. .u 9. x 4 III). ’ 4 .4 Mn. N .9344!“ mi- ‘3 . LEM... .4 l. . . n. I34: v.33... a. c . .. . v.8 3 3;.) . va...:.l.J. 0 I‘IIS‘II .0: vtrlrvl asp - v V“ II «.....v-&ur.l" .11 .l IIIIVQQOIOIW c.1VplbW'1lunl «)tL. a. l .11-.- . - n juofhflryfff writ, ....v....io. £225.... .-7«.! AA! A IFIEL In! 18:31:: 1) 3. '3) Date 0-7639 HHHHHH Illllllllllllllllllllllllllllllllllllllllllllllllllll 3 1293 01694 This is to certify that the thesis entitled TECHNIQUES FOR IMPROVEMENT OF THE AXISYMMETRIC FINITE ELEMENT MODEL USED IN MICHPAVE presented by John C. Anderson has been accepted towards fulfillment of the requirements for Master of Science degree in Civil Engineering fl/M t/~_/._ Major professor Il/s'l/ag MS U is an Affirmative Action/Equal Opportunity Institution LIBRARY Mlchlgan State ' Unlverslty PLACE IN RETU to remove this checkout RN BOX from your record. before date due. TO AVOID FINES return on or DATE DUE DATE DUE DATE DUE .1 _ an.» \gfiw%% Lou... ______.__.__—— / ________.__——— / /—.—— _____._.__._—- _____._._._——— _____,.____——— _,______._.._—— / _____,._._.———— / ma worming-m4 TECHNIQUES FOR IMPROVEMENT OF THE AXISYMMETRIC FINITE ELEMENT MODEL USED IN MICHPAVE By John C. Anderson Submitted to Michigan State University in partial firlfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil and Environmental Engineering 1996 ABSTRACT TECHNIQUES FOR INIPROVEMENT OF THE AXISYMMETRIC F INITE ELEMENT MODEL USED IN MICHPAVE By John C. Anderson An attempt is made to reduce the errors arising from axisymmetric finite element modeling of flexible pavements using MICHPAVE. Three improvement techniques are implemented. First, the use of infinite elements to model the roadbed soil in place of the flexible boundary in MICHPAVE is investigated. Second, the width of the finite element mesh is increased by extending the distance between the load and the lateral boundary. Third, the aspect ratios of the finite elements are improved by increasing the number of elements used in the finite element mesh. Although infinite elements have the potential to yield good results with a coarse mesh, calibration requirements introduce an undesirable feature into the analysis. Further, when infinite elements are used, computed displacements do not improve uniformly as the mesh is refined. On the other hand, when the flexible boundary is used computed displacements do improve uniformly as the mesh is refined. Hence, if the available computer memory is sufficient for a refined mesh, use of the flexible boundary is recommended. ACKNOWLEDGMENTS The author would like to express his deepest gratitude to his graduate advisor Professor Ronald Harichandran for his guidance, encouragement, and infinite patience throughout this research. Gratitude is also expressed to the Master’s Committee members, Professors Gilbert Baladi and Frank Hatfield for their useful comments. The author expresses special thanks to his mother and father for encouraging him to finish school. Finally, the author gratefully acknowledges the financial support provided by the Michigan Department of Transportation for this research. TABLE OF CONTENTS LIST OF TABLES ........................................................................................................ vii LIST OF FIGURES ........................................................................................................ ix CHAPTER 1 INTRODUCTION ............................................................................................... 1 2 LITERATURE REVEIW ..................................................................................... 3 2.1 GENERAL ................................................................................................... 3 2.2 DESIGN METHODS FOR FLEXIBLE PAVEMENT STRUCTURES ........ 4 2.2.1 Empirical Methods ................................................................................... 5 2.2.2 Mechanistic-Empirical Methods ................................................................ 6 2.3 PAVEMENT ANALYSIS PROGRAMS CAPABLE OF NONLINEAR ANALYSIS .................................................................................................. 7 2.3.1 KENLAYER ............................................................................................ 7 2.3.2 Commercial Multipurpose Finite Element Programs .................................. 8 2.3.3 MICHPAVE ............................................................................................ 9 3 INFINITE ELEMENTS ..................................................................................... 12 3.1 GENERAL ................................................................................................. 12 3.2 AXISYMMETRIC INFINITE ELEMENTS ............................................... 15 3.2.1 Formulation of Radial Axisymmetric Infinite Element Stiffness Matrix 15 iv V 3.2.2 Formulation of Vertical Axisymmetric Infinite Element Stifi‘ness Matrix. 23 3.2.3 Coupling of Infinite Elements to Finite Elements ..................................... 24 3.3 COMPARISON OF ANALYSIS RESULTS .............................................. 25 3.3.1 Problem Description ............................................................................... 25 3.3.2 The Depth Ratio ...................................................................... i ............... 27 3.3.3 Comparison of Deflections ..................................................................... 29 3.3.4 Comparison of Strains ............................................................................ 34 3.3.5 Comparison of Stresses .......................................................................... 35 3.4 SUMMARY ............................................................................................... 41 4 MESH REFINEMENTS .................................................................................... 42 4.1 GENERAL ................................................................................................. 42 4.2 MESH CHANGES WITH A FLEXIBLE BOUNDARY ............................. 44 4.2.1 Deflections ............................................................................................. 44 4.2.2 Strains .................................................................................................... 63 4.2.2.1 Radial Strain at the Bottom of the Asphalt Layer ............................... 63 4.2.2.2 Vertical Strain at the Top of the Subgrade ........................................ 63 4.2.3 Stresses .................................................................................................. 74 4.2.3.1 Radial Stress at the Bottom of the Asphalt Layer .............................. 74 . 4.2.3.2 Vertical Stress at the Top of the Subgrade ........................................ 74 4.2.4 Conclusions ............................................................................................ 82 4.3 MESH CHANGES WITH INFINITE ELEMENTS .................................... 83 4.3.1 The Depth Ratio ..................................................................................... 83 4.3.2 Deflections ............................................................................................. 83 vi 4.3.3 Strains .................................................................................................... 91 4.3.3.1 Radial Strain at the Bottom of the Asphalt Layer .............................. 91 4.3.3.2 Vertical Strains at the Top of the Subgrade ....................................... 91 4.3.4 Stresses ................................................................................................ 105 4.3.4.1 Radial Stress at the Bottom of the Asphalt Layer ............................ 105 4.3.4.2 Vertical Stress at the Top of the Subgrade ...................................... 105 4.3.5 Conclusions .......................................................................................... 113 4.4 COMPARISON OF MPFB AND MPIE VERTICAL WITH REFINED MESHES .................................................................................................. 113 4.5 SUMMARY AND CONCLUSIONS ........................................................ 117 5 CONCLUSIONS AND RECOMIVIENDATIONS ............................................ 123 LIST OF REFRENCES ............................................................................................... 125 LIST OF TABLES TABLE 3-1 : Deflections Found Using Chevronx, MPFB, MPIE Vertical, and MPIE Radial (Lateral Boundary at 10a, Standard Mesh Size) ....................................... 30 3-2 : Strain Values Found Using Chevronx, MPFB, MPIE Vertical, and MPIE Radial (Lateral Boundary at 10a, Standard Mesh Size) ....................................... 35 3-3 Stress Values Found Using Chevronx, MPFB, MPIE Vertical, and MPIE Radial (Lateral Boundary at 10a, Standard Mesh Size) ....................................... 38 4-1a: Number of Elements in the Horizontal Direction for Test Pavement Sections ...... 45 4-lb: Number of Elements in the Vertical Direction for Test Pavement Sections .......... 46 4-1c: Total Number of Finite Elements for Test Pavement Sections ............................. 46 4-2 Deflections Under the Wheel Load for Varying Meshes Found Using MPFB ...... 50 4-3 Values of Radial Strain at the Bottom of the Asphalt Layer Under the Wheel Load Found Using MPFB .................................................................................. 67 4-4 : Values of Vertical Strain at the Top of the Subgrade Under the Wheel Load Found Using MPFB .................................................................................. 67 4-5 : Values of Radial Stress at the Bottom of the Asphalt Layer Under the Wheel Load Found Using MPFB .................................................................................. 75 4-6 : Values of Vertical Stress at the Top of the Subgrade Under the Wheel Load Found Using MPFB .................................................................................. 75 4-7 : Computational Time for Test Pavement Sections ................................................ 82 4-8 : Values of Radial Strain at the Bottom of the Asphalt Layer Under the Wheel Load Found Using MPIE Vertical ...................................................................... 98 vii 4-9 : 4—10: viii Values of Vertical Strain at the Top of the Subgrade Under the Wheel Load Found Using MPIE Vertical ...................................................................... 98 Values of Radial Stress at the Bottom of the Asphalt Layer Under the Wheel Load Found Using MPIE Vertical .................................................................... 106 Values of Vertical Stress at the Top of the Subgrade Under the Wheel Load Found Using MPIE Vertical .................................................................... 106 LIST OF FIGURES FIGURE 3-1a Infinite Elements in rz-Space ........................................................................... l3 3-lb Square Element in Err-Space ............................................................................ 13 3-2a Radial Infinite Elements ................................................................................... 14 3-2b Vertical Infinite Elements ................................................................................ 14 3-3 A Typical Axisymmetric Infinite Element ......................................................... 18 3-4 The Six-Node Isoparametric Element .............................................................. 18 3-5 Pavement Section Used in Analysis ................................................................. 26 3-6 Alpha Values for Pavements wit Different Asphalt Thickness .......................... 29 3-7a Deflection Basins; 1” AC Thickness ................................................................ 31 3-7b Deflection Basins; 5” AC Thickness ................................................................ 32 3-7c Deflection Basins; 10” AC Thickness .............................................................. 33 3-8 Comparison of Radial Strain at Bottom of Asphalt Layer ................................. 36 3-9 Comparison of Vertical Strain at Top of Subgrade ........................................... 37 3-10 Comparison of Radial Stress at Bottom of Asphalt Layer ................................ 39 3-11 Comparison of Vertical Stress at Top of Subgrade .......................................... 4O 4-1a Mesh 1 with a Lateral Boundary at 20a; 5 in. Asphalt Layer ............................ 47 4-1b Mesh 2 with a Lateral Boundary at 20a; 5 in. Asphalt Layer ............................ 48 iii: 4-1c 4-2a 4-21) 4-2c 4-3a 4-3b 4-3c 4-4a 4-4b 4-4c 4-5a 4-5b 4-5c 4-6a 4-6b 4-6c 4-7a X Mesh 3 with a Lateral Boundary at 20a; 5 in. Asphalt Layer ............................ 49 Pavement Deflections Under the Wheel Load Found Using MPFB, 1” Asphalt Layer ............................................................................................. 51 Pavement Deflections Under the Wheel Load Found Using MPFB, 5” Asphalt Layer ............................................................................................. 52 Pavement Deflections Under the Wheel Load Found Using MPFB, 10” Asphalt Layer ........................................................................................... 53 Deflection Basin of 1” Asphalt Layer Found Using MPFB, Lateral Boundary at 10a .................................................................................. 54 Deflection Basin of 1” Asphalt Layer Found Using MPFB, Lateral Boundary at 200 .................................................................................. 55 Deflection Basin of 1” Asphalt Layer Found Using MPFB, Lateral Boundary at 400 .................................................................................. 56 Deflection Basin of 5” Asphalt Layer Found Using MPFB, Lateral Boundary at 10a .................................................................................. 57 Deflection Basin of 5” Asphalt Layer Found Using MPFB, Lateral Boundary at 20a .................................................................................. 58 Deflection Basin of 5” Asphalt Layer Found Using MPFB, Lateral Boundary at 40a .................................................................................. 59 Deflection Basin of 10” Asphalt Layer Found Using MPFB, Lateral Boundary at 10a .................................................................................. 6O Deflection Basin of 10” Asphalt Layer Found Using MPFB, Lateral Boundary at 20a .................................................................................. 61 Deflection Basin of 10” Asphalt Layer Found Using MPFB, Lateral Boundary at 40a .................................................................................. 62 Deflection Under the Wheel Load Found Using MPFB, 1” Asphalt Layer ........ 64 Deflection Under the Wheel Load Found Using MPFB, 5” Asphalt Layer ........ 65 Deflection Under the Wheel Load Found Using MPFB, 10” Asphalt Layer ...... 66 Radial Strain at Bottom of 1” Asphalt Layer Found Using MPFB .................... 68 4-7b 4-7c \4-8a 4-8b 4-8c 4-9a 4-9b 4-9c 4-10a : 4-10b: 4-10c: 4-11a: 4-11b : 4-llc: 4-12a : 4-12b : 4-12c : 4-13a: xi Radial Strain at Bottom of 5” Asphalt Layer Found Using MPFB .................... 69 Radial Strain at Bottom of 10” Asphalt Layer Found Using MPFB .................. 70 Vertical Strain at Top of Subgrade Found Using MPFB, 1” Asphalt Layer ...... 71 Vertical Strain at Top of Subgrade Found Using MPFB, 5” Asphalt Layer ...... 72 Vertical Strain at Top of Subgrade Found Using MPFB, 10” Asphalt Layer 73 Radial Stress at Bottom of 1” Asphalt Layer Found Using MPFB .................... 76 Radial Stress at Bottom of 5” Asphalt Layer Found Using MPFB .................... 77 Radial Stress at Bottom of 10” Asphalt Layer Found Using MPFB .................. 78 Vertical Stress at Top of Subgrade Found Using MPFB, 1” Asphalt Layer ...... 79 Vertical Stress at Top of Subgrade Found Using MPFB, 5” Asphalt Layer ...... 80 Vertical Stress at Top of Subgrade Found Using MPFB, 10” Asphalt Layer 81 Effect of Mesh Fineness on Alpha Values of Pavements with Different Asphalt Thicknesses; Lateral Boundary at 10a ................................................. 84 Effect of Mesh Fineness on Alpha Values of Pavements with Different Asphalt Thicknesses; Lateral Boundary at 20a ................................................. 84 Effect of Mesh Fineness on Alpha Values of Pavements with Different Asphalt Thicknesses; Lateral Boundary at 400 ................................................. 84 Pavement Deflections Under the Wheel Load Found Using MPIE Vertical, 1” Asphalt Layer ............................................................................................. 85 Pavement Deflections Under the Wheel Load Found Using MPIE Vertical, 5” Asphalt Layer ............................................................................................. 86 Pavement Deflections Under the Wheel Load Found Using MPIE Vertical, 10” Asphalt Layer ........................................................................................... 87 Deflection Basin of l” Asphalt Layer Found Using MPIE Vertical, Lateral Boundary at 10a .................................................................................. 88 4-13b: 4-l3c: 4-14a : 4-14b : 4-14c : 4-15a: 4-15b: 4-15c: 4-16a: 4-16b: 4-16c : 4-17a: 4-17b: 4-17c : 4-18a: 4-18b: 4-18c : xii Deflection Basin of l” Asphalt Layer Found Using MPIE Vertical, Lateral Boundary at 20a .................................................................................. 89 Deflection Basin of l” Asphalt Layer Found Using MPIE Vertical, Lateral Boundary at 40a .................................................................................. 90 Deflection Basin of 5” Asphalt Layer Found Using MPIE Vertical, Lateral Boundary at 10a .................................................................................. 92 Deflection Basin of 5” Asphalt Layer Found Using MPIE Vertical, Lateral Boundary at 20a .................................................................................. 93 Deflection Basin of 5” Asphalt Layer Found Using MPIE Vertical, Lateral Boundary at 40a .................................................................................. 94 Deflection Basin of 10” Asphalt Layer Found Using MPIE Vertical, Lateral Boundary at 10a .................................................................................. 95 Deflection Basin of 10” Asphalt Layer Found Using MPIE Vertical, Lateral Boundary at 20a .................................................................................. 96 Deflection Basin of 10” Asphalt Layer Found Using MPIE Vertical, Lateral Boundary at 400 .................................................................................. 97 Radial Strain at Bottom of 1” Asphalt Layer Found Using MPIE Vertical ........ 99 Radial Strain at Bottom of 5” Asphalt Layer Found Using MPIE Vertical ...... 100 Radial Strain at Bottom of 10” Asphalt Layer Found Using MPIE Vertical... 101 Vertical Strain at Top of Subgrade Found Using MPIE Vertical, 1” Asphalt Layer ........................................................................................... 102 Vertical Strain at Top of Subgrade Found Using MPIE Vertical, 5” Asphalt Layer ........................................................................................... 103 Vertical Strain at Top of Subgrade Found Using MPIE Vertical, 10” Asphalt Layer ......................................................................................... 104 Radial Stress at Bottom of 1” Asphalt Layer Found Using MPIE Vertical ..... 107 Radial Stress at Bottom of 5” Asphalt Layer Found Using MPIE Vertical ..... 108 Radial Stress at Bottom of 10” Asphalt Layer Found Using MPIE Vertical.... 109 4-19a : 4-19b: 4-19c: 4-20a : 4-20b : 4-20c : 4-2 1 4-22 4-23 4-24 xiii Vertical Stress at Top of Subgrade Found Using MPIE Vertical, 1” Asphalt Layer ........................................................................................... 1 10 Vertical Stress at Top of Subgrade Found Using MPIE Vertical, 5” Asphalt Layer ........................................................................................... l l 1 Vertical Stress at Top of Subgrade Found Using MPIE Vertical, 10” Asphalt Layer ......................................................................................... 112 Deflection Basins of 1” Asphalt Layer Found Using Chevronx, MPFB, MPIE Vertical, Mesh 3, Lateral Boundary at 20a .......................................... 114 Deflection Basins of 5” Asphalt Layer Found Using Chevronx, MPFB, MPIE Vertical, Mesh 3, Lateral Boundary at 20a .......................................... 1 15 Deflection Basins of 10” Asphalt Layer Found Using Chevronx, MPFB, MPIE Vertical, Mesh 2, Lateral Boundary at 40a .......................................... 116 Comparison of Radial Strains at the Bottom of the Asphalt Layer Found Using MPFB and MPIE Vertical ................................................................... 118 Comparison of Vertical Strains at the Top of the Subgrade Found Using MPFB and MPIE Vertical ................................................................... 119 Comparison of Radial Strains at the Bottom of the Asphalt Layer Found _Using MPFB and MPIE Vertical ................................................................... 120 Comparison of Vertical Strains at the Top of the Subgrade Found Using MPFB and MPIE Vertical ................................................................... 121 CHAPTER 1 INTRODUCTION In 1989, the MICHPAVE pavement analysis program was developed at Michigan State University by Harichandran, Yeh and Baladi. The goal of this program was to create a “user-fiiendly” non-linear finite element pavement analysis program that could be used on personal computers by Michigan Department of Transportation (MDOT) engineers. At that time, the MS-DOS operating system on personal computers could acess only 640 KB of memory, thus finite element meshes with a large number of degrees-of—fi'eedom could not be used to model pavement sections. This problem was overcome by placing a relatively shallow finite element mesh on a flexible boundary. This technique produced reasonable results, but significant errors were present in the values of several key pavement responses. This thesis explores techniques to reduce these errors. Chapter 2 contains a synopsis of relevant design methods for analyzing and 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 relevant mechanistic-empirical pavement analysis programs are briefly reviewed. These are MICHPAVE, KENLAYER, and a commercial multipurpose finite 2 element package named ABAQUS. The advantages and disadvantages/limitations of each program are discussed. Chapter 3 presents infinite elements as an alternative to the flexible boundary method of modeling the roadbed soil. The first part of the chapter discusses the formulation of the element stiffness matrix, and the coupling of the infinite elements to the finite elements. Two types of infinite elements are considered at the base: 1) quadrilateral elements that extend radially outward from the load, and 2) rectangular elements that extend vertically downward. The second part of the chapter compares the results found using the original version of MICHPAVE with a flexible boundary with those found using radial and vertical infinite elements. Included is a discussion of the depth ratio used to calibrate the infinite element model. Chapter 4 explores the effects of; 1) extending the radial distance between the center of the wheel load and the lateral boundary, and 2) increasing the fineness of the finite element mesh, on results found using MICHPAVE. Both a flexible boundary and vertical infinite elements are used to model the roadbed soil, and the values of several key pavement responses are compared for both methods. The final chapter, Chapter 5, presents the conclusions of the research and recommendations for firture work in this area. CHAPTER 2 LITERATURE REVIEW 2.1 GENERAL Flexible pavements are constructed of bituminous, granular, and cohesive materials. As stated by Yoder and Witczak (1975), a flexible pavement consists of a relatively thin wearing surface built over a base course, subbase course, and compacted roadbed soil. The thickness of the flexible pavement is meant to include all components of the pavement above the roadbed soil. The load-carrying capacity of flexible pavements is brought about by the load- distributing characteristics of the layered system. The layer with the highest quality materials is placed at the top of the pavement. The strength of the flexible pavement is the result of building up thick layers, thereby distributing the load over the subgrade (Yoder, et a1. 1975). The thickness design of the pavement is influenced, in part, by the strength of the subgrade. The structural design of flexible pavements has gradually evolved from art to science. Prior to the 1920’s, the thickness of a pavement was based purely on experience (Huang, 1993). The same thickness was used for a section of highway even though widely different soils were encountered. A concerted effort to address this problem was made during the 1920’s when the US. Bureau of Public Road (now known as the Federal Highway Administration) developed a soil classification system based upon field 4 observations of soil behavior under highway pavements. This system helped the pavement engineer in correlating pavement performance with roadbed soil type (Yoder, et a1. 1975). With the initiation of the National System of Interstate Highway and the Federal Aid Secondary System in 1944, pavement systems were subjected to greater loads and frequencies than ever before (Baladi and Snyder, 1989). In order to produce pavement systems to meet these needs, highway engineers introduced and implemented new empirical and mechanistic-empirical design procedures. These new approaches resulted in a better design process, although severe breakup was still a common phenomenon (Yoder, et a1. 1975). The 1960’s saw the emergence of new analytical pavement design techniques. Elastic and viscoelastic layered pavement models were developed as were finite element models. These new structural models provided the pavement engineer with a better understanding of pavement behavior and performance. With advancements in computer capabilities and an ever growing quantity of pavement performance data, pavement engineers continue to seek alternative design methods which yield faster and more accurate results. 2.2 DESIGN METHODS FOR FLEXIBLE PAVEMENT STRUCTURES Pavement failure and/or distress can be divided into two categories: structural and functional. Structural failure occurs when the pavement can no longer carry the design load. Functional failure occurs when ride quality diminishes and/or when a pavement becomes unsafe to drive on. Pavements that exhibit structural distress and/or failure will 5 also exhibit functional distress and/or failure. Pavements exhibiting functional distress and/or failure may be structurally sound. Beginning in the early 1950’s several empirical and mechanistic-empirical pavement design methods were developed. The empirical methods were developed on the basis of firnctional failure criteria, while the mechanistic-empirical methods were based upon various structural failure criteria. Efforts to improve both empirical and mechanistic- empirical design methods have focused on two areas. The first of these is a proper characterization of the paving materials. The second is based on limiting strains and deflections in the pavement sections. Further, in order to calculate the stresses, strains, and deflections of the pavement layers, several theoretical analysis methods have been developed. These include: elastic, viscoelastic, transfer function, and finite element methods. 2.2.1 Empirical Methods Empirical design methods have been conceived using performance data from existing pavement sections to develop relationships between pavement thickness and load repetitions until failure. Each method has its own philosophy and is based upon certain assumptions, experience, and criteria. Thus, it is not uncommon for different empirical methods to yield different layer thickness even when the same inputs are used. The major advantage in using empirical methods is that they tend to be simple and easy to use. The disadvantage in using empirical methods is that they can only be applied to a given set of environmental, material, and loading conditions for which they were developed. If these conditions are changed, the design is no longer valid and a new 6 method must be developed through trial and error to be commensurate with the new conditions (Huang, et al. 1993). Empirical methods also give little insight on the mechanical behavior of pavements. Three empirical design methods have been reviewed by Yeh (1989). These methods are the AASHTO, the National Stone Association (NSA), and the California methods of design. 2.2.2 Mechanistic-Empirical Methods Mechanistic-empirical methods of pavement design consist of structural analysis of the pavement system along with the incorporation of empirical distress or performance functions. Structural analysis refers to the calculation of stresses, strains, and deflections that are caused in a pavement section due to traffic, temperature, and/or moisture loads. Once these values are determined at critical locations, comparisons can be made with maximum allowable values obtained from experimental or theoretical studies based on predictions of pavement distress. Such distress might include cracking, rutting, and roughness. The pavement’s thickness can then be adjusted so that calculated stresses, strains, and deflections do not exceed maximum allowable values. Mechanistic flexible pavement design procedures make the assumption that a pavement can be modeled as a multilayered elastic or viscoelastic structure resting on a rigid, elastic, or viscoelastic foundation. If this assumption holds true, it is possible to calculate stresses, strains, and displacements due to traffic loading and environmental conditions at any point in the pavement section. Unfortunately, a pavement’s performance is influenced by many factors that cannot be precisely modeled by mechanistic methods. 7 Thus, empirical regression equations that include many parameters including strains, stresses, and/or displacements were calibrated using field pavement performance data. This is why the methods are referred to as mechanistic-empirical design procedures. The major advantage of this design philosophy is its ability to design a pavement based on more than one failure criteria and to include mechanical responses that contribute to these failures in a rational way; it allows the highway engineer to produce a durable yet economic pavement. The disadvantage, however, is that the method requires more comprehensive data than empirical methods and an appropriate computer program for the analysis. Extensive laboratory and field testing may be required to determine design parameters such as resilient moduli and creep compliance. The essential ingredients of mechanistic-empirical methods have been reviewed by Yeh (1989). These include elastic layer analysis programs, the shell method, the Asphalt Institute design, the VESYS II computer program, and nonlinear finite element programs. 2.3 PAVEMENT ANALYSIS PROGRAMS CAPABLE OF NONLINEAR ANALYSIS 2.3.1 KENLAYER KENLAYER is a nonlinear elastic layer pavement analysis/design program developed at the University of Kentucky by Huang. The program was explicitly designed for analysis of flexible pavements with no joints or rigid layers. The KENLAYER program can be applied to a multilayered pavement system under stationary or moving wheel loads with each layer being either linear elastic, nonlinear elastic or viscoelastic. For linear analysis, dual, dual-tandem or dual-tridem axle loads can be modeled in addition to a single wheel load. Damage analysis can be conducted by dividing each year into a maximum of 24 periods, each with a different set of material properties. Each period can have a maximum of 24 load groups, either single or multiple. The damage caused by fatigue cracking and permanent deformation in each period over all load groups is summed up to evaluate the design life. KENLAYER has one significant limitation. The use of the stresses at a single point in each nonlinear layer to compute the modulus of the layer is not theoretically correct. Since the stresses vary in the radial direction, the modulus should also change in the radial direction, but KENLAYER is unable to accommodate this. Another minor limitation is that KENLAYER relies on an external data input file and is not interactive. This makes the analysis process less “user-friendly”. 2.3.2 Commercial Multipurpose Finite Element Programs A study was done at the University of Texas at Austin using a commercial multipurpose finite element package termed ABAQUS (Cho, McCullough and Weissman, 1996). Three finite element approaches were applied to model multilayer pavement structures; plane strain, axisymmetric, and three dimensional. The plane strain model was inadequatedue to its inability to correctly represent a single wheel load. The axisymmetric model produced reasonably accurate solutions when infinite elements were used to model the roadbed soil and the infinite horizontal extent of layers. Considering the current limitations on computational resources (speed and memory) and current design methods that are based on single axle effects, the axisymmetric modeling approach is a very good alternative for modeling a pavement structure using the finite element method (Cho, et a1. 9 1996). The three dimensional model yielded a reasonable approximation when geometry and boundary conditions were well controlled. 2.3.3 MICHPAVE In 1989, a nonlinear finite element analysis program was developed at Michigan State University by Harichandran, Yeh, and Baladi. The program, named MICHPAVE, used an axisymmetric finite element mesh and a circular wheel load. Three major achievements were associated with MICHPAVE. First, the program introduced the concept of a flexible boundary to model the subgrade. Second, an extremely “user- friendly” nonlinear finite element program for pavement analysis and design was created. Third, two empirical equations to predict fatigue life and rut depth were developed for use with the nonlinear finite element analysis. A major advantage of the MICHPAVE pavement analysis program is its use of a flexible boundary. The subgrade below the flexible boundary is considered as a homogenous half-space, whose stifihess matrix can be determined and superimposed to the stiffness matrix of the pavement above the flexible boundary to form the overall stiffness matrix. The use of a flexible boundary at a limited depth beneath the surface of the subgrade, instead of a rigid boundary at a large depth below the surface of subgrade, greatly reduces the number of finite elements required. Consequently, the storage requirement is significantly reduced and the program can be implemented on personal computers. The use of a flexible boundary also reduces the number of oblong elements at the bottom of the mesh. This, coupled with the decrease in simultaneous equations to be solved, helps to improve the accuracy of the MICHPAVE program. 10 Although the MICHPAVE program produces reasonable results, there are some concerns on its accuracy (Huang, et al. 1993). If the pavement is considered to be linear elastic, the responses obtained by NHCHPAVE and those obtained by a linear elastic layer program, such as ELSYMS, should match very closely. Huang found that this was not the case. The two programs were compared using the following pavement responses: vertical deflection on the surface, the four components of strain at the bottom of the asphalt layer, and the same strains at the top of the subgrade. Comparisons were made at the axis of symmetry as well as at a distance of 0.7 inches from the axis which is the center of the first column of finite elements. The results of the comparison indicated that the surface deflections predicted by MICHPAVE were only 90% of those predicted by ELSYMS. The compressive strains at the top of the subgrade obtained by MICHPAVE were only 80% of those obtained by ELSYMS. When the asphalt layer thickness is greater than 2.5 inches, the tensile strains at the bottom of the asphalt layer agree reasonably well. However, when the asphalt layer thickness is less than 2.5 inches the tensile strains obtained by MICHPAVE are as much as 10% greater than those obtained by ELSYMS. There are several possibilities that may cause inaccurate results in the finite element method (Huang, et al. 1993): 1. The shape of finite elements has a significant effect on the accuracy. The use of oblong elements can yield inaccurate results. ll 2. The stresses and strains are evaluated most accurately at the center of elements. Evaluations at element boundaries, particularly at element comers, are more prone 10 error. 3. The stresses at layer interfaces are obtained by linear extrapolation from those the center of the two elements below or above the interface. This procedure can lead to error if the mesh is not fine enough. CHAPTER 3 INFINITE ELEMENTS 3.1 GENERAL One problem encountered when modeling pavement sections with finite elements is that of effectively modeling the roadbed soil. In the pavement system, the roadbed soil is typically an unbounded medium. Thus, in order to adequately model the pavement the finite element mesh would have to extend to a great depth. Often, simple truncation at a sufficiently deep rigid boundary is used to represent the roadbed soil. However, the depth at which the rigid boundary should be placed is problem dependent, and the analysis may be expensive, in terms of time and memory, because many elements will be required. MICHPAVE dealt with this problem by implementing a flexible boundary (Harichandran and Yeh, 1988). Finite elements were used to model the roadbed soil in the vicinity of the loaded area. The bottom boundary was placed at a depth below which displacements and stresses were not of interest. This bottom boundary was assumed to be flexible, and the half-space below the boundary was assumed to be homogeneous. Displacements that occur in the soil below the boundary were therefore considered in the analysis. An alternative to the flexible boundary is the use of infinite elements. Pioneered by Bettess (1981), the use of infinite elements involves mapping an element which stretches to infinity onto a plane bilinear isoparametric element as shown in Figure (3-1). A series 12 13 of these elements can then be coupled to the original finite element mesh. In the case of MICHPAVE such elements would replace the flexible boundary. Figures (3-2a) and (3-2b) illustrate the two types of infinite elements that are investigated in this study. tie 'xw-C % . . .... . 22% a“: Figure 3-1 : (a) Infinite Element in rz—Space, (b) Square Element in En—Space 14 IR ux AC Layer I \ \ l I \ N \ I l \ \ I 1 \ \ I I ~ \ Base I l \ \ I \ \ I k \ \ l I \ \ I l \ \ I \ L \ I I \ \ i k \ I I \ ‘ . \ \ Subgrade I \ \ \ I I \. \ \ I \ \ I \ \ \ I I ~ infinite elements Figure 3-2a : Radial Infinite Elements AC Layer Subgrade ~ inthlte clematis Figure 3-2b : Vertical Infinite Elements . 15 3.2 AXISYMMETRIC INFINITE ELEMENTS 3.2.1 Formulation of Axisymmetric Radial Infinite Element Stiffness Matrix Cook (1981) formulates the plane linear isoparametric element which can be revised to obtain the axisymmetric infinite 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 six-noded infinite element is shown in Figures (3-3) and (3-4). The global coordinates (radial and vertical) at an internal point can be related to the corresponding nodal quantities through mapping functions: r {z}=[M]{C} (3-1) where {C }T = {r, z,- r, z, n. 2,. r,,, 2”,} are the coordinates of the nodes and [M] maps the square region in Figure (3-1b) onto the infinite region in Figure (3-1a). The displacements at an internal point are interpolated from the nodal displacements through shape functions: u { }= [N1{U} (3-2) w where {U}T = { u, w,- u, w, u]. w u... Wm} are the nodal displacements. The mapping function matrix is MOMOMkOMmO] [Mlzlo M o M,- 0 Mk 0 M... 16 in which individual map functions are M. = 0-5(1-§)(1-n)/(1+n) M. = 0.5(1+£)(1-n)/(1+n) (3-3) M = 0.5(1+§)(2n)/(1+n) Mm = 0.5(1-§)(2n)/(1+n) Nodes g and h are not mapped since displacements at these nodes are assumed to be zero. The shape function matrix is NgONhOMOMONkONmO] [N]=[0N30NhONiON,ONk0Nm in which the individual shape functions are N. = 0.25(1-§)(-n+n’) M = 0.25(1+§)(-n+n2) N. = 0.5(1-£)(1-n2) (3-4) M = 0-5(1+€)(1-n2) M. = 0.25(1+:)(n+n’) N». = 0.25(1-§)(n+n2) Since the displacements at nodes g and h are assumed to be zero, Ng and N}, need not be used in the element formulation. These shape functions are consistent with a six-noded l7 isoparametric element as is shown in Figure (3-4), rather than the more traditional four-noded isoparametric element. Sometimes it is convenient to write Equation (3-1) and (3-2) in the scalar forms 7:2Ml‘l u=ZN1uI (3-5) z=ZMzzi w=ZN1wi where I = i, j, k, m If ¢ is some function of r and 2, then applying the chain rule of differentiation yields flzflfl+§$f§ d5 drdé 61sz fl=§ifl+fl£ dn drdn dzdn .. {131% {1:} M where [J] is the Jacobian matrix J11 J12 = Jar J12 (3-7) 3 ll fissile gleam whose elements are J11 = 7,: = Mi,§ri+ Mg"; + Mk.§rk + Mmerm l8 ‘V ‘Y 8.4 Figure 3-3 : A Typical Axisymmetric Infinite Element Lo ‘1 n ml'NAI 1‘.” in 1 i I g 1 h R,u Figure 3-4 : The Six-Node Isoparametric Element 19 J12 = 2.: = Muézl + Mjszi + M11521- + Mmszm (3-8) J21 = rm = Munr, + A/Imr, + Minn. + Mnmrm J22 = Zm = Mm": + Mm"; + Mknzk + Mm.an and M1,: = -0.5(1-n)/(1+n) Mm = -(1-E,)/(1+n)2 ...etc. (3-9) The inverse relation of Equation (3-7) is ¢,r $39 = F 3-10 {1,2} {NW} 1 1 where [r] = [r11 1‘12] = [J]! 1 |: J22 —J12:| (3-11) 1‘21 1‘22 =J11J22-J21J12 -J21 J11 The relationships between the strain and displacement vectors are ’83 . du/dr 89 u/r 312 =< _ a} dw/dz i ( ) k'erJ kdu/dz+dw/dr} It is convenient to treat the tangential strain 89 separately. Considering 5,, 82, and )1” only, the strain-displacement relations can be expressed as 20 (11,3 er 1 0 0 0 11,: {e}: a: = o 0 011 > (3-13) W,r 8r: 0 I 1 O \M’,/ The displacement derivatives in the global coordinates can be related to those in the local isoparametric coordinates through ru r\ “—11 1‘12 0 O _ u,;‘ u,.- 1‘21 r22 0 0 11,11 1 1 = < > (3-14) w r O 0 1‘11 r12 W,§ L 0 O 1‘21 1‘22‘ W11 The displacement derivatives in the local coordinates are expressed in terms of the derivatives of the shape functions and the nodal displacements, as ru,§‘ FNng 0 116.: 0 NLE, 0 Nut; 0 I 11,11 Mn] 0 Mm O Nk.n 0 Nm.n 0 i i: 1 3-15 W; O M“: 0 Mi 0 Nk.§ 0 Nut: {f} ( ) 9 W n 0 Nut] 0 Mn] 0 Nk.n O Nmflld ’ J _ Combining Equations (3-13), (3-14), and (3-15), we obtain the strain-displacement relations {8} = [B]{U} Matrix [B] is the product of the three successive rectangular matrices in equations (3-13), (3-14), and (3-15). The tangential strain may also be expressed in terms of the shape functions and nodal displacements, as Se = [N]{U}/r (3-16) 21 where r = M,r, + Mr] + Mm.- + Mmrm Thus, incorporating an into the strain-displacement relations, we can write (8: P811 B12 B13 B14 B15 B16 B17 B181 86 Bzr B22 1323 B24 325 B26 827 28 8: B31 B32 B33 B34 B35 B36 B37 B38 k'erJ B41 B42 B43 B44 B45 B46 B47 B48 where 811: DIM; + 11sz = B42 313 = FHA/1,: + 13sz = 344 BIS = Fir/V1,: + Fllem = B46 317 = rlleJ; + r12Nm.n = B48 321: Ni/I? ; 323 = 116”;st = N1” ; 327 = Nm/r B32 = r21N1.§ '1’ rzsz 2 B41 334 = F 21115.: + 13sz = B43 B36 = F 2111/11.: + r22N1.n = B45 B38 = rerm.E, + rzsz.n = B47 andBrz =Br4 =Brs = 318 = 322 =Bz4 = 326 = 328 = B31 = 333 =B35 = B37 = 0 The element stiffness matrix is then (3-17) 22 [11.]:ij [13]T [D] [B]rdrdB dz (3-18) =21: H [B]T [D] [B]rdrdz = 21:1],1', [BJT [D] [B] r IlJll dF. dn where the matrix of elastic constants [D] is d b b 0 [D]: E b d b o (349) l+v b b d o _0 0 0 0.5_ The constants in the above matrix are -1-2v 1—v ;b= v 1-2v where E = elastic modulus, and v = Poisson’s ratio Equation (3-18) must be integrated numerically. Using Gauss quadrature in two dimensions, the integral of a filnction 11) (13,11) can be expressed as 1:.11. ‘1’ (5,91% 0'" = 22 w. w..¢ (£1.11) (320) where W,- and W, are weights associated with the Gauss-points (gm). Each element in the upper triangular part of [kc] is integrated in this way and symmetry is used to fill in the lower triangular part. 23 3.2.2 Formulation of Axisymmetric Vertical Infinite Element Stiffness Matrix The mapping in Equations (3—1) and (3-3) maps the rectangular element shown in Figure (3-4) to the radial infinite element shown in Figure (3-2a). In order to change the mapping so that elements map onto the vertical infinite elements shown in Figure (3-2b), .the mapping function matrix must be changed to M’g 0 M'h 0 M'i 0 M'». 0 0 M O M- 0 M1 0 Mm [M] = l in which individual map fimctions are M '.= 0.25(1-§)(1-n> M3. = 0.25(1+§)(l-n) I (321) M’k= 0.25(1+E,)(1+n) M3,, = O.25(1-§)(1+n) and M,, M}, Mk, and M... are the same as those found in Equation (3-3). This changes Equation (3-5) to r=ZM'I'7'1' 11:le ll] (3-22) z=ZMlzl w=ZN1w1 whereI=i,j, k,m and1’=g, h, k, m The elements of the J acobian matrix [J] found Equation (3-8) become J11: ’3: = M’g.§rg+ M'hgrh + M,k.§rk + M'mgrm 24 T i = M1221 1” Mai-’1' + Magi-'1' + Ming-Tm (3-23) J21 = rm = M,g.nrg + Mhnrh + Minn”: + M‘mnjrm J22 : 2,7] : M1,nzt+ MIMI-y} + kaizk + Mm.nzm M’gé = -O.25(I-T]) M1,... = -o.25(1-g) ...etc. (3-24) The remainder of the formulation follows the procedure outlined in section 3.2. 1. 3.2.3 Coupling of Infinite Elements to Finite Elements Once the stiffness matrix of the half-space has been determined by assembling the stiffness matrices of the infinite elements, it is necessary to couple it with the stiffness matrix corresponding to the finite elements above the half-space. It is beneficial at this point to consider the physical meaning of the infinite element stiffness matrix. Due to the axisymmetric nature of the problem, the d.o.f. along the boundary are really the vertical and radial displacements of the rings. At the origin the ring degenerates to a point. If there are n rings, there will be (2n-1) d.o.f. since there is no radial d.o.f. at the origin (the radial displacement at the origin is zero due to symmetry). The stiffness matrix will then have dimensions of [(2n-1) x (Zn-1)]. The element k9 (i'h row and 1‘” column) of the global stiffiress matrix K, is the total force required along the ring at d.o.f. 1' when d.o.f. j is displaced by a unit amount while all other d.o.f. are held fixed. 25 The stiffness matrix of the finite elements, denoted by King, may be partitioned as follows: K _ KFF KF‘B (3 25) FE KBF K33 where the (212-1) x (2n-1) matrix K33 corresponds to the d.o.f. along the bottom boundary. The stiffness matrix of the half-space is denoted by KHS (also a (2n-1) x (211-1) matrix). The stiffness matrix of the combined system is then KF‘F‘ KFB K: (3%) KBF K33 + KHS If the nodal displacements D are partitioned corresponding to K as D- D” 327 ’08 (-) Then the solution of the stiffness equations KD = Q (3-28) will yield the displacement at all nodes, including those at the boundary, DB. 3.3 COMPARISON OF ANALYSIS RESULTS 3.3.1 Problem Description The effectiveness of the infinite elements was examined using three pavement sections. Asphalt layer thicknesses of 1 in., 5 in, and 10 in. were used in combination with a 20 in. base thickness and a varying subgrade thickness to bring the total finite element mesh depth to 52 in. In reality the subgrade depth is infinite, but deflection, 26 stress, and strain values are only found to a depth of 52 in. A pavement cross section with material properties is shown in Figure (3-5). A 9,000 lb wheel load was applied with a tire pressure of 100 psi. g’? 13/»? xix/M ' / .’- , >40 ’/././ 1’! r ' -' / / .' W‘V‘i/ 5‘!,,’,/:5{" ”/0 ’3‘ - ' '7 ~- .5: amt/w; :. aw?- ” ”.4922?” 3"- ‘..r:: ‘1’ ’5’. ’W/w/Iflg/eW/flr/ W/xfi/xis/ifl/ . , . . __ ii" 1' ' , . I “_ y’fiMx . ' fl -‘ l ' 2’ I“: {16" ' x’ 4’%' ' ., ’. -. :/- a? //’ 3',“ ’ f 'I r f'. ' .' ' f I" . .-4 o’er??? .r/ ' _ Figure 3-5 : Pavement Section Used in Analysis The results found using the revised MICHPAVE with both radial and vertical infinite elements (henceforth referred to as MPIE radial and MPIE vertical) were compared with results found using the elastic layer program Chevronx, and the original MICHPAVE with a flexible boundary (henceforth referred to as MPFB). The two infinite element types are illustrated in Figures (3-2a) and (3-2b). Five responses were used in the comparisons. All values were taken at a radial distance of 0 inches. The five responses are surface deflection, radial strain at the bottom of the asphalt, vertical strain at the top 27 of the subgrade, radial stress at the bottom of the asphalt. and vertical stress at the t0p of the subgrade. 3.3.2 The Depth Ratio Infinite elements are defined using six nodes; two at the bottom boundary of the finite element mesh, two at an infinite depth, and two somewhere in between. The problem that arises is that of defining the depth at which these middle nodes should be placed. If they are placed at too shallow a depth, the subgrade becomes too stiff and the deflections are under-predicted. If they are placed at too great a depth, the subgrade becomes too soft and deflections are over-predicted. In the case of MICHPAVE, the depths of the two middle nodes were defined using a depth factor, or, defined as, or = .37 (3-29) where 21 = vertical distance from the pavement surface to the top node of the infinite element Z; = vertical distance from the pavement surface to the middle node of the infinite element By choosing an or that is greater than 1, the depth of the middle node can be set as 22 = or 21 (3'30) 28 In the case of the radial infinite element the horizontal distance from the center of the circular load to the middle nodes must also be determined. This can be done using the equation R2 = or R1 (3-31) where R1 = horizontal distance from the center of the wheel load to the top node of the infinite element R2 = horizontal distance from the center of the wheel load to the middle node of the infinite element The depth ratio to be used for a particular mesh was determined by trial and error using the surface deflection at a radial distance of 0 inches as a calibration criteria. The or value that resulted in this deflection matching the deflection found using the Chevronx program, was used in the comparisons found in subsequent sections. It was impossible to match this deflection value using the radial infinite elements, so the a value that gave the closest match of the deflections to those computed with the Chevronx program was used. The depth ratio for each of the three pavement sections analyzed was different, as is shown in Figure (3-6). This presents a calibration problem. Anytime a pavement sections properties change, a new or must be found. Thus, a new variable is introduced into the pavement analysis process. 29 25 .. 20 I: l +MPIE ii i vertical 1 2 15 ,_ l 1 a 1: i .. .. 1 +MPIE < 10 j: , T radial 0 SE 5 l . . . . 0 1 2 3 4 5 6 7 8 9 1O 11 Asphalt Thickness (in.) Figure 3-6 : Alpha Values for Pavements with Different Asphalt Thickness 3.3.3 Comparison of Deflections It is misleading to compare the value of surface deflection at a radial distance of 0 in. since the infinite element model was calibrated by matching the deflection at this point. A more reasonable comparison is that of the deflection basins found using the four analyses. Table (3-1) contains the values of surface deflection and Figures (3-7a) through (3-7c) show the deflection basins found using Chevronx, MPFB, MPIE vertical, and MPIE radial, for the three pavement sections. The results for each of the three pavement sections indicate that when prOperly calibrated, MPIE vertical predicts deflections close to the load quite well, but over- predicts vertical deflections as radial distance from the load increases. MPIE radial greatly under-predict the surface deflections of the pavement and changing the depth ratio or only worsens the results. Although the flexible boundary seems to do the best in predicting vertical deflections at large radial distances from the load, the shape of the deflection basin is about the same for both the MPIE vertical and MPFB methods of 30 3%? 338? 3.38? 383. 38? 38? 358? 2.08? 38? 2.3? 438? S33? 2:. 33¢? 33? 33? 33? 38? :23. $38? $38? 38? 2.2.3. :33- 333- n: :33? 35? 3:3. :88? .33. 8:3. 38? 33? 8:8? 33? $3? 38? 4.3 2.23- 323- :3? 3.3. 23? £23. 33? 8:3. 383? 83? 33? 23? SN :83- :5? 3:3. 2:3. 83? 823. 3:3. 33? :33. 2:3. 33? 5.8? S: 33:? :3? $3? 83? 38? $3? $3? 8:3. £3? 23? 223. 323. S; 225? :3? 323. R3? 883. 323. $23. $23. :83. 2.3? 5:3. 33. o: «33? S3? 3:3. :23. 38? 3:3. 33? £23. 823. .323. 83? 33? 3 333? 33? :3? :23. 2:3. 2:3. :3? 2:3. 38? £3? 33? 38? 3 83? 23? :3? $3? $3? :3? 83? 33? 2.3? 33? 38? 38? S. 32.? 83? £23. 3.8? 3.3? $23. $3? $3? 23? 3.3? 3.3? 83.3? 2 83¢? 23? :23. £23. 83? 33? 23? :3? $3? $3? 83? 2.3? 3 .38? 23? 2.29? 223. 223. 5.8? :3? :38? 3:8? 33.3? 33? 38? 3 38? 23? :23. 223. :3? 2.3? 283. 2.3? 38? 38? M33? 83? 3 .53.. .323» .33.. .aumtg .33.. .333» :5 3:82: 32 3:2 9:: £8.55 was HE: 932 22.55 am: was 552 2235 3.5. 2:13. $.13. .._uo< :5 3:83: Anna :8: 328m .8. a 528m Band .23: was Ea fine; was «E: £8526 men: use": 288:8 H TM 03.; 31 $05.2... U< ... £58m 20:00:09 H «Wm 0.3m... .5. 8:8“... 33¢ 0o? 5.8. was. . I . I as...) was. ...... mum—2 I...I.I xco.>0...o \\ IO‘C‘Cl I‘l. r .P rL > F P P h a < a q 4 4 . < 4 Al .. a. | r q «hi-- -0; “I .a1» . 4 u u . 9mm odn 9mm “Only! “’ . A a . 0.8 0.9 odv (u!) uounouaa 32 305.2... U< zm acamm 50000.5 H Sum 0.5m...— .=_. 8.5»... .23. I mNod- L \ . I \\! Noo \ ‘1 \ .. 4 I'll .llll I1 .500. was. . I . I 90.0. .‘o‘lfi. \ 3...; was. ...... \ \\.. 0&2 I I I . . I I. I I\ \ .A 5? 20.5.5 I I I... .l \ \. .ll, :liII!|s -IL \ ‘\ \ .\ 1. ‘\ I‘. LY I .\K 83. a \ 1.. .\ ‘I.‘o\ln\ A I ‘ a ‘ ‘ 1 O ‘ .31....34...” .«.. ..J.+ ..+. 0.4. 0.1.4. .... . r 0 0.0m 0.0V 0.0V 0.00 0.00 0.mN 0.0m 0.0— 0.0. 0.0 0.0 (a!) uouasuaa 33 .52... was. . I . I 32:? was. ...... mum—2 I I I x5320 In: | .IIIlIII.¢|IlII|II| E: 85:5 3.5. $05.25 U< ..o_ £53m 508000 N oné 82mm... 0 10‘ I‘L J 1:1IJ Flaw“- . ‘lV ..|.l‘.| IA nI“ 0.00 0.0V 0.mm 0.0m 9mm 0.9 0.09 0.m _‘0.0- I 30.? N50. 1. 5o. wood. mood. V000. N000- In!) 00906900 34 modeling the half-space. Thus, by changing the depth ratio a, the deflection basin of the vertical infinite elements could be shifted up or down to approximately match that of the flexible boundary. The deflection basin could be altered to more closely approximate the deflection basin found using Chevronx by varying the depth ratio for each infinite element rather than holding it constant. This, however, would fiirther complicate the process of calibrating the depth ratio. 3.3.4 Comparison of Strains Two values of strain were chosen for comparison: radial strain at the bottom of the asphalt layer and vertical strain at the top of the subgrade. The former was chosen because of its significance in the fatigue life prediction model used in MICHPAVE. The latter value was chosen because of its significance in the rut depth prediction model used in MICHPAVE. Table (3-2) contains these strain values for each of the pavement sections. The values of radial strain at the bottom of the asphalt layer found using a flexible boundary and those found using the two types of infinite elements are virtually the same. This is illustrated in Figure (3-8). All three cases over-predict the value in pavement sections with a thin asphalt layer and under-predict the value in sections with a thick asphalt layer. Values of vertical strain at the top of the subgrade found using both infinite element types are substantially better than those found using a flexible boundary, although all three methods under-predict the value for the three pavement sections. This is shown 35 in Figure (3-9). On the basis of strain, the infinite elements seem to perform better than the flexible boundary. Table 3-2 : Strain Values Found Using Chevronx, MPFB, MPIE Vertical, and MPIE Radial (Lateral Boundary at 10a, Standard Mesh Size) Asphalt Radial Strain at Vertical Strain at Thickness Program Bottom of Asphalt Percent Top of Subgrade Percent (in) (10‘) Error (104') Error Chevronx 422.8 - -732.4 — l MPFB 530.4 25.4 -64l.3 -12.4 MPIE vertical 525.8 24.4 -658.7 -10.1 MPIE radial 525.5 24.3 -655.5 -10.5 Chevronx 300.8 - -3 77.1 - 5 MPFB 285.4 -5.1 -293.0 -22.3 MPIE vertical 284.4 -5.5 -32l.9 -l4.6 MPIE radial 284.5 -5.4 -318.5 -15.5 Chevronx 122.8 - -l8l.7 - 10 MPFB 110.9 -9.7 -1 13.7 -37.4 MPIE vertical 110.2 -10.3 -l42.1 -21.8 MPIE radial 110.3 -10.2 -l38.8 -23.6 3.3.5 Comparison of Stresses Values of radial stress at the bottom of the asphalt layer and vertical stress at the top of the subgrade were compared for each method of modeling the half-space. These values are contained in Table (3-3). In all results presented in this work, stresses are computed at the centerline (i.e., at r = O in.). For finite element analyses with first-order interpolation functions, the optimal location for computation of stresses is the center of elements. For example, the radial stress at the center of the element located at the left edge and at the bottom of the AC layer has an error of 6.5%, while the same stress at the node located at r = 0 in. at the bottom of the AC layer has an error of 9.0%. Apart from 36 Eva. NEED EOE? MEE- again 8.03 =2%< Mo 5880 3 Exam 3..va 0o somtanU H w-m 050E 0_. E... «852.: 5.3 .332 m 0..- mo. .mw 10113 )08310d 37 E02 MES—D _m0_to> w_n=2- ovfimnsw he no... a Eabw 180.35 00 53.3500 “ mé 230E Ad: macs—2.: 3:5 =u_nm< 9 0 F 10113 :uazuad 38 this improvement, the general trends in the stresses as the location of the lateral boundary is varied or as the mesh is refined are similar for stresses computed at nodes and at element centers. Table 3-3 : Stress Values Found Using Chevronx, MPFB, MPIE Vertical, and MPIE Radial (Lateral Boundary at 10a, Standard Mesh Size) Asphalt Program Radial Stress at Percent Vertical Stress at Percent Thickness Bottom of Asphalt Error Top of Subgrade Error (in) (psi) (psi) Chevronx 272. 1 - -5.5 1 - 1 MPFB 331.3 21.8 -5.26 -4.6 MPIE vertical 328.5 20.7 -5.59 1.4 MPIE radial 328.3 20.6 -5.54 0.5 Chevronx 220.2 - -2.93 - 5 MPFB 200.4 -9.0 -2.68 -8.5 MPIE vertical 199.6 -9.3 -3. 14 7.2 MPIE radial 199.7 -9.3 -3.08 5.2 Chevronx 91.0 - - l .5 - 10 MPFB 83.1 -8.7 -1.4 -6.4 MPIE vertical 82.4 -9.5 -1 .8 23.7 MPIE radial 82.5 -9.3 -1.8 19.6 The values of radial stress at the bottom of the asphalt layer found using a flexible boundary and those found using infinite elements are virtually the same. This is illustrated in Figure (3-10). Both cases over-predict the value in the pavement sections with a thin asphalt layer and under-predict the value in sections with a thick asphalt layer. The values of vertical stress at the top of the subgrade found using a flexible boundary and those found using infinite elements are also virtually the same. As shown in Figure (3-11), the flexible boundary tends to under-predict these values, while the infinite elements tend to over-predict them. Although it appears that the infinite elements greatly over-predict this 39 i: iliJ .508 wEED _wo_to> NEE I mum—2. 303 6:33 .«o Eozom 8 $25 308: he scam—3500 H on “Ear. 0— ...... 8232.: a»... .333 m iiil Iii «t— 10113 manna 008005 00 nob a 32% _mo_:o> .6 comtmanU H :-m 950.". :5 «853.: is .332 2 m db- 40 .908 NEED .020? mi:- mum—2' 0' mp 10113 messed 41 vertical stress for the 10 inch asphalt layer section based on Figure (3-11), the percent error is misleading since this vertical stress is small for the pavement with the thick asphalt layer. Thus, the infinite elements models are not substantially much better or worse than the flexible boundary model on the basis of stress. 3.4 SUMMARY Infinite elements can be used in place of the flexible boundary to model the half- space below the finite element mesh. Infinite elements are superior to the flexible boundary in predicting vertical strain at the top of the subgrade, while showing no substantial improvement in any of the other criteria used. The radial infinite elements are unable to accurately predict surface deflections using a constant depth ratio a, and thus is not considered any further. The depth ratio of the infinite element should be calibrated prior to the final analysis, especially if accurate displacements are required. As the properties of the pavement section change, calibration of this depth ratio could become quite involved. Thus, unless depth ratios can be accurately predicted by means other than trial and error, it is computationally more efficient to use the flexible boundary when analyzing pavements with MICHPAVE. CHAPTER 4 MESH REF INEMENTS 4.1 GENERAL When the original version of MICHPAVE was released in 1989, the memory limit of MS-DOS, the operating system on IBM-compatible personal computers, to have the operating system and the instruction and data space requirements of an application not exceed 640 KB, restricted the number of elements that could be used to model a pavement section. With advances in technology, these memory restrictions have been overcome for 32-bit personal computers (386’s and later) and applications can now access all installed memory. Hence, much finer meshes can be used in the finite element analysis of MICHPAVE, which should serve to improve the program’s accuracy. At the time of this work a 32-bit implementation of MICHPAVE for personal computers had not yet been performed. However, the old MICHPAVE had been ported to 32-bit Sun workstations running the Unix operating system, which also does not suffer from memory limitations, and had been enhanced to allow many more elements (Harichandran, 1995). The work described in this thesis is based on the Unix version of MICHPAVE Version 1.3. Two improvement strategies were tested using the pavement sections established in Chapter 3 to see what effect they had on the key responses selected in that chapter. These strategies were: 1) increasing the lateral boundary of the finite element mesh, and 2) increasing the number of elements used in the finite element mesh. Both strategies were 42 43 employed using a flexible boundary as well as vertical infinite elements. Results from the various analyses are reported in the subsequent sections. The original MICHPAVE used a finite element mesh that extended horizontally to a distance of 10a, where a represents the radius of the circular wheel load. At this distance MICHPAVE consistently over-predicts the surface deflections of the pavement near the lateral boundary. In an effort to improve this inaccuracy, the location of the lateral boundary was extended to distances of 20a and 40a. Another problem with the original MICHPAVE was that of poor element aspect ratios due to the limited number of elements used in the finite element analysis. The original MICHPAVE produced a mesh with favorable aspect ratios under the wheel load, but as radial and vertical distance from the load increased, the aspect ratios also increased to 4:1 and greater. These long, slender elements can yield inaccuracies in the computed results, especially for strains and stresses. An effort was made to reduce these errors by improving the aspect ratios of these slender elements. Three mesh types were used: Mesh 1 - The original mesh chosen by MICHPAVE by default. Mesh 2 - Uses default mesh under the load but improves the remaining part of the mesh so no element has an aspect ratio of greater than 3:1. Mesh 3 - Improves Mesh 2 by decreasing the size of elements at large horizontal and vertical distances from the load. 44 When the lateral boundary location was extended, elements were added to the mesh to keep the element size near the boundary the same. Meshes are created in MICHPAVE by defining the number of elements in the horizontal direction within four regions (extending from 0 to a, a to 3a, 3a to 6a, 6a to the lateral boundary), and in the vertical direction within each pavement layer. The number of elements in each region is variable. Table (4-1a) displays the number of elements in the horizontal direction in each region, Table (4-1b) displays the number of elements in the vertical direction in each region, and Table (4-lc) displays the total number of elements in the finite element mesh for each of the pavement sections analyzed. Figures (4-1a) through (4-Ic) show meshes used for the 5 in. AC pavements with a lateral boundary at 20a. 4.2 MESH CHANGES WITH A FLEXIBLE BOUNDARY 4.2.1 Deflections The original MICHPAVE displayed deficiencies in predicting surface deflection under the wheel load and at radial distances near the lateral boundary. Table (4-2) displays the surface deflections under the center of the wheel load computed using MPFB for all mesh types and lateral boundary locations, and the percentage errors relative to the Chevronx program. Figures (4-2a) through (4-2c) illustrate the deflection basins directly under the wheel load. These figures show that deflection under the wheel load increases as the mesh becomes finer. Extending the distance to the lateral boundary also improves the accuracy of the deflection basin by increasing deflections under the wheel load for the l in. and 5 in. AC pavements and decreasing deflection in the 10 in. AC pavement. we N: w v wN N: w v m N. w e m :82 em a v v M: v v w n v e v N :82 o— : m w v n m w v N m e v _ :82 mo N. w v ”N N: w v w N: w v m :82 3 v v v w: v v v m w v w N :82 m 2 m a v n m e e N m e v _ :82 mm 2 2 m mm m: o: m o: 2 2 m m :82 3 w v m 2 v v m e .V v m N :82 : E m .0 v N. m e v N n v v _ :82 new - no no - um um - 3 2 - uoN - so no - um um - 3 2 - o no: - no no - um um - 3 2 - c on»... 0:0 805.020. new a 96:80 38:3 EN 3 @3550 3805 no: a .Cgasom :88“: :82 “BEEN $552M Ecomtom .«o 8:852 2600an 82880 :80. :8 02885 03:302.: 0:: E 8580...: mo 8:822 H «TV 2:2. 46 Table 4-1b : Number of Elements in the Vertical Direction for Test Pavement Sections Asphalt Mesh Number of Vertical Elements Thickness (in) Type Asphalt Layer Base Roadbed Soil Mesh l 3 4 5 1 Mesh 2 3 8 10 Mesh 3 3 20 15 Mesh 1 3 5 5 5 Mesh 2 3 5 7 Mesh 3 3 12 14 Mesh 1 6 5 5 10 Mesh 2 6 5 6 Mesh 3 6 12 13 Table 4-1c : Total Number of Finite Elements for Test Pavement Sections Asphalt Mesh Total Number of Elements Thickness (in.) Type Lateral Boundary at 10a Lateral Boundary at 200 Lateral Boundary at 400 Mesh 1 156 216 336 l Mesh 2 357 567 987 Mesh 3 1520 2470 4370 Mesh l 169 234 364 5 Mesh 2 255 450 690 Mesh 3 928 1508 2668 Mesh 1 208 288 448 10 Mesh 2 289 510 782 Mesh 3 992 1612 2852 ' Figures (4-3a) through (4-3c) display the entire deflection basins of the 1 in. AC pavement for all mesh types at each lateral boundary location. With the lateral boundary located at 100, Mesh 3 gives a good approximation of the deflection basin found using Chevronx close to the load, but at large radial distances the deflection is over-predicted. This problem is resolved when the lateral boundary is extended to distances of 20a and 40a. The deflection basin computed using Mesh 3 with a lateral boundary located at 20a 47 803 :m:qm< .E m. SON 8 ba0csom .823 a :23 _ :82 H 24. 230E E and u o .85 880. .0 8:85 zoom s 887.5 .28”. O O 00 on :O NO . . . . . . . . . . . . . . . . . . . . uuuuuuuuu .onnouocncunoaciunourooounonab . . . . . . . . . . . . . . . . . . . . . . . . nnnnnnnnn finnuunuvrauunouusrtfilava-iona . . . . . . . . . . . . a a n c . . . . uuuuuuuuu rt:utuoluounaonunultra-viuonuL . . . . . . . . . . . . . . . . o p c 0 rrrrrrrrr .anuuluiuubtuvuuuosuf1-1.1-1... . . . . o n o o . . . . . . . . . . . . . . . . . ..flvmwN. u u u u . . . . . . . . . . ..... C ....... C ttttttttt .uuroouoncdnuuc . cod . . . . . . . . . . . . nnnnnnnnn front-nun.---u-suoafnunouoao. . . . . . . . . . . . . . . . . uuuuuuuuu Tani-aicnoununicuucTnnuonuono . . . . . . . . o c c g . . . . nnnnnnnnn .uaocuuoouuuuosicouJIcnuInuuia . . . . . . . . . . . . . p p . - :0 m . . . . uuuuuuuuu .1natiooutbnuiiuutournuclaoitu . . . . IIIIIIIII .IuosiilotbutltuluuIrlltlooolb . . . . F P i IF :Avuhu £80 . . . . . . . ... . . . . . . . ... . . . . . . . ... . . . . . . . ... . . . . . . . ... ooooooooo rout-annoy-..touts-tultuoftooscnu- tutornsautrrs. . . . . . . . ... . . . . . . . ... . . . . . . . ... . . . . . . . ... . . . . . . . ... . . . . . . . ... IIIIIIIII WIIIIOIOIJDIIIIIIOIYIIIJIIIIJIIII IJII‘IJIII‘NdA . . . . . . . ... . . . . . . . ... . . . . . . . ... m 00 . . . . . . . ... “V an m . . . . . . . ... nnnnnnnnn rntnuuuooauauununuu‘inuLounahunuu nLitfnsotofrs. . . . . . . . ... . . . . . . . ... . . . . . . . ... . . . . . . . ... . . . . . . . ... ......... Puts-niacbauuuuuuvu.uuun.nutvuuoti oa.nufohaun.tru. . . . o . . . ... . . . . . . . ... . . . . . . . ... u o o 0 0 u 0 .0. . . . . . . . ... . . . . . . . ... ‘ ‘ ‘ ‘ ‘ ‘ ‘ “‘ . . . . . . . ... . . . . . . . ... . . . . . . . ... ......... r-.u--.-..tconuuuonvounuuouo.c.a- -u--r..:-nw.*. . .- . a . . a .q . . . . . . . ... . . . . . . . ... . . . . . . . ... IIIIIIIII r0IIIIIOIGIIIIICIIIYIIILIIOOLCIII l‘llflh'lhr'ba . . . . . . . ... . . . . . . . ... . . . . . . . ... ”mom . . . . . . . ... ........ .IIII|"--‘-'-'I-'I-Yl--..-"-‘-"- -‘Il'-.'--',.1 . . . . . . . ... . . . . . . . ... . . . . . . . ... . . . . . . . ... ......... ‘--'-.--'d-'-'---l-"-.'-----J--" -J"fi-d-.-1~‘A . . . . . . . ... . . . . . . . ... . . . . . . . ... p . L p p b p h? . . . . . . . ... ......... r--iii..i-i-iuioi.-i-..-i- --.--r-..---rp. . . . . . . . .... O< ttttttttt FinalltttbtllltlluletILllluhltll Iktarubtuifrb. . . . . . . . ... h b p 5 F b - PP? Il_’l_’l—!d :coEoG 48 803‘— :m:%< .E m SON E b09500 0:23 a :03 N :82 H :_.v 050E C: 00.0 H o .090 0m0oo. :0 8:005 80m 2 8805 .068... O O 00 on ..0.mm . . . . . . . . . . . . o . . . . . . o o . . ... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 OI0IIIYII0III0III0III II0 I0O0Ill000a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 . 0 0 u o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 I-l'--|.||-“l"|'lr-l‘I-I.I-I'I'I‘I-'.-'I'---r-l‘-.l'--.’I-“l..---T--‘.-I"I‘.'. .L-.'-.-Clr'.A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 O'U‘ICIQOUCJl'Ig'IUMIIJIUU‘--I‘IIJI.-‘II"---.II'J--I~C'U-I.CJ---‘I.-Y--JI"~-.JI-. IJ--‘-‘-II‘~‘A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 DO 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 n DD m OOI'IILIIILIIIPIIIrDILlClioOI'OILIOIhIIIPIOt-IIILIOI'OOIrIILOII'IIITIILIII'lILlII lLII'IbIIt'Phg 0 0 0 0 0 0 0 0 0 0 0 0 0 - 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 Iinfill-IIILaIIPltcrovLIvt0|Inf-0L00000IIP00LIIIBAIIFIIcrllbnolhlllTIIholIPOILOOI ItuolflblllPPba 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 _ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 O.mN 0 0 0 . 0 o 0 . 0 g 0 0 0 0 u 0 0 0 0 0 0 0 0 000 ‘ ‘ 1‘ ‘ 4 ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ I‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘I ‘i‘l‘ .. . . . . . . . . . . . . . . . . . . . . . . . ... 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 o 0 . - 0 u 0 u 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 --I..'- O C I. --'.I-I..-" . -‘-."-l.-'- ---'. ---.--. .I--. .‘I.’ -. .'-.. '-'Y-- .--'v--.. ..... . --.'- .--‘. *OA . -.a-u..u- a . . q . . u a . a a . J 0 J . . . a g. a 0 0 0 0 c . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 . . 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 u 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 IIIPIILIIILIIIPItI-clotulntutorsILIuILtIcPIcI-llttnoopolIrIILclIDIIIIIILIIIPuILIII oLclflhlttffbx . o c 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 u . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 ”mom 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 000 IIIVIIIIIIon:IVIIITIIJIIIoutnYutlllodolcvItloltoduluoillvuIIAIOIQOIIleAtlI'tIlllt OJIIYIAOII'QAA . . . 0 0 o 0 u 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0.0 p . u n . p p p p p p p p L p p p p F . p p . p.- 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 .‘Irl-l-'-ISII-P'-I.IDIL---PI-IP-‘tl--~l.-'---.-|-~---'IIIr0-5--0’0IlI--b-'."I.-I'. IL.-'-~IIIPP.J . . . . . . . . . . . . . . . . . . . . . . . ... U< IOOPIILIOIbIIIPIOorIILIOIPoIlrltklln0II0POIIrIILIOIPOIIrIILIIIPIIIWIIBOIOPIOLIIO ILIIFOblllp-Psa 0 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0.0 > b p b b i b b b b b h P b b b + r h b b h h bbh 00 £80 £82 2885 BE... Depth te Element Mesh ini F 0.0" 5.0" 49 I I I I I I I I I I I I I I I I I I I I I I I I I I .-L-'.-....'.---.---'--0--'-.'-J...|-J-J-J--~--L-1-.I-----L-.I-.'--'.-I-J--'-- - I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I "P‘I'""I"I"I‘fi'fl’fi‘fi'fi'fi'1‘1“ 0‘00'0'0‘00F0'0‘0q00r0'0‘0fi00r- I I I I I I I I I I I I I I I I I I I I I I I I I I hubobchobob0I0-I00I0J0J0J0J0J0J010J00L0‘odook-‘O‘Oluohc‘c‘od-0&0 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I p0.00.00I-0‘00.00.00.00'00.0-'00.-..0‘010100.00 0'0‘00.00r010-.-0'00r010-.00r- I I I I I I I I I I I I I I I I I I I I I I I I rOOCP-F-P-P'I'~-q'q-~’fl-Q-~-‘-I CCU.-.COOQOOP-’-.-‘-.’-O-Q.q.-P. I I I I I I I I I I I I I I I I I I I I I I I I I I L-L-L-I—-L-'--'--‘--'-J-J-J-J-J-J-a-J--\.-L-J--I.-L.J-J-.I.-L-J.JooL- I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I l I I I I I I I I I I I I I 0r0r000‘00.00.00.00.00.0‘0‘0‘0‘010 0...-rg'0‘00‘00'0‘0‘00r0'0‘00.00r0 I I I I I I I I I I I I I I I I I I I I I I I I bub-h. 050b00-0‘0dod0d0d0d0‘0d040‘0050O0‘00h0b0‘0d--b-Qa‘odo0h— I I I I I I I I I I I I I I I I I I I I I I I I I j.000.0000'00'00'00‘00'00'00‘00.00.00.00.0400'00'00:00.00'00I00J00'00'00'00J00'00'00 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I l I I I I I I I I I I I I I I I I I P'P'P‘”.P'P-I'.I'q-q-q-q-g-‘-Q-1‘q'O'0'0‘--P-P.Q-W-.P"-Q....P. I I I I I I I I l I I I I I I I I I I I I I I I I I r-L-I--I—-I.-I.-I--l-.l-J-J-J-J-J-J-J-d--5-5-d--|.-l..dcdoobcbododcoh- I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I r-r0r0.0.00.00.00.00'00.00b010%-‘0‘0100'00r0'0‘00.00r0100'00r0r0100.00r. I I I I I I I I I I I I I I I I I I I I I I I I I I hop-.000p0h0000000.0‘0-00‘0‘0‘0001 00.--..-00‘00hqp000g0ap000000.00..- I I I I I I I I I I I I I I I I I I I I I I I I I I I.-L.L-I.-I..'-.'--'--I--°-J-J-J-J.J.q-.'--I..L-J--'.-L.J.J--I--II-J.-'-.I..- I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I --P.r..-P.P-I-.I..I-fi-q.fi-‘-‘-‘-I-fi--'.'.‘-.P-'-‘-‘--P-'-‘.‘-.r. I I I I I I I I I I I I I I I I I I I I I I.-.0b0I-0I00h-0I00600I0-l0d0d040d0-‘01 0‘005000d00b0b000d00500000d-0|.- I I I I I I I I I I I I I I I l I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I b0.0-0000.00.00.00.00.00.00.00.000‘0‘0100.00.00000I00.00'00100.00.0000700.00'00 I I I I I I I I I I l I I I I I I I I I I I I b0p0p000p0p0p0pq0q0q0‘0‘0g0q0400'00p0'0‘00p0’000q00'0'0‘00‘00’0 I I I I I I I I I I I I I I I I I I I I I I I I I p0Lchub0b0L0I0J-0IOJ0J0J0J0J0J-40J0050‘0docho‘0‘0Joogo‘OJ0J.050 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I .r0rcr0P0‘00.00.00.00.0‘0‘0‘0‘0‘010‘--'0'0‘-0.00'010‘OUFO'0‘00.00'0 I I I I I I I I I I I I I I I I I I I I I I I I I I pcfioh0rchohokdododod0‘0‘0‘0‘04 -‘00000000000000.000050000003000. I I I I I I I I I I I I I I I I l I I I I I I I I I cho'.. 0.00.00.00.00.00.0-.00.0J0J040400.00 0‘0J00‘00LOJIO'OO'IQL0J00IQ0L0 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I hopopcphpopo'cyq0‘0‘0‘0‘0‘0‘-{-‘oqp0'0‘--r0'0‘oqooru'oqo0.0-'0 I I I I I I I I I I I I I I I I I I I I I I I I I I p0hob0 .b0b0I00I00I0-I0J0J-‘0J0J0q0‘005000dgchufio‘cduobob000d00.0 I I I I I I I I I I I I l I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I bororop-.....-.0-.00.00.00.00.0‘-‘-1-1-0..- 0'01.-..0r0‘0‘00.0-r0‘0..0or. I I I I I I I I I I I I I I I I I I I I I I I I POOOPODoP-Pm-IOQOCIO-o-Q-Qo-o-0-1nqo-ro.oqoor-r-Q-q--h-o-Q-q--P- I I I I I I I I I I I I I I I I I I I I I I I I I p0L0L0p0I0050'00'00'0J0J0J0J0J0J01-0’0050‘0‘00'00‘0‘0J00~0‘0J00I00L0 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 1 fi 1 j I I V 1 V V V fl Y 1 V V I V I ' 1 0p0p000p0p0'00'0”0‘0‘0‘0q0‘0q010‘00'0v0‘00'00'0‘0-‘00'0'0‘0‘00'0 0.0.000.0h060-I000-0‘0d0d0‘0d0401 0d00b060d00h080‘0‘00h050‘0d00b. 0L0B0p05050'0-I0J-J0J0J0J0J0J0‘0J0QL0‘0J0050L0‘0‘005000J0J00L0 0k0'0000'00'00'00'00'00'00'0J00'00‘0J01 00.00.00L0J00I00L0J0.‘00'.0L0J00'00.00 I I I I I I I I I I I I I I I I I I I I ’0'0'0 .P.’.Fq-q-q-q-q-‘oqo‘o4-qooro'o‘ocpo'.‘OqCOPD'-‘.q--'- hofioho .b-b0hd‘dod0dcfi0‘0‘0‘010‘00.o’cflqubop0.0‘00’000Q0‘00’0 05-h0b0b0I00I00I0J0J0J0J0J0J0J040J0-50‘0J00h050‘0d005-‘0‘0d005. E-L0t0 0'00'00‘00'00'00'00'00.00'0J040400.00 0‘0J00'0-‘0‘00'00'00L0J00'00‘0 I I I I I I I I I I I l I I I I I 0r0r0b0'00'00'00.00'00'0‘0‘0‘01010400‘00r0'0‘00'00r0‘0‘00‘00r0‘00.00’0 b'r"-.-P-P-I.-V..I-q-‘.‘-‘-“-‘-4-q'-r-'-‘--r-r-‘-q--P-'-‘.fi--P. pnhobap-b.b.‘.*d-d0d040d0d0‘o+cinch0‘odoohcfio‘0do0bo‘oJ0d0050 b-‘0‘00b0'¢0'-0.00.00.00.00.00.00‘0J0J0400.0-L0‘0Joo'ooL0‘00'00I-Ot.J00'00L0 b0'0000h0'00'00'0000‘00'0 .00.00.000.0 00'00'00'00‘0000'00.00.00.00.0 00.00.00 I I I I I I I I .I I I I 3 j I I I I I I I I' 1 I I tor-rob0P0r0'0-.00'0‘0‘01‘0‘0-‘010 0‘00r0'0‘00r0r010‘00r-r-10fi00r0 p0.0,-..0p0p0P-Q00'0q0g0q0q0q0004 0g00p0.0g00p0.0.0g00p0v0goq00'0 huh..--0-0b-I0d0d0d0d0J-‘0‘0‘010d--.0.04-0“-.0‘0d005000‘0dosh. p-L.L-p-5-I.-'.-I.-I--'-J-J-J.J-J- -J-oL-&.J-.I.-L.J-J-.L-L-J.J--L. I I I I I I I I I I I I L I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I bar-rah-r0.00.0-..0.--.050‘0‘0‘0104can-r0'0‘00‘00r0‘0‘00fc'0100.00r0 p-'-P-h-F-P-P.'-fi-qO‘O‘Q‘-‘-‘-*-q--'-'-q--p-'.‘.q--'-'-‘QQOOPO nob-bobcbchoIcuod-‘odododoflc‘O‘odoob-.-‘--h-§ofiodoohobo‘0d00.0 A A A 4 A A A A A A A L A J A A A .lll a) (I) 0 CD Roadbed 60 30 Radio! Distance in Rodii (Radius of Iooded area, 0 = 5.35 in) Asphalt Layer m. 5 h a Lateral Boundary at 200; Mesh 3 wit Figure 4- l c 50 is very close to that computed using Chevronx, and that found using Mesh 3 with a lateral boundary located at 40a is almost exact. Table 4-2 : Deflections Under the Wheel Load for Varying Meshes Found Using MPFB Lateral Boundary at 100 Lateral Boundary at 200 Lateral Boundary at 40a Asphalt Mesh Deflection Percent Deflection Percent Deflection Percent Thickness Type (in) Error (in) Error (in) Error (in.) Chevronx -0.05050 - -0.05050 - -0.05050 - l Mesh 1 -0.04798 -5.0 -0.04795 -5.0 -0.04868 -3.6 Mesh 2 -0.04864 -3.7 -0.04880 -3.4 -0.049l6 -2.7 Mesh 3 -0.05033 -0.3 -0.0503l -O.4 -0.05062 0.2 Chevronx -0.02445 - ~0.02445 - -0.02445 - 5 Mesh 1 -0.02333 46 -0.02325 -4.9 ~0.02369 ~3.l Mesh 2 -0.02401 -l.8 -0.02371 -3.0 -0.02405 -l.6 Mesh 3 -0.02436 -O.4 -0.02391 -2.2 ~0.02427 ~0.7 Chevronx -0.01527 - -0.01527 - -0.01527 - 10 Mesh 1 -0.01551 1.6 -0.0l442 -5.6 -0.01496 -2.0 Mesh 2 -0.01598 4.6 -0.01474 -3.5 -0.01516 -0.7 Mesh 3 -0.01614 5.7 -0.0l478 -3.2 -0.01519 -0.5 Figures (4-4a) through (4-4c) display the entire deflection basins of the 5 in. AC pavement for all mesh types at each lateral boundary location. Like that of the 1 in. AC pavement, the deflection basin of the 5 in. AC pavement benefits from the extension of the lateral boundary location. In this case Mesh 3 with a lateral boundary located at 40a does a much better job predicting deflections than does Mesh 3 with a lateral boundary located at 20a. Figures (4-5a) through (4-5c) display the entire deflection basins of the 10 in. AC pavement for all mesh types at each lateral boundary location. At a lateral boundary Emma» ..... «8.35.8.2 ...... movua :30: ........... 89358.2 , I . z womamwcmoz ...... 8% 2.8.2 1 I i 8:3..82.I.I 823:8: ...... no, a 28.2 I I I c2520 5&3 :2%< i .952 9.63 venom 284 6055 2: $95 mcozoocoo EoEo>mm H amé 8sz E... 85:5 33¢ ('0!) 00996080 his =2am< ..m .952 $33 canon 284 .855 of 395 mcocoocoa EoEo>wm H nmé oSwE IJ 109100535 . . movemNcmwi ...... 035'ch ........... 8~.mm;82.3.1 3999322 ...... 823:8: a: a 823582.I.I 8.528: ...... 82328.2 III 5320 E... 853.: 32¢ Il'L mwod- "No.0- No.0. I'll!) uoumuaa 0905 2933 ..o_ .932 $53 venom 304 .855 2: $95 mcouoocon E2523 H omé 8&5 lavas. . «Stung—2 ...... nova—2mm: .......... momfimcmos..i,3 «090N532 ,,,,,, momfipzmos. 3.3.... wo...mmzmo§.l.l Stunnmos. ...... 8.6.502 Ill c2550 2... 85:5 33¢ I‘I‘IIIOII mm «0.0. IIIIIIIOIIOOI II .......... 20.0- IIIII II 85.9 90.0. I in" )‘.\rl" .‘ Q.""\"\C ‘ ......................... . Qi’itiaz mv-od. V5.0. mnpod- m 5.0- ('0!) 001136030 S: E bawcsom =28“:— ammnaz wEmD ncsom .993 “Ring 1 mo Emmm 508009 N mmé 05E”— SQHWH m2 5 N 58.2 ...... 823.3% III 5520 0:: 009.530 .303— lfitlllii' in..." 00.0. (1'!) uouoouoa No.0. 5.0. com 3 bauczom 1283 .952 $53 venom .934 fish—fix .. _ mo Emmm concocoa H nmé 833m 5 5 _ mow «In-» mmmm . I1.“ mom a m 58.2 ...... mow «m — :wmi I. .I I. 53ch E... 8535 .23. no.0. «0.0. 5.0. ('0!) uonoouoa cow 8 bacczom E83 .95: 353 9:5”— 603 52394. .. _ («0 53m 508009 N omé “:sz 2... 85:5 33¢ 8o. - - 8.? I} I. - , 3.0. 6 S i i mod. «8.3535. . I . I now .m N :85. ...... Sin _ 522 I II c2550 I no.0. .III‘I 5o. L F I r Ii. Ir » I 0 < a w «J» d If n O (11!) "0933030 57 S: S bauczom .283 .952 was: 250m baa.— 2293 ..m 00 £25 cocoa—«on H 31v 83w:— mo_sm;$2.l.l 82358.2 ...... 823522 III :0;ch 3... 8:53 33¢ ('0!) museum 000.0. 0m 9 cc mm on mm on m... 58 com 8 bauczom .833 dun—E wEmD 950m .993 :2%< ..m .00 58m cocoa—ova H nvé 8sz 823522 . I . I 8m 3 N 522 ...... mom a 28.2 I I I :03on 2... 8:36 33¢ mNoo. m 5.0. «0.0. mood. mp or ('0!) nomad 59 now 3 bauczom .8034 .932 9%: ucsom $35 2233‘ ..m no 535 cosooooQ U 31‘ Eswi «8.35% .I. I wow 5 N 522 ...... Sim . 58.2 I I I :03sz 2... 83:5 33¢ ‘.|.\I 9400. No.0. m 5.0. 5.0. mood. 0.. I'll!) Housman 60 so. 3 bawcsom .823 dun—E was: 950m “931— =afim< :2 mo Emmm couoocon H mmé QSME E... 8:33 5.5. No.0. 4 .\....I.h .\‘.u «t. \‘l IIIEI: i , A... \ 90.? ...\U\ .1 .. \ 4 III IIIIIIIa; \. .. ...\\\ «Ovuumzwozulul \uu\.\n\\\ a 40“ 1“ 823582 ...... ca... \\\ : QMAJ‘. .x\ 8% £82 III \...&.1..\\\\ :05ch 1.3!...) 1...\.\ \ \ : £11.! n...a~ar..u.\.... \ a 3.....1..H.nl I... 6.9 infil‘l IH-‘Illll'l \ a. I 1+4 II If r II II IILI . wood. ('0!) “03130030 61 com S Emcczom .8023 dug—2 $53 950.,— bbfi 293m< :2 m0 525 5:8ch H nmé oSwE 2... 8520 3.5. 8.0. \I... m5? .I.... Moms m :msz. I-. H mom a N 522 ...... mom a 28.2 I I I :03ch \ l \ .\\‘\ \ 5o. ‘ \.\\ \ | \ \“u\\\ \\\\\\ \ . \V \ \\ .I..“ \ \‘hi 7 E“ n. 1 \I‘ o o y a ,1 1 o I» r moody! om mv 9. mm 8 mw ow m. 2 m o In!) uogmuaa 62 cow 3 bmvcsom .8023 .932 $53 .520... 893‘— ..9.%< ..o. ..o 2.85 5.80.39 H omé 83m... 3... 85:5 .23. no.0. In!) flamenco ammfiwm .. I . I new .m N 58.2 ...... 83m _ 5.8.2 I I l c2550 63 located at 40a, all three meshes do extremely well in predicting surface deflections for the 10 in. AC pavement. Figures (4-6a) through (4-6c) show the percent error associated with all mesh types and lateral boundary locations for each asphalt layer thickness. For the 1 in. AC pavement the mesh finess has the greatest effect on the surface deflection under the wheel load, while for the 10 in. AC pavement the location of the lateral boundary has the greatest effect. 4.2.2 Strains 4.2.2.1 Radial Strain at the Bottom of the Asphalt Layer Table (4-3) displays the values of radial strain at the bottom of the AC layer for all the pavement sections analyzed using MPFB. As is shown in Figures (4-7a) through (4-7c), Mesh 3 with a lateral boundary located at 20a improves the estimate of radial strain in the l in. and 5 in. AC pavements. In the l in. AC pavement the error is reduced from 25.4% to 22.1%, and in the 5 in. AC pavement the error drops from -S. 1% to -3. 1%. For the 10 in. AC pavement, Mesh 2 with a lateral boundary located at 40a reduces the error from -10.3% to -2.4%. The results found using Mesh 3 with a lateral boundary at 40a are not significantly more accurate for any asphalt thickness. 4.2.2.2 Vertical Strain at the Top of the Subgrade Table (44) displays the values of vertical strain at the top of the subgrade for all the pavement sections analyzed using MPFB. As shown in Figures (4-8a) through (4-8c), for the 1 in. AC pavement, the use of Mesh 3 with a lateral boundary located at 20a reduced the error in the computed vertical strain from -12.4% to -0.5°/o. For the 5 in. AC 64 80.404 0.8.03. ... .952 was: 0:00”. 0004 .0055 05 80.045 80:00:03 H 09v 05w.“— buucaom .820.— .o 5.300.. 09. 8m 0.0. mcmoiu memos.- 232.. song guessed .0 :82—w N 5.0.2- 2002:. is .303 .0 .0002 05.5 258 33 02.3 2: .025 2008000 N no. b.6500 .283 .0 3.38.. 000 80 8. v 200E mug mama 66 80.84 0.8.034 ..o. dams. wEmD 0:00“. .0004 .00....» 0... 000.5 000.000.40Q u 00% 05w.”— raucaom .33.... .0 8:33 8.. 8m 8. 0.7 0N- -0- 0.0 ON of ob 10113 )Uaafld 67 Table 4-3 : Values of Radial Strain at the Bottom of the Asphalt Layer Under the Wheel Load Found Using MPFB Lateral Boundary at 100 Lateral Boundary at 200 Lateral Boundary at 400 Asphalt Mesh Radial Strain Percent Radial Strain Percent Radial Strain Percent Thickness Type ( 10") Error ( 10") Error (10") Error (in.) ‘ Chevromt 422.8 - 422.8 - 422.8 - 1 Mesh 1 530.4 25.4 516.7 22.2 523.2 23.7 Mesh 2 525.8 24.4 518.9 22.7 518.5 22.6 Mesh 3 523.1 23.7 516.2 22.1 515.7 22.0 Chevronx 300.8 - 300.8 - 300.8 - 5 Mesh 1 285.4 -5.1 287.2 4.5 287.0 46 Mesh 2 286.3 —4.8 288.1 -4.2 288.0 43 Mesh 3 289.9 -3.6 291.5 -3.1 291.2 -3.2 Chevronx 122.8 - 122.8 - 122.8 - 10 Mesh 1 110.2 ~10.3 118.8 -3.3 119.1 -3.0 Mesh 2 112.2 -8.6 119.6 -2.6 119.9 -2.4 Mesh 3 113.0 -8.0 120.2 -2.1 120.4 -2.0 Table 4-4 : Values of Vertical Strain at the Top of the Subgrade Under the Wheel Load Found Using MPFB Lateral Boundary at 10a Lateral Boundary at 200 Lateral Boundary at 40a Asphalt Mesh Vertical Percent Vertical Percent Vertical Percent Thickness Type Strain Error Strain Error Strain Error (in.) (10") (10‘) (10‘) Chevronx -732.4 - -732.4 - -732.4 - l Mesh 1 -64l.3 -12.4 -678.8 -7.3 -690.8 -5.7 Mesh 2 -669.4 -8.6 -714.7 -2.4 -718.7 -l.9 Mesh 3 -684.3 -6.6 -728.4 -0.5 -732.4 0.0 Chevronx -3 77.1 - -377.l - -377. l - 5 Mesh 1 -293.0 -22.3 -347.8 -7.8 -354.3 -6.0 Mesh 2 -308.3 -l8.2 -359.5 -4.7 -365.3 -3.1 Mesh 3 -317.0 -15.9 -367.5 -2.5 -373.8 -0.9 Chevronx -181.7 - -181.7 - -l81.7 - 10 Mesh 1 -113.7 -37.4 -l65.0 -9.2 -l74.7 -3.9 Mesh 2 -121.8 -33.0 -l70.5 -6.2 -l79.2 -1.4 Mesh 3 -124.9 -31.3 -172.6 -5.0 -182.1 0.2 68 mag wEmD 9:5..— 8wa 223.2 .. _ mo Eozom 8 £85 331 H and Sam:— Eaucaom .283 .o .3330.— m fiozn N =82- . £8.23 8.. 0. pm QVN 10113 1113:1104 69 ME: was 28» :3 =25 =20 59:5 a sea .32 H a: 2%: rune—.8 .283 .o c0830.. so P 0.6. m :moZU N cum—2 I £0: a 10113 iuazuad 70 mus: was: 250.2 .93 222.2 :0. 00 528m 3 sea 33. H a: 2&5 E0253 .233 .0 00:30.. now now no— oN T -mflwzn N 2002 I _ :82! 10113 wanna 71 .293 :23< i .mmmz wEmD ccsom 00835 00 no... 8 Eubm 32:“; M £4 8:?“— ?aucaom .333 .0 $2300.. 00? now 00— 0.3.- 0N T 0.07 M. £00.20 N £00.2- pawns—B 0.0. 10113 1000103 72 .934 €233 ..m .9702 $33 250"— oufimnsm .00 00,—. a £85 .8015 > H £40 ocswfi 0.2.80 .22.... .o 8.084 00v mom 00.. 0.mN- 0.m P- 10113 1000103 0.0 P. 73 .993 220?. ..o_ .9702 $5.3 058.0 ovflwnzm 00 non. 3 52m .3005 U 33» 2:3”— Euucaom 3.81:0 000003 00v «ON 00.. 0.91 9mm- 0.m F- 0.0 T 10113 1000103 74 pavement the error drops from -22.3% to -2.5% when using Mesh 3 with a lateral boundary located at 20a. The 10 in. AC pavement is best served using Mesh 2 with a lateral boundary at 40a. Error is reduced from -37.4% to -1.4%. For all the AC layer thickness, the results found using Mesh 3 with a lateral boundary at 40a is almost exact, but such a mesh contains many elements and the increase in accuracy is not worth the increase in computational effort. 4.2.3 Stresses 4.2.3.1 Radial Stress at the Bottom of the Asphalt Layer Table (4-5) displays the values of radial stress at the radial stress at the bottom of the AC layer for all the pavement sections analyzed using MPFB. As can be seen in Figures (4-9a) through (4-9c), there is no significant improvement to the radial stress prediction for the l in. and 5 in. AC pavement sections. However, when the 10 in pavement is analyzed using Mesh 2 with a lateral boundary at 40a, the error is reduced from -8.7% to -1.3%. 4.2.3.2 Vertical Stress at the Top of the Subgrade Table (4-6) displays the values of vertical stress at the top of the subgrade found using MPFB. As is shown in Figures (4-10a) through (4-10c), when Mesh3 with a lateral boundary at 20a is used in the analysis, the error in predicting vertical stress for the l in. AC pavement reduces from -4.6% to 0.2%. For the 5 in. AC pavement, using Mesh 3 with a lateral boundary at 20a reduces the error from -8.5 to -1.6%. The 10 in. AC pavement vertical stress prediction error reduces from -6.4% to -O.9% when using Mesh 2 with a lateral boundary at 40a. 75 Table 4-5 : Values of Radial Stress at the Bottom of the Asphalt Layer Under the Wheel Load Found Using MPFB Lateral Boundary at 10a Lateral Boundary at 200 Lateral Boundary at 40a Asphalt Mesh Radial Percent Radial Percent Radial Percent Thickness Type Stress Error Stress Error Stress Error (in.) (psi) (psi) (psi) Chevronx 272.1 - 272.1 - 272.1 - l Mesh 1 331.3 21.8 321.5 18.2 325.8 19.7 Mesh 2 333.6 22.6 328.4 20.7 328.0 20.5 Mesh 3 330.8 21.6 325.4 19.6 325.0 19.4 Chevronx 220.2 - 220.2 - 220.2 - 5 Mesh 1 200.4 -9.0 201.8 -8.4 201.7 -8.4 Mesh 2 201.1 -8.7 202.4 -8.1 202.3 81 Mesh 3 203.6 -7.5 204.8 -7.0 204.6 -7.1 Chevronx 91.0 - 91.0 - 91.0 - 10 Mesh 1 83.1 -8.7 89.0 -2.1 89.3 -1.8 Mesh 2 83.9 -7.7 89.6 -1.5 89.8 -1.3 Mesh 3 84.5 -7.1 90.0 -1.0 90.2 -0.8 Table 4-6 : Values of Vertical Stress at the Top of the Subgrade Under the Wheel Load Found Using MPFB Lateral Boundary at 100 Lateral Boundary at 20a Lateral Boundary at 40a Asphalt Mesh Vertical Percent Vertical Percent Vertical Percent Thickness Type Stress Error Stress Error Stress Error (in.) (psi) (psi) (psi) Chevronx -5.51 - -5.51 - -5.51 - 1 Mesh l -5.26 -4.6 -5.26 -4.6 -5.37 -2.6 Mesh 2 -5.34 -3.1 -5.41 -l.8 -5.44 -1.3 Mesh 3 -5.47 -0.7 -5.52 0.2 -5.55 0.7 Chevronx -2.93 - -2.93 - -2.93 - 5 Mesh 1 -2.68 -8.5 -2.74 -6.4 -2.78 -5.1 Mesh 2 -2.80 -4.4 -2.83 -3.3 -2.87 -2.0 Mesh 3 -2.86 -2.3 -2.88 -1.6 -2.92 -0.3 Chevronx -1.46 - -1.46 - -1.46 - 10 Mesh 1 -1.37 -6.4 -l.34 -8.4 -l.40 -4.3 Mesh 2 -1.45 —0.9 -l.40 -4.3 -1.45 09 Mesh 3 -1.48 1.2 -1.41 -3.6 -1.47 0.5 76 mug—2 wEmD venom 0931— :2%< .. _ ,«o Ectom 8 $25 33.0 H no-0 05w:— b-vcaon .220.— .0 5530.. SN 212”. .ll. N £005.. yrcmwil 0.mN 10113 1000103 77 mafia). was: .055”. 093-. swim-<1 ..m .0 Eosom .0 mumbm .063. H no-0 05E". E00500 .283 .0 00:000.. 00N mlfi-mzn l . 202- 28.20 - 10113 1000103 78 0&0). $55 .055”. 5.0-. 22.02 :2 .0 Eozom .0 895m .063. H oo-v 20mm E00500 .233 .0 00030.. new m cumin N5002- — zmoil 10113 1000103 79 n. ma... N :8:- 50.03.] 5.3-. :20... ... dung. wEmD 0.50”. 0.00.3.6 .o no... .0 $26 .8.to> H «2.0 2:»... 200.500 .285 .0 00:33 000 8N no. 0.m- . of v °.nl 0.N- . 0..- . 0.0 0.. 10113 1000103 80 028-. 09.090 ..m .95.). 0:03 .0000". 0.0835 00 000. .0 32% .8.t0> 0.00.500 .231. .0 00:30.. 000 08 no _ 4 23mm 00. 0.0.- 10113 1000103 593 2933 ..o_ .932 $53 250m owfiwnzm mo nah 3 $on _8_to> H 03% oSwE Eaucaom .233 3 3233.. wow gm 00.. 0.9- 81 ”flows—u m 58.2- _ cumin mug wanna 82 4.2.4 Conclusions A lateral boundary located at a radial distance of 10a from the center of the wheel load is not satisfactory when analyzing pavement sections with MICHPAVE. Analysis results are significantly improved by increasing the distance to the lateral boundary by a factor of two or greater. Analysis results are also significantly improved by refining the finite element mesh. The 1 in. and 5 in. AC pavement sections are accurately modeled using Mesh 3 with a lateral boundary located at 20a, while the 10 in. AC pavement is more effectively modeled using Mesh 2 with a lateral boundary located at 40a. While modeling all pavement sections using Mesh 3 with a lateral boundary located at 40a produces very accurate results, it is not computationally efficient to use a finite element mesh with so many elements, as is shown in Table (4-7), which displays computational times on a Sun Sparc Station 20 for each of the meshes analyzed. Table 4-7 : Computational Time for Test Pavement Sections Asphalt Mesh Computational Time (min:sec) Thickness (in.) Type Lateral Boundary at 100 Lateral Boundary at 200 Lateral Boundary at 400 Mesh 1 0:02 0:04 0:04 1 Mesh 2 0:03 0:05 0:17 Mesh 3 0:17 1:05 4:54 Mesh 1 0:02 0:02 0:04 5 Mesh 2 0:03 0:05 0:13 Mesh 3 0:09 0:29 2:22 Mesh 1 0:02 0:03 0:05 10 Mesh 2 0:03 0:05 0:14 Mesh 3 0:09 0:29 2:29 83 4.3 MESH CHANGES WITH INFINITE ELEMENTS 4.3.1 The Depth Ratio As described in Chapter 3, when infinite elements are used in conjunction with MICHPAVE, a depth ratio 0t must be employed. As is shown in Figures (4-1 la) through (4-11c), both mesh size and location of the lateral boundary affect the value of the depth ratio. The value of the depth ratio decreases as mesh fineness increases. As the radial distance between the load and the lateral boundary location increases the depth ratio also increases. The depth ratio was once again determined by trial and error by trying to match the surface deflection at a radial distance of 0 in. with that computed using the Chevronx program. 4.3.2 Deflections Figures (4-12a) through (4-12c) illustrate the deflection basins directly under the wheel load. As these figures show, mesh fineness and lateral boundary location have very little effect on the prediction of deflection under the wheel load for all of the AC thicknesses tested. Further, these predictions are quite accurate, which should be expected since the depth ratio a was calibrated to give the correct deflection under the load. Figures (4-13a) through (4-13c) display the entire deflection basins for the 1 in. AC pavement for all mesh types at each lateral boundary location. While the prediction of deflection close to the load is quite accurate, at large radial distances from the load MPIE over-predicts the deflection values. This is true for all mesh types at all lateral boundary 84 +MeshT '+Mesh212 +Mash3I 012345678910 Asphalt Thickness (In.) Figure 4-1 1a : Effect of Mesh Fineness on Alpha Values of Pavements with Different Asphalt Thicknesses; Lateral Boundary at 10a + Nbsh 1 I + Mesh 2 -¢— Nbsh 3 . t _.____—._J Alpha 012345678910 Asphalt Thlckness (In.) Figure 4-11b Effect of Mesh Fineness on Alpha Values of Pavements with Different Asphalt Thicknesses; Lateral Boundary at 20a +Nbsh1 +msh 2 -o—lVesh 3 012345678910 Asphalt Thickness (In.) Figure 4-11c : Effect of Mesh Fineness on Alpha Values of Pavements with Different Asphalt Thicknesses; Lateral Boundary at 40a ES €33 .._ ._ao_:o> ma: wEmD venom EBA .855 2: $ch mcozoocoa “563$ H «NT.» 833m 2... 85:6 33¢ ii Ii. ...Wi.fl 833522. «9.69.on ...... woven P2922 ........... 893282,... . i Swanson-2 ...... 82m . :82 a .3 a 8..~mfio2.l.l 8236821.... 82358.2 III c9520 (w) nomad 86 @9528? movumszOE ...... 83358.2 .. 8235821.: mogmwzmoi. . . . .... 82358.2 E t s. 82mgmmz.l.l 823582...... 8235...: Ill 5320 .93 =89< a E... 8535 .85. 48:85 EA: wEmD venom 304 6055 05 .025 mcouoocon 258025 N anv 823m as...» «8.... .. mNod. nNod. «No.0. (1!!) 001336000 87 Nov an mm: .. ....I. .523 =2%< :2 .8302, MES. mEmD venom two; .855 2: 825 macroecoa Eon—gum H om.-.» 83m...— NQvuumcmoz ...... movaupcmos. .......... 82328.2 . ti... 89328.2 ...... 8m... 2.8.2 2. it 823282.I.l 82328.2 ...... 8.528s. III c9220 VIII .2... 8586 3.5. 88 2: 8 Svcsom .823 ._mo_to> mum—2 wEmD venom 5.3..— =SEm< .. _ mo Emum cocoocoa . am _ -v 233m .2... 8:85 3.5. 8.0. . 8.9 .. 3.9 82328.2.L.I T . 8. a N 282 ...... 8 o. 8:3..82III .. c9520 8.? u....I.I...I. M Huh. I . t . - IIIII “.1... So. I I a .I .I w I a I I .I o 8 9. 9. mm on 8 8 m. o. m o ('0!) “003930 89 woman 23.2 . V.-. .I now .u N 58.2 ...... NON an r saw: I. l I 5.550 com a Emucsom .88.... 28.2.; was 8.5 .855“. .93 28%... ... .0 .225 8.888. M 2m . -v 2.8.“. .2... 35.3.0 53¢ "n‘fl IIIIlIlHHHHHHNHHHXI 1. IIIII Ilill'll'lol” Illl om 9. 8 mm 8 mm m. o. o mod. 5.0. ('9!) "0938000 90 cow .8 bmvcsom .883 ..8.to> mfg. wEmD .855... 6.81.2833 .. . .0 imam cotoocon. H 02% 95w... .2... 85:5 32¢ \ ('0!) nonnauao 833..82.|.H . h 8.0. 8.. a N 28.2 ...... \~ 8.. a 28.2 I I I \. c9520 \\ Nod. . u...” .I.”... H ”IHNAH» 1a.”... “H.” IIIIII 8.0. I II I Iv I I I I I o 91 locations. Figures (4-14a) through (4-14c) and Figures (4-15a) through (4-15c) illustrate similar results for the 5 in. and 10 in. AC pavements respectively. 4.3.3 Strains 4.3.3.1 Radial Strain at the Bottom of the Asphalt Layer Table (4-8) displays the values of radial strain at the bottom of the AC layer for all the pavement sections analyzed using MPIE vertical. As shown in Figures (4-16a) through (4-16c), in the 1 in. AC pavement, increasing the fineness of the finite element mesh improves the estimate of radial strain only slightly, while increasing the distance to the lateral boundary worsens the estimate. Mesh 3 with a lateral boundary located at 10a shows the most improvement, reducing the error slightly from 24.4% to 24.2%. In the 5 in. AC pavement increasing the mesh fineness improves the estimate of radial strain, however increasing the distance to the lateral boundary has virtually no effect on this value. Using Mesh 3 with a lateral boundary located at 20a reduces error from -5.5% to - 3.6%. The estimate of radial strain in the 10 in. AC pavement is improved both by increasing mesh fineness and by increasing the distance to the lateral boundary, however there is no difference in the estimates found using a lateral boundary at 20a and 40a. Mesh 3 with a lateral boundary located at 20a reduces error from -10.3% to -5.7%. 4.3.3.2 Vertical Strain at the Top of the Subgrade Table (4-9) displays the values of vertical strain at the top of the subgrade for all pavement sections analyzed using MPIE vertical. As is shown in Figures (4-17a) through (4-17c) for the 1 in., 5 in., and 10 in. AC pavements, increasing the fineness of the finite element mesh increases the estimated value of vertical strain. Increasing the distance to 92 no. .2. baccsom .8823 ..8.to> NEE $5: .250... 8...... ..§...m< ..m ..o 5.8m 2.0.82.2. H .3 . -v 23m... .2... 8..qu 33¢ mmod. I- 8.... 0. I 19...\ . . g .900. m 8:3..82.1.I m 82328.2 ...... 111...... m 8.3.2.85. III Illlllllpl l?“1..w.....un....1 .. 3.0. m. ....I....n|... Mural- ( £95050 III-III! .- \\ \\ .. mood. I I .I- II I u I I .II I .I a h .I .. 0 mm on mm ON 9 0.. o 93 88328.2.I.I 883282.... 38.3.8.2 111 :05sz ..8.E> 8...). 8.5 38... 8.3 22...... ..m .o 38 8.888 H 2.34 as»... com .a bmvczom .883 .2... 8.5.... 53¢ \\I. 83- I: 8.... 8...... 4 ‘l‘l‘I‘H-h‘hli “‘0 2. IIIJWMMnHIJVI \ 5.... \\ 8...... I. n n I. h n a I. I . I U I n n o .8 on 8 m. o. m o ('01) 001100900 94 Gov 3 >395an .8054 I_3_to> mum—2 wEmD 250m “981— éfiafix ..m mo :anm cozoocoa H 03% Esmfi E... 85:5 .23. Rod- \\ 8.9 C. .. IR , I flame mad. uduv \ : §.um;$2.l.l A. 8.10 u :82 ...... 83“ v 58.2 I I I huh“ .1...“ , 5o. . I 53on .hhu. “Wuhan “MIN...“ fl...n rh.|\\ . \\ mood- I 0 Ir 0 III 0 m. o. o I'm) uomauaa 95 2: 8 baucsom 1223 ._wo_to> mam—2 wEmD ucsom his =a€m< :2 mo Emmm cocoa—ED H mm 7v Esmfi E... 8:53 3.5. So- 4 i \h- mPOdn Imozmmfius..l.l : no. a m 58.2 ...... \i‘ ., mozmzmozlll xx ‘ c9520 \ 11¢.“ a .II r\“fl III‘I.‘ ‘0 . IIII..I.I.H..UI.\I 8o. ail-OMI-IO‘I III-.I.).- \ i I II .II , I II I II 83. om mv 0v mm on mm ON m9 ('0!) “0933050 96 mom 3 bmncsom .883 ,_8Eo> ma: mag: 28“. is fifié .2 mo 53m aeséom H pm 3 05mm 2... 85.»... 3.5. -8~.amc$2.l.l «83:82 ...... 89353.2 III 53on ‘I, \\ No.0- ,. 20.0. 5.0- mp o— mood- ('0!) “WK! 97 39. a baucsom 2283 ._8_E> ma: mam: 28¢ .93 :93? ..2 96 £QO Susana H on _ -v 2&5 E... 855.. 33¢ No.0- 4, 4 f \‘NI m—Iod- f If 833:8:.I.H I, «3 an N 50.2 ...... \\\1\ : MD? «a — £82 I. I I comwoco I 3 III \\ 5.0. , ‘Itll'll‘fifljlljl‘h\ .T 4 1 r [1 L 4 I v I L 4 4p 1 m8.0| ('0!) “0939030 Table 4-8 : 98 Values of Radial Strain at the Bottom of the Asphalt Layer Under the Wheel Load Found Using MPIE Vertical Lateral Boundary at 10a Lateral Boundary at 200 Lateral Boundary at 400 Asphalt Mesh Radial Percent Radial Percent Radial Percent Thickness Type Strain Error Strain Error Strain Error (in.) (10‘) (10‘) (10") Chevronx 422.8 - 422.8 - 422.8 - l Mesh 1 525.8 24.4 527.9 24.9 534.5 26.4 Mesh 2 528.0 24.9 530.0 25.4 530.0 25.4 Mesh 3 525.3 24.2 527.2 24.7 527.2 24.7 Chevronx 300.8 - 300.8 - 300.8 - 5 Mesh 1 284.4 -5.5 285.5 -5.1 285.4 -5.1 Mesh 2 285.4 -5.1 286.5 -4.8 286.5 48 Mesh 3 289.0 -3.9 290.1 -3.6 290.0 -3.6 Chevronx 122.8 - 122.8 - 122.8 - 10 Mesh 1 110.2 -10.3 114.3 -6.9 114.9 -6.4 Mesh 2 111.4 -9.3 115.2 -6.2 115.3 -6.1 Mesh 3 112.2 -8.6 115.8 -5.7 115.9 -5.6 Table 4-9 : Values of Vertical Strain at the Top of the Subgrade Under the Wheel Load ' Found Using MPIE Vertical Lateral Boundary at 100 Lateral Boundary at 20a Lateral Boundary at 40a Asphalt Mesh Vertical Percent Vertical Percent Vertical Percent ‘ Thickness Type Strain Error Strain Error Strain Error (in.) (10") (10") (10‘) Chevronx -732.4 - -732.4 - -732.4 - l Mesh 1 -658.7 -10.1 -693.8 -5.3 -705.1 -3.7 Mesh 2 -692.6 -5.4 -728.4 -0.5 -729.2 —0.4 Mesh 3 -706.2 -3.6 -742.0 1.3 -742.0 1.3 Chevronx -377.1 - -377.l - -3 77. l - 5 Mesh 1 -321.9 -14.6 -364.2 -3.4 -37l.9 -l.4 Mesh 2 -335.0 -1 1.2 -375.7 -0.4 -377.0 0.0 Mesh 3 -342.9 -9.1 -383.5 1.7 -383.8 1.8 Chevronx -l81.7 - -l81.7 - -181.7 - 10 Mesh l -l42.l -21.8 -185.0 1.8 -l89.4 4.2 Mesh 2 -l47.9 -18.6 -l90.0 4.6 -l90.6 4.9 Mesh 3 -150.1 -17.4 -l92.l 5.7 -l92.3 5.8 99 Eomto> m5: 953 950"— 534 “—29.3. 1 mo Eozom 8 Exam 33M N no Tv 05m:— Eaucaom .283 .0 .3830; no F 59. wow mug waxed 100 38> m2: was 28¢ .93 2.232 .20 528m a £85 362 H 82. 23mm E23509 .283 3 c2280.. «9. now now 0.0. 10113 wanna 101 IMesht IMesh2 DMeshB Joua waned 10a Location 0! Lateral Boundary Radial Strain at Bottom of 10" Asphalt Layer Found Using MPIE Vertical a 0 Figure 4-16c 102 .83 2.2%.... .._ ._8_to> NEE mammD 950m 3335 we non. an 585 Homto> ” «:6 Saw:— zaucaom .23... .o 5.88.. 84 8a 8. m 522 D N cum—2 I P 522' q»- ON? 06 F. 10113 wanna 103 n 82D N £8.2- . £85.: a .93 22%... ..m _ao_to> NEE wEmD 250m 3835 .«o QoI—I S Eabm _¢o_to> H 92.,» oSwE Eon—Son .233 .o .3330.— 8N mop «r- 06 F- 06 P- 0N P- 0.0..- 0.0. Cd. 0.? ON. 06 ON mug weaned 104 .931— :m:%< zo— ._¢o_to> NEE mammD 250m ocmcwnsm mo nob 3 58m .momto> H only 8:3..— Euvcaom .233 .o .8330.— «on 8.. 9mm- Tll.'tl:lil r O.ONI - o.m P- - QB- - 9m. - 0.0 o.m 0.0. 10113 waxed 105 the lateral boundary improves the accuracy of the estimated value substantially, although the estimates found using a lateral boundary located at 20a and those found using a lateral boundary located at 40a are virtually the same. In the 1 in. and 5 in. AC pavements, Mesh 2 with a lateral boundary located at 20a reduces error from -10.1% to 1.3% and -14.6% to 1.7% respectively. In the 10 in. AC pavement, Mesh 1 with a lateral boundary located at 20a reduces error from -21.8% to 5.7%. 4.3.4 Stresses 4.3.4.1 Radial Stress at the Bottom of the Asphalt Layer Table (4-10) displays the values of radial stress at the bottom of the AC layer for all pavement sections analyzed using MPIE vertical. As is shown in Figures (4-183) through (4-18c), for the 1 in. AC pavement, neither increasing the fineness of the finite element mesh nor increasing the distance to the lateral boundary improves the accuracy of the estimate of radial stress. In the case of the 5 in. AC pavement, increasing the fineness of the finite element mesh improves the estimate of radial stress slightly, while increasing the distance to the lateral boundary has seemingly no effect on the predicted value of radial stress. The radial stress prediction for the 10 in. AC pavement improves with both increase in the fineness of the finite element mesh and extension of the lateral boundary location. There is little benefit in increasing the lateral boundary location from 20a to 40a. 4.3.4.2 Vertical Stress at the Top of the Subgrade Table (4-11) displays the values of vertical stress at the top of the subgrade for all pavement sections analyzed using MPIE vertical . As is shown in Figures (4-19a) through 106 Table 4-10 : Values of Radial Stress at the Bottom of the Asphalt Layer Under the Wheel Load Found Using MPIE Vertical Lateral Boundary at 10a Lateral Boundary at 20a Lateral Boundary at 40a Asphalt Mesh Radial Percent Radial Percent Radial Percent Thickness Type Stress Error Stress Error Stress Error (in.) (psi) (psi) (psi) Chevronx 272.1 - 272.1 - 272.1 - 1 Mesh 1 328.5 20.7 330.2 21.3 334.5 22.9 Mesh2 335.4 23.3 336.9 23.8 336.9 23.8 Mesh 3 332.4 22.2 333.9 22.7 333.9 22.7 Chevromt 220.2 - 220.2 - 220.2 - 5 Mesh 1 199.6 -9.3 200.4 -9.0 200.3 -9.0 Mesh 2 200.3 -9.0 201.1 -8.7 201.1 -8.7 Mesh 3 202.8 -7.9 203.7 -7.5 203.6 -7.5 Chevronx 91.0 - 91.0 - 91.0 - 10 Mesh 1 82.4 -9.5 85.4 -6.1 85.9 -5.6 Mesh 2 83.3 -8.5 86.0 -5.4 86.1 -5.3 Mesh 3 83.8 -7.8 86.5 -4.9 86.6 -4.8 Table 4-11 : Values of Vertical Stress at the Top of the Subgrade Under the Wheel Load Found Using MPIE Vertical Lateral Boundary at 100 Lateral Boundary at 200 Lateral Boundary at 40a Asphalt Mesh Vertical Percent Vertical Percent Vertical Percent Thickness Type Stress Error Stress Error Stress Error (in.) (psi) (psi) (psi) Chevronx -5.51 - -5.51 - -5.51 - 1 Mesh 1 -5.59 1.4 -5.82 5.6 -5.94 7.8 Mesh 2 -5.73 4.0 -5.96 8.1 -5.97 8.3 Mesh 3 -5.84 6.0 -6.07 10.1 -6.07 10.1 Chevronx -2.93 - -2.93 - -2.93 - 5 Mesh 1 -3.14 7.2 -3.43 17.1 -3.50 19.5 Mesh 2 -3.24 10.7 -3.51 19.9 -3.52 20.2 Mesh 3 -3.29 12.4 -3.55 21.2 -3.56 21.6 Chevronx -1.46 - -1.46 - -1.46 - 10 Mesh 1 -l.81 23.7 -2.05 40.1 -2.10 43.5 Mesh2 -l.86 27.1 -2.10 43.5 -2.11 44.2 Mesh 3 -l.87 27.8 -2. 11 44.2 -2. 11 44.2 107 m 582D N 28.2- F. :82: _mo_to> NEE was: Esau his: swim/x .._ mo Eozom S $0.5m 33% H uwTv 83mm.”— Euucaom .283 .o 5:33 0. .— N mug ruaarad 108 _8Eo> NEE mEmD 250m 553 2933‘ ..m mo Eozom “a mmobm 33¢ H 3:. 22m:— buvcaom .283 .o .3233 Nov NON as 0.9- 0.0. I 06. . of 0.0 roua wanna 109 .mo_to> mg: 9.75 950m .993 239% :3 mo Eozom E mmohm 3qu H ow 7v oSmE Eat—Sam .223 .0 5:33 wow mom «or 0.0 T mflMED J! N :92- . 522! 10113 waxed 110 .93)— :a:%< .._ ._8_to> mam—z wEmD unsou— ovfiwasm he no... 3 9.65m 183.5 > H «07¢ 8sz 9953 .22.... .o 8:83 8a n :82D N cums.- 2.322! 0.0 0.0.. ON- .Ioua wowed 111 n, swam N 58.2- @223 his :2%< ..m ._~o_to> MES mEmD 250m ocewnsm we mob 3 $85 _8_to> H paTv 2sz E5550 .233 .o 5:33 mow mug wanna 112 n fiosfi N 50.2- $823 .934 :~:%< ..o_ 48E“; 9% 963 950m onSwnsm mo mob “a 38m 185.5 > n 09% 2:3..— .t-ucaom .283 .0 5:33 8N JOLIE mauled 113 (4-19c), increasing the fineness of the finite element mesh increases the error in the predicted value of vertical stress, as does increasing the distance to the lateral boundary. 4.3.5 Conclusions When using MICHPAVE with vertical infinite elements to analyze pavement responses to wheel loadings, increasing mesh fineness and extending the radial distance between the center of the wheel load and the lateral boundary reduces error in predicted vertical strain at the top of the subgrade, but shows no significant improvement in any of the other key responses. Furthermore, the depth ratio a must be determined and calibrated based on a known response of the pavement. In this case the model was calibrated by trial and error using the surface deflection under the center of the wheel load found using the Chevronx program. The depth ratio on was held constant for all infinite elements. Varying the depth ratio for each infinite element may improve the accuracy of the tested responses, but calibration of such a model would increase the computational effort greatly. 4.4 COMPARISON OF MPFB AND MPIE VERTICAL WITH REFINED MESHES The results found using MPFB and MPIE vertical were compared to see which method of modeling the roadbed soil produced more accurate results. For the 1 in. and 5 in. AC pavements, Mesh 3 with a lateral boundary located at 20a was used to model the pavement section, while Mesh 2 with a lateral boundary located at 40a was used to model the 10 in. pavement section. Figures (4-20a) through (4-20c) illustrate the deflection basins associated with each of the three asphalt layer thicknesses analyzed. Figure (4-21) compares the error in the radial strain at the bottom of the asphalt layer, Figure (4-22) 114 f . ._8_to> ma: can dam—2 £5320 mammD 250m 534 2933.. .. _ m0 235 sauce—«on ” uomé Esmfi com 3 5950mm .823 .m :82 A c: 00:55.0 3.0-1 1 _m0_to> was. .mom 3 m cam—2 ...... mum—2 .mom a m 28.2 I I I c0320 III]. .." ........ ...i...... 8 q (1!!) 00990000 5.0- m— 0.. 115 32t9>w:$‘ .mom a m 58.2 . mum: .mom a m :82 I I I 8320 l llllj ._mo_to> NEE use .932 £20325 wEmD vcsom c993 zeaam< ..m we mEmmm :ozoocoa H nomé Eswi com a baucsom 3823 .m :82 :5 85:5 i=5. om ('0!) normal: 116 III _8_5> was. «$5292.. . . . . Ems. .833 522 I I I £95020 f. Eli l. -lllll 28 .952 080325 $53 255 .993 220?. ..o_ 00 SEE couoocon H com-v oSmE 8.. a $2.5m .833 .N :32 .aoEmS ma: 2... 8:85 .33. 0F 0 5.0- v 5.0- N 5.0- 5.0- 000.0- 000.0- I'UII unnamed 117 compares the error in the vertical strain at the top of the subgrade, Figure (4-23) compares the error in the radial stress at the bottom of the asphalt layer, and Figure (4-24) compares the error in the vertical stress at the top of the subgrade. In each case, the values found using MPFB are more accurate than those found using MPIE vertical. 4.5 SUMMARY AND CONCLUSIONS Two improvement strategies were tested to see what effect they had on the values of several key pavement responses found using MICHPAVE. These strategies were: 1) increasing the lateral boundary of the finite element mesh, and 2) increasing the number of elements used in the finite element mesh. It was found that computation error in stiff pavements (e.g., the 10 in. AC pavement) is sensitive to the location of the lateral boundary because the stiff AC layer transmits load in the lateral direction far from the point of application through its bending action. Thus, for stiff pavements, the lateral boundary should be placed at a radial distance of 40a. For flexible pavements (e. g., the 1 in. and 5 in. AC pavements), placing the lateral boundary at a distance of 200 is sufficient. However, computation error in pavements with a very thin AC layer is sensitive to the fineness of the finite element mesh and a mesh such as the one used in Mesh 3 is recommended. The flexible boundary performed better than did the vertical infinite elements when the depth ratio was held constant for all infinite elements. Calibrating the infinite elements using different depth ratios for each infinite element may improve the results yielded by 118 _8_E> m5: 25 9:: ma: 28m .93..— 2294 2: mo Eotom 2: E 25% Easy— mo 538800 N _N-v Eswi E... 32:25 =53: 2 m . _m0_t0> NEE- mun—2D mug waned 119 __8_:o> mas.- f mum—25 _8_E> was as 9&2 was: 2.8m 2:233 05 no gob of a 285m _8_to> mo 533800 N mmé 059m 2... «852.: .532 o_. m F NI ~0- Iona wanna 120 _8_to> BE 28 man: was: venom 5.34 2232 of mo Eozom 2: 8 $0ch 3me mo 538800 N mmé 8sz fig: moo—.803... zuzam< o P m — _mo_to> MES.- mum—en OT mN 10113 maxed 121 .mo_to> MES.- mums—D 35> ma: 23 FE: mam: 258 ocacwnsm 2: we no... 05 “a $25 _8_to> «0 538800 ” vmé «Cami :5 «85¢...» asaé S m «>- A A A fif fiy Mm. A AA A v A A A A v V v v ow mug moaned 122 this method of modeling the subgrade, but would be cumbersome and computationally inefficient. CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS In an attempt to reduce the error in predicted values of pavement responses found when using MICHPAVE, three improvement techniques were employed. The first technique involved replacing the flexible boundary used to model the roadbed soil with infinite elements. The second technique involved increasing the radial distance between the center of the wheel load and the lateral boundary. The final technique involved increasing the number of finite elements in the mesh used to model the pavement section in order to improve the aspect ratios of the elements. Each technique was tested on its own as well as in combination with the other techniques to see what effect they had on five chosen pavement responses. These responses were surface deflection, radial stress and strain at the bottom of the asphalt layer under the wheel load, and vertical stress and strain at the top of the subgrade under the wheel load. The results were compared with results found using the Chevronx elastic layer program. The main findings of this research are: (1) While vertical infinite elements with a constant depth ratio for all element produce reasonable results when used to model the roadbed soil, the results found using a flexible boundary are more accurate for a refined mesh with a lateral boundary located at 20a or 40a. Further, the flexible boundary has the advantage of not requiring any calibration. This is more desirable because the depth ratio calibration that must be 123 124 done when using infinite elements is tedious and inefficient, especially if it must be done manually. In general, radial infinite elements did not perform well. (2) The location of the lateral boundary with respect to wheel load has a significant effect on the accuracy of the results found by MICHPAVE. A finite element mesh with a lateral boundary located at a radial distance of 10a, as is used in the original version of MICHPAVE, is not adequate, especially for stiff pavements. It is recommended that a lateral boundary located at 20a be used when modeling a flexible pavement, while a lateral boundary located at 40a be used when modeling a stiff pavement. (3) Increasing the number of elements in the finite element mesh to improve the aspect ratios of the finite elements significantly improves the results found by MICHPAVE. It is recommended that no element have an aspect ratio of greater than 3: 1. Although the infinite elements did not perform better than the flexible boundary for refined meshes, they have the potential to yield better results for a coarse mesh with a lateral boundary located at 10a, such as that used in the original MICHPAVE program. In such a situation, however, it would be better if the depth ratio is varied from one infinite element to another in an appropriate manner. If such a model is to be implemented then it is recommended that an automatic method of calibrating the depth ratios be developed. In general, if the available computer memory is sufficient for a refined mesh and a lateral boundary placement at a distance of 20a to 40a from the center of the load, the use of the flexible boundary produces good results and is recommended. LIST OF REFRENCES LIST OF REF RENCES Baladi, G.Y., and Snyder, M., “Highway Pavements,” A Four Week Course Developed for the FH WA, January, 1989. Bathe, KI, and Wilson, E.L., “Numerical Methods in Finite Element Analysis,” Prentice Hall, Engelwood Cliffs, NJ, 1976. Bettess, P., and Bettess, J.A., “Infinite Elements for Static Problems,” Engineering Computations, Vol. I, No. 1, 1984, pp. 4-16. Cho, Y.H., McCullough, BF, and Weissman, J., “Considerations on the Finite Element Method Application in Pavement Structural Analysis,” Transportation Research Board, 75th Annual Meeting, January, 1996. Cook, R.D., “Concepts and Applications of Finite Element Analysis,” John Wiley & Sons, Inc., 2.... Ed., New York, 1981. pp. 2945, 113-124. Harichandran, R.S., Oral Communications, Dept. of Civil and Environmental Engineering, Michigan State University, East Lansing, MI, 1995. Harichandran, R., and Yeh, M.S., “Flexible Boundary in Finite Element Analysis of Pavements,” Transportation Research Board, 67'” Annual Meeting, Paper No. 870116, January, 1988. Huang, Y.H., “Pavement Analysis and Design,” Prentice Hall, Engelwood Cliffs, NJ, 1993. 126 Yeh, M.S., “Nonlinear Finite Element Analysis and Design of Flexible Pavements,” Department of Civil and Environmental Engineering, Michigan State University, East Lansing, MI, 1989. Yoder, E.J., and Witczak, M.W., “Principles of Pavement Design,” John Wiley & Sons, Inc., 2"cl Ed, New York, 1975. "Illlllllllllllllllllllll