Determining the support reactions

 

Let us consider some examples.

Example 1. Determine the support reactions of the cantilever beam (Fig. 5.3 a).

Solving. Represent the reaction of the patching up in the form of two forces and directed as indicated in Fig. 5.3 a and the reaction moment .

a) b)

Fig. 5.3.

 

Compose the beam equilibrium equation.

1. Equal the projection sum of all forces on the axis Z acting on the beam to zero: . We obtain . The horizontal component is equal to zero without any horizontal load.

2. The same is for The resultant force replaces the uniform load q as applied in the middle of the region

 

 

from where we have

 

.

 

The vertical component reaction of the cantilever beam is equal to the force sum applied to the beam.

3. Compose the third equilibrium equation. Equal the moment sum of all forces about the some point to zero, for example, about the point A:

 

 

from which

 

The minus sign indicates that the accepted previously direction of the reaction moment has to be changed in the opposite direction. So, the reaction moment is equal to the moments sum of the external forces about the patching up.

Example 2. Determine the support reaction of the simple beam (Fig. 5.3 b).

Solving. Since the horizontal load is lacking, then Az = 0.

 

1. .

2.

.