EGR 236Lecture22Transvers Shear (1)

7/28/2019 EGR 236Lecture22Transvers Shear (1)

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EGR 236 Properties and Mechanics of Materials Spring 2013

Lecture 22: Transverse Shear Stress in Beams

Today:

-- Homework questions:-- New Topics:

-- Shear Stress in beams

-- Homework: Read Section 7:1-3

Work Problems from Chap 7: 13, 16, 24, 30 (modified)

Following today's class you should be able to:

-- explain how shear stress is set up in a beam subjected to shearing loads

-- be able to identify the location of the largest shear stress in a beam

-- be able to calculate the shear stress in straight beams of symmetric cross

section.

Shear Stress in Beams:

So far you have learned how to determine the stress caused by the internal

bending moment that is set up in beams subjected to loads. While usually not

as large as the bending stresses, there are also shear stresses set up by the

internal shearing force, V, that may contribute to the failure of the beam.

To calculate Bending Stress: To calculate Shear Stress

in a beam: in a beam:

Mc

I =

VQ

It =

where M= internal moment V= internal shearing force

c = distance from NA Q = 1st moment of area

I= moment of inertia t= width of section

w

FB

L - xx

M

V VFA


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Yard Stick Demo:

Draw picture of

Slats working as a whole Slats working individually:

To cause the body to deform in the fashion shown below, how must shear

stress be applied along the length of the beam:

F


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Shear Stress Distribution over the Cross Section:

In the Equation:

VQ

It =

Q represents the 1st moment of the area above the

line of interest about the Neutral Axis.

Q A y=trepresents the width of material that separates the areaA from the rest of the

material of the cross section. In this case, tis equal to the width b.

Notice that for typical cross sections, the maximum shear stress occurs at the

Neutral Axis.

To calculate this for a rectangular cross section:

Q A y ( bh )( h ) bh= = = 21 1 1

2 4 8

I bh= 31

12

t b= so

max

rect

V ( bh )VQ V V

It bh A( bh )( b )

= = = =

2

3

1

3 38

1 2 2

12

V

b

h

b

NAyA

yh

A

NA

b


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To calculate this for a circular cross section: d = 2r

r

Q A y ( r )( ) r

= = =2 31 4 2

2 3 3

I r= 414

t r= 2 so

max

circle

V ( r )VQ V V

It r A( r )( r )

= = = =

3

24

2

4 43

1 3 32

4

To calculate shear stress across the section of an I-beam:

(using an approximate method which assumes all mass is concentrated in the flange)

I beam i beamQ A y ( A )( d ) A d = =1 1 1

2 2 4

I beam I beamI A ( d ) A d =2 21 1

2 4

webt t= so

I beam

max

webI beam

V ( A d )VQ V V

It d t AA d ( t )

= = =2

1

4

1

4

d

NA

y

A

d


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Limitations to the Transverse Shear formula:

The shear formula assumes presumes that the shear stress has a parabolic

distribution up and down the face of the section. From side to side the shear

stress is assumed constant.

This is not quite true, but can be used as an approximate formula is the section

is relatively high compared to the width.

Rule of thumb: Valid for rectangular sections if

VQ

It = b h

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EGR 236Lecture22Transvers Shear (1)