Clamped-free beam under uniform load and endpoint force
= Clamped-free beam under uniform load and endpoint force
For
$$
E[y]=\int_0^L\left(\frac A2(y'')^2+\rho gy\right)\,dx
$$
with the <boundary conditions> $y(0)=y'(0)=0$, $y''(L)=0$, and $-Ay'''(L)=F$, the <higher-order Euler-Lagrange equation> is
$$
Ay''''+\rho g=0.
$$
Its solution splits as
$$
y=y_0+y_F,
\qquad
y_0=-\frac{\rho g}{24A}x^2(6L^2-4Lx+x^2),
\qquad
y_F=\frac{F}{6A}x^2(3L-x).
$$