Second Order Differential Equations - Beam Deflections

1. Jul 27, 2017

Saracen Rue

1. The problem statement, all variables and given/known data
A cantilever of length $L$ is rigidly fixed at one end and is horizontal in the unstrainted position. If a load is added at the free end of the beam, the downward deflection, $y$, at a distance, $x$, along the beam satisfies the differential equation: $$\frac{d^2y}{dx^2}=k\left(L-x\right) \ for\ 0\le x\le L$$
Where $k$ is a constant. Find the deflection, $y$, in terms of $x$ and hence find the maximum deflection of the beam.

2. Relevant equations
Basic knowledge of integration

3. The attempt at a solution
After double integrating both sides I'm left with $y=\frac{k}{2}Lx^2-\frac{k}{6}x^3+cx+d$ As the question tells us the beam is horizontal when their is no weight on it, we know that $y=0$ when $x=0$.
$0=\frac{k}{2}L\left(0\right)^2-\frac{k}{6}\left(0\right)^3+c\left(0\right)+d$
$d=0$
$y=\frac{k}{2}Lx^2-\frac{k}{6}x^3+cx$
This is where I get stuck - I can't find any part of the question which helps me calculate the value of the constant $c$. The answer says that $c$ should be zero, but I don't understand how they know this.

2. Jul 28, 2017

ehild

The beam is horizontal when unloaded. What do you think the direction of the tangent of the loaded beam is at the fixed end?