Two carbon fibre laminates use the same fibre with E11 = 132.4 GPa. Laminate 1 has 8 layers stacked 0/-45/+45/0/0/+45/-45/0 and Laminate 2 has 10 layers stacked +45/-45/+45/-45/0/0/-45/+45/-45/+45. Using the Hart-Smith 10 percent rule, by how much does the stiffness of Laminate 1 exceed that of Laminate 2?

Step-by-step solution

1. 1

   State the rule and compute the off-axis contribution once. Under the Hart-Smith $10\%$ rule a $0^\circ$ ply contributes its full $E_{11}$ and every off-axis ply contributes $10\%$ of it. Since both laminates use the same fibre,

   $0.10E_{11}=0.10\times 132.4=13.24\ \text{GPa}$

   per off-axis ply, for both laminates.

2. 2

   Count the plies in Laminate 1. The sequence $0/-45/+45/0/0/+45/-45/0$ has $8$ plies, of which the $0^\circ$ entries are positions $1,4,5,8$ — that is $4$ on-axis and $4$ off-axis.

3. 3

   Average over the 8 plies of Laminate 1.

   $E_{1}=\frac{4(132.4)+4(13.24)}{8}=\frac{529.6+52.96}{8}=\frac{582.56}{8}=72.82\ \text{GPa}.$

4. 4

   Count and average for Laminate 2. The sequence $+45/-45/+45/-45/0/0/-45/+45/-45/+45$ has $10$ plies, of which only positions $5$ and $6$ are $0^\circ$ — that is $2$ on-axis and $8$ off-axis:

   $E_{2}=\frac{2(132.4)+8(13.24)}{10}=\frac{264.8+105.92}{10}=\frac{370.72}{10}=37.07\ \text{GPa}.$

5. 5

   Subtract.

   $E_{1}-E_{2}=72.82-37.072=35.748\approx 35.75\ \text{GPa}.$

   Carrying the unrounded $37.072$ matters here: rounding each laminate to $72.82$ and $37.07$ first would give $35.75$ too, but on other numbers that shortcut can shift the second decimal.

6. 6

   Read the design lesson. Laminate 2 has more layers yet is barely half as stiff, because what the rule tracks is the fraction of plies aligned with the load — $50\%$ against $20\%$. Adding off-axis plies adds thickness and weight while contributing only a tenth of their stiffness in the load direction.