Intuitive approach to cantilever deflection based on beam length

Question

A high schooler like me asked a question on here regarding the equations used for cantilever deflection, inquiring about derivations. It turns out the derivation is learnt in the first or second year of a phx/engineering degree at university, and also needs an understanding of differential equations which I don't yet have. I am conducting an experiment where I change the length of a cantilever, applying a constant end load and measuring the deflection. From the equations I know that as length increases, deflection increases, but I was wondering how one would explain this in terms of forces? Some sort of intuitive approach using Newton's laws or compression/tension without having to understand how the formula is derived?

Thank you :]

Answer 1

The physical starting point for the derivation is that the curvature of the beam at any point is proportional to the bending moment: $$M = EI/\rho$$ where $M$ is the bending moment, $E$ is Young's modulus, $I$ is the moment of inertia of the beam, and $\rho$ is the radius of curvature at the point.

You can demonstrate that experimentally if you apply a moment (not a shear force) at the end of a cantilever beam. The moment will then be constant along the length of the beam and the deflected shape will be an arc of a circle.

If you apply a shear force at the end of the beam, the moment varies along the length, and there isn't an easy way to find the deflection without using calculus.

You might be able to devise an experiment to verify the equation above directly, if you take a very flexible beam (e.g. a long thin strip of metal) and measure the moment at each end which will bend it into a complete circle.

Another idea would be to make a cantilever where I varies along the length in the same way as M. If you apply a shear load at the tip, you could do that by making the beam a triangular shape (wide at the base and tapering to a point at the end) with constant thickness. The deflected shape when you apply a shear load at the tip would again be the arc of a circle.