Self-weight deflection of a square cantilever ($y(l)=3\rho g{l}^4/(2E{a}^2)$)

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.