where h is the laminate thickness, superscripts u and I denote upper and lower limits of
associated quantities, and E1, 82, and yl2 are the ply strains along fiber direction,
transverse to fiber direction, and shear strain, respectively. The stack thickness of plies
with ply-angle 97, which is allowed to have matrix cracking, is tl. The stack thickness of
the plies with ply-angle 82, which are not allowed to crack and must provide in total a
minimum intact thickness of 0.04 inch to prevent hydrogen leakage, is t2. The four design
variables are the ply angles 6, and 62 and their stack thickness tl and t2. The individual
stack thickness from a continuous optimizer (SQP in MATLAB) is rounded up to the
nearest multiple of 0.005 inch.
Optimizations without Matrix Cracking
In order to see the effect of mechanical and thermal loads, it is instructive to
compare designs for different operational temperatures. Table 4-4 shows the optimum
laminates at these temperatures. In the last row of Table 4-4, the numbers in the
parentheses are the continuous thickness before rounding. Without thermal strains, a
cross-ply laminate with thickness of 0.04 inch can easily (with 0.1% transverse strain as
the margin of safety) carry the mechanical loads. When thermal strains are taken into
account, the angle between the + 6 plies must decrease in order to reduce the thermal
strains. The ply angles do not vary monotonically because both the residual strains and
the stiffness of the laminate increase with the decrease of temperature. At cryogenic
temperatures the angle decreases to 25.50, and at that angle the axial loads cannot be
carried efficiently and the thickness increases to 0.1 inch. Figure 4-4 shows that the
thickness of optimum laminates for temperature dependent and constant material
properties at 770F changes substantially with the working temperature for a strain limit