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OCR for page 3
3
Although classical viscoelastic fundamentals suggest a log E * = -1.249937 + 0.02923 200 - 0.001767( 200 )
2
linear relationship between log a(T) and T (in degrees Vbeff
-0.002841 4 - 0.058097Va - 0.802208
Fahrenheit), experience has shown that for precision, a Vbeff + Va (Eq. 8)
second-order polynomial relationship between the logarithm 3.871977 - 0.0021 4 + 0.003958 38 - 0.000017( 38 ) + 0.00547 34
2
of the shift factor, that is log a(Ti), and the temperature in + [ -0.603313 - 0.313351 log( f ) - 0.393532 log( )]
1+ e
degrees Fahrenheit (Ti) should be used. The relationship can
thus be expressed as follows:
where
|E*| = dynamic modulus, 105 psi
Log a(Ti) = aTi + bTi + c
2
(Eq. 5)
= binder viscosity at the age and temperature of inter-
est, 106 Poise
where
f = loading frequency, Hz
a(Ti) = shift factor as a function of temperature Ti
Va = air void content, %
Ti = temperature of interest, °F
Vbeff = effective binder content, % by volume
a, b and c = coefficients of the second-order polynomial
34 = cumulative % retained on 19-mm sieve
38 = cumulative % retained on 9.5-mm sieve
If the value of the coefficient a approaches zero, the shift
4 = cumulative % retained on 4.76-mm sieve
factor equation collapses to the classic linear form.
200 = % passing 0.075-mm sieve
The Witczak predictive equation (Equation 8) can be pre-
2.1.2. Dynamic Modulus: Levels sented in the same form as Equation 3 for a mixture-specific
of Analysis master curve as follows:
The Mechanistic-Empirical Pavement Design Guide uses
the laboratory-measured E* data for the Level 1 design Log E* = + (Eq. 9)
1 + e+ (log t )
r
analysis; it uses E* values predicted from the Witczak E*
predictive equation in Levels 2 and 3. The master curve for
where
the Level 1 analysis is developed using numerical optimiza-
|E*| = dynamic modulus, 105 psi
tion to shift the laboratory mixture test data into a smooth
master curve. Before shifting the test data, the relationship = -1.249937 + 0.02923200 - 0.001767(200 ) 2
between binder viscosity and temperature must be estab- Vbeff (Eq. 9a)
-0.0028414 - 0.058097Va - 0.802208
lished. This is done by first converting the binder stiffness Vbeff + Va
data at each temperature to viscosity using Equation 6. The
parameters of the ASTM Ai-VTSi equation are then found = 3.871977 - 0.00214 + 0.00395838
by linear regression of Equation 7 after log-log transforma- - 0.000017(38)2 + 0.0054734 (Eq. 9b)
tion of the viscosity data and log transformation of the tem- = - 0.603313 - 0.393532 log (Tr) (Eq. 9c)
perature data. = 0.313351 (Eq. 9d)
G * 1 4.8628
= (Eq. 6) tr = reduced time of loading at reference temperature
10 sin Va = air void content, %
Vbeff = effective binder content, % by volume
log log = A + VTS log TR (Eq. 7) 34 = cumulative % retained on 19-mm sieve
38 = cumulative % retained on 9.5-mm sieve
where 4 = cumulative % retained on 4.76-mm sieve
= binder viscosity, cP 200 = % passing 0.075-mm sieve
G* = binder complex shear modulus, Pa Tr = binder RTFOT viscosity at the reference tempera-
= binder phase angle, degree ture, 106 Poise
A, VTS = regression parameters
TR = temperature, ēRankine
2.2 FLOW NUMBER (Fn)
The master curve for the Level 2 analysis is developed
using the Witczak Dynamic Modulus Predictive Equation An approach to determine the permanent deformation char-
(Equation 8) from specific laboratory test data. The Level 3 acteristics of paving materials is to employ a repeated dynamic
analysis requires no laboratory test data for the AC binder, load test for several thousand repetitions and record the cumu-
but requires those mixture properties included in the Witczak lative permanent deformation as a function of the number of
predictive equation. cycles (repetitions) over the testing period. This approach was
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employed by Monismith and coworkers in the mid-1970s using
uniaxial compression tests. Several research studies conducted
by Witczak and coworkers used a temperature of 100oF or
130oF at 10, 20, or 30 psi deviator stress level. A haversine
pulse load of 0.1 sec and 0.9 sec dwell (rest time) is applied for
the test duration of approximately 3 hours. This approach
results in approximately 10,000 cycles applied to the specimen.
Several parameters describing the accumulated permanent
deformation response can be obtained from the Fn test. Fig-
ure 2 illustrates the typical relationship between the total
cumulative permanent strain and number of load cycles. Like
the creep test, the cumulative permanent strain curve is gener-
ally defined by three zones: primary, secondary, and tertiary.
In the primary zone, permanent deformations accumulate
rapidly. The incremental permanent deformations decrease
Figure 3. Regression constants "a" and "b" when
reaching a constant value in the secondary zone. Finally,
plotted on a log-log scale.
the incremental permanent deformations again increase and
permanent deformations accumulate rapidly in the tertiary
zone. The starting point, or cycle number, at which tertiary
flow occurs is referred to as the flow number. The intercept a represents the permanent strain at N = 1,
Typical permanent deformation parameters, which are whereas the slope b represents the rate of change in perma-
obtained and analyzed from the repeated load permanent nent strain as a function of the change in loading cycles [log
deformation test, include the intercept (a, ) and slope (b, ) (N)]. An alternative form of the mathematical model used to
parameters. The permanent deformation properties (, ) characterize the permanent strain per load repetition (pn)
have been used as input for predictive design procedures. All relationship can be expressed by
of the parameters derived from the linear (secondary) portion
of the cumulative permanent strainrepetitions curve ignore
p ( aN b )
the tertiary zone of material deformability. Thus, all four of = pn = or, pn = abN ( b-1) (Eq. 10)
the parameters noted (, , b, a) are regression constants of N N
a statistical model that is only based on the "linear" sec-
ondary phase of the permanent strainrepetition curve. The resilient strain (r) is generally assumed to be indepen-
The log-log relationship between the permanent strain and dent of the load repetition value (N). As a consequence, the
the number of load cycles can be expressed by the classical ratio of permanent-to-elastic strain components of the material
power model: p = aNb, where a and b are regression con- in question can be defined by
stants depending on the material-test combination condi-
tions. Figure 3 illustrates the relationship when plotted on a pn ab b-1
= N (Eq. 11)
log-log scale. r r
ab
By letting = and = 1 - b, one obtains
r
pn
= N - (Eq. 12)
r
In the above equation, pn is the permanent strain resulting
from a single load application; that is, at the Nth application.
is a permanent deformation parameter representing the
constant of proportionality between permanent strain and
elastic strain (i.e., permanent strain at N = 1). is a perma-
nent deformation parameter indicating the rate of decrease in
incremental permanent deformation as the number of load
applications increases.
Figure 4 illustrates the above relationship and the occur-
Figure 2. Typical relationship between total cumulative rence of the flow point when the rate of decrease in perma-
permanent strain and number of load cycles. nent strain is constant.
