principal stresses and strains to vsr
TRANSCRIPT
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rincipal Stresses & StrainsConceptual Perspect ive
By,Bharath V G
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Principal Planes:--The planes which carry only Direct Stressesand no Tangential
Stressesare called Principal Planes
Principal Stresses:--
The intensity of stress on the Principal Planes are known as
Principal Stresses
1
1
2
pppn
pt
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1
1 2
2
pppn
pt
All planes parallel to this planes are
principal plane with zero principalstress
All planes parallel to this planes arePrincipal plane with principal stress p
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!"portance of #inding Principal Planes:--
1$ The Principal Planes carry the %a&i"'" Direct Stress
(Tensile or )o"pressi*e+$
2$ The Principal Planes helps to locate the Planes of%a&i"'" Shear(!nclined at , with Principal Planes+
,
1
1 2
2
pp
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)ase:- 1.le"ent S'/0ected to Direct Stress Shear
Stresses :--
)onsider an ele"ent s'/0ected to the direct stresses p p
(/oth tensile+ alongwith a shear stress of intensity 3 as
shown in fig're$ The shear stress intensity on all the sides will
/e 3 for e3'ili/ri'" of the ele"ent$ 4et t /e the
thickness of the ele"ent$
)onsider a plane 5. inclined at angle with 5)$
A 5
)
pp
p6
p6
3
3
D .#orce on face 5) P1 p 7 5) 7 t
#orce on face 5) P2 3 7 5) 7 t
#orce on face ).
P8 3 7 ). 7 t
#orce on face ). P, p 7 ). 7 t
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#orce on face 5)
P1 p 7 5) 7 t
#orce on face 5) P2 3 7 5) 7 t
#orce on face ).
P8 3 7 ). 7 t
#orce on face ). P, p 7 ). 7 tP2si
n
P2co
s
P,
5
).
P1
Pn
Pt
P1cos
P1sin
P,cos
P,sin
P8
P2
P8cos
P8sin
9esol*ing forces parallel perpendic'lar to 5.
Pn (P1;P8+cos ; (P2;P,+sin Pt (P1;P8+ sin - (P2;P,+cos
A 5
)
p
p
p6
p6
3
3
D .
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9esol*ing forces perpendic'lar parallel to 5.
Pn (P1;P8+cos ; (P2;P,+sin
Pt (P1;P8+ sin - (P2;P,+cos
pn Pn < (5.$t+
=(p75)7t ;37).7t+ )os;(375)7t ; p 7).7t+
Sin> < (5.$t+
p cos2; 3 sincos; 3 sincos; p sin2 p
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To locate Principal Planes:--
pn ?(p ; p+
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tan2 23 < (p - p+ There are two
*al'es of which will satisfy the
a/o*e e3$ 1 2 where 2CB; 1
sin2123
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CB
All planes parallel to this plane are principal
planes and principal stress p1 inclination
All planes parallel to this plane are principal planes
and principal stress p2 inclination ; CB deg$
p
p
3
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4et t /e the thickness of the ele"ent$
)onsider a plane A) inclined at angle with A5$
A
5)
p1p1
p2
p2
pn
pt
p2
A
5)
p1pn
pt
p1cos
p1sin
p2cosp2sin
To locate Planes of %a&i"'" Shear Stress:--p1 (p;p+
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)onsider a plane A) inclined at angle with A5$ ow #orce on faceA5 P1 p17A57t
#orce on face 5) P2 p275)7t
.3'ating all the forces along to A)Pt P1sin- P2cos
p1 A5 t sin- p2 5) t cos
pt Pt < (A)$ t+
p1 A5 t sin- p2 5) t cos
A)$ t
p1sin(A5
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The Shear stress pt (p1- p2+sincosE
pt (p1- p2+sincosE
(p1- p2+
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p1 (p;p+
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p1 (p;p+
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%ohrs Stress )ircle %ethod:--
Ghen principal stresses planes at a point in a two
di"ensional stresses are gi*en we can 'se the %ohrsStress )ircle %ethod to o/tain the nor"al and
tangential stress-intensities across a gi*en plane
thro'gh the point$
!n %ohrs )ircle E
The or"al Stress (pn+ is plotted on L-a&is$ The Tangential Stress (pt+ is plotted on M-a&is$
Angle on %ohrs circle is taken as twice the gi*en
angle
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A 5
)
pp
p6
p6
3
3
D .
Sign )on*entions:--1$ Tensile Stresses are taken as Positi*e (;*e+
2$ )o"pressi*e Stresses are taken as egati*e (-*e+
8$ )lockwise Shear is taken as (;*e+
,$ Anticlockwise Shear is taken as (-*e+
Nor"al Stress (pn+
She
ar
Str
ess
(pt+
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%ohrs Stress )ircle %ethod:--
)ase:-1 !f an ele"ent is s'/0ected to the direct stresses (p
p+ shear stress (3+ then the principal stresses are (p1 p2+and the nor"al and tangential stresses (pn pt+ across a plane
inclined at angle to the plane carrying stress p are
p1 (p;p+
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)
P()D+
Npn
Op6
3pt
P(5)+
O
p
3
1$ Select origin N and draw a&is NL$
2$ Draw NO p $ Nn faces 5) DA the shear stress 3 is
anticlockwise ( -*e+ draw OP 3 (downwards+8$ NO p and on faces A5 )D the shear stress 3 is
clockwise hence it is ;*e so draw OP 3 ( 'pwards as it is
;*e+
,$ oin the points P P which will intersect NL at )$
A 5
)pp
p6
3
3
DSteps to Draw %ohrs )ircle:-
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$ Point ) is the )entre of %ohrs )ircle$ Gith ) as centre and radi's
e3'al to )P or )P draw a circle which will intersect NL at points A 5$
Q$ %eas're NA which gi*es "a&i"'" direct stress and no Shear stress
it is %a0or principal stress (p1+ NAp1
%eas're N5 as %inor principal stress (p2+ N5 p2$
p2A
)
5
()D+P
N
p1
pnOp6
3
pt
P(5)+
O
p
3
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F$ #ro" plane )P "eas're angle as 2anticlockwise'p to )A therefore the principal plane will /e located
at anticlockwise fro" face 5)$
p2A
)
5
()D+P
N
p1
pn2
Op6
3pt
P(5)+
O
p
3
A 5
)pp
p6
3
3
D
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R$ %eas're )T which gi*es "a&i"'" Shear stress $ Ghich is inclined
at CB 2FB in %ohrs circle so in act'al case it is inclined at , 18
with principal planes$
p2A
)
5
()D+P
T
N
p1
pn2O
p6
3
pt
P(5)+
O
p3
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)
()D+P9
Npn
O
3pt
P(5)+
O
3
A 5
)
pp
p6
3
3
D
2
9pn
pt
p6
p
C$ #ro" point P (5)+ "eas're angle 2in anticlockwise direction anddraw a line )9$ #ro" 9 draw 99 on NL$ %eas're N9 as pn and 99
as pt$
f
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Proof:-
OO NO - NO p - p
)O )O (p - p+
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p2
A
)
5
P
T
N
p1
pn2O
p6
3
pt
p -p
P
O
p
3
T
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let O)P 2 then
tan2 OP
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)ase:- 2 .le"ent S'/0ected to direct stress
Stresses only and no shear stress:--
)onsider an ele"ent s'/0ected to only Direct Stresses p
p (/oth tensile+ as shown in fig're$ Iere the shear stress
intensity 3 is zero$
A
5)
pp
p6
p6
pn
pt
1 .l t S /0 t d t Di t Sh St
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p1 (p;p+
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p1 (p;p+
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A
5)
p
p6
p6
pn
pt
p
)ase:- 2.le"ent S'/0ected to Direct Stress only and
no shear stress:--
pn (p ; p+
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%ohrs Stress )ircle %ethod:--
)ase:-!! !f the tangential stresses acting on ele"ent
is zero then the nor"al and tangential stresses acrossa plane inclined at angle to the plane carrying p are
pn (p ; p+
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1$ Set off the a&is NL on this set off NA p N5 p
2$ 5isect 5A at )$
8$ Gith ) as centre and )A or )5 as radi's draw a circle$
This circle is known as %ohrs circle$
Steps to Draw %ohr s )ircle:-
p6
A
)
5N
p
pn
pt
P
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,$ Draw line )P at angle 2 with NL in anticlockwise so that thepoint P is on the circle$$ #ro" P drop perpendic'lar PO on L a&is$ %eas're NO as pn
"eas're PO as pt$
Q$ oin and "eas're NP which gi*es res'ltant stress pr$
p6
A
)
5N
p
pn
ptP
2
O
pn
pt
pr
T
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F$ oin "eas're )T which represents the plane of "a&i"'" shear
stress$ !t inclined at CB 2FB in %ohrs circle with principal plane
hence in act'al case they are inclined at , 18 with principal planes$
p6
A
)
5N
p
pn
ptP
2
O
pn
pt
pr
T
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p6
A
)
5
P T
N
p
pn
2
O
pn
ptpr
pt
p - p
) 8 St i N Di ti N l
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)ase :-8 Stress in Nne Direction Nnly:--
)onsider an ele"ent s'/0ected to stress p in only
one direction as shown$Iere the p B 3 B$
1
1 2
2
pp
1 .le"ent S'/0ected to Direct Shear Stresses
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p1 (p;p+
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p1 (p;p+
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)ase :-8Stresses in Nne Direction only :--
pn p cos2
pt p
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Principal Strain:--!f p1 p2 p8 are principal stresses then the principal
strains e1 e2 e8 are gi*en /yE
e1 p1
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ALL THE BEST
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