# Why is the derrivative of a displacement at the end of a rod equal to 0?

I'm studying a book on finite elements method and well, simply, I don't understand how they get this boundary condition.

if we integrate the differential equation: $$\int_{L}^0 (AE \frac{d^2u}{dx^2}-q=0) dx \quad \text{we get} \ AE \frac{du}{dx}-qx=c$$ By inspection we note c is zero and strain is proprtional to qx: $$\quad \frac{du}{dx}\propto qx$$

Note the strain du/dx is zero at L and maximum at x=0 due to triangular loading starting from zero at L and ending at qL at x=0

It is not a typo so you can disregard the previous answer. This is not a bending problem, it is normally-loaded element. The first boundary condition just means that the displacement at the fixed end is zero. du/dx is the strain, therefore the second boundary condition simply means that the strain at the free end of the rod is zero.

Cheers

• du/dx is the rate of change of strain. – Tiger Guy Jun 3 at 17:38
• @TigerGuy Your comment (and you answer) are both wrong. du/dx is the strain, not the rate of change of strain. – alephzero Jun 3 at 22:38
• no, u is the strain. Which is why u(0) = 0. du/dx is clearly the derivative. Source: I looked it up. – Tiger Guy Jun 4 at 6:04
• Can you link the source where you looked it up? I thought u is the displacement... (which works with it as zero at the LHS) – Jonathan R Swift Jun 4 at 7:19
• @JonathanRSwift, from googling du/dx strain, ocw.mit.edu/courses/mechanical-engineering/… – Tiger Guy Jun 5 at 13:33

Simple. It's wrong, a typo.

This is a cantilever beam, so we know the deflection at the support and the rotation at the support are both zero:

$$u(0) = 0 \quad \left.\dfrac{\partial u}{\partial x}\right|_{x=0} = 0$$

That equation means the rate of change of the strain (the derivative) at length L = 0. This is because they have defined the force as being evenly distributed along the length of the bar. As such, the strain along the bar will increase as the distance from the support descreases. At the unsupported end, the force does not have anything additive to it from the forces past it, so the rate of change of strain = 0.