Theory of Simple Bending
What is simple bending?
Consider a basic example:
Which way is it easier to bend the rule?
Why?
What is a Section?
We are referring to the shape of a Cross-Section
Remember - we considered the cross-section
area when calculating the direct stress (σ).
Properties of Sections
We need to understand how the distribution of
area, i.e. the cross-section shape, affects the
ability to resist externally applied loads.
First of all we need to describe the distribution of
area about an axis.
First Moment of Area
First Moment of
Area
Centroid
(“Centre of Area”)
Simple shapes
For regular, symmetrical shapes the centroid (also
known as the ‘Centre of Area’) lies at the geometrical
centre :
For non-symmetrical sections we need to work it out.
Assumptions made in
Simple Bending Theory
1. The beam is slender (it’s sectional
dimensions are small compared to
length).
2. Initially straight with uniform cross section.
3. Material is homogeneous, isotropic and
linearly elastic.
4. The beam bends in a single plane.
5. Plane sections remain plane after
bending.
6. Lateral (Poisson's) contraction is
negligible.
When we talk about
7. The radius of curvature is large.
‘beams’ we will be implying
these assumptions
Consider a longitudinal
section of a beam.
Consider two crosssections, AC and BD ,
close together.
After bending length AB
will extend to A’B’, and
length CD will reduce to
C’D’.
A
B
Longitudinal
Axis
C
D
A’
B’
C’
D’
If AB increases in
length and CD
decreases…
A
NS
…then there must be a
transverse plane that
remains unchanged?
This plane is called the
‘neutral plane’ (or the
‘neutral surface’).
NS
B
F
NS
G
C
D
A’
B’
F’
G’
NS
C’
D’
We will now consider a
small element.
NS
It is defined by HJ at
distance y from the NS.
NS
A
B
H
J
F
y
G
C
D
A’
B’
H’
J’
F’
G’
NS
y
NS
C’
D’
NS
NS
A
B
H
J
F
y
G
C
D
A’
B’
H’
J’
G’
F’
y
NS
D’
C’
θ
NS
R
A
B
σ
y
NS
C
NA
NS
D
NA
Tensile
NA
NA
Compressive
Tensile
NA
=
NA
Compressive
is identified as the
‘Second Moment of Area’ (I)
Euler-Bernoulli Theory of Simple Bending
Recalling that
and,
we can rearrange...
We can rearrange our previous expressions to
obtain:
Moments of Area
1st moment of area
describes the
distribution of area
about an axis.
If a cross section is
symmetrical, it is
straight forward to
determine the centroid
position, and thus the
NA by observation.
Doubly symmetric
Singly symmetric
Moments of Area
To apply the 2nd moment of
area (I ) in the bending
equation we need to
evaluate it.
 For a given cross section
 About the neutral surface
(‘neutral axis’ NA)
If the section is symmetrical
about it's NA, a simple
solution exists.
d
x
x
b
2nd Moment of Area (I)
h/3
d x
x
b
x
h x
b
x
x
D
Hollow Sections
If we consider two different hollow sections:
B
d
x
x
x
x
b
D
Di
Do
2nd moments of area are summative if (only if) they are
about the same axis.
Then, we can add them together to find the 2nd moment
of area for the whole section.
Parallel Axis Theorem
x
x
h
NA
NA
Parallel Axis Theorem
2nd moments of area are
summative if (only if) they are
about the same axis.
In this example we can transfer
the 2nd moment of area of each
of the rectangles about its own
centroidal axis (s1 and s2
respectively) to the centroidal
axis of the overall T-section,
and then add up.
s1
NA
NA
s2
What is the Perpendicular Axis Theorem ?
For example, did you observe that for a
circular section, J = 2I ?
(We have already used J in the Simple
Torsion Theory)