L-3
Bending of Beams
A beam is a rod or bar of uniform cross–section (circular or rectangular) whose length is
very much greater than its thickness as shown in Fig. 1.8.
The beam is considered to be made up of a large number of thin plane layers called
surfaces placed one above the other. Consider a beam to be bent into an arc of a circle by the
application of an external couple as shown Fig. 1.9 Taking the longitudinal section ABCD of the
bent beam the layers in the upper half are elongated while those in the lower half are compressed.
Fig. 1.8 A beam
In the middle there is a layer (MN) which is not elongated or compressed due to bending
of the beam. This layer is called the ‘neutral surface’ and the line (MN) at which the neutral
layer intersects the plane of bending is called the ‘neutral axis’.
Fig.1.9 Bending of a beam
It is obvious that the length of the filament increases or decreases in proportion to its
distance away from the neutral axis MN.
The layers below MN are compressed and those above MN are elongated and there will
be such pairs of layers one above MN and one below MN experiencing same forces of elongation
and compression due to bending and each pair forms a couple.
Properties of Matter and Sound
1.2
The resultant of the moments of all these internal couples are called the internal bending
moment and in the equilibrium condition, this is equal to the external bending moment.
1.12
Bending Moment of a Beam
Consider the section PBCP’ (Fig. 1.10), the extended filaments lying above the neutral
axis MN are in state of tension and exert an inward pull on the filament adjacent to them towards
the fixed end of the beam. In the same way the shortened filaments lying below the neutral axis
MN are in a state of compression and exert an outward push on the filaments adjacent to them
towards the loaded end of the beam. As a result tensile and compressive stresses develop in the
upper and lower halves of the beam respectively and form a couple which opposes to bending of
the beam. The moment of this couple is called the moment of the resistance. When the beam is
in equilibrium position the bending moment and restoring moment or moment of resistance
should be equal. To find an expression for the moment of the restoring couple consider a fiber
DC at a distance r from the neutral axis MN as shown in Fig.1.11. Let the radius of curvature be
R and ф be the angle subtended by it at the centre of curvature. In unstrained position of the
beam, the length of the fiber DC = MN = Rф. In the strained position the length of the fibre,
DC = (R + r) ф.
P
A
M
B
P’
D
N
C
Load
Fig. 1.10 Calculation of bending moment of a beam
A
B
M
N
C
D
ф
Fig.1.11 Strained position
Properties of Matter and Sound
Strain in the fiber DC, =
or Strain 
1.3
Change in the length
Original length
(R  r)   R
r

R
R
(1)
i.e., strain is proportional to the distance from the neutral axis.
Let the area of the fiber be A and its neutral axis be at a distance r from neutral axis of
the beam and the strain produced be r/R. We have
Stress = Y x Strain = Y r / R
(2)
Hence, force on the area A
F = Y(r/R) x A
(3)
Therefore the moment of this force about MN
= Y(r/R) x A x r = Y A r2 / R
(4)
As the moment of the forces acting on both the upper and lower halves of the section are
in the same direction, the total moment of the forces acting on the filaments due to straining
 Y
ar2 Y
Y
 ar 2  I g
R
R
R
(5)
where Ig is the geometrical moment of inertia and is equal to AK2, A beding the total area of the
section and K being the radius of gyration of the beam
:. moment of the forces 
Y
Ig
R
(6)
In equilibrium bending moment of the beam is equal and opposite to the moment of
bending couple due to the load on one end.
:. Bending moment of the beam =
Y
Ig
R
(7)
The quantity YIg (=Y A K2) is called the flexural rigidity of the beam. Flexural rigidity is
defined as the bending moment required to produce a unit radius of curvature.
1.13
Uniform Bending
The beam is loaded uniformly on its both ends, the bent beam forms an arc of a circle.
The elevation in the beam is produced. This bending is called uniform bending.
Properties of Matter and Sound
1.4
Consider a beam (or bar) AB arranged horizontally on two knife – edges C and D
symmetrically so that AC = BD = a as shown in Fig. 1.12
Fig. 1.12 Uniform Bending
The beam is loaded with equal weights W and W at the ends A and B.
The reactions on the knife edges at C and D are equal to W and W acting vertical upwards.
The external bending moment on the part AF of the beam is
= W x AF – W x CF = W (AF – CF)
= W x AC = W x a
Internal bending moment =
(1)
YI g
R
(2)
where
Y - Youngs’ modulus of the material of the bar
Ig - Geometrical moment of inertia of the cross-section of beam
R - Radius of curvature of the bar at F
In the equilibrium position,
external bending moment = internal bending moment
Wa 
YI g
R
(3)
Since for a given value of W, the values of a, Y and Ig are constants, R is constant so that
the beam is bent uniformly into an arc of a circle of radius R.
CD = l and y is the elevation of the midpoint E of the beam so that y = EF
Then from the property of the circle as shown in Fig. 1.13
Properties of Matter and Sound
F
C
F
1.5
y
C
D
l/2
E
O
R
Fig. 1.13 Circle Property
EF (2R – EF) = (CE)2
(4)
l
2
y (2R – y) =   2
y 2R =
y=
or
l2
( y2 is negligible)
4
(6)
l2
8R
(7)
1 8y

R l2
From (3) and (8), Wa =
or
(5)
Y
(8)
8y
YI g
2
W l2 a
8I g y
If the beam is of rectangular cross-section, I g 
(9)
bd 3
, where b is the breath and d is the
12
thickness of beam. If M is the mass, the corresponding weight W = Mg
Properties of Matter and Sound
Hence Y 
3 Mgl 2 a
2 bd 3 y
1.6
(10)
from which Y the Young’s modulus of the material of the bar is determined.
Experiment
A rectangular beam (or bar) AB of uniform – section is supported horizontally on two
knife – edges A and B as shown in Fig. 1.14
Fig. 1.14 Uniform bending experiment
Two weight hangers of equal masses are suspended from the ands of the beam. A pin is
arranged vertically at the mid-point of the beam. A microscope is focused on the tip of the pin.
Initial reading of the microscope in the vertical scale is noted.
Equal weights are added to both hangers simultaneously and the reading of the
microscope in the vertical scale in noted.
The experiment is repeated for decreasing order of magnitude of the equal masses.
The observations are then tabulated and the mean elevation (y) at the mid point of the bar
is determined.
Table 1.3 Uniform bending experimental observation
Microscope reading elevation
Mean elevation(y) for
Load in Kg
Load increasing
W
W+50 gms
W+100 gms
W+150 gms
W+200 gms
W+250 gms
Load decreasing
Mean
a load of M.
Properties of Matter and Sound
1.7
The length of the bar between the knife edges ‘l’ is measured. The distance of one of the
weight hangers from the nearest knife edge p is measured. The breadth (b) and thickness (d) of
the bar are measured using vernier calipers and screw gauge.
The young’s modulus of the material of the beam is determined by the relation
Y
3 M g al2
2 bd 3 y
Nm-2
Worked Example 1.11: Uniform rectangular bar 1 m long 2 cm broad and 0.5 cm thick is
supported on its flat face symmetrically on two knife edges 70 cm
apart. If loads of 200 g are hung from the two ends, find the elevation
at the center of the bar. Young’s modulus of the material of the bar is
18 x 1010 Pa.
The distance between the nearer knife edge and the point of suspension
a=15x10-2 m
Elevation at the centre,
y
3 M g al2
2 Yb d 3
3 200 10 3  9.8  15 10 2  0.7 2

2 18  1010  2  10 2  (0.5 10 2 ) 3
= 4.802 x 10-4 m