Page 420 - Mechanics of Asphalt Microstructure and Micromechanics

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412 Ch a p t e r w e l v e

12.14b). Two extensometers were mounted on both the top and bottom of the beam to
measure the compressive and tensile displacements (strains) simultaneously. A nomi-
nal 9.5-millimeter gradation asphalt mixture with PG64-22 binder and anti-stripping
agent (0.5% by weight of binder Adhere HP+) was used for their study. The asphalt
content is 5.7%. It should be noted that 15% of RAP is used in this mix. Testing at both
25°C and 40°C was performed. During the test, three parameters were recorded at ev-
ery 0.1 s. The load was measured with a 2-kilonewton capacity load cell; the compres-
sive deformation on the top of beam (Δ c ) and the tensile deformation at the bottom of
the beam (Δ t ) were measured with strain-gage based extensometers with a gauge length
of 25 mm. Both extensometers were attached at the mid-span of the beam. The measure-
ment range of the two extensometers is ±2 mm.
A ramping load was applied to the beam at the rate of 5 N/s. This loading rate was
chosen based on the experiences that the target peak load can be reached in a reasonable
amount of time without introducing significant creep deformation at 25°C. Depending
on the temperature and thus the modulus, the maximum load was varied such that the
maximum incurred strain (tensile and compressive strains) was comparable to the previ-
ous study (Secor and Monismith, 1965). At 25°C, the maximum load was 200N corre-
sponding to the data collection time of 40 s. The self-weight of the beam was also consid-
ered as it may induce significant strains at relatively high temperatures.
Beam specimens were kept at the target temperature in an environmental chamber
for four hours before testing. This minimized the temperature gradient within the as-
phalt specimen and consequent modulus gradient, which can invalidate the following
analysis of the stress and strain in a beam with bi-modulus.

12.7.2 Elastic Analysis of a Bernoulli-Euler Beam of Bi-Modulus
The model of testing is a simply supported beam subjected to two equal, concentrated
loads symmetric to the center of the span, as shown in Figure 12.15. The span of the
beam is L and the distance between the support and the location at which a concen-
trated load is applied is d. The cross-section of the beam is shown in Figure 12.15a, with
h denoting the height and w the width. The self-weight of the beam is also considered
and denoted as q with unit of N/m.
The moments from concentrated loads and uniformly distributed self-weight can
be obtained with the classical beam theory as described in many structural mechanics
books, e.g., Gere and Timoshenko (1984). The moment diagrams from both loads are
plotted separately in Figure 12.15. The total momentum in the beam is obtained using
the superposition principle. Let the longitudinal direction of the beam be the same as
the direction of the x coordinate. For convenience, the origin of the coordinates is set at
the mid-span of the beam in the following analysis.
The moment between two concentrated loads is described as:
Pd q qL 2 s s
M= − x + for − ≤ x≤ (12-9)
2
2 2 8 2 2
Where s is the distance between the two concentrated loads.
The Bernoulli-Euler beam theory assumes that the transverse plane sections remain
plane and normal to the longitudinal fibers after bending. Thus, the following relation
on strain distribution always exists for a continuous beam.
ε =− κ y (12-10)
x