Technical information

4-14
3
21
3
2,22
3
1,21
2
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
⋅+⋅
=
hh
EhEh
E
com
...................................................................... (4.3.2)
()()
2
2
2
1
* comcom
com
EEE += .............................................................................. (4.3.3)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
com
com
com
E
E
1
2
arctan φ
....................................................................................... (4.3.4)
in which
0.1
1
=h in. (25.4 mm) is the lift thickness of Mix 1, 0.4
2
=h in. (102 mm) is
the lift thickness of Mix 2 (refer to Figure 3.2.1),
i
E
,1
and
i
E
,2
(2,1=i ) are the
components of the complex modulus for Mix
i at a given test temperature and
frequency (refer to Table 2.3.5, Table 2.3.6 and equation 3.3.3), and
*
com
E and
com
φ
are
the combined dynamic modulus and phase angle at a given test temperature and
frequency.
The results from these computations are shown in Table 4.3.1 from which
master curves were constructed for a reference temperature of 15.5ºC (60ºF) using the
approach developed by Levenberg and Shah (2008). Referring to equations 3.3.4, 3.3.5,
3.3.6 and 3.3.7 (Subsection 3.3.2), the derived parameters for the combined properties
were:
328,2
1
=a
MPa (337.6 ksi),
2
2
1003.1
−
⋅=a ,
5
3
10889.6
−
⋅=a s,
164=
∞
E
MPa
(23.8 ksi),
0.33
1
=c , and 1.302
2
=
c ºC (575.8ºF). The resulting dynamic modulus and
phase angle master curves (vs. reduced frequency
r
f ) are plotted in Figure 4.3.1. The
corresponding time-temperature shift factor (
T
a ) vs. physical temperature is plotted in
Figure 4.3.2.
Next step, referring to Table 4.2.1, recall that the backcalculated modulus of the
HMA layers was 350,000 psi (2,412 MPa) in the initial part of the experiment. This
value is suitable for a temperature of 15.5ºC (60ºF) and a loading speed of 5 mph (~2.2
m/s). Using Figures 4.3.1 and 4.3.2, it may be seen that this stiffness level is paired with
a reduced frequency of 0.0232 Hz and a time-temperature shift factor of 1.0. Using the
same figures a new HMA modulus can be computed for any given loading speed and
temperature by adjusting
r
f and
T
a relative to the APT conditions. For example, at