B Beam on an inclined plane

[SeniorGara]

TL;DR Summary

I analyzed a beam resting on two supports attached to an inclined plane. My equations suggest equal reaction forces, but my intuition says otherwise. Is my reasoning correct? What are the conditions when the beam starts sliding or rotating?

Hello!

I have a question regarding a beam on an inclined plane. I was considering a beam resting on two supports attached to an inclined plane. I was almost sure that the lower support must be more loaded. My imagination about this problem is shown in the picture below.

Here is how I wrote the condition of equilibrium forces:
$$
\begin{cases}
F_{g\parallel}=F_{t1}+F_{t2}, \\
F_{g\perp}=F_{r1}+F_{r2}
\end{cases}.
$$
On the other hand, the equilibrium of moments relative to the center of mass can be written as:
$$
\frac{1}{2}LF_{r1}=\frac{1}{2}LF_{r2},
$$
which of course leads to the equation:
$$
F_{r1}=F_{r2}.
$$
As I mentioned above, I'm not sure that this result is correct. So my question is: is everything fine and my intuition wrong, or did I make some mistake somewhere?If everything is correct, I have a second question: as alpha increases, the system reaches a point where the beam either starts to move along the plane or begins to rotate. What should the conditions look like for the first and the second case?

 

[Baluncore]

If everything is correct, I have a second question: as alpha increases, the system reaches a point where the beam either starts to move along the plane or begins to rotate.

You show two fulcrums that carry the load. The load will not slide over a fulcrum, that needs to be a roller.
If FR1 and FR2 are due to weight, then they should be vertical, not diagonal.