Cantilever

A beam fixed at one end and free at the other. A transverse end force $F$ gives bending moment $M(x)=F(L-x)$ and, for a rectangular section, maximum bending stress $6FL/(w{h}^2)$ at the fixed end. Longitudinal tension and compression are greatest on opposite outer fibres.

A uniform square section has second moment of area $a^4/12$ and gravitational load per length $\rho g{a}^2$. Euler-Bernoulli bending therefore gives $y^{\prime \prime \prime \prime }=12\rho g/(E{a}^2)$, with zero displacement and slope at the clamp and zero moment and shear at the free end. Integrating gives the displayed tip deflection and $y(x)=\rho g{x}^2(x^2-4lx+6l^2)/(2E{a}^2)$. Small slope and strain are required.