1 u n . MSU RETURNING MATERIALS: Place in book drop to “33.411155 remove this checkout from ——. your record FINES win be charged f book is retuinn! fitter the date stamped ne‘uow. MAY 2 3 1994' r I. new 9 199A . r.» “ ——%§~\ 'Q NONLINEAR FINITE ELEMENT ANALYSIS AND DESIGN OF FLEXIBLE PAVEMENTS By } Ming-Shan Yeh A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil and Environmental Engineering 1989 ua/ooox ABSTRACT NONLINEAR FINITE ELEMENT ANALYSIS AND DESIGN OF FLEXIBLE PAVEMENTS BY Ming-Shah Yeh A nonlinear finite element program called MICK-PAVE has been developed for use on personal computers to aid the routine design of flexible pavements. Three major achievements have been accomplished in this research. First, a new concept of utilizing a flexible boundary in pavement analysis has been introduced, and its characteristics fully ‘investigated. Second, an extremely "user-friendly" nonlinear finite element program for pavement analysis and design has been implemented on personal computers. Third, two empirical equations to predict fatigue life and rut depth have been developed for use with nonlinear finite element analysis. In the HIGH-PAVE program, the pavement is represented by an axisymmetric finite element model, and the resilient modulus model together with the Mohr-Coulomb failure criterion is used to characterize the nonlinear material response of granular and cohesive soils. Extrapolation and interpolation techniques have been used to improve stresses and strains at layer boundaries. Results from a variety of analyses have been compared with exact solutions (when available), and with the results from existing computer programs. Extensive sensitivity analyses have also been performed to explore the capabilities and limitations of the program. ACKNOWLEDGMENTS The author would like to express his deepest gratitude to his major adviser Prof. Ronald Harichandran and co-advisor Prof. Gilbert Baladi for their guidance and encouragement throughout this research. Gratitude is also expressed to the doctoral committee members Professors Robert Wen and David Yen for their useful comments. The auther expresses special thanks to his colleague C.M. Lee for his valuable suggestion during the early development of the finite element program, and to his wife, Seok Yan Yeh, for her many encouragements, support, and help during this study. Finally, the author gratefully acknowledges the financial support provided by the Michigan Department of Transportation for this research. ii LIST OF TABLES TABLE OF CONTENTS LIST OF FIGURES ................................................. CHAPTER 1. INTRODUCTION ............................................ 2. LITERATURE REVIEW ....................................... 2. 2 2. l .2 2. 3 2. : GENERAL ........................................... : DESIGN METHODS FOR FLEXIBLE PAVEMENT STRUCTURE .... .l : Empirical Methods ............................. .2.1.1 : AASHTO Flexible Pavement Design ........... .2.1.2 : National Stone Association Design Method .. .2.1.3 : California Method of Design ............... .2 : Mechanistic-Empirical Methods ................. .2.2.1 : Chevron Program ........................... .2.2.2 : The Shell Method .......................... .2.2.3 : The Asphalt Institute Design .............. .2.2.4 : VESYS (Visco-Elastic System) II Computer Program ................................... .2.2.5 : Nonlinear Finite Element Method ........... : MATERIAL CHARACTERIZATION ......................... 3 .1 : Hyperbolic Stress-Strain Relationship ......... iii 14 15 16 18 19 20 22 23 24 25 iv 2.3.2 : The Resilient Modulus ......................... 2.3.3 : Shear and Volumetric Stress-Strain Relationship 2.3.4 : Third Order Hyperelastic model ................ 2.4 PAVEMENT EVALUATION METHODS ....................... 2.4.1 : Transfer Function Theory ...................... 2.4.2 : Back-calculation Methods ...................... 2.5 SUMMARY ........................................... LINEAR FINITE ELEMENT ANALYSIS ........................... 3.1 : GENERAL ........................................... 3.2 : AXISYMMETRIC FINITE ELEMENT ANALYSIS .............. 3.2.1 Formulation of Axisymmetric Element Stiffness Matrix .............................. 3.2.2 : Assembling the Global Stiffness Matrix ........ 3.2.3 Formulation of the Edge Loads ................. 3.2.4 : Gauss Elimination for the Solution of the Stiffness Equations ........................... 3.3 USE OF FLEXIBLE BOTTOM BOUNDARY ................... 3.3.1 Introduction .................................. 3.3.2 : Modeling of Flexible Boundary ................. 3.3.3 Flexibilities for Homogeneous Half-Space ...... 3.4 : COMPARISON OF THE FLEXIBLE BOUNDARY WITH OTHER LINEAR METHODS .................................... 3.5 SENSITIVITY OF FLEXIBLE BOUNDARY IN LINEAR ANALYSIS 3.5.1 Effect of the Depth of the Flexible Boundary .. 3.5.2 Effect of the Tire Pressure on the Location of the Flexible Boundary ......................... 3.5.3 Effect of the Wheel Load on the Location of the Flexible Boundary ......................... 27 29 3O 3O 3O 31 31 34 34 37 37 42 43 44 46 46 47 51 54 68 68 75 81 3. 6 DISCUSSION AND SUMMARY .............................. NONLINEAR FINITE ELEMENT ANALYSIS ....................... 4. 4. 4. 1 : INTRODUCTION ...................................... 2 : MATERIAL NONLINEARITY ............................. 3 : SENSITIVITY OF FLEXIBLE BOUNDARY IN NONLINEAR ANALYSIS .......................................... 4.3.1 : Equivalent Modulus for Halfspace below the Flexible Boundary ............................. 4.3.2 : Effect of the Depth of the Flexible Boundary .. 4.3.3 : Effect of the Wheel Load on the Location of the Flexible Boundary ......................... .4 : THE MICH-PAVE PROGRAM ............................. 4.4.1 : Mesh Generation ............................... 4.4.2 : Gravity and Lateral Stresses of Pavement Materials ..................................... 4.4.3 : Recovering Global Stresses from Modified Principal Stresses ............................ 4.4.4 : Interpolation and Extrapolation of Stresses and Strains at Layer Boundaries ............... COMPARISONS WITH OTHER PROGRAMS ......................... 5 5 5. 5 6 .1 : COMPARISONS WITH SAP-IV RESULTS ................... .2 : COMPARISONS WITH CHEVSL RESULTS ................... 5.2.1 : Linear Elastic Analysis using the CHEVSL and MICH-PAVE Programs ............................ 5.2.2 : Equivalent Resilient Moduli for Linear Analysis 3 : COMPARISONS WITH ILLI-PAVE RESULTS ................ .4 : SUMMARY ........................................... .l : GENERAL ........................................... 87 89 89 89 99 99 100 107 112 112 116 117 119 124 124 126 126 131 141 142 148 148 6.2 6.3 6.4 : 6.5 6.6 6.7 7.2 : APPENDIX vi FATIGUE LIFE MODEL ................................ : RUT DEPTH MODEL ................................... SENSITIVITY ANALYSIS .............................. .1 : .2 : .1 .2 Fatigue Model ................................. Rut Depth Model ............................... : ANALYSIS OF MICH-PAVE INPUT/OUTPUT ................ : Thickness and Modulus of the Asphalt Concrete Percent Air Voids in the Asphalt Concrete ..... : Thickness of Granular Layers .................. : Modulus of the Granular layer ................. Elastic Modulus of Roadbed Soil ............... DETAILED ANALYSIS OF THE FATIGUE LIFE EQUATION .1 .2 : Thickness of AC ............................... : Air Voids in AC ............................... : Thickness of Granular Layer ................... : Resilient Modulus of the Granular Layer ....... : Resilient Modulus of the Roadbed Soil ......... OUTLINE OF COMPUTER PROGRAM ................................. LIST OF REFERENCES OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 150 156 157 157 158 159 160 164 168 173 174 178 182 182 185 185 188 190 193 196 196 200 202 211 TABLE 3-1 3-2 3-3 3-4 3-5 3-6 3-7 3-8 3-9 3-10 3-11 3-12 LIST OF TABLES Errors in surface deflections ............................ Errors in vertical displacements beneath center of load Errors in vertical stresses beneath center of load ....... Errors in radial stresses beneath center of load ......... Sensitivity of surface deflections to the depth of the flexible boundary ........................................ Sensitivity of radial stresses to the depth of the flexible boundary (at a radial distance of .67" from the center of the loaded area) ............................... Sensitivity of vertical stresses to the depth of the flexible boundary (at a radial distance of .67" from the center of the loaded area) ............................... Sensitivity of surface deflections to increase in the tire pressure from 100 to 120 psi ............................. Sensitivity of the radial stresses to increase in the tire pressure from 100 to 120 psi (at a radial distance of .61" from the center of the loaded area) ...................... Sensitivity of vertical stresses to increase in the tire pressure from 100 to 120 psi (at a radial distance of .61" from the center of the loaded area ) ..................... Sensitivity of surface deflections to increase in the wheel load from 9 to 12 kip .............................. : Sensitivity of radial stresses to increase in the wheel load from 9 to 12 kips (at a radial distance of .77" from the center of the loaded area) ........................... vii 61 62 62 63 69 71 73 75 77 79 81 83 3-13 3-14 4-1 4-2 4-3 4-4 4-6 4-7 4-8 4-9 4-10 5-1 5-2 5-3 5-4 5-5 viii Sensitivity of vertical stresses to increase in the wheel loads from 9 to 12 kips (at a radial distance of .77" from the center of the loaded area ) .......................... Capabilities of the finite element models ................ Resilient moduli of elements just above flexible boundary The properties of the pavement section ................... Sensitivity of the surface deflections to the depth of the flexible boundary ................................. Sensitivity of the radial and vertical stresses to the depth of the flexible boundary (at a redial distance of .5" from the center of loaded area) ...................... Sensitivity of surface deflections to increase in the wheel load from 9 to 11 kip .............................. Sensitivity of the radial and vertical stresses to increase in the wheel load from 9 to 11 kips (at a redial distance of .74" from the center of loaded area) ......... Comparison of radial and tangential stresses at 67" from the center of the loaded area ....................... Comparison of vertical and shear stresses at 67" from the center of the loaded area ....................... Comparison of radial and tengential strains at 67" from the center of the loaded area ....................... Comparison of vertical and shear strains at 67" from the center of the loaded area ....................... Comparisons of the vertical displacements at the center of the loaded area ................................ Design data for a 12" full-depth AC on the roadbed soil Design data for a 3" AC and a 12" granular material on the roadbed soil ...................................... Comparisons of surface deflections between CHEVSL and MICH-PAVE programs for a 12" full-depth AC section ....... Comparisons of surface deflections between CHEVSL and MICH-PAVE programs for the three layer section ........... 85 87 100 101 102 104 107 109 122 123 123 124 126 127 127 128 5-6 ' 5-7 5-8a : 5-8b 5-9 5-10 6-1 6-2 6-3 6-4 6-5 6-6 6-7 6-8 6-9 : ix Comparisons of vertical and radial stresses at .75 inch from the center of the loaded area for a 12" full-depth AC section ............................................... Comparisons of vertical and radial stresses at .75 inch from the center of the loaded area for the three layer section .................................................. Comparison of the equivalent resilient moduli for linear analysis by three different approaches (section 1) ....... : Comparison of the equivalent resilient moduli for linear analysis by three different approaches (section 2) ....... Linear and nonlinear material properties of the 12 inches full-depth AC section .......................... Linear and nonlinear material properties for the three layer section ............................................ Fatigue lives and rut depths of field data ............... Parameters for calibrating the fatigue life and rut depth equations used in MICH-PAVE .............................. Tensile strain at the bottom of the AC and compressive strain at the top of the roadbed soil for varying AC thicknesses ........................................... Tensile strain at the bottom of the AC and compressive strain at the top of the roadbed soil for varying values of air voids in the AC ............................ Tensile strain at the bottom of the AC and compressive strain at the top of the roadbed soil for varying thicknesses of the granular layer ........................ Tensile strain at the bottom of the AC and compressive strain at the top of the roadbed soil for varying material properties of the granular layer ................ Tensile strain at the bottom of th AC and compressive strain at the top of the roadbed soil for varying elastic moduli of the roadbed soil ....................... Effect of the thickness of the AC on the fatigue life of pavements ........................................ Effect of the air voids in the AC on the fatigue life of pavements ............................................. 130 131 137 137 140 155 164 168 173 174 178 182 185 6-10 6-11 6-12 6-13 6-14 : Effect of the thickness of the granular layer on the fatigue life of pavements ................................ : Effect of the material properties of the granular layer on the fatigue life of pavements ......................... : Effect of the modulus of the roadbed soil on the fatigue life of pavements ................................ : Comparison of the equivalent wheel load factor between Equation 6-6 and the AASHTO method ....................... Sensitivity of some response measures to key properties of pavement sections ..................................... 185 188 190 193 194 FIGURE 3-1 3-2 3-3 3-4 : 3-5 3-6a 3-6b 3-7 3-8 3-9 3-10 3-11 3-12 3-l3a: 3-l3b: 3-14a: : Tranditional finite element mesh LIST OF FIGURES A typical axisymmetric finite element .................... The four—node isoparametric element ...................... Normal and tangential loads/unit length applied to an isoparametric element .................................... Typical conversion of tire pressure into node forces ..... Finite element mesh on flexible boundary ................. : Vertical ring loads ...................................... : Radial ring loads ........................................ Typical nodes and degree-of-freedom ...................... Vertical load on thin annulus ............................ Uniform vertical loading to estimate f Linear radial load to estimate f 11 ....................... Finite element mesh used with flexible boundary .......... Comparison of surface deflections with and without flexible boundary for the homogeneous material ........... Comparison of surface deflections with and without flexible boundary for the multilayer pavement ............ Comparison of vertical displacements beneath center of load with and without flexible boundary for the homogeneous material ..................................... xi kk ................. 39 39 45 45 48 50 50 55 55 56 56 58 59 64 64 65 3-l4b: 3-15a: 3-le: 3-l6a: 3-l6b: 3-17 3-18 3-19 3-20 3-21 3-22 3-23 3-24 : 3-25 : 4-1 : xii Comparison of vertical displacements beneath center of load with and without flexible boundary for the multilayer pavement ...................................... Comparison of vertical stresses beneath center of load with and without flexible boundary for the homogeneous material ................................................. Comparison of vertical stresses beneath center of load with and without flexible boundary for the multilayer pavement ................................................. Comparison of radial stresses beneath center of load with and without flexible boundary for the homogeneous material ................................................. Comparison of radial stresses beneath center of load with and without flexible boundary for the multilayer material ................................................. Sensitivity of surface deflection to the depth of the flexible boundary ........................................ Sensitivity of radial stress to the depth of the flexible boundary ........................................ Sensitivity of vertical stress to the depth of the flexible boundary ........................................ Sensitivity of surface deflection to increase in the tire pressure from 100 to 120 psi ............................. Sensitivity of radial stress to increase in the tire pressure from 100 to 200 psi ............................. ' Sensitivity of vertical stress to increase in the tire pressure from 100 to 200 psi ............................. Sensitivity of surface deflection to increase in the wheel load from 9 to 12 kip .................................... Sensitivity of radial stress to increase in the wheel load from 9 to 12 kip .................................... Sensitivity of vertical stress to increase in the wheel load from 9 to 12 kip .................................... Complete pavement response ............................... 65 66 66 67 67 70 72 74 76 78 80 82 84 86 90 4-2 4-3 4-8 4-9 4-10 : 4-12 4-13 4-14 4-15 4-16 5-1 5-2 5-3 : 5-4 : xiii Typical variation of resilient modulus with repeated stress for the granular material ......................... Typical variation of resilient modulus with repeated stress for the roadbed soil .............................. Examples for stress modification at end of iteration ..... Stress modification procedures for given iteration ....... Sensitivity of surface deflection to the depth of the flexible boundary ........................................ Sensitivity of radial stress to the depth of the flexible boundary ........................................ Snsitivity of vertical stress to the depth of the flexible boundary ........................................ Sensitivity of surface deflection to increase in the wheel load from 9 to 11 kip .............................. Sensitivity of radial stress to increase in the wheel load from 9 to 11 kip .............................. Sensitivity of vertical stress to increase in the wheel load from 9 to 11 kip .............................. Finite element mesh for section 1 ........................ Finite element mesh for section 2 ........................ : Calculation of gravity and lateral stresses .............. ’ Stress and principal stresses in r-z plane ............... Interpolation and extrapolation of stresses and strains at layer boundary ........................................ Finite element mesh and material properties for a typical section .......................................... Comparison of surface deflection between CHEVSL and MICH-PAVE ................................................ Comparison of vertical stress between CHEVSL and MICH-PAVE (3" AC and 12" base) ..................................... Comparison of radial stress between CHEVSL and MICH-PAVE (3" AC and 12" base) ..................................... 94 94 96 98 103 105 106 108 110 111 114 115 118 118 121 125 129 132 133 5-5 5-6 5-7 5-8 5-9 5-10 5-11 5-12 5-13 6-1 6-2 6-3 6-4 : 6-5 6-6 6-7 6-8 6-9 xiv Three different approaches to calculate equivalent resilient moduli ......................................... Material properties of pavement section 1 ................ Material properties of pavement section 2 ................ Comparison of surface deflections duo to the different equivalent resilient moduli (section 1) .................. Comparison of surface deflections duo to the different equivalent resilient moduli (section 2) .................. : Comparison of surface deflection (12" of full-depth AC) Comparison of surface deflection (3" of AC and 12" of base) ................................................. Comparison of vertical stress along three programs (12" of full-depth AC) ................................... ° Comparison of radial stress along three programs (12 " of full-depth AC) .................................. Tensile strain at the bottom of the AC layer to thickness of AC .................................................... Compressive strain at the top of the roadbed soil to thickness of AC .......................................... Surface deflection at the center of loading to thickness of AC .................................................... Tensile strain at the bottom of the AC layer to air voids of AC .................................................... Compressive strain at the top of roadbed soil to air voids of AC .................................................... Surface deflection at the center of loading to air voids of AC .................................................... Tensile strain at the bottom of the AC layer to thickness of granular layer ........................................ Compressive strain at the top of the roadbed soil to thickness of granular layer .............................. Surface deflection at the center of loading to thickness of granular layer ........................................ 135 136 136 138 139 143 144 145 146 161 162 163 165 166 167 170 171 172 6-10 : 6-11 : 6-12 6-13 6-14 : 6-15 6-16 6-17 6-18 6-19 6-20 XV Tensile strain at the bottom of the AC layer to material properties of granular layer ............................. Compressive strain at the top of the roadbed soil to material properties of granular layer .................... ' Surface deflection at the center of loading to material properties of granular layer ............................. modulus of roadbed Compressive strain modulus of roadbed Surface deflection modulus of roadbed at the soil .. at the soil .. Effect of AC thickness on Effect of air voids in AC : Tensile strain at the bottom of the AC layer to elastic soil .. eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee fatigue life ................... on fatigue life ................ Effect of granular layer thickness on fatigue life ....... 1 : Effect of K parameter of granular layer on fatigue life : Effect of roadbed soil modulus on fatigue life ........... 175 176 177 179 180 181 183 184 186 187 189 CHAPTER 1 INTRODUCTION In recent years pavement design is being done more and more based on mechanistic analysis. The migration from empirical methods to mechanistic analysis has been facilitated by the availability of relatively inexpensive microcomputers that can be used in daily practice. Early mechanistic analysis computer programs (such as CHEVSL, BISAR, ELSYMS, etc.) modeled pavemments as being composed of linear elastic layers, and computed deflections, stresses, and strains within a pavement arising from a single circular wheel load. Each pavement layer is assumed to extend infinitely in the horizontal directions, allowing the three-dimensional problem to be reduced to an axisymmetric two- dimensional problem. Due to the linear elastic assumption, multiple wheel loads can be analyzed by superposing the results due to single wheel loads. The main drawbacks of the linear elastic layer programs are that: (a) they cannot represent the nonlinear resilient behavior of granular and cohesive soils; (b) they normally assume weightless material; (c) they may yield tensile stresses in granular material, which cannot physically occur; and (d) they do not represent "locked-in" stresses due to compaction during construction. In order to overcome these shortcomings, nonlinear analysis programs based on the finite element method have been developed (e.g. ILLI-PAVE, etc.). However, due to the large memory and computational effort requirements, they have been implemented on mainframe computers. Further, the interaction of the user 2 with these programs, in terms of data input and interpretation of the output, are not "friendly" precluding their use in daily practice. For most state highway agencies, a "user-friendly" flexible pavement program can be used for the design or rehabilitaion of flexible pavements in daily practice is desired. And this program should consider all the major factors affecting the design or rehabilitation of the flexible pavements. In order to achieve this, the main goal of this research was to review existing analysis and design methods, and then to develop a "user-friendly" program that can be used on personal computers in daily practice. Since current personal computers have limited memory capacities, the traditional finite element method which requires large amount of memory cannot be suitably implemented on them for nonlinear pavement analysis. In order to overcome this, Harichandran and Yeh (1988) proposed a new technique of placing a relatively shallow finite element mesh on a flexible boundary. This technique substantially reduces the memory and computational requirements of the nonlinear finite element method, without significantly sacrificing accuracy. In this research, this technique is implemented together with extremely user-friendly input and output features, to develop a nonlinear finite element flexible pavement analysis program on personal computers, for daily use by state highway agencies. The program has been named MICH- PAVE. Chapter 2 contains a review of relevant design methods and material models for designing flexible pavements. In general, there are two different design approaches, empirical methods, and mechanistic- empirical (rational) methods. The former are developed on the basis of functional failure criteria, while the latter are based upon various structural failure criteria. Three empirical methods are briefly reviewed at the beginning. These are the AASHTO, the National Stone Association (NSA), and the California methods of design. Following this, eight mechanistic-empirical analysis and design computer programs are briefly reviewed. These are the Chevron, CHEVSL, ELSYMS, CHEVIT, BISAR, DAMA, VESYS-II, and ILLI-PAVE programs. The advantages and disadvantages/limitations of each design method/program is discussed. Four material models such as the hyperbolic stress-strain relationship, the resilient modulus model, shear and volumetric stress- strain relationship (also called contour method), and the third order hyperelastic model are briefly reviewed. The advantages and disadvantages/1imitations of these material models are also discussed. Chapter 3 presents an overview of linear finite element analysis. The advantages and disadvantages of four finite element models are discussed here. They are the sandwich plate theory model, the plane strain model, the three dimensional finite element model, and the axisymmetric model. The reasons for selecting the axisymmetric model for implementation in the MICH-PAVE program are also outlined. The first part of this chapter discusses the formulation of the axisymmetric element stiffness matrix, assembling of the global stiffness matrix, formulation of the edge loads, and the solution of the stiffness equations. The second part covers the new concept of locating a flexible boundary at the bottom of the finite element mesh. The technical representation of the flexible boundary is discussed, and results obtained by utilizing it in linear analyses are compared with other linear methods. Finally, the sensitivity of the linear analysis 4 results to the location of the flexible boundary, the tire pressure, and the wheel load is investigated. Chapter 4 deals with nonlinear finite element analysis including: (a) the selection of suitable material models; (b) use of the Mohr- Coulomb failure criterion to modify the principal stresses in the granular layers and roadbed soil (Raad et al., 1980); (c) sensitivity studies of the results of nonlinear analysis to the location of the flexible boundary, and magnitude of the tire pressure and wheel load; (d) the automatic generation of the finite element mesh; (e) the incorporation of the gravity stress and lateral "locked-in" stress due to compaction; (f) recovery of the global stresses from the modified principal stresses; (g) the use of interpolation and extrapolation to improve the accuracy of stresses and strains at layer interfaces. In chapter 5, the linear elastic part of the MICH-PAVE program is validated by comparison of its results with those obtained from the commercial finite element program, SAP-IV (Bathe, et al.,l973). The results are also compared with those obtained from the linear elastic layer program, CHEVSL. In addition, the nonlinear part of MICH-PAVE program is compared with the nonlinear finite element program, ILLI- PAVE. A method of estimating "equivalent resilient moduli" for granular layers and roadbed soil based on nonlinear finite element analysis is introduced, These equivalent moduli can subsequently be used in linear analysis programs such as CHEVSL. In Chapter 6, two empirical equations for estimating the fatigue life and rut depth of flexible pavements based on the results obtained from mechanistic analysis are presented (Baladi, 1989). The sensitivity of the fatigue life and rut depth due to variations in key properties 5 and response values, such as the tensile strain at the bottom of the asphalt concrete layer, the compressive strain at the top of the roadbed soil, and the surface deflection, etc., are examined. The final chapter, Chapter 7, presents the conclusions of the research and recommendations for future work in this area. CHAPTER 2 LITERATURE REVIEW 2 ° N RAL The classical definition of flexible pavements, as stated by Yoder and Witczak (1975), includes those pavements that have an asphalt concrete surface. An asphalt pavement may consist of a thin wearing surface course built over a base course, subbase course, and compacted roadbed soil. Thus, the term pavement herein implies all the layers (courses) in the pavement structure. The load carrying-capacity of a flexible pavement is brought about by the load distribution characteristics of the layered system. The highest quality layer is placed at or near the surface. Hence, the strength of the pavement is the result of building up thick layers and, thereby, distributing the load over the relatively weak roadbed soil (Yoder, et a1. 1975). The structural design of flexible pavements is an evolutionary process which is continually changing as new data becomes available. This process involves the design of the supporting foundation (roadbed soil), subbase, base (may be asphalt treated), and the asphalt course. In the early stages of development, design and/or evaluation of flexible pavements consisted of rule-of-thumb procedures based on judgement and past experience. In the 1920's, the U.S. Bureau of Public Road (BPR)1 developed a soil classification system based upon field observation of soil behavior under highway pavements (Baladi, et al., 1989). This system, in conjunction with the accumulated data, helped the highway l. The Bereau of Public Road is now known as the Federal Highway administration. engineer to correlate pavement performance with roadbed soil types. After World War II, highway engineers were faced with the need to design and predict the performance of pavement systems subjected to greater loads and frequencies than they had ever experienced (Yoder, et al., 1975; Baladi, et al, 1989). Therefore, new empirical/rational pavement design procedures were introduced and implemented in the early 1950's that resulted in a better pavement design process, although severe breakup was still a common phenomenon on some highways (Yoder, et al., 1975; Baladi, et a1, 1989). Beginning in the early 1960’s, new analytical pavement design techniques started to emerge. Elastic and viscoelastic layered pavement models and finite element models were developed, and are slowly being tried across the country. These new structural models provide the pavement engineer with a better understanding of pavement behavior and performance. One drawback (as perceived by some engineers) is that the models require new types of data to be collected prior to their use. Consequently, combinations of analytical, empirical, and statistical pavement design methods were developed and implemented by some highway agencies. Other agencies adopted a standard cross-section for the design of flexible pavements. Still others directed their engineers and researchers to look for better solutions (Baladi, et a1, 1989). W In general, two classes of pavement failure and/or distress can be found: structural and functional. The former is associated with the inability of the pavement to carry the design load. The latter deals mainly with ride quality (a smooth and comfortable ride at the posted speed limits) and safety (loss of skid resistance and hydroplaning due to rutting) issues (Baladi, et al., 1989). Pavements that exhibit structural distress and/or failure (e.g. severe alligator cracking) will also exhibit functional distress and/or failure. Functionally distressed and/or failed pavements (e.g. very rough) may nevertheless be structurally sound. Each class (functional or structural) of pavement distress and/or failure usually contains several types of distress and/or failure (e.g. fatigue failure, rutting, block cracking, etc.) (Baladi, et al.,1989). Beginning in the early 1950's, several rational and empirical pavement design methods were developed. The former were based upon various structural failure criteria, while the empirical methods were developed on the basis of functional failure criteria. Efforts to perfect both empirical and rational design methods have been focused in two areas. The first of these is a proper characterization of the paving materials. The second is based on limiting the deflections and strains in the pavement structure. Further, in order to calculate the stresses, strains, and deflections of the pavement layers, several theoretical analysis methods were developed. These include : elastic, viscoelastic, transfer function, and nonlinear finite element methods. A brief literature review of the empirical and theoretical pavement design methods are presented in the subsequent sections. Wm As noted above, several empirical methods for the design of flexible pavements were developed. Each method has its own philosophy and is based upon certain assumptions, experience, and criteria. Hence, it is not uncommon, for the same input variables, to obtain different layer thicknesses using different methods. Nevertheless, three of these methods: the AASHTO, the National Stone Association (NSA), and the California method of design are briefly reviewed. The major advantage in using empirical mehtods is that they tend to be simple and easy to use. Unfortunately, they are usually only accurate for the exact conditions for which they have been developed. They may be invalid outside the range of variables used in the development of the methods. 2.2 l l ' AASHTO Flexible Pavement Design The AASHO flexible pavement design procedure is based upon the results of the AASHO Road Test conducted in Ottawa, Illinois, in the late 1950's and early 1960's. Based upon the Road Test results, the AASHO Committee on Design published an Interim Design Guide in 1961 and issued a revised edition in 1972. A new revision of the AASHTO Design Guide was published in 1986. The current guide retains modified AASHO Road Test performance prediction equations as the basic models for use in pavement design. Major flexible pavement design procedure changes have been made in several areas including: 1. Incorporation of a design reliability factor, based upon a variation in the magnitude of the design traffic to allow the designer to use the concept of risk analysis for various classes of highways. 2. Replacement of the soil support number with the resilient modulus (AASHTO T274) to provide a rational testing procedure for 10 defining material properties. . Use of the resilient modulus test for assigning layer coefficients to both stabilized and unstabilized material. . Provision of guidance for the construction of subsurface drainage systems and modifications to the design equations to take advantage of improvements in performance that result from good drainage. . Replacement of the subjective regional factor with a rational approach to the adjustment of designs to account for environmental conditions such as moisture and temperature variations. The 1986 AASHTO flexible pavement design method considers the following general design variables: time constraints (include performance and analysis periods), traffic, reliability, environmental impacts, and performance criteria. 1. Time constraints - The selection of various performance and analysis periods forces the designer to consider several design strategies which range from a low-maintenance structure to staged construction options. The performance period is the period of time that elapses as a new or rehabilitated pavement structure deteriorates from its initial serviceability to its terminal serviceability and requires rehabilitation. The designer must select minimum and maximum allowable bounds on the performance period. The analysis period is the period of time that any design strategy must cover. The analysis period may be identical to the selected performance period. However, realistic practical performance limitations for some pavement designs may necessitate 11 the consideration of staged construction or planned rehabilitation to achieve the desired analysis period. The AASHTO design guide recommends that the analysis period be selected to include at least one rehabilitation of the pavement. . Traffic - The AASHTO Flexible pavement design methods are based on the cumulative number of expected l8-kip equivalent single- axle loads (ESAL), during the analysis period. . Reliability - Design reliability refers to the degree of certainty that a given design alternative will last for the entire analyis period. The AASHTO design-performance reliability is controlled through the use of a reliability factor that is multiplied by the design period traffic prediction to produce design load applications for use in the design equations. For a given reliability level, the reliability factor is a function of the overall standard deviation which accounts for standard variation in material properties and construction practices, the probable variation in the traffic prediction, and the normal variation in pavement performance for a given design traffic. . Environmental Impacts - Temperature and moisture changes have substantial effects on the strength, durability, and load- carrying capacity of the pavement and roadbed materials through the mechanics of swelling soils, frost heave, and other phenomena. Criteria for modifying the input requirements and for adjusting the pavement performance period due to environmental conditions are provided in the AASHTO design guide. . Performance criteria - The serviceability of a pavement is defined as its ability to serve the type of traffic that uses the 12 facility. The primary measure of serviceability used by the AASHTO procedures is the present serviceability index (PSI), which ranges from a minimum of 0 (representing impossible conditions) to a maximum of 5 (representing perfect conditions). The AASHTO flexible pavement design equation is given below: loglo(APSI/(4.2-l.5)) .4 + lO94/(SN+1)5'19 +2.32*log10(MR) -8.07 (Eq.2-1) logloW18 - ZR*SO + 9.36 *1og10(SN+l) - .20 + Where : ths - the number of l8-kip single-axle load repetitions; SN - the structural number; APSI - the design serviceability loss; ZR - reliability factor; So - standard deviation; and MR - effective roadbed soil resilient modulus. In addition, the AASHTO flexible pavement design procedure provides means to adjust the layer coefficients to take into account the effects of certain levels of drainage on pavement. Therefore, the structural number equation modified for drainage becomes : SN - alh1 + azhzm2 + a3h3m3 (Eq.2-2) Where: a1, a2, a3 - layer coefficients for surface, base, and subbase, respectively; hl’ h2, h3 - layer thicknesses for surface, base, and subbase, respectively; m2, m3 - drainage modifying factors for base, and subbase, respectively. A set of nomographs and computer program (DNPSBG) were developed to aid the pavement designer in evaluating the influence of design 13 variables on the final thickness and to examine the various design (AASHTO, 1986). The 1986 AASHTO design procedures have several limitations including (ERES, l. 1987; Baladi, et al., 1989) Limited Materails and Subgrade - The AASHO Road Test used a specific set of pavement materials and one roadbed soil. The extrapolation of the performance of these specific materials to general applications may be dangerous. . No Mixed Traffic - The AASHTO Road Test accumulated traffic on each test section by operating vehicles with identical axle loads and axle configurations. In-service pavements are exposed to many different axle configurations and loads. . Short Road Test Performance Period - The number of years and heavy axle load applications upon which the design procedure is based represents only a fraction of the design age and load applications that many pavements must endure. . Load Equivalency Factors - The load equivalency factors used to determine cumulative l8-kip ESAL pertain specifically to the road test materials, pavement composition, climate and subgrade soils. The accuracy of extrapolating them to other regions, material and environment, is not known. . Variability - A serious limitation of the AASHTO design procedure is that it is based upon very short pavement sections where construction and material quality were highly controlled. Typical highway projects are normally several miles in length and contain much greater construction and material variability. 14 6. Lack of Guidance on Some Design Input - Structural coefficients and drainage modifying factors are very significant in influencing flexible pavement layer thickness and there is very little guidance given in the guide. Successful use of the AASHTO Guide requires considerable experience and knowledge of the assumptions and underlying basis for design. It is strongly recommended that the resulting design be checked using other procedures and mechanistic analysis (Baladi, et al., 1989). 2.2.1.2 ; National StoneLAssociation Design Method The National Stone Association (NSA, 1972) design method is based upon the U.S. Corps of Engineers (1961) pavement design procedure which uses a modified California Bearing Ratio (CBR) to determine the strength of roadbed soil (Baladi, et al., 1989). The NSA method incorporates crushed stone-base course as a part of the pavement system. The basis of the method is to provide adequate thickness and material quality to prevent repetitive shear deformation within any layer and to minimize the effects of frost-action. The NSA design method uses only two basic input criteria to determine the total thickness of the pavement structure. The first is the strength of the roadbed soil as determined by its CBR. The second is the amount of traffic (l8-kip ESAL applications) estimated to travel the roadway over a twenty year design life. The total pavement thickness is obtained from a design table using the CBR and Traffic values. The total thickness is then divided into asphalt concrete (AC) and granular base layers. A minimum thickness of AC is required for each particular level 15 of traffic. The advantages of this method are (Baladi, et al., 1989): 1. The method is simple to use. 2. The input requirements are minimal and usually easy to obtain. 3. The method has been revised as necessary through long-term monitoring of performance of in-service pavements. The disadvantages of this method are: 1. The strength properties of each layer above the roadbed soil are not considered. 2. The time and temperature dependence of the AC layer is ignored. 3. A stabilized base layer is not an option using this method. 4. Uncertainty and variability in performance may result from the application of a generalized design procedure to a site-specific condition. 5. The minimum thickness of AC is not always sufficient to withstand the design traffic. 2.2.1.3 ; California Method of Design The California method of design is based on two properties of the paving materials: Cohesion which is obtained using a cohesionmeter, and stabilometer resistance value (R) obtained using a stabilometer. The R value is used along with equivalency factors to design the pavement structure (Baladi, et al., 1989). The required thickness of gravel equivalent (GE) above each material is determined using the following equation: GE - 0.0032(TI)(100-R) (Eq.2-3) where 16 GE - gravel equivalent; TI - traffic index which is a function of the amount of equivalent 5 kip wheel loads; R - stabilometer value. The actual thickness of each layer is then determined by dividing the GE by an appropriate equivalency factor. The advantages of this method is that it is simple and easy to use. However, the simple form becomes a drawback because the AC properties are neglected. Another disadvantage of this method is that the equivalent 5 kip wheel load is far less than the current wheel loads on the highways (Baladi, et al., 1989). 2.2.2 ; Mechanistic-Empirical Methods The basic components of mechanistic-empirical or rational methods consist of a structural analysis of the pavement system and the incorporation of distress or performance functions into the method. Structural analysis refers to the calculation of stress, strain, and deflection developed in a pavement section due to traffic loads, temperature, and/or moisture. Once these values are determined at critical locations in the pavement structure, comparisons can be made with the maximum allowable values obtained from experimental or theoretical studies based on predictions of pavement distress such as cracking, rutting, or roughness. The pavement can then be designed by adjusting the different layer thicknesses so that the calculated stresses, strains, and deflections are less than the maximum allowable values. l7 Mechanistic flexible pavement design procedures are typically based on the assumption that a pavement can be modeled as a multi-layered elastic or visco-elastic structure on an elastic or visco-elastic foundation. Assuming that pavements can be modeled in this manner, it is possible to calculate the stress, strain, or deflection due to traffic loadings and environmental conditions at any point within or below the pavement structure. However, researchers have recognized that pavement performance is influenced by a number of factors that cannot be precisely modeled by mechanistic methods. Therefore, these methods were calibrated using field observations of pavement performance. Thus, the xnethods are referred to as a mechanistic-empirical design procedures. An important advantage of this design philosophy is the ability to aarmalyze a pavement for several different failure modes such as cracking aarmd.permanent deformation (rutting). This allows the engineer to adjust t:11£e pavement design and to produce a cost-effective pavement section that does not fail prematurely. The main disadvantage of this design method is that it requires more ‘3<>En13rehensive and sophisticated data than empirical design mehtods. E“tensive laboratory and field testing may be required to determine the deSign parameters such as the resilient modulus, creep compliance, and Otnlfiflrs. lievertheless, most of the mechanistic-empirical design methods have ‘beetl computerized. Several of these methods are presented in the subsequent sections . 18 2.2.2.1 ; Chevron Program The Chevron program (developed by the Chevron Oil Company) is based on Burmister's linear layered elastic solution. The basic assumptions behind Burmister’s theory in relation to pavement structures are: 1. Single-wheel loads that are vertical, uniformly distributed over a circular area, and statically applied. 2. Each pavement layer consists of a homogeneous, isotropic, and linear elastic material. 3. There is continuous contact between each interface of the layered pavement. 4. Each layer is infinite in the horizontal directions and has a finite depth, except for the bottom layer which has an infinite depth. 5. Deformations of the pavement are small. 6. Temperature effects are neglected. 7. The pavement is weightless. The Chevron program has been widely used in the analysis and design (’13 flexible pavement structures. The program is relatively easy to use z1r1<1 'requires little computer time. The disvantages of the Chevron pro gram include: 1. It neglects the effects of the pavement weight. 2. It cannot model the nonlinear behavior of the granular layers and roadbed soil. 3. It is limited to a single wheel load. Several modifications of the Chevron program were made. These include: l9 1. The CHEVSL program which is capable of handling dual loads and up to 5 layers. 2. The ELSYMS program which is designed for a microcomputer, and can handle up to 5 layers and 10 wheel loads (Kopperman, et al., 1985). 3. The CHEVIT program which includes iterative procedures to determine the stress-dependent moduli of pavement materials and a superposition subroutine for multiple-wheel loads (Chou, 1976). However, the CHEVIT program assumes that the material moduli in the horizontal direction are constant. This is true if the applied stresses are small such as in the roadbed soil. In the base layer where the stresses are high, the assumption leads to a certain degree of error. 22.. 2 2 ° Th Shell Method The shell method uses a set of design charts for the design of the f?1.£3: 8 e [1/Ei + Rfe/(a1 - a3)f] Equation 2-8 can be simplified by using Eq.(2-4) to express 6 in terms of stresses (01-03), and substituting Ei and (al - a3)f from Eq.(2-5) and (2-7),respectively. R (1 - sin ¢)(a - a ) f l 3 n Et - [l - ] K Pa (a3/Pa) (Eq.2-9) 2c cos ¢ + 203 sin d This equation can be used to calculate the appropriate value of the tangent modulus for any stress conditions[a and (a1 - 03)], if the 3. values of the parameters K, n, c, d, and Rf are known (Chen, et al., 1982). The limitations of the hyperbolic stress-strain relationship include: 1: The relationships must be suitable for analysis of stresses and strains prior to failure. When elements have already failed, the results will no longer be reliable. In most elastic analysis of pavement structures, the granular layer will fail in tension. Therefore, the model cannot be used without modification. Also, the model is more appropriate for monotonic loading conditions than for repeated loadings that are encountered in pavement analysis. 2: The hyperbolic relationships do not include volume changes due to changes in shear stresses or "shear dilatancy". 2.2.2 ; The Regilieng Modulus Hicks and Monismith (1972) developed nonlinear models that expresses the resilient modulus of granular material in term of 28 confining pressure (lateral stress) and of cohesive soils in term of the deviatoric stress (the principal stress difference 01 - 03). The resilient modulus is a dynamic test response defined as the ratio of the repeated axial deviator stress to the recoverable axial strain as follow: M _ (Eq.2-10) where Mr - the resilient modulus (psi); ad - the repeated axial deviator stress (psi); 6a - the recoverable axial strain (in./in.). For granular materials, the resilient modulus can be expressed either in terms of the bulk stress or lateral stress as follows (Young, et a1. , 1977): Mr - K1(0)K2 (Eq.2-11) or M - K '(0 )K2' (Eq.2-l2) r l 3 where Mr - resilient modulus (psi); 0 - confining pressure (psi); 3 0 - bulk stress (- 01+02+a3) (psi); and K1,K2,K1 ,K2 - experimental test constants. As with static testing on granular materials, Mr modulus values increase with increasing density, decreasing saturation and increasing angularity of the particles. For fine-grained soils, the resilient modulus can be expressed as Mr - K2 + (K1 - ad)K3, for ad < K1 (Eq.2-l3a) 29 Mr - K2 + (ad - K1)k4, for ad > K1 (Eq.2-l3b) K1, K K3, K - material constants determined by least squares curve fitting methods; and a - deviator stress (01 - 03) (psi). For granular material, Equation 2-11 was found to be more accurate than equation 2-12. Analysis of test data revealed a higher correlation coefficient and a lower standard error for (Eq.2-ll) than for (Eq.2-12). The explanation for this is believed to be that (Eq.2-ll) accounts for all 3 principal stresses, whereas (Eq.2-12) accounts for only 2 principal stresses. Some limitations of using the resilient modulus models in the pavement analysis include: 1. It only considers a very limited range of stress paths. 2. The models may not be suitable for a three dimensional system, since they are based on laboratory tests with only a two dimensional state of stress. 2.3.3 ; Shear and Volumetric Stress-Strain Relationship To overcome the first limitation of the resilient modulus models and to be able to account for the effects of shear and volumetric strains, Brown and Pappin introduced shear and volumetric stress-strain relationships (Brown, et al., 1981). Their relationships can more accurately simulate the deformation and stresses within pavements. However, these relationships require seven material constants in order to compute the shear and volumetric strains. For most state highway 30 agency laboratories, the lack of sophisticated equipment to control the stress paths in order to estimate all the constants, has resulted in a very limited application of these relationships. 2.3.4 ; Third Order Hyperelastic model. Ko and Mason (1976) used a complete three-dimensional third-order hyperelastic model to simulate the behavior of medium-loose Ottawa sand under loading. This classical continuum mechanics model accounts for material nonlinearity, its dependence on the hydrostatic stress, and its dilatancy and stress-induced anisotropy. The model requires nine material constants that are difficult to estimate in practice. Hence, its application remain very limited. 2.4 ; PAVEMENT EVALUATION METHODS Several in-service pavement structural evaluation methods were developed based on nondestructive deflection testing. These include transfer functions and back-calculation of layer moduli. 2.4.1 : Transfer Function Theory Although transfer function theory has been applied for some time by mechanical and electrical engineers, Swami, et a1. (1970) were the first to use transfer functions to characterize the time-dependent behavior of asphalt concrete in the laboratory. Boyer (1972), Highter, et a1. (1974), and Baladi (1979) applied transfer function theory to in- service pavements. They measured the pavement surface deflection due to moving wheel loads and calculated the transfer function of various air field and highway pavements. They showed that the parameters of the 31 transfer function represented the pavement. global properties that are independent of the type of load input and that the temperature of the pavement had the greatest single effect on the transfer function. Due to the nature of the transformation, however, the nonlinear behavior of the pavement materials was not accounted for. 2.4.2 Back-calculation Methods Recently many methods for the back-calculation of layer moduli using nondestructive deflection testing have been developed. Some of the methods use the elastic layer theory while others use the finite element method. Conceptually, all methods are based on iterative routines whereby layer moduli are assumed and the the pavement surface deflection is computed. If the computed values match the field measured ones, then the calculation is terminated. It should be noted that (for most methods) the calculated layer moduli are not unique, they depend on the assumed values of the seed moduli. Moreover, various combinations of layer moduli values may exist such that the calculated deflections match the measured ones. Nevertheless, the methods are still in the developmental stage and they can be used to estimate the material properties of each layer of an existing flexible pavement. ills—Um Two basic methods are currently being used to determine the required layer thicknesses for flexible pavement structures: empirical, and mechanistic-empirical or rational methods. Empirical methods are derived from experience or observation, often without regard to system behavior or pavement theory. The advantage of 32 using empirical models is that they tend to be simple and easy to use. Unfortunately, they are usually only accurate for the exact conditions under which they were developed. Mechanistic-empirical design methods utilize calculated stresses, strains, and deflections and pavement distress or performance prediction models. Mechanistic: approaches are, in general, capable of analyzing a pavement structure using several different failure modes. One disadvantage is that they typically require more comprehensive and sophisticated data inputs than empirical design methods. Several factors influence the response of pavements including temperature, material properties, water table, tire pressure and magnitude of the applied load. Several pavement design procedures were developed and computerized whereby one or several of these factors were accounted for. Each program has it own advantages and limitations. The main limitation of the Chevron, CHEVSL, ELSYMS, and BISAR programs is that they do not account for nonlinear material responses. The CHEVIT program attempts to partially account for nonlinearities, but suffers from the drawback that any given sublayer must have the same modulus even though stresses vary with radial distance away from the load. The ILLI-PAVE program, is perhaps the only one that is capable of reasonably representing nonlinear materials. However, the large number of finite elements required for a typical analysis requires a significant amount of memory and computational time. Four nonlinear material models can be found: the hyperbolic stress- strain, the resilient modulus, the shear and volumetric stress-strain, and the third order hyperelastic models. 33 The principal objectives of flexible pavement thickness design are to minimize compressive strains in the roadbed soils and to minimize the tensile strains at the bottom of the asphalt layer. CHAPTER 3 LINEAR FINITE ELEMENT ANALYSIS 3 1 ‘ GENERAL A comprehensive analysis of flexible pavements should include soil overburden, inelastic behavior of granular and cohesive material, the finite width of the pavement, and the lack of bonding between the asphalt and granular layers. Four kinds of finite element methods that may be applied in the analysis of flexible pavements are reviewed here. None of these models, however, is capable of incorporating all the above effects, therefore the method that incorporates the most important factors should be chosen. One approach is to use a sandwich plate model of an asphalt layer with a granular base. Its main advantage is its capatility of modeling the finite width of the pavement (i.e., the edge effect). Since the nodes of the plate elements lie in the center of the thickness, it is difficult to connect them to the nodes of the brick elements representing the granular base. The unbound behavior of the granular base cannot be easily represented in this method. Kujawski and Wiberg (1982) used this technique to study rigid pavements. They assumed that the displacement vector u - u(x,y) can be expressed by the sum of zero order components which are constant along the z-direction, and two angles of rotation due to pure bending and shear deformations of the plate. They also found the interaction between the plate and the three dimensional element. The vertical displacements based on their approach showed good accuracy but the horizontal displacements did not. 34 35 Furthermore, the elastic modulus of asphalt is only about one fifth of that of concrete, making this model less suitable for flexible pavements. Another approach is to use a two dimensional plane strain model of the pavement cross section, in which both the edge effect and inelastic behavior can be modeled. However, the wheel loading would have to be considered as infinitely long in the longitudinal (out of plane) direction. This would be unrealistic because most trucks have few axles with fairly wide spacings, rather than many closely spaced axles. The problem is therefore three dimensional and cannot be simplified to a two dimensional problem if all the conditions to be investigated are retained. The most comprehensive approach would be one that uses three- dimensional finite element (FE) analysis. In a three dimensional model, suitable boundary conditions must be imposed at some reasonable distance away from the loaded region in all three directions. Ioannides and Donnelly (1988) used the radius of relative stiffness for a slab on an elastic foundation (18) to decide the vertical and lateral subgrade extent. From Barkdales and Hicks's (1973) report, the tranverse distance of one side from the loading should be about 90 inches. Assuming the distance between the two wheels to be 72 inches, and allowing 90 inches outside the wheel loads, the total transverse distance is 252 inches. E h3 (1_”s)2 1/3 2 6(l-p ) ES where E : Young's modulus of slab h : Slab thickness p: : Poisson's ratio for elastic foundation p : Poisson's ratio for slab Es : Young's modulus of elastic foundation 36 Using a depth of 270 inches, and dividing this region into eight-node brick elements results in approximately 3240 degrees-of-freedom (d.o.f.) per lO-inch length of pavement. Dysli and Fontana (1982) used a three dimensional model to simulate a field excavation which had 3153 d o.f.. This 3-D model was processed on a VAX 11-780 computer with 2 Mbytes of core storage and the computation required over 30 hours of CPU time. Therefore, from a pratical point of view, the use of a three dimensional model is time consuming and uneconomical. Furthermore, it should be noted that the solution of an inelastic system by the FE method requires many iterative or incremental solutions of linear problems with the stiffness matrix having to be reassembled many times. This makes the problem even longer. Thus, unless supercomputers become readily accessible, a 3-D FEM. model cannot be used for day-to day design. The most commonly used 2-D model is the axisymmetric one. This model assumes that the pavement geometry and loading are both axisymmetric. With these assumptions, a three dimensional problem can be reduced to a two dimensional one. However, the asphalt surface must be assumed to be infinitely wide and therefore the edge effect cannot be considered. Although non-axisymmetric multiple wheel loads can be analyzed by superposition for linear elastic material, this cannot be done for nonlinear materials. Apart from multiple wheel loads and consideration of the edge effect, all other effects can be included. In comparison to the other three approaches, the axisymmetric model has significant advantages in the analysis of flexible pavements. 37 3.2 ; AXISYMMETRIC FINITE ELEMENT ANALYSIS 3 ' ormulation of Axis etric Element Stiffness Matrix Cook (1981) formulates the plane linear isoparametric element which can be revised to obtain the axisymmetric element. Bathe and Wilson (1976) outline a computer program (called subroutine QUADS) which can implement plane stress, plane strain, and axisymmetric analysis. The geometry and shape of a four-node element is shown in Figures (3-1) and (3-2). The global coordinates (radial and vertical), and the corresponding displacements at an internal point can be related to the corresponding nodal quantities through shape functions: r U - [N]{C) and - [N]{U} (Eq.3-1) Z W T where {C} - {ri zi rj Zj rk zk rm T and {U} - {ui wi uj wj uk wk um wm} are the nodal displacements. zm) are the coordinates of the nodes The shape function matrix is 1 j k 0 N 0 N 0 N 0 N N 0 N 0 N 0 N 0 [N1 - m 1 j k Sometimes it is convenient to write Equation (3-1) in the form r - E N r u - 2 N u I I I I (Eq.3-2) z - 2 NI zI w - 2 NI wI where I - i, j, k, m The individual shape functions are Ni - 0.25(l-£)(l-n) N - 0.25(l+€)(1-n) (Eq.3-3) j Nk - 0.25(1+5)(1+n) 38 Nm - 0.25(l-£)(l+n) If d is some function of r and 2, then applying the chain rule of differentiation yields 6¢ do 6r a¢ az __.. + 86 8r BE 62 85 8¢ do 6r 8¢ dz ———— - + an ar an 62 an ’ ¢P or { 6} - [J]{ r} (Eq 3'4) ¢." ¢’z Where [J] is the Jacobian matrix — 6r dz 1 66 36 J J 11 12 [J] ' ' [ ] (Eq 3'5) 3r az J21 J22 .37) 3'11 The inverse relations of equation ( 3-5 ) is ¢. . { r} - [Fl{ ’3} (Eq.3-6> 45.2 M F F 1 J —J a... m - [11 12] - [ 22 12] (3.3-7) I‘21 P22 J11J22'J21J12 ‘J21 J22 J11 - r,€ - Ni,£ ri + Nj,€ rj + Nk,£ rk + Nm,£ rm (Eq.3-8) There are similar expressions for J12, J21, and J22, where Ni,£ - -0.25(1-n) N1 0 - -0.25(l-€) ...etc. (Eq.3-9) \ The relationships between the strain and displacement vectors are er au/ar co - u/r (Eq.3-10) ez aw/az Irz au/az + aw/ar It is convenient to treat the tangential strain 6 separately. 9 39 FR. F S ....«-/ :1 \ b 1 ’\ \4 / l J \ / i / / In." / 1 Figure 3—1 : A Typical Axisymmetric Finite Element Figure 3-2 : The Four-Node Isoparametric Element 40 Considering er, :2 and 7rz only, the strain-displacement relations can be expressed as e l 0 0 0 (e) - er - 0 0 o 1 £2 0 1 1 o rz The displacement derivatives in the global coordinates can be related (Eq.3-ll) , 9 s s c c N n N H D 9 to those in the local isoparametric coordinates through ' r r“'r1 P11 P12 0 O 1 u’€1 u, - F F 0 0 u, + 2L 21 22 4 7)} (Eq.3-12) w,r O 0 F11 F12 w,€ Lw,zj _ 0 0 F21 F22‘ kw’fl‘ Finally, the displacement derivatives in the local coordinates are expressed in term of the derivatives of the shape functions and the nodal displacements, as 'u,€l ”N1 6 o Nj’s o Nk’g o Nm,€ o ‘ ‘u,"> - N1," 0 NJ," 0 Nk,n o Nm’n 0 {U} (Eq.3-l3) w,6 0 Ni,§ Nj,€ 0 Nk,€ 0 Nm,§ 1"an . 0 Ni," 0 NJ," 0 Nk,q 0 Nmnrtma Combining equations (3-11), (3-12) and (3-13), we obtain the strain- displacement relations {6} - [BllUl Matrix [B] is the product of the three succesive rectangular matrices in equations (3-11) through (3-13). The tangential strain may also be expressed in terms of the shape functions and nodal displacements, as ‘0 - [N]{U)/r (Eq.3-14) where r - N r + N r + N r + NmrIn 1 1 j j k k Thus, incorporating 6 into the strain-displacement relations, we can 0 41 write fer 1 P311 B12 B13 314 BIS B16 B17 B18 1 {e} _ .‘o p ' B21 B22 B23 324 B25 B26 B27 B28 {U} (Eq_3_15) ‘2 B31 B32 B33 B34 B35 B36 B37 B33 17:24 ~341 842 843 Baa 345 Bus 347 B48 ‘4x8 where B11 ' r11111,: + I12111," ‘ 342 B13 ' P11113.5 + P12Nj,n ' Baa B15 ' P11Nk,5 + P12Nk,n ‘ B46 B17 ' I'11"'m,g + F12Nm,n I 548 821 - Ni/r ; 323 = Nj/r ; 825 = Nk/r ; 827 = Nm/r B32 ' I‘21”‘145 I F22N1,q " 341 334 ' FZINj,£ + P22Nj,n ' 343 B36 ' PZINk,£ + I‘22Nk,n ‘ 345 B38 ' r21Nm,5 I P22Nm,n ' 347 and B =B =B = 0. B12'314'315'318'322'324‘326= 26‘331'333 35 37 The element stiffness matrix is [ke] - f f [V [B]T[D][B]rdrd0dz (Eq.3-l6) - 2x I f [B]T[D][B]rdrdz 1 1 Ike] - 2« J J [BITIDIIBJrIIJIIdsdn (Eq.3-17> -1 -1 where the matrix of elastic constants [D] is E [D] --— 0 0 (Eq.3-18) l+v 0 OU‘O‘O- OU‘O-U‘ OO-O‘O‘ 0.5 The constants in the above matrix are: l-v u d - ; b - l-2v l-2v iIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII"""""""" 42 where E - elastic modulus, and v - Poisson's ratio Equation (3-17) must be integrated numerically. Using Gauss quadrature in two dimensions, the integral of a function ¢(€,n) can be expressed as l 1 [Re] - ¢(€.n)d£dn = Z X wi.wj.¢ (Eq.3-l9) -1 -1 1 3 where W1 and Wj are weights associated with the Gauss-points (€i,nj). 3.2 2 ' Assembling the Global Stiffness Matrix The global stiffness matrix is usually a banded matrix where all the nonzero elements cluster about the diagonal. For efficiency in storage and computation the zeros outside the band need not be stored or processed. Since the global stiffness matrix is also symmetric,it is only necessary to store the elements on the diagonal and on one side of the diagonal. The semibandwidth of a symmetric banded matrix BW, can be found by the following scheme: (a) find the column number j of the last nonzero entry in row i of [K] and compute bi = 1+(j-i); (b) repeat this for all rows and indentify the largest bi as BW. The elements in the upper semibandwidth can be stored into a compressed matrix by shifting rows to the left. 1 space for row 2; 2 spaces for row 3; and i-l spaces for row i. Thus all diagonal coefficients of the matrix appear in the first column of the matrix. The entire information content of a symmetric banded matrix resides in the (N x BW) coefficients of the semiband, where N is the total d.o.f.. In pratice, N may greatly exceed BW, so there is obvious merit in storing (N x BW) coefficients rather than all N2 matrix coefficients. 43 There are other efficient methods to assemble the global stiffness matrix, such as the skyline method which stores the nonzero elements of the matrix (Bathe, et al., l976)in a one-dimensional array. However, the semibandwidth method is both easy implement and sufficient for the problem being considered. 3.2.3 ; Formulation of the Edge Loaga At the upper edge of an element n - l, and the shape functions reduce to N1 - 0.5(1 - 6) N2 - 0.5(1 + 5) N115 - -0.5 N2.€ - +0.5 The edge loads applied along the upper edge of an element need to be transformed into equivalent nodal loads. This is performed by integrating along the surface S T fS - 2nr J N 8 d8 (Eq 3-20) S E8 is the forces along surface. Changing to the 6 coordinate, the radial and vertical nodal loads can be expressed as (see figure (3-3)) 1 8r az Pr1 - 2nr Ni Pt - Pn d£ (Eq.3-21) _1 65 as 1 ar 62 P21 - 2xr Ni Pn + PC d5 (Eq.3-22) _1 65 86 where Pn is the normal load, and Pt is the tangential load. In pavement analysis, wheel loads are considered to act vertically and hence Pt - 0. Therefore equations (3-21) and (3-22) reduce to 44 1 dz Pri - 2xr J Ni { - Pn 35 } d5 - 0 (Eq.3-22) -1 Since dz/df - 0 for local and global axes in the same direction (as in this case). 1 dr Pzi - 2nr Ni Pn d5 (Eq.3-23) -1 35 Using Gaussian quadrature, one dimensional integration of a function ¢(£) becomes Pz - § wi ¢(si) (Eq.3-23a) where W1 is the weight associated with the Gauss-point £1 The sum of the nodal forces at the same node from the adjacent elements is the equivalent nodal load due to the tire pressure (See Figure 3-4). 3.2.4 ; Gauss Elimination for the Solution of the StiffnesaaEquations In the Gauss elimination of the stiffness equations [K]{U} - {R} the first equation is symbolically solved for the unknown U1, then substituted into the subsequent equations. The second equation is similarly treated, then the third, and so on. This forward reduction process alters {R} and changes [K] to an upper triangular form, with 1's on the diagonal. Finally, the unknown displacements are found by back- substitution, so that the numerical value of U is found last. 1 45 Z l a Pn '-_ Pt . l 1'" ' l o 11...... 1» 41—441.. 1 1 n I2 3 [4 f L. . I g 15 17 l 8 f g . -1 ,. r .r I ¢_ R Figure 3-3 : Normal and Tangential Loads/unit Length Applied to an Isoparametric Element 11 2 Total Loads - wa P Tire Pressure - P HI 1@[2@3@ z. @ . 5e h—a/4 4.5.74.1...1/4 4+ r o o o L Er and ' (1+v) (l-2u) pr, for r s ro uv(r;ro;p) - 4 2E (Eq.3-29) (1+v) (l-2v) 2 pro , for r > ro L 2Er respectively, where r - horizontal distance from the center of the load; E, V - Elastic modulus and Poisson's ratio of half-space. F(a,fi;1;x) is the Hypergeometric function with parameters a, fl and 7, the series representation of which is ad u(a + 1) 1306 + 1) F(a,fi;1;x) - l +-—————- x + x + ... (Eq.3-30) (1)1 (1) (2) 1(1 + 1) For a radial (outward) load varying linearly from zero at the center to p at distance r0, acting on a circular area of radius r0, the vertical and radial displacements are: (1 + v) (1 - 2v) r 2 WR(r;ro;p) - 1 - [—-] pro, for r < r 2E r0 o(Eq.3-3l) 0 , for r 2 r o and ' 2 (Luz) F(1.5,—0.5;2;(r/ro) ) -———————— pr , for r < r0 E 4(l-u2) uR(r;ro;p) - < -————————— pro, for r = r0 (Eq.3-32) 3nE 2 3 2 p(1-v ) ro F(1.5,0.5;3;(r /r) ) [ ],for r > r o 2 o L 4B 1' respectively. Consider now the two boundary nodes shown in Figure (3-7), with d.o.f i and j at node A, and d.o.f. k and 2 at node B. By approximating the vertical ring load at k by a uniform vertical load over a very thin annulus of width 25 (see Figure 3-8), the flexibility coefficients fik’ f. and f can be estimated as follows: jk 2k fik - wv(r1; r2+e; p1) - wv(r1; rZ-e; p1) (Eq.3-33) fjk - uv(r1; r2+e; p1) - uv(rl; rz-e; p1) (Eq-3-34) f1k - uv(r2; r2+e; p1) - uv(r2; rZ-e; p1) (Eq-3-35) l where p1 - -———————— (Eq.3-36) 4nero With p1 defined as in equation (3-35), the total load on the annulus is unity. As long as e is small, these coefficients are not sensitive to the exact magnitude of e. Due to the Maxwell-Betti reciprocal theorem fki - f1k - fkj - fjk and sz - flk' This technique cannot be used to estimate fkk or f££' These flexibilities are very sensitive to the magnitude of e. In fact, as mentioned before, as e 4 O, f 4 w and f e m. To estimate the kk £1 54 diagonal flexibilities, therefore, we assume that a uniform load ( with unit total load ) is applied in an annulus from midway between nodes A and B to midway between nodes B and C in Figure (3-7). The loading approximations used for the vertical and radial loads are illustrated in Figure (3-9) and (3-10). fkk z wv(r2;(r2+r3)/2;p2) - wv(r2;(rl+r2)/2;p2) (Eq 3-37) and f1, z uR(r2;(r2+r3)/2;p4) - uR(r2;(r1+r2)/2;P3) (EQ.3-38) where l p - (Eq.3-39) 2 1r[(r2+r3)2 - (r1+r2)2] 6(r +r ) p3 - 31 2 3 (Eq.3-40) «[(r2+r3) ' (r1+r2) 1 and (r +r ) P4 - p3 --Z-§- (Eq.3-4l) (r1+r2) The expression for p2, p3 and p4 given above ensure the load patterns illustrated in Figure (3-9) and (3-10), with the total load in each case being unity. All the diagonal terms of the flexibility matrix can be estimated as in equations (3-37) and (3-38). 3.3 ; COMPARISON OF THE FLEXIBLE BOUNDARY WITH OTHER LINEAR METHODS Analysis using the flexible boundary were performed for homogeneous and multilayered (Three-layered) half-spaces. in both cases, a load of 100 psi was applied on a circular area of redius 10", and the flexible boundary was placed at a depth of 50", above which a finite element mesh was used. The side boundary was placed at 100" from the centerline (10 times the radius of the loaded area) for both cases. The material 55 z A . f3 _. 4 r? if r. .1 s i k T , T 1 :—\> _.—> A \r A B C Homogeneous half-space Figure 3-7 : Typical Nodes and Degree—of-Freedom Figure 3-8 : Vertical Load on Thin Annulus 56 lling'(2 Ring 13 \ a 4’ \ fl’ \ \ Ring A \ \ \ \ P p Figure 3-9 : Uniform Vertical Loading to Estimate fkk 4; Ring C Ring 8 \‘4 \ \ \ Ring A \ P‘. g ‘ r -—P 1 I ll , / / / I I Figure 3-10 : Linear Radial Load to Estimate f11 57 properties were as follows: Homogeneous : E - 5000 psi; v - 0.45. Multilayer : layer 1 (asphalt) - E - 200,000 psi; v - 0.35; depth - 10"; layer 2 (base) - E - 15,000 psi; v - 0.40; depth - 20"; layer 3 (roadbed soil)- E - 5,000 psi; v - 0.45; infinite depth. In order to compare the results with the traditional finite element approach, a mesh of depth 510" was uesed with a fixed boundary. (As noted by Duncan et. a1. (1968) a deep mesh is required in traditional finite element analysis.) The number of elements was kept the same in both meshes to facilitate a direct comparison, while keeping the computational effort approximately the same. This meant that in the traditional mesh the deeper elements had very large length to width ratios. The mesh used with the flexible boundary and the traditional mesh (finite element only) are shown in Figures 3-11 and 3-12, respectively. The same meshes were used for the homogeneous and multilayered cases. The vertical displacements along the top free surface and the variation of the vertical dsiplacement with depth beneath the center of the loaded area shown in Figures 3-13 and 3-14. The percentage errors in both finite element approaches, as compared with the exact results, are tabulated in Tables 3-1 and 3-2. (The abbreviations "FE+FB" and "FE only" are used to denote "finite elements plus flexible boundary" and "finite elements only", respectively, in the figures and tables.) It is apparent that use of the flexible boundary gives significantly better results, especially for the multilayered case where displacements within the subgrade contribute significantly toward the total displacements. The flexible boundary approach is more accurate for the homogeneous case than for the multilayered case, but in both cases it is more accurate than traditional finite element approach. 58 r” 100” Homogeneous Elastic Half-Space Figure 3-11 : Finite Element Mesh used with Flexible Boundarv 59 G. I 00 " Load ' I ‘1 :lo I 2 O " 4. O " J \ \‘WW‘\\M\\\\ ' C Fixed Boundary Figure 3-12 : Traditional Finite Element Mesh 60 The variation of vertical and radial stresses with depth beneath the center of the loaded area are presented in Figures 3-15 and 3-16, respectively. The percentage errors, as compared with exact results, are tabulated in Tables 3-3 and 3-4. Again, use of the flexible boundary gives better results than the tranditional approach. The differences in the stresses, however, are less significant than those in the displacements. For the homogeneous case, at depths below 30" the actual radial stresses are very small. Due to this, a comparison of the percentage errors (which were very large) are somewhat meaningless and have been omitted from Table 3-4. For the same reason, the percentage errors are very large for the tranditional approach at large depths. The percentage errors should be compared with the value of the actual stresses in mind. One point worth noting is the lack of accuracy of the finite element method (both with and without the flexible boundary) when stresses are evaluated near the corners of elements (i.e., near nodes). Stresses are most accurate at the middle of elements and are reasonable at the middle of element edges, but not accurate near element cornors. For homogeneous material the stresses from finite element solutions are not continuous across element boundaries (as they should be). This is the reason for the large errors in the radial stresses at depths of 10" and 30" (see Table 3-4). These depths represent the interfaces between layers for the multilayered case. Also, when elements with very large aspect ratios (such as the deeper element in the traditional mesh (Figure 3-12) are used, the results tend to be poor. A better mesh than the one in Figure 3-12 would require many more elements and hence would result in a much greater computational effort. 61 A number of case studies were performed using the flexible boundary approach, varying the moduli and thicknesses of the base and subbase layers. In all cases the results compared favorably with the exact solutions. Table 3-1 : Errors in surface deflections Homogeneous Multilayer Radial Dist. (inches) Exact Percentage Error Exact Percentage Error displ. displ. (inch) FE + FB FE only (inch) FE + FB FE only 0 .3190 -1.9 -9.3 .0807 -6.6 -l7.4 2.5 .3140 -2.6 -10.1 .0802 -6.8 -l7.6 5 .2980 -3.3 -ll.l .0788 -6.9 -17.9 7.5 .2677 -4.6 -l3.1 .0763 -7.1 -18.3 10 .2031 -6.8 -l7.7 .0725 -7.5 -l9.0 15 .1136 -2.5 -19.8 .0653 -8.0 -20.2 20 .0825 -2.8 -23.6 .0592 -8.4 ~20.9 25 .0652 -2.5 -25.4 .0537 -8.7 -21.4 30 .0539 -3.0 -25.1 .0488 -8.9 -2l.5 40 .0401 -4.6 -24.4 .0406 -9.1 -21.0 50 .0321 -5.6 -21.9 .0342 -8.9 -l9.4 60 .0266 -4.7 -l7.8 .0292 -7.3 -16.0 80 .0201 -4.8 -12.0 .0220 0.9 -4.6 100 .0160 6.4 3.1 .0173 19.3 15.2 62 Table 3-2 : Errors in vertical displacements beneath center of load Homogeneous Multilayer Depth (inches) Exact Percentage Error Exact Percentage Error displ. displ. (inch) FE + FB FE only (inch) FE + FB FE only 0 .3190 -l.9 -9.3 .0807 -6.6 -l7.4 1.25 .3134 -2.5 -10.3 .0810 -6.7 -l7.5 3.75 .2916 -2.4 -11.1 .0809 -6.6 -l7.5 6.25 .2620 -2.3 -12.3 .0803 -6.7 -17.7 8.75 .2314 -l.9 -13.6 .0792 -6.8 -18.0 10 .2171 -2.0 -14.5 .0784 -6.8 -17.2 12.5 .1913 -0.6 -15.2 .0749 -6.7 -18.5 17.5 .1516 -0.4 -l9.4 .0691 -7.0 -19.9 22.5 .1239 -0.6 -23.9 .0646 -7.1 -20.9 27.5 .1042 -0.8 -26.1 .0608 -7.2 -20.9 30 .0964 -l.4 -26.4 .0588 -7.1 -20.5 32.5 .0897 -0.9 -22.2 .0563 -6.8 -l8.0 37.5 .0785 -0.6 -14.l .0518 -6.3 -l3.3 42.5 .0698 -0.4 -6.7 .0480 -5.8 -9.1 47.5 .0628 0.0 0.0 .0447 -4.9 -5.2 50. .0598 0.0 +2.7 .0433 —4.5 -3.4 Table 3-3 : Errors in vertical stresses beneath center of load Homogeneous Multilayer Depth (inches) Exact Percentage Error Exact Percentage Error stress stress (psi) FE + FB FE only (psi) FE + FB FE only 1.25 99.81 -3.2 -3.0 97.62 -4.5 -4.4 3.75 95.67 -2.0 -l.8 80.43 -1.3 -0.7 6.25 85.11 0.0 0.6 53.54 1.1 2.7 8.75 71.45 1.8 2.9 30.11 15.3 19.7 10 64.64 ~2.3 -O.5 24.62 10.7 16.9 12.5 52.39 2.3 5.2 20.26 ~2.3 6.3 17.5 34.55 1.7 8.6 13.68 -4.6 10.7 22.5 23.69 -0.6 11.8 9.25 -6.2 16.8 27.5 17.00 -2.7 11.7 6.44 -6.8 17.9 30 14.62 -l.6 -25.5 5.65 -6.1 -18.2 32.5 12.69 -3.7 -110. 5.13 -10.3 ~112. 37.5 9.79 -3.9 -109. 4.30 -ll.3 -109. 42.5 7.76 -4.6 -107. 3.66 -l3.0 -106. 47.5 6.30 -14.9 ~103. 3.15 ~23.6 -100. 50. 5.71 22.8 ~101. 2.94 1.1 -97. 63 Table 3-4 : Errors in radial stresses beneath center of load Homogeneous Multilayer Depth (inches) Exact Percentage Error Exact Percentage Error stress stress (psi) FE + F8 FE only (psi) FE + F8 FE only 1.25 77.11 ~4.5 -7.6 169.9 -2.2 -7.8 3.75 46.25 -3.3 -7.0 69.39 -3.7 -9.7 6.25 25.59 7.9 3.6 -19.36 3.0 -1.7 8.75 13.79 31.1 27.5 -111. -1.8 -7.3 10 (+) 10.15 159.0 158.5 -163.5 9.9 3.7 10 (-) 10.15 176.0 181. 2.05 -217. -128. 12.5 5.58 62.2 68.6 .54 25.7 352. 17.5 1.83 46.4 145. -1.63 -9.3 150. 22.5 .65 —9.7 602. -3.66 -8.9 103. 27.5 .23 -95.3 3197. -6.27 ~9.0 -97.2 30 (+) .13 -8.04 -8.6 121. 30 (-) .13 3.28 -131. —165. 32.5 .07 .28 6.4 -821. 37.5 .00 .22 25.1 -949. 42.5 -.03 .17 44.6 47.5 -.04 .14 -106. 50. -.04 .12 403. Vertical Displ. ( inches ) 0————OEMKH a——AFE+ FLEXIBLE BOUNDARY D-- -n FE ONLY l ' l ' I . r , . 0 10 20 30 40 50 Radial Distance ( inches ) Figure 3-13a : Comparison of Surface Deflections with and without Flexible Boundary for the Homogeneous Material .00 I ' I l I I I I 1 r 1 I I I I l I I I A .01- 3 .02- 5 a .5 .03: V .04- ."5. 05- .fi - Q .06- 3 1r. £3 .O7i 1:; .08-1 > ‘ h—omcw ‘ .09- h—A 93+ mam BOUNDARY . o-«oFBONLY .10 .....T.I.,.,.,.,. o 10 20 so 40 50 so 70 so 9'0'100 Radial Distance ( inches ) Figure 3-13b : Comparison of Surface Deflections with and without Flexible Boundary for the Multilayer Pavement Depth ( inches ) 50 . IITfIIIIIfiYI IIIITIIIrfIIIwIXer / 1 l_1 , a-n & 1 U 1 d l 9 q I —1 l r o———0EUCT b——~AF8+ mm BOUNDARY ‘ 0""05'3 ONLY .. 0.00 Figure 3-14a : 1' I Y T V I 'l 0.10 fil’Tj'ljUI' 0.85 6.151 '656'325 0.39 0.35 Vertical Displacement ( inches ) Comparison of Vertical Displacements beneath Center of Load with and without Flexible Boundary for the Homogeneous Material 0 1 I I I I I I I r I I I I 1:! I I Ix I [A I I I . Asphalt ‘ ,‘ /1 j j I 10 3' /- A d I .4 (I) ‘ 4 J: 1 Subbase 4 0 20" "‘ G 'l . .C: 30 ..J 4 0. £3 : Subgrade ‘ 40- ~ ‘ Humor ‘ ‘ I A-"HAFE+IIEHBUBBOUNDARY‘ . f/ UF-4353(nflx ‘ so 44.,L..r,..a,..e,...1... 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Figure 3-14b : Vertical Displacement ( inches ) Comparison of Vertical Displacements beneath Center of Load with and without Flexible Boundary for the Multilayer Pavement 66 o ' 1 r, 10- s (I) . '2 . U 20" —‘ .5 * - I: 30- _ A ’I O. ‘ 7 OJ " I Q . a: . .404 . _ I ‘ T o—QEXACT ‘ " ‘ A—-—AFE+ FLEXIBLE BOUNDARY ‘ 9 I3""‘43FEONLY ‘ so..Ie..,...r...,...1...],. -20 0 20 40 60 80 100 120 Vertical Stress ( psi ) Figure 3-15a : Comparison of Vertical Stresses beneath Center of Load with and without Flexible Boundary for the Homogeneous Material o I U T I I I 1 r I f I l' I U I I I I I I I 1' . J ‘ Asphalt « 10 A " a U) 1 if: « J U 20" Subbase “ .9. * . V l ‘ J: 30 i 44 ‘ .’U D: I Q) " I i ‘ Q " l: 1 Subgrade 40- : - ' I 1’ .__. EXACT ‘ . l .— —al-‘E+ macaw BOUNDARY ‘ - I ll 0- - -0 FE ONLY « 50 .7. 44% . .<., . . . I . .. I . . 1 1.»- . I r . -20 O 20 40 60 80 100 120 Vertical Stress ( psi ) Figure 3-15b : Comparison of Vertical Stresses beneath Center of Load with and without Flexible Boundary for the Multilayer Pavement Depth ( inches ) 67 o . I . 10- q i 4 20- _ . ., ," 1 30‘ 3’ _ . 7’“ . . l I 1 90 . 401 i _ ‘ 7" o—omc'r ‘ ‘ '1 ...—firm P'lEXIBLE BOUNDARY ‘ J at u--op3UIQ I T I F I a I r- r I l. [mFII I IOPI H. .....- . 4 IF.P._. .L+_b. . p-.. . . _ . ...—LL.p_ .. .._... ”I O 73 Table 3-7 : Sensitivity of vertical stresses to the depth of the flexible boundary (at a radial distance of .67" from the center of the loaded area) Depth Vertical Stresses (psi) (inches) Case 1 Case 2 Case 3 CHEVSL 0 100. 100. 100. 99.7 0.5 98.72 98.72 98.71 98.9 1.5 90.43 90.44 90.44 90.14 2.5 73.96 73.97 74.00 73.9 3.5 53.34 53.37 53.42 53.67 4.5 33.53 33.56 33.65 34.10 5.5 20.22 20.26 20.37 20.26 6. 17.49 17.54 17.66 17.3 7. 14.03 14.08 14.21 14.82 9. 10.19 10.25 10.42 10.9 11. 7.33 7.41 7.62 7.94 13. 5.21 5.29 5.53 5.73 15. 3.67 3.76 4.02 4.10 17. 2.71 2.80 3.09 3.07 18. 2.49 2.58 2.88 2.8 19. 2.26 2.34 2.65 2.64 21 2.01 2.09 2.41 2.37 23. 1.80 1.88 2.21 2.14 25. 1.62 1.71 2.04 1.95 27. 1.47 1.56 1.89 1.79 29. 1.29 1.43 1.75 1.64 30. 1.20 1.35 1.60 1.58 Table 3-5 and Figure 3—17, show that the flexible boundary placed at about 3 feet beneath the surface of the roabed soil yields the most accurate surface deflections. However, Tables 3-6 to 3-7 and Figures 3- 18 to 3-19, indicate that the radial and vertical stresses are relatively insensitive to the location of the flexible boundary. For linear analysis, it is recommended that the flexible boundary be placed at about 3 feet beneath of the surface of the roadbed soil, unless stresses and displacements are required beneath this depth. 74 9%.an @338 226: ms. .5 5&8 9: 2 88% _8_to> to $2.123 ”m7... 83: 9va 38% _oo_to> oS om on on 8 on 9. on cm 2 o . F $00 0.6 wonl H m 080 min ... . n 88 4 a . mm I 1 I . .5510 I . f . r v a! TONI.- . T O . . o n h m 1 r9... u. . H n7. 1 . u .- .. ( .1 \ me... n fl .r m? T .I I O 75 3.5.2 ; Effect of the Tire Pressure on the Location of the Flexible fioundary Here all input parameters are kept the same except the tire pressure, which is increased from 100 psi to 120 psi. Comparisons of the surface deflections, three cases and the CHEVSL program are presented in Tables 3-8 and Figures 3-20 to 3-22, respectively. and the radial and vertical stresses between the to 3-10, tire Table 3-8 : Sensitivity of surface deflections to increase in the pressure from 100 to 120 psi. radial dist. Surface Deflection (inch) (inches) Case 1 Case 2 Case 3 CHEVSL 0 .02598 .02664 .03016 .02736 0.6 .02593 .02658 .03010 .02734 1.8 .02574 .02639 .02991 .02717 3.1 .02538 .02603 .02955 .02679 4.3 .02478 .02544 .02896 .02622 6.1 .02374 .02440 .02793 .02476 8.6 .02252 .02320 .02674 .02355 11 .02145 .02214 .02569 .02246 13.4 .02045 .02115 .02471 .02135 17.1 .01913 .01985 .02344 .01972 22. .01760 .01836 .02199 .01775 26.9 .01639 .01719 .02086 .01602 34.2 .01522 .01607 .01978 .01380 44. .01436 .01526 .01901 .01141 76 fig GE 3 00? E0: 8385 8:. 2: c. omootog 3 cozoozoo mootam .6 3323mm Homlm 8:9... Ace 83 B 28.50 9: E0: mocoyma 660$ 00 o¢ on ON 0 P o erer-n—-nP-lP-bbP—hl-bbelbnn—prLyb-b-n—pnub-runny mm .. .5)on ... n 300 r N 300 . _. omoo bbbfbpbb-L-plbn-nlan—pbthLp-Dth-b-nn-p—Pb-Bbbbbr 004* I (U! ooow) uonoeueo Gowns 77 Table 3-9 : Sensitivity of the radial stresses to increase in the tire pressure from 100 to 120 psi (at a radial distance of .61" from the center of the loaded area) Depth Radial Stresses (psi) (inches) Case 1 Case 2 Case 3 CHEVSL 0 -251.90 -250.35 -248.57 -265.20 0.5 -209.30 -207.96 -206.41 -217.60 1.5 -124.10 -123.17 -122.07 -131.60 2.5 -52.41 -51.88 -51.22 -57.63 3.5 11.06 11.19 11.42 8.58 4.5 73.35 73.08 72.87 73.45 5.5 142.92 142.24 141.60 146.40 6. (+) 177.70 176.82 175.96 188.10 6. (-) .68 .60 .48 .48 7. 1.00 .91 .79 1.09 9. 1.64 1.52 1.42 2.17 11. 2.32 2.19 2.10 3.25 13 3.19 3.04 2.94 4.50 15. 4.38 4.20 4.10 6.05 17 6.06 5.85 5.74 8.09 18. (+) 6.90 6.68 6.55 9.37 18. (-) -.38 -.51 -.77 -.22 19. -.40 -.54 -.78 -.21 21. -.43 -.59 -.81 -.20 23. -.46 -.64 —.85 -.18 25. -.48 -.68 -.88 -.17 27. -.50 -.71 -.92 -.16 29. -.46 -.74 -.97 -.16 30. -.45 -.75 -1.00 -.15 78 com Seam E as com 2 02 :8: 838cm 2:. of c. 0.0.8.65 3 38% Baum 9: to 92:28 owe 88a 663.. ”Elm 239... oo. o 00 P I com! com... b hp P. .L'p — p p n p uh hp - bbbb Pb . P. — n - pip - u p p - —ka - p n u p n on- r 4 romeo olo . n w 800 one u .I n 300 4 .4 . .v. :5an I IWN w Howl m .... .. mm? H h H. wot. m N n. A... . n r.- p-bl-bb nbb p . - Pbb P-L P?pl- - n n p p —#b .LL - p P. HI 79 Table 3-10 : Sensitivity of vertical stresses to increase in the tire pressure from 100 to 120 psi (at a radial distance of .61" from the center of the loaded area ) Depth Vertical Stresses (psi) (inches) Case 1 Case 2 Case 3 CHEVSL 0 120. 120. 120. 119.6 0.5 118.39 118.40 118.39 118.7 1.5 108.11 108.12 108.13 108.1 2.5 87.62 87.64 87.67 87.97 3.5 62.18 62.22 62.27 62.81 4.5 38.06 38.10 38.19 38.67 5.5 21.94 21.99 22.10 21.77 6. 18.73 18.79 18.92 18.19 7. 14.65 14.72 14.86 15.48 9. 10.51 10.61 10.79 11.23 11. 7.51 7.63 7.85 8.13 13. 5.31 5.44 5.69 5.83 15. 3.72 3.86 4.15 4.15 17. 2.74 2.87 3.19 3.08 18. 2.52 2.65 2.98 2.81 19. 2.28 2.41 2.75 2.65 21 2.03 2.15 2.51 2.38 23. 1.82 1.94 2.32 2.15 25. 1.65 1.77 2.15 1.96 27. 1.50 1.63 2.01 1.79 29. 1.32 1.50 1.89 1.65 30. 1.24 1.42 1.74 1.58 When the tire pressure is increased from 100 to 120 psi, the flexible boundary placed at about 3 feet from the surface of roadbed soil still has the most accurate surface deflections. As before, the radial and vertical stresses are quite close for these three cases and the CHEVSL method. Thus the optimal location of the flexible boundary is not sensitive to the tire pressure, and placement at a depth of about 3 feet still yields the best results. 80 .Eo p.511 3 as owe 2 OS : 838.5 9:. 9: c. @8802 3 32% _8_t..> 9: to €588 "Sun 8:3 ON TjTVIITVFITII'IUVUUIUYTI[IIUI you away mmobm _oo_to> Om 0m 0% ON Dr F 300 olo _ on... N 8.00 nlu n n 800 a a n. I 403.10 I wow wow: mm... m9... (U!) Lndecn 3.5.3 ' Effect of the Wheel Load on the Location 0 Boundary Here all input parameters are kept the same except the wheel load which is increased from 9,000 surface deflections, three cases and the CHEVSL program are presented in Tables 3-11 to 3-13, 81 to 12,000 pounds. and Figures 3-23 to 3-25, respectively. Comparisons the Flexible and the radial and vertical stresses between the Table 3-11 : Sensitivity of surface deflections to increase in the wheel load from 9 to 12 kips. radial dist. Surface Deflection (inch) (inches) Case 1 Case 2 Case 3 CHEVSL 0 .03246 .03278 .03532 .03523 0.8 .03237 .03269 .03523 .03519 2.3 .03206 .03239 .03493 .03491 3.9 .03151 .03183 .03438 .03434 5.4 .03064 .03096 .03352 .03351 7.7 .02910 .02943 .03200 .03181 10.8 .02719 .02754 .03013 .02999 13.9 .02544 .02581 .02843 .02811 17.0 .02383 .02422 .02687 .02631 21.6 .02172 .02215 .02486 .02386 27.8 .01932 .01981 .02260 .02096 34.0 .01745 .01800 .02089 .01848 43.3 .01565 .01629 .01929 .01541 55.6 .01434 .01506 .01814 .01231 of the 82 8: S 3 m 88.. n03 60;; 9: E 38.65 2 cozoozmo mootzm co bSEmcmm ”main 959.1 Ac; boom to ..cho 9: So: 8565 660m 00 00 0... on ON or O uh-PkbPC—nthP-DKPPP-bnbh-b-DPS-P.Pun—b—P-xPPFanb_P-bb--hhr— .8 r 0% m mm... W. \q... on: ... . \\ n. \ .\ le m - .. \\ ..l .. ... 1 \\\. ON... n q i I ... 1 ..\\\\ \1 n ...! ..Q \\. .... 0 Fl. u .5)on ! .... n 800 4 . .... a $00 0 ml m F 030 o Dub-DbPPb—Ppnbb-hbh—b-n-PBL-n-b-thn-Ib—bD-bhppb—nnpbb-P#! O (U! ooou) uonoeueo Gowns 83 Table 3-12 : Sensitivity of radial stresses to increase in the wheel load from 9 to 12 kips (at a radial distance of .77" from the center of the loaded area) Depth Radial Stresses (psi) (inches) Case 1 Case 2 Case 3 CHEVSL 0 -273.71 -272.18 -269.48 -282.30 0.5 -229.73 -228.40 -226.04 -234.90 1.5 -141.76 -l40.84 -139.16 ~147.10 2.5 -63.13 -62.62 -61.59 -67.34 3.5 10.37 10.49 10.86 7.87 4.5 83.85 83.57 83.29 83.35 5.5 163.92 163.25 162.30 164.70 6. (+) 203.96 203.08 201.81 209.50 6. (-) .59 .53 .39 .10 7. 1.21 1.13 .98 1.06 9. 2.44 2.32 2.17 2.70 11. 3.67 3.52 3.35 4.28 13. 5.09 4.91 4.70 6.01 15. 6.88 6.66 6.43 8.10 17 9.27 9.02 8.74 10.80 18. (+) 10.47 10.20 9.90 12.40 18. (-) -.24 -.33 -.59 -.27 19. -.25 -.34 -.61 -.25 21. -.26 -.38 -.63 -.23 23. -.28 -.41 -.67 -.21 25. -.29 -.44 -.71 —.19 27. -.29 -.47 -.75 -.17 29. -.24 -.49 -.81 -.16 30. -.22 -.51 -.84 -.15 84 swung 6v 8: S 2 m 80.: boos Box; or: E 0896:. ' 03V mmobm 650m cow 2: o . 8 T com! 3.326 6631 .6 3388 $1.. 8:9... ,——-—-——————..-.—-..rb-IP-hbFur—berrbbetbr-pbnDb _ IIlerjT'rrUIt'ItlrIU‘VlI —P-b-bb.-—-phrP-P-b—bPPLnb|-.Ppb1bbh...-n! oonl — 030 I .on... N 080 one n n 800 c a . .635 I nmm Woml WT. .....oT n (U!) HldGG 85 Table 3-13 : Sensitivity of vertical stresses to increase in the wheel loads from 9 to 12 kips (at a radial distance of .77" from the center of the loaded area ) Depth Vertical Stresses (psi) (inches) Case 1 Case 2 Case 3 CHEVSL 0 100. 100. 100. 99.70 0.5 98.75 98.75 98.75 98.83 1.5 90.78 90.79 90.79 90.12 2.5 75.09 75.10 75.13 74.66 3.5 55.46 55.48 55.53 55.58 4.5 36.42 36.43 36.51 37.08 5.5 23.66 23.68 23.79 23.96 6. 20.88 20.90 21.01 21.00 7. 17.33 17.35 17.49 18.27 9. 12.82 12.86 13.04 13.70 11. 9.37 9.42 9.62 10.14 13. 6.73 6.78 7.02 7.40 15. 4.79 4.85 5.12 5.36 17. 3.58 3.64 3.93 4.05 18. 3.31 3.36 3.66 3.71 19. 3.00 3.05 3.36 3.51 21. 2.67 2.72 3.04 3.16 23. 2.39 2.45 2.78 2.86 25. 2.16 2.22 2.55 2.60 27. 1.95 2.02 2.34 2.39 29. 1.70 1.85 2.16 2.20 30. 1.58 1.73 1.95 2.11 When the pavement structure is subjected to high wheel loads (such as 12 kips), Table 3-11 and Figure 3-23 shows that the deeper flexible boundary results in the most accurate surface deflections. The above tables and figures, show that when flexible boundary is placed deeper, the radial stresses decrease slightly, and the vertical stresses increase slightly. The radial and vertical stresses from the CHEVSL program are always slightly higher than the results of the finite element plus flexible boundary solution. 86 imam é a: S 8 m Eo: boos 692, m5 E 08805 2 32% _8_tm> to 332mm ”31.. 8:9... cmav mmobm _oo._.tm> 00 F on On 0* ON TVTU'IIVU'IIVUlr'flrrUITIIITT b-prr-b-Pb-PP#-bhn—L-nvapbbb—h-thb-ppF-L-LLPPP — omoo ole u 080 one n 300 4 h .v 300 I lenbbbb-bPLbIP-nbbLrr-DDPLLbn—prb-nubI-DPPbLb-npnn 87 3.6 DISCUSSION AND SUMMARY Four FE methods are reviewed to ascertain which ones are capable of simulating the behavior of asphalt pavements observed in the field and the laboratory. While a 3-D FEM is the most suitable technique, present computational capabilities limit its use in practice. Table 3-14 summarizes the effects that can and cannot be accounted for by each of the four models. Table 3-14 : Capabilities of the Finite Element Models Sandwich Plane strain 3-D FE Axisymme- plate theory model model tric model model Soil overburden Yes Yes Yes Yes Inelastic beha- vior of granu- Yes Yes Yes Yes lar and cohe- sive material Edge effect Yes Yes Yes No Unbound behavi- or between asp- No No No No halt and granu- lar layer Economical CPU No Yes No Yes time Accuracies of computational Not good Not good Good Good results From the above table, the major disadvantages of the sandwich plate theory which is sometimes used for the analysis of rigid pavements, is that it cannot accurately calculatethe horizontal displacements of pavement. Also it is not well suited for the analysis of flexible 88 pavements. The major disadvantage of the two dimensional plane strain model is that it assumes an infinitely long loading which is unrealistic. The 3-D FE model is the most versatile but needs large storage and a large amount of computation time. At present, due to their limitations, it is practically impossible to implement this model on personal computers. The obvious disadvantages of the axisymmetric model are that it cannot consider the edge effect of pavements and multiple- wheel loadings in nonlinear problems. Nevertheless, the axisymmetric model appears to be the most suitable at the present time. If flexible pavements can be assumed to be composed of linear elastic material, the Chevron series programs are perhaps the best. These can quickly and accurately compute the required displacements and stresses. However, the Chevron series programs cannot be used to model the inelastic behavior of granular or cohesive material. The axisymmetric finite element model is selected as the basic foundation for the development of a nonlinear finite element program to be implemented on personal computers. The validation of this with the general purpose SAP IV program, will be discussed in Chapter 5. A flexible boundary concept is used in the static finite element analysis of pavements. Such modeling enables the bottom boundary to be placed at any depth below which displacements and stresses are not of interest, while accurately representing the displacements occuring in the material below the boundary. The principal advantage of this new technique is its computational efficiency, especially when used with nonlinear finite element approaches requiring iterative solutions. CHAPTER 4 NONLINEAR FINITE ELEMENT ANALYSIS 4.1 : INTRODUCTION In a pavement, repeated vehicle loads cause permanent deformations as well as resilient (recoverable) deformations. At the present time, the computational capacity to follow the stress-strain curve through millions of load repetitions is not readily available. In practice, mechanistic models are used only to compute stresses and resilient strains, while permanent (plastic) deformation is empirically related to the resilient strains, magnitude of load, number of load applications, material properties, etc. (see chapter 6). The resilient response is characterized by the resilient modulus of the material (see Figure 4-1), which is defined as the ratio of the repeated axial deviator stress to the recoverable axial strain as in Eq.2-10. 4.2 : MATERIAL NONLINEARITY There are many nonlinear material models that may be used in the finite element analysis. Four suitable models, the hyperbolic, the resilient modulus, the shear and volumetric stress-strain (also called the contour model), and the third hyperelastic models are reviewed here. The advantages and limitations of each model is briefly discussed, and the reasons for choosing the resilient modulus model with the Mohr- Coulomb failure criterion are discussed. The advantage of the hyperbolic model is that it is simple and easy to use (Duncan, et al., 1970). However, it is primarily suited for 89 O. Deviator Stress f 90 / /II I [I I, I / l 1 ReSion of Interest for // ‘ztl”’/,.___ Resilient Deformations / I I ' / l I ’ / I I, ’ I ’ I / / L _> | 6P I 67—.1 e - Total Strain adf - Deviator stress at failure "7 e - Permanent (plastic) strain a: - Recoverable (resilient) strain Figure 4-1: Complete Pavement Response 91 monotomic loading problems than for repeated loading problems. It is therefore unsuitable in estimating pavement responses under repeated loads. The contour model is an advanced and sophisticated model which can successfully predict the nonlinear responses of granular materials, and is especially suited to three dimensional problem (Brown, et al., 1981). However, due to the limitation of laboratory equipment in most state highway agencies, it is difficult to estimate the necessary material constants. Therefore, the model is unsuitable for the development of a daily design program for the state highway departments. The advantages of the hyperelastic model are that it can simulate the nonlinearity, shear dilatancy, stress-induced anisotropy, effect of the confining stress, and the effect of the third stress invariant of soils. However, the limitations of the model include (Chen, et a1. 1982): 1. Best results from the model are generally expressed at low stress levels below failure. 2. Although the nine material constants required for the model can be estimated, it is still not easy to apply. In addition, the nine material constants have no direct physical interpretation. 3. If monotonic loading without unloading or reloading is applied, then the model satisfies all the requirements such as uniqueness, stability, and continuity. However, when the unloading or reloading occurs, it fails to satisfy continuity at or near material loadings. Actually, pavement structures are subjected to many loadings and reloadings. Therefore, the model is not very suitable to simulate the granular materials and roadbed soil in 92 the pavement structure. The advantages of the resilient modulus model include: 1. The model reduces the complicated nonlinear response to a simple form and is easy to use. 2. The resilient moduli of granular materials and roadbed soil can be determined by most state highway agencies. 3. The granular materials and roadbed soil still maintain their resilient behavior under repeated loads even after the occurrence of large permanent deformations (Haynes, et al., 1963; Seed, et al., 1962). However, there are some limitations of the resilient modulus model: 1. The model does not consider the loading and unloading stress paths and the shear dilatancy of granular materials and roadbed soil. 2. The model is accurate only in the range of relatively low stresses. Due to practical constraints, the resilient modulus model is selected in this study. If only the FEM with the resilient modulus model is used, it will converge extremely slowly. Therefore, Raad and Figueroa (1980) applied the resilient modulus model with the Mohr-Coulomb failure criterion. The Mohr-Coulomb failure criterion was used to modify the principal stresses of each element in the granular layers and roadbed soil after each iteration such that the Mohr-Coulomb envelope was not exceeded. Before introducing the Mohr-Coulomb criterion, the resililent moduli of granular materials and roadbed soil are briefly reviewed. For granular materials, the resilient modulus can be expressed as 93 Mr - K1(9)K2 (Eq.2-ll) or M - K '(0 )K2' (Eq.2-12) r 1 3 Where Mr - resilient modulus (psi); 6 .= bulk stress (- 01+02+a3) (p51); 03 - confining pressure (psi); 0 v _ ' K1,K2,K1 ,K2 material constants. The relationship between the bulk stress and the resilient modulus is shown in Figure 4-2. Since the regression equation expressed in equation 2-11 is found to be more accurate than that expressed in equation 2-12. The former is chosen for this analysis. For fine-grained soils subjected to repeated deviatoric stresses at low values of confining pressure, the resilient modulus can be expressed as Mr = K2 + K3[Kl - (al - 03)]; for K1 > (01 - a3) (Eq.2-13a) and Mr = K2 + K4[(01 - 03) - K1]; for K1 < (01 - a3) (Eq.2-l3b) where (01 - a3) - deviator stress (psi); K1, K2, K3, K4 - material constants determined by linear regression. The relationship between the resilient modulus and the deviator stress is shown in Figure 4-3. The nonlinear FE analysis of the flexible pavement structure can be divided into the following steps: 94 The Resilient Modulus (Log Mr, psi) l,‘_ Log(K1) __’ Bulk Stress (a1 + 02 + 03) Figure 4-2 : Typical Variation of Resilient Modulus with Repeated Stress for Granular Material C: In 3‘ 3 '3 K3 '0 O I: u I C Q, ”:4 H .3 M a? T I " 0 .C H N 3 .#__.K1 .l ’ ‘\ The Deviator stress (01 - 03, psi) Figure 4-3 : Typical Variation of Resilient Modulus with Repeated Stress for Roadbed Soil (cohesive soil) stress 95 . The pavement is discretized into a set of elements connected at nodal points. . Nonlinear properties of the granular materials and roadbed soil are included by means of successive iteration. . The principal stresses in the granular layers and roadbed soil are modified at the end of each iteration so that they do not exceed the strength of the material as defined by the Mohr- Coulomb envelope. At the end of each iteration, the maximun and minimum allowable principal stresses in each element are calculated as US 45 2 (01)max - avtan [45 + - J + 2c tan[45 + —] (Eq.4-1) 2 2 2 ¢ ¢ (03)min - avtan [45 - - J + 2c tan[45 - —] (Eq.4-2) 2 2 where (01)max - max1mum allowable pr1nc1pal stress (p31); (a3)min - minimum allowable principal stress (psi); 01, a3 - major and minor principal stress, respectively (psi); av - vertical stress which includes gravity (psi); c, o - cohesion and angle of friction (degrees) of the soil, respectively. However, 01 should not be greater than 01', the major principal associated with 03 at failure (see Fig 4-4). where 2 ¢ ¢ 0' = a tan 45 + — + 2c tan 45 + — (Eq.4-3) 1 3 2 2 Shear Stress (r) Shear Stress (r) 96 Shear Stress (r) Unmodified Stress State (01, a Shear Stress (7) Example 1 A (a) f C " B a a a vb 3 v 1' (al)max Normal Stress (a) Unmodified Stress State (01, a3) 0 j (b) B A“: a (a3)m1n v (al)max Normal Stress (a) Modified Stress State [(03)min’ av] Example 2 L (a) a (a l 1)max Normal Stress (a) 3) (b) ~\Qr(3 B. A 1’ \l <03)min 03 0V 01' (a1)max Normal Stress (a) Modified Stress State (03, 01') Figure 4-4 : Examples for Stress Modification at End of Iteration -. - \I 97 The tensile stress that can be resisted by cohesive soils is 2 ¢ 0' - -2c tan [4S - -] (Eq,4-4) T 2 If the maximum and/or minimum principal stresses exceed these limits then they are set to the corresponding extreme value. Figure 4-4a illustrates typical principal stresses before modification, and Figure 4-4b illustrates the modified principal stresses. 4. For the next iteration, the stresses determined in the preceeding iteration are used to calculate the resilient moduli (using Eq. 2-11 and Eqs. 2-l3a, 2-13b) of elements in the granular layers and the roadbed soil. Furthermore, the strains, and resilient deformation of the pavement structure are determined. 5. The convergence error in each iteration can be expressed as (Brown, et al., 1981): a - 2 (an - E0)/ 2 En2 (Eq.4-5) Where EO - the current value of resilient modulus (psi); En - the resilient modulus from the previous iteration (psi); 2 - summation over all nonlinear elements. If the convergence error between the two successive iterations is smaller than .001, then the iterations are terminated. A detailed flow chart of how the principal stresses, 0 and a are 1 3’ modified is illustrated in Figure 4-5. 98 Input of a , a , a 1 3' c' [2’ and d for Each Elem ent // (01) max - avtan2(45+¢/2) + 2c tan(45+¢/2) (”3)m1n - avtan2(4S-¢/2) - 2c tan(4S—¢/2) No Yes 3)min - T a3)min>aT Yes (a3)min $0350 No a - a 7‘7 a: - aTtan2(45+¢/2) 01 - a tan2(45+¢/2) a _ (a ) + 2c tan(45+¢/2) + 2c tan(45+¢/2) a: _ a 3 min Yes 01201 0 01-01 I No \ IL I a I \ Yes No gUCPut of Stresses, Z) \ onvergence trains & DiSplacemenj7/ “‘K\\££:::;ff;29L/zzr (::Stop ) Figure 1, - 5‘ : Stress Modification Procedures for Given Iteration 99 4.3 : SENSITIVITY OF FLEXIBLE BOUNDARY IN NONLINEAR ANALYSIS When the flexible boundary is used in nonlinear analysis, the nonlinear properties of the material above the boundary may be accounted for through the use of finite elements, but the halfspace below the boundary is assumed to be linear elastic. The boundary should therefore be placed at a sufficient depth below which material nonlinearities can be neglected due to low stresses. The depth at which the boundary is placed should depend on the strength of the upper layers of the pavement section, as well as on the magnitude of the wheel load, since these factors affect the stresses in the roadbed soil. Some reasonable method of obtainng an equivalent modulus for the halfspace beneath the boundary must also be developed. In this section, the sensitivity of the displacements and stresses with respect to the location of the flexible boundary is studied for pavements with nonlinear materials. The estimation of the equivalent modulus of the halfspace is also discussed. 4.3.1 : Equivalent Modulus for Halfspace below Flexible Boundary It is necessary to find an equivalent modulus for the halfspace below theflexible boundary in order to approximately account for the displacements below it. A typical section is given in Table 4-2, and two approaches of estimating an equivalent modulus are discussed. In the first, the equivalent modulus for the halfspace is calculated by averaging the resilient moduli of all elements immediately above the boundary; in the second, the equivalent modulus is calculated by averaging the resilient moduli of all bottom elements except for the three elements which are closest to the right vertical boundary. 100 The resilient moduli of elements just above the flexible boundary are tabulated in Table A-1. becomes The change in the modulus sharp close to the right vertical boundary, indicating that the boundary has an undersirable effect on the estimated moduli. In order to minimize the influence of the boundary, it is suggested that the last three elements be excluded when estimating an equivalent modulus of the halfspace as the average modulus of the elements immediately above the halfspace. Table 4-1 : Resilient moduli of elements just above flexible boundary Element* 1 2 3 4 5 6 7 8 Modulus 8300 8300 8300 8310 8320 8310 8320 8360 (psi) Distance+ 0.5 1.5 2.5 3.5 4.5 5.5 7.5 10.5 (inch) Element* 9 10 11 12 13 14 15 Modulus 8410 8440 8520 8660 8780 8950 8920 (psi) Distance+ 13.5 16.5 21 27 33 42 54 (inch) * Elements are numbered sequentially in the outward radial direction starting at r - 0; + Distance is to the center of element. 4.3,2 ; Effect of the Depth of the Flexible Boundary The sensitivity of the depth of the flexible boundary with respect to surface deflections and stresses is discussed below. Results obtained with the flexible boundary placed at depths of 12, 35 and 85 inches beneath the surface of roadbed soil are compared with the results obtained using the ILLI-PAVE program (see Tables 4-3, 4-4 and Figures 4- 101 6 to 4-8). The ILLI-PAVE program uses finite elements extending to large depths and has a fixed boundary at the bottom. The properties of the pavement section are given in Table 4-2 Table 4-2 : The properties of the pavement section Layer Thick. E v Den. K K K K K 0 l 2 3 4 (in) (P81) (pcf) AC 3 500,000 .4 150 .67 Base 12 .38 140 .60 5000 .5 R.S.* 285 .45 115 .82 6.2 3021 1110 -l78 * R. S. a Roadbed soil The cases with the flexible boundary at depth of 12, 35 and 85 inches are henceforth denoted as cases 1, 2 and 3, respectively. Based on the comparisons for linear material, it would be expected that a deep fixed boundary would still give rise to smaller surface deflections than an exact solution, with the flexible boundary solution being closer to the exact solution (see Figures 3-13 and Table 3-l in section 3.4). Thus, the surface deflection computed with ILLI-PAVE being smaller than those estimated with the flexible boundary in Table 4-3 and Figure 4-6 is not surprising. 102 Table 4-3 : Sensitivity of the surface deflections to the depth of the flexible boundary Rad. Surface Deflection (in) Rad. Surf. Def. % of Diff. Dist. Dist. (Case 2 - ILLI)1 (in) Case 1 Case 2 Case 3 (in) ILLI-PAVE / ILLI-PAVE 0. .03852 .03515 .03515 0 .03184 10.4 .5 .03843 .03506 .03506 1 .03175 1.5 .03813 .03476 .03477 2 .03137 2.5 .03758 .03422 .03423 3 .03074 3.5 .03678 .03344 .03346 4 .02986 4.5 .03575 .03242 .03245 5 .02876 5.5 .03449 .03117 .03122 6 .02742 7.5 .03172 .02845 .02853 7.5 .02533 12.3 10.5 .02771 .02454 .02468 9 .02335 13.5 .02407 .02102 .02124 12 .01966 16.5 .02092 .01801 .01830 15.8 .01583 21. .01722 .01456 .01495 19.5 .01296 27. .01339 .01106 .01156 27 .00936 18.2 33. .01080 .00878 .00938 42 .00549 28.4 42. .00871 .00705 .00776 57 .00360 54. .00729 .00592 .00671 72 .00306 103 boucaom 03:8: 85 to 59.5 .2: 3 5:02.60 mootam 9.390 xtztmcow ”ole. 8:9... c5 noon .8 .350 or: 80...”. 8:035 663. on on 8 on 04 on cm 2 o P-nbn-bnn—D-Pbb-ppp—pPFbeb—ph-p-n.bb—IPbP-nbbPL—ttbb:phpbbbLthn—P-an-P— o¢| m .I on... r a .w x on: m \\ w \x mm: ... x .1 \ own. I a n \\ n. \\ m7. n \\ m .1. .. . .. ..... m\ “>515: x . um...“ 1...... n 800 a ... N 800 n n... T P 88 o TbePP-nhL—P-CPbbbn—LPD-EnblbeEp-pbP- .bpnbpbhb-pr-phbbbpbbtbthl—hbbein-b. o (U! 0001*) 11011391190 aooyns 104 Table 4-4 : Sensitivity of the radial and vertical stresses to the depth of the flexible boundary (at a redial distance of .5" from the center of loaded area) Depth Case 1 Case 2 Case 3 LLI-PAVE (in) 0r 02 Or 02 Or Oz 0r Oz 0. -360.78 79.6 -354.95 79.6 -350.47 79.6 .75 -200.68 72.6 -197.24 72.6 -l94.59 72.8 -185. 70.9 2.25 119.5 53.6 118.18 53.7 117.18 54.1 108. 51.7 3(+) 279.59 48.1 275.89 48.2 273.07 48.7 3(-) -7.57 48.1 -7.59 48.2 -7.7 48.7 4 -6.67 38.2 -6.7 38.4 -6.81 39. -6.11 35.6 6 -4.85 27.8 -4.9 28.1 -5.03 28.9 -4.58 26.7 8 -3.47 19.9 -3.53 20.3 -3.65 21. -3.36 19.6 10 -2.50 14.4 -2.56 14.7 -2.68 15.5 -2.5 14.6 12 -1.84 10.7 -1.9 11. -2.01 11.6 -1.91 11.1 14 -1.45 8.4 -1.5 8.7 -1.57 9.1 -1.54 8.95 15(+) -1.25 8.1 -1.3 8.1 -1.36 8.1 15(-) -2.94 8.1 -2.92 8.1 -2.57 8.1 16.5 -2.7 7.6 -2.5 7.69 19.5 -2.22 6.6 -2.46 6.84 22.5 -2.02 6.0 -2 32 6 2 25.5 -l.78 5.3 27 -1.66 4.9 From Table 4-3 and Figure 4-6, the increase the depth of the flexible boundary from 35 to 85 inches does not change the surface deflections significantly. Therefore, from the surface deflection point of view, the flexible boundary located about 3 feet beneath the surface of roadbed soil is recommended. Furthermore, from Table 4-4, and Figures 4-7 and 4-8, the radial and vertical stresses are similar for all three cases. Therefore, from the stress point of view, the flexible boundary located deeper than 1 foot beneath the surface of the roadbed soil is adequate. 105 q 3...”. .3 . boncsom 85.8.... 2.. .0 580 ms. 3 0.85m .28”. e... .0 3.3.2.8 ” 08V 80ch .263. a... was... com com 8. o o2... 08.. 8?. 8.1. r....p.p.L..:...P._:.:p:._:.:8:.....pCCLFCth.“www.c0nb:Ion! ... a u 080 one ... . . n 800 4 a .. m fl . 951...... ... 1mm: .. . ...- H _ H m. H mm... H . H . . H H .. . ...- HolflHIIIL ... .- I'lllllllll‘ .- EW- 0 (U!) undeo 106 Azm."m “UV @858 83.8.... 2... ..o 5.80 2.. o. 88% .82.; 9... .0 3.)...mcmm,nmns $8.... a . emu. «mobm _oo_to> on on .8 on 8 on on o. o Drip-PPEPhhbbhbb_bPanDin-thbPPPbbh—Bbhrhanbe#b-b-h-PPE-ant-b:P-nhbb ' . F 030 o 0 .on n N 800 one . .. . n 300 4 .4 . . .. .mw . m> .0 335.28 H:1. ea... 28. mmobm .oo_to> panb-1P-—pbn-h--b—B-n-P-P—pnnnbpb-n—p-an-phD-Dpptnh- PrmbbP- nn—nbtbbb-b—ub-bhb-Pb—Ptbbbb: 2: 8 on on 8 on -9. on .8 9 o P 300 ole N 300 aim n 800 a a m .. . .. a TIT'IIII'ITIIITITI'IITI'IIII O '1’ In ‘1‘ , IO ... l IIII‘IIII'IIII'TIII O ‘7‘ TI' 0 F I I I In I (U!) mdeo 112 It should be noted that when the wheel load was increased up to 12,000 pounds, the program did not converge even after 25 iterations. This indicates that this pavement section becomes significantly nonlinear at this very high load, and is too weak for practical purposes. It also indicates that the algorithm being used converges well only for moderate levels of nonlinearity and may not converge for strongly nonlinear problems. This, however, should not be a cause for concern in normal practice. 4.4 1 THE MICH-PAVE PROGRAM The nonlinear FE program with the flexible boundary developed in this work is named MICE-PAVE. Details such as the choice of the FE mesh, effects of gravity and compaction ("locked-in") stresses of pavement materials, the modification of stresses, and the interpolation and extrapolation of stresses at layer boundaries, that are employed in this program are discussed below. 4.4-1 ; Mesh Generation The FEM needs to satisfy some basic requirements as far as the mesh is concerned, such as how far the vertical and bottom boundaries should be located, the size and shape of elements, and the distribution of elements in the various regions. Duncan and Monismith (1968) showed comparisons between displacements and stresses computed using the FEM and those computed using elastic half-space and layered system analysis to establish criteria for locating the boundaries in the FEM. For an elastic half-space subjected to a uniform circular load, displacements and stresses computed by the FEM compare well with those 113 determined from the Boussinesq solution when the boundary nodal points in the FEM are fixed at a depth of about 18 radii of the loaded area for the bottom boundary, and constrained from moving radially on the vertical side boundary at a distance of about 12 radii from the center. However, to obtain a reasonable comparison between the two procedures for a three-layered system, it was necessary to move the bottom boundary in the FEM to a depth of about 50 radii while maintaining the same radial constraint as for the half-space analysis for the side boundary. From previous experience, stresses based on quadrilateral elements will be accurate provided that the length-to-width ratio for the elements do not exceed five to one. Furthermore, smaller elements may be used close to the loaded area, and progressively larger elements may be used in the regions away from the loaded region. Based on the above considerations and experience, the following mesh generation rules were used. In the radial direction, a mesh with a total width of 10 radii was divided into four zones. The first zone, which is between 0 and l radius, is equally divided into four elements; the second zone which is between 1 radius and 3 radii, is equally divided into four elements; the third zone which is between 3 radii and 6 radii, is equally divided into three elements; the fourth zone that is between 6 radii and 10 radii, is equally divided into 2 elements. The MICH-PAVE program will automatically generate the default values of the finite element mesh along the radial and vertical direction (see Figure 4-11, 4-12). In the MICH-PAVE program, a flexible boundary is used instead of a fixed bottom boundary. Therefore, it does not need a mesh that is 50 radii deep. Normally when a mesh depth of about 10 radii is used, it 114 m =8 . mm =a.~d :8. sense ea H cowuoom you 5mm: ucoaoam ouucum u NH-< musmfim Asa w I a .m0u< oovooa mo mauoomv «Nana ca oocmumwa Magnum .... a a d _ . 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I. sesame: s~_==~a .o .. a 116 will achieve reasonably accurate results. In general, if the flexible boundary is located too close to the top of roadbed soil, then the displacements of elements may still be accurate, but the stresses may not be. If the flexible boundary is placed too deep, the primary advantages of using the flexible boundary is lost. The depth at which the boundary is placed is a function of tire pressure, wheel load, and material properties of each layer. Also, the requirement of how many elements should be used in each layer along the depth is a function of the thickness and material properties of each layer. In general, the thicker the layer, the more elements must be used. 4.4.2 : Gravity and Lateral Stresses of Pavement Materials The MICH-PAVE program includes the gravity stress arising from the weight of the materials and considers the lateral stress between elements. At a point at depth 2 located within the ith layer, the vertical gravity stress is computed as i—l i-l 0g - Z yjtj + (z - Z t.) 71 (Eq.4-6) j-l j-l 3 where 11 and t are the unit weight and thickness of the jth layer. .1 The lateral stress, ah, is calculated from the coefficient of earth pressure at rest, K0, and vertical gravity stress as (see Figure 4-14) ah - KO 08 (Eq.4'7) The final radial, tangential and vertical stresses are expressed as or - or + ah (Eq.4-8) 117 a. I r , a ' and av' are the radial, tangential, and vertical stresses due t to the wheel load, respectively. In order to account for "locked-in" stresses due to compaction, a value for K0 higher than the coefficient of earth pressure at rest may be used. 4,4,3 ; Recovering Global Stresses from Modified Principal Stresses As mentioned in Section 4.2, in MICH-PAVE the principal stresses are modified after each iteration to avoid Mohr-Coulomb failure within any element. It is necessary to obtain the stresses in radial, tangential and vertical coordiate directions (global stresses) from the modified principal stresses. The technique used for this for each element is outlined below. 1. Find the angle of rotation from the calculated global stresses to the principal stresses. Since fro is zero, we only need to consider the r-z plane since the tangential stress is always a principal stress. (see Figure 4-15). The angle of rotation is a - 0.5 can'larrz/(ar - 02)) (Eq.4-9) 2. Generate the rotational transformation matrix [R] - 0 1 0 -sin a 0 cos a cos a 0 sin a (Eq.4-10) which relates the global and principal stresses. 3. Modify the principal stresses as outlined in Section 4.2 such that the Mohr-Coulomb failure criterion is not violated. The modified principal stress matrix may be written as 01 0 0 [a ] - 0 02 0 (Eq.4-ll) p O O 03 118 ; ——i: r—— Lave: l Density(l) !1\ . - ‘ Laver 2 Density(Z) N t ' 2 1!! 4 ___L____ 08 Lave: 3 Density(3) 1 t—Oh ' s Figure 4-14: Calculation of GraVity and Lateral Stresse a z ‘1 r rz -—————-» rz a or r -—-’- <(——— r rz «r—————- Figure 4-15 : Stresses and Principal Stresses in r-z Plane 119 4. Find the global stress matrix corresponding to [Up] by using the rotational transformation matrix: T [as] - [R] [UPJIR] (Eq.4-12> where or 0 r6 [as] - 0 at 0 - modified global stress matrix (Eq.4-l3) r0 0 oz and or, at, oz, and Trfi are the radial, tangential, vertical, and shear stresses, respectively. Note that since [R] is an orthogonal matrix, i.e., [R].1 - [R]T, the inverse relation of that in Eq. 4-12 is [op] - 1811a31181T (Eq.4-141 4.4.4 ;.Interpolation and Extrapolation of Stresses and Strains at Layer Boundaries For a pavement section in which different layers are fully bonded (i.e., no slip at layer boundaries), quantities such as the vertical and shear stresses, and the radial and tangential strains should be continuous across layer interfaces. However, due to the low order interpolation functions chosen in the FE method, these quantities are not continous across element boundaries. Thus if these quantities are estimated by FE approach at two adjacent points across an interface, the results will show an apparent discontinuity that is an artifact arising from the error in the FE method. This is undesirable and can be overcome by using interpolation to estimate these quantities at an interface from those at the middle of the adjacent elements. This is illustrated in Figure 4-16 (a). For example, if a is the stress at the center of an 1 element immediately above the interface and 02 is the stress at the 120 center of an element immediately below the interface, then the stress at the interface obtained by linear interpolation is 024-02 00 - 1 2 2 1 (8.1.4-15) 21 + 22 where 21 and 22 are the depths of the points at which 01 and 02 are evaluated. Since the FE approach gives accurate estimates of stresses and strains at the center of elements, but can yield significant error at element edges, even those stresses and strains at the interfaces that are discontinuous across the interface can deviate significantly. For quantities that are discontinuous across an interface, it is possible to estimate their values at one side of the interface by linear extrapolation of the values from the center of two elements on that side of the interface. For example, if 01 and 02 are stresses at the center of two adjacent elements below (or above) an interface, then the stress at the lower (upper) side of the interface can be obtained through linear extrapolation as 02 - 01 00 =' T;— (20 - 21) + 0‘1 (Eq.4-l6) 2 l where 21 and 22 are the depths of the points at which 01 and 02 are estimated (see Figure 4-16 (b) and (c)). Finally, based on prior knowledge of the solution, the surface stresses are arbitrarily set to their proper values in HIGH-PAVE. Vertical and shear stresses must be zero at the surface, except for the vertical stress below the wheel load, which must be identical to the tire pressure. 121 (a): Interpolation for Vertical and Shear Stresses Between Two Layers 8—01—8_ Z1 Interface I‘ ”2.51 2 (b): Extrapolation for Radial and Tangential Stresses, and Vertical Strain Below the Interface Interface I 00 ' 2 ”o ' X (20 - 21) + a1 I“°1-‘I (c): Extrapolation for Radial and Tangential Stresses, and Vertical Strain Above the Interface ' 1901—4 Interface P—ao —’1 0 Figure 4-16: Interpolation and Extrapolation of Stresses and Strains at Layer Boundary The improvement in accuracy through interpolation and extrapolation is illustrated by analysis using the MICH-PAVE properties of the Layer 1: AC Layer 2: Base considering a typical pavement section for linear - E - 300,000 psi; -E- Layer 3: Roadbed soil - E - Comparison after of 122 20,000 psi; 8,000 psi; stresses and strains and CHEVSL section are given below. 1". programs. The thickness - 8" v - .38; thickness - 12" u - .45; semi-infinite depth between MICH-PAVE interpolation/extrapolation and adjustment of and CHEVSL are given in Tables 4-7 to 4-10. surface (before Table 4-7 : Comparison of radial and tangential stresses at 67" from the center of the loaded area Depth Radial stress (psi) Tangential stress (psi) (in) Before After CHEVSL Before After CHEVSL 0 -199.02 -l67.91 -172.8 -199.02 -l67.9l -l73.0 8(+) 124.65 106.31 111.6 124.65 106.31 111.9 8(-) 1.84 .35 .22 1.84 .35 .24 20(+) 2.53 2.49 3.52 2.53 2.49 3.53 20(-) -.66 -.47 -.23 -.66 -.47 -.23 Table 4-8 : Comparison of vertical and shear stresses at 67" from the center of the loaded area Depth Vertical stress (psi) Shear stress (psi) (in) Before After CHEVSL Before After CHEVSL 0 143.57 100. 99.7 2.62 0. 0. 8(+) 0. 12.25 12.25 3.29 2.31 .55 8(-) 9.15 12.25 12.25 .65 2.31 .55 20(+) 3.03 2.99 3.36 .12 .10 .07 20(-) 1.61 2.99 3.36 .07 .10 .07 material stresses) 123 Table 4-9 : Comparison of radial and tengential strains at 67" from the center of the loaded area Depth Radial strain (micro) Tangential strain (micro) (in) Before After CHEVSL Before After CHEVSL 0 207 207 212 207 207 213 8(+) 231 203 239 231 203 241 8(-) 231 203 239 231 203 241 20(+) 136 124 173 136 124 173 20(-) 136 124 173 136 124 173 Table 4-10 : Comparison of vertical and shear strains at 67" from the center of the loaded area Depth Vertical strain (micro) Shear strain (micro) (in) Before After CHEVSL Before After CHEVSL 0 52 108 129 24 164 8(+) 287 316 339 31 24 8(-) 528 582 621 90 90 20(+) 248 242 302 16 14 20(-) 275 315 394 26 26 It can therefore be seen that the interpolation and extrapolation used in MICH-PAVE to obtain stresses and strains at layer boundaries results in significant improvement. CHAPTER 5 COMPARISONS WITH OTHER PROGRAMS 5,1 : COMPARISONS WITH SAP-IV RESULTS In order to validate the linear elastic part of MICH-PAVE its results were compared with those obtained from the commercial FE program SAP-IV (Bathe, et al., 1973). The mesh used was 100 inches wide in the radial direction and 160 inches deep (see Figure 5-1). The same mesh was used with both programs and the displacements and stresses in every element due to a 31415.9 pound wheel load and 100 psi of tire pressure were computed. The results were essentially identical and the displacements under the load are shown in Table 5-1. Therefore, the FE axsymmetrical model is verified to be correct. Table 5-1 : Comparisons of the vertical displacements at the center of the loaded area 124 Depth Vertical Displacement (in.) (in.) SAP-IV MICH-PAVE without F.B.* 0. .043477 .043476 2.5 .043626 .043625 5.0 .043149 .043147 7.5 .042293 .042292 10. .040974 .040973 17.5 .031967 .031967 25. .026018 .026018 32.5 .021862 .021863 40. .018656 .018657 52.5 .013914 .013915 : F.B. denotes flexible boundary. 125 100 psi 0" R F . E - 200,000 si 10" ' 1 p v - .35 m I 2 15,000 psi U - .4 40" [-11 I 3 10,000 psi v - .4 90” E4 - 5,000 psi v - .45 160" On 10" 30" 45" 60" 80" 100" Figure 5-1: Finite Element Mesh and Material Properties for a Typical Section 126 5.2 ' COMPARISONS WITH CHEVSL RESULTS Since the CHEVSL program is a linear elastic layer analysis program, it cannot account for the nonlinear behaviour of granular material and roadbed soils. However, the MICH-PAVE program can account for both linear and nonlinear behavior of granular material and roadbed soils. The linear elastic part of the MICH-PAVE program utilizing the flexible boundary is compared here to the CHEVSL program which gives essentially exact solutions for elastic materials. Also the use of equivalent resilient moduli in the CHEVSL program, which are estimated from the results of the MICH-PAVE program, are investigated. 5.2,1 ; Linear Elasgic Analysis using the CHEVSL and MICH-PAVE Programs Two pavement sections, a full-depth asphalt concrete on roadbed soil, and a section with 3 inches of AC and 12 inches of granular material, overlying the roadbed soil, are analyzed. Since the CHEVSL program assumes weightless material, zero material density is also used in the MICH-PAVE program. Tables 5-2 and 5-3 show the basic design data. A wheel load of 9002.55 pound and a tire pressure of 79.6 psi is used. Table 5-2 : Design data for a 12" full-depth AC on roadbed soil Layer Thickness Elastic Modulus Poisson’s Type (in.) (psi.) Ratio AC 12 500,000 0.40 Roadbed Semi-infin 8,753 0.45 Soil ite (52)* * : In the MICH-PAVE program, the flexible boundary was placed at a depth of 52 inches. 127 Table 5-3 : Design data for a 3" AC and a 12" granular material on the roadbed soil Layer Thickness Elastic Modulus Poisson’s Type (in.) (psi.) Ratio AC 3 500,000 0.40 Granular 12 21,696 0.38 Roadbed Semi-infin 7,387 0.45 Soil ite (60)* Figures 4-12 and 4-13 show the finite element mesh corresponding to Tables 5-2 and 5-3, respectively. show comparisons of surface deflections between the two programs. Table 5-4 : Comparisons of surface deflections between CHEVSL and MICH- Tables 5-4 and 5-5 and Figure 5-2 PAVE programs for a 12" full-depth AC section Radial Distance from Surface Deflections (in.) * the Center of the Error Loaded Area (in.) CHEVSL MICH-PAVE (%) 0. .01322 .01250 -5.5 0.75 .01321 .01247 -5.6 2.25 .01313 .01238 -5.7 3.75 .01295 .01220 -5.8 5.25 .01266 .01188 -6.2 7.25 .01203 .01134 -5.7 10.5 .01158 .01079 -6.8 13.5 .01110 .01036 -6.7 16.5 .01061 .00996 -6.1 21. .00989 .00940 -5.0 27. .00896 .00872 -2.7 33. .00809 .00816 .9 42. .00692 .00758 9.5 54. .00562 .00715 27.2 * : The percentage errors are calculated assuming that the CHEVSL results are exact. 128 Table 5-5 : Comparisons of surface deflections between CHEVSL and MICH —PAVE programs for the three layer section Radial Distance from Surface Deflections (in.) * the Center of Loaded Error Area (in.) CHEVSL MICH-PAVE (%) 0. .03496 .03259 -6.8 0.75 .03487 .03237 -7.2 2.25 .03420 .03167 -7.4 3.75 .03291 .03043 -7.5 5.25 .03109 .02870 -7.7 7.25 .02812 .02577 -8.4 10.5 .02384 .02216 -7.0 13.5 .02060 .01916 -7.0 16.5 .01801 .01675 -7.0 21. .01503 .01413 -6.0 27. .01219 .01149 -5.7 33. .01014 .00972 -4.1 42. .00798 .00825 3.4 54. .00611 .00724 18.5 Tables 5-4 and 5-5 show that the surface deflections calculated by the MICH-PAVE program are less than those calculated by the CHEVSL program by about 5 to 7 percent. However, at the right vertical boundary region, the error is much higher due to the interaction of the boundary itself. Tables 5-6 and 5-7, and Figure 5-3 and 5-4 show comparisons of vertical and radial stresses at .75 inch from the center of the loaded area between CHEVSL and MICH-PAVE for the two pavement sections. These tables show that the stresses predicted by both programs are very close. om m>mzo Coosuon GONuooHuoo oomuusm mo Comwumaeoo “N-m muzwwm AGNV mouc popooA oSu mo noucoo on» aoum monoumwa Hmwpmm Fa om ow on em OF 0 kph—P—LhPerPbpbbbp—b_prPF-__—P-__PP—pb-Frppr—b—p_-__F~._ om! 39918 .o.< 58.013. ..N_ 010 . 35815.2. .o.< 5881.3. ..N. ... ... 3.1.318 88m :N. a. .o.< ..m 018 - $23.15.... 8%. ..N. a .o.< ..n ...- 10.7 ........\ on .. x m 1 xix T ... .... .81 (xxx... . 1-3.8.1T5m 1 1.... 111 ...... 1 1 1 IO?! .r\...............!:.. .-.... H ...mph~—b.~hb-L~—»__t-~_b—bpb_bppr____.~b____.p_h__hp_I ('U! 0001*) uonoeuea Booms 130 Table 5-6 : Comparisons of vertical and radial stresses at .75 inch from the center of the loaded area for a 12" full-depth AC section Depth Vertical Stress(psi.) Error Radial Stress(psi.) Error (in.) (’3) H) CHEVSL HIGH-PAVE CHEVSL MICH-PAVE 0 79.36 79.60 .3 124.7 111.95 -10.7 1 78.73 77.51 - 1.5 95.43 89.25 - 6.5 3 68.49 66.75 - 2.5 47.61 43.96 - 7.7 5 49.59 48.27 - 2.6 15.19 13.54 -10.9 7 30.12 29.22 - 3. - 9.35 - 8.85 - 5.3 9 14.34 14.01 - 2.3 -32.8 -29.85 - 9. 11 4.06 4.96 11.2 -60.8 -54.97 - 9.6 l2(+) 2.82 3.76 64.9 -78.5 -67.53 -14. 12(-) 2.82 3.76 64.9 .77 .65 ~15.6 14.9 2.40 2.14 -10.8 .62 .62 0 20.6 1.85 1.64 -ll.4 .42 .57 35.7 26.3 1.5 1.31 -12.7 .30 .56 87. 32 1.25 1.11 -11.2 .23 .55 139. 37.7 1.06 .96 - 9.4 .18 .53 200. 43.4 .91 .86 - 5.5 .14 .51 271. 49.1 .79 .76 - 3.8 .12 .47 267. 52 .74 .71 - 4 .11 .44 264. (+) and (-) indicate locations just above and just below the interface. 131 Table 5-7 : Comparisons of vertical and radial stresses at .75 inch from the center of the loaded area for the three layer section Depth Vertical Stress(psi.) Error Radial Stress(psi.) Error (in.) (is) (‘3) CHEVSL MICH-PAVE CHEVSL MICH-PAVE 0. 79.4 79.6 .2 395.5 380.6 - 3.8 .5 76.76 75.52 - 1.6 274.1 267.3 - 2.5 1.5 59.92 58.7 - 2. 44.4 40.7 - 8.4 2.5 41.68 40.64 - 2.5 -183. -183.7 .4 3.(+) 37.11 36.84 - .7 -30l.8 -295.9 - 2. 3.(-) 37.11 36.84 - .7 9.03 7.29 -19.3 4. 32.95 30.77 - 6.6 5.80 5.36 - 7.6 6. 25.3 23.53 - 7. 1.63 1.51 - 7.4 8. 19. 17.67 - 7. - 1.02 - .88 -l3.7 10. 14.2 13.12 - 7. - 3.16 - 2.74 ~13.4 12. 10.5 9.7 - 7.6 - 5.38 - 4.65 -13.6 14. 7.95 7.33 - 7.8 - 8.18 - 7.04 -13.9 15.(+) 7.18 6.37 -11.3 - 9.97 - 8.24 -17.4 15.(-) 7.18 6.37 —11.3 .36 .32 -11.1 18.2 5.63 5.12 - 9. .22 .27 24.6 3.72 3.31 -11. .09 .18 31.1 2.62 2.32 -ll.4 .05 .27 37.5 1.96 1.73 -ll.7 .03 .36 43.9 1.51 1.36 - 5.6 .03 .40 50.4 1.20 1.12 - 8.3 .03 .42 56.8 .98 .93 - 8.2 .03 .39 60. .89 .83 -10.1 .03 .38 It is suggested that stresses in the FEM be computed at the center of each element. This yields the most accurate results. 5.2,2 ; Equivalent Resilient Moduli for Linear Analysis Sometimes, it is desirable to compare the results of the nonlinear FEM with that of elastic layer programs such as CHEVSL. However, the input material constants are different between these two methods. It is therefore necessary to obtain an equivalent resilient modulus which can be used in elastic layer programs from the different resilient moduli in each element of the nonlinear FEM. 132 Amman .NH new o< .mV m>mzo Sassoon mmouum HmoNuuo> mo comaumdeoo ”m-m ouzwflm cm on em p—phb-thnbphhppp—n Agony mmouum Hmofiuu0> om on on ON 0— o -—-——p——-hnpn—-I-—-b._—-n———_-h-b——~bp-P—-p—.pb—b—-—L TTTjTIITIfrIIleITIIrllIT 'TrTI f. r 1.0510 «:0 m>mzo cmmzuon mmmuum assume we comauaaeoo He-m muswaa Aumav mmmuum amazes 00¢ 00m 00m 00_ 0 00—l 00ml 000.1 00¢! p—prp-PP—1bpbhthPP—FPh-bbbel——1ntt1PFL-hFLP~hlnb—Pb-uhb-P-hhpbP-bpIPhL-prbbe . .6510 4:8 - T m m>wavm acouommaa 0:» cu one mcoNuoonon communm on» no mcomwuumaoo " m-m «woman Acfiv mou< commoq mo Houcoo on» Eoum mocmumHn Hmavom CC. .-.,1.I.1 1-_1 . _ ., m>anvm ucoummmqn mzu ou man mooauooamon oomwudw on» no acomwuaaaoo . m-m unawfim Acfiv mmu< commoq mo Hoodoo on» Eoum oocsumfin Howvum m. 0 .4 00 ON 0.. 0 1111...-.1:1_.1--._1- 117111—1117 u p F _ c r _ F _ _ _ b _ _1 _ p p L Omlo . l1 . m>§io§ ... + - - - N .0238: 01¢ . .- H conga: 0:0 .811... . m 89.08: , \ o 1... c 1 . Pd .. 4 . . «1:8 1 (seqour) uotnoatgea eosgzns deflections are quite similar for However, recommended for linear analysis. The analysis use for the 140 all three sets of equivalent moduli. reason mentioned earlier, the calculation of equivalent utilizing the MICH-PAVE program, in the CHEVSL program is further investigated through typical examples. Tables properties of a 12 with 3 inches of AC and 12 The earlier in section 4-2. A cohesion of 6.45 psi and an angle of friction of 0 were used for 5-9 and 5-10 three layer section, c show the the roadbed linear and nonlinear material - 0 psi and ¢ - 45° the first approach is material constants and nonlinear stress-strain model were resilient moduli for of equivalent resilient moduli obtained through nonlinear inch full-depth AC section and a three layer section inches of granular material on roadbed soil. outlined soil in the full-depth section. For the was used for the granular material and c = 6.45 psi and d - 0° was used for the roadbed soil Table 5-9 : Linear and nonlinear material properties of the 12 inches full-depth AC section Layer Thick Modulus v K K K K K densi . 0 l. 2. 3 4 Type ness (p51) (p51) (p31) ty (in.) (Pcf) AC 12 500,000 .4 .67 150 R.S. 40 8,753 .45 .82 6.2 3021 1110 -178 115 141 Table 5-10 : Linear and nonlinear material properties for three layer section Layer Thick Modulus u K0 K1 K2 K3 K4 densi Type ness (psi) (psi) (psi) ty (in.) (pcf) AC 3 500,000 .4 .67 150 Gran. 12 21,696 .38 .6 5000 .5 140 R.S 45 7,387 .45 .82 6.2 3021 1110 -178 115 Figures 5-10 and 5-11 show comparisons of surface deflections for nonlinear analysis using MICH-PAVE and the linear with equivalent moduli. vertical area. It the and radial stresses at nonlinear Figures 5-12 .5 inch from the center of the and 5-13 Some of this is clear that there is a greater difference in stresses difference analysis using CHEVSL show comparisons of the loaded between arises and linear analyses. because CHEVSL neglects the weight of the material. In particular, CHEVSL can give tensile stresses in the granular material. 5.3 ° COMPARISONS WITH ILLI-PAVE RESULTS Since the finite element meshes between ILLI-PAVE and MICH-PAVE are different, the surface deflection, and vertical and radial stresses between the two programs cannot be computed at every point. The output of the ILLI-PAVE and percentage differences between the results from the two programs are computed here at some identical points. The percentage difference is calculated taking ILLI-PAVE results as the reference ‘Value. When the percentage difference is a positive value, then the result calculated by MICH-PAVE is higher than that calculated by ILLI- 142 PAVE. Comparisons of surface deflections between the two programs are shown in Figures 5-10 and 5-11, and comparsions of the vertical and radial stresses at .5 inch from the center of the loaded area are shown in Figures 5-12 and 5-13. Figures 5-12 and 5-13 show that the vertical and radial stresses between the two programs are very close. However, Figures 5-10 and 5-11 show the surface deflections calculated by the MICH-PAVE are about 12 percent higher than those computed by ILLI-PAVE. However, as discussed in Section 4.3.2, the surface displacements in linear analysis are better when a flexible boundary is used, as in MICH-PAVE, than a deep fixed boundary is used, as in ILLI-PAVE (see Figures 3-13 and Table 3—1 in Section 3.4). Extending this observation to the nonlinear case, it is to be expected that MICH-PAVE would yield slightly larger surface displacements than ILLI-PAVE. Further, based on the results for linear analysis, one would be inclined to assume that the surface deflections obtained through MICH-PAVE are more accurate than those obtained by ILLI-PAVE. 5,4 : SUMMARY Comparison of results obtained from linear elastic analysis using MICH-PAVE with those of SAP-IV, a general purpose finite element program, vertified that the linear elastic part of MICH-PAVE is performing without error. The errors obtained from FE analysis are estimated based on the essentially exact solutions obtained through the CHEVSL linear layer analysis program. 143 A0< £ud0p--3m mo =NHV :oHuoonoo ooomuSm no somuumdeoo ” 0H-m ouswwm AGNV mou< mommoq 0:» mo umucoo onu scum oucmumwa Hmwpmm 00 on 00 00 0v on 0N OF 0 .u....u_.:..C_P__~...up:_.h..u...._u...P-p:—.p~.p._.._...u.:.u_.:.....p DWI H .m>m:o 018 H H mimic... Um - ... mail: 414 wom1 w me1 w womI .1. .1 .4. - F T m Hrrrhrrhnr... ('U! 0001*) uonaeueo 800mg 144 Amman mo .Na can o< mo .mV cofiuooamma oommusm mo conflummaoo ” HH-m muswwu AGHV mou< topmoA 0;» mo moucoo as» aoum mocmumwo Hmwnmm 00 0x. 00 0m 0w 0m 0m 0— 0 pp-bpp-pLL-LbbprbL—pk»pp—pp-—--LCbP—pp_-F1~—-—-PhL..nLF—hhpptnnr—n-ppbbpp H .6510 018 w €514.92 aim. m m> mo comaumdeoo Aamav mmouum Hmowuum> ”NH-m mcsmua 0m 00 00 0¢ on 0m 0— 0 hr—p-thn—bPPbppbhpbpnhn-1-b1—~_Cphhpn—P-bb-nbbb—-nbblbpLr—hb-hnbnt—FbePh-hrrOhl IIIIIIIIII r w: I _..1 a r1 a . T- I I r I I I I'IIIQLailI... r ‘11!-$¥IIIIIIIII r - o I._.—; ........ ..I. ............. ...hh::.:_u ........ ..C..:Lt::p: .5an ole m>mzo o1¢ . ... “$51.12: elm ... - mini: .--. . w M mom- I. 4 fl % H m 4 m w won! r. T f - 1 10m! i - ................. m H. .13.. r 1 m WIFLl‘Il—lh» VFW—LL _ F. _ .Lbh .FL _ b _L _ _ _ hbr— PP. O (M!) mdea 147 A technique of estimating equivalent elastic moduli of layers from MICE-PAVE for use with linear analysis programs such as CHEVSL is presented. Use of the equivalent moduli in CHEVSL indicate that fairly accurate stresses can be obtained from a linear analysis once the equivalent moduli are known. However, the surface displacements from the linear analysis tend to be about 5 to 7 higher than those obtained through nonlinear analysis. Note, however, that a nonlinear program such as MICH-PAVE is required in order to estimate equivalent moduli for the layers. Comparison of results from MICH-PAVE and ILLI-PAVE indicate that the stresses obtained from both programs are very close. However, the displacements from MICH-PAVE are about 12 % larger than those from ILLI- PAVE. Based on exact Solutions from linear analysis, the flexible boundary used in MICE-PAVE is expected to give better deflection estimates than the deep fixed boundary used in ILLI-PAVE. CHAPTER 6 FATIGUE LIFE AND RUT DEPTH MODELS 6,; GENERAL Laboratory fatigue life data for asphalt mixes has been accumulated in large quantities since the early 1950's. Traditionally, the data are plotted as stress or strain amplitude versus the resulting life, commonly known as S-N curves. For asphalt mixes, as for most other materials, fatigue life steadily increases with decreasing stress or strain amplitude until the stress or strain level of the fatigue limit is reached, below which the life apparently becomes infinitely long. In general, stresses at or below the fatigue limit cause only elastic strains. It should be emphasized that cyclic and/or cumulative plastic strains are ultimately responsible for fatigue damage and the consequent fatigue failure. In addition, since asphalt mixes are weaker in tension than in compression, the magnitude of the induced tensile strain due to the applied load is typically used to estimate the fatigue life of the mixes. Two tests are generally used in the laboratory to assess the fatigue life of compacted asphalt mixes: the beam test and the indirect tensile test. For the beam test, the fatigue life of a beam specimen is typically defined by the number of load application at which the flexural modulus of the beam decreases to 50 percent of its original value. For the indirect tensile test, the fatigue life is defined by the number of load applications at which a crack is observed along the 'vertical diameter of the test specimen (Baladi,et al.,l989). 148 149 Using stress-controlled cyclic load indirect tensile tests and Marshall size specimens, Baladi (1989) developed empirical models using statistical methods, relating the fatigue life and the cumulative plastic compressive deformations along the vertical diameter of the test specimens to the various asphalt mix and test variables. These models are presented in equations 6-1 and 6-2. ln(N ) - 36.631 - 0.1402 x TT - 2.300 x ln(CL) - FL 0.5095 x AV - 0.001306 x xv + 0.06403 x ANG (Eq.6-l) ln(CDl) = -ll.615 + 0.07028 x TT + 0.5000 x ln(N) + 1.148 x ln(CL) + 0.3326 x AV - 0.001007 x KV (Eq.6-2) where: ln - natural log; NFL - fatigue life - number of load applications to fatigue; CD1 - cumulative plastic deformation along the vertical diameter (inch x 10 a); TT - test temperature (OF); CL - the magnitude of the cyclic load (pounds); AV - the percent air voids in the asphalt mix; (AV = 3 to 7); KV - the kinematic viscosity of the asphalt binder (centistokes); and ANG - the angularity of the aggregate in the mix, (ANG = a for 100% crushed aggregate, 2 for rounded river deposited natural aggregate); Based on limited field observations of in-service flexible pavements in the States of Michigan and Indiana, Baladi noted that the fatigue life of asphalt pavements is about 20 times larger than that estimated using equation 6-1. He also noted that equations 6-1 and 6-2 can be used to predict the fatigue life and rut depth of in-service pavements 150 provided that (for each pavement in question) the values of the coefficients in the equations are adjusted to reflect the properties of the different pavement layers and their thicknesses. Based upon these observations, equations 6-1 and 6-2 were empirically calibrated using the modulus of the various pavement layers of several in-service pavements and the outputs (stresses, strains, and surface deflection) obtained from the MICH-PAVE computer program. The calibration process and the resulting fatigue and rut depth models are presented in the next section. 6.2 FATIGUE LIFE MODEL As noted in the previous section, equation 6-1 was empirically calibrated using the actual modulus and thicknesses of the different pavement layers and the output of the HIGH-PAVE computer program. The calibration process was accomplished in four steps as outlined below: Step 1 - The actual fatigue lives of 10 pavement sections (see table 6- 1) with known layer thicknesses and properties were tabulated. The fatigue life of the pavement sections is defined by the estimated number of 18 kips equivalent single axle load (ESAL) that traveled the pavement sections prior to the initiation of low severity alligator cracking (one or two disconnected longitudinal hair cracks in the wheel path). It should be noted that, for most cases, accurate traffic data (ESAL) was not available. Hence, visual trafficcount and several assumptions were made to convert the average daily traffic (ADT) data to ESAL. Step 2 - In this step, the actual pavement cross section (layer thicknesses and moduli) were used as input to the MICH-PAVE program 151 and the compressive and tensile strains and stresses in the asphalt concrete were calculated. These values along with the properties of the pavement layers were then correlated to the fatigue life of the 10 pavement sections using the same equation form as that of equation 6-1. This step yielded a fatigue life equation which accurately predicted the actual fatigue life of only four pavement sections. The predicted fatigue life of the remaining six sections, however were not as accurate. Step 3 - Thirty arbitrary selected pavement sections (see table 6-2) were analyzed using the 1986 AASHTO design guide and the HIGH-PAVE program. Using the same equation form as that obtained in step 1 above, the resulting life in terms of ESAL obtained from the 1986 AASHTO design guide were then statistically correlated to the modulus, stresses, and strains of the various pavement layers that were calculated using the MICH-PAVE program. This step resulted in a second fatigue life equation that accurately predicted the life of the 30 arbitrary selected pavement sections. the value of the constants in front of each variable of this last equation were slightly different than those of the previous equation obtained in step 2. Step 4 - The two equations obtained from steps 2 and 3 were then combined by averaging the values of the constants of each variable from the two equations. Finally, the sensitivity of the fatigue life of the resulting equation to each variable of the equation was studied relative to the sensitivity of equation 6-1 and to the outputs (stresses and strains) of the MICH-PAVE program. The purpose of the study was to make final adjustment in the value of the 152 constant in front of each variable in the equation to yield similar sensitivity to that of equation 6-1 and to the output of MICH-PAVE program. Equation 6-3 is the final fatigue life equation. log(ESAL) - where: log ESAL SD TBEQ TAC AV TS CS KV 2.416 - 2.799(log(SD)) + 0.00694(TBEQ) + 0.917(log(MRb)) 0.154(TAC) - 0.261(AV) + 0.0000269(MRS) l.096(log(TS)) + l.l73(log(CS)) - 0.001(KV) (Eq.6-3) base 10 logarithmic operator; number of equivalent 18 kips single axle load traveling the pavement section prior to failure; surface deflection under the center of the load (inch); base thickness plus the equivalent thickness of the subbase (inches); equivalent thickness of the subbase is calculated by multiplying the actual thickness of the subbase by the ratio of the modulus of the subbase to that of the base material; resilient modulus of the base material (psi); thickness of the AC layer (inch); the percent air voids of the AC layer (%); effective resilient modulus of the roadbed soil as defined in the AASHTO 1986 design guide (psi); the tensile strain at the bottom of the AC layer; the average compressive strain of the AC layer; and kinematic viscosity of the asphalt concrete binder (centistrokes). 153 It should be noted that an average annual air temperature of 75°F was assumed and used in all steps. Nevertheless, the fatigue life calculated using equation 6-3 of the 10 in-service pavement sections and the 30 arbitrary selected sections are listed in tables 6-1 and 6-2, respectively. 154 Table 6-1: Fatigue lives and rut depths of field data. Site TAG/MRAC TB/MRb TSB/MrS MRS FL(1) FL(2) AASHTO RD(l) RD(2) in/ksi in/ksi in/ksi ksi ESAL ESAL ESAL in in 1N1 4/500 12/60 6/12 6 1900000 2200000 750000 .3 .123 INl 5/500 12/60 6/12 6 5500000 5300000 1750000 .2 .101 INl 6/500 12/60 6/12 6 9700000 11500000 3700000 - .087 1N2 3/150 12/25 4 9000 4245 13500 .2 .303 1N3 3/150 6/30 6 12200 2054 5200 ~ .347 1N4 3/150 10/25 4 4000 2226 7000 - .316 MIUl 4/150 10/20 36/9 3 82000 66589 620000 <.1 .047 MIU2 2.5/150 20/20 10/9 3 41000 26892 280000 <.1 .086 MIU3 6/350 16/20 13/9 3 500000 535550 1000000 .5-1 .029 MIU4 4.5/350 12/20 70/9 3 5475000 4960000 450000000 <.1 .030 1N1 TAC TB TSB MRAC one inch thick AC; resilient resilient resilient thickness of the asphalt concrete course (inch) thickness of the base layer (inch); thickness of the subbase (inch); : resilient modulus of the asphalt concrete (ksi) modulus of the base layer (ksi); modulus of the subbase layer (ksi); modulus of the subgrade (ksi); fatigue life of the pavement section (ESAL); Indiana (1N1 through 1N4) an average annual air temperature of 75 : calculated fatigue life (ESAL); : measured rut depth (inch); and o 3 OF was used, for Michigan sections (MIUl through MIU4), used. : a pavement which was overlayed twice. Each overlay consist of : calculated rut depth (inch); for all pavement sections located in 66°F was 155 Table 6-2: Parameters for calibrating the fatigue life equations used in MICH-PAVE. and rut depth C TAC AV TB MR3c MRb MRS TS CS SD FL(1) FL(2) Ratio RD (in)(%)(in)(ksi)(ksi) (ksi) (in) AASHTO (ESAL) (in) 1 2 7 6 150 25.9 3 532 143 .1074 212 134 .63 0.58 2 4 7 6 150 19.2 3 809 302 .0836 778 631 .81 0.32 3 8 7 6 150 12.0 3 495 318 .0550 8767 9970 1.14 0.22 4 2 7 6 150 73.6 3 ~10 57 .0826 1261 * * * 5 8 7 6 150 25.0 3 383 320 .0518 23340 30622 1.31 0.18 6 8 7 6 150 38.2 3 304 322 .0497 45325 66288 1.46 0.16 7 2 5 6 300 24.5 3 654 121 .1012 446 328 .74 0.50 8 2 3 6 500 23.2 3 667 98 .0955 835 933 1.12 0.44 9 8 5 6 300 10.1 3 317 164 .0472 47196 32677 .69 0.21 10 8 3 6 500 8.9 3 216 100 .0420 210720 114230 .54 0.19 11 2 7 9 150 24.2 3 444 239 .0955 784 629 .80 0.57 12 2 7 12 150 22.7 3 426 293 .0879 1865 1606 .86 0.54 13 4 7 9 150 56.1 3 239 322 .0599 26190 28509 1.09 0.24 14 4 7 12 150 54.5 3 220 354 .0550 87556 69635 .80 0.22 15 4 7 6 150 57.9 3 300 274 .0677 6051 8318 1.38 0.26 16 2 7 6 150 27.4 6 490 238 .0685 1351 1190 .88 0.58 17 2 7 6 150 28.3 9 474 290 .0533 3631 3891 1.07 0.56 18 8 5 6 300 11.9 9 278 168 .0234 680635 467143 .69 0.17 19 8 3 12 500 10.6 9 194 103 .0199 5.31e6 4.79e6 .90 0.11 20 8 3 12 500 12.7 25 179 105 .0122 59.0e6 66 9e6 1.13 0.10 21 8 3 12 500 35.6 9 146 108 .0176 34.4e6 29.6e6 .86 0.08 22 8 3 12 500 39.2 25 131 112 .0099 444 e6 512 e6 1.15 0.07 23 8 3 18 500 10.6 9 193 103 .0203 8.41e6 11.9e6 1.42 0.10 24 8 3 18 500 11.9 25 184 104 .0129 95. e6 132 e6 1.39 0.08 25 8 3 18 500 33.8 9 139 110 .0172 95.4e6 84.5e6 .89 0.07 26 8 3 18 500 37.2 25 130 112 .0099 2880e6 1286e6 .45 0.06 27 8 5 18 300 23.4 9 213 177 .0207 11.3e6 11.8e6 1.05 0.09 28 8 5 18 300 26.3 25 198 181 .0129 193 e6 148 e6 .77 0.08 29 8 5 18 300 37.4 9 171 183 .0191 44.3e6 30.1e6 .68 0.08 30 8 5 18 300 40.3 25 160 188 .0115 640 e6 399 e6 .62 0.07 Where: FL<1> fatigue life calculated using the 1986 AASHTO design guide; FL(2) - fatigue life calculated using the MICHPAVE program; TS tensile microstrain at the bottom of the AC; CS average compressive microstrain within the AC; Ratio - ratio of FL /F ; TB - equivalent 6%; hizkness; RD - rut depth is calculated by Eq.6-4 (in); and * - pavement failed in shear. 156 NOTE: The rut depth is calculated at the fatigue life 8f the pavement section and an average annual temperature of 75 F. 6.3 RUT DEPTH MODEL The rut depth model (equation 6.2) was calibrated using field data from 7 different pavement sections (see table 6-1). First, each section was analyzed using the MICH-PAVE computer program. The calculated stresses, strains, and surface deflection, and the layer thicknesses and properties were then used to calibrate equation 6.2. This resulted in the following model: log(RD) = - 1.6 + .067(AV) - l.4(log(TAC)) + .07(AAT) - .000434(KV) + .15(1og(ESALC)) - .4(log(MRS)) - .50 log(HRb) + .1(log(SD)) + .01(log(CS)) - .7(log(TBEQ)) + .O9(log(50-(TAC+TBEQ))) (Eq.6-4) where: RD - rut depth (inches); ESALt = the number of ESAL at time t where the rut depth is being calculated; AAT = average annual air temperature (OF); 50 -(TAC+TBEQ)) = the affected depth of the roadbed soil; and all other variables are as before. It should be noted that since the number of variables in equation 6- 4 is higher than the number of pavement sections where rut depth data wereavailable,theva1ue 0f thecoefficient infrontofthe average annual temperature (AAT) term was kept the same as that of equation 6-2. and a constant tire pressure of 100 psi was assumed. In addition, a step-wise statistical analyses were conducted whereby a maximum of 7 variables were used in each step. This resulted in several equations each containing only seven variables beside the AAT. It was noted that for all equations the values of the constants in front of the AV, TAC, 157 and ESAL were almost the same while the values of the other constants changed slightly from one equation to another. Nevertheless, Equation 6-4 was obtained by averaging the values of each constant in front of each variable obtained from all equations. Measured and calculated (using equation 6-4) rut depths for 7 in-service pavement sections are listed in table 6-1. 6.4 ' SENSITIVITY ANALYSIS 6,4,1 Fatigue Model Sensitivity analysis of the fatigue life equation (Eq. 6-3) has indicated that: 1. The fatigue life of a flexible pavement decreases as the surface deflection at the center of the loaded area increases. Increasing the surface deflection from 0.001 to 0.01 inch causes a decrease in the fatigue life by a factor of 630. It should be noted that the surface deflection is an intrinsic function of the thicknesses and properties of all pavement layers. 2. Increasing the thickness of the granular layer from 6 to 18 inches causes an increase in the fatigue life by a factor of about 6.8. 3. Increasing the modulus of granular layer from 10 to 50 ksi causes an increase in the fatigue life by a factor of about 4.4. 4. Increasing the thickness of AC from 2 to 8 inches causes an increase in the fatigue life by a factor of about 8.4. 5. The fatigue life decreases as the percent air voids in the mix increases. Increasing AV from 3 to 7 percent yields a decrease in the fatigue life by a factor of about 11. 158 Increasing the modulus of roadbed soil from 3 to 25 ksi causes an increase in the fatigue life by a factor of about 4. An increase in the tensile strain at the bottom of AC layer from 90 to 600 micro strain causes a decrease in the fatigue life by a factor of about 8. An increase in the average compressive strain within the AC layer from 100 to 300 micro strain causes an increase in the fatigue life by a factor of about 3.6. Decreasing the kinematic viscosity of the AC from 270 to 159 centistokes causes an increase in the fatigue life by a factor of about 1.3. 6.4.2 Rut Depth Model Sensitivity analysis of the rut depth equation (Eq. 6-4) has indicated that: 1. Increasing the percent air voids in the mix from 3 to 7 percent yields an increase in the rut depth by a factor of about 1.9. Increasing the thickness of the AC from 2 to 8 inches causes a decrease in the rut depth by a factor of about 7. The rut depth equation is very sensitive to the average annual temperature. Increasing the average annual temperature from 60 to 77°F yields an increase in the rut depth by a factor of about 15.5. Increasing the kinematic viscosity of the AC from 159 to 270 centistokes causes a decrease in the rut depth by a factor of about 1.1. This indicates that the rut depth equation is relatively insensitive to the kinematic viscosity of the AC. 159 5. Increasing the number of load repetitions from 100,000 to 1,000,000 ESAL causes an increase in the rut depth by a factor of about 1.4. 6. Increasing the modulus of roadbed soil from 3 to 25 ksi causes a decrease in the rut depth by a factor of about 2.3. 7. Increasing the modulus of granular layer from 10 to 50 ksi causes a decrease in the rut depth by a factor of about 2.2. 8. An increase in the pavement surface deflection at the center of the loaded area from 0.001 to 0.01 inch causes an increase in the rut depth by a factor of 1.3. 9. An increase in the average compressive strain within the AC from 100 to 300 micro strain causes an increase in the rut depth by a factor of about 1.01. 10. Increasing the equivalent base thickness from 6 to 18 inches causes a decrease in the rut depth by a factor of about 2.2. 11. Increasing the affected depth of the roadbed soil from 20 to 40 inches causes a increase in the rut depth by a factor of about 1.1. 6.5 : ANALYSIS OF MICH-PAVE INPUT/OUTPUT The sensitivity of three important outputs (tensile strain at the bottom of the AC layer, compressive strain at the top of the roadbed soil, and pavement surface deflection) of MICH-PAVE computer program to the input variables was studied using several pavement sections. The results of the study are summarized in the following sections. 160 6.5.1 : Thickness and Modulus of the Asphalt Concrete The thickness and modulus of the AC course have significant effects upon the tensile strain at the bottom of the AC layer. For example, a 2 inch thick and soft (MR < 150 ksi) AC course located on top of a stiff base layer whose modulus is larger than about one third of that of the AC (i.e., cement or asphalt treated base layer) will cause no tensile strain at the bottom of the AC layer. The reason is that the AC course and the base layer will act as one layer whose nutral axis is located within the base layer (well below the bottom of the AC course). Hence, the entire AC layer will be in compression (see Table 6-3). Figure 6-1 depicts the effect of the thickness of the AC on the tensile strain at the bottom of the AC layer. Increasing the thickness of the AC surface causes a significant decrease in the compressive strain at the top of the roadbed soil (see Figure 6-2). This is especially true when the modulus of the AC layer is significantly higher than that of the granular base material. Hence, one way to reduce the compressive strain at the top of the roadbed soil (which contributes most to high surface deflection and rut), is to increase the thickness of the AC. Nevertheless, Figure 6-3 shows the pavement surface deflection as a function of the thickness of the AC layer. The numerical results are listed in Table 6-3. 161 10 UTI'TI‘TtinT ITITIUIIIITT'IUWIITIITU'ITITITTITIWWITIFU II ... i-C) t. . ~00 / -.._ / —l\ .3 N / ...." 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O O O O O O O (uaug) uogaeueg GODIJHS Thickness of AC (in) —3: Surface Deflection at the Center of Loading to Thickness of AC Figure 6 164 Table 6-3: Tensile strain at the bottom of the AC and compressive strain at the top of the roadbed soil for varying AC thicknesses. Case Thickness Tensile Compressive Surface (in) Micro Strain Micro Strain Def. (in) 4 2 - 10 3430 0.08258 15 4 300 2462 0.06769 6 8 304 1226 0.04970 6.5.2 : Percent Air Voids in the Asphalt Concrete For a 2 inch thick AC layer, decreasing the percent air voids from 7 to 3 percent (increasing the modulus from about 150 to about 500 ksi) results in an increase in the tensile strain at the bottom of the AC layer. For an 8 inch thick AC layer (on the other hand), similar decrease in the percent air voids causes a decrease in the tensile strain (see Table 6—4 and Figure 6-4). These observations were expected because of the stress distribution within the AC layer and its flexibility. Thicker AC will spread the load over wider area than thin AC layer. The significant of these observations is that (since the tensile strain is not an independent variable) a successful predictive fatigue model for flexible pavements should not be strictly based on the stiffness of the AC or the percent air voids. It must accounts for all the variables that affect the tensile strain at the bottom of the AC course. The percent air voids of the AC layer has minor effects upon the compressive strain at the top of the roabed soil as well as on the pavement surface deflection as shown in Figure 6-5 and 6-6, respectively. Again, these observations were expected because surface deflection and compressive strain at the top of the roadbed soil are 165 (D o< e0 mEo> .__< 9. .634 04. 06 Eofiom of to Eobm ozmcmp ”elm 0.3911 i—I\ (D E 0:0 m20> E m e L m g N h F b O VIIIlTIiITTIrTII1tIrlrfiI1rII \ \ \ :N .II... Q< *0 mmwcxumrc. 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The percent air voids and/or the stiffness of the AC layer will cause no significant increase and/or decrease in the magnitude of these stresses. The stresses delivered to the roadbed soil are significantly affected by the thicknesses of the protective layers (AC, base, and subbase). Table 6-4: Tensile strain at the bottom of the AC and compressive strain at the top of the roadbed soil for varying values of air voids in the AC. Case Air Voids Modulus Tensile Compressive Surface ( % ) (ksi) Micro Strain Micro Strain Def. (in) l 7 150 532 5247 0.1074 7 5 300 654 4767 0.1012 8 3 500 667 4508 0.0955 3 7 150 495 1440 0.0550 9 5 300 317 975 0.0472 10 3 500 216 716 0.0420 6.5.3 : Thickness of Granular Layers The effects of the thickness of the granular layer upon the tensile strain at the bottom of the AC layer, the compressive strain at the top of the roadbed soil, and the surface deflection were also analyzed using the MICH-PAVE program. The results are listed in Table 6-5 and plotted in Figures 6-7 through 6-9. Examination of these results have indicated that: 1) Increasing the thickness the granular layer causes a reduction in the tensile strain at the bottom of the AC layer (see Figure 6-7) until an optimum thickness is reached beyond which the magnitude of the reduction is insignificant. For example, in reference to Table 6-5, increasing the thickness of the granular layer from 6 to 9 inches causes a 26 percent reduction in the tensile strain. 2) 3) 169 Adding three more inches result in a decrease of only 8 percent. The optimum thickness of the granular layer for any typical pavement design can be determined using economical analysis. It should be noted that, in general, lower tensile strain yield higher fatigue life). Increasing the thickness of the granular layer from 6 to 9 inches results in a 29 percent decrease in the compressive strain at the top of the roadbed soil (see Table 6-5 and Figure 6-8). Further increase in the thickness of the granular layer from 9 to 12 inches will cause an additional 24 percent decrease. Thus, as far as the compressive strain is concerned, thicker granular layer is beneficial. It should be noted that lower compressive strains at the top of the roadbed soil yield lower rut potential of the soil until the elastic strain limit is reached. This limit is mainly a function of the type of roadbed soil and its degree and time duration of saturation. Hence, the optimum thickness of the granular layer to reduce the rut potential of the roadbed soil should be analyzed on a case by case basis relative to the type of the roadbed soil and drainage conditions. Finally, Figure 6-9 indicates that increasing the thickness of the granular layer from 6 to 12 inches causes a reduction in the surface deflection by about 20 percent. This percentage decrease is about half of that due to an increase in the thickness of the AC from 2 to 8 inches (see section 6.5.1). 170 .664 63:80 .6 6656:: B .664 2 6 536m 2: 6 :65 888 K16 839.1 Ace .664 63:80 .6 66:42:; 2 3 2 S m m A e m t _ t _ . t _ 8N . 6v. 9 ...1. C. no... to z... .1- 4 .. ...1 1 .1. I 4 r I I 1 1 lo ... / ... / room 1 / / T - / . T / 4. .. / T T x. .8... e . . 1 1 r _ _ _ r _ P 1 own (ugans 0.10114) ugaJig eusuei 171 6.84 636.6 .6 mmmcxofe B :om omnnoom .6 gap 65 6 Eobm o>_mm6._ano ”mic 6.3914 Ace .664 636.5 .6 mmoconH .2 N4 : 04 m m \1 m m P L L L _ 4% _ 000—. r fix NF ...... 4v. .o< .6 :4. To 4 4 . 4 ./ 4 1 / .. l / 1 / loomp T / 4. 4. / III 1 . / - - / 4 4. / 4.08m .. / . r / . 4 / / 4 . t _ .1 _ _ t _ J 4 (mogul) ugaJig eAiSSSJdLUOQ 172 .664 63:90 6 66567:. 3 96004 6 6660 65. 6 £262.60 mootnm ”mlm 6.39.1 V) NP _ Ac; .664 63:80 6 66:425. 2 p 04 _ m 4 m _ N w _ L I!) IITrTrTII—rTIITIrrII— 6.. S 3. no< 6 ...v Ill IIIIIITTTjjIITTTIjI omod wood 0 LO 0. 0 1:) co 0. O Ohod (uaug) uonaeueg eaapng 173 Table 6-5: Tensile strain at the bottom of the AC and compressive strain at the top of the roadbed soil for varying thickness of the granular layer. Case Thickness Tensile Compressive Surface of Base(in) Micro Strain Micro Strain Def. (in) 15 6 300 2462 0.0677 13 9 239 1744 0.0599 14 12 220 1317 0.0550 6.5.4 : Modulus of the Granular layer The effects of the modulus of the granular layer on the tensile strain at the bottom of the AC course, the compressive strain at the top of the roadbed soil, and on the pavement surface deflection vary and depend upon the thickness of the AC and its modulus. Recall that (see section 4-2) the nonlinear behavior of the granular layer was modeled using the K1(6)K2 equation. The sensitivity of the outputs of MICH-PAVE program to the modulus of the granular layer was studied by varying the value of K1. The results of the study are summarized in Table 6—6 and plotted in Figure 6-10 through 6-12. Examination of the results indicated that: 1) For a pavement with 2 inch thick AC layer, increasing the value of K1 from 4 to 12 ksi causes the tensile strain at the bottom of the AC layer to become compressive (see Figure 6~10) and causes tensile strain in the granular layer. The reason for this was explained in section 6.5.1 above. 2) For a 4 inch thick AC layer, increasing the K1 value from 4 to 12 ksi results in a reduction in the tensile strain by a factor of about 2.7 (see Figure 6-10). 174 3) For an 8 inch thick AC layer, increasing the K1 value from 4 to 12 ksi causes no significant effects in the tensile strain. Table 6-6 and Figures 6-11 and 6-12 show that (except in the case of the 2 inch thick AC layer) increasing the K1 value of the granular layer from 4 to 12 ksi does not significantly reduce the compressive strain or the pavement surface deflection. Therefore, one can concluded that (except for thin asphalt layers) the compressive strain at the top of the roadbed soil and the pavement surface deflection are insensitive to the modulus of the granular layer. Table 6-6: Tensile strain at the bottom of the AC and compressive strain at the top of the roadbed soil for varying material properties of the granular layer. Case K1 Value Modulus* Tensile Compressive Surface (ksi) (psi) Micro Strain Micro Strain Def. (in) 1 4 25886 532 5247 0.1074 4 12 73597 - 10 3430 0.0826 2 4 19153 809 3323 0.0836 15 12 57933 300 2462 0.0677 3 4 12002 495 1440 0 0550 5 8 24960 383 1334 0.0518 6 12 38216 304 1226 0.0497 Modulus represents the equivalent resilient modulus of the granular layer. 6.5.5 : Elastic Modulus of Roadbed Soil Due to the lack of an appropriate nonlinear model for roadbed soil in the State of Michigan, the nonlinear behavior of the roadbed soil is modeled in the MICH-PAVE program (see section 4-2) using an available model which require four material constants as input. For most State Highway Agency laboratories, these constants cannot be obtained. ‘Therefore for simplicity, the following analyses were conducted using a 175 .664 63:80 6 6626.5. 64.662 3 664 64.. 6 686m 8.. 6 :65 883 5416 84.9.4 9me .6484 63:80 6 63_o> 54 <— l. 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NF 0.— Lm Lm 4.4 N _ o< .0 ..m I one o 4 o< .0 ...4 ....- 4 r. 0< *0 :m I 4..O.VO.O 1 fl 1 llllnuolll'ilnlllll. 4. 1 romod 1 l 1. l l l 1 1060.0 / . .lo 4. // T r / rooto 11.1.. ll- 11L...1 1.6 .l lllrl.l....l_ll11-i-L1.-. _ l b . _ _ ON _..0 (uaul) uoliaegeg eaopng 178 linear model. Table 6-7 and Figure 6-13 show that the tensile strain at the bottom of the AC layer is insensitive to variations (from 3 to 9 ksi) in the modulus of the roadbed soil. Figures 6-14 and 6-15, on the other hand, show that the compressive strain at the top of the roadbed soil and the surface deflection are very sensitive to the modulus of the roadbed soil. For example, increasingthe modulus of the roadbed soil from 3 to 9 ksi results in a decrease of about 39 percent in the compressive strain, and a decrease of about 50 percent in the surface deflection. Therefore, the modulus of the roadbed soil and the thickness of the AC layer are the most significant factors affecting the pavement surface deflection. Table 6-7: Tensile strain at the bottom of th AC and compressive strain at the top of the roadbed soil for varying elastic moduli of the roadbed soil. Case Modulus Tensile Compressive Surface (ksi) Micro Strain Micro Strain Def. (in) 9 3 317 975 0.0472 18 9 278 591 0.0230 6 6 ' D D ANALY S OF THE FATIGUE LIFE E UATION The following sections, discuss the influence of the thickness and air voids in the AC, the thickness and modulus of the granular layer, and the modulus of the roadbed soil on the fatigue life of pavements. It should be noted that the fatigue life is always calculated using equation 6-3. 179 6m .6863. 6 86822 28m. 3 66.. 94 6 836m. 8.. 6 66.6. 88...; ”216 24.6.... 46... :8 888m 6 83462 6.48m. 04 w m .6 N . _ . t 1 t 1 8m - .9. con 11.. .o< 6 minnoz um o< 6 ...v Ill . .. 4. r TomN 4 .. fl .1 1 1 H 1 .1 l I 1 .1. I -8... 4. 1.1 In 4. 4. fi .- .1 100m .. 4 1 T T I T b p lF _ p P s 4. 00.? (UlDJlS 0.101u.1) UlDJlS ensuei 180 .3 nmnnoom 6 $3622 265 3 6m 6268”. .6 aop 9: #0 £85 m>_mmo._ E00 2110 830E o— m r w r r 9me 6m nmnnoom 6 62262 362m h m m ¢ m b p L b b / / / /- _mv__oon H w< *0 mJSnoz um. 04. *0 flu III N +oom 100m room 100m fioom loco— OOP— (ongw) 1110.115 eAgssedeog 181 :8 862$ 6 62262 265 B 3663 . .6 1.3ch m5 #0 cozomzmo mootnm "91.0 @139... O ems :3 68681. 6 12382 260a m m n o n .v n N _ b _ L _ r _ H E con .1 2 6 3.282 a 2 6 =1 ..1- . m. .x m. 1 / m 1 / fl / W n / w 1 / 1 ... / I n. / m. .l / 1 fin / fl 1 / 1 w. / m. r / 1 n / m w / n .1. x. m _ 1F 11 _ _ _ .1 1 0No.o mNod o n O. o Ln 1*) O. O o #- O. o 95.0 omod (113111) 11014091190 aooyng 182 6.6 1 ' Thickness of AC Table 6-8 and Figure 6-16 show that increasing the thickness of the AC from 2 to 8 inches causes an increase in the fatigue life by a factor of about 74 for an AC with 150 ksi modulus, and by a factor of 122 for an AC with 500 ksi modulus. The proper thickness of the AC to be used in any pavement design will naturally depends on the pavement type, traffic level, and economic analysis. Table 6-8: Effect of the thickness of the AC on the fatigue life of pavements. Case Thickness K value of Fatigue Lige FaCCOr* of AC (in) Granular Layer (ksi) (ESAL) 1 2 134 2 4 4 631 3 3 9,970 74 7 2 4 328 9 8 32.677 99.6 8 2 4 933 1° 8 114,230 122 * Factor is calculated by dividing the fatigue life for 8 inches of AC by the fatigue life for 2 inches of AC; ** ESAL represents the 18 kip equivalent single axle load. 6.6.2 ; Air Voids in AC Table 6-9 and Figure 6-17 show that when the thickness of the AC is 2 inches, decreasing its air voids from 7 to 3 percent only results in an increase in the fatigue life by a factor of about 7.0. When the thickness of the AC is 8 inches, decreasing its air voids from 7 to 3 percent causes an increase in the fatigue life by a factor of about 11.4. Since the percent air voids in a typical asphalt mix is mainly a function of compaction, higher fatigue life can be achieved by using 183 O 6.5 896... e 8 $865 2 6 665 H $.16 8%: 295 2 6 665.2,: m m to N o n — b — b b 'P P [b 00F m \\. m m \ m l \ \. 1.000— r \ \ r \\\x\ \ \ w m \ \ W .... \\\ \\ 1:15 r \ \ \\ 1. W .\ \ w m , \ m n \\ ma 2.. E ... 1.. 3. a. 3 com u 9.. 6 13262 I .. 8+“: h E 1 1.. C. a. E can ...... 3. 6 2382 «L. .. m 6.. ... u 36> 5. a. no. of .11. 9.. 6 3.262 o... W n _ lb PI 51 . _ P 11 — “11 . QO+MF (71783) 9111 99590.1 184 1n 3: 896... . :0 2 5 66> E 6 686 H :16 6.55 as 2 5 m66> .__< U’T r :m .& ll 2 6 «$52,: ..1- 9.. 6 $652,: I l l L s m m ... n N t _ _ _ t 09 I [ll/ . a. .u. l/ m m. 1.- 111. 182 m. . . n r r e ... n. .... .1 I 333 m... j I l 1 \./ m l l m V 1 1 1..... mo+mr j I T 185 better compaction specifications and quality control. Table 6-9: Effect of the air voids in the AC on the fatigue life of pavements. Case Air Voids Modulus of Thickness of Fatigue life Factor of AC (%) AC (ksi) AG (in) (ESAL) l 7 150 134 7 5 300 2 328 8 3 500 933 7.0 3 7 150 9,970 9 5 300 8 32,677 10 3 500 114,230 11.4 6.6.3 : Thickness of Granular Layer Figure 6-18 and Table 6-10 show that increasing the thickness of the granular layer from 6 to 9 to 12 inches cause an increase in the fatigue life by factors of 3.4 and 2.4, respectively. Table 6-10: Effect of the thickness of the granular layer on the fatigue life of pavements. Case Thick. of Thick. E of K Value Fatigue Life Factor G.L. (in) of AC(in) AC(ksi) G.L.(ksi) (ESAL) 15 6 4 150 12 8,318 13 9 4 150 12 28,509 3.4 14 12 4 150 12 69,635 2.4 * G.L. is an abbreviation for the granular layer. 6.6,4 : Resilient Modulus of the Granular Layer In general, higher resilient modulus of the granular layer causes higher fatigue life. However, the functional relationship between the resilient modulus of the granular material and the fatigue life is dependent upon the thickness of the AC layer. For example, Table 6-11 and Figure6-l9 showthat increasing the resilient modulus of the granular material by a factor of 3 causes an increase in the fatigue life of a 4 186 3: mammal co mmmcxfif. 1.98.. 16.2616 do Lootm M E19 83me 395 1.984 63:80 .6 mmocon._. : N: 0.9 w m +6 Ll t b t b L E 9 .1. 36> § .2 6 E 2: s. .6 ..1- T T IIIITIT TTIrI—rI— I 1. ITTTI I—U' FT‘I—T'IIT L \ 1 r 1. coop ¢o+.1..: mo+me (7vs3) 9111 89590.1 187 6.5 639611 :0 .664 63:80 6 ..BmEotod fix .6 635 ” Elm 050E 0me 1.664 636.5 .6 m2o> C. b F b P F — L — b 00—. in n m 1\ \h m P \1 \ n. 000—. \ \ .l ... \1 \1 \\ 1 W. \\ \. \\ m \ I \ \\ I .1. 1111. 11.1 \\ 1 ....11... 1.1 \\ \\ :m ....u 0< $0 mmmcxomrfiu I m 1.1 ...V 1|" o< $0 meCv-Omnh .Il- Pmo+m —. P —1 . — L| b b — L P b1 (11753) 9.111 9'15!le 188 inch thick AC layer by a factor of 13.2 and by a factor of 6.6 for an 8 inch thick AC. Table 6-11: Effect of the material properties of the granular layer on the fatigue life of pavements. Case K1 Value of Equiv. MR. Thick. of Fatigue Life Factor G.L. (ksi) of G.L.(psi) AG (in) (ESAL) 2 4 19,153 4 631 15 12 57,933 4 8,318 13.2 3 4 12,002 8 9,970 5 8 24,960 8 30,622 6 12 38,216 8 66,288 6.6 * Equiv. MR represents the equivalent resilient modulus of the granular layer (see section 5.2.2) 6.6.5 : Resilient Modulus of theiRoadbed Soil Although the resilient modulus of the roadbed has no significant effects on the tensile strain at the bottom of the AC layer (see section 6.5.5), it has significant effects on the pavement surface deflection and an the pavement fatigue life. Table 6-12 and Figure 6-20 show that for an 8 inch thick AC layer with 300 ksi modulus, increasing the resilient modulus of the roadbed soil from 3 to 9 ksi causes an increase in the fatigue life by a factor of 14.3. One important point should be noted is that the resilient modulus of the roadbed soil varies from season to season and relative to the moisture content. The resilient modulus to be used as an input to the fatigue life model should be the effective resilient modulus. The AASHTO 1986 design guide for flexible pavements contains detailed explanation concerning the calculation of the effective resilient modulus. 189 6.: 6:96... 6 12:82 6m 868m 6 66.6 H 8.16 8:9. as; :8. 6868.. 6 12382 OF m o .1 b In b 1—1 [h I. b 1P fix .v fl 03:5 —v_ a. omom 6 2m .o< 6 fix com on ..w III T [UTFI I T 1— 1 [IITU I f—I \ Ith1TT VfiT‘ITTI ¢o+m~ mo+mF mo+w_ (was) 9111 9951103 190 Table 6-12: Effect of the resilient modulus of the roadbed soil on the fatigue life of pavements. Case Modulus of Thick. of Modulus of Fatigue Life Factor R.S. (ksi) AG (in) AC (ksi) (ESAL) 9 3 8 300 32,677 18 9 8 300 467,143 14.3 R.S. represents the roadbed soil. 6,6 ; EQUIVALENT WHEEL LOAD FACTOR An equivalent wheel load factor (ELF) defines the damage per pass caused to a specific pavement section caused by an arbitrary vehicle relative to the damage per pass caused by a selected vehicle with an 18- kip single axle load moving on the same pavement section (Yoder, et al., 1975). Deacon (1971) and Witczak (1973) showed that the equivalent wheel load factor Fj for any vehicle can be expressed by the following equation: a. ' FJ. - [—L] (3116-5) where : c - constant between 3 and 6 with common values of 4 to 5; a - the tensile strain at the bottom of the AC under an 18-kip single axle load; cj - the tensile strain at the bottom of the AC course due to any axle load. Based on elastic layer analysis using CHEVSL program, Eq.6-5 cannot be applied for pavements with very thin AC layers in which the radial strain at the bottom of the AC may be compressive or only marginally 191 tensile. In addition, the damage delivered to a pavement section by a passing wheel load is functions of the thicknesses and properties of all pavement layers. Since, the pavement surface deflection is also functions of the properties and thicknesses of all pavement layers including the roadbed soil, it was thought that expressing the ELF in terms of the pavement surface deflection will yield more accurate results. Consequently, the surface deflection for several pavement sections and axle loads were calculated using the MICE-PAVE program (see Table 6-13). The results were then correlated to the AASHTO equivalent wheel load factor (EWLF). This resulted in the following equation: so. 4.25 ELF = [—1] (Eq.6-6) where: SDs - surface deflection of pavement in inches due to an 18 kips single axle load or any other standard axle load; SDj - surface deflection of pavement in inches under an arbitrary wheel load. The data (surface deflection) in Table 6-13 was obtained using the linear CHEVSL computer program with various wheel loads to compute the surface deflections of several pavement sections. The AASHTO equivalent wheel load factors were obtained using the structural number (SN) of each pavement section. Nevertheless, it can be seen that Equation 6-6 compares well with the AASHTO estimate. The advantages of equation 6- 6 is that the ELF can be calculated at any time and for any season by simply conducting a nondestructive deflection testing. Since pavement deflection varies from season to season, with moisture content, and with pavement age, the ELF can be obtained for any time and any moisture or 192 seasonal conditionas. The AASHTO EWLF method, on the other hand, assumes, for each pavement section and axle load, a constant value throughout the pavement life and for all seasonal and/or moisture conditions. 193 Table 6-13 : Comparison of the equivalent wheel load factor between Equation 6-6 and the AASHTO method. Axle Load Surface Deflection EWLF Case (kip) (inch) Eq.6-6 AASHO 4 0.01291 0.004 0.004 10 0.02761 0.11 0.12 6 18 0.04636 1. l. 30 0.07439 7.46 7.94 40 0.09739 23.4 8.51 4 0.00487 0.003 0.003 10 0.01079 0.097 0.09 23 18 0.01868 1. 1. 30 0.03024 7.75 6.9 40 0.03957 24.3 1.6 4 0.01002 0.0005 0.004 10 0.02236 0.11 0.1 10 18 0.03791 1. l. 30 0.06176 7.96 6.83 40 0.08178 26.2 2.50 4 0.01252 0.004 0.004 10 0.02713 0.12 0.11 14 18 0.04455 1. l. 30 0.06989 6.78 7.5 40 0.09062 20.5 5.0 4 0.00493 0.003 0.004 10 0.01088 0.01 0.1 19 18 0.01884 1. l. 30 0.03053 7.78 6.83 40 0.03996 24.4 2.50 4 0.00345 0.005 0.002 10 0.00726 0.12 0.08 28 18 0.01187 1. l. 30 0.01825 6.22 7.79 40 0.02325 17.4 3.04 6,7 ; SQMMARX The dominant factor in the fatigue life equation (Eq.6-3) for flexible pavements is the surface deflection under the wheel load rather than the tensile strain at the bottom of the AC. The surface deflection 194 under the wheel load is an intrinsic function of all input parameters, such as the thickness of the AC, air voids in the AC, thickness of the granular layer, modulus of the granular layer, modulus of the roadbed soil, wheel load, and tire pressure. Of these, the thickness of the AC and the modulus of the roadbed soil are the two dominant factors that affect the surface deflection. A summary of the sensitivity of the tensile strain at the bottom of the AC layer, the compressive strain at the top of the roadbed soil, and the pavement surface deflection due to the various pavement variables are presented in Table 6—14. Table 6-14 : Sensitivity of some response measures to key properties of pavement sections. Tensile Strain Compressive Strain Surface Deflection Thickness of the sensitive very sensitive very AC sensitive Air Voids in the sensitive insensitive insensitive AC Thickness of the sensitive to an sensitive sensitive Granular Layer optimum thick. Modulus of the sensitive insensitive insensitive Granular Layer Modulus of insensitive very sensitive very Roadbed Soil sensitive Equations 6-3 and 6-4 are strictly applicable to three- and/or four- layerpavement sections in which the AC is the toplayer that underlain by a granular (base and/or subbase) layer and a roadbed soil. The model should not be applied to pavement sections where a second asphalt layer is sandwiched between the base and subbase layers without verification and calibration. Nevertheless, equations 6-3 and 6-4 can be improved by 195 further calibration using field data. It is strongly recommended that the equations be checked using field data prior to their uses as predictive models. When a thin AC layer is used in a pavement section, a compressive radial strain may occur at the bottom of the AC rather than a tensile radial strain. This is contrary to the normal design assumption. For such pavements, it is recommended that a response measure such as the surface deflection be used, instead of the the tensile strain at the bottom of the AC for designing the pavement. CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS 7.1 ' SUMMARY AND CONCLUSIONS Two major achievements have been accomplished in this research. First, a new concept of utilizing a flexible boundary in flexible pavement analysis has been introduced, and its characteristics fully investigated. Second, an extremely "user-friendly" nonlinear finite element program for pavement analysis and design, named MICH-PAVE, has been implemented on personal computers. The main findings of this research are outlined below: (1) In the linear analysis of a multilayer pavement, the finite element mesh with a flexible boundary results in a decrease in the percentage error of the surface deflection when compared to the results obtained using a finite element mesh yielding about the same level of computational efforts. Other response quantities such as strains and strains are also improved. (2) Sensitivity studies of the flexible boundary in linear analysis show that: (a) for normal conditions (9 kip wheel load and 100 psi tire pressure), the optimal location of the flexible boundary is about 3 feet from the top of the roabed soil; (b). when the tire pressure is increased to 120 psi, the optimal location of the flexible boundary is still about 3 feet from the top of the roadbed soil; and 196 197 (c). When the wheel load is increased to 12 kips, the optimal location of the flexible boundary is about 7 feet from the top of the roadbed soil. Qualitatively, the flexible boundary should be placed deeper when the wheel load is large. (3) Sensitivity studies of the flexible boundary in nonlinear analysis show that locating it at about 3 feet from the top of the roadbed soil yields the required accuracy (see Tables 4-5 and 4-6). However, It should be noted that when the wheel load was increased up to 12 kips, the HIGH-PAVE program did not converge even after 25 iterations. This indicates that this pavement section becomes significantly nonlinear at this very high load, and is too weak for practical purposes. It also indicates that the algorithm being used converges well only for moderate levels of nonlinearity and may not converge for strongly nonlinear problems. This, however, should not be a cause for concern in normal practice. (4) The axisymmetric finite element model is selected as the basic foundation for the development of a nonlinear finite element program to be implemented on personal computers. (5) The resilient modulus model is selected to characterize the nonlinearity in granular and cohesive soils. The reasons for this are: (a) the model reduces the complicated nonlinear response to a simple form and is easy to use; (b) the resilient moduli of granular materials and roadbed soil can be determined by most state highway agencies; and 198 (c) the granular materials and roadbed soil still maintain their resilient behavior under repeated loads even after the occurrence of large permanent deformations. (6) If only the FEM with the resilient modulus model is used, it will converge extremely slowly. Therefore, the Mohr-Coulomb failure criterion is applied with the resilient modulus model. The Mohr-Coulomb failure criterion is used to modify the principal stresses in each element in the granular layers and roadbed soil after each iteration. (7) In order to minimize the influence of boundary interactions, the equivalent modulus for the halfspace below flexible boundary is calculated by averaging the resilient moduli of all bottom elements except for the three elements which are closest to the right vertical boundary. The equivalent modulus approximately accounts for the displacements of the halfspace. (8) The MICH-PAVE program automatically generates the finite element mesh along both the radial and vertical directions. The default mesh may, however, be changed by the user. (9) The MICH-PAVE program includes the gravity stress arising from the weight of the materials and approximately accounts for the lateral "locked in" stress due to compaction. (10) The MICH-PAVE program uses extrapolation to improve the radial and tangential stresses, and vertical strains at layer boundaries, and uses interpolation to improve the vertical, and shear stresses at layer boundaries (see Section 4.4.4). (11) An equivalent resilient modulus for each layer is obtained as the average of the moduli of the finite elements in the layer that lie 199 within an assumed 2:1 load distribution zone. These equivalent moduli may be used in any analyses utilizing linear elastic layer programs. (12) Comparison of results from the MICH-PAVE and ILLI-PAVE programs indicate that the stresses obtained from both programs are very close. However, the displacements from MICH-PAVE are about 12% larger than those from ILLI-PAVE. Based on exact solutions from linear analysis, the flexible boundary used in MICH-PAVE is expected to give better deflection estimates than the deep fixed boundary used in ILLI-PAVE. (13) The fatigue life (Eq.6-3) and rut depth (Eq.6-4) equations offer a preliminary concept of how empirical equations may be used together with the FEM to design flexible pavements. At the present time, these equations have only been calibrated on three and four layer pavement sections consisting of an asphalt concrete upper layer, a granular middle layer (base, or base with subbase), and a cohesive roadbed soil bottom layer. It may not be accurate to use these equations for different pavement sections. (14) The dominant factor in the fatigue life equation (Eq.6-3) of flexible pavements is the surface deflection under the wheel load rather than the tensile strain at the bottom of the AC. The surface deflection under the wheel load is an intrinsic function of all input parameters, such as the thickness of the AC, air voids in the AC, thickness of the granular layer, modulus of the granular layer, modulus of the roadbed soil, wheel load, and tire pressure. Of these, the thickness of the AC and the modulus of the roadbed soil are the two dominant factors that affect the surface deflection. However, in the rut depth equation (Eq.6- 3), the average annual temperature is the dominant factor. 200 (15). When a thin AC layer is used in a pavement section, a compressive radial strain may occur at the bottom of the AC rather than a tensile radial strain. This is contrary to the normal design assumption. For such pavements, it is recommended that a response measure such as the surface deflection be used, instead of the fatigue life, for designing the pavement. (16). The tensile strain failure model (Eq. 6-5) used to predict the equivalent wheel load factor (EWLF) cannot be applied for the pavement with a very thin AC layer in which the radial strain at the bottom of the AC may be compressive or only marginally tensile. However, a reasonable estimate of the EWLF can still be obtained by using the surface deflection of the pavement section instead of the tensile strain at the bottom of the AC (see Eq. 6-6). 7.2 ; RECOMMENDATIONS FOR FUTURE RESEARCH Based on the results of this research, the following recommendations for future research are suggested. (1) At the present time, the MICH-PAVE program considers only the linear response of the asphalt concrete. In reality, asphalt concrete is a nonlinear viscoplastic material. It is very difficult to simulate all of the complex behavior of asphalt concrete. However, the viscoelastic model can be used to closely simulate the response of asphalt concrete. Also, temperature is a very important factor affecting the behavior asphalt concrete, but it is not intrinsically considered in the current version of MICH-PAVE. Therefore, it is recommended that the viscoelastic model and a temperature sensitive description of material properties for 201 the asphalt concrete be incorporated in the future. (2) The major limitations of the axisymmetric FE model are that it cannot represent the multiple wheel loads and consider the edge effects. Only three-dimensional FE analysis can fully represent these effects. However, a three-dimensional analysis requires large amounts of memory and computational time. If the concept of the flexible boundary is used with the new generation of engineering workstations that are under development, then it may be possible to make available a three- dimensional analysis program for daily use by design engineers. (3) The shear and volumetric stress-strain relationship (also called the coutour model) is a more accurate material model for granular layers and roadbed soils, because it considers the stress path dependent responses of these materials. However, most state highway agencies do not presently own the sophisticated equipment required to estimate the material constants required for such a model. When suitable equipment becomes available, more sophisticated material models may be used in the analysis. (4) Both the fatigue life and rut depth equations need to be improved considerably when additional field data becomes available. At the present time, equations 6-3 and 6-4 are strictly applicable to three or four layer pavement sections consisting of an asphalt concrete upper layer, a granular middle layer (base, or base with subbase), and a cohesive roadbed soil bottom layer. It is desirable to have fatigue life and rut depth equations that can be applied on more general pavement sections. Futher research is required to establish these. APPENDIX OUTLINE OF COMPUTER PROGRAM 202 The MICH-PAVE program is written in the FORTRAN language and includes several source files. The source files are written for version 4 of the Microsoft FORTRAN compiler. The graphics part uses the GRAFMATIC package available from Microcompatibles, Inc., 301 Prelude Drive, Silver Spring, Maryland 20901, Phone: (301) 593-0683. The program is made up of two parts. The first contains subroutines for user- friendly screen manipulations, and the second contains subroutines for the finite element analysis. For convenience, a "make" file named MICHPAVE.MAK has been prepared to compile each source file into an object file, and to link all object files together into an executable file (MICHPAVE.EXE). The function of each subroutine is explained in the following section. Subroutines Function 1. THE MAIN Call appropriate subroutines to perform the following PROGRAM functions: 1. Show the overview screen; 2. Create a new data file; 3. Change a current data file; 4. Modified an existing data file; 5. Perform analysis; 6. Type summary results on screen; 7. Plot results on screen; 8. Print results on printer; 9. Exit-return to DOS. 2. 3. 4. 10. ll. 12. 13. 14. 15. 16. FORM SCREAD CTGRY . DRAWMU . CLEAR . ERROR . BOX . CHKFIL CHKDAT CHKLAY INIMU2 OPENFI OPENPT TOPSCR MENU 203 Move cursor to the field IFLD and display alphanumeric keys. It also determines the next field to move to if a cursor movement key is pressed. Read a variable of length LNTH located at screen position (ICOL,IROW). Determine the category of a single keyboard entry. Functions - l; Cursor movement (except left and right) - 2; Numbers (also ., +, -, e and E) = 3; Letters - 4; Ins, Del, Backspace, Cursor left and right = 5; Others - 0. Draw boxes and write texts for the called menus. Clear the screen and color it blue. Display an error message on the bottom screen line. Draw a single or double line box. Check for the existence of an input/output file. Check the crucial part of the input data before analysis and plotting. Check the input sequence. Initialize all input or calculated data. Check whether or not input file is opened. Check whether the data file required for plotting figures on the screen exists. Show the initial screen. Generate the main menu. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. HELP PLOTMS INIDA INIDAl INIDAZ WRINIDA FALIFE FTLFZ WRFL FTSUM MATTYP MATPl MATP2 WRMAT MATPRP 204 Create the screen showing an overview of the program. Draw the finite element mesh on the screen. Generate the screens for the required initial data, such as input and output filenames, wheel load, etc., and/or subcreen for the fatigue life and rut depth data. Define the input form for the initial data, such as wheel load, tire pressure, etc.. Read the initial data from INIDAl. Write the initial data, such as wheel load, tire pressure, etc., on the screen. Define the input form for fatigue life and rut depth data. Read the fatigue life and rut depth data from FALIFE. Write the data of fatigue life and rut depth on the screen. Generate the screen for the results of fatigue life and rut depth. Define the layer type for each layer. Define the input form for the layer type. Read the layer type from MATPl. Write the layer type for each layer on the screen. Define up to three material properties as outlined below 1. Subroutine ELSPRP for linear elastic material (generally asphalt concrete); 2. Subroutine GRAPRP for granular materials (base and subbase); 3. Subroutine COHPRP for cohesive materials (roadbed soils). 32. 33. 34. 35. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 50. COHPRP COHlPRP WRCOH GRAPRP GRAlPRP WRGRA RDELPR ELSPRP WRELS CROSSVH WRCRVH CROSV CROSVl WRCROV CROSH CROSHl 205 Define the input form for properties of cohesive materials. Read properties of cohesive materials from COHPRP. Write properties of cohesive materials on the screen. Define the input form for properties of granular materials. Read properties of granular materials from GRAPRP. Write properties of granular materials on the screen. Read the linear elastic material properties from ELSPRP. Define the input form for linear elastic material properties. Write properties of linear elastic material on the screen. Generate the menu for the number and locations of required horizontal and vertical sections. Write the number of horizontal and vertical sections on the screen. Define the input form for the locations of required vertical sections. Read the locations of required vertical sections from CROSV. Write the locations of the vertical sections on the screen. Define the input form for the locations of required horizontal sections. Read the locations of required horizontal sections from CROSH. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. WRCROH CROSS CROl OPTIMAL MODFM MOFMV MOFMVl WRMFMV MOFMH MOFMHl WRMFMH DEFMESH MICHPAVE GENTN 206 Write the locations of the horizontal sections on the screen. Define the input form for the number of required horizontal and vertical sections. Read the number of horizontal and vertical sections from CROSS. Find the optimal locations for the required vertical and horizontal sections. Modify the finite element mesh in the vertical and horizontal directions. Define the locations and number of elements in the vertical direction. Read the number of elements in the vertical direction from MOFMV. Write the number of elements in the vertical direction on the screen. Define the locations and number of elements in the horizontal direction. Read the number of elements in the horizontal direction from MOFMH. Write the number of elements in the horizontal direction on the screen. Define the default values of the finite element mesh. This is main computation part which reads and writes all necessary data, and performs the analysis. Calculate the total nodal points, elements, and boundary nodal points of the finite element mesh. 65. EDGELD 66. MATERIAL 67. GRAVITY 68. YGMl 69. YGM2 70. AUTOGEN 71. BAND 72. PRCALC 73. SPRING 74. SMATINV 75. SPF 207 Convert the wheel load to the equivalent nodal forces. Find the initial elastic moduli and gravity stresses for every element. Calculate the gravity stresses of every element in the asphalt concrete layer. Calculate the gravity stresses and initial elastic moduli of every element in granular layers. Calculate the gravity stresses and initial elastic moduli of every element in roadbed soils. The initial elastic modulus of each element is set equal to the value of K2. FInd the nodes [LNODS(I,J)], nodal coordinates [COORD(I,J)], boundary nodes [NOFIX(I)], boundary conditions [IFPRE(I,J); 0 - free, 1 = fixed], and equation number of every nodes in the FE mesh. Calculate the bandwidth of the global stiffness matrix. Calculate Poisson's ratio for every element. Compute the flexibility and stiffness matrices for a half- space loaded with a finite number of ring loads. The rings are approximated by a thin annulus. Loading on tributary area is used for diagonal terms. The first and last nodes have only a vertical d.o.f., while all other nodes have both radial and vertical d.o.f. Invert a real symmetric matrix. Subroutine is adapted from "Computer Program for Filtering and Spectral Analysis," by M.T. Silvia and E.A. Robinson, Elsevier, 1979, pp. 190. Calculate the nodal forces of the flexible boundary, and store those values in the global force vector. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. DCALC SHAPE QUAD4 ASSEM ADD GAUSSEL STRMDF AVERE CHECMR RMl 208 Calculate the material matrix D(4,4). Calculate the shape functions and their derivatives, and Jacobian matrix, its inverse and determinant, and [B] matrix in the r-z coordinates. Generate the element stiffness matrix [k], and store it in the array SE. Assemble the local element stiffness matrices into the global stiffness matrix [S], and the local force matrices [ELOAD] into the global force vector {RRG}. Store the global stiffness matrix in the banded form. Add the flexible boundary stiffness matrix [SPSTIFF] into the global stiffness matrix. Use Gauss elimination to solve the stiffness equations, and then find the displacement vector {RC}. Add the gravity stress to the vertical stress, and add the lateral stress to the radial and tangential stresses. Find the equivalent modulus for the halfspace below the flexible boundary. The modulus is calculated by averaging the resilient moduli of all bottom elements except for the three elements closest to the right vertical boundary. Check that the calculated principal stresses don‘t exceed the Mohr-Coulomb failure criterion. Calculate the resilient moduli of all elements in the granular layers. Calculate the resilient moduli of all elements in the roadbed soils. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. FORCE RECSTR PRST STRl STR2 EXTRAPO INTERPO STRCAL EAVER PLINE FATIGUE 209 Calculate the strain vector {STN}, stress vector {STRS}, and nodal force vector {F} for all elements, and calculate the convergence error. Recover the global stresses which are used in subroutines FORCE, STRl, and STR2, from the modified principal stresses. Calculate the principal stresses and the principal directions in one element. Calculate the displacements, stresses, and strains at the middle point of elements for the selected horizontal sections. Calculate the displacements, stresses, and strains at the middle point of elements and at layer boundaries for the selected vertical sections. Use extrapolation to modify the radial and tangential stresses, and vertical strains at interfaces. Use interpolation to modify the vertical and shear stresses. Calculate the displacements, stresses, and strains at a specific point in the selected vertical sections. Use the 2:1 load distribution zone to calculate the equivalent resilient moduli of granular layers and roadbed soils. Draw a horizontal line in the output file. Use the empirical equations to calculate the fatigue life and rut depth of the design pavement section. 98. PLOTVH 99. PLOTV 100. PLOTH lOl. PLTVERT 102. PLTHORI 103. PLTHSD 104. STRSSPLT 105. AUTOSCL 210 Generate the screen menu for plotting the results of required vertical and horizontal sections. Generate the screen submenu for plotting the stresses, strains, and displacements along the vertical sections. Generate the screen submenu for plotting the stresses, strains, and displacements along the horizontal sections. Read the stresses, strains, and displacements along the vertical sections from data files. Read the stresses, strains, and displacements along the horizontal sections from data files. Plot the stresses, strains, and displacements along the horizontal sections. Plot the displacements, stresses, and strains along the selected vertical sections. Determine the starting and ending values to be used in subroutines STRSSPLT and PLTHSD. 211 List of References American Association of State Highway Officials, " AASHO Intermin Guide For Design of Pavement Structures 1972," AASHO, AASHO committee on Design, Wahington, D. C., 1972. American Association of State, Highway, and Transportation Officials, "AASHTO Guide for Design of Pavement Structures," AASHTO, Washington, D. C., 1986. American Association of State, Highway, and Transportation Officials, "DNPS86, a Computer Program to Perform the AASHTO Design Procedure, 1986," AASHTO, Wahington,D.C., 1986. Baladi, G.Y., "In-Service Performance of Flexible Highway Pavement," International Air Transportation Conference, Vol. 1, ASCE, 1979, PP. 16- 32. 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Brown, S.F., and Pappin, J.W., ”Analysis of Pavements with Granular Bases", Transportation Research Record 810, 1981, PP. l7-23. Chen, W.F. and Saleeb, A.F., "Constitute Equations for Engineering Materials," John Wiley & Sons, Inc., Vol. 1, New York, 1982, PP. 516- 525. 212 Chou, Y.T., "An Interative Layered Elastic Computer Program for Rational Pavement Design," Federal Aviation Administration, FAA-RD-75-226, Feburary 1976. Claessen, A.I.M., Edwards, J.M., Sommer, P., and Uge, P., "Asphalt Pavement Design, The Sell Method," Proceedings of Fourth International Conference on the Structural Design of Asphalt Pavements, Vol. 1, University of Michigan, August 1977. Cook, R.D., "Concepts and Applications of Finite Element Analysis," John Wiley & Sons, Inc., Second Ed., New York, 1981. PP. 29-45, 113-124. Corps of Engineers, ”Revised Method of Thickness Design for Flexible Highway Pavements at Military Installations," Waterways Experiment Station, Technical Resport 3-582, August, 1961. Deacon, J.A., "Equivalent Passages of Aircraft with Respect to Fatigue Distress of Flexible Airfield Pavements," Proceedings, AAPT, Vol. 40, 1971. Dehlen, G.L., and Monismith, G.L., "Effect of Nonlinear Material Response on the Bahaviour of Pavements under Traffic," Highway Research Board 310, Wahington, D. C., 1970. De Jong, D.L., Peutz, M.G.F., and Korswagen, A.R., "Computer Program BISAR Layered Systems under Normal and Tangential Surface Loads," Kominklijke/Shell-Laboratorium, Amsterdam, External Report AMSR.0006.73, 1973. Duncan, J.M., Monismith, G.L., and Wilson, E.L., "Finite Element Analysis of Pavements," Highway Research Record, No. 228, Highway Research Board, Washington, D. C., 1968, PP. 18-33. Duncan, J.M., and Chang, G.Y., "Nonlinear Analysis of Stress and Strain in Soils," Journal of the Soil Mechanics and Foundation, Proceeding of the American Society of Civil Engineers, September 1970, PP. 1629-1653. Dysli, M., and Fontana, A., "Deformations around the Excavations in Clayey Soil," International Symposium on Numerical Mehtods in Geomechanics, September 1982, PP. 634-642. ERES Consultants, Inc. etc., "Pavement design Principal and Practices," ERES Cbnsultants, Inc., Champaign, Illinois, National Highway Institute, Washington, D. C., 1987. Harichandran, R., and Yeh, M.S., "Flexible Boundary in Finite Element Analysis of Pavements," Transportation Research Board, 67th Annual Meeting, Paper No. 870116, January 1988. Haynes, H.J., and Yoder, E.J., ”Effects of Repeated Loading on Gravel and Crushed Stone Base Course Materials Used in the AASHO Road Test," Highway Research Record, No. 39, Highway Research Board, Washington, D. C., 1963, PP. 82-96. 213 Hicks, R.G., and Monismith, G.L., "Prediction of the Resilient Response of Pavements Containing Granular Layers Using Non-Linear Elastic Theory," Proceedings of Third International Conference on the Structural Design of Asphalt Pavements, London, English, 1972. Highter, W.H., and Harr, M.E., "Predicting Pavement Performance," Journal of Transportation, ASCE, May, 1974. Huffred, W.L., and Lai, J.S., "Analysis of N-Layered Viscoelastic Pavement System," Report No. FHWA-RD-78-22, Federal Highway Administration, Washington, D. C., January 1978. Ioannides, A.M., and Donnelly, J.P., "Three-Dimensional Analysis of Slab on stress dependent Foundation," Transportation Research Board, 67th Annual Meeting, Paper No. 870092, January 1988. Janbu, N., "Soil Compressibility as Determined By Oedometer and Triaxial Tests," Proceedings European Cbnference on Soil Mechanics and Foundation Engineering, Wiesbaden, Germany, Vol. 1, 1963, PP. 19-25. Ko, H.Y., and Masson, R.M., "Nonlinear Characterization and Analysis of Sand," Numberical Methods in Geomechanics, Vol. 1, Edited by Desai, C. S., 1976. PP. 294-305. Kopperman, S., Tiller, G., and Tseng, M.T., "ELSYMS Interactive Microcomputer Version, User's Manual, IBM-PC and Compatible Version," FHWA, Final Report, Contract No. DTFH61-8S-C-00051, September, 1985. Kujawski, J., and Wibery, N.E., "Thick Plates on Multi-Layered Soil, a semianalytical 3D FEM - Solution," International on Numerical Models in Geomechanical, Zurich, September 1982, PP. 693-702. National Crushed Stone Association, ”Flexible Pavement Design Guide for Highways", NCSA Publication, Washington, D. C., 1972. Poulos, H.S., and Davis, E.H., "Elastic Solutions for Soil and Rock Mechanics," Wiley, New York, 1974. Raad, L., and Figueroa, J.L., "Load Response of Transportation Support System," Transportation Engineering Journey, ASCE, Vol. 106,1980, PP. 111-128. Seed, H.B., Chan, C.K., and Lee, C.E., "Resilience Characteristics of subgrade Soils and Their Relation to Fatigue Failures in Asphalt Pavements," Proceedings, First International Conference on the Structural Design of Asphalt Pavements, Ann Arbor, August 1962, PP. 77- 113. Soussou, T.E., Moavenzadeh, F., and Findakly, H.K., "Synthesis of Rational Design of Flexible Pavements," Civil Engineering Dept. Report, Massachusetts Institute of Technology, January 1973. 214 Swami, S.A., , Goetz, W.H., and Herr, M.E., "Time and Load Independent Properties of Bituminous Mixtures,” Highway Research Board 313, 1970, PP. 63-78. The Asphalt Institute, ”Thickness Design-Asphalt Pavements for Highways and Streets," The Asphalt Institute, Manual Series No. l (MS-1), September 1981. The Asphalt Institute, "Computer Program DAMA: User's Manuel," The Asphalt Institute, CP-l, October, 1983. Thompson, M.R., "ILLI-PAVE, User's Manual," Transportation Facilities Group, Dept. of Civi Engineering, university of Illinois, November 1986. Witczak, M. W., "Prediction of Eqvivalent Damage Repetitions from Aircraft Traffic Mixtures for Full Depth Airfield Pavements," Proceedings", AAPT, Vol. 42, 1973. Yoder, E.J., and Witczak, M.W., "Principal of Pavement Design," John Wiley and Sons, Inc., 2nd. Edition, New York, 1975. Young, M.A., and Baladi, G.Y., "Repeated Load Triaxial Testing, State of the Art," Michigan State University, March 1977.