Principal Stresses and Strains to Vsr

8/12/2019 Principal Stresses and Strains to Vsr

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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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)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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)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

THANK YOU

Principal Stresses and Strains to Vsr

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