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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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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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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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· 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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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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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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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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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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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.
