PH8151 Engineering Physics Chapter 1

30 CHAPTER 1. PROPERTIES OF MATTER

1.10 Cantilever: Theory and Experiment

A cantilever is a beam fixed horizontally at one end and loaded at the free end. Consider
a cantilever of lengthlloaded with a weightW =mgat the free end. (Heremis the
mass suspended at the free end of cantilever andg= 9.8 m/s^2. The mass of the cantilever
is assumed to be negligible here so that we do not need to consider it here.) Due to the
loading, the neutral filament AB of the cantilever forms an arc ABÕof radiusRand the
free end is depressed by a vertical distanceBBÕ=yas shown in Figure1.25. Consider two
points P and Q on the bent neutral filament ABÕat distancesxandx+dx, respectively
from the fixed end A. As can be seen from Figure1.25, PQ subtends an angled◊at the
centre (the point O) of the arc.

)PQ=(x+dx)≠x=dx=R◊

Draw two tangents PC and QD to ABÕat points P and Q respectively such that the

A

B’

B

W

C

D

P

Q

O

dy

y

x

dx

dθ

dθ

R

R

l-x

l

Figure 1.25: Neutral surface of a bent cantilever

vertical distance CD =dy. The angle between PC and QD isd◊since PC and QD are
respectively perpendicular to OP and OQ. If the depressionyis small compared to the
radius of curvature, we can make the approximation,PC¥QD¥(l≠x).

)CD=dy=(l≠x)d◊

Sincedx=R◊,
dy
dx

=

(l≠x)d◊
Rd◊
Hence,
1
R

=

1

(l≠x)

dy

dx

(1.30)

The internal bending moment at point P of the neutral filament is given by the expression
(1.29):

bending moment of the beam =

YIg

R

=

YIg

(l≠x)

dy

dx

(1.31)

30