ch. 7 flexural loading : stresses in beamcontents.kocw.or.kr/document/1.solid mechanics -...
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1
Ch. 7 Flexural Loading : Stresses in Beam
Ø Beam : member subjected to loads applied transverse to the long dimension, which cause the member to bend.
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Ø Beams are classified on their supports or reactions.
2
7-1. Introduction
(a) Simple beam- supported by a roller- having on span- normal reaction and no couple
(b) Simple beam with overhang
(c) Continuous beam- with more than two simple supports
(d) Cantilever beam
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(d) Cantilever beam- one end is built into a wall- neither move transversely nor rotate
(e) Beam fixed at left end and simply supported near the other end
(f) Beam fixed at both ends
3
7-1. (continued)
:rV Resisting shear at the section:rM Resisting moment at the section
:R Reaction at the support
å = 0yF
l From a free body diagram of entire beam
Rl From
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
o is any axis perpendicular to xy plane
rV
å = 0oMl From
rM
òò-=
-=
A xr
A xyr
dAyM
dAV
s
t
4
7-2. Flexural strains
l Neutral surface : surface of which longitudinal elements undergo no change in length
l Plane section before bending remains a plane after bending
l Neutral axis : intersection of neutral surface with any cross section
Þ-
==i
ifx L
LLLde
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Þ-
==i
ifx L
LLLde
yyx
xxx rqr
qrqre 1-=
DD-D-
=D
D-D=
)()())(('
Strain developed in a fiber is directly proportional to the distance of the fiber from the neutral surface of the beam
5
7-3. Flexural stresses
l For linear elastic action
Normal stress on transverse cross section of the beam varies linearly with distance y from the neutral surface
yEE xx res -==
òò -=-=A xAr dAydFyM s
l Equilibrium equation for pure bending
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
0
0
=-=-=
÷÷ø
öççè
æ-=
===
ò
òò
òòå
AyEdAyE
dAyEdA
dAdFF
cA
AA x
A xAx
rr
rs
s
Yc is the distance from the neutral axis to the centroidal axis c-c of the cross section
Neutral axis passes through the centroid of the cross section
6
7-3. (continued)
where c is the distance to the surface of the beam farthest from the neutral surface
cEr
s -=max
cx cy
cyyE ss
rs ==-= max
yEE xx res -==by using
ycE xssr
==- max
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
òò -=-=A
cA xr dAy
cdAyM 2ss
cx cy
cyyE ss
rs ==-= max
Second moments of more complex areas can be derived from combinations of simple shapes
: second moment of area (Izz)
7
7-4. Elastic flexural formula
l For elastic flexural formula
yIMr
x -=s
l For the section about y-axis
where S is the section modulus of the beam
SM
IcM rr -=-=maxs
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
l For the section about y-axis
0==== òòò yzc
Ac
Ac
A x Ic
zydAc
dAyc
zdAz ssss
where Iyz is the mixed second moment of cross sectional area with respect to the centroidal y- and z-axes.
By considering equilibrium equation, å = 0yM
8
7-4. (continued)
yIM
IyM x
rr
xss -=Û-=
(Example 7-2) The maximum flexural stress at a given is 15 MPa. Determine (a) the resisting moment developed at the section, (b) percentage decrease in resisting moment if the dotted portion of the cross section is removed
(a) By using
464633
10676610676612
20010012
mmmbhI )(.)(. -==´
==
mkNmNcIM x
r -=-=== -
-
10101010100
1067661015 33
66
)()()(.)(s
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
mkNmNcIM x
r -=-=== -
-
10101010100
1067661015 33
66
)()()(.)(s
(b) 4646333
106521065212
1505012
20010012
mmmbhI )(.)(. -==´
-´
==
mkNmNcIM x
r -=-=== -
-
8971089710100
106521015 33
66
.)(.)()(.)(s
%.. 12110010
89710=´
-== Ddecreasepercent
9
7-4. (continued)
(Example 7-3) On a section where resisting moment is -75kN-m, determine (a) maximum tensile flexural stress, (b) maximum compressive flexural stress.
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
AMyAydAy A
ccA=Û=ò
The neutral axis is horizontal and passes through the centroid of the cross section
mmAMy
mmMmmA
Ac
A
9550012
5001871500187120075501002515051225200
50012755025150252003
2
===
=++=
=++=
,,,
,,))(())(().)((,)()()(
10
7-4. (continued)
The second moment of area of the part w.r.t. horizontal centroidal axis of the total cross section
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
For part A :
2CxCx yAII +='
4623
2 1013421055075125075 mmyAII CAAxCAAx )(.))(()(
' =+=+=
Similarly
4623
2
4623
2
1029345822520012
25200
101375150251215025
mmyAII
mmyAII
CCBxCCCx
CBBxCBBx
)(.).)(()(
)(.))(()(
'
'
=-+=+=
=+=+=
4646 105583105583 mmmIIII CxBxAxx )(.)(.''''-==++=
(Parallel axis theorem)
11
7-4. (continued)
(a) Since resisting moment is negative, maximum tensile flexural stress occurs at the top of the beam
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
MPamNIyM tr 7116107116
105583101301075 26
6
33
./)(.)(.
))()()((max ==
--=-= -
-
s
(a) Since resisting moment is negative, maximum tensile flexural stress occurs at the top of the beam
(b) Maximum compressive flexural stress occurs at the bottom of the beam
MPamNIyM br 385102885
10558310951075 26
6
33
./)(.)(.
))()()((max -=-=
---=-= -
-
s
12
7-5. Shear forces and bending moments
l Force equilibrium equation,
rr VVorVPwxR ==--
where V is the resultant of the external transverse forces called as transverse shear at the section
l Moment equilibrium equation,
rr MMorMhxPwxRx ==--- )(2
2
where M is the algebraic sum of the moments of the external forces called as bending moment at the section
å = 0yF
å = 0oM
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
where M is the algebraic sum of the moments of the external forces called as bending moment at the section
13
7-5. (continued)
(Example 7-4) Write equations for the shear force V and the bending moment M (a) in the interval AB, (b) in the interval BC, (c) in the interval CD.
åå
=--=
=--=
062000564008
022000364008
2
1
)())(()(
)())(()(
RM
RM
A
D
lbRandlbR 30001400 21 ==
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
14
7-5. (continued)
(a)
)( ftxlbVVFy
201400
01400
<<=
=-=å
)( ftxlbftxMMxMo
201400
01400
<<-=
=+-=å
(b)
)()(
ftxlbxVVxFy
622200400
024001400
<<+-=
=---=å
)(
)(
ftxlbftxxM
MxxxMo
628002200200
02
224001400
2 <<--+-=
=+÷øö
çèæ -
-+-=å
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
)(
)(
ftxlbftxxM
MxxxMo
628002200200
02
224001400
2 <<--+-=
=+÷øö
çèæ -
-+-=å
(c)
)()(
ftxlbxVVxFy
86200400
0200024001400
<<+-=
=----=å
)(
)()(
ftxlbftxxM
MxxxxMo
8611200200200
0620002
224001400
2 <<-++-=
=+-+÷øö
çèæ -
-+-=å
15
7-5. (continued)
(c)
)()(
ftxlbxVVxFy
86200400
0200024001400
<<+-=
=----=å
)(
)()(
ftxlbftxxM
MxxxxMo
8611200200200
0620002
224001400
2 <<-++-=
=+-+÷øö
çèæ -
-+-=å
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
)(
)()(
ftxlbftxxM
MxxxxMo
8611200200200
0620002
224001400
2 <<-++-=
=+-+÷øö
çèæ -
-+-=å
similarly
)()(
ftxlbxVxVFy
86200400
030008400
<<+-=
=+--=å
)(
)()(
ftxlbftxxM
xxxMMo
8611200200200
0830002
88400
2 <<-++-=
=-+÷øö
çèæ -
---=å
16
7-5. (continued)
(Example 7-5) On a section 3m to the right of A, determine (a) flexural stress at a point 25mm below the top of the beam, (b) maximum flexural stress on the section.
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
kNRRM
A
AB
62
081210768220
=
=-++=å )()())(()(
S152X19
mkNMMMo
-=
=+-+=å16
07103625258 )()().)((
mdmSmI
1524.0)10(121)10(20.9
36
46
==
=-
-
17
7-5. (continued)
(a) At a point 25mm below the top of the beam,
MPamNIyM
mmmdy
08910048910209
051201016
05120251252
4152252
266
3
./)(.)(.).)((
...
-=-=-=-=
==-=-=
-s
(b) Maximum flexural stress
)(.)(.
/)(.)()(
max
CMPaTMPa
mNSM
IMc
2313223132
1023132101211016 26
6
3
==
==== -s
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
)(.)(.
/)(.)()(
max
CMPaTMPa
mNSM
IMc
2313223132
1023132101211016 26
6
3
==
==== -s
on the bottomon the top
18
7-6. Load, Shear force and Bending moment
l Mathematical relationships between loads, shear forces, and bending moments
å = 0yF
l From force equilibrium
xwPVVVPxwV
avg
LavgL
D+=D
=D+-+D+ 0)(
(1) If P=0 and w=0, shear force is constant
RL VVV =Þ=D 0
(2) If , shear force jumps by the concentrated load as
0¹P0®Dx
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
PVVPV LR +=Þ=D
(3) If , the slope of shear force graph is equal to the intensity of loading as
wdxdV
xVxwV
xavg ==DD
ÞD=D®D 0lim
òò ==-Þ==DD
®D
2
1
2
1120
x
x
V
VxwdxdVVVw
dxdV
xVlim
Change in shear between sections is equal to the area under the load diagram
0=P0®Dx
19
7-6. (continued)
å = 0centerM
l From moment equilibrium
022
=D+D
D+-D
---D+ )()()( xwaxVVxVCMMM avgLLLL
wwandaxas
xaxforxwaxVxVCM
avg
avgL
®®®D
D<<
D-D-
DD+D+=D
00222
,
)(
(1) If , bending moment jumps by C as 0¹C 0®Dx
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(1) If , bending moment jumps by C as
CMMCM LR +=Þ=D
(2) If , the slope of bending moment is equal to the value of shear force at that section as
)( xwaxVxVM avgL D-D
D+D=D2
0¹C
00 == PC ,
VdxdM
xM
x==
DD
®D 0lim
òò ==-2
1
2
112
x
x
M
MVdxdMMM
0®Dx
20
7-6. (continued)
l Shear and bending moment diagrams provide a convenient method for obtaining maximum values of shear and bending moment.
(2) algebraic equations in case of uniformly distributed or varies according to a known equation
(3) drawing shear diagram from the load diagram and bending moment diagram from shear diagram
(1) calculating values of shear and bending moment at various sections
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
21
7-6. (continued)
(Example 7-6) (a) write equations for shear and bending moment in AB, (b) write equations for shear and bending moment in BC, (c) draw complete shear and bending moment diagrams for the beam
(a) Section in AB
lbftMandlbVMM
VF
CC
CC
Cy
×-=-=Þ
=++-=
=---=
åå
86001700
03620010500
06200500
))(()(
)(
:0=å yFftxforlbV
VFy
40500
0500
<<-=Þ
=--=å
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
:0=å yFftxforlbV
VFy
40500
0500
<<-=Þ
=--=å
:0=å OMftxforlbftxM
MxMO
40500
0500
<<×-=Þ
=+=å )(
(b) Section in BC
:0=å yFftxforlbxV
VxFy
104200300
04200500
<<-=Þ
=----=å )(
:0=å OM
ftxforlbftxxM
MxxxMO
1041600300100
02442005002 <<×-+-=Þ
=+--+=å /))(()(
22
7-7. Shear stresses in beams
l Beam by stacking flat slabs one on top of another without fastening them together
Relative motion of the ends of the cards with respect to each otherSolid beam does not exhibit this relative movementIndication of presence of shearing stress on longitudinal planes
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
23
stressflexuralI
MywheredAF
dytdAdF
:-==
==
ò ss
ss
òò
òòD+
-=D+
-=
-=-=
c
yA
c
yA
dytyI
MMdAyI
MMF
dytyIMdAy
IMF
1
1
2
1
)()()(
)(
l Differential force on area dA
l Resultant normal forces
7-7. (continued)
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
òD
-=-=c
yH dytyIMFFV
112 )(
òò
ò
÷ø
öçè
æ-=÷
ø
öçè
æ-
DD
=
DD
-==
®D
c
y
c
yx
c
ys
Havg
dytytIdx
dMdytytIx
M
dytyxtI
MAV
11
1
110
)()(lim
)(
t
t
tIQV
H -=t
l Summation in horizontal direction
: Q is first moment of area
24
úúû
ù
êêë
é-÷
øö
çèæ==
==
ò
ò
21
22
221
1
yhIVdyty
tIV
dytytIV
tIQV
h
y
c
yt
l Transverse shear stress at a point of section
7-7. (continued)
( ) AV
htV
hthV
IhV
23
23
1288 3
22
max ====t
l This equation is used for rectangular cross sectionl Maximum shear stress is 1.5 times average shear stress
l This equation is worthless for I-beam or T-sections
For rectangular cross sections3% error : beam with depth having twice width
12% error : beam with square cross section100% error : beam with width having four times depth
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
l This equation is used for rectangular cross sectionl Maximum shear stress is 1.5 times average shear stress
l This equation is worthless for I-beam or T-sections
For rectangular cross sections3% error : beam with depth having twice width
12% error : beam with square cross section100% error : beam with width having four times depth
25
inyC 4)2(10)10(2)1)(2(10)7)(10(2=
++
=
7-7. (continued)
l Transverse shear stress at a point of section
( ) ( ) )( 82822 1
21
221
21
1
1
<<--=-=
== ò
yyI
VycI
V
dytytI
VtIQV c
yt
l Location of centroid
AMy A
c =Ü
l Variations of shearing stresses on an inverted T-shaped transverse section
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
( ) ( ) )( 82822 1
21
221
21
1
1
<<--=-=
== ò
yyI
VycI
V
dytytI
VtIQV c
yt
l Discontinuity at a junction of flange and stem because of abrupt change in thickness
l Maximum longitudinal and transverse shear stresses occur at the neutral surface at a section where transverse shear V is maximum.
( ) ( ) )( 24422 1
21
221
22
2
1
-<<--=-=
== ò
yyI
VycI
V
dytytI
VtIQV c
yt
26
7-7. (continued)
Let shear V be 37.5 kN for W203 X 22 section (I=20 X 106mm4)
at the neutral axis
MPamNtIQV
mmmQ
w
NANA
NA
9.32/)10(89.32
)10)(2.6)(10(0.20)10)(76.108)(10(5.37)10(76.108)10(76.108
)5.47)(2.6(95)99)(8(102
26
36
63
3633
@=
==
==
+=
--
-
-
t
l Detremination of the shearing stress in an I-beam
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
MPamNtIQV
mmmQ
w
NANA
NA
9.32/)10(89.32
)10)(2.6)(10(0.20)10)(76.108)(10(5.37)10(76.108)10(76.108
)5.47)(2.6(95)99)(8(102
26
36
63
3633
@=
==
==
+=
--
-
-
t
at the junction between web and flange
MPamNtIQV
mmmQ
w
JJ
J
4.24/)10(43.24
)10)(2.6)(10(0.20)10)(78.80)(10(5.37
)10(78.80)10(78.80)99)(8(102
26
36
63
3633
@=
==
===
--
-
-
t
27
7-7. (continued)
- Shear force V causes shear stress, yxxy tt =
- Resolve this shear stress into n-t coordinate system
xnxtxy and ttt Þ
0== nxxn tt- Outside surface is a free surface
- Any shearing stress at point A must be tangential to the surface of the shaft and not in the direction of shear force
l Beam with a solid circular cross section subjected to shaer load V
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
- At neutral axis, shear stress in the direction of V is
AV
rrrrV
ItVQ
NA
NANA 3
424
3424
2
====)()())((
max ppptt
- Any shearing stress at point A must be tangential to the surface of the shaft and not in the direction of shear force
32
34
2
32 rrrQNA =´=Üp
p
28
7-7. (continued)
(Example 7-10) Determine
(a) average shearing stress on a horizontal plane 4” above the bottom of the beam and 6’ from the left support
Shear force V on 6’ from the left support is 900 lbFirst moment Q4 is
3444 48426 inAyQ C === ))((
Second moment of area about the neutral axis is
Average shearing stress on a horizontal plane 4” above is
42323 253332102101213102102
121 inINA .))(())(())(())(( =+++=
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Average shearing stress on a horizontal plane 4” above is
42323 253332102101213102102
121 inINA .))(())(())(())(( =+++=
psitI
VQNA
54022533
489004
44 .
)(.)(
===t
(b) Maximum transverse shearing stress in the beam
Maximum shearing stress will occur at the neutral axis on the cross section occurring largest shear force V.
364824 inQNA == ))((
psitI
VQsNA
NA 05422533
64900 .)(.)(
max ===t
29
7-7. (continued)
(c) Average shearing stress in the joint between the flange and the stem at 6’ from the left support
psitI
VQinAyQ
sNA
FJ
FCFF
65022533
60900602103 3
.)(.)())((
===
===
t
(d) Force transmitted from the flange to the stem by the glue in a 12” length of the joint at a section 6’ from the left support
lbAV JJg 112152126350 .))((. === t
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
lbAV JJg 112152126350 .))((. === t
(e) Maximum tensile flexural stress in the beam
)(..))((max
max TpsiI
cM 112153533
8126750===s
30
7-8. Principal stresses in flexural members
l To find principal stresses and maximum shearing stresses at the selected points on the sections of maximum shear V and maximum bending moment M
l flexural stress is maximum
l transverse and longitudinal shearing stresses are zero
flexural stresses are principal stress
20-
= pstmax
l At point on the top and bottom edge of the section
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
20-
= pstmax
l transverse and longitudinal shearing stressesare maximum
l flexural stress is zero
Shearing stresses are maximum shearing stresses
l At point on neutral axis
31
7-8. (continued)
(Example 7-10) Cantilever beam carries a uniformly distributed load of 160kN/m on a span of 2.5m. Determine maximum normal and shearing stresses in the beam.
Maximum bending moment and shearing transverse shear are
kNwLV
mkNwLM
400)5.2(160
5002)5.2(160
2
22
===
×-=-=-=
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
W610x145I=1243x106 mm2
S=4079x103 mm3
32
7-8. (continued)
At neutral axis, flexural stress is zero and
MPamNtI
VQmmQNA
06110986001190101243
10255210400
102552614291112852957198304
266
33
36
./)(.).)(())(.)((
)(.).)(.(.))(.(.
@===
=+=
-
-
t
In the web at the junction with top flange
)(./)(.)(
).)()(( TMPamNIyM 71141068114
1012432851010500 26
6
3
@=-
-=-= -s
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
MPamNtI
VQmmQJ
94710894701190101243
10771110400
1077112957198304
266
33
36
./)(.).)(())(.)(()(.))(.(.
@===
==
-
-
t
)(./)(.)(
).)()(( TMPamNIyM 71141068114
1012432851010500 26
6
3
@=-
-=-= -s
At the top surface, transverse shearing stress is zero and
)(./)(.)()( TMPamN
SM
IyM 61221058122
10409710500 26
6
3
@=-=-=-= -s
33
7-8. (continued)
For the points at the junction of web and flange
deg..
).(tantan
..).(.)(....)(....
..
).(..
minmaxmax
,
9319068114
89472212
21
47441742
3717051322
3417341741743457
1132051324174345741743457
89472
0681142
068114
22
11
2
1
22
22
21
-=-
-=
-=
@=--
=-
=
@-=-=
@=+=±=
-+÷øö
çèæ -
±+
=
+÷÷ø
öççè
æ -±
+=
--
yx
xyp
p
p
xyyxyx
pp
MPaMPa
CMPaMPaTMPaMPa
sst
q
sst
s
s
tssss
s
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
deg..
).(tantan
..).(.)(....)(....
..
).(..
minmaxmax
,
9319068114
89472212
21
47441742
3717051322
3417341741743457
1132051324174345741743457
89472
0681142
068114
22
11
2
1
22
22
21
-=-
-=
-=
@=--
=-
=
@-=-=
@=+=±=
-+÷øö
çèæ -
±+
=
+÷÷ø
öççè
æ -±
+=
--
yx
xyp
p
p
xyyxyx
pp
MPaMPa
CMPaMPaTMPaMPa
sst
q
sst
s
s
tssss
s
Stress Top edge Junction Neutral axis
sp1 122.6 MPa (T) 132.1 MPa (T) 61.0 MPa (T)
sp2 0 17.37 MPa (C) 61.0 MPa (C)
tmax 61.3 MPa 74.7 MPa 61.0 MPa
34
7-9. Flexural stresses – Unsymmetric bending
zkyka 21 ++=s
l stress variation is
òò
ò
-=
=
==
Arz
Ary
A
dAyM
dAzM
dAR
s
s
s 0
l force and moment equilibrium are
òòòòòò
òòò
---=
++=
=++=
AAArz
AAAry
AAA
dAyzkdAykdAyaM
dAzkdAyzkdAzaM
zdAkydAkdAaR
22
1
221
21 0
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
òòòòòò
òòò
---=
++=
=++=
AAArz
AAAry
AAA
dAyzkdAykdAyaM
dAzkdAyzkdAzaM
zdAkydAkdAaR
22
1
221
21 0
l The origin of coord. is at the centroid of cross section
òòòòò
===
==
AyzAyAz
AA
dAyzIdAzIdAyI
dAzdAy
,, 22
0
Iy, Iz : second moment of areaIyz : mixed second moment of area
35
7-9. Flexural stresses – Unsymmetric bending
yzzrz
yyzry
IkIkMIkIkM
AaR
21
21
0
--=
+===
22
21
0
yzzy
yzrzzry
yzzy
yzryyrz
IIIIMIM
k
IIIIMIM
k
a
-+
-=
-+
-=
=
l elastic flexural formula for unsymmetric bending is
rzyzzy
yzyry
yzzy
yzz
yzzy
yzrzzry
yzzy
yzryyrz
MIII
zIyIM
IIIyIzI
zIII
IMIMy
IIIIMIM
úû
ùêë
é
-+-
+úû
ùêë
é
--
=
úû
ùêë
é
-+
+úû
ùêë
é
-+
-=
22
22s
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
rzyzzy
yzyry
yzzy
yzz
yzzy
yzrzzry
yzzy
yzryyrz
MIII
zIyIM
IIIyIzI
zIII
IMIMy
IIIIMIM
úû
ùêë
é
-+-
+úû
ùêë
é
--
=
úû
ùêë
é
-+
+úû
ùêë
é
-+
-=
22
22s
l orientation of neutral axis is
0=+++- zIMIMyIMIM yzrzzryyzryyrz )()(
zIMIMIMIM
yyzryyrz
yzrzzryúû
ùêë
é
++
=
l slope of neutral axis is
yzryyrz
yzrzzry
IMIMIMIM
dzdy
++
== btan
36
7-9. (continued)
(Example 7-12) (a) Flexural stress at A
)(781.0781.0)300()108()240(135)5.4(108)4(1352
2
22
Cksiksi
MIIIzIyI
MIIIzIyI
MIIIyIzI
A
rzyzzy
yzy
rzyzzy
yzyry
yzzy
yzz
=-=úû
ùêë
é-
+-=
úúû
ù
êêë
é
-
+-=
úúû
ù
êêë
é
-
+-+
úúû
ù
êêë
é
-
-=
s
s
(b) Orientation of neutral axis
deg7.38
800.0135108tan
=
===+
+=
b
by
yz
yzryyrz
yzrzzry
II
IMIMIMIM
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
deg7.38
800.0135108tan
=
===+
+=
b
by
yz
yzryyrz
yzrzzry
II
IMIMIMIM
(c) Maximum tensile and compressive flexural stresses
)(16.1016.10)300()108()240(135)5.1(108)4(135
)5.1,4(
)(16.1016.10)300()108()240(135)5.1(108)4(135
)5.1,4(
2
2
Tksiksi
CAt
Cksiksi
BAt
C
B
==úû
ùêë
é-
---=
-
=-=úû
ùêë
é-
---=
-
s
s
37
7-10. Stress concentrations under flexural loadings
l flexural stress in smooth member
IMy
-=s
l flexural stress in the vicinity of discontinuity
÷øö
çèæ-=
IMyKs K : stress concentration factor that depends on geometry of the member
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
38
7-10. (continued)
494333
1016510165122550
12
41252
10301030
mmmbhI
MPaFS
MPa
uall
u
)(.)(.)(.
-====
===
=
ss
s
(Example 7-13) Cantilever is made of SAE 4340 steel and is 50mm wide. Determine the maximum safe moment M if FS of 2.5 with respect to failure by fracture is specified.
(1) For r=5mm,5112025532575 .., =Þ==== tKhrhw
(2) For r=10mm
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(2) For r=10mm
28140251032575 .., =Þ==== tKhrhw
(3) For r=15mm18160251532575 .., =Þ==== tKhrhw
ttt
a
KKyKIM 2146
105121016510412
3
96
=-
-=-= -
-
))(.())(.)((s
yI
KyIM
IMy
t
allsss -=-=Þ-=
(1) For r=5mm mNK
Mt
×=== 1421511
21462146.
using