Mechanical Engineering Principles

6

Bending moment and shear force

diagrams

At the end of this chapter you should be

able to:

• define a rigid-jointed framework


• define bending moment


• define sagging and hogging


• define shearing force


• calculate bending moments


• calculate shearing forces


• plot bending moment diagrams


• plot shearing force diagrams


• define the point of contraflexure

6.1 Introduction

The members of the structures in Chapter 4 with-
stood the externally applied loads in either tension
or compression; this was because they were not sub-
jected to bending.
In practise many structures are subjected to bend-
ing action; such structures include beams and rigid-
jointed frameworks. A rigid-jointed framework is
one which has its joints welded or riveted or bolted
together; such structures are beyond the scope of
this text (see reference [1], on page 54). Prior to
calculating bending moments and shearing forces,
we will first need to define bending moment and
shearing force.

6.2 Bending moment (M)

The units of bending moment are N mm, N m,
kN m, etc. When a beam is subjected to the couples

(a) Sagging moment (+) (b) Hogging moment (−)

MMMM

Figure 6.1 Bending moments

shown in Figure 6.1, the beam will suffer flexure

due to the bending moment of magnitudeM.

If the beam is in equilibrium and it is subjected to

a clockwise couple of magnitudeMon the left of

the section, then from equilibrium considerations,

the couple on the right of the section will be of

exactly equal magnitude and of opposite direction

to the couple on the left of the section. Thus, when

calculating the bending moment at a particular point

on a beam in equilibrium, we need only calculate

the magnitude of the resultant of all the couples on

one side of the beam under consideration. This is

because as the beam is in equilibrium, the magnitude

of the resultant of all the couples on the other side

of the beam is exactly equal and opposite. The beam

in Figure 6.1(a) is said to besaggingand the beam

inFigure6.1(b)issaidtobehogging.

Thesign conventionadopted in this text is:

(a) sagging momentsare said to bepositive

(b) hogging momentsare said to benegative

6.3 Shearing force (F)

Whereas a beam can fail due to its bending moments

being excessive, it can also fail due to other forces

being too large, namely the shearing forces; these

are shown in Figure 6.2. The units of shearing force

areN,kN,MN,etc.

It can be seen from Figures 6.2(a) and (b) that the

shearing forcesF act in a manner similar to that

exerted by a pair of garden shears when they are

used to cut a branch of a shrub or a plant through