Unit 1 Engineering Physics

1.13 I-shaped Girders

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 (1.44) which can
be written in the form

S=(BM)‡ max

max

=(BM)‡max

w

(1.45)

In this equation,‡max=‡wis the working stress given by expression (1.2), 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 (1.45).

For a bar of rectangular cross-section with thicknessdand breadthbas shown in Figure1.35,
distancecof the top (and bottom) layer from neutral axisNNÕisd/ 2. Therefore, the section modulus
of a rectangular beam is

Srectangle=Icg=

(^1) bd 3
12
2
!d
2
" =bd
2
6 =
(bd)d
6
Here,bd=Ais the area of the rectangle. Therefore,
Srectangle=Ad 6 =0. 167 Ad (1.46)
Since,A=fir^2 andc=rfor a cylindrical beam of radiusr, its section modulus is
Scircle=Icg=
fir^4
4
r =
(fir^2 )r
4 =0.^25 Ar

1.13 I-shaped Girders

A/2

A/2

d/2

d/2

(a) (b)

d/2

d/2

b

(c)

A = bd

Figure 1.36: (a)Section modulus cal-

culation of a solid rectangular beam and

(b)Ideal(but impossible!) cross-section

with highest section modulus for a given

areaA.

Let us now consider the relative eciency in
bending of various cross-sectional shapes. In
general, a beam is more ecient if the material
is located farther from the neutral axis, where
it is more highly stressed and provides a larger
section modulus.

As given in the equation (1.46), the section modulus
of rectangular beam (Figure1.36(a)) is

Srectangle=0. 167 Ad

This equation shows that a rectangular cross section
of given area becomes more ecient as the heightdis
increased (and the widthbis decreased to keep the area
constant). However, there is a limit to the increase in height, because the beam becomes laterally unstable
when the ratio of height to width becomes too large. Thus, a beam of very narrow rectangular section
may fail because of lateral (sideways) buckling rather than insucient strength of the material.

PH8151 39 LICET