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# Cantilever beam bearing reactions

## Task

A beam fixed on one side at point A is loaded with forces F1 and F2. Which bearing reactions occur in A?

## Solution

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See the free body diagram of the cantilever beam below. We consider left turning moments as positive.

In this exercise the only bearing of the beam is the fixation in A. The bearing loads can be determined as follows:

$\require{cancel}$ $\newcommand{\myvec}[1]{{\begin{pmatrix}#1\end{pmatrix}}}$

The equilibrium of forces in the x-direction is

$\tag{1} \sum F_x = 0 = F_{Ax}$

The equilibrium of forces in the y-direction is

$\tag{2} \sum F_y = 0 = F_1 -- F_2 + F_{Ay}$

$\tag{3} F_{Ay} = F_2 -- F_1$

The equilibrium of moments around point A is

$\tag{4} \sum M(A) = 0 = -F_1 \cdot (a+b) + F_2 \cdot b + M_A$

$\tag{5} M_A = F_1 \cdot (a+b) -- F_2 \cdot b$