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40 Proposed Standard Practice for Developing Dynamic Modulus Master Curves for Hot-Mix Asphalt Concrete Using the Simple Performance Test System NCHRP 9-29: PP 02 1. SCOPE 1.1 This practice describes testing and analysis for developing a dynamic modulus master curve for hot-mix asphalt concrete using the Simple Performance Test System. This practice is intended for dense- and gap- graded mixtures with nominal maximum aggregate sizes to 37.5 mm. 1.2 This standard may involve hazardous materials, operations, and equipment, This standard does not purport to address all of the safety problems associated with its use. It is the responsibility of the user of this procedure to establish appropriate safety and health practices and to determine the applicability of regulatory limitations prior to its use. 2. REFERENCED DOCUMENTS 2.1 AASHTO Standards NCHRP 9-29 PP 01, Preparation of Cylindrical Performance Test Specimens Using the Superpave Gyratory Compactor. NCHRP 9-29 PT 01, Determining the Dynamic Modulus and Flow Number for Hot Mix Asphalt (HMA) Using the Simple Performance Test System 2.2 Other Publications Equipment Specification for the Simple Performance Test System, Version 3.0, Prepared for National Cooperative Highway Research Program (NCHRP), October 16, 2007. 3. TERMINOLOGY 3.1 Dynamic Modulus Master Curve a composite curve constructed at a reference temperature by shifting dynamic modulus data from various temperatures along the log frequency axis.

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41 3.2 Reduced Frequency The computed frequency at the reference temperature equivalent to the actual loading frequency at the test temperature. 3.3 Reference Temperature The temperature at which the master curve is constructed. 3.4 Shift Factor- Shift in frequency associated with a shift from a test temperature to the reference temperature. 4. SUMMARY OF PRACTICE 4.1 This practice describes the testing and analysis needed to develop a dynamic modulus master curve for hot-mix asphalt concrete mixtures. It involves collecting dynamic modulus test data at specified temperatures and loading rates, then manipulating the test data to obtain a continuous function describing the dynamic modulus as a function of frequency and temperature. 5. SIGNIFICANCE AND USE 5.1 Dynamic modulus master curves can be used for mixture evaluation and for characterizing the modulus of hot-mix asphalt concrete for mechanistic-empirical pavement design. 6. APPARATUS 6.1 Specimen Fabrication Equipment - Equipment for fabricating dynamic modulus test specimens as described in NCHRP 9-29 PP 01, Preparation of Cylindrical Performance Test Specimens Using the Superpave Gyratory Compactor. 6.2 Dynamic Modulus Test System - A dynamic test system meeting the requirements of Equipment Specification for the Simple Performance Test System, Version 3.0. 6.3 Analysis Software Software capable of performing numerical optimization of non- linear equations. Note 1 - The Solver Tool included in Microsoft Excel is capable of performing the numerical optimization required by this practice. 7. HAZARDS 7.1 This practice and associated standards involve handling of hot asphalt binder, aggregates and asphalt mixtures. It also includes the use of sawing and coring

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42 machinery and servo-hydraulic testing equipment. Use standard safety precautions, equipment, and clothing when handling hot materials and operating machinery. 8. STANDARDIZATION 8.1 Items associated with this practice that require calibration are included in the documents referenced in Section 2. Refer to the pertinent section of the referenced documents for information concerning calibration. 9. DYNAMIC MODULUS TEST DATA 9.1 Test Specimen Fabrication 9.1.1 Prepare at least two test specimens to the target air void content and aging condition in accordance with NCHRP 9-29 PP 01, Preparation of Cylindrical Performance Test Specimens Using the Superpave Gyratory Compactor. Note 2 A reasonable air void tolerance for test specimen fabrication is 0.5 %. Note 3 The coefficient of variation for properly conducted dynamic modulus tests is approximately 13 %. The coefficient of variation of the mean dynamic modulus for tests on multiple specimens is given by Table 1. Table 1. Coefficient of Variation for the Mean of Dynamic Modulus Test on Replicate Specimens. Specimens Coefficient of Variation For the Mean 2 9.2 3 7.5 4 6.5 5 5.8 6 5.3 7 4.9 8 4.6 9 4.3 10 4.1 Use Table 1 to select an appropriate number of specimens based on the uncertainty that can be tolerated in the analysis. 9.1.2 Record the following volumetric properties for each test specimen: Voids in the mineral aggregate (VMA)

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43 Voids filled with asphalt concrete (VFA) 9.2 Testing Conditions 9.2.1 Measure the dynamic modulus and phase angle of each specimen using the dynamic modulus test system at each of the temperatures and loading frequencies given in Table 2. Begin testing at the lowest temperature and highest frequency. Test all frequencies in descending order before moving to the next highest temperature. Table 2. Recommended Testing Temperatures and Loading Frequencies. PG 58-XX and softer PG 64-XX & PG 70-XX PG 76 XX and stiffer Temperature Loading Temperature Loading Temperature Loading C Frequencies C Frequencies C Frequencies Hz Hz Hz 4 10, 1, 0.1 4 10, 1, 0.1 4 10, 1, 0.1 20 10, 1, 0.1 20 10, 1, 0.1 20 10, 1, 0.1 35 10, 1, 0.1, 40 10, 1, 0.1, 45 10, 1, 0.1, and 0.01 and 0.01 and 0.01 Note 4 The dynamic modulus testing may be performed with or without confinement. The same confining stress conditions must be used at all temperatures and loading rates. An unconfined dynamic modulus master curve is typically used in mechanistic-empirical pavement analysis methods. 9.2.2 Accept only test data meeting the data quality statistics given in Table 3. Repeat tests as necessary to obtain test data meeting the data quality statistics requirements. Table 3. Data Quality Statistics Requirements. Data Quality Statistic Limit Load standard error 10 % Deformation standard error 10 % Deformation uniformity 30 % Phase uniformity 3 degrees Note 5 The data quality statistics in Table 3 are reported by the Simple Performance Test System software. If a dynamic modulus test system other than the Simple Performance Test System is used, refer to Equipment Specification for the Simple Performance Test System, Version 3.0 for algorithms for computation of dynamic modulus, phase angle, and data quality statistics. 9.3 Dynamic Modulus Data Summary 9.3.1 Prepare a summary table of the dynamic modulus data. At each temperature and frequency, compute:

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44 1. Average dynamic modulus 2. Average phase angle 3. Dynamic modulus coefficient of variation 4. Standard deviation of phase angle Figure 1 presents an example summary data sheet. Conditions Specimen 1 Specimen 2 Specimen 3 Average Modulus Average Std Dev Temperature Frequency Modulus Phase Angle Modulus Phase Angle Modulus Phase Angle Modulus CV Phase Phase C Hz Ksi Degree Ksi Degree Ksi Degree Ksi % Deg Deg 4 0.1 1170.9 18.8 1214.8 19.6 1443.2 18.5 1276.3 11.5 19.0 0.5 4 1 1660.8 12.0 1743.5 12.5 2027.0 11.6 1810.5 10.6 12.0 0.4 4 10 2107.3 8.1 2245.6 8.4 2596.1 8.2 2316.3 10.9 8.2 0.2 20 0.1 259.1 33.9 289.9 33.5 315.2 34.6 288.1 9.8 34.0 0.6 20 1 604.1 27.4 657.3 26.8 711.2 27.0 657.5 8.1 27.1 0.3 20 10 1065.1 21.0 1181.5 18.8 1231.4 19.8 1159.3 7.4 19.9 1.1 40 0.01 17.2 18.6 16.5 18.8 18.8 19.2 17.5 6.7 18.9 0.3 40 0.1 26.5 24.8 26.4 26.1 30.6 26.0 27.8 8.6 25.6 0.7 40 1 62.9 31.5 63.9 32.1 74.5 32.7 67.1 9.6 32.1 0.6 40 10 180.1 35.2 197.6 35.1 220.6 35.2 199.4 10.2 35.2 0.1 Figure 1. Example Dynamic Modulus Summary Sheet. 10. DATA ANALYSIS 10.1 Dynamic Modulus Master Curve Equation 10.1.1 General Form. The general form of the dynamic modulus master curve is a modified version of the dynamic modulus master curve equation included in the Mechanistic Empirical Design Guide (MEDG) (Applied Research Associates, Inc., 2004) log E * = + (Max - ) (1) 1 + e + log f r where: E* = dynamic modulus, psi fr = reduced frequency, Hz Max = limiting maximum modulus, psi , , and = fitting parameters 10.1.2 Reduced Frequency. The reduce frequency in Equation 1 is computed using the Arrhenius equation. Ea 1 1 log f r = log f + - (2) 19.14714 T Tr where: fr = reduced frequency at the reference temperature, Hz f = loading frequency at the test temperature, Hz

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45 Tr = reference temperature, K T = test temperature, K Ea = activation energy (treated as a fitting parameter) 10.1.3 Final Form. The final form of the dynamic modulus master curve equation is obtained by substituting Equation 2 into Equation 1. log E * = + (Max - ) (3) E a 1 1 + log + - T Tr 1+ e 19.14714 10.2 Shift Factors. The shift factors at each temperature are given by Equation 4, E a 1 1 log[a (T )] = - (4) 19.14714 T Tr where: a(T) = shift factor at temperature T Tr = reference temperature, K T = test temperature, K Ea = activation energy (treated as a fitting parameter) 10.3 Limiting Maximum Modulus. The maximum limiting modulus is estimated from mixture volumetric properties using the Hirsch model (Christensen, et. al, 2003) and a limiting binder modulus of 1 GPa (145,000 psi), Equations 5 and 6. VMA VFA x VMA 1 - Pc (5) | E* | max = Pc 4,200,0001 - + 435,000 + 100 10,000 VMA 1 - 100 + VMA 4,200,000 435,000( VFA) where 0.58 435,000(VFA ) 20 + Pc = VMA (6) 0.58 435,000(VFA) 650 + VMA E*max = limiting maximum mixture dynamic modulus, psi VMA = Voids in mineral aggregates, % VFA = Voids filled with asphalt, %

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46 10.4 Fitting the Dynamic Modulus Master Curve 10.4.1 Step 1. Estimate Limiting Maximum Modulus 10.4.1.1 Using the average VMA and VFA of the specimens tested, compute the limiting maximum modulus using Equations 5 and 6. 10.4.1.2 Compute the logarithm of the limiting maximum modulus and designate this as Max 10.4.2 Step 2. Select a the Reference Temperature 10.4.2.1 Select the reference temperature for the dynamic modulus master curve and designate this as Tr. Usually 20 C (293.15 K) is used as the reference temperature. 10.4.3 Step 3. Perform Numerical Optimization 10.4.3.1 Substitute Max computed in Section 10.4.1.2 and Tr selected in Section 10.4.2.1 into Equation 3. 10.4.3.2 Determine the four fitting parameters of Equation 3 (, , , and Ea) using numerical optimization. The optimization can be performed using the Solver function in Mircosoft EXCEL. This is done by setting up a spreadsheet to compute the sum of the squared errors between the logarithm of the average measured dynamic moduli at each temperature/frequency combination and the values predicted by Equation 3. The Solver function is used to minimize the sum of the squared errors by varying the fitting parameters in Equation 3. The following initial estimates are recommended: = 0.5, = -1.0, =-0.5, and Ea = 200,000. 10.4.4 Step 4. Compute Goodness of Fit Statistics 10.4.4.1 Compute the standard deviation of the logarithm of the average measured dynamic modulus values for each temperature/frequency combination. Designate this value as Sy. 10.4.4.2 Compute the standard error of estimate using Equation 7. ( ) 0.5 1 10 ^ * - log E * 2 S e = log E (7) 6 1 i i where: Se = standard error of estimate ^ * i = value predicted by Equation 3 after optimization for each log E temperature/frequency combination log E * i = logarithm of the average measured dynamic modulus for each temperature/frequency combination.

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47 10.4.4.3 Compute the explained variance, R2, using Equation 8. 2 8S e R 2 = 1- 2 (8) 9S y where: R2 = explained variance Se = standard error of estimate from Equation 7. Sy = standard deviation of the logarithm of the average dynamic modulus values 10.5 Evaluate Fitted Master Curve 10.5.1 The ratio of Se to Sy should be less than 0.05 10.5.2 The explained variance should exceed 0.99 10.6 Determine AASHTO Mechanistic-Empirical Pavement Design Guide Inputs 10.6.1 Substitute the logarithm of the limiting maximum modulus (Max) determined in Section 10.4.1.2 and the fitting parameters (, , , and Ea) determined in Section 10.4.3.2 into Equation 3 and compute the dynamic modulus at the following temperatures and loading frequencies. A total of 30 dynamic modulus values will be calculated. Temperatures Frequencies -10, 4.4, 21.1, 37.8, and 54.4 C 25, 10, 5, 1, 0.5, and 0.1 Hz (14, 40, 70, 100, 130, C) 11. REPORT 11.1 Mixture identification 11.2 Measured dynamic modulus and phase angle data for each specimen at each temperature/frequency combination 11.3 Average measured dynamic modulus and phase angle at each temperature/frequency combination 11.4 Coefficient of variation of the measured dynamic modulus data at each temperature/frequency combination

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48 11.5 Standard deviation of the measured phase angle data at each temperature/frequency combination. 11.6 VMA and VFA of each specimen tested 11.7 Average VMA and VFA for the specimens tested 11.8 Reference temperature 11.9 Parameters of the fitted master curve (Max, , , , and Ea) 11.10 Goodness of fit statistics for the fitted master curve (Se, Sy, Se/Sy, R2) 11.11 Plot of the fitted dynamic modulus master curve as a function of reduced frequency showing average measured dynamic modulus data 11.12 Plot of shift factors as a function of temperature 11.13 Plot of average phase angle as a function of reduced frequency. 11.14 Tabulated temperature, frequency, and dynamic modulus for input into MEPDG 12. KEYWORDS 12.1 Dynamic modulus, phase angle, master curve 13. REFERENCES 13.1 Applied Research Associates, Inc., ERES Consultants Division Guide for Mechanistic-Empirical Design of New and Rehabilitated Pavement Structures, Final Report Prepared for the National Cooperative Highway Research Program, March, 2004. 13.2 Christensen, D.W., Pellinen, T.K., Bonaquist, R.F., "Hirsch Model for Estimating the Modulus of Asphalt Concrete," Journal of the Association of Asphalt Paving Technologists, Vol 72, 2003.