SOLUTION:
The maximum normal stress in a plate under
bending is given by
Stress = M t/(2 I)
where
M = bending moment (lb-in)
t = thickness of plate (in)
I = second moment of area (in^4)
For a rectangular cross-section, the second
moment of area is
I = b t^3/12
where
b = width of plate (in).
Using a factor of safety, F = 2 and given
b = 4 inches,
then the thickness of the plate using the
Maximum Stress criterion is
t= 0.9871 in
Now the designer wants to replace the 0.9871
inches thick aluminum plate by a plate of
maximum thickness 0.4935 inches (half that of
aluminum) made of laminated composites.
The bending moment per unit width is 3,250 lb-in.
Using the factor of safety of two, the plate is
designed to take a bending moment per unit
width of 6250 lb.
The simplest choices are to replace the
aluminum plate by an all Graphite/Epoxy
laminate or an all Glass/Epoxy laminate. Using
PROMAL program, the strength ratio for a single
0 degree ply for the above load for Glass/Epoxy
ply is
SR = 5.4942 E-5
Since the bending moment per unit width is
inversely proportional to the square of the
thickness of the plate, the minimum number of
plies required would be 135 which gives the
thickness of the all Glass/epoxy laminate as
t = 135 x 0.0049213
= 0.664 in.
The thickness of an all Glass/Epoxy laminate is
more than 0.4935 inches and is hence not
acceptable. Similarly for an all
Graphite/Epoxy a laminate, the minimum number
of plies required would be
n = 87 plies
which gives the thickness of the plate as
t = 87 x 0.0049213
= 0.42815 in.
The thickness of an all Graphite/Epoxy laminate
is less than 0.4935 inches and is hence
acceptable.
Even if an all graphite/epoxy laminate is
acceptable, Graphite/Epoxy is 2.5 times
costlier than a Glass/Epoxy, one would suggest
the use of a hybrid laminate. The question
which arises now is in what sequence the plies
should be stacked. In a plate under a bending
moment, the magnitude of ply stresses is
maximum on the top and bottom face. Since the
longitudinal tensile and compressive strengths
are larger in the Graphite/Epoxy ply then in a
Glass/Epoxy ply, one would put them as the
facing material and the Glass/Epoxy in the
core.
The maximum number of plies allowed in the
hybrid laminate is
= 0.4936/0.0049213
= 100 plies
Several combinations of 100 ply symmetric
hybrid laminates were subjected to the applied
bending moment. Minimum strength ratios in
each laminate stacking sequence were found. If
the strength ratios are greater than one, the
cost of the stacking sequence was determined.
The hybrid laminate with the lowest cost is [16
plies of Graphite/Epoxy, 68 plies of
Glass/Epoxy and 16 plies of Graphite/Epoxy].
The above example dictated the use of
unidirectional laminates. How will the design
change if there were multiple loads? Examples
of multiple loads include a leaf spring
subjected to bending moment as well as torsion,
a thin pressure vessel subjected to an internal
pressure to yield a biaxial state of stress.
In such cases, one may have a choice not only
of material systems or their combination but
that of orientation of plies as well. Since
there can be infinite combinations of angle
plies, attention may be focussed on angle plies
of 0, 90, 45 and -45 angle plies and their
combination thereof. This reduces the
possibilities to a finite number for a limited
number of material systems, but still the
number can be quite large to handle.