Beams: Shear flow, thin walled members

8/10/2019 Beams: Shear flow, thin walled members

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A. J. Clark School of Engineering Department of Civil and Environmental Engineering A. J. Clark School of Engineering Department of Civil and Environmental Engineering A. J. Clark School of Engineering Department of Civil and Environmental Engineering A. J. Clark School of Engineering Department of Civil and Environmental Engineering A. J. Clark School of Engineering Department of Civil and Environmental Engineering

Third Edition

LECTURE

15

6.6 6.7

Chapter

BEAMS: SHEAR FLOW, THIN

WALLED MEMBERS

by

Dr. Ibrahim A. Assakkaf

SPRING 2003ENES 220 Mechanics of Materials

Department of Civil and Environmental Engineering

University of Maryland, College Park

LECTURE 15. BEAMS: SHEAR FLOW, THIN-WALLED MEMBERS (6.6 6.7) Slide No. 1ENES 220 Assakkaf

Shear on the Horizontal Face of a Beam Element

Consider prismatic beam

For equilibrium of beam element

( )

=

+==

CD

ADDx

dAyI

MMH

dAHF 0

xVxdx

dMMM

dAyQ

CD

A

==

=

Note,

flowshearI

VQ

x

Hq

xI

VQH

==

=

=

Substituting,


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Shear on the Horizontal Face of a Beam Element

flowshearI

VQ

x

Hq ==

=

Shear flow,

where

sectioncrossfullofmomentsecond

aboveareaofmomentfirst

'

2

1

=

=

=

=

+AA

A

dAyI

y

dAyQ

Same result found for lower area

HH

QQ

qI

QV

x

Hq

=

=

=+

=

=

=

axisneutralto

respecthmoment witfirst

0


Shearing Stress in Beams

Example 16

The transverse shear Vat a certainsection of a timber beam is 600 lb. Ifthe beam has the cross section

shown in the figure, determine (a) thevertical shearing stress 3 in. belowthe top of the beam, and (b) themaximum vertical stress on the crosssection.


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Example 16 (contd)

12 in.8 in.

4 in.

8 in.



Example 16 (contd)

From symmetry, the neutral axis is located

6 in. from either the top or bottom edge.

( ) ( )

( )( ) ( )( )[ ]

( )( ) ( )( )[ ] 3

3

3

4

33

in0.1124222528

in0.945.3212528

in3.98112

84

12

128

=+=

=+=

==

NAQ

Q

I

8 in.

4 in.

8 in.

12 in.

8 in.

4 in.

8 in.

N.A.

3 in.

( )( )

( )( )

psi2.17143.981

1126000)b(

psi7.14343.981

946000)a(

maxmax

3

3

===

===

It

VQ

It

VQQ


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Longitudinal Shear on a Beam

Element of Arbitrary Shape

Consider a box beam obtained bynailing together four planks as shown in

Fig. 1.

The shear per unit length (Shear flow) q

on a horizontal surfaces along which the

planks are joined is given by

flowshearI

VQq == (1)




Figure 1


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But couldBut could qq be determined if the planksbe determined if the plankshad been joined alonghad been joined along vertical surfacesvertical surfaces,,

as shown in Fig. 1b?as shown in Fig. 1b?

Previously, we had examined the

distribution of the vertical components

xyof the stresses on a transverse

section of a W-beam or an S-beam as

shown in the following viewgraph.


Shearing Stresses xy in Common

Types of Beams For a narrow rectangular beam,

A

V

c

y

A

V

Ib

VQxy

2

3

12

3

max

2

2

=

==

For American Standard (S-beam)

and wide-flange (W-beam) beams

web

ave

A

V

It

VQ

=

=

max


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Longitudinal Shear on A Beam


But what about theBut what about the horizontalhorizontalcomponentcomponent xzxz of the stresses in theof the stresses in the

flanges?flanges?

To answer these questions, the

procedure developed earlier must be

extended for the determination of the

shear per unit length q so that it will

apply to the cases just described.


Longitudinal Shear on a Beam Element

of Arbitrary Shape We have examined the distribution of

the vertical components xy on a

transverse section of a beam. We

now wish to consider the horizontal

components xz of the stresses.

Consider prismatic beam with an

element defined by the curved surface

CDDC.

( ) +==a

dAHF CDx 0

Except for the differences in

integration areas, this is the same

result obtained before which led to

I

VQ

x

Hqx

I

VQH =

==


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Example 17

A square box beam is constructed from

four planks as shown. Knowing that the

spacing between nails is 1.75 in. and thebeam is subjected to a vertical shear of

magnitude V= 600 lb, determine the

shearing force in each nail.

SOLUTION:

Determine the shear force per unit

length along each edge of the upper

plank.

Based on the spacing between nails,

determine the shear force in each

nail.


Example 17 (contd)

For the upper plank,

( )( )( )3

in22.4

.in875.1.in3in.75.0

=

== yAQ

For the overall beam cross-section,

( ) ( )

4

31213

121

in42.27

in3in5.4

=

=I

SOLUTION:

Determine the shear force per unit

length along each edge of the upper

plank.

( )

lengthunitperforceedgein

lb15.46

2

in

lb3.92

in27.42

in22.4lb6004

3

=

==

===

qf

I

VQq

Based on the spacing between nails,

determine the shear force in each

nail.

( )in75.1in

lb15.46

== lfF

lb8.80=F


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Shearing Stress in Thin-Walled

Members

It was noted earlier that Eq. 1 can beused to determine the shear flow in an

arbitrary shape of a beam cross section.

This equation will be used in this section

to calculate both the shear flow and the

average shearing stress in thin-walled

members such as flanges of wide-

flange beams (Fig. 2) and box beams orthe walls of structural tubes.



Members

Figure 2

Wide-Flange Beams


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Members

Consider a segment of a wide-flangebeam subjected to the vertical shear V.

The longitudinal shear force on the

element is

x

I

VQH = (2)



Members

Figure 3


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Members

The corresponding shear stress is

Previously found a similar expression

for the shearing stress in the web

It

VQ

xt

Hxzzx =

= (3)

ItVQ

xy = (4)



Members

The variation of shear flow across the

section depends only on the variation of

the first moment.

I

VQtq ==

For a box beam, q grows smoothly from

zero at A to a maximum at Cand Cand

then decreases back to zero at E.

The sense of q in the horizontal portions

of the section may be deduced from the

sense in the vertical portions or the

sense of the shear V.


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Members

For a wide-flange beam, the shear flowincreases symmetrically from zero at A

and A, reaches a maximum at Cand the

decreases to zero at Eand E.

The continuity of the variation in q and

the merging of q from section branches

suggests an analogy to fluid flow.



Members

Example 18Knowing that the

vertical shear is 50

kips in a W10 68

rolled-steel beam,determine the

horizontal shearing

stress in the top

flange at the point a.


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Members

Example 18 (contd)

SOLUTION:

For the shaded area,

( )( )( )3in98.15

in815.4in770.0in31.4

=

=Q

The shear stress at a,

( )

( )( )in770.0in394in98.15kips50

4

3

==

It

VQ

ksi63.2=

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Beams: Shear flow, thin walled members