Transcript Chapter 6 Section 3,4 Bending Deformation, Strain and

Chapter 6
Section 3,4
Bending Deformation, Strain and
Stress in Beams
6.2 Bending Deformation and Strain
Key Points:
1. Bending moment
causes beam to
deform.
2. X = longitudinal axis
3. Y = axis of symmetry
4. Neutral surface –
does not undergo a
change in length
Key Points:
1. Internal bending moment causes beam to deform.
2. For this case, top fibers in compression, bottom in
tension.
Key Points:
1. Neutral surface – no change in length.
2. All cross-sections remain plane and perpendicular to longitudinal axis.
3. Neglect deformation of cros section within its own plane.
Radius of
curvature
Length
decreases
Length =
constant
s's
(   y )  
y
  lim
 lim

s 0
s 0
s


 
y

Says normal strain is
linear
Maximum at outer surface
(where y = c)
 y
    max
c
6.2 Bending Stress – The Flexure Formula
What about Stress????
Recall from section 6.1:
 y
    max
c
Therefore, it follows that
 y
    max
c
Sum moments about cut:
 y
M   ydF   y (dA)   y  maxdA
c
A
A
A 
M
 max
c
2
y
 dA
A
This is the
moment of
inertia, I
The Flexure Formula:
Internal bending
moment, lb-in
Max bending stress,
psi
Mc
 max 
I
Or in general:
Distance from NA to
outer fiber, in
Moment of inertia, in4
My

I
Examples:
• Find maximum moment
• Find area properties, I and c
• Calculate stress
WHERE IS
BENDING STRESS
MAXIMUM???
Answer:
•Outer surface
(furthest away from
Neutral Axis)
•Value of x along
length where moment
is maximum!!
Example: The T-shape beam is subjected to the loading below.
1. Draw shear and moment diagram. Identify location and magnitude of Mmax.
2. Determine location and magnitude of maximum bending stress and draw stress
profile. Is the beam safe if the material is aluminum w/ y = 15 ksi?
3. What is the largest internal moment the beam can resist if allow = 2 ksi?
45K
Statics: Example 1 - Pliers
Given: Typical
household pliers as
shown.
Find: Force applied to
wire and force in pin
that connects the two
parts of the pliers.
Do this for homework.
See solution Link
Side: what is the shear
stress in pin and bending
stress in handle? SofM
Statics: Example 2 – Crane Structure
Given: Crane structure
as shown.
Find: Forces and
FBD’s for cables A-B
and A-E, boom DEF
and post AFC.
Do this for homework.
See solution Link
Side: what is the normal stress
in cables (average normal
only) and normal stress in
boom and post (combined
loading)? SofM
Example 4: Determine the resultant internal
loadings acting on the cross sections located
through points D and E of the frame. (1-114)