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 E cient
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 e ciency in bend-
ing of various cross-sectional
shapes. In general, a beam
is more e cient 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 e cient 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 insu cient 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