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MECHANICS OF

MATERIALS

CHAPTER

6 Shearing Stresses in Beams and Thin-Walled MembersPROF. AHMED B. SHURAIM

STRUCTURAL ENGINEERING

CIVIL EN GINEERING DEPT.

KING SAUD UNIVERSITY

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6 - 2

Introduction

Distribution of normal and shearing

stresses satisfies

Transverse loading applied to a beam

results in normal and shearing stresses intransverse sections.

When shearing stresses are exerted on the

vertical faces of an element, equal stressesmust be exerted on the horizontal faces

Longitudinal shearing stresses must exist

in any member subected to transverse

loading.

( )

( ) !!

!

!!

====

===== ===

xzxzz

xyxyy

xyxzxxx

yMdAF

dAzMVdAF

dAzyMdAF

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shear formula

6 - 3

olving for

oting that

hear formula

Area#

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&etermination of the Shearing Stre'' in a(eam

The averageshearing stress on the horizontal

face of the element is obtained by dividing theshearing force on the element by the area of

the face.

"n the upper and lo#er surfaces of the beam,

yx$ !. %t follo#s that xy$ ! on the upper and

lo#er edges of the transverse sections.

%f the #idth of the beam is comparable or large

relative to its depth, the shearing stresses at

edges &D'andD() are significantly higher than

at the middle &D.

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the di'tri)ution of 'hearing 'tre''e' in a tran'ver'e 'ection of arectangular )eam i'parabolic. *a+ingy # 0 in E,. 6./ e o)tain thevalue of the ma1imum 'hearing 'tre'' in a given 'ection of a narrowrectangular beam:

6 - 6

Shearing Stre''e' xyin *ommon Types of +eams

or -merican tandard &/beam)

and #ide/flange &W/beam) beams

web

ave

A

V

It

VQ

=

=

max

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E1am4le 6.05

A )eam i' made of three4lan+' nailed together.noing that the '4acing

)eteen nail' i' 2% mm andthat the vertical 'hear in the)eam i'V# %00 ! determine the'hear force in each nail.

"L0T%"12

Determine the horizontal force per

unit length or shear flo# qon the

lo#er surface of the upper plan3.

*alculate the corresponding shear

force in each nail.

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E1am4le 6.05

"L0T%"12

Determine the horizontal force per

unit length or shear flo# qon the

lo#er surface of the upper plan3.

*alculate the corresponding shear

force in each nail for a nail spacingof (4 mm.

( ) ( )

( ) ( )

( ) ( )

( ) ( )

56

(

'('

'(

'

6

m'!(!.'6

7m!6!.!m'!!.!m!(!.!

m!(!.!m'!!.!8(

m'!!.!m!(!.!

m'!'(!

m!6!.!m'!!.!m!(!.!

=

+

+

=

=

==

I

yAQ

m19!5

m'!'6.(!

)m'!'(!)&14!!&

56/

6

=

==

I

VQq

mNqF 9!5)&m!(4.!&)m!(4.!& ==

16.:(=F

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Sam4le Pro)lem 6.2

A tim)er )eam i' to 'u44ort thethree concentrated load''hon. noing that for thegrade of tim)er u'ed

determine the minimumre,uired de4th dof the )eam.

"L0T%"12

Develop shear and bending moment

diagrams. %dentify the maximums.

Determine the beam depth based on

allo#able normal stress.

Determine the beam depth based on

allo#able shear stress.

;equired beam depth is equal to the

larger of the t#o depths found.MPaMPa

allall

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Sam4le Pro)lem 6.2

"L0T%"12

Develop shear and bending momentdiagrams. %dentify the maximums.

mkNM

kNV

.:4.'!

4.'5

max

max

==

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Sam4le Pro)lem 6.2

Determine the beam depth based on allo#able

normal stress.

Determine the beam depth based on allo#ableshear stress.

;equired beam depth is equal to the larger of the t#o.

;

.6((mmd =

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7urther &i'cu''ion of the &i'tri)ution ofStre''e' in a !arro 8ectangular (eam

*onsider a narro# rectangular cantilever beam

subected to loadPat its free end2

hearing stresses are independent of the distance

from the point of application of the load.

1ormal strains and normal stresses are unaffected

by the shearing stresses.

rom aint/=enant>s principle, effects of the load

application mode are negligible except in immediate

vicinity of load application points.

tress?strain deviations for distributed loads are

negligible for typical beam sections of interest.

=(

(

'(

c

y

A

Pxy

I

Pxyx +=

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9ongitudinal Shear on a (eam Elementof Ar)itrar Sha4e

We have examined the distribution of

the vertical components xyon a

transverse section of a beam. We

no# #ish to consider the horizontal

components xzof the stresses.

*onsider prismatic beam #ith anelement defined by the curved surface

*DD>*>.

@xcept for the differences inintegration areas, this is the same

result obtained before #hich led to

( ) +==a

dAHF CDx !

I

VQ

x

Hqx

I

VQH =

==

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E1am4le 6.0$

A ',uare )o1 )eam i'con'tructed from four 4lan+' a''hon. noing that the'4acing )eteen nail' i' 5.% in.and the )eam i' 'u)ected to avertical 'hear of magnitude V#600 l) determine the 'hearing

force in each nail.

"L0T%"12

Determine the shear force per unit

length along each edge of the upper

plan3.

+ased on the spacing bet#een nails,determine the shear force in each

nail.

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E1am4le 6.0$

7or the u44er4lan+

7or the overall )eam cro''-'ection

"L0T%"12

Determine the shear force per unitlength along each edge of the upper

plan3.

+ased on the spacing bet#een nails,

determine the shear force in eachnail.

( ) ( ) ( )

in((.5

.in

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Shearing Stre''e' in

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Shearing Stre''e' in

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Shearing Stre''e' in

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The section becomes fully plastic &y!$ !) at

the #all #hen

orP"BM! , yield is initiated at$and$.

or an elastoplastic material, the half/thic3ness

of the elastic core is found from

Pla'tic &eformation'

;ecall2

or M % P" & M! , the normal stress does

not exceed the yield stress any#here along

the beam.

Caximum load #hich the beam can support is

'! MMP" ==(

=

(

(

''

(

c

yMPx !!

momentelasticmaximum== !!c

IM

"

MP

'=max

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Pla'tic &eformation'

Areceding discussion #as based on

normal stresses only

*onsider horizontal shear force on an

element #ithin the plastic zone,

Therefore, the shear stress is zero in theplastic zone.

hear load is carried by the elastic

core,

-sAdecreases, maxincreases and

may exceed

!

( ) ( ) !=== dAdAH !!DC

A

P

byA

y

y

A

P!

!

xy

=

=

=

(

(#here'

(

max

(

(

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6 - 23

Sam4le Pro)lem 6.3

noing that the vertical'hear i' %0 +i4' in a =5016rolled-'teel )eam determinethe hori?ontal 'hearing 'tre''in the to4 >ange at the 4ointa.

"L0T%"12

or the shaded area,

The shear stress at a,

( ) ( ) ( )

in:

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@n'mmetric 9oading of

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When the force A is applied at a distance e to theleft of the #eb centerline, the member bends in

a vertical plane #ithout t#isting.

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E1am4le 6.0%

Determine the location for the shear center of the

channel section #ith b$ 5 in., ($ 6 in., and t$ !.'4 in.

#here

*ombining,

I

(Fe=

I

Vt(b

d)(

)t

I

Vd)

I

VQd)qF

b bb

5

((

! !!

=

===

( )(bt(

(btbtt(III fla*geweb

+

++=+=

6

('(

'(

'(

'(

('('

(

( ).in5

.in6(

in.5

( +

=+

=

b

(

be .in6.'=e

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6 - 2

E1am4le 6.06

Determine the shear stress distribution for

V$ (.4 3ips.

hearing stresses in the flanges,

hearing stress in the #eb,

ItVQ

tq ==

( )

( )( ) ( )( ) ( )

( ) ( ) ( )3si((.(

in6in56in6in'4.!

in53ips4.(6

66

6(

((

('('

=+

=

+=

+=

===

(bt(Vb

(bt(

V(b

)I

V(()t

It

V

It

VQ

$

( )( )( )

( )

( )

( ) ( )

( ) ( ) ( )3si!6.

in6in66in6in'4.!(

in6in553ips4.(

6(

5

6

5

('('