1、Standard Practice for Developing Dynamic Modulus Master Curves for Asphalt Mixtures AASHTO Designation: R 62-131American Association of State Highway and Transportation Officials 444 North Capitol Street N.W., Suite 249 Washington, D.C. 20001 TS-2d R 62-1 AASHTO Standard Practice for Developing Dyna
2、mic Modulus Master Curves for Asphalt Mixtures AASHTO Designation: R 62-1311. SCOPE 1.1. This practice describes testing and analysis for developing a dynamic modulus master curve for hot mix asphalt (HMA). This practice is intended for dense- and gap-graded mixtures with nominal-maximum aggregate s
3、izes up to 37.5 mm. This practice is intended for use with T 342 and addresses the wide range of test temperatures and frequencies specified therein. 1.2. This standard may involve hazardous materials, operations, and equipment. This standard does not purport to address all of the safety concerns as
4、sociated 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
5、Specimens Using the Superpave Gyratory Compactor (SGC) R 35, Superpave Volumetric Design for Asphalt Mixtures T 342, Determining Dynamic Modulus of Hot Mix Asphalt (HMA) Mechanistic-Empirical Pavement Design Guide (MEPDG) 3. TERMINOLOGY 3.1. dynamic modulus master curvea composite curve constructed
6、at a reference temperature by shifting dynamic modulus data from various temperatures along the “log frequency” axis. 3.2. reduced frequencythe computed frequency at the reference temperature, equivalent to the actual loading frequency at the test temperature. 3.3. reference temperaturethe temperatu
7、re at which the master curve is constructed. 3.4. shift factorthe 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 HMA. It
8、 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. 2015 by the American Association of State Highway and Transportation
9、Officials.All rights reserved. Duplication is a violation of applicable law.TS-2d R 62-2 AASHTO 5. SIGNIFICANCE AND USE 5.1. Dynamic modulus master curves can be used for mixture evaluation and for characterizing the modulus of HMA for mechanistic-empirical pavement design. 6. APPARATUS 6.1. Specime
10、n Fabrication EquipmentFor fabricating dynamic modulus test specimens as described in PP 60. 6.2. Dynamic Modulus Test SystemMeeting the requirements of T 342. 6.3. Analysis SoftwareCapable of performing numerical optimization of nonlinear equations. Note 1The Solver tool included in Microsoft Excel
11、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 HMA. It also includes the use of sawing and coring machinery and servohydraulic testing equipment. Use standar
12、d 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 referen
13、ced 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. Use Table 1 to select an app
14、ropriate number of specimens based on the uncertainty that can be tolerated in the analysis. Note 2The coefficient of variation for properly conducted dynamic modulus tests is approximately 13 percent. The coefficient of variation of the mean dynamic modulus for tests on multiple specimens is given
15、in Table 1. 2015 by the American Association of State Highway and Transportation Officials.All rights reserved. Duplication is a violation of applicable law.TS-2d R 62-3 AASHTO Table 1Coefficient of Variation for the Mean of Dynamic Modulus Test on Replicate Specimens Number of Specimens Coefficient
16、 of Variation for 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 9.1.2. Record the following volumetric properties for each test specimen: Air void content, Binder content, Voids in the mineral aggregate (VMA), and Voids filled with asphalt (VFA). 9.1.2.1. Refer to R 35 for methods for
17、computing volumetric properties. 9.2. Testing Conditions: 9.2.1. Determine the dynamic modulus and phase angle for each specimen as specified in T 342. 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 qu
18、ality statistics requirements. Table 2Data Quality Statistics Requirements Data Quality Statistic Limit Load standard error 10% Deformation standard error 10% Deformation uniformity 30% Phase uniformity 3 degrees Note 3Algorithms for computation of the data quality statistics are given in T 342. 9.3
19、. 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.
20、Figure 1 presents an example of a summary data sheet. 2015 by the American Association of State Highway and Transportation Officials.All rights reserved. Duplication is a violation of applicable law.TS-2d R 62-4 AASHTO Conditions Specimen 1 Specimen 2 Modulus Phase Angle Temperature, F Frequency, Hz
21、 Modulus, ksi Phase Angle, degree Modulus, ksi Phase Angle, degree Average, ksi CV, % Average, degree Std Dev, degree 14 25 3100.0 6.1 3320.0 9.1 3210.0 4.8 7.6 2.1 14 10 3080.0 5.6 2990.0 4.2 3035.0 2.1 4.9 0.9 14 5 2970.0 6.8 2930.0 7.9 2950.0 1.0 7.4 0.8 14 1 2700.0 7.1 2670.0 8.6 2685.0 0.8 7.9
22、1.1 14 0.5 2580.0 6.7 2540.0 8.9 2560.0 1.1 7.8 1.5 14 0.1 2080.0 9.6 2250.0 11.9 2165.0 5.6 10.8 1.6 40 25 2650.0 18.0 2410.0 19.3 2530.0 6.7 18.7 0.9 40 10 2430.0 13.2 2170.0 10.0 2300.0 8.0 11.6 2.3 40 5 2290.0 10.1 2070.0 9.6 2180.0 7.1 9.8 0.3 40 1 1890.0 12.4 1730.0 10.5 1810.0 6.3 11.4 1.3 40
23、 0.5 1690.0 15.2 1630.0 12.9 1660.0 2.6 14.0 1.6 40 0.1 1360.0 17.6 1330.0 15.1 1345.0 1.6 16.4 1.8 70 25 1060.0 26.6 1220.0 29.4 1140.0 9.9 28.0 2.0 70 10 890.0 24.8 1010.0 27.0 950.0 8.9 25.9 1.5 70 5 763.0 26.3 856.0 27.6 809.0 8.1 27.0 0.9 70 1 514.0 28.3 573.0 28.8 543.5 7.7 28.5 0.4 70 0.5 420
24、.0 25.2 492.0 26.4 456.0 11.2 25.8 0.8 70 0.1 270.0 36.1 317.0 36.6 293.5 11.3 36.4 0.4 100 25 308.0 36.4 416.0 39.6 362.0 21.1 38.0 2.2 100 10 240.0 34.8 308.0 36.7 274.0 17.5 35.8 1.4 100 5 192.0 34.5 232.0 35.7 212.0 13.3 35.1 0.8 100 1 111.0 34.6 132.0 35.0 121.5 12.2 34.8 0.3 100 0.5 87.5 32.9
25、104.0 34.2 95.8 12.2 33.5 0.9 100 0.1 55.3 34.6 64.0 35.7 59.7 10.3 35.2 0.8 130 25 90.7 40.1 109.0 41.1 99.9 13.0 40.6 0.7 130 10 64.0 35.0 78.0 34.6 71.0 13.9 34.8 0.3 130 5 50.9 32.3 59.0 32.5 55.0 10.4 32.4 0.2 130 1 31.4 27.9 36.1 28.8 33.8 9.8 28.4 0.7 130 0.5 25.7 26.0 30.1 26.9 27.9 11.2 26.
26、4 0.6 130 0.1 18.7 26.8 20.2 27.6 19.5 5.5 27.2 0.5 Figure 1Example Dynamic Modulus Summary Sheet 10. DATA ANALYSIS 10.1. Dynamic Modulus Master Curve Equation: 10.1.1. General FormThe general form of the dynamic modulus master curve is the sigmoid function included in the Mechanistic-Empirical Pave
27、ment Design Guide (MEPDG). ( )loglog *1rfEe+=+(1) where: *E = the dynamic modulus, psi; , , , and = the fitting parameters; and fr= the reduced frequency, Hz. 2015 by the American Association of State Highway and Transportation Officials.All rights reserved. Duplication is a violation of applicable
28、law.TS-2d R 62-5 AASHTO 10.1.2. Reduced FrequencyThe reduced frequency in Equation 1 is computed using timetemperature shift factors. Several time temperature shift functions are available in the literature. This practice presents two shift factor equations: shift factors from the MEPDG and second-o
29、rder polynomial. 10.1.2.1. MEPDG Shift Factors: ( )log log log logRrTf fc= + (2) where: fr= the reduced frequency at the reference temperature; f = the loading frequency at the test temperature; c = a fitting coefficient; = the viscosity of the binder at the test temperature, cP; TR= the reference t
30、emperature, R; and RT = the viscosity of the binder at the reference temperature, cP. Viscosities in Equation 2 are determined using Equation 3 as follows: ( )loglog 10A VTS T+= (3) where: = the viscosity, cP; A and VTS = the parameters of the binder viscosity-temperature susceptibility relationship
31、; and T = the test temperature, R (F + 459.67). 10.1.2.2. Second-Order Polynomial: ( ) ( )212log logr RRf faTTaTT= + + (4) where: fr= the reduced frequency at the reference temperature; f = the loading frequency at the test temperature; a1, a2= the fitting coefficients; TR= the reference temperature
32、, F; and T = the test temperature, F. 10.1.3. Final FormThe final form of the dynamic modulus master curve equation is obtained by substituting the selected shift factor relationship into Equation 1. 10.1.3.1. MEPDG Shift Factors: ( )( ) ( ) log loglog 10 10log *1A VTS T A VTS TRfcEe+ + =+(5) 10.1.3
33、.2. Second-Order Polynomial Shift Factors: ( )( ) ( )212loglog *1RRfaTT aTTEe+ + + =+(6) 2015 by the American Association of State Highway and Transportation Officials.All rights reserved. Duplication is a violation of applicable law.TS-2d R 62-6 AASHTO 10.1.4. Fitting the Dynamic Modulus Master Cur
34、ve: 10.1.4.1. Select Reference Temperature: Select the reference temperature for the dynamic modulus master curve and designate this value as TR. Usually 70F is used as the reference temperature. 10.1.4.2. Perform Numerical Optimization: Determine the fitting parameters of Equation 5 or 6 using nume
35、rical 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 v
36、alues predicted by Equation 5 or 6. See Equation 7. ( )221error log * log *niiEE= (7) where: 2error= the sum of squared errors; n = the number of temperature/frequency combinations used in the testing; log * = the value predicted by Equation 5 or 6 for each temperature/frequency combination; andiElo
37、g * = the logarithm of the average measured dynamic modulus for eachtemperature/frequency combination.iEThe Solver function is used to minimize the sum of the squared errors by varying the fitting parameters. Recommended initial estimates are listed in Table 3. Note that these estimates depend on th
38、e units used as shown in Table 3. Table 3Recommended Initial Estimates MEPDG Shift Factors TR= 529.67R * ksiE = f = Hz Polynomial Shift Factors TR= 70F * ksiE = f = Hz Fitting Parameter Initial Estimate Fitting Parameter Initial Estimate 3.0 3.0 1.0 1.0 0.5 0.5 0.5 0.5 c 1.0 a10.1 a20.0001 10.1.4.3.
39、 Compute “Goodness of Fit” Statistics: Compute the standard deviation of the logarithm of the average measured dynamic modulus values for all temperature/frequency combinations. Designate this value as Sy. ( )21log * log *1niyEESn=(8) where: 2015 by the American Association of State Highway and Tran
40、sportation Officials.All rights reserved. Duplication is a violation of applicable law.TS-2d R 62-7 AASHTO Sy= the standard deviation of the logarithm of the average measured data; log * = the logarithm of the average measured dynamic modulus for each temperature/frequency combination;iElog *E = the
41、 average of the logarithm of the average measured dynamic moduli = 1log *niEn; and n = the number of temperature/frequency combinations used in the testing. Compute the standard error of estimate using Equation 7 as follows: ( )( )0.5211log * log *1neiiS EEnp= (9) where: Se= the standard error of es
42、timate; n = the number of temperature/frequency combinations used in the testing; p = the number of fitting parameters; log * = the value predicted by Equation 5 or 6 after optimization for each temperature/frequency combination; andiElog * = the logarithm of the average measured dynamic modulus for
43、 each temperature/frequency combination.iECompute the explained variance, R2, using Equation 10 as follows: ( )( )222111eynp SRnS= (10) where: R2= the explained variance; n = the number of temperature/frequency combinations used in the testing; p = the number of fitting parameters; Se = the standard
44、 error of estimate from Equation 9; and Sy= the standard deviation of the logarithm of the average measured dynamic modulus values from Equation 8. 10.1.5. Evaluate Fitted Master Curve: 10.1.5.1. The ratio of Seto Syshould be less than 0.05. 10.1.5.2. The explained variance should exceed 0.99. 2015
45、by the American Association of State Highway and Transportation Officials.All rights reserved. Duplication is a violation of applicable law.TS-2d R 62-8 AASHTO 11. REPORT 11.1. Report the following: 11.1.1. Mixture identification; 11.1.2. Measured dynamic modulus and phase angle data for each specim
46、en at each temperature/frequency combination; 11.1.3. Average measured dynamic modulus and phase angle at each temperature/frequency combination; 11.1.4. Coefficient of variation of the measured dynamic modulus data at each temperature/frequency combination; 11.1.5. Standard deviation of the measure
47、d phase angle data at each temperature/frequency combination; 11.1.6. Air voids, binder content, VMA, and VFA of each specimen tested; 11.1.7. Average air voids, average binder content, average VMA, and average VFA for the specimens tested; 11.1.8. Reference temperature; 11.1.9. Parameters of the fi
48、tted master curve; 11.1.10. “Goodness of fit” statistics for the fitted master curve (Se, Sy, Se/Sy, R2); 11.1.11. Plot of the fitted dynamic modulus master curve as a function of reduced frequency showing average measured dynamic modulus data; 11.1.12. Plot of shift factors as a function of tempera
49、ture; and 11.1.13. Plot of average phase angle as a function of reduced frequency. 12. KEYWORDS 12.1. Dynamic modulus; master curve; phase angle. 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, M
