Chapter 6  Bending
I
My−
=σ
EI
M
=
ρ
1
Objectives
 Develop relationship between stress distribution, σ, and applied bending moment, M.
 Develop relationship between radius of curvature, ρ, and applied bending moment, M.
 Solve problems and calculate stress and strain for beams in pure bending
 Calculate moment of inertia for various beams
M
M
Prismatic Members in Pure Bending
So far we have studied bars under axial or torsional loading. Now we will study prismatic
members in bending.
Pure Bending 
Cut bar at arbitrary location, C
M is the _____________________________, which is the result of ______________
The moment here is _______________
In general beam bending, the internal bending moment
Pure bending is a special case where M is
What is the only nonzero stress in pure bending?
Deformations in a Symmetric Member in Pure Bending
In pure bending, where M and M' act on a plane of
symmetry, what is the shape that all horizontal lines
(such as AB on the top surface) will be deformed
into?
Does the length of line AB increases or decrease?
Does the length of line A'B' increases or decrease?
What are σ
x
and ε
x
on the neutral surface?
Neutral axis coincides with
A
A'
B'
B
Derivation of ε
x
Line DE is
Let ρ be
Let y be
Original length of DE and JK are both
New length of arc JK
Deformation of arc JK
Strain of arc JK is
Longitudinal, normal strain varies linearly with the distance y from the neutral axis.
Stresses and Deformations in the Elastic Range
From geometry, strain is defined as
Distance from neutral axis
Radius of curvature of neutral axis
In the case where bending moment, M, about the zaxis causes stresses
yieldx
σ
σ
<
everywhere in
the beam
We can relate the stress due to bending and the moment by
Moment of inertia, I, of the beam crosssection
1.
Can find for simple shapes using formulas in Table 81 (page 419).
2.
We will find the centroid and use the parallel axis theorem find the moment of interia of the
cross section.
M M
x
y
Example of Pure Bending  BC is in pure bending
Entire structure
Consider section BC.
Equilibrium
10 m
2
m
2
m
30 kN 30 kN
A
D
B
C
Example
Given  A bending moment of M = 4000 lbft. is applied to a "T" shaped beam
Find  a) the location of the neutral axis
b) the moment of inertia with respect to the neutral axis
c) the max tensile and compressive stresses
d) the curvature of the beam
end side
M
Example
The channel section shown is subjected to a bending moment of M = 5000 Nm. Determine the
maximum tensile bending stress, the maximum compressive bending stress and the curvature.
Given
Find
Assume
Analysis
160 mm
100 mm
10 mm
M
Chapter 6 (cont.)  Transverse Loading of Beams
It
VQ
=τ
I
VQ
q =
Objectives

Determine shear force, V(x), on horizontal beam under transverse load

Calculate the shear flow, q, in a beam under transverse load

Calculate the shear stress,
τ
, in a beam under transverse load
Transverse Loading of Beams
Typically in bending, internal shear force, V(x) exists and internal bending moment, M(x) varies
along the length of the beam.
When a beam is subjected to vertical loads
Consider a cantilever beam, length L
FBD Equilibrium
V(x) and M(x)
Moment is moment arm crossed with force
Shear force is
P
x
y
Goal here is to determine
τ
xy
vs. y for a given crosssection
Determination of Shear on Horizontal Plane
Need to make assumption regarding normal stresses
Returning to cantilever beam
Consider chunk of cantilever beam
Q is first area moment
q is horizontal shear per unit length
x
y
P
z
y
Shear Stress
How does shear flow relate to
τ
xy
?
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