# Extension of a cantilever beam composed of unbonded layers

The question of the magnitude of the vertical deflection of a cantilever beam composed of unbonded layers has been posted here, but I was wondering if there is an analytical way to solve for the end extension of the sliding layers of the composite beam as well.

If the force is applied to the end of the top beam, I am interested in figuring out how far the ends of the other beams slide past the end of the top beam. I'm assuming all layers are the same material and full contact is maintained at the interface. The point of interest is circled in the figure.

Any help is greatly appreciated. If this is not analytically solvable, can this be done in a way other than plugging it into FEA and seeing the result?

If the layers are unbonded, their deflection can be treated independently. The amount of sliding from layer to layer is $$t\tan \theta$$, where $$t$$ is the beam thickness and $$\theta$$ is the end angle. You can prove this to yourself by drawing a little triangle (with legs $$t$$ and $$t\tan\theta$$) at the end of any of the beams.