# Flexure Deflection - Both ends fixed (one translation free)

I have a flexure problem. One of its ends is fixed to a frame (say F1) which is stationary (no rotation & no translation) and the other end is also fixed to another frame (say F2) which can move in translatory motion (rotation fixed & translation free). The flexure here is assumed to be a slender column.

On top of the flexure, F2 has a weight $$W$$ and is moved with a force $$F$$ in the horizontal direction causing a displacement of $$y$$ whereas at the bottom the flexure is held rigidly by F1 and doesn't allow any rotation or translation. I want to find the displacement $$y$$ caused due to the forces $$F$$ and $$W$$.

Please help me solve this. Also, I request you to write the moment equation I need to consider here... and anything else I need to know.

Depending on the slenderness of your beam you may need to iterate this several times to get close to the final deflection, Y.

First, we find the deflection due to lateral load F and then we apply the Pdelta moment and find the secondary shear, v and add it.

The deflection, Y due to force F is

$$Y= \frac{Fl^3/8}{3EI}=\frac{Fl^3}{24EI}$$

Same as the deflection of a cantilever beam with half the length subjected to F.

$$\Sigma M_{base}=0\quad W * Y = V * L$$
$$V = W * Y / L$$
$$Y= \frac{Vl^3}{24EI}$$