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