Hand Written Notes for THEORIES OF FAILURE SOM- II

1. STRENGTH OF MATERI ALS - I
ToPIc 2
HEORIEES
OF
FAILURE

2. OF FAILURE
THEORIES
axamples Shaiks
Du elag number
actong K bodues eginantng prolnca.
Cam omanml
Condiiows nqnd
t m dar
t h u a Clarnn thadiu
be nd
i lwm e u t a
fonoLng oe
that expan
Condt ons .
theDhues laikune hat expkaim
a kun etins Cnoo dkuut
Condutiens
C Maxlmum Pueipal 3toss Ihesaj
Theo
2 Maxumum Shoa 3Kass Sres
ohe
RM TheoA
)Maximum Ynne pal 30am
Thad
bam Thaon.
4)To Sta &nns tho0
5) Sho Sham Enna Thoy
t t 4laodeng e tcns we u duts
olaal. S al
Tende &hets oat the
eloic mit m Aimpe
tinAon Comresienhorp
Oet ec

3. Mo i 'neipal Shass Thuo
Tho
he &implut amd slalast heas
a wne Jho hey io also Callad RankIA
Accehduug o i theoy
Faulwe O Ccw nshim he maxi mLm
einp le bus( t h
- -
_ ysbm Teathas Val
sbazs at t eladic mu
Cownplex
mple Dnsiem e m maxumu
webol Comphrsive
he elastic mit (e)
e s hache4
Aimpe
W
e
ecm Limpke empo
Ne th max mum e pal 8tesK
te deb t a on, the MoLR.
nee
exCesd e w
mg
Maas muat net
S ress mathial
e Dueise)
ilt Br M)
e

4. (3
TKio koAy elUlminalos ke
ako u'nepl Asos amd
o o bamoo
hesdas
Fe britla matou
which olo net hail by ielding, th
max. princ palAress hoA s CmaidJ
th
NES
as
ut Aasf by becaus ,'b na
Jaie by bte bare
mai als
his thoon appeans be plooxi'maTly
ntleo CeerE ondi nay CLs and
bu to melal
Flaos Cobaduettens The thaehy
. Ow a muld 2 l 2pecimaw whm Zimble
Ca ed aut, h
2lidt
0ccs
aphoimataly 4stoka axio o
puime. Thio shus ot fa du
t modimLm e o hossNathr han
diect
Jthas been tnat a malal wic a
eNew thougw t eon m Ame ComT
eNew
Ca
exc tu int m imate

5. n a eallic bodr the pn ipal stresses ne+5MN/m (nd
95 MN/n, the
mple18
al stress Peng e hr iasti nil stress in simple tensin
is cqualanud is >201/V/m nd the tactor f afrin hased
f faihure fer the maie rual is thu mavimum prim ial strss throry
the rlast
limit if h
f f a i l u n
5 MN/m
S p e ( G v e n
) , and 'inipal stresses
9 MN/m
220 MN/m
=Elastic limil stress(/ension).and
herc.
=lastic limit stress(compression).
= , (working stress in tension)
(F.O.S. means factor of safcty)
F.O.S.
220
= 6-28
35
FO.S
Also. o =
G, (working stress in compression)
F.O.S.
ec
-95=
F.O.S.
220
F.O.S.= = 23
95
So. the material according to the maximum principal stress theory will fail due to the
mpressive principal stress.
FO.S. 23 (Ans.)
Example 182. In a cast-iron body the principal stresses are + 40 MN/m* and -
100 MN/m
third principal stress being zero. The elastic limit stresses in simple tension and in simple
mpression are 80 MNIm? and 400 MN/* respectively. Find the factor of safery based on the
ste limit if the criterion offailure is the maximum principal stress theory.
olution. Given: o, = 40 MN/m?,
Principal stresses
o, = 0
- 100MN/m?
80 MN/m* (elastic limit stress in tension)
400 MN/m- (elastic limit stress in compression)
Now, , 0, (working stress in tension)
80
o F.O.S.
or, 40
F.O.S.
FO.S. A =2
or
Also o (working stress in compression)
4o0
4 Fo S 4
F.O.S.

6. MA K,
y
Pt
Shan E Sam
Detile Malhal
yedd o Ductl Matuls
m
FOS
Iec
FoS
B tt
Motuas
foS
ec
FoS

7. Shaa SbiasThoey
2. Ma Lmum
OR
SboLs DiAeomt huoy
Thus thoy Called Guauo
ThosCasThony
thot
OCcw Kan
maxi
MaxL mum Kian sDs Omax
tht Comblx AyK&n aachos hi abua
MALXLum hao hoss sLmple
m
nsigr at tke elastic inuto t
dt t a a aD Compo nswl m
whi c amo ach axiall
a
stresses, 0f Comhsi ne
Bix
Zer

8. muidl be
CA
oluctile
ma leuals mde Lhoo
mnd hat tals
weakast
dig due tile mauals e
hei b
.3a4 Considv,Max. hoa tTes Theo
alays
JAL Lhaa
A hear Anoas darlapad vu based on th badn
el vaues and
M
Sbusses n a l le each dh
bo
Shear 8hels olevelsped t boc.
hw be
Mokas Cucle Cau
se
phuneip as
+ ( +
2
I+(*
Mat Sha 2hs
2
Noud
'

9. m a x
2
Y
Cmoy -()
Sbtae gne re ( .
Max &hatskres,( Max
Cowplexyt
Max. Ahea
reks i elsbi
m i w
2
Ma mat col RLTon
Jh mAx
( ot eld punlv
mo-x alue mahual
divi de h
max Aheat ArAs
Sot (Fo, t matoual willbe Lafe.
Codhon
J v
FoS naial.

10. hlds t ductle
malua
tas b t e rimipal Atessos
CopreAsie a n t must be TabAn as
e ( imph ansion
2 2
et
FoS
- )
,b O Max shaa gtre Tohy
r: A
mi'ld stul shtt
6amnaxImum g 20kNm and mAXImum
bendun
20 mm diameb Anbsetd
bendng moment 1 2kNm at a þarbola
faels Saaly(Fos)acdig
moALMLm
mpe nim 22oMNM
2
SA
GLven Dala
Damair Aht, d=
12om
Radw 6o mm
Tong,T= 2o KNm
aimum bending momLnt, M= 12Nn
Et t m Aimtle sm, t= 220 M
MaAmum
Fac S (Fos
FoS
(-
TT

11. 10 Ae hane
ins
n dhot bendun4 4hu amd
Hham
iven
max dhoo res a n
wipa Touss
C
Noud nd ut T
e
nud +hat Bendng bah
ve
R
M beuoln mam
M. o.
E YoungsMadulus n Elste
R Radie CwhNw
Ferw abeve
M M
M z Sehon mod

12. :)
Z
T
TT
4
2
M
3 od3
0.73
F0.73 MN/
Llen
Ce
Me dao emod t Tu
ToA Epni
le aldo
s ToRAm Eqmaton
Ge
TE MaxLmum ToRg
J Polar M.0.2 = d
32
Modul
Ang Auet m oda n
Ahot
Ahart mabral
R Radius t kt
a b i ToAL0r e n a n

13. R
Tol
T T. d
C
16
32
16T 1X o o x 10
TT d3 TTx (o.12)
s8.9s MNm
b
e
(239)to+(699
2
3S 34) t (3s.36)+ (sa.5
3S.36 + 68.74
= 04.12 MN
2
33.38 38 MN mn
AcLhdin T Max.
res The
-3)=

14. 13
2
137.5 MN
2
220 M
FOS=
- ) 1375MN/
F.OSS-16 A.
3 MaxI
MAXIMUNM PRINCIPAL STRAIN_THEOl
Jhao Theo so callad SE. Venanl theo4.
- - -
dhio thaoky als that
Tho mataal Occwl whn
L mum uincipal tanala Aram mh
maleal Naachas tuhe A n eloktc
u'wei pal tomphuss ve sngm reches Hhe sa'
m Bimla Compsioa.
b
e -lGt -
A ne Sa diaeton 1 PS
m
Nao Condi Cawc a e
CCndi na Mak. riwe Shain tho 4?
e, 7 6et (6,ust se +ve)

15. and (Emust be- Ve)
e Ca Aa hat
te
(3)
e,u +ve)
E
Sec
eg7 (¬, - Ve
TRat wLans
-
-(5+5) E
amd
C6)
E (+7 6)
E
et 7)
- ( + ) 7
(8.)
- ( +)7
.
Tophadeutalura
- C+)< (o)
At e JUST lasiie aikua
+ ) =
Ond
F D631GN PuRPoES

16. Aoa Auos t amd egns
L
h i c h
CaCam ba danat by and
and C12 an
pkacad by &oe 8hassas,
daagdua wpraas
(+)= E
(t=
ToNa oin we
imperlant foA tuo thuoy
aloos net gwe exact Aoulo
duetla matalshe (AL, MS eE)
e hoony oes met f t wel w-T Tha
pel mumlal herulK excopt ouly bout6
maials ke () f biaxial n im
Cmprosson & (-f sdichh omeTmy
h mmom dud)
Jwi hoy t wped muchv
haaticad poAas
ADA
PTO

17. 19
STRAIN ENERGY. TH6RY
ToTAL
J theo Ay ha Ou
lheimodynaMLC
nalay ad as a loqical load
Omd r en b Belbami Ho h
Smokoy tats that
i ue malual
maual wl_occu
mawal ho_arhas tt Am anst
-
mati al elasti mitE m
pleGaiom
A dimungional Ahs gn, mu lkom
na wit veum
iven b
U ++2v(+ +S)
2E
md awe Sama
Ne at t toua
2-E
o
+5)-a»ic *"s+5] =
w
douaw braci cas, Th e elashe mE
oheglaro b

18. G -
2. t + )= -- -0)
FoS
N (A o able Sveus)
FoS
we tet, db Cau (= o) an n (i) aduos
FoS
2
( 2337)<
poi n a a conolndud abat thup thaa
() J esudi 1tkoshy Q imilsl T
expman rlb duet matuals(usmch
ndrtialdi ng as i
(2 h thioy an nat be abpeed la mathuals uhorr
emd na
theoy does not e exact houl a
Comporad aapai'mantl nasu eNen les duche
mawual fa i c dy o La to be most
iabe Btt v e v ooe adulla nst exaeat.
5SHE AR STRAN ENEGY TaeORY
i thoy also CalldDiskahon Eng
Thao
kuony Mies-AenRyTkaon
AcchdLng o

19. l u n e
mabual ocou
whhe thi ha a a
u u t
Vume a sasad mawunSaachas
veme
he_haah_ha Onea1
at_te elosbc nt bet m simke
andion, "
ha &Mam enua dua ti bt'neip
eskas and e ww.t vuma
t h &bussed aua
2
U -)+(%-)+(
-()
Nou Aimpetmsin tst m laic
p'nap
pent than
s ie pt Th Aha Aham enui* b
unit Vobuwne tvan
2
E+-(-
12c
et et
2 2
Equtno ws U) amd 2)
p
c-a +C-+(g-| 2

20. 2
(- + )+ C-)= a
Sn ackal odsdign aplacod by ba
valat Atnoas im
2
( ( -) +
(-5) = 32
Fos
h ab eve thao Kae beew Bund k ne bost
nesulb5
nasulo dethle motials r wic
e appoimot.3ha das o
ec
mmatal.
boinz aL Comeludad about th thaouy
The thaoy does nat agne uta Tki xpaiemanlal
hsul t he matials
o quu d khut.
am
ec
(11) F zdro Ge pYeksws bnaim hiw kao
V e s
at=
O, w mans T mawuk
fal da amy noa k Psme o
icw
a
obvi Oy nw Prsb
A cua n tue enal n S'onaA
nei p dasehens, bitl
o cchs aunol h maxLum
iwe'p Sres hes
T ive Alalol
neslp.
gandad
Ce s t t magt olurtenalnals undr
vau pus t s adinj .