Past exam of the materials science course of the University of Cambridge / 2017 / Paper 1 / 5

Measure $x$ from the clamped end and let downward deflection $y$ be positive. In the small-slope elastic beam model, the bending moment has magnitude $M(x)=F(l-x)$ and $EI{y}^{\prime \prime }=M(x)$. Integrating with $y(0)=y^{\prime }(0)=0$ givesFor the rectangular cross-section, $I=w{h}^3/12=6.75\times 10^{-7}{\mathrm{m}}^4$, and thereforeRequiring $\delta \le 0.2\mathrm{m}$ with $F=1079.1\mathrm{N}$ yieldsThis calculation neglects beam self-weight as stipulated, and also transverse shear deformation and geometric nonlinearities. The end slope at the limit is $3\delta /(2l)=0.10$, so the small-slope approximation is reasonable for the intended estimate. Meeting a stiffness specification alone does not establish safe strength or fatigue life.

The optimum axial-stiffness arrangement has continuous straight fibres parallel to the beam axis with good fibre–matrix bonding. For a common axial strain $\epsilon$, force balance on a section gives $F=E_f\epsilon A_f+E_m\epsilon A_m$. Divide by total area and by $\epsilon$ to obtain the longitudinal rule of mixtures,This is an isostrain upper idealization, consistent with the Voigt elastic approximation. The glass fraction required by the deflection constraint isAbout 95.8% glass by volume is impractical for an ordinary glass-fibre/polymer board. It leaves little matrix for wetting, load transfer and defect tolerance. Identical circular parallel fibres pack at most $\pi /(2\sqrt{3})=0.9069$ in a regular hexagonal array; even that zero-void ideal gives only $68.3\mathrm{GPa}$. Polydisperse or noncircular fibres can change that geometric limit, so it is not an absolute theorem about every possible composite. Waviness, voids, incomplete bonding and nonaxial fibres further reduce the achievable modulus. A thicker section or a stiffer reinforcement is a more plausible design change.

Let $V_c$ be the carbon-fibre volume fraction of the whole composite. Then $V_m=0.4$ and $V_g=0.6-V_c$. Under the same longitudinal rule of mixtures,Setting $E=71.94\mathrm{GPa}$ givesThus carbon occupies 17.8% of the complete composite, or about $29.6\%$ of its total fibre volume. The resulting $60\%$ total fibre fraction is much more plausible than the glass-only estimate. This remains an ideal stiffness calculation requiring aligned, effectively loaded fibres; strength and fatigue checks are separate.