# Stress in Beams

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15 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

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Chapter 6

Section 3,4

Bending Deformation, Strain and
Stress in Beams

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

6.2 Bending Deformation and Strain

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.

y
y
s
s
s
s
s

)
(
lim
'
lim
0
0
Radius of
curvature

Length =
constant

Length
decreases

y

Says normal strain is
linear

Maximum at outer surface
(where y = c)

max

c
y
max

c
y
6.2 Bending Stress

The Flexure Formula

What about Stress????

Recall from section 6.1:

Therefore, it follows that

max

c
y
dA
y
c
M
dA
c
y
y
dA
y
ydF
M
A
A
A
A

2
max
max
)
(

Sum moments about cut:

This is the
moment of
inertia, I

The Flexure Formula:

I
Mc

max

I
My

Or in general:

Max bending stress,
psi

Internal bending
moment, lb
-
in

Distance from NA to
outer fiber, in

Moment of inertia, in
4

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.

Side: what is the shear
stress in pin and bending
stress in handle?
SofM

Do this for homework.

See solution Link

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.

Side: what is the normal stress
in cables (average normal
only) and normal stress in
boom and post (combined
loading)?
SofM

Do this for homework.

See solution Link

Example 4:

Determine the resultant internal
loadings acting on the cross sections located
through points D and E of the frame. (1
-
114)