A laminate is made from five unidirectional glass fibre layers each with longitudinal stiffness 106 GPa, stacked 0/+45/90/-45/0. Use the Hart-Smith 10 percent rule to calculate the stiffness of the laminate, to one decimal place.

Step-by-step solution

1. 1

   State the rule. The Hart-Smith $10\%$ rule is a quick design estimate: a ply aligned with the load direction contributes its full $E_{11}$, while any off-axis ply — $+45^\circ$, $-45^\circ$ or $90^\circ$ alike — contributes only $10\%$ of $E_{11}$. The laminate stiffness is the average contribution per ply.

2. 2

   Count the plies in each group. The sequence $0/+45/90/-45/0$ has

   $2\ \text{plies at }0^\circ,\qquad 3\ \text{off-axis plies }(+45,\ 90,\ -45).$

3. 3

   Add the on-axis contribution.

   $2\times 106=212\ \text{GPa}.$

4. 4

   Add the off-axis contribution at ten percent each.

   $3\times 0.10\times 106=3\times 10.6=31.8\ \text{GPa}.$

   Note that $90^\circ$ and $\pm 45^\circ$ are treated identically — that is the deliberate simplification of the rule.

5. 5

   Average over all five plies.

   $E_{\text{laminate}}=\frac{212+31.8}{5}=\frac{243.8}{5}=48.76\ \text{GPa}\approx 48.8\ \text{GPa}.$

6. 6

   Sanity-check with the fraction shortcut. With $40\%$ of plies on-axis and $60\%$ off-axis,

   $E=106\left(0.40+0.10\times 0.60\right)=106(0.46)=48.76\ \text{GPa},$

   the same number. The result is below half of $E_{11}$, as expected for a layup where most plies do not point along the load.