28 CHAPTER 1. PROPERTIES OF MATTER
1.9 Bending Moment of a Beam
Consider a beam whose neutral filament MN is bent into an arc of radiusRwith centre
at O as shown in Figure1.24. Then,
Stress due to bending in beams
x
The neutral axis is an axis
in the cross section of a
beam along which there are
no longitudinal stresses or
strains.
Radius of curvature of the
Neutral surface = R
Strain in layer EF = x/R
Stress due to bending at
layer EF= σ
E F
R
Young’s modulus = Y = (Stress due to bending at layer EF) / (Strain in layer EF)
Or Y = σ /(x/R) = (σR) /x
Hence, stress due to bending = σ = (Yx)/R
O
P x
P’ Q’
Q
N N’θ
δAMDCompressive
forceTensile
forceFigure 1.24:A beam whose neutral filament is bent into an arc of radiusROP=OQ=R and \POQ=◊such thatPQ=R◊Consider another filament DE at a distancexabove MN and two pointsPÕandQÕon DE
as shown in Figure1.24. Then,
PÕQÕ=(R+x)◊
The strain at the layer DE =change in length
original length=
PÕQÕ≠PQ
PQ
So,linear strain =[(R+x)◊]≠(R◊)
R◊=
x◊
R◊=
x
R(1.25)
Since,stress = (YoungÕs modulus(Y)◊strain), we have from equation (1.25),
stress =Y(^3) x
R
4
(1.26)
Hence, the force acting on the elemental area”Aof the cross-section of the beam which
is at a distancexfrom the neutral axisNNÕis
stress ◊ area =
Yx
R
◊”A
Moment of the force”Fabout the neutral axisNNÕis
(Force)◊(‹ distance from axis to line of action of force) =Yx”A
R◊x=Y
R
(”Ax^2 )