Mechanics of Materials chp9
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Transcript of Mechanics of Materials chp9
1
Chapter-9
Transverse Shear
In this chapter we will develop a method of finding the
shear stress in a beam.
Also, shear flow, will be discussed and examples will be
worked.
V: result of a transverse
shear-stress distribution that
acts over the beam’s cross
section.
2
Shear in a beam subject to bending may be longitudinal and transverse.
Longitudinal can be illustrated by the bending beam below:
If the boards are bonded then shear
stresses build up and the cross section
warps. This condition violates our
assumption of sections remaining plane
when bent but warping is relatively small
especially for a slender beam.
We will now use the assumptions or
homogeneity and prismatic cross section to
develop a shear formula similar to the
flexure formula. . .
3
It is important to recall that shear stress is
complimentary meaning transverse and
longitudinal shear stresses are numerically equal.
Shear stress
tI
QV
.
.
The shear stress in the member at the point located
y’ from the neutral axis. This stress is assumed to be
constant and therefore averaged across the width t
of the member.
The Internal resultant shear force, determined from the
method of sections and the equations of equilibrium.
V
The moment of inertia of the entire cross sectional area computed
about the neutral axis.
I
The width of the members cross sectional area, measured at the point
where is to be determined.
t
Where A’ is the top or bottom portion of the member’s
cross sectional area, defined from the section where t
is measured, and is the distance to the centroid of
A’, measured from the neutral axis 'y
4
Shear formula
tI
QV
.
.
It is necessary that the material behave in a linear elastic
manner and have a modulus of elasticity that is the same
in tension as it is in compression.
5
Shear Stresses in Beams
Applying the shear formula for common beam cross-sectional situations:
Rectangular: tI
QV
.
.
12
3hbI
t = b
byh
byh
yh
yAyydAQA
22
'42
1
222
1'''
2
2
3 4
6
.
.y
h
bh
V
tI
QV
6
2
2
3 4
6
.
.y
h
bh
V
tI
QV
This result indicates that the shear-stress distribution over the cross section is
parabolic.
The intensity varies from zero at the top and bottom, y = ± h/2, to a maximum value at
the neutral axis, y = 0. Specifically, since the area of the cross section is A=b.h, then at
y=0 we have:
So that it can be shown that integrating the shear stress, τ , over the entire cross-
sectional area A yields the shear force V.
7
Wide Flange Beams:
A wide flange beam consists of two flanges and a web. An analysis of the shear in a
wide flange beam results in the illustration below:
8
9
Solution: y
z
w=125mm From the bottom:
A
Ayy
.~
Distance from the bottom to
the C.G of each element.
A A A
z
A d2
tI
QV
.
.
10
yz
C
A
C
B
y
y
A
A
'y : Distance from the center of gravity C to the center of gravity of the studied
element.
z
z
11
12
Support reactions:
Solution
13