Bending

Mechanics

Jul 18, 2012 (5 years and 10 months ago)

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Chapter 6 - Bending

I
My−

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 non-zero 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 z-axis 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 cross-section
1.
Can find for simple shapes using formulas in Table 8-1 (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 lb-ft. 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 N-m. 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

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 cross-section

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
?