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Bearing loads and hinged rod force

A beam is hold by a fixed bearing on the left and a hinged rod (pendulum support). The beam itself is stiff. The force F acts on the beam.

Determine the bearing loads in A, B and C and the force in the hinged rod.

F = 15 kN
α = 75°

Solution

The following video is in German language, but English subtitles are available.

Step by step

The whole beam is divided into three sections to determine the reaction forces.

tm1-21

Section I: Balance of forces in x-direction

$\tag{1} 0={F_{\mathit{Bx}}}+{F_{\mathit{Ax}}}$

Section I: Balance of forces in y-direction

$\tag{2} 0={F_{\mathit{By}}}+{F_{\mathit{Ay}}}-F$

Section I: Balance of moments around A

$\tag{3} 0={F_{\mathit{By}}} l-2 F l$

Section II: Balance of forces in x-direction

$\tag{4} 0=-{F_S} \cos{\left( \alpha \right) }-{F_{\mathit{Bx}}}$

Section II: Balance of forces in y-direction

$\tag{5} 0={F_S} \sin{\left( \alpha \right) }-{F_{\mathit{By}}}$

Section III: Balance of forces in x-direction

$\tag{6} 0={F_S} \cos{\left( \alpha \right) }+{F_{\mathit{Cx}}}$

Section III: Balance of forces in y-direction

$\tag{7} 0={F_{\mathit{Cy}}}-{F_S} \sin{\left( \alpha \right) }$

Step by step solution of the found equations

$\tag{8} {F_{\mathit{By}}}=2 F$

$\tag{9} {F_{\mathit{By}}}=30 \mathit{kN}$

$\tag{10} {F_{\mathit{Ay}}}=F-{F_{\mathit{By}}}$

$\tag{11} {F_{\mathit{Ay}}}=-15 \mathit{kN}$

$\tag{12} {F_S}=\frac{30 \mathit{kN}}{\sin{\left( \alpha \right) }}$

$\tag{13} {F_S}=31.06 \mathit{kN}$

$\tag{14} {F_{\mathit{Bx}}}=-{F_S} \cos{\left( \alpha \right) }$

$\tag{15} {F_{\mathit{Bx}}}=-8.04 \mathit{kN}$

$\tag{16} {F_{\mathit{Ax}}}=-{F_{\mathit{Bx}}}$

$\tag{17} {F_{\mathit{Ax}}}=8.04 \mathit{kN}$

$\tag{18} {F_{\mathit{Cx}}}=-{F_S} \cos{\left( \alpha \right) }$

$\tag{19} {F_{\mathit{Cx}}}=-8.04 \mathit{kN}$

$\tag{20} {F_{\mathit{Cy}}}={F_S} \sin{\left( \alpha \right) }$

$\tag{21} {F_{\mathit{Cy}}}=30.0 \mathit{kN}$