# AASHTO Designation PP 61-13

TS-2d PP 61-1 AASHTO Standard Practice for Developing Dynamic Modulus Master Curves for Asphalt Mixtures Using the Asphalt Mixture Performance Tester (AMPT) AASHTO Designation: PP 61-13 1,21. SCOPE 1.1. This practice describes testing and analysis for developing a dynamic modulus master curve for asphalt mixtures using the Asphalt Mixture Performance Tester (AMPT). This practice is intended for dense- and gap-graded mixtures with nominal-maximum aggregate sizes up to 37.5 mm. This practice accounts for the temperature limitations of the AMPT and provides guidance to the user for the selection of appropriate test temperatures. 1.2. This standard may involve hazardous materials, operations, and equipment. This standard does not purport to address all of the safety concerns associated with its use. It is the responsibility of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory limitations prior to use. 2. REFERENCED DOCUMENTS 2.1. AASHTO Standards: PP 60, Preparation of Cylindrical Performance Test Specimens Using the Superpave Gyratory Compactor (SGC) TP 79, Determining the Dynamic Modulus and Flow Number for Asphalt Mixtures Using the Asphalt Mixture Performance Tester (AMPT) Mechanistic-Empirical Pavement Design Guide (MEPDG) 2.2. Other Publication: 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. 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. © 2013 by the American Association of State Highway and Transportation Officials. All rights reserved. Duplication is a violation of applicable law. Copyright American Association of State Highway and Transportation Officials Provided by IHS under license with AASHTO Licensee=University of Texas Revised Sub Account/5620001114 Not for Resale, 05/28/2014 10:02:25 MDT No reproduction or networking permitted without license from IHS --`,,,`,`,`,``,`,,,,,,,`,`,``,`-`-`,,`,,`,`,,`---TS-2d PP 61-2 AASHTO 4. SUMMARY OF PRACTICE 4.1. This practice describes the testing required using the AMPT and the analysis needed to develop a dynamic modulus master curve for asphalt 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 asphalt mixtures for mechanistic-empirical pavement design. 6. APPARATUS 6.1. Specimen Fabrication Equipment—For fabricating dynamic modulus test specimens as described in PP 60. 6.2. Dynamic Modulus Test System—Meeting the requirements of the equipment specification for the Simple Performance Test (SPT) System, Version 3.0. 6.3. Analysis Software—Capable of performing numerical optimization of nonlinear 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 machinery and servohydraulic 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 or verification are included in the documents referenced in Section 2. Refer to the pertinent section of the referenced documents for information concerning calibration or verification. 9. DYNAMIC MODULUS TEST DATA 9.1. Test Specimen Fabrication: 9.1.1. Prepare at least two test specimens at the target air void content ±0.5 percent and with the aging condition in accordance with PP 60. Note 2—The number of specimens to test depends on the desired accuracy of the analysis. Refer to Table 4 in TP 79 for guidance on the reproducibility of dynamic modulus and phase angle measurements. 9.1.2. Record the following volumetric properties for each test specimen: Voids in the mineral aggregate (VMA), and Voids filled with asphalt (VFA). © 2013 by the American Association of State Highway and Transportation Officials. All rights reserved. Duplication is a violation of applicable law. Copyright American Association of State Highway and Transportation Officials Provided by IHS under license with AASHTO Licensee=University of Texas Revised Sub Account/5620001114 Not for Resale, 05/28/2014 10:02:25 MDT No reproduction or networking permitted without license from IHS --`,,,`,`,`,``,`,,,,,,,`,`,``,`-`-`,,`,,`,`,,`---TS-2d PP 61-3 AASHTO 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 1. Begin testing at the lowest temperature and highest frequency. Test all frequencies in descending order before moving to the next highest temperature. Table 1—Recommended Testing Temperatures and Loading Frequencies PG 58-XX and Softer PG 64-XX and PG 70-XX PG 76-XX and Stiffer Temperature, °C Loading Frequencies, Hz Temperature, °C Loading Frequencies, Hz Temperature, °C Loading Frequencies, 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, and 0.01 40 10, 1, 0.1, and 0.01 45 10, 1, 0.1, and 0.01 Note 3—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 2. Repeat tests as necessary to obtain test data meeting the data quality statistics requirements. Table 2—Data Quality Statistics Requirements Data Quality Statistic Limit Load standard error 10% Deformation standard error 10% Deformation uniformity 30% Phase uniformity 3 degrees Note 4—The data quality statistics in Table 2 are reported by the AMPT software. If a dynamic modulus test system other than the AMPT is used, refer to the equipment specification for the SPT System, Version 3.0, for algorithms for the 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 the following: 1. Average dynamic modulus; 2. Average phase angle; 3. Coefficient of variation of the dynamic modulus; and 4. Standard deviation of the phase angle. Figure 1 presents an example summary data sheet. © 2013 by the American Association of State Highway and Transportation Officials. All rights reserved. Duplication is a violation of applicable law. Copyright American Association of State Highway and Transportation Officials Provided by IHS under license with AASHTO Licensee=University of Texas Revised Sub Account/5620001114 Not for Resale, 05/28/2014 10:02:25 MDT No reproduction or networking permitted without license from IHS --`,,,`,`,`,``,`,,,,,,,`,`,``,`-`-`,,`,,`,`,,`---TS-2d PP 61-4 AASHTO Conditions Specimen 1 Specimen 2 Specimen 3 Average Modulus, ksi Modulus CV, % Average Phase Angle, degree Std Dev Phase Angle, degree Temperature, °C Frequency, Hz Modulus, ksi Phase Angle, degree Modulus, ksi Phase Angle, degree Modulus, ksi Phase Angle, degree 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 Pavement Design Guide (MEPDG). ( ) log Max log * 1 r f E e βγ δ δ + − = + +(1) where: * E = the dynamic modulus, psi; δ, β, and γ = the fitting parameters; Max = the limiting maximum modulus, psi; and f r= the reduced frequency, Hz. 10.1.2. Reduced Frequency—The reduced frequency in Equation 1 is computed using the Arrhenius equation. 11 log log 19.14714 a r r E ff TT ∆ = +− (2) where: f r= the reduced frequency at the reference temperature, Hz; f = the loading frequency at the test temperature, Hz; ΔE a= the activation energy (treated as a fitting parameter); T = the test temperature, °K; and T r= the reference temperature, °K. 10.1.3. Final Form—The final form of the dynamic modulus master curve equation is obtained by substituting Equation 2 into Equation 1. © 2013 by the American Association of State Highway and Transportation Officials. All rights reserved. Duplication is a violation of applicable law. Copyright American Association of State Highway and Transportation Officials Provided by IHS under license with AASHTO Licensee=University of Texas Revised Sub Account/5620001114 Not for Resale, 05/28/2014 10:02:25 MDT No reproduction or networking permitted without license from IHS --`,,,`,`,`,``,`,,,,,,,`,`,``,`-`-`,,`,,`,`,,`---TS-2d PP 61-5 AASHTO ( ) 11 log 19.14714 Max log * 1 a r E f TT E e ∆ β+γ + − −δ =δ+ +(3) 10.2. Shift Factors—The shift factors at each temperature are given in Equation 4 as follows: ( ) 11 log 19.14714 a r E aT TT ∆ = − (4) where: a(T) = the shift factor at temperature T; ΔE a= the activation energy (treated as a fitting parameter); T = the test temperature, °K; and T r= the reference temperature, °K. 10.3. Limiting Maximum Modulus—The maximum limiting modulus is estimated from asphalt mixture volumetric properties using the Hirsch model (Christensen et al., 2003) and a limiting binder modulus of 1 GPa as provided in Equations 5 and 6. ( ) ( ) max 100 1 * 4, 200, 000 1 435, 000 1 100 10, 000 4, 200, 000 435, 000 VMA c c P VMA VFA VMA EP VMA VFA − × = −+ + − + (5) where: ( ) ( ) 0.58 0.58 435,000 20 435,000 650 c VFA VMA P VFA VMA + = + (6) max * E = the limiting maximum asphalt mixture dynamic modulus, psi; VMA = the voids in the mineral aggregate, percent; and VFA = the voids filled with asphalt, percent. 10.4. Fitting the Dynamic Modulus Master Curve: 10.4.1. 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 value as max. 10.4.2. Select the Reference Temperature: 10.4.2.1. Select the reference temperature for the dynamic modulus master curve; and designate this value as T r. Usually 20°C (293.15°K) is used as the reference temperature. 10.4.3. Perform Numerical Optimization: © 2013 by the American Association of State Highway and Transportation Officials. All rights reserved. Duplication is a violation of applicable law. Copyright American Association of State Highway and Transportation Officials Provided by IHS under license with AASHTO Licensee=University of Texas Revised Sub Account/5620001114 Not for Resale, 05/28/2014 10:02:25 MDT No reproduction or networking permitted without license from IHS --`,,,`,`,`,``,`,,,,,,,`,`,``,`-`-`,,`,,`,`,,`---TS-2d PP 61-6 AASHTO 10.4.3.1. Substitute “max,” computed in Section 10.4.1.2, and T r , selected in Section 10.4.2.1, into Equation 3. 10.4.3.2. Determine the four fitting parameters of Equation 3 (δ, β, γ, and ΔE a ) using numerical optimization. The optimization can be performed using the Solver function in Microsoft Excel ® . This calculation is performed by 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: δ = 0.5, β = –1.0, γ = –0.5, and ΔE a= 200 000. 10.4.4. 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 S y . 10.4.4.2. Compute the standard error of estimate using Equation 7. ( ) 0.5 10 2 1 1 ˆ log * log * 6 e i i S EE = − ∑(7) where: S e= the standard error of estimate; ˆ log * i E = the value predicted by Equation 3 after optimization for each temperature/frequency combination; and log * i E = the logarithm of the average measured dynamic modulus for each temperature/frequency combination. 10.4.4.3. Compute the explained variance, R 2 , using Equation 8. 2 2 2 8 1 9 e y S R S = − (8) where: R 2= the explained variance; S e= the standard error of estimate from Equation 7; and S y= the standard deviation of the logarithm of the average dynamic modulus values. 10.5. Evaluate Fitted Master Curve: 10.5.1. The ratio of S eto S yshould be less than 0.05. 10.5.2. The explained variance should exceed 0.99. 10.6. Determine AASHTO Mechanistic-Empirical Pavement Design Guide (MEPDG) Inputs: © 2013 by the American Association of State Highway and Transportation Officials. All rights reserved. Duplication is a violation of applicable law. Copyright American Association of State Highway and Transportation Officials Provided by IHS under license with AASHTO Licensee=University of Texas Revised Sub Account/5620001114 Not for Resale, 05/28/2014 10:02:25 MDT No reproduction or networking permitted without license from IHS --`,,,`,`,`,``,`,,,,,,,`,`,``,`-`-`,,`,,`,`,,`---TS-2d PP 61-7 AASHTO 10.6.1. Substitute the logarithm of the limiting maximum modulus (max) determined in Section 10.4.1.2 and the fitting parameters (δ, β,