Drawing bending moment and shearing force diagrams

The diagrams M and Q are drawn to clearly imagine the character of the bending moment and the shearing force change along the bar length and to find the dangerous sections. The drawing technique of the diagrams is considered by the following examples.

Example. Draw the diagram M and Q for the cantilever beam (Fig. 5.8 a).

Solving. There are two regions ( and BC), differing by the load character, hence, also by and the change law and Q.

а)   b)   c)

Fig. 5.8.

The bending moment at the section of the region a distance z₁ to the right of А is found by the moment of the left force. For that purpose the distributed load to the left of the section is replaced by the resultant force qz₁ as applied in the middle of the region over the length z_1. We get

(5.6)

The minus sign is necessary because the beam tends to bend concave downward. It is the parabola equation. The parabola is drawn by approximately three points:

(5.7)

Do a section on the region ВС at a distance z₂ to the free bar end. Replace the distributed load along the length а₁ by the resultant force applied in the middle of the region АВ. The moment at the section is

(5.8)

This is the equation of the straight line. Determine for two values z:

and

Then we get

(5.9)

The diagram М is given in Fig. 5.8 b.

The shearing force at the section / — / is equal to as the sum of the projection on the vertical line of the force located to the left of the section.

The shearing force at the section //—// is equal to

The diagram is given in Fig. 5.8 c.