Unit 1 Engineering Physics

38 CHAPTER 1. PROPERTIES OF MATTER

that produces the depression (M) = 0.1 kg. Then, Young’s Modulus is given by,

Y=

Mgl^3

4 bd^3 y

=

(0. 1 kg)(9.8m/s^2 )(1m)^3

4(5◊ 10 ≠^3 m)(5◊ 10 ≠^3 m)^3 (1. 96 ◊ 10 ≠^3 )m

=20.204 GPa

1.12 Stress due to bending in beams

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

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

N N’

δA

d

b

b

d

d/2

F

Figure 1.35:Stress due to bending of beams.

There are no longitudinal stresses or strains at the neutral axisNNÕ(Figure1.35)of
a bent beam. If radius of curvature of the neutral surface =R, then strain in layer EF
at a distancexfrom the neutral axisNNÕ=x/R. Stress due to bending at layer EF is

‡(x) = (YoungÕs Modulus)◊(Strain in layer EF) =Y◊

(^3) x
R
4
From equation (1.28),
1
R

=

(BM)

YIg

(1.43)

Hence,stress due to bending,‡(x)=

(BM)x

Ig

Therefore, stress is zero at the neutral axis (sincex=0) and as we move away from the
neutral axis, the stress increases and reach maximum at the top layer and at the bottom
layer of the beam. In other words, the layers near the neutral layer experience very little
stress while the layers at the top and bottom have the highest stress during bending.

If(BM)maxis the maximum possible bending moment for a beam and ifcis the distance from neutral
axis to the farthest layer of the beam^6 (i.e., the topmost and bottommost sides in the case of rectangular
beams), then the maximum normal stress at a cross-section is given by

‡max=(BM)maxc

Ig

=(BM)max

(Ig/c)

=(BM)max

S

(1.44)

In the above equation,(Ig/c)©Sis called theSection Modulusof the cross-sectional area. The section
modulus has unit m^3 since the unit ofIgis m^4. The above expression (1.44) shows that‡maxis inversely

(^6) Note that,c=d/ 2 for a rectangular beam whereasc=rfor a cylindrical beam of radiusr.

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