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EN2701 Mechanics of Solids
Shear Stresses in Beams
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Shear Stresses in Beams
EN2701 Mechanics of Solids
School of Engineering, RGU
Slide 2 of 60
Shear Stresses in Beams
Introduction
The Shear Formula
Shear Stresses in Rectangular Beams !or"ed E#am$le 1
Shear Stresses in !ide%Flange &I' Beams
(imitations in the )se of the Shear Formula !or"ed E#am$le 2
Shear Stresses in *ircular Beams
Shear Stresses in Built%)$ Beams !or"ed E#am$le +
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Shear Stresses in Beams
EN2701 Mechanics of Solids
School of Engineering, RGU
Slide of 60
Shear Stresses in Beams % Intro Beams generall, su$$ort -oth shear and moment loadings.
Normal or -ending stresses/ / associated ith -endingmoments are found from the -ending euation
!hen -eam is su-3ected to non%uniform -ending in e4er, case e#ce$t hen -ending moment/ M/ constant along
length of uniform -eam
-oth -ending moments/ M/ and shear forces/ 5/ act on *S6
remem-er shear force and -ending moment diagrams from le4el 1
Shear force/ 5/ is result of trans4erse shear stress distri-utionthat acts o4er -eams cross%section.
Relationshi$ needed to allo distri-ution of shear stresses/ /associated ith shear force to -e determined.
R
E
,I
M=
=
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Shear Stresses in Beams
EN2701 Mechanics of Solids
School of Engineering, RGU
Slide ! of 60
Shear Stresses in Beams % Intro *om$lementar, $ro$ert, of shear.
Results in associated longitudinal shear stresses act alonglongitudinal $lanes of -eam.
Element of material from interior of -eam ill -e su-3ected to-oth trans4erse and longitudinal shear stress.
"
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Shear Stresses in Beams
EN2701 Mechanics of Solids
School of Engineering, RGU
Slide # of 60
E#istence of longitudinal shear stresses illustrated -,considering -eam made u$ of to -oards.
If to$ and -ottom surfaces of -oards are smooth and not-onded together then a$$lication of a load ill cause -oards toslide relati4e to each other.
Each -oard ill -e in com$ression a-o4e its neutral a#is and intension -elo its neutral a#is.
Shear Stresses in Beams % Intro
(oer longitudinal fi-res ofu$$er -oard ill sliderelati4e to u$$erlongitudinal fi-res of loer
-oard.
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Shear Stresses in Beams
EN2701 Mechanics of Solids
School of Engineering, RGU
Slide 6 of 60
If -oards are -onded together to ma"e a solid -eam/longitudinal shear stresses e#ist hich $re4ent relati4e sliding.
6lso longitudinal shear stresses at free surfaces on to$ and-ottom of -eam ill not e#ist.
8ue to com$lementar, nature/ trans4erse shear at free surfaceill also -e 9ero.
Shear Stresses in Beams % Intro
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Shear Stresses in Beams
EN2701 Mechanics of Solids
School of Engineering, RGU
Slide 7 of 60
The Shear Formula Formula for shear stress is de4elo$ed indirectl, using
-ending euation/
relationshi$ -eteen -ending and shear &5 : dM;d#'/
consideration of longitudinal shear stress.
*onsider an element of length d# cut from a -eam.
FB8 shos shear forces 5 and -ending moments M and
&M < dM' acting on element.
$d $
"
"
MM % d M
d $
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Shear Stresses in Beams
EN2701 Mechanics of Solids
School of Engineering, RGU
Slide & of 60
The Shear Formula Then consider to$ segment of
element that has -een
sectioned at a distance of ,from the neutral a#is.
Segment has a *S6 of 6 anda idth of t at section.
t' () (
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Shear Stresses in Beams
EN2701 Mechanics of Solids
School of Engineering, RGU
Slide * of 60
The Shear Formula No consider hori9ontal
stresses acting on segment.
(inearl, 4ar,ing normalstresses and due to Mand &M < dM'
Shear stress acting on-ottom surface.
*onsidering force euili-riumof segment
0+d$t,d)d)--)-)
=
M M % d M
d $
(
%%%%=1>
shear stress acts o4er an area t d#assuming is constant through thic"ness t
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Shear Stresses in Beams
EN2701 Mechanics of Solids
School of Engineering, RGU
Slide 10 of 60
From -ending euation
and
Su-stituting into euation =1>
E#$and first term
The Shear Formula
IM,= ( )
I,dMM +=-
0+-- =+
d#td6,IM
d6,I
dMM66
0+---
=+ d#td6,I
Md6,
I
dMd6,
I
M666
0+,-
= d#td6,IdM
6
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Shear Stresses in Beams
EN2701 Mechanics of Solids
School of Engineering, RGU
Slide 11 of 60
Therefore
Rearranging to o-tain
Sim$lified further
: first moment of area 6 a-out neutral a#is : ?
@ence
The Shear Formula+,
-d#td6,
I
dM6
=
= -1
6d6,
d#
dM
It
d#
dM5=
-) d)'
It
5?
=E./ation non as
the Shear orm/la
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EN2701 Mechanics of Solids
School of Engineering, RGU
Slide 12 of 60
8eri4ation considered shear stresses acting on -eams
longitudinal $lane. Because of com$lementar, nature of shear also used to find
trans4erse shear stress on -eams *S6.
6s deri4ation used -ending euation shear formula onl, 4alid if
material -eha4es in a linear%elastic manner/ material is homogeneous and isotro$ic/
material has same Aoungs modulus in tension and com$ression.
The Shear Formula
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EN2701 Mechanics of Solids
School of Engineering, RGU
Slide 1 of 60
The Shear Formula
: shear stress at distance , from neutral a#is.
6ssumed constant across idth/ t/ of -eam.5 : internal resultant shear force/ determined from sectioning
-eam and considering euili-rium.
I : 2nd moment of area of entire*S6 a-out neutral a#is.
t : idth of -eam/ at $osition here to -e determined.? : 1st moment of area of to$ &or -ottom' $ortion of *S6/defined from section here t is measured.
? calculated using
6 : area of to$ &or -ottom' $ortion of *S6/ defined from sectionhere t is measured
: distance to centroid of 6 measured from neutral a#is.
It
5?=
-)-'3=
-'
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Shear Stresses in Beams
EN2701 Mechanics of Solids
School of Engineering, RGU
Slide 1! of 60
Shear Stresses in Rectangular Beams *onsider a -eam ith rectangular
cross%section.
idth : - and height : h.
8istri-ution of shear stressthroughout cross%section found -,determining shear stress at
ar-itrar, distance , from neutrala#is.
"
4
h
'
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EN2701 Mechanics of Solids
School of Engineering, RGU
Slide 1# of 60
Shear Stresses in Rectangular Beams First ste$ is to calculate ?
1st moment of area of 6
a-out neutral a#is.
) (
' 'h2
h2
4
-)'3=
= '
2
h4-)
''2h
21' +
=
+
= '
2
h4''
2
h
2
13
4'!
h
2
13 2
2
=
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Shear Stresses in Beams
EN2701 Mechanics of Solids
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Slide 16 of 60
Shear Stresses in Rectangular Beams Second ste$/ calculate I.
For rectangular -eam of idth
- and height h/ I/ a-outneutral a#is is
6$$l,ing shear formula
) (
' 'h2
h2
4
12
-hI=
--h
-,h
5
It
5?
12
!2
1
22
==
= 2
2
'
!
h
4h
"6
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Shear Stresses in Beams
EN2701 Mechanics of Solids
School of Engineering, RGU
Slide 17 of 60
Shear Stresses in Rectangular Beams
Shos shear stress distri-utiono4er cross section isparabolic.
Intensit, 4aries from 0 at to$
and -ottom here , : h;2.
To ma#imum at neutral a#ishere , : 0.
= 2
2
'
!
h
4h
"6
h2
h2
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Shear Stresses in Beams
EN2701 Mechanics of Solids
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Slide 1& of 60
Shear Stresses in Rectangular Beams To calculate maximum4alue
Remem-er area of total crosssection/ 6 : -h
Ma# shear stress is 1. timesa4erage 4alue calculated fromsim$le formula : 5;6
h2
h2
= 0Ch
-h5D
2
+ma#
-h2
5+ma# =
6
5
2
+ma# =
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EN2701 Mechanics of Solids
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Slide 1* of 60
!or"ed E#am$le 1 Beam shon made of ood.
Su-3ected to resultant internal
4ertical shear force/ 5 : +0"N.
8imensions in mm.
&a'8etermine shear stress in-eam at $oint .
&-' *alculate ma#imum shearstress in the -eam.
"
5# 0
2 0
1 #! 0
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EN2701 Mechanics of Solids
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Slide 20 of 60
!or"ed E#am$le 1 % Solution
&a' First calculate section
$ro$erties I and ?.
Remem-er I is for wholesection.
12
-hI
+
=
( )12
0B.00C.0I
+=
CF m107.C1DI =
! 0
2 #
2 #
N )
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Slide 21 of 60
!or"ed E#am$le 1 % Solution
? is first moment of area of
shadedarea.
Shaded area is area a-o4e .
is distance from neutral a#isto centroid of area.
) (
'' 7 1 #
! 0
2 #
2 #
2 0
5
N )
26 m10&000200!0-) ==
( ) m01#001000#0' =+=
D
10G0001B.0H6,?
==+D m1012? =
,
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Shear Stresses in Beams
EN2701 Mechanics of Solids
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Slide 22 of 60
!or"ed E#am$le 1 % Solution
!idth of the section &t' at is
C0 mm. 6$$l,ing shear formula
) (
'' 7 1 #
! 0
2 #
2 #
2 0
5
N )
tI
5? =
MaD.21 =
0C.0107.C1D101210+0
D+
=
Ea10D.21 D =
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Slide 2 of 60
!or"ed E#am$le 1 % Solution
&-' Shear stress is ma#imum at
neutral a#is.*onsider area of -eam a-o4eneutral a#is ) (
' 7 1 2 #
! 0
2 #
2 #
N )
2 m10102#00!0-) ==
m012#0'=
+101012.0H6,? ==
+D m10.12? =
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EN2701 Mechanics of Solids
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Slide 2! of 60
!or"ed E#am$le 1 % Solution
I and t are as -efore.
6$$l,ing shear formula
) (
' 7 1 2 #
! 0
2 #
2 #
N ) a10.22 Dma# =
M5a#22ma$ =
0C.0107.C1D
10.1210+0
tI
5?
D+
ma#
==
Ma# shear stress can also -e calculated from
5a10#220#00!0
100
2
)
"
2
6
ma$ =
==
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Slide 2# of 60
Tutorial ?uestions
Aou can no attem$t ? 1 % C on the tutorial sheet.
6s" if ,ou need hel$.
@el$ each other
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Slide 26 of 60
Shear Stresses in !ide Flange &I' Beams
!ide%flange -eam &or I%-eam'consists of to &ide'
Jflanges and a Je-.
!hen su-3ected to shear force5/ shear stresses de4elo$edthroughout cross%section.
f l a n g e s
e 4
8istri-ution of stresses much more com$licated than inrectangular -eam.
ossi-le to determine stresses using same techniues as forrectangular -eams
h d l & '
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Slide 27 of 60
Shear Stresses in !ide Flange &I' Beams
6s ith rectangular -eam/ shear stress 4aries $ara-olicall, o4erde$th.
For flanges thic"ness/ t/ in shear formula is idth of flange.
For e-/ thic"ness/ t/ is thic"ness of e-.
Shear stresses in flanges small com$ared to those in e-.
5ariation of shear stress o4er de$th of e- is small.
8 a r a 4 o l a
f l a n g e
e 4
f l a n g e
Sh S i id l & '
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Shear Stresses in Beams
EN2701 Mechanics of Solids
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Slide 2& of 60
Shear Stresses in !ide Flange &I' Beams
Most of 4ertical shearforce is carried -, e-.
Ma#imum shear stressoften a$$ro#imated -,di4iding shear force 5 -,area of e-
8 a r a 4 o l a
a 9 gf l a n g e
e 4
f l a n g e
e4a9g
)
"=
(i it ti i ) f Sh F l
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Slide 2* of 60
(imitations in )se of Shear Formula
Kne assum$tion used in deri4ationof shear formula is shear stress is
uniforml, distri-uted o4er idth/ t. 6ccurac, tested -, com$aring
results ith more rigorousmathematical anal,sis -ased ontheor, of elasticit,.
If -eams cross%section isrectangular/ shear stressdistri-ution calculated from theor,is shon
Ma#imum 4alue/ ma#/ occurs at
edges of cross%section. Magnitude de$ends on ratio -;h
&idth;de$th'.
(m a $m a $
4
hN )
(i it ti i ) f Sh F l
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Slide 0 of 60
(imitations in )se of Shear Formula
For sections ha4ing a -;h ratio of0./ ma#onl, a-out +L greater
than calculated from shearformula.
For flat sections ith -;h : 2/ma#is C0L greater than ma#.
Error e4en greater as -;h ratioincreases.
For flanges of ide%flange -eamsa realistic 4alue of shear stress isnot e4en a$$ro#imated.
( m a $ m a $
4 7 0 # h
hN )
( m a $ m a $
4 7 2 h
h
N )
(i it ti i ) f Sh F l
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Shear Stresses in BeamsEN2701 Mechanics of Solids
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(imitations in )se of Shear Formula
!ill not gi4e accurate results for shearstress at flange;e- 3unction of I%
-eam. Inner regions of flanges are free
-oundaries and shear stresses must -e9ero.
)sing shear formula a non%9ero 4alue is
found. (imitations for flanges of I%-eams not
im$ortant in engineering $ractice.
Kften/ engineers onl, calculatema#imum shear stress.
This occurs at neutral a#is/ here -;hratio is 4er, small.
*alculated result 4er, close to actualma#imum shear stress.
! " d E l 2
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!or"ed E#am$le 2 Steel ide%flange -eam
has dimensions shon.
Su-3ected to shearforce 5 : G0 "N.
lot shear stressdistri-ution acting o4er-eams cross%sectionalarea.
0 0 m m
2 0 0 m m
2 0 m m
2 0 m m
1 # m m
! " d E l 2 S l ti
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!or"ed E#am$le 2 % Solution Shear stress
distri-ution ill -e
$ara-olic and ha4eform shon.
8ue to s,mmetr,/onl, shear stresses at$oints B/ B and *
needed. Second moment of
area/ I/ ill -e samefor each $osition.
B
:
B ( B ( B
:
! " d E l 2 S l ti
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Shear Stresses in BeamsEN2701 Mechanics of Solids
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!or"ed E#am$le 2 % Solution For flanged cross section I is
found in one of to a,s
1. *alculate I for large rectangle
0 0 m m
2 ! 0 m m
( )12 2C.0+.0I+
=
!o "ed E am$le 2 Solution
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Shear Stresses in BeamsEN2701 Mechanics of Solids
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!or"ed E#am$le 2 % Solution For flanged cross section I is
found in one of to a,s
1. *alculate I for large rectangleand su-tract I for 2 smallrectangles.
I : 1.D # 10%DmC
0 0 m m
2 0 0 m m
2 0 m m
2 0 m m
1 # m m
1 ! 2 # m m 1 ! 2 # m m
( )12 2.01C2B.02
+
( )12 2C.0+.0I+
=
!or"ed E#am$le 2 Solution
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!or"ed E#am$le 2 % Solution
2. *alculate I for all + rectangles
and use $arallel a#is theorem.I for flanges must -e mo4edso that neutral a#es coincide.
I : 1.D # 10%DmC
0 0 m m
2 0 0 m m
2 0 m m
2 0 m m
1 # m m
( ) ( )( )
+
+
= 2
110020012
02002
12
2001#0I
++=
2++
6"12
-h
212
-h
I
!or"ed E#am$le 2 Solution
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!or"ed E#am$le 2 % Solution Stress at $oint B
tB: 0.+ m
6 : area of flange
02.0+.011.0H6,? HB ==
+.010D.1
10DD.010G0
tI
5?D
++
HB
==
0 0 m m
1 0 0 m m
2 0 m m
' 7 1 1 0 m m ) (
N )++
HB m10DD.0? =
Ma1+.1HB =
!or"ed E#am$le 2 Solution
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!or"ed E#am$le 2 % Solution Stress at $oint B
tB: 0.01 m
?B: ?B
01.010D.1
10DD.010G0
tI
5?D
++
B
==
0 0 m m
1 0 0 m m
2 0 m m
' 7 1 1 0 m m ) (
N )
MaD.22B =
!or"ed E#am$le 2 Solution
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!or"ed E#am$le 2 % Solution Stress at $oint *
t*: 0.01 m
6 : area of section a-o4eneutral a#is.
? for this area is sim$l,sum of ?s for 2 rectanglesthat ma"e u$ sha$e.
?*: ?1< ?2
0 0 m m
1 0 0 m m
2 0 m m
1 # m m' 1
' 2
) ( 1
) (2
N )
2211* 6,6,? +=
?*: &0.11 # 0.+ # 0.02' < &0.0 # 0.01 # 0.1'
?*: 0.7+ # 10%+m+
MEatI
5?* 22#
01#01061##
107#010&06
=
==
!or"ed E#am$le 2 Solution
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!or"ed E#am$le 2 % Solution
B: 1.1+ Ma
B : 22.D Ma
* : 2.2 Ma
B
:
B ( 1 1 2 2 6
2 # 2
T t i l ? ti
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Tutorial ?uestions
Aou can no attem$t ? % D on the tutorial sheet.
6s" if ,ou need hel$.
@el$ each other
Shear Stresses in *ircular Beams
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Shear Stresses in *ircular Beams This t,$e of -eam im$ortant in
the transmission of $oer.
E.g. -ending and shearing loadsare induced in shafts -, forces atgears/ -earings and $ulle,s.
*onsider -eam ith solid circular
cross%section su-3ected to shearload 5.
6ccording to shear formula ashear force/ 5/ causes a shearstress/ / in same direction as 5.
"
Shear Stresses in *ircular Beams
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Shear Stresses in *ircular Beams 6t $oint this stress can -e
resol4ed into to com$onents
normal &n' and tangential &t' tosurface. Kutside surface of shaft is free
surface and so nmust -e 9ero. Indicates shear stress at $oint
must -e tangential to surface andnot in direction of shear force.
6t neutral a#is shear stress ill -ein direction of shear force
Shear formula can -e used.
6lso $osition of ma#imum shearstress.
n t
5
Shear Stresses in *ircular Beams
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Shear Stresses in BeamsEN2701 Mechanics of Solids
School of Engineering, RGUSlide !! of 60
Shear Stresses in *ircular Beams
For a semi%circle
and
For circular cross%section
Therefore
tI
5?=ma$
&
d-)
2=
=
d2'
12
d
d2
&
d
-)'3
2
=
==
6!
!dI =
!;
;
16;
;
6!;
12;2!
ma$6
5
d
5
dd
d5=
=
=
6
5
!ma$ =
m a $
N )
) (
'
Shear Stresses in Built%)$ Beams
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Shear Stresses in BeamsEN2701 Mechanics of Solids
School of Engineering, RGUSlide !# of 60
Shear Stresses in Built%)$ Beams Fa-ricated from to or more $ieces of material 3oined together
to form single/ solid -eam.
*onstructed in a great 4ariet, of sha$es to meet s$ecial needsor to $ro4ide larger cross%sections that are ordinaril, a4aila-le.
!ooden box beamconstructed of to $lan"s/
as flanges/ connected -,$l,ood e-s.
ieces 3oined together ithnails/ scres or glue.
f l a n g e1 e 4
Sh St i B ilt ) B
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Shear Stresses in BeamsEN2701 Mechanics of Solids
School of Engineering, RGUSlide !6 of 60
Shear Stresses in Built%)$ Beams lued/ laminated -eam "non
as glulam beam.
Boards glued together to formmuch larger -eam than could-e cut from solid $iece oftim-er.
!elded steel plate girder/fa-ricated from three steel$lates elded together.
!ide flange -eamstrengthened -, ri4etingchannel section to each flange.
Shear Stresses in Built%)$ Beams
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Shear Stresses in BeamsEN2701 Mechanics of Solids
School of Engineering, RGUSlide !7 of 60
Shear Stresses in Built%)$ Beams
8esign must ensure that entire cross%section acts as a single/solid unit.
*alculations in4ol4e to $hases
1. -eam anal,sed as if it as solid cross%section/ ta"ing intoaccount -oth -ending and shear stressesO
2. connecting elements &nails/ -olts/ glue/ elds' anal,sed toensure the, are strong enough and suita-l, $ositioned.
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Shear Stresses in BeamsEN2701 Mechanics of Solids
School of Engineering, RGUSlide !* of 60
Shear Stresses in Built%)$ Beams
: shear flo/ measured as force $er unit length along -eam.
5 : internal resultant shear force/ determined from sectioning-eam and euili-rium.
I : 2nd moment of area of entirecross%sectional a-out neutral
a#is. ? : 1stmoment of area of to$ &or -ottom' $ortion of cross%
sectional area/ defined from section here shear flo to -ecalculated.
I
5? =
Shear Stresses in Built )$ Beams
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Shear Stresses in BeamsEN2701 Mechanics of Solids
School of Engineering, RGUSlide #0 of 60
Shear Stresses in Built%)$ Beams 6$$lication of shear flo formula follos same $rocedure as for
shear stress formula.
5er, im$ortant to identif, correct 4alue for ? hendetermining shear flo at $articular 3oint in cross%section.
Reuired 4alue of ? calculated from shaded sections.
Note shear flo ill -e resisted -,
single ro of fasteners in &a' and &-'
to ros of fasteners in &c' and &d'
three ros of fasteners in &e'
, a + , 4 + , c + , e +, d +
!or"ed E#am$le +
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Shear Stresses in BeamsEN2701 Mechanics of Solids
School of Engineering, RGUSlide #1 of 60
!or"ed E#am$le + !ood -o# -eam % 2 -oards as flanges and 2 e-s of $l,ood.
Each -oard : C0 mm # 1G0 mm. l,ood : 1 mm thic".
Total height of -eam : 2G0 mm. l,ood fastened to flanges -, scres ha4ing alloa-le load in
shear of 1100 N $er scre.
If shear force acting on cross%section is 10. "N/ determinema#imum $ermissi-le longitudinal s$acing/ s/ of scres.
! 0
! 0
2 & 0
1 # 1 & 0 1 #s s s
!or"ed E#am$le + Solution
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Shear Stresses in BeamsEN2701 Mechanics of Solids
School of Engineering, RGUSlide #2 of 60
!or"ed E#am$le + - SolutionShear force transmitted -eteen one of flanges and to e-s
found from shear flo formula
? for to$ flange &shaded'
6 : 0.1G # 0.0C : 7.2 # 10%+m2
? : 0.12 # 7.2 # 10%+
? : GDC # 10%Dm+
m12.0, =
( ) ( )12
201&0
12
2&0210
12
==
-hI
!610226! mI =
I
5? =
! 0
1 ! 0
1 # 1 & 0 1 #
'
) (
!or"ed E#am$le + - Solution
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Shear Stresses in BeamsEN2701 Mechanics of Solids
School of Engineering, RGUSlide # of 60
!or"ed E#am$le + Solution
: +C.+ "N;m
Shear flo $er metre of length that must -e carried -, scres.
If scres are s$aced a distance Js a$art load ca$acit, of scresis
F : load carried -, one scre
2 -ecause there are to lines of scres
6
6
10226!
10&6!10#10
==I
5?
s
2
!or"ed E#am$le + - Solution
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Shear Stresses in BeamsEN2701 Mechanics of Solids
School of Engineering, RGUSlide #! of 60
!or"ed E#am$le + SolutionEuating load ca$acit, of scres to shear flo
Rearranging
s : DC.1 mm
Therefore lines of scres must -e no more than DC.1 mm a$art.
.s
2
=
m06!1010!
11002
.
2s
=
==
Tutorial ?uestions
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Shear Stresses in BeamsEN2701 Mechanics of Solids
School of Engineering, RGUSlide ## of 61
Tutorial ?uestions
Aou can no attem$t ? 7 % on the tutorial sheet.
6s" if ,ou need hel$.
@el$ each other
Summar,
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Shear Stresses in BeamsEN2701 Mechanics of Solids
School of Engineering, RGUSlide #6 of 61
Summar,
Summar, !or"ed E#am$le C
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Shear Stresses in BeamsEN2701 Mechanics of Solids
School of Engineering, RGUSlide #7 of 61
Summar, !or"ed E#am$le C&com-ined loading'
To forces :1G "N and F:1 "N are a$$lied to the shaft ith a radius ofR:20 mm as shon. 8etermine the ma#imum normal and shear stressesde4elo$ed in the shaft.
!or"ed E#am$le C % Solutions
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Shear Stresses in BeamsEN2701 Mechanics of Solids
School of Engineering, RGUSlide #& of 61
!or"ed E#am$le C Solutions&com-ined loading'
To forces :1G "N and F:1 "N are a$$lied to the shaft ith a radius ofR:20 mm as shon. 8etermine the ma#imum normal and shear stressesde4elo$ed in the shaft.
!or"ed E#am$le C % Solutions
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Shear Stresses in BeamsEN2701 Mechanics of Solids
School of Engineering, RGUSlide #* of 61
!or"ed E#am$le C Solutions&com-ined loading'
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!or"ed E#am$le C % Solutions
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!or"ed E#am$le C Solutions&com-ined loading'