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

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.