PH8151 Engineering Physics Chapter 1

40 CHAPTER 1. PROPERTIES OF MATTER

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

Srectangle=

Ig

c =

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 (1.47)

Since,A=fir^2 andc=rfor a cylindrical beam of radiusr, its section modulus is

Scircle=

Ig

c =

fir^4

4

r =

(fir^2 )r

4 =0.^25 Ar

Learning Resource : Understanding Stresses in Beams

Yet another very informative presentation fromThe Ecient

EngineerYoutube channel - about bending moment, uniform

(pure) bending, stresses in beams, section modulus, etc. and

applications.

URL:https://youtu.be/f08Y39UiC-o

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 calculation 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 bend-
ing of various cross-sectional
shapes. In general, a beam
is more ecient if the ma-
terial is located farther from
the neutral axis, where it is
more highly stressed and pro-
vides a larger section modu-
lus.

As given in the equation
(1.47), the section modulus of rect-
angular 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 height
dis 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.

In order to maximise the section modulus for a given cross-sectional area A and heightd,wedistribute
each halfA/ 2 of the area at a distanced/ 2 from the neutral axis, as shown in Figure1.36(b). The

40