity for a complete range of strain levels in realistic prediction of ground deformations and surface settlements in geotechnical structures such as footings, braced excavations etc. (Jardine et al. (1986), Atkinson (1991), Elhakim and Mayne (2008) Lehane et al. (2008)) Jardine et al. (1986) studied the influence of nonlinear stress strain characteristics on soil structure interaction. They used laboratory measured nonlinear stress strain properties for finite element analysis of footi ngs, piles excavations and pressure meter tests, to assess the influence of smallstrain nonlinearity in comparison with linear elastic behavior. From their study they concluded that although linear elasticity remains a convenient tool for expressing measurements of soil stiffness, unless the nonlinear nature of soil is taken account, interpretation of field measurements can be misleading. Also, small strain nonlinearity has a significant influence on interpretation of equivalent elastic moduli of in situ d eformation tests. Elhakim and Mayne (2008) developed a logarithmic modulus reduction algorithm to model and calculate soil modulus at different strain levels, using Gmax as the benchmark. By incorporating these nonlinear moduli at different strain levels i n settlement analysis calculations, PAGE 52 52 they were able to predict load displacement response of footing on soft clay very close to actual field measured load displacement curve (Figure 2 19) Lehane et al. ( 2008) conducted load test s on four different size fo oting s on Perth sand. For their settlement predictions, they incorporated modulus nonlinearity for different strain levels including small level strains via Plaxis: HS small model. They found that, foundation settlements predicted with Plaxis: HS small mod el are much better than existing standard linear design methods and in some cases, close to actual measured load test settlements. (Figure 2 20 ) In their analysis they observed that settlement predictions performed with insitu measured parameters are more accurate than predictions performed with lab tested parameters. Settlements predicted with lab tested parameters are larger than settlements predicted using measured parameters and in some cases 4 times larger. They explained that age and structure have a strong influence on Gmax, but the reference strain for in situ and reconstituted sand are similar. Because of this difference in Gmax, settlement predictions with in situ parameter are bett er than lab tested parameters. Based on their experience with Plaxis: HS Small model, some limitations of the model were reported and they are: 1. 0.7) is constant at all stress levels 2. Gmax varies with 3 only and is not dependent on the major principal effective stress. 2.6 Pavement Response Models 2.6.1 MEPDG Nonlinear Response Model MEPDG recommends considering modulus nonlinearity of base soils for level 1 material parameter inputs. Nonlinear finite element response model for base layer analysis developed in MEPDG calculates nonlinear resilient mod ulus using the following equation: PAGE 53 53 (2 13) bulk stress at the peak of loading oct octahedral shear stress at the peak of loading paatmospheric pressure k1k7 material parameters material parameters k1, k2 and k3 should be determined from MR laboratory data. Equation 213 calculates diff erent MR values for different stress levels based on corresponding stress state. T hese MR values are calculated via constitutive equation, in which material parameters k1, k2 and k3 are determined from MR laboratory data that is corresponding to limited strain range Therefore these moduli values generated from nonlinear response model are appropriate for the limited strain range that can be measured in lab testing but not a complete range of strains including small level strains required for accurate mate rial nonlinear modeling. MEPDG nonlinear response model for level 1 material nonlinear parameter inputs is not calibrated yet for complete designs analysis, and is not much useful for practical design purposes where a single effective moduli value is needed for initial design calculations. 2.6.2 Single Effective Moduli Determination Based on Nonlinear Versus Linear Analysis Responses Comparison Roque et al. (1992) performed a comprehensive analysis to determine whether linear elastic layer analysis can be u sed to accurately predict the nonlinear response of pavements. They performed no nlinear analysis using f inite element computer program ILLIPAVE to predict nonlinear response of pavement structure. Surface deflections obtained from nonlinear analysis were u sed as matching factors to back calculate equivalent elastic moduli, using BISAR. 40 kN was used as design load for the analysis. PAGE 54 54 For nonlinear analysis, asphalt surface layer was considered elastic and both base and subgrade were considered nonlinear. ILL IPAVE uses stress dependent nonlinear moduli for nonlinear analysis. For better predictions and comparison, the upper 0.30 m of the subgrade was modeled as a separate layer. Comparison of surface deflections, stresses and strains between nonlinear analysis and back calculated linear analys is was made. But, these stresses and strain s comparisons were made for surface layer only. They found that, within the surface layer, fairly accurate predictions of deflections, stress and strain equal to nonlinear analysi s can be made, when a single effective layer modulus was used to represent the surface and base layers. Nonlinear response of subgrade can be represented by two elastic layers and corresponding effective layer moduli. It may be very difficult to accurately represent the nonlinear response of weak pavement structures or silty sand subgrade. This e ffective moduli determined at design wheel load can be applied to predict nonlinear pavement response at load levels of 25% of design load. 2.6.3 PLAXIS HSsmall M odel Plaxis HSsmall model is the only commercially available and readily usable software, which can implement modulus nonlinearity in soils, with respect to both stress confinement and strain magnitude. HSsmall model considers nonlinear modulus at different strain levels (as high as 101) including very small level strain (i.e. 6). Lehane et al (2008) used for nonlinear modeling of soil under footing for settlement predictions and obtained fairly accurate results. Required input parameters for nonlinear modulus modeling can be obtained from resonant column testing and it is proposed to use Plaxis HSsmall for development of nonlinear pavement response model. HSsmall model characteristics, back ground and required input parameters are explained in detail in Chapter 6. PAGE 55 55 2.7 C losing Remark s Conclusions of this literature review are 1) Gmax of dry soils is primarily dependent on effective c ), and void ratio (e) Gradation and max aggregate size are some of other various secondary influencing parameters. 2) Different empirical relationships were developed to calculate Gmax and hyperbolic models were developed to calculate G at different strain levels. Normalization of G with Gmax nullifies the effect of confinement. 3) Increase in suction due to drying of aggregate can increase modulus significantly. 4) Consideration of nonlinear smal l strain modulus in geotechnical design and analysis improves soil deformation/settlements predictions. Gmax can be considered as benchmark for soil modulus nonlinearity. 5) Nonlinear small strain stiffness of soils can have significant influence on determ ination of equivalent elastic modulus used for linear design and analysis. 6) It may be possible to determine single effective modulus which can approximate base layer nonlinearities, for a range of pavement structures and loads. PAGE 56 56 Figure 2 1. Characteri stic modulus strain behavior of soils with typical strain ranges for laboratory tests and structures ( Reprinted by permission from Atkinson and Sallfors ( 1991) ) . Figure 2 2. G/Gmax (Reprinted by permission from Seed et al. (1986) ) PAGE 57 57 Figure 2 3. G/Gmax undisturbed and reconstituted riverbed gravel (a) for cc=0.59 ( Reprinted by permission from Yasu da et al. (1996)) PAGE 58 58 Figure 2 4. max = 1.27 cm) ( Reprinted by permission from Li n et al. (2000)) Figure 2 5. Variation of normalized shear modulus with shear strain (dmax= .127 cm) ( Reprinted by permission from Li n et al. (2000)) PAGE 59 59 Figure 2 6. G 3 = 58.7 kN/m2) ( Reprinted by permission from Li n et al. (2000)) Figure 2 7. Variation of Gmax 3 ( Reprinted by permission from Li n et al. (2000)) PAGE 60 60 Figure 2 8. G/Gmax lo ( Reprinted by permission from Menq (2003)) Figure 2 9. Comparison of calculated and measured values of normalized shear modulus ( Reprinted by permission from Hardin and Drnevich ( 1972) ) PAGE 61 61 Figure 2 10. Data points def ining G/Gmax by all 15 investigators along with best fit curve and one standard deviation bounds for entire data set ( Reprinted by permission from Rollins et al. ( 1998)) PAGE 62 62 a b Figure 2 11. a) G max containing 0, 20, 40 and 60% gravel size particles ( Reprinted by permission from Rollins et al. ( 1998)). PAGE 63 63 Figure 2 12. Mean curves defining G/Gmax avelly soils at various confining pressures along with standard deviation boundaries for reduced data set ( Reprinted by permission from Rollins et al. ( 1998)). Figure 2 13. Shear modulus degradation curve proposed by Santos and Correia (2001) (Reprinted by permission from Santos and Correia (2001) ) PAGE 64 64 Figure 2 14. Shear wave velocity versus degree of saturation for different materials : a) C lean glass beads ; b) M ixture of Kaolinite and glass beads; c) Granite powder; d) S andboil sand ( Reprinted by pe rmission from Cho and Santamarina (2001)) PAGE 65 65 0 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 Time (days) E (ksi) ___ LOXAHATCHEE NEWBERRY OCALA MIAMI GEORGIA Figure 2 15. FFRC test results of specimen exposed to constant moisture ( Reprinted by permission from Toros (2008)) Figure 2 16. The FFRC test results of specimens exposed to laboratory ambient ( R eprinted by permission from Toros (2008)) PAGE 66 66 Figure 2 17. The FFRC test results of each material during first cycle of oven drying ( Reprinted by permission from Toros (2008)) Figure 2 18. The FFRC test results of each material during first cycle of wet ting ( Reprinted by permission from Toros (2008)) PAGE 67 67 Figure 2 19. Load displacement curves at Bothkennar footing test, fitted using the proposed model (footing performance data after High et al. ( 1997), Jardine et al. ( 1995) ) (Reprinted by permission from E lhakim and Mayne (2008)) Figure 2 20. Comparison of measured foundation load settlement response at Shenton Park to prediction using Plaxis HS Small model ( Reprinted by permission from Lehane et al. (2008)) PAGE 68 68 CHAPTER 3 EXPERIMENTS The first objective o f this research study is to characterize modulus nonlinearity of selected base soils via a suitable laboratory testing program. Laboratory testing methodology should be selected such that it can effectively account for confinement stress, loading strain magnitude and moisture content effects on nonlinear modulus, including small strain nonlinear modulus. In literature review of Chapter 2, various research studies conducted to determine nonlinear small strain shear modulus of sands and gravelly soils, via re sonant column testing are discussed. Based on literature review and testing goals. Fixed Free Resonant Column test and Free Free Resonant Column test are selected for our laboratory testing, for nonlinear modulus characterization. Background, mechanism and limitations of these testing methods are discussed in this chapter. 3.1 Fixed Free Resonant Column Testing Fixed Free Resonant Column test (Fixed Free RC) is a dynamic testing method, which can determine material nonlinear shear modulus (G) under differe nt confinement pressures and at very small to medium level strains. 3.1.1 Background and Testing Mechanism Resonant Column method is based on one dimensional wave equation derived from the theory of line a r elastic vibration. For our testing, FixedFree Re sonant Column Torsional Shear (Fixed Free RCTS) testing device is used. Using this equipment, both resonant column and torsional shear tests can be preformed. For our testing FixedFree Resonant Column (Fixed Free RC) mechanics only is utilized. In Fixed F ree RC device, the soil column is fixed at the base and free to rotate at top. With this apparatus, an external cyclic torsional load is applied to top of the specimen. The loading frequency is gradually changed until maximum response (strain PAGE 69 69 amplitude) is found. The lowest frequency at which maximum strain amplitude obtained is the n) of the soil specimen and driving system, for that specific applied torsional load. This fundamental frequency is function of soil modulus, specimen ge ometry and characteris tics of resonant column device. 3.1.2 Shear Modulus (G) The governing equation for motion for the F ixed F ree RC test as idealized in Figure 3 1a, for torsional vibration with a KelvinVoigt soil model is derived as follows. For a torque ( T ) applied to an elastic soil cylinder which generates angle of twist ( ) along an incremental length of specimen ( dz ) can be expressed as: (3 1) where T = torque, G = shear modulus of the soil and J = polar moment of iner tia of the cross sectional area From Figure 3 1b, the torque on two faces of t he soil element are T and T+ Using the torque ( T ) from E quation 31, we obtain: (3 2) Applying Newtons second law to the motion of the soil column and equating this net torque to the product of the mass polar moment of inertia and the angular acceleration: (3 3) where I = mass moment of inertia = = soil mass density From soil mass density ( ) and shear wave velocity ( Vs) Shear modulus ( G ) can be calculated as: s 2 (3 4) Substituting from E quation 33 and using E quation 34, we obtain the wave equation in to rsion for an elastic rod: PAGE 70 70 (3 5) The general solution to E quation 35, is found using separation of variables as : (3 6) n=natural circular frequency and A and B=constant s dependent on t he boundary conditions of the soil column. The boundary conditions in this FixedFree RC system are: 1. The angular displacement at the bottom (fixed end ) is zero 2. The torque at the top of the soil specimen (free end) is equal to the inertia torque of the dri ve system but opposite. By solving the E quation 31, by substituting these known boundary equations, final resulting expression is: (3 7) where I mass moment of inertia, I0mass moment of inertia of drive system, h height of specimen Once the shear wave velocity ( Vs) is determined, shear modulus ( G ) can be calculated from E quation 34. 3.1.3 The shear strain in a cylindrical resonant column loaded in torsion varies from zero at the center line of specimen to a maximu m value at its outer edge as shown in Figure 3 2. Since a single or unique value of shear strain amplitude with the measured shear modulus ( G ) is required, conventionally requ is assumed as 2/3 ( r0) for solid specimens with radius r0. Shear Strain ( ) is calculated as follows: h requ r max ) ( (3 8) Where equr equivalent radius of specimen = 03 2 r PAGE 71 71 0r Radius of solid specimen, max maximum angle of twist= sensorr x sensorr distance of target center (or fiber optic sensor) from specimens central axis x radial displacement of target. The radial displacement of target is measured by fiber optic sensor system, which can measure strains in the rang e of 105% to 101%. 3.1.4 Equipment Setup Fixed F ree RC equipment consists of a perpex glass cylindrical chamber with leak proof top and bottom covering plates at both ends of the chamber (Figure 33 and Figure 34). Bottom cap of the specimen is attached firmly to the chambers bottom plate, and top end of the specimen is left free to rotate, thus bottom end becomes fixed end and top end becomes free end. Torsional loading motor is attached to specimens top cap with proper supporting system. A fiber opti c sensor target to measure radial displacements is attached to specimens top cap. This target extends out from the top cap, in specimens radial direction. Fiber optic sensor cable is attached to one of chamber supporting rods and positioned facing the ta rget. This sensor system measures the radial displacements of target caused due to torsional loading. Entire testing chamber unit is fixed firmly to a loading frame for stability. Tests can be run and controlled through a software program, which sends cont rol signals to resonant column interface and servo amp unit (Figure 35a), control ling torsional loading motor and amount of load to be applied. The fiber optics sensor system (Figure 3 5c) sends target displacement information back to the controlling soft ware, which eventually ca lculates the resonant frequency ( n) internally using software (Figure 3 6). Air confinement can be applied via pressure control panel (Figure 35b) 3.1.5 Calibration of the Drive System: When torsional load is applied during tes ting, torsional motor rotates top specimen cap along with the testing specimen. Weight of the cap and motor, along with specimens weight, PAGE 72 72 becomes part of total weight of the testing system. To nullify the effects of top specimen cap and motor weight and t o determine the properties of specimen alone, calibration of driving system is required. Calibration of driving system is performed using a metallic specimen instead of real soil specimen (Figure 3 7). The metallic specimen is assumed to have zero or close to zero damping and a constant torsional stiffness ( k) Then from Newtons second law, the mass moment of inertia is related to the natural or resonant frequency ( n) as follows: (3 9) Recommended procedure to find the mass moment of inertia of the driving system ( I0) is to pe rform two resonant column tests with the metal calibration specimen, one by itself and the other with an added mass. After performing frequency sweep with constant force amplitude, the solutions will be: without added mass: (3 10) with added mass : (3 11) where I0=mass moment of inertia of the drive system and any other fixture that will be used during actual soil testing Ical = mass moment of inertia of the calibration sp ecimen Imass = mass moment of inertia of the added mass 1 = resonant frequency of calibration specimen without the added mass 2 = resonant frequency of calibration specimen with the added mass By combining equations 310 and 311, we get: (3 12) PAGE 73 73 3.1.6 Limitations (We Experienced) of This Testing Equipment: The maximum frequency that can be applied using the torsional loading motor of our Fixed Free RC equipment is 300 Hz. In other words, specimens with resonant frequency less than 300 Hz only can be tested with this equipment. Part of our testing program (explained in detail in Chapter 5) is to run tests on specimens compacted at optimum moisture content and then dried to different water contents, thus the modulus of material can be determined at different water content s and at different strain levels. But, after drying the specimen to below certain water content (as explained in later chapters), its resonant frequency is reaching beyond 300 Hz due to increase in its modulus. As a result, we are able to determine material modulus up to certain water content below OMC, but not all the way dried to zero water content. 3.1.7 Equipment Credibility Verification After installation, equipment is subjected to verification to make sure that the testing measurements are correct and credible. This verification is performed by conducting tests on Ottawa sand. For Ottawa sand, experimental results determined for shear modulus are available in literature, and thus our testing results can be compared w ith literature data for verification. Verification test is conducted on 4 inch diameter and 8 inch height Ottawa sand specimen compacted to minimum void ratio (emin) of 0.435. This compacted specime n is subjected to six c) 50, 100, 150, 200, 250 and 300 KPa. Shear modulus (G) at different strain levels in the strain range of 104% to 101% are measured at each confinement. These testing results are shown in Figures 3 8 and 39. With the very smallstrain modulus (Gmax4%) data at different confining pressures, Gmax c) plot i s d eveloped and shown in Figure 3 9. From this plot, an empirical relationship to calculate Gmax is developed in the form of (proposed by Hardin and Richart (1963) ) PAGE 74 74 (2 5) where AG and nG are material constants and F(e) = (2.17e)2/ (1+e), Gmax c l are in kPa. From our testing data, obtained value for nG=0.5165 and e=0.435, F(e)=2.098 By subst ituting values of F(e) and nG in Equation 2 5, and solving it with the testing data, empirical equation obtained to calculate Gmax is: Gmax = (5350) (2.098) ( c l) 0.5165 (3 13) nG= 0.5165 obtained from our testing matches well with the literature da ta for Ottawa sand (Hardin and Richart ( 1963) Menq ( 2003) ). Also, value of AG=5350 fall in the range of 3300 to 9000 for sands (Menq ( 2003) ). Thus, by comparing our Ottawa sand testing results with literature data, the credibility of Fixed Free RC equipmen t data is successfully verified. 3.2 Free Free Resonant Column Testing (F reeF ree RC) F ree F ree RC testing method (Kim and Stokoe (1992), Menq (2003), Toros(2008)) is used to determine very smallstrain modulus (Gmax or Emax) (i.e. 5%) of Florida base materials at different water contents. This test can be conducted very quickly on laboratory compacted specimens. Further, F ree F ree RC test is nondestructive, and thus can be conducted on the same specimen at different water contents by drying it as required. 3.2.1 Back Ground and Mechanism Two different types of stress wave measurements can be conducted on a solid rod with F ree F ree RC testing: 1) resonance measurements and 2) direct arrival measurement. With PAGE 75 75 known specimen n), unconstrained modulus or Youngs modulus (E) can be determined using e quations 314 and 315. Unconstrained compression wave velocity Vc n l (3 14) where l length of specimen, Youngs modulus E c 2 n l)2 (3 15) of constrained compression wave is determined via direct arrival measurement, from which constrained compression wave velocity ( Vp) is calculated as (3 16) where: l = length of the s pecimen, t = measured travel time of constrained compression wave With known constrained compression wave velocity ( Vp) and unit mass of the specimen ( ) constrained modulus ( M ) can be calculated as 2 2 t Mp (3 17) With known constrained and unconstraine d wa ve velocities, Poissons ratio ( ) can be calculated as: 2 2 2 2 2 24 1 8 1 1 c p c p c p c p c p ME (3 18) With known Poissons ratio ( ), Youngs modulus ( E ), and constrained modulus ( M ); shear modulus ( G ) can be calculated as: PAGE 76 76 ) 1 ( 2 E G (3 19) ) 1 ( 2 2 M M G (3 20) For our F reeF ree RC testing, we used the same equipment used by Toros (2008) to measure very small strain modulus of Florida base materials. A detailed back ground and mechanism explanation about F ree F ree RC testing and equipment verification was given in Toros (2008) P hD. dissertation. 3.2.2 Equipment Set up: Modified Proctor compacted specimens without any confinement of casing or membrane are tested with this equipment. The same testing procedure, which was verified by Toros (2008) is followed. Our F ree F ree RC testing system consists of a dynamic signal analyze r (DSA) or (oscilloscope), an instrumented impact hammer and a n accelerometer (transducer). S pecimens are oriented horizontally and suspended with flexible straps to achieve free free boundary conditions (Figure 310). E xcitation point with impact hammer is at the center of one end of the specimen, and location of accelerometer is at the center of other end of the specimen. A ccelerometer is glued to the center of one end of the specimen. The main difference between our testing procedure and Toros (2008) tes ting procedure is confinement with casing around specimen (Figure 311). Toros (2008) tested 6 inch diameter and 12 inch height specimens with plastic casing around them, where as we tested 4 inch diameter and 8 inch height naked specimens without any conf ining c asing PAGE 77 77 (a) (b) Figure 3 1. Fixed Free RC testing specimen ( a) I dealized testing specimen (b) D ifferential soil element Figure 3 2. Shear Strain in soil specimen PAGE 78 78 Figure 3 3. Fixed Free RC equipment used for our testing (Photo courtesy of Ayithi (2011)) PAGE 79 79 Figure 34. Sectional view of Fixed Free RC testing equipment PAGE 80 80 a) (b) (c) Figure 35. Controlling units of Fixed Free RC equipment a) Resonant Column interface unit and servo amp unit b) Pr essure control panel c) F iber optic sensor placed facing the target attached to specimen (Photo courtesy of Ayithi (2011)) PAGE 81 81 Figure 3 6. Real time test execution window, showing resonant frequency reaching loop (Photo courtesy of Ayithi (2011)) PAGE 82 82 Figur e 37. Calibration specimen with added mass (Photo courtesy of Ayithi (2011)) PAGE 83 83 Figure 38. Shear modulus versus % shear strain curves at different confining pressures for Ottawa sand. Figure 39. Gmax c relationship curve for Ottawa sand PAGE 84 84 (a) (b) Figure 3 10. Free Free Resonant Column test equipment se tup a) Overall setup b) S pecimen with transducer and instrumented impact hammer (a) (b) Figure 3 11. Free Fre e resonant column testing specimen a) W ithout confinement casing b) W ith confinement casing ( tested by Toros (2008)) (Photo courtesy of Ayithi (2011)) PAGE 85 85 CHAPTER 4 MATERIALS AND SPECIMEN PREPARATION In the State of Florida, limerock type aggregates are commo nly used for base layer construction. For our testing, it is proposed to select one limerock material and one nonlimerock material for our testing. Newberry limerock and Georgia graded aggregate base a re used for our testing representing one from each category. A detailed explanation about their source, particle size distribution, basic parameters and specimen preparation are given in this chapter 4.1 Materials 4.1.1 Sources Newberry limerock and Georgia granite graded aggregate base a re selected for our testing. Material s ources for these two soils are given in Table 41. Representative samples of these selected materials are collected by FDOT SMO (Florida Department of Transportation State Materials Office) staff from mines mentioned in Table 41, foll owing FDOT standard method, i.e. Florida Methods 1 (FM 1) T 002 that is similar to AASHTO T2 4.1.2 Characterization These collected samples are transported in bags, to FDOT SMO laboratory for further characterization and lab testing. Before further testin g, transported sample bags a re placed in a thermostatically controlled drying oven at a temperature of 110F until the samples a re friable. After letting these oven dried samples to cool down, sieve anal ysis and basic parameter tests are conducted. Represe ntative samples of these two mat erials are shown in Figure 41. Sieve analysis i s performed following the procedure AASHTO T 27. G rain size distribution for particles larger than #200 size (i.e. 0.075 mm) is shown in Figure 42. PAGE 86 86 Specific gravity of fine and coarse aggregates a re per formed following the procedure FM 1 T 084 and T 085, which are similar to AASHTO T084 and T085 procedures, respectively. Atterberg limits a re determined by following AASHTO T90 for plastic limit and plasticity index and AASHTO T89 for liquid limit. These basic material parameters are presented in T able 4 2. 4.2 Specimen Preparation and Installation Fixed Free RC tests are conducted on both dry and wet (partially saturated) compacted cylindrical sp ecimens and FreeFree RC tests are conducted on wet compacted specimens. 4 inch (10.16 cm) diameter and 8 inc h (20.32 cm) height specimens a re prepared following standard method for specimen preparation and detailed explanation is given in later sections. Aggregate pas sing through inch si eve only i s used for specimen preparation. After oven drying of mine collected samples, aggregate that does not pass through inch sieve i s separated and send through limerock crusher, so that all the aggregate is smaller than inch. This crushed aggrega te i s mixed with uncrushed aggregate and the whole material is used for specimen preparation satisfying gr a dation requirements. 4.2.1. Dry Specimens 4.2.1.1 Dry s pecimen p reparation Dry specimens are used in Fixed Free RC testing only. Dry specimens a re prepared at three different void ratios for each material as presented in Table 4 3. Dry specimens are prepared following split mold and membrane method. Initially, a split mould with me mbrane on inner wall of mould is attached to specimens bottom cap (pe destal) (Figure 4 3a). Dry aggregate i s placed inside the split mold and compacted by dropping a loading hammer manually (Figure 4 3b). The amount of energy required (i.e. height of drop, no of layers and weight of hammer) to compact the dry aggregate to obtain targeted void ratio i s determined by trial and error method. After completing compaction, to keep the specimen intact without PAGE 87 87 falling off after removal of split mould, vacuum i s applied through bottom drainage line (Figure 43c). Our p rimary goal in dry compaction i s to achieve targeted void ratio. 4.2.1.2 Dry s pecimen i nstallation For dry specimens, after completing compaction, top cap with vertical and horizontal serrations is placed firmly such that there is proper contact grip between soil and cap F riction due to serrations on both top and bottom caps provides a good grip between soil and cap. Due to this contact grip, entire soil column rotates as a single integral unit, when torsional load is applied. After attaching top cap and loadi ng motor, confinement chamber i s placed and required air confinement i s applied through pressure control panel (Figure 43d). Once the dry compacted specimen stands stable by externally applied air co nfinement, vacuum application is disconnected and released off. 4.2.2 Wet (Partially Saturated) Specimens 4.2.2.1 Wet s pecimen p reparation Wet specimens a re compacted at optimum moisture content (OMC) using modified proctor compactor (Figure 4 4a) Before compaction, the soil water mixture i s soaked for at least 12 hour s in a nylon sheet covered container (Figure 4 4b) for uniform distribution of water throughout the soil mixture. After soaking, the mixture i s compacted in a standard size mold used for making 4 inch diameter and 8 inch height specimens (Figure 4 4c) Nu mber of layers and number of blows required per each layer a re estimated based on AASHTO D1557 (Standard test method for laboratory compaction characteristics of soils using modified effort). Specimens are compacted in total six layers and 36 blows per lay er, which meet s the required compaction effort of 2700 kN m/m3, according to AASHTO D1557. A fter extruding from compaction mould, specimen i s covered with a 0.012 thick rubber membrane and allowed to sit for 12hours in an air tight container at room temp erature before testing. PAGE 88 88 4.2.2.2 Wet s pecimen installation For wet OMC compacted specimens, s pecimens top and bottom caps a re glued to the specimen using bondo. Glue material i s selected such that it should not influence the testing measurements (i.e. reso nant frequency n) and material properties. Bottom cap of the specimen i s attached firmly to the chambers bottom plate, and top end of the specimen i s left free to ro tate. Torsional loading motor i s attached to specimens top cap with proper supporting s ystem PAGE 89 89 Table 4 1. List of materials used for testing and their sources Material Mine Number or Source Type Newberry limerock 26 002 Newberry Quarry Limerock Georgia g ranite Graded Aggregate Base (GAB) GA 178 Macon Quarry Granite Table 4 2. Basic mat erial parameters Parameter Georgia g ranite Newberry l imerock Unified soil classification GW GM GM D50 (mm) mean grain size 5 3 D10 (mm) effective grain size 0.16 0.13 Cuthe coefficient of uniformity 50 61.5 Ccthe coefficient of curvature 1.76 0.15 G Specific gravity 2.74 2.72 Maximum dry density (kN/m 3 dry max ) 22.08 18.21 Optimum moisture content (OMC) (%) 5.5 13 Void ratio at OMC (e OMC ) 0.20 0.45 Plastic limit NP NP Plasticity index NP NP Liquid limit NP NP where: Cu = D60 / D10, Cc = (D30)2/ (D60 D10) Table 4 3. Selected void ratios for dry specimen testing Material Void Ratio (e) 1 (e OMC ) 2 3 Newberry l imerock 0.45 0.50 0.55 Georgia g ranite 0.20 0.25 0.29 PAGE 90 90 (a) (b) Figure 4 1. Representative samples a) Georgia granite b) Newberry limerock Figure 4 2. Grain size distribution PAGE 91 91 (a) (b) (c) (d) Figure 4 3. Dry specimen compaction and installation a) S plit mould with membrane inside b) S plit mou ld with dry compacted limerock c) A fter application of vacuum d) A fter installation of chamber and external confinement application. PAGE 92 92 (b) (a) (c) Figure 4 4. Wet specimen compaction a ) M odified proctor compactor with 4 inch diameter and 8 inch height specimen mould b) S oaked aggregate trays with nylon cover sheet c) C ompacted wet specimens of 4 inch diameter and 8 inch heig ht PAGE 93 93 CHAPTER 5 EXPERIMEN TAL RESULTS AND ANAL YSIS The first objective of this research work is characterization of base soils modulus nonlinearity with respect to confinement stress, loading strain and moisture content, via laboratory testing. To achieve this objective, it is proposed to conduct Fixed Free Resonant Column tests (Fixed Free RC) and Free Free Resonant column (Free Free RC) tests on selected representative base soils. Testing mechanisms and their background are explained in detail in Chapter 3. Base soils selected for our testing, their basic properties characterization and specimen preparation methods are explained in Chapter 4. Fixed Free RC tests are conducted on both dry and unsaturated (wet) compacted specimens of Newberry limerock and Geo rgia granite to determine shear modulus (G) under different confinement pressures and at strain levels as small as 105% (very small level strain s) and as high as 101% (medium level strain s ). Free Free RC tests are conducted on unsaturated (wet) compacted specimens of Newberry limerock and Georgia Granite to determine very smallstrain modulus ( Emax or Gmax) under no confinement Specimen testing conditions such as water content, confinement pressure and strain magnitude are presented in follow ed sections. Experimental results are discussed and analyzed. 5.1 Fixed Free Resonant Column Torsional Shear Testing 5.1.1 Dry Specimen Testing Results Dry compacted specimens of Newberry limerock and Georgia granite are tested at three different void ratios as report ed in Table 4 3, to investigate the effect of void ratio. Specimen at each void ratio is subjected to four different confining pressures 50, 100, 150 and 200 kPa, to inves tigate confinement effect. s for PAGE 94 94 eOMC (i.e. e=0.45 for Newberry limerock and e=0.20 for Georgia granite) are shown in Figures 51 and 52. and plots corresponding to remaining two void ratios are presented in Appendix A. Ba sed on literature (Seed et al. ( 1986) Menq (2003), Atkinson ( 2000) ), for dry gravels strains less than 104% are referred as very small level strains and corresponding shear modulus is at its maximum (i.e. Gmax). Thus 5% as G max, normalized shear modulus (G/Gmax) versus % shear strain curves are developed and shown in Figures 53 and 54. From Figures 5 1 and 5 2, it can be observed that At constant confining pressure, modulus decreases with increase in strain magnitude. At constant strain magnitude, modulus increases with increases in confinement and rate of increase is maximum at very smallstrain (i.e. 105%). Modulus is maximum (i.e. Gmax) and elastic at strains lower than 104% and s tarts decreasing thereafter with increase in strain. Modulus decreases linearly in the strain range of 104% to 103% and nonlinearly thereafter. Similar results for gravelly soils were reported by Menq (2003). Seed et al. (1986) and Rollins et al. (1998) also observed that in gravelly soils G starts decreasing from 104% strain with increase in strain magnitude. Based on Ottawa sand testing results (Figure 3 7a) and literature database, modulus of sands are maximum and elastic at strains smaller than 103% and decrease thereafter, with increase in strain. By comparing results of gravelly soils with sands, it can be concluded that presence of gravel size aggregates increases nonlinearity in modulus reduction. This confirms with the findings of Seed et al (1 986) and Menq (2003) testing results on gravels. The difference in moduli at different confining pressures and for same strain magnitude, decreases with increase in strain. This indicates that confinement effect on modulus decreases with increase in strain magnitude. At strains higher than 101%, modulus does not vary significantly with respect to either increase in strain or increase in confinement. This observation indicates that at larger PAGE 95 95 strains (i.e. 101% and higher) pressure confinement and strain ma gnitude have no significant influence on moduli of gravelly soils. For both soils, modulus reduction curves at constant confinement are flatter for 50 kPa and 100 kPa compared to 150 kPa and 200 kPa. This behavior implies that nonlinearity of modulus reduc tion increases with increases in confinement. Similar behavior in sands was reported by Pestana and S alvati (2006) for Monterey sand. G/Gmax normalized curves for any confining pressure ( Figures 5 3 and 54) are falling on each other and behaving very simi lar. This indicates that rate of decrease in modulus with increase in strain does not depend on confining pressure. In Figure 5 5, normalized data points for both materials lie well within the maximum and minimum ranges of gravel soils reported by Seed et al. (1986). Normalized curves at different confining pressures are falling on each other and this behavior indicates that normalization of modulus nullifie s the confinement effect on modulus. Lin et al. (2000) (Figure 2 5) and Yasuda and Matsumoto (1993) (Figure 2 3) reported same behavior for gravelly soils. But Rollins et al. (1998) reported that, G/Gmax are different for different confinements and move from low end of the data range towards high end with increase in confinement (Figure 2 12). However, they also pointed out that the deviation between various confinements curves are relatively small and use of same best fit hyperbolic curve for any confining pressure would not likely cause significant error. Gmax values (G at 105% strain) at different void ratios (e) and different confining pressures are plotted on a log log plot of modulus versus confining pressure a nd shown in Figure 56 and Figure 57. Empirical e quations in the form of Eq uation 25 (shown below) are derived PAGE 96 96 for each mat erial; by plotting trend line for whole set of Gmax values at different void ratios as shown in Figure 58. (2 5) where AG and nG are material constants. Gmax and 0 are in kPa, Derived empirical equations for each mat erial are given below: For Newberry limerock, at any given void ratio (e), Gmax = (2575) F ( e c )0.7 0 2 ( 51) For Georgia granite, at any given void ratio (e), Gmax = (816) F ( e c )0.6 389 ( 52) Shear modulus of Newberry limerock is proportio nal to pressure confinement to the power of 0.702 and of Georgia granite is proportional to pressure confinement to the power of 0.6389. Values of nG obtained from this testing complies well with the values given in literature, i.e. in the range of 0.5 0.85 for gravel type soils (Menq, ( 2003) ) These equations can be used to calculate Gmax of dry material at any confining pressure c ) and void ratio ( e ). 5.1.2 Unsat urated (W et) Specimen Testing Results Primary goal of conducting Fixed Free RC tests on unsaturated specimens i s to characterize modulus nonlinearity at different water contents and different strain levels ; and to evalu ate suction effect due to drying, on modulus Based on previous research investigations conducted by Toros(2008), very small strain modulus (Gmax) of base course soils increases with decrease in water content. He dried OMC compacted specimens in different environments and tested them at different water contents during the process of drying starting from OMC to all the way close to zero water content. Cho and Santamarina (2001) reported that, specimen drying simulates actual dr ying in real filed conditions and makes the material stiffer compared to compacting the specimen at required water content. So, the modulus of material is higher in case PAGE 97 97 of dried specimen than in case of compacted specimen at targeted water content. Hence, to simulate actual filed con ditions for our testing, it was decide to achieve different water contents by drying OMC compacted specimens. For our testing program, 4 inch diameter and 8 inch height cylindrical specimens a re compacted at OMC using modified proctor compactor. These OMC compacted specimens a re allowed to dry in lab environment (i.e. at room temperature), by leaving them open in lab, without any membrane cover. Specimens dried to different water contents in this method ar e tested to determine modulus at different shear strain levels, as low as 105% (ver y smallstrain) and as high as 101% (medium to large strains). 5.1.2.1 Equipment l imitations During our testing, a problem related to Fixed Free RC equipments inability to test a specimen dried to below certain water co ntent i s encountered. The basic mechanism of Fixed Free RC n) corresponding to applied torsional load, in order to calculate the shear wave velocity (Vs) from which shear modulus (G) can b e estimated. Thus the resonant frequency of the testing specimen should be within the motor applicable frequency capacity, i.e. less than 300 Hz (also explained in 3.1.6). In other words, specimens shear modulus value should be such that its resonant frequency is not more than 300 Hz. But, during our testing process it is realized that modulus of specimen is increasing with decrease in water content A fter reducing to certain water content, modulus is increas ing such that its resonant frequency is going beyond 300 Hz, which is the maximum limit of torsinal loading motor frequency Hence, it is decided to conduct tests at water contents, for which the specimens resonant frequency is less than 300 Hz. Based on this limitations, specimen tested water content s for both materials are given in Table 5 2. PAGE 98 98 5.1.2.2 Confinement effect on unsaturated s pecimens Similar to dry specimen testing, initially it is planned to tests unsaturated specimens under zero (i.e. no confine ment), 50, 100, 150 and 200 kPa confinement pressures. But, when tests are run on unsaturated specimens under confinement, no increase in modulus is observed with increase in confinement. Possible reason for this phenomenon may be explained through unsaturated soil mechanics. In the science of soil mechanics, it is well known that modulus of soil increases with increase in confinement. It is also known in the science of unsaturated soil mechanics that under normal loading conditions, effective stress does not change with increase in total stress due to pore pressure effect in unsaturated soils and so the modulus. Based on these reasons, it is presumed that due to pore pressure effect inside the testing specimens, no increase in modulus is observed with increase in confinement. Thus to determine actual modulus of unsaturated specimen under confinement, it is necessary to measure pore pressure inside the specimen with proper pore pressure measuring equipment. But pore pressure measurement was not part of our i nitial testing plan, hence it is decided to conduct test s on unsaturated specimens with no confinement only. Based on these unconfined testing results, unsaturated material modulus under confinement can be calculated approximately by an indirect method, which is explained in later sections of this ch apter. 5.1.2.3 Results and a nalysis Shear modulus versus strain plots of unsaturated specimens tested at different water contents, under no confinement are shown in Figure 59 for Newberry limerock and in Figure 5 10 for Georgia granite. Dry specimen tes ting curve corresponding to 50 kPa confining pressure is also included for comparison purpose. From these plots it can be observed that a t constant strain magnitude, modulus increases with decrease in water content and is maximum (i.e. Gmax) at very smallstrain (i.e. 105%). curves at different water contents are behaving similar to PAGE 99 99 dry curves at different pressure confinement s presented in Figure 5 1 and Figure 5 2 Since the void ratio is same for both dry and wet specimens (i.e. @ eOMC), increase in G with decre ase in water content is probably due to increase in effective confinement with decrease in water content, which is probably due to increase in suction with decrease in water content. Though, we did not make any pore pressure measurements to prove the prese nce of suction, research investigations by Wu et al. (1984), Qian et al (1993) and Cho and Santamarina (2001) reported that decrease in water content increases suction and thus additional confinement. This phenomenon is also well explained in unsaturated soil mechanics (Lu and Likos ( 2004) ) At different water contents, the difference in moduli at same strain magnitude decreases with increase in strain and similar phenomenon is also observed in dry soils at different confinements. This implies that suctio n confinement effect decreases with increase in strain G water content specimens. This is probably due to failure of suction menisci at lower strains in low water content specimens and at higher strains in high water c ontent specimens Cho and Santamarina (2001), from their microscale particle menisci studies, reported that the strain at menisci failure decreases with decrease in water content and small menisci in relatively dry specimens may fail before the strain at p eak strength of soil. Hence, at low water contents small suction menisci cause high small strain modulus and modulus decreases at faster rate due to early menisci failure. In dry condition, shear modulus is maximum (i.e Gmax) at 104% strain and decrease t o 0.15Gmax at 101% strain. Whereas, in unsaturated condition, shear modulus is maximum at 105% strain and decrease to 0.075 Gmax at 101% strain. This implies that in unsaturated gravelly soils, presence of moisture increases nonlinearity in modulus reduction and modulus starts PAGE 100 100 decreasing from 105% strain. This same behavior was observed by Menq (2003) in gravelly soils. Increase in modulus nonlinearity is probably due to water lubrication effect, which reduces inter particle frictional contact forces, a llowing soil particle movement easier and irreversible, even at strains lower than 104%. Similar to dry specimen s in wet specimens also, at strains near 101% and higher, the difference in moduli at different water contents is very minimal. This implies that suction confinement effect on modulus becomes less significant at higher strains. In case of Newberry limerock, Gmax at 13% (OMC) and 12% water content is lower than that of dry case at 50 kPa pressure confinement, which implies that additional conf inement due to suction is less than 50 kPa at 13% and 12% water content s and higher than 50 kPa at water content s 11% and lower. Similarly, in case of Georgia Granite, Gmax at 5.5% (OMC) water content is almost equal to Gmax of dry soil at 50 kPa, which me ans at water contents lower than 5.5%, confinement due to suction is greater than 50 kPa. Additional confinement provided due to suction is calculated and corresponding results are reported in later sections. Normalized curves of G/Gmax versus logarithmic for both dry and wet specimens together on the same plot are shown in Figure 511 and Figure 5 12. From these plots, it can be observed that in the case of dry specimens, normalized curves for any confining pressure are falling on each other and behaving very similar. This indicates that rate of decrease in modulus with increase in strain does not depend on confining pressure. In the case of unsaturated specimens, normalized curves for any water content are falling on each other. This in dicates that rate of decrease in modulus of unsaturated specimens, with increase in strain, does not depend on water content and its suction effect. Entire set of unsaturated specimens normalized curves are falling below that of dry specimens and t his in dicates that rate of decrease in modulus with increase in strain is higher in unsaturated soils compared to dry soils, PAGE 101 101 and hence the modulus nonlinearity is higher in unsaturated condition compared to dry condition. Based on microscale particle level studi es of Cho and Santamarina (2001), higher modulus nonlinearity in unsaturated specimens is due to early failure of small menisci at very small level strains. Menq (2003) observed the same difference in case of dry and partially saturated gravelly soils beha vior. Shear modulus (G) normalized with shear modulus at OMC (GOMC) versus water content plots for different strain levels are shown in Figures 513 and Figure 5 14. Curves of G/GOMC versus water content, for different strain magnitude s are approximately falling on each other and behaving more or less similar. A t any given strain magnitude increase in G / GOMC with decrease in water content is approximately same. In other words, additional confinement provided due to suction effect (various magnitudes at di fferent water contents) is not changing, i.e. same at different strain levels. This behavior indicates that in unsaturated soils, for the range of strain magnitude s measured in this testing, at given water content, additional confinement due to suction eff ect remains same irrespective of strain magnitude. It appears that, unsaturated material under additional confinement due to suction, behaves similar to dry material under external constant confinement. Cho and Santamrina (2001) reported in their particle level studies that, in strain magnitudes are smaller than threshold strains, Cho and Santamarinas findings support our observations from the plots that suc tion effect remains same irrespective of increase in strain magnitude. Since its an indirect conclusion, a further detailed particle study is required to analyze the strain level effect on menisci failure and its suction effect. PAGE 102 102 5.1.2.4 Additional e ffec tive s tress or additional confinement pressure provided due to suction, at 105% strain magnitude By comparing wet specimen testing results with that of dry specimens, additional effective stress or confinement pressure provided due to suction can be eval uated. A detailed evaluation procedure is explained here. From dry specimens testing results, E quations 51 and 52 are derived to calculate Gmax (i.e. G at =105%) of dry material at different confining pressures. At eOMC Gmax for any chosen confining pressure can be calculated using equations 51 and 52. By substituting unsaturated specimens Gmax value obtained at given water content into these equations, confinement pressure required to be applied over a dry specimen to produce the equivalent modul us can be calculated. Since it is already known that all wet specimens are tested under no confinement, earlier calculated confinement pressure magnitude becomes the additional confinement pressure or effective stress being provide by suction in a wet spec imen at corresponding selected water content. Following this procedure, additional effective stress or confinement provide due to suction at different water contents or degree of saturation is determined and is shown in Figures 515 and Figure 516. From Figure 5 15 and Figure 516, it can be observed that increase in additional confinement for same amount of decrease in water content is relatively very high in Georgia gr anite compared to Newberry lime rock. This is due to low eOMC and higher reduction in degree of saturation (Sr) in Georgia granite. In soils with low void ratio, degree of saturation goes down faster compared to soils with high void ratio, for the same amount of reduction in water content. As the degree of saturation decreases faster, suctio n magnitude increases proportionally faster and hence increase in additional confinement at faster rate. PAGE 103 103 5.2 Free Free Resonant Column Testing Free Free resonant column (F ree F ree RC) tests are run on modified proctor compacted specimens of Newberry limer ock and Georgia Granite, under no confinement to measure very smallstrain Youngs modulus (Emax). OMC compacted specimens are dried in laboratory environment and tested at different water contents during drying process, starting at OMC to all the way clos e to zero. These results are presented and discussed in followed sections. Toros (2008) conducted same tests on relatively bigger specimens at different water contents similar to our drying process. He used a plastic cylindrical case around the specimens, which provides some confinement. Toros (2008) results are compared with our testing results. 5.2.1 Results and Discussion F ree F ree RC with no confinement test results for both Newberry lime rock and Georgia granite are shown in Figures 5 17 and 5 18 res pectively. These results show that in both materials, very smallstrain Youngs modulus (Emax) increase with decrease in water content. Their trends are similar to results reported by Toros(2008). Toros (2008) concluded that, increase in modulus with decre ase in water content is probably due to increase in additional confinement due to increase in suction and suction increase with decrease in water content. Cho and Santamarina (2001) also concluded the same for sands. Free Free RC testing results are compar ed with Fixed Free RC test results at very smallstrain (i.e. 105%) and Free Free RC results of Toros (2008) and presented in Figures 5 19 and 520. From the results in Figures 5 19 and 520, it can be observed that Fixed Free RC modul i values at 105% s train are more or less equal to that of Free Free RC results at corresponding water contents This implies that strain magnitudes of Free Free RC testing are nearly equal to 105%. A close observation of these plots tells that Fixed Free RC moduli value ar e slightly higher than Free Free moduli values. This difference is probably due to the weight of motor and PAGE 104 104 top specimen cap attached to specimen top in FixedFree RC testing, which might prov ide some vertical confinement, hence result ing i n little bit high er moduli values compared to Free Free RC moduli values at no confinement. At same water content, T oros (2008) Free Free RC moduli value s are higher than that of our Free Free RC moduli value. The materials and method of compaction are same for both tests and the only difference is casing confinement. Our Free Free RC tests are run on specimen s with no casing and Toros (2008) Free Free RC tests were run on specimens with plastic casing. Toros compacted specimens in plastic cylind rical moulds and tested speci mens with the mould on. T hese plastic cylinders might be providing some confinement around the side surface of the specimen, which might have resulted in higher moduli values compared to that of specimens with no confinement. 5.2.2 A Method to Estimate App roximate Modulus at Given Conditions of Water Content, Confining Pressure and Strain Magnitude: Gmax of dry material can be determined via Equations 51 and 52, which are derived based on dry material Fixed Free RC testing results. By knowing Gmax at any water content under no confinement (either from Fixed Free RC or Free Free RC) and substituting this value in Equations 51 or 52, additional effective confinement stress provided due to suction can be calculated by following the procedure explained in S ection 5.1.2.4. By adding this additional confinement stress to the initially chosen confinement where we need to estimate modulus approximate Gmax value can be calculated, via E quations 51 and 52. From Figure s 511 and 5 12, it is understood that G/Gm ax versus strain cu rves behave similar at any water content. Thus after calculating Gmax at given confi nement and water content using PAGE 105 105 E quations 51 and 52, value of G/Gmax at required strain level can be determined through Figure s 511 and 512, and finally approximate value of G can be calculated. 5.3 Laboratory Testing Results Closing Remarks Gmax of dry Newberry limerock and dry Georgia Granite at eOMC and known confi nement can be determined using E quations 51and 52. Shear modulus of dry gravell y soils is maximum and elastic at strains lower than 104% and starts decreasing linearly in the strain range of 104% 103% and nonlinearly thereafter. Presence of gravel size aggregate makes modulus of gravelly soils more nonlinear compared to sands. I n unsaturated gravelly soils, capillary suction confinement increases with decrease in water content and has significant effect on Gmax. Shear modulus of unsaturated gravelly soils is maximum at 105% strain and start decreasing thereafter with increase in strain. Presence of moisture increases modulus nonlinearity in unsaturated gravelly soils compared to dry gravelly soils. In dry gravelly soils, rate of decrease in G with increase in strain is independent of confinement pressure magnitude. In unsaturated gravelly soils, rate of decrease in G with increase in strain is independent of water content and its suction confinement. In unsaturated gravelly soils, at constant water content, additional confinement due to does not change with increase in strain. St rain magnitudes generated in FreeFree RC testing are approximately in the range of 105% and corresponding moduli are nearly equal to very small strain moduli obtained from Fixed Free RC testing Based on testing results, at 105% strain magnitude, additi onal confinement provided due to suction can be as high as 900 kPa, which is equivalent to 39m of overburden PAGE 106 106 pressure confinement. Since confinement due to suction can influence soil modulus, it is necessary to consider suction effects in determination of base layer design modulus. PAGE 107 107 Table 5 1. nG and AG values for dry Newberry lime rock and Georgia granite. Material nG AG Newberry limerock 0.702 3166.00 Georgia granite 0.6389 750.70 Table 5 2a List of unsaturated specimen tested water contents for N ewberry Limerock Material void ratio at OMC (e OMC ) Tested water contents (%) Degree of saturation, Sr (%) Newberry l imerock 0.45 13 78.6 12 72.5 11 66.5 10 60.4 Table 5 2b. List of unsaturated specimen tested water contents for Georgia Granite Material void ratio at OMC (e OMC ) Tested water contents (%) Degree of saturation, Sr (%) Georgia Granite 0.2 5.5 75.3 4.5 61.7 3.5 48 PAGE 108 108 Figure 5 1. Shear Modulus (G) versus % Shear strain eOMC (i.e. e = 0.45) Figure 5 2. Shear Modulus (G) versus dry Georgia granite at eOMC (i.e. e =0.20) PAGE 109 109 Figure 53. G/Gmax versus % shear strain OMC for Newberry limerock Figure 5 4. G/Gmax OMC for Georgia granite PAGE 110 110 Figure 5 5. Comparison of G/Gmax r ry lime rock and Georgia granite with Seed and Idriss (1986) maximum and minimum limits Figure 5 6. Gmax versus confining pressure curve Newberry limeroc k at different void ratios PAGE 111 111 Figure 5 7. Gmax versus confining pressure curve Georgia granite at different void ratios Figure 5 8. Gmax empirical equations for Newberry limerock and Georgia granite PAGE 112 112 Figure 5 9. Shear moduli (G) versus % shear curves for Newberry limerock at different water contents Figure 5 10. Shear modulus (G) versus % curves for Georgia granite at different water contents PAGE 113 113 Figure 511. G/Gmax versus shear strain curves for Newberry limerock f or both dry & wet specimens Figure 5 12. G/Gmax versus % shear strain curves for Georgia granite, for both dry and wet specimens PAGE 114 114 Figure 5 13. G/GOMC versus water content (%) at different strain levels for Newberry limerock Figure 5 14. G/GOMC ver sus water content (%) at different strain levels for Georgia granite PAGE 115 115 Figure 5 15. Additional confinement provided due to suction at 105% strain magnitude at different water contents Figure 5 16. Additional confinement provided due to suction at 105% strain magnitude at different degree of saturation PAGE 116 116 Figure 5 17. Youngs Modulus versus % water content results obtained from Free Free RC test on Newberry l imerock Figure 5 18. Youngs Modulus versus % water content results obtained from Free Fre e RC test on Georgia Granite PAGE 117 117 Figure 5 19. Comparison of Newberry limerock Free Free RC test data with Fixed Free RC very small strain data and T oros (2008) data Figure 5 20. Comparison of Georgia granite Free Free RC test data with Fixed Free RC ve ry smallstrain data and T oros (2008) data PAGE 118 118 CHAPTER 6 NONLINEAR FINITE ELE MENT MODELING OF BASE LAYER One of the main objectives of this research study is to develop a nonlinear pavement response model utiliz ing laboratory testing results presented in Cha pter 5 and incorporate base modulus nonlinearity with respect to effective stress confinement, loading strain and moisture content; and analyze under performance conditions. To perform a nonlinear analysis using stress and strain dependent nonlinear modul i for unbound base materials, it requires using a nonlinear finite element model that can compute pavement response s such as stress, strain and displacement ; at any stress point of finite element pavement model. P LAXIS H ardening S oil s mall model (P laxis H s small) a nonlinear finite element model software is selected for our nonlinear analysis. A detailed discussion about its features, suitability for our analysis and analysis methodology are presented in this chapter. 6.1 PLAXIS : Hardening Soil Small M ode l ( Plaxis H s s mall M odel) PLAXIS is a special purpose two dimensional finite element computer program used to perform deformation and stability analysis for various types of geotechnical applications. Real situations may be modeled either by a plain strain or an axisymmetric model. H s s mall model is an elastoplastic type of hyperbolic model and incorporates strain dependent stiffness moduli, simulating the different reaction of soils to small strain (i.e. strains below 103%) and large strains (i.e. strains above 101%). Soil modulus behaves elastic at very smallstrains (i.e. lower than 104%) and decreases nonlinearly with increase in strain amplitude. Figure 6 1 shows an example of typical s shaped soil modulus reduction curve. The most frequently used hyperbolic model to estimate nonlinear modulus reduction in soils (Figure 6 1), including both small strains and large strains, is Hardin Drnevich relationship and described as: PAGE 119 119 (61) where threshold strain is quantified as (62) H s small model uses a more straight forward and less prone to error modified HardinDrnevich relationship proposed by Santos and Correia (2001), and is described as: (63) 0.7 is the shear strain at G = 0 .7 Gmax Figure 6 2 shows fit of modified HardinDrnevich relationship (Equation 63) with Santos and Correia (2001) testing data. Therefore, two parameters are needed to know to describe the modulus behavior at small strains, and they are: Initial or very smallstrain modulus Gmax shear strai 0.7 at which secant shear modulus G is reduce d to 70% of Gmax Some basic characteristics of H s small model are: Stress dependent stiffness according to power law Input parameter m Plastic straining due to primary deviatoric loading Input para meter E50 ref Plastic straining due to primary compression Input parameter Eoed ref Elastic unloading/reloading Input parameter Eur ref, ur Failure according to the Mohr Coulomb model Nonlinear reduction of s mallstrain modulus Input parameter Gmax0.7 6.1.1 Parameters of H s small M odel Input parameters required for H s small model are presented in Table 6 1. All stiffness related parameters (i.e. Gmax ref, E50 ref, Eoed ref and Eur ref) are defined at a particular reference stress, at which they are determined in laboratory. Based on the stiffness value defined PAGE 120 120 at a particular reference stress, Plaxis H s small model calculates stiffness values at any required stress for given conditions, using the following equations (6 4) (6 5) (6 6) (6 7) w 1 = vertical stress, at which test was conducted 3 = confining stress or minor principal stress at which test was conducted. Based on parameter values for different soils found in literature (Benz ( 2006) Lehane et al. ( 2008) ) it ca n be approximated that Eur = Emax/3 = Gmax H s small model manual suggests to use E50 = Eoed = Eur/3, by default. 6.1.2 Compatibility of Plaxis H s small Model Modulus Reduction Model to Our Laboratory Determined Base Soils Modulus R eduction Behavior In order to use Plaxis H s small model to model Newberry limerock and Georgia granite, it is necessary to cross check the compatibility between H s small models nonlinear modulus reduction model curve and lab testing results modulus reduction curve s. For both materials, G/Gmax versus following Santos and Correia (2 001) modified hyperbolic curve E quation 63. These material curves are compared with theoretical and shown in Figure 6 3. From the curves in Figure 63, it can be concluded that H s s mall model hyper bolic model curve can be used to well represent the nonlinear smallstrain modulus reduction behavior of Newberry limerock and Georgia granite. PAGE 121 121 6.2 Flexible Pavement Nonlinear Response Model The primary basis of our finite element model for flexible pavement is adapted from MEPDG (2004), Appendix RR (Finite Element Procedures for Flexible Pavement Analysis). Some of the main features of MEPDG (2004) finite element model are: 1) Axisymmetric model 2) Static single wheel load with 150 mm radius circular cross sectional area and 550 kPa tire contact stress 3) Vertical side boundaries are 10 to 12 radii far from center of wheel load and horizontal bottom boundaries are 50 times of radii below from top of surface layer 4) Linear surface asphalt concrete (AC) la yer and linear subgrade layer 5) Nonlinear base layer These above mentioned MEPDG features are considered for our pavement model and their implementation is explained in detail in next followed sections. 6.2.1 Axisymmetric Model Axisymmetric model with 15 node triangular ele ments i s chosen for our pavement modeling and a typical pavement cross section and FE mesh form ation is shown in Figures 64 a and 64b. Size of the e lement, i.e. fineness of mesh i s selected such that i) there is a smooth continuity of resulting stresses and strains between two adjacent elements and ii) time taken f or processing is not too long. Since surface AC layer and subgrade layer are considered as elastic, fineness of mesh is critical for base layer only. Vertical side boundaries are at least 12 radii (i.e. > 1.8 m) from load center and bottom horizontal boundary is at least 50 radii (i.e. 7.5 m) below from the top of AC layer. It i s also made sure that location of boundaries has no influence on load resulting deformations, by cros s checking the deformations near boundaries are either zero or almost zero. Vertical side boundaries are fixed horizontally and allowed to move vertically; whereas horizontal boundaries are fixed both in horizontal and vertical directions. PAGE 122 122 6.2.2 Pavement C ross Sections T otal six different pavement cross sections with different layer thicknesses are considered for nonlinear analysis, which are presented in Table 6 2 and also shown in Figure 65. 6.2.3 Loading Conditions Circular single wheel with 150 mm radi us and 550 kPa of static contact stress is the only loading condition considered for our entire analysis. 6.2.4 Input Parameters for Surface and Subgrade Layers of Flexible Pavement Since our focus is on modulus nonlinearity of base layer, to single out i ts influence on performance of whole pavement, it is decided to model all remaining layers, i.e. AC surface and subgrade as linearly elastic. Various parameters used for A C surface and subgrade layers are presented below. Following the MEPDG Finite Elemen t Analysis material property selection, both asphalt concrete (AC) surface layer and subgrade layer are considered elastic for our FE analysis. Three different elastic moduli for AC and four different elastic moduli for subgrade are chosen and presented in Table 6 3. 6.3 Initial Plaxis HSsmall Pavement Model Runs and Recalibration Before conducting a full length pavement response analysis via a nonlinear response model, it is desirable to check the models applicability and accuracy for analysis by compari ng some model analysis results against known insitu measured pavement behavior results. From our literature survey, it is found that no insitu measured actual pavement results, with proper nonlinear material data that can be utilized for our model analys is, are available in literature. Since there are no known results of pavement structures are available in literature, we looked for any other geotechnical structures in situ measured known results that can be useful to serve our purpose of nonlinear response model applicability and accuracy check. It is found that insitu PAGE 123 123 measured results are available from Perth sand footing field studies conducted by Lehane et al. (2008) and these results are selected for our model verification. Lehane et al. (2008) conducted load tests on four different si ze footings built in Perth sand and made in situ measurement of actual settlements for each footing. Comparison of settlements obtained from their actual in situ measurements and settlement predictions from HSsmall model, for footing 1, is considered for our verification purpose 6.3.1 Footing Model Analysis and Verification Footing1 (of dimensions 1.51.51 m ) nonlinear response model is developed via Plaxis HSsmall model, similar to Lehanes finite element model as sho wn in Figure 66. Same regular material properties used by Lehane are used for our modle and shown in Table 64.This footing model is subjected to loads similar to actual field test loads. Obtained settlement predictions are compared with actual measured s ettlements and shown in Figure 6 7a. This initial nonlinear analysis comparison revealed that the HSsmall model is soft and predictions are overestimated. Similar observations were reported by Lehane et al. (2008). This comparison exercise concludes that it is required to recalibrate our HSsmall response model such that its analysis res ults are equal to nearly equal or actual field results. 6.3.2 Model Recalibration From Section 6.1.1, it is understood that Gmax ref and 0.7 are the basic model input paramet ers and should be determined from lab testing. Other stiffness parameters, i.e. E50 ref, Eoed ref Eur ref and m (po wer for stress level dependency ) are useful in analysis at larger strains. Plaxis recommends using default values for these stiffness parameter s, which are based on and calculated from Gmax ref 0.7 One possible to way to recalibrate our response model is to modify these input parameters such that actual footing settlements are equal or nearly equal to model predicted settlements. PAGE 124 124 Since Gmax ref,0.7 and m are determined from our lab testing and represent the true material behavior; its not allowed to modify them. Hence, it is decide to modify o ther nonmeasured stiffness parameters, i.e. E50 ref, Eoed ref and Eur ref to re calibrate the response model. These nonmeasured sti ffness parameters are modified by trial and error method such that model predicted and actual in situ settlements are matching. After trial and error method, finally arrived calibrated parameters are shown in Table 6 5 comparing with regular parameters (i. e. noncalibrated). Comparison of settlement prediction after recalibration is shown in Figure 6 7b. 6.4 Recalibrated Nonlinear Input Parameters of Base Soils for Plaxis Based on recalibration carried out in Section 6.3.2, input parameters for different ma terials used in our analysis are modified accordingly and reported in Table 6 7, 6 8 and 69. These recalibrated input parameters are obtained based on our experimental results presented in C hapter 5. 6.4.1 Recalibrated Parameters of Newberry Limerock and Georgia Granite As explained in C hapters 3 and 4, compacted unsaturated specimens of Newberry limerock and Georgia granite a re tested at different water contents under no confinement. Since the shear moduli values for different strain amplitudes are obtai ned at no confinement, reference stress (pref) for model input is taken as 1 kPa. Value of m, which indicates stiffness dependence on stress, is obtained from dry material testing. From confinement stress versus shear moduli plots for dry materials (Figure 56), for Newberry limerock, m=0.702 and for Georgia granite m=0.6389. Fixed Free RC tests are run on Newberry limerock at 13, 12, 11 and 10 % water contents only. Gmax for water contents lower than 10% water content are obtained from Free Free RC test results (Figure 5 15) PAGE 125 125 Final recalibrated input parameters for Newberry limerock at 13, 12, 11, 10, 8 and 5.5% w.c.; and for Georgia granite at 5.5, 4.5 and 3.5% w.c. are presented in Table 66 and Table 67 respectively. 6.4.2 Input Parameters for Miami Limerock Miami limerock is another base layer material commonly used in the State of Florida. Toros (2008) research investigation on Miami limerock report ed that its modulus behavior is relatively different from other limerocks available in Florida and increases tremendously with drying compared to other limerocks. Therefore, in order to understand and compare its behavior with Newberry limerock, it i s decided to perform nonlinear model analysis on Miami limerock also. Since no experimental study is con duct ed on Miami limerock, it is decided to obtain approximate Gmax versus % water content behavior curve f rom Free Free RC testing results of Toros (2008). Since Toros (2008) test specimens have some confining due to casing on side surface area, approximat e FreeFree test moduli values for Miami limerock with no casing confinement are calculated and presented in Figure 6 8. Based on this estimated approximate Emax versus water content curve for Miami limerock, HS Small model recalibrated parameters are developed and reported in Table 68 for selected water contents 6.5 Demonstration of R esponse Model Nonlinear Behavior One of the main objectives of developing this response model and recalibrated material parameters is to incorporate both stress dependent and strain dependent modulus nonlinearity in base layer nonlinear analysis. Hence, it is necessary to demonstrate that i) Recalibrated parameters are capable of incorporating nonlinearity and ii) Developed pavement model (Figure 64a and Figure 6 4b) is capa ble of incorporating nonlinearity in base layer. PAGE 126 126 6.5.1 Demonstration of Input Parameters Nonlinearity For nonlinear soils, deformations or strains increase nonlinearly with increase in load and our parameters also supposed to exhibit same behavior. For pa rameters nonlinearity demonstration, the same footing used for model recalibration (Figure 66) is selected and subjected to different loads. 10% w.c Newberry limerock HSsmall parameters are assigned for soil under the footing. Settlement predictions obta ined from this analysis are plotted in Figure 6 9. From Figure 69, it can be observed that footing settlements increase nonlinearly with load. Rate of increase in settlement with load is low at lower loads and high at higher loads, and demonstrates materi al nonlinearity. 6.5.2 Demonstration of Pavement Models Nonlinearity Pavement model shown in Figure 64a is subjected to different wheel loads in the range of 350800 kPa. Surface deflection profiles obtained from this analysis are plotted in Figure 610. From Figure 6 10, it can be observed that the deflection basin is varying nonlinearly with load, which demonstrates the nonlinearity of pavement model. PAGE 127 127 Table 6 1. Input parameters of Plaxis HSsmall model Failure parameters c cohesion angle of internal friction angle of dilatancy Basic parameters for nonlinear small strain stiffness G max ref The initial or very smallstrain modulus at reference stress 0.7 shear strain level at which shear modulus G is reduced to 70% of G max Basic parameters for soil stiffness E 50 ref Secant stiffness in standard drained triaxial test E oed ref Tangent stiffness for primary oedometer loading E ur ref Unloading /reloading stiffness (default E ur ref = 3 E 50 ref ) m Po wer for stress level dependency Advanced parameters ur Poissons ratio for unloading ur = 0.2) p ref reference stress for stiffness K 0 nc K 0 value for normal consolidation (default K 0 nc = 1 sin ) R f failure ratio q f /q a (default R f = 0.9) tension Tensile strength (defa tesnsion = 0 stress units) C increment As in Mohr coulomb model (default c incrememt = 0) Table 6 2. Different types of pavement structures considered for analysis S tructure Number Asphalt Concrete Surface Thickness (mm) Base Thickness (mm) 1 200 450 2 200 300 3 100 450 4 100 300 5 100 200 6 50 300 PAGE 128 128 Table 6 3. Material input parameters for surface asphalt concrete and subgrade layers Elastic Modulus (MPa) Poisson's ratio Unit weight (kN/m3) Asphalt Concrete 12500 0.39 23 3000 1000 Subgrade 125 0.32 18 70 50 30 Table 6 4. I nput parameters used in HSsmall model for footing settlement predictions (Lehane et al. 2008) Parameter Unit Value kN/m 3 18 P ref kPa 100 E 50 ref MPa 20 E oed ref MPa 20 E ur ref MPa 45 G 0 ref MPa 160 m 0.5 c kPa 1 35 o 0 ur 0.2 0.7 2.50E 05 PAGE 129 129 Table 6 5. Comparison of Actual and Modified calibration parameters Stiffness Parameter Suggested values Modified values through Calibration The initial or very smalls train modulus at reference stress Gmax Gmax Secant stiffness in standard drained triaxial test ( E50) Eur/ 3 Eur/2 Tangent stiffness for primary oedometer loading ( Eoed ref) Eur/ 3= E50 Eur/2 Unloading /reloading stiffness (default Eur ref = 3 E50 ref) Eur 0.85 Emax Note: Emax = max Table 6 6. Recalibrated HSsmall input parameters for Newberry limerock Newberry l imerock Water content (%) 13 12 11 10 8 5.5 G max (Mpa) 47.38 84.48 190.3 221.39 449.61 906.536 5.E 06 5.E 06 5.E 06 5.E 06 5.E 06 5.E 06 P ref (kPa) 1 1 1 1 1 1 E max (MPa) 132.66 234.85 525.23 606.61 1227.44 2465.78 E ur (MPa) 112.76 199.63 446.44 515.62 1043.32 2095.91 E oed (MPa) 56.38 99.81 223.22 257.81 521.66 1047.96 E 50 (MPa) 56.38 99.81 223.22 257.81 521.66 1047.96 m 0.702 0.702 0.702 0.702 0.702 0.702 3 ) 21.42 20.45 20.2 20 19.64 18.95 0.4 0.39 0.38 0.37 0.365 0.36 PAGE 130 130 Table 6 7. Recalibrated HS s mall input parameters for Georgia granite Georgia g ranite Water content (%) 5.5 4.5 3.5 G max (Mpa) 46.67 162.65 238.22 5.E 06 5.E 06 5.E 06 P ref (kPa) 1 1 1 E max (MPa) 133.48 455.42 657.49 E ur (MPa) 113.45 387.11 558.86 E oed (MPa) 56.73 193.55 279.43 E 50 (MPa) 56.73 193.55 279.43 m 0. 638 9 0.6389 0. 6389 3 ) 21.42 20.45 20.2 0.43 0.4 0.38 Table 6 8. Recalibrated HS s mall input p arameters for Miami limerock Miami l imerock Water content (%) 8 6 4 Gmax (Mpa) 12.22 226.95 1286.8 5.E 06 5.E 06 5.E 06 P ref (kPa) 1 1 1 Emax (MPa) 35.19 640.00 3500.10 Eur (MPa) 29.91 544.00 2975.08 Eoed (MPa) 14.96 272.00 1487.54 E50 (MPa) 14.96 272.00 1487.54 m 0.702 0.702 0.702 3) 21.42 20.45 20.2 0.44 0.41 0.36 PAGE 131 131 Figure 6 1. Characteristic modulus strain behavior of soil with typical strain ranges for laboratory tests and structures. (Atkinson and Sallfors, 1991) Figure 6 2. Results from modified Hardin Drnevich relationship compared to test data by Santos and Correia (2001) PAGE 132 132 Figure 6 3. Comparison of Newbery limerock and Georgia granite actual testing data with HS s mall models Santos and Correia (2001) modified hyperbolic curve PAGE 133 133 Figure 6 4. Typical Plaxis HS s mall model cross section model used for nonlinear analysis. PAGE 134 134 Figure 6 5. Typical HS s mall model finite element mesh of a pavement cross section Asphalt con crete layer base subgrade PAGE 135 135 Structure1 Structure2 200 mm 200 mm 450mm 300 mm (a) S tructure3 S tructure4 100mm 100 mm 450 mm 300 mm (c) (d) S tructure5 S tructure6 50 mm AC 100 mm 200 mm 300 mm Note: A C Asphalt Concrete Figure 6 6. Cross sections considered for nonlinear base pavement analysis AC Base Subgrade AC Base Subgrade AC Base Subgrade AC Base Subgrade AC Base Subgrade Base Subgrade PAGE 136 136 Figure 6 7. Finite element mesh us ed for model recalibration analysis ( footing 1 by Lehane et al. (2008) ) Figure 6 8. Comparison of actual settlements with settlements predicted using regular parameters via Plaxis HSsmall footing model. PAGE 137 137 Figure 6 9. Comparison of actual settlements with settlements predicted using regular parameters and calibrated settlements via Plaxis HSsmall footing model. Figur e 6 10. Approximate Miami l imerock Gmax versus water content plot. PAGE 138 138 Figure 6 11. Load versus settlement curve to demonstrate material input parameters nonlinearity. Figure 6 12. Pavement surface deflection basin for different load demonstrating pavem ent model nonlinearity. PAGE 139 139 CHAPTER 7 BASE LAYER NONLINEAR MODELING ANALYSIS AN D RESULTS Characterization of modulus nonlinearity of base soils via lab testing program is discussed in Chapter 5. Using these laboratory test results, a workable nonlinear response model for pavement analysis is developed via PLAXIS. Nonlinear response model development procedure, model characteristics and calibration, pavement cross sections proposed for analysis, material parameters, and verification and demonstration of models nonlinearity are presented in Chapter 6. Now we have a functional operational nonlinear pavement response model ready for analysis. Next objective is using the responses obtained from response model nonlinear analysis of pavement ; develop a methodology to calculate nonlinear equivalent single effective modulus for the whole base layer that can approximate known nonlinearities of base layer and can be used as MEPDG level 2 and level 3 parameter inputs for practical design applications. Pavement responses, obtained from nonlinear analysis, such as deformations, strains and stresses at various critical locations of pavement cross section can be considered for determining single effective base modulus that can approximately produce same pavement responses as in nonlinear analysis. Effective modulus determination methodology and its verification, influence of moisture content, subgrade modulus and overall structural cross section on single effective modulus, comparison of nonlinear and linear analyses responses, and effect of base nonlinearity on pavement performance are discussed in this chapter. 7.1 Nonlinear Equivalent Single Effective Base Modulus Determination 7.1.1 Methodology To derive the single effective linear modulus value for whole base layer, it is pr oposed to use pavement surface deflection basin as the single matching factor between nonlinear and linear PAGE 140 140 analysis For nonlinear analysis, both AC surface and subgrade layers are considered linear and base layer is considered nonlinear. In other words, t he moduli of surface and subgrade layer are linear (elastic) and the modulus of base layer is nonlinear. Using the maximum surface deflection value obtained from nonlinear analysis as the matching factor an equivalent elastic modulus value for whole base layer is determined by trial and error method. Surface deflection profile obtained from single effective modulus base case should match well with nonlinear base case surface deflection profile. During this process, elastic moduli of AC surface layer and subgrade are kept same as in corresponding nonlinear case Once the equivalent single effective elastic modulus value for base layer is determined, pavement responses obtained at critical locations (Figure 7 1) for both linear and nonlinear cases can be com pared for further comparison analysis 7.1.2 An Example of Surface Deflection Basin Matching Between Nonlinear Modulus Base Case and Single Effective Elastic Modulus Base Case Nonlinear analysis via Plaxis HSsmall nonlinear pavement response model (developed in Chapter 6) is performed on Structure 1 (refer to Figure 6 5) with 10% w.c. base. AC surface layer and subgrade layer are considered elastic with 1000 MPa and 50 MPa moduli, respectively. From nonlinear analysis, maximum surface deflection of 0.582 mm is obtained under a wheel load of 550 kPa. Keeping the elastic moduli of surface and subgrade layers same as in nonlinear base modulus case, equivalent single effective modulus for base layer that can produce the same maximum surface deflection of 0.582.m m is determined by trial and error method. Obtained surface deflection basins for both nonlinear modulus base and single effective elastic modulus base cases are plotted in Figure 7 2 for comparison purpose. From this figure, it can be observed that surfac e deflection basin obtained from single effective base modulus case matches very well with the surface deflection basin of nonlinear base modulus case. PAGE 141 141 7.1.3 Demonstration of Nonlinear Reduction of Single Effective Modulus with Increase in Load In the science of nonlinear material mechanics, it is well known that modulus of particulate material decreases nonlinearly with increase in strain. For particulate materials like soil, permanent strain or deformation increases nonlinearly with increase in load, and modulus decreases nonlinearly with increase in strain. One of the main objectives of our research work is to incorporate this modulus nonlinearity in pavement base layer design and analysis. Hence it is interesting to verify that equivalent single effectiv e modulus of base layer, determined from nonlinear analysis surface deflection, reflects this nonlinearity. In other words, single effective modulus of base layer expected to decrease nonlinearly with increase in load. A demonstration exercise is performed on strcture 1 and structre 4, with Newberry limerock base at 13% and 10% moisture contents and Georgia granite base at 5.5% and 3.5% moisture contents. Moduli of AC surface layer and subgrade layer are kept as 1000 MPa and 50 MPa, respectively. After cond ucting nonlinear analysis at different loads via our Plaxis HSsmall nonlinear response model, single effective base modulus for each load is determined following the single effective modulus determination methodology explained in section 7.1.1. Results of these analyses are presented in Figures 7 3 and 74. From these plots it can be primarily demonstrated that single effective modulus of base material decreases with increase in load, in a nonlinear manner. It can also be observed that for both Newberry lim erock and Georgia granite, modulus nonlinearity increases with decrease in water content. The rate of reduction in modulus with increase in load is higher in base layers with lower moisture contents than in higher moisture content. This tells us that as th e water content starts decreasing, material starts behaving more nonlinear. PAGE 142 142 7.2 Single Effective Moduli Data for Base Layer 7.2.1 Nonlinear Equivalent Single Effective Design Modul i Value s for Base Layer Nonlinear analyses are run via Plaxis HSsmall nonlin ear response model, on six different pavement structures presented in Figure 6 5. Nonlinear material parameters presented in Tables 66, 67 and 68 for Newberry limerock, Georgia granite and Miami limerock at different water contents are assigned to base layer. AC surface layer is considered elastic with 1000 MPa modulus. Different subgrade elastic moduli presented in Table 63 are assigned to subgrade layer, to analyze the effect of subgrade elastic modulus on resulting base layer single effective modulus Methodology explained in section 7.1.1 is followed to determine single effective design modulus for base layer. Single effective moduli values determined for different structures are presented in Tables 71 to 73. Table 71 presents effective moduli val ues obtained for Newberry limerock base, and so Table 7 2 and Table 7 3 for Georgia granite and Miami limerock respectively. Following the procedure explained in section 5.2.2 and using Equations 51 and 52, maximum Youngs modulus (Emax) of base soil is calculated at in situ overburden stress or confinement at middle depth of base layer. Single effective moduli values along with corresponding Emax and ratio of effective modulus to maximum Youngs modulus (i.e. E/Emax) are presented in Appendix B This da tabase would be primarily helpful in prior approximate estimation of effective moduli, from known Emax. It can be primarily observed that E/Emax decreases with decrease in water content, which indicates that modulus nonlinearity increases with decrease in water content. 7.2.1.1 Influence of moisture content on base layer single effective design modulu s For structures 1 to 6, at constant subgrade modulus, variation of single effective modulus with decrease in base material water content is shown in Figures 7 5, 7 6 and 77, for Newberry PAGE 143 143 limerock, Georgia granite and Miami limerock, respectively. From these column plots it can be observed that for same subgrade modulus, effective base modulus increases significantly with decrease in water content. Our laborato ry test data showed that nonlinear small strain modulus of base material increase significantly with decrease in water content. Decrease in water content increases additional confinement caused by suction, which in turn increases small strain modulus. This increase in small strain modulus is being reflected on equivalent single effective modulus and hence single effective modulus increases with decrease in water content. Among all three different base materials, Miami limerock base has the highest single ef fective moduli values. From lab testing results, it is observed that increase in smallstrain modulus due to suction also is highest for Miami limerock, among all three base materials. This indicates that highest increase in smallstrain modulus due to suc tion, compared to other two materials, is reflected in equivalent single effective modulus. This also confirms the importance of smallstrain modulus nonlinearity in determining single effective modulus for linear design methods. 7.2.1.2 Influence of subgr ade modulus on single effective base design modulus From the column plots in Figures 75, 7 6 and 77, it can be observed that for any given structure and base water content, single effective base modulus increases with increase in subgrade modulus. In our case, highest effective modulus for any base water content corresponding to a given structure occurs at 125 MPa. This indicates that single effective design modulus of base layer at any given water content depends on modulus of subgrade, which supports ba se layer. This behavior can be explained by basics of pavement mechanics. It is well known that pavement permanent deformation is mainly dependent on subgrade modulus and deformation decreases with increase in subgrade modulus. As the subgrade modulus incr eases, magnitude of deviatoric stresses acting on base layer decreases. Since soil modulus is nonlinearly PAGE 144 144 stress dependent and increases with decrease in deviatoric stress, base layer effective modulus increases with increase in subgrade modulus. Therefore as the subgrade modulus increases, single effective base modulus increases at faster rate with decrease in moisture content. 7.2.1.3 Influence of structure type on single effective base design modulus For same base moisture content, variation of base single effective design modulus for different structures with different subgrade moduli is shown in Figures 78, 79 and 710 for Newberry limerock, Georgia granite and Miami limerock, respectively. From these column plots, it can be observed that at any give n moisture content and subgrade modulus combination, base layer single effective modulus value is dependent on structure type. In other words, single effective modulus is dependent on structure layers thicknesses. As the thicknesses of different layers in pavement structure varies, the magnitude of wheel load deviatoric stresses being transferred from top layers to bottom layers also varies. Usually, as the thickness of a layer increases, deviatoric stresses being transferred to the layer beneath it decreases and corresponding modulus increases. Thus, since soil modulus is nonlinear with respect to stress, as the magnitude of deviatoric stress varies; its effective modulus also varies. 7.2.1.4 C omparison of single effective design moduli for different materials Single effective design moduli values obtained at 8% w.c. for Newberry limerock, 3.5% w.c. for Georgia granite and 4% for Miami limerock are compared with their respective optimum moisture contents (OMC), for structure 1 and str u cture4 in Figure 711 a and Figure 7 11b, respectively. Effective moduli values obtained with both 50 MPa and 125 MPa subgrade moduli are shown in these column plots. These column plots can provide an idea about how fast and by how much magnitude, the effective modulus of each material increases due to drying, compared to other two materials? From these comparison column plots, it can be primarily observed that PAGE 145 145 among all three materials Miami limerock behaves very different and its effective modulus increases at faster rate ve ry significantly as the material dries out. At OMC, effective modulus of Miami limerock is much lower than that of Newberry limerock and Georgia granite. But as the material dries out, its modulus increases at a very faster rate compared to other two mater ials and reaches maximum among all three materials. For structure 1 and structure 4 with 50 MPa and 125 MPa subgrade modulus, normalized single effective base modulus obtained at different base water contents, for all the three materials are compared in Fi gure 7 12a for structure 1 and Figure 712b for structure 4. Effective modulus (E) at different water contents normalized with effective modulus at optimum moisture content (Eopt) is shown on Y axis and reduction in degree of saturation due to drying is shown on X axis. From these plots also, it can be observed that approximately for the same amount of decrease in degree of saturation, moduli of Newberry limerock and Georgia granite are increasing more or less at same rate with respect to Eopt, where as Miami limerock is increasing very significantly and behaving much dif ferent. Toros (2008), from his F ree F ree RC test data, also observed very significant increase in Emax of Miami limerock as the material dries out, compared to Newberry limerock and Georgia granite. For a decrease of 45% in degree of saturation, effective moduli of Newberry limerock and Georgia granite are increasing by around 3 3.5 times for both structure 1 and structure 4, where as effective modulus of Miami limerock is increasing by arou nd 17 times for structure 1 and around 45 times for strucutre 4. These observations indicate that increase in design modulus with material drying is material specific. Since modulus tests are generally conducted at OMC, it is very important to be aware of this behavior, or else a pavement designer can get mislead ed very easily. PAGE 146 146 7.2.1.5 C omparison of equivalent single effective design moduli with mepdg moisture effect model design moduli MEPDG(2004) recommends below given generalized regression model to incorporate moisture effect (suction effect) on resilient modulus (MR). (7 1) For coarse grain ed soils: a=0.0401 and ks=6.8157. It would be interesting to see if MEPDG moisture effect model can accurately estimate design modulus of our base soils or not? In other words, can MEPDG moisture model determine design modulus at different moisture contents accurately for our base soils? For this purpose, single effective moduli values derived for our base soils (Tables 7 1, 72 and 73) are compared with MEPDG model derive d moduli values. Effective moduli values for selected structures with 50 MPa and 125 MPa subgrade moduli and at different base moisture contents are compared with MEPDG moisture model calculated moduli values, and shown in F igures 7 13, 714 and 715 for Newberry limerock, Georgia granite and Miami limerock, respectively. Effective modulus (E) or resilient modulus (MR) normalized with modulus at optimum moisture content (Eopt or MR opt) is on Y axis, and decrease in degree of saturation at different water contents is on X axis. Legend indicates the structure number and its subgrade modulus. From these figures, it can be observed that for MEPDG model, as the moisture content decreases, modulus can increase maximum up to two ti mes of modulus at optimum moisture content. Whereas, effective base moduli values derived for our materials does not meet this criteria and in the case of Miami limerock it can increase up to 45 times. E/Eopt values for our PAGE 147 147 materials are determined only up to 40% decrease in degree of saturation and can increase further with further decrease in degree of saturation. In general, for structures with low subgrade modulus (50 MPa for our plots), effective moduli are falling slightly near MEPDG moisture model va lues. As the subgrade modulus increases, E/Eopt values are moving away from moisture model curve and increases up to 3.5 times for Newberry limerock, 2.5 times for Georgia granite and 47 times for Miami limerock. From these observations, it can be conclude d that MEPDG moisture effect model cannot incorporate moisture /suction effect accurately for Florida base materials. 7.3 Evaluation of Applicability of Single Effective Modulus in place of Nonlinear Modulus Design methodology to determine equivalent single effective design modulus for base layer is developed by considering surface deflection basin as the only matching factor. It is also shown via an example in section 7.1.2 that surface deflection basins generated from nonlinear base analysis and correspond ing equivalent linear base analysis are matching well. Now, as a next step, it is interesting to compare pavement responses generated from nonlinear and equivalent linear analyses models to assess the applicability of single effective modulus for base laye r in lieu of nonlinear modulus. For this purpose, various pavement responses obtained from nonlinear and corresponding equivalent linear analysis are compared, at various critical locations (Figure 7 1) 7.3.1 Compari son of Nonlinear and Equivalent Linear A nal ysis R esponses Below given responses obtained from nonlinear analysis and corresponding equivalent linear analysis at different locations of pavement structure (as shown in Figure 7 1) are compared 1) Surface deflection 2) Horizontal stress ( xx, tensile stress) at top of AC layer PAGE 148 148 3xx, tensile strain) at top of AC layer 4xx, tensile stress) at bottom of AC layer 5xx, tensile strain) at bottom of AC layer 6) Vertical stress ( yy, compressive stress) at top of base layer 7yy, compressive strain) at top of base layer 8yy, compressive stress) at bottom of base layer 9yy, compressive strain) at bottom of base layer 10yy, compressive stress) at top of subgrade layer 11yy, compressive strain) at top of subgrade layer The basis of selecting above responses for comparison is related to basics of pavement mechanics. In general, s urface cracking and rutting are two most important distresses that occur xx, tensile yy, compressive strain) at top of subgrade layer. Comparison of these two responses for nonlinear and linear analysis can illustrate whether adopting single effective modulus in place of nonlinear modulus would influence the rutting and cracking performance of pa vement? Next, surface deflection is the basic matching factor between nonlinear and equivalent linear analysis. Remaining stress and strain responses at different locations can provide basic information about how are the stresses and strains at different layer intersections varying and matching up, for nonlinear and equivalent linear analyses?, and does considering equivalent single effective modulus in place of nonlinear modulus would influence these responses? Pavement responses obtained from nonline ar and equivalent linear analysis of few selected pavement structures with 1000 MPa AC surface and 50 MPa subgrade modulus, and different base materials with different water contents, are compared. List of structures analyzed for responses comparison is gi ven in Table 74. For all the cases given in Table 74, nonlinear and equivalent linear response curves comparison plots for all above mentioned pavement responses xx, tensile strain) at bottom of AC PAGE 149 149 layer, obtained from nonlinear and equivalent linear analysis of structure 1 and structure 4, with different base water contents are compared in Figure 7 16a for Newberry limerock and Figure 716b for Georgia granite. Same set of comparison plots are yy, compressive strain) at top of subgrade layer and shown in Figure 717a for Newberry limerock and Figure 717b for Georgia granite. From Surface deflection comparison plots (Figures C 1 to C 8 for Newberry limerock and F igures C 89 to C 92 for Georgia granite, from Appendix C); it can be observed that the surface deflection profiles for nonlinear and equivalent linear analyses are matching well. Hence, considering single effective modulus in place of nonlinear modulus doe s not affect the real surface deflections. From the remaining response plots shown in Appendix C, except for vertical yy) at the top of subgrade plots (Figures C 81 to C 88 and C 129 to C 132), no significant differences in curves of nonlinear and corresponding linear analyses are observed and hence, single effective modulus is acceptable with respect to those responses as well. From the comparison plots of horizontal tensile str ain ( xx) at bottom of AC laye r (Figure 716a to 716b), for both structure 1 and structure 4 with different base water contents, it can be observed that horizontal str ain ( xx) at bottom of AC layer for nonlinear base modulus case is equal or nearly equal to equivalent linear case with single effective base mod ulus. In the science of pavement mechanics, it is well known that pavement surface cracking is generally dependent on horiz ontal tensile str ain ( xx) at bottom of AC surface layer. Hence, t his implies that considering equivalent single effective modulus for design of base layer thickness may not significantly affect the overall cracking performance of pavement. Comparison plots of vertical strain ( yy) at the top of subgrade for nonlinear and equivalent linear cases are compared in Figures 7 17a to 717b for structure 1 and structure 4, with different PAGE 150 150 base water contents. In the case of structure1, for both materials at different water contents, y y at top of subgrade (Figures C 81 to C 84 and Figure 717a) for nonlinear analysis is equal or nearly equal to that of corresponding equivalent linear analysis. As the water content decreases, yy of nonlinear case becomes very slightly higher than that of equivalent linear case, but not of significant magnitude. Whereas, in the case of structure4, in general, yy at top of subgrade (Figures C 85 to C 88 and Figure 717b) for nonlinear analysis is higher than corresponding equivalent linear analysis, and this difference increases with decrease in water content. This indicates that the base material is behaving more and more nonlinear with decreases in water content. Since structre 4 is thinner than structure 1, this indicates that material nonlinearity inc reases as the structure thickness and base layer water content decreases, together. According to pavement mechanics, yy at top of subgrade is critical for pavements rutting performance. So, as the base material nonlinearity increases with decrease in structure thickness and base layer water content together, use of equivalent single effective design modulus may over es timate pavements rutting performance. Hence, a new rutting performance criterion may be required to use single effective design modulus in place of nonlinear modulus, when, both the structure thickness and base material water content decreases together. In overall, for thick structures like structure 1, at different base water contents, no significant differences are observed in various pavement responses including yy at top of subgrade obtained from nonlinear and equivalent linear analysis. Hence, equivalent single effective modulus for base layer, which is derived based on maximum surface deflection, can be adopted for practical design purposes for thick structur es similar to structure 1. Whereas, for thin structures like structure 4, with the decrease of base layer thickness and water content together, it appears that modulus nonlinearity of base soil becomes more significant and affects pavements PAGE 151 151 rutting perfor mance and hence, equivalent linear method may overestimate rutting performance. Hence, in order to use equivalent effective modulus for base layer design calculations of thin structures, it may be required to develop a new rutting performance criterion. 7.3.2 Analysis of Nonlinear and L i near Responses in the Perspective of Rutting Performance C riteri a Based on observations made in last section, it is understood that using equivalent single effective modulus in place of nonlinear modulus may affect the rutt ing performance of pavement. Thus, it would be interesting to investigate the differences in pavement rutting performance, when a base layer is designed by i) nonlinear method considering modulus nonlinearity and ii) equivalent linear design method consider single effective modulus. Vertical strain ( yy) at top of subgrade layer is considered for this rutting performance comparison investigation It is well acknowledged in literature that pavement permanent deformation or settlement is primarily dependent on subgrade modulus (Huang ( 1993) ) Several empirical equations have been developed for permanent deformation criteria. These equations are helpful in predicting allowable number of load repetitions related to yy at top of subgrade. Asphal t Institutes (AI) empirical equation for permanent deformation failure criterion is frequently used for pavement rutting performance analysis. AIs equation assumes elastic behavior for all layers. Asphalt Institutes empirical equation for rutting failur e criteria is: (7 2) Where Nd allowable number of load repetitions to limit permanent deformation c yyfor our case) f4 = 1.365 109 f5 = 4.477 PAGE 152 152 It should be noted that E quation 72 was developed based on elastic criteria, i.e. all the layers are assumed to be elastic. Analyzing the vertical strai yy) at top of subgrade, obtained for both nonlinear and linear cases, to determine Nd using E quation 72, would help to identify the importance of considering base modulus nonlinearity in performance analysis. For structure 1 and structure 4 with selected base layer moisture contents, Nd yy at top of subgrade layer and corresponding analyzed data is presented in Tables 7 5 and 76. From the analysis data presented in Table 7 5 for structure 1, it can be observed that the ratio of Ndl inear/Ndnonlinear is close to one which means allowable number of load repetitions is nearly same. This implies that adopting equivalent single effective modulus for base layer thickness design in place of nonlinear modulus, may not significantly affect t he overall rutting performance of structure 1. So, elastic criteria based empirical equation can be used for rutting performance analysis of thick structures, similar to structure 1. From the data presented in Table 7 6 for structure 4, it can be observed that the ratio of Ndlinear/Ndnonlinear is varying between 1.5 to 3 times, which means considering equivalent single effective modulus in place of nonlinear modulus for base layer may overestimate allowable number of load repetitions as many as three times. This implies that considering equivalent single effective modulus for base layer thickness design for structure 4, may overestimate the overall rutting performance of the pavement by as many as three times. It also imp lies that elastic design method may overestimate the actual rutting performance of the pavement str ucture4. Hence, i t may not be appropriate to use elastic criterion based empirical equations for rutting performance analysis of thin structures, similar to structure 4. In the case of structur e 4 with Georgia granite base (Table 7 6) a little differen ce of 0.03% yy at top of subgrade, between linear and nonlinear case, is overestimating the rutting PAGE 153 153 performance by three times. We know that the above estimated difference (i.e. over estimation) in rutting performance is due to considering equivalent s ingle effective modulus in place of nonlinear modulus, just for base layer only. This implies that considering nonlinearity in other soil layers of a pavement structure may influence pavement performance more significantly due to cumulative nonlinearity ef fect, compared to considering nonlinearity only in base layer. Since pavement su bgrade layers are generally built with soils and hence its nonlinearity also yy at top of subgrade and so rutting performance. Thus a n investigation into subgrade nonlinearity may provide more information about importance of considering nonlinearity in soil layers and its influence on rutting performance PAGE 154 154 Table 7 1. Equivalent elastic moduli values obtained for different pavement structures, with different subgrade moduli for Newberry limerock base Subgrade Modulus (Mpa) Structure Moisture Content (%) 13 12 11 10 8 5.5 30 1 65 79 112 118 142 171 2 58 70 92 94 118 139 3 54 67 90 101 108 130 4 49 60 69 70 76 78 50 1 79 102 140 153 196 230 2 74 92 120 125 155 178 3 66 89 112 124 148 168 4 62 80 108 112 123 132 5 73 100 102 6 76 90 107 70 1 90 117 170 175 230 267 2 85 109 143 149 184 207 3 73 100 135 142 195 205 4 71 96 129 135 139 144 125 1 108 157 227 241 310 387 2 107 147 210 220 278 308 3 93 141 214 220 281 311 4 84 127 196 206 228 240 5 105 189 209 6 103 180 187 PAGE 155 155 Table 7 2. Effective equivalent elastic moduli values obtained for different pavement structures, with different subgrade moduli for Georgia granite base Subgrade Modulus (Mpa) Structure Moisture Content (%) 5.5 4.5 3.5 30 1 63 1 04 117 2 56 94 104 3 54 88 100 4 46 74 75 50 1 76 135 152 2 70 112 124 3 66 115 119 4 60 100 106 70 1 86 155 178 2 82 135 150 3 73 133 162 4 67 123 137 125 1 112 210 243 2 105 195 220 3 90 207 220 4 84 184 220 PAGE 156 156 Table 7 3. Effective equivalent elastic moduli values obtained for different pavement structures and with different subgrade moduli for Miami l imerock base Subgrade Modulus (Mpa) Structure Moisture Content (%) 8 6 4 50 1 30 168 337 4 6 54 159 125 1 31 260 545 4 7 122 320 Table 7 4. List of structures analyzed for pavements various responses comparison Base Material Type of Structure Water Contents (%) Newberry l imerock 1 and 4 13 10 8 5.5 Georgia granite 1 and 4 5.5 3 .5 PAGE 157 157 Table 7 5. Asphalt Institutes rutting criteria analysis for structure 1 (200mm AC surface, 450 mm base) Top of subgrade base layer water content Analysis method % yy yy (kN/m2) ND (AI empirical equation) Nd linear/ Nd nonlinear Newberry Limerock 10.0% nonlinear 0.0415 31.60 1888562 1.25 linear 0.0395 32.41 2363985 13.0% nonlinear 0.0458 32.08 1219731 1.19 linear 0.0440 35.70 1446650 Georgia Granite 3.5% nonlinear 0.0405 31.05 2099273 1.12 linear 0.0395 33.57 2346634 5.5% nonlinear 0.0442 32.99 1422634 1.01 linear 0.0441 36.49 1434653 AC layer modulus = 1000 MPa Subgarde Modulus = 50 MPa PAGE 158 158 T able 7 6. Asphalt Institutes rutting cr iteria analysis for structure 4 (100 mm AC surface, 300 mm base) base layer water content Analysis method % yy yy (kN/m2) N D (AI empirical equation) Nd linear/ Nd nonlinear Newberry Limerock 10.0% nonlinear 0.10303 58.809 32218 2.18 linear 0.08653 50.073 70378 13.0% nonlinear 0.1349 73.247 9640 1.50 linear 0.1232 69.16 14470 Georgia Granite 3.5% nonlinear 0.131926 66.955 10652 3.13 linear 0.102238 58.406 33350 5.5% nonlinear 0.141 68.023 7908 2.19 linear 0.11838 64.193 17301 AC layer modulus = 1000 MPa Subgarde Modulus = 50 MPa PAGE 159 159 1 AC Surface 2 3 Base 4 5 Subgrade Figure 7 1. Critical locations for pavement response analysis 1 Top of AC surface layer 2 Bottom of AC surface layer 3 Top of Base layer 4 Bottom of Base layer 5 Top of Subgrade layer Figure 72. S urface deflection basin c omparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer. PAGE 160 160 Figure 7 3. Nonlinear variation of single effective modulus with increas e in load, for Newberry limerock base at 13% and 10% moisture contents. Figure 74. Nonlinear variation of effective modulus with increase in load, for Georgia granite base at 5.5% and 3.5% moisture contents PAGE 161 161 a b c d e f Figure 7 5. Newberry liemroc k effective base modulus versus subgrade elastic modulus relationship with decrease in moisture content, for different structures a) Structure 1 b) S tructure 2 c) S trcutrue 3 d) S trcutrue 4 e) S trcutrue 5 f) S trcutrue 6. PAGE 162 162 a b c d Figure 7 6. Gerogia gran ite effective base modulus versus subgrade elastic modulus relationship with decrease in moisture content, for different structures a) S tructure 1 b) S tructure 2 c) S trcutrue 3 d) S trcutrue 4. PAGE 163 163 a b Figure 7 7. Miami lim e rock effective base modulus vers us subgrade elastic modulus relationship with decrease in moisture content, for different structures a) Structure 1 b) S tructure 4. PAGE 164 164 a b c d e f Figure 7 8. Newberry lim e rock effective base modulus versus subgrade elastic modulus relationship for diffe rent struct ures, at constant moisture content a) 13% w.c. b) 12% w.c. c) 11% w.c. d) 10% w.c. e) 8% w.c. f) 5.5% w.c PAGE 165 165 a b c Figure 7 9. Georgia granite effective base modulus versus subgrade elastic modulus relationship for different structres, at cons tant moisture content a) 5.5% w.c. b) 4.5% w.c. c) 3.5% w.c. PAGE 166 166 a b c Figure 7 10. Miami limerock effective base modulus versus subgrade elastic modulus relationship for different structres, at constant moisture content a) 8% w.c. b) 6% w.c. c) 4% w.c. PAGE 167 167 a b Figure 7 11. Comparsion of effe c tive moduli of all three materials at different base water contents and with subgrade modulus of 50 MPa and 125 MPa, obtained for a) S tructure 1 b) S tructure 4 PAGE 168 168 a b Figure 7 12. Comparsion of normalized effec tive moduli of all three materials obtained with subgrade modulus of 50 MPa and 125 MPa, a) S tructure 1, b) S tructure 4 PAGE 169 169 Figure 7 13. Comparison of Newberry limerock effective base moduli va l ues with MEPDG mositure model PAGE 170 170 Figure 7 14. Comparison of Georgi a granite effc e tive base moduli va l ues with MEPDG modulus mositure model PAGE 171 171 Figure 7 15. Comparison of Miami l imerock effective base moduli v a l ues with MEPDG mositure model PAGE 172 172 a Figure 7 16. Com pa rison of horizontal te ns ile strain at bottom of AC layer xx) obtaine d for structure 1 and str ucture 4 with different base water contents of a) Newberry l imerock b)Georgia granite PAGE 173 173 a b Figure 7 17. Compa rison of vertical compressive strain at top of subgrade yy) obtaine d for structure 1 and str u ct ure 4 with different base water contents of a) Newberry l imerock b) Georgia granite PAGE 174 174 CHAPTER 8 SUBGRADE LAYER NONLI NEAR MODELING ANALYS IS AND RESULTS Primary focus of t his research work is regarding modulus nonlinearity of base layer and determining single effective modulus of base layer for level 2 and level 3 design parameter inputs. A methodology to determine nonlinear equivalent single effective design modulus for base layer that can approximate the nonlinearities of base soil is developed in Chapter 7. Single effective moduli values for all three materials at different water contents, and for different structures with different subgrade moduli are derived following this methodology By comparing nonlinear analysis with equivalent linear analysis, it is observed that nonlinear analysis which is more accurate than equivalent linear analysis results in more strain at top of subgrade layer than that of equivalent linear analysis with single effective base modulus. Since rutting is primarily based on vertical strain at top of subgrade, it indicates that equivalent linear design method for base layer overestimates pavement rutting performance compared to nonlinear design method. It is concluded from Chapter 7 that by keeping both AC surface and subgrade layers elastic, and incorporating nonlinearity just in base layer; results in more vertical strain at top of subgrade layer, compared to equivalent linear design method which is commonly practiced. Since subgrade soils are also nonlinear in real filed conditions, it is expected that incorporating modulus nonlinearity in subgrade layer also may result in further more vertical strain at top of subgrade, which can influence the rutting performance more significantly. Hence, out of our professional interest, it is pro posed to conduct a nonlinear analysis by incorporating nonlinearity in both base and subgrade layers. This investigation will help us in obtaining some basic idea about the effect of subgrade nonlinearity on pavement rutting and cracking performances. Also, the inadequacies of elastic based design methods and performance criteria can be addressed. PAGE 175 175 8.1 Nonlinear Equivalent Single Effective Modulus Determination for Base and Subgrade Layers 8.1.1 Methodology Main objective of this task is to investigate the influence of accounting nonlinearity in both and subgrade layers on pavement performance, by comparing nonlinear and equivalent linear analyses results. In other words, compare and evaluate pavement responses obtained from modeling analysis of following two cases: i) Linear AC, nonlinear base, nonlinear subgrade ii) Linear AC, equivalent elastic base and equivalent elastic subgrade. Similar to nonlinear base analysis in Chapter 7, to derive equivalent single effective modul i value s for both nonlinear base and nonlinear subgrade, it is proposed to use pavement surface deflection basin as the single matching factor. Since both base and subgrade layers are considered nonlinear, deriving equivalent single effective modulus for both layers by tria l and error metho d based on surface deflection is not feasible. Hence, it is proposed to use Falling W eight D eflecto meter (FWD) analysis procedure that can back calculate equivalent elastic modulus for multiple layers using surface deflection basin and layers thicknesses a s inputs. Falling Weight Deflectometer (FWD) method is a well known pavement in situ testing and analysis method to back calculate equivalent moduli value s for different layers of an existing pavement based on deflection zone created by falling weight It is mainly used to assess the working condition of an existing pavement. By analyzing the surface deflection profile obtained from nonlinear response model analysis (i.e. nonlinear analysis via Plaxis HS small model for nonlinear base and nonlinear subgrad e) via FWD analysis technique, equivalent elastic modulus for surface, base and subgrade layers can be determined. Once th e equivalent single effective modulus value for each layer is determined, pavement responses can be obtained from equivalent PAGE 176 176 linear an alysis Then the pavement responses obtained from nonlinear and equivalent linear analyses can be compared for further evaluation. 8.1.2 Material Parameters and Structural Inputs For this analysis, both base and subgrade layers are considered nonlinear, where as AC surface layer is considered elastic. For this brief investigation, only selected pavement structures with selected base moisture contents and with one nonlinear subgrade modulus are analyzed. Two types of structures, i.e. structure 1 and structur e 4 (refer Figure 6 5) are considered for this analysis. Material moduli used for different layers are presented in Table 8 1. For subgrade layer, only Ottawa sand is considered as nonlinear material. Based on Ottawa sand lab testing results pres ented in F igure 3 7, re calibrated HSs mall model input parameters for Ottawa sand are developed and presented in Table 82. It should be noted that 1000 MPa is the only elastic modulus selected for AC surface layer and Ottawa sand is the only nonlinear material selected for subgrade. Nonlinear and equivalent linear analys es are performed following the procedure explained in 8.1.1. 8.1.3 An Example of Surface Deflection Basin Matching Between Nonlinear Base and Subgrade Case and Equivalent Linear Base and Subgrade Case Nonlinear analysis via Plaxis HSsmall response model (developed in Chapter 6) is performed on Structure 1 (refer to Figure 6 5) with 10% w.c. base and Ottawa sand subgrade. AC surface layer is considered elastic with 1000 MPa modulus. From Plaxis HSsmall nonlinear analysis, maximum surface deflection of 0.62 mm is obtained under wheel load of 550 kPa. Keeping the moduli of surface layer same, equivalent single effective modulus for base and subgrade layers that can produce the same surface deflection basin is determined vi a FWD analysis as explained in S ection 8.1.1. Obtained surface deflection basins for both nonlinear case and equivalent linear case are compared in Figure 8 2. From this figure, it can be observed that PAGE 177 177 surface deflection basin obtained from equivalent effective elastic base and subgrade moduli case matches very well with the deflection basin obtained from nonlinear base and nonlinear subgrade moduli case. 8.2 Comparis on of Nonlinear and Equivalent L inear Analysis Results 8.2.1 Nonlinear Equivalent Single Effective Design Modul i Value s for Base and Subgrade FWD back calculated single effective moduli values obtained for structure 1 and s tructure 4 with different water contents of Newberry limerock base and Ottawa sand subgrade are presented i n Table 8 3 for structure 1 and in Table 8 4 for structure 4. Similar results with different water contents of Georgia granite base and Ottawa sand subgrade are presented in Table 85 for structure 1 and in Table 8 6 for structure 4. From the data presented in Tables 83 to 86; it can be primarily observed that for both structures equivalent single effective moduli values for base layer increases with material drying. For structure 1, FWD back calculated effective modulus of base layer is greater than subgrade modulus. Whereas for structure 4, FWD back calculated effective modulus of base layer is lower than that of subgrade layer. This behavior is probably due to decrease in thickness of strcture 4 compared to structure 1, which might be causing greater de viatoric stress act on base layer of structure 4. Due to decrease in structure thickness and greater deviatoric stresses, modulus nonlinearity increases and hence, greater strain magnitude and eventually lower equivalent single effective modulus. When we compare the back calculated base effective moduli results with that of in Chapter 7, e quivalent e ffective modulus of base layer for any water content is smaller than the corresponding equivalent elastic modulus obtained with elastic subgrade of 50 MPa (refe r to Table 7.1 to 7.3). This decrease in base equivalent effective modulus might be due to the PAGE 178 178 additional effect of subgrade nonlinearity along with base nonlinearity, which eventually causes greater strain in base and lower equivalent effective moduli. In the case of structure1, back calculated effective moduli for subgrade layer is in the range of 55 60 MPa, where as for strcutre 4 it is 42 44 MPa. This decrease in subgrade effective modulus in structure 4 clearly indicates increase in subgrade nonlinear ity due to decrease in structure thickness. Thus as the structure thickness decreases, material modulus nonlinearity increases. Back calculated equivalent elastic modulus for AC layer is in the range of 800 MPa for thick structure and in the range of 1050 MPa for thin structure, where as the actual modulus value used for AC layer in nonlinear analysis is 1000 MPa. FWD analysis back calculates elastic moduli based on surface deflection zone input. This variation in AC layer back calculated modulus from actual value (i.e. 1000 MPa) may also have little bit influence on back calculated equivalent e ffective moduli value of base and subg rade layer. 8.2.2 Comparison of Nonlinear and Equivalent Linear Analyses Pavement R esponses Various pavement responses (as give n S ection 7.3.1) obtained at different locations of pavement structure (as shown in Figure 7 1 ) from nonlinear analysis and corre sponding equivalent linear analysis are compared for further analysis. Pavement responses obtained from nonlinear and equivalent linear analyses of few selected pavement structures with 1000 MPa AC surface, different nonlinear base materials at different w ater contents and Ottawa sand nonlinear subgrade are compared. List of structures analyzed for responses comparison is given in Table 87. For all the cases given in Table 87, nonlinear and equivalent linear response curves comparison plots for all paveme nt responses mentioned in section 7.3.1 are presented in Appendix D. PAGE 179 179 xx, tensile strain) at bottom of AC layer, obtained from nonlinear and equivalent linear analysis of structure 1 with different base water contents are compared in Figure 8 3a for Newberry limerock and Figure 8 3b for Georgia granite. Same set yy, compressive strain) at top of subgrade layer and shown in Figure 84a for Newberry limerock and Figure 8 4b for Georgia granite. From Surface deflection comparison plots (Figures D 1 to D 3 for New berry limerock and Figures D 34 to D 35 for Georgia granite, from Appendix D); it can be observed that the surface deflection profiles for nonlinear and equivalent linear analyses are matching well. Hence, considering back calculated effective moduli for base and subgrade layers in place of nonlinear moduli does not affect the actual surface deflections. From the comparison column plots of horizontal tensile strain ( xx) at bottom of AC laye r (Figure 8 3a and 83b), for structure 1 with different base water contents and Ottawa sand subgrade modulus, it can be observed that horizontal strain ( xx) at bottom of AC layer for nonlinear moduli case is equal or nearly equal to equivalent linear case with back calculated single effective moduli. This indicates that considering back calculated effective modulus for both base and subgrade layers in place of nonlinear moduli for pavement design may not significantly affect the o verall cracking performance of pavement structure Hence, subgrade nonlinearity may not influence the cracking performance of a pavement structure. From the comparison plots of vertical strain ( yy) at top of subgrade (Figure 84a to 84b), for both Newberry limerock and Georgia granite, it can be observed that yy at top of subgrade for nonlinear analysis is almost two times greater than corresponding equivalent linear analysis. This indicates that the subgrade nonlinearity may influence pavements rutting performance very significantly. Since the vertical strain ( yy) at top of subgrade calculated by nonlinear analysis is PAGE 180 180 greater than equivalent linear analysis, currently practiced elastic desi gn methods might be overestimating rutting performance. Thus, a new rutting performance criterion may b e required to design pavement structures using single effective design modulus in place of nonlinear modulus, for both base and subgrade layers. Since subgrade nonlinearity can significantly influence rutting performance, it may be necessary to consider subgrade nonlinearity in pavement design procedures and nonlinear response models. 8.2.3 Analysis of Nonlinear and L i near responses in the Perspective of Cracking and Rutting Performance C riteria Main purpose of this exercise is to investigate the effect of base and subgrade nonlinearity at top of subgrade layer for structure 1with Newberry limerock and Georgia granite as base layers for selected water contents are given in Table 8 8. Nd for rutting failure is calculated based on Asphalt Institute rutting criterion (Equation 7 2) From the analysis data presented in Table 8 8 for structure 1 bottom of AC layer for nonlinear case is almost equal to that of linear case. It implies that subgrade nonlinearity may not affect the cracking performance of structure 1 and consideration of equivalent elastic modulus for both base and subgrade layers is acceptable in the aspect of cracking performance criterion. yy at top of subgrade layer for nonlinear case is almost two times that of linear case. It implies that subgrade nonlinearity may affect the rutting performance o f pavement structure, significantly. Asphalt Institutes empirical equation for rutting failure criteria is: (7 2) Where Nd allowable number of load repetitions to limit permanent deformation PAGE 181 181 c compressive strain on the top of yyfor our case) f4 = 1.365 109 f5 = 4.477 Allowable number of load repetitions (Ndyy using E quation (72) and are presented in Table 8 8. The value of Nd linear/ Nd nonlinear for high water contents goes up to 15.7. This indicates that, by considering elastic modulus for subgrade layer in design procedures, rutting performance of pavement is being over estimated by almost 15 times to actual value. It also implies that elastic design methods overestimate the actual rutting performance of structure It may not be appropriate to use elastic criterion based empirical equations for rutting performance analysis of pavement structures Based on nonlinear base analysis results obtained for structure 4 from Chapter 7, it is understood that as the structure thickness decreases modulus nonlinearity influence on pavement performance increases. Since modulus nonlinearity increases with decrease in structure thickness, it is expected that considering nonlinear base and subgrade fo r strcuture 4 would influence rutting performance more than that of structure 1. Therefore, it may be necessary to consider subgrade nonlinearity in design and analysis of pavement structures and further research investigation is required for complete unde rstanding of its influence on pavement performance. PAGE 182 182 Table 81. Different material properties used for nonlinear base and nonlinear subgrade analysis Pavement Layer Material Modulus (MPa) Surface AC 1000 Base Newberry l imerock 13, 12, 11 & 10% w. c. Georgia g ranite 5.5, 4.5 & 3.5% w.c. Subgrade Ottawa sand HS Small parameters are given in Table 82. Table 8 2. HS Small model nonlinear material inputs for Ottawa sand subgrade layer Parameter Value G max (Mpa) 120 5.E 04 P ref (kPa) 100 E max (MPa) 316.80 E ur (MPa) 269.28 E oed (MPa) 134.64 E 50 (MPa) 134.64 m 0.5 3 ) 18.11 0.32 PAGE 183 183 Table 8 3. Equivalent elastic moduli values obtained for structure 1 with Newberry l imerock base and Ottawa sand subg rade Layer Non linear Plaxis HSsmall model Analysis (M P a) Linear FWD Analysis Modulus (MP a) AC Surface Layer 1000 862 Base 13% w.c. 67.75 Subgrade Ottawa sand 54.95 AC Surface Layer 1000 858.12 Base 12% w.c. 86.60 Subgrade Ottawa sand 55.64 AC Surfa ce Layer 1000 795.10 Base 11% w.c. 124.24 Subgrade Ottawa sand 53.92 AC Surface Layer 1000 780.686 Base 10% w.c. 125.69 Subgrade Ottawa sand 58.67 Table 8 4. Equivalent elastic moduli values obtained for structure 4 with Newberry l imerock base and Ottawa sand subgrade Layer Nonlinear Plaxis HS s mall model Analysis (Mpa) Linear FWD Analysis (Mpa) AC Surface Layer 1000 1073.79 Base 13% w.c. 24.82 Subgrade Ottawa sand 44.61 AC Surface Layer 1000 1099.37 Base 12% w.c. 31.44 Subgrade Ottawa sand 43.85 AC Surface Layer 1000 1012.98 Base 11% w.c. 42.06 Subgrade Ottawa sand 43.85 AC Surface Layer 1000 1019.66 Base 10% w.c. 42.33 Subgrade Ottawa sand 44.26 PAGE 184 184 Table 8 5. Equivalent elastic moduli values obtained for structure 1 with Georgia g ranit e base and Ottawa sand subgrade Layer Nonlinear Plaxis HSs mall Analysis (Mpa) Linear FWD Analysis (Mpa) AC Surface Layer 1000 (elastic) 851.77 Base 5.5% w.c. 61.36 Subgrade Ottawa sand 55.80 AC Surface Layer 1000 (elastic) 823.99 Base 4.5% w.c. 114.0 4 Subgrade Ottawa sand 58.67 AC Surface Layer 1000 (elastic) 820.82 Base 3.5% w.c. 124.00 Subgrade Ottawa sand 56.67 Table 8 6. Equivalent elastic moduli values obtained for structure 4 with Georgia Granite base and Ottawa sand subgrade Layer Nonli near Plaxis HS s mall Analysis (MP a) Linear FWD Analysis (M P a) AC Surface Layer 1000 (elastic) 1058 Base 5.5% w.c. 24.54 Subgrade Ottawa sand 44.5 AC Surface Layer 1000 (elastic) 1005.32 Base 4.5% w.c. 40.47 Subgrade Ottawa sand 44.68 AC Surface Laye r 1000 (elastic) 944.44 Base 3.5% w.c. 46.19 Subgrade Ottawa sand 45.02 Table 8 7. List of structures analyzed for pavements various responses comparison Base Material Type of Structure Water Contents Newberry l imerock 1 13 10 4 10 Geo rgia granite 1 5.5 3.5 PAGE 185 185 Table 8 8. Asphalt Institutes rutting criteria analysis for structure 1 with nonlinear base and nonlinear subgrade base layer water content Design method XX ( ) XX (kN/m2) ( ) % YY YY (kN/m2) ND (AI emperical equation) ND linear / ND nonlinear Newberry Limerock 10.0% nonlinear 0.0127 217.12 0.0724 22.51 155959 14.34 linear 0.0121 137.40 0.0400 36.14 2236444 13.0% nonlinear 0.0179 304.57 0.0803 24.30 98558 15.52 linear 0.0179 251.68 0.0435 37.74 152 9735 Georgia g ranite 3.5% nonlinear 0.0126 214.05 0.0711 23.42 169236 12.59 linear 0.0121 147.80 0.0404 36.65 2129920 5.5% nonlinear 0.0191 325.58 0.0814 26.47 92731 15.71 linear 0.0192 267.20 0.0440 39.23 1456394 AC surface = 1000 MPa Ottawa sand subgrade PAGE 186 186 Figure 8 1. FWD analysis of s tructure 4 PAGE 187 187 Figure 8 2. Comparison of nonlinear and linear surface deflections for structure 1 with 10% w.c. base and Ottawa sand subgrade. PAGE 188 188 a b Figure 83. Comaprison of horiz ontal tensile strain at bottom of AC layer (% xx) obt ain ed for structure 1 with Ottawa sand subgrade and different bas e water contents of a)Newberry l imerock b)Georgia granite PAGE 189 189 a b Figure 84yy) obtai ne d for structure 1 with Ot tawa sand subgrade and different bas e water contents of a)Newberry l imerock b)Georgia granite PAGE 190 190 CHAPTER 9 CLOSURE 9.1 Summary of Findings Main goal of this research investigation is to develop a design methodology to determine single effective modulus of ba se layer that can approximate known material modulus nonlinearities and can be used as MEPDG level 2 or level 3 material parameter input. In order to achieve this goal it is proposed to conduct a lab testing program on base soils and develop a nonlinear r esponse model using the lab testing results, which can assist in developing effective modulus determination methodology. First, a laboratory testing program is conducted on two base layer soils used in the State of Florida, to characterize their modulus nonlinearity with respect to stress, strain s including small level strains and moisture content. Fixed Free resonant column (FixedFree RC) tests that can measure shear modulus in the strain range of 105% 101% while varying pressure confinement and moist ure content, are conducted on modified proctor compacted cylindrical specimens of Newberry limerock and Georgia granite. Second, utilizing these laboratory tested parameters and results, a nonlinear finite element pavement response model that can account f or the above mentioned modulus nonlinearities is developed via Plaxis HSsmall model. Various types of pavement structures with nonlinear base and elastic AC surface and subgrade layers; are analyzed for single wheel loading via nonlinear response model. Ba sed on pavement responses obtained from this nonlinear analysis, a practical design methodology is developed to determine nonlinear equivalent single effective elastic design modulus for whole base layer that can approximately account for nonlinearity in b ase layer under pavement loading conditions and can be used for MEPD G level 2 and level 3 material parameter inputs. Influence of base layer nonlinearity on pavement cracking and rutting performance is evaluated. Out of professional interest, influence of both base and subgrade modulus nonlinearity on pavement PAGE 191 191 performance is also evaluated for few selected pavement structures. Credibility of lab testing results and nonlinear response model is verified. 9.1.1 Laboratory Testing on Unbound Aggregate Base Soil s Limerock is commonly used for pavement base layer construction in the State of Florida. Newberry limerock and Georgia granite are selected for lab testing, representing one limerock category and one nonlimerock category. Fixed Free Resonant C olumn tests (Fixed Free RC) and FreeFree R esonant C olumn (Free Free RC) test methods are followed. Fixed Free RC equipment is upgraded to measure strains as small as 105% strain using fiber optic sensor for strain measurement. Credibility of this up graded apparatus is evaluated and verified by conducting tests on Ottawa sand specimens and comparing sand test results with the Ottawa sand data available in literature. Fixed Free RC test are run on modified proctor specimens of Newberry limerock and Georgia granite to investigate the influence of effective confinement stress, loading strain including small level strains and moisture content. Modulus in the strain range of 105% to 101% is measured to analyze small strain modulus nonlinearity of these soils. Testing spe cimens are dried in lab environment and tested at different water contents in the process of drying. Additional effective confinement provided due to suction at different water contents is evaluated. Empirical equations to calculate maximum shear modulus ( Gmax) of dry soils are developed. An approximate methodology to calculate shear modulus at any given effective confinement, strain level and water content is proposed. Free Free R C tests are run on compacted specimens to determine very small strain modulus (i.e. Gmax at 5% strain) at different water contents ranging from OMC to near dry condition. From above mentioned testing program, following findings have been derived: 1) Based on tests on dry specimens, shear modulus of Newberry limerock is proportional to pre ssure confinement to the power of 0.702 and of Georgia granite is proportional to pressure confinement to the PAGE 192 192 power of 0.6389. 2) For dry materials, s hear modulus is maximum and elastic at strains lower than 104% and starts decreasing linearly in the strain range of 104% 103% and decrease nonlinearly thereafter. Presence of gravel size aggregate makes modulus of dry gravelly soils more nonlinear compared to dry sands. 3) Gmax of dry Newberry limerock and dry Georgia Granite at known void ratio(e) and con finement c) can be determined using following empirical equations : For Newberry limerock, Gmax c)0.7 0 2 For Georgia granite, Gmax c)0.638 9 where 4) In unsaturated condition, s hear modulus is maximum at 105% strain and start decreasing thereafter with increase in strain. 5) In dry condition, shear modulus is maximum (i.e Gmax) at 104% strain and decreases to 0.15 Gmax at 101% strain. Whereas, in unsaturated condition, shear modulus is maximum at 105% strain and decrease to 0.075 Gmax at 101% strain. C ompared to dry gravelly soils moisture presence increases modulus nonlinearity in unsaturated gravelly soils. 6) In unsaturated soils, decrease in water content due to material drying provides capillary suction conf inement This suction confinement increases with decrease in water content and increase Gmax very significantly. 7) Additional confinement due to suction at different moisture contents is material specific and need to be evaluated separately for each mater ial for accurate modulus nonlinearity analysis. 8) In dry soils, rate of decrease in G (i.e. G/Gmax) with increase in strain is independent of confinement pressure magnitude. In unsaturated soils, rate of decrease in G with increase in strain is independe nt of water content and its suction confinement. 9) In unsaturated condition, at constant water content, additional confinement magnitude due to suction effect does not change with increase in strain. 10) Strain magnitudes generated in Free PAGE 193 193 Free RC testing are approximately in the range of 105% and corresponding moduli are nearly equal to very small strain moduli of Fixed Free RC tests 9.1.2 Development of Nonlinear Response Model and Base Layer Nonlinear Modeling and Analysis Utilizing above laboratory t esting data as material nonlinear inputs, a nonlinear finite element response model that can account for modulus nonlinearity is developed via Plaxis HSsmall model. This response model can incorporate modulus nonlinearity with respect to effective stress confinement, strain magnitude including small level strains and moisture content. From initial pavement modeling and analysis exercises via Plaxis HSsmall, it is observed that the response model behaves too soft and as a result produces high deformations. H ence the response model is recalibrated using Lehane et al. ( 2008) footing analysis results. Input parameters of the HSsmall model are recalibrated such that Lehanes footing analysis via recalibrated model, pr edicts deformations matching with actual field measured footing settlements. This recalibrated nonlinear response model is followed for our further pavement base layer modeling and analysis. Based on laboratory testing data of Newberry limerock, Georgia granite and Miami limerock, calibrated material input parameters for nonlinear response model are developed. Required functional capabilities of the response model to incorporate modulus nonlinearity with respect to input parameters and loading are verified. To single out the influence of base layer modulus nonlinearity on pavement responses and performance, AC surface layer and subgrade layer are considered elastic; and base layer is considered nonlinear. Considering the surface deflection basin as the single matching factor between nonlinear and linear analysis, a new design methodology to determine nonlinear equivalent single effective modulus that can approximate the nonlinearities for whole base layer is developed. Single effective moduli data base for different structures with different subgrade PAGE 194 194 moduli and at different base water contents is developed. Influence of moisture content, subgrade modulus and structure type (thickness) on equivalent single effective modulus is evaluated. Effective moduli values of all three materials are compared with each other. Effective moduli data is also compared with MEPDG moisture (suction) model determined moduli data to verify the suitability of MEPDG moisture model for our base materials. Nonlinear and equivalent linear analyses pavement responses obtained at crit ical locations are compared to evaluate the applicability of base layer single effective modulus in place of nonlinear modulus. Pavement responses are analyzed to evaluate the effect of nonlinear base modulus on pavement performance in the perspective of r utting and cracking, in comparison with single effective modulus responses. Following findings have been derived from base layer nonlinear modeling and analysis 1) For same subgrade modulus, effective base modulus increases significantly with decrease in w ater content. Rate of increase in effective modulus with decrease in moisture content is material specific. 2) Our lab testing data showed that decrease in water content increases additional confinement caused by suction, which in turn increases small stra in modulus. This increase in smallstrain modulus is being reflected on equivalent single effective modulus and hence single effective modulus increases with decrease in water content. 3) For any given structure and base water content, single effective bas e modulus increases with increase in subgrade modulus. Single effective design modulus of base layer at any given water content depends on modulus of subgrade which supports base layer. 4) At any given moisture content and subgrade modulus combination, bas e layer single effective modulus value i s dependent on structures layers thicknesses. As the thicknesses of different layers in pavement structure varies, the magnitude of wheel load deviatoric stresses being transferred from top layers to bottom layers a lso varies. PAGE 195 195 Since soil modulus is nonlinear with respect to stress, as the magnitude of deviatoric stress varies; its effective modulus also varies. 5) Am ong all three materials, Miami limerock behaves very different and its effective modulus increases at faster rate very significantly as the material dries out. At OMC, effective modulus of Miami limerock is mu ch lower than that of Newberry l imerock and Georgia granite. But as the material dries out, its modulus increases at a very faster rate compared to o ther two materials and reaches maximum among all three materials. Therefore increase in design modulus with material drying is material specific. Since modulus tests are generally conducted at OMC, it is very important to be aware of this behavior, or else a pavement designer can get mislead ed very easily. 6) For MEPDG moisture model, with decrease in water content, modulus can increase maximum up to two times of modulus at optimum moisture content. But, effective base moduli values derived for our material s do not meet this criterion E/Eopt values increase up to 3.5 times for Newberry limerock, 2.5 times for Georgia granite and 47 times for Miami limerock. E/Eopt values for our materials are determined only up to 40% decrease in degree of saturation and ca n increase further with further decrease in degree of saturation. MEPDG moisture effect model cannot incorporate moisture/suction effect accurately for Florida base materials. 7) Surface deflection profiles for nonlinear and equivalent linear analyses are matching well and hence considering single effective modulus in place of nonlinear modulus does not affect the actual surface deflections. 8) H orizontal str ain ( xx) at bottom of AC layer for nonlinear base modulus case is equal or nearly equal to equivalent linear case with single effective base modulus. Hence it appears that considering equivalent single effective modulus for design of base layer thickness may not affect the overall cracking performance of pavement. 9) For thick structures, yy) at the top of subgrade for nonlinear base modulus is equal or nearly equal to equivalent linear case. For thin structures, the PAGE 196 196 difference in vertical str yy) at the top of subgrade between nonlinear base modulus and equivalent linear case increases as the base material water content decreases. Hence it appears that as the structure thickness and bas e layer water content decreases together material nonlinearity increases. 10) In overall, for both thick and thin structures, considering equivalent single effective design modulus in lieu of nonlinear modulus for base layer does not affect pavements cracking performance and elastic based cracking performance empirical equation can be used for cracking performance evaluation. 11) For thick structures, use of single effective modulus does not affect rutting performance. As the structure thickness and base moisture content decreases together, equivalent linea r design based on single effective design modulus of base layer may over estimate pavements rutting performance. A s the structure thickness and base moisture content decrease together, modulus nonlinearity of base soil becomes more significant and elastic analysis based performance criteria may not be applicable. Development of new performance criterion compatible with nonlinearity may be required for accurate performance analysis. 9.1.3 Nonlinear Modeling and Analysis of Base and Subgrade Layers From nonlinear modeling and analysis of pavement base layer, it is observed that nonlinear equivalent linear design methods may over estimate rutting performance of a pavement. Since rutting performance is mainly dependent on subgrade vertical strain and subgrade soils are also nonlinear, it is expected that subgrade nonlinearity may also influence pavement rutting performance significantly. Hence out of professional interest, it is decided to conduct nonlinear modeling and analysis of base and subgrade layers toge ther for few selected structures and evaluate effect of subgrade nonlinearity on pavement performance. For this nonlinear modeling, both base and subgrade soils are considered nonlinear and AC layer is considered elastic. Ottawa sand is chosen as the only subgrade material. Considering PAGE 197 197 surface deflection basin obtained from nonlinear analysis as matching factor, FWD back calculation method is followed to back calculate equivalent elastic moduli for all layers. Elastic moduli obtained from FWD analysis are used for equivalent linear analysis. Pavement responses obtained from nonlinear analysis and equivalent linear analysis are compared to evaluate the influence of nonlinear subgrade nonlinearity on pavement performance in the perspective of rutting and crac king. Following findings have been derived from nonlinear modeling and analy sis of base and subgrade layers: 1) FWD back calculated subgrade moduli values for nonlinear subgrade structures with different base water contents are in the range of 40 44 MPa. B y comparing the back calculated equivalent effective base moduli values of these nonlinear subgrade structures with that of same structures with 50 MPa elastic subgrade modulus, it can be observed that effective modulus of base layer with nonlinear subgrade is significantly lower than effective modulus of base layer with 50 MPa elastic subgrade layer. This significant decrease in base equivalent effective modulus for nonlinear subgrade structures is might be due to the additional effect of subgrade nonlinea rity along with base nonlinearity, which eventually causes greater strain and lower equivalent effective moduli. 2) S urface deflection profiles for nonlinear and equivalent linear analyses are matching well. Hence, considering FWD back calculated effective moduli for base and subgrade layers in place of nonlinear moduli does not affect the actual surface deflections. 3) H orizontal strain ( xx) at bottom of AC layer for nonlinear base and subgrade moduli case is equal or nearly equal to equivalent linear case with back calculated single effective moduli. This indicates that considering back calculated effective modulus for both base and subg rade layers in place of nonlinear modulus, for pavement design may not significantly affect the overall cracking performance of pavement structure Hence subgrade PAGE 198 198 nonlinearity may not influence the cracking performance of a pavement structure. 4) V ertical str yy) at top of subgrade f rom nonlinear analysis is almost two times greater than corresponding equivalent linear analysis. This indicates that the subgrade nonlinearity may influence pavements rutting performance very significantly. Since the verticayy) at top of subgrade calculated by nonlinear analysis is greater than equivalent linear analysis, it appears that currently practiced elastic design methods might be overestimating rutting performance. 5) From number of load repetitions (Nd) c alculations, it is found that equivalent elastic design method overestimates the rutting performance by approximately 17 times, compared to nonlinear analysis with nonlinear base and subgrade. Thus, a new rutting performance criterion may be required to de sign pavement structures using single effective design modulus in place of nonlinear modulus, for both base and subgrade layers. Since subgrade nonlinearity can significantly influence rutting performance, it may be necessary to consider subgrade nonlinear ity in pavement design procedures and nonlinear response models. 9.2 Conclusions Based on above findings, following conclusions are drawn. First, nonlinear small strain modulus of unbound aggregate base soils increases significantly with decrease in water content. It appears that decrease in water content due to material drying creates suction effect, which provides significant additional effective confinement. Suction effect created additional effective confinement, can increase nonlinear modulus signific antly up to strain levels as high as 102%. Second, it appears that single effective modulus, calculated based on surface deflection obtained from nonlinear analysis, can approximate modulus nonlinearities of base layer and can be used for MEPDG level 2 an d level 3 material parameter inputs. Single effective modulus of base layer can be influenced by decrease in moisture content and its suction effect, subgrade modulus and structure thickness. Suction effect on effective modulus is material specific and need to be PAGE 199 199 evaluated for each material separately. It appears that increase in small strain modulus due to suction effect is being reflected on nonlinear equivalent single effective modulus at different moisture contents, and hence small strain modulus nonlinearity should be considered for thickness design analysis. Third, based on comparison of pavement responses obtained from nonlinear and equivalent linear analysis, it appears that base layer nonlinearity does not affect cracking performance, but affects ru tting performance as the structure thickness and moisture content decreases together. As the structure thickness and moisture content decreases together, use of base layer single effective modulus in design analysis may overestimate rutting performance and hence it may be required to develop a suitable rutting performance criterion that can fit for the nonlinear rutting analysis. Last, subgrade nonlinearity appears to influence rutting performance very significantly and use of elastic modulus for whole subgrade layer overestimates rutting performance. 9.3 Recommendations The following recommendations are suggested after reviewing all of the findings and conclusions previously discussed: 1) Base course soils small strain modulus nonlinearity and moisture suct ion effect on it, need to be implemented in pavement design guidelines. 2) Nonlinear equivalent single effective base modulus determination methodology can be implemented in MEPDG for determining level 2 and level 3 material parameter inputs. 3) Suction ef fect on base material nonlinear modulus is material specific and need to be analyzed for each material separately. MEPDG moisture model for suction effect need to modified or improved for local base course soils. 4) Nonlinear based rutting performance crit eria needs to be developed for accurate nonlinear base and subgrade dependent rutting performance evaluation. 5) Influence of subgrade nonlinearity is analyzed for few selected structures and one subgrade material only. This analysis could be extended to m ore types of structures and different subgrade PAGE 200 200 soils for complete understanding of subgrade nonlinearity influence on pavement performance. 6) Our research work is limited to axisymmetric modeling of pavement structure subjected to single wheel loading. It may be further extended to three dimensional modeling of pavement structure, under multiple wheel and axle loads for accurate nonlinear modeling of soil element stress state and further performance analysis. 7) A fully fledged field testing program to mea sure in situ pavement responses is needed to verify computer based nonlinear response model results and modify the model accordingly for future pavement research. Field testing data will provide more insight into accurate pavement stress state and further fine tuning of nonlinear modeling. 8) Our research work is based on lab tested parameters only. Further research work based on insitu tested material parameters may provide more accurate pavement responses and performance evaluation. PAGE 201 201 APPENDIX A FIXED FRE E RESON ANT COLUMN TESTING DATA FOR DIFFERENT BASE SOILS Figure A 1. e = 0.5 Figure A 2. e=0. 55 PAGE 202 202 Figure A 3. G/Gmax Figure A 4. G/Gmax 55 for Newberry limerock PAGE 203 203 Figure A 5. a granite at e=0.25 Figure A 6. e=0.29 PAGE 204 204 Figure A 7. G/Gmax 25 for Georgia granite Figure A 8. G/Gmax s at e=0. 29 for Georgia granite PAGE 205 205 APPENDIX B NONLINE A R EQUIVALENT LINEAR EFFECTIVE BASE MODULI DATA FOR DIFFERENT TYPES OF BASE SOILS Table B 1. For structure 1, nonlinear equivalent linear effective moduli data for Newberry limerock base layer. Base Layer Water Content 13% 12% 11% 10% 8% 5.5% E max (Mpa)* 179.82 272.05 548.75 631.12 1249.55 2487.10 Case 1: 50 Mpa subgrade base E (Mpa) 79 102 140 153 196 230 E/E max 0.44 0.37 0.26 0.24 0.16 0.09 Case 2: 30 Mpa subgrade ba se E (Mpa) 65 79 112 118 142 171 E/E max 0.36 0.29 0.20 0.19 0.11 0.07 Case 3: 70 Mpa subgrade base E (Mpa) 90 117 170 175 230 267 E/E max 0.50 0.43 0.31 0.28 0.18 0.11 Case 4: 125 Mpa subgrade base E (Mpa) 108 157 227 2 41 310 387 E/E max 0.60 0.58 0.41 0.38 0.25 0.16 E max Maximum Young's modulus is calculated for in situ overburden stress at middle height of base layer PAGE 206 206 Table B 2. For structure 2, nonlinear equivalent linear effective moduli data for Newberry limerock base layer. Base Layer Water Content 13% 12% 11% 10% 8% 5.5% E max (Mpa)* 170.45 264.114 542.403 625.074 1245.22 2483.69 Case 1: 50 Mpa subgrade base E (Mpa) 74 92 120 125 155 178 E/E max 0.43 0.35 0.22 0.20 0.12 0.07 Case 2: 30 Mpa subgrade base E (Mpa) 58 70 92 94 118 139 E/E max 0.34 0.27 0.17 0.15 0.09 0.06 Case 3: 70 Mpa subgrade base E (Mpa) 85 109 143 149 184 207 E/E max 0.50 0.41 0.26 0.24 0.15 0.08 Case 4: 125 Mpa subgrade base E (Mpa) 107 147 210 220 278 308 E/E max 0.63 0.56 0.39 0.35 0.22 0.12 Table B 3. For structure 3, nonlinear equivalent linear effective moduli data for Newberry limerock base layer. Base Layer Water Content 13% 12% 11% 10% 8% 5.5% E max (Mpa)* 166.819 260.58 539.487 622.215 1242.78 2481.57 Case 1: 50 Mpa subgrade base E (Mpa) 66 89 112 124 148 168 E/E max 0.40 0.34 0.21 0.20 0.12 0.07 Case 2: 30 Mpa subgrade base E (Mpa) 54 67 90 101 108 130 E/E max 0.32 0.26 0.17 0.16 0.09 0.05 C ase 3: 70 Mpa subgrade base E (Mpa) 73 100 135 142 195 205 E/E max 0.44 0.38 0.25 0.23 0.16 0.08 Case 4: 125 Mpa subgrade base E (Mpa) 93 141 214 220 281 311 E/E max 0.56 0.54 0.40 0.35 0.23 0.13 PAGE 207 207 Table B 4. For structure 4, nonl inear equivalent linear effective moduli data for Newberry limerock base layer. Base Layer Water Content 13% 12% 11% 10% 8% 5.5% E max (Mpa)* 158.3 253.46 533.84 616.7 1238.44 2478.16 Case 1: 50 Mpa subgrade base E (Mpa) 62 80 108 112 123 13 2 E/E max 0.39 0.32 0.20 0.18 0.10 0.05 Case 2: 30 Mpa subgrade base E (Mpa) 49 60 69 70 76 78 E/E max 0.31 0.24 0.13 0.11 0.06 0.03 Case 3: 70 Mpa subgrade base E (Mpa) 71 96 129 135 139 144 E/E max 0.45 0.38 0.24 0.22 0.11 0.06 Case 4: 125 Mpa subgrade base E (Mpa) 84 127 196 206 228 240 E/E max 0.53 0.50 0.37 0.33 0.18 0.10 Table B 5. For structure 5, nonlinear equivalent linear effective moduli data for Newberry limerock base layer. Base Layer Wat er Content 13% 11% 5.5% E max (Mpa)* 152.538 530.054 2457.67 Case 1: 50 Mpa subgrade base E (Mpa) 73 100 102 E/E max 0.48 0.19 0.04 Case 2: 125 Mpa subgrade base E (Mpa) 105 189 209 E/E max 0.69 0.36 0.09 PAGE 208 208 Table B 6. For structure 6, nonli near equivalent linear effective moduli data for Newberry limerock base layer. Base Layer Water Content 13% 11% 5.5% E max (Mpa)* 152.112 529.529 2457.19 Case 1: 50 Mpa subgrade base E (Mpa) 76 90 107 E/E max 0.50 0.17 0.04 Case 2: 125 Mpa su bgrade base E (Mpa) 103 180 187 E/E max 0.68 0.34 0.08 PAGE 209 209 Table B 7. For structure 1, nonlinear equivalent linear effective moduli data for Georgia granite base layer. Base Layer Water Content 5.5% 4.5% 3.5% E max (Mpa) 136.23 444.71 647.68 Case 1: 50 Mpa subgrade base E (Mpa) 76 135 152 E/E max 0.56 0.30 0.23 Case 2: 30 Mpa subgrade base E (Mpa) 63 104 117 E/E max 0.46 0.23 0.18 Case 3: 70 Mpa subgrade base E (Mpa) 86 155 178 E/E max 0.63 0.35 0.27 Case 4: 125 Mpa subgrade base E (Mpa) 112 210 243 E/E max 0.82 0.47 0.38 Table B 8. For structure 2, nonlinear equivalent linear effective moduli data for Georgia granite base layer. Base Layer Water Content 5.5% 4.5% 3.5% E max (Mpa) 134.45 443.70 646.91 Case 1: 50 Mp a subgrade base E (Mpa) 70 112 124 E/E max 0.52 0.25 0.19 Case 2: 30 Mpa subgrade base E (Mpa) 56 94 104 E/E max 0.42 0.21 0.16 Case 3: 70 Mpa subgrade base E (Mpa) 82 135 150 E/E max 0.61 0.30 0.23 Case 4: 125 Mpa subgrade base E (Mpa) 105 195 220 E/E max 0.78 0.44 0.34 PAGE 210 210 Table B 9. For structure 3, nonlinear equivalent linear effective moduli data for Georgia granite base layer. Base Layer Water Content 5.5% 4.5% 3.5% E max (Mpa) 133.87 443.37 646.63 Case 1: 50 Mpa subgrade base E (M pa) 66 115 119 E/E max 0.49 0.26 0.18 Case 2: 30 Mpa subgrade base E (Mpa) 54 88 100 E/E max 0.40 0.20 0.15 Case 3: 70 Mpa subgrade base E (Mpa) 73 133 162 E/E max 0.55 0.30 0.25 Case 4: 125 Mpa subgrade base E (Mpa) 90 207 220 E/E max 0.67 0.47 0.34 Table B 10. For structure 4, nonlinear equivalent linear effective moduli data for Georgia granite base layer. Base Layer Water Content 5.5% 4.5% 3.5% E max (Mpa) 132.08 442.40 645.84 Case 1: 50 Mpa subgrade base E (Mpa) 60 100 106 E/E max 0.45 0.23 0.16 Case 2: 30 Mpa subgrade base E (Mpa) 46 74 75 E/E max 0.35 0.17 0.12 Case 3: 70 Mpa subgrade base E (Mpa) 67 123 137 E/E max 0.51 0.28 0.21 Case 4: 125 Mpa subgrade base E (Mpa) 84 184 220 E/E max 0.64 0.42 0.34 PAGE 211 211 Table B 11. F or structure 1, nonlinear equivalent linear effective moduli data for Maimi limerock base layer. Base Layer Water Content 8.0% 6.0% 4.0% E max (Mpa) 32.99 612.77 3474.36 Case 1: 50 Mpa subgrade base E (Mpa) 30 168 337 E/E max 0.91 0.27 0.10 Case 2 : 125 Mpa subgrade base E (Mpa) 31 260 545 E/E max 0.94 0.42 0.16 Table B 12. For structure 4, nonlinear equivalent linear effective moduli data for Miami limerock base layer. Base Layer Water Content 8.0% 6.0% 4.0% E max (Mpa) 32.99 612.77 3474.36 Case 1: 50 Mpa subgrade base E (Mpa) 6 54 159 E/E max 0.18 0.09 0.05 Case 2: 125 Mpa subgrade base E (Mpa) 7 122 320 E/E max 0.21 0.20 0.09 PAGE 212 212 APPENDIX C COMPARISON OF NONLINEAR AND EQUIVALENT LINEAR RESPONSES OBTAINED FROM NONLINEAR BASE ANALYS IS C.1 Newberry Limerock C.1.1 Surface Deflection Profiles C.1.1.1 Structure1 Figure C 1. S urface deflection c omparison for nonlinear and linear cases, for structure1 with 13% w.c. base layer PAGE 213 213 Figure C 2. S urface deflection c omparison for nonlinear and linear cases, for structure1 with 10% w.c. base layer. Figure C 3. Surface deflection comparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer. PAGE 214 214 Figure C 4. Surface deflection comparison for nonlinear and linear case s, for structure 1 with 5.5% w.c. base layer C.1.1.2 Structure4 Figure C 5. S urface deflection c omparison for nonlinear and linear cases, for structure4 with 13% w.c. base layer. PAGE 215 215 Figure C 6. S urface deflection c omparison for nonlinear and linear c ases, for structure 4 with 10% w.c. base layer. Figure C 7. Surface deflection comparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer. PAGE 216 216 Figure C 8. S urface deflection c omparison for nonlinear and linear cases, for structure4 with 5.5% w.c. base layer. PAGE 217 217 C.1.2 xx, Tensile Stress) at Top of AC Layer C.1.2.1 Structure 1 Figure C 9xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer. Figure C 10xx at top of AC layer compariso n for nonlinear and linear cases, for structure 1 with 10% w.c. base layer. PAGE 218 218 Figure C 11. xx at top of AC layer c omparison for nonlinear and linear cases, for structure 1 with 8 % w.c. base layer. Figure C 12xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. PAGE 219 219 C.1.2.2 Structure 4 Figure C 13xx at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 13% w.c. base layer. Figure C 14. xx at top of AC layer c omp arison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer. PAGE 220 220 Figure C 15. xx at top of AC layer c omparison for nonlinear and linear cases, for structure 4 with 8 % w.c. base layer. Figure C or nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. PAGE 221 221 C.1.3 xx, Tensile Strain) at Top of AC Layer C.1.3.1 Structure 1 Figure C 17. xx at top of AC layer c omparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer Figure C 18. xx at top of AC layer c omparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer PAGE 222 222 Figure C 19. xx at top of AC layer c omparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer Figure C xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. PAGE 223 223 C.1.3.2 Strcuture 4 Figure C 21. xx at top of AC layer comparison for nonlinear and linear cases, for structure4 with 1 3% w.c. base layer. Figure C 22xx at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer. PAGE 224 224 Figure C 23xx at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer Figure C 24. xx at top of AC layer c omparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer PAGE 225 225 C.1.4 xx, Tensile Stress) at Bottom of AC Layer C.1.4.1 Structure1 Figure C 25. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 1 3% w.c. base layer. Figure C 26. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer. PAGE 226 226 Figure C 27. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer. Figure C 28. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. PAGE 227 227 C.1.4.2 Strcuture 4 Figure C 29. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 13% w.c. base layer. Figure C 30. xx at bottom of AC layer c omparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer. PAGE 228 228 Figure C 31. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer. Figure C 32. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. PAGE 229 229 C.1.5 xx, Tensile Strain) at Bottom of AC layer C.1.5.1 Structure 1 Figure C 33. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer. Figure C 34xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer. PAGE 230 230 Figure C 35. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer. Figure C 36xx at bottom of AC layer co mparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. PAGE 231 231 C.1.5.2 Structure 4 Figure C 37. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 13% w.c. base layer. Figure C 38. xx at b ottom of AC layer c omparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer. PAGE 232 232 Figure C 39. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer. Figure C 40. xx at bott om of AC layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. PAGE 233 233 C.1.6 yy, compressive stress) at top of base layer C.1.6.1 Structure 1 Figure C 41. yy vertical stress at top of base layer compari son for nonlinear and linear cases, for structure 1 with 13% w.c. base layer. Figure C 42yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer. PAGE 234 234 Figure C 43yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer Figure C 44. yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. PAGE 235 235 C.1.6.2 Strcuture 4 Figure C 45yy at top of base layer comparison for nonlinear and linear cases, for structure 4 with 13% w.c. base layer Figure C 46. yy vertical stress at top of base layer c omparison for nonlinear and linear cases, for structure 4 wit h 10% w.c. base layer. PAGE 236 236 Figure C 47. yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer. Figure C 48. yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. PAGE 237 237 C.1.7 yy, Compressive Strain) at Top of Base Layer C.1.7.1 Structure 1 Figure C 49. yy at top of base layer comparison for nonlinear and linear cases, for structure 1 with 1 3% w.c. base layer. Figure C 50yy at top of base layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer. PAGE 238 238 Figure C 51. yy at top of base layer comparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer. Figure C 52. yy at top of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. PAGE 239 239 C.1.7.2 Structure 4 Figure C 53. yy at top of base layer comparison for nonlinear and linear cases, for structure 4 w ith 1 3% w.c. base layer. Figure C 54. yy at top of base layer c omparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer. PAGE 240 240 Figure C 55. yy at top of base layer comparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer. Figure C 56. yy at top of base layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. PAGE 241 241 C.1.8 Vertical S tress ( yy, Compressive Stress) at Bottom of Base Layer C.1.8.1 Structure 1 Figure C 57. yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 1 3% w.c. base layer. Figure C 58. yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer. PAGE 242 242 Figure C 59. yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer. Figure C yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. PAGE 243 243 C.1.8.2 S trcuture 4 Figure C 61. yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 13% w.c. base layer. PAGE 244 244 Figure C 62. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer. Figure C 63. yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer. Figure C 64. yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w. c. base layer. PAGE 245 245 C.1.9. yy, Compressive Strain) at Bottom of Base Layer C.1.9.1 Strcuture 1 Figure C 65. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer. PAGE 246 246 Figure C 66yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer Figure C 67. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer. PAGE 247 247 Figure C 68. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. C.1.9.2 Structure 4 Figure C 69. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure 4 with 13% w.c. base layer. PAGE 248 248 Figure C 70. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer. Figure C 71. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer. Figure C 72yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer PAGE 249 249 C.1.10 yy, Compressive Stress) at Top of Subgrade Layer C.1.10.1 Structure 1 Figure C 73. yy a t top of subgrade c omparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer. PAGE 250 250 Figure C 74yy at top of subgrade comparison for nonlinear and linear cases, for structure 1 with 1 0% w.c. base layer Figure C 75. yy at top of subgrade c omparison for nonlinear and linear cases, for structure 1 with 8% w.c. base layer. PAGE 251 251 Figure C 76. yy at top of subgrade c omparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. C.1.10.2 Structure 4 Figure C 77. yy at top of subgrade c omparison for nonlinear and linear cases, for structure 4 with 13% w.c. base layer. PAGE 252 252 Figure C 78. yy at top of subgrade c omparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer. Figure C 79. yy at t op of subgrade c omparison for nonlinear and linear cases, for structure 4 with 8% w.c. base layer. Figure C 80yy at top of subgrade comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. PAGE 253 253 C.1.11 yy, Compressive Strain) at Top of Subgrade Layer C.1.11.1 Structure 1 Figure C 81. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 1 with 13% w.c. base layer. PAGE 254 254 Figure C 82. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 1 with 10% w.c. base layer. Figure C 83. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 1 with 8 % w.c. base layer. PAGE 255 255 Figure C 84. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 1 with 5.5% w.c. base layer. C.1.11.2 Structure 4 Figure C 85. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 4 with 13% w.c. base layer. PAGE 256 256 Figure C 86. yy at top of subgr ade layer c omparison for nonlinear and linear cases for structure 4 with 10% w.c. base layer. Figure C 87. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 4 with 8% w.c. base layer. PAGE 257 257 Figure C 88. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 4 with 5.5% w.c. base layer. PAGE 258 258 C.2 Georgia Granite C.2.1 Surface Deflection Profiles C.2.1.1 Structure 1 Figure C 89. S urface deflection c omparison for nonlinear and linear cases, for str ucture 1 with 5.5% w.c. base layer. Figure C 90. Surface deflection comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer. PAGE 259 259 C.2.1.2 Structure 4 Figure C 91. S urface deflection c omparison for nonlinear and linear cases, f or structure 4 with 5.5% w.c. base layer. Figure C 92. S urface deflection c omparison for nonlinear and linear cases, for structure 4 with 3.5% w.c. base layer. PAGE 260 260 C.2.2 xx, Tensile Stress) at Top of AC Layer C.2.2.1 Structure 1 Figur e C 93. xx at top of AC layer c omparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. Figure C 94xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer PAGE 261 261 C.2.2.2 Strcut rue 4 Figure C 95. xx at top of AC layer c omparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. Figure C 96. xx at top of AC layer c omparison for nonlinear and linear cases, for structure 4 with 3.5% w.c. base layer. PAGE 262 262 C.2.3 xx, Tensile Strain) at Top of AC Layer C.2.3.1 Structure 1 Figure C 97. xx at top of AC layer c omparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer Figure C 98. xx at top of AC layer c ompar ison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer PAGE 263 263 C.2.3.2 Structure 4 Figure C 99. xx at top of AC layer c omparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer Figure C 100. xx at top of AC layer c omparison for nonlinear and linear cases, for structure 4 with 3.5% w.c. base layer PAGE 264 264 C.2.4 xx, Tensile Stress) at Bottom of AC Layer C.2.4.1 Structure 1 Figure C 101. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. Figure C 102xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer. PAGE 265 265 Figure C 103xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer Figure C xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 3.5% w.c. base layer. PAGE 266 266 C.2.5. xx, Tensile Strain) at Bottom of AC Laye r C.2.5.1 Structure 1 Figure C 105. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. Figure C 106. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 wit h 3.5% w.c. base layer. PAGE 267 267 C.2.5.2 Structure 4 Figure C 107. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. Figure C 108. xx at bottom of AC layer comparison for nonlinear and linear case s, for structure 4 with 3.5% w.c. base layer. PAGE 268 268 C.2.6 yy, Compressive Stress) at Top of Base Layer C.2.6.1 Structure 1 Figure C 109. yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 wit h 5.5% w.c. base layer. Figure C 110. yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer. PAGE 269 269 C.2.6.2 Structure 4 Figure C 111. yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. Figure C 112. yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 4 with 3.5% w.c. base layer. PAGE 270 270 C.2.7 yy, Compres sive Strain) at Top of Base Layer C.2.7.1 Structure 1 Figure C 113. yy at top of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. Figure C 114. yy at top of base layer comparison for nonlinear and linear cases, for structure1 with 3.5% w.c. base layer. PAGE 271 271 C.2.7.2 Structure 4 Figure C 115. yy at top of base layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. Figure C 116. yy at top of base layer comparison f or nonlinear and linear cases, for structure 4 with 3.5% w.c. base layer. PAGE 272 272 C.2.8 yy, Compressive Stress) at Bottom of Base Layer C.2.8.1 Structure 1 Figure C 117. yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. Figure C 118yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer PAGE 273 273 C.2.8.2 Structure 4 Figure C 119. yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. Figure C 120yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 3.5% w.c. base layer. PAGE 274 274 C.2.9 yy, Compressive Strain) at Bottom of Base Layer C.2.9.1 Structure 1 Figure C 121. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer. Figure C 122. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer. PAGE 275 275 C.2.9.2 Structure 4 Figure C 123. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. Figure C 124. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure 4 with 3.5% w.c. base layer. PAGE 276 276 C.2.10 yy, Compressive Stress) at Top of Subgrade Layer C.2.10.1 Structure 1 Figure C 125. yy at top of subgrade c omparison for nonlinear and linear cases, for st ructure 1 with 5.5% w.c. base layer. Figure C 126. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 1 with 3.5% w.c. base layer. PAGE 277 277 C.2.10.2 Structure 4 Figure C 127. yy at top of subgrade layer c omparison for nonlinear and linear cases, for structure 4 with 5.5% w.c. base layer. Figure C 128. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 4 with 3.5% w.c. base layer. PAGE 278 278 C.2.11 yy, Compressive Strain) at Top of Subgrade Layer C.2.11.1 Structure 1 Figure C 129. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 1 with 5.5% w.c. base layer. Figure C 130. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 1 with 3.5% w.c. base layer. PAGE 279 279 C.2.11.1 Structure 4 Figure C 131. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 4 with 5.5% w.c. base layer. Figure C 132. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 4 with 3.5% w.c. base layer. PAGE 280 280 APPENDIX D COMPARISON OF NONLINEAR AND EQUIVALENT LINEAR RESPONSES OBTAINED F0R NONLINEAR BASE AND NONLINEAR SUBGRADE ANALYSIS D.1 Newberry Limerock D.1.1 Surface Deflection Profiles Figure D 1. Surface deflection comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade PAGE 281 281 Figure D 2. Surface deflection comparison for nonlinear and linear cases, for structure 1 with 10% w .c. base layer and nonlinear Ottawa sand subgrade. Figure D 3. Surface deflection comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer and nonlinear Ottawa sand subgrade PAGE 282 282 D.1.2 xx, Tensile Stress) at T op of AC Layer Figure D 4xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade. Figure D 5. xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 1 0% w.c. base layer and nonlinear Ottawa sand subgrade. PAGE 283 283 Figure D xx at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer and nonlinear Ottawa sand subgrade. D.1.3 Horizontal Strain xx, Tensile Strain) at Top of AC Layer Figure D 7. xx at top of AC layer c omparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade PAGE 284 284 Figure D 8. xx at top of AC layer c omparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer and nonlinear Ottawa sand subgrade Figure D 9xx at top of AC layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer and nonlinear Ottawa sand subgra de. PAGE 285 285 D.1.4 xx, Tensile Stress) at Bottom of AC Layer Figure D 10. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 1 3% w.c. base layer and nonlinear Ottawa sand subgrade. Figure D 11. xx a t bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 1 0% w.c. base layer and nonlinear Ottawa sand subgrade. PAGE 286 286 Figure D 12. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 4 with 1 0% w.c. base layer and nonlinear Ottawa sand subgrade. D.1.5 xx, Tensile Strain) at Bottom of AC layer Figure D 13. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer and nonlinear O ttawa sand subgrade. PAGE 287 287 Figure D 14. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 1 0% w.c. base layer and nonlinear Ottawa sand subgrade. Figure D 15xx at bottom of AC layer comparison for nonlinear and li near cases, for structure4 with 1 0% w.c. base layer and nonlinear Ottawa sand subgrade. PAGE 288 288 D.1.6 yy, compressive stress) at top of base layer Figure D 16. yy vertical stress at top of base layer comparison for nonlinear and linear case s, for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade Figure D 17. yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer and nonlinear Ottawa sand subgrade PAGE 289 289 Figure D 18. yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 4 with 10% w.c. base layer and nonlinear Ottawa sand subgrade D.1.7 yy, Compressive Strain) at Top of Base Layer Figure D 19yy at top of base layer comparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade. PAGE 290 290 Figure D 20. yy at top of base layer comparison for nonlinear and linear cases, for structure 1 wit h 10% w.c. base layer and nonlinear Ottawa sand subgrade. Figure D 21yy at top of base layer comparison for nonlinear and linear cases, for structure 4 with 1 0% w.c. base layer and nonlinear Ottawa sand subgrade. PAGE 291 291 D.1.8 yy, Compress ive Stress) at Bottom of Base Layer Figure D 22. yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 1 3% w.c. base layer and nonlinear Ottawa sand subgrade. Figure D 23. yy at bottom of base layer comparison f or nonlinear and linear cases, for structure 1 with 1 0% w.c. base layer and nonlinear Ottawa sand subgrade. PAGE 292 292 Figure D 24. yy at bottom of base layer comparison for nonlinear and linear cases, for structure 4 with 1 0% w.c. base layer and nonlinear Ottaw a sand subgrade. D.1.9. yy, Compressive Strain) at Bottom of Base Layer Figure D 25. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade PAGE 293 293 Figure D 26. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer and nonlinear Ottawa sand subgrade Figure D 27. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure4 with 10% w.c. base layer and nonlinear Ottawa sand subgrade PAGE 294 294 D.1.10 yy, Compressive Stress) at Top of Subgrade Layer Figure D 28. yy at top of subgrade c omparison for nonlinear and linear cases, for structure 1 with 13% w.c base layer and nonlinear Ottawa sand subgrade Figure D 29. yy at top of subgrade c omparison for nonlinear and linear cases, for structure 1 with 10% w.c. base layer and nonlinear Ottawa sand subgrade PAGE 295 295 Figure D 30. yy at top of subgrade c omparis on for nonlinear and linear cases, for structure 4 with 10% w.c. base layer and nonlinear Ottawa sand subgrade D.1.11 yy, Compressive Strain) at Top of Subgrade Layer Figure D 31. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 1 with 13% w.c. base layer and nonlinear Ottawa sand subgrade PAGE 296 296 Figure D 32. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 1 with 10% w.c. base layer and nonlinear Ottawa sand subgrade Figure D 33. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 4 with 10% w.c. base layer and nonlinear Ottawa sand subgrade PAGE 297 297 D.2 Georgia Granite D.2.1 Surface Deflection Profiles Figure D 34. Surface deflection c omparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade Figure D 35. Surface deflection comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ot tawa sand subgrade. PAGE 298 298 D.2.2 xx, Tensile Stress) at Top of AC Layer Figure D 36xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade Figure D 37xx at top of A C layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade PAGE 299 299 D.2.3 xx, Tensile Strain) at Top of AC Layer Figure D 38xx at top of AC layer comparison for nonlinea r and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade. Figure D 39xx at top of AC layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade PAGE 300 300 D.2.4 xx, Tensile Stress) at Bottom of AC Layer Figure D 40. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade. Figure D 41. xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade. PAGE 301 301 D.2.5. xx, Tensile Strain) at Bottom of AC Layer Figure D 42xx at bottom of AC layer compa rison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade. Figure D 43xx at bottom of AC layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade. PAGE 302 302 D.2.6 yy, Compressive Stress) at Top of Base Layer Figure D 44. yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa s and subgrade Figure D 45. yy vertical stress at top of base layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade PAGE 303 303 D.2.7 yy, Compressive Strain) at Top of Base L ayer Figure D 46yy at top of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade. Figure D 47yy at top of base layer comparison for nonlinear and linear cases, for s tructure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade. PAGE 304 304 D.2.8 yy, Compressive Stress) at Bottom of Base Layer Figure D 48. yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade. Figure D 49. yy at bottom of base layer comparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade. PAGE 305 305 D.2.9 yy, Compres sive Strain) at Bottom of Base Layer Figure D 50. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade Figure D 51. yy at bottom of base layer c omparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade PAGE 306 306 D.2.10 yy, Compressive Stress) at Top of Subgrade Layer Figure D 52. yy at top of subgrade c omparison for nonlinear and li near cases, for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade Figure D 53. yy at top of subgrade c omparison for nonlinear and linear cases, for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade PAGE 307 307 D.2.11 yy, Compressive Strain) at Top of Subgrade Layer Figure D 54. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 1 with 5.5% w.c. base layer and nonlinear Ottawa sand subgrade Figure D 55. yy at top of subgrade layer c omparison for nonlinear and linear cases for structure 1 with 3.5% w.c. base layer and nonlinear Ottawa sand subgrade. PAGE 308 308 LIST OF REFERENCES Atkinson, J.H. (2000). Nonliner Soil Stiffness in Routine Design, Geotechnique, 50(5), 487508. Atkinson, J.H., and Sallfors, J. (1991). Experimental Determination of Stress Strain Time Characteristics in Laboratory and Insitu Tests Proceedings of Tenth European Conference on Soil Mechanics and Foundation Engineer, Rotterdam, Netherlands, 915958. C ho, G.C., and Santamarina, J.C. (2001). Unsaturated Particulate Materials Particle Level Studies, Journal of Geotechincal and Geoenvironmental Engineering, 127 ( 10) 8496. Hardin, B.O., and Richart, F.E., Jr. (1963). Elastic Waves in Granular S oils, Journal of Soil Mechanics and Foundation Division, 89( SM1 ) 3365. Hardin, B.O., and Drnevich, V.P. (1972) Shear Modulus and Damping in Soils: Measurement and Parameter Effects, Journal of Soil Mechanics and Foundations Division, 98( SM6 ) 603624. Hardin, B.O., and Drnevich, V.P. (1972) Shear Modulus and Damping in Soils: Design Equations and Curves, Journal of Soil Mechanics and Foundations Divison, 98( SM7 ) 667692. Hardin, B.O., and Kalinski, M.E., (2005). Estimating Shear Modulus of Gravell y Soils, Journal of Geotechincal and Geoenvironmental Engineering, 131( 7) 867873. Hunag, W., Yang, S, Kung, J.H.S., and Lin, H. (2006). Effect of Matric Suction on resilient Modulus of Compacted Subgrade Soils, Transportation Research Board 85th Annual Meeting Compendium of Papers CD ROM 21 pages Kokusho, T. (1987) InSitu Dynamic Soil Properties and Their Evaluations, Proceedings of the 8th Asian Regional Conference on Soil Mechanics and Foundation Engineering, Kyoto, vol. 2, 215240. Lehane, B.M., Doherty, J.P., and Schneider, J.A. ( 2008). Settlement Prediction for Footings on Sand, Proceedings of the fourth international symposium on deformation Characteristics of Geomateriasl, Atlanta (133150). Lin, S., Lin, P.S., Luo, H., and Juang, C.H. (20 00). Shear Modulus and Damping Ratio Characteristics of Gravelly Deposits, Canadian Geotechnical Journal 37, 638651. Kim, M.K., Tutumluer, E., and Kwon, J. (2009). Nonlinear Pavement Foundation Modeling for ThreeDimensional Finite Element Analysis of Flexible Pavements, International Journal of Geomechanics 9(5), 195208. PAGE 309 309 Picornell, M ., and Nazarian, S. (1998). Effect of Soil Suction on the Low Strain Shear Modulus of Soils, Proceedings of 2nd International Conference on Unsaturated Soils : UNSAT 8, Beijing China, 102 107. Pestana, J.M., and Salvati, L.A. (2006). Small Strain Behavior of Granular Soils.1:Model for Cemented and Uncemented Sands and Gravels, Journal of Geotechincal and Geoenvironmental Engineering, 132 ( 8) 10711081. Qian, X., Gr ay, D.H., and Richard, W. (1993). Voids and Granulometry: Efefcts on Shear Modulus of Unsaturated Sands, Journal of Geotechnical Engineering, 119 ( 2) 295314. Rollins, K.M., Evans, M.D., Dielhl, N.B., and Daily, W.D. (1998). Shear Modulus and Damping R elationships for Gravels, Journal of Geotechincal and Geoenvironmental Engineering, 124( 5) 396 405. Roque, R., Romero, P., and Hiltunen, D.R. (1992) The Use of Linear Elastic Analysis to Predict the Nonlinear Response of Pavements, Proceedings of 7th I nternational Conference on Asphalt Pavements Nottingham University, U.K., 295310. Santos, J.A., and Correia, A.G. (2001) Reference Threshold Shear Strain for Soil. Its application to Obtain an Unique StrainDependent Shear Modulus Curve for Soil Proceedings of the 15th International Conference on Soil Mechanics and Geotechnical Engineering, Istanbul, Vol 1, 267270. Seed, H.B., Wong, R.T., Idriss, I.M., and Tokimatsu, K (1986), Moduli and Damping Factors for Dynamic Analyses of Cohesionless Soils, J ournal of Geotechnical Engineering 112( 12) 10161032. Singh, A., Roberson, R., Ranaivoson, A., Siekmeir, J., and Gupta S. (2006), Water Retention Characteristics of Aggregates and Granular Materials, Proceedings of the Fourth International Conference on Unsaturated Soils Carefree, AZ, 13261337. Wu, S., Gray, D.H., and Richart, F.E., Jr. (1984), Capillarity Effects on Dynamic Modulus of Sands and Silts, Journal of Geotechnical Engineering, 110 ( 9) 11881203. Yasuda, N., Matsumoto, N. (1993), Dyna mic Deformation Characteristics of Sands and Rockfill Materials, Can adian Geotechnical Journal, 30, 745757. Yasuda, N., Ohat, N., and Nakamura, A. (1996), Dynamic Deformation Characteristics of Undisturbed Riverbed Gravels, Canadian Geotechnical Journal,, 33, 237249. PAGE 310 310 BIOGRAPHICAL SKETCH Aditya Ayithi was born in 1978 in Vinukonda India; and remained in Vinukonda until he graduated from L oyola High School in 1993. He enrolled in Pragathi residential college for Intermediate college education and gradu ated in 1995. He enrolled in Jawaharlal Nehru Technological University, Anantapur; in 1996 and with a Bachelor of Technology in civil engineering in summer 2000. After working in civil engineering private industry for one year, he enrolled in Indian Instit ute of Technology, Delhi for graduate studies in 2001 fall and graduated with Master of Technology in Soil Mechanics and Foundation engineering in f all 2002. Before pursuing his Doctoral education, he worked for more than two and half years in civil engineering in various positions, such as research management scientist at Department of Science and Technology, Government of India, for two years and project engineer at RADISE International Inc., FL; for 7 months. He enrolled at the University of Florida in Gainesville, FL in August, 2005 where he worked as a graduate research assistant under Dr. Dennis R. Hiltunen. He completed his Master of Science in May 2008 and Doctor of Philosophy in A ugust 2011. 