PH8151EPUnit1

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.

From equation (1.29),
1
R

=

(BM)

YIg

(1.44)

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^5 (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

(1.45)

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 (1.45) shows that‡maxis inversely
proportional to the section modulus of the beam. In other words, for a given external force, a beam with
higher section modulus will develop less stress than a beam with lower section modulus. Hence, beams
with higher section moduli have better load bearing ability.

The overall process of designing a beam requires consideration of numerous factors, such as the
materials, loads, environmental conditions, and type of structure. However, in many cases, the task
eventually reduces to the selection of a particular beam shape and size, subject to the constraint that the
actual stresses in the beam must not exceed the allowed maximum stresses. Such suitable beam shapes
and sizes can be selected by knowing the required section modulus from the expression (1.45) which can
be written in the form
S=(BM)max
‡max
=(BM)max
‡w
(1.46)

In this equation,‡max=‡wis the working stress given by expression (1.3), which is based upon the
properties of the material and the magnitude of the desired factor of safety. To ensure that the allowable
stresses are not exceeded, the selected beam must have a cross-sectional area that provides a section
modulus at least as large as that obtained from equation (1.46).

(^5) Note that,c=d/ 2 for a rectangular beam whereasc=rfor a cylindrical beam of radiusr.

PH8151 39 LICET