In order to reduce material costs, a composite panel is to be made by placing fibers in the matrix material in an X-pattern of \(\pm \alpha\) as shown in Figure 6.46, instead of randomly distributing the fibers over all angles. The X-pattern composite is to be designed so that it has at least \(90 \%\) of the stiffness of the randomly oriented fiber composite along the longitudinal \((L)\) axis.

From tensile tests of a unidirectional composite consisting of the same fiber and matrix materials and the same fiber volume fraction, it is found that the off-axis Young's modulus of the composite can be described by the equation \[E_{x}(\theta)=100-90 \sin \theta(\mathrm{GPa})\left(0 \leq \theta \leq \frac{\pi}{2}\right)\]
whereas the Young's modulus of the matrix material is \(E_{\mathrm{m}}=3.5 \mathrm{GPa}\). Determine the angle \(\alpha\) in Figure 6.46 such that the longitudinal Young's modulus of the X-pattern composite is equal to \(90 \%\) of the Young's modulus of the randomly oriented fiber composite.

Transcribed Image Text:

Fiber T and matrix Matrix 8 L Matrix FIGURE 6.46 Composite panel with fibers arranged in X-pattern.