How determine the compression of a beam with a concentrated end load in the x axis?

Question

I'm studying Electronics Engineering so I have almost no background in mechanics, just a little of statics and dynamics. I have recently started an internship in an investigation center where they make microelectromechanical systems (MEMS). I was asked to know the deformation in $y$- and $x$-axes for a cantilever beam with a concentrated end load. I have more interest in the vectorial mechanics rather than the material mechanics.

I have done research in Minhang Bao's book Analysis and Design Principles of MEMS Devices, where I found that the maximum displacement at the free end is: $$w(L) = \dfrac{4FL^3}{EI}$$ if the load is on the $y$-axis.

But I couldn't find any information related to the compression in the $x$-axis. I came up with the idea of using a similar triangles approach but think that won't be so accurate.

Answer 1

It's not entirely clear what you mean, but I'm assuming you mean this:

In this case, the axial deflection at the free end is equal to

$$ w_x(L) = \dfrac{FL}{EA} $$

This can be trivially obtained by noting that the compressive stress is equal to $$\sigma = \dfrac{F}{A}$$

And, by Hooke's Law we know that the relationship between stress and deformation (assuming linear elasticity) is equal to $$\begin{align} \sigma &= E\epsilon \\ \therefore \epsilon &= \dfrac{\sigma}{E} \end{align}$$ where $E$ is the elastic modulus of the material.

The deflection at the free end is equal to $$w(L) = \int_0^L\epsilon(x)\text{d}x$$ which, for a member under uniform stress such as this one, simplifies to $$w(L) = \epsilon L$$ therefore, using the equations above, we get $$ w_x(L) = \dfrac{FL}{EA} $$