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Clamped-free beam under uniform load and endpoint force

For

E [y] = \int_{0}^{L} (\frac{A}{2} (y^{''})^{2} + ρ g y) d x

with the boundary conditions

y (0) = y^{'} (0) = 0

,

y^{''} (L) = 0

, and

- A y^{'''} (L) = F

, the higher-order Euler-Lagrange equation is

A y^{''''} + ρ g = 0.

Its solution splits as

y = y_{0} + y_{F}, y_{0} = - \frac{ρ g}{24 A} x^{2} (6 L^{2} - 4 Lx + x^{2}), y_{F} = \frac{F}{6 A} x^{2} (3 L - x) .

Table of contents 49 1
- Endpoint force derivative of clamped-free beam energy Clamped-free beam under uniform load and endpoint force 22

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