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.

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

$$\require{cancel}$$

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

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

So the result from equation 3 is

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

Section 2

$$\require{cancel}$$

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

$$\tag{12} M_C = - F \cdot a$$