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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.

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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. . .

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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

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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.

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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.

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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:

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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

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Support reactions:

Solution

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