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14 second 10 cycles are then collected and used to compute the eters for the binder were: A= 10.299 and VTS = -3.426. The test dynamic modulus and phase angle. The loading frequencies specimens were compacted at the optimum asphalt content of recommended in this document will require approximately 5.5 percent to 4.0 percent air voids. For this condition, the per- 40 min per specimen at each testing temperature, including cent VMA was 15.8 and the percent voids filled with asphalt was time for specimen instrumentation and chamber tempera- 76.2. Table 5 presents dynamic modulus data measured on ture equilibrium. Thus, a testing program including three replicate samples using the combination of temperatures and replicate specimens will require approximately 2 hours per tem- loading rates recommended in the abbreviated testing protocol. perature for data collection. The first step for fitting the master curve is to estimate the limiting maximum modulus using Equation 10. For a mixture with a VMA of 15.8 percent and a VFA of 76.2 per- 2.6 Arrhenius Shift Factor cent, the limiting maximum modulus from Equation 10 is Relationship 3,376,743 psi. Using this value of the limiting maximum mod- Equation 7 presented the modified form of the MEPDG ulus, the viscosity-temperature susceptibility parameters, and master curve equation used to generate a dynamic modulus the measured data, the master curve parameters can be master curve using the proposed abbreviated testing. This obtained through numerical optimization of Equation 7. The equation requires knowledge of the viscosity-temperature optimization can be performed using the Solver function in relationship of the binder used in the mixture. For mixture Microsoft EXCEL. This is done by setting up a spreadsheet evaluation, the binder viscosity-temperature relationship to compute the sum of the squared errors between the loga- may not be known. A dynamic modulus master curve can still rithm of the measured dynamic moduli and the values pre- be developed using an alternative shift factor relationship dicted by Equation 7. The Solver function is used to minimize based on the Arrhenius equation given in Equation 12. the sum of the squared errors by varying the fitting parame- ters in Equation 7. The following initial estimates are recom- Ea 1 1 mended: = 0.5, = -1.0, = -0.5, and c = 1.2. The master log [ a(T )] = - (12) 19.14714 T Tr curve developed from this example data is shown in Figure 10. The goodness of fit statistics show Equation 7 provides an where excellent fit to the measured data with an R2 greater than 0.99 a(T) = shift factor at temperature T; and an Se/Sy less than 0.04. Using the abbreviated tempera- Tr = reference temperature, K; tures and loading rates, the measured data cover approxi- T = test temperature, K; and mately 80 percent of the range defined by the fitted limiting Ea = activation energy (treated as a fitting parameter). minimum and computed limiting maximum moduli. Using Equation 12 for the shift factors, the dynamic mod- ulus master curve equation for use with proposed abbreviated 2.8 Summary and Draft Standard testing procedure becomes: Practice ( Max - ) An abbreviated testing protocol for developing dynamic log E * = + Ea 1 1 (13) + log + - modulus master curves for routine mixture evaluation and 1+ e 19.14714 T Tr flexible pavement design was developed in Phase IV of NCHRP where E * = dynamic modulus; Table 5. Dynamic modulus test data = loading frequency, Hz; collected using the abbreviated dynamic Tr = reference temperature, K; modulus master curve testing. T = test temperature, K; Specimen 1 Specimen 2 Max = specified limiting maximum modulus; and Frequency, Modulus, Phase, Modulus, Phase, , , and Ea = fitting parameters. Temp, F Hz ksi Degree ksi Degree 40 0.01 771.6 25.0 901.1 23.7 40 0.1 1274.9 19.0 1496.1 17.9 2.7 Example Using the Abbreviated 40 1 1861.7 13.9 2164.1 12.8 40 10 2458.2 9.6 2811.0 8.6 Dynamic Modulus Master Curve 70 0.01 161.0 30.0 174.6 29.6 Testing 70 0.1 362.7 29.2 398.3 29.2 70 1 771.7 24.5 844.4 24.3 This section illustrates the development of master curves 70 10 1332.1 18.0 1446.0 18.0 104 0.01 23.3 24.4 28.3 22.9 using the proposed procedure. The mixture that was tested 104 0.1 50.8 27.8 53.4 27.9 was a coarse graded 9.5 mm limestone mixture made with a PG 104 1 137.8 29.4 140.9 29.5 70-22 binder. The viscosity-temperature susceptibility param- 104 10 336.2 29.7 352.8 29.9

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15 10,000,000 Max = 3,376,744 psi 1,000,000 2 R = 0.9978 Se/Sy = 0.038 40 F E* , psi 100,000 3 70 F Log Shift Factor 2 1 104 F 0 -1 10,000 -2 FIT -3 Min = 4,259 psi 30 50 70 90 110 Temperature, F 1,000 1.0E-06 1.0E-04 1.0E-02 1.0E+00 1.0E+02 1.0E+04 1.0E+06 Reduced Frequency, Hz Figure 10. Example master curve using abbreviated testing sequence. Project 9-29. This abbreviated testing protocol requires testing cost of the environmental chamber for the testing system, and at 40, 70, and 104F using loading frequencies of 10, 1, 0.1, and increases the complexity of testing. Moisture condensation and 0.01 Hz. The data can be fit to the MEPDG dynamic modulus icing make testing at this temperature challenging even for equation after an estimate of the limiting maximum modulus highly experienced technicians. is made using the Hirsch model. The abbreviated dynamic To aid in implementation of the abbreviated dynamic modulus master curve testing protocol eliminates the lowest modulus testing protocol, a draft standard practice titled temperature in the AASHTO TP62 testing sequence and opti- "Developing Dynamic Modulus Master Curves for Hot-Mix mizes the temperatures and loading frequencies for minimal Asphalt Concrete Using the Simple Performance Test Sys- overlap to the modulus data. Testing at the lowest temperature tem" was prepared. This draft standard practice is presented in the AASHTO TP62 sequence, 14 F, greatly increases the in Appendix A.