Som ii {theories of failure}

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
hu a Clarnn
thadiu
be nd
i lwm e
uta
fonoLng oe
that ex pan
Cond t 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 4la odeng
et cns 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
Accehdu ug o i theoy
Faulwe O Ccw nshim he maxi mLm
einp le bus (
th
--
_ ysbm
Teathas Val
sbazs
at t
eladic mu
Cownplex
mple
Dnsiem e m maxumu
webol Comphrsive
he elastic mit (e)
es hache4
Aimpe
W
e
ecm Limpke empo
Ne th max mum e pal 8tesK
te deb ta on, the
MoLR.
nee
exCesd e w
mg
Maas muat
net
S ress mathial
e Duei se)
ilt Br M)
e

4. (3
TKio koAy elUlminalos ke
ako u'nepl Asos amd
oo bamoo hesdas
Fe britla matou
which olo net
hail by ielding,
th
max. princ palAress hoA s CmaidJ
th NE S
as
ut Aasf by becaus ,'b na
Jaie by bte bare
mai als
his thoon appeans be
ploo xi'maTly
ntleo
CeerE ondi
nay CLs and
bu to
melal
Flaos Cobaduettens The thaehy
. Ow a muld 2 l
2peci maw whm
Zimble
Ca ed aut,
h
2lidt
0ccs
aphoimataly 4stoka axio o
puime. Thio shus ot fa du
t
modimLm eo hossNathr han
diect
Jthasbeen tnat a malalwic 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
mple 18
al stress Peng e hr iasti nil stress in simple tensin
is cqual anud
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 failun
5 MN/m
Spe(Gven
), 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 criterionof failure
is the maximum principal stress theory.
olution. Given:
o,
= 40 MN/m?,
Principal stresses
o, = 0
-100 MN/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 Guau o
Thos CasThony
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 inut o
t
dt ta a aD Compo nswl m
whi c amo ach axiall a
stresses, 0f Comhsine
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 nal
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. max
2
Y
Cmoy -()
Sbtae gne
re (
.
Max &hatskres, ( Max
Cowplex yt
Max. Ahea
reks i elsbi
mi 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 an t must
be TabAn as
e ( imph
ansion
2 2
et
FoS
-)
,b O
Max shaa gtre Tohy
r: A mi'ld stulshtt
6amnaxImum g 20 kNm and mAXImum
bendun
20 mm diameb Anbsetd
bendng moment 12 kNm at a þarbola
faels Saaly(Fos) acdig
moALMLm
mpe nim 22o MNM
2
SA
G Lven Dala
Damair
Aht,
d=
12o m
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
an
wipa Touss
C
Noud nd ut T
e
nud +hat Bendng bah
ve
R
M beuoln mam
M. o.
E YoungsMadulusn 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 M N/
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
abi ToAL0r enan

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
(2 39)to+ (6 99
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.OS S- 16 A.
3 MaxI MAXIMUNM PRINCIPAL STRAIN_THEOl
Jhao Theo so callad SE. Venanl theo 4.
---
dhio thaoky als that
Tho mataal
Occwl whn
L mum uincipal tanala Aram mh
maleal Naachas tuhe An 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 ae
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
. To phadeutalura
-C+)< (o)
At e JUST lasiie aikua
+)=
Ond
F D631GN PuRPoE S

16. Aoa Auos t amd egns L
hich Ca Cam
ba danat
by and
and C12
an
pkacad by &oe 8hassas,
daag dua wpraas
(+)= E
(t=
ToNaoinwe
imperlantfoA
tuo
thuoy
aloos net gwe exact Aoulo
duetla matals he (AL, MS eE)
e hoony oes met ft 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 di mungional 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
( 233 7) <
poi n aa 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 nas u eNen les duche
mawual fa i c dy o La to be most
iabe Btt vev ooe adulla nst exaeat.
5SHE AR STRAN ENEGY TaeORY
i thoy also CalldDiskahonEng
Thao kuony Mies-AenRyTkaon
AcchdLng o

19. lune
mabual o cou
whhe
thi ha a a
uut
Vume a sasad mawun
Saachas
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 v uma
th &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
Eq utno 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.3hadas o
ec
mmatal.
boinz aL Comeludad about th
thaouy
The thao y does nat agne uta Tki
xpaiemanlal
hsul t he matials
o quu d khut.
am
ec
(11) F zdro Ge pYeksws bnaim
hiw kao
Ves
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 daseh ens, bitl
o cchs aunol h
maxL um
iwe'p Sres hes T
ive Alalol
neslp.
gandad
Ce s tt magt olurtenalnals undr
vau pus ts adi nj
.