UFDC Home  myUFDC Home  Help 



Full Text  
PAGE 1 1 EFFECTS OF TRUCK TIRE TYPE AND TIRE PAVEM E NT INTERACTION ON TOP DOWN CRACKING AND INSTABILITY RUTTING By GUANGMING WANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 PAGE 2 2 2009 Guangming Wang PAGE 3 3 T o the memory of my dearest Mom Meihua Lin PAGE 4 4 ACKNOWLEDGE MENTS It is my great pleasure to acknow ledge and thank those individuals who have contributed to this dissertation. First and foremost thanks go to my advisor and committee chairman Dr. Rey naldo Roque, for his willingness to shar e his knowledge and experiences through invaluable guidance and co nstant encouragement. Special thanks go to Dr. Mang Ti a committee co chairman, who brought me to the University of Florida and gives me selfless support throughout my Ph.D studies. I would also like to thank other committee members, Dr. Larry Muszynski and Dr. Dennis Hiltunen, for their support in accomplishing my work. They are all great mentors and advisors. I would like to ex tend my thanks and appreciation to Dr. Michael C. McVay for lending his valuable advice and Dr. Morris de Beer for generously providing me with measured truck tire pavement interface stress data and technical papers I would also like to thank Mr. James H. Greene and other individuals from FDOT for their technical support and cooperation. Thanks also go to Mr. George Lopp for his support in the laboratory and his valuable advice. I would thank Patrick Dunn, Alvaro Guarin, Aditya Ayithi, Chulseung Koh and other individuals in materials group for their friendship. I would like to thank Weitao Li, Zhou Jian and all other Chinese students in our department for spending time and having fun together. I w ill not forget every moment we had. Finally, I would like to express my deepest love to my wife Ping Chen, for her faith love, selfless support and constant encouragement. I would also like to thank my parents Meihua Lin and Keting Wang, brothers Jian Wang and Min Wang and all other family members who gave me strength and confidence to conquer challenges that I faced along the way. PAGE 5 5 TABLE OF CONTENTS page AC KNOWLEDGE MENTS .............................................................................................................4 LIST OF TABLES ...........................................................................................................................8 ABSTRACT ...................................................................................................................................20 CHAPTER 1 INTRODUC TION ..................................................................................................................22 1.1 Background .......................................................................................................................22 1.2 Hypothesis ........................................................................................................................23 1.3 Objectives .........................................................................................................................23 1.4 Scope .................................................................................................................................24 1.5 Research Approach ...........................................................................................................24 2 LITERATURE REVIEW .......................................................................................................27 2.1 Introduction .......................................................................................................................27 2.2 Tire pavement Interface Stress .........................................................................................30 2.3 Tire Modeling ...................................................................................................................36 2.4 Mechanisms of Fracture and Rutting in Asphalt Pavements ............................................45 2.4.1 Rutting Models .......................................................................................................45 2.4.2 Traditional Fatigue Approach .................................................................................47 2.4.3 Fracture Mechanics Method ...................................................................................48 2.4.4 HMA Fracture Mec hanics ......................................................................................50 2.5 Summary ...........................................................................................................................50 3 THEORETICAL ANALYSIS OF SURFACE STRESSES ...................................................53 3.1 Point Load .........................................................................................................................53 3.2 Line Load ..........................................................................................................................55 3.3 Vertical Strip Load ...........................................................................................................55 3.4 Horizontal Strip Load .......................................................................................................58 3.5 Circular Vertical Load ......................................................................................................60 3.6 Circular Horizontal Load ..................................................................................................61 3.7 Combination of Normal and Transverse Load .................................................................63 3.8 Contact Between Elastic Solids ........................................................................................65 3.9 Summa ry ...........................................................................................................................67 4 EFFECTS OF TIRE TYPE AND PAVEMENT CROSS SECTION PROFILE ON TOP DOWN CRACKING AND INSTABILITY RUTTING ...............................................69 4.1 Introduction .......................................................................................................................69 PAGE 6 6 4.2 Evaluate Tire Types on Pavement Responses Based on BISAR Analysis .......................74 4.2.1 Modeling Tire Pavement Interface Stresses in BISAR ..........................................74 4.2.2 Effects of Tire Types on Pavement Responses ......................................................76 4.2.3 Damage Ratio .........................................................................................................83 4.2.4 Summary .................................................................................................................85 4.3 Development of 2D Axle Tire Pavement Contact Model ..............................................85 4.3.1 Tire Nomenclature ..................................................................................................85 4.3.2 Finite Element Analysis .........................................................................................89 4.3.2.1 Introduction of ADINA program .................................................................89 4.3.2.2 Governing equation ......................................................................................89 4.3.2.3 Contact analysis ............................................................................................89 4.3.2.4 Contact algorithms .......................................................................................90 4.3.3 Tires to Be Modeled ...............................................................................................92 4.3.4 Mesh of Tire Model ................................................................................................93 4.3.5 Development of Axle tire pavement Co ntact Model .............................................95 4.3.6 Element Selection ...................................................................................................98 4.3.7 Loading and Boundary Conditions .......................................................................100 4.3.8 Model Verification ...............................................................................................101 4.4 Effects of Rutted Surface on Near surface Stress Distributions .....................................104 4.4.1 St ructure and Material Properties of Model .........................................................104 4.4.2 Contact Stress Distributions of a Tire on Rutting Surface ...................................104 4.4.3 Critical Locations for Top Down Cracking ( TDC ) and Instability Rutting .........109 4.4.4 Effects of Rut on Near Surface Stress Distributions ............................................111 4.5 Evaluate Effects of Tire Types on Near surface Stress Distributions ............................115 4.5.1 Contact Stress Distributions .................................................................................115 4.5.2 Near su rface Stress Distributions .........................................................................115 4.6 Parametric Study .............................................................................................................119 4.6.1 Pavement Structure ...............................................................................................119 4.6.2 Influence of the Normal Axle Load ......................................................................120 4.6.3 Influence of the Inflation Pressure .......................................................................125 4.7 Summary .........................................................................................................................128 5 3D FINITE ELEMENT MODELING OF TIRE PAVEMENT INTERACTION .............130 5.1 Introduction .....................................................................................................................130 5.2 3D Modeling o f Tire pavement Interaction ..................................................................131 5.2.1 The Tire to be Modeled ........................................................................................131 5.2.2 Meshing of Tire Mode l .........................................................................................131 5.2.3 Meshing of Tire Pavement Contact Model ..........................................................132 5.2.4 Element Selection .................................................................................................132 5.2.5 Loading and Boundary Condition ........................................................................139 5.2.6 Mesh Convergence Analysis ................................................................................140 5.3 Model Calibration ...........................................................................................................141 5.4 Contact Patch Analysis ...................................................................................................152 5.5 Modeling of Wide base Tires .........................................................................................160 5.6 Summary and Conclusion ...............................................................................................167 PAGE 7 7 6 INVESTIGATE NEAR SURFACE STRESS STATES BASED ON 3 D TIRE PAVEMENT CONTACT MODEL .....................................................................................169 6.1 Introducti on .....................................................................................................................169 6.2 Development of 3D Tire pavement Interaction Model .................................................171 6.2.1 Pavement Structure Information ...........................................................................171 6.2.2 Element Mesh .......................................................................................................172 6.2.3 Model Verification ...............................................................................................174 6.3 Predicted Near surface Stress States ..............................................................................177 6.3.1 Vertical Stress States ............................................................................................178 6.3.2 Shear Stress States ................................................................................................179 6.3.3 Principal Stress States ...........................................................................................180 6.3.4 Mohr Coulomb Envelops and pq Space .............................................................184 6.3.5 Location of Critical Stress Poi nts .........................................................................186 6.3.6 Yield Percentage ...................................................................................................190 6.4 Effects of Tire Types on Near surface Stress Distributions Based on 3 D Analysis .....192 6.4.1 Contact Stress Distributions .................................................................................192 6.4.2 Near surface Stress Distributions .........................................................................195 6.5 Parametric Study .............................................................................................................195 6.5.1 Effects of Pavement Structure ..............................................................................197 6.5.2 Environmental Conditions ....................................................................................200 6.5.3 Tire Structure ........................................................................................................205 6.5.3.1 Effects of radial ply ....................................................................................205 6.5.3.2 Effects of tread ...........................................................................................207 6.5.4 Loading Conditions ..............................................................................................207 6.5.4.1 Effects of load ............................................................................................209 6.5.4.2 Effects of tire inflation pressure .................................................................212 6.6 Summary .........................................................................................................................215 7 CONCLUSIONS AND RECOMMENDATIONS ...............................................................217 7.1 Conclusions .....................................................................................................................217 7.2 Recommendations ...........................................................................................................218 APPENDIX A BISAR INPUT FOR TIRE PAVEMENT CONTACT STRESSES ....................................220 B SELECTED PREDICTED 3 D TIRE PAVEMENT CONTACT STRESSES ...................223 C SELECTED ADINA CODE FOR 3 D TIRE PAVEMENT CONTACT MODEL ............232 LIST OF REFERENCES .............................................................................................................248 BIOGRAPHICAL SKETCH .......................................................................................................257 PAGE 8 8 LIST OF TABLES Table page 21 Subgrade strain criteria used by various agencies (Huang, 1993) ....................................46 22 Fatigue cracking models used by various agencies (Huang, 1993) ..................................48 41 Relative damage between wide base and dual tires (Bonaquist, 1992) ............................70 42 Relative damage between wide base and dual tires (Sebaaly et al. 1989) .......................71 43 Pavement structural characteriscs used in BISAR ............................................................75 44 Damage ratio betwe en any tire and dual tire ....................................................................84 45 Specifications for Goodyear G149 RSA ......................................................................92 46 Specifications for Goodyear G286 A SS ......................................................................93 47 Specifications for Michelin X One XDA HT Plus ....................................................93 48 Material Properties for Tire Models .................................................................................94 49 Pavement structure and material properties ....................................................................104 410 Statistic results of the comparisons .................................................................................105 411 Statistic results of the comparisons .................................................................................111 412 Statistic results of the comparisons .................................................................................115 413 Pavement structur e information used for parametric study ............................................119 414 Summary of peak SIGMA 1 (unit: psi) .........................................................................120 415 Summary of peak maximum shear stress (unit: psi) ......................................................120 51 Material properties for tire pavement model ..................................................................142 52 Summary of 3 D contact stresses (11R22.5) ..................................................................160 53 Vertical tire axle displacement ........................................................................................162 54 Material properties for tire Models .................................................................................162 61 Material properties and layer thickness of the pavement ................................................171 62 Statistic results of the comparisons .................................................................................192 PAGE 9 9 63 Parametric variations used for study ...............................................................................197 64 Loading conditions used for the study ............................................................................209 65 Statistic results due to the effects of tire normal load .....................................................210 66 Statistic results due to the effects of tire inflation pressure ............................................213 A 1 3D tire contact stresses used in BISAR for 445/50R22.5..............................................220 A 2 3D tire contact stresses used in BISAR for 11R22.5 .....................................................221 A 3 3D tire contact stresses used in BISAR for 425/65R22.5..............................................222 B 1 Predicted 3 D vertical contact stress for 425/65R22.5 (9000 lb, 115 psi) ......................223 B 2 Predicted 3 D lateral contact stress for 425/65R22.5 (9000 lb, 115 psi) ........................224 B 3 Predicted 3 D longitudinal contact stress for 425/65R22.5 (9000 lb, 115 psi) ...............225 B 4 Predicted 3 D vertical contact stress for 445/50R22.5 (9000 lb, 100 psi) ......................226 B 5 Predicted 3 D lateral contact stress for 445/50R2 2.5 (9000 lb, 100 psi) ........................227 B 6 Predicted 3 D longitudinal contact stress for 445/50R22.5 (9000 lb, 100 psi) ...............228 B 7 Predicted 3D vertical contact stress for dual 11R22.5 (4500 lb, 110 psi) .....................229 B 8 Predicted 3 D lateral contact stress for dual 11R22.5 (4500 lb, 110 psi) .......................230 B 9 Predicted 3 D longitudinal contact stress for dual 11R22.5 (4500 lb, 110 psi) ..............231 PAGE 10 10 LIST OF FIGURES Figure page 11 Re search approach diagram ..............................................................................................26 21 TDC observed and core extracted from field section .......................................................28 22 Transverse stress distributions at AC surface (Jacobs, 1995) ...........................................28 23 Schematic of instability rutting .........................................................................................30 24 Schematic of Pressure Sensitive Film (Marshek et al ., 1985). .........................................32 25 Configuration of pressure film test (Marshek et al., 1985). ..............................................32 26 Schematic of system developed by Smither s Scientific Service ......................................33 27 Structural characteristics of radial and bias ply truck tires and their effects on the pavement surface (after Roque et al. 1998) .......................................................................34 28 Schematic of VRSPTA (after De Beer et al., 1997) .........................................................35 29 VRSPTA in measurement (after De Beer et al., 1997) .....................................................35 210 3D contact stress distribution for radial truck tire (after De Beer et al., 1997) ...............36 211 Flexible ring tire model (Loo, 1985) .................................................................................37 212 2D thin shell finite element models .................................................................................38 213 Geometry of tire model (Nakajim and Padovan 1986) .....................................................39 214 Extended elastic ring on elastic foundation (Bhm, 1991) ...............................................39 215 Geometry of tire model (Wang, 1990) ..............................................................................40 216 Membrane P195/75R14 tire model (Rhyne et al. 1994) ...................................................41 217 Detailed view of tire cleat enveloping situation (Kao et al., 1997) ..................................42 218 (a) Meshing of cross section (b) 2 D model (c) 3 D model (Zhang, 2001) ......................42 219 Element mesh of tire cross section (J Pelc, 2002) ............................................................44 220 Fatigue Crack Growth Behavior (after Jacobs, 1995) ......................................................49 221 Dissipated creep strain energy (after Roque et al., 1997) .................................................51 222 Crack propagation in asphalt mixture (after Roque et al., 1999) ......................................51 PAGE 11 11 31 Point load and normalized surface radial stress distributions ...........................................54 32 Shear stress distributions along depth at different constant radial distance ......................54 33 Line load ...........................................................................................................................55 34 Vertical strip load (a) A Uniform stress and (b) A triangular stress .................................56 35 Surface horizontal stresses due to uniform vertical strip load ..........................................57 36 Surface horizontal stresses due to triangular vertical strip load .......................................57 37 Shear stress distributions along depth ...............................................................................58 38 Horizontal Strip Load (a) A Uniform Stress and (b) A Triangular Stress ........................59 39 Horizontal Surface Stresses Due to Uniform Horizontal Strip Load ................................59 310 Horizontal Surface Stresses Due to Triangular Horizontal Strip Load .............................59 311 Horizontal normal stresses on the surface du e to circular vertical load ...........................60 312 Uniform Circular Horizontal Loads ..................................................................................61 313 Horizontal Surface Stresses Due to a) Anti sym metric Horizontal Load; b) Uni directional Horizontal load; c) Vertical Load (Jill M. Holewinski et al., 2003) ................62 314 Normal and transverse loading conditions ........................................................................64 315 Transverse stress at the edge of the load (x = a) as a function of distance below the pavement surface for different transverse contact pressure distributions (A.C. Collop and D. Cebon, 1993). .........................................................................................................64 316 Schematic diagram of Hertz contact .................................................................................65 317 Tensile and compressive stress contour map ( A. Franco Jr et al., 2004) .........................66 318 Distributions of normalized radial stress at the surface ....................................................67 41 Vertical contact stress distributions for different tire size (Markstal ler et al, 2000) ........73 42 Typical tire footprints from left to right: dual, super single and NGWB .........................75 43 Tire loading configur ations used in BISAR ......................................................................75 44 xx distributions along AC surface for dual 11R22.5 .............................................76 45 xx distributions along AC surface for super single 425/65R22.5 .........................76 PAGE 12 12 46 xx distributions along AC surface for NGWB 445/50R22.5 .................................77 47 Compar isons of maximum tensile stress at AC surface ....................................................78 48 xx distributions along AC surface due to unbalanced dual tires ...........................79 49 xx distributions along AC surface due to unbalanced dual tires ...........................79 410 xx distributions along AC surface due to unbalanced dual tires ...........................80 411 Comparisons of maximum shear strain along top AC layer .............................................80 412 Comparisons of maximum shear strain due to unbalanced dual tires ...............................81 413 Comparisons of maximum tensile strain at AC bottom ....................................................81 414 Comparisons of maximum tensile strain due to unbalanced dual tires .............................82 415 Comparisons of vertical compressive strain at subgrade ..................................................82 416 Comparisons of vertical compressive strain due to unbalanced dua l tires .......................83 417 Tire Sign Convention ........................................................................................................86 418 Components of a unisteel radial tire ( after Goodyear 2004) .............................................87 419 Tire Dimension ( after Goodyear 2004) .............................................................................88 420 Contactor and target selection ...........................................................................................90 421 Constraint function for normal contact .............................................................................91 422 Constraint functions for tangential contact .......................................................................92 423 Goodyear G1 49 RSA 11R22.5 .....................................................................................92 424 Goodyear G286 A SS 425/65R22.5 ..............................................................................93 425 Michelin X One XDA HT Plus 445/50R22.5 ............................................................93 426 Developed 2D finite element tire models ........................................................................95 427 Axle Tire Pavement Interactions ......................................................................................96 428 2D Axle tire pavement contact model for dual tire 11R22.5 ..........................................96 429 2D Axle tire pavement contact model for super single 425/65R22.5 .............................97 430 2D Axle tire pavement contact model for NGWB 445/50R22.5 ....................................97 PAGE 13 13 431 Flat pavement surface .......................................................................................................97 432 Pavement surface with rut .................................................................................................97 433 Contact surfaces between tire and pavement surface .......................................................98 434 9node Biquadratic El ements ............................................................................................98 435 Loading and boundary conditions ...................................................................................100 436 Comparison of vertical contact stress for dual 11R22.5 .................................................102 437 Comparison of vertical contact stress for dual 11R22.5 .................................................102 438 Comparison of vertical contact stress for NGWB 445/50R22.5.....................................102 439 Comparison of vertical contact stress for Super Single 425/65R22.5 ............................103 440 Predicted vertical contact stress for wide b ase tires .......................................................103 441 Predicted transverse contact stress for widebase tires ...................................................103 442 Forces acting on a tire on one side of a rut cross section ...............................................105 443 Comparison of vertical contact stress for dual 11R22.5 .................................................106 444 Comparison of transverse contact stress for dual 11R22.5 .............................................106 445 Comparison of vertical contact stress for wide base 425/65R22.5 .................................107 446 Comparison of tra nsverse contact stress for wide base 425/65R22.5 ............................107 447 Comparison of vertical contact stress for wide base 445/50R22.5 .................................108 448 Comparison of transverse contact stress for wide base 445/50R22.5 ............................108 449 Distributions of bending stress and principal tensile stress along AC surface ...............110 450 Distributions of principal tensile stress along AC depth ................................................110 451 Distributions of principal tensile stress at AC surface for 11R22.5 ...............................112 452 Distributions of principal tensile stress at AC surface for 425/65R22.5 ........................112 453 Distributions of principal tensile stress at AC surface for 445/50R22.5 ........................113 454 Distributions of maximum shear stress at AC depth for 11R22.5 ..................................113 455 Distributions of maximum shear stress at AC depth for 425/65R22.5 ...........................114 PAGE 14 14 456 Distributions of maximum shear stress at AC depth for 445/50R22.5 ...........................114 457 Comparisons of vertical contact stress among different tires .........................................116 458 Comparisons of transverse contact stress among different tires .....................................116 459 Comparisons of principal tensile stresses among tires for flat surface ...........................117 460 Comparisons of principal tensile stresses among tires for rutting surface ......................117 461 Comparisons of maximum shear stresses among tires for flat surface ...........................118 462 Comparisons of maximum shear stresses among tire s for rutting surface ......................118 463 Comparisons of peak SIGMA 1 due to different AC thickness .....................................121 464 Comparisons of peak SIGMA 1 due to different AC thickness .....................................121 465 Comparisons of peak maximum shear stress due to different AC thickness ..................122 466 Com parisons of peak maximum shear stress due to different AC thickness ..................122 467 Comparisons of peak maximum shear stress due to different base moduli ....................123 468 Comparisons of peak maximum shear stress due to different base moduli ....................123 469 Influence of axle load on vertical contact stress .............................................................124 470 Influence of axle load on transverse contact stress .........................................................124 471 Influence of axle load on peak max. shear stress and principal tensile stress ................125 472 Influence of tire inflation on tire deformation ................................................................126 473 Influence of inflation pressure on vertical contact stress ................................................126 474 Influence of inflation pressure on transverse contact stress ............................................127 475 Influence of inflation pressure on maximum shear stress ...............................................127 476 Peak SIGMA 1 as functions of inflation pressure ..........................................................128 51 Element group mesh and loading conditions ..................................................................133 52 Element group mesh and loading conditions ..................................................................134 53 Element group mesh .......................................................................................................135 54 Tire pavement contact group ..........................................................................................136 PAGE 15 15 55 Whole tire pavement model ............................................................................................137 56 8node El ement ...............................................................................................................138 57 Mesh convergence curve .................................................................................................140 58 Computer r unning time vs. circular divisions .................................................................141 59 Sensitivity analysis: effects of radial ply ........................................................................144 510 Sensitivity analysis: effects of sidewall ..........................................................................144 511 Sensitiv ity analysis: effects of belt 1 ..............................................................................145 512 Sensitivity analysis: effects of belt 2 ..............................................................................145 513 Sensitivity analysis: effects of tread 1 ............................................................................146 514 Sensitivity analysis: effects of tread 2 ............................................................................146 515 Time function vs. time step for normal axle load ...........................................................147 516 Load deflection curves for test and model (90 psi) ........................................................147 517 Load deflection curves for test and model (100 psi) ......................................................147 518 Load deflection curves for test and model (110 psi) ......................................................148 519 Z displacement contours at vertical axle load of 2250 lbs .............................................148 520 Z displacement contours at vertical axle load of 4500 lbs .............................................149 521 Z displacement contours at vertical axle load of 6750 lbs .............................................149 522 Z displacement contours at vertical axle load of 9000 lbs .............................................150 523 Comparisons of measured and predicted vertical contact stress across tire width .........150 524 Comparisons of measured and predicted transverse contact stress across tire width .....151 525 Comparisons of measured and predicted longitudinal contact stress .............................151 526 Vertical contact stress contour map (110 psi; 2250 lbs) .................................................153 527 Vertical contact stress contour map (110 psi; 4500 lbs) .................................................153 528 Vertical contact stress contour map (110 psi; 6750 lbs) .................................................154 529 3D Vertical contact stress distribution (110 psi; 6750 lbs) ...........................................154 PAGE 16 16 530 3D Vertical contact stress distribution (110 psi; 4500 lbs) ...........................................155 531 3D Vertical contact stress distribution (110 psi; 2250 lbs) ...........................................155 532 3D Lateral contact stress distribution (110 psi; 6750 lbs) .............................................156 533 3D Lateral contact stress distribution (110 psi; 4500 lbs) .............................................156 534 3D Lateral contact stress distribution (110 psi; 2250 lbs ) .............................................157 535 3D Longitudinal contact stress distribution (110 psi; 6750 lbs) ....................................157 536 3D Longitudinal contact stress distrib ution (110 psi; 4500 lbs) ....................................158 537 3D Longitudinal contact stress distribution (110 psi; 2250 lbs) ....................................158 538 Vertical contact stress contour along midplane depth (6,000 lbs) .................................159 539 Vertical contact stress contour along mid plane depth (9,000 lbs) .................................159 540 Vertical contact stress contour along mid plane depth (13,500 lbs) ...............................159 541 Vertical contact stress contour along mid plane depth (18,000 lbs) ...............................160 542 3D model for 425/65R22.5 ............................................................................................161 543 3D model for 445/50R22.5 ............................................................................................161 544 Comparisons of load displacement curves for 425/65R22.5 ..........................................162 545 Comparisons of loaddisplacement curves for 445/50R22.5 ..........................................163 546 Z displac ement contours for 425/65R22.5 (12 kips, 115 psi) ........................................163 547 Z displacement contours for 455/65R22.5 (12 kips, 100 psi) ........................................164 548 3D Vertical contact stresses for 445/50R22.5 (100 psi; 9000 lbs) ................................164 549 3D Lateral contact stresses for 445/50R22.5 (100 psi; 9000 lbs) ..................................165 550 3D Longitudinal contact stresses for 445/50R22.5 (100 psi; 9000 lbs) ........................165 551 3D Vertical contact stresses for 425/65R22.5 (115 psi; 9000 lbs) ................................166 552 3D Lateral contact stresses for 425/65R22.5 (115 psi; 9000 lbs) ..................................166 553 3D Longitudinal contact stresses for 425/65R22.5 (115 psi ; 9000 lbs) ........................167 61 3D Tire pavement contact group mesh .........................................................................173 PAGE 17 17 62 3D Tire pavement contact finite element mesh .............................................................173 63 Plan view of the pavement surface .................................................................................174 64 Comparisons of measured and predicted vertical contact stresses .................................175 65 Comparisons of measured and predicted transverse contact stresses .............................175 66 Comparisons of measured and predicted longitudinal contact stress es ..........................176 67 Predicted tire footprints ...................................................................................................176 68 Deformed pavement surface ...........................................................................................177 69 3D model without tire ....................................................................................................178 610 Line contours of vertical contact stress for the tire pavement model .............................179 611 Line contours of vertical contact stress for the uniform vertical load ............................179 612 Maximum shear stress e (in psi) distributions for uniform load .....................................180 613 Maximum shear stress (in psi) distributions for tire pavement ......................................180 614 Schematic of sign convention and maximum shear stress direction ...............................181 615 Maximum shear stress distribution as a function of depth at tire edge ...........................181 616 Mohrs circle ...................................................................................................................182 617 pavement model .................................182 618 .................................182 619 Maximum principal stress distribution along lateral distance to tire edge .....................183 620 Critical stress locations ...................................................................................................183 621 Stress states at critical location .......................................................................................184 622 Mohr Coulomb envelop ..................................................................................................185 623 Stress states in p q space wi th Mohr Coulomb failure envelopes for 30 psi cohesion and different internal friction angles ................................................................................186 624 Stress states in p q space with Mohr Coulomb failure envelopes for 20 psi cohesion and different internal friction angles ................................................................................186 625 Divided pavement zones for locating critical stress points purpose ...............................187 PAGE 18 18 626 Locat ions of stress points in pq diagram for radial tire model ......................................187 627 Locations of stress points in pq diagram for uniform load model .................................188 628 Locations of critical stress points and their fitted curve .................................................189 629 Direction of maximum shear stress along fitted curve ...................................................189 630 Illustration of yield percentage .......................................................................................190 631 ) .............................................191 632 ) .............................................191 633 Tire pavement contact models ........................................................................................193 634 Comparisons of vertical contact stress among different tires .........................................193 635 Comparisons of lateral contact stress among different tires ...........................................194 636 Comparisons of longitudinal contact stress among different tires ..................................194 637 Comparisons of principal tensile stresses among tires ...................................................196 638 Comparisons of maximum shear stresses among tires ...................................................196 639 Comparisons of average shear yield percentage among tires .........................................197 640 Effects of AC thickness and base moduli on vertical contact stress distributions (Tire middle yz plane) ...............................................................................................................198 641 Effects of AC thickness and base moduli on transverse contact stress distributions (Tire middle yz plane) ......................................................................................................198 642 Effects of AC thickness and base moduli on transverse contact stress distributions (Tire middle yz plane) ......................................................................................................199 643 Effects of AC thickness and base moduli on peak maximum shear stress .....................199 644 Effects of AC thickness and base moduli on principal tensile st 1 .........................200 645 Temperature gradient cases used to determine moduli gradients in the AC layer (Myers, 2000) ...................................................................................................................201 646 Modulus profiles for AC layer used in ADINA analysis ................................................202 647 Effects of temperature induced moduli gradients on tire pavement interface vertical contact stress distributions ...............................................................................................203 PAGE 19 19 648 Effects of temperature induced moduli gradients on tire pavement interface transverse shear contact stress distributions ....................................................................203 649 Effect s of temperature induced moduli gradients on tire pavement interface longitudinal shear contact stress distributions .................................................................204 650 Effects of temperature induced moduli gradients on peak maximum shear stress and principal tensile stres 1 .................................................................................................204 651 Effects of radial ply moduli on vertical contact stresses(middle yz plane) ....................206 652 Effects of radial ply moduli on principal tensile stress ...................................................206 653 Effects of radial ply moduli on maximum shear stress ...................................................207 654 Effects of tread modul i on vertical contact stress ( m iddle yz plane) ..............................208 655 Effects of radial ply moduli on principal tensile stress ...................................................208 656 Ef fects of tread moduli on maximum shear stress ..........................................................209 657 Effects of load on vertical contact stress distributions ....................................................210 658 Effects of load on transverse contact stress distributions ...............................................211 659 Effects of load on longitudinal contact stress distributions ............................................211 660 Significant levels of statistic parameters due to variable load ........................................212 661 Effects of inflation pressure on vertical contact stress distributions ...............................213 662 Effects of inflation pressure on transverse contact stress distributions ..........................214 663 Effects of inflation pressure on longitudinal contact stress distrib utions .......................214 664 Effects of inflation pressure on maximum shear stress ..................................................215 665 Effects of inflation pressure on principal te nsile stress ..................................................215 PAGE 20 20 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 EFFECTS OF TRUCK TIRE TYPE AND TIRE PAVEM E NT INTERACTION ON TOP DOWN CRACKING AND INSTABILITY RUTTING By Guangming Wang December 2009 Chair: Dr. Reynaldo Roque Major: Civil Engineering Top down cracking (TDC) and instability rutting are two of the main failures in the fl exible pavements across the world. Earlier studies have show n that the se are near surface distresses that are greatly affected by tire pavement interaction. The overall objective of this study was to develop and validate 2D and 3 D tire models a s well as tire pavement interaction model s based on finite element code ADINA and use them to investigate the effects of tire type, tire pavement interact ion and pavement cross section profiles on topdown cracking and instability rutting performance. A range of r adial truck tire types including dual, super single and new generation wide base tires were modeled based on tire geometry and structure information provided by the tire manufacturers. Tire models were calibrated based on load deflection data and verified by comparing measured and predicted contact stresses The results indicated that the tire models developed can accurately capture both vertical and horizontal contact stress characteristics and thus can be used for further evaluation purpose s In this stud y, it was found that there is a significant difference i n contact stress distributions among different tire types based on finite element simulation. Accordingly, near surface stress states in terms of principal tensile stress and maximum shear stress were quite different among the different tires Analysis results indicated that super single wide base tire PAGE 21 21 might produce greater damage to the pavement in terms of both topdown cr acking and instability rutting than either dual tires or new generation wideb ase tire. Analysis results also indicated that p avement cross sectional profiles with rutting or cross slopes increase d the propensity for topdown cracking and severity of instability rutting Parametric studies showed that pavement structure had relatively little effect on the stress states affecting topdown cracking. However, weak base or subgrade increase d the propensity for instability rutting. Also, both overloads and under inflation would likely increase the initiation of topdown cracking and insta bility rutting. Increasing the flexibility of tire sidewall or decreasing the rigidity of the tire tread may reduce the propensity for topdown cracking and instability rutting. Future studies including dynamic effects of vehicle and degree of wear on th e tire pavement interaction are needed. PAGE 22 22 CHAPTER 1 INTRODUCTION 1.1 Background Radial truck tire contact stresses are highly non uniform and depend strongly on the structural characteristics of the individual tire. The nonuniformity is due to bending mo duli in the tire structure and is therefore influenced by tire design (Tielking J.T. et al ., 1982, 1987). Also, pavement stress/strain distributions throughout the pavement thickness are sensitive to nonuniformity of tire contact stress as well as the tir e axle load. Clearly, knowledge of tire contact stress distribution is very important for pavement analysis, design and performance. Furthermore, proliferation of new tire technology including super wide based radial truck tires as substitute for dual tire loads has heightened the need for increased understanding of the effects of tire geometry, structure and axle load on tire pavement contact stress distributions It also poses significant concerns in terms of their potential impact on pavement performance Long term field studies to evaluate these effects are costly, timeconsuming, and may be of limited value because it will be difficult to isolate the performance effects of specific tire types from the many other effects involved in field studies. The He avy Vehicle Simulator (HVS) can provide results in a relatively short period of time regarding the impact of tire types on rutting performance, but even for this full scale experiment, expenses are high and results are limited to the individual tires, mixt ures, and other conditions used in the experiments selected. For example, different experiments would need to be performed for each tire type, inflation pressure, and load level. Effects of tire wear on performance would be difficult to handle. Furthermore evaluation of cracking performance using the HVS remains a major challenge, so effects of cracking cannot be immediately evaluated. Finally, the effect of surface condition, which affects tire pavement interaction, would be difficult to study either in t he field or with the HVS. PAGE 23 23 On the other hand, t heoretical modeling of tire behavior, considering the effects of the individual characteristics and properties of different tire types, offers an alternate, less costly, and more generalized approach to evaluat e these effects. In addition, development of tire and tire pavement interaction models will be supplementary and supportive to any ongoing and future field or HVS experiments by providing a tool that can help isolate the tire and tire pavement interaction effects from the other confounding effects that are generally present in these types of experiments. 1.2 Hypothesis The main hypothesis of this study is that tire structural characteristics and pavement cross section profiles caused by rutting resulting in nonuniform tire contact stresses may play a significant role in the development of surface tensile stresses associated with top down cracking and critical stress states associated with surface instability rutting. 1.3 Objectives The overall objective of this st udy is to develop and validate 2D and 3 D tire model s as well as tire pavement interaction model s and use them along with existing mixture and/or pavement performance criteria to evaluate the effect of tire type and tirepavement interaction on pavement top down cracking and instability rutting performance. Specific tasks associated with this overall objective are presented below: Review of literature and available tire data. Develop, c alibrate, and v alidate 2 D and 3 D f inite e lement m ethod (FEM) t ire m odels and t ire pavement i nteraction m odels Evaluate effect of tire type and condition on pavement response and performance. Evaluate effect of pavement structure and cross sectional profile on pavement response and performance. PAGE 24 24 1.4 Scope Modeling of a tire is really a challenging job due to its geometrical and material complexity. Tire materials are basically composite and rubber compounds which vary throughout tire structure. Unfortunately, exact material properties and actual structural makeup of these tires used by industry are not available to the general public, which makes tire modeling even more challenging. In addition, due to limitation of technology and reducing computational time, the axletire pavement mode l will be limited to 2 D finite element analysis. Some other limitations of the study are listed as follows: The study was restricted to radial truck tire models and static loading condition. Materials were modeled as linear elastic which seems appropriate for behavior under a single lo ad application, i.e., no wear or permanent shape change. Plane strain conditions were assumed for 2D analyses and only the stress strain response of the AC layer was considered in the study. The accuracy of stress/strain distributions within the base and subgrade layers was not considered. Tire contact stress measurements used in this study were obtained from a tire placed on an instrumented steel bed. Therefore, the pavement was modeled as a relatively stiff single layer support in 3D tire pavement inter action model. 1.5 Research Approach The research was mainly an analytical study supplemented by the acquisition of measured tire pavement interface stresses and other specific truck tire data from tire manufacturers. The tire manufacturers normally provide information related to tire behavior. The information can be divided into two groups. One group concerned about specifications related to tire geometry. The other group describe d the relationship between applied axle load and tire responses such as tire deflections. The overall research approach is presented in Figure 11, which is detailed below: PAGE 25 25 Literature Review: A review of how tire characteristics affect topdown cracking and instability rutting was conducted The history of tire contact stress meas urements was also reviewed. Recent progress related to modeling tire s and the mechanism of fracture and rutting in asphalt pavement was also covered Tire data, including make up, response characteristics, and measured contact stresses were collected from manufacturers and other sources. Findings from these efforts were used to define the most promising approaches for modeling and validating tire models. Theoretical Analysis of Surface Stresses: A theoretical knowledge of the elastic contact stress fields generated by point loads, line loads and area loads and various indenter geometries was examined, followed by comparisons of horizontal surface stresses from typical loading configurations. The Hertzian (H. Hertz, 1981, 1982) contact equations were also in troduced. The purpose of this analysis was to give some theoretical background on how topdown cracking and instability rutting are initiated The analysis also provides some benchmarks for verifying FEM solutions late r 2D Modeling of Tire pavement Inter action: BISAR (de Jong et al. 1973) analysis was first conducted to study the effects of tire types on pavement responses. A 2 D axletire pavement interaction model was then established to study how tire structural properties and pavement cross sectional profiles affect near surface stress states related to top down cracking and instability rutting 3D Modeling of Tire pavement Interaction: To further study tire pavement interaction, a 3D tire pavement interaction finite element model was established and verified, followed by contact stresses analysis. Evaluate Near surface Stress States Based on 3 D Tire pavement Interaction Model : N ear surface stress states predicted by 3 D tire pavement interaction model were compared with those predicted by the unifor m vertical loading condition ; T he effects of tire types on near surface stress states related to top down cracking and instability rutting were evaluated P arametric studies were conducted to evaluate pavement structures, material properties tire properti es and loading conditions on critical pavement responses Conclusions and Recommendations : Key points were summarized and conclusions were drawn. Finally, recommendations were made for future study. PAGE 26 26 Lliterature Review Tirepavement Interface Stress Modeling of Tire Mechanism of Fracture in AC Introduction Theoretical Analysis of Surface Stress Rutting Surface vs. Flat Surface WidebaseTire vs. Dual Tire 2D Modeling of Tirepavement Interaction 3D Modeling of Tirepavement Interaction 3D Modeling of Tire 3D Modeling of Tirepavement Interaction Model Calibration Contact Stress Analysis Evaluate Nearsurface Stress States Based on 3D Tirepavement Interaction Effects of Tire Types on Nearsurface Stress States Parametric StudyPavement Structure Conclusion and Recommendation 2D Modeling of Tirepavement Interaction BISAR Analysis Tirepavement Interaction Model vs. Uniform Vertical Load Model Parametric StudyTire Properties Parametric StudyLoading Conditions Figure 11. Research a ppro ach diagram followed in this study PAGE 27 27 CHAPTER 2 LITERATURE REVIEW 2.1 Introduction Top down cracking (TDC) and pavement instability rutting are two major types of distresses in asphalt pavements. In contrast with conventional bottom up fatigue cracking, TD C initiates at the pavement surface and propagates downward as shown in Figure 21. Since the early 1980s, TDC has been recognized in many countries, including France (Dauzats M and Linder R, 1982), South Africa (Hugo F and Kennedy T W, 1985), Netherlands ( A Gerritsen et al. 1987), Japan (Himeno K et al. 1987) and UK (Nunn M E et al. 1997). In the United States, TDC is increasingly reported as a major distress in asphalt pavements in several states, including Florida (Roque and Meyers, 1998, 1999, 2001), Michigan (Svasdisant et al. 2001), and Washington (Uhlmeyer, 2000). The causes of TDC are complicated yet most scholars believe it is load associated cracking since the majority of TDC is found under or near wheel paths. Tire pavement interface stresses and their corresponding effects on TDC are the most commonly studied issues. As early as 1984, Molenaar (1984) used the CIRCLY multi layer computer program to analyze the effects of tire pavement vertical and lateral contact stresses (inward shear stress es) on TDC. He found that high surface tensile strains induced at the edge of the tire are the cause of TDC. Later, Gerritsen et al. (1987) did similar analysis and drew the same conclusions as Molenaar (1984). In 1995, Jacobs (1995) employed estimated me asured threedimensional (3D) tire pavement contact stresses in linear elastic multilayer program BISAR (de Jong et al. 1973) and calculated transverse normal stress distributions along the asphalt concrete (AC) surface. He found that significant tensile stresses at the edge of the tire were sufficient to cause fracture. The PAGE 28 28 tensile stresses dissipated rapidly with increasing depth as shown in Figure 22. However, it should be noted here that Figure 21. TDC observed and core extracted from field sectio n Figure 22. Transverse stress distributions at AC surface (Jacobs, 1995) PAGE 29 29 estimated tire pavement contact stresses might not have been accurate enough. Myers (1997) used more realistic measured tirepavement contact stresses in BISAR and found critical tensile stresses under the middle of outer ribs. Groenendijk (1998) used 3D finite element program, CAPA, to analyze the load induced surface tensile stresses in the pavement. He employed the tire pavement contact stresses measured from Road Surface Press ure Transducer Array (VRSPTA), to simulate different loads and different inflation pressures, and concluded that the combined influence of the nonuniform tensile contact stress and the aging of the AC at the surface could result in critical tensile stress at the surface rather than the bottom of the AC. In the United Kingdom, Bensalem et al. (2000) modeled the behavior of un cracked and cracked pavements using 3D finite element analysis. The loading conditions were idealized as a uniform vertical contact stress, along with a lateral contact stress having maximum at the tire edge and zero in the center of tire. They found that the shear strain at the edge of the wheel in the vertical plane was higher than the maximum tensile strain at both the bottom of the surface layer and the pavement surface for all the pavements analyzed. They considered this to be of great importance in both initiation an d propagation of topdown cracking in flexible pavements. On the other hand, rutting especially instability rutting in asphalt pavement has generated more and more concern among scholars and pavement engineers, due to its increasing severity and extent in many states (Drakos, 2003). Instability rutting, as shown in Figure 23, occurs within wheel paths and is due to the lateral displacement of material within the pavement layer. It occurs when the structural properties of the compacted pavement are inadequate to resist the stresses imposed upon it (Novak, 2003). Studies by Dawley et al. (1990) and Drakos (2003) showed t hat instability rutting was primarily a near surface phenomenon, affecting only top 13 inches of the AC layer with visible slip surfaces associated with rutting failure. Novak (2003) PAGE 30 30 used 3D finite element program, ADINA, to analyze near surface shear st resses on different planes. He employed tire pavement contact stresses from Smithers Scientific Services, Inc., in Ravenna, Ohio, and found that maximum shear stress developed on a transverse plan, which was perpendicular to travel direction. Figure 2 3. Schematic of instability rutting Obviously, both TDC and instability rutting are near surface phenomena, which are greatly influenced by the tire pavement contact stresses. Measurements of the tire pavement contact stresses helped identify possible mechanisms for surface initiated cracking and near surface rutting. 2.2 Tire pavement Interface Stress Over the past two decades, tire technology has developed quickly The type of truck tires has changed from bias ply tires to conventional width radial tires a nd has been moving toward wider super single wide base tires. Correspondingly, tire inflation pressures have increased from 7080 psi to 100120 psi, and even up to 130140 psi for super single wide base tires. Research indicates that the changes of tire structure and tire inflation would alter tire pavement interface stress characteristics (Roque et al. 1998; De.Beer et al. 1997; Al Qadi et al. 2000, 2004; PAGE 31 31 Emmanuel G. Fernando et al. 2006). As a result, the knowledge of tire pavement interface stress is extremely important to understand the mechanisms of pavement distress. Measurements of tirepavement interface stress under different tire configurations have been conducted in the laboratory by many researchers. Back to early 1962, Bode ( 1962) employe d electronic pick ups embedded into the pavement and recorded local forces of a rolling bias ply tire Later, Seitz and Hussman (1971) used a rotating steel drum to measure automobile tire contact stress. Both cases showed that tire pavement interface stres ses varied across the tire footprint. In 1985, Marshek et al. (1985) developed a measurement system called pressure sensitive film to measure tire pavement contact stresses using pressure. The schematic of pressure sensitive film and configuration of pres sure film test are presented in Figure 24 and Figure 2 5, respectively. The main characteristics of this measurement system are ( Emmanuel G. Fernando et al. 2006) : Measurements were done in the laboratory using pressure sensitive film that produced a prin t with a color intensity pattern proportional to the applied pressures. The developed print was interpreted using a densitometer and the calibration curve established for the film used in the test. The tire load was applied statically onto the pressure fil m. Only vertical contact pressures were measured. To overcome the shortcomings of previous measurement systems that could not measure tire pavement shear contact stresses, the tire industry developed triaxial load pins, which allowed measurements of tire pavement interface stresses and displacements for vertical, longitudinal and transverse axes. Dr. Marion Pottinger of Smithers Scientific Services, Inc. (Ravenna, Ohio, Pottinger, 1992) developed a device to measure the tire pavement contact stresses under truck tires. As shown in Figure 26, a series of 16 triaxial load pin transducers were inserted into a flat PAGE 32 32 steel bed to measure vertical, transverse, and longitudinal forces and displacements under a rolling tire. The tire was held at one location while the steel bed was moved in the longitudinal direction, F igure 24. Schematic of Pressure Sensitive Film ( Marshek et al. 1985). Figure 2 5. Configuration of pressure film test ( Marshek et al ., 1985). causing the tire to roll over a row of 16 transducer s. Stresses and displacements were recorded every 0.20 inch longitudinally (parallel to wheel path) and every 0.15 inch transversely (perpendicular to wheel path) by varying the transverse position of the sensors. The PAGE 33 33 measurements provided a higher definit ion of actual tire pavement contact stresses than previously obtained. F igure 26. Schematic of system developed by Smithers Scientific Service (Myers, 2000) Based on the measurements of tirepavement interface stress, Pottinger (1992) also examined dif ferent contact effects between radial truck tires and bias ply truck tires. These effects are generally referred to as the Pneumatic effect and Poissons effect. Poissons effect dominates radial truck tires while Pneumatic effect dominates bias ply truck tires. Later, Roque et al. (1998) extended this concept and attributed these differences to the tire structure. As shown in Figure 27, Radial tires are constructed to have stiff treads and flexible sidewalls, to minimize deformation of the tire during rolling. Thus, the lateral stresses induced on the road by the radial truck tire will tend to push out from the center of the tire ribs. In contrast, bias ply tires tend to have high wall moduli and a flexible tread, resulting in smaller lateral contact stresses. PAGE 34 34 F igure 27. Structural characteristics of radial and bias ply truck tires and their effects on the pavement surface (after Roque et al. 1998) In South Africa, De Beer et al. (1997) developed a more sophisticated system called VehicleRoad Pressur e Transducer Array (VRSPTA) that can measure 3 D stresses under bias ply, conventional and wide based radial truck tires under different loads and inflation pressures in the laboratory or field. VRSPTA consisted of an array of calibrated triaxial strain ga uged pins mounted on a steel plate and fixed flush with the road surface. A schematic of VRSPTA is shown in Figure 28. Figure 29 shows the VRSPTA being used to obtain measurements. Some PAGE 35 35 results from VRSPTA are presented in Figure 2 10, which indicates that both vertical and transverse contact stresses were highly nonuniform. F igure 28. Schematic of VRSPTA (after De Beer et al., 1997) F igure 29. VRSPTA in measurement (after De Beer et al., 1997) PAGE 36 36 Vertical contact stress distribution Lateral co ntact stress distribution F igure 210. 3 D contact stress distribution for radial truck tire (after De Beer et al., 1997) 2.3 Tire Modeling Accurate tire m odeling is the key to relevant tire pavement contact modeling analysis However, tire modeling is v ery challenging due to its structural and material complexity. For many years tire modeling has remained in the numerical analytical level. The earlier tire models employed the simple lumped mass spring parameter model approach, i.e. the elastic string/ring/beam with pre tension on uniformly distributed elastic/viscoelastic foundations. For instance, Loo (1985) developed an analytical model for a pneumatic tire, which consisted of a flexible circular ring under tension with a nest of radially arranged line ar springs and dampers as PAGE 37 37 shown in Figure 211. The ring represented the tread band of the tire and was assumed to be massless and completely flexible, while springs and dampers stood for tire inflation pressure and their moduli was obtained from experiments by conducting contact patch length measurements and static point load tests on the specific tire model. Later some semianalytical models were also introduced. However, none of these simplified models could fully explain and simulate the complicated pro cesses of the true tire mechanics behavior. Extensive experiments were required to calibrate the lumped equivalent properties and therefore their accuracy and range of application and va lidity were highly questionable F igure 211. Flexible ring tire mo del (Loo, 1985) With the development of computer technology, the finite element method (FEM) began to dominate industry and has become a main analytical tool in tire modeling. Padovan (1977) developed a 2 D curved axisymmetric thin shell element model to e xamine the power dissipation rolling resistance and thermo viscoelastic problem of steady state rolling tires. Trivisonno (1977) also examined a similar non steady state thermal problem with a 2 D finite element model. He divided the tire into circumferential rings composed of 300 elements. Figure PAGE 38 38 212 shows these kinds of simple 2D thin shell finite element models. Tielking (1982) developed a finite element tire model using orthotropic, nonlinear shell elements that can F igure 212. 2 D thin shell finite element models respond to asymmetric load. The tire was modeled by an assemblage of axisymmetric curved shell elements, which were connected to form a meridian of curvature and were located at the carcass mid surface. Each element formed a complete ring and was connected at nodes. The elements are homogeneous and orthotropic with a set of modulus values specified for each element group. Nakajim and Padovan (1986) developed a 2D contact model using finite element program ADINA. An HR7815 was modeled usi ng 225 2D elements with 1350 degrees of freedom. The tread and sidewalls were modeled by a viscoelastic ring on an elastic foundation as shown in Figure 213. The material and geometric data were based on experimental values. The predicted values matched experimental results well. Bhm (1991) develop a ring onelastic foundation model to analyze tire dynamics in the frequency range up to 1000 Hz. Tread and belt were elastically connected by the sidewalls to the wheel rim. Contact area and pressure distribu tion were simulated using empirically determined parameters. To compute the shear forces, he extended the original simple elastic model to several PAGE 39 39 circumferential lines for tread and sidewall as shown in Figure 214. This model took a lot of computational time and only dealt with shear forces and moments. F igure 213. Geometry of tire model (Nakajim and Padovan 1986) F igure 214. Extended elastic ring on elastic foundation ( B hm, 1991) Wang (1990) developed a 3D tire model to investigate tire terrai n interaction issue. The tire was built based on the tire data provided by tire manufactures and calibrated with load  PAGE 40 40 deflection curves To simpl if y the analysis, the carcass was assumed to be homogenous, isotropic elastic material with a Poissons ratio of 0.48. The model was first inflated and then loaded against the ground. Results showed that the tire ground contact geometry predicted by the model was in reasonable agreement with that found in the Generalized Deflection Chart provided by the manufactu re r Figure 215 s hows the geometry of the tire model. F igure 215. Geometry of tire model (Wang, 1990) To study the influence of tire/rim assembly non uniformity on vehicle ride quality, Rhyne et al. (1994) developed a 3D membrane tire model. A P195/ 75R14 passenger truck tire was modeled as a ring attached to the wheel by a pure membrane than can carry load in the radial direction only, as shown in Figure 216. The ring represented the tread and belt package of the tire, rigid transversely but flexible circumferentially. PAGE 41 41 F igure 216. Membrane P195/75R14 tire model (Rhyne et al. 1994) Kao and Muthukrishnan (1997) simulated a simple tire test using LS DYNA 3D and demonstrated that it was possible to predict tire transient dynamic responses from tire d esign data. For the first time, the tire model incorporated geometry, material properties of various components, fiber reinforcement, layout, and other features of a commercial passenger radial truck tire s P205/65R15. Before that, almost all tire finite element models were developed using only a single type of element using reasonable assumptions and simplifications which might result in the loss of accuracy and detailed information regarding the tire structure. In this model, tire carcass composite proper ties were calculated from a strain energy function derived for fiber reinforced rubber. The Mooney model was used for rubber material. The model was coupled with a rigid model and inflated to a specified inflation pressure. The whole model had 9600 element s and 70,080 degrees of freedom. The predicted tire center vertical and horizontal forces showed good correlation with experiments. Figure 217 shows a detailed view of tire cleat enveloping situation. PAGE 42 42 F igure 217. Detailed view of tire cleat envelopin g situation (Kao et al. 1997) F igure 218. (a) Meshing of cross section (b) 2D model (c) 3 D model (Zhang, 2001) Zhang (2001) developed a nonlinear finite element model for a radial truck tire using ANSYS (1998) based on its composite structural ele ments to analyze the various stress fields with particular focus on inter ply shear stresses in the belt and carcass layers as functions of the PAGE 43 43 structural parameters and the normal load. The model incorporates the geometry and orientation of the cords in belt and carcass layers, staking sequence of different layers, large magnitude and nonlinear deformations of the multiple layers and the nearly incompressible property of the tread block, as shown in Figure 218. The model was validated in a qualitative sense through a comparison of the normal force deflection characteristics and the contact patch geometry derived from the model with the laboratory measured data. To study interactions between a rolling truck tire and a rigid pavement structure, Meng (2002) de veloped a 3 D finite element model using ABAQUS finite element code. A Goodyear 295/75R22.5 Unisteel G167A Low Profile Radial smooth tire was modeled to roll over the rigid pavement. Fiber reinforced composite model and rubber material model was utilized to simulate the real tire structure. Different tire pressure levels, load levels and slip angles were applied in the simulations. Results for static loaded against a rigid flat plate were compared with experimental data from the literature. Contact stress distributions and contact areas for the rolling tire over the rigid pavement at various conditions were also presented. At the same time, J Pelc (2002) also developed a static 3 D model of a pneumatic tire based on finite element program MARC. The cord ru bber composite was modeled by overlaying the elements characterizing the cord and those representing incompressible rubber, as shown in Figure 219. A 2D model was used to simulate the tire mounting and inflation process. The model was then developed in 3D. The displacements, radial moduli and delaminating stresse s caused by the vertical load were determined. The shape of the tire footprint and pressure distribution in this zone were also predicted. Good agreement between measured and computed moduli char acteristics was observed. The results indicated that the proposed technique of PAGE 44 44 element overlaying yields a tire model that is numerically more stable than those using only orthotropic elements. F igure 219. Element mesh of tire cross section (J Pelc, 20 02) Recently, M. Zamzamzadeh et al (2006) modeled a 205/60R14 radial tire under differe nt loading conditions using ABAQUS Code. They developed a 2D axisymmetric model first and then generated a full 3 D finite element model by revolving the 2 D mesh. The model comprised one body ply, two steel belts and two cap plies. Different hyper elastic models for rubber as Mooney Rivlin, Ogden, Arruda Boyce were compared to determine the best one for simulation. Through footprint dimension comparison, they claimed that the predicted results were very close to the experimental results. It should be noted that this model did not consider tire grooves. Based on above review, it is clear that much work has been done on tire modeling analysis, especially by the tire indu stry. However, it should be pointed out that most analyses were focused on the tire itself. To date, no studies were focused on how tire structure characteristics affect pavement performance. This study will be focused on how the tire affects pavement nea r PAGE 45 45 surface stress states, particularly related to top down cracking and instability rutting. No doubt, accurately modeling tires will be the key to success and reviewing those previous tire models will help to generate some guiding principal s on modeling ti res using the finite element method. 2.4 Mechanisms of Fracture and Rutting in Asphalt Pavements It is generally agreed that fatigue cracking and rutting are the two principal types of distress to be considered for asphalt pavement design (Huang, 1993). F atigue cracking is based on the maximum horizontal tensile strain at the bottom of HMA, which is considered as an indicator for the damage potential due to traffic load repetitions acting on the pavement. Rutting is based on the maximum vertical compressiv e strain on the top of the subgrade, which is considered as an indicator for the permanent deformation potential of the pavement subgrade. Both Asphalt Institute and Shell design methods employ these two criteria to determine the thickness of pavement laye rs. Clearly, neither method accounts for either topdown cracking nor instability rutting that is related to near surface stress states. Different models have been developed to evaluate cracking and rutting performance of asphalt pavement. Though some of these models may not directly relate to surface initiated cracking or instability rutting, reviewing those models will definitely help understand mechanism related to near surface distress. The following subjects were examined: Rutting models Classical fat igue approach Fracture mechanics method Dissipated creep strain energy approach 2.4.1 Rutting Models As stated earlier, instability rutting in asphalt pavement has been cited as the main distress in many states and has generated more and more concern amon g pavement researchers and engineers as well (Drakos, 2003). Instability rutting is primarily a near surface phenomenon, PAGE 46 46 affecting only the top 13 inches of the AC layer with visible slip surfaces associated with rutting failure. In both Asphalt Institute and Shell design methods, however, the rutting model is related to the vertical compressive strain c on the top of the subgrade by: 5) (4 f c df N ( 21) This equation is also used by several other agencies with different values of 4f and 5f as shown in Table 21. Table 21. Subgrade strain criteria used by various agencies (Huang, 1993) Age ncy 4f 5f Rut depth (in) Asphalt Institute 1.365 10 9 4.477 0.5 Shell (Revised 1985) 50% reliability 6.15 10 7 4.0 85% reliability 1.94 10 7 4.0 95% reliability 1.05 10 7 4.0 U.K. Transport & Road Research Laboratory (85% reliability) 6.18 10 8 3.95 0.4 Belgian Road Research Center 3.05 10 9 4.35 Clearly, the subgrade strain method does not account for instability rutting in the AC material Evaluation of surface rutting based on the subgrade strain does not appear to be reasonable especially under heavy traffic with thicker HMA, in which most of the permanent deformat ion occurs, rather than in the subgrade. Therefore, it is more reasonable to determine the permanent deformation in each layer and sum up the results, since rutting is caused by the accumulation of the permanent deformation over all layers. To account for this, Ohio State ( Majidzadeh et al, 1980) developed following model: m pN A1) ( ( 22) Where, A is an experimental constant dependi ng on material type and state of stress and m is another experimental constant depending on material type. PAGE 47 47 Recently, studies by Drakos (2003) and Novak (2003) showed that instability rutting may be caused by high shear flow near the tire edge, which are st rongly influenced by tire type However, they did not develop rutting models associated with this shear flow. 2.4.2 Traditional Fatigue Approach The traditional fatigue approach assumes that the maximum tensile strains are located at the bottom of the asphalt concrete layer. Cracks will be initiated by these tensile strains and propagate from the bottom upward into the AC layer. This approach was based on the assumption that tire pavement contact stresses are uniform and vertical. In practice, tirepavemen t contact stresses are threedimensional, as verified by many studies ( Pottinger, 1992; de Beer et al. 1997; Roque et al ., 1998). By introducing shear contact stresses, studies by Jacobs (1995) and Myers et al. (1998) showed that critical tensile strain might be located at AC surface near tire edge rather than at the bottom of AC. Several fatigue models have been developed to explain bottom up cracking Monismith et al. (1985) developed the fatigue behavior of a particular mixture by: c mix b t fS A N 1 1 ( 2 3) where, Nf is the number of load applications to failure, A is a factor based on asphalt content and degree of compaction, t is the tensile strain, Smix is the mixture moduli and b and c are con stants determined from beam fatigue tests. In the Asphalt Institute and Shell design methods, the allowable number of load repetitions Nf to cause fatigue cracking is related to tensile strain t at the bottom of the HMA and HMA modulus E1 by: 3 2* 1 f f t fE f N ( 2 4) PAGE 48 48 For the standard mix used in design, the Asphalt Institute equation for 20 % of area cracked is: 854 0 291 30796 0 E Nt f ( 2 5) And the Shell equation is : 363 2 671 50685 0 E Nt f (2 6) The E1 term may be neglected due to the fact that exponent f2 is much greater than f3, which indicates that the effect of t on Nf is much greater than that of E1: 21 f t ff N (2 7) Equation (27) has been used by several agencies with the values of f1 and f2 shown in Table 22. The variance of exponen t coefficients was due to the difference in materials, test methods, field conditions, and structural models. Table 22. Fatigue cracking models used by various agencies (Huang, 1993) Agency 1f 2f Illinois Department of Transportation 5 10 6 3.00 Transport and Road Research Laboratory 1.66 10 10 4.32 Belgian Road Research Center 4.92 10 14 4.76 2.4.3 Fracture Mechanics Method Unlike traditional fatigue approach, this method addressed the possibility of a critical condition existing in pavements. Fracture will occur when the stress intensity factor computed at the crack tip under repeated loading exceeds the fracture toughness of the material. Th e rate of crack propagation can be defined by: nK A dN da ) ( ( 2 8) where A and n are parameters depending on the material and experimental condition; a is the crack length, N is the number of load repetitions, and K is the difference between maximum PAGE 49 49 and minimum stress intensity factors during repeated loading. Ewalds and Wanhill (1986) defined crack development as having three different phases as shown in Figure 220: initiation phase where micro cracks develop, the propagation phase where the micro cracks develop into macro cracks and crack growth becomes stable, and the disintegration phase where the material fail s, and crack growth is unstable. Collop and Cebon (1995) carried out a study on the surface crack growth in the pavement using facture mechanics method. They developed an axisymmetric finite element model to investigate stresses ahead of the longitudinal cracks in the surface layer and found that increasing the modulus ratio and re ducing asphalt thickness will tend to reduce the transverse tensile stress at the pavement surface. Also, increasing the asphalt thickness and decreasing the elastic modulus ratio will increase the magnitude of the stress intensity factor and depth to whic h the crack propagates. Myers (2000) also developed a 2D finite element model to investigate longitudinal surface crack growth. She found that Mode I cracking dominated crack growth and temperature induced moduli gradients contributed to tensile stress d evelopment around the crack tip. F igure 220. Fatigue Crack Growth Behavior (after Jacobs, 1995) PAGE 50 50 2.4.4 HMA Fracture Mechanics The theory of HMA fracture mechanics was developed by Roque et al. (1997, 1999, and 2001) at the University of Florida. Basica lly, fracture energy (FE) of HMA consists of D issipated C reep S train E nergy (DCSE) and elastic energy, as shown in Figure 221. Roque found that there is a fundamental damage threshold existing in HMA that controls cracking performance in asphalt pavemen t Be low threshold, only healable micro damage develops. Once the threshold is exceeded, damage becomes macro and un healable. Under this rule, the crack grows in a stepwise manner rather than in a continuous manner, as shown in Figure 2 22. In the Superpav e fracture test under cyclic loading conditions, the accumulated DCSE can be easily calculated using creep compliance parameters obtained from the Superpave IDT creep test and the characteristics of the applied load. For example, for a haversine lo ad appli ed in 0.1 s, with a 0.9 s rest period, the DCSE per cycle is determined as follows: dt t t cycle DCSEp AVE) sin( ) sin(1 0 0 ( 1 9) where AVEis the average stress in the zone of interest and pis the creep strain rate. Details regarding the determination of the stress distributions, average stresses, and accumulated DCSE within each process zone may be found elsewhere ( Z hang et al. 2001). 2.5 Summary The discussion presented in this chapter indicates that both topdown cracking and instability rutting are near surface phenomena, which is greatly affected by tirepavement contact stresses. Both topdown cracking and instability rutting occur near tire edge, where critical shear stresses are developed. PAGE 51 51 F igure 221. Dissipated c reep s train e nergy (after Roque et al., 1997) F igure 222. Crack propagation in asphalt mixture (after Roque et al., 1999) The measurement of tire pavement contact stress shows that tire pavement contact stress is th ree dimensional. Also, tire structural characteristics affect tire pavement contact stress distributions. Poissons effect dominates radial truck tires while Pneumatic effect dominates bias ply truck tires. Though 3D modeling of tires is complicated, muc h work has been done on the tire modeling analysis, especially in tire industry, which provides some guidance for further model development in this project. However, it should be pointed out that most analyses were focused on tire itself. Very few studies were carried out how tire structure characteristics affect PAGE 52 52 pavement performance. This study will be focused on how the tire affects pavement near surface stress states which are related to top down cracking and instability rutting. Traditional rutting mod el s do not account for instability rutting in the AC layer, neither classical fatigue approach es account for topdown cracking. Fracture mechanics provides a solid foundation for understanding cracking in asphalt pavements. It introduces the concept of ini tiation, propagation, and disintegration. HMA fracture mechanics has been found to reasonably explain mechanism of topdown cracking, which believes there is a fundamental damage threshold existing in HMA that controls cracking performance in asphalt pavem ent. And the crack grows in a stepwise manner rather than continue manner, as shown in Figure 222. PAGE 53 53 CHAPTER 3 THEORETICAL ANALYSIS OF SURFACE STRESS ES Tire pavement interaction is a contact problem A theoretical knowledge of the elastic contact stress fields generated by point loads, line loads and various indenter geometries such as spheres is critically important to understand mechanisms of topdown cracking and instability rutting. Here, the term indenter refers to the body to which the loading for ce is applied and will be used throughout the text. The chapter begins with analytical modeling of lateral stresses induced by normal and shear tractions acting on the pavement surface, followed by comparisons of horizontal surface stresses from typical l oading configurations. The discussion then turns to the formation of Hertzian contact equations and contact analysis between a sphere indenter and a flat surface. All theoretical solutions are determined by assuming the pavement is a half space, homogeneous, linear, isotropic and elastic material. And most of these solutions are taken from the literature ( Fischer Cripps and Anthony, 2000; Holewinski et al. 2003; Muni Budhu, 2000; Johnson, 1987; Poulos and Davis, 1974; Love, 1929; Dean, 1994). 3.1 Point L oad Boussinesq (1885) presented a solution for the distribution of stresses within a pavement due to a point load. The radial, hoop and shear stresses on an element located at point A (Figure 31) are given in cylindrical pol ar coordinates as f ollows : 2 / 1 2 2 2 2 2 / 5 2 2 2) ( 2 1 ) ( 3 2 z r z z rv z r z r Pr (3 1) 2 / 1 2 2 2 2 2 / 3 2 2) ( 1 ) ( ) 2 1 ( 2 z r z z r z r z v P (3 2) 2 / 5 2 2 2)( 2 3 z r rz Prz (3 3) PAGE 54 54 A stress singularity o ccurs for both radial and hoop stresses at the origin. Radial stress r is the tensile stress at the surface and decreases quickly as inverse square of distance away from loading center, as shown in Figure 3 1. Figure 3 2 shows shear st ress distributions along the depth at different constant radial distances (r1=0.05, r2=0.1, r3=0 .2), which indicates that a maximum value exists near the surface. The shear stress decreases dramatically with both depth and radial distance. z r P z r A Normalized Surface Radial Stress Distribution 140 120 100 80 60 40 20 0 0 0.2 0.4 0.6 0.8 1 1.2 r r/ Q Figure 3 1. Point l oad and normalized s urface r adial s tress distributions 0 0.5 1 1.5 2 0 10 20 30 40 DepthNormalized Shear StressT r 1 z T r 2 z T r 3 z z Figure 3 2. Shear s tress distributions along depth at different c onstant r adial distance PAGE 55 55 3.2 Line Load The problem of the two dimensional case of a uniformly distributed force acting along a line is known as Flamants problem. With reference to Figure 33, the horizontal stresses and shear stresses due to a line load, P (force/length), are: 2 2 2 2) ( 2 z x z x Px (3 4) 2 2 2 2) ( 2 z x Pxzxz (3 5) Notice that horizontal stresses are independent of elastic parameters, and zero on the surface except for the point x=0 where a stress singularity occurs. The distributions of shear stress are similar to point load case. zP z x x Figure 3 3. Line l oad 3.3 Vertical Strip Load A strip load is the load transmitted by a structure of finite width and infinite length on a pavement surface. Two t ypes of strip loads are common ly used in civil engineering. One is uniform stress (Figure 3 4a), and the other is triangular stress (Figure 34b). PAGE 56 56 B x Z x z B x Z x z (a) (b) R1R2 Figure 3 4. Vertical s trip l oad (a) A Uniform s tress and (b) A t riangular s tre ss The horizontal stresses and shear stresses due to surface stress qs (force/area) are as follows: (a) Area transmitting a uniform stress (Figure 3 4a) ) 2 cos( sin s xq (3 6) )) 2 sin( (sin s xzq (3 7) (b) Area transmitting a triangular stress (Figure 3 4b) 2 sin 2 1 ln2 2 2 1R R B z B x qs x (3 8) ) 2 2 cos 1 ( 2 B z qs xz (3 9) The angel and are defined in Figure 3 4. Again, the horizontal stresses are independent of elastic parameters. The distributions of surface horiz ontal stresses for both cases are shown in Figure 35 and Figure 36 respectively. The typical shear stress distributions along depth outside the loading area for a given B and x are shown in Figure 3 7. Again, the maximum PAGE 57 57 shear stress occurs near surface, which might be responsible for instability rutting of asphalt pavement. 2 1 0 1 2 2 1 0 1 2 x/B xx/qs Figure 3 5. Surface horizontal s tresses due to uniform vertical s trip l oad 2 1 0 1 2 2 1 0 1 2 x/B xx/qs Figure 3 6. Surface horizontal s tresses due to t riangular v ertical s trip l oad PAGE 58 58 0 0.5 1 1.5 2 2.5 0 0.05 0.1 0.15 0.2 0.25 0.3 xz/qs z Uniform Triangular Figure 3 7. Shear s tress distributions along depth 3.4 Horizontal Strip Load Similar to the vertical strip load case, two types of loading conditions are applied here. One is uniform stress (Figure 3 8a), and the other is triangular stress (Figure 3 8b). The horizontal stres ses due to surface stress qs (force/area) are as follows: (a) Area transmitting a uniform stress (Figure 3 8a) ) 2 sin( sin ln2 2 2 1 R R qs x (3 7) (b) Area transmitting a triangular stress (Figure 3 8b) 2 ( 3 ln ln ) 2 ( 22 1 1 2 2 2 1 0z r r B r r r B x B qx (3 8) Where R1, R2, r1 r2 are defined in Figure 38. Once again, the horizontal stresses are independent of elastic parameters. The distributions of horizontal surface stre sses for uniform case are shown in Figure 3 9, which shows discontinuity (infinite value) at the edges. For triangular case, however, the horizontal surface stresses are finite and continuous everywhere as shown in Figure 310. PAGE 59 59 B x Z x z (a) R1R2 B x Z x z (b) r1r2 2 1r Figure 3 8. Horizontal Strip Load (a) A Uniform Stress and (b) A Triangular Stress 3 2 1 0 1 2 3 2 1.5 1 0.5 0 0.5 1 1.5 2 x/(B) xx/q Figure 3 9. Horizontal Surface Stresses Due to Uniform Horizontal Strip Load 3 2 1 0 1 2 3 2 1.5 1 0.5 0 0.5 1 1.5 2 x/(B) xx/q Figure 3 10. Horizontal Surface Stresses Due to Triangular Horizontal Strip Load PAGE 60 60 3.5 Ci rcular Vertical Load The problem of uniform circular vertical load can be solved via integrating Boussinesqs equations from point load case. The horizontal normal stress on the surface under the circularly loaded area is given as: ) 2 1 ( 2 v qs x (3 9) Outside the circularly loaded area, the resulting formula is lengthy and is presented in the following form: ] ) 2 1 ( 2 [ C v B vA qs x (3 10) Where the coefficients A, B and C can be found in literature ( Poulos and Davis, 1974). Software such as Wesl ea ( 1999 ) or Circly (1977 ) can also be used to solve this problem. The results of a sample calculation for v = 0.35 a re shown in Figure 311. Notice that inside the loaded area, the horizontal normal stress on the surface are compressive and become tensile outside the loaded area. The compressive stresses at the center have a maximum value of (1+2 v ) qs/2 and the tensile stresses at the edge have the value of (1 2v ) qs/3. The maximum shear stress is qs/3, located at about 0.48r (r is the contact circular radi us ) below surface. 0.2 0 0.2 0.4 0.6 0.8 1 2 1 0 1 2 x/B xx/qs Figure 3 11. Horizontal normal s tresses on the s urface due to c ircular vertical l oad PAGE 61 61 3.6 Circ ular Horizontal Load Wardle et al. ( Wardle, 1976; Gerrard and Harrison, 1971) applied integral transform methods to solve stresses distributions due to circular horizontal load. The solutions are then evaluated using software CIRCLY ( 1977). For uni directi onal circular horizontal load, the horizontal surface stresses are singular along the boundary with infinite compression at A and infinite tension at B, Figure 312. For axisymmmetrical circular horizontal load with finite qs at the boundary, whether it is inward or outward, the horizontal surface stresses are also singular along the boundary. However, the horizontal surface stresses are finite if qs is zero at the boundary of loaded region, with compression for qs acting outward and tension for qs acting i nward. Figure 313 summarized distributions of horizontal surface stresses due to three loading conditions: a) anti symmetric horizontal load; b) uni directional horizontal load; c) vertical load ( Jill M. Holewinski et al., 200 3). qs y x z A B Figure 3 12. Uniform Circular Horizontal Loads PAGE 62 62 Figure 3 13. Horizontal Surface Stresses Due to a) Anti symmetric Horizontal Load; b) Unidirectional Horizontal load; c) Ver tical Load (Jill M. Holewinski et al., 2003) PAGE 63 63 3.7 Combination of Normal and Transverse Load In a plane strain elastic half space, the horizontal stress xx due to a normal contact pressure ) (p and a horizontal contact pressure ) (q acting ove r a strip a a are given by Johnson ( Johnson, K L, 1987) as: d z x x q d z x x p z z xa a a a xx 2 2 2 3 2 2 2 2] ) [( ) )( ( 2 ] ) [( ) )( ( 2 ) ( (3 11) Assuming uniform normal contact pressure 0) ( p p and transverse contact pressure decrease linearly from 0q at a x to 0q at a x as show in Figure 314, E quation 311 can be integrated in closed form at edges of the loaded strip as follows: ) 2 ( tan 4 2 ) 2 ( tan 3 4 ) 10 3 ( 2 4 ln ) (1 2 2 0 1 2 2 2 2 2 2 2 0z a a z az p z a a z a z a z z a z q z axx (3 12) At the surface of the half space, limiting above equation gives: 2 5 ) 4 ln( ) ln( 2 ) (0 2 0 0 0limp a q z q z axx z (3 13) From equation 313, it can be observed that transverse surface stresses are singular at both end of contact strip due to l oad transverse load discontinuity at this point. Figure 315 shows equation 312 plotted on semi logarithmic scales as a function of depth though the pavement ( A.C. Collop and D. Cebon, 1993). Another two corresponding curves for different triangular distr ibutions of transverse contact pressure with no discontinuity at the edges of the loaded strip, which represents more realistic tire pavement contact pressure distributions, are also given in Figure 315. As stated early, the stress is no longer singular for both triangular distributions of contact pressure. The maximum values are determined by the loading geometries. It can also be observed PAGE 64 64 that loading geometries only affect the transverse stress near to the pavement surface ( mm z 10 ) (. A.C. Collop and D. Cebon, 1993). P0+a a +a x x q0 Figure 3 14. Normal and transverse loading conditions Figure 3 15. Transverse stress at the edge of the load ( x = a) as a function of distance below the pavement surface for different transv erse co ntact pressure distributions ( A.C. Collop and D. Cebon, 1993). PAGE 65 65 3.8 Contact Between Elastic Solids The stresses arising from the contact between two elastic solids have many practical applications in industry such as impact damage of engineering ce ramics, design of gear teeth and so on. For pavement engineering, the contact between a sphere indenter and a flat surface is of particular interest. With reference to Figure 3 16, Hertz ( 1881 and 1882) formulated mathematical relationships between indente r load P, indenter radius R, contact area, and maximum tensile stress as follows: 3 1 *) 4 3 ( E PR a (3 14) Where E v E v E2 1 2 1 *1 1 1 (3 15) Where, E1, E v1, and v were defined in Figure 316. Hertz also found that the maximum tensile stress occurs at the edge of the contact circle at the specimen surface and is given by: 2 max2 ) 2 1 ( a P v (3 16) a R E1,v1E,v P Figure 3 16. Schematic diagram of Hertz contact Assuming pm is average contact pressure, the radial stress induced within the interior of the specimen is given by ( M.T. Huber 1904; B.R. Lawn et al., 1974): PAGE 66 66 2 tan ) 1 ( 1 1 3 2 1 2 32 / 1 1 2 / 1 2 2 / 1 2 2 2 2 3 2 / 1 3 2 / 1 2 2u a a u v u a v u u z z a u u a u z u z r a v pm r (3 17) Where: 2 / 1 2 2 2 2 2 2 2 2 24 ) () ( 2 1 z a a z r a z r u (3 18) The stress contour map is given in Figure 317. At the surface, the radial stress distribution inside the contact circle is given as ( Fischer Cripps and Anthony, 2000): 2 / 1 2 2 2 / 3 2 2 2 21 23 1 1 2 2 1 a r a r r a vpm r a r (3 19) And outside the contact circle: 2 22 2 1 r a v pm r a r (3 20) Figure 3 17. Tensile and c ompressive s tress c ontour m ap ( A. Franco Jr et al. 2004) The normalized radial stress distribution at the surface is given in Figure 3 18. PAGE 67 67 1.5 1 0.5 0 0.5 0 0.5 1 1.5 2 2.5 r/a r/pm Compressive Tensile Figure 3 18. Distributions of normalized r adial s tress at the s urface From Figures 3 17 to 318, it can be seen that tensile stress occurs on or close to the specimen surface near the edge of the contact circle. This tensile stress is usually responsible for the initia tion of so called Hertzian cone cracks. From the pavement engineering perspective, this tensile stress might be responsible for the initiation of top down cracking and/or near surface damage. 3.9 Summary Some key points about this chapter are summarized as follows: For point load, radial stress r is tensile at the surface. It is singular at the origin but decreases quickly as inverse square of distance away from loading center. For line load, horizontal stress is zero on the surface e xcept for the point r=0 where a stress singularity occurs. For uniform vertical strip load, the horizontal surface stresses are compressive and constant within the loaded area. For triangular vertical strip load, the horizontal surface stresses are also co mpressive but triangular within the loaded area. There is no horizontal s tress outside the loaded area for either case. For circular vertical load, the horizontal normal stresses on the surface are compressive within the loaded area and become tensile outs ide the loaded area. The compressive stresses at the center have a maximum value of (1+2 v ) qs/2 and the tensile stresses at the edge have the value of (1 2v ) qs/3. PAGE 68 68 For uni directional circular horizontal load with finite qs at the boundary, the horizontal s urface stresses are singular along the boundary with infinite compression at one side and infinite tension on another. However, the horizontal surface stresses are finite if qs is zero at the boundary of the loaded region. A xi symm etric circular horizontal load has the same characteristics For the contact between a sphere indenter and a flat surface, tensile stress occurs on or close to the specimen surface near the edge of the contact circle. This tensile stress is usually responsible for the initiation of so called Hertzian cone cracks. From the pavement engineering perspective, this tensile stress might be responsible for the initiation of topdown cracking and/or near surface damage. PAGE 69 69 CHAPTER 4 EFFECTS OF TIRE TYPE AND PAVEMENT CROSS SECTION PROFILE ON TOP DOWN CRACKING AND INSTABI LITY RUTTING 4.1 Introduction During past decades, new technologies were applied by the tire industry to increase efficiency and increasingly wider radial tires ( wide base or super single ) were introduced to replace conventio nal dual tire systems Compared to conventional dual tire system, wide base tire assembl ies decrease gross vehicle weight by 880lbs to 1,272 lbs (allowi ng more cargo weight), increase fuel economy by 2 to 5% and low er tire repair and replacement costs (Ki lcare, 2001; Wide base, 2002). Although the economical benefits of wide base tire to trucking industry sound pretty attractive, the relative damage to pavement caused by wide base tires over dual tires arouse big concerns among pavement engineers as well as tire engineers and have become a hot research topic in recent years. In the early 1960s, Zube and Forsyth (1965 ) performed an experiment to compare the vertical deflections and transverse strains of a flexible pavement surface due to wide base tires a nd dual wheels. Their results indicated that pavement deflection was equivalent for 27 kN carried on a single tire or 40 kN carried on a dual pair. Christ i son et al (1978) conducted an experiment to measure asphalt layer interface strains and surface def lections under different axle and tire configurations which gave similar results. Their studies showed that pavement damage in terms of measured stains caused by wide base tire could be theoretically 7 10 times worse than a dual pair for an equal load. This was also theoretically confirmed later by Treybig (1983). Eisenmann et al (1987) reported that the measured strain under wide base tires were 50 % greater than those under dual tires carrying the same load, and this would increase pavement fatigue dama ge by as much as a factor of 2.5. PAGE 70 70 Considering more potential damage to pavement might be caused by wide base tires than dual tires, the Federal Highway Administration (FHWA) initiated a study in 1989 to assess the impact of wide base tires, specifically th e 425/65R22.5, on conventional flexible pavement damage by using accelerated pavement testing (APT) at the Turner Fairbanks Research Center (Bonaquist, 1992). This study found that measured pavement strain and stress significantly increased under the singl e widebase tires, and both the fatigue and rutting life of the pavement decreased dramatically. Table 4 1 illustrated the results of this analysis. Table 4 1. Relative d amage between w ide base and dual t ires (Bonaquist, 1992) Damage Relative Damage 89 m m HMA (T = 14C) 178mm HMA (T = 23C) Ratio Damage Ratio Damage HMA Rutting 1.23 1.23 1.31 1.31 Base Rutting 1.40 1.40 1.31 1.31 Subgrade Rutting 1.53 1.53 1.09 1.09 Fatigue 1.44 4.30 1.37 3.52 Almost at the same time, Sebaaly and Tabatabaee (1989) also evaluated the effects of tire configurations on the responses of flexible and rigid pavements using pavement instrumentation. Four tire types were considered in their study: the duals 11R22.5 and 245/75R22.5, and the wide base 385/R65R22.5 and 425/65R 22.5. Results of the study showed that tire type has almost no effects on the rigid pavement For the flexible pavement, the authors calculated the number of loading cycles for 10 % and 40 % fatigue damage based on the following equations (Finn, 1986): ) 10 log( 854 0 log 291 3 947. 15 %) 10 ( log3E Nf (4 1) ) 10 log( 854 0 log 291 3 086 16 %) 45 ( log3 E Nf (4 2) Where, f N Number of load applications to cause 10 or 45 % fatigue cracking in the wheel path; PAGE 71 71 Tensile strain at the bottom of the HMA layers (microstrain); and E Resilient modulus of the HMA material (psi). The damage factor was then calculated as follows: Damage Factor ) ( ) 5 22 11 ( tire any N R Nf f (4 3) The results are given in T able 4 2. It should be noted that the relative damage between wide base tire and dual tires were also calculated based on the fourth power law. From the table, it can b e observed that the damage caused by wide base tire is always greater than that caused by dual tires, but the relative damage was much lower than was reported by FHWA. Other studies found similar conclusions by using computer models (Gillespie et al 1993; Siddharthan et al. 1998) Table 4 2. Relative d amage between w ide base and dual t ires ( Sebaaly et al. 1992) Tire Type Relative Damage Microstrain(70 F) Damage (10 %) Damage (45 %) Strain Ratio Damage 11R22.5 145 1.00 1.00 1.00 1.00 245/75R22.5 156 1.30 1.30 1.07 1.34 425/65R22.5 159 1.50 1.50 1.09 1.44 385/65R22.5 164 1.50 1.50 1.13 1.64 However, Joseph Ponniah (2003) pointed out that past studies might have overstated the adverse effects of widebase tires on pavement damage due to following re asons: Unbalanced loads between tires of a dual set due to unequal tire pressures, uneven tire wear, and pavement crown. Pavement deterioration increases as loads on two dual tires become more unbalanced. Wander effect. The effect of wander is considered beneficial to pavement deterioration because the repetitive loads are reduced particularly for single tires as the load is distributed over wider areas of pavement surface. Wander is expected to have a smaller beneficial effect on dual tires because the reduction in the number repetitive loadings is expected to be marginal due to the potential overlapping of the dual tire load distribution (M.S. Hurtle et al. 1992) Dynamic loading caused by surface roughness. Pavem ent damage from dynamic loading is typica lly localized and is approximately 2 to 4 times more severe than the damage due to PAGE 72 72 static loading. It is commonly believed that wide base tires having only two sidewalls are expected to absorb more of the dynamic loading than a pair of dual tires with four sidewalls. From the perspective of mechanical model, most analyses have the following limitations which might also overestimate the effects of wid e base tires on pavement damage: Loading condition was assumed to be uniform, vertical and circular. Actuall y, the distributions of contact stress between tire and pavement are extremely non uniform and threedimensional. The contact area under truck tire load is closer to rectangular than circular. Using power law damage relationship like fourth power to evalua te pavement fatigue damage based on measured strain is questionable. For thin pavement, using high power to evaluate fatigue damage sounds reasonable since tirepavement contact stress has great effects on the tensile strain at the bottom of AC layer. For thick AC layer, tensile strain at AC bottom will be affected more by the total load than by the contact stress. Therefore, using high power might not be appropriate. Due to above concerns plus introduction of new generation wide base tire (455 and 445), se veral studies were conducted to reevaluate the effects of new generation wi de base tire on pavement damage The new generation of wide base tires uses a new crown design crown architecture design that allows for lower aspect ratio geometry. Compared to co nventional wide base tires (385 and 425), the new wide base tires are 15 % to 18 % wider and have less average contact stress and more uniform stress distributions, as shown in Figure 4 1 ( Al Qadi et al. 2005). Al Qadi et al.(2005 ) conducted a study using both finite element (FE) analysis and instrumented field test sections at the Virginia Smart Road on two new generation wide base tires, the 445/50R22.5 and 455/55R22.5. Fatigue cracking, top down cracking, and rutting failure mechanisms were evaluated. They found that the 455/55R22.5 induced approximately the same pavement response or damage as the standard dual assembly tested (275/80R22.5). The other wide base tire tested (445/50R22.5) was found to slightly increase the induced damage. Later, Priest et al. (2005) also got similar results. PAGE 73 73 Figure 4 1. Vertical c ontact stress distributions for different tire size (Markstaller et al. 2000) Although numerous computer models have been conducted to study the effects of tire types on pavement fatigue and general rutting, to date few models have been developed to assess effects of tire types on top down cracking and instability rutting, neither do they consider the real interaction between tire and pavement. Most of models only applied measured cont act stre ss from instrumentation such as VRSPTA rather than build a tire pavement contact model. In fact, tire structure characteristics could affect the tire pavement contact stress and might play a key role in understanding the mechanism of topdown cracking and instability rutting. In addition, pavement surfaces with a cross slope (such as the side of a rut in the wheel path) might also contribute to topdown cracking. Therefore, the purpose of this chapter is to investigate how tire types and pavement surface pr ofiles affect top down cracking and instability rutting. The main objectives of the chapter are listed as follows: PAGE 74 74 Conduct BISAR analysis to evaluate how tire types affect pavement topdown cracking and instability rutting Develop 2D axletire pavement fi nite element contact model to evaluate how pavement surface profiles as well as tire types affect top down cracking and instability rutting. 4.2 Evaluate Tire Types on Pavement Responses Based on BISAR Analysis To roughly assess how tire types affect pavem ent responses, a stress/strain analysis based on BISAR (de Jong et al. 1973) was conducted in this section. Surface tensile stress distributions, maximum shear stress distributions along top AC depth, critical tensile strain at AC bottom and compressive st rain on the top of subgrade were evaluated in the follows. 4.2.1 Modeling TirePavement Interface Stresses in BISAR Generally, the elastic multilayer analysis program BISAR has been used to model uniform circular loads. Since the tirepavement interface stresses are actually non uniform, a concept of superposition was employed to model more realistic tire pavement contact stresses. The idea is that a nonuniform load can be achieved by using a series of small uniform circular loads of different pressure t o represent a nonuniform stress condition. The arrangement of the circular loads input to BISAR is based on the actual tire footprint. T ypical tire footprints of dual tire 11R22.5, super single tire 425/65R22.5 and new generation wide base tire (NGWB) 445 /50R22.5 are given in Figure 42. Correspondingly, the loading configurations for these three tires are illustrated in Figure 4 3. Each input file to BISAR was generated with less than 10 input loading stresses. The coordinates (x, y, z) at which each loa ding stress is acting, the magnitude of the stresses (vertical and horizontal), and the angle at which the horizontal stress is acting in the x y plane were reasonably defined based on previous studies ( Jacobs, 1995; Myers et al. 1997). Details were illus trated in Appendix Table A 1, Table A 2 and Table A 3 respectively. Multiple runs of BISAR were required since BISAR can only handle a limited amount of input. Results of PAGE 75 75 different outputs were superimposed. For comparision purpose, all tires were loaded on the same pavement. The structure and material properties of the loaded pavement are given in Table 4 3. Table 4 3. Pavement structural characteriscs used in BISAR E (psi) H (in) AC 500,000 0.40 8 Base 40,000 0.35 8 Subbase 20, 000 0.35 12 Subgrade 20,000 0.35 211.5 Figure 4 2. Typical tire footprints from left to right: dual, super single and NGWB 15.8" 7" 1.40" 1.20" 12.6" 8.0" 1.6" 8" 6" 1.5" 1" 22" Dual 11R22.5 Super Single 425/65R22.5 NGWB 445/50R22.5 Figure 4 3. Tire loading configurations used in BISAR PAGE 76 76 4.2.2 Effects of Tire Types on Pavemen t Responses To investigate how tire types affect pavement responses, stress xx at AC surface, maximum shear stress along top AC layer, maximum tensile strain at AC bottom and maximum comressive strain at subgrade were evaluated. It should be noted, however, when calculating stress and strain at the surface, one must stay at least 0.5mm away from the edges of the loaded area due to numerical problems in BISAR ( Jacobs, 1995). 150 100 50 0 50 100 20 15 10 5 0 5 10 15 20 Tire Transverse Location (in) xx (psi) Dual Figure 4 4. xx distributions along AC surface for d ual 11R22.5 350 300 250 200 150 100 50 0 50 100 15 10 5 0 5 10 15 Tire Transverse Location (in) xx (psi) 425/65R22.5 Figure 4 5. xx distributions along AC surface for super single 425/65R22.5 PAGE 77 77 300 250 200 150 100 50 0 50 100 15 10 5 0 5 10 15 Tire Transverse Location (in) xx (psi) 445/50R22.5 Figure 4 xx distributions along AC surface for NGWB 445/50R22.5 Figures 4 4 to 4xx along AC surface for t hree different tires repectively. It can be seen that tensile stresses principally occur beside the longitudinal edges of loading strip of the 3 D contact stress distibutions at AC surface for all cases. The maximum tensile stress can be reached at almost 80 psi, which might be responsible for topdown cracking. Comparing with the wide base tire, dual tire has four critical locations for tensile stress xx. The maximum values of tensile stress at AC surface caused by these three tires are given in Figure 4 7, which shows that super single tire 425/65R22.5 has the highest tensile stress among the three, while the new generation tire 455/50R22.5 has the sm allest value. This might be due to fact that the average contact stress for 455/50R22.5 is much less than that for 425/65R22.5. However, the propensity for topdown cracking might increase as loads on two dual tires become more unbalanced. Figures 4 8 to 410 show the effects of propensity. A maximum tensile stress of almost 150 psi could be reached at the AC surface when dual tires are completely unbalanced. Maximum shear strain along top AC layer was one of the causes to be responsible for instability rutting. Studies by Drakos (2003) and Novak (2007) showed that critical locations for PAGE 78 78 maximum shear strain were along the longitudinal edges of the tire. The distributions of maximum shear strain along top 2 in AC layer at tire edges are given in Figure 4 11. From the figure, it can be observed that, in all cases, shear strain is extremely large at surface, then decreases rapidly, and increases with depth until reaching the maximum value at a depth between 0.5 to 1.5 in, after which it decreases from the peak value. Large shear strains at AC surface might be caused by tire pavement contact shear traction. Comparing maximum shear strain among three types of tire, it clearly showed that super single tire produced much higher shear stress than both dual and new ge neration wide base tires. Again, this might be due to higher average contact stress caused by super single tire, which greatly affects near surface stress/strain distributions. However, when unbalanced load between dual tires is considered, the maximum she ar strain increased with the severe degree of unbalanced load, which might exceed the shear strain caused by super single tire, as shown in Figure 412. 60 62 64 66 68 70 72 74 76 78 80 425/65R22.5 DUAL 445/50R22.5 Tire Type Maximum Tensile Stress (psi) Figure 4 7. Comparisons of maximum tensile stress at AC surface PAGE 79 79 200 150 100 50 0 50 100 150 40 20 0 20 40 60 80 Tire Transverse Location (in) xx (psi) 75%L+125%R Figure 4 8. xx distribu tions along AC surface due to unbalanced dual tires 250 200 150 100 50 0 50 100 150 40 20 0 20 40 60 80 Tire Transverse Location (in) xx (psi) 50%L+150%R Figure 4 9. xx distributions along AC surface due to unbalanced dual tires PAGE 80 80 300 250 200 150 100 50 0 50 100 150 200 40 20 0 20 40 60 80 Tire Transverse Location (in) xx (psi) 0%L+200%R Figure 4 10. xx distributions along AC surface due to unbalanced dual tires 0 0.5 1 1.5 2 2.5 3 0 20 40 60 80 100 120 140 Maximum Shear Microstrain Depth to Surface (in) 445/50R22.5 425/65R22.5 DUAL Figure 4 11. Comparisons of maximum shear strain along top AC layer PAGE 81 81 0 0.5 1 1.5 2 2.5 3 0 50 100 150 200 250 Maximum Shear Microstrain Depth to Surface (in) 100%l+100%R 75%l+125%R 50%l+150%R 0%l+200%R Figure 4 12. Comparisons of maximum shear strain due to unbalanced dual tires Comparisons of maximum tensile strain at AC bottom among different tires are presented in Figure 4 13, which shows that 425/65R22.5 has the highest value while dual tires has the lowest value. However, if unbalanced load between dual tires is considered, the maximum tensile strain can increase to near 140 u for completely unbalanced load, as shown in Figure 414. 0 20 40 60 80 100 120 425/65R22.5 DUAL 445/50R22.5 Tire Type Maximum Tensile Microstrain Figure 4 13. Comparisons of maximum tensile strain at AC bottom PAGE 82 82 0 20 40 60 80 100 120 140 160 DUAL(100%L+100%R) DUAL (75%L+125%R) DUAL (50%L+150%R) DUAL (0%L+200%R) Unbalanced Dual Tires Maximum Tensile Microstrain Figure 4 14. Comparisons of maximum tensile strain due to unbalanced dual tires Figure 4 15 shows the comparisons of compressive strain on the subgrade among different tires. Compressi ve strain on subgrade was considered as a key parameter in both AI and Shell methods to control pavement rutting. Although 425/65R22.5 caused highest compressive strain, there was little difference among tires. For the unbalanced dual tires, the difference of compressive strain is also very limited, as shown in Figure 4 16. This indicates that compressive strain on subgrade is less affected by average contact stress than by total tire load. 120 100 80 60 40 20 0 425/65R22.5 DUAL 445/50R22.5 Tire Type Subgrade Microstrain Figure 4 15. Comparisons of vertical compressive strain at subgr ade PAGE 83 83 120 100 80 60 40 20 0 DUAL(100%L+100%R) DUAL (75%L+125%R) DUAL (50%L+150%R) DUAL (0%L+200%R) Unbalanced Dual Tires Subgrade Microstrain Figure 4 16. Comparisons of vertical compressive strain due to unbalanced dual tires 4.2.3 Damage Ratio To compare the aggressiveness of the any tire to dual tire 11 R 22.5, the damage ratio for each failure mechanism was calculated. For fatigue fail ure, the damage ratio can be defined as: ) ( ) 5 22 11 ( tire any N R N DRf f (4 4) Where, DR = damage ratio between any tire and dual tire 11 R 22.5 Nf (11 R 22.5) = number of cycles to failure for the 11R 22.5; and Nf (any tire) = number of cycles to failure for any tire Nf were calculated based on Eq. 4 1 and 42 for 10 % and 45 % damage respectively. For topdown cracking and instability rutting, the damage ratio was calcu lated based on fourth power law as follows: 4 5 22 11 any R tireDR (4 5) PAGE 84 84 Where, tire any = maximum tensile/shear strain for any tire 5 22 11 R= maximum tensile/shear strain for 11 R 22.5 For subgrade rutting, it is assumed that rutting is related linearly to vertical compressive strain on the top of subgrade (Bonaquist, 1992) and the damage ratio was calculated as follows: 5 22 11 R tire anyDR (4 6) Where, tire any = maximum vertical compressive strain for any tire 5 22 11 R= maximum vertical compre ssive strain for 11 R 22.5 All the results were summarized in Table 4 4. For all cases without considering unbalance dual tires, super single (425/65R22.5) produced the worst damage to pavement, while new generation wide based tire (445/50R22.50 caused the l east damage to pavement except for fatigue cracking which was about four times more damage than balanced dual tires. If unbalance load between dual tires was considered, however, dual tires might produce much more damage to pavement than the super single t ire. Table 4 4. Damage ratio between any tire and dual tire Tire Type Fatigue Cracking Topdown Cracking Subgrade Rutting AC Instability Rutting Damage (10%) Damage (45%) 11R22.5 100%L+100%R 1.00 1.00 1.00 1.00 1.00 75%L+125%R 2.47 2.47 2.45 1.06 2.02 50%L+150%R 5.02 5.02 5.08 1.08 3.69 00%L+100%R 14.73 14.73 16.08 1.11 9.84 425/65R22.5 6.72 6.72 1.46 1.11 1.81 445/50R22.5 4.27 4.27 0.61 0.98 0.45 PAGE 85 85 4.2.4 Summary The effects of tire types on pavement response were evaluated via BISAR static a nyalysis. The loading configurations for three types of tires (11R22.5, 425/65R22.5 and 445/50R22.5) for BISAR input were created based on these three tire footprints. The magnitude and orientation of 3D contact stresses was reasonably defined for different tires based on measured contact stresses and previouse studies. Some key points are summarized as follows: Pavement near surface stress/strain is greatly affected by overall average tirepavement contact stress. The higher the contact stress, the more damage to pavement surface. Subgrade rutting is more affected by total load than by average contact stress. For all cases without considering unbalance dual tires, super single (425/65R22.5) produced the worst damage to pavement, while new generation wi de based (NGWB) tire (445/50R22.5) caused the least damage to pavement except for fatigue cracking which was about 4 times damage more than balanced dual tires. If unbalance load between dual tires was considered, however, dual tires might produce much more da mage to pavement than super single tire. 4.3 Development of 2D Axle TirePavement Contact Model 4.3.1 Tire Nomenclature Given the interdisciplinary nature of this work, some of t ire nomenclature and sign conventions are presented here, which will be used throughout the text. Tire sign convention used here is based on the SAE (Society of Automobile Engineering)standards, which define tire longitudinal axis as vehicle traveling direction and the lateral or transverse axis as perpendicular to travel, as shown in Figure 417. The tires modeled are modern radial tires, which are consisted of numerous parts as shown in Figure 418 (Goodyear, 2004). The functions and main components of those parts are described as follows: PAGE 86 86 Sidewal l : M ade of rubber must withstand flexing and weathering and provide protection for the ply. Belts : Steel cord belt plies provide strength, stabilize the tread, and protect the air chamber from punctures. Vertical Lateral Longitudinal Figure 4 17. Tire Sign Convention Radial Ply : T he radi al ply, together with the belt plies, withstands the loads of the tire under operating pressure. The plies must transmit all load, driving, braking and steering forces between the wheel and the tire tread. Tread : M ade of rubber and provides the interface b etween the tire and the road. Its primary purpose is to provide traction and wear. Figure 419 shows tire dimension. Some of nomenclatures are defined as follows (Goodyear, 2004): Outside Diameter (OD) : The unloaded diameter of the tire/ rim combination. Section Width (SW) : The maximum width of the tire section, excluding any lettering or decoration. PAGE 87 87 Section Height (SH) : The distance from the rim to the maximum height of the tire at the centerline. Figure 4 18. Components of a unisteel radial tire ( after Goodyear 2004) Aspect Ratio (AR) : The section height divided by the section width, expressed as a percentage (SH/SW x 100 % ). Loaded Section (LS) : The width of the cross section at the Tire and Rim Associations dual tire load and inflation pressure. St atic Loaded Radius (SLR) : The distance from the road surface to the horizontal centerline of the wheel, under dual load. Minimum Dual Spacing : The minimum dimension recommended from rim centerline to rim centerline for optimum performance of a dual wheel i nstallation. PAGE 88 88 It should be noted that tire deflection is the key response of tire structure to its applied load. Tire deflection is the difference between loaded section height and unloaded section height. Tire deflection load relationship is often used to calibrate tire model. Figure 4 19. Tire Dimension ( after Goodyear 2004) PAGE 89 89 4.3.2 Finite Element Analysis 4.3.2.1 Introduction of ADINA p rogram The multipurpose finite element program ADINA version 8.3 was used here to model 3D tire pavement interaction i n this study. The capability of the ADINA program for 2D and 3D finite element analysis, its versatility in modeling materials behaviors under load and temperature effects, and its capability in modeling the contact condition between tire and pavement m ake this program very appropriate to model complicated tire pavement interaction. The ADINA program has a very friendly user interface to build the needed models for specific applications. It has a routine to automatically create finite element meshes bas ed on the boundary definitions and density specifications. The program also has a complete post process ing routine to generate the results both numerically and graphically. 4.3.2.2 Governing e quation For static analysis, the governing equation for finite element analysis is given by: } { } ]{ [ f d K (4 7) Where ] [ K is the moduli matrix, } { d the vector of displacements at the nodal points of t he mesh and } { f the corresponding vector of forces and/or moments. 4.3.2.3 Contact analysis Tire pavement interaction is contact issue. Contact in ADINA is modeled using contact groups. Each contact group is composed of one or more conta ct surfaces. Contact pairs are then de fined between contact surfaces. Contact pair consists of target and contactor. It is recommended that stiffer surface be target surface and contractor surface should not penetrate into target surface, as shown in Figur e 420. PAGE 90 90 Figure 4 20. Contactor and target selection For static analysis and dynamic implicit analysis, three general types of contact model can be carried out for contact analysis in ADINA 1) segment; 2) node to node;3) Rigidtarget. Segment contact is generally applicable to the most analysis. It can be used for analysis involving: Self contact (contact surface acts as both target and contactor) Contact between flexible surfaces Contact between rigid target surfaces and flexible contactor surfaces Node to node contact is used only very limited applications, for example in pin joints. In general, consider node to node contact only if segment contact cannot be used. Rigidtarget contact is applicable for the analysis where the target surfaces are always as sumed rigid, as in metal form problems. In the present work, the segment contact has been used. 4.3.2.4 Contact algorithms One of the most important areas of the tire pavement contact analysis is to determine the state of the stress and strain under contac t loading. ADINA offers three contact algorithms 1) constraint function method; 2) Lagrange multiplier (segment) method; and 3) rigid target method. Each contact group must belong to one of these three contact algorithms. However, different contact groups can use different algorithms. Constraint function algorithm works better in most cases and should normally be selected. Lagrange multiplier algorithm may work well in some cases involving friction. And rigid method is a significantly simplified contact alg orithm PAGE 91 91 intended primarily for metal forming applications. Therefore, constraint function has been selected in the present work and introduced as follows. In constraint function algorithm, constraint functions are used for nopenetration and frictional cont act condition. The normal constraint function is given by Ng g g w 2) 2 ( 2 ) ( (4 8) Where g is gap, and is the normal contact force. N is a small user defined parameter The function is shown in Fig ure 4 21. It involves no inequalities, and is smooth and differentiable. Figure 4 21. Constraint function for normal contact The frictional constraint function ) u v( is defined by 0 ) v arctan( 2 v Tu ( 49) Where u is sliding velocity and T is a small parameter which can provide some "elasticity" to the Coulomb friction law. is a n ondimentional friction variable given by T F (4 10) PAGE 92 92 Where T F is a tangential force and is coefficient of friction The frictional constraint function is illustrated in Figure 422. Figure 4 22. Constraint functions for tangential contact 4.3.3 Tires to Be Modeled To compare the effects of tire types on pavement response, three tires were modeled here: Goodyear G149 RSA 11R22.5, Goodyear G286 A SS 425/65R22.5 and Michelin X One XDAHT Plus 445/50R22.5. Their pictures were shown in Figures 423 to 425, respectively. Accordingly, their manufactures specifications were given in Tables 4 5 to 47. Figure 4 23. Goodyear G149 RSA 11R22.5 Table 4 5. Specifications for Goodyear G149 RSA Size Tread Depth 32nds Load Radius Overall Diameter Overall Width Min. Dual Spacing Dual load Dual Inflation in. mm. in. mm. in. mm. in. mm. lbs. kg. psi k P a 11R22.5 20 19 .4 493 41.6 1057 10.9 277 12.5 318 5,840 2650 105 720 PAGE 93 93 Figure 4 24. Goodyear G286 A SS 425/65R22.5 Table 4 6. Specifications for Goodyear G286 A SS Size Tread Depth 32nds Load Radius Overall Diameter Overall Width Max. Tire Load (Single) in. mm. in. mm. in. mm. lbs. psi kg. k P a 425/65R22.5 20 20.3 516 44.2 1124 16.1 410 11,400 120 5150 830 Figure 4 25. Michelin X One XDA HT Plus 445/50R22.5 Table 4 7. Specifications for Michelin X One XDAHT Plus Size Tread Depth 32nds Load Radius Overal l Diameter Overall Width Max. Tire Load (Single) in. mm. in. mm. in. mm. lbs. psi kg. k P a 445/50R22.5 28 18.7 47.5 40.5 1028 17.1 435 10,200 120 4625 830 4.3.4 Mesh of Tire Model Modeling a truck tire is really a challenge job due to its structural a nd material complexity. Typically, a radial tire mainly consists of the following parts: ply, belts, tread and sidewall, as shown previously in Figure 418. Tire materials are basically composite and rubber compounds PAGE 94 94 vary through tire structure. However, exact material properties and actual structural make up of these tires used by industry are not available to general public and thus make tire model extremely difficult. Fortunately, some basic response data regarding the behavior of typical radial truck t ires and their structural makeup provided by Smithers Scientific Services along with previous job did by Myers (2000) make the development of twodimensional tire model possible. Table 4 8. Material Properties for Tire Models 11R22.5 425/65R22.5 445/50R2 2.5 Tire Parts Modulus (psi) Possion Ratio, V Modulus (psi) Possion Ratio, V Modulus (psi) Possion Ratio, V Rim 1.00E+12 0.1 1.00E+12 0.1 1.00E+12 0.1 Radial Ply 2 .00E+0 5 0.4 2.00E+05 0.3 3.00E+04 0.3 Belt 5 .00E+0 9 0.4 5.00E+08 0.2 1.00E+08 0.2 Sidewall 5 .00E+0 2 0.495 5.00E+02 0.495 1.16E+03 0.495 Skirt Tread 1.16E+03 0.495 1.16E+03 0.495 1.16E+03 0.495 Shoulder 1.16E+03 0.495 1.16E+03 0.495 1.16E+03 0.495 Tread 1.16E+03 0.495 1.16E+03 0.495 1.16E+03 0.495 Grove 9.80E 06 0.499 9.80E 06 0.499 9.80E 06 0.499 A proper tire model should catch both geometrical characters and structural behaviors. The structural behavior of radial truck tires is governed by a low stiff wall structure and a high stiff tread structure resulting from the steel reinforcemen t embedded in tread (Roque et al. 1999). Given the cross section view in Figure 4 18 and specifications in Tables 4 5 to 47, the twodimensional ( 2D ) finite element models for these three tires are developed as shown in Figure 4 26. All models are meshe d with 2D solid plane strain elements. Different colors represent different element groups with corresponding moduli The rim is made of alloy and modeled as rigid body with very high moduli Ply and belts can be modeled as reinforcement. Sidewall, tread skirt, shoulder and tread are made of rubber with different moduli All materials are treated as isotropic elastic material. The determination of tire material properties is based on the procedure developed by Myers (2000) through adjusting the tire materi al properties to match measured the PAGE 95 95 radius of curvature of the tire tread. Also, the material properties of previous tire models developed by other scholars ( Meng 2002; Zhang 2001; Shoop 2001) will be taken as a reference. Finally, the material properties for these three tires are presented in Table 4 8. 11R22.5 425/65R22.5 455/50R22.5 Figure 4 26. Developed 2D finite element tire models 4.3.5 Development of Axle tire pavement Contact Model As stated early, most of above studies didnt consider axle tire pavement interaction when they did comparison analysis between dual tire and wide base tire. Actually, axletire pavement interaction cant be captured using only surface stress (i.e. measured stress). To capture the rutting characteristics of pavement surface, an axle has to be used to link tires together, otherwise tire will be unstable on unlevel surface, see Figure 427. Therefore, a comprehensive axle tir e  PAGE 96 96 pavement finite element model was developed using ADINA, which can capture not only the loading characteristics, but also tire geometry and pavement surface conditions as well. Unstable Structure Stable Structure Figure 4 27. Axle Tire Pavement Interaction s The axletire pavement interaction was modeled with three types of tires, i.e. dual tire 11R22.5, super single tire 425/65R22.5 and new generation wide base tire 445/50R22.5. The whole meshes of these three models were presented in Figures 4 28 to 430. Each model considered two surface conditions, i.e. flat surface and rutting surface, see Figures 431 to 432. And contact surfaces were defined in each model to model the contact conditions between tire and AC surface, which was shown in Figure 4 33. F igure 4 28. 2 D Axle tire pavement contact model for dual tire 11R22.5 PAGE 97 97 Figure 4 29. 2 D Axle tire pavement contact model for super single 425/65R22.5 Figure 4 30. 2 D Axle tire pavement contact model for NGWB 445/50R22.5 Figure 4 31. Flat paveme nt surface Figure 4 32. Pavement surface with rut PAGE 98 98 Figure 4 33. Contact surfaces between tire and pavement surface 4.3.6 Element Selection All 2 D solid elements were modeled as 9 node plane strain elements, with two translational degrees of freed om per node, as shown in Figure 434. This type of node configuration has been shown to give a high level of accuracy in combination with an acceptable computing time demand. All elements are treated as isoparametric element. x y Figure 4 34. 9 node Biquadratic Elements The element matrices can be derived as follows: dV B C B KT] ][ [ ] [ ] [ (4 11) Where, ] [ C is a matrix of elastic coefficients giv en by 2 2 1 0 0 0 1 0 1 ) 2 1 )( 1 ( ] [ v v v v v v v E C (4 12) PAGE 99 99 ] [ B is matrix of strain displacement relations given by ] [ ... ] [ ] [ ] [9 2 1B B B N B (4 13) And y N x N y N x N Bi i i i i0 0 i =1 to 9 (4 14) N is shape function given by : 9 2 1 9 2 10 0 0 0 0 0 N N N N N N N (4 15) Assuming the variations of the displacements in between the nodes to be linear, the displacement can be expressed by Q N v v u v u N y x v y xu ] [ ] [ ) ( ) (9 2 2 1 1 (4 16) Where Q is the vector of nodal displacement degrees of freedom and ) (i i v u denote the displacements of node i i = 1 to 9. The two dimensional strain displacement relations can be derived from Eq. (416) and expressed as : PAGE 100 100 Q B x v y u y v x uxy y x ] [ (4 17) Stress strain relations can be expressed as : ] [ C (4 18) 4.3.7 Loading and Boundary Conditions The tire pavement contact model can be simulated by two steps. First, the inflation pressure is applied on the inner surface of the tire model and different pressure levels can be set to satisfy the requirements of the analysis. Second, a vertical load is applied on the rim axle under given inflation pressure and the load can be gradually loaded under assigned time steps. The detail loading configuration is shown in Figure 435. The boundary conditions for static analysis are also shown in Figure 4 35. It can be observed that the bottom of pavement is fixed at Z translation while sides of pavement and rim axle are restricted with Y translation. Figure 4 35. Loading and boundary conditions PAGE 101 101 4.3.8 Model Verification In order to further verify developed tire pavement contact models, a comparison was made between predicted contact stresses and measured contact stresses. Predicted contact stresses were obtained at the nodes of the pavement surface under the tire, and measured stresses were provided by Smithers Scientific Services, Michelin (2009) and literature references (Markstaller e t al ., 2000; Al Qadi et al. 2004; M.de Beer et al. 1997). Figures 4 36 to 437 show comparisons of contact stresses for Dual 11R22.5 between measured and predicted at same load and inflation level which clearly indicate that predicted vertical and horizontal shear contact stresses are similar to those measured under the real tire, except for some variation in magnitude. The variation might be caused by different loading conditions (FEM was running under static load, while the measurement was conducted under moving steel bed), tread groves (FEM didnt consider longitudinal grove) and element mesh. The overall errors are within 20 %. Unfortunately, there are no detail measured contact stresses for both wide base 425/65R22.5 and 445/50R22.5, except for measured maximum contact stresses under each rib, as shown in F igure 438 and Figure 439 respectively. Both figures show that predicted maximum contact stresses agree well with measured contact stresses The detail distributions of predicted contact stresses for both wide base 425/65R22.5 and 445/50R22.5 are given in Figure 4 40 and Figure 441. It is noted that the transverse contact stresses show the some asymmetric distribution under each rib, either compression or tension, and the smallest shear stress is found at the center of each rib. And Poissons effect is dominant over pneumatic effect. Those characteristics also agree well with previous studies (Roque et al ., 1998; Al Al Qadi et a .l 2004). The most important characteristic is the models ability to capture the patterns of both vertical contact stress and horizontal shear contact stress distributions. PAGE 102 102 200 150 100 50 0 50 6 4 2 0 2 4 6 Tire Lateral Distance (in) Vertical Contact Stress (psi) Predicted Measured Figure 4 36. Comparison of vertical contact stress for dual 11R22.5 80 60 40 20 0 20 40 60 80 6 4 2 0 2 4 6 Transverse Contact Stress (psi) Predicted Measured Figure 4 37. Comparison of vertical contact stress for dual 11R22.5 0 20 40 60 80 100 120 140 160 0 2 4 6 8 10 Tire Ribs Maximum Vertical Contact Stress (psi) Predicted Measured Figure 4 38. Comparison of vertical contact stress for NG WB 445/50R22.5 PAGE 103 103 0 50 100 150 200 250 0 1 2 3 4 5 6 7 Tire Ribs Maximum Vertical Contact Stress (psi) Predicted Measured Figure 4 39. Comparison of vertical contact stress for Super Single 425/65R22.5 250 200 150 100 50 0 50 10 5 0 5 10 Tire Lateral Distance (in) Vertical Contact Stress (psi) 455/50R22.5 425/65R22.5 Figure 4 40. Predicted vertical contact stress for widebase tires 80 60 40 20 0 20 40 60 80 10 5 0 5 10 Tire Lateral Distance (in) Transverse Contact Stress (psi) 455/50R22.5 425/65R22.5 Figure 4 41. Predicted transverse contact stress for widebase tires PAGE 104 104 4.4 Effects of R utted Surface on Nearsurface Stress Distributions 4.4.1 Structure and Material Properties of Model To evaluate effects of a rutting surface on pavement stress distribution, a typical pavement structure is used here. Its physical and material properties ar e listed in Table 4 9. T he tire properties are got from Table 48. Table 49. Pavement structure and material properties Layer Thickness (in) Modulus (psi) (E) Poisson's Ratio ( v ) AC Layer 1 2 5 .00E+0 5 0.35 Layer 2 2 5 .00E+06 0.35 Layer 3 2 5 .00E+0 5 0.35 Layer 4 2 5 .00E+06 0.35 Base 12 4.00E+04 0.4 Subgrade 180 2.00E+04 0.4 4.4.2 Contact Stress Distributions of a Tire on Rutting Surface It is wellknown that pavement transverse surface is not flat but with some cross slope for drainage pur pose, let alone a pavement with rut. When a vertically oriented tire operates on a side of rut in the wheel path, the horizontal component of its load will push the wheel toward the lowest part of the rut as shown in Figure 442. The lateral force per unit load can be expressed as: sin W Fx Where: W = Load on the tire = Inclination angle of rut side Thus for one degree of inclination angle, a lateral force of 0.0174 lb/lb is produced in the downhill direction by the gravitational component. For radial tire, this lateral force might be PAGE 105 105 responsible for creating rut or increasing severity of rut in asphalt pavement surface (Gillespie et al., 1993). W FyFx Figure 4 42. Forces acting on a tire on one side of a rut cross section A comparison of contact stress distribution between flat surface and rutting surface for different tires is given in Figures 4 43 to 448. The comparisons were conducted at the same load and inflation level. Comparing with contact stress distributions on flat surface, it is noted that contact stress distributions on a rutted surface are not symmetric along the centre of the tire. Rather, contact stresses are more concentrated on the tire shoulder and decrease along the downhil l direction. And the maximum contact stress induced on a rutted surface are much higher than that induced on flat surface for all tires, except for transverse contact stress of 425/65R22.5, which the maximum values induced by flat and rutting surface are pretty close. Table 4 10 summarizes statistic results of the comparisons. Table 4 10. Statistic results of the comparisons Max. Contact Stress (psi) Max. Contact Stress (psi) Max. Contact Stress (psi) 11R22.5 425/65R22.5 445/50R22.5 Vertical Lateral V ertical Lateral Vertical Lateral Flat 128 46 165 83 147 40 Rut 208 56 317 82 210 117 Rut/Flat 1.63 1.22 1.92 0.99 1.43 2.93 PAGE 106 106 250 200 150 100 50 0 50 6 4 2 0 2 4 6 Tire Lateral Location (in) Vertical Contact Stress (psi) Flat Rut Figure 4 43. Comparison of vertical contact stress for dual 11R22.5 80 60 40 20 0 20 40 60 6 4 2 0 2 4 6 Tire Lateral Location (in) Transverse Contact Stress (psi) Flat Rut Figure 4 44. Comparison of transverse contact stress for dual 11R22.5 PAGE 107 107 350 300 250 200 150 100 50 0 50 8 6 4 2 0 2 4 6 8 Tire Lateral Location (in) Vertical Contact Stress (psi) Flat Rut Figure 4 45. Comparison of vertical contact stress for wide base 425/65R22.5 100 80 60 40 20 0 20 40 60 80 100 8 6 4 2 0 2 4 6 8 Tire Lateral Location (in) Transverse Contact Stress (psi) Flat Rut Figure 4 46. Comparison of transverse contact stress for wide base 425/65R22.5 PAGE 108 108 250 200 150 100 50 0 50 10 5 0 5 10 Tire Lateral Location (in) Vertical Contact Stress (psi) Flat Rut Figure 4 47. Comparison of vertical contact stress for wide base 445/50R22.5 140 120 100 80 60 40 20 0 20 40 60 80 10 5 0 5 10 Tire Lateral Location (in) Transverse Contact Stress (psi) Flat Rut Figure 4 48. Comparison of transverse contact stress for wide base 445/50R22.5 PAGE 109 109 4.4.3 Critical Locations for Top Down Cracking ( TDC) and Instability Rutting T o initiate top down cracking, there must be some critical positions where ma ximum surface tensile stress occur s. F or a 2 D finite element model, two kinds of stresses, i.e. bending stress and principal tensile stress, could lead to the initiation of top down cracking. Therefore, clearly understanding the distributions of bending s tress and principal tensile stress (SIGMA 1) along AC surface has become extremely important. A wide base tire (425/65R22.5) model with rutting surface was used here to get distributions of these two types of stresses, which were shown in Figure 449. Figure 4 49 shows that there are two critical locations that could be responsible for topdown cracking: one is about 30 inches far from the edge of tire and the other is right at the edge of tire. The maximum principal tensile stress SIGMA 1 is almost double than the maximum bending stress Stress_yy which shows that principal tensile stress at the tire edge could potentially be the most critical stress for top down cracking. This makes sense since a lot of topdown cracks occur right near the edge of wheel pa th in the fields. In addition, the distributions of SIGMA 1 along AC depth were also plotted as shown in Figure 4 50, which also shows that maximum tensile stress only occurred near AC surface. Therefore, the following analysis will be focused on the compa risons of principal tensile stress distributions along AC surface between flat surface and rutt ed surface for different types of tire respectively As for instability rutting, previous studies (Novak et al. 2004; Kai Su et al ., 2008) showed that a critica l shear plane was developed near the edge of the tire and the critical location for maximum shear stress was located near the surface zone. Therefore, the comparison will be focused on the maximum shear stress distributions along AC depth near the edge of the tire between flat surface and rutted surface for different types of tire respectively. PAGE 110 110 Transverse Bending stress Vs. Principal Tensile Stress 100 80 60 40 20 0 20 40 60 100 80 60 40 20 0 20 40 60 80 100 Transverse distance (in) Stress_yy/SIGMA1 (psi) Stress_yy SIGMA1 Figure 449. Distributions of bending stress and principal tensile stress along AC surface 1.20 1.00 0.80 0.60 0.40 0.20 0.00 150 100 50 0 50 100 Principle Tensile Stress SIGMA1 (psi) Depth (in) 0.11 in to Tire Edge 0.23 in to Tire Edge 0.36 in to Tire Edge 0.50 in to Tire Edge 0.66 in to Tire Edge Figure 450. Distributions of principal tensile stress along AC d epth PAGE 111 111 4.4.4 Effects of Rut on Near Surface Stress Distributions Figures 4 51 to 453 show the comparisons of principal tensile stress (SIGMA 1) distributions along AC surface between flat and rut ted surface conditions for different types of tires. It can be seen that principal tensile stress decreases rapidly with the distance to the tire edge, almost no tensile stress after 0.5 in away from the tire edge. This might explain that why topdown cracking occurs most often at the tire edge. And peak SIGMA 1 for rutting surface is significantly higher than that for flat surface for all cases, with a factor about 1.6 to 2.0. This in dicates that a pavement with some rutting might significantly increase the propensity of top down cracking. The comparisons of maxi mum shear stress distributions along top AC depth at the edge of the tire are given in Figures 4 54 to 456 between flat and rut surface conditions for each tire. It can be observed that, for all cases, shear stress increases initially with depth, reaches the maximum value at a depth around 0.5 in and decreases from the peak value after that. This implies that there exists critical locations right beneath the tire edge in the near pavement surface. If the shear stress at the critical location is large enoug h, it might easily cause instability rutting and topdown cracking /damage And the rutting surface produces much higher maximum shear stresses than flat surface does, with a factor ranging from 1.2 to 2.1. Again, this indicated that pavement with rut ting m ight increase the severity of rutting and propensity of top down cracking. Table 4 11 shows the statistic results of the comparisons. Table 4 11. Statistic results of the comparisons 11R22.5 425/65R22.5 445/50R22.5 Peak SIGMA1 (psi) Peak Max. Shear Str ess (psi) Peak SIGMA1 (psi) Peak Max. Shear Stress (psi) Peak SIGMA1 (psi) Peak Max. Shear Stress (psi) Flat 17.5 51.1 27.8 79.7 19.5 54.5 Rut 35.6 109.0 46.9 115.5 31.3 1 04.2 Rut/Flat 2.03 2.14 1. 69 1.45 1.60 1.91 PAGE 112 112 Dual 11R22.5 20.00 0.00 20.00 40.00 60.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Distance to tire edge (in) SIGMA1 (psi) Flat Rut Figure 4 51. Distributions of principal tensile stress at AC surface for 11R22.5 Super Single 425/65R22.5 20.00 0.00 20.00 40.00 60.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 Distance to Tire Edge (in) SIGMA1 (psi) Flat Rut Figure 4 52. Distributions of principal tensile stress at AC surface for 425/65R22.5 PAGE 113 113 NGWB 445/50R22.5 20.00 0.00 20.00 40.00 60.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Distance to tire edge (in) SIGMA1 (psi) Flat Rut Figure 4 53. Distributions of principal tensile stress at AC surface for 445/50R22.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 20 40 60 80 100 120 Maximum Shear Stress (psi)11R22.5 Depth to AC surface(in) Flat Rut Figure 4 54. Distributions of maximum shear stress at AC depth for 11R22.5 PAGE 114 114 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 20 40 60 80 100 120 Maximum Shear Stress (psi)425/65R22.5 Depth to AC surface(in) Flat Rut Figure 4 55. Distributions of maximum shear stress at AC depth for 425/65R22.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 20 40 60 80 100 120 Maximum Shear Stress (psi)445/50R22.5 Depth to AC surface(in) Flat Rut Figure 4 56. Distributions of maximum shear stress at AC depth for 445/50R22.5 PAGE 115 115 4.5 Evaluate Effects of Tire Types on Nearsurface Stress Distributions 4.5.1 Contact Stress Distributions Comparisons of contact stress distributions among different tires at their recommended inflation pressures under the same load level are presented in Figures 4 57 to 458. Obviously, the super single 425/65R22.5 causes the highest maximum vertical contact stress, while dual 11R22.5 and NGWB 445/50R22.5 have very close maximum vertical contact stress, as shown in Figure 4 57. This makes sense since the new wide base tire is 15 % to 18 % wid er than conventional wide base tires (385 and 425) and thus has more average contact area, which makes it much comparable to that of dual tire assembly. Accordingly, the super single 425/65R22.5 also produces much higher transverse contact stresses than both dual 11R22.5 and NGWB 445/50R22.5 do. This characteristic of contact stress distribution will greatly affect the near surface stress /strain distribution. Table 412 gives the statistic results of the comparisons. Table 4 12. Statistic results of the co mparisons Contact Stress (psi) Contact Stress (psi) Contact Stress (psi) 11R22.5 425/65R22.5 445/50R22.5 Vertical Lateral Vertical Lateral Vertical Lateral Maximum 161 57.5 213 69.5 147 40.1 Tire/Dual 1.00 1.00 1.32 1.21 0.91 0.70 4.5.2 Near sur face Stress Distributions Figures 4 59 to 460 show the comparisons of principal tensile stresses distributions along AC surface among different types of tire for flat and rutting surface respectively. As expected, super single 425/65R22.5 produces much hi gher peak SIGMA 1 than both dual 11R22.5 and NGWB 445/50R22.5 do. So do maximum shear stress as shown in Figures 461 to 462. This makes sense since near surface stress distributions are greatly affected by contact stresses, while super single 425/65R22.5 produces the highest average contact stresses. However, the difference PAGE 116 116 between super single 425/65R22.5 and dual 11R22.5 is less for a rutted surface than that for flat surface, which might be due to unbalanced load in dual tire set 250 200 150 100 50 0 50 10 5 0 5 10 Tire Lateral Distance (in)Vertical Contact Stress (psi) 455/50R22.5 425/65R22.5 11R22.5 Figure 4 57. Comparisons of vertical contact stress among different tires 80 60 40 20 0 20 40 60 80 10 5 0 5 10 Tire Lateral Distance (in) Transverse Contact Stress (psi) 455/50R22.5 425/65R22.5 11R22.5 Figure 4 58. Comparisons of transverse contact stress among different tires PAGE 117 117 Flat Surface 20.00 0.00 20.00 40.00 60.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 Distance to Tire Edge (in) SIGMA1 (psi) 425/65R22.5 445/50R22.5 11R22.5 Figure 4 59. Comparisons of principal tensile stresses among tires for flat surface Rutting Surface 20.00 0.00 20.00 40.00 60.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 Distance to Tire Edge (in) SIGMA1 (psi) 425/65R22.5 445/50R22.5 11R22.5 Figure 4 60. Comparisons of principal tensile stresses among tires for rutted surface PAGE 118 118 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 20 40 60 80 100 Maximum Shear Stress (psi)Flat Surface Depth to AC surface(in) 11R22.5 425/65R22.5 445/50R22.5 Figure 4 61. Comparisons of maximum shear stresses among tires for flat surface 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 20 40 60 80 100 120 140 Maximum Shear Stress (psi)Rut Surface Depth to AC surface(in) 11R22.5 425/65R22.5 445/50R22.5 Figure 4 62. Comparisons of maximum shear stresses among tires for rutt ed surface PAGE 119 119 4.6 Parametric S tudy 4.6.1 Pavement S tructure A comprehensive parametric study was conducted to investigate how pavement structure affect s the peak principal tensile stress (SIGMA 1) at AC surface and peak maximum shear stress along top 3 in of AC layer. T wo pavement thickness value s were chosen to encompass the range typically found on highvolume interstate highways. Typical AC moduli were selected to represent actual field AC pavement s Base and subgrade moduli included a range of values found in field sections to capture the effects of pavement bending. All specific parametric values were presented in Table 4 13. The axletire pavement contact model used for parametric study throughout this section was NGWB 445/50R22.5 and the condition of AC surface was flat. Table 4 13. Pavement structure information used for parametric study Variable Parameter Values AC Thickness 4 in 8 in AC Moduli 500 ksi Base Moduli 20 ksi 40 ksi Subgrade Moduli 1 ksi 5 ksi 1 0 ksi 20 ksi Base Thickness 12 in (Constant) Subgrade Thickness 6 0 in (Constant ) Possion Ratio AC:0. 40, BS:0. 35,SB:0. 35 Loading Condition 18k lbs axle load, 1 0 0 psi inflation pressure The study focused on the comparisons of peak SIGMA 1 and maximum shear stress in following three aspects: AC thickness, base moduli and subgrade moduli The results are given in Tables 4 14 to 415 and Figures 4 63 to 468. Figures 4 63 to 464 clearly show that AC thickness, base moduli and subgrade moduli have no significant effects on the peak values of SIGMA 1. In other words, peak SIGMA 1 is a lo cal effect and pavement structure may have little effect on it. However, for peak maximum shear stress, things are different. It can be observed from Figures 465 to 468 that base and subgrade moduli have significant effects on the values of peak maximum shear stress. Typically, the weaker the support, the higher the peak PAGE 120 120 maximum shear stress, which means that weak support might increase the propensity of instability rutting and top down cracking. AC thickness has relatively less effects on peak maximum sh ear stress. However, it seems that thick AC layer produced relatively higher maximum shear stress that the thin one, which indicates that thick AC might be more prone to instability rutting and top down cracking. Table 4 14. Summary of peak SIGMA 1 (unit: psi) 8 inch AC Base Modulus (psi) Subgrade Modulus psi) 20000 10000 5000 1000 20000 19.80 20.20 20.60 21.30 40000 19.50 19.80 20.40 21.20 4 inch AC Base Modulus (psi) Subgrade Modulus psi) 20000 10000 5000 1000 20000 19.90 20.30 20.80 21.40 40000 19.60 20.10 20.70 21.30 Table 4 15. Summary of peak maximum shear stress (unit: psi) 8 inch AC Base Modulus (psi) Subgrade Modulus psi) 20000 10000 5000 1000 20000 86.80 107.90 131.70 176.70 40000 72.00 90.20 107.60 136.20 4 inch AC Base Modulus (psi) Subgrade Modulus psi) 20000 10000 5000 1000 20000 69.90 89.40 120.30 174.80 40000 61.90 73.90 93.60 131.20 4.6.2 I nfluence of the Norma l Axle L oad The normal tire axle load directly affects contact path geometry and contact stress distribut ions in the tire pavement interface. Moreover, the normal axle load strongly influences the tire deformation and stress fields developed in the tire layers. The influences of variations of normal axle load on the tire pavement interface contact stress dist ributions are given in Figures 469 to 470. It was clearly demonstrated that both maximum vertical contact stress and transverse contact stress changed from tire centre to the tire edges when the tire was heavily PAGE 121 121 loaded. This characteristic is very consis tent with the measured contact stress reported by de Beer et al (1997). Figure 471 illustrates the peak principal tensile stresses and maximum shear stress as functions of axle normal load for specific inflation pressure condition. The results show that both SIGMA 1 and maximum shear stress increase with increase in normal axle load in a liner manner. Base Moduli:20 ksi 0 5 10 15 20 25 20000 10000 5000 1000 Subgrade Moduli (psi) Principle Tensile Stress (psi) 4 in AC 8 in AC Figure 4 63. Comparisons of peak SIGMA 1 due to different AC thickness Base Moduli:40 ksi 0 5 10 15 20 25 20000 10000 5000 1000 Subgrade Moduli (psi) Principle Tensile Stress (psi) 4 in AC 8 in AC Figure 4 64. Comparisons of peak SIGMA 1 due to different AC thickness PAGE 122 122 Base Moduli:20 ksi 0 20 40 60 80 100 120 140 160 180 200 20000 10000 5000 1000 Subgrade Moduli (psi) Max. Shear Stress (psi) 4 in AC 8 in AC Fig ure 4 65. Comparisons of peak maximum shear stress due to different AC thickness Base Moduli:40 ksi 0 20 40 60 80 100 120 140 160 20000 10000 5000 1000 Subgrade Moduli (psi) Max. Shear Stress (psi) 4 in AC 8 in AC Figure 4 66. Comparisons of peak maximum shear stress due to different AC thickness PAGE 123 123 AC Thickness: 8 in 0 20 40 60 80 100 120 140 160 180 200 20000 10000 5000 1000 Subgrade Moduli (psi) Max. Shear Stress (psi) BS40ksi BS20ksi Figure 4 67. Comparisons of peak maximum shear stress due to different base moduli AC Thickness: 4 in 0 20 40 60 80 100 120 140 160 180 200 20000 10000 5000 1000 Subgrade Moduli (psi) Max. Shear Stress (psi) BS40ksi BS20ksi Figure 4 68. Comparisons of peak maximum shear stress due to different base moduli PAGE 124 124 200 150 100 50 0 50 10 5 0 5 10 Tire Lateral Distance (in) Vertical Contact Stress (psi) 13.5k lbs 18k lbs 22.5k lbs Figure 4 69. Influence of axle load on vertical contact stress 60 40 20 0 20 40 60 10 5 0 5 10 Tire Lateral Distance (in) Transverse Contact Stress (psi) 13.5k lbs 18k lbs 22.5k lbs Figure 4 70. Influence of axle load on transverse contact stress PAGE 125 125 0 20 40 60 80 100 13.5k 18k 22.5k Axle Load (lbs) Stress (psi) Max. Shear Stress Principle Tensile Stress Figure 4 71. Influence of axle load on peak max. shear stress and principal tensile stress 4.6.3 I nfluence of the Inflation Pressure Similar as normal axle load, the inflation pressure also directly affects contact patch geometry, contact stress distribution and tire deformatio n. Figure 472 shows the influence of inflation pressure on tire deformation under the same axle load. When the tire is over inflated, more loads are concentrated on the tire center. And the loads will shift from tire center to tire edges when the tire is under inflated. This could be attributed to the change of sidewall flexibility. It was said the sidewall flexing increases noticeably when a tires inflation drops 15 to 20 % below recommended ( Goodyear, 2004). Obviously, either case will reduce the tire pavement contact area and thus change distributions of tire contact stress. Figures 4 73 to 474 show how tire inflation pressure affects tire pavement contact stress distributions. As expected, both maximum vertical contact stress and transverse contact st ress changed from tire centre to the tire edges when the tire was under inflated. PAGE 126 126 Figure 4 75 shows how tire inflation pressure affects the distributions of maximum shear stress in AC layer. The maximum shear stress increases as the decrease of the tire in flation pressure, though the increase is limited. The peak principal tensile stress (SIGMA 1) at AC surface as functions of inflation pressure is given in Figure 476. It can be observed that SIGMA 1 decreases as the increase of inflation pressure. Both fi gures indicate that under inflation might slightly increase the propensity of topdown cracking and topdown cracking. Over Inflation Under Inflation Correct Inflation Figure 4 72. Influence of tire inflation on tire deformation 200 150 100 50 0 50 10 5 0 5 10 Tire Lateral Distance (in) Vertical Contact Stress (psi) 80 psi 100 psi 125 psi Figure 4 73. Influence of inflation pressure on vertical contact stress PAGE 127 127 60 40 20 0 20 40 60 10 5 0 5 10 Tire Lateral Distance (in) Transverse Contact Stress (psi) 80 psi 100 psi 125 psi Figure 4 74. Influence of inflation pressure on transverse contact stress 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 50 55 60 65 70 75 80 Maximum Shear Stress (psi) Depth to AC surface(in) 80 psi 100 psi 125 psi Figure 4 75. Influence of inflation pressure on max imum shear stress PAGE 128 128 0 5 10 15 20 25 80 100 125 Tire Inflation Pressure (psi) Peak SIGMA1 (psi) Figure 4 76. Peak SIGMA 1 as functions of inflation pressure 4.7 Summary The main points are summarized as follows: BISAR analyses indicate that, without considering unbalance d dual tires, super single (425/65R22.5) produced the wors t damage to pavement, while new generation wide based tire (445/50R22.5) caused the least damage to pavement except for fatigue cracking which was about 4 times damage more than balanced dual tires. If unbalanced load between dual tires was considered, how ever, dual tires might produce much more damage to pavement than super single tire. The developed axle tire pavement finite element contact model can successfully capture patterns of both vertical contact stress and horizontal shear contact stress distributions for all tires, which indicates that the model can be used for evaluation purpose. Based on developed 2 D models, under the same axle load, the super single 425/65R22.5 causes the highest contact stress, while dual 11R22.5 and NGWB 445/50R22.5 have ve ry close contact stress at their recommendation inflation levels. Comparing with a flat AC surface, contact stresses induced on the rut surface are more concentrated on the tire shoulder and decrease along the downhill direction. And the maximum contact stress induced on rutting surface are much higher than that induced on flat surface. Significant principal tensile stress es (SIGMA 1) are located at the edges of the tire for both flat and rutt ed surface, and decreases dramatically with the increase distan ce to the tire edge. The critical location for the maximum shear stress is at the depth aproximately 0.5 in beneath the AC surface near the edge of the tire. PAGE 129 129 Comparing with flat surface, both peak SIGMA 1 and maximum shear stress due to a rutted surface is increased significantly for all tires, which clearly shows that rutting surface might play a significant effect on top down cracking (TDC) and instability rutting in AC layer Evaluation of tire types on near surface stress states indicated that super sin gle 425/65R22.5 produces much higher peak SIGMA 1 than both dual 11R22.5 and NGWB 445/50R22.5 do. So do maximum shear stress. However, the difference between super single 425/65R22.5 and dual 11R22.5 is less for a rutted surface than that for flat surface, which might be due to unbalanced load in dual set. Parametric studies indicate that pavement structure sound s play a relatively small role on the development of peak SIGMA 1. However, thick AC layer and weak base and subgrade support produce high maximum shear stress. The weaker the support, the higher the maximum shear stress, which indicates that weak support might increase the propensity of instability rutting and top down cracking. The maximum contact stress changed from the tire center towards the ti re edges when the tire was heavily loaded or under inflated. Both peak SIGMA 1 and maximum shear stress increase with the axle load. The peak SIGMA 1 and maximum shear stress slightly increase with the decrease of the inflation pressure, which indicates th at under inflation might increase the propensity of topdown cracking and instability rutting. PAGE 130 130 CHAPTER 5 3D FINITE ELEMENT MODELING OF TIRE PAVEMENT INTERACTION 5.1 Introduction In C hapter 4, the tire pavement interaction was modeled based on two dimensi onal (2D) finite element model (FEM). Although the 2 D tire pavement interaction model was simple and efficient, there are some limitations due to following reasons: 1) 2D model can only model one cross section of the whole geometry and cant capture the complexities associated with three dimensional ( 3D ) tire pavement interaction behavior; 2) using 2D planar element is proper for pavement but might not be reasonable for tire structure. Therefore, to overcome those limitations, it is necessary to develo p more sophisticated 3D model to further explore the mechanism of topdown cracking (TDC) and instability rutting. The primary objective of this chapter is to develop and evaluate a 3 D FEM based approach to predict tire pavement interaction, including ti re pavement interface stresses, using tire characteristics reported by tire manufacturers. The objective involved pursuit of the following tasks: Develop a 3 D FEM based model of a radial dual tire using tire geometry and structure information provided by tire manufacturer. Develop a 3 D FEM based model of the tire and pavement contact surfaces. Calibrate the 3 D tire model and tire pavement interface contact model using tire deflection data provided by tire manufacturer. Use the calibrated models to predi ct tire pavement interface contact stresses for multiple loads and inflation pressures. Evaluate the models by comparing predicted stresses to stresses measured from the actual radial truck tire. Develop 3D FEM for wide base tires including super single and new generation wide base (NGWB) tire based on developed methodology for establishment of 3D FEM tire and tire pavement interaction models. PAGE 131 131 5.2 3D Modeling of Tire pavement Interaction 5.2.1 The Tire to b e Modeled The tire to be modeled is Goodyear G149 RSA 11R22.5, as shown previously in Figure 423. The typical cross section for this kind of tires was also shown previously in Figure 418. The specifications of this tire were presented in Table 4 5. 5.2.2 Meshing of Tire Model Threedimensional (3 D) m odeling of a radial truck tire involves significant challenges because of the numerous components and different material properties involved (from rubber to steel). As shown previously in Figure 418, the main components involved are: radial ply, belts, tread and sidewall. Tire materials are basically composite and rubber compounds which vary throughout tire structure. However, exact material properties and actual structural make up of these tires used by industry are not available to the general public, which makes tire modeling even more challenging. However, some basic response data, including loaddeflection response and general structural make up, which are generally available, make the development of three dimensional tire model possible. A proper t ire model should accurately capture both geometrical characteristics and structural behavior. The structural behavior of radial truck tires is governed by a low moduli wall structure and a high moduli tread structure resulting from the steel reinforcement embedded in the tread (Roque et al. 1999). Based on the developed 2D tire model in C hapter 4, the 3D tire model was created by revolving the 2D model about the axis of symmetry The 2 D solid element was then converted to 3D solid element during model generation. To save computer resource and computing time, a relatively fine mesh was chosen around the tread zone while a relatively coarse mesh was created around the rim. Totally, the whole tire model has 10 element PAGE 132 132 groups and each element group represe nts different tire component associated with particular material property. Figures 51 to 53 show meshes of these element groups. 5.2.3 Meshing of TirePavement Contact Model A surface surface contact group between tire and pavement was created. The pavem ent was defined as rigid surface and treated as target contact surface while tire tread was defined as contactor surface. Figure 5 4 showed the contact group mesh. The whole tire pavement contact model was presented in Figure 5 5, which has eleven elemen t groups, one contact group with one target surface and one contactor surface. 5.2.4 Element Selection All 3 D solid elements were modeled as 8 node elements, with three translational degrees of freedom per node, as shown in Figure 56. The 8 node ele ment has a hexahedral shape, with one node at each of its 8 vertices. This type of node configuration has been shown to give a high level of accuracy in combination with an acceptable computing time demand. All elements are treated as isoparametric elemen t. The element matrices can be derived as follows: dV B C B KT] ][ [] [ ] [ (5 1) Where, ] [ C is a matrix of elastic coefficients given as : 2 2 1 0 0 0 0 0 0 2 2 1 0 0 0 0 0 0 2 2 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 ) 2 1 )( 1 ( ] [ v v v v v v v v v v v v v v E C (5 2) ] [ B is the matrix of strain displacement relations given by PAGE 133 133 Rim Belt1 Radial Ply Figure 5 1. Element group mesh and loading conditions PAGE 134 134 Sidewall Belt2 Skirt Tread Figure 5 2. Elem ent g roup m esh and l oading c onditions PAGE 135 135 Shoulder Tread Tread Rib Tread Grove Figure 5 3. Element g roup m esh PAGE 136 136 Contractor Surface Tire Tread Targe t Surface Pavement Surface Figure 5 4. Tire pavement contact group PAGE 137 137 Figure 5 5. Whole tire pavement model PAGE 138 138 z x y Figure 5 6. 8 node El ement ] [ ... ] [ ] [ ] [8 2 1B B B N B (5 3) And x N z N z N x N x N y N z N y N x N Bi i i i i i i i i i0 0 0 0 0 0 0 0 0 ] [ i =1 to 8 (5 4) N is shape function given by 8 3 2 2 1... 0 0 0 0 ... 0 0 0 0 ... 0 0 ] [ N N N N N N (5 5) And ); 1 )( 1 )( 1 ( 8 1 ) ,, (i i i itt ss rr t s r N i =1 to 8 (5 6) ri,si and ti are the coordinates of node under natural coordinate systems. PAGE 139 139 Assuming the variations of the displacements in between the nodes to be linear, the displac ement can be expressed by Q N w u w v u N z y x w z y x v z y x u ] [ ] [ ) ( ) ( ) (8 2 1 1 1 (5 7) Where Q is the vector of nodal displacement degrees of freedom and ) (i i iw v u denote the displacements of node i i = 1 to 8. The three dimensional strain displacement relations can be derived from Eq. (57) and expressed as Q B z u x w y w z v x v y u z w y v x uzx yz xy zz yy xx ] [ (5 8) Stress strain relations can be expressed as ] [ C (5 9) 5.2.5 Loading and Boundary Condition The tire pavement contact model can be simulated by two steps. First, the inflation pressure was applied on the inner surface of the tire model and different pre ssure levels were set to satisfy the requirements of the analysis. Second, a vertical load was applied on the rim axle PAGE 140 140 under given inflation pressure and the load was gradually loaded under assigned time steps. The detail loading conditions were shown in F igures 5 1 to 52.The boundary conditions for static analyses were also shown in Figure 5 5. It can be observed that the bottom of pavement was all fixed while sides of rim axle were restricted with all d egrees of freedom except for Z translation. 5.2.6 Mesh Convergence Analysis As stated earlier, t o optimize computer resource and computing time, fine mesh was chosen around the tread zone while coarse mesh was created around the rim. Sixty divisions were uniformly created along the tire circumference except for the tread zone, which was contact zone in the model. For the tread zone, a higher density mesh was chosen. To make sure the selected circular divisions for tread zone were accurate enough for analysis, a mesh convergence analysis was conducted. A convergence curve of maximum contact stress against circular divisions for tread zone was plotted, as shown in Figure 57. At the same time, the program running time against circular divisions for tread zone was also plotted, as shown in Figure 58. It can be observed that difference of maximum contact stress between 240 divisions and 300 divisions is within 5 %, while running time for 240 divisions is less than half of that for 300 divisions. Therefore, 240 divisions were selected for tread zone. The full tire pavement model had 46,800 elements in total as presented in Figure 55. 0 100 200 300 400 500 600 0 50 100 150 200 250 300 350 Tread zone circular divisions Max. Contact Stresses (Psi) Figure 5 7. Mesh convergence curve PAGE 141 141 0 100 200 300 400 500 60 120 180 240 300 Tread zone circular divisions Running time (Min.) Figure 5 8. Computer r unning time vs. circular divisions 5.3 Model Calibration Once the 3D tire pavement contact model was created, the next s tep was to calibrate model to determine values of material properties. The load deflection curves have been widely used for both numerical and experimental analysis (Meng, 2002) and were the primary means used for model calibration in this study The defle ction of tire is determined as displacement of tire axle after applying load on it. The calibration process is carried by the following procedures: 1) assigning initial moduli to different element groups; 2) doing sensitivity analysis to determine critical tire part that affects the tire deflection most; 3) adjusting moduli of critical tire part to get different tire deflection values; 4) comparing tire model deflections with experimental values to determine the moduli of the critical tire part; 5) validating model via contact stress distribution analysis. The initial stillness for different element groups is set based on previous tire models studied by other researchers (Myers and Roque 2000; Meng 2002; Zhang 2001; Shoop 2001). They are listed in Table 5 1. Considering the characteristics of tire structure, the main factors that affect tire deflection are radial ply, belt, sidewall and tread. Therefore, sensitivity analysis was focused on these PAGE 142 142 factors. The sensitivity analysis was performed by varying one parameter while maintaining other parameters constant. The relationship between each parameter and tire deflection was then plotted to evaluate the sensitivity of each parameter. The steeper curve, the higher degree of sensitivity to the changes of particul ar parameter. Table 5 1. Material properties for tire pavement model Element Group # Tire Parts Modulus (psi) E Possion Ratio, V 1 Rim 1.00E+12 0.3 2 Radi al Ply 6 .00E+0 3 0.4 3 Belt 1 1 .00E+0 4 0.4 4 Belt 2 3 .00E+0 6 0.4 5 Sidewall 1 .00E+02 0.495 6 Ski rt Tread 4 .00E+0 2 0.495 7 Shoulder 4 .00E+03 0.495 8 Tread 1 4 .00E+02 0.495 9 Tread 2 2 .00E+02 0.495 10 Grove 9.80E 06 0.499 11 Pavement 1.00E+12 0.4 Figures 5 9 to 514 show the sensitivity analysis results. All sensitivity analyses were conducted under 4 ,400 lbs normal axle load and 115 psi inflation pressure. From those figures, it clearly shows that radial ply has the most influence on tire deflection while tread and belt have little influence on it. Therefore, the calibration process will be focus ed on radial ply in this study A set of experimental deflection data was used for calibrating tire model. The deflection test was loaded at a range of normal axle load (0 to 9000 lbs) at three tire inflation pressure: 90 psi, 100 psi and 110 psi. The FEM for tire was also loaded under the same loading condition as experimental test. Considering the results of sensitivity analysis and characters of sidewall, the material properties from Table 5 1 were still used in FEM, except for radial ply whose elastic m odulus was changed at three levels: 3 ,000 psi, 5, 000 psi and 7,000 psi. In order to get tire deflection under different loading levels, two time functions were used. One was a constant time function; the other was a linear time function. The purpose of ti me function is to control how a load varies with time. The constant time function was applied to PAGE 143 143 inflation pressure, in other words, the inflation pressure will not change with time. The linear time function was applied to normal axle load. A total 9 ,000 l bs normal axle load was applied on the model under four time steps. The relationship between time function value and time step was presented in Figure 5 15. Figure 5 16 to Figure 5 18 show the comparisons of tire deflection between FEM and experimental te st at three inflation pressure levels with different radial ply moduli Good agreement was observed for a radial ply modulus of 5,000 psi. Therefore, 5,000 psi was set for radial ply modulus. Deflection contour maps under inflation pressure of 110 psi are presented in Figures 5 19 to 522. To further verify model, a comparison was made between contact stresses predicted by tire model and contact stresses measured under the real tire by Smithers Scientific Services, Inc. Figures 5 23 to 525 clearly indicated that both vertical and horizontal contact stresses predicted by FEM were quite similar to those measured under real tire. Although some variation in magnitude is observed, the tire FEM captures the trends of both vertical and horizontal contact stress di stributions. The models ability to capture the horizontal contact stress reversals under the individual tire ribs is particularly important. Further contact patch analysis will be given in the next section. PAGE 144 144 0.4 0.5 0.6 0.7 0.8 0.9 5.0E+03 6.0E+03 7.0E+03 8.0E+03 9.0E+03 1.0E+04 Modulus of Radial Ply (psi) Tire Deflection (in) Figure 5 9. Sensitivity analysis: effec ts of radial ply 0.4 0.5 0.6 0.7 0.8 0.9 1.0E+02 2.0E+02 3.0E+02 4.0E+02 5.0E+02 Modulus of Sidewall (psi) Tire Deflection (in) Figure 5 10. Sensitivity analysis: effects of sidewall PAGE 145 145 0.4 0.5 0.6 0.7 0.8 0.9 0.0E+00 2.0E+05 4.0E+05 6.0E+05 8.0E+05 1.0E+06 Modulus of Belt 1 (psi) Tire Deflection (in) Figure 5 11. Sensitivity analysis: effects of belt 1 0.4 0.5 0.6 0.7 0.8 0.9 1.0E+06 1.5E+06 2.0E+06 2.5E+06 3.0E+06 Modulus of Belt 2 (psi) Tire Deflection (in) Figure 5 12. Sensitivity analysis: effects of belt 2 PAGE 146 146 0.4 0.5 0.6 0.7 0.8 0.9 1.0E+02 2.0E+02 3.0E+02 4.0E+02 5.0E+02 Modulus of Tread (psi) Tire Deflection (in) Figure 5 13. Sensitivity analysis: effects of tread 1 0.4 0.5 0.6 0.7 0.8 0.9 4.0E+02 4.5E+02 5.0E+02 5.5E+02 6.0E+02 Modulus of Rib Tread (psi) Tire Deflection (in) Figure 5 14. Sensitivity analysis: effects of tread 2 PAGE 147 147 0 0.25 0.5 0.75 1 0 1 2 3 4 Time Step Time Function Value Figure 5 15. Time function vs. time step for normal axle load 0 2000 4000 6000 8000 10000 0.000 0.500 1.000 1.500 2.000 2.500 Tire Deflection (in) Normal Load (lbs) Test Model3k Model5k Model7k Figure 516. Load deflection c urves for t est and m odel (90 psi) 0 2000 4000 6000 8000 10000 0.000 0.500 1.000 1.500 2.000 2.500 Tire Deflection (in) Normal Load (lbs) Test Model3k Model5k Model7k Figure 517. Load deflection curves for test and model (100 ps i) PAGE 148 148 0 2000 4000 6000 8000 10000 0.000 0.500 1.000 1.500 2.000 2.500 Tire Deflection (in) Normal Load (lbs) Test Model3k Model5k Model7k Figure 518. Load deflection curves for test and model (110 psi) Figure 5 19. Z displacement contours at vertical axle load of 2,250 lbs PAGE 149 149 Figure 5 20. Z displacement contours at vertical axle load of 4,500 lbs Figure 5 21. Z displacement contours at vertical axle load of 6,750 lbs PAGE 150 150 Figure 5 22. Z displacement contours at vertical axle load of 9,000 lbs Lateral Vertical Contact Stress Distributions ( Middle Plane) 0.00 50.00 100.00 150.00 200.00 250.00 6.00 4.00 2.00 0.00 2.00 4.00 6.00 Tirewidth (in) Vertical Stress (psi) Test FEM Poly. (FEM) Poly. (Test) Figure 523. Comparisons of m easured and predicted vertical c ontact s tress across tire width PAGE 151 151 Transverse Contact Stress Distributions (Middle Plane) 80.00 60.00 40.00 20.00 0.00 20.00 40.00 60.00 80.00 6.00 4.00 2.00 0.00 2.00 4.00 6.00 Transverse Distance (in) Transverse Contact Stress (psi) Test FEM Figure 524. Comparisons of m eas ured and predicted t ransverse c ontact s tress across tire width Longitudinal Contact Stress Distributions (Middle Plane) 15 10 5 0 5 10 15 4 3 2 1 0 1 2 3 4 Longitudinal Distance (in) Longitudinal Contact Stress (psi) Test FEM Figure 525. Comparisons of m easured and predicted longitudinal c ontact s tress PAGE 152 152 5.4 Contact Patch Analysis Figures 5 26 to 528 show the typical vertical contact stress distributions in con tour map and footprint shapes of tire under normal loads ranging from 2,250 lbs to 6 ,750 lbs at inflation pressure of 110 psi. The legends in the figures show the contour map representing different ranges of the averaged values of the nodal contact stresse s in psi. The results indicate that the contact area increases with normal load under given inflation pressure condition. The three dimensional contact stress fields, generated from the average values of stress acting on the surface nodes under normal loa ds ranging from 2,250 lbs to 6 ,750 lbs at inflation pressures of 110 psi, are given in Figures 529 to 537. It clearly shows that maximum vertical contact stress was located at the center of tire in all cases. However, the maximum vertical contact stress will gradually shift from inner ribs to outer ribs when the tire is heavily loaded, as demonstrated in Figures 5 38 to 541. This agrees well with other studies (M. de Beer et al. 1997; Zhang, 2001; Meng, 2002). Lateral contact stress diagrams clearly in dicate a phenomenon referred to as the Poisson effect that has been pointed out by other researchers (Roque et al. 1998; M.G.Pottinger et al. 1999). The individual ribs attempt to expand laterally when the tire is loaded. However, transverse stresses are produced as the pavement surface restrains the expansion, and thus transverse stresses are induced in the pavement surface which tries to pull the pavement apart underneath each individual rib. It should be noted although the pneumatic effect exists to re duce tensile stresses under ribs, the Poisson effect dominates transverse contact stress due to structural characters of radial tire (Myers et al, 1999). Longitudinal contact stress is generally less than lateral contact stress and shows antisymmetry under static loading condition, which also agrees well those predicted by other studies (Clark 1981; Tielking et al. 1987; Lippmann et al. 1974). PAGE 153 153 Some important predicted data are summarized in table 5 2. It can be seen that contact stresses increase with the normal load and inflation pressure. The ratio among vertical, lateral and longitudinal contact stresses is about 1: 0.3:0.2, similar to findings by de Beer (1997) and Tielking (1987). Figure 5 26. Vertical contact stress contour map (110 psi; 2,250 lbs ) Figure 5 27. Vertical contact stress contour map (110 psi; 4,500 lbs) PAGE 154 154 Figure 5 28. Vertical contact stress contour map (110 psi; 6,750 lbs) 4 2 0 2 4 5 0 5 0 50 100 150 200 250 300 Longitudinal (in) 110 psi; 6750 lbs Lateral (in) Vertical Contact Stress (psi) Figure 5 29. 3 D Vertical contact stress distribution (110 psi; 6 ,750 lbs) PAGE 155 155 2 1 0 1 2 5 0 5 0 50 100 150 200 250 300 Longitudinal (in) 110 psi; 4500 lbs Lateral (in) Vertical Contact Stress (psi) Figure 5 30. 3 D Vertical contact stress distribution (110 psi; 4,500 lbs) 2 1 0 1 2 5 0 5 0 50 100 150 200 250 300 Longitudinal (in) 110 psi; 2250 lbs Lateral (in) Vertical Contact Stress (psi) Figure 5 31. 3 D Vertical contact stress distribution (110 psi; 2 ,250 lbs) PAGE 156 156 4 2 0 2 4 5 0 5 100 50 0 50 100 Longitudinal (in) 110 psi; 6750 lbs Lateral (in) Lateral Contact Stress (psi) Figure 5 32. 3 D Lateral contact stress distribution (110 psi; 6,750 lbs) 2 1 0 1 2 5 0 5 100 50 0 50 100 Longitudinal (in) 110 psi; 4500 lbs Lateral (in) Lateral Contact Stress (psi) Figure 5 33. 3 D Lateral contact stress di stribution (110 psi; 4,500 lbs) PAGE 157 157 2 1 0 1 2 5 0 5 100 50 0 50 100 Longitudinal (in) 110 psi; 2250 lbs Lateral (in) Lateral Contact Stress (psi) Figure 5 34. 3 D Lateral contact stress distribution (110 psi; 2,250 lbs) 4 2 0 2 4 5 0 5 60 40 20 0 20 40 60 Longitudinal (in) 110 psi; 6750 lbs Lateral (in) Longitudinal Contact Stress (psi) Figure 5 35. 3 D Longitudinal contact stress distribution (110 psi; 6,750 lbs) PAGE 158 158 2 1 0 1 2 5 0 5 60 40 20 0 20 40 60 Longitudinal (in) 110 psi; 4500 lbs Lateral (in) Longitudinal Contact Stress (psi) Figure 5 36. 3 D Longitudinal contact stress distribution (110 ps i; 4 ,500 lbs) 2 1 0 1 2 5 0 5 60 40 20 0 20 40 60 Longitudinal (in) 110 psi; 2250 lbs Lateral (in) Longitudinal Contact Stress (psi) Figure 5 37. 3 D Longitudinal contact stress distribution (110 psi; 2,250 lbs) PAGE 159 159 Figure 538. Vertical c ontact s tress c ontour along m id plane depth (6,000 lbs) Figure 539. Vertical c ontact s tress c ontour along m id plane depth (9,000 lbs) Figure 540. Vertical c ontact s tress c ontour along m id plane depth ( 13,500 lbs) PAGE 160 160 Figure 541. Vertical c ontact s tress c ontour along m id plane depth (18,000 lbs ) Table 5 2. Summary of 3 D contact stresses (11R22.5) Norma l Load (lbs) Inflation Pr essure (psi) Contact Area (in^2) Maximum Contact Stress (psi) Ratio of Maximum Stress Stress zz Stress yz Stress xz Z Y X 6750 110 48.33 274.58 82.50 56.79 1.00 0.30 0.21 90 48.33 267.31 79.15 54.72 1.00 0.30 0.20 70 48.33 261.09 71.71 53.12 1.00 0.27 0.20 4500 110 32.22 226.20 66.50 47.60 1.00 0.29 0.21 90 32.22 218.75 63.20 45.56 1.00 0.29 0.21 70 32.22 211.89 56.07 43.81 1.00 0.26 0.21 2250 110 22.69 146.62 42.66 32.04 1.00 0.29 0.22 90 22.69 139.66 39.51 30.20 1.00 0.28 0.22 70 22.69 132.72 32.29 28.40 1.00 0.24 0.21 5.5 Modeling of Wide base Tires 3D finite element models for wide base tires (super single 425/65R22.5 and new generation wide base 445/50R22.5) were developed in this section based on methodologies developed for modeling dual 11R22.5 in previous sections, as shown in Figures 542 and 5 43 respectively The load displacement measurements under different loading levels at various inflation pressures were first got from FDOT, which were then used for model calibration to determine material properties of the tire. The detail load displacement measurements are presented in Table 5 3. The material properties for calibrated tire models were given in Table 5 4. Figures 544 and 545 show the comparisons of loaddisplacement curv es between measured and calibrated models. Typical z displacement contours maps for 425/65R22.5 and 445/50R22.5 PAGE 161 161 were given in Figures 5 46 and 547, respectively. The three dimensional contact stress field generated based on calibrated models were shown i n Figures 5 48 to 553. Although, there are no detail contact stress measurements for wide based tires especially for new generation wide base tires, the magnitude and characteristics of the predicted contact stress agree well with other studies (Markstall er et al. 2000; Al Qadi et al. 2005). Figure 5 42. 3 D model for 425/65R22.5 Figure 5 43. 3 D model for 445/50R22.5 PAGE 162 162 Table 5 3. Vertical tire axle displacement Tire Inflation Pressure, psi Vertical Tire Deformation, inch 9 kip 12 kip 15 kip 18 kip 425/65R22.5 80 1.91 2.46 3.78 4.18 425/65R22.5 115 1.45 1.90 2.26 3.08 425/65R22.5 125 1.41 1.81 2.18 2.68 445/50R22.5 80 1.62 2.17 2.65 NA 445/50R22.5 100 1.38 1.83 2.23 NA 445/50R22.5 125 1.18 1.58 1.95 2.46 Table 54. Material properties fo r t ire Model s 425/65R22.5 445/50R22.5 Tire Parts Modulus (psi) Poisson Ratio, V Modulus (psi) Poisson Ratio, V Rim 1.00E+12 0.1 1.00E+12 0.1 Radial Ply 2.00E+03 0.3 1.00E+03 0.3 Belt 2.00E+06 0.2 2.00E+06 0.2 Sidewall 6.00E+02 0.495 6.00E+02 0.495 S kirt Tread 6.00E+02 0.495 6.00E+02 0.495 Shoulder 6.00E+02 0.495 6.00E+02 0.495 Tread 5.00E+02 0.495 5.00E+02 0.495 Grove 9.80E 06 0.499 9.80E 06 0.499 Loaddisplacement curve for 425/65R22.5 0 0.5 1 1.5 2 2.5 6 9 12 15 18 Axle Normal Load (kips) Zdisplacement (in) Test FEM Figure 5 44. Comparisons of loaddisplacement curves for 425/65R22.5 PAGE 163 163 Loaddisplacement curve for 445/50R22.5 0 0.51 1.5 2 2.5 6 912 15 18 Axle Normal Load (kips) Zdisplacement (in) Test FEM Figure 5 45. Comparisons of loaddisplacement curves for 445/50R22.5 Figure 5 46. Z displacement contours for 425/65R22.5 (12 kips, 115 psi) PAGE 164 164 Figure 5 47. Z displacement contours for 455/65R22.5 (12 kips, 100 psi) 4 2 0 2 4 10 5 0 5 10 50 0 50 100 150 200 Longitudinal (in) Lateral (in) Vertical Contact Stress (psi) Figure 5 48. 3 D Vertical contact stresses for 445/50 R22.5 (100 psi; 9,000 lbs) PAGE 165 165 4 2 0 2 4 10 5 0 5 10 40 20 0 20 40 Longitudinal (in) Lateral (in) Lateral Contact Stress (psi) Figure 5 49. 3 D Lateral contact stresses for 445/50R22.5 (100 psi; 9,000 lbs) 4 2 0 2 4 10 5 0 5 10 30 20 10 0 10 20 30 Longitudinal (in) Lateral (in) Longitudinal Contact Stress (psi) Figure 5 50. 3 D Longitudinal contact stresses for 445/50R22.5 (100 psi; 9, 000 lbs) PAGE 166 166 4 2 0 2 4 10 5 0 5 10 50 0 50 100 150 200 250 Longitudinal (in) Lateral (in) Vertical Contact Stress (psi) Figure 5 51. 3 D Vertical contact stresses for 425/65R22.5 (1 15 psi; 9 ,000 lbs) 4 2 0 2 4 10 5 0 5 10 50 0 50 Longitudinal (in) Lateral (in) Lateral Contact Stress (psi) Figure 5 52. 3 D Lateral contact stresses for 425/65R22.5 (1 15 psi; 9 ,000 lbs) PAGE 167 167 4 2 0 2 4 10 5 0 5 10 40 20 0 20 40 Longitudinal (in) Lateral (in) Longitudinal Contact Stress (psi) Figure 5 53. 3 D Longitudinal contact stresses for 425/65R22.5 (1 15 psi; 9 000 lbs) 5.6 Summary and Conclusion A 3 D FEM based tire pavement int er action model for dual 11R 22.5 was developed using tire geometry and structure information provided by the tire manufacturer. The model was calibrated based on tiredeflection data and validated with measured contact stresses from the actual radial truck tire. Excellent correspondence between predicted and measured contact stresses was observed. Some important findings are summarized as follows: Sensitivity analyses clearly show that both radial ply and sidewall moduli greatly influenced tire deflection, w hile tread and belt moduli had relatively little influence. Evaluation of predicted responses agreed well with expected trends: Contact area increases with the tire load under given inflation pressure. The maximum vertical contact stress was located at the center of tire when the tire load was small. However, it would be gradually shifted to out ribs when the tire was heavily loaded. PAGE 168 168 Lateral contact stress diagrams clearly indicate a phenomenon referred to as the Poisson effect that dominates later al contact stress distribution. Longitudinal contact stress was generally less than lateral contact stress and shows antisymmetry under static loading condition. The ratio of vertical, lateral and longitudinal contact stress was about 1:0.3:0.2. Based on methodolog ies developed for modeling dual 11R22.5, 3D finite element models for wide base tires (super single 425/65R22.5 and new generation 445/50R22.5) were also developed. The magnitude and characteristics of contact stresses agrees well with other studies (Mark staller et al. 2000; Al Qadi et al. 2005). Therefore it might be concluded that geometry, structural characteristics and load deflection data reported by tire manufacturers can be used to develop 3D finite element models to predict tire pavement interface stresses and their effects on pavement performance. This can be of significant benefit for further studies on how super wide based tires affect pavement analysis, design and performance. PAGE 169 169 CHAPTER 6 INVESTIGATE NEAR SURFACE STRESS STATE S BASED ON 3 D TI RE PAVEMENT CONTACT MODEL 6.1 Introduction Top down cracking (TDC) and instability rutting are two key forms of failure observed at the surface of the AC pavements. Both are near surface phenomenon and greatly affected by tirepavement interface stresses. Studies by de Beer et al (1997) and Myers et al (1999) have shown that tire pavement interface stresses are significantly influenced by tire structure characteristics. Roque et al (1998) found that Poissons effect dominates radial truck tires while Pne umatic effect dominates bias ply truck tires. However, for specific tire such as radial type, how the material properties affecting tire pavement interface stress distributions is till unknown, neither scholars have conducted such kinds of studies. On the other hand, traditional methods of pavement analysis assume that tire pavement interface stress is vertical, circular and uniform. However, tire pavement interface stresses measured by Pottinger (1992) and de Beer et al (1997 and 1999) indicated that the distributions of contact stresses are threedimensional (3 D), nonuniform and noncircular. The effects of these complex tire pavement interface stresses on pavement response have not been widely analyzed. Roque et al. (2000) and Myers et al. (2001) conducted a series of two dimensional (2D) finite element analyses where a 2 D cross section of measured tire pavement interface stresses was applied on a layered half plane. The results indicate that the predicted stress state was significantly different from that of a uniform vertical strip load. Similar results were also reported based on 3D layered elastic theory solutions. Those results were based on a tire pavement contact measurement system developed by Pottinger (1992), which consisted of 1,200 distinc t measurement points recording contact stresses in three directions, namely x, y, and z direction. Marc Novak et al. (2004) employed the finite element code ADINA to identify the PAGE 170 170 threedimensional stress states in a typical flexible pavement configuration using the measured radial tire contact stresses. The predictions show that measured radial tire contact stresses result in stress states being both larger in magnitude and more focused near the surface than those obtained from traditional uniform vertical loading conditions. In addition, other scholars such as Jacob (1995), Groenendijk (1998), Jill M. Holewinski (2003), etc. did similar studies either using analytical programs such as BISAR or finite element codes such as ABAQUS. They all applied measured contact stresses on the models while neglecting the influence of tire structures on the pavement. To date, no detailed studies have been conducted regarding placing a 3D tire model on the pavement. Applying measured contact stresses on the pavement rather than placing a real tire model on the pavement has following limitations: Hard to convert the measured contact stresses to the input load format that finite element models or analytical approaches require, especially for transverse and longitudinal shear contact stresses. Neglecting tire pavement interface contact condition. Neglecting interaction of axle load and inflation pressure. Neglecting the influence of tire structural characteristics on the pavement. Applying load one time one condition only. On t he contrary, placing a 3 D tire model on the pavement has following advantages: Dont need to consider load transfer. Easily define tire pavement interface contact condition. Fully consider interaction of axle load and inflation pressure. Fully consider the influence of tire structural characteristics on the pavement. Easily adjust loading conditions. Therefore, in this chapter, a 3D dual tire model was first placed on a 3 layer pavement system for illustration and verification purpose. The wide base tire models were then placed on the pavement to investigate the effects of tire types on near surface stress states. The 3 D tire PAGE 171 171 models used here were developed from Chapter 5. A contact group between tire and pavement was defined. Although flexible pavement m aterials are generally non linear in nature, a firm understanding of the linear elastic stress states should precede any further analysis. All pavement layers are assumed to be linear elastic, and dynamic effects are ignored in favor of promoting a basic u nderstanding of static stress states before complicating the analysis with dynamic effects. The main tasks are: Develop and verify 3D tire pavement interaction model using ADINA. Investigate near surface stress states based on verified 3 D tire pavement i nteraction model Evaluate effects of types on near surface stress states Conduct parametric study tire structure. Conduct parametric study pavement structure. Conduct parametric study loading conditions. 6.2 Development of 3D Tirepavement Interaction Mod el 6.2.1 Pavement Structure Information A typical three layer pavement system, namely asphalt concrete, base and subgrade, was used in the analysis. This system was previously used by Myers (2000) and Drakos (2003). The thickness of asphalt concrete layer was 8 in., overlying a 12 in. thick base. The subgrade was assumed to be 50 in. thick. The properties of each layer were defined as isotropic, homogenous and linear elastic material. Table 6 1 presents the elastic moduli, Poissons ratio and layer thicknes s for each layer of the pavement. The elastic modulus of asphalt concrete corresponds to cold winter day, and the base and subgrade values were chosen based on typical measured values in the State of Florida (Drakos et al., 2001; Myers et al., 2001). Table 61. Material properties and layer thickness of the pavement Pavement Layer Elastic Modulus (psi) Poissons Ratio Thickness (in.) Asphalt Concrete 800,000 0.40 8 Base 40,000 0.35 12 Subgrade 20,000 0.35 50 PAGE 172 172 6.2.2 Element Mesh A part of the challenge wi th this finite element modeling is how to properly determine model size and element mesh of the pavement system. An initial assessment of the model size based on uniform pressure loading condition indicated that the 3D pavement system should be at least 6 0 in. in vertical direction and extend horizontally at least 60 in. in each direction from the center of tire contact load to adequately represent the semi infinite half space conditions associated with pavement problems. Further assessment showed that st resses near the footprint (area of interest) were not affected by the extent of the boundaries. Therefore, a dimension of 120 in. 120 in. 0 in. in x, y and z direction was chosen for the pavement model. The x axis represents the tire travel direction an d y axis represents the transverse direction. To optimize computer resource and save program running time, a finer mesh was chosen near tire footprint while coarse mesh was created far away from the footprint. The resolution of the mesh was gradually decre ased in three dimensions, namely x, y and z direction, from the loading center. However, in order to capture characteristics of tire tread pattern, the mesh of pavement contact zone under the tire must be fine enough to match the element size of the tire t read zone whose element dimensions are 0.25 in. x 0.52 in. x 0.1 in. in x, y and z direction respectively. Elements in top 4 in. pavement layers were modeled as 8node 3D solid element, while the rest elements were modeled as 4 node 3 D solid element. All nodes have three translational degrees of freedom. A contact group between tire tread and pavement section under the tire was defined, as shown in Figure 61. Figure 62 presents the final 3 D mesh and Figure 63 shows the plan view of the pavement surfac e. The boundary conditions for the four sides (faces) of the FEM were fixed in the horizontal (X and Y) direction and free in the vertical (Z) direction, whereas the bottom of the FEM was fixed in all directions. The model consisted of 68,751 nodes with 206,253 degrees of freedom. PAGE 173 173 The memory required for analysis exceeded 1 ,200 megabytes and took over 2 hours to complete a single run of the program. Figure 6 1. 3 D Tire pavement contact group mesh Figure 6 2. 3 D Tire pavement contact finite element mesh PAGE 174 174 F igure 63. Plan view of the pavement surface 6.2.3 Model Verification In order to further evaluate the potential of the 3D tire pavement contact model, a comparison was made between predicted contact stresses and measured contact stresses. Predicted contact stresses were obtained at the nodes of the pavement surface under the tire, and measured stresses were provided by Smithers Scientific Services, Inc. for comparison purpose. Figures 6 4 to 66 show comparisons of contact stresses between measured and predicted at tire mid plane, which clearly indicate that predicted vertical and horizontal shear contact stresses are similar to those measured under the real tire, except for some variation in magnitude. The variation might be caused by different loa ding conditions (FEM was running under static load, while the measurement was conducted under moving steel bed), tread groves (FEM didnt consider longitudinal grove) and element mesh. The overall errors are within 20 %. And the most important thing is th e models ability to capture the patterns of both vertical contact stress PAGE 175 175 and horizontal shear contact stress distributions. Figure 67 shows predicted tire footprint, whose shape is pretty similar to the measured one. The predicted contact area was about 48 in2 under normal load 4,500 lbf at 110 psi inflation pressures, which is very close to the actual measured contact area 47 in2 under the same loading condition. The deformed pavement surface was presented in Figure 6 8. 250 200 150 100 50 0 50 6.00 4.00 2.00 0.00 2.00 4.00 6.00 Tire Transverse Distance (in.) Vertical Contact Stress (psi) Measured Predicted Figure 6 4. Comparisons of meas ured and predicted vertical contact stresses 80 60 40 20 0 20 40 60 6.00 4.00 2.00 0.00 2.00 4.00 6.00 Tire Transverse Distance (in) Transverse Contact Stress (psi) Measured Predicted Figure 6 5. Comparisons of measured and predicted transverse contact stresses PAGE 176 176 15 10 5 0 5 10 15 6.00 4.00 2.00 0.00 2.00 4.00 6.00 Tire Longitudinal Distance (in.) Longitudinal Contact Stress (psi) Measured Predicted Figure 6 6. Comparisons of measured and predicted longitudinal contact stresses Figure 6 7. Predicted tire footprints PAGE 177 177 F i gure 68. Deformed pavement surface 6.3 Predicted Nearsurface Stress States As stated earl ier both topdown cracking and instability rutting are near surface phenomenon, affecting only top 13 in. of the asphalt concrete layer (Dawley et al., 1990; Drako s, 2003). Clearly, near surface distresses are greatly associated with near surface stress states. And identifying near surface states will help to under mechanism of near surface distresses related to top down cracking and instability rutting. Therefore, in the following, near surface vertical stresses, horizontal stresses, shear stresses, principal stresses, and confining stresses will be presented and evaluated. Those stresses were obtained under loading condition of 4,500 lbf normal load with 110 psi tire inflation. For evaluation purpose, a comparison was performed between models with tire and without tire under the same pavement structure and normal load. For the model without tire, a uniform vertical pressure was applied on a contact square area nearl y equal to 47 in2. with 4,500 lbf in total, as shown in Figure 69. PAGE 178 178 Figure 6 9. 3 D model without tire 6.3.1 Vertical Stress States Figure s 610 and 611 show the near surface vertical stress contours from the tire pavement contact model and uniform ver tical load case, respectively. The contour plots are obtained under the tire at middle yz plane for the tire pavement contact model, and the corresponding location for the uniform vertical load case The tire pavement contact model produces new surface ver tical stress es that are higher in magnitude than those produced by the uniform vertical load. The magnitude of the maximum vertical stress for tire pavement contact model was approximately two times more than that for uniform vertical load. The stress dist ributions for tire pavement contact model are much more nonuniform across tire cross section than those for uniform vertical load. For the tire pavement contact model, t he vertical stresses under each rib are both higher in magnitude and more intense in t he top 0.5 in. asphalt layer but dissipate quickly along the depth. The plot also shows the effects of transverse groves PAGE 179 179 on the distributions of stress, i.e., the stresses between ribs are almost zero but extremely high at the edge of each grove, which mig ht develop high shear stresses there And these high shear stresses developed at groves might be responsible for near surface distresses such as top down cracking and instability rutting. Figure 6 10. Line contours of vertical contact stress for the tire pavement model Figure 6 11. Line contours of vertical contact stress for the uniform vertical load 6.3.2 Shear Stress States max, is defined as the difference between the first and third (largest and smallest principal stresses in magnitude, respectively) divided by two: max 1 3)/2 Studies by Novak (2004) showed that maximum shear stress occurs within transverse plane yz plane, which suggests that the most critical plane for investigating instability rutting is the transverse plane adjacent to the tire loading center. The maximum shear stress distributions under the tire at middle yz plane fo r tire pavement contact model and uniform model are given in Figure s 612 and 613 respectively, with directional arrows showing direction of the smallest PAGE 180 180 angel formed between the maximum shear stress and horizontal plane The sign convention and the metho d for calculating angle are presented in Figure 6 14. F igure 612. Maximum shear stress e (in psi) distributions for uniform load F igure 613. Maximum shear stress (in psi) distributions for tire pavement As expected, the high shear stresses develop at the tire edges for both models For tire pavement contact model, however, additional high shear stresses develop at the groves and concentrate on the top 0.5 in. asphalt layer. Again, the tire pavement contact model produces much higher shear stres ses than uniform vertical load and the maximum value for tire pavement contact model is almost double than that for uniform vertical load. At the tire edge, the shear stress increases initially with depth, reaching the maximum value at a depth near 0.5 in., after which it decreases from the peak value rapidly, as shown in Figure 615. This agrees well with studies by Novak et al. (2004) and Su et al. (2008). 6.3.3 Principal Stress States Principal stress refers to the magnitudes of stress that occur on cert ain planes on which no shear stresses exist. To be consistent with ADINA sign convention, here positive principal stress PAGE 181 181 is defined as tension while negative principal stress is defined as compression. As shown in Figure 6 16, high shear stress might drive compressive stress state into tension. zz () zz () yy (+) yy (+) zy () yz (+) yz (+) zy () yy, yz zz, zy Pole 13 maxmax () Horizontal Plane Tire Load Element Tire Load Shear DirectionWhen yz < 0, yy < zz F igure 614. Schematic of sign convention and maximum shear stress direction 0.00 0.50 1.00 1.50 2.00 2.50 0.00 10.00 20.00 30.00 40.00 50.00 60.00 Maximum Shear Stress (psi.) Depth (in.) TirePavement Uniform Load F igure 615. Maximum shear stress distribution as a function of depth at tire edge PAGE 182 182 F igure 616. Mohrs circle Figures 6 17 and 618 show line contours of principal stress under the tire at middle yz plane for tire pavement contact model and uniform vertical load model, respectively. Comparing with tire pavement model, stress distributions for uniform vertical load are more uniform and less magnitude. There is no tensile stress occurring near surface for uniform load. However, for tire pavement model, tensile stress was produced due to high shear st ress at tire groves. The maximum princip al 1 could reach more than 30 psi, which might be a critical factor when evaluating top down cracking. F igure 617. Line Contours of principal stress 1 for the tire pavement model F igure 618. Line Cont ours of principal stress 1 for the uniform vertical load PAGE 183 183 Since high shear stress also develops at the edge of tire, a typical maximum principal stress distribution along lateral distance away from tire edge is given in Figure 6 19. It was found that stre ss reaches highest at locations about 0.30.5 in. away from tire edge for both models. And stress for tire pavement model was much higher in magnitude than that for uniform vertical load. A further investigation of stress states revealed that the critical location for maximum principal stress was located at surface about 0.30.5 in. away from the tire loading, as shown in Figure 620. 60.0 50.0 40.0 30.0 20.0 10.0 0.0 10.0 20.0 30.0 40.0 0.00 0.50 1.00 1.50 Distance to tire edge (in.) SIGMA1 (psi) Uniform Load TirePavement F igure 619. Maximum principal stress distribution along lateral distance to tire edge 0.30.5" Rib Rib Rib F igu re 620. Critical stress locations Stress analyses via Mohrs circles at the critical location for both models were presented in Figures 6 21. It clearly shows that the normal and shear stresses induced by tires are driving the PAGE 184 184 stress state into tensile planes comparing with the uniform load. The figures also show that the maximum principal 1 acts on a plane that is less than 45 with the horizontal axle. These conclusions are quite insistent with studies by Novak et al. (2004). 60 40 20 0 20 40 60 60 40 20 0 20 40 60 TirePavement SIGMAP1 Plane Uniform Load SIGMAP1 Plane 3=29 psi 1=9.4 psi 1=27.5 psi 3=38 psi F igure 621. Stress states at critical location 6.3.4 Mohr Coulomb Envelops and pq Space It is wellknown that asphalt mixtures are pressure dependent materials and often been modeled as Mohr Coulomb materials (Mcleod, N.W, 1950; Krutz et al. 1990). Mohr Coulomb envelop can be obtained based on asphalt mixture strength test at different levels of confine ment, as shown in Figure 6 22. Asphalt mixtures with instability rutting were believed to behave under low confinement levels (Novak et al., 2004). Though Mohr circle can clearly describe stress states at particular location, it is hard to plot Mohr circle s for different locations in a single diagram. To overcome such a disadvantage, an alternative scheme called p q space is used here to plot different stress points. The pq space is defined by (Lee W. Abramson et al., 2001): PAGE 185 185 23 1 p (6 1) 23 1 q (6 2) Correspondingly, the classical Mohr Coulomb failure envelope with c and strength parameters can be plotted in p q space using following equations: tan p a q (6 3) Where, cos c a and sin tan (6 4) Using p q space method, a range of stress states that occur at points within middle transverse yz plane under the loading center are presented in Figures 6 23 to 624, with about 2 in. off the tire edges laterall y and 2 in. depth to the surface. Two different angles of internal friction, namely 20 and 40, with two different cohesion strengths, namely 20 psi and 30 psi, were used here to define different Mohr Coulomb envelopes. The figures clearly show that tire pavement contact model producing stress states that are much closer to the Mohr Coulomb failure envelope than a uniform vertical loading. And asphalt mixtures with high cohesion strength and internal friction angle may help to resist instability rutting an d topdown cracking. c f3f2f1 3 2 1 F igure 622. Mohr Coulomb envelop PAGE 186 186 0 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 p (psi) q (psi) TirePavement Uniform 30C40D 30C20D F igure 623. Stress states in p q space with Mohr Coulomb failure envelopes for 30 psi cohesion and different internal friction angles 0 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 p (psi) q (psi) TirePavement Uniform 20C40D 20C20D F igure 624. Stress states i n pq space with Mohr Coulomb failure envelopes for 20 psi cohesion and different internal friction angles 6.3.5 Location of Critical Stress Points To locate stress points close to failure envelops in pq diagram, the top 2 in. AC layer was divided into se veral zones as shown in Figure 6 2 5. Since the model was symmetric, only half PAGE 187 187 AC layer was used here. Pavements at tire edge, under rib and between ribs were classified as three main zones, each with three layers. Stress points from those zones were picke d and plotted in pq diagrams again. A1 A2 A3 A4 A5 A6 A7 A8 A9 A4 A5 A6 A7 A8 A9 A4 A5 A6 2" 2" 1.5" 1" 0.5" 1" Rib Grove Figure 6 25. Divided pavement zones for locating critical stress points purpose pq disributions for radial tire case 0 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 p (psi) q (psi) A1 A2 A3 A4 A5 A6 A7 A8 A9 20C40D 20C20D Figure 6 26. Locations of stress points in pq diagram for radial tire model PAGE 188 188 pq distributions for uniform load case 0 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 p (psi) q (psi) A1 A2 A3 A4 A5 A6 A7 A8 A9 20C40D 20C20D Figure 6 27. Locations of stress points in pq diagram for uniform load model Figures 6 26 and 627 show the locations of stress points in pq diagram for radial tire model and uniform load model, respectively. It can be observed that, for both models, stress points from tire/loa ding edges and bottom of top 2 in. AC layer are much closer to Mohr Coulomb failure envelopes than stress points from other zones. The critical stress points which were close to Mohr Coulomb failure envelopes were plotted in Figure 628 Based on these poi nts, a curve was fitted as shown in Figure 6 28, which clearly indicated that there might be a shear failure plane developed in the top 2 in. AC layer. The approximate direction of maximum shear stress along the fitted curve was presented in Figure 6 29, w hich clearly shows there was a shear flow developed. This shear flow might be responsible for topdown cracking and instability rutting. PAGE 189 189 y = 0.078x2 0.0678x + 2.0756 R2 = 0.8547 0.50 0.00 0.50 1.00 1.50 2.00 2.50 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00 Tire Lateral Locations (in.) Distance to Surface (in.) Critcal Points Fitted Curve Figure 6 28. Locations of critical stress points and their fitted curve 0.50 0.00 0.50 1.00 1.50 2.00 2.50 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00 Tire Lateral Locations (in) Distance to Surface (in) Figure 6 29. Direction of maximum shea r stress along fitted curve PAGE 190 190 6.3.6 Yield Percentage To determine the degree of criticality of certain stress state, the concept of yield percentage was used here. The yield percentage can be defined as shear stress at certain stress state divided by yielde d shear stress in terms of Mohr Coulomb criterion. Mohr Coulomb criterion assumes that material will fail or yield if the shear stresses in the material exceed the yield shear stress of the material. As illustrated in Figure 6 30, the yield percentage can be calculated as follows ( Jacob Groenendijk, 1998): cos 2 sin ) (3 1 3 1c ST SR MQ MR YDT R (6 5) Where: : angle of internal friction of material. c: cohesion strength of material 13 c M S R T Q Figure 6 30. Illustration of yield percentage ( Jacob Groenendijk, 1998) A yield percentage of 100 % or more indicates that local shear stress state equals or exceeds the yield criterion and the material will fail or yield locally with 1st loading cycle A yield percentage below 100 % means that material will not fail at the first loading application. However, the material might fail after certain number of loading repetitions. The higher the yield PAGE 191 191 0 20 40 60 80 100 120 140 6.02 4.93 4.00 3.67 3.00 2.67 2.33 2.00 1.67 1.33 1.00 0.33 0.00 Tire Lateral Location (in.) Shear Failure Percentage (%) TireModel UniformModel Figure 6 31. Comparisons of shear yield p ) 0 10 20 30 40 50 60 70 80 90 6.02 4.93 4.00 3.67 3.00 2.67 2.33 2.00 1.67 1.33 1.00 0.33 0.00 Tire Lateral Location (in.) Shear Yield Percentage (%) TireModel UniformModel Figure 6 32. Comparisons of shear yield percentage (c: 340) PAGE 192 192 percentage, the lower number the loading repetitions to failure. Figures 631 to 632 show that tire pavement contact model produces much higher yield percentage than the uniform vertical loading model does. And asphalt mixtures with high cohesion strength and internal friction angle will have low yield percentage and may help to resist instability rutting and top down cracking. 6.4 Effects of Tire Types on Nearsurface Stress Distributions Based on 3 D Analysis 6.4.1 Contact Stress Distributions The effects of tire types on near surface stress distributions based on 2D analysis were conducted in C hapter 4. To further study the effects of the tire types on pavement near surface stress states, a 3 D analysis was conducted in this section. Three tires, 11R22.5, 425/65R22.5 and 445/50R22.5, were put on a typical three layer pavement system, as shown in Figure 633. The tire models used here were developed fr om C hapter 5. The material properties and physical dimensions of the pavement were given in Table 6 1. Comparisons of contact stress distributions among different tires at their recommend ed inflation pressures (11R22.5: 100 psi; 425/65R22.5: 115 psi; 445/50R22.5: 100 psi) under the same load level of 9,000 lbf (4,500 lbf for single 11R22.5) are presented in Figures 6 34 to 636. Obviously, the super single 425/65R22.5 causes the highest maximum vertical contact stress, while dual 11R22.5 and NGWB 445/50R22.5 have very close maximum vertical contact stress, as shown in Figure 6 34. Accordingly, the super single 425/65R22.5 also produces much higher transverse contact stresses than both dual 11R22.5 and NGWB 445/50R22.5 do. The results are very consistent with 2D analysis. Table 6 2 gives the statistic results of the comparisons. Table 6 2. Statistic results of the comparisons 11R22.5 425/65R22.5 445/50R22.5 Contact Stress (psi) Contact Stress (psi) Contact Stress (psi) Vertical Lateral Longitudinal V ertical Lateral Longitudinal Vertical Lateral Longitudinal Maximum 217 39.3 24.1 299 42.5 24.9 194 30.0 17.4 Tire/Dual 1.00 1.00 1.00 1.32 1.08 1.03 0.89 0.76 0.72 PAGE 193 193 Dual Tire 11R22.5 Super Single 425/65R22.5 New Generation 445/50R22.5 Figure 6 33. Tire pavement contact models 350 300 250 200 150 100 50 0 15.00 10.00 5.00 0.00 5.00 10.00 15.00 Tire Lateral Distance (in) Vertical Contact Stress (psi) 445/50R22.5 425/65R22.5 11R22.5 Figure 6 34. Comparisons of vertical contact stress among different tires PAGE 194 194 50 40 30 20 10 0 10 20 30 40 50 10.00 5.00 0.00 5.00 10.00 Tire Lateral Distance (in) Lateral Contact Stress (psi) 445/50R22.5 425/65R22.5 11R22.5 Figure 6 35. Comparisons of lateral contact stress among diffe rent tires 30 20 10 0 10 20 30 3.00 2.00 1.00 0.00 1.00 2.00 3.00 Tire Longitudinal Distance (in) Longitudinal Contact Stress (psi) 445/50R22.5 425/65R22.5 11R22.5 Figure 6 36. Comparisons of longitudinal contact stress among different tires PAGE 195 195 6.4.2 Near surface Stress Distributions Figure 637 shows the comparisons of principal tensile stresses distributions along AC surface among different types of tire. As expected, super single 425/65R22.5 produces much higher peak SIGMA 1 than both dual 11R22.5 and NGWB 445/50R22.5 do. So does maximum shear stress as shown in Figure 6 38. This makes sense since near surface stress distributions are greatly affected by contact stresses, while super single 425/65R22.5 produces the highest average contact stresses. Figure 6 39 shows the comparisons of average shear yield percentage along the critical shear p lane developed in the top 2 in. AC layer. The average shear yield percentage can be defined as: L li i AVG Where, i: Shear yield percentage of one particular small section; il: Length of the small section; : L Length of the whole section. Again, super single 425/65R22.5 produces the highest average shear yield percentage than dual 11R22.5 and new generation 445/50R22.5, which indicates that super single 425/50R22.5 might cause the worst damage to the pavement in terms of topdown cracking and ins tability rutting. 6.5 Parametric Study In this section, a comprehensive parametric study was conducted to investigate how pavement structure, environmental conditions, tire structure and loading conditions affect near surface stress states related to top down cracking and instability rutting. The parametric study PAGE 196 196 was carried under 4,500 lbf normal loads with 110 psi inflation pressure. The dual tire model (11R22.5) was selected for parametric study. 80 60 40 20 0 20 40 60 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Distance to Tire Edge (in) SIGMA1 (psi) 445/50R22.5 425/65R22.5 11R22.5 Figure 6 37. Comparisons of principal tensile stresses among tires 0.00 0.50 1.00 1.50 2.00 2.50 0 10 20 30 40 50 60 70 80 Maximum Shear Stress (psi) Depth to Surface (in) 445/50R22.5 425/65R22.5 11R22.5 Figure 6 38. Comparisons of maximum shear stresses among tires PAGE 197 197 0 10 20 30 40 50 60 70 11R22.5 425/65R22.5 445/50R22.5 Tire Type Average Shear Yield Percentage (%) Figure 6 39. Comparisons of average shear yield percentage among tires 6.5.1 Effects of Pavement Structure The two pavement thickness values that were chosen encompassed the range typically found on highvolume interstate highways in Florida. Base and subgrade thickness were fixed as constant. Typical AC and subgrade moduli and were selected to represent actual field AC pavement. The base moduli were varied between 20 ksi and 40 ksi to capture the effects of pavement bending. All the specific parametric values used here are given in T able 6 3. Table 6 3. Parametric variations used for study Variable Parameter Values AC Thickness 4 in 8 in AC Moduli 800 ksi Base Moduli 20 ks i 40 ksi Subgrade Moduli 20 ksi Base Thickness 12 in (Constant) Subgrade Thickness 50 in (Constant) Possion Ratio AC:0. 40, BS:0. 35,SB:0. 35 Loading Condition 4,500 lbs axle load, 110 inflation pressure Figures 6 40 to 642 show the tire pavement inter face contact stress distributions under different combination of AC thickness and moduli ratio E1:E2. It can be clearly observed that PAGE 198 198 AC thickness and moduli ratio E1:E2 had almost no effect on the tire pavement interface contact stress distributions. Figu res 6 43 to 644 show how AC thickness and base moduli affect peak maximum shear stress and principal 1 developed at top 2 in. AC layer. Again, both AC thickness and base moduli have little effect on peak maximum shear stress and principal 1. 250 200 150 100 50 0 50 6.00 4.00 2.00 0.00 2.00 4.00 6.00 Tire Transverse Distance(in) Vertical Contact Stress (psi) 8" ACE1:E2=800:40ksi 8" ACE1:E2=800:20ksi 4" ACE1:E2=800:40ksi 4" ACE1:E2=800:20ksi Figure 6 40. Effects of AC thickness and base moduli on vertical contact stress distributions (Tire middle yz plane) 40 30 20 10 0 10 20 30 40 5.00 4.00 3.00 2.00 1.00 0.00 1.00 2.00 3.00 4.00 5.00 Tire Transverse Distance (in) Lateral Contact Stress (psi) 8" ACE1:E2=800:40ksi 8" ACE1:E2=800:20ksi 4" ACE1:E2=800:40ksi 4" ACE1:E2=800:20ksi Figure 6 41. Effects of AC thickness and base moduli on transverse contact stress distributions (Tire middle yz plane) PAGE 199 199 20 15 10 5 0 5 10 15 20 4.00 3.00 2.00 1.00 0.00 1.00 2.00 3.00 4.00 Tire Longitudinal Distance (in) Longitudinal Contact Stress (psi) 8" ACE1:E2=800:40ksi 8" ACE1:E2=800:20ksi 4" ACE1:E2=800:40ksi 4" ACE1:E2=800:20ksi Figure 6 42. Effects of AC thickness and base moduli on transverse contact stress distributions (Tire middle yz plane) 0 10 20 30 40 50 60 70 80 20 40 Base Moduli (ksi) Maximum Shear Stress (psi) 4" AC 8" AC Figure 6 43. Effects of AC thickness and base moduli on peak maximum shear stress PAGE 200 200 0 7 14 21 28 35 20 40 Base Moduli (ksi) SIGMA1(psi) 4" AC 8" AC Figure 6 44. Effects of AC thickness and base moduli on principal t 1 6.5.2 Environmental Conditions In current AASHTO design, the AC layer of a flexible pavement system is generally modeled as uni form moduli throughout its cross section. However, asphalt material is a temperature sensitive material and the moduli of the AC layer is almost never uniform due to effects of environmental conditions such as climatic and solar conditions. The effects of moduli gradients induced by temperature on the near surface stress states were particularly investigated in thi s section. How to determine temperature gradients in AC layer and correspondingly build temperaturemoduli relationships is not a trivial ex er cise. Fortunately, Myers (1997) developed a comprehensive method of which key points are listed below : Temperature gradients in AC layer were analytically determined by using the FHWA Environmental Effects Model (Lytton et al. 1990) with air temperature data for northcentral Florida from NOAA. Four temperature gradient cases were created to represent typical climatic conditions, as shown in Figure 6 4 5. The AC temperaturemodulus relationship developed by Ruth et al. (1981) was utilized to generate a moduli gradient corresponding to the temperature gradients determined. PAGE 201 201 A detailed description of this method can be found in studies conducted by Myers (1997, 2000). F igure 645. Temperature gradient cases used to determine moduli gradients in the AC layer (Myers, 2000) Based on above method, four different modulus profiles for AC layer were created with each correspond ing to particular temperature gradients to represent a typical time in a day, as shown in Figure 646. Case 1 was selected to represent time at about 11 am when no gradient would occur due to warm temperature condition. The uniform modulus was determined based on the temperature calculated at onethird of the depth of the AC layer (Huang 1993). Case 2 represented a situation around 7 pm when the sun goes down and then pavement is cold at the surface but remains very warm inside of the AC layer. Case 3 was c hose to represent pavement temperature conditions at around 5 am when the surface temperature is at its lowest and the bottom temperature still remains relatively warm. Case 4 represented the time when a sudden rainstorm causes rapid cooling of the pavemen t surface, while the rest of AC layer still remains high temperature. PAGE 202 202 Figures 6 47 to 649 show the effects of moduli gradients on tire pavement interface contact stress distributions. In this study, it was observed that moduli gradients of AC layer have no virtually effects on the tire pavement interface contact stress at all. The effects of moduli gradients of AC layer on peak maximum shear stress and principal 1 were also given in Figures 650, which also shows that moduli gradients have almost no effect on critical shear stress and principal tensile stress developed within top 1 in. AC layer. It should be noted that all the analyses were conducted under 40 ksi base moduli and 20 ksi subgrade moduli. 2" 2" 2" 2" 8" E=800 ksi E=800 ksi E=800 ksi E=800 ksi 2" 2" 2" 2" 8" E=1200 ksi E=400 ksi E=200 ksi E=600 ksi 2" 2" 2" 2" 8" E=1000 ksi E=500 ksi E=300 ksi E=700 ksi 2" 2" 2" 2" 8" E=800 ksi E=300 ksi E=300 ksi E=300 ksi Case 1: Uniform Stiffness (11 am) Case 2: Sharpest Gradient (7 pm) Case 3: Greatest Differential (5 am) Case 4: Rapid Cooling (raining) F igure 646. Modulus profiles for AC layer used in ADINA analysis PAGE 203 203 250 200 150 100 50 0 50 5.00 4.00 3.00 2.00 1.00 0.00 1.00 2.00 3.00 4.00 5.00 Tire Transverse Distance (in.) Vertical Contact Stress (psi) Case1 Case2 Case3 Case 4 F igure 647. Effects of temperature induced moduli gradients on tire pavement interface vertical co ntact stress distributions 40 30 20 10 0 10 20 30 40 6.00 4.00 2.00 0.00 2.00 4.00 6.00 Tire Transverse Distance (in) Transverse Contact Stress (psi) Case 1 Case 2 Case 3 Case 4 F igure 648. Effects of temperature induced moduli gradients on tire pavement interface transverse shear contact stress distributions PAGE 204 204 20 15 10 5 0 5 10 15 20 4.00 3.00 2.00 1.00 0.00 1.00 2.00 3.00 4.00 Tire Longitudinal Distance (in.) Longitudinal Contact Stress (psi) Case1 Case2 Case3 Case3 F igure 649. Effects of temperature induced moduli gradients on tire pavement interface l ongitudinal shear contact stress distributions 0 10 20 30 40 50 60 70 80 1 2 3 4 Case # Stress (psi) Maximum Shear Stress Principal Tensile Stress F igure 650. Effects of temperature induced moduli gradients on peak maximum shear stress and principal 1 PAGE 205 205 6.5.3 Tire Structure A radial tire structure can be described as a non rigid inflate d torus composed of flexible carcass of high tensile cords restricted by stiff belts and fastened to steel beads (Zhang, 2001). The tire pavement interface responses due to an applied load are inherently dependent on tire structural characteristics and mat erial properties. Two parameters, i.e. radial ply moduli and tread moduli which were considered as the most effects on tire pavement interface response were particularly selected for the analysis. The influences of these parameters on the peak maximum she ar stress and principal tensile stress developed within the top 2 in of the AC layer, which were considered as the main cause for near surface distresses such as instability rutting and topdown cracking, were investigated for static radial truck tire with 110 psi inflation pressure and 4,500 lbs normal load. It should be noted, however, the variations in a single parameter may produce variations in the loaddeflection characteristics. Therefore, the selected parameters are varied within a relatively small range such that the resulting influence on the load deflection characteristics is minimal. The tire model used here for parametric study was dual 11R22.5. 6.5.3.1 Effects of r adial p ly Radial ply, which acts as transmitting all load, driving, braking and s teering forces between wheel and tire tread, is among the most important structural characteristics of a radial tire. Radial ply is also the main part of side wall. The variations in moduli of radial ply would affect flexibility and bending performance of sidewall, which further affects tire pavement interface stress distributions. Figure 6 51 shows how the moduli of radial ply affects vertical contact stress distributions under tire at transverse middle plane (yz plane). It can be observed that magnitude o f vertical contact stress was increased with the moduli of radial ply. Figures 652 and 653 show how the effects of the moduli of radial ply on the peak maximum shear stress and principal 1 developed within top 2 in. of AC layer. It clearly shows that both PAGE 206 206 maximum shear stress and principal 1 were increased dramatically with the moduli of radial ply, which means that increasing flexibility of sidewall may de crease propensity of topdown cracking and instability rutting. 350 300 250 200 150 100 50 0 50 100 6 4 2 0 2 4 6 Tire Transverse Distance (in.) Vertical Contact Stress (psi) 1000 3000 5000 Figure 6 51. Effects of radial ply moduli on vertical contact stresses ( m iddle yz plane) 0 10 20 30 40 50 1000 3000 5000 Radial Ply Moduli (psi) Principal Tensile Stress (psi) Figure 6 52. Effects of radial ply moduli on principal tensile stress PAGE 207 207 0 20 40 60 80 100 1000 3000 5000 Radial Ply Moduli (psi) Maximum Shear Stress (psi) Figure 6 53. Effect s of radial ply moduli on maximum shear stress 6.5.3.2 Effects of t read As stated early, tread is made of rubber which provides the interface between the tire and the road. Its primary purpose is to provide traction and wear. Changing material properties of tr ead may induce the change of traction between tire and pavement and thus influence the tire pavement interface stress distributions. The effects of tread moduli on the vertical contact stress under the tire at the middle transverse plane (yz plane) are giv en in Figure 654. It was observed that vertical contact stresses were increased significantly when tread moduli increased from 100 psi to 300 psi but increased little when the moduli changed from 300 to 500. Figures 655 and 656 show the effects of tread moduli on the peak maximum shear stress and principal tensile stress 1 developed within top 2 in. of AC layer. It clearly shows that both maximum shear stress and principal 1 were increased with tread moduli ; however, the increase is not significant when the moduli increased from 300 psi to 500 psi. 6.5.4 Loading Conditions The increasing variety of truck types and traffic volumes in modern road networks make it necessary to evaluate the effects of radial tire on pavement response due to differen t loads and PAGE 208 208 inflation pressures. The various loading conditions used for the study were listed in Table 64. The tire pavement interface contact stresses, tire footprint area and critical stresses developed within top 2 in. of AC layer were analyz ed to evaluate the effects of load and inflation pressure levels. 350 300 250 200 150 100 50 0 50 100 6 4 2 0 2 4 6 Tire Transverse Distance (in.) Vertical Contact Stress (psi) 100 psi 300 psi 500 psi Figure 6 54. Effects of tread moduli on vertical contact stress ( m iddle yz plane) 0 10 20 30 40 100 300 500 Tread Moduli (psi) Principal tensile Stress (psi) Figure 6 55. Effects of radial ply moduli on principal tensile stress PAGE 209 209 0 20 40 60 80 100 120 140 100 300 500 Tread Moduli (psi) Maximum Shear Stress (psi) Figure 6 56. Effect s of tread moduli on maximum shear stress Table 6 4. Loading conditions used for the study Condition Load (lbf) Inflation Pressure (psi) Condition Load (lbf) Inflation Pressure (psi) 1 2 250 110 4 4 500 70 2 4 500 110 5 4 500 110 3 6 750 110 6 4 500 140 6.5.4.1 Effects of l oad The normal tire axle load directly affects contact path geometry and contact stress distributions in the road pavement interface. Moreover, the normal axle load strongly influences the tire deformation and stress fields developed in tire l ayers. Figures 6 57 to 659 show the effects of varying load at the given inflation pressure 110 psi on the tire pavement interface contact stress distributions. It was found that both vertical contact stress and horizontal contact stress were increased wi th the load. Comparing with lower loads, however, higher loads had relatively small effect on the tire pavement contact stresses. This might attribute to the fact that the tire tends to fatten and distribute the contact stress more evenly at higher load le vel. These findings are very consistent with the field measurements (Drakos, 2003). PAGE 210 210 Table 6 5 summarizes the statistic results of the effects of varying load on the maximum contact stress, critical maximum shear stress and principal tensile stress developed within top 2 in. of AC layer, and footprint area. In order to compare the significant levels of those statistic values affected by load, all the statistic values were normalized within their groups as follows: Normalized Value = (value of one load level) / (value of normal load level) And the normalized values of those statistic parameters vs. load level s were plotted in Figure 6 60. The figure illustrates that the load influences footprint area most while affects maximum shear stress least. Table 6 5. Statistic results due to the effects of tire normal load Load Level (lb) Maximum Contact Stress Maximum Shear Stress(psi.) Principal Tensile Stress (psi.) Footprint Area (in.^2) Stress_zz (psi.) Stress_yz (psi.) Stress_xz (psi.) 2 250 159.69 18.11 13.85 5 8.63 23.62 24.01 4 500 227.05 26.65 17.59 66.99 30.23 46.70 6 750 273.17 30.25 20.72 74.87 38.26 60.84 300 250 200 150 100 50 0 50 5 4 3 2 1 0 1 2 3 4 5 Tire Transverse Location (in.) Vertical Contact Stress (psi.) Lower Load2250 lb Normal Load4500 lb Higher Load6750 lb Figure 6 57. Effects of load on vertical contact stress distributions PAGE 211 211 40 30 20 10 0 10 20 30 40 5 4 3 2 1 0 1 2 3 4 5 Tire Transverse Location (in.) Transverse Contact Stress (psi.) Lower Load2250 lb Normal Load4500 lb Higher Load6750 lb F igure 658. Effects of load on transverse contact stress distributions 25 20 15 10 5 0 5 10 15 20 25 4 3 2 1 0 1 2 3 4 Tire Longitudinal Distance (in.) Longitudinal Contact Stress (psi.) Lower Load2250 lb Normal Load4500 lb Higher Load6750 lb F igure 659. Effects of load on longitudinal contact stress distributions PAGE 212 212 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1000 2000 3000 4000 5000 6000 7000 8000 Load Levels (lbf) Normalized Value Contact Stress_ZZ Contact Stress_YZ Contact Stress_XZ Max. Shear Stress Prin. Tensile Stress Footprint Area Figure 6 60. Significant levels of statistic parameters due to variable load 6.5.4.2 Effects of tire inflation pressure Similar as normal axle load, the tire inflation pressure a lso directly affects contact patch geometry, contact stress distribution and tire deformation. Figures 661 to 663 show the effects of inflation pressure on the tire pavement interface contact stress distributions. It was found that inflation pressure has little effect on horizontal shear contact stress. For vertical contact stress, a substantial increase was observed around middle rib as inflation pressure was increased. The increase of vertical contact stresses was caused by the reduced the tire footprin t area due to the increasing inflation pressure. The peak principal tensile stress and maximum shear stress developed within top 2 in. AC layer as functions of inflation pressure were shown in Figures 664 to 665 respectively. It was observed that both pr incipal tensile stress and maximum shear stress were decreased as the increase of the tire inflation pressure, which means that under inflation would increase the propensity of topdown cracking and instability rutting. This could PAGE 213 213 be attributed to the incr ease of sidewall flexibility. It was said the sidewall flexing increases noticeably when a tires inflation drops 15 to 20% below recommended (Goodyear, 2004). The statistic results of the effects of varying load on the maximum contact stress, critical max imum shear stress and principal tensile stress developed within top 2 in. of AC layer, and footprint area were summarized in Table 6 6. Table 6 6. Statistic results due to the effects of tire inflation pressure Inflation Pressure Level ( psi ) Maximum Contac t Stress Maximum Shear Stress (psi.) Principal Tensile Stress (psi.) Footprint Area (in.^2) Stress_zz (psi.) Stress_yz (psi.) Stress_xz (psi.) 70 227.13 26.20 18.60 80.23 36.64 48.63 110 237.05 26.65 17.59 66.99 30.23 46.70 140 260.66 26.70 16.71 6 0.09 26.96 43.30 300 250 200 150 100 50 0 50 100 5 4 3 2 1 0 1 2 3 4 5 Tire Transverse Location (in.) Vertical Contact Stress (psi.) Lower Pressure70 psi Normal Pressure110 psi Higher Pressure140 psi Figure 6 61. Effects of inflation pressure on vertical contact stress distributions PAGE 214 214 30 20 10 0 10 20 30 5 4 3 2 1 0 1 2 3 4 5 Tire Transverse Location (in.) Transverse Contact Stress (psi.) Lower Pressure70 psi Normal Pressure110 psi Higher Pressure140 psi F igure 662. Effects of inflation pressure on transverse contact stress distributions 25 20 15 10 5 0 5 10 15 20 25 4 3 2 1 0 1 2 3 4 Tire Longitudinal Location (in.) Longitudinal Contact Stress (psi.) Lower Pressure70 psi Normal Pressure110 psi Higher Pressure140 psi Figure 6 63. Effects of inflation pressure on longitudinal contact stress distributions PAGE 215 215 0 10 20 30 40 50 60 70 80 90 50 70 90 110 130 150 Inflation Pressure (psi) Maximum Shear Stress (psi) F igure 664. Effects of inflation pressure on maximum shear stress 0 5 10 15 20 25 30 35 40 45 50 70 90 110 130 150 Inflation Pressure (psi) Principle Tensile Stress (psi) F igure 665. Effects of inflation pressure on principal tensile stress 6.6 Summary The main points of this chapter are summarized as follows: A 3 D tire pavement interaction model was developed. The pavement was modeled as 3 layer linear elastic system rather than a rigid plate. The verification of the 3D tire pavement model indicates that model has ability to capture the characteristics of tire pavement interface stress distributions, not only vertical but also horizontal, which means that model can be used for studying pavement response under real tire. PAGE 216 216 Comparing with uniform vertical load model, tire pavement model produce contact stress not only highe r in magnitude but also more variable in distributions. High shear stress was expected to develop at the tire edge for both models. Again, tire pavement contact model produces about double maximum shear stress than the uniform model. Near surface stress st ates investigation revealed that there existed a critical location for principal tensile stress, which is about 0.30.5 in. away from the tire edge T he pq space method clearly shows that tire pavement contact model producing stress states that are much closer to the Mohr Coulomb failure envelope than a uniform vertical loading. A critical shear plane was developed in the top 2 in. AC layer. The tire pavement contact model produces much higher shear yield percentage than the uniform vertical loading model does A sphalt mixtures with high cohesion strength and internal friction angle may help to resist instability rutting and top down cracking. Investigating the effects of tire types on near surface stress states indicates that the super single 425/65R22.5 ca uses the highest contact stress es while dual 11R22.5 and NGWB 445/50R22.5 have very close contact stress Accordingly, super single 425/65R22.5 might cause the worst damage to the pavement in terms of topdown cracking and instability rutting. Pavement st ructure and moduli gradient due to temperature have almost no effect on tire pavement contact stress distributions and critical maximum shear stress and principal tensile stress developed within top 2 in. of AC layer. Both maximum shear stress and principa l 1 were increased dramatically with the moduli of radial ply, which means that increasing flexibility of sidewall may decrease propensity of top down cracking and instability rutting. Parametric study on tread indicates that both maximum s hear stress and principal tensile 1 were increased with tread moduli ; however, the increase is not significant when the tread moduli increased from 300 psi to 500 psi. This means that increasing tire tread rigidity might increase the propensity of topdown cracking and instability rutting. The normal tire axle load directly affects contact path geometry and contact stress distributions in the road pavement interface. Tire pavement contact stress, critical maximum shear stress and principal tensile s tress developed within top 2 in. of AC layer and tire footprint area increased with increase in load. Both principal tensile stress and maximum shear stress decrease with increase in inflation pressure, which means that under inflation would increase the propensity of top down cracking. This could be attributed to the increase of sidewall flexibility PAGE 217 217 CHAPTER 7 CO N C LUSIONS AND RECOMMENDATIONS 7.1 Conclusions Top down cracking and instability rutting are two of the main failures in the flexible pavements across the world. Earlier studies have shown that these are near surface distresses that are greatly affected by tirepavement interaction. The overall objective of this study was to develop and validate 2D and 3D tire models as well as tire pavement int eraction models based on finite element code ADINA and use them to investigate the effects of tire type, tire pavement interaction and pavement cross section profiles on topdown cracking and instability rutting performance. A range of radial truck tire types including dual, super single and new generation wide base tires were modeled based on tire geometry and structure information provided by the tire manufacturers. Tire models were calibrated based on load deflection data and verified by comparing meas ured and predicted contact stresses. The results indicated that the tire models developed can accurately capture both vertical and horizontal contact stress characteristics and thus can be used for further evaluation purposes. There was a significant diff erence in contact stress distributions among different tire types based on finite element simulation. Accordingly, near surface stress states in terms of principal tensile stress and maximum shear stress were quite different among the different tires. Analysis results indicated that super single wide base tire might produce greater damage to the pavement in terms of both topdown cracking and instability rutting than either dual tires or new generation wide base tire. Analysis results also indicated that p avement cross sectional profiles with rutting or cross slopes increased the propensity for topdown cracking and severity of instability rutting Some key conclusions from this study are summarized as follows: Based on t he successful modeling of 2D and 3D tire and tire pavement interaction in the study it may be conclude d that geometry and structural characteristics reported by tire PAGE 218 218 manufacturers along with measured load deflection data can be used to develop 3 D finite element models to predict tire pav ement interface stresses and their effects on pavement performance. This could be a significant benefit for any ongoing and future field or HVS experiments by providing a tool that can help isolate the tire and tire pavement interaction effects from the ot her confounding effects that are generally present in these types of experiments. Shear induced principal tensile stress at AC surface near the tire edge is higher in magnitude than bending stress produced at AC surface, which indicates that this principal tensile stress might be more likely responsible for the initiation of top down cracking. Comparing with dual 11R22.5 and new generation wide base 445/50R22.5, super single 425/65R22.5 might produce the worst damage to the pavement in terms of topdown cracking and instability rutting without considering unbalanced dual tires. However, i f unbalance load between dual tires was considered, dual tires might produce much more damage to pavement than super single tire. Pavement cross sectional profiles with rutting or cross slopes can increase the propensity for topdown cracking and severity of instability rutting. Pavement structures were found to have little effect on topdown cracking. However, weak base or subgrade might increase the propensity for instability rutting. Both overloads and under inflation would likely increase the initiation of top down cracking and instability rutting. Increasing the flexibility of tire sidewall or decreasing the tire tread moduli might help to reduce the propensity for topdown cracking and instability rutting. Comparing with traditional uniform vertical loading conditions, the 3 D tire pavement interaction analysis showed that radial tires produce high near surface shear stresses and principal tensile stresses. A critical sh ear plane was developed in the top 2 in. AC layer The tire pavement contact model produces much higher shear yield percentage than the uniform vertical loading model does A sphalt mixtures with high cohesion strength and internal friction angle may help to resist instability rutting and top down cracking. 7.2 Recommendations Based on conclusions presented above, the recommendations for future work were made as follows: E fforts to accurately modeling tires need to be continued. The exact material properties of tire parts should be studied and tested. More contact stress measurements on new generation wide based tires should be obtained for model verification purpose s PAGE 219 219 The influence of dynamic effects of a moving vehicle/tire should be considered in further modeling of tire pavement interaction, which might help to better understand the mechanism of topdown cracking and instability rutting. The effects of viscoelastic behavior of asphalt concrete on the near surface stress states should be considered along w ith tire pavement interaction. New fatigue model related to top down cracking and its effects on pavement design needs to be proposed, studied and established. Further studies need to re evaluate the mechanism of instability rutting and the model for insta bility rutting. Advanced imaging facilities such as computerized tomography need to be employed to help detect the initiation of top down cracking and predicted shear failure plane in the shallow AC layer under the tire. PAGE 220 220 APPENDIX A BISAR INPUT FOR TIRE P AVEMENT CONTACT STRESSES Table A 1. 3D tire contact stresses used in BISAR for 445/50R22.5 Coordinates (in) Contact Stresses (psi) Angle (deg.) x y z Vertical Lateral Longitudinal Lateral Longitudinal 7.20 2.80 0.00 120.00 35.00 20.00 180.00 90.00 7.2 0 2.80 0.00 120.00 35.00 20.00 180.00 270.00 7.20 1.40 0.00 130.00 40.00 25.00 180.00 90.00 7.20 1.40 0.00 130.00 40.00 25.00 180.00 270.00 7.20 0.00 0.00 150.00 45.00 0.00 180.00 0.00 7.20 2.80 0.00 120.00 35.00 20.00 0.00 90.00 7.20 2.80 0.00 1 20.00 35.00 20.00 0.00 270.00 7.20 1.40 0.00 130.00 40.00 25.00 0.00 90.00 7.20 1.40 0.00 130.00 40.00 25.00 0.00 270.00 7.20 0.00 0.00 150.00 45.00 0.00 0.00 0.00 5.40 2.40 0.00 110.00 30.00 20.00 180.00 90.00 5.40 2.40 0.00 110.00 30.00 20.00 1 80.00 270.00 5.40 1.20 0.00 120.00 35.00 25.00 180.00 90.00 5.40 1.20 0.00 120.00 35.00 25.00 180.00 270.00 5.40 0.00 0.00 140.00 40.00 0.00 180.00 0.00 5.40 2.40 0.00 110.00 30.00 20.00 0.00 90.00 5.40 2.40 0.00 110.00 30.00 20.00 0.00 270.00 5 .40 1.20 0.00 120.00 35.00 25.00 0.00 90.00 5.40 1.20 0.00 120.00 35.00 25.00 0.00 270.00 5.40 0.00 0.00 140.00 40.00 0.00 0.00 0.00 3.70 2.40 0.00 120.00 35.00 25.00 180.00 90.00 3.70 2.40 0.00 120.00 35.00 25.00 180.00 270.00 3.70 1.20 0.00 130. 00 40.00 30.00 180.00 90.00 3.70 1.20 0.00 130.00 40.00 30.00 180.00 270.00 3.70 0.00 0.00 150.00 45.00 0.00 180.00 0.00 3.70 2.40 0.00 120.00 35.00 25.00 0.00 90.00 3.70 2.40 0.00 120.00 35.00 25.00 0.00 270.00 3.70 1.20 0.00 130.00 40.00 30.00 0.00 90.00 3.70 1.20 0.00 130.00 40.00 30.00 0.00 270.00 3.70 0.00 0.00 150.00 45.00 0.00 0.00 0.00 1.90 2.80 0.00 130.00 40.00 25.00 180.00 90.00 1.90 2.80 0.00 130.00 40.00 25.00 180.00 270.00 1.90 1.40 0.00 150.00 45.00 30.00 180.00 90.00 1.90 1.40 0.00 150.00 45.00 30.00 180.00 270.00 1.90 0.00 0.00 170.00 50.00 0.00 180.00 0.00 1.90 2.80 0.00 130.00 40.00 25.00 0.00 90.00 1.90 2.80 0.00 130.00 40.00 25.00 0.00 270.00 1.90 1.40 0.00 150.00 45.00 30.00 0.00 90.00 1.90 1.40 0.00 150. 00 45.00 30.00 0.00 270.00 1.90 0.00 0.00 170.00 50.00 0.00 0.00 0.00 0.00 2.80 0.00 140.00 0.00 30.00 0.00 90.00 0.00 2.80 0.00 140.00 0.00 30.00 0.00 270.00 0.00 1.40 0.00 160.00 0.00 35.00 0.00 90.00 0.00 1.40 0.00 160.00 0.00 35.00 0.00 270.00 0.00 0.00 0.00 180.00 0.00 0.00 0.00 0.00 PAGE 221 221 Table A 2. 3D tire contact stresses used in BISAR for 11R22.5 Coordinates (in) Contact Stresses (psi) Angle (deg.) x y z Vertical Lateral Longitudinal Lateral Longitudinal Left Tire 10.25 2.25 0.00 120.00 40.00 40.00 0.00 90.00 10.25 2.25 0.00 120.00 40.00 40.00 0.00 270.00 10.25 0.75 0.00 140.00 60.00 30.00 0.00 90.00 10.25 0.75 0.00 140.00 60.00 30.00 0.00 270.00 8.50 2.50 0.00 140.00 30.00 40.00 0.00 90.00 8.50 1.50 0.00 160.00 40.00 30.00 0 .00 90.00 8.50 0.50 0.00 180.00 60.00 20.00 0.00 90.00 8.50 0.50 0.00 180.00 60.00 20.00 0.00 270.00 8.50 1.50 0.00 160.00 40.00 30.00 0.00 270.00 8.50 2.50 0.00 140.00 30.00 40.00 0.00 270.00 7.00 2.50 0.00 160.00 0.00 40.00 0.00 90.00 7.00 1.50 0.00 180.00 0.00 30.00 0.00 90.00 7.00 0.50 0.00 200.00 0.00 20.00 0.00 90.00 7.00 0.50 0.00 200.00 0.00 20.00 0.00 270.00 7.00 1.50 0.00 180.00 0.00 30.00 0.00 270.00 7.00 2.50 0.00 160.00 0.00 40.00 0.00 270.00 5.50 2.50 0.00 140.00 30 .00 40.00 180.00 90.00 5.50 1.50 0.00 160.00 40.00 30.00 180.00 90.00 5.50 0.50 0.00 180.00 60.00 20.00 180.00 90.00 5.50 0.50 0.00 180.00 60.00 20.00 180.00 270.00 5.50 1.50 0.00 160.00 40.00 30.00 180.00 270.00 5.50 2.50 0.00 140.00 30.00 40 .00 180.00 270.00 3.75 2.25 0.00 120.00 40.00 40.00 180.00 90.00 3.75 2.25 0.00 120.00 40.00 40.00 180.00 270.00 3.75 0.75 0.00 140.00 60.00 30.00 180.00 90.00 3.75 0.75 0.00 140.00 60.00 30.00 180.00 270.00 Right Tire 10.25 2.25 0.00 120.00 40 .00 40.00 0.00 90.00 10.25 2.25 0.00 120.00 40.00 40.00 0.00 270.00 10.25 0.75 0.00 140.00 60.00 30.00 0.00 90.00 10.25 0.75 0.00 140.00 60.00 30.00 0.00 270.00 8.50 2.50 0.00 140.00 30.00 40.00 0.00 90.00 8.50 1.50 0.00 160.00 40.00 30.00 0.00 90.0 0 8.50 0.50 0.00 180.00 60.00 20.00 0.00 90.00 8.50 0.50 0.00 180.00 60.00 20.00 0.00 270.00 8.50 1.50 0.00 160.00 40.00 30.00 0.00 270.00 8.50 2.50 0.00 140.00 30.00 40.00 0.00 270.00 7.00 2.50 0.00 160.00 0.00 40.00 0.00 90.00 7.00 1.50 0.00 180 .00 0.00 30.00 0.00 90.00 7.00 0.50 0.00 200.00 0.00 20.00 0.00 90.00 7.00 0.50 0.00 200.00 0.00 20.00 0.00 270.00 7.00 1.50 0.00 180.00 0.00 30.00 0.00 270.00 7.00 2.50 0.00 160.00 0.00 40.00 0.00 270.00 5.50 2.50 0.00 140.00 30.00 40.00 180.00 90.00 5.50 1.50 0.00 160.00 40.00 30.00 180.00 90.00 5.50 0.50 0.00 180.00 60.00 20.00 180.00 90.00 5.50 0.50 0.00 180.00 60.00 20.00 180.00 270.00 5.50 1.50 0.00 160.00 40.00 30.00 180.00 270.00 5.50 2.50 0.00 140.00 30.00 40.00 180.00 270.00 3.75 2.25 0.00 120.00 40.00 40.00 180.00 90.00 3.75 2.25 0.00 120.00 40.00 40.00 180.00 270.00 3.75 0.75 0.00 140.00 60.00 30.00 180.00 90.00 3.75 0.75 0.00 140.00 60.00 30.00 180.00 270.00 PAGE 222 222 Table A 3. 3D tire contact stresses used in BISAR for 425/65R22.5 Coordinates (in) Contact Stresses (psi) Angle (deg.) x y z Vertical Lateral Longitudinal Lateral Longitudinal 5.50 3.20 0.00 120.00 30.00 20.00 180.00 90.00 5.50 1.60 0.00 130.00 40.00 25.00 180.00 90.00 5.50 0.00 0.00 140.00 45.00 0.00 180.00 0. 00 5.50 1.60 0.00 130.00 40.00 25.00 180.00 270.00 5.50 3.20 0.00 120.00 30.00 20.00 180.00 270.00 3.30 3.20 0.00 130.00 40.00 25.00 180.00 90.00 3.30 1.60 0.00 140.00 45.00 30.00 180.00 90.00 3.30 0.00 0.00 150.00 50.00 0.00 180.00 0.00 3.30 1.60 0.00 140.00 45.00 30.00 180.00 270.00 3.30 3.20 0.00 130.00 40.00 25.00 180.00 270.00 1.10 3.20 0.00 140.00 45.00 30.00 180.00 90.00 1.10 1.60 0.00 150.00 50.00 40.00 180.00 90.00 1.10 0.00 0.00 160.00 60.00 0.00 180.00 0.00 1.10 1.60 0.00 150.00 5 0.00 40.00 180.00 270.00 1.10 3.20 0.00 140.00 45.00 30.00 180.00 270.00 1.10 3.20 0.00 140.00 45.00 30.00 0.00 90.00 1.10 1.60 0.00 150.00 50.00 40.00 0.00 90.00 1.10 0.00 0.00 160.00 60.00 0.00 0.00 0.00 1.10 1.60 0.00 150.00 50.00 40.00 0.00 270.00 1.10 3.20 0.00 140.00 45.00 30.00 0.00 270.00 3.30 3.20 0.00 130.00 40.00 25.00 0.00 90.00 3.30 1.60 0.00 140.00 45.00 30.00 0.00 90.00 3.30 0.00 0.00 150.00 50.00 0.00 0.00 0.00 3.30 1.60 0.00 140.00 45.00 30.00 0.00 270.00 3.30 3.20 0.00 130.00 40.00 25.00 0.00 270.00 5.50 3.20 0.00 120.00 30.00 20.00 0.00 90.00 5.50 1.60 0.00 130.00 40.00 25.00 0.00 90.00 5.50 0.00 0.00 140.00 45.00 0.00 0.00 0.00 5.50 1.60 0.00 130.00 40.00 25.00 0.00 270.00 5.50 3.20 0.00 120.00 30.00 20.00 0.00 270.00 PAGE 223 223 APPENDIX B SELECTED PREDICTED 3D TIRE PAVEMENT CONTACT STR ESSES Table B 1. Predicted 3 D vertical contact stress for 425/65R22.5 (9000 lb, 115 psi) Coordinate STRESSZZ y x 3.67 3.00 2.33 1.67 1.00 0.33 0.33 1.00 1.6 7 2.33 3.00 3.67 6.60 0.03 14.33 40.20 55.06 55.42 52.37 52.37 55.42 55.06 40.20 14.33 0.03 6.30 50.29 54.54 59.33 67.42 79.19 63.12 63.12 79.19 67.42 59.33 54.54 50.29 6.00 5.08 39.32 62.04 76.09 83.32 83.19 83.19 83.32 76.09 62.04 39.32 5.08 5.70 4.91 32.86 71.10 89.88 89.46 85.67 85.67 89.46 89.88 71.10 32.86 4.91 5.40 0.15 4.95 49.37 75.62 82.21 74.53 74.53 82.21 75.62 49.37 4.95 0.15 5.10 0.40 0.41 25.28 52.72 66.60 69.52 69.52 66.60 52.72 25.28 0.41 0.40 4.80 1.35 12.00 72.59 117.27 136.59 117.54 117.54 136.59 117.27 72.59 12.00 1.35 4.50 7.98 43.40 82.38 101.84 118.35 143.29 143.29 118.35 101.84 82.38 43.40 7.98 4.20 1.88 3.15 39.70 85.62 119.56 104.47 104.47 119.56 85.62 39.70 3.15 1.88 3.90 5.71 38.77 7 2.98 104.34 127.95 163.08 163.08 127.95 104.34 72.98 38.77 5.71 3.60 5.34 42.13 108.46 140.00 169.31 183.76 183.76 169.31 140.00 108.46 42.13 5.34 3.30 5.02 42.16 109.41 156.98 192.37 195.64 195.64 192.37 156.98 109.41 42.16 5.02 3.00 4.97 41.6 6 113.80 154.52 197.09 246.40 246.40 197.09 154.52 113.80 41.66 4.97 2.70 5.14 40.47 108.77 145.03 178.20 186.87 186.87 178.20 145.03 108.77 40.47 5.14 2.40 5.42 36.53 78.21 99.25 149.01 146.40 146.40 149.01 99.25 78.21 36.53 5.42 2.10 1.76 3.24 45.38 134.73 138.57 159.89 159.89 138.57 134.73 45.38 3.24 1.76 1.80 5.81 37.50 79.44 102.54 147.90 153.64 153.64 147.90 102.54 79.44 37.50 5.81 1.50 4.63 36.08 107.74 144.01 177.96 190.97 190.97 177.96 144.01 107.74 36.08 4.63 1.20 4.33 35.28 109.82 155.37 197.20 218.06 218.06 197.20 155.37 109.82 35.28 4.33 0.90 4.23 34.39 108.29 155.51 196.37 214.72 214.72 196.37 155.51 108.29 34.39 4.23 0.60 4.34 33.56 105.58 142.77 177.03 189.93 189.93 177.03 142.77 105.58 33.56 4.34 0.30 5.18 3 3.35 76.83 102.11 144.63 153.68 153.68 144.63 102.11 76.83 33.35 5.18 0.00 1.51 2.69 44.89 126.67 144.18 158.40 158.40 144.18 126.67 44.89 2.69 1.51 0.30 5.18 33.35 76.83 102.11 144.63 153.68 153.68 144.63 102.11 76.83 33.35 5.18 0.60 4.34 33.56 105 .58 142.77 177.03 189.93 189.93 177.03 142.77 105.58 33.56 4.34 0.90 4.23 34.39 108.29 155.51 196.37 214.72 214.72 196.37 155.51 108.29 34.39 4.23 1.20 4.33 35.28 109.82 155.37 197.20 218.06 218.06 197.20 155.37 109.82 35.28 4.33 1.50 4.63 36.08 1 07.74 144.01 177.96 190.97 190.97 177.96 144.01 107.74 36.08 4.63 1.80 5.81 37.50 79.44 102.54 147.90 153.64 153.64 147.90 102.54 79.44 37.50 5.81 2.10 1.76 3.24 45.38 134.73 138.57 159.89 159.89 138.57 134.73 45.38 3.24 1.76 2.40 5.42 36.53 78.21 9 9.25 149.01 146.40 146.40 149.01 99.25 78.21 36.53 5.42 2.70 5.14 40.47 108.77 145.03 178.20 186.87 186.87 178.20 145.03 108.77 40.47 5.14 3.00 4.97 41.66 113.80 154.52 197.09 246.40 246.40 197.09 154.52 113.80 41.66 4.97 3.30 5.02 42.16 109.41 15 6.98 192.37 195.64 195.64 192.37 156.98 109.41 42.16 5.02 3.60 5.34 42.13 108.46 140.00 169.31 183.76 183.76 169.31 140.00 108.46 42.13 5.34 3.90 5.71 38.77 72.98 104.34 127.95 163.08 163.08 127.95 104.34 72.98 38.77 5.71 4.20 1.88 3.15 39.70 85.62 119.56 104.47 104.47 119.56 85.62 39.70 3.15 1.88 4.50 7.98 43.40 82.38 101.84 118.35 143.29 143.29 118.35 101.84 82.38 43.40 7.98 4.80 1.35 12.00 72.59 117.27 136.59 117.54 117.54 136.59 117.27 72.59 12.00 1.35 5.10 0.40 0.41 25.28 52.72 66.60 69. 52 69.52 66.60 52.72 25.28 0.41 0.40 5.40 0.15 4.95 49.37 75.62 82.21 74.53 74.53 82.21 75.62 49.37 4.95 0.15 5.70 4.91 32.86 71.10 89.88 89.46 85.67 85.67 89.46 89.88 71.10 32.86 4.91 6.00 5.08 39.32 62.04 76.09 83.32 83.19 83.19 83.32 76.09 62.04 39.32 5.08 6.30 50.29 54.54 59.33 67.42 79.19 63.12 63.12 79.19 67.42 59.33 54.54 50.29 6.60 0.03 14.33 40.20 55.06 55.42 52.37 52.37 55.42 55.06 40.20 14.33 0.03 PAGE 224 224 Table B 2. Predicted 3 D l ateral contact stress for 425/65R22.5 (9000 lb, 115 psi) Co ordinate STRESS YZ y x 3.67 3.00 2.33 1.67 1.00 0.33 0.33 1.00 1.67 2.33 3.00 3.67 6.60 12.60 15.91 20.28 24.26 27.67 23.61 23.61 27.67 24.26 20.28 15.91 12.60 6.30 2.66 12.41 14.37 16.47 19.64 19.29 19.29 19.64 16.47 1 4.37 12.41 2.66 6.00 13.12 1.11 10.88 17.03 15.38 18.41 18.41 15.38 17.03 10.88 1.11 13.12 5.70 0.38 9.36 3.00 12.35 12.34 10.86 10.86 12.34 12.35 3.00 9.36 0.38 5.40 0.85 9.36 6.57 2.69 7.12 8.38 8.38 7.12 2.69 6.57 9.36 0.85 5.10 0.70 1.25 8.80 19.36 24.76 22.53 22.53 24.76 19.36 8.80 1.25 0.70 4.80 2.13 9.62 17.42 22.33 25.87 29.92 29.92 25.87 22.33 17.42 9.62 2.13 4.50 1.21 4.54 11.30 9.74 5.80 6.01 6.01 5.80 9.74 11.30 4.54 1.21 4.20 0.42 0.46 1.42 6.44 11.13 15.81 15.81 11.13 6.44 1.42 0.46 0.42 3.90 2.38 12.92 24.96 24.57 28.43 40.76 40.76 28.43 24.57 24.96 12.92 2.38 3.60 0.37 3.60 17.43 23.16 35.51 33.24 33.24 35.51 23.16 17.43 3.60 0.37 3.30 0.11 0.17 3.66 5.76 21.79 32.82 32.82 21.79 5.76 3.66 0.1 7 0.11 3.00 0.28 1.03 1.68 2.27 10.89 20.65 20.65 10.89 2.27 1.68 1.03 0.28 2.70 0.05 3.97 14.96 15.19 27.98 46.42 46.42 27.98 15.19 14.96 3.97 0.05 2.40 1.89 12.38 22.58 8.44 22.28 23.13 23.13 22.28 8.44 22.58 12.38 1.89 2.10 0.29 0.15 0.12 0.57 0.25 1.98 1.98 0.25 0.57 0.12 0.15 0.29 1.80 2.22 10.70 21.99 8.78 22.68 24.72 24.72 22.68 8.78 21.99 10.70 2.22 1.50 0.13 1.55 13.38 15.51 29.48 39.09 39.09 29.48 15.51 13.38 1.55 0.13 1.20 0.03 0.52 1.30 1. 90 16.83 24.79 24.79 16.83 1.90 1.30 0.52 0.03 0.90 0.16 0.71 3.47 0.12 17.77 25.78 25.78 17.77 0.12 3.47 0.71 0.16 0.60 0.00 2.50 14.18 16.61 30.32 38.36 38.36 30.31 16.61 14.18 2.50 0.00 0.30 1.80 10.16 21.73 10.83 21.07 24.75 24.75 21.07 10.83 21.73 10.16 1.80 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.30 1.80 10.16 21.73 10.83 21.07 24.75 24.75 21.07 10.83 21.73 10.16 1.80 0.60 0.00 2.50 14.18 16.61 30.32 38.36 38.36 30.31 16.61 14.18 2.50 0.00 0.90 0.16 0.71 3.47 0.12 17.77 25.78 25.78 17.77 0.12 3.47 0.71 0.16 1.20 0.03 0.52 1.30 1.90 16.83 24.79 24.79 16.83 1.90 1.30 0.52 0.03 1.50 0.13 1.55 13.38 15.51 29.48 39.09 39.09 29.48 15.51 13.38 1.55 0.13 1.80 2.22 10 .70 21.99 8.78 22.68 24.72 24.72 22.68 8.78 21.99 10.70 2.22 2.10 0.29 0.15 0.12 0.57 0.25 1.98 1.98 0.25 0.57 0.12 0.15 0.29 2.40 1.89 12.38 22.58 8.44 22.28 23.13 23.13 22.28 8.44 22.58 12.38 1.89 2.70 0.05 3.97 14.96 15.19 27.98 46 .42 46.42 27.98 15.19 14.96 3.97 0.05 3.00 0.28 1.03 1.68 2.27 10.89 20.65 20.65 10.89 2.27 1.68 1.03 0.28 3.30 0.11 0.17 3.66 5.76 21.79 32.82 32.82 21.79 5.76 3.66 0.17 0.11 3.60 0.37 3.60 17.43 23.16 35.51 33.24 33.24 35.51 23.16 17.43 3.60 0.37 3.90 2.38 12.92 24.96 24.57 28.43 40.76 40.76 28.43 24.57 24.96 12.92 2.38 4.20 0.42 0.46 1.42 6.44 11.13 15.81 15.81 11.13 6.44 1.42 0.46 0.42 4.50 1.21 4.54 11.30 9.74 5.80 6.01 6.01 5.80 9.74 11.30 4.54 1.21 4 .80 2.13 9.62 17.42 22.33 25.87 29.92 29.92 25.87 22.33 17.42 9.62 2.13 5.10 0.70 1.25 8.80 19.36 24.76 22.53 22.53 24.76 19.36 8.80 1.25 0.70 5.40 0.85 9.36 6.57 2.69 7.12 8.38 8.38 7.12 2.69 6.57 9.36 0.85 5.70 0.38 9.36 3.00 12.35 12.34 10.86 10.86 12.34 12.35 3.00 9.36 0.38 6.00 13.12 1.11 10.88 17.03 15.38 18.41 18.41 15.38 17.03 10.88 1.11 13.12 6.30 2.66 12.41 14.37 16.47 19.64 19.29 19.29 19.64 16.47 14.37 12.41 2.66 6.60 12.60 15.91 20.28 24.26 27.67 23.61 23.61 27.67 24.26 20.28 15.91 12.60 PAGE 225 225 Table B 3. Predicted 3 D longitudinal contact stress for 425/65R22.5 (9000 lb, 115 psi) Coordinate STRESS XZ y x 3.67 3.00 2.33 1.67 1.00 0.33 0.33 1.00 1.67 2.33 3.00 3.67 6.60 1.81 5.09 5.51 2.65 0.10 0.34 0.34 0.10 2.65 5.51 5.09 1.81 6.30 9.76 3.70 3.42 3.41 1.12 2.12 2.12 1.12 3.41 3.42 3.70 9.76 6.00 4.36 10.11 7.87 5.04 1.18 2.38 2.38 1.18 5.04 7.87 10.11 4.36 5.70 3.3 8 12.77 14.40 6.56 3.61 3.47 3.47 3.61 6.56 14.40 12.77 3.38 5.40 1.12 6.33 13.20 9.10 2.31 3.30 3.30 2.31 9.10 13.20 6.33 1.12 5.10 0.60 3.10 8.81 8.11 0.99 1.22 1.22 0.99 8.11 8.81 3.10 0.60 4.80 1.89 9.86 18.57 14.41 1.74 4.58 4.58 1.74 14.41 18.57 9.86 1.89 4.50 5.05 12.66 11.27 7.61 5.08 2.46 2.46 5.08 7.61 11.27 12.66 5.05 4.20 1.57 5.38 11.88 12.54 3.92 0.92 0.92 3.92 12.54 11.88 5.38 1.57 3.90 4.95 11.90 12.94 14.61 12.09 4.68 4.68 12.09 14.61 12. 94 11.90 4.95 3.60 5.43 18.24 24.45 24.91 13.97 3.73 3.73 13.97 24.91 24.45 18.24 5.43 3.30 5.67 18.95 29.69 34.04 18.59 3.73 3.73 18.59 34.04 29.69 18.95 5.67 3.00 5.67 19.39 30.43 32.05 31.75 15.78 15.78 31.75 32.05 30.43 19.39 5. 67 2.70 5.34 18.26 26.58 30.59 15.57 3.33 3.33 15.57 30.59 26.58 18.26 5.34 2.40 4.83 12.65 13.18 18.26 12.55 0.67 0.67 12.55 18.26 13.18 12.65 4.83 2.10 1.68 6.56 18.00 16.44 5.96 2.85 2.85 5.96 16.44 18.00 6.56 1.68 1.80 4.86 13.06 13.46 17.69 12.74 1.73 1.73 12.74 17.69 13.46 13.06 4.86 1.50 4.89 17.49 26.53 29.85 15.46 3.56 3.56 15.46 29.85 26.53 17.49 4.89 1.20 5.00 18.01 30.26 33.41 20.23 6.07 6.07 20.23 33.41 30.26 18.01 5.00 0.90 4.90 17.74 30.19 33.68 19.93 5.72 5.72 19.93 33.68 30.19 17.74 4.90 0.60 4.62 16.93 26.28 29.64 15.39 3.56 3.56 15.39 29.64 26.28 16.93 4.62 0.30 4.38 12.48 13.69 17.42 12.61 2.06 2.06 12.61 17.42 13.69 12.48 4.38 0.00 1.52 6.37 17.19 16.91 6.51 2 .22 2.22 6.51 16.91 17.19 6.37 1.52 0.30 4.38 12.48 13.69 17.42 12.61 2.06 2.06 12.61 17.42 13.69 12.48 4.38 0.60 4.62 16.93 26.28 29.64 15.39 3.56 3.56 15.39 29.64 26.28 16.93 4.62 0.90 4.90 17.74 30.19 33.68 19.93 5.72 5.72 19.93 33. 68 30.19 17.74 4.90 1.20 5.00 18.01 30.26 33.41 20.23 6.07 6.07 20.23 33.41 30.26 18.01 5.00 1.50 4.89 17.49 26.53 29.85 15.46 3.56 3.56 15.46 29.85 26.53 17.49 4.89 1.80 4.86 13.06 13.46 17.69 12.74 1.73 1.73 12.74 17.69 13.46 13.06 4 .86 2.10 1.68 6.56 18.00 16.44 5.96 2.85 2.85 5.96 16.44 18.00 6.56 1.68 2.40 4.83 12.65 13.18 18.26 12.55 0.67 0.67 12.55 18.26 13.18 12.65 4.83 2.70 5.34 18.26 26.58 30.59 15.57 3.33 3.33 15.57 30.59 26.58 18.26 5.34 3.00 5.67 19. 39 30.43 32.05 31.75 15.78 15.78 31.75 32.05 30.43 19.39 5.67 3.30 5.67 18.95 29.69 34.04 18.59 3.73 3.73 18.59 34.04 29.69 18.95 5.67 3.60 5.43 18.24 24.45 24.91 13.97 3.73 3.73 13.97 24.91 24.45 18.24 5.43 3.90 4.95 11.90 12.94 14. 61 12.09 4.68 4.68 12.09 14.61 12.94 11.90 4.95 4.20 1.57 5.38 11.88 12.54 3.92 0.92 0.92 3.92 12.54 11.88 5.38 1.57 4.50 5.05 12.66 11.27 7.61 5.08 2.46 2.46 5.08 7.61 11.27 12.66 5.05 4.80 1.89 9.86 18.57 14.41 1.74 4.58 4.58 1.74 14.41 18.57 9.86 1.89 5.10 0.60 3.10 8.81 8.11 0.99 1.22 1.22 0.99 8.11 8.81 3.10 0.60 5.40 1.12 6.33 13.20 9.10 2.31 3.30 3.30 2.31 9.10 13.20 6.33 1.12 5.70 3.38 12.77 14.40 6.56 3.61 3.47 3.47 3.61 6.56 14.40 12.77 3.38 6.00 4.36 10.11 7.87 5.04 1.18 2.38 2.38 1.18 5.04 7.87 10.11 4.36 6.30 9.76 3.70 3.42 3.41 1.12 2.12 2.12 1.12 3.41 3.42 3.70 9.76 6.60 1.81 5.09 5.51 2.65 0.10 0.34 0.34 0.10 2.65 5.51 5.09 1.81 PAGE 226 226 Table B 4 Predicted 3 D vertical contact stre ss for 445/50R22.5 (9000 lb, 100 psi) Coordinate STRESSZZ y x 3.00 2.33 1.67 1.00 0.33 0.33 1.00 1.67 2.33 3.00 8.40 3.37 18.38 43.51 48.86 46.98 46.98 48.86 43.51 18.38 3.37 8.00 4.84 29.18 58.05 80.81 72.73 72.73 8 0.81 58.05 29.18 4.84 7.60 13.31 49.68 76.64 94.34 110.79 110.79 94.34 76.64 49.68 13.31 7.20 15.78 40.80 62.89 68.89 66.38 66.38 68.89 62.89 40.80 15.78 6.80 11.06 41.79 67.80 79.25 87.79 87.79 79.25 67.80 41.79 11.06 6.40 21.47 65.14 100.29 123. 51 138.73 138.73 123.51 100.29 65.14 21.47 6.00 20.26 72.12 107.15 131.75 167.28 167.28 131.75 107.15 72.12 20.26 5.60 21.75 65.94 97.83 134.79 128.19 128.19 134.79 97.83 65.94 21.75 5.20 11.43 41.08 67.61 79.23 90.11 90.11 79.23 67.61 41.08 11.43 4.80 11.41 42.57 71.98 82.57 90.29 90.29 82.57 71.98 42.57 11.41 4.40 20.58 66.47 99.62 135.10 133.81 133.81 135.09 99.62 66.47 20.58 4.00 17.54 71.96 108.58 130.48 172.58 172.58 130.48 108.58 71.96 17.54 3.60 19.72 65.99 98.62 134.92 133.30 133.30 1 34.92 98.62 65.99 19.72 3.20 10.68 41.22 70.77 81.07 88.66 88.66 81.07 70.77 41.22 10.68 2.80 10.65 41.44 70.83 81.56 89.06 89.06 81.56 70.84 41.44 10.65 2.40 19.45 65.72 98.96 134.83 133.03 133.03 134.83 98.95 65.72 19.45 2.00 16.81 71.40 108.35 1 29.85 172.53 172.54 129.85 108.35 71.40 16.81 1.60 19.28 65.59 98.27 134.85 132.95 132.95 134.85 98.27 65.59 19.28 1.20 10.45 40.94 70.54 80.83 88.43 88.43 80.83 70.54 40.94 10.45 0.80 10.47 41.07 70.54 81.04 88.56 88.59 81.02 70.54 41.06 10.47 0.40 19.22 65.51 98.48 134.69 132.90 132.68 134.88 98.44 65.52 19.22 0.00 16.66 71.26 129.75 172.35 172.58 129.55 108.28 108.23 71.25 16.67 0.40 19.22 65.51 98.48 134.69 132.90 132.68 134.88 98.44 65.52 19.22 0.80 10.47 41.07 70.54 81.04 88.56 88.59 81.02 70.54 41.06 10.47 1.20 10.45 40.94 70.54 80.83 88.43 88.43 80.83 70.54 40.94 10.45 1.60 19.28 65.59 98.27 134.85 132.95 132.95 134.85 98.27 65.59 19.28 2.00 16.81 71.40 108.35 129.85 172.53 172.54 129.85 108.35 71.40 16.81 2.40 19.45 65.72 98.96 134.83 133.03 133.03 134.83 98.95 65.72 19.45 2.80 10.65 41.44 70.83 81.56 89.06 89.06 81.56 70.84 41.44 10.65 3.20 10.68 41.22 70.77 81.07 88.66 88.66 81.07 70.77 41.22 10.68 3.60 19.72 65.99 98.62 134.92 133.30 133.30 134.92 98.62 65.99 19.72 4.00 17.54 71.96 108.58 130.48 172.58 172.58 130.48 108.58 71.96 17.54 4.40 20.58 66.47 99.62 135.10 133.81 133.81 135.09 99.62 66.47 20.58 4.80 11.41 42.57 71.98 82.57 90.29 90.29 82.57 71.98 42.57 11.41 5.20 11.43 41.08 67.61 79.23 90.11 90.11 79.23 67.61 41.08 11.43 5.60 21.75 65.94 97.83 134.79 128.19 128.19 134.79 97.83 65.94 21.75 6.00 20.26 72.12 107.15 131.75 167.28 167.28 131.75 107.15 72.12 20.26 6.40 21.47 65.14 100.29 123.51 138.73 138.73 123.51 100.29 65.14 21.47 6.80 11.06 41.79 67.80 79.25 87.79 87.79 79.25 67.80 41.79 11.06 7.20 15.78 40.80 62.89 68.89 66.38 66.38 68.89 62.89 40.80 15.78 7.60 13.31 49.68 76.64 94.34 110.79 110.79 94.34 76.64 49.68 13.31 8.00 4.84 29.18 58.05 80.81 72.73 72.73 80.81 58.05 29.18 4.84 8.40 3.37 18.38 43.51 48.86 46.98 46.98 48.86 43.51 18.38 3.37 PAGE 227 227 Table B 5. Predicted 3 D lateral contact stress for 445/50R22.5 (9000 lb, 100 psi) Coordinate STRESSYZ y x 3.00 2.33 1.67 1.00 0.33 0.33 1.00 1.67 2.33 3.00 8.40 5.51 1.09 7.59 12.84 12.57 12.57 12.84 7.59 1.09 5.51 8.00 2.39 7.26 12.71 17.54 20.43 20.43 17.54 12.71 7.26 2.39 7.60 2.13 0.97 1.48 1.79 0.97 0.97 1.79 1.48 0.97 2.13 7.20 1.38 3.29 5.50 6.64 6.87 6.87 6.64 5.50 3.29 1.38 6.80 1.89 9.18 14.74 19.27 23.90 23.90 19.27 14.74 9.18 1.89 6.40 2.90 10.31 15.49 21.69 30.71 30.71 21.69 15.49 10.31 2.90 6.00 0.13 0.62 0.09 6.77 1.98 1.98 6.77 0.09 0.62 0.13 5.60 3.17 9.95 13.52 14.48 25.31 25.31 14.48 13.52 9.95 3.17 5.20 2.80 8.48 10. 93 16.16 15.87 15.87 16.16 10.93 8.48 2.80 4.80 2.17 8.60 12.79 18.30 17.55 17.55 18.30 12.79 8.60 2.17 4.40 2.26 9.76 12.20 22.30 29.96 29.96 22.30 12.20 9.76 2.26 4.00 0.25 0.27 0.39 0.25 0.17 0.17 0.25 0.39 0.27 0.25 3.60 2.53 10.23 12.59 22.62 30.26 30.26 22.62 12.59 10.23 2.53 3.20 2.35 8.52 11.92 17.62 17.43 17.43 17.62 11.92 8.52 2.35 2.80 2.16 8.38 11.87 17.61 17.35 17.35 17.61 11.87 8.38 2.16 2.40 2.28 9.85 12.22 22.26 30.05 30.05 22.26 12.22 9.85 2.28 2.00 0.05 0.10 0.26 0.10 0.05 0.05 0.10 0.26 0.10 0.05 1.60 2.36 10.05 12.46 22.47 30.20 30.20 22.47 12.46 10.05 2.36 1.20 2.22 8.42 11.84 17.63 17.37 17.37 17.63 11.84 8.42 2.22 0.80 2.19 8.41 11.85 17.63 17.37 1 7.33 17.66 11.84 8.41 2.19 0.40 2.31 9.94 12.34 22.36 30.12 30.13 22.35 12.35 9.94 2.31 0.00 0.00 0.00 0.00 0.01 0.04 0.04 0.01 0.00 0.00 0.00 0.40 2.31 9.94 12.34 22.36 30.12 30.13 22.35 12.35 9.94 2.31 0.80 2.19 8.41 11.85 17.63 17. 37 17.33 17.66 11.84 8.41 2.19 1.20 2.22 8.42 11.84 17.63 17.37 17.37 17.63 11.84 8.42 2.22 1.60 2.36 10.05 12.46 22.47 30.20 30.20 22.47 12.46 10.05 2.36 2.00 0.05 0.10 0.26 0.10 0.05 0.05 0.10 0.26 0.10 0.05 2.40 2.28 9.85 12.22 22.26 30.05 30.05 22.26 12.22 9.85 2.28 2.80 2.16 8.38 11.87 17.61 17.35 17.35 17.61 11.87 8.38 2.16 3.20 2.35 8.52 11.92 17.62 17.43 17.43 17.62 11.92 8.52 2.35 3.60 2.53 10.23 12.59 22.62 30.26 30.26 22.62 12.59 10.23 2.53 4.00 0.25 0.27 0.39 0.25 0.17 0.17 0.25 0.39 0.27 0.25 4.40 2.26 9.76 12.20 22.30 29.96 29.96 22.30 12.20 9.76 2.26 4.80 2.17 8.60 12.79 18.30 17.55 17.55 18.30 12.79 8.60 2.17 5.20 2.80 8.48 10.93 16.16 15.87 15.87 16.16 10.93 8.48 2.80 5.60 3.17 9.95 13.52 14.48 25.31 25.31 14.48 13.52 9.95 3.17 6.00 0.13 0.62 0.09 6.77 1.98 1.98 6.77 0.09 0.62 0.13 6.40 2.90 10.31 15.49 21.69 30.71 30.71 21.69 15.49 10.31 2.90 6.80 1.89 9.18 14.74 19.27 23.90 23.90 19.27 14.74 9.18 1.89 7 .20 1.38 3.29 5.50 6.64 6.87 6.87 6.64 5.50 3.29 1.38 7.60 2.13 0.97 1.48 1.79 0.97 0.97 1.79 1.48 0.97 2.13 8.00 2.39 7.26 12.71 17.54 20.43 20.43 17.54 12.71 7.26 2.39 8.40 5.51 1.09 7.59 12.84 12.57 12.57 12.84 7.59 1.09 5.51 PAGE 228 228 Table B 6. Predicted 3 D longitudinal contact stress for 445/50R22.5 (9000 lb, 100 psi) Coordinate S TRESS XZ y x 3.00 2.33 1.67 1.00 0.33 0.33 1.00 1.67 2.33 3.00 8.40 2.26 7.02 5.04 1.34 1.92 1.92 1.34 5.04 7.02 2.26 8.00 3.89 10.50 9.90 0.40 2.85 2.85 0.40 9.90 10.50 3.89 7.60 7.72 11.92 9.48 3.59 0.15 0.15 3.59 9.48 11.92 7.72 7.20 6.27 9.08 5.72 0.83 0.30 0.30 0.83 5.72 9.08 6.27 6.80 6.36 8.50 7.02 2.45 0.59 0.59 2.45 7.02 8.50 6.36 6.40 10.23 16.09 17.45 14.35 5.58 5.58 14.35 17.45 16.09 10.23 6.00 11.42 19.69 22.03 15.74 3.99 3.99 15.74 22.03 19.69 11.42 5.60 10.23 15.72 20.93 0.83 4.28 4.28 0.83 20.93 15.72 10.23 5.20 6.20 8.65 6.25 3.07 1.25 1.25 3.07 6.25 8.65 6.20 4.80 6.50 9.09 7.21 2.49 0.68 0.68 2.49 7.21 9.09 6.50 4.40 10.20 16.16 20.12 9.34 0.62 0.62 9.34 20.12 16.16 10.20 4.00 10.94 20.22 22.29 16.18 5.42 5.42 16.18 22.29 20.22 10.94 3.60 10.06 16.01 19.97 9.04 0.40 0 .40 9.04 19.97 16.01 10.06 3.20 6.25 8.98 7.12 2.49 0.68 0.68 2.49 7.12 8.98 6.25 2.80 6.28 9.01 7.17 2.54 0.67 0.67 2.54 7.17 9.01 6.28 2.40 9.99 16.08 20.11 9.27 0.52 0.52 9.26 20.11 16.08 9.99 2.00 10.75 20.17 22.25 16.30 5.60 5.60 16.30 22.25 20.17 10.75 1.60 9.95 15.96 19.96 9.31 0.53 0.53 9.31 19.96 15.96 9.95 1.20 6.19 8.97 7.13 2.49 0.68 0.68 2.49 7.13 8.97 6.19 0.80 6.21 8.97 7.14 2.51 0.68 0.68 2.51 7.13 8.97 6.21 0.40 9.94 16.00 20.01 9.30 0 .52 0.52 9.29 20.03 16.00 9.94 0.00 10.71 20.14 16.29 5.63 5.62 16.31 22.22 22.24 20.14 10.71 0.40 9.94 16.00 20.01 9.30 0.53 0.53 9.30 20.01 16.00 9.94 0.80 6.21 8.97 7.14 2.51 0.68 0.68 2.51 7.13 8.97 6.21 1.20 6.19 8.97 7.13 2.49 0.68 0.68 2.49 7.13 8.97 6.19 1.60 9.95 15.96 19.96 9.31 0.53 0.53 9.31 19.96 15.96 9.95 2.00 10.75 20.17 22.25 16.30 5.60 5.60 16.30 22.25 20.17 10.75 2.40 9.99 16.08 20.11 9.27 0.52 0.52 9.26 20.11 16.08 9.99 2.80 6.28 9.01 7.17 2 .54 0.67 0.67 2.54 7.17 9.01 6.28 3.20 6.25 8.98 7.12 2.49 0.68 0.68 2.49 7.12 8.98 6.25 3.60 10.06 16.01 19.97 9.04 0.40 0.40 9.04 19.97 16.01 10.06 4.00 10.94 20.22 22.29 16.18 5.42 5.42 16.18 22.29 20.22 10.94 4.40 10.20 16.16 20. 12 9.34 0.62 0.62 9.34 20.12 16.16 10.20 4.80 6.50 9.09 7.21 2.49 0.68 0.68 2.49 7.21 9.09 6.50 5.20 6.20 8.65 6.25 3.07 1.25 1.25 3.07 6.25 8.65 6.20 5.60 10.23 15.72 20.93 0.83 4.28 4.28 0.83 20.93 15.72 10.23 6.00 11.42 19.69 22. 03 15.74 3.99 3.99 15.74 22.03 19.69 11.42 6.40 10.23 16.09 17.45 14.35 5.58 5.58 14.35 17.45 16.09 10.23 6.80 6.36 8.50 7.02 2.45 0.59 0.59 2.45 7.02 8.50 6.36 7.20 6.27 9.08 5.72 0.83 0.30 0.30 0.83 5.72 9.08 6.27 7.60 7.72 11.92 9.48 3.59 0.15 0.15 3.59 9.48 11.92 7.72 8.00 3.89 10.50 9.90 0.40 2.85 2.85 0.40 9.90 10.50 3.89 8.40 2.26 7.02 5.04 1.34 1.92 1.92 1.34 5.04 7.02 2.26 PAGE 229 229 Table B 7. Predicted 3 D vertical contact stress for dual 11R22.5 (4500 lb, 110 psi) Coord. STRESSZZ y x 1.96 1.66 1.38 1.12 0.87 0.63 0.41 0.20 0.00 0.20 0.41 0.63 0.87 1.12 1.38 1.66 1.96 4.11 35.87 48.01 35.54 54.38 37.35 47.45 46.35 34.19 65.08 34.19 46.35 47.45 37.35 54.38 35.54 48.01 35.87 3.70 61.84 83.25 53.93 93.51 60.79 81.09 77.97 59.81 110.02 59.81 77.97 81.09 60.79 93.51 53.93 83.25 61.84 3.31 83.36 103.47 92.76 118.58 100.57 115.10 109.77 79.91 158.69 79.91 109.77 115.10 100.57 118.58 92.76 103.47 83.36 2.94 93.29 112.95 86.89 134.79 92.86 120.32 117.29 85.19 170.08 85.19 117.29 120.32 92.86 134.79 86.89 112.95 93.29 2.60 68.15 103.04 68.95 118.13 77.68 107.34 99.05 74.03 150.93 74.03 99.05 107.34 77.68 118.13 68.95 103.04 68.15 2.27 50.01 65.96 56.94 78.72 64.69 74.52 76.92 6 4.06 98.55 64.06 76.92 74.52 64.69 78.72 56.94 65.96 50.01 1.96 61.68 92.12 71.09 107.40 81.43 105.32 94.07 75.44 140.64 75.44 94.07 105.32 81.43 107.40 71.09 92.12 61.68 1.66 84.20 122.26 87.99 149.55 100.88 134.43 133.18 92.91 199.55 92.91 133.18 134.43 100.88 149.55 87.99 122.26 84.20 1.38 76.13 116.88 87.05 143.40 100.59 131.77 129.99 92.63 193.85 92.63 129.99 131.77 100.59 143.40 87.05 116.88 76.13 1.12 69.32 91.96 73.22 108.74 84.45 102.55 101.58 77.92 144.31 77.92 101.58 102.55 84.45 108.74 7 3.22 91.96 69.32 0.87 48.91 78.97 51.33 95.37 60.10 88.26 78.03 64.86 114.53 64.86 78.03 88.26 60.10 95.37 51.33 78.97 48.91 0.63 45.28 72.65 50.02 88.83 58.79 84.44 74.25 62.38 108.31 62.38 74.25 84.44 58.79 88.83 50.02 72.65 45.28 0.41 58.31 74.86 71.37 87.54 82.28 88.83 90.33 71.97 122.97 71.97 90.33 88.83 82.28 87.54 71.37 74.86 58.31 0.20 68.35 107.51 74.41 135.65 86.59 121.29 117.96 86.28 174.51 86.28 117.96 121.29 86.59 135.65 74.41 107.51 68.35 0.00 80.47 121.41 89.55 150.53 105.02 138.63 1 37.85 91.75 211.89 91.75 137.85 138.63 105.02 150.53 89.55 121.41 80.47 0.20 68.35 107.51 74.41 135.65 86.59 121.29 117.96 86.28 174.51 86.28 117.96 121.29 86.59 135.65 74.41 107.51 68.35 0.41 58.31 74.86 71.37 87.54 82.28 88.83 90.33 71.97 122.97 71.97 90.33 88.83 82.28 87.54 71.37 74.86 58.31 0.63 45.28 72.65 50.02 88.83 58.79 84.44 74.25 62.38 108.31 62.38 74.25 84.44 58.79 88.83 50.02 72.65 45.28 0.87 48.91 78.97 51.33 95.37 60.10 88.26 78.03 64.86 114.53 64.86 78.03 88.26 60.10 95.37 51.33 78.97 48.91 1.12 69.32 91.96 73.22 108.74 84.45 102.55 101.58 77.92 144.31 77.92 101.58 102.55 84.45 108.74 73.22 91.96 69.32 1.38 76.13 116.88 87.05 143.40 100.59 131.77 129.99 92.63 193.85 92.63 129.99 131.77 100.59 143.40 87.05 116.88 76.13 1.66 84.20 122.26 87.99 149.55 100.88 134.43 133.18 92.91 199.55 92.91 133.18 134.43 100.88 149.55 87.99 122.26 84.20 1.96 61.68 92.12 71.09 107.40 81.43 105.32 94.07 75.44 140.64 75.44 94.07 105.32 81.43 107.40 71.09 92.12 61.68 2.27 50.01 65.96 56.94 78.72 64.69 74.52 76.92 64.06 98.55 64.06 76.92 74.52 64.69 78.72 56.94 65.96 50.01 2.60 68.15 103.04 68.95 118.13 77.68 107.34 99.05 74.03 150.93 74.03 99.05 107.34 77.68 118.13 68.95 103.04 68.15 2.94 93.29 112.95 86.89 134.79 92.86 120.32 117.29 85.19 170.08 85.19 117. 29 120.32 92.86 134.79 86.89 112.95 93.29 3.31 83.36 103.47 92.76 118.58 100.57 115.10 109.77 79.91 158.69 79.91 109.77 115.10 100.57 118.58 92.76 103.47 83.36 3.70 61.84 83.25 53.93 93.51 60.79 81.09 77.97 59.81 110.02 59.81 77.97 81.09 60.79 93.51 53.93 83.25 61.84 4.11 35.87 48.01 35.54 54.38 37.35 47.45 46.35 34.19 65.08 34.19 46.35 47.45 37.35 54.38 35.54 48.01 35.87 PAGE 230 230 Table B 8. Predicted 3 D lateral contact stress for dual 11R22.5 (4500 lb, 110 psi) Coord. STRESSYZ y x 1.96 1.66 1.38 1.12 0.87 0.63 0.41 0.20 0.00 0.20 0.41 0.63 0.87 1.12 1.38 1.66 1.96 4.11 20.84 27.88 18.34 31.19 20.21 26.85 25.95 20.03 36.33 20.03 25.95 26.85 20.21 31.19 18.34 27.88 20.84 3.70 22.28 26.83 25.09 30.78 27.71 30.78 29. 30 22.74 41.03 22.74 29.30 30.78 27.71 30.78 25.09 26.83 22.28 3.31 18.42 20.53 20.13 26.27 21.19 25.23 24.86 18.00 35.96 18.00 24.86 25.23 21.19 26.27 20.13 20.53 18.42 2.94 0.13 8.09 0.74 11.00 0.84 8.27 6.98 5.74 12.95 5.74 6.98 8.27 0.84 11.00 0. 74 8.09 0.13 2.60 13.71 14.90 6.80 16.46 5.48 12.15 9.64 2.45 20.54 2.45 9.64 12.15 5.48 16.46 6.80 14.90 13.71 2.27 0.92 1.72 3.55 0.59 4.89 3.06 1.88 4.68 0.33 4.68 1.88 3.06 4.89 0.59 3.55 1.72 0.92 1.96 14.79 26.50 14.88 33.92 18.15 29.46 27.57 15.15 48.28 15.15 27.57 29.46 18.15 33.92 14.88 26.50 14.79 1.66 5.56 11.91 7.53 18.32 9.76 14.38 17.85 8.96 27.30 8.96 17.85 14.38 9.76 18.32 7.53 11.91 5.56 1.38 7.95 16.05 8.19 20.83 9.12 16.51 16.36 7.66 28.59 7.66 16.36 16.51 9.12 20.83 8.19 16.05 7.95 1.12 14.97 22.01 19.17 27.88 22.05 25.18 28.65 15.72 44.43 15.72 28.65 25.18 22.05 27.88 19.17 22.01 14.97 0.87 12.74 13.21 11.93 14.60 13.38 12.79 15.97 8.90 23.30 8.90 15.97 12.79 13.38 14.60 11.93 13.21 12.74 0.63 5.28 0.93 10.36 0.25 11.98 3.83 8.77 4.44 10.05 4.44 8.77 3.83 11.98 0.25 10.36 0.93 5.28 0.41 13.55 22.45 15.00 30.49 17.94 24.81 27.94 15.69 43.22 15.69 27.94 24.81 17.94 30.49 15.00 22.45 13.55 0.20 13.02 27.95 11.96 38.94 15.36 31.50 30.44 13.00 56.07 13.00 30.44 31.50 15.36 38.94 11.96 27.95 13.02 0.00 0.03 0.00 0.07 0.08 0.14 0.08 0.14 0.08 0.15 0.09 0.14 0.08 0.14 0.08 0.07 0.00 0.03 0.20 12.90 28.24 12.08 39.21 15.50 31.72 30. 65 13.16 56.34 13.17 30.65 31.72 15.50 39.21 12.08 28.24 12.90 0.41 13.45 22.63 15.17 30.63 18.08 24.96 28.08 15.86 43.35 15.86 28.08 24.96 18.08 30.63 15.17 22.63 13.45 0.63 5.73 1.00 10.61 0.39 12.24 3.97 9.01 4.60 1 0.17 4.60 9.01 3.97 12.24 0.39 10.61 1.00 5.73 0.87 12.28 13.29 11.69 14.44 13.18 12.60 15.81 8.76 23.16 8.76 15.81 12.60 13.18 14.44 11.69 13.29 12.28 1.12 15.38 22.30 19.13 27.83 21.98 25.07 28.59 15.83 44.45 15.83 28.59 25.07 21.98 27.83 19.13 22.30 15.38 1.38 9.26 15.89 8.20 20.70 9.04 16.27 16.18 7.48 31.29 7.48 16.18 16.27 9.04 20.70 8.20 15.89 9.26 1.66 5.81 12.35 7.71 18.70 10.01 13.56 19.51 9.85 30.61 9.85 19.51 13.56 10.01 18.70 7.71 12.35 5.81 1.96 15.93 26.77 15. 22 34.24 18.45 30.07 27.75 16.21 53.07 16.21 27.75 30.07 18.45 34.24 15.22 26.77 15.93 2.27 1.08 1.36 3.85 0.38 5.20 4.55 1.04 4.92 0.56 4.92 1.04 4.55 5.20 0.38 3.85 1.36 1.07 2.60 13.43 14.34 6.62 16.04 5.20 11.98 9.25 2.52 21.09 2.52 9.25 11.98 5.20 16.04 6.62 14.34 13.43 2.94 0.81 8.46 0.47 11.31 1.07 8.51 7.19 5.84 13.25 5.84 7.19 8.51 1.07 11.31 0.47 8.46 0.81 3.31 18.75 20.66 20.22 26.37 21.18 25.33 24.78 18.06 35.90 18.06 24.78 25.33 21.18 26. 37 20.22 20.66 18.75 3.70 22.56 26.84 25.27 30.82 27.86 30.81 29.46 22.85 41.14 22.85 29.46 30.81 27.86 30.82 25.27 26.84 22.56 4.11 21.04 27.99 18.58 31.39 20.47 27.05 26.25 20.22 36.69 20.22 26.25 27.05 20.47 31.39 18.58 27.99 21.04 PAGE 231 231 Table B 9. Predicted 3 D longitudinal contact stress for dual 11R22.5 (4500 lb, 110 psi) Coord. STRESS XZ y x 1.96 1.66 1.38 1.12 0.87 0.63 0.41 0.20 0.00 0.20 0.41 0.63 0.87 1.12 1.38 1.66 1.96 4.11 5.46 0.45 3.52 0.85 2.51 4.39 6.47 10.75 0.00 10.75 6.47 4.39 2.51 0.85 3.52 0.45 5.46 3.70 12.90 1.45 6.41 3.32 4.31 8.20 10.63 18.34 0.00 18.34 10.63 8.20 4.31 3.32 6.41 1.45 12.90 3.31 14.09 7.67 10.23 5.67 0.62 4.94 17.68 28. 25 0.00 28.25 17.68 4.94 0.62 5.67 10.23 7.67 14.09 2.94 20.53 3.04 14.28 5.74 3.13 13.16 17.12 30.95 0.00 30.95 17.12 13.16 3.13 5.74 14.28 3.04 20.53 2.60 17.98 3.89 10.39 5.74 2.03 11.42 16.08 30.16 0.00 30.16 16.08 11.42 2.03 5. 74 10.39 3.89 17.98 2.27 10.65 4.93 7.66 4.79 0.22 6.72 4.52 12.76 0.00 12.76 4.52 6.72 0.22 4.79 7.66 4.93 10.65 1.96 14.32 7.39 10.81 7.04 2.22 7.36 13.80 26.93 0.00 26.93 13.80 7.36 2.22 7.04 10.81 7.39 14.32 1.66 25.85 8.83 18. 50 10.50 1.74 18.33 19.42 39.21 0.00 39.21 19.42 18.33 1.74 10.50 18.50 8.83 25.85 1.38 23.77 11.11 18.06 10.85 0.17 17.06 18.17 37.80 0.00 37.80 18.17 17.06 0.17 10.85 18.06 11.11 23.77 1.12 18.43 6.43 11.75 8.22 0.60 10.18 11.17 25.18 0.00 25.18 11.17 10.18 0.60 8.22 11.75 6.43 18.43 0.87 15.28 3.60 9.99 5.15 0.75 9.11 10.59 20.78 0.00 20.78 10.59 9.11 0.75 5.15 9.99 3.60 15.28 0.63 13.97 4.47 9.71 5.26 0.34 7.99 9.92 19.40 0.00 19.40 9.92 7.99 0.34 5.26 9.71 4.47 13.97 0.41 13.66 9.32 9.62 7.87 3.51 5.75 7.35 19.35 0.00 19.35 7.35 5.75 3.51 7.87 9.62 9.32 13.66 0.20 24.24 9.05 18.31 9.96 1.65 17.66 16.03 33.44 0.00 33.44 16.03 17.66 1.65 9.96 18.31 9.05 24.24 0.00 26.78 11.50 20.09 12.39 0.33 19.07 22.05 43.81 0.00 43.81 22.04 19.07 0.33 12.39 20.09 11.50 26.78 0.20 24.24 9.05 18.31 9.96 1.65 17.66 16.03 33.44 0.00 33.44 16.03 17.66 1.65 9.96 18.31 9.05 24.24 0.41 13.66 9.32 9.62 7.87 3.51 5.75 7.35 19.35 0.00 19.35 7. 35 5.75 3.51 7.87 9.62 9.32 13.66 0.63 13.97 4.47 9.71 5.26 0.34 7.99 9.92 19.40 0.00 19.40 9.92 7.99 0.34 5.26 9.71 4.47 13.97 0.87 15.28 3.60 9.99 5.15 0.75 9.11 10.59 20.78 0.00 20.78 10.59 9.11 0.75 5.15 9.99 3.60 15.28 1.12 18.43 6.43 11.75 8.22 0.60 10.18 11.17 25.18 0.00 25.18 11.17 10.18 0.60 8.22 11.75 6.43 18.43 1.38 23.77 11.11 18.06 10.85 0.17 17.06 18.17 37.80 0.00 37.80 18.17 17.06 0.17 10.85 18.06 11.11 23.77 1.66 25.85 8.83 18.50 10.50 1.74 18.33 19.42 39.21 0.00 39.21 19.42 18.33 1.74 10.50 18.50 8.83 25.85 1.96 14.32 7.39 10.81 7.04 2.22 7.36 13.80 26.93 0.00 26.93 13.80 7.36 2.22 7.04 10.81 7.39 14.32 2.27 10.65 4.93 7.66 4.79 0.22 6.72 4.52 12.76 0.00 12.76 4.52 6.72 0.22 4 .79 7.66 4.93 10.65 2.60 17.98 3.89 10.39 5.74 2.03 11.42 16.08 30.16 0.00 30.16 16.08 11.42 2.03 5.74 10.39 3.89 17.98 2.94 20.53 3.04 14.28 5.74 3.13 13.16 17.12 30.95 0.00 30.95 17.12 13.16 3.13 5.74 14.28 3.04 20.53 3.31 14.09 7.67 10.23 5.67 0.62 4.94 17.68 28.25 0.00 28.25 17.68 4.94 0.62 5.67 10.23 7.67 14.09 3.70 12.90 1.45 6.41 3.32 4.31 8.20 10.63 18.34 0.00 18.34 10.63 8.20 4.31 3.32 6.41 1.45 12.90 4.11 5.46 0.45 3.52 0.85 2.51 4.39 6.47 10.75 0.00 10.75 6.47 4.39 2.51 0.85 3.52 0.45 5.46 PAGE 232 232 APPENDIX C SELECTED ADINA CODE FOR 3D TIRE PAVEMENT CONTACT MODEL PAGE 233 233 Command file created from session file information stored within AUI database Database created 1 May 2009, 00:00:00 * by ADINA: AUI version 8.3.1 * DATABASE NEW SAVE=NO PROMPT=NO FEPROGRAM ADINA CONTROL FILEVERSION=V83 FEPROGRAM PROGRAM=ADINA CONTROL PLOTUNIT=PERCENT VERBOSE=YES ERRORLIM=0 LOGLIMIT=0 UNDO=5, PROMPTDE=UNKNOWN AUTOREPA=YES DRAWMATT=YES DRAWTEXT=EXACT, DRAWLINE=EXACT DRAWFILL=EXACT AUTOMREB=YES ZONECOPY=NO, SWEEPCOI=YES SESSIONS=YES DYNAMICT=YES UPDATETH=YES AUTOREGE=NO, ERRORACT=CONTINUE FILEVERS=V83 INITFCHE=NO SIGDIGIT=6, AUTOZONE=YES PSFILEVE=V0 COORDINATES POINT SYSTEM=0 @CLEAR 1 0.00000000000000 9.00000000000000 19.2000000000000 0 2 0.00000000000000 9.00000000000000 20.0000000000000 0 3 0.00000000000000 7.40000000000000 20.0000000000000 0 4 0.00000000000000 7.40000000000000 19.2000000000000 0 5 0.00000000000000 6.60000000000000 19.2000000000000 0 6 0.00000000000000 6.60000000000000 20.0000000000000 0 7 0.00000000000000 5.40000000000000 20.0000000000000 0 8 0.00000000000000 5.40000000000000 19.2000000000000 0 9 0.000 00000000000 4.60000000000000 19.2000000000000 0 10 0.00000000000000 4.60000000000000 20.0000000000000 0 11 0.00000000000000 3.40000000000000 20.0000000000000 0 12 0.00000000000000 3.40000000000000 19.2000000000000 0 13 0.00000000000000 2.600000000 00000 19.2000000000000 0 14 0.00000000000000 2.60000000000000 20.0000000000000 0 15 0.00000000000000 1.40000000000000 20.0000000000000 0 16 0.00000000000000 1.40000000000000 19.2000000000000 0 17 0.00000000000000 0.600000000000000 19.2000000000000 0 18 0.00000000000000 0.600000000000000 20.0000000000000 0 19 0.00000000000000 0.600000000000000 20.0000000000000 0 20 0.00000000000000 0.600000000000000 19.2000000000000 0 21 0.00000000000000 1.40000000000000 19.2000000000000 0 22 0.00000000000000 1 .40000000000000 20.0000000000000 0 23 0.00000000000000 2.60000000000000 20.0000000000000 0 24 0.00000000000000 2.60000000000000 19.2000000000000 0 25 0.00000000000000 3.40000000000000 19.2000000000000 0 26 0.00000000000000 3.40000000000000 20.00000000 00000 0 27 0.00000000000000 4.60000000000000 20.0000000000000 0 28 0.00000000000000 4.60000000000000 19.2000000000000 0 29 0.00000000000000 5.40000000000000 19.2000000000000 0 30 0.00000000000000 5.40000000000000 20.0000000000000 0 31 0.00000000000000 6.60000000000000 20.0000000000000 0 32 0.00000000000000 6.60000000000000 19.2000000000000 0 33 0.00000000000000 7.40000000000000 19.2000000000000 0 34 0.00000000000000 7.40000000000000 20.0000000000000 0 35 0.00000000000000 9.00000000000000 20.0000000000000 0 36 0.00000000000000 9.00000000000000 19.2000000000000 0 37 0.00000000000000 9.00000000000000 18.8000000000000 0 38 0.00000000000000 9.00000000000000 18.8000000000000 0 39 0.00000000000000 7.40000000000000 18.8000000000000 0 40 0.000000000000 00 7.40000000000000 18.8000000000000 0 41 0.00000000000000 9.00000000000000 18.0000000000000 0 42 0.00000000000000 9.00000000000000 18.0000000000000 0 43 0.00000000000000 8.20000000000000 18.1000000000000 0 44 0.00000000000000 8.20000000000000 18.10 00000000000 0 45 0.00000000000000 0.00000000000000 18.6000000000000 0 46 0.00000000000000 8.00000000000000 11.0000000000000 0 47 0.00000000000000 8.00000000000000 11.0000000000000 0 48 0.00000000000000 9.00000000000000 11.0000000000000 0 49 0.00000000000000 9.00000000000000 11.0000000000000 0 PAGE 234 234 50 0.00000000000000 9.80000000000000 15.0000000000000 0 51 0.00000000000000 9.80000000000000 15.0000000000000 0 52 0.00000000000000 8.80000000000000 11.0000000000000 0 53 0.00000000000000 8.80000000000000 11.0000000000000 0 54 0.00000000000000 9.40000000000000 15.2000000000000 0 55 0.00000000000000 9.40000000000000 15.2000000000000 0 56 0.00000000000000 9.20000000000000 15.0000000000000 0 57 0.00000000000000 9.20000000000000 15.0000000000000 0 58 0.00 000000000000 7.70000000000000 17.5000000000000 0 59 0.00000000000000 7.70000000000000 17.5000000000000 0 60 0.00000000000000 0.00000000000000 18.3000000000000 0 61 0.00000000000000 8.00000000000000 9.00000000000000 0 62 0.00000000000000 8.00000000000 000 9.00000000000000 0 63 0.00000000000000 9.00000000000000 9.00000000000000 0 64 0.00000000000000 9.00000000000000 9.00000000000000 0 65 0.00000000000000 11.0000000000000 9.00000000000000 0 66 0.00000000000000 11.0000000000000 9.00000000000000 0 67 0.00000000000000 11.0000000000000 8.00000000000000 0 68 0.00000000000000 11.0000000000000 8.00000000000000 0 69 0.00000000000000 9.00000000000000 8.00000000000000 0 70 0.00000000000000 9.00000000000000 8.00000000000000 0 71 0.00000000000000 8.00000 000000000 8.00000000000000 0 72 0.00000000000000 8.00000000000000 8.00000000000000 0 73 15.0000000000000 12.0000000000000 20.0000000000000 0 74 15.0000000000000 12.0000000000000 20.0000000000000 0 75 15.0000000000000 12.0000000000000 20.000000000000 0 0 76 15.0000000000000 12.0000000000000 20.0000000000000 0 77 15.0000000000000 12.0000000000000 23.0000000000000 0 78 15.0000000000000 12.0000000000000 23.0000000000000 0 79 15.0000000000000 12.0000000000000 23.0000000000000 0 80 15.0000000000000 12.0000000000000 23.0000000000000 0 81 15.0000000000000 15.0000000000000 20.0000000000000 0 82 15.0000000000000 15.0000000000000 23.0000000000000 0 83 15.0000000000000 15.0000000000000 20.0000000000000 0 84 15.0000000000000 15.0000000000000 23.0000000000000 0 85 15.0000000000000 15.0000000000000 20.0000000000000 0 86 15.0000000000000 15.0000000000000 23.0000000000000 0 87 15.0000000000000 15.0000000000000 20.0000000000000 0 88 15.0000000000000 15.0000000000000 23.0000000000000 0 @ LINE POLYLINE NAME=1 TYPE=SEGMENTED @CLEAR 43 0.00000000000000 0.00000000000000 0.00000000000000 45 0.00000000000000 0.00000000000000 0.00000000000000 44 0.00000000000000 0.00000000000000 0.00000000000000 @ LINE POLYLINE NAME=2 TYPE=SPLINE @CLEAR 48 0.00000000000000 0.00000000000000 0.00000000000000 50 0.00000000000000 0.00000000000000 0.00000000000000 41 0.00000000000000 0.00000000000000 0.00000000000000 @ DELETE LINE ALL FIRST=1 LAST=1 LINE POLYLINE NAME=3 TYPE=SPLINE @CLEAR 43 0.00000000000000 0.00000000000000 0.00000000000000 45 0.00000000000000 0.00000000000000 0.00000000000000 44 0.00000000000000 0.00000000000000 0.00000000000000 @ LINE POLYLINE NAME=4 TYPE=SPLINE @CLEAR 46 0.00000000000000 0.00000000000000 0.00000000000000 56 0.00000000000000 0.00000000000000 0.00000000000000 58 0.00000000000000 0.00000000000000 0.00000000000000 @ PAGE 235 235 LINE POLYLINE NAME=5 TYPE=SPLINE @CLEAR 47 0.00000000000000 0.00000000000000 0.00000000000000 57 0.00000000000000 0.00000000000000 0.00000000000000 59 0.00000000000000 0 .00000000000000 0.00000000000000 @ LINE POLYLINE NAME=6 TYPE=SPLINE @CLEAR 58 0.00000000000000 0.00000000000000 0.00000000000000 60 0.00000000000000 0.00000000000000 0.00000000000000 59 0.00000000000000 0.00000000000000 0.00000000000000 @ DELETE LINE A LL FIRST=4 LAST=4 DELETE LINE ALL FIRST=6 LAST=6 DELETE LINE ALL FIRST=3 LAST=3 DELETE LINE ALL FIRST=2 LAST=2 DELETE LINE ALL FIRST=5 LAST=5 SURFACE VERTEX NAME=1 P1=1 P2=2 P3=3 P4=4 SURFACE VERTEX NAME=2 P1=4 P2=3 P3=6 P4=5 SURFACE VERTEX NAME=3 P1=5 P2=6 P3=7 P4=8 SURFACE VERTEX NAME=4 P1=8 P2=7 P3=10 P4=9 SURFACE VERTEX NAME=5 P1=9 P2=10 P3=11 P4=12 SURFACE VERTEX NAME=6 P1=12 P2=11 P3=14 P4=13 SURFACE VERTEX NAME=7 P1=13 P2=14 P3=15 P4=16 SURFACE VERTEX NAME=8 P1=16 P2=15 P3=18 P4=17 SURFACE VERTEX NAME=9 P1=17 P2=18 P3=19 P4=20 SURFACE VERTEX NAME=10 P1=20 P2=19 P3=22 P4=21 SURFACE VERTEX NAME=11 P1=21 P2=22 P3=23 P4=24 SURFACE VERTEX NAME=12 P1=24 P2=23 P3=26 P4=25 SURFACE VERTEX NAME=13 P1=25 P2=26 P3=27 P4=28 SURF ACE VERTEX NAME=14 P1=28 P2=27 P3=30 P4=29 SURFACE VERTEX NAME=15 P1=29 P2=30 P3=31 P4=32 SURFACE VERTEX NAME=16 P1=32 P2=31 P3=34 P4=33 SURFACE VERTEX NAME=17 P1=33 P2=34 P3=35 P4=36 SURFACE VERTEX NAME=18 P1=38 P2=1 P3=4 P4=39 SURFACE VERTEX NA ME=19 P1=39 P2=4 P3=33 P4=40 SURFACE VERTEX NAME=20 P1=40 P2=33 P3=36 P4=37 SURFACE VERTEX NAME=21 P1=41 P2=38 P3=39 P4=43 SURFACE VERTEX NAME=22 P1=44 P2=40 P3=37 P4=42 LINE POLYLINE NAME=67 TYPE=SPLINE PAGE 236 236 @CLEAR 43 0.00000000000000 0.00000000000000 0.00000000000000 45 0.00000000000000 0.00000000000000 0.00000000000000 44 0.00000000000000 0.00000000000000 0.00000000000000 @ LINE POLYLINE NAME=68 TYPE=SPLINE @CLEAR 58 0.00000000000000 0.00000000000000 0.00000000000000 60 0.00000000000000 0.0000000000 0000 0.00000000000000 59 0.00000000000000 0.00000000000000 0.00000000000000 @ SURFACE PATCH NAME=23 EDGE1=62 EDGE2=58 EDGE3=64 EDGE4=67 LINE STRAIGHT NAME=69 P1=58 P2=43 LINE STRAIGHT NAME=70 P1=59 P2=44 LINE POLYLINE NAME=71 TYPE=SPLINE @CLEAR 48 0.00000000000000 0.00000000000000 0.00000000000000 50 0.00000000000000 0.00000000000000 0.00000000000000 41 0.00000000000000 0.00000000000000 0.00000000000000 @ LINE POLYLINE NAME=72 TYPE=SPLINE @CLEAR 52 0.00000000000000 0.00000000000000 0.0000000000000 0 54 0.00000000000000 0.00000000000000 0.00000000000000 43 0.00000000000000 0.00000000000000 0.00000000000000 @ LINE POLYLINE NAME=73 TYPE=SPLINE @CLEAR 46 0.00000000000000 0.00000000000000 0.00000000000000 56 0.00000000000000 0.00000000000000 0.00000000 000000 58 0.00000000000000 0.00000000000000 0.00000000000000 @ LINE POLYLINE NAME=74 TYPE=SPLINE @CLEAR 47 0.00000000000000 0.00000000000000 0.00000000000000 57 0.00000000000000 0.00000000000000 0.00000000000000 59 0.00000000000000 0.00000000000000 0.000 00000000000 @ LINE POLYLINE NAME=75 TYPE=SPLINE @CLEAR 53 0.00000000000000 0.00000000000000 0.00000000000000 55 0.00000000000000 0.00000000000000 0.00000000000000 44 0.00000000000000 0.00000000000000 0.00000000000000 @ LINE POLYLINE NAME=76 TYPE=SPLINE @CLEAR 49 0.00000000000000 0.00000000000000 0.00000000000000 51 0.00000000000000 0.00000000000000 0.00000000000000 42 0.00000000000000 0.00000000000000 0.00000000000000 @ LINE STRAIGHT NAME=77 P1=48 P2=52 LINE STRAIGHT NAME=78 P1=52 P2=46 LINE STRAI GHT NAME=79 P1=47 P2=53 LINE STRAIGHT NAME=80 P1=53 P2=49 SURFACE PATCH NAME=24 EDGE1=71 EDGE2=63 EDGE3=72 EDGE4=77 PAGE 237 237 SURFACE PATCH NAME=25 EDGE1=72 EDGE2=69 EDGE3=73 EDGE4=78 SURFACE PATCH NAME=26 EDGE1=67 EDGE2=70 EDGE3=68 EDGE4=69 SURFACE PATCH NAME=27 EDGE1=74 EDGE2=70 EDGE3=75 EDGE4=79 SURFACE PATCH NAME=28 EDGE1=76 EDGE2=66 EDGE3=75 EDGE4=80 SURFACE VERTEX NAME=29 P1=63 P2=48 P3=46 P4=61 SURFACE VERTEX NAME=30 P1=61 P2=46 P3=47 P4=62 SURFACE VERTEX NAME=31 P1=62 P2=47 P3=49 P4=64 SUR FACE VERTEX NAME=32 P1=67 P2=65 P3=63 P4=69 SURFACE VERTEX NAME=33 P1=69 P2=63 P3=61 P4=71 SURFACE VERTEX NAME=34 P1=71 P2=61 P3=62 P4=72 SURFACE VERTEX NAME=35 P1=72 P2=62 P3=64 P4=70 SURFACE VERTEX NAME=36 P1=70 P2=64 P3=66 P4=68 VOLUME REVOLVE D NAME=1 MODE=AXIS SURFACE=32 ANGLE=360.000000000000, SYSTEM=0 AXIS=YL PCOINCID=YES PTOLERAN=1.00000000000000E 05, NDIV=1 @CLEAR 36 @ VOLUME REVOLVED NAME=3 MODE=AXIS SURFACE=33 ANGLE=360.000000000000, SYSTEM=0 AXIS=YL PCOINCID=YES PTOLERA N=1.00000000000000E 05, NDIV=1 @CLEAR 35 @ VOLUME REVOLVED NAME=5 MODE=AXIS SURFACE=34 ANGLE=360.000000000000, SYSTEM=0 AXIS=YL PCOINCID=YES PTOLERAN=1.00000000000000E 05, NDIV=1 SUBDIVIDE VOLUME NAME=1 MODE=DIVISIONS NDIV1=1 NDIV2=2 NDI V3=30, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS3=NO SUBDIVIDE VOLUME NAME=2 MODE=DIVISIONS NDIV1=1 NDIV2=2 NDIV3=30, RATIO1=1.00000000000000 RAT IO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS3=NO SUBDIVIDE VOLUME NAME=3 MODE=DIVISIONS NDIV1=1 NDIV2=5 NDIV3=30, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.000 00000000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS3=NO @CLEAR 4 @ SUBDIVIDE VOLUME NAME=5 MODE=DIVISIONS NDIV1=1 NDIV2=8 NDIV3=30, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETR IC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS3=NO VOLUME REVOLVED NAME=6 MODE=AXIS SURFACE=29 ANGLE=360.000000000000, SYSTEM=0 AXIS=YL PCOINCID=YES PTOLERAN=1.00000000000000E 05, NDIV=1 @CLEAR PAGE 238 238 31 @ SUBDIVIDE VOLUME NAME=6 MODE=DIVISIONS NDIV1=2 NDIV2=5 NDIV3=30, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS3=NO @CLEAR 7 @ VOLUME REVOLVED NAME=8 MODE=AXIS SURFACE=30 ANGLE=360.000000000000, SYSTEM=0 AXIS=YL PCOINCID=YES PTOLERAN=1.00000000000000E 05, NDIV=1 SUBDIVIDE VOLUME NAME=8 MODE=DIVISIONS NDIV1=2 NDIV2=8 NDIV3=30, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS3=NO MATERIAL ELASTIC NAME=1 E=1.00000000000000E+12 NU=0.120000000000000, DENSITY=0.00000000000000 ALPHA=0.00000000000000 MDESCRIP='NONE' MATERIAL ELASTIC NAME=2 E=400.000000000000 NU=0.495000000000000, DENSITY= 0.00000000000000 ALPHA=0.00000000000000 MDESCRIP='NONE' MATERIAL ELASTIC NAME=3 E=2000.00000000000 NU=0.300000000000000, DENSITY=0.00000000000000 ALPHA=0.00000000000000 MDESCRIP='NONE' MATERIAL ELASTIC NAME=4 E=2000.00000000000 NU=0.30000000000000 0, DENSITY=0.00000000000000 ALPHA=0.00000000000000 MDESCRIP='NONE' MATERIAL ELASTIC NAME=5 E=2000000.00000000 NU=0.100000000000000, DENSITY=0.00000000000000 ALPHA=0.00000000000000 MDESCRIP='NONE' MATERIAL ELASTIC NAME=6 E=400.000000000000 NU= 0.495000000000000, DENSITY=0.00000000000000 ALPHA=0.00000000000000 MDESCRIP='NONE' MATERIAL ELASTIC NAME=7 E=400.000000000000 NU=0.495000000000000, DENSITY=0.00000000000000 ALPHA=0.00000000000000 MDESCRIP='NONE' MATERIAL ELASTIC NAME=8 E=400. 000000000000 NU=0.495000000000000, DENSITY=0.00000000000000 ALPHA=0.00000000000000 MDESCRIP='NONE' MATERIAL ELASTIC NAME=9 E=500.000000000000 NU=0.495000000000000, DENSITY=0.00000000000000 ALPHA=0.00000000000000 MDESCRIP='NONE' MATERIAL ELAST IC NAME=10 E=9.80000000000000E 06 NU=0.499000000000000, DENSITY=0.00000000000000 ALPHA=0.00000000000000 MDESCRIP='NONE' MATERIAL ELASTIC NAME=11 E=1.00000000000000E+12 NU=0.100000000000000, DENSITY=0.00000000000000 ALPHA=0.00000000000000 MDESCR IP='NONE' EGROUP THREEDSOLID NAME=1 DISPLACE=DEFAULT STRAINS=DEFAULT MATERIAL=1, RSINT=DEFAULT TINT=DEFAULT RESULTS=STRESSES DEGEN=NO FORMULAT=0, STRESSRE=GLOBAL INITIALS=NONE FRACTUR=NO CMASS=DEFAULT, STRAIN F=0 UL FORMU=DEFAULT LVUS1=0 L VUS2=0 SED=NO RUPTURE=ADINA, INCOMPAT=DEFAULT TIME OFF=0.00000000000000 POROUS=NO, WTMC=1.00000000000000 OPTION=NONE DESCRIPT='NONE' PRINT=DEFAULT, SAVE=DEFAULT TBIRTH=0.00000000000000 TDEATH=0.00000000000000 EGROUP THREEDSOLID NAME=2 DISP LACE=DEFAULT STRAINS=DEFAULT MATERIAL=2, RSINT=DEFAULT TINT=DEFAULT RESULTS=STRESSES DEGEN=NO FORMULAT=0, STRESSRE=GLOBAL INITIALS=NONE FRACTUR=NO CMASS=DEFAULT, STRAIN F=0 UL FORMU=DEFAULT LVUS1=0 LVUS2=0 SED=NO RUPTURE=ADINA, INCOMPAT =DEFAULT TIMEOFF=0.00000000000000 POROUS=NO, WTMC=1.00000000000000 OPTION=NONE DESCRIPT='NONE' PRINT=DEFAULT, SAVE=DEFAULT TBIRTH=0.00000000000000 TDEATH=0.00000000000000 EGROUP THREEDSOLID NAME=3 DISPLACE=DEFAULT STRAINS=DEFAULT MATERIAL=3, PAGE 239 239 RSINT=DEFAULT TINT=DEFAULT RESULTS=STRESSES DEGEN=NO FORMULAT=0, STRESSRE=GLOBAL INITIALS=NONE FRACTUR=NO CMASS=DEFAULT, STRAIN F=0 UL FORMU=DEFAULT LVUS1=0 LVUS2=0 SED=NO RUPTURE=ADINA, INCOMPAT=DEFAULT TIME OFF=0.00000000000000 POROUS= NO, WTMC=1.00000000000000 OPTION=NONE DESCRIPT='NONE' PRINT=DEFAULT, SAVE=DEFAULT TBIRTH=0.00000000000000 TDEATH=0.00000000000000 EGROUP THREEDSOLID NAME=4 DISPLACE=DEFAULT STRAINS=DEFAULT MATERIAL=4, RSINT=DEFAULT TINT=DEFAULT RESULTS=STR ESSES DEGEN=NO FORMULAT=0, STRESSRE=GLOBAL INITIALS=NONE FRACTUR=NO CMASS=DEFAULT, STRAIN F=0 UL FORMU=DEFAULT LVUS1=0 LVUS2=0 SED=NO RUPTURE=ADINA, INCOMPAT=DEFAULT TIME OFF=0.00000000000000 POROUS=NO, WTMC=1.00000000000000 OPTION=NONE DESCRIPT='NONE' PRINT=DEFAULT, SAVE=DEFAULT TBIRTH=0.00000000000000 TDEATH=0.00000000000000 EGROUP THREEDSOLID NAME=5 DISPLACE=DEFAULT STRAINS=DEFAULT MATERIAL=5, RSINT=DEFAULT TINT=DEFAULT RESULTS=STRESSES DEGEN=NO FORMULAT=0, STRESSRE=G LOBAL INITIALS=NONE FRACTUR=NO CMASS=DEFAULT, STRAIN F=0 UL FORMU=DEFAULT LVUS1=0 LVUS2=0 SED=NO RUPTURE=ADINA, INCOMPAT=DEFAULT TIME OFF=0.00000000000000 POROUS=NO, WTMC=1.00000000000000 OPTION=NONE DESCRIPT='NONE' PRINT=DEFAULT, SAVE= DEFAULT TBIRTH=0.00000000000000 TDEATH=0.00000000000000 EGROUP THREEDSOLID NAME=6 DISPLACE=DEFAULT STRAINS=DEFAULT MATERIAL=6, RSINT=DEFAULT TINT=DEFAULT RESULTS=STRESSES DEGEN=NO FORMULAT=0, STRESSRE=GLOBAL INITIALS=NONE FRACTUR=NO CMASS=DEFAU LT, STRAIN F=0 UL FORMU=DEFAULT LVUS1=0 LVUS2=0 SED=NO RUPTURE=ADINA, INCOMPAT=DEFAULT TIME OFF=0.00000000000000 POROUS=NO, WTMC=1.00000000000000 OPTION=NONE DESCRIPT='NONE' PRINT=DEFAULT, SAVE=DEFAULT TBIRTH=0.00000000000000 TDEATH=0.0 0000000000000 EGROUP THREEDSOLID NAME=7 DISPLACE=DEFAULT STRAINS=DEFAULT MATERIAL=7, RSINT=DEFAULT TINT=DEFAULT RESULTS=STRESSES DEGEN=NO FORMULAT=0, STRESSRE=GLOBAL INITIALS=NONE FRACTUR=NO CMASS=DEFAULT, STRAIN F=0 UL FORMU=DEFAULT LVUS1 =0 LVUS2=0 SED=NO RUPTURE=ADINA, INCOMPAT=DEFAULT TIME OFF=0.00000000000000 POROUS=NO, WTMC=1.00000000000000 OPTION=NONE DESCRIPT='NONE' PRINT=DEFAULT, SAVE=DEFAULT TBIRTH=0.00000000000000 TDEATH=0.00000000000000 EGROUP THREEDSOLID NAME=8 DISPLACE=DEFAULT STRAINS=DEFAULT MATERIAL=8, RSINT=DEFAULT TINT=DEFAULT RESULTS=STRESSES DEGEN=NO FORMULAT=0, STRESSRE=GLOBAL INITIALS=NONE FRACTUR=NO CMASS=DEFAULT, STRAIN F=0 UL FORMU=DEFAULT LVUS1=0 LVUS2=0 SED=NO RUPTURE=ADINA, INCO MPAT=DEFAULT TIMEOFF=0.00000000000000 POROUS=NO, WTMC=1.00000000000000 OPTION=NONE DESCRIPT='NONE' PRINT=DEFAULT, SAVE=DEFAULT TBIRTH=0.00000000000000 TDEATH=0.00000000000000 EGROUP THREEDSOLID NAME=9 DISPLACE=DEFAULT STRAINS=DEFAULT MATERIAL= 9, RSINT=DEFAULT TINT=DEFAULT RESULTS=STRESSES DEGEN=NO FORMULAT=0, STRESSRE=GLOBAL INITIALS=NONE FRACTUR=NO CMASS=DEFAULT, STRAIN F=0 UL FORMU=DEFAULT LVUS1=0 LVUS2=0 SED=NO RUPTURE=ADINA, INCOMPAT=DEFAULT TIME OFF=0.00000000000000 POR OUS=NO, WTMC=1.00000000000000 OPTION=NONE DESCRIPT='NONE' PRINT=DEFAULT, SAVE=DEFAULT TBIRTH=0.00000000000000 TDEATH=0.00000000000000 EGROUP THREEDSOLID NAME=10 DISPLACE=DEFAULT STRAINS=DEFAULT, MATERIAL=10 RSINT=DEFAULT TINT=DEFAULT RESULTS=STRESSES DEGEN=NO, FORMULAT=0 STRESSRE=GLOBAL INITIALS=NONE FRACTUR=NO, CMASS=DEFAULT STRAIN F=0 UL FORMU=DEFAULT LVUS1=0 LVUS2=0 SED=NO, RUPTURE=ADINA INCOMPAT=DEFAULT TIME OFF=0.00000000000000, POROUS=NO WTMC=1.00000000000000 OPTIO N=NONE DESCRIPT='NONE', PRINT=DEFAULT SAVE=DEFAULT TBIRTH=0.00000000000000, TDEATH=0.00000000000000 EGROUP THREEDSOLID NAME=11 DISPLACE=DEFAULT STRAINS=DEFAULT, MATERIAL=11 RSINT=DEFAULT TINT=DEFAULT RESULTS=STRESSES DEGEN=NO, FORMULAT=0 STRESSRE=GLOBAL INITIALS=NONE FRACTUR=NO, CMASS=DEFAULT STRAIN F=0 UL FORMU=DEFAULT LVUS1=0 LVUS2=0 SED=NO, RUPTURE=ADINA INCOMPAT=DEFAULT TIME OFF=0.00000000000000, POROUS=NO WTMC=1.00000000000000 OPTION=NONE DESCRIPT='NONE', PAGE 240 240 PRINT =DEFAULT SAVE=DEFAULT TBIRTH=0.00000000000000, TDEATH=0.00000000000000 GVOLUME NODES=8 PATTERN=0 NCOINCID=BOUNDARIES NCFACE=123456 NCEDGE=, '123456789ABC' NCVERTEX=12345678 NCTOLERA=1.00000000000000E 05, SUBSTRUC=0 GROUP=1 MESHING=MAPPED PREFSH AP=AUTOMATIC, DEGENERA=YES COLLAPSE=NO MIDNODES=CURVED METHOD=DELAUNAY, BOUNDARY=ADVFRONT @CLEAR 1 2 3 4 5 6 7 8 @ VOLUME REVOLVED NAME=9 MODE=AXIS SURFACE=24 ANGLE=360.000000000000, SYSTEM=0 AXIS=YL PCOINCID=YES PTOLERAN=1.00000000000000E 05, NDIV=1 @CLEAR 28 @ VOLUME REVOLVED NAME=11 MODE=AXIS SURFACE=25 ANGLE=360.000000000000, SYSTEM=0 AXIS=YL PCOINCID=YES PTOLERAN=1.00000000000000E 05, NDIV=1 @CLEAR 27 @ LOAD PRESSURE NAME=1 MAGNITUD=100.000000000000 BETA=0.00000000000000, LINE=0 APPLY LOAD BODY=0 @CLEAR 1 'PRESSURE' 1 'SURFACE' 59 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 @ APPLY LOAD BODY=0 @CLEAR 1 'PRESSURE' 1 'SURFACE' 59 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 @ SUBDIVIDE VOLUME NAME=9 MODE=DIVISIONS NDIV1=5 NDIV2=2 NDIV3=30, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS 3=NO @CLEAR 10 @ SUBDIVIDE VOLUME NAME=11 MODE=DIVISIONS NDIV1=5 NDIV2=2 NDIV3=30, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS3=NO @CLEAR 12 @ GVOLUME NODES=8 PATTERN=0 NCOINCID=BOUNDARIES NCFACE=123456 NCEDGE=, '123456789ABC' NCVERTEX=12345678 NCTOLERA=1.00000000000000E 05, SUBSTRUC=0 GROUP=2 MESHING=MAPPED PREFSHAP=AUTOMATIC, DEGENERA=YES COLLAPSE=NO MIDNODES=CURVED METHOD=DELAUNAY, BOUNDARY=ADVFRONT @CLEAR PAGE 241 241 9 10 @ GVOLUME NODES=8 PATTERN=0 NCOINCID=BOUNDARIES NCFACE=123456 NCEDGE=, '123456789ABC' NCVERTEX=12345678 NCTOLERA=1.00000000000000E 05, SUBSTRUC=0 GROUP=3 MESHING=MAPPED PREFSHAP=AUTOMATIC, DEGENERA=YES COLLAPSE=NO MIDNODES=CURVED METHOD=DELAUNAY, BOUNDARY=ADVFRONT @CLEAR 11 12 @ APPLY LOAD BODY=0 @CLEAR 1 'PRESSURE' 1 'SURFACE' 59 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 @ APPLY LOAD BODY=0 @CLEAR 1 'PRESSURE' 1 'SU RFACE' 59 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 2 'PRESSURE' 1 'SURFACE' 73 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 @ APPLY LOAD BODY=0 @CLEAR 1 'PRESSURE' 1 'SURFA CE' 59 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 2 'PRESSURE' 1 'SURFACE' 71 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 3 'PRESSURE' 1 'SURFACE' 69 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 @ APPLY LOAD BODY=0 @CLEAR 1 'PRESSURE' 1 'SURFACE' 59 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 2 'PRESSURE' 1 'SURFACE' 71 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 3 'PRESSURE' 1 'SURFACE' 69 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 @ VOLUME REVOLVED NAME=13 MODE=AXIS SURFACE=26 ANGLE=360.000000000000, SYSTEM=0 AXI S=YL PCOINCID=YES PTOLERAN=1.00000000000000E 05, NDIV=1 SUBDIVIDE VOLUME NAME=13 MODE=DIVISIONS NDIV1=2 NDIV2=8 NDIV3=30, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS3=NO SUBDIVIDE VOLUME NAME=13 MODE=DIVISIONS NDIV1=8 NDIV2=2 NDIV3=30, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS3=NO GVOLUME NODES =8 PATTERN=0 NCOINCID=BOUNDARIES NCFACE=123456 NCEDGE=, '123456789ABC' NCVERTEX=12345678 NCTOLERA=1.00000000000000E 05, SUBSTRUC=0 GROUP=4 MESHING=MAPPED PREFSHAP=AUTOMATIC, DEGENERA=YES COLLAPSE=NO MIDNODES=CURVED METHOD=DELAUNAY, BOUNDARY= ADVFRONT @CLEAR 13 @ PAGE 242 242 APPLY LOAD BODY=0 @CLEAR 1 'PRESSURE' 1 'SURFACE' 59 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 2 'PRESSURE' 1 'SURFACE' 71 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 3 'PRESSURE' 1 'SURFACE' 69 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 @ APPLY LOAD BODY=0 @CLEAR 1 'PRESSURE' 1 'SURFACE' 59 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.000000 00000000 1 0 2 'PRESSURE' 1 'SURFACE' 71 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 3 'PRESSURE' 1 'SURFACE' 69 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 4 'PRESSURE' 1 'S URFACE' 75 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 @ APPLY LOAD BODY=0 @CLEAR 1 'PRESSURE' 1 'SURFACE' 59 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 2 'PRESSURE' 1 'SURF ACE' 71 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 3 'PRESSURE' 1 'SURFACE' 69 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 4 'PRESSURE' 1 'SURFACE' 75 0 1 0.00000000000000 0 1 0 0 0 'NO', 0.00000000000000 0.00000000000000 1 0 @ VOLUME REVOLVED NAME=14 MODE=AXIS SURFACE=23 ANGLE=360.000000000000, SYSTEM=0 AXIS=YL PCOINCID=YES PTOLERAN=1.00000000000000E 05, NDIV=1 VOLUME REVOLVED NAME=15 MODE=AXIS SURFACE=21 ANGLE=360.000000000000, SYSTEM=0 AXIS=YL PCOINCID=YES PTOLERAN=1.00000000000000E 05, NDIV=1 @CLEAR 22 @ SUBDIVIDE VOLUME NAME=14 MODE=DIVISIONS NDIV1=2 NDIV2=8 NDIV3=30, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.000000 00000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS3=NO SUBDIVIDE VOLUME NAME=15 MODE=DIVISIONS NDIV1=2 NDIV2=2 NDIV3=30, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETRIC EXTEND=NON E CBIAS1=NO, CBIAS2=NO CBIAS3=NO @CLEAR 16 @ VOLUME REVOLVED NAME=17 MODE=AXIS SURFACE=19 ANGLE=360.000000000000, SYSTEM=0 AXIS=YL PCOINCID=YES PTOLERAN=1.00000000000000E 05, NDIV=1 VOLUME REVOLVED NAME=18 MODE=AXIS SURFACE=18 ANGLE=360. 000000000000, SYSTEM=0 AXIS=YL PCOINCID=YES PTOLERAN=1.00000000000000E 05, NDIV=1 @CLEAR 20 @ PAGE 243 243 SUBDIVIDE VOLUME NAME=17 MODE=DIVISIONS NDIV1=1 NDIV2=37 NDIV3=30, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS3=NO SUBDIVIDE VOLUME NAME=18 MODE=DIVISIONS NDIV1=1 NDIV2=4 NDIV3=30, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1= NO, CBIAS2=NO CBIAS3=NO @CLEAR 19 @ GVOLUME NODES=8 PATTERN=0 NCOINCID=BOUNDARIES NCFACE=123456 NCEDGE=, '123456789ABC' NCVERTEX=12345678 NCTOLERA=1.00000000000000E 05, SUBSTRUC=0 GROUP=5 MESHING=MAPPED PREFSHAP=AUTOMATIC, DEGENERA=YES COL LAPSE=NO MIDNODES=CURVED METHOD=DELAUNAY, BOUNDARY=ADVFRONT @CLEAR 14 @ GVOLUME NODES=8 PATTERN=0 NCOINCID=BOUNDARIES NCFACE=123456 NCEDGE=, '123456789ABC' NCVERTEX=12345678 NCTOLERA=1.00000000000000E 05, SUBSTRUC=0 GROUP=6 MESHING=MAPPED PREFSHAP=AUTOMATIC, DEGENERA=YES COLLAPSE=NO MIDNODES=CURVED METHOD=DELAUNAY, BOUNDARY=ADVFRONT @CLEAR 15 16 @ VOLUME REVOLVED NAME=20 MODE=AXIS SURFACE=1 ANGLE=360.000000000000, SYSTEM=0 AXIS=YL PCOINCID=YES PTOLERAN=1.00000000000000E 05, NDIV=1 @CLEAR 3 5 7 9 11 13 15 17 @ SUBDIVIDE VOLUME NAME=20 MODE=DIVISIONS NDIV1=2 NDIV2=4 NDIV3=180, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS3=NO SUBDIVIDE VOLUME NAME=28 MODE=DIVISIONS NDIV1=2 NDIV2=4 NDIV3=180, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS3=NO SUBDIVIDE VOLUME NAME=21 MODE=DIVIS IONS NDIV1=2 NDIV2=3 NDIV3=180, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS3=NO @CLEAR 22 23 24 25 26 27 @ VOLUME REVOLVED NAME=29 MODE=AXIS SURFACE=2 A NGLE=360.000000000000, PAGE 244 244 SYSTEM=0 AXIS=YL PCOINCID=YES PTOLERAN=1.00000000000000E 05, NDIV=1 @CLEAR 4 6 8 10 12 14 16 @ SUBDIVIDE VOLUME NAME=29 MODE=DIVISIONS NDIV1=2 NDIV2=2 NDIV3=180, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=1.00000000000000 PROGRESS=GEOMETRIC EXTEND=NONE CBIAS1=NO, CBIAS2=NO CBIAS3=NO @CLEAR 30 31 32 33 34 35 36 @ GVOLUME NODES=8 PATTERN=0 NCOINCID=BOUNDARIES NCFACE=123456 NCEDGE=, '123456789ABC' NCVERTEX=12345678 NCTOLERA=1.00000000000000E 05, SUBSTRUC=0 GROUP=7 MESHING=MAPPED PREFSHAP=AUTOMATIC, DEGENERA=YES COLLAPSE=NO MIDNODES=CURVED METHOD=DELAUNAY, BOUNDARY=ADVFRONT @CLEAR 17 @ GVOLUME NODES=8 PATTERN=0 NCOINCID=BOUNDARIES NCFACE=123456 NCEDGE=, '123456789ABC' NCVERTEX=12345678 NCTOLERA=1.00000000000000E 05, SUBSTRUC=0 GROUP=8 MESHING=MAPPED PREFSHAP=AUTOMATIC, DEGENERA=YES COLLAPSE=NO MIDNODES=CURVED METHOD=DELAUNAY, BOUNDARY=ADVFRONT @CLEAR 18 19 @ GVOLUME NODES=8 PATTERN=0 NCOINCID=BOUNDARIES NCFACE=12345 6 NCEDGE=, '123456789ABC' NCVERTEX=12345678 NCTOLERA=1.00000000000000E 05, SUBSTRUC=0 GROUP=9 MESHING=MAPPED PREFSHAP=AUTOMATIC, DEGENERA=YES COLLAPSE=NO MIDNODES=CURVED METHOD=DELAUNAY, BOUNDARY=ADVFRONT @CLEAR 20 21 22 23 24 25 26 27 28 @ GVOLUME NODES=8 PATTERN=0 NCOINCID=BOUNDARIES NCFACE=123456 NCEDGE=, '123456789ABC' NCVERTEX=12345678 NCTOLERA=1.00000000000000E 05, SUBSTRUC=0 GROUP=10 MESHING=MAPPED PREFSHAP=AUTOMATIC, DEGENERA=YES COLLAPSE=NO MIDNODES=CURVED METHOD=DELAUNAY, BOUNDARY=ADVFRONT @CLEAR 29 30 PAGE 245 245 31 32 33 34 35 36 @ VOLUME VERTEX NAME=37 SHAPE=HEX VERTEX1=76 VERTEX2=87 VERTEX3=83, VERTEX4=74 VERTEX5=80 VERTEX6=88 VERTEX7=84 VERTEX8=78 VOLUME VERTEX NAME=38 SHAPE=HEX VERTEX1=75 VERTEX2=76 VERTEX3=74, VERTEX4=73 VERTEX5=79 VERTEX6=80 VERTEX7=78 VERTEX8=77 VOLUME VERTEX NAME=39 SHAPE=HEX VERTEX1=85 VERTEX2=75 VERTEX3=73, VERTEX4=81 VERTEX5=86 VERTEX6=79 VERTEX7=77 VERTEX8=82 SUBDIVIDE VOLUME NAME=37 MODE=DIVISIONS NDIV1=6 NDIV2=15 NDIV3=3, RATIO1=0.200000000000000 RATIO2=1.00000000000000, RATIO3=0.200000000000000 PROGRESS=GEOMETRIC EXTEND=NONE, CBIAS1=NO CBIAS2=NO CBIAS3=NO SUBDIVIDE VOLUME NAME=38 MODE=DIVISIONS NDIV1=6 NDIV2=15 NDIV3=3, RATIO1=5.00000000000000 RATIO2=1. 00000000000000, RATIO3=0.200000000000000 PROGRESS=GEOMETRIC EXTEND=NONE, CBIAS1=NO CBIAS2=NO CBIAS3=NO SUBDIVIDE VOLUME NAME=38 MODE=DIVISIONS NDIV1=6 NDIV2=15 NDIV3=3, RATIO1=5.00000000000000 RATIO2=1.00000000000000, RATIO3=0.2000000 00000000 PROGRESS=GEOMETRIC EXTEND=NONE, CBIAS1=NO CBIAS2=NO CBIAS3=NO @CLEAR 39 @ SUBDIVIDE VOLUME NAME=38 MODE=DIVISIONS NDIV1=60 NDIV2=45 NDIV3=3, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=0.200000000000000 PROGRESS=GEOMET RIC EXTEND=NONE, CBIAS1=NO CBIAS2=NO CBIAS3=NO SUBDIVIDE VOLUME NAME=38 MODE=DIVISIONS NDIV1=60 NDIV2=30 NDIV3=3, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=0.200000000000000 PROGRESS=GEOMETRIC EXTEND=NONE, CBIAS1=NO CBIA S2=NO CBIAS3=NO SUBDIVIDE VOLUME NAME=38 MODE=DIVISIONS NDIV1=60 NDIV2=45 NDIV3=3, RATIO1=1.00000000000000 RATIO2=1.00000000000000, RATIO3=0.200000000000000 PROGRESS=GEOMETRIC EXTEND=NONE, CBIAS1=NO CBIAS2=NO CBIAS3=NO GVOLUME NODES=8 PA TTERN=0 NCOINCID=NO SUBSTRUC=0 GROUP=11, MESHING=MAPPED PREFSHAP=AUTOMATIC DEGENERA=YES COLLAPSE=NO, MIDNODES=CURVED METHOD=DELAUNAY BOUNDARY=ADVFRONT @CLEAR 38 @ GVOLUME NODES=8 PATTERN=0 NCOINCID=BOUNDARIES NCFACE=123456 NCEDGE=, '123456789AB C' NCVERTEX=12345678 NCTOLERA=1.00000000000000E 05, SUBSTRUC=0 GROUP=11 MESHING=MAPPED PREFSHAP=AUTOMATIC, DEGENERA=YES COLLAPSE=NO MIDNODES=CURVED METHOD=DELAUNAY, BOUNDARY=ADVFRONT @CLEAR 37 39 @ FIXITY NAME=XYZ @CLEAR 'X TRANSLATION' PAGE 246 246 'Y TRANSLATION' 'Z TRANSLATION' 'OVALIZATION' @ FIXITY NAME=XY @CLEAR 'X TRANSLATION' 'Y TRANSLATION' 'OVALIZATION' @ FIXBOUNDARY SURFACES FIXITY=ALL @CLEAR 155 'XYZ' 150 'XYZ' 145 'XYZ' @ MASTER ANALYSIS=STATIC MODEX=EXECUTE TSTART=0.0000000 0000000 IDOF=111, OVALIZAT=NONE FLUIDPOT=AUTOMATIC CYCLICPA=1 IPOSIT=STOP, REACTION=YES INITIALS=NO FSINTERA=NO IRINT=DEFAULT CMASS=NO, SHELLNDO=AUTOMATIC AUTOMATI=OFF SOLVER=SPARSE, CONTACT =CONSTRAINT FUNCTION TRELEASE=0.0000000000000 0, RESTART =NO FRACTURE=NO LOADCAS=NO LOADPEN=NO MAXSOLME=0, MTOTM=2 RECL=3000 SINGULAR=YES STIFFNES=1.00000000000000E 09, MAP OUTP=NONE MAP FORM=NO NODAL DE='' POROUS C=NO ADAPTIVE=0, ZOOM LAB=1 AXIS CYC=0 PERIODIC=NO VECTOR S=GEOMETRY EPSI FIR=NO, STABILIZ=NO STABFACT=1.00000000000000E 12 RESULTS=PORTHOLE, FEFCORR=NO CGROUP CONTACT3 NAME=1 FORCES=YES TRACTION=YES NODETONO=NO, FRICTION=0.00000000000000 EPSN=0.00000000000000, EPST=0.00000000000000 DIRECTIO=NORMAL CONTINUO=YES, INITIAL =ALLOWED PENETRAT=ONE DEPTH=0.00000000000000, OFFSET=0.00000000000000 OFFSET T=CONSTANT CORNER C=NO, TBIRTH=0.00000000000000 TDEATH=0.00000000000000 TIED=NO, TIED OFF=0.00000000000000 HHATTMC=0.00000000000000, FCTMC=0.500000000000000 FTTMC=0.500000000000000 RIGID TA=NO, NORMAL S=1.00000000000000E+11 TANGENTI=0.00000000000000, PTOLERAN=1.00000000000000E 08 RESIDUAL=0.00100000000000000, LIMIT FO=1.00000000000000 ITERATIO=2 TIME PEN=0.00000000000000 CONSISTE=DEFAULT USER FRI=NO DESCRIPT='NONE', CFACTOR1=0.00000000000000 CS EXTEN=0.00100000000000000, ALGORITH=DEFAULT RTP CHEC=NO RTP MAX=0.00100000000000000, XDAMP=NO XNDAMP=0.100000000000000 CONTACTSURFA NAME=1 PRINT=DEFAULT SAVE =DEFAULT SOLID=NO BODY=0, ORIENTAT=AUTOMATIC MARQUEEB=0 DESCRIPT='NONE' @CLEAR 146 1 0 @ FIXBOUNDARY SURFACES FIXITY=ALL @CLEAR 145 'XYZ' 150 'XYZ' 155 'XYZ' 37 'XY' 43 'XY' @ COPY ZONE NAME1=WHOLE_MODEL NAME2=MESHPLOT00001 POSITION=1 ZONE NAME=MESHPLOT00001 NODEATTA=YES GEOMATTA=YES @CLEAR 'whole model' 'SUBTRACT CONTACT GRO 1 OF RE 1 OF SUB 0 OF PR ADINA' @ PAGE 247 247 CONTACTSURFA NAME=2 PRINT=DEFAULT SAVE=DEFAULT SOLID=NO BODY=0, ORIENTAT=AUTOMATIC MARQUEEB=0 DESCRIPT='NONE' @CLEAR 91 1 0 1 24 1 0 94 1 0 126 1 0 98 1 0 128 1 0 102 1 0 130 1 0 106 1 0 132 1 0 110 1 0 134 1 0 114 1 0 136 1 0 118 1 0 138 1 0 122 1 0 @ CONTACTPAIR NAME=1 TARGET=1 CONTACTO=2 FRICTION=0.200000000000000, TBIRTH=0.00000000000000 TDEATH=0.00000000000000, H HATTMC=0.00000000000000 FCTMC=0.00000000000000, FTTMC=0.00000000000000 NX=1 NY=1 NZ=1 COPY ZONE NAME1=WHOLE_MODEL NAME2=MESHPLOT00001 POSITION=1 ZONE NAME=MESHPLOT00001 NODEATTA=YES GEOMATTA=YES @CLEAR 'whole model' 'SUBTRACT CONTACT GRO 1 OF RE 1 OF SUB 0 OF PR ADINA' @ ZONE NAME=MESHPLOT00001 NODEATTA=YES GEOMATTA=YES @CLEAR 'whole model' 'SUBTRACT contact group 1' 'CONTACT GRO 1 OF RE 1 OF SUB 0 OF PR ADINA' @ APPLY CONCENTRATED SUBSTRUC=0 REUSE=1 THERMOST=0 @CLEAR 94 3 4500.0000000000 0 1 0.00000000000000 0 274 3 4500.00000000000 1 0.00000000000000 0 @ **** *** ADINA OPTIMIZE=SOLVER FILE=, *** 'C: \ wgm \ Graduate dissertation \ wide base \ 455N \ cal \ 2e6B 2e3R 400 *** S 400T1 500T2.dat' FIXBOUND=YES MIDNODE=NO OVERWRIT=YES PAGE 248 248 LIST OF REFERENCES 1. Gerritsen, A. Gurp, C.V., van der Heide, J., Molenaar, A. and Pronk, A. Prediction and Prevention of Surface Cracking in Asphaltic Pavements, Proceedings of the 6th International Conference on Structural Design of Asphalt Pavements ,1987, pp.378392. 2. Al Qadi, I.L., Loulizi, A., Elseifi, M.A, and Lahouar, S. Effect of tire type on flexible pavements response to truck loading. Final Report submitted to Michelin Americas Research and Development Corporation, 515 Michelin Road, South Carolina 2000. 3. Al Qadi I.L., M. Elseifi, P.J. Yoo, Pavement Damage due to Different Tire Tires and Vehicle Configurations, Final Report Submitted to : Michelin Americas Research And Development Corporation. 2004. 4. Collop, A.C. and Cebon, D., A Theoretical Analysis of Fatigue Cracking in Flexible Pavements, Proceedings of the Institution of Mechanical Engineers. Part C, vol. 209, no. 5, 1995, pp.345361. 5. Love, A.E.H. The Stress Produced in a Semi Infinite Solid by Pressure on Part of the Boundary, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, vol. 228, 1929, pp.377420. 6. Franco Jr., A. and Roberts, S. G. Surface mechanical analyses by Hertzian indentation, Cermica vol.50 no.314 So Paulo Apr./ June 2004. 7. Anderson, T.L., Fracture Mechanics, Fundamentals and Applications, Third Edition, CRC Press, Inc., Ann Arbor, 2005. 8. ADINA 8.3 Manuals, ADINA R&D, Inc.2005. 9. ANSYS Structural Analysis Guide, Release 5.5, ANSYS, Inc. Southpointe, 1998. 10. Al Qadi, I. L., M. A. Elseifi, and P. J. Yoo, Characterization of Pavement Damage due to Different Tire Configurations, The Journal of AAPT, Vol. 74, 2005, pp. 921962. 11. Angela L. Priest, David H. Timm and William E. Barrett, Mechanistic Comparison of Wide base S ingle Vs. Standard Dual Tire Configurations, NCAT Report 0503, 2005 12. Bensalem, A., A. J., Broen, M. E. Nunn, D. B. Merrill, and W. G. Lloyd, Finite Element Modeling of fully Flexible Pavements: Surface Cracking and Wheel Interaction, Proceedings of the Second International Symposium on 3D Finite Element For Pavement Analysis, Design, and Research, West Virginia 2000, pp. 103 121 13. Bode, G., Forces and Movements Under Rolling Truck Tires, Automobiltechnische Zeitschrift, Vol. 64, No. 10, 1962, pp. 300306. 14. Lawn, B.R. Wilshaw, T.R. and Hartley, N.E.W. A computer simulation study of hertzian cone crack growth, Int.J.Fract. 10 1, 1974, pp.116. PAGE 249 249 15. Bonaquist R, An assessment of the increased damage potential of wide based single tires. 7th Int. Conf. on As phalt Pavements, Ed. Brown SF and Hicks RG. 4 vols. Nottingham, UK, International Society for Asphalt Pavements, 1992. 16. B hm, F., Tire Models for Computational Car Dynamics in the Frequency Range Up to 1000 Hz, International Colloquium on Tire Models for Vehicle Dynamics Analysis, 1991, pp.8291. 17. Boussinesq, J., Application des Potentiels of l Etude de le l Equilibre et du Mouvement des Solides Elastiques, Gauthier Villars, Paris, 1885. 18. Christison JT, Anderson KO and Shields BP, 'In situ measurements of strains and deflections in a full depth asphaltic concrete pavement.' Proc. Assoc. Asphalt Paving Technology 47, 1978, pp 398430. 19. Christos Andrea Drakos, Identification of Physical Model to Evaluate Rutting Performance of Asphalt Mixtures, PhD Dissert ation, University of Florida, Gainesville, 2003. 20. CIRCLY version 2, Computer Program for the Analysis of Multiple Complex Circular Loads on Layered Anisotropic Media by L.J. Wardle, Division of Applied Geomechanics, CSIRO, Australia, 1977. 21. Gerrard C.M. and Harrison, W. J. The Analysis of a Loaded Half Space Comprised ofAnisotropic Layers, CSIRO Aust. Div. Appl. Geomechanics Tech. Pap., no. 10, 1971. 22. Clark S.K., Mechanics of Pneumatic Tires; Edited by S.K. Clark. Washington, D.C.: US Department of Transpor tation, National Highway Traffic Safety Administration, 1981. 23. Dauzats M and Linder R. A method for evaluation of the structural condition of pavements with thick bituminous road bases. 5th International Conference on the Structural Design of Asphalt Pavements. Delft, Netherlands, 1982. 24. de Jong, D., Peutz, M., and Korswagen, A. Computer Program BISAR Layered Systems Under Normal and Tangential Surface Loads, AMSR, 0006.73, Shell Research BV,1973. 25. de Beer, M. and Fisher, C. Contact Stresses of Pneumatic Ti res Measured with the VehicleRoad Surface Pressure Transducer Array (VRSPTA) System for the University of California at Berkeley (UCB) and the Nevada Automotive Test Center (NATC) Confidential Contract Research Report CR 97/053, Vol. 1, Transportek, CSIR Pretoria, South Africa, 1997. 26. de Beer, M. and Fisher, C Tire Contact Stress Measurements with the Stress In Motion (SIM) Mk IV System for the Texas Transportation Institute (TTI), USA: [Part of TxDOT Project 0 4361] Restricted Contract Report CR 2002/82, CSIR Transportek, South Africa, 2002. PAGE 250 250 27. de Beer, M., Kannemeyer, L. and Fisher, C. Towards Improved Mechanistic Design of Thin Asphalt Layer Surfacings Based on Actual Tire/Pavement Contact Stress in Motion (SIM)Data in South Africa. 7th Conference on Asp halt Pavements in Southern Africa, 1999. 28. de Beer, M., Fisher, C. and Jooste, F.J. Determination of Pneumatic Tyre/Pavement Interface Contacts Stresses Under Moving Loads and Some Effects on Pavements with Thin Asphalt Surfacing Layers, Proceeding of the 8th International Conference on Asphalt Pavements vol. 1, 1997, pp.179227. 29. Dawley, C.B ., Hogewiede, B.L. and Anderson, K.O. Mitigation of Instability Rutting of Asphalt Concrete Pavements in Lethbridge, Alberta, Canada, Journal of the Association of As phalt Paving Technologists, Vol. 59, 1990, pp. 481508. 30. Perdomo, D. and Nokes, B. Theoretical Analysis of the Effects of WideBase Tires on Flexible Pavement Using CIRCLY, Transportation Research Record 1388, Transportation Research Board (TRB), National Research Council Washington D.C., 1993, pp.108119. 31. Development of pavement structural subsystems. NCHRP Report 291, Transportation Research Board, Washington, D.C. 32. Ewalds, H.L., and R.J.H. Wanhill, Fracture Mechanics, Delftse Uitgevers Maatschappij, D elft, Netherlands, and Edward Arnold Publishers, London, 1986. 33. Emmanuel G. Fernando, Dilip Musani, Dae Wook Park, and Wenting Liu. Evaluation of Effects of Tire Size and Inflation Pressure on Tire Contact Stresses and Pavement Response, FHWA Report TX 06/0 43611, Texas Transportation Institute the Texas A&M University System College Station, Texas 2006. 34. Eisenmann J and Hilmer A, 'Influence of wheel load and inflation pressure on the rutting effect at asphalt pavements experiments and theoretical inve stigations. Sixth international conference on the structural design of asphalt pavements, Ann Arbor, 1987. 35. Ervin, R.D., et al. Influence of Truck Size and Weight Variables on the Stability and Control Properties of Heavy Trucks. Volume II. Report: FHWA R D 83030, UMTRI8310/2. University of Michigan, Tra nsportation Research Institute Federal Highway Administration. 1983. 36. Finn, F., C. Saraf, R. Kulkarni, K. Nair, W. Smith and A. Abdullah. Development of Pavement Structural Subsystems. NCHRP Report 291, National Cooperative Highway Research Program, Transportation Research Board. Washington D.C. 1986. 37. Finn, F., C. Saraf, R. Kulkarni, K. Nair, W. Smith and A. Abdullah. The Use of Distress Prediction Subsystems for the Design of Pavement Structures. Proc eedings 4th International Conference on the Structural Design of Asphalt Concrete Pavement Structures, Vol. 1. 1977. pp. 338. PAGE 251 251 38. Fischer Cripps and Anthony C, Introduction to Contact Mechanics (New York: Springer, Inc., 2000). 39. Gillespie, T D ; Karamihas, S M ; S ayers M W ; Nasim, M A ; Hansen, W ; Ehsan, N ; Cebon, D Effects of Heavy vehicle Characteristics on Pavement Response and Perf ormance. NCHRP Report 353. Transportation Research Board, 1993. 40. Goodyear. Radial truck tire and retread service manual. 2004. 41. Groenendijk, J., Accelerated testing and Surface Cracking of Asphaltic Concrete Pavements, A PhD dissertation. Department of Civil Engineering, Delft University of Technology, the Netherlands, 1998. 42. Hugo F and Kennedy T W. Surface cracking of asphalt mixtures in southern Africa. Proceedings of the Association of Asphalt Paving Technologists, Vol.54, February, 1985. 43. Himeno K, Watannabe T and Maruyama T. Estimation of the fatigue life of asphalt pavement. 6th International Conference on the Structural Design of Asphalt Pavements.Ann Arbor, USA, 1987. 44. Huang, Y.H., Pavement Analysis and Design, Prentice Hall, Englewood Cliffs, NJ, 1993. 45. Poulos H.G. and Davis, E.H. Elastic Solutions for Soil and Rock Mechanics. New Yor k:John Wiley & Sons, Inc., 1974. 46. H. Hertz, On the contact of elastic solids, J. Reine Angew. Math. 92 (1881), pp. 156 171. 47. H.Hertz, On hardness, Verh. Ver. Beforderung Gewerbe Fleisses 61, 1882, P.410. 48. Jill M. Holewinski, See Chew Soon, Andrew Drescher, and Henryk Stolarski. Investigation of Factors Related to Surface Initiated C racks in Flexible Pavements. Final Report, Report No.: MN/RC 200307, Minnesota Department of Transportation, St. Paul, Minnesota, 2003. 49. Jacobs, M.M. Crack Growth in Asphaltic Mixes, Ph.D Dissertation, Delft Institute of Technology, the Netherlands, 1995. 50. J Pelc, Static T hreedimensional Modeling of Pneumatic Tires U sing the T echnique of E lement O verlaying Proc Instn Mech Engrs Vol 216 Part D: J Automobile Engineering Poland, 2002, pp.709716. 51. Johnson, K L 'Contact mechanics', Cambridge University P ress, 1987. 52. Joseph Ponniah. Use of New Technology Single Wide Base Tires: Impact on Pavements. Final Report, Ontario Ministry of Transportation, 2003 PAGE 252 252 53. Kao, B.G. and Muthukrishnan, M., Tire Transient Analysis with an Explicit Finite Element Program, Tire Science and Technology. Vol.25, No.4, 1997, pp.230244. 54. Kilcarr, Sean. Single Life May Be Best for Linehaul Tires. FleetOwner (Online), PRIMEDIA, Inc, 2001, Available at: http://fleetowner.com/news/fleet_single_life_may/, Accessed 9 February 2005. 55. Krut z, N.C., and P.E. Sebaaly. The Effects of Aggregate Gradation on Permanent Deformation of Asphalt Concrete. Journal of Association of Asphalt Paving Technologists Vol. 59, 1990, pp. 481508. 56. Kai Su, Lijun Sun, Yoshitaka Hachiya, Tyota Maekawa. Analysis o f Shear Stress in Asphalt Pavements under Actual Measured Tire Pavement Contact Pressure. 6th ICPT, Sapporo, Japan, July 2008. 57. Lan Meng, Truck Tire/Pavement Interaction Analysis by the Finite Element Method, Ph.D. Dissertation, Michigan State University USA, 2002. 58. Loo.M. A Model Analysis of Tire Behavior Under Vertical Loading and Straight line Free Rolling, Tire Science and Technology. Vol.13. No.2, 1985, pp.6790. 59. Wardle, L.J. Integral Transform Methods for Multilayered Anisotropic Elastic Systems CSIRO Aust. Div. Appl. Geomechanics Tech. Pap., no. 27, 1976. 60. Lippmann, S.A., Oblizajek, K.L., The Distribution of Stress between the Thread and the Road. In: Automotive Engineering Congress. Detroit, Michigan, February 25March 1, 1974. 61. Lee W. Abramson ,Thomas S. Lee Sunil Sharma Glenn M. Boyce Slope Stability and Stabilization. 2nd edition, Wiley, New Yo rk, 2001. 62. Myers, L.A., Roque, R., and B.E. Ruth, Mechanisms of Surface Initiated Longitudinal Wheel Path Cracks in High Type Bituminous Pavements, Proceedings of the Association of Asphalt Paving Technologists, Volume 67, 1998, pp 401432. 63. Myers, L.A., R oque, R., Ruth, B.E., and C. Drakos, Measurement of Contact Stresses for Different Truck Tire Types to Evaluate Their Influence on Near Surface Cracking and Rutting, In Transportation Research Record 1655, TRB, National Research Council, Washington, D.C., 1999, pp. 175184. 64. Myers, L.A., Roque, R., and B. Birgisson, Propagation Mechanisms for Surface Initiated Longitudinal Wheel Path Cracks, In Transportation Research Record 1778,TRB, National Research Council, Washington, D.C., 2001, pp. 113121. 65. Myers, L.A. Mechanism of Wheel Path Cracking That Initiates At the Surface of Asphalt Pavements, Masters Thesis, University of Florida Gainesville, December 1997. PAGE 253 253 66. Myers, L. A., Development and Propagation of Surface Initiated Longitudinal WheelPath Cracks in Flexible Highway Pavements PhD Thesis, University of Florida, 2000. 67. Marc E. Novak, Creation of A Laboratory Testing Device to Evaluate Instability Rutting In Asphalt Pavements, Ph.D Dissertation, University of Florida 2007. 68. Molenaar, A.A.A., Fatig ue and Reflective Cracking due to Traffic (With Discussion), Proceedings of the Association of Asphalt Paving Technologists, Volume 53, 1984, pp 440474. 69. Marshek, K. M., W. R. Hudson, R. B. Connell, H. H. Chen, C. L. Saraf and R. B. Connell. Effect of Tru ck Tire Inflation Pressure and Axle Load on Pavement Performance Research Report 3862F, Center for Transportation Research, The University of Texas at Austin, Austin, Tex., 1985. 70. Zamzamzadeh M. and Negarestani, M. A 3D Tire/Road Interaction Simulation by a Developed Finite Element Model, Tire Society Conference, Akron, USA, 2006. 71. Marc Novak, Bjorn Birgisson, Reynaldo Roque, Near surface S tress S tates in F lexible P avements U sing M easured R adial T ire C ontact S tresses and ADINA, Computers and Structure s 81, 2003, pp.859870. 72. McLeod, N.W. A Rational Approach to the Design of Bituminous Paving Mixtures. Journal of Association of Asphalt Paving Technologists Vol. 19, 1950, pp. 83187. 73. Majidzadeh, K., Aly, M.G., Bayomy, F. and El Laithy, A. (1980) Impleme ntation of a Pavement Design System, Vol. 1 and 2, Final Report, EES 578, The Ohio State University Engineering Experiment Station, June 1980. 74. Monismith, C.L., J.A. Epps, and F.N. Finn, Improved Asphalt Mix Design, Proceedings of the Association of Aspha lt Paving Technologists, Vol. 55, 1985, pp. 124132. 75. Muni Budhu,Soil Mechanics and Foundations,John Wiley & Sons, Inc. 2000. 76. Huber, M.T. Zur Theorie der Beruhrung fester elastischer Korper, Ann. Phys. 43 61, 1904, pp.153163. 77. Hurtle, M.S., Pihlajamk i, J.T. and Miettinen, V.V. The Effect of Wide Base Tires on Pavements, Third International Symposium on Vehicle Weights and Dimensions, Cambridge, UK, June 28July 2, 1992. 78. Markastaller, M., Pearson, A., and Janajreh, I.On vehicle testing of Michelin new wide base tire. Paper No. 01 3432. Presented at Annual ASE International Truck and Bus Conference, Portland, Ore.2000. 79. Michelin. Michelin Truck Tire Service Manual 2006. PAGE 254 254 80. Nunn M E, A Brown, D Weston and J C Nicholls. Design of longlife flexible pavements for heavy traffic. Department of Transport TRL Report 250. Transport Research Laboratory, Crowthorne, 1997. 81. Nakajima, Y. and Padovan, J., Numerical Simulation of Tire Sliding Events Involving Impacts with Holes and Bumps, Tire Science and Technolog y. Vol.14, No.2. 1986, pp.125136. 82. Novak, M., B. Birgi sson, R. Roque and B. Choubane, One Way and Two Way Directional Heavy Vehicle Simulator Loading. In Transportation Research Record: Journal of the Transportation Research Board, No. 1896, TRB, Nationa l Res earch Council, Washington, D.C., 2004, pp 208214. 83. Pottinger, M.G., The Three dimensional Contact Stress Field of Solid and Pneumatic Tires, Tire Science and Technology, Vol. 20(1), 1992, pp. 332. 84. Padovan, J., Numerical Simulation of Rolling Tires Tire Rolling Losses and Fuel Economy, An R&D Planning Workshop, Series SAE Conference Proceedings P 74. 1977, pp.8794. 85. Padovan, J., On Standing Waves in Tire, Tire Science and Technology. Vol. 5, No.2. 1977, pp. 83101. 86. Pottinger, M.G. and Mclmtyre, J.E., Effect of Suspension Alignment and Modest Cornering on Footprint Behavior of Performance Tires and Heavy Duty Radial Tires, Tire Science and Technology. TSTCA, Vol.27. No.3, 1999, pp.128160. 87. Roque, R., Myers, L.A., and B. Birgisson, Evaluating Me asured Tire Contact Stresses to Predict Pavement Response and Performance. In Transportation Research Record 1716, TRB, National Research Council, Washington, D.C., 2000, pp 7381. 88. Roque, R., L. Myers, and B.E. Ruth, Loading Characteristics of Modern Truck Tires and Their Effects on Surface Cracking of Asphalt Pavements, Proceedings of the Fifth International Conference on the Bearing Capacity of Roads and Airfields, Trondheim, Norway, 1998, pp. 93102. 89. Roque, R. W.G. Buttlar, B.E. Ruth, M. Tia, S.W. Dic kison, B.Beid, Evaluation of SHRP Indirect Tension Tester to Mitigate Cracking In Asphalt Pavements and Overlays, Finial Report to the Florida Department of Transportation. August 1997. 90. Roque, R., Z. Zhang, and B. Sankar Determination of Crack Growth Ra te Par ameters of Asphalt Mixtures Using the Superpave Indirect Tension Test (IDT). Journal of the Association of Asphalt Paving Technologists, Vol. 68, 1999, pp. 404433. 91. Roque, R., and B.E. Ruth, Mechanisms and Modeling of Surface Cracking of Asphalt Pavemen ts,Assoiciation of Asphalt Paving Technologists, Vol.59, 1990, pp. 396431. PAGE 255 255 92. Rhyne T.B., Gall, R., and Chang, L. Y., Influence of Rim RunOut on the Nonuniformity of Tire Wheel Assemblies Tire Science and Technology. Vol.22, No.2. 1994, pp.125136. 93. Ruth, B., Bloy, L., and A. Avital, Low Temperature Asphalt Rheology as Related to Thermal and Dynamic Behavior of Asphalt Pavements, Final Report, Project 245U20, Department of Civil Engineering, University of Florida, 1981. 94. Seitz, N. and A. Hussman, Forc es and Displacement in Contact Area of Free Rolling Tires, Proceedings of Society of Automotive Engineers, Montreal, Canada, 1971, pp. 23232329. 95. S. Matsuno and T. Nishizawa, Mechanism of Longitudinal Surface Cracking in Asphalt Pavement, Proceedings of 7th International Conference on Asphalt Pavements vol. 2, 1992, pp.277291. 96. Sebaaly, P.E. Pavement damage as related to tires, pressures, axle loads, and configurations. Vehicle, tire, pavement interface. ASTM STP 1164, J.J. Henry and J.C. Wambold, Eds ., American Society for Testing and Materials, Philadelphia, 1992, pp.5468. 97. Siddharthan, R. V., Yao, J., and Sebaaly, P.E Pavement strain from moving dynamic 3D load distribution. J. Transp. Eng., 124(6), 1998, pp.557566. 98. Svasdisant T., Schorsch, M ., Baladi, G.Y., and S. Pinyosunun, Mechanistic Analysis of Top down Cracks in Asphalt Pavement, Accepted for presentation and publication by the Transportation Research Board, Washington DC, 2001 99. Sally Annette Shoop, Finite Element Modeling of Tire Ter rain Interaction, Dissertation, University of Michigan, USA, 2001. 100. Trivisonno, N. M., Application of Tire Thermograph to Rolling Resistance, Tire Rolling Losses and Fuel Economy, An R&D Planning Workshop, Series SAE Conference Proceedings P 74. 1977, pp.103109. 101. Tielking, J.T., A Tire Contact Solution Technique, Tire Modeling, Proceedings of A Workshop Held at Langley Research Center. Hampton. Virginia, Series NASA Conference Publication 2264, 1982, pp.95121. 102. Tielking, J.T., Roberts, F.L., Tire Contac t Pressure and Its Effect on Pavement Strain, Journal of Transportation Eng ineering. Volume 113, Number 1, 1987, pp.56 71. 103. Treybig HJ, Equivalency factor development for multiple axle configurations. Transp. Res. Rec. TRB 949, 1983 pp. 3244. 104. Uhlmeyer, J.S., Willoughby, K., Pierce, L.M., and J. P. Mahoney, TopDown Cracking in Washington State Asphalt Concrete Wearing Courses, In Transportation Research Record 1730, TRB, National Research Council, Washington, D.C., 2000, pp. 110116 PAGE 256 256 105. W.R. Dean, H.W. Pa rsons, and I.N. Sneddon, A Type of Stress Distribution on the Surface of a Semi Infinite Elastic Solid, Proceeding of the Cambridge Philosophical Society, vol. 40, 1994, pp.519. 106. Wang, Lei, Tire Modeling Using a ThreeDimensional Finite Element Method, Masters Thesis, Carleton University, Ottawa, Ontario, Canada, 1990 107. Weslea for Windows version 3.0, A Mechanistic Multi layers Pavement Analysis Computer Program by D. Timm, B. Birgisson, and D. Newcomb, Department of Civil Engineering, University of Minnesota, 1999. 108. Widebase Wheel Fills Low profile Needs. Modern Bulk Transporter (Online), PRIMEDIA, Inc, 2002, Available at: http://bulktransporter.com/mag/transportation_widebase_wheel_fills/index.html, Accessed 9 February 2005. 109. Xiong Zhang, Nonlinear F inite Element Modeling and Incremental Analysis of A Composite Truck Tire Structure, Ph.D. Dissertation, Concordia University, Canada, 2001. 110. Zhang, Z., R. Roque, and B. Birgisson. Evaluation of Laboratory Measured Crack Growth Rate for Asphalt Mixtures. In Transportation Research Record: Journal of the Transportation Research Board, No. 1767, TRB, National Research Council, Washington, D.C., 2001, pp. 6775. 111. Zhang, Z., R. Roque, B. Birgisson, and B. Sangpetngam. Ident ification and Verification of a Suitab le Crack Growth Law (with Discussion). Journal of the Association of Asphalt Paving Technologists, Vol. 70, 2001, pp. 206241. 112. Zube E and Forsyth R, 'An investigation of the destructive effect of floatati on tires on flexible pavement.' Highway Res. Rec., H RB, N71 1965. pp 129150. PAGE 257 257 BIOGRAPHICAL SKETCH Guangming Wang was born in Linhai, Zhejiang province, China. He received a Bachelor of Science in e ngineering and Master of Science in c ivil e ngineering from Changsha University of Science and Technology in 2000 and 2003, respectively. He worked for Shantou Highway Station as a test engineer and construction inspector during the year 2003 to 2004. And then he joined Zhejiang University of Science and Technology and became a college lecturer from 2004 to 2005. After that, he decided to across Pacific to pursue his Ph.D in the United States. In Aug ust 2005, he was admitted to University of Florida and worked as a research assistant in the D epartment of Civil Engineering. After completing his doctoral study at U niversity of Florida Guangming Wang intends to work in academia, government agencies, or industrial companies in c ivil e ngineering to conti nue his service to the society. 