Our
purpose is to explain why highways in California seem
to be damaged by the heavy trucks more
than routes in
Brittany by les camions.
For the
solution to the Five
Axles puzzle, we shall refer to
sketches and make comparisons
of semi
trailer-trucks. The most common
configurations plying the respective roadways
are the
C-rig, an
18-wheeler typically observed in California
and the B-rig,
which is a 12-wheeler typically observed in
Brittany.Both rigs have five axles. Normalized
dimensions will be used, wherein the length of
the load-carrying compartments of both rigs
are postulated as the same, setting xC =
50 feet (15.2 meters ). By
coincidence, the
length of the wheel-base for the B-rigxW = 50 feet, and for the C-rigthe
wheel-basexW = 63 feet, or about
25% longer. Complying with
regulations, the ‘roadway footprint’
of the B-rigis designed
to be 13% shorter than the
C-rig
(62
feet bumper-to-bumper vs 70 feet).
Axle-Loads
The
center of gravity (CG) for the uniformly
loaded trailer in each rig is shown in the
sketch below. Its location relative to
the Kingpin is represented
by the symbol xL. One
expects that for both rigs, the loaded
weight of trailers wL
must be much greater than the weights of
tractors wT.
Let wL/wT
≈ 8, say. The CG
for the B-rig
tractor relative to its KingpinxT
is shifted slightly forward (3
feet) from that in the C-rig,
which is explainable by the cabine
avancé (cab-over-engine) configuration
on the B-rig
versus cabine
conventionelle (conventional cab) on
the C-rig. The
location of the Kingpin
on the B-rig
trailer is shown to be shifted rearward by 5
feet compared to that in the C-rig trailer.
Dimensions are summarized in the table below
the sketches.
Table ofDimensions
Symbol
Description
C-rig
B-rig
(alpha)
(relative
to Kingpin*)
feet
meters
feet
meters
xB
Highway
Footprint
70.0
21.3
62.0
18.8
xC
Load
Capacity
50.0
15.2
50.0
15.2
xF
Front
Axle*
19.0
5.7
19.0
5.7
xL
Load
CG*
22.0
6.8
15.0
4.6
xR
Rear
Axles*
41.0
12.6
26.0
7.8
xS
Axle
Separation
6.0
1.7
6.0
1.7
xT
Tractor
CG*
8.0
2.2
10.0
3.1
xW
Wheelbase
63.0
19.2
50.0
15.2
Weight
regulations for all kinds of roadway
vehicles are administered based on the
distribution of static loads among
axles. Accordingly, for the
solution to the Five Axles
puzzle, we must estimate the
individual axle-loadings (forces
f1
to f5), and that takes
some algebra (algebra lovers are
invited to request the derivations here).
For
the sketch below, consider yourself to
be a stationary observer positioned at
the far left of the horizontal line
taking measurements while either the C-rig
truck and the B-rig
truck rolls
over your observation point from right
to left. For comparing static
loads, we will consider fully loaded
(80,000 lbs) trucks to be moving ever so
slowly over a weighing station.
The forces supported by the
roadway have
been depicted in the sketch
below as bell-shaped curves
superimposed upon the axle
locations. These forces
are used for regulatingload
limits for trucks. Each
force curve reaches its peak as
the respective axle passes
directly over the observation
point. Our reasoning is
elementary. The force is
'felt' in the pavement structure
below the observation
point in pavement layers both
ahead of and behind the axle.
The dashed curves represent
estimates for the axle-loads for an
unloaded trailer.
The
most pronounced outcome shown in the
sketch below, is that for
the B-rig
the peak axle-load fB2 =
34,240 lbs which is
greater than any of the axle loads for
the C-rig
-- by a factor of about 1.75-to-one.
The C-rig
axle-loadings on fC4 and fC5
are carried by eight trailer
wheels, while the B-rigloadings
on fB3, fB4 and fB5
are carried on only six
wheels. For
load-per-wheel, that
represents a ratio of more
than 1.4-to-one in favor
of the C-rig. Clearly, our solution to the
Five
Axles
puzzle
will allow us to
invoke neither axle
nor wheel loadings for
the B-rig.
Static
vs Dynamic Loads
In common
with
regulations of
truck loading,
our attempted solution to the Five
Axles
puzzle has
made a tacit
assumption
that static
loads will
provide the
explanation.
Accordingly,
we have not
considered dynamic
forces
attributable
to truck
wheels in
contact with
pavements at
highway
speeds.
Of
course, even a
perfect
roadway
experiences dynamic
forces imposed
by wheels as
vehicles of
all kinds come
rolling along.
A typical
segment of
flat and level
pavement is
called upon to
support the
rather sudden
application of
a truck's
static weight
under pressure
from its
wheels.
After a period
of time, which
depends on the
speed of the
truck, that
weight is just
as suddenly
relieved.
Until the next
axle-load
arrives at
that
segment.
For a given
segment along
the path of a
truck wheel,
pavement
impacts take
place five
times for
either the C-rig or the B-rig.
Solvers
may want to review how axle-load,
which is a force,
creates stress
in the roadway materials under the
wheels and necessarily results in
strain.
Two types of stress
are especially important: compression
and shear.
There are a half-dozen
conventional failure
theories,
including stress (normal or shear),
energy (strain or distortion),
fracture or creep.
Damaging Roadways
Roadways are seldom perfect. Imperfections
take many
forms, and
our analytic
efforts must
figure out how
they can be
affected
differently
between two
truck types --
more
specifically
explaining why
more damage
seems to be
caused by C-rigs than B-rigs.
One
supposes that
a small
imperfection
will be made
worse by
vehicle
wheels, trucks
more than
cars. Indeed,
one
authoritative
U.S.
reference
estimates
that...
One
80,000-pound
(36,000 kg)
truck does as
much damage to
roads as 750
3,800-pound
(1,700 kg)
cars
...which
is hard to reconcile
on a per-wheel
basis. Using
those numbers,
a car imposes
3,800 / 4 =
950 lbs on the
pavement under
each of its
four wheels,
while an
18-wheel truck
carries 80,000
/ 18 = 4,444
lbs per
wheel.
That's a big
difference,
but 4,444
/ 950
is only 4.68
times heavier
not 750
times.
Tire pressures
might be a
factor.
Car-tire
inflation may
be as low as
32 psi, while
truck-tires
may be
pressurized as
high as 120
psi.
That's 120 /
32 = 3.75
times
higher.
Even so, the
product of
4.68 and 3.75
gives us a
ratio of 17.5
not 750.
It is clear
that
differences in
static wheel
loadings will
not provide an
explanation
and therefore
a solution to
the Five
Axles
puzzle.
What
are we missing here?
Solvers who
have tried to
come up with the
reason for why
one of the
axles on the B-rig is retracted may have a clue.
It's called a
'dead axle' or
'lazy axle' in
one
reference.
Fully
loaded, the
trailer on the
B-rig has three axles (fB3, fB4, fB5)
instead of
just two axles
on the C-rig (fC4, fC5). WfC4, (fC4, fCheels
B-rig
B-rigB-rigon
the B-rig
are dragged
onto circles
that have
incompatible
radii. Then
too,
retracting the
axle saves
tire wear when
the truck is
lightly
loaded.
The B-rigB in
the photograph
is shown with
its lazy axle
retracted
while
apparently
operating on
the open
highway.
The
truck may be
lightly loaded
or even
unloaded.
Also,
with an
unloaded
trailer, the
retraction of
fB3 some
amount of
normal force
is preserved
on fB2.
Come
to think of
it, trucks do
operate
unloaded much
of the time,
'dead heading'
between
deliveries.
A popular
subject in
assuring
roadability of
vehicle
designs is
unsprung
mass.
The main
concerns are
handling by
drivers and
riding comfort
for
passengers.
An empty trailer does get bounced around a lot
without a
payload on
board.
Solvers will discover the explanation for whyC-rigs
do more damage
to roadways
than B-rigs in the solution to...