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Beam with joint and offset load

This exercise is about the following questions:

  • How to calculate the forces in a joint?
  • How to calculate the bearing loads if a force acts with an offset?

Task

A force F acts with an offset on a beam. The beam is fixed on one side and connected to a floating bearing with a joint. Find the reaction forces in A, B and C.

Beam with joint and an offset force
Beam with joint and an offset force

Solution

The following video is in german language.

Reaktionskräfte bestimmen - Technische Mechanik 1, Übung 18

The beam is divided into two sections to determine the bearing reactions and the joint forces.

Section 1

Free-cut area 1 of the beam
Free-cut section 1 of the beam
\[ \require{cancel} \]

The equilibrium of forces in the x-direction is

\[\tag{1} \sum F_x = 0 = F + F_{Bx} \]

\[\tag{2} F_{Bx} = -F \]

The equilibrium of forces in the y-direction is

\[\tag{3} \sum F_y = 0 = F_{Ay} + F_{By} \]

The equilibrium of moments around point A gives

\[\tag{4} \sum M(A) = 0 = -F \cdot a + F_{By} \cdot 2 \cdot a \]

\[\tag{5} F_{By} = \frac{F \cdot \bcancel{a}}{2 \cdot \bcancel{a}} \]

So the result from equation 3 is

\[\tag{6} F_{Ay} = -- \frac{F}{2} \]

Section 2

Free-cut area 2 of the beam
Free-cut section 2 of the beam
\[ \require{cancel} \]

The equilibrium of forces in the x-direction is

\[\tag{7} \sum F_x = 0 = -- F_{Bx} + F_{Cx} \]

\[\tag{8} F_{Cx} = -F \]

The equilibrium of forces in the y-direction is

\[\tag{9} \sum F_y = 0 = -- F_{By} + F_{Cy} \]

\[\tag{10} F_{Cy} = \frac{F}{2} \]

The equilibrium of moments around point C gives

\[\tag{11} \sum M(C) = 0 = F_{By} \cdot 2a + M_C \]

\[\tag{12} M_C = -- F \cdot a \]