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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?

Beam with a fixation bearing and two external forces
Beam with a fixation bearing and two external forces

Solution

The following video is in german language. Please scroll down for the written solution.

Lagerreaktionen der festen Einspannung bestimmen - Technische Mechanik 1, Übung 13

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

Free cut of the beam
Free cut of the beam

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

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

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