DEVELOPMENT OF A FAST AND COST - EFFECTIVE ASPHALT MIXTURE FATIGUE TEST SYSTEM By Aksel Seitllari A DISSERTATION Submitted to Michigan State University In partial fulfilment of the requirements for the degree of Civil Engineering Doctor of Philosophy 20 20 ABSTRACT DEVELOPMENT OF A FAST AND COST - EFFECTIVE ASPHALT MIXTURE FATIGUE TEST SYSTEM By Aksel Seitllari Fatigue cracking is one of the critical distress modes in asphalt pavements. Accurate prediction and evaluation of fatigue performance are crucial to extending the service life of asphalt mixtures. Naturally, laboratory testing methods for fatigue characte rization are time - consuming and require sophisticated procedures. Any effort to improve the speed and quality of the information gained from laboratory fatigue tests is valuable. This research work presents the results of a study investigating the possibil ity of implementing a new approach to characterize asphalt mixture fatigue behavior. This new approach includes cyclic tests run on cylindrical asphalt specimens in three - point beam mode (herein referred to as three - point bending cylinder (3PBC) geometry). Timoshenko beam theory along with the viscoelastic continuum damage (VECD) theory w as implemented to model the mechanical response of the specimens. An excellent correlation between the results of 3PBC tests and uniaxial push - pull fatigue tests were obser ved. The 3PBC setup possesses the most advantages of uniaxial push - pull tests and includes more advantages such as not requiring a saw to cut the ends of the sample, not requiring gluing operation (and the gluing jig) and the possibility of estimating Pois 3PBC approach was evaluated through laboratory tests conducted on various asphalt mixtures with varying binder types, mix components and volumetric properties. T he approach proposed herein was validated through finit e element analysis. In addition, ruggedness evaluation of the 3PBC testing approach through varying factors and their levels were investigated and presented. Copyright by AKSEL SEITLLARI 20 20 iv To My Beloved Selma Të Shtrenjtës Sime Selma v A CKNOWLEDGMENTS All praise due to God, the Most Merciful, and the All - that devices the will and materializes it. Little did I know that at one point in time I will be writing this pa ragraph, concluding the end of my doctorate studies. Praised be the Almighty for all the blessings. This research work was performed under the guidance and supervision of Dr. M. Emin Kutay. I would like to thank Dr. Kutay for his continuous support, guidan ce, patience, and dedication through my graduate school. I am lucky to have had him as an example and mentor. I am also grateful to my dissertation advisory committee members; Dr. Karim Chatti, Dr. Neeraj Buch, and Dr. Sara Roccabianca for the feedback the y have provided to improve this work. This experience has been a lot smoother with the support and care of technical staff and department secretaries. This research work was funded by the National Cooperative Highway Research Progra m Innovations Deserving Exploratory Analysis ( NCHRP IDEA) . The program is sponsored by the member states of the American Association of State Highway and Transportation Officials. Their support is greatly appreciated. Family is the place of rest and tranquility, knowledge and lo ve. It is t he safest bay to anchor before the sail. The support and spiritual lectures from my mother (Manushaqe) , teaching notes from my father (Kujtim) , peace (Selma) and trust (Inar) from my siblings instilled in me the desire to pursue this road. It has been much easier with my soul companion (Jonida) and her never - ending support when far and near to complete this dissertation. It is the warmth of my that I could make it so far. vi My doctor broad conversations with my dear friend Dr. Yogesh Kumbargeri. Bits of advice from Dr. Michele Lanotte and Dr. Ilker Boz. The vividness and fellowship of the pavement research g roup (soon to be Drs.) Mahdi Ghazavi, M. Munum Masud, Angela Farina, Hao Ye, Mumtahin Hasnat, and many others who I had the pleasure of knowing. I am thankful as this journey introduced me to great people, with brilliant minds and high spirits. One journe perhaps none. Eager, curious, persistent and thankful for every moment the Almighty blesses me with, until the twinkle that the wayfarer resumes the sail, I will pursue my dream and send good ahead. vii TABLE OF CONTENTS LIST OF TABLES ................................ ................................ ................................ ....................... ix LIST OF FIGURES ................................ ................................ ................................ ...................... x 1. INTRODUCTION ................................ ................................ ................................ ................. 1 2. LITERATURE REVIEW ................................ ................................ ................................ .... 4 2.1 Non - homogenous test methods ................................ ................................ ......................... 7 2.1.1 Four - point bending beam fatigue (4PBB) ................................ ................................ . 7 2.1.2 Trapezoidal beam fatigue (a.k.a. two - point bending beam test) ............................... 8 2.1.3 Supported flexure fatigu e test ................................ ................................ ................... 9 2.1.4 Diametral test ................................ ................................ ................................ .......... 10 2.1.5 Loaded wheel tester ................................ ................................ ................................ 10 2.2 Homogenous test methods ................................ ................................ .............................. 12 2.3 Screening tests ................................ ................................ ................................ ................ 13 2.3.1 Texas overlay tester ................................ ................................ ................................ 13 2.3.2 Indirect tension test ................................ ................................ ................................ . 14 2.3.3 Semi - circular beam test ................................ ................................ ........................... 16 2.3.4 Disc - shaped compact tension test ................................ ................................ ........... 17 3. OBJECTIVES AND RESEARCH PLAN ................................ ................................ ........ 23 3.1 Task 1: Development of the Three - Point Bending Cylinder (3PBC) Test Setup ........... 23 3.2 Task 2: Application of Timoshenko Beam Theory to analyze the 3PBC test results ..... 24 3.3 Task 3: Application of Viscoelastic Continuum Damage (VECD) theory to 3PBC test 24 3.4 Task 4: Ruggedness evaluation of the developed 3PBC test system .............................. 25 4. EXPERIMENTAL PROGRAM AND MATERIALS ................................ ..................... 26 4.1 Experiments for Stage I. Development of the Thre e - Point Bending Cylinder (3PBC) fatigue test system (Tasks 1 through 3) ................................ ................................ ..................... 26 4.2 Stage II. Ruggedness evaluation ................................ ................................ .................... 28 5. DEVELOPMENT OF THE THREE - POINT BEAM CYLINDER (3PBC) TEST SETUP ( Task 1 ) ................................ ................................ ................................ ........................... 31 6. APPLICATION OF TIMO SHENKO BEAM THEORY TO 3PBC SETUP ( Task 2) . 35 6.1 ................................ ............ 35 7. VERIFICATION AND VALIDATION OF 3PBC TEST ( Task 2) ................................ 43 7.1 Verification of Applicability of Timoshenko Beam Formulations to 3PBC setup using 3D Finite Element Analysis ................................ ................................ ................................ ............ 43 7.2 Timoshenko Beam Model Validation Using Laboratory Tests ................................ ...... 47 7.3 Summary of chapter findings ................................ ................................ .......................... 49 viii 8. APPLICATION OF VISCOELASTIC CONTINUUM DAMAGE (VECD) THEORY TO 3PBC TEST ( Task 3) ................................ ................................ ................................ ............ 53 8.1 Results of Uniaxial |E*| Tests using the AMPT ................................ ............................. 55 8.2 Comparison of 3PBC and PP fatigue test results ................................ .......................... 58 8.3 Summary of chapter findings ................................ ................................ .......................... 63 9. RUGGEDNESS STUDY OF THREE - POINT BEAM CYLINDER (3PBC) TEST ( Task 4 ) ................................ ................................ ................................ ................................ ............... 65 9.1 Mixture sampling and sample preparation ................................ ................................ .... 67 9.2 Uniaxial Dynamic Modulus (|E*|) Test ................................ ................................ ......... 68 9.3 Factors and levels of ruggedness analysis ................................ ................................ ..... 70 9.3.1 3PBC test setup designs ................................ ................................ .......................... 72 9.3.2 Air void content ................................ ................................ ................................ ...... 73 9.3.3 Span length ................................ ................................ ................................ .............. 83 9.3.4 Specimen diameter ................................ ................................ ................................ .. 85 9.4 Summary of chapter findings ................................ ................................ .......................... 97 10. CONCLUSIONS ................................ ................................ ................................ ................. 99 11. RECOMMENDATIONS ................................ ................................ ................................ .. 102 APPENDICES ................................ ................................ ................................ ........................... 104 APPENDIX A ................................ ................................ ................................ .......................... 105 APPENDIX B ................................ ................................ ................................ .......................... 123 REFERENCES ................................ ................................ ................................ .......................... 141 ix LIST O F TABLES Table 2.1 Summary of laboratory fatigue test methods ................................ ................................ 18 Table 4.1 Mixture volumetric properties and gradations for asphalt mixtures ............................. 28 Table 7.1 Prony series coefficients for (a) 4E30SBS, (b) 4E3SBS and (c) 4E3DVR at 10 °C and 20 °C ................................ ................................ ................................ ................................ ............. 51 Table 8.1 Dynamic modulus master curve and shift factor coefficients ................................ ....... 58 Table 9.1 Factors and levels of ruggedness analysis ................................ ................................ .... 66 Table 9.2 Experimental design for ruggedne ss testing ( ASTM E1169) ................................ ....... 67 Table 9.3 Dynamic modulus master curve and shift factor coefficients for 5E3 and 3E3 mixtures ................................ ................................ ................................ ................................ ....................... 70 Table 9.4 Factors and corresponding levels for the ruggedness evaluation of 3PBC test ............ 72 Table 9.5 Statistical evaluation on the effect of air void content on fatigue life of asphalt mixture ................................ ................................ ................................ ................................ ....................... 82 Table 9.6 Statistical evaluation on the effect of span length and diameter on fatigue life of asphalt mixture ................................ ................................ ................................ ................................ .......... 82 Table 9.7 Training data set and testing data set input statistics ................................ .................... 93 Table 9.8 Model statistic results ................................ ................................ ................................ ... 94 Table B. 1 MLR model database ................................ ................................ ................................ 123 x LIST OF FIGURES Figure 4.1. Experimental flow chart. ................................ ................................ ............................ 30 Figure 5.1. (a) General schematic of the 3PBC fixture, (b) side view and (c) top view with a loaded specimen. ................................ ................................ ................................ ................................ ...... 32 Figure 5.2. 3PBC test setup with a loaded specimen in the (a) Material Testing System (MTS) and (b) Asphalt Mixture Perf ormance Tester (AMPT). ................................ ................................ ...... 33 Figure 6.1. Exaggerated deflection of a fixed beam with a central load. ................................ ..... 36 and (c) |E*| values with cycles during 3PBC tests. ................................ ................................ ................................ ................................ .............. 41 Figure 7.1. Deformation of the 3PBC test sample simulated in 3D FE (ABAQUS) (Deformation scale factor = 1000). ................................ ................................ ................................ ...................... 43 Figure 7.2. Comparison of (a) |E*| values input to 3D FE and those computed by the Timoshenko - viscoelastic modes. ................................ ................................ ................................ ....................... 46 Figure 7.3. Comparison of (a) |E*| values obtained from |E*| master curve (|E*| AMPT ) and those computed by the Timoshenko beam theory ( |E*| 3PBC ) and (b) the back - compared to the |E*| - (Maher & Bennert, 2008) . . 49 Figure 8.1. Linear viscoelastic properties of the mixtures: (a) log - log scale and (b) semi - log scale plots of dynamic modulus master curves for 4E30SBS, 4E3SBS, and 4E3DVR, and (c) shift factor coefficients as a function of temperature. ................................ ................................ ..................... 57 Figure 8.2. Damage characteristic curves of 4E30SBS for (a) 3PBC and (b) PP test results. ...... 59 Figure 8.3. Damage characteristic curves of 4E3SBS for (a) 3PBC and (b) PP test results . ........ 60 Figure 8.4. Damage characteristic curves of 4E3DVR for (a) 3PBC and (b) PP test results. ...... 61 Figure 8.5. Best fit damage characterization curves for 3PBC and PP test results. ...................... 62 Figure 8.6. Comparisons of (a) the number of cycles to failure (N f ) versus strain relationship of 4E30SBS, 4E3SBS and 4E3DVR asphalt mixtures for 3PBC and PP test at f = 10 Hz, T = 20 o C, and (b) direct comparisons of N f values. ................................ ................................ ...................... 63 Figure 9.1. Linear viscoelastic properties in (a) log - log scale, (b) semi - log scale plots of dynamic modulus master curves and (c) shift factor coefficients as a function of temperature for 5E3 and 3E3 asphalt mixtures . ................................ ................................ ................................ .................... 69 xi Figure 9.2. Ruggedness study flow chart. ................................ ................................ ..................... 71 Figure 9.3. 3PBC test setup with loaded specimens Material Testing System (MTS) with a diameter of (a) 38 mm and (b) 100 mm. ................................ ................................ ....................... 73 Figure 9.4. Lateral displacement limit for 3PBC test. Vertical axis shows the lateral displacement divided by the vertical displac ement in the central clamp, in percentage. ................................ ... 74 Figure 9.5. (a) (c) Damage characteristic curves for 6 %, 7 % and 8 % air void conte nts, respectively and (d) number of cycles to failure (N f ) results at frequency of 5 Hz, temperature of 15 °C and strain level of 150 microstrain, for 68 mm - 125 mm (reference) geometry (5E3 mix). ................................ ................................ ................................ ................................ ....................... 76 Figure 9.6. (a) (c) Damage characteristic curves for 6 %, 7 % and 8 % air void contents, respectively and (d) number of cycles to failure (N f ) results at frequency of 5 Hz, temperature of 15 °C and strain level of 150 microstrain, for 68 mm - 135 mm geometry (5E3 mix). ............... 77 Figure 9.7. (a) (c) Damage characteristic curves for 6 %, 7 % and 8 % air void contents, respectively and (d) number of cycles to failure (N f ) results at frequency of 5 Hz, temperature of 15 °C and strain level of 150 microstrain, for 38 mm - 100 mm geometry (5E3 mix). ............... 78 Figure 9.8. (a) (c) Damage characteristic curves for 6 %, 7 % and 8 % air void contents, respectively and (d) number of cycles to failure (N f ) results at fr equency of 5 Hz, temperature of 15 °C and strain level of 150 microstrain, for 38 mm - 125 mm geometry (5E3 mix). ............... 79 Figure 9.9. (a) (c) Damage characteristic curves for 6 %, 7 % and 8 % air void contents, respectively and (d) number of cycles to failure (N f ) results at frequency of 5 Hz, temperature of 15 °C and strain level of 150 microstrain, for 100 mm - 135 mm (5E3 mix). ............................. 80 Figure 9.10. Damage characteristic curves (C - S) of all geometries plotted for (a) 6 %, (b) 7 % and (c) 8 % air void contents. ................................ ................................ ................................ .............. 84 Figure 9.11. Number of cycles to failure (N f ) results for different length and diameter combinations ................................ ................................ ................................ ......... 85 Figure 9.12. (a) Damage characteristic curves and (b) the number of cycles to failure N f results at frequency of 5 Hz, temperature - 100 mm geometry. ................................ ................................ ................................ ................................ ...... 89 Figure 9.13. (a) Damage characteristic curves and (b) the number of cycles to failure results N f - 125 mm (reference) geometry. ................................ ................................ ................................ ... 90 Figure 9.14. (a) Damage characteristic curves and (b) the number of cycles to failure N f results at - 135 mm geometry. ................................ ................................ ................................ ................................ ...... 91 xii Figure 9.15. Performance of the developed MLR model for (a) training data set and (b) testing data set. ................................ ................................ ................................ ................................ ................. 96 Figure A.1. Mix ID: 5E3, sample dimensions: diameter = 38 mm, length = 100 mm. ............. 105 Figure A.2. Mix ID: 5E3, sample dimensions: diameter = 38 mm, length = 125 mm. ............. 108 Figure A.3. Mix ID: 5E3, sample dimensions: diameter = 68 mm, length = 125 mm. ............. 111 Figure A.4. Mix ID: 5E3, sample dimensions: diameter = 68 mm, length = 135 mm. ............. 114 Figure A.5. Mix ID: 5E3, sample dimensions: diameter = 100 mm, length = 135 mm. ........... 117 Figure A.6. Mix ID: 3E3, sample dimensions: diam eter = 38 mm, length = 100 mm. ............. 120 Figure A.7. Mix ID: 3E3, sample dimensions: diameter = 68 mm, length = 125 mm. ............. 121 Figure A.8. Mix ID: 3E3, sample dimensions: diameter = 100 mm, length = 135 mm. ........... 122 1 1. INTRODU CTION Increasing service life, reducing cost , and improving sustainability of asphalt pavements have always been the goal s of engineers and researchers over the years. E merging new construction technologies and materials provide solutions satisfying the long - term performance goals of asphalt pavement . Typically , the long - term performance of asphalt pavements is evaluated based o n different means . The recent FHWA Distress Identification Manual list s fifteen different distresses in asphalt pavements among which , cracking shares the major part (Miller & Bellinger, 2014) . Since 1948, cracking, caused by the repeated bending distresses in pavements, has been a primary concern for the engineers involved in pavement management (Carl L Monismith, 1994) . In 1955, Hveem (Hveem, 1955) acknowledged the need for assessing the fatigue characteristics of asphalt concrete w hile Monismith et al. (C L Monismith, Secor, & Blackmer, 1961) emphasize d the importance of material characterization on fatigue performance of asphalt pavements. Fatigue phenomenon in asphaltic layers is caused by repeated traffic loading applications and predominantly happen s at intermed iate temperatures. Excessive tensile strains at the bottom and top of the asphalt layers lead to microcracks, which eventually grow, coalesce and lead to serious structural deterioration. Generally , t wo types of fatigue c racking can occur , depending o n the place the cracks initiate. While bottom - up fatigue cracking is mostly observed in relatively thin asphalt layers because of the flexural bending, top - down fatigue cracks can be seen in both thick and thin asphalt layers on the wheel path evolving between the tire edge and the asphalt layer. To better understand and assess the resistance of asphalt mixtures to fatigue cracking, numerous laboratory tests have been developed to simulate the traffic load applications in the field. Traditionally, the four - poin t bending fatigue test mode has been the most common test method to characterize the fatigue resistance of asphalt mixtures (Huurman & Pronk, 2012) . United States 2 practice for four - point bending beam fatigue ( 4 PBB) test follows AASHTO T321 and ASTM D7460 testing protocols . However, in general, these tests are lengthy, cumbersome and expensive. As an alternative, uniaxial fatigue tests (Kutay, Gibson, Youtcheff, & Dongré, 2009; Aksel Seitllari, Lanotte, & Kutay, 20 19b; Zeiada, Kaloush, Underwood, & Mamlouk, 2016) have been gaining wide acceptance for fatigue evaluation of asphalt pavements because of their advantages. Ease of application of c onstitutive models ( i.e., Viscoelastic Continuum Damage Theory) to uniaxial testing geometry has been a great advantage. However , the most challenging issues with the uniaxial testing are (i) two ends of the sample need to be cut parallel and (ii) the gluing end platens using a gluing jig can be cumbersome. As a result, m any end - failures are experienced when sample ends are not cut parallel or gluing is not done properly leading to excessive sample preparation time and consumption of material. While the uniaxial testing is still superior to 4 PBB testing, it is currently no t suitable for as a routine testing alternative for balanced and performance - based mix design approaches. Hence, there is a need for a test which addresses not only the above challenges but is also simple, sensitive to asphalt mix design, repeatable and pr actical. This research study introduces a more practical fatigue testing alternative to the uniaxial fatigue tests and corresponding analyses based on the Viscoelastic Continuum Damage (VECD) theory. E ach aspect of test development is presented individua lly in separate chapters. Chapter 2 presents a comprehensive literature review on various methods of fatigue evaluation of asphalt mixtures . Chapter 3 presents the objectives of this research study followed by a research plan. The experimental program and materials used in this study are presented in chapter 4. The new test set up that was introduced in this study has been explained in chapter 5. Chapters 6 focus es on the application of Timoshenko Beam Theory while c hapter 7 deal s with verification and valid ation of 3 this theory. In chapter 8 the implementation of Viscoelastic Continuum Damage Theory is presented . Chapter 9 discusses the ruggedness results for the test t hat has been developed in this study. In c hapter 10 are presented the summary and conclusions of this study. 4 2. LITERATURE REVIEW This section includes a literature review on cracking characterization methods of asphalt mixtures. Also, a tabulated summary (see Table 2 . 1 ) is provided with the details of each testing approach. Federal Aid Highway Act of 1956 lead to the construction of t he United States I nterstate network , which consequently increase d the truck traffic volume and truck load ing , threatening the in - service life of the pavement structures. WASHO program and AASHTO Road Test shift ed the primary focus on the influence of pavement layers and their effects on the general performance of the pavement (AASHTO, 1962; WASHO, 1955) . Among the outcomes of th e s e project s , cracking was observed to be strongly related to surface (Hveem, 1955) investigate d the relation ship between surface deflection and pavement cracking , acknowledging the necessity for assessing the fatigue characteristic of asphalt concrete . The p ave ment community struggle d with the cracking phenomena for decades and acknowledged the cracking failure to be the primary concern in asphalt pavements (Carl L Monismith, 1994) . Fatigue cracking is one of the main failure modes of pavement structures, which results in degradation of the pavement materials and eventually pavement structure. Repeated traffic loading applications in asphaltic layers cause f atigue phenomenon which primarily happen s in intermediate temperatures. Typically , the fatigue phenomenon manifests itself in two types of c racking . B ottom - up cracking (a.k.a. alligator cracking) initiates from underside the asphalt layer and propagates upward . This type of c racking adhere s to two phases; crack formation and crack propagation. The first phase consists of the formatio n of hairline microcracks leading to stiffness reduction and the second phase is the coupling of the formed micro cracks leading to macro - cracks. The formation 5 and propagation of cracks are caused by the presence of tensile and shear stresses generated by t raffic loads and environmental effects (Prowell et al., 2010) . The s econd type of fatigue cracking is top - down which is mainly observed on the wheel path evolving between the tire edge and the asphalt layer . This failure generally initiates at the surface of the asphalt pavement and propagates through it . This failure type is commonly addressed as a combination of shear strains and surface tension at the tire edge aided by aging and thermal stresses (Lytton, Zhang, Gu, & Luo , 2018) . Proper mix design, structural design, and enhanced material selection can significantly slow down the fatigue cracking and lengthen the life cycle of asphalt pavement s . In recent years, h owever, the growing use of Reclaimed Asphalt Pavement (RAP), warm mix asphalt (WMA) technologies, use of alternative aggregates, binder additives (Ground Tire Rubber (GTR) materials) and other technological advances are changing asphalt concrete beyond the traditional Superpave mix design method and character ization . For these developments to be beneficial , the incorporation of new materials should be well characteri zed to enhance pavement field performance. Performance - based testing of various mixture properties as part of mix design procedures can provide mo re details reflecting the pavement field performance (S. D. Diefenderfer & Bowers, 2019) . Yet , not all of the performance characteristics can be accuratel y characterized using performance - based tests and not many of performance - based tests are ready for implementation as part of mix design methods. According to a recent survey done as part of the NCHRP 20 - 07 project, 36 DOTs (out of 43 responded) consider f atigue cracking as the most common failure mode that the agency wants to address (West, Rodezno, Leiva, & Yin, 2018) . However, in the same survey, 34 of the DOTs also indicated that fatigue cracking test is not required in their mix design specifications. This is primarily because a simple, practical, and robust fatigue test is not available. State DOTs and r oad agencie s are interested in ways to specify asphalt 6 mix designs better in an effort to improve their pavements fatigue life, make the roadway network more sustainable, longer - lasting, and more economical. By developing fatigue performance criteria through a practi cal fatigue test , this goal can be achieved. In the design process of the asphalt mixtures, address ing fatigue performance is crucial . To this end , numerous laboratory tests have been developed to simulate the traffic load applications in the field and pr ovide general information . Usually , the tests are subjected to two types of loading modes ; stress - controlled mode and strain - controlled mode. It is anticipated that thick pavements with high modulus materials behave more like stress controlled mode while relatively thin pavements with softer modulus materials behave more like the strain - controlled mode (C.L Monismith, 1966) . The number of cycles to failure (N f ) has traditi onally been used to quantify the fatigue resistance of the tested material regardless of the loading mode. Considerable effort has been done to describe the point of failure (a.k.a. failure criteria) for characterizing the fatigue performance in laboratory tests. Several failure criteria have investigated by different researchers (Aksel Seitllari, Boz, Habbouche, & Diefenderfer, 2020; Aksel Seitllari et al., 2019b; Soltani & Anderson, 2005; Underwood, B. Shane, 2006; Y. D. Wang, Keshavarzi, & Kim, 2018; Zeiada et al., 2016; Zhang, Sabouri, Guddati, & Kim, 2013) . Sabouri and Kim proposed new energy - based failure criteria denoted as G R which represents the re lationship between the number of cycles to failure and the average rate of the p se udo - strain energy that is released until failure (Sabouri & Kim, 2014) . Kutay et al. (Kutay, Gibson, & Youtcheff, 2008) used numerous commonly used failure criteria to define the point of failure in laboratory tests using both stress - controlled and strain - controlled modes. While for strain - controlled tests 50 % reduction in stiffness was recommended, it was noted that the s tress - controlled tests experienced an inverse relationship with the field data and was not recommended for fatigue characterization. 7 The s election of a laboratory fatigue testing method is crucial in addressing the fatigue distress encountered in constru cted asphalt pavements. The following section provides a detailed discussion o f the current state of practice for most recognized test methods to characterize load - related cracking (fatigue cracking) . The discussion provides characteristics associated with each test method with a detailed s ummar y provided in Table 2 . 1 . 2.1 Non - homogenous test methods These types of fatigue testing are commonly referred to as non - homogeneous test s due to the varying internal stress - strain distribution throughout specimen geome try (Ning Li, Molenaar, Van De Ven, & Wu, 2013) . 2.1.1 Four - point bending beam fatigue (4PBB) Traditionally, the four - point bending beam (4PBB) fatigue test mode has been the most common test method to characterize the fatigue resistance of asphalt mixtures (Huurman & Pronk, 2012; Mateos, Wu, Denneman, & Harvey, 2018) . Upon the com pletion of the SHRP program, the test was standardized hence improving the reliability o f the test results. United States practice for the 4PBB test follows both testing protocols AASHTO T321 1 and ASTM D7460 10 2 . Later, the AASHTO T321 was refined after the original work proposed by Professor Monismit h at the University of California Berkley (Tayebali, Deacon, Coplantz, Harvey, & Monismith, 1994) . While the loading mode of the ASTM standard is haversine, the recent version is due and revisions are required to match the loading mode with AASHTO protocol (Braham & Underwood, 2016) . 1 Standard Method of Test for Determining the Fatigue Life of Compacted Asphalt Mixtures Subjected to Repeated Flexural Bending 2 Standard Test Method for Determining Fatigue Failure of Compacted Asphalt Concrete Subjected to Repeated Flexural Bending 8 According to AASHTO T321, the 4PBB test s pecimen dimensions are 380 ± 6 mm in length, 50 ± 6 in height and 63 ± 6 mm in width. The test is performed in strain - controlled or stress - controlled mode at a certain frequency (i.e., 10Hz) . After mounted in the apparatus, the beam is clamped in four loca tions and loaded in the two inner clamps. As a result , the center portion of the beam is subjected to pure bending deformation. For each loading cycle, t he change in local stiffness varies for every unit volume from a maximum compression to maximum tension , which is believed to be similar behavior of an asphalt layer in the field. However, in general, flexure tests are lengthy, cumbersome and expensive. The extensive material requirement for sample preparation, difficulty in meeting air void target, a large number of samples needed for testing, and excessive equipment cost are some of the challenges encountered when running these tests (Chiangmai, 2010; Zhou et al., 2016) . The 4PBB test has proven to be sensitive to mix design properties and testing conditions. Further, it serves as a key test in mechanistic - empirical fatigue pavement design approaches, which are used to estimate the pavement performance for various distress mechanisms through its design life (ARA Inc. ERES Consultants Division, 2004; Ullidtz, Harvey, Tsai, & Monismith, 200 6) . 2.1.2 Trapezoidal beam fatigue ( a.k.a. t wo - point bending beam test ) The flexural bending fatigue test on trapezoidal beam specimens has been common ly used as a fatigue testing approach in Europe . Several research groups have conducted extensive research on trapezoidal beam fatigue tests under both stress and strain - controlled modes (Van Dijk 1975; Verstraeten 1972; Rowe 1993) . This practice is detailed by the European standard EN 12697 3 . The size of tested trapezoidal beam specimens depend s upon the nominal maximum aggregate size of the gradation mixture. Generally, the dimensions at major base cross - section are 55 mm by 20 mm, 3 Bituminous mixtures Test methods for hot mix asphalt. Part 24: Resistance to fatigue 9 the minor base cross - section is 25 mm by 25 mm and height of 250 mm. The specimens are sawed from compacted asphalt mixture slabs. However, the fabrication of req uired quality test specimens with the correct dimensions is a challenging task in this test . Prior to testing, the major base is attached to a metal plate in an upright position while a thin metallic plate is glued to a minor base and connected to a load cell. The specimen is then subjected to sinusoidal loading in strain - controlled mode. Similar to the 4PBB test, the number of cycles to failure is usually recorded when stiffness value reduces by half of its initial value. Typically, the test requires a mi nimum of 18 samples to be tested. In addition to difficulties in meeting the volumetric criteria, p remature failure due to improper gluing operations and geometry imperfections demand a higher number of replicates. The test was proven sensitive to the visc oelastic properties of the asphalt mixtures and binder content and is of widespread use in the French pavement design method (Verstraeten, 1972) . 2.1.3 Supported flexure f atigue test The flexure fatigue tests are supported in multiple ways intending to simulate similar behavior of an asphalt layer in the field. Researchers have applied different geometries and supports to investigate any potential relationship between the stresses/strains and fatigue life of the asphalt mixtures. Researchers at Ohio State University developed a circular slab specimen setup supported on a rubber mat (Majidzadeh & Kauffmann, E. M Ramsamooj, 1971) . The specimens were subjected to re peated load at the center of the specimen. Similarly, in another study rectangular asphalt beams were supported on a rubber mat (Barksdale & Miller, 1977) . Before testing, the setup was conditioned using the environmental chamber and subjected to a haversine load pulse. T hese test methods were better representing the field conditions but at the same time challenging for routine use. The complexity of the test, setup high cost, testing time and 10 complicated test machine were some of the serious concerns constraining further development of these methods. 2.1.4 Diametral test The diametral test (a.k.a. indirect tensile test) is an alternative fatigue testing method to evaluate the fatigue performance of asphalt mixtures using cylindrical specimens. The dimension s of the test specime n are 100 mm diameter and 64 mm thick which make s it easy to prepare and/or extract from road cores. Typically, the cylindrical samples are subjected to sine/haversine diametral load pulse at a frequency of 20 - 120 cycles per minute. The test can be run using any hydraulic or pneumatic system and is relatively fast. The Center for Highway Research at the University of Texas at Austin conducted extensive research on the implementation of this test method in evalua ting fundamental properties of asphalt mixtures (Cowher & Kennedy, 1975; Kennedy, 1977; Moore & Kennedy, 1971) . The geometry of this test provide s a biaxial stress system which might be a better representative of the field conditions. Nonetheless, several concerns were raised including the potential permanent deformation of the test specimen and underestimation of fatigue life o f asphalt pavements if principal tensile stress is used as a fatigue evaluation criteri on . 2.1.5 Loaded wheel tester The loaded wheel tester (LWT) includes laboratory and in situ wheel test tracks. This test is usually used to evaluate the rutting performance , moisture susceptibility and fatigue resistance of asphalt mixtures . The concept was implemented in Hamburg, Germany. The so - called Hamburg Wheel Tracking Device (HWTD) device was manufactured by Helmut - Wind, Inc. and used to evaluate the rutting and stri pping potential of asphalt mixtures (Aschenbrenter, 1995) . Similar ly, the same concept w as explored by different agencies over the years including the French Rutting 11 Tester (FRT), Third - Scale Model Mobile Load Simulator (MMLS3) and Asphalt Pavement Analyzer (APA). The lat ter was a joint effort of the Georgia Department of Transportation (GDOT) and the Georgia Institute of Technology (GIT) to evaluate rutting susceptibility of asphalt mixture. The differences among these approaches are mainly on the dimensions of te sted specimens, tire type , and pressure , and testing conditions, however, the concept is similar. Generally, a loaded wheel rolls back and forth over an asphalt specimen ( i. e . , beam or cylindrical ) and the rut depth is measured after a certain number of loading cycles. Limited studies on evaluating the fatigue performance of asphalt mixtures using the LWT concept exist in the literature . Van Dijk used a small wheel tracking device to measure the f atigue performance of asphalt mixtures in laboratory condition s (Van Dijk, 1975) . The important outcome of this study was the understanding of fatigue characterization of asphalt mixtures in terms of crack initiation and real cr ack development. A s imilar study was performed by (G. M. Rowe & Brown, 1997) . Researchers at Worcester Polytechnic Institute explored MMLS3 in characterizing the fatigue performance of the asphalt mixtures. The e xperimental result agreed we ll with the numerical analysis (Bhattacharjee, Gould, Mallick, & Hugo, 2004) . Other studies modif ied APA to test the fatigue performance of the asphalt mixtures (Wu, Huang, & Shu, 2014) . The fatigue performance of the mixtures measu red through LWT would rank similar to other fatigue tests (i.e., 4PBB). Nonetheless, the concept had numerous issues including testing time, wheel speed, special equipment requirement and associated cost, failure to measure fundamental properties and rutti ng deformation in low stiffness asphalt mixtures prompt ed its development . Alternatively, accelerated loading test (ALT) facilities were introduced. These facilities can simulate field conditions with actual pavement structure. However, in addition to chal lenges 12 encountered in LWT, ALT has extremely initial, operational and maintenance cost s hampering its advancement . 2.2 Homogenous test methods Homogenous test methods refer to un iaxial fatigue tests on cylindrical specimens. Unlike the non - homogenous tests where the stress - strain application is not uniform thorough the tested specimen, t his approach assumes constant stress state across the specimen section which can be related to fundamental properties of the tested materials . The uniaxial fatigue test syst em was developed in the Transport Research Laboratory (TRL) in tension only and the University of Nottingham in tension - compression loading (Pell & Cooper, 1975; Rai thby & Sterling, 1972) . The uni axial fatigue loading was further explored by different research groups over the years (N Li, Molenaar, Van De Ven, & Wu, 2013; Nguyen, Pouget, Di Benedetto, & Sauzéat, 2009) . In the United States , this approach was utilized by several institutions as an alternative to the standardized 4PBB test (Christensen & Bonaquist, 2009; B. K. Diefenderfer, Bowers, & Diefenderfer, 2015; Kim, Hyon - Jong, & Little, 1997; Kutay et al., 20 08; Prowell et al., 2010; Soltani & Anderson, 2005) . Research programs were developed to characterize the fatigue performance by simply relating the stress/strain to fatigue life (number of cycles to failure) or investigate fundamental theories (i.e., f racture and damage mechanics) which could be used for pavement design and field performance predictions . U niaxial push - pull (Arambula & Kutay, 2009; Kutay et al., 2009) and pull - pull tests (Kim et al., 2018; Zeiada et al., 2016) have been gaining wide acceptance for fatigue evaluation of asphalt pavements because of their advantages. These advan tages include homogenous stress - strain distribution throughout the sample geometry, ability to produce samples using the Gyratory compactor and straightforward application of the constitutive models to predict fatigue 13 performance of asphalt pavements, such as the Viscoelastic Continuum Damage (Chehab, Kim, Schapery, Witczak, & Bonaquist, 2002; Lee & Kim, 1998; Luo, Luo, & Lytton, 2013; Soltani & Anderson, 2005; N. Tapsoba, Sauzéat, Di Benedetto, Baaj, & Ech, 2015; Walubita, Martin, Cleveland, & Lytton, 2006) . However, one of the most challenging issues with the uniaxial testing is that two ends of the sample need to be cut parallel and the gluing end platens using a gluing jig can be cumbersome (Zeiada et al., 2016) . As a result, many of the end - failures are experienced when sample ends are not cut parallel or gluing is not done properly. Since the samples are expected to fail in the center, many of the samples and their results are discarded, leading to excessive sample preparation time and consumption of material. While the uniaxial testing is still superior to FPBB testing, it is currently not suitable for as a routi ne testing alternative for balanced and performance - based mix design approaches. 2.3 Screening tests There exist several tests commonly used to provide a hierarchical classification of cracking susceptibility of asphalt mixtures. The features of these tests d iffer and they are usually employed for screening purposes to reduce the crack related risks for particular asphalt mix design (Aksel Seitllari et al., 20 20) . While these tests are useful in ranking mixtures, they are monotonic fracture tests (or low frequency cyclic in the case of overlay tester) and they cannot be ( been) used to calibrate the field - scale fatigue models ( i.e ., the one in the AASHTOWare Pavement ME Design software). Hence, there is a need for a test which addresses not only the above challenges but is also simple, sensitive to asphalt mix design, repeatable and practical. 2.3.1 Texas overlay tester The Texas overlay test (OT) was initially designed by Germann and Lytton at Texas A&M Transportation Institute (TTI) (Germann & Lytton, 1979) . The primary objective of this test was 14 to evaluate the types of anti - reflection cracking measures. The test setup and procedure were later updated and besides reflective cracking, OT serve s as a test in a balanced mix design system and pavement design for TxDOT (Zhou, Hu, Hu, & Scullion, 2009) . The test specimen size is 76 mm wide, 38 mm thic k and 150 mm long, and can be prepared from SGC compactor or extracted from field cores. Before testing , the test specimen is glued on two steel plates; one fixed and the second one movable in horizontal direction simulating the movements of cracks beneath the overlay. A cyclic triangular waveform to a maximum displacement of 0.006 mm is applied on the horizontally moving steel plate at a frequency of 10 seconds per cycle. The tests are usually conducted at a temperature of 25 ºC. Generally, the crack initiates at the bottom of the specimen and propagates upward. The test stops when a 93 % reduction of initial load occurs or within 1200 cycles whichever develops first . Several studies have indicated a good correlation between the OT and FHWA - ALF fatigue cracking and the structural test sections at NCAT 2006 test track (Hu, Zhou, Scullion, & Leidy, 2012; Ozer et al., 2018; Zhou, Hu, Chen, & Scullion, 2007) . Nonetheless, issues were aroused regarding the high variability of the test attributed mainly to the fabrication quality and precise test speci mens , and improper gluing operation ; necessitating to repeat some of the tests to acquire reliable data . 2.3.2 Indirect tension test The indirect tension test (IDT) has was originally developed and applied to evaluate the thermal cracking performance of asphalt mixtures under the SHRP - A - 407 program (Roque & Buttlar, 1992) . Creep compliance and tensile strength were two parameters derived to distinguish between the mixtures. Subsequently, the test was standardized (AASHTO T322 4 ) and currently is 4 Determining the Creep Compliance and Stren gth of Hot Mix Asphalt (HMA) Using the Indirect Tensile Test Device 15 a key input in mechanistic - empirical fatigue pavement design approaches to estimate the pavement thermal cracking performance through its design life. According to AASHTO T322, the IDT specimens are subjected to a constant deformation rate until failure . Given its simplicity and the relatively inexpensive testing equipment, the IDT test was further investigated to evaluate the asphalt concrete performance at intermediate temperatures . The testing rates and conditioning temperatures vary dep ending on the local practice of the test. Various a nalysis methods were developed to interpret the fatigue cracking performance of asphalt mixtures at intermediate temperatures (Witzcak, Kaloush, Pellinen, El - Basyouny, & Von Quintus, 2002) . Several studies performed monotonic IDT tests on laboratory and field cores. Fracture energy (are a under the normalized stress - strain curve in the loading portion) was shown to correlate well with field fatigue performance of the asphalt pavements (Kim & Wen, 2002; Roque, Birgisson, Drakos, & Dietrich, 2004; Roque & Buttlar, 1992; C. Wang, Castorena, Zhang, & Richard Kim, 2015) . Recently, a new analysis approach for the IDT test, named the indirect tensile asphalt cracking test (IDEAL - CT), was introduced to investigate the cracking resistance of asphalt mixtures at intermediate temperatures (Zhou, Im, Sun, & Scullion, 2017) . In this test, a monotonic load at a constant displacement rate (50 mm/min) is applied to a cylindrical specimen (usually with a 150 - mm diameter), and t he resulting load - displacement curve is analyzed to determine the crack performance index of asphalt mixtures: the CT index . This cracking index has shown good correlations with the cracking performance of asphalt mixtures in the field. Several state agenci es are currently evaluating this test as a potential tool to screen their mixtures for cracking susceptibility as part of their mix design process and/or quality control process (Benner t, Haas, & 16 Wass, 2018; S. D. Diefenderfer & Bowers, 2019) . It should be noted that ASTM D8225 - 19 5 is available for this test. 2.3.3 Semi - circular beam test The s emi - circular beam (SCB) test method was originally suggested only for asphalt mixture fracture pro perty characterization. The SCB test for low temperature is run as per the AASHTO TP105 protocol ; Determining the Fracture Energy of Asphalt Mixtures Using the Semicircular Bend Geometry (SCB) . Since its introduction, the SCB test has emerged as a simple, quick, and reliable test to evaluate the cracking susceptibility of asphalt mixtures . This test uses a monotonic loading mode at a constant displacement rate, and the resulting load - displacement curve is further consi dered to evaluate the susceptibility of an asphalt mixture to load - related cracking. Although there are several different versions of this test ( i. e., tests using different loading rates and notch depths), LTRC method and IFIT method are the commonly imple mented practices (Al - Qadi, Ozer, Lambros, Khatib, & Singhvi, 2015; Mohammad, Kim, & Challa, 2016) . The LTRC method is typically run in accordance with ASTM D8044 6 and the critical strain energy release rate is measured. High values of critical strain energy releas e rate suggest tougher materials which are desirable for fracture resistant mixtures. The test was proven to be sensitive to mix design properties (Nsengiyumva, Kim, & You, 2015) . For the IFIT method , there exists a provisional AASHTO TP124 7 testing protocol. This protocol was 5 Determination of Cracking Tolerance Index of Asphalt Mixture Using the Indirect Tensile Cracking Test at Intermediate Temperature 6 Evaluation of Asphalt Mixture Cracking Resistance usi ng the Semi - Circular Bend Test (SCB) at Intermediate Temperatures 7 Determining the Fracture Potential of Asphalt Mixtures Using Semicircular Bend Geometry (SCB) at Intermediate Temperature 17 introduced in 2016 as the Illinois flexibility index (FI) test. While the FI test has shown the capability to distinguish the fatigue cracking performance of asp halt mixtures, numerous studies have highlighted the high variability of this approach as well as its lack of sensitivity with regard to some mix design variables (Kaseer et al., 2018; Nemati, Haslett, Dave, & Sias, 2019; Zhou et al., 2017) . Further, it has been reported that the post - peak response of brittle materials cannot be captured , hence hindering the calculation of the FI parameter (Zhu, Dave, Rahbar - Rastegar, Sias Daniel, & Zofka, 2017) . 2.3.4 Disc - shaped compact tension test The disc - shaped compact tension (DCT) test is used to determine the fracture energy of the asphalt mixtures at low temperatures by using a disc - shaped specimen. This test has become an ASTM standardized test method (ASTM D7313 8 ) . Even though there is no curren t application of DCT fracture energy on predicting the fatigue cracking performance of asphalt pavements, researchers have shown a positive but we a k correlation with the ALF performance results (Ozer et al., 2018) . DCT sample preparation requires boring holes and notching which, compared to IDT and SCB tests involves a longer time for sample preparation , but its greatest advantage is that the crack path is longer and provides more robust information on the fracture behavior of the mixtures . A t esting time for r unning the test is relatively short and a minimum of technical training is required to perform the test . 8 Determining Fracture Energy of Asphalt - Aggregate Mixtures Using the Disk - Shaped Compact Tension Geometry 18 Table 2 . 1 Summary of laboratory fatigue test methods Test effort Test class Test ID Testing protocol and specimen dimensions Advantages Limitations Equipmen t price Test control mode Performanc e measure criteria Sample preparati on and test duration (days) Test validat ion Non - Homogeneous Four - point bending beam fatigue (4PBB) AASHTO TP 107 L = 380 mm, W = 63 mm, T = 50 mm - Established test method, - Similar to field behavior of asphalt pavement, - Medium training required, - Implemented in pavement design, - Easy data analysis and interpretation - Lengthy, - Costly - Extensive material required, - Troublesome specimen fabrication and precision - High variability ~ $45,000 - Stress controlled - Strain controlled - Fracture mechanics - Fundamental - Breaking stress - Breaking strain - Reduction in stiffness > 20 Field Trapezoid al beam fatigue EN 12697 B = 55 mm x 20 mm, b = 25 mm x 25 mm, L or H = 250 mm - Common testing method in Europe, - Medium training required, - Implemented in pavement design, - Easy data analysis and interpretation - Lengthy, - Costly - Extensive material required, - Troublesome specimen fabrication and precision - High variability - Gluing operation required ~ $45,000 - Stress controlled - Strain controlled - Fundamental - Breaking stress - Breaking strain - Reduction in stiffness > 20 4PBB 19 Supported flexure fatigue test Barksdale, 1977 - Good simulation of field conditions - The test is more time consuming than many other fatigue tests - Lengthy, - Costly - Relatively expensive - Intermediate loading - Fundamental - Breaking stress - Breaking strain - Reduction in stiffness > 20 4PBB Diametral test Kennedy, 1977 T = 64 mm, D = 100 mm - Simple test method, - Similar to field behavior of asphalt pavement, - Low training required, - Same equipment for other purposes - Easy data analysis and interpretation - Easy run on lab & field samples - Repeatable - Underestimate fatigue life - Accumu lation of permanent deformation ~ $50,000 - Stress controlled - Fundamental - Breaking stress - Reduction in stiffness > 20 4PBB Loaded wheel tester Van Dijk, 1975 - Good simulation of field conditions - Fatigue behavior under wheel loads - Crack evolution can be monitored - Lengthy, - Costly - Special equipment required - Accumulation of permanent deformation > $50,000 - Stress controlled - Strain controlled - Breaking stress - Breaking strain > 10 20 Homogenous Uniaxial fatigue test AASHTO TP 107 L = 130 mm, D = 100 mm Texas A&M L = 150 mm, D = 100 mm MSU* L = 130, 150 mm, D = 100, 75, 68 mm - Established test method, - Implemented in pavement design, - Simpler, faster, cheaper if compared to 4PBB - *in Te ns ion - Compre s sion, similar to field behavior of asphalt pavement, - Lengthy, - Costly - Special equipment required - Long training required, - Troublesome specimen fabrication and precision - High variability - Gluing operation required - Complex data analysis and interpretation > $50,000 - Stress controlled - Strain controlled - Fracture mechanics - Fundamental - Breaking stress - Breaking strain - Reduction in stiffness > 10 Field 4PBB Diame tral test Screening Texas overlay tester TxDOT Standard Tex - 248 - F L = 150 mm, W = 76 mm, T = 38 mm - Good field correlation - Medium training required - Implemented in pavement design - Easy data analysis and interpretation - Lengthy, - High variability - Glui ng operation required ~ $45,000 - Displacement controlled - Fracture mechanics - Reduction in load > 3 Field 21 Indirect tension test AASHTO T 322 T = 50 mm, D = 150 mm ASTM D 8225 T = 62 mm, D = 150 mm - Simple test method, - Low training required, - Same equipment for other purposes - Easy data analysis and interpretation - Easy run on lab & field samples - Repeatable - Numerous fatigue performance evaluation criteria - Under development for fatigue evaluation - Not able to implement in pavement design > 10,000 - Fracture mechanics - CT index - TI - N flex Factor > 1 Field Semi - circular beam test AASHTO TP124 T = 50 mm, D = 150 mm, Notch = 15 mm LTRC T = 57 mm, D = 150 mm, Notch = 25.4, 31.8, 38.0 - Rel a tively simple test method, - Medium training required, - Same equipment for other purposes - Easy data analysis and interpretation - Can be run on lab & field samples - Well correlated with fi eld data - High variability - Not sensitive to some of mix design properties - Not able to implement in pavement design > 10,000 - Fracture mechanics - FI - Jc > 3 Field 22 Disc - shaped compact tension test ASTM D 7313 T = 50 mm, D = 150 mm, Notch = 35 mm - Rel a tively simple test method, - Medium training required, - Same equipment for other purposes - Easy data analysis and interpretation - Can be run on lab & field samp les - Well correlated with field data at low temper a ture - High variability - Not sensitive to some of mix design properties - Not able to implement in pavement design > 10,000 - Fracture mechanics - Fracture energy > 3 Field 23 3. OBJECTIVES AND RESEARCH PLAN The o verall goal of this study is to develop a practical fatigue testing alternative to characterize the fatigue performance of asphalt mixtures, sensitive to various aspects of asphalt mix design, repeatable and simple . In order to achieve this goal, the following specific objectives were established: Development of practical laboratory equipment and protocol to characterize the fatigue performance of the asph alt mixtures . Implementation of Timoshenko Beam Theory to analyze the 3PBC test results . Investigate the applicability of Viscoelastic Continuum Damage theory formulations to new test protocol and estimate fatigue live s (N f ) of asphalt mixtures at various strain levels, temperatures , and frequencies from a single test run at one strain level/temperature/frequency combination. s ratio using the test data . P erform ruggedness evaluation of the test . A resear ch plan was devised to materialize these objectives. The tasks for the preliminary research plan are summarized in the sub - section below: Stage I. Development of the Three - P o int Bending Cylinder (3PBC) fatigue t est s ystem As part of this stage, the develop ment of the 3PBC fatigue testing system was discussed . D ata analysis methodologies were detailed and validated for different testing conditions. The specific tasks are summarized below: 3.1 Task 1: Development of the Three - Poin t Bending Cylinder (3PBC) Test S e tu p The objective of Task 1 is the design of a robust 3PBC test setup for testing cylindrical asphalt specimens. The test setup was designed for its seamless integration into the Material 24 Testing System (MTS) and Asphalt Mixture Performance Tester (AMPT) units. Of particular interest was the speed of mounting of asphalt sample into the fixture, which is important to reduce temperature equilibration time. The test setup was made of high - strength steel to avoid any undesirable bending while testing the specimen. 3.2 Task 2: Application of Timoshenko Beam Theory to analyze the 3PBC test results Mathematical formulations were developed based o n the Timoshenko Beam Theory to calculate the dynamic modulus (damaged) for each loading cycle. The developed formulations were validated through a series of 3D f inite e lement ( 3D FE) analyses. Furthermore, numerous 3PBC laboratory tests were performed to validate the theoretical results obtained from the application of the Timoshenko Beam Theory . Also, the developed formulations were used to 3.3 Task 3 : Application of Viscoe lastic Continu um Damage (VECD) t heory to 3PBC t est One of the mai n objectives of this work was to develop a test method to reduce the testing burden. The a daptation of VECD theory in fatigue characterization of asphalt mixtures significantly reduces the experimental bur den required to calibrate phenomenological fatigue life formulation. I mplementation of VECD theory to model the mechanical response of the cylindrical specimens tested using the 3PBC test results as part of this task . Various mixtures were tested at different strain levels, temperatures and frequencies and their N f were measured. The VECD constitutive model w as implemented to construct the damage characteristic curve (C - S), which is a unique curve that can be used to predict the fatigue lif e of asphalt mixtures at different frequencies and temperatu res at a required strain level. Stage I I. Ruggedness e valuation 25 This stage focuses on performing the ruggedness test of the 3PBC test. T o achieve this goal, the task for the execution of ruggednes s testing is presented below: 3.4 Task 4: Ruggedness evaluation of the developed 3PBC test system The important results and inferences obtain from the previous tasks were utilized to investigate the ruggedness of the developed 3PBC test system. This task was performed in general accordance with ASTM E 1169 9 . The ruggedness testing plan executed in this project involves (i) i dentifying the major test factors that may influence the 3PBC test and (ii) d e veloping a statistically sound yet efficient laboratory experimental design . 9 Standard Practice for Conducting Ruggedness Tests 26 4. EXPERIMENTAL PROGRAM AND MATERIALS A laboratory e xperimental program was developed includ ing 3PBC fatigue, linear viscoelastic dynamic modulus (|E*| LVE ) and push - pull (tension - compression) fatigue tests. The work plan was divided into two stages: Stage I involves the examination of the design and robustness of the 3PBC test system, and data analysis methodology. Stage II focuses on ruggedness evaluation of the developed 3PBC test system to identify the effects of certain parameters that influence the test results and determine acceptable limits. 4.1 Experiments for Stage I . Development of the Three - P o int Bending Cylinder (3PBC) fatigue test system (Tasks 1 th rough 3) As part of this stage, three different mixtures with two different binder types, aggregate gradations, and mix designs were tested. The nominal maximum aggregate size (NMAS) of the considered mixtures is 12.5 mm (see Table 4 . 1 ) . Crumb rubber and polymer modified binders with different asphalt contents were used for each mixture. A summary of the mix design characteristics and volumetric properties are provided in Table 4 . 1 . The Superpave Gyratory Compactor (SGC) was used to prod uce all the performance testing samples (i.e., 3PBC, |E*| and PP). All samples had 7 ± 0.5 % air void content after coring and/or cutting. Figure 4.1 illustrates the overall experimental flowchart. In Stage I, t he linear viscoelastic dynamic modulus (|E*| LVE ) tests were conducted using the AMPT, and the |E*| master curves were developed in general accordance with the AASHTO T378 and R84 protocols. All test specimens were compacted to a height of 180 mm using the SGC with a minimum of two replicates per mix ture. The |E*| LVE tests in AMPT were conducted at temperatures of 4 , 20 , 40 and 5 4 º C, and at loading frequencies of 10, 1 and 0.1 Hz. The diameters of specimens for both 3PBC and PP tests were 68 mm. The heights of the PP samples were 150 mm. The heig hts of the 3PBC samples were the same as the gyratory 27 compactor, which was 180 mm. It should be noted that this height may be reduced to save material in the future for 3PBC samples but for consistency between PP and 3PBC test samples, the SGC height was k ept the same. The PP and 3PBC tests were conducted at varying levels of frequency, temperature and strain level combinations ( see Figure 8.2 - Figure 8.4 ). This was done to investigate and validate the applicability of the Viscoelastic Continuum Damage (VECD) theory on the 3PBC samples. The axial deformations of PP samples were measured by means of three LVDTs mounted on the samples at 120 o intervals over t he middle 70 mm of the sample, whereas the central deflections needed in 3PBC formulations were measured using 2 LVDTs placed in the middle clamp (see Figure 5.1 ). 28 Table 4 . 1 Mixture volumetric properties and gradation s for asphalt mixtures Mixture ID Stage I Stage II 4E3DVR (1) 4E3SBS (2) 4E30SBS (2) 5E3Neat 3E3Neat Aggregate type A A B C D Mix Design ESALs (Millions) 3 3 30 3 3 NMAS (mm) 12.5 12.5 12.5 9.5 19.0 Asphalt PG PG70 - 28 PG70 - 28 PG70 - 28 PG64 - 28 PG58 - 28 Asphalt cement (%) 5.7 5.5 5.8 6.3 5.2 N design 86 86 109 86 86 D e s i g n air voids (%) 3.0 3.0 3.0 3.5 3.0 VMA (3) (%) 18.5 19.6 18.8 16.4 13.8 VFA (4) (%) 64.8 63.5 65.2 78.7 78.3 RAP (5) (%) 20 20 15 22 28 Sieve size Percent passing, % inch mm 3/4" 19 100.0 100.0 100.0 100 .0 100.0 1/2" 12.5 91.2 91.2 99.3 99.9 89.2 3/8" 9.5 88.3 88.3 89.4 99.3 82.1 #4 4.75 78.2 78.2 65.9 78.4 61.7 #8 2.36 53.8 53.8 53.3 63.5 51.5 #16 1.18 36.9 36.9 40.9 46.3 37.9 #30 0.6 25.8 25.8 28.4 31.2 25.3 #50 0.3 15.8 15.8 12.9 16.2 13.1 #100 0.15 7.8 7.8 6.2 7.9 7.2 #200 0.075 5.1 5.1 4.5 5.0 4.7 Note: ( 1 ) DVR = De - vulcanized Rubber modified asphalt binder , ( 2 ) SBS = Styrene - Butadiene - Styrene modified asphalt binder , (3) VMA = Voids in Mineral Aggregate, (4) Voids Filled With Asphalt, (5) RAP = Recycled Asphalt Pavement 4.2 Stage II . Ruggedness e valuation As part of this stage, two new asphalt mixture s , particularly 3E3 and 5E3 , were selected and tested. The 3E3 is a base mixtur e with 19.0 mm NMAS and PG58 - 28 binder grade, whereas, 5E3 is a surface mixture with 9.5 mm NMAS and PG64 - 28 binder grade . M ix design details and volumetric properties are shown in Table 4 . 1 . Sample preparation and |E*| test procedure were performed in accordance with the same procedures followed in Stage I. The purpose of the ruggedness testing is to identify the factors that significantly influence the results of the 3PBC test (i.e., that influence number of cycles to failure and damage characteris tics (C - S) curve). Based on previous experiences in fatigue testing of asphalt mixtures 29 and the literature review conducted at the time of preparing this document , several general factors have been identified as factors that may potentially influence the 3PBC test results. These tentative factors and their levels for consideration in the ruggedness experimental design are summarized and discussed . To fulfill this goal, the analysis of the data w as performed in general accordance with ASTM E 1169 standard. Factor effects are calculated as the difference between average responses obtained after conducting 3PBC testing using high and low levels of the factor in question. 30 Figure 4 . 1 . Experimental flow char t. 31 5. DEVELOPMENT OF THE THREE - POINT BEAM CYLINDER (3PBC) TEST SETUP ( Task 1 ) The objective of this part of the study was the design of the 3PBC fatigue testing system. Factors including production cost, easy to use and compatibility w ith different testing units were considered during the design process. Figure 5.1 illustrates the latest Three - Point Bending Cylinder (3PBC) . Test setup, which is composed of a solid base, two fixed end supports used to clamp the sample and a central clamp for application of cyclic (zero - mean) vertical load. Supports and loading clamps are composed of two C - shaped pieces, which are screwed together to hold the asphalt sample in - place. The lower C - shaped pieces are connected (screwed or welded) to the base - strength steel to prevent any undesirable deformation during the test. T he base plate includes four orienting knobs to provide proper placement of the central clamp to ensure that the load is applied at the center of the specimen . The tests in this study were conducted using a servo - hydraulic Material Testing System (MTS) unit as presented in Figure 5 . 2 a. Nevertheless, the 3PBC test setup is very well suited to be placed within the chambers of the Asphalt Mixture Performance Tester (AMPT) unit ( Figure 5 . 2 b). It should be noted that , for verification purposes, multiple 3PBC tests were r u n in AMPT using the uniaxial fatigue protocol in AMPT and very promising results were obtained. A testing protocol well suited with 3PBC test requirem ents is still under development, therefore no results are presented herein. 32 Figure 5 . 1 . (a) General schematic of the 3PBC fixture, (b) s ide view and (c) t op view w ith a loaded specimen. The testing temperature in MTS was controlled through a forced - air draft conditioning chamber. A sinusoidal strain - controlled vertical load (with zero mean) is applied at the center of the specimen through the vertically free - moving clamp as illustrated in Figure 5.1 and Figure 5 . 2 . (a) TOP VIEW SIDE VIEW (b) (c) 33 The displacement measurements required for calculating the maximum strain at the bottom - center (or top - center) of the specimen were obtained using two linear variable displacement transducers (LVDT s ) attached on both sides of the central clamp. The difference between the two LVDTs was observed to be very low. However, during the test, the strain level was controlled through the actuator strain gauge and not LVDTs. This was done to avoid equipment damage due to the potential instabilities of actuator caused by the slow feedback of LVDT measurements. This problem is less pronounced in mos t AMPTs, therefore, on - specimen LVDT controlled testing is quite possible with the AMPTs. Besides , a third LVDT was attached to the top of the fixed supports to measure any potential lateral displacement due to steel bending, which is not desirable. As men tioned earlier, the 3PBC is a zero - mean cyclic strain - controlled test and the test ends when the microcracks propagate through the entire sample diameter (which can be observed visually). The stresses and strains for each cycle were computed using the Timo shenko Beam T heory. Figure 5 . 2 . 3PBC test setup with a loaded specimen in the (a) Material Testing System (MTS) and (b) Asphalt Mixture Performance Tester (AMPT). 34 The current design of the 3PBC test setup possesses the most advantages of the current state of practice tests and includes more improvements . The new test system proposed in this research study is a cyclic three - point beam test setup, where the sample is a cylindrical asphalt specimen prepared using a gyratory compactor. Samples do not need to be cut, i.e., a diamond saw is not necessary. Also, there is no n eed for end - platens and gluing jig. These characteristics reduce the cost of the equipment, amount of material needed to run the test, and overall test duration (from sample preparation to the end of testing) is much shorter than the traditional fatigue te sting alternatives. 35 6. APPLICATION OF TIMOSHENKO BEAM THEORY TO 3PBC SETUP ( Task 2) The Euler beam theory commonly used for the analysis of slender isotropic beams considers the beam kinematics in term s of flexural stiffness. The low aspect ratio of 3PBC requires considerations of shear - induced deformations in the so - - Beam T heory (Cowper, 1966; Timos henko & Gere, 1972) . The a nalytical formulations presented herein for the stiffness of thick 3PBC are based on the following assumptions: The 3PBC is considered as a short beam with both ends clamped The central clamp restrains the sample from bending and moves parallel to end - clamps at a given cycle, during a particular test at a certain frequency/temperature combination (see the end of this section for a discussion 6.1 Timoshenko Bea Initially, the Timoshenko beam theory has been presented herein by considering 3PBC as a linear elastic material. The formulations are then extended to viscoelastic behavior (for a given frequency/temperature combinat ion) using the elastic - viscoelastic correspondence principle (Schapery, 1984) . For a linear elastic, isotropic homogenous slender beam with both ends fixed and loaded at the center by a force , the Euler theory states that the maximum vertical deflection ( ) can be c alculated as follows (Timoshenko, 1940) : [ 6 . 1 ] 36 where, is the maximum vertical deflection, L is the span length, E is the moment of inertia along axis xx. The schematic view of a fixed beam with a central load is illustrated in Figure 6.1 . Figure 6 . 1 . Exaggerated deflection of a fixed beam with a central load. When the aspect ratio (length to diameter ratio) of a cylindrical beam is less than 6, the shear - induced deflection becomes significant and Euler theory does not apply (Du, Lin, Lu, & Zhang, 2010) . For such beams, Timoshenko beam theory needs to be used to calculate the deflection as follows (Timoshenko & Gere, 1972) ; [ 6 . 2 ] w here , is the shear coefficient, A is the cross - sectional area. Hutchinson (Hutc hinson, 2001) derived the following expression for the shear coefficient ( ) for beams with low aspect ratio: 37 [ 6 . 3 ] w here , Equation [ 6.2 ] can be rearranged to yield elastic modulus as a function of the vertical load ( ) and measured deflection at the center ( ): [ 6 . 4 ] For viscoelastic materials that are exposed to cyclic load at a constant frequency, Equation [6.4] can be used to calculate the magnitude of the dynamic modulus at each cycle N (i.e., ) as follows: [ 6 . 5 ] [ 6 . 6 ] w here , is the (damaged or undamaged) dynamic modulus at each cycle N, and are the peak - to - peak load and deflection, respectively. in Equation [ 6 . 5 ] to the dynamic modulus (|E*|) (Maher & Bennert, 2008; N. Tapsoba et al., 2015) . Maher and Bennert (Maher & Bennert, 2008) showed that the following relationship can be used to compute [ 6 . 7 ] w here , and are the slope and intercept of the relationship. One of the important advantages of the 3PBC test is that one can determine the and constants of 38 achieved by running the 3PBC test at a relatively low load level (so that the sample is within linear viscoelastic range) at a few temperatures/frequencies. Then the error between the computed using the Equations [ 6 . 5 ] through [ 6.7 ] (herein referred to as ) and corresponding obtained from the traditional uniaxial dynamic modulus test using the Asphalt Mixture Performance Tester (herein referred to as ) i s minimized by varying the and constants (Equation [6.7 ]) as follows: [ 6 . 8 ] - |E*| relationship from 3PBC tests can be summarized as follows: 1) Run Dynamic Modulus (|E*|) tests following AASHTO T378 and generate the |E*| master curve in accordance with AASHTO R84. 2) Calculate the |E*| corresponding to the frequency and temperature combination for the planned 3PBC test. This |E*| is herein called . 3) Run the 3PBC test at the planned frequency and temperature combination. 4) Assume initial a and b [6.7 ] ). 5) [6.7 ] , by using |E*| computed from the dynamic modulus master curve obtained in step 1 (i.e., ) 6) [ 6 . 6 ] to compute K. Then plug in the computed K (as well as and ) to Equation [ 6 . 5 ] to compute the |E*|. This |E*| is herein called |E*| 3PBC . 7) Compute the difference between and . 39 8) Vary the a and b values, repeat the steps 5, 6 and 7 until the error between and is minimized (Equation [ 6 . 8 ] ). In order to verify this procedure to estimate and constants, a 3D finite element analysis ( 3D FE) was performed to simulate a perfect 3PBC test, using the exact geometry of the 3PBC sample and the fixtures. The results of the 3D FE analyses are shown in the next section. In addition, actual 3PBC tests at a low strain (load) level were also conducted and the and constants computed and compared against the measured and constants reported in (Maher & Bennert, 2008) . Resu lts are presented in the later sections . Once and - |E*| relationship) are estimated for the linear viscoelastic (undamaged) state, they can be used in 3PBC fatigue tests in the damaged state s known not to change significantly during the fatigue tests (N. Tapsoba et al., 2015) . Since both left - and r ight - hand side of the Equation [ 6 . 6 ] includes the |E*|, another from one cycle to another can be used to simplify the computational steps. A summary of the steps of computing |E*| at each cycle (i.e., |E*| N ) as damage grows are summarized below: 1) Using the known values of and Equation [ 6 . 7 ] for cycle N = 1 (initial condition), by using |E*| computed from the dynamic modulus master curve (i.e. ( )). This condition is the undamaged state. 2) (N. Tapsoba et al., 2015) , i t is assumed that . 3) Calculate using in Equation [ 6 . 3 ] and using Equation [ 6 . 6 ] . 40 4) For the next cycle N+ N, compute the damaged modulus, , using Equation [ 6 . 5 ] . 5) For the next cycle N+ Equation [ 6 . 7 ] , by using |E*| computed from the previous cycle (i.e. ( )) 6) Repeat steps 2 through 5 for all subsequent cycles. 41 Figure 6 . 2 and (c) |E*| values with cycles during 3PBC tests . 42 Figure 6.2 s ), (b) and (c) |E*| N values with cycles during 3PBC tests. As shown in Figure 6.2 a , with cycles (as expected), however, the total increase in (at the end of 100,000 cycles) is about 10% of the initial value, which is relatively low. The incr ease in is much lower than that of , where the total increase at the end of 100,000 cycles is less than 1%. Therefore, these values may be assumed as constant to further simplify the formulations of 3PBC. However, this simplification was not done in this particular study for the purpose of keeping the level of accuracy as high as possible. Lastly, it should be noted that the maximum tensile and compressive stresses in the 3PBC sample occur on the farthest point from the neutral axis. For a circular c ross - section the maximum tensile and compressive stresses are equal and can be obtained from the following expression: [ 6 . 9 ] where, is the normal stress, is the maximum bending moment at the center of the beam ( for the 3PBC setup), is the section modulus, which (for a solid circular cross - section) is calculated as (where d=diameter). Even though Equation [6.9 ] was developed for pure bending with no shear presence, studies have shown that the effect of shear on normal stresses is negligible (American Wood Council, 2005; Timosh enko & Gere, 1972) . 43 7. VERIFICATION AND VALIDATION OF 3PBC TEST ( Task 2) In order to be able to verify the applicability of Timoshenko Beam Theory on different asphalt mix samples, t hree - dimensional f inite e lement ( 3D FE) analyses were performed to simulate the 3PBC test, using the exact geometry of the 3PBC sample and the fixture. In addition, t he Timoshenko formulations were validated using the 3PBC laboratory tests and calculate the laboratory - measured |E*|values. The corresponding dy namic moduli at the respective temperature/frequency combinations were compared with the |E*| master curve of the asphalt samples measured using the AMPT. 7.1 Verification of Applicability of Timoshenko Beam Formulations to 3PBC setup using 3D Finite Element A nalysis Three - dimensional f inite element ( 3D FE) analyses were performed to validate the theoretical results obtained from the application of the Timoshenko beam theory. For this purpose, the 3PBC test setup and loading were modeled in ABAQUS . Figure 7.1 s hows a view of the 3D FE mesh used . Figure 7 . 1 . Deformation of the 3PBC test sample simulated in 3D FE (ABAQUS) (Deformation scale factor = 1000). 44 The simulations were run for two intermediate temperatures: 10 ºC and 20 ºC at the following frequencies; 10 Hz, 1 Hz , and 0.1 Hz . The asphalt sample s in the 3D FE simulations were modeled in two modes using (i) elastic and (ii) viscoelastic properties . The first mode (elastic) was selected for its simplicity and computational efficiency . It is worth not ing that the viscoelastic materials (i.e., asphalt) can be treated as elastic materials when subjected to a cyclic zero - mean test with a constant frequenc y. This practice is theoretically valid when the peak - to - peak results (i.e., force, displacement) are used. The respective moduli for each temperature/frequency combination were computed from the laboratory - measured dynamic modulus ( |E*| AMPT ) master curve of the asphalt mixtures utilized in this study, and used as input elastic modulus to 3D FE analysis . and temperature and related to the mixture |E*| (Maher & Bennert, 2008; Nouffou Tapsoba et al., 2014) . In this part of the study, Equation [6.7 ] input to each ABAQUS simulation and also in the application of the Timoshenko beam theory (i.e., Equation [6.6] ). For the ABAQUS simulations, a and b values of - 0.024 and 0.45 were used, respectively. Thes e values were obtained from the reference (Maher & Bennert, 2008) for mixtures with PG76 - 22, which is the closest PG to the mixtures used in this study. Simulations were in displacement control and corresponding forces in the center clamp were retrieved , then used in Equation [6.5] to calculate the |E*|. Since Equation [6.5] performed using the Equations [6.7] and [6.8] to back - calculate the a and b values and these values were compared against the inp Equation [ 7.1 ] : 45 [ 7 . 1 ] In addition, the viscoelastic 3D FE analys e s were performed as an attempt to better simulate the real mechanical behavior of an asphalt mixture under cyclic loading. Therefore, the viscoelastic properties of the asphalt mixtures were assigned using Prony series coefficients. The relaxation modulus Prony series coefficients for each asphalt mixture at a certain testing temperature were obtained through interconversion from |E*| AMPT master curve. Table 7 . 1 shows the relaxation times i ) and dimensionless elastic coefficients (g i ) of the generalized Maxwell model (Prony series) at temperature s of 10 °C and 2 0 °C. Figure 7.2 a illustrates the input |E*|, which were obtained from the measured master curve (i.e. , |E*| AMPT ) and the corresponding |E*| (i.e., |E*| 3PBC ) calculated after ABAQUS simulations using Timoshenko beam theory. As shown in the figure, both the computed elastic |E*| 3PBC as well as the viscoelastic |E*| 3PBC values match very well with the values obtained from |E*| AMPT master curve. While t he in computed elastic |E*| 3PBC ranges from 0.2 % to 1 6 % , the computed for viscoelastic |E*| 3PBC is less than 12 %. The change in error range between the two simulation modes is primarily related to the assigned material properties and their effects on the mechanical response of the asphalt mixture when subjected to loading. The results also indicate that typically, the error increases with the decrease in loading frequency. However, at high frequency (i.e., 10 Hz), the error for both elastic and viscoelastic simulations is smaller than 5 % which is very low considering all the errors that can emanate from sample preparation. As a result , the 3PBC tests are recommended to be performed at a frequency of 5 Hz and above, where the maximum error is smaller compared to lower frequencies . Figure 7.2 ve r sus the back - calculated elastic and viscoelastic for different simulations run at 46 different frequencies and temperatures. As shown, both the back - match very well io . Figure 7 . 2 . Comparison of (a) |E*| values input to 3D FE and those computed by the Timoshenko beam theory and ( - in elastic and viscoelastic modes . 47 7.2 Timoshenko Beam Model Validation Using Laboratory Tests The Timoshenko formulations were further evaluated using the 3PBC laboratory tests. After the 3PBC tests, the Equa tions [ 6 . 5 ] through [6.7 ] were used to calculate the laboratory - measured |E*| 3PBC . The tests were run under cyclic loading with zero - mean at three intermediate temperatures: 10, 15, and 20 ° C at the following frequencies; 5 Hz and 1 Hz. The corresponding dynamic moduli at these temperature/frequency combinations were extracted from the |E*| master curve of the asphalt samples measured using the AMPT. - calculated using the same principle as described in the previous secti on. Figure 7.3 a sh ows the |E*| values that were obtained from the measured master curve (i.e., |E*| AMPT ) and the corresponding |E*| (i.e., |E*| 3PBC ) calculated using Timoshenko beam theory. As shown, the |E*| values match reasonably well s from 0.4 % to 16 %, which are low considering the errors that can be caused by the sample - to - sample variability. Generally, it 48 Figure 7.3 b illustrates the back - and b coefficients retrieved from (Maher & Bennert, 2008) match well with the back - cal cul ated values from 3PBC setup. 49 Figure 7 . 3 . Comparison of (a) |E*| values obtained from |E*| master curve ( |E*| AMPT ) a nd those computed by the Timoshenko beam theory ( |E*| 3PBC ) and ( b) the back - compared to the |E*| - relationships retrieved from (Maher & Bennert, 2008) . 7.3 Summary of chap ter findings The objective of this part of the study was to evaluate the applicability of the Timoshenko theory on 3PBC test results via 3D finite element ( 3D FE) analysis. Also , several laboratory test s 50 were performed to validate the Timoshenko b eam m odel results. The r esearch findings from this study have been summarized as follows: Based on 3D FE analyses , it was shown that the error in elastic |E*| 3PBC ranges from 0.2 % to 1 6 % , whereas for viscoelastic |E*| 3PBC is less than 12 %. The results also indicate that typically, the error increases with the decrease in loading frequency. The 3PBC tests are recommended to be performed at 5 Hz and above, where the maximum error is less than 5 %. Additionally , the back - tio values match very well. The laboratory results indicated that the |E*| values obtained from the measured master curve (i.e., |E*| AMPT ) and the corresponding |E*| (i.e., |E*| 3PBC ) calculated using Timoshenko beam theory match reasonably well . The calculated error ranges from 0.4 % to 16 %. Generally, it was observed that the error increases with the decreasing frequency and increasing temperature. Also, the back - coefficients retrie ved from (Maher & Bennert, 2008) match well with the back - calculated values from 3PBC setup. Overall, this part of the study showed the applicability of the Timoshenko beam t heo ry to analyze data obtained from the 3PBC tests. Also, an i ndirect estimation the 3PBC test data was confirmed (Aksel Seitllari & Kutay, 2019) . 51 Table 7 . 1 Prony series coefficients for ( a) 4E30SBS, ( b) 4E3SBS and ( c) 4E3DVR at 1 0 °C and 20 °C (a) T (ºC), f (Hz), 10, 10, 0.11 10, 1, 0.13 10, 0.1, 0.14 20, 10, 0.12 20, 1, 0.14 20, 0.1, 0.15 G o (Pa) 9.05E+09 8.89E+09 8.74E+09 9.57E+09 9.40E+09 9.32E+09 g i 0.0361 0.0361 0.0361 0.1013 0.1013 0.1013 0.0443 0.0443 0.0443 0.1058 0.1058 0.1058 0.0887 0.0887 0.0887 0.1709 0.1709 0.1709 0.1532 0.1532 0.1532 0.2137 0.2137 0.2137 0.2212 0.2212 0.2212 0.2013 0.2013 0.2013 0.2282 0.2282 0.2282 0.1275 0.1275 0.1275 0.1463 0.1463 0.1463 0.0535 0.0535 0.0535 0.0569 0.0569 0.0569 0.0171 0.0171 0.0171 0.015 0.015 0.015 0.0047 0.0047 0.0047 0.0094 0.0094 0.0094 0.0036 0.0036 0.0035 i (s) 1.0E - 07 1.7E - 06 2.8E - 05 4.6E - 04 7.7E - 03 1.3E - 01 2.2E+00 3.6E+01 6.0E+02 1.0E+04 (b) T (ºC), f (Hz), 10, 10, 0.12 10, 1, 0.13 10, 0.1, 0.14 20, 10, 0.13 20, 1, 0.14 20, 0.1, 0.16 G o (Pa) 8.52E+09 8.45E+09 8.37E+09 8.97E+09 8.89E+09 8.73E+09 g i 0.0732 0.0732 0.0732 0.1624 0.1624 0.1624 0.0716 0.0716 0.0716 0.1333 0.1333 0.1333 0.1132 0.1132 0.1132 0.1706 0.1706 0.1706 0.1528 0.1528 0.1528 0.177 0.177 0.177 0.1823 0.1823 0.1823 0.157 0.157 0.157 0.1752 0.1752 0.1752 0.1086 0.1086 0.1086 0.1263 0.1263 0.1263 0.0566 0.0566 0.0566 0.0663 0.0663 0.0663 0.0229 0.0229 0.0229 0.0246 0.0246 0.0246 0.0077 0.0077 0.0077 0.0139 0.0139 0.0139 0.0032 0.0032 0.0032 i (s) 1.0E - 07 1.7E - 06 2.8E - 05 4.6E - 04 7.7E - 03 1.3E - 01 2.2E+00 3.6E+01 6.0E+02 1.0E+04 52 Table 7.1 (c) T (ºC), f (Hz), 10, 10, 0.11 10, 1, 0.12 10, 0.1, 0.13 20, 10, 0.12 20, 1, 0.14 20, 0.1, 0.15 G o (Pa) 1.20E+10 1.19E+10 1.18E+10 9.14E+09 8.98E+09 8.90E+09 g i 0.0931 0.0931 0.0931 0.1169 0.1169 0.1169 0.0881 0.0881 0.0881 0.1095 0.1095 0.1095 0.1324 0.1324 0.1324 0.1608 0.1608 0.1608 0.1666 0.1666 0.1666 0.1902 0.1902 0.1902 0.1836 0.1836 0.1836 0.1839 0.1839 0.1839 0.1605 0.1605 0.1605 0.1311 0.1311 0.1311 0.1022 0.1022 0.1022 0.0669 0.0669 0.0669 0.0467 0.0467 4.67E - 02 0.0262 0.0262 0.0262 0.0156 0.0156 1.56E - 02 0.0084 0.0084 0.0084 0.0107 0.0107 1.0700E - 02 0.0055 0.0055 0.0055 i (s) 1.0E - 07 1.7E - 06 2.8E - 05 4.6E - 04 7.7E - 03 1.3E - 01 2.2E+00 3.6E+01 6.0E+02 1.0E+04 53 8. APPLICATION OF VISCOELASTIC CONTINUUM DAMAGE (VECD) THEORY TO 3PBC TEST ( Task 3) Adaptation of VECD theory in fatigue characterization of asphalt mixtures significantly reduced the experimental burden required to calibrate phenomenological fatigue life formulation (i.e., ). Extensiv e literature exists on the VECD and its practical applications in characterizing uniaxial fatigue behavior of asphalt mixtures (Christensen Jr & Bonaquist, 2005; Kim, Lee, & Little, 1997; Kutay et al., 2008; Kutay & Lanotte, 2017; Park, Kim, & Schapery, 1996; Underwood, B. Shane, 2006) . Several re cent studies have presented the application of the VECD theory on the analysis of fatigue behavior on flexural tests (Mello, Farias, & Kaloush, 2018; Tarefder, Bateman, & Swamy, 2013) . - viscoelastic correspondence (E - VC) principle, which can be applied to bo th linear and non - linear viscoelastic materials, and work potential theory (Schapery, 1990) . The E - VC principle states that the constitutive equations for a particular viscoelastic media are equivalent to equations of elastic media when the concept of pseudostrain is used in stead of actual physical s train. The pseudostrain in the time domain can be computed using the following convolution integral: [ 8 . 1 ] w here , is the pseudostrain, is a reference modulus often set as unity. Once is assumed as unity, corresponds to linear viscoelastic stress, E(t) is the linear viscoelastic relaxation modulus, t is the time and is the time variable of integration. The amount of work req uired for initiation and coalescing of microcracks is conveniently determined by the use of damage parameters (internal state variables). The mathematical approach 54 to describe these phenomena includes the simplest form of pseudostrain energy density functi on and a single internal state variable S described as follows (Park et al., 1996) : [ 8 . 2 ] [ 8 . 3 ] [ 8 . 4 ] w here , is the stress, is the pseudostrain energy density function, C(S is pseudostiffness as a function of a single damage parameter, represents the damage evolution rate, is an initial stiffness parameter used to eliminate the sample to sample variability, is time and is a constant related to the rate of damage growth in viscoelastic media ( , where is the maximum slo pe of the relaxation modulus master curve in log - log scale). In the case of cyclic loading at a constant frequency with no rest periods which is also applied in this study, the pseudostrain ( ) and the pseudostiffness (C) values at each cycle can be ca lculated as follows: [ 8 . 5 ] [ 8 . 6 ] where, is the linear viscoelastic dynamic modulus (i.e., ), is the dynamic modulus and is the peak strain measured at the cycle. The damage parameter (S) at the peak of each cycle was calculated using a practical procedure proposed by Kutay et al. (Kutay et al., 2008) : [ 8 . 7 ] 55 w here , is the reduced frequency. It is noted that the damage characteristic curve (C - S) is a unique curve that can be used to predict the fatigue life of asphalt mixtures at different frequencies and temperatures at a required strain level. The C - S curve is computed from the peak - to - peak stress - strain data retrieved for each cycle. However, the C - S curve should not be used to rank the fatigue performance of different mixtures since it is a normalized curve. Mixture classification should be done based on their fatigue life (number of cycles to failure (N f )) at a certain temperature, frequency and strain level. It is important to select a fatigue failure criterion, i.e., the N f . Numerous failure criteria have investigated by different researchers (Soltani & Anderson, 2005; Underwood, B. Shane, 2006; Y. D. Wang et al., 2018; Ze iada et al., 2016; Zhang et al., 2013) . Kutay et al. (Kutay et al., 2008) summarized different failure criteria existing in the litera ture and compared the fatigue testing facility (APT). It was concluded that the selected failure criteria had the same trends with data. Hence, a 50 % reduction in stiffness was recommended, therefore applied in this study. The fatigue life ( ) was calculated using the following expression derived for the specific case of cyclic tests at constant frequency with no rest periods (Kutay et al., 2009) : [ 8 . 8 ] w here , is the damage parameter value corresponding to C = 0.5. 8.1 Results of Uniaxial |E*| Tests using the AMPT One of the prerequisite steps of the VECD - based characterization is the determination of the linear viscoelastic |E*| master curve. Therefore, |E*| test needs to be run for analysis of both 56 3PBC and PP test results. In this study, the following shift factor and sigmoidal relationships were used to const ruct the |E*| master curve: [ 8 . 9 ] [ 8 . 10 ] w here , is the reference temperature, and are the shift factors polynomial coefficients, are the sigmoidal coefficients, is the reduced frequency ( The was selected as 20 º C. Figure 8.1 a and Figure 8.1 b illustrate the |E*| master curves in log scale and semi - log scale for each mixture, respectively. Also, the relationship between the shift factors and the temperature is presented in Figure 8.1 c. The dynamic modulus master curve and shift factor coeffici ents are presented in Table 8.1 . Overall, all mixtures had similar |E*| values. The 4E3DVR was slightly stiffer than the two SBS modified mixtures. 57 Figure 8 . 1 . Linear viscoelastic properties of the mixtures: ( a) log - log scale and ( b) semi - log scale plots of dynamic modulus master curves for 4E30SBS, 4E3SBS, and 4E3DVR, and ( c) shift factor coefficients as a function of temperature . 58 Table 8 . 1 Dynamic modulus master curve and shift factor coefficients Mixture ID Shift Factor (a(T)) Coefficients 4E30SBS 4E3SBS 4E3DVR a 1 0.0009 0.0006 0.0007 a 2 - 0.1568 - 0.1456 - 0.1508 Sigmoidal Coefficients c 1 1.0892 - 0.5296 - 0.0166 c 2 3.2607 4.9230 4.5466 c 3 - 1.0345 1.4746 1.2303 c 4 - 0.5102 0.3417 0.3227 8.2 Comparison of 3PBC and PP fatigue test results The damage characteristic (C - S) curves were constructed for each replicate of PP and 3PBC tests run at various temperature s and frequency combinations. Equations [ 8 . 5 ] through [ 8 . 7 ] were used to compute C and S values at each loading cycle. According to the VECD theory, C - S curves computed from the data of each replicate should collapse to form a unique curve. The C - S curves of 4E30SBS mixture are shown in Figure 8.2 a and Figure 8.2 b, for 3PBC and PP testing methodologies, respectively. It is clear that the C - S curve collapsed to form a single curve (with slight variations due to sample - to - sample variability) regardless of strain level, temperature, and frequencies. The C - S curves of 4E3SBS and 4E3DVR mixtures followed a similar trend, as shown in Figure 8.3 and Figure 8.4 . 59 Figure 8 . 2 . Damage characteristic curves of 4E30SBS for ( a) 3PBC and ( b) PP test results. 60 Figure 8 . 3 . Damage characteristic curves of 4E3SBS for ( a) 3PBC and ( b) PP test results . 61 Figure 8 . 4 . Damage characteristic curves of 4E3DVR for ( a) 3PBC and ( b) PP test results. Figure 8.5 shows the best fit C - S curves for all three mixtures tested using 3PBC and PP testing methods. Overall, it can be observed that the C - S curves from 3PBC tests seem to be shifted upwards, by about the same amount for all mixtures. Since the C - S is a normalized curve, it is better to analyze the 3PBC and PP tests result in terms of the number of cycles to failure (N f ). 62 Figure 8 . 5 . Best fit damage characterization curves for 3PBC and PP test results. In this study, the failure criterion is selected as a 50 % reduction of pseudo - stiffness (C = 0.5). The C - S curves shown in Figure 8 . 2 through 11 were employed to calculate the asphalt mixture fatigue life under the following conditions: frequency = 10 Hz, temperature = 10, 15 , and 20 º C , and strain level=100, 200, 300, 400, 500 and 600 microstrain (mm/mm x 10 - 6 ) . The N f values for each temperature/frequency/strain level combination were computed using Equation [ 8 . 8 ] . The N f values for the 3PBC test and PP test are illustrated in Figure 8.6 a , where N f values obtained from 3PBC analyses were generally higher than those obt ained the PP test results. This finding is consistent with literature comparing uniaxial and flexural fatigue tests (Ning Li et al., 2013) . Figure 8.6 b shows a direct comparison between the N f values computed from 3PBC and PP tests. As shown, there is a strong correlation between the PP test results and the 3PBC results. Overall, 3PBC tests resulted in about N f values 4.5 times those obtained from PP tests. These results demonstrate that the 3PBC is a promising alt ernative test method for fatigue characterization of asphalt mixtures. 63 Figure 8 . 6 . Comparisons of (a) the number of cycles to failure (N f ) versus strain relationship of 4E30SBS, 4E3SBS and 4E3DVR asphalt mixtures for 3PBC and PP test at f = 10 Hz, T = 20 o C, and (b) direct comparisons of N f values. 8.3 Summary of chapter findings The objective of this part of the research study was to investigate the implementation of VECD theory to model the fatigue response of the 3PBC test results at multiple frequency - temperature - strain levels . In order to achieve this objective, laboratory cylindrical samples from 64 three different asphalt mixtures w ere prepared and subjected to 3PBC tests. Also, push - pull (PP) tests were performed on the same mixture s. Based on the results presented in this part, the following conclusions can be made: It was shown that the viscoelastic continuum damage theory can be used to model the fatigue life of an asphalt mixture at many temperature /frequency/strain level combinations, by simply running the 3PBC tests at a few temp eratures/strain levels. The 3PBC testing results were analyzed using the same VECD formulations deve loped for PP data analysis. The damage characteristic curves (C - S) were generated and compared. It was shown that the C - S curves obtained from the 3PBC test collapsed to form a unique curve, just like the C - S curves of PP tests. However, the C - S curves of 3PBC and PP tests were not the same. The C - S curves of 3PBC tests were generally shifted upwards compared to the PP - based C - S curve for a given mixture. This difference is understandable since the mode of loading is different. Asphalt mixture hierarchical ranking obtained from 3PBC and PP tests agreed very well with each other. Overall, the number of cycles to failure obtained from 3PBC test results were about 4.5 times those obtained from the PP tests. The results presented in this part of the resear ch study are quite promising, however, the 3PBC test method should still be considered to be under development. A ruggedness study should be performed to develop a proper test specification (Aksel Se itllari & Kutay, 2019) . 65 9. RUGGEDNESS STUDY OF THREE - POINT BEAM CYLINDER (3PBC) TEST ( Task 4 ) As reported in the previous chapters, promising experimental results of the 3PBC test were obtained for characterizing the fatigue behavior of asphalt pavements . The next step was a ruggedness evaluation of the 3PBC test setup , where the effects of certain parameters that influence the test results were identified and quantified. An effort to establish acceptance limits for these parameters is presented in this ch apter . The Three - Point Bending Cylinder (3PBC) test is a promising method that can be used for routine testing practices by state DOTs and roadway agencies. However, it should be noted that this test is still considered to be under development. Hence, i den tifying and investigating the potential sources of variability in this test is crucially important to improve its accuracy and to develop test specifications . According to (Bonaquist, 2008) , any developed test method is expected to provide accurate results that can only be achieved through decent control of the sources of varia bility in the test procedure. Ruggedness evaluation is the appropriate method to identify important sources of variability by intentionally varying the most influential parameters. Such ex amination provides evidence on the reliability of the test technique during ordinary usage and helps establish control thresholds. Hence, ruggedness testing is a crucial part of the development of a new test method and as such, was performed in this study. ASTM E1169, Standard Practice for Conducting Ruggedness Test provides a practical procedure for concurrently evaluating the effects of certain changes in each of the operating conditions in an independent manner. This practice recommends the ruggedness te sting should involve a single laboratory on uniform material and potentially followed up by an inter - laboratory (round - robin) study. The inter - 66 laboratory study was not within the scope of this wor k , but it should be considered in a future study . The detai ls of a proper ruggedness study are comprehensively described in ASTM E1169. Typically, the primary considerations in the design of a ruggedness test are (i) selection of the potential factors and their levels and (ii) selection of the test conditions. The general procedure requires seven factors and two different levels ( low and high ) usually determined based on experience. In this study, however, three factors were selected with several corresponding levels. To meet the protocol requirement of seven facto rs, factor s were duplicated. Table 9 . 1 illustrates the potential factors affecting 3PBC tests , and the ranges of the parameters selected for this study. Table 9 . 1 Factors and levels of ruggedness analysis Factor Variable Lower Level Value Upper Level Value A Air void [%] 6 8 B Span length [mm] 100 135 C Specimen diameter [mm] 38 100 Based on ASTM E1169, t he level setting is indicated by 1 and - 1 for the expected high level and low level , respectively . An orthogonal design approach is followed to determine the effect of ea ch factor on the final results. To achieve this ta sk , a specific combination of factor variables and their levels is required , as detailed in Table 9 . 2 . According to this condition , two combinations for each level will yield a total of 16 tests to complete the experimental work. 67 Table 9 . 2 Experimental design for ruggedness testing ( ASTM E1169) Factor s Determination number Air void [%] Span length [mm] Span length [mm] Span length [mm] Specimen diameter [mm] Specimen diameter [mm] Specimen diameter [mm] 1 1 1 1 - 1 1 - 1 - 1 2 - 1 1 1 1 - 1 1 - 1 3 - 1 - 1 1 1 1 - 1 1 4 1 - 1 - 1 1 1 1 - 1 5 - 1 1 - 1 - 1 1 1 1 6 1 - 1 1 - 1 - 1 1 1 7 1 1 - 1 1 - 1 - 1 1 8 - 1 - 1 - 1 - 1 - 1 - 1 - 1 Note: the level cells were color - coded for better visualization 9.1 Mixture sampling and sample preparation The asphalt mixture u sed for the ruggedness evaluation was obtained as a loose mix from a local asphalt plant in Lansing, MI . The mix design and volumetric propertie s of the loose mixture were shown in Table 4 . 1 ( 5E3 Neat mixture ) . The loose mix was carefully sampled in 5 - gallon metallic pails to avoid segregation. Usually, from a 5 - gallon metallic pail , 3 Superpave Gyratory Compactor ( SGC ) specimens can be prepared . Prior to loose mixture separation, the 5 - gallon metallic pail is reheated at 110 º C until the loose mix is easily spreadable. This process usually takes less than 3 hours. The loose mixture is poured in a metallic pan and mixed to provide homogeneity. Then, three smaller pans are filled with the required mass of the specimen and placed in a preheated oven at the compaction temperature. To minimize the aging of the loose mix, the temperature of the material wa s frequently checked using thermocouples. Then the material wa s poured in compaction molds and a n SGC Pine compactor wa s used to compact the specimen at a specific height to meet required target air voids. Before any further mechanical processing, the compacted samples are stored to cool down at room temperature overnight . A ll the SGC samples were compacted at a height of 180 mm and a diameter of 150 mm . Depending on the performance test, cylindrical samples were cored and cut at desired diameter s and height s . Subsequently, the 68 physical properties of the resized specimens were measured and recorded. The specimens falling out of the predefined air void rang e were discarded from further testing. 9.2 Uniaxial Dynamic Modulus (|E*|) Test Samples for uniaxial dynamic modulus (|E*|) were obtained from cylindrical samples (150 mm diameter, 1 8 0 mm height ) compacted with the Superpave Gyratory Compactor (SGC) in gene ral accordance with the AASHTO R83 protocol . Cores were extracted from those samples using a diamond - coring stand, while the ends of the cores were trimmed using a masonry saw in order to obtain smooth and parallel end surfaces. The final height of the testing samples was set to 150 mm in all cases. The diameters of the cored samples were 100 mm. The mass of mixture compacted by the SGC was determined to reach the target air voids content of 7 % ± 0.5 % at the sample core . At leas t two replicates for each test and mix were prepared. The |E*| measurements were carried out in accordance with the AASHTO R84 p rotocol using the Asphalt Mixture Performance T esting (AMPT) device. Samples were subjected to uniaxial sinusoidal compressive s tress at four temperatures (4, 20, 40 and 5 4 ° C) and three frequencies ( 10, 1, and 0.1 Hz) at each temperature . The dynamic modulus master curve was obtained using the time - temperature superposition (TTS) principle (Kutay & Lanotte, 2017) (Aksel Se itllari, Lanotte, & Kutay, 2019a) . As described in Section 8.1 , raw data were shifted horizontally at a reference temperature (T ref = 21 °C) using shift factor coefficients (a (T)). A second - order polynomial function ( Equation [ 8 . 9 ] ) and a sigmoidal model ( Equation [ 8 . 10 ] ) w ere used to fit shift factors and the |E*| master curve, respectively. 69 Figure 9 . 1 . Linear viscoelastic properties in (a) log - log scale, (b) semi - log scale plots of dynamic modulus master curves and (c) shift factor coefficients as a function of temperature for 5E3 and 3E3 asphalt mixtures . 70 Figure 9.1 a and Figure 9.1 b illustrate the |E* | master curves in log scale and semi - log scale for each respective replicate . Also, the relationship between the shift factors and the temperature is shown in Figure 9.1 c. More information for |E*| master curve and shift factor coefficients are detailed in Table 9 . 3 . Table 9 . 3 Dynamic modulus master curve and shift factor coefficients for 5E3 and 3E3 mixtures Shift factor (a(T)) coefficients a 1 a 2 5E3 0.0006 - 0.1478 3E3 0.0006 - 0.1417 Sigmoidal Coefficients c 1 c 2 c 3 c 4 5E3 0.9344 3.4658 1.0157 0.4397 3E3 0.8882 3.4522 0.8942 0.4932 9.3 Factors and levels of ruggedness analysis It is significant to investigate the influence of certain parameters on the 3PBC test results and establish acceptance limits . A general test matrix designed to perform the ruggedness study is visualized in Figure 9 . 2 . The presented test matrix includes the linear viscoelastic characterization of the asphalt mixture, variation s in specimen geometry (d iameter and span length) and the air void content for each combination along with the estimated replicate number. 71 Figure 9 . 2 . Ruggedness study flow chart . The number of cycles to failure (N f ) was measured using a closed - loop servo - hydraulic material testing system (MTS) containing an environmental chamber. Prior to testing, the temperature of the test specimens was checked using a dummy specimen of the same size as the |E*| samples with a therm ocouple inside. The displacement level at the actuator was initially selected through a trial and error process to ensure the on - specimen LVDT measurements showed the strain level on the sample was only about 200 microstrains ( ) . The frequency of the 3PB C tests was 5 Hz , and the tests were conducted at a temperature of 15 º C . The testing conditions were kept the same for all the evaluated test combinations. Viscoelastic Continuum Damage (VECD) theory was then used to analyze the data from the 3PBC test results . This allow ed determination of fatigue life (N f ) at target strain level (i.e. 15 0 , 3 00 ) , the temperature of 15 º C and frequency of 5 Hz. The following subsection presents the arrangements incorporated in the 3PBC test setup to accommodate the r equired changes for the ruggedness study. Further, the selected factors and the 72 two levels for the ruggedness testing as well as the sensitivity of the 3PBC test toward these factors were discussed . 9.3.1 3PBC test setup designs Three different 3PBC test setup designs were optimized to ad o pt the factors and their levels considered for the ruggedness study as presented in Table 9.4 . In this table, the initial geometry of the 3PBC test setup herein referred to as reference geometry, and other geometry combinations are detailed. As previously mentioned, the air void sensitivity for each combination was investigated. It is worth noting that all test samples were cored in a vertical direction. Table 9 . 4 Factors and corresponding levels for the ruggedness evaluation of 3PBC test Reference g eometry 1 st 2 nd 3 rd 4 th Air Voids (%) 6, 7, 8 - - - - SGC specimen height (mm) 180 - - - - Test specimen height (mm) 68 68 38 38 100 Span length (mm) 125 135 100 125 135 Coring Direction Vertical Vertical Vertical Vertical Vertical Test temperature 15 - - - - Test frequency (Hz) 5 - - - - 150 - - - - Figure 9 . 3 illustrate s the test setups and cored test specimens for additional designs. The reference design was not shown here for brevity (see Figure 5 . 2 ). As shown in Figure 9 . 3 a, for 38 mm diameter samples , the 3PBC test setup components were kept the same as the reference setup designed for 68 mm samples. F or 100 mm diameter samples , as shown in Figure 9 . 3 b , the thickness of the base plate and side claps were increased. This was mainly done to avoid any undesired deformation of the fixed ends which could influence the test results. 73 Figure 9 . 3 . 3PBC test setup with loaded specimens Material Testing System (MTS) with a diameter of (a) 38 mm and (b) 100 mm. 9.3.2 Air void content The effect of air void content o n fatigue performance of the mixtures is important , as demonstrated in several studies (Harvey & Tsai, 1996; West, Rodezno, Leiva, & Taylor, 2018) . C lose control of this parameter is a crucial task. The current state of practice addresses ± 0.5 % air void toleranc e. A wider tolerance range is helpful to optimize the specimen preparation time and resources. Generally, the required field air void content is 7 % and used in this study as control air void content for the tested mixtures. (a) (b) 74 To investigate test sensitivit y to the air void content and establish rugged limits; 6 %, 7 % and 8 % ± 0.5 % were selected as shown in Table 9.4 . Loose material was used to prepare SGC samples as described in the preceding sections. While several trial samples for each combination wer e used to ensure the on - specimen strain level is close to 200 microstrains, the lateral LVDT was used to measure the lateral displacement of the side clamps. The results obtained from this task are presented in Figure 9.4 . As shown in this figure, in general , the lateral displacement is less than 10 % of the vertical displacement except for two cases for 6 % air void content . Since the lateral displacement at 7 % and 8 % , air void contents w ere below 10 % the vertical displ acement, the lateral displacement limit was preliminary set at 10 % of the vertical displacement . However, for future reference, further detailed FE analysis will be performed to quantify if the lateral displacement limit threshold of 10 % affects the stre ss - strain results on the loaded specimen. Figure 9.5 - Figure 9.9 present the damage characteristic (i.e., C vs S) curves and ( an example ) the number of cycles to failure ( at the frequency of 5 Hz, temperature of 15 ° C and strain level of 150 ) obtained from specimens tested at 6 %, 7 % , and 8 % air void contents , for different Figure 9 . 4 . Lateral displacement limit for 3PBC test . Vertical axis shows the lateral displacement divided by the vertical displacement in the central clamp, in percentage. 75 specimen geometries . As shown, in all geometries, variability between the individual tests for each air void content is low, as evidenced by the clear collapse of the C ( pseudostiffness) versus S (damage parameter) curves. There were few exceptions to this (see Figure 9 . 8 a), presumably caused by issues with the sample preparation and testing. It is interesting to note that, in most of t he samples, effect of air voids on the number of cycles to failure (N f ) results was somewhat minimal , within the narrow air void range from 6 % to 8 % . S tatistical analysis is presented in the la t er part of this section. 76 Figure 9 . 5 . (a) (c) Damage characteristic curves for 6 %, 7 % and 8 % air void content s, respectively and (d) number of cycles to failur e (N f ) r esults at frequency of 5 Hz, temperature of 15 ° C and strain level of 150 microstrain , for 68 mm - 125 mm (reference) geometry (5E3 mix) . 77 Figure 9 . 6 . (a) (c) Damage characteristic curves for 6 %, 7 % and 8 % air void content s, respectively and (d) number of cycles to failur e (N f ) r esults at frequency of 5 Hz, temperature of 15 ° C and strain level of 150 microstrain, for 68 mm - 135 mm geometry (5E3 mix) . 78 Figure 9 . 7 . (a) (c) Damage characteristic curves for 6 %, 7 % and 8 % air void content s, respectively and (d) number of cycles to failur e (N f ) r esults at frequency of 5 Hz, temperature of 15 ° C and strain level of 150 microstrain, for 38 mm - 100 mm geometry (5E3 mix) . 79 Figure 9 . 8 . (a) (c) Damage characteristic curves for 6 %, 7 % and 8 % air void content s, respectively and (d) number of cycles to failur e (N f ) r esults at frequency of 5 Hz, temperature of 15 ° C and strain level of 150 microstrain, for 38 mm - 125 mm geometry (5E3 mix) . 80 Figure 9 . 9 . (a) (c) Damage characteristic curves for 6 %, 7 % and 8 % air void content s, respectively and (d) number of cycles to failur e (N f ) r esults at frequency of 5 Hz, temperature of 15 ° C and strain level of 150 microstrain, for 100 mm - 135 mm (5E3 mix) . 81 O ne - way analysis of variance (ANOVA) results are shown in Table 9 . 5 . P - values were used to signify the magnitude of the differences with a 90 % confidence interval. The difference in N f is defined as statistically significant when the p - value is lower than 0.1. Also, Tukey test results were examined to determine the likely ranges for the differences and to assess the practical significance of thos e differences. This test is usually used as a post hoc test especially when results and highlight s the differences. Factors that do not share a letter have a stat istically significant difference in their mean value. P - values for each combination are presented in Table 9 . 5 . A comparison of the P - values indicate s the effect of air void content on N f values , generally, is not statistically significant. Likewise, Tuke It should be realized that the objective of this exercise is to identify controllable factors that potentially affect the test results and establish threshold limits for their control. O verall, it can be said that the e ffect of air voids on the number of cycles to failure (N f ) results was minimal and there are no statistically significant differences in fatigue life within the air void range from 6 % to 8 %. Conforming to the results obtained from all the observed scenarios, t he ± 1 % air void tolerance can be considered as a reasonable practice for the 3PBC test method. 82 Table 9 . 5 Statistical evaluation on the effect of air void content on fatigue life of asphalt mixture One - way ANOVA Selected geometry d = 68 mm l = 125 mm d = 68 mm l = 135 mm d = 38 mm l = 100 mm d=38 mm l=125 mm d=100 mm l=135 mm Air void (%) 6 7 Gr. + 6 7 Gr. 6 7 Gr. 6 7 Gr. 6 7 Gr. 6 P - Value * - A - A - A - A - A 7 P - Value 0.2 33 - A 0. 879 - A 0. 879 - A 0. 013 - B 0. 892 - A 8 P - Value 0. 089 0.5 29 A 0. 588 0. 381 A 0.37 3 0.188 A 0. 004 0.575 B 0. 001 0.001 B * N f is significant ly different if P - Value < 0.1 + N f that do not share a letter are significantly different ; Gr. Grouping Table 9 . 6 Statistical evaluation on the effect of span length and diameter on fatigue life of asphalt mixture One - way ANOVA Geometry combination d = 68 mm l = 125 mm d = 68 mm l = 135 mm d = 38 mm l = 100 mm d=38 mm l=125 mm Grouping + d = 68 mm l = 125 mm - B \ C d = 68 mm l = 135 mm P - Value * 1.000 - B \ C d = 38 mm l = 100 mm P - Value 0.06 4 0.0 66 - A d = 38 mm l = 125 mm P - Value 0. 001 0.0 01 0.379 - A d = 100 mm l = 135 mm P - Value 1.000 1.000 0.04 3 0.0 01 C * N f is significant ly different if P - Value < 0.1 + N f that do not share a letter are significantly different 83 9.3.3 Span length Span length is considered to be one of the parameters that might affect the stress - strain distribution on the specimen when subjected to loading. This effect becomes more pronounced when the so - called low aspect ratio or - are used, which is the case for the 3PBC test. Different span lengths were adopted and their effect on fatigue life ( i.e. N f ) of asphalt mixture s w as explored. It is worth noting that from the practical point of view, the span length should be compatible with horizontal - coring from field cores ( i.e., 150mm diameter cores) or slabs. The effects of span length on the fatigue performance of the asphalt mixture using the 3PBC test were explored. Figure 9 . 10 shows the damage characteristic curves (C - S) of all geometries (different span lengths and diameters) and all air void levels. As shown, in general, different span lengths of a given diameter collapsed in a single curve, with slight differences. The N f value s obtained at two different microstrain levels of 150 3 are presented in Figure 9 . 11 a and Figure 9 . 11 b , respec t ively . It can be seen that the change in span length affects the N f value slightly . The differences are a bit more pronounced for the 38 mm diameter specimens when compared to the 68 mm diameter specimens . This could be because of the higher variability observed in the 38mm diameter, 125 mm long samples . One - way ANOVA was performed to assess the effect of span length for a given diameter on the N f . The measured N f values at different air void contents (i.e., 6 %, 7 % , and 8 %) were , however, combined for each geometry, respectively. P - values for each scenario are presented in Table 9.6. From the table, i t is shown that the effect of span length on N f is not statistically significant. Overall, the effects of span length on the 3PBC test results are minimal . 84 Figure 9 . 10 . Damage characteristic curves (C - S) of all geometries plotted for (a) 6 %, (b) 7 % and (c) 8 % air void contents. 85 Figure 9 . 11 . Number of cycles to failure (N f ) results for different length and diameter combinations at (a) 15 3 . 9.3.4 Specimen diameter The effects of specimen thickness are known to influence the fatigue resistance of asphalt mixtures when the so - called - Therefore, f or this ruggedness study, various specimen diameter sizes mainly 38 mm , 68 mm and 100 mm , were used to investigate diameter effects on the predicted fatigue life when the 3PBC 86 te st is adopted. Looking into the applicability of 3PBC test to small sample diameters is important use for thin lifts of asphalt mixture layers. Figure 9 . 10 shows the effect of the diameter on the d amage characteristic curves (C - S) . As shown , C vs S curves of larger diameter samples are below those of the smaller - diameter samples. Figure 9 . 11 shows the number of cycles to failure based on the C versus S curves computed for each diameter . From the figure, differences in N f values for the different scenarios are easily observed , where the N f values tended to decrease with increasing diameter. The fatigue life for 38 mm diameter 3PBC test samples is higher compared to the reference geometry (68 mm) diameter and the 100 mm diameter has the shortest fatigue life. This trend is consistent in all a ir void contents and in strain level s (see Figure 9.11 a and Figure 9.11 b ) . P - values for each scenario are presented in Table 9.6. From the table, it is confirmed that the effect of sample diameter on N f is statistically significant. No n - homogenous distribution of stress es and strain s are seen as one of the potential reasons affecting these changes in the N f values . Also, the high presence of shear in larger diameter samples and incapability of the current VECD formulations to fully comprehend its e ffects might be influencing the results. Numerous techniques were explored to relate the fatigue performance in terms of numbers of cycles to failure for the selected specimen diameters. In this context , the applicati on of traditional analysis (i.e., multi - linear regression) and advanced novel computing techniques (i.e., multigene genetic programming and artificial neural network ) were utilized to develop predictive models to relate fatigue performance among varying sp ecimen diameters. A major advantage of such approaches is their potential to train complex patterns and develop statistically accurate and 87 sound models (Naser & Seitllari, 2019; A. Seitllari & Naser, 2019; Aksel Seitllari & Kutay, 2018) (Haider, Masud, & Chatti, 2020; Masud, 2018) . Before any action s to implement the modeling methods, expansion of the variable database (i.e., a set of inputs and outputs) is important to statistically reinforce the predictive models. On this basis, a new 3E3 mixture was tested, in addition to the existing 5E3 mixture. The 3E3 mixtures consist of 19.0 mm NMAS typically used for the construction of base layers in asphalt pavement s. This is important because the bottom - up fatigue cracking performance of asphalt pavements heavily rel ies on the fatigue cracking characteristi cs of the base (lowermost) asphalt layer . On the other hand , the 5E3 is a surface mixture with NMAS of 9.5 mm and mainly used as the surface (top) layer, where the top - down fatigue cracking initiates . The 3E3 loose asphalt mixture was collected from a local asphalt plant in Lansing, MI. The mix design properties and volumetric details of the 3E3 loose mixture are summarized in Table 4 . 1 Mixture sampling, processing and sample preparation procedures were similar to the 5E3 mixture. Nonetheless , the 3E3 specimens were prepared only at 7 % ± 0.5 % target air void conten t (i.e., samples were not prepared at different air void contents) . The |E*| measurements were carried out in accordance with the AASHTO R84 p rotocol using the Asphalt M ixture Performance T esting (AMPT) device at three temperatures (4, 20, and 40 °C) and three frequencies ( 10, 1, and 0.1 Hz) at each temperature . Figure 9.1 a and Figure 9.1 b show the |E*| master curves in log scale and semi - log scale for each respective replicate . Also, the relationship between the shift factors and the temperature is shown in Figure 9.1 c. The |E*| master curve and shift factor coefficients are summarized in Table 9 . 3 . One span length with one corresponding diameter (i.e, 38 mm 100 mm, 68 mm 125 mm and 100 mm 135 mm) w as used to investigate t he relation of fatigue performance of the 3E3 88 asphalt mixture . The 3E3 samples for each respective specimen dimensions were cored from the SGC compacted material and process ed similar to the 5E3 mixture described in section 9.1 . A minimum of two replicates were tested at temperatures of 1 0 °C and 20 °C at 5 Hz, with on - specimen initial strain level ranging from 200 - 350 microstrains ( ) (see APPENDIX A ) . While for 68 mm 100 mm and 100 mm 135 mm one test specimen was extracted for each SGC compacted material , for 38 mm 100 mm combination , four replicates were e xtracted from one SGC sample having a total of eight repl i cates (see Figure 9 . 3 a) . In addition to high laboratory productivity, t he 38 mm 100 mm geometry combination is more advantageous as it is also able to facilitate the laboratory fatigue evaluation of thin asphalt pavements. Viscoelastic Continuum Damage (VECD) theory was then adopted to analyze the data from the 3PBC test results . It is important to note that the higher the NMAS of an asphalt mixture, the more is the s ample - to - sample variability. In this regard , the C - S curves for each testing conditions were constructed and individual fatigue life (N f ) at target strain level (i.e. 150 ) , the temperature of 1 0 °C and 20 °C , and frequency of 5 Hz was de termined as presented in Figure 9 . 12 to Figure 9 . 14 . As shown in the figures, the C - S curves o f different temperatures collapsed for all the tested dimensions regardless of the variable nature of the 3E3 mixture caused by its large NMAS (i.e., 19 mm) , which is a crucial factor affecting the continuum medium . In the 3 8 mm diameter geometr y , however, one of four replicates (i.e., 3PBC - 3E3 - 38 - 100 - 4 - 10C - 5Hz and 3PBC - 3E3 - 38 - 100 - 1 - 2 0C - 5Hz ) extracted from each SGC compacted material, created variability and statistically affected the results. Likewise, for 68 mm 125 mm geometry re plicate 3PBC - 3E3 - 6 8 - 125 - 1 - 2 0C - 5Hz , the obtained C - S curve shifted upwards (see Figure 9 . 13 a) . In general, the observed abnormal shifts in C - S curves could be attributed to various factors but not limited to such as sample preparation related issues, machine compliances and \ or other 89 technical related practices. Such cases were excluded from further considerations , but i t is critical to be recorded as they , indee d , help improve the 3PBC test practice. The N f values were shown to change with the variations in sample dimensions as illustrated in Figure 9 . 12 b to Figure 9 . 14 b. The revealed observations appear to agree with the conclusions derived f rom the 5E3 mixture. Figure 9 . 12 . (a) Damage characteristic curves and (b) the number of cycles to failure N f results at frequency of 5 Hz, temperature of 10 ° C & 20 ° C , and strain level of 150 for 3 8 mm - 1 00 mm geometry. 90 Figure 9 . 13 . (a) Damage characteristic curves and (b) the number of cycles to failure results N f results at frequency of 5 Hz, temperature of 10 ° C & 20 ° C , and strain level of 150 for 68 mm - 125 mm (reference) geometry. 91 Figure 9 . 14 . (a) Damage characteristic curves and (b) the number of cycles to failure N f results at frequency of 5 Hz, temperature of 10 ° C & 20 ° C , and strain level of 150 for 100 mm - 135 mm geometry. T o build the ground for the investigation of the fatigue life (N f ) relationship among different geometry combinations of 3PBC test, t he C - S curves developed for 5E3 and 3E3 asphalt mixtures w ere used to calculate the N f under the following conditions: frequency = 1, 5, 10 Hz, temperature = 5, 10, 15 , 20, 25 and 3 0 ºC, and s train level = 100, 150, 200, 250, 300, 350, 400 , 450, 500, 550 a nd 600 microstrain (mm/mm x 10 - 6 ). The N f values for each 92 temperature/frequency/strain level combination were computed using Equation [ 8 . 8 ] and sum marized in APPENDIX B Initially, the application of multi - linear regression (MLR) analysis was performed to inspect if a ny cogent relation could be achieved among the N f values for different 3PBC sample diameters. In addition to the number of cycles to failure of the tested 3PBC sample diameter N f (D i ) , strain level ( microstrain ); 3PBC test sample diameter D i (inch); and dynamic modulus |E*| (psi); were selected as input variables to correlate with the number of cycles t o failure for the reference geometry N f (D R ) (i.e., D R = 2.68 inch es ) . A total of 792 data points obtained from the analysis of the C - S curve were used to develop and validate the MLR model. The data points were randomly divided into two data sets; training data set (80 %) and testing data set (20 %). Even though the division process was random, statistical parameters w ere used to ensure clarity regarding the range of the variables as presented in Table 9 . 7 . It can be seen in the tab le that t he strain level shows the highest skewed di stribution. The data statistics for other input variables have a balanced distribution. It should be noted that the selected data were converted o n a logarithmic scale with base 10. This practice was particularly implemented to minimize the large range that exists among the selected variables and avoid any potential overtraining of the developed models. 93 The predictive strength and accuracy of the models w ere determined based on mean square error (MSE), mean absolute relative error (MARE) and coefficient of determination (R 2 ) as defined in Equations [ 9 . 1 ] to [ 9 . 3 ] . While R 2 represents how close the data are to the fitted line , the MSE and MARE parameters were used to determine if the relationship is signific ant (Aksel Seitllari, 2014; Aksel Seitllari, Kumbargeri, Biligiri, & Boz, 2018) . [ 9 . 1 ] [ 9 . 2 ] [ 9 . 3 ] Table 9 . 7 Training data set and testing data set input statistics Training Phase Input s x c sx x min x max N f for the tested sample diameter (N f (D i )) - 3.510 0.234 - 0.706 - 4.000 - 3.222 Strain level ( 0.383 0.210 0.019 0.175 0.595 3PBC test sample diameter D i (inch) 5.776 0.299 - 0.533 5.048 6.218 Dynamic modulus |E*| (psi) 3.097 1.898 0.332 - 0.329 8.791 Output N f for the reference diameter (N f (D R =2.68)) 3.231 1.415 0.648 1.017 7.471 Testing Phase Input s x c sx x min x max N f for the tested sample diameter (N f (D i )) - 3.525 0.248 - 0.555 - 4.000 - 3.222 Strain level ( 0.390 0.211 - 0.051 0.175 0.595 3PBC test sample diameter D i (inch) 5.752 0.292 - 0.364 5.048 6.218 Dynamic modulus |E*| (psi) 3.151 1.886 0.116 - 0.060 8.134 Output N f for the reference diameter (N f (D R =2.68)) 3.350 1.468 0.465 1.016 7.471 Note: =overall mean, s x = standard deviation, c sx =skewness coefficient, x min =minimum and x max = maximum. 94 where, is the observed output, is the predicted output, is the averaged predicted output, n is the total number of data sets and is the error for each input set. As previously mentioned, t he MLR model was developed using the training data set (i.e., 80 % = 634 data points). The final version of the MLR equat ion is in the following form: [ 9 . 4 ] where, Figure 9 . 15 a illustrates the measured and the predicted N f for the reference diameter (N f (D R =2.68)) in the training phase . As shown in the figure, the data points align to the line of equality for the MLR training set scatterplot. The calculated statistical measures used to evaluate the MLR mod el are presented in Table 9 . 8 . According to the table, the training data set revealed the followings statistics : R 2 = 0.99, MSE = 0.02 and MARE = 4.50 . Figure 9 . 15 b shows the performance of the MLR model when applied to the testing data set. Note that the testing data set is independent and was not used to train the MLR model. The predicted versus measured values of N f (D R =2.68) for the testing data set exhibit a similar trend to the training data set. Likewise, the calculated MLR statistics of the testing data set exhibited the following measures R 2 = 0.99, MSE = 0.02 and MARE = 4.22 . Table 9 . 8 Model statistic results Training data set Testing data set Statistics R squared Mean Absolute Relative Error Mean Squared Error R squared Mean Absolute Relative Error Mean Squared Error MLR (Log scale) 0.99 4.50 0.02 0.9 9 4.22 0.02 In addition, multigene genetic programing (MGGP) and artificial neural network (ANN) analysis were performed to check if better fit can be achieved. Both these techniques, however, 95 revealed similar results to the MLR model. For clarity and ease of applicat ion, only the MLR model was pursued excluding the details of the other two techniques. In essence, the obtained results clearly support the efficiency of the Equation [ 9 . 4 ] to correlate N f of different sample diameters to N f (D R =2.68) with high confidence. It is envisioned that and \ or road agencies ' capacities to evaluate the fatigu e performance of asphalt cores despite the lay er thickness or aggregate size. 96 Figure 9 . 15 . Performance of the developed MLR model for (a) training data set and (b) testing data set. 97 9.4 Summary of chapter findings The o bjective of this part of the study was to perform a ruggedness evaluation of the 3PBC test. For this purpose, several factors that were believed to influence the 3PBC test results were i dentif ied. Cylindrical samples were prepared to materialize these factors and subjected to 3PBC tests. The research findings from this study have been summarized as follows: Three different fixture s were designed to address the influence of selected factors on the 3PBC test results. The general lateral displacement of these fixtures w as measured using the lateral LVDT, and the maximum displacement limit of 10 % of the vertical displacement was recommended . The VECD theory was successfully applied to all selected geometries. The C - S curv es were used to obtain the N f value of each sample to get information about the sample - to - sample variability for each air void content. Based on the received results, t he a ir void content tolerance range of ± 1 % was considered a reasonable practice for t he 3PBC test method . Th is range is in alignment with the current state of practice for performance specimens. The variation in the span length on the 3PBC test did not have considerable effects on the fatigue life of the asphalt mixture. It was observed that the changes in specimen diameter affect ed the N f values . Typically, specimens with smaller diameter exhibited longer fatigue life. This was consistent for all the air void contents and/or strain level s . Various techniques were explored to relate the fatigue performance in terms of numbers of cycles to failure for the selected specimen diameters. Results revealed that MLR developed equation can capture the relation between the N f (D R =2.68) and other input variables with high 98 confidence. This allows the 3PBC test to run at various diameters to correlate with the reference geometry (i.e., N f (D R =2.68) ) . Overall, this part of the research study led to ruggedness evaluation of the 3PBC testing a pproach through varying factors and their levels. The proposed research addresses an important long - standing issue for fatigue cracking characterization of asphalt pavements. Certainly, the proposed methodology possesses great potential to significantly re duce the testing cost and substantially increase the testing speed. A future objective of the study plan to improve by further modifying the VECD formulations to capture the diameter effects on fatigue life of the mixture. In that case, the 3PBC test will provide a mechanistic, yet practical protocol to evaluate cracking resistance of asphalt mixtures. 99 10. CONCLUSIONS The objective of this study was to introduce a more practical fatigue testing approach to the uniaxial fatigue tests and corresponding analyses based on the Viscoelastic Continuum Damage (VECD) theory. T his new approach referred to as the t hree - pint bendin g cylinder (3PBC) test, aimed at addressing the challenges present in the current state of practice, with a focus on test procedure simpl icity , sensitive to asphalt mix design, repeatable and efficient. The new test setup was designed and data analysis met hodology was developed , validated and verified using three - dimensional finite element (3D FE) analysis and laboratory results. Furthermore, the VECD theory was adopted and implemented to model the mechanical response of the specimens. Finally, ruggedness e valuation of the 3PBC test approach through varying factors and their levels were investigated and presented. The findings of this study can be summarized as follows: Based on 3D FE analyses , it was shown that the error in elastic |E*| 3PBC ranges from 0.2 % to 1 6 % , whereas for viscoelastic |E*| 3PBC is less than 12 %. The results also indicate that typically, the error increases with the decrease in loading frequency. The 3PBC tests are recommended to be performed at 5 Hz and above, where t he maximum error is less than 5 %. Additionally, the back - The laboratory results indicated that the |E*| values obtained from the measured master curve (i.e., |E*| AMPT ) and the corresponding |E*| (i.e., | E*| 3PBC ) calculated using Timoshenko beam theory match reasonably. The calculated error ranges from 0.4 % to 16 %. Generally, it was observed that the error increases with the decreasing frequency and increasing temperature. 100 Also, the back - calculated Poiss coefficients retrieved from (Maher & Bennert, 2008) match well with the back - calculated values from 3PBC setup. It was sh own that the viscoelastic continuum damage theory can be used to model the fatigue life of an asphalt mixture at many temperature /frequency/strain level combinations, by simply running the 3PBC tests at a few temp eratures/strain levels. The 3PBC testing re sults were analyzed using the same VECD formulations developed for PP data analysis. The damage characteristic curves (C - S) were generated and compared. It was shown that the C - S curves obtained from the 3PBC test collapsed to form a unique curve, just lik e the C - S curves of PP tests. However, the C - S curves of 3PBC and PP tests were not the same. The C - S curves of 3PBC tests were generally shifted upwards compared to the PP - based C - S curve for a given mixture. This difference is understandable since the mo de of loading is different. Asphalt mixture hierarchical ranking obtained from 3PBC and PP tests agreed very well with each other. Overall, the number of cycles to failure obtained from 3PBC tests were about 4.5 times those obtained from the PP tests. Three different fixture s were designed to address the influence of selected factors on the 3PBC test results. The general lateral displacement of these fixtures w as measured using the lateral LVDT, and the maximum displacement limit of up to 10 % of the ve rtical displacement was set. The VECD theory was successfully applied to all selected geometries. The C - S curves were used to obtain the N f value of each sample to get information about the sample - to - sample variability for each air void content. 101 Based on the received results, t he a ir void content tolerance range of ± 1 % was considered a reasonable practice for the 3PBC test method . The variation in the span length on the 3PBC test did not have considerable effects on the fatigue life of the asphalt mixt ure. It was observed that the changes in specimen diameter affected the N f values. Typically, specimens with smaller diameter exhibited longer fatigue life. This was consistent for all the air void contents and/or strain levels. Various techniques were e xplored to relate the fatigue performance in terms of numbers of cycles to failure for the selected specimen diameters. Results revealed that MLR developed equation can capture the relation between the N f (D R =2.68) and other input variables with high confidence. This allows the 3PBC test to run at various diameters to correlate with the reference geometry (i.e., N f (D R =2.68) ). 102 11. RECOMMENDATIONS The proposed research addresses an important long - standing issue for fatigue cracking characterization of asphalt pavements. Certainly, the proposed methodology possesses great potential to significantly reduce the testing cost and substantially increase the testing speed. Nevertheless, enhancements can be made to the p ractice presented in this study. A list of future works are listed below: - empirical pavement design is important measured during the AASHTO R83 test procedure, if additional radial LVDTs are pavements is often not a measured property, rather a default value is used in pavement design when a design is done using the AASHTOWare Pavement ME Design software. the 3PBC test is a calculated property, rather than a direct measurement, there is a need for its va lidation. Diameter effects on 3PBC fatigue life: The general purpose of this study is to provide a mechanistic, yet practical protocol to evaluate cracking resistance of asphalt mixtures. The obtained results show that the proposed approach possesses grea t potential to achieve this diameters in the fatigue life of the tested asphalt samples. Hence, further investigations should be performed to improve the analytical for mulations to capture the diameter effects on the fatigue life of the mixture. 103 Inter - laboratory study: t he results presented in this research study are quite promising, however, the 3PBC test method should still be considered to be under development. A n in ter - laboratory (round - robin) study should be performed to determine the accuracy and precision ( i.e., ASTM C802, ASTM C670) of the develop ed test practice. 104 APPENDICES 105 APPENDIX A Figure A. 1 . Mix ID: 5E3, sample dimensions: diameter = 38 mm, length = 100 mm . 106 Figure A.1. 107 Figure A.1. 108 Figure A. 2 . Mix ID: 5E3, sample dimensions: diameter = 38 mm, length = 125 mm . 109 Figure A.2. 110 Figure A.2. 111 Figure A. 3 . Mix ID: 5E3, sample dimensions: diameter = 68 mm, length = 125 mm . 112 Figure A.3. 113 Figure A.3. 114 Figure A. 4 . Mix ID: 5E3, sample dimensions: diameter = 68 mm, length = 135 mm . 115 Figure A.4. 116 Figure A.4. 117 Figure A. 5 . Mix ID: 5E3, sample dimensions: diameter = 100 mm, length = 135 mm . 11 8 Figure A.5. 119 Figure A.5. 120 Figure A . 6 . Mix ID: 3E3, sample dimensions: diameter = 38 mm, length = 100 mm . 121 Figure A . 7 . Mix ID: 3E3, sample dimensions: diameter = 68 mm, length = 125 mm . 122 Figure A . 8 . Mix ID: 3E3, sample dimensions: diameter = 100 mm, length = 135 mm . 123 APPENDIX B Table B. 1 MLR model database f (Hz) T (F ) Strain ( ) D i (inch) |E*| psi N f (D i ) (inch) N f (D R = 2.68 inch) Mix ID: 3E3 1 41 0.0001 1.50 1048381 3423720 436953 1 41 0.0002 1.50 1048381 386386 49313 1 41 0.0002 1.50 1048381 82182 10488 1 41 0.0003 1.50 1048381 24737 3157 1 41 0.0003 1.50 1048381 9275 1184 1 41 0.0004 1.50 1048381 4047 516 1 41 0.0004 1.50 1048381 1973 252 1 41 0.0005 1.50 1048381 1047 134 1 41 0.0005 1.50 1048381 594 76 1 41 0.0006 1.50 1048381 356 45 1 41 0.0006 1.50 1048381 223 28 1 50 0.0001 1.50 773444 2639539 336872 1 50 0.0002 1.50 773444 297887 38018 1 50 0.0002 1.50 773444 63359 8086 1 50 0.0003 1.50 773444 19071 2434 1 50 0.0003 1.50 773444 7150 913 1 50 0.0004 1.50 773444 3120 398 1 50 0.0004 1.50 773444 1521 194 1 50 0.0005 1.50 773444 807 103 1 50 0.0005 1.50 773444 458 58 1 50 0.0006 1.50 773444 274 35 1 50 0.0006 1.50 773444 172 22 1 59 0.0001 1.50 524549 3487425 445084 1 59 0.0002 1.50 524549 393576 50230 1 59 0.0002 1.50 524549 83711 10684 1 59 0.0003 1.50 524549 25197 3216 1 59 0.0003 1.50 524549 9447 1206 1 59 0.0004 1.50 524549 4122 526 1 59 0.0004 1.50 524549 2009 256 1 59 0.0005 1.50 524549 1066 136 1 59 0.0005 1.50 524549 605 77 1 59 0.0006 1.50 524549 362 46 1 59 0.0006 1.50 524549 227 29 1 68 0.0001 1.50 328959 7650815 976438 1 68 0.0002 1.50 328959 863438 110197 1 68 0.0002 1.50 328959 183648 23438 1 68 0.0003 1.50 328959 55278 7055 1 68 0.0003 1.50 328959 20726 2645 124 Table B . 1 1 68 0.0004 1.50 328959 9043 1154 1 68 0.0004 1.50 328959 4408 563 1 68 0.0005 1.50 328959 2339 299 1 68 0.0005 1.50 328959 1327 169 1 68 0.0006 1.50 328959 795 101 1 68 0.0006 1.50 328959 497 63 1 77 0.0001 1.50 194511 25099981 3203393 1 77 0.0002 1.50 194511 2832675 361521 1 77 0.0002 1.50 194511 602492 76893 1 77 0.0003 1.50 194511 181350 23145 1 77 0.0003 1.50 194511 67995 8678 1 77 0.0004 1.50 194511 29666 3786 1 77 0.0004 1.50 194511 14462 1846 1 77 0.0005 1.50 194511 7674 979 1 77 0.0005 1.50 194511 4353 556 1 77 0.0006 1.50 194511 2607 333 1 77 0.0006 1.50 194511 1632 208 1 86 0.0001 1.50 111784 104580592 13347130 1 86 0.0002 1.50 111784 11802513 1506299 1 86 0.0002 1.50 111784 2510320 320380 1 86 0.0003 1.50 111784 755605 96434 1 86 0.0003 1.50 111784 283304 36157 1 86 0.0004 1.50 111784 123606 15775 1 86 0.0004 1.50 111784 60257 7690 1 86 0.0005 1.50 111784 31972 4080 1 86 0.0005 1.50 111784 18137 2315 1 86 0.0006 1.50 111784 10861 1386 1 86 0.0006 1.50 111784 6800 868 5 41 0.0001 1.50 1266889 5890037 841192 5 41 0.0002 1.50 1266889 664724 94933 5 41 0.0002 1.50 1266889 141383 20192 5 41 0.0003 1.50 1266889 42556 6078 5 41 0.0003 1.50 1266889 15956 2279 5 41 0.0004 1.50 1266889 6962 994 5 41 0.0004 1.50 1266889 3394 485 5 41 0.0005 1.50 1266889 1801 257 5 41 0.0005 1.50 1266889 1022 146 5 41 0.0006 1.50 1266889 612 87 5 41 0.0006 1.50 1266889 383 55 5 50 0.0001 1.50 1007293 3035533 433523 5 50 0.0002 1.50 1007293 342577 48926 5 50 0.0002 1.50 1007293 72864 10406 5 50 0.0003 1.50 1007293 21932 3132 5 50 0.0003 1.50 1007293 8223 1174 5 50 0.0004 1.50 1007293 3588 512 125 Table B . 1 5 50 0.0004 1.50 1007293 1749 250 5 50 0.0005 1.50 1007293 928 133 5 50 0.0005 1.50 1007293 526 75 5 50 0.0006 1.50 1007293 315 45 5 50 0.0006 1.50 1007293 197 28 5 59 0.0001 1.50 744457 2525658 360705 5 59 0.0002 1.50 744457 285035 40708 5 59 0.0002 1.50 744457 60625 8658 5 59 0.0003 1.50 744457 18248 2606 5 59 0.0003 1.50 744457 6842 977 5 59 0.0004 1.50 744457 2985 426 5 59 0.0004 1.50 744457 1455 208 5 59 0.0005 1.50 744457 772 110 5 59 0.0005 1.50 744457 438 63 5 59 0.0006 1.50 744457 262 37 5 59 0.0006 1.50 744457 164 23 5 68 0.0001 1.50 509918 3447217 492318 5 68 0.0002 1.50 509918 389038 55561 5 68 0.0002 1.50 509918 82746 11817 5 68 0.0003 1.50 509918 24906 3557 5 68 0.0003 1.50 509918 9338 1334 5 68 0.0004 1.50 509918 4074 582 5 68 0.0004 1.50 509918 1986 284 5 68 0.0005 1.50 509918 1054 151 5 68 0.0005 1.50 509918 598 85 5 68 0.0006 1.50 509918 358 51 5 68 0.0006 1.50 509918 224 32 5 77 0.0001 1.50 326060 7422059 1059990 5 77 0.0002 1.50 326060 837621 119626 5 77 0.0002 1.50 326060 178157 25444 5 77 0.0003 1.50 326060 53625 7659 5 77 0.0003 1.50 326060 20106 2871 5 77 0.0004 1.50 326060 8772 1253 5 77 0.0004 1.50 326060 4276 611 5 77 0.0005 1.50 326060 2269 324 5 77 0.0005 1.50 326060 1287 184 5 77 0.0006 1.50 326060 771 110 5 77 0.0006 1.50 326060 483 69 5 86 0.0001 1.50 198270 22822529 3259426 5 86 0.0002 1.50 198270 2575652 367845 5 86 0.0002 1.50 198270 547825 78238 5 86 0.0003 1.50 198270 164895 23550 5 86 0.0003 1.50 198270 61825 8830 5 86 0.0004 1.50 198270 26974 3852 5 86 0.0004 1.50 198270 13150 1878 126 Table B . 1 5 86 0.0005 1.50 198270 6977 996 5 86 0.0005 1.50 198270 3958 565 5 86 0.0006 1.50 198270 2370 338 5 86 0.0006 1.50 198270 1484 212 10 41 0.0001 1.50 1352618 8691675 1109278 10 41 0.0002 1.50 1352618 980905 125188 10 41 0.0002 1.50 1352618 208632 26627 10 41 0.0003 1.50 1352618 62798 8015 10 41 0.0003 1.50 1352618 23545 3005 10 41 0.0004 1.50 1352618 10273 1311 10 41 0.0004 1.50 1352618 5008 639 10 41 0.0005 1.50 1352618 2657 339 10 41 0.0005 1.50 1352618 1507 192 10 41 0.0006 1.50 1352618 903 115 10 41 0.0006 1.50 1352618 565 72 10 50 0.0001 1.50 1105596 3860400 492685 10 50 0.0002 1.50 1105596 435668 55602 10 50 0.0002 1.50 1105596 92664 11826 10 50 0.0003 1.50 1105596 27892 3560 10 50 0.0003 1.50 1105596 10458 1335 10 50 0.0004 1.50 1105596 4563 582 10 50 0.0004 1.50 1105596 2224 284 10 50 0.0005 1.50 1105596 1180 151 10 50 0.0005 1.50 1105596 670 85 10 50 0.0006 1.50 1105596 401 51 10 50 0.0006 1.50 1105596 251 32 10 59 0.0001 1.50 844854 2683764 342516 10 59 0.0002 1.50 844854 302878 38655 10 59 0.0002 1.50 844854 64420 8222 10 59 0.0003 1.50 844854 19390 2475 10 59 0.0003 1.50 844854 7270 928 10 59 0.0004 1.50 844854 3172 405 10 59 0.0004 1.50 844854 1546 197 10 59 0.0005 1.50 844854 820 105 10 59 0.0005 1.50 844854 465 59 10 59 0.0006 1.50 844854 279 36 10 59 0.0006 1.50 844854 175 22 10 68 0.0001 1.50 600374 3005124 383530 10 68 0.0002 1.50 600374 339145 43284 10 68 0.0002 1.50 600374 72134 9206 10 68 0.0003 1.50 600374 21712 2771 10 68 0.0003 1.50 600374 8141 1039 10 68 0.0004 1.50 600374 3552 453 10 68 0.0004 1.50 600374 1731 221 10 68 0.0005 1.50 600374 919 117 127 Table B . 1 10 68 0.0005 1.50 600374 521 67 10 68 0.0006 1.50 600374 312 40 10 68 0.0006 1.50 600374 195 25 10 77 0.0001 1.50 398030 5326749 679828 10 77 0.0002 1.50 398030 601154 76722 10 77 0.0002 1.50 398030 127862 16318 10 77 0.0003 1.50 398030 38486 4912 10 77 0.0003 1.50 398030 14430 1842 10 77 0.0004 1.50 398030 6296 804 10 77 0.0004 1.50 398030 3069 392 10 77 0.0005 1.50 398030 1628 208 10 77 0.0005 1.50 398030 924 118 10 77 0.0006 1.50 398030 553 71 10 77 0.0006 1.50 398030 346 44 10 86 0.0001 1.50 249551 13893534 1773166 10 86 0.0002 1.50 249551 1567964 200112 10 86 0.0002 1.50 249551 333496 42563 10 86 0.0003 1.50 249551 100382 12811 10 86 0.0003 1.50 249551 37637 4803 10 86 0.0004 1.50 249551 16421 2096 10 86 0.0004 1.50 249551 8005 1022 10 86 0.0005 1.50 249551 4248 542 10 86 0.0005 1.50 249551 2410 308 10 86 0.0006 1.50 249551 1443 184 10 86 0.0006 1.50 249551 903 115 1 41 0.0001 3.94 1048381 15246 436953 1 41 0.0002 3.94 1048381 1721 49313 1 41 0.0002 3.94 1048381 366 10488 1 41 0.0003 3.94 1048381 110 3157 1 41 0.0003 3.94 1048381 41 1184 1 41 0.0004 3.94 1048381 18 516 1 41 0.0004 3.94 1048381 9 252 1 41 0.0005 3.94 1048381 5 134 1 41 0.0005 3.94 1048381 3 76 1 41 0.0006 3.94 1048381 2 45 1 41 0.0006 3.94 1048381 1 28 1 50 0.0001 3.94 773444 11754 336872 1 50 0.0002 3.94 773444 1327 38018 1 50 0.0002 3.94 773444 282 8086 1 50 0.0003 3.94 773444 85 2434 1 50 0.0003 3.94 773444 32 913 1 50 0.0004 3.94 773444 14 398 1 50 0.0004 3.94 773444 7 194 1 50 0.0005 3.94 773444 4 103 1 50 0.0005 3.94 773444 2 58 128 Table B . 1 1 50 0.0006 3.94 773444 1 35 1 50 0.0006 3.94 773444 1 22 1 59 0.0001 3.94 524549 15530 445084 1 59 0.0002 3.94 524549 1753 50230 1 59 0.0002 3.94 524549 373 10684 1 59 0.0003 3.94 524549 112 3216 1 59 0.0003 3.94 524549 42 1206 1 59 0.0004 3.94 524549 18 526 1 59 0.0004 3.94 524549 9 256 1 59 0.0005 3.94 524549 5 136 1 59 0.0005 3.94 524549 3 77 1 59 0.0006 3.94 524549 2 46 1 59 0.0006 3.94 524549 1 29 1 68 0.0001 3.94 328959 34070 976438 1 68 0.0002 3.94 328959 3845 110197 1 68 0.0002 3.94 328959 818 23438 1 68 0.0003 3.94 328959 246 7055 1 68 0.0003 3.94 328959 92 2645 1 68 0.0004 3.94 328959 40 1154 1 68 0.0004 3.94 328959 20 563 1 68 0.0005 3.94 328959 10 299 1 68 0.0005 3.94 328959 6 169 1 68 0.0006 3.94 328959 4 101 1 68 0.0006 3.94 328959 2 63 1 77 0.0001 3.94 194511 111772 3203393 1 77 0.0002 3.94 194511 12614 361521 1 77 0.0002 3.94 194511 2683 76893 1 77 0.0003 3.94 194511 808 23145 1 77 0.0003 3.94 194511 303 8678 1 77 0.0004 3.94 194511 132 3786 1 77 0.0004 3.94 194511 64 1846 1 77 0.0005 3.94 194511 34 979 1 77 0.0005 3.94 194511 19 556 1 77 0.0006 3.94 194511 12 333 1 77 0.0006 3.94 194511 7 208 1 86 0.0001 3.94 111784 465703 13347130 1 86 0.0002 3.94 111784 52557 1506299 1 86 0.0002 3.94 111784 11179 320380 1 86 0.0003 3.94 111784 3365 96434 1 86 0.0003 3.94 111784 1262 36157 1 86 0.0004 3.94 111784 550 15775 1 86 0.0004 3.94 111784 268 7690 1 86 0.0005 3.94 111784 142 4080 1 86 0.0005 3.94 111784 81 2315 1 86 0.0006 3.94 111784 48 1386 129 Table B . 1 1 86 0.0006 3.94 111784 30 868 5 41 0.0001 3.94 1266889 28708 841192 5 41 0.0002 3.94 1266889 3240 94933 5 41 0.0002 3.94 1266889 689 20192 5 41 0.0003 3.94 1266889 207 6078 5 41 0.0003 3.94 1266889 78 2279 5 41 0.0004 3.94 1266889 34 994 5 41 0.0004 3.94 1266889 17 485 5 41 0.0005 3.94 1266889 9 257 5 41 0.0005 3.94 1266889 5 146 5 41 0.0006 3.94 1266889 3 87 5 41 0.0006 3.94 1266889 2 55 5 50 0.0001 3.94 1007293 14795 433523 5 50 0.0002 3.94 1007293 1670 48926 5 50 0.0002 3.94 1007293 355 10406 5 50 0.0003 3.94 1007293 107 3132 5 50 0.0003 3.94 1007293 40 1174 5 50 0.0004 3.94 1007293 17 512 5 50 0.0004 3.94 1007293 9 250 5 50 0.0005 3.94 1007293 5 133 5 50 0.0005 3.94 1007293 3 75 5 50 0.0006 3.94 1007293 2 45 5 50 0.0006 3.94 1007293 1 28 5 59 0.0001 3.94 744457 12310 360705 5 59 0.0002 3.94 744457 1389 40708 5 59 0.0002 3.94 744457 295 8658 5 59 0.0003 3.94 744457 89 2606 5 59 0.0003 3.94 744457 33 977 5 59 0.0004 3.94 744457 15 426 5 59 0.0004 3.94 744457 7 208 5 59 0.0005 3.94 744457 4 110 5 59 0.0005 3.94 744457 2 63 5 59 0.0006 3.94 744457 1 37 5 59 0.0006 3.94 744457 1 23 5 68 0.0001 3.94 509918 16802 492318 5 68 0.0002 3.94 509918 1896 55561 5 68 0.0002 3.94 509918 403 11817 5 68 0.0003 3.94 509918 121 3557 5 68 0.0003 3.94 509918 46 1334 5 68 0.0004 3.94 509918 20 582 5 68 0.0004 3.94 509918 10 284 5 68 0.0005 3.94 509918 5 151 5 68 0.0005 3.94 509918 3 85 5 68 0.0006 3.94 509918 2 51 5 68 0.0006 3.94 509918 1 32 130 Table B . 1 5 77 0.0001 3.94 326060 36175 1059990 5 77 0.0002 3.94 326060 4083 119626 5 77 0.0002 3.94 326060 868 25444 5 77 0.0003 3.94 326060 261 7659 5 77 0.0003 3.94 326060 98 2871 5 77 0.0004 3.94 326060 43 1253 5 77 0.0004 3.94 326060 21 611 5 77 0.0005 3.94 326060 11 324 5 77 0.0005 3.94 326060 6 184 5 77 0.0006 3.94 326060 4 110 5 77 0.0006 3.94 326060 2 69 5 86 0.0001 3.94 198270 111236 3259426 5 86 0.0002 3.94 198270 12554 367845 5 86 0.0002 3.94 198270 2670 78238 5 86 0.0003 3.94 198270 804 23550 5 86 0.0003 3.94 198270 301 8830 5 86 0.0004 3.94 198270 131 3852 5 86 0.0004 3.94 198270 64 1878 5 86 0.0005 3.94 198270 34 996 5 86 0.0005 3.94 198270 19 565 5 86 0.0006 3.94 198270 12 338 5 86 0.0006 3.94 198270 7 212 10 41 0.0001 3.94 1352618 38705 1109278 10 41 0.0002 3.94 1352618 4368 125188 10 41 0.0002 3.94 1352618 929 26627 10 41 0.0003 3.94 1352618 280 8015 10 41 0.0003 3.94 1352618 105 3005 10 41 0.0004 3.94 1352618 46 1311 10 41 0.0004 3.94 1352618 22 639 10 41 0.0005 3.94 1352618 12 339 10 41 0.0005 3.94 1352618 7 192 10 41 0.0006 3.94 1352618 4 115 10 41 0.0006 3.94 1352618 3 72 10 50 0.0001 3.94 1105596 17191 492685 10 50 0.0002 3.94 1105596 1940 55602 10 50 0.0002 3.94 1105596 413 11826 10 50 0.0003 3.94 1105596 124 3560 10 50 0.0003 3.94 1105596 47 1335 10 50 0.0004 3.94 1105596 20 582 10 50 0.0004 3.94 1105596 10 284 10 50 0.0005 3.94 1105596 5 151 10 50 0.0005 3.94 1105596 3 85 10 50 0.0006 3.94 1105596 2 51 10 50 0.0006 3.94 1105596 1 32 10 59 0.0001 3.94 844854 11951 342516 131 Table B . 1 10 59 0.0002 3.94 844854 1349 38655 10 59 0.0002 3.94 844854 287 8222 10 59 0.0003 3.94 844854 86 2475 10 59 0.0003 3.94 844854 32 928 10 59 0.0004 3.94 844854 14 405 10 59 0.0004 3.94 844854 7 197 10 59 0.0005 3.94 844854 4 105 10 59 0.0005 3.94 844854 2 59 10 59 0.0006 3.94 844854 1 36 10 59 0.0006 3.94 844854 1 22 10 68 0.0001 3.94 600374 13382 383530 10 68 0.0002 3.94 600374 1510 43284 10 68 0.0002 3.94 600374 321 9206 10 68 0.0003 3.94 600374 97 2771 10 68 0.0003 3.94 600374 36 1039 10 68 0.0004 3.94 600374 16 453 10 68 0.0004 3.94 600374 8 221 10 68 0.0005 3.94 600374 4 117 10 68 0.0005 3.94 600374 2 67 10 68 0.0006 3.94 600374 1 40 10 68 0.0006 3.94 600374 1 25 10 77 0.0001 3.94 398030 23720 679828 10 77 0.0002 3.94 398030 2677 76722 10 77 0.0002 3.94 398030 569 16318 10 77 0.0003 3.94 398030 171 4912 10 77 0.0003 3.94 398030 64 1842 10 77 0.0004 3.94 398030 28 804 10 77 0.0004 3.94 398030 14 392 10 77 0.0005 3.94 398030 7 208 10 77 0.0005 3.94 398030 4 118 10 77 0.0006 3.94 398030 2 71 10 77 0.0006 3.94 398030 2 44 10 86 0.0001 3.94 249551 61869 1773166 10 86 0.0002 3.94 249551 6982 200112 10 86 0.0002 3.94 249551 1485 42563 10 86 0.0003 3.94 249551 447 12811 10 86 0.0003 3.94 249551 168 4803 10 86 0.0004 3.94 249551 73 2096 10 86 0.0004 3.94 249551 36 1022 10 86 0.0005 3.94 249551 19 542 10 86 0.0005 3.94 249551 11 308 10 86 0.0006 3.94 249551 6 184 10 86 0.0006 3.94 249551 4 115 Mix ID: 5E3 1 41 0.0001 1.50 1235891 11029925 527307 132 Table B . 1 1 41 0.0002 1.50 1235891 951211 45474 1 41 0.0002 1.50 1235891 167166 7992 1 41 0.0003 1.50 1235891 43393 2075 1 41 0.0003 1.50 1235891 14416 689 1 41 0.0004 1.50 1235891 5678 271 1 41 0.0004 1.50 1235891 2534 121 1 41 0.0005 1.50 1235891 1243 59 1 41 0.0005 1.50 1235891 658 31 1 41 0.0006 1.50 1235891 370 18 1 41 0.0006 1.50 1235891 218 10 1 50 0.0001 1.50 923935 11929009 570289 1 50 0.0002 1.50 923935 1028747 49181 1 50 0.0002 1.50 923935 180792 8643 1 50 0.0003 1.50 923935 46930 2244 1 50 0.0003 1.50 923935 15591 745 1 50 0.0004 1.50 923935 6141 294 1 50 0.0004 1.50 923935 2740 131 1 50 0.0005 1.50 923935 1345 64 1 50 0.0005 1.50 923935 711 34 1 50 0.0006 1.50 923935 400 19 1 50 0.0006 1.50 923935 236 11 1 59 0.0001 1.50 649823 19873936 950112 1 59 0.0002 1.50 649823 1713911 81937 1 59 0.0002 1.50 649823 301203 14400 1 59 0.0003 1.50 649823 78187 3738 1 59 0.0003 1.50 649823 25975 1242 1 59 0.0004 1.50 649823 10231 489 1 59 0.0004 1.50 649823 4565 218 1 59 0.0005 1.50 649823 2240 107 1 59 0.0005 1.50 649823 1185 57 1 59 0.0006 1.50 649823 666 32 1 59 0.0006 1.50 649823 394 19 1 68 0.0001 1.50 433274 48700894 2328240 1 68 0.0002 1.50 433274 4199923 200785 1 68 0.0002 1.50 433274 738094 35286 1 68 0.0003 1.50 433274 191596 9160 1 68 0.0003 1.50 433274 63653 3043 1 68 0.0004 1.50 433274 25072 1199 1 68 0.0004 1.50 433274 11186 535 1 68 0.0005 1.50 433274 5489 262 1 68 0.0005 1.50 433274 2904 139 1 68 0.0006 1.50 433274 1632 78 1 68 0.0006 1.50 433274 965 46 1 77 0.0001 1.50 278141 159857337 7642288 1 77 0.0002 1.50 278141 13785960 659064 133 Table B . 1 1 77 0.0002 1.50 278141 2422743 115824 1 77 0.0003 1.50 278141 628902 30066 1 77 0.0003 1.50 278141 208935 9989 1 77 0.0004 1.50 278141 82297 3934 1 77 0.0004 1.50 278141 36718 1755 1 77 0.0005 1.50 278141 18018 861 1 77 0.0005 1.50 278141 9531 456 1 77 0.0006 1.50 278141 5358 256 1 77 0.0006 1.50 278141 3167 151 1 86 0.0001 1.50 175580 618663867 29576417 1 86 0.0002 1.50 175580 53353041 2550645 1 86 0.0002 1.50 175580 9376256 448250 1 86 0.0003 1.50 175580 2433915 116358 1 86 0.0003 1.50 175580 808600 38657 1 86 0.0004 1.50 175580 318499 15226 1 86 0.0004 1.50 175580 142103 6794 1 86 0.0005 1.50 175580 69733 3334 1 86 0.0005 1.50 175580 36888 1763 1 86 0.0006 1.50 175580 20735 991 1 86 0.0006 1.50 175580 12255 586 5 41 0.0001 1.50 1531662 15078089 720837 5 41 0.0002 1.50 1531662 1300321 62164 5 41 0.0002 1.50 1531662 228518 10925 5 41 0.0003 1.50 1531662 59319 2836 5 41 0.0003 1.50 1531662 19707 942 5 41 0.0004 1.50 1531662 7762 371 5 41 0.0004 1.50 1531662 3463 166 5 41 0.0005 1.50 1531662 1700 81 5 41 0.0005 1.50 1531662 899 43 5 41 0.0006 1.50 1531662 505 24 5 41 0.0006 1.50 1531662 299 14 5 50 0.0001 1.50 1222672 10970079 524446 5 50 0.0002 1.50 1222672 946050 45228 5 50 0.0002 1.50 1222672 166259 7948 5 50 0.0003 1.50 1222672 43158 2063 5 50 0.0003 1.50 1222672 14338 685 5 50 0.0004 1.50 1222672 5648 270 5 50 0.0004 1.50 1222672 2520 120 5 50 0.0005 1.50 1222672 1236 59 5 50 0.0005 1.50 1222672 654 31 5 50 0.0006 1.50 1222672 368 18 5 50 0.0006 1.50 1222672 217 10 5 59 0.0001 1.50 922624 11944736 571041 5 59 0.0002 1.50 922624 1030104 49246 5 59 0.0002 1.50 922624 181030 8655 134 Table B . 1 5 59 0.0003 1.50 922624 46992 2247 5 59 0.0003 1.50 922624 15612 746 5 59 0.0004 1.50 922624 6149 294 5 59 0.0004 1.50 922624 2744 131 5 59 0.0005 1.50 922624 1346 64 5 59 0.0005 1.50 922624 712 34 5 59 0.0006 1.50 922624 400 19 5 59 0.0006 1.50 922624 237 11 5 68 0.0001 1.50 658577 19383902 926685 5 68 0.0002 1.50 658577 1671651 79916 5 68 0.0002 1.50 658577 293776 14045 5 68 0.0003 1.50 658577 76259 3646 5 68 0.0003 1.50 658577 25335 1211 5 68 0.0004 1.50 658577 9979 477 5 68 0.0004 1.50 658577 4452 213 5 68 0.0005 1.50 658577 2185 104 5 68 0.0005 1.50 658577 1156 55 5 68 0.0006 1.50 658577 650 31 5 68 0.0006 1.50 658577 384 18 5 77 0.0001 1.50 447932 44866064 2144909 5 77 0.0002 1.50 447932 3869211 184975 5 77 0.0002 1.50 447932 679975 32507 5 77 0.0003 1.50 447932 176510 8438 5 77 0.0003 1.50 447932 58640 2803 5 77 0.0004 1.50 447932 23098 1104 5 77 0.0004 1.50 447932 10305 493 5 77 0.0005 1.50 447932 5057 242 5 77 0.0005 1.50 447932 2675 128 5 77 0.0006 1.50 447932 1504 72 5 77 0.0006 1.50 447932 889 42 5 86 0.0001 1.50 294341 136235454 6512998 5 86 0.0002 1.50 294341 11748829 561675 5 86 0.0002 1.50 294341 2064738 98709 5 86 0.0003 1.50 294341 535970 25623 5 86 0.0003 1.50 294341 178061 8513 5 86 0.0004 1.50 294341 70136 3353 5 86 0.0004 1.50 294341 31292 1496 5 86 0.0005 1.50 294341 15356 734 5 86 0.0005 1.50 294341 8123 388 5 86 0.0006 1.50 294341 4566 218 5 86 0.0006 1.50 294341 2699 129 10 41 0.0001 1.50 1651947 19095547 912899 10 41 0.0002 1.50 1651947 1646784 78728 10 41 0.0002 1.50 1651947 289406 13836 10 41 0.0003 1.50 1651947 75125 3591 135 Table B . 1 10 41 0.0003 1.50 1651947 24958 1193 10 41 0.0004 1.50 1651947 9831 470 10 41 0.0004 1.50 1651947 4386 210 10 41 0.0005 1.50 1651947 2152 103 10 41 0.0005 1.50 1651947 1139 54 10 41 0.0006 1.50 1651947 640 31 10 41 0.0006 1.50 1651947 378 18 10 50 0.0001 1.50 1352178 11939111 570772 10 50 0.0002 1.50 1352178 1029619 49223 10 50 0.0002 1.50 1352178 180945 8650 10 50 0.0003 1.50 1352178 46970 2246 10 50 0.0003 1.50 1352178 15605 746 10 50 0.0004 1.50 1352178 6146 294 10 50 0.0004 1.50 1352178 2742 131 10 50 0.0005 1.50 1352178 1346 64 10 50 0.0005 1.50 1352178 712 34 10 50 0.0006 1.50 1352178 400 19 10 50 0.0006 1.50 1352178 236 11 10 59 0.0001 1.50 1049741 10949560 523465 10 59 0.0002 1.50 1049741 944281 45143 10 59 0.0002 1.50 1049741 165948 7933 10 59 0.0003 1.50 1049741 43077 2059 10 59 0.0003 1.50 1049741 14311 684 10 59 0.0004 1.50 1049741 5637 269 10 59 0.0004 1.50 1049741 2515 120 10 59 0.0005 1.50 1049741 1234 59 10 59 0.0005 1.50 1049741 653 31 10 59 0.0006 1.50 1049741 367 18 10 59 0.0006 1.50 1049741 217 10 10 68 0.0001 1.50 771841 14856204 710229 10 68 0.0002 1.50 771841 1281186 61250 10 68 0.0002 1.50 771841 225156 10764 10 68 0.0003 1.50 771841 58446 2794 10 68 0.0003 1.50 771841 19417 928 10 68 0.0004 1.50 771841 7648 366 10 68 0.0004 1.50 771841 3412 163 10 68 0.0005 1.50 771841 1675 80 10 68 0.0005 1.50 771841 886 42 10 68 0.0006 1.50 771841 498 24 10 68 0.0006 1.50 771841 294 14 10 77 0.0001 1.50 539810 29054283 1388996 10 77 0.0002 1.50 539810 2505616 119786 10 77 0.0002 1.50 539810 440337 21051 10 77 0.0003 1.50 539810 114304 5465 10 77 0.0003 1.50 539810 37974 1815 136 Table B . 1 10 77 0.0004 1.50 539810 14958 715 10 77 0.0004 1.50 539810 6674 319 10 77 0.0005 1.50 539810 3275 157 10 77 0.0005 1.50 539810 1732 83 10 77 0.0006 1.50 539810 974 47 10 77 0.0006 1.50 539810 576 28 10 86 0.0001 1.50 362933 76818988 3672480 10 86 0.0002 1.50 362933 6624804 316711 10 86 0.0002 1.50 362933 1164242 55659 10 86 0.0003 1.50 362933 302217 14448 10 86 0.0003 1.50 362933 100403 4800 10 86 0.0004 1.50 362933 39548 1891 10 86 0.0004 1.50 362933 17645 844 10 86 0.0005 1.50 362933 8659 414 10 86 0.0005 1.50 362933 4580 219 10 86 0.0006 1.50 362933 2575 123 10 86 0.0006 1.50 362933 1522 73 1 41 0.0001 3.94 1235891 23816 527307 1 41 0.0002 3.94 1235891 2054 45474 1 41 0.0002 3.94 1235891 361 7992 1 41 0.0003 3.94 1235891 94 2075 1 41 0.0003 3.94 1235891 31 689 1 41 0.0004 3.94 1235891 12 271 1 41 0.0004 3.94 1235891 5 121 1 41 0.0005 3.94 1235891 3 59 1 41 0.0005 3.94 1235891 1 31 1 41 0.0006 3.94 1235891 1 18 1 41 0.0006 3.94 1235891 0 10 1 50 0.0001 3.94 923935 25758 570289 1 50 0.0002 3.94 923935 2221 49181 1 50 0.0002 3.94 923935 390 8643 1 50 0.0003 3.94 923935 101 2244 1 50 0.0003 3.94 923935 34 745 1 50 0.0004 3.94 923935 13 294 1 50 0.0004 3.94 923935 6 131 1 50 0.0005 3.94 923935 3 64 1 50 0.0005 3.94 923935 2 34 1 50 0.0006 3.94 923935 1 19 1 50 0.0006 3.94 923935 1 11 1 59 0.0001 3.94 649823 42913 950112 1 59 0.0002 3.94 649823 3701 81937 1 59 0.0002 3.94 649823 650 14400 1 59 0.0003 3.94 649823 169 3738 1 59 0.0003 3.94 649823 56 1242 1 59 0.0004 3.94 649823 22 489 137 Table B . 1 1 59 0.0004 3.94 649823 10 218 1 59 0.0005 3.94 649823 5 107 1 59 0.0005 3.94 649823 3 57 1 59 0.0006 3.94 649823 1 32 1 59 0.0006 3.94 649823 1 19 1 68 0.0001 3.94 433274 105158 2328240 1 68 0.0002 3.94 433274 9069 200785 1 68 0.0002 3.94 433274 1594 35286 1 68 0.0003 3.94 433274 414 9160 1 68 0.0003 3.94 433274 137 3043 1 68 0.0004 3.94 433274 54 1199 1 68 0.0004 3.94 433274 24 535 1 68 0.0005 3.94 433274 12 262 1 68 0.0005 3.94 433274 6 139 1 68 0.0006 3.94 433274 4 78 1 68 0.0006 3.94 433274 2 46 1 77 0.0001 3.94 278141 345174 7642288 1 77 0.0002 3.94 278141 29767 659064 1 77 0.0002 3.94 278141 5231 115824 1 77 0.0003 3.94 278141 1358 30066 1 77 0.0003 3.94 278141 451 9989 1 77 0.0004 3.94 278141 178 3934 1 77 0.0004 3.94 278141 79 1755 1 77 0.0005 3.94 278141 39 861 1 77 0.0005 3.94 278141 21 456 1 77 0.0006 3.94 278141 12 256 1 77 0.0006 3.94 278141 7 151 1 86 0.0001 3.94 175580 1335857 29576417 1 86 0.0002 3.94 175580 115203 2550645 1 86 0.0002 3.94 175580 20246 448250 1 86 0.0003 3.94 175580 5255 116358 1 86 0.0003 3.94 175580 1746 38657 1 86 0.0004 3.94 175580 688 15226 1 86 0.0004 3.94 175580 307 6794 1 86 0.0005 3.94 175580 151 3334 1 86 0.0005 3.94 175580 80 1763 1 86 0.0006 3.94 175580 45 991 1 86 0.0006 3.94 175580 26 586 5 41 0.0001 3.94 1531662 32558 720837 5 41 0.0002 3.94 1531662 2808 62164 5 41 0.0002 3.94 1531662 493 10925 5 41 0.0003 3.94 1531662 128 2836 5 41 0.0003 3.94 1531662 43 942 5 41 0.0004 3.94 1531662 17 371 5 41 0.0004 3.94 1531662 7 166 138 Table B . 1 5 41 0.0005 3.94 1531662 4 81 5 41 0.0005 3.94 1531662 2 43 5 41 0.0006 3.94 1531662 1 24 5 41 0.0006 3.94 1531662 1 14 5 50 0.0001 3.94 1222672 23687 524446 5 50 0.0002 3.94 1222672 2043 45228 5 50 0.0002 3.94 1222672 359 7948 5 50 0.0003 3.94 1222672 93 2063 5 50 0.0003 3.94 1222672 31 685 5 50 0.0004 3.94 1222672 12 270 5 50 0.0004 3.94 1222672 5 120 5 50 0.0005 3.94 1222672 3 59 5 50 0.0005 3.94 1222672 1 31 5 50 0.0006 3.94 1222672 1 18 5 50 0.0006 3.94 1222672 0 10 5 59 0.0001 3.94 922624 25792 571041 5 59 0.0002 3.94 922624 2224 49246 5 59 0.0002 3.94 922624 391 8655 5 59 0.0003 3.94 922624 101 2247 5 59 0.0003 3.94 922624 34 746 5 59 0.0004 3.94 922624 13 294 5 59 0.0004 3.94 922624 6 131 5 59 0.0005 3.94 922624 3 64 5 59 0.0005 3.94 922624 2 34 5 59 0.0006 3.94 922624 1 19 5 59 0.0006 3.94 922624 1 11 5 68 0.0001 3.94 658577 41855 926685 5 68 0.0002 3.94 658577 3610 79916 5 68 0.0002 3.94 658577 634 14045 5 68 0.0003 3.94 658577 165 3646 5 68 0.0003 3.94 658577 55 1211 5 68 0.0004 3.94 658577 22 477 5 68 0.0004 3.94 658577 10 213 5 68 0.0005 3.94 658577 5 104 5 68 0.0005 3.94 658577 2 55 5 68 0.0006 3.94 658577 1 31 5 68 0.0006 3.94 658577 1 18 5 77 0.0001 3.94 447932 96878 2144909 5 77 0.0002 3.94 447932 8355 184975 5 77 0.0002 3.94 447932 1468 32507 5 77 0.0003 3.94 447932 381 8438 5 77 0.0003 3.94 447932 127 2803 5 77 0.0004 3.94 447932 50 1104 5 77 0.0004 3.94 447932 22 493 5 77 0.0005 3.94 447932 11 242 139 Table B . 1 5 77 0.0005 3.94 447932 6 128 5 77 0.0006 3.94 447932 3 72 5 77 0.0006 3.94 447932 2 42 5 86 0.0001 3.94 294341 294168 6512998 5 86 0.0002 3.94 294341 25369 561675 5 86 0.0002 3.94 294341 4458 98709 5 86 0.0003 3.94 294341 1157 25623 5 86 0.0003 3.94 294341 384 8513 5 86 0.0004 3.94 294341 151 3353 5 86 0.0004 3.94 294341 68 1496 5 86 0.0005 3.94 294341 33 734 5 86 0.0005 3.94 294341 18 388 5 86 0.0006 3.94 294341 10 218 5 86 0.0006 3.94 294341 6 129 10 41 0.0001 3.94 1651947 41232 912899 10 41 0.0002 3.94 1651947 3556 78728 10 41 0.0002 3.94 1651947 625 13836 10 41 0.0003 3.94 1651947 162 3591 10 41 0.0003 3.94 1651947 54 1193 10 41 0.0004 3.94 1651947 21 470 10 41 0.0004 3.94 1651947 9 210 10 41 0.0005 3.94 1651947 5 103 10 41 0.0005 3.94 1651947 2 54 10 41 0.0006 3.94 1651947 1 31 10 41 0.0006 3.94 1651947 1 18 10 50 0.0001 3.94 1352178 25780 570772 10 50 0.0002 3.94 1352178 2223 49223 10 50 0.0002 3.94 1352178 391 8650 10 50 0.0003 3.94 1352178 101 2246 10 50 0.0003 3.94 1352178 34 746 10 50 0.0004 3.94 1352178 13 294 10 50 0.0004 3.94 1352178 6 131 10 50 0.0005 3.94 1352178 3 64 10 50 0.0005 3.94 1352178 2 34 10 50 0.0006 3.94 1352178 1 19 10 50 0.0006 3.94 1352178 1 11 10 59 0.0001 3.94 1049741 23643 523465 10 59 0.0002 3.94 1049741 2039 45143 10 59 0.0002 3.94 1049741 358 7933 10 59 0.0003 3.94 1049741 93 2059 10 59 0.0003 3.94 1049741 31 684 10 59 0.0004 3.94 1049741 12 269 10 59 0.0004 3.94 1049741 5 120 10 59 0.0005 3.94 1049741 3 59 10 59 0.0005 3.94 1049741 1 31 140 Table B . 1 10 59 0.0006 3.94 1049741 1 18 10 59 0.0006 3.94 1049741 0 10 10 68 0.0001 3.94 771841 32078 710229 10 68 0.0002 3.94 771841 2766 61250 10 68 0.0002 3.94 771841 486 10764 10 68 0.0003 3.94 771841 126 2794 10 68 0.0003 3.94 771841 42 928 10 68 0.0004 3.94 771841 17 366 10 68 0.0004 3.94 771841 7 163 10 68 0.0005 3.94 771841 4 80 10 68 0.0005 3.94 771841 2 42 10 68 0.0006 3.94 771841 1 24 10 68 0.0006 3.94 771841 1 14 10 77 0.0001 3.94 539810 62736 1388996 10 77 0.0002 3.94 539810 5410 119786 10 77 0.0002 3.94 539810 951 21051 10 77 0.0003 3.94 539810 247 5465 10 77 0.0003 3.94 539810 82 1815 10 77 0.0004 3.94 539810 32 715 10 77 0.0004 3.94 539810 14 319 10 77 0.0005 3.94 539810 7 157 10 77 0.0005 3.94 539810 4 83 10 77 0.0006 3.94 539810 2 47 10 77 0.0006 3.94 539810 1 28 10 86 0.0001 3.94 362933 165872 3672480 10 86 0.0002 3.94 362933 14305 316711 10 86 0.0002 3.94 362933 2514 55659 10 86 0.0003 3.94 362933 653 14448 10 86 0.0003 3.94 362933 217 4800 10 86 0.0004 3.94 362933 85 1891 10 86 0.0004 3.94 362933 38 844 10 86 0.0005 3.94 362933 19 414 10 86 0.0005 3.94 362933 10 219 10 86 0.0006 3.94 362933 6 123 10 86 0.0006 3.94 362933 3 73 141 REFERENCES 142 REFERENCES AASHTO. 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