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5
CHAPTER 2
Abbreviated Dynamic Modulus Master
Curve Testing
2.1 Introduction ing at this temperature can be eliminated, the cost of the
equipment, the complexity of the procedure, and the over-
An abbreviated dynamic modulus master curve testing and all time required to develop a master curve can be signifi-
analysis procedure was developed in Phase IV of NCHRP cantly reduced.
Project 9-29 to reduce the effort and equipment costs associ- The approach taken in this project takes advantage of the
ated with developing dynamic modulus master curves for fact that all asphalt binders reach approximately the same
pavement structural design. The dynamic modulus test glassy modulus at very low temperatures (6). Using this
protocol was developed in NCHRP Projects 9-19 and 1-37A binder modulus and recently developed relationships to pre-
and has been standardized as AASHTO Provisional Standard dict mixture dynamic modulus from binder modulus and
TP62, Standard Method of Test for Determining Dynamic volumetric data, an estimate of the limiting maximum mod-
Modulus of Hot-Mix Asphalt Concrete Mixtures. The recom- ulus of the mixture can be made and used in the development
mended test sequence in AASHTO TP62 for the development of the dynamic modulus master curve (7).
of a master curve for pavement structural design consists of
testing a minimum of two replicate specimens at tempera-
2.2 MEPDG Dynamic Modulus
tures of 14, 40, 70, 100, and 130 °F at loading frequencies of
Master Curve
25, 10, 5, 1.0, 0.5, and 0.1 Hz. This testing provides a database
of 60 dynamic modulus measurements from which the To account for temperature and rate of loading effects on
parameters of the master curve are determined by numerical the modulus of asphalt concrete, the MEPDG uses asphalt
optimization. concrete moduli obtained from a master curve constructed
It is desirable that equipment for performing such testing at a reference temperature of 70oF (2). Master curves are
be available to highway agencies at a reasonable cost and constructed using the principle of time-temperature super-
the test procedure be appropriate for agency laboratories. position. First a standard reference temperature is selected,
A recently completed FHWA pooled-fund study identified in this case 70oF, then data at various temperatures are
several issues associated with the test protocol and con- shifted with respect to loading frequency until the curves
cluded that the overall time required to perform the testing merge into a single smooth function. The master curve of
must be shortened if highway agencies are going to use it for modulus as a function of frequency formed in this manner
routine testing (3). NCHRP Project 9-19 included a study of describes the frequency dependency of the material. The
the minimum testing required to develop the dynamic amount of shifting at each temperature required to form
modulus master curves and concluded that reasonable mas- the master curve describes the temperature dependency of
ter curves can be developed using tests at three tempera- the material. Thus, both the master curve and the shift fac-
tures: 14, 70, and 130°F at loading frequencies of 33, 2.22, tors are needed for a complete description of the rate and
0.15, and 0.01 Hz (5). This reduced sequence still requires temperature effects. Figure 1 presents an example of a mas-
testing at 14°F that the FHWA pooled fund study found dif- ter curve constructed in this manner. The shift factors are
ficult due to moisture condensation and ice formation (3). presented in the inset figure.
Additionally, low temperature testing significantly increases In the MEPDG, the sigmoidal function in Equation 1 is
the cost of the environmental chamber and increases the used to describe the frequency dependency of the modulus
loading capacity and cost of the testing equipment. If test- master curve.

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6
1.0E+07
1.0E+06
14 F
40 F
E* , psi
70 F
1.0E+05
100 F
130 F
6 Fit
Log Shift Factor
4
2
0
1.0E+04 -2
-4
-6
0 50 100 150
Temperature, F
1.0E+03
1.0E-07 1.0E-05 1.0E-03 1.0E-01 1.0E+01 1.0E+03 1.0E+05 1.0E+07
Reduced Frequency, Hz
Figure 1. Example dynamic modulus master curve and shift factors.
to provide a rational explanation for the temperature depen-
log E * = + (1)
1+ e + (log r )
dency of the shift factors (6). In the MEPDG, the shift factors
where were expressed as a function of the binder viscosity to allow
E * = dynamic modulus; aging over the life of the pavement to be considered using the
r = reduced frequency, Hz; Global Aging Model developed by Mirza and Witczak (8).
= minimum value of E * ; Equation 3 presents the shift factor relationship used in the
+ = maximum value of E * ; and MEPDG (2).
, = parameters describing the shape of the sigmoidal log [ a(T )] = c [ log() - log(70RTFOT )] (3)
function.
where
Research has shown that the fitting parameters and a(T) = shift factor as a function of temperature and age
depend on aggregate gradation, binder content, and air void = viscosity at the age and temperature of interest
content. The fitting parameters and depend on the char- 70RTFOT = viscosity at the reference temperature of 70 °F,
acteristics of the asphalt binder and the magnitude of and . AASHTO T240 residue
The temperature dependency of the modulus is incorporated c = fitting parameter
in the reduced frequency parameter, r, in Equation 1. Equa-
tion 2 defines the reduced frequency as the actual loading The viscosity as a function of temperature can be expressed
frequency multiplied by the time-temperature shift factor, a(T). using the viscosity-temperature relationship given in ASTM
D 2493.
r = a(T) × (2a)
log log = A + VTS log TR (4)
log(r) = log() + log[a(T)] (2b)
where
where = viscosity, cP;
r = reduced frequency, Hz; TR = temperature, Rankine;
= loading frequency, Hz; A = regression intercept; and
a(T) = shift factor as a function of temperature; and VTS = regression slope of viscosity-temperature relationship.
T = temperature.
Combining Equations 3 and 4 yields the shift factor as a
The shift factors are a function of temperature. Various function of temperature relationship used in the MEPDG for
equations such as the Arrhenius function and the Williams- the construction of dynamic modulus master curves from
Landel-Ferry equation have been recommended in an attempt laboratory test data.