Download - CE302 Beams Ch6
-
7/25/2019 CE302 Beams Ch6
1/27
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
-
7/25/2019 CE302 Beams Ch6
2/27
KingSaudUnive!i"#$
!%u
ai&
CE 302:Prof.
AhmedShuraim
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
-
7/25/2019 CE302 Beams Ch6
3/27
KingSaudUnive!i"#$
!%u
ai&
CE 302:Prof.
AhmedShuraim
shear formula
6 - 3
olving for
oting that
hear formula
Area#
-
7/25/2019 CE302 Beams Ch6
4/27
KingSaudUnive!i"#$
!%u
ai&
CE 302:Prof.
AhmedShuraim
6 - $
-
7/25/2019 CE302 Beams Ch6
5/27
KingSaudUnive!i"#$
!%u
ai&
CE 302:Prof.
AhmedShuraim
6 - %
&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.
-
7/25/2019 CE302 Beams Ch6
6/27
Ki
ngSaudUnive!i"#$
!%u
ai&
CE 302:Prof.
AhmedShuraim
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
-
7/25/2019 CE302 Beams Ch6
7/27
Ki
ngSaudUnive!i"#$
!%u
ai&
CE 302:Prof.
AhmedShuraim
6 -
-
7/25/2019 CE302 Beams Ch6
8/27
Ki
ngSaudUnive!i"#$
!%u
ai&
CE 302:Prof.
AhmedShuraim
6 -
C 302
-
7/25/2019 CE302 Beams Ch6
9/27
Ki
ngSaudUnive!i"#$
!%u
ai&
CE 302:Prof.
AhmedShuraim
6 -
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.
CE 302
-
7/25/2019 CE302 Beams Ch6
10/27
Ki
ngSaudUnive!i"#$
!%u
ai&
CE 302:Prof.
AhmedShuraim
6 - 50
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
CE 302
-
7/25/2019 CE302 Beams Ch6
11/27
Ki
ngSaudUnive!i"#$
!%u
ai&
CE 302:Prof.
AhmedShuraim
6 - 55
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
-
7/25/2019 CE302 Beams Ch6
12/27
Ki
ngSaudUn
ive!i"#$
!%u
ai&
CE 302:Prof.
AhmedShuraim
6 - 52
Sam4le Pro)lem 6.2
"L0T%"12
Develop shear and bending momentdiagrams. %dentify the maximums.
mkNM
kNV
.:4.'!
4.'5
max
max
==
CE 302
-
7/25/2019 CE302 Beams Ch6
13/27
Ki
ngSaudUn
ive!i"#$
!%u
ai&
CE 302:Prof.
AhmedShuraim
6 - 53
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 =
CE 302
-
7/25/2019 CE302 Beams Ch6
14/27
Ki
ngSaudUn
ive!i"#$
!%u
ai&
CE 302:Prof.
AhmedShuraim
6 - 5$
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 +=
CE 302:
-
7/25/2019 CE302 Beams Ch6
15/27
Ki
ngSaudUn
ive!i"#$
!%uai&
CE 302:Prof.
AhmedShuraim
6 - 5%
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 =
==
CE 302:
-
7/25/2019 CE302 Beams Ch6
16/27
Ki
ngSaudUn
ive!i"#$
!%uai&
CE 302:Prof.
AhmedShuraim
6 - 56
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.
CE 302:
-
7/25/2019 CE302 Beams Ch6
17/27
Ki
ngSaudUn
ive!i"#$
!%uai&
CE 302:Prof.
AhmedShuraim
6 - 5
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
CE 302:
-
7/25/2019 CE302 Beams Ch6
18/27
Ki
ngSaudUn
ive!i"#$
!%uai&
CE 302:Prof.
AhmedShuraim
6 - 5
Shearing Stre''e' in
-
7/25/2019 CE302 Beams Ch6
19/27
Ki
ngSaudUn
ive!i"#$
!%uai&
CE 302:Prof.
AhmedShuraim
6 - 5
Shearing Stre''e' in
-
7/25/2019 CE302 Beams Ch6
20/27
Ki
ngSaudUn
ive!i"#$
!%uai&
CE 302:Prof.
AhmedShuraim
6 - 20
Shearing Stre''e' in
-
7/25/2019 CE302 Beams Ch6
21/27
KingSaudUn
ive!i"#$
!%uai&
CE 302:Prof.
AhmedShuraim
6 - 25
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
CE 302:
-
7/25/2019 CE302 Beams Ch6
22/27
KingSaudUn
ive!i"#$
!%uai&
CE 302:Prof.
AhmedShuraim
6 - 22
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
(
(
CE 302:
-
7/25/2019 CE302 Beams Ch6
23/27
KingSaudUn
ive!i"#$
!%uai&
CE 302:Prof.
AhmedShuraim
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:
-
7/25/2019 CE302 Beams Ch6
24/27
KingSaudUn
ive!i"#$
!%uai&
CE 302:Prof.
AhmedShuraim
6 - 2$
@n'mmetric 9oading of
-
7/25/2019 CE302 Beams Ch6
25/27
KingSaudUn
ive!i"#$
!%uai&
CE 302:Prof.
AhmedShuraim
6 - 2%
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.
@n'mmetric 9oading of
-
7/25/2019 CE302 Beams Ch6
26/27
KingSaudUn
ive!i"#$
!%uai&
CE 302:Prof.
AhmedShuraim
6 - 26
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
CE 302:
-
7/25/2019 CE302 Beams Ch6
27/27
KingSaudUn
ive!i"#$
!%uai&
Prof.Ahmed
Shuraim
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
('('