plane stress loading

7/29/2019 Plane Stress Loading

1/19

Plane StressLoading

1

x

Therefore, the state

of stress at a pointcan be defined by

the three

independent

stresses:x; y; and xy

x

y x

y

A

A


2/19

REFERENCES:

http://en.wikipedia.org/wiki/

Principal_stress#Overview

http://www.learnengineering.

org/2013/01/principal-stress-

plane-mohr-circle.html#

www.enduratec.com
http://en.wikipedia.org/wiki/Principal_stresshttp://en.wikipedia.org/wiki/Principal_stresshttp://www.learnengineering.org/2013/01/principal-stress-plane-mohr-circle.htmlhttp://www.learnengineering.org/2013/01/principal-stress-plane-mohr-circle.htmlhttp://www.learnengineering.org/2013/01/principal-stress-plane-mohr-circle.htmlhttp://www.learnengineering.org/2013/01/principal-stress-plane-mohr-circle.htmlhttp://www.learnengineering.org/2013/01/principal-stress-plane-mohr-circle.htmlhttp://www.learnengineering.org/2013/01/principal-stress-plane-mohr-circle.htmlhttp://www.learnengineering.org/2013/01/principal-stress-plane-mohr-circle.htmlhttp://www.learnengineering.org/2013/01/principal-stress-plane-mohr-circle.htmlhttp://www.learnengineering.org/2013/01/principal-stress-plane-mohr-circle.htmlhttp://www.learnengineering.org/2013/01/principal-stress-plane-mohr-circle.htmlhttp://www.learnengineering.org/2013/01/principal-stress-plane-mohr-circle.htmlhttp://www.learnengineering.org/2013/01/principal-stress-plane-mohr-circle.htmlhttp://www.learnengineering.org/2013/01/principal-stress-plane-mohr-circle.htmlhttp://en.wikipedia.org/wiki/Principal_stresshttp://en.wikipedia.org/wiki/Principal_stress


3/19

IntroductionStress at a point is resolved into normal

and shear components

When there is no shear stresses acting on

the planes giving the maximum normal

stress acting on the planes.

These planes are called the principal

planes, and stresses normal to these planes

are the principal stresses 1, 2 and 3 whichin general do not coincide with the cartesiancoordinate

axes x, y, z.

When a material is deformed by an external loading,

work done during elastic deformation is stored as

elastic energy and will be recovered when the load is

released.


4/19

Objective

4

A

x

y State ofStress at A

whatwould be

x, y,

and xy?

x


5/19

opyright

001

Transformation

5

x

y

x

A

State of

Stress at A

x

y x

y xyx=

?

xy=?


6/19

Transformation

6

2sin2cos22

' xyyxyxx

2os2in2

' cs xyyxxy

Solving equilibrium

equations for the wedge


7/19

Principal Planes & Principal

Stresses

7

PrincipalPlanes

~ are the two planes where the

normal stress () is themaximum or minimum

~ the orientations of the

planes (p) are given by:

yx

xy

p

2tan2

1 1

gives two values

~ there are no shear stresses on

principal planes

~ these two planes are mutuallyperpendicular


8/19

Principal Planes &

Principal Stresses

8

x

p

1

p

2

90

Orientation ofPrincipal Planes


9/19

Principal Planes & Principal

Stresses

9

PrincipalStresses

~ are the normal stresses () actingon the principal planes

Ryx

21max

Ryx

2

2min

2

2

2xy

yx

R


10/19

Maximum Shear(max)

10

~ maximum shear stress occurs on

two mutually perpendicular

planes

xy

yxs

2

tan2

11

gives two values

(s1 and s2)

~ orientations of the two planes

(s) are given by:

max = R2

2

2xy

yx

R


11/19

Maximum Shear

11

x

s

1

s

2

90

Orientation of MaximumShear Planes


12/19

Principal Planes & Maximum

Shear Planes

12

x

Principal

plane

Maximum

shear plane

p = s45

45


13/19

Plane Strain Loading

13

x

~ where all elements

of the body aresubjected to normal

and shearstrains

acting along a plane

(x-y); noneperpendicular to the

plane (z-direction)

z

= 0; xz

=

0; zy = 0


14/19

Plane Strain

Transformation

14

Similar to previous

derivations. Just replace by ,and by/2


15/19

Plane Strain

Transformation

15

SignConvention:

Shear strain ( ): decreasingangle positive

e.g

.,

Normal strains (x and y):extension positive

x

y

bef

ore

x

y

afte

r

xpositi

veynegativepositi

ve


16/19

Plane Strain

Transformation

16

2sin2

2cos22

' xyyxyxx

2os2

2in22

'cs xyyxxy

Same format as

the stress

transformatione uations


17/19

Principal Strains

17

yx

xy

p

1tan

2

1Gives two values

(p1 and p2)

~ maximum (1) and minimum (2)principal strains

~ occur along two mutuallyperpendicular directions, given by:

Ryx

21

Ryx

21

22

22

xyyxR


18/19

Maximum Shear Strain

(max)

18

max/2

= R22

22

xyyxR

p = s45


19/19

References: http://www.google.co.in/url?sa=t&rct=j&q=ignition%

20system%20pdf&source=web&cd=1&cad=rja&ved=0CCkQFjAA&url=http%3A%2F%2Fwww.ignou.ac.i

n%2Fupload%2Funit%25204.pdf&ei=N7wTUufENsi

XrAfn24EQ&usg=AFQjCNEqTMr1qLrKhrtETcfCqS

2Fh88vIQ&bvm=bv.50952593,d.bmk

http://en.wikipedia.org/wiki/Ignition_system#Modern_

ignition_systems

I.C Engines By V.Ganesan ,TMH Punlication.
http://www.google.co.in/url?sa=t&rct=j&q=ignition%20system%20pdf&source=web&cd=1&cad=rja&ved=0CCkQFjAA&url=http://www.ignou.ac.in/upload/unit%204.pdf&ei=N7wTUufENsiXrAfn24EQ&usg=AFQjCNEqTMr1qLrKhrtETcfCqS2Fh88vIQ&bvm=bv.50952593,d.bmkhttp://www.google.co.in/url?sa=t&rct=j&q=ignition%20system%20pdf&source=web&cd=1&cad=rja&ved=0CCkQFjAA&url=http://www.ignou.ac.in/upload/unit%204.pdf&ei=N7wTUufENsiXrAfn24EQ&usg=AFQjCNEqTMr1qLrKhrtETcfCqS2Fh88vIQ&bvm=bv.50952593,d.bmkhttp://www.google.co.in/url?sa=t&rct=j&q=ignition%20system%20pdf&source=web&cd=1&cad=rja&ved=0CCkQFjAA&url=http://www.ignou.ac.in/upload/unit%204.pdf&ei=N7wTUufENsiXrAfn24EQ&usg=AFQjCNEqTMr1qLrKhrtETcfCqS2Fh88vIQ&bvm=bv.50952593,d.bmkhttp://www.google.co.in/url?sa=t&rct=j&q=ignition%20system%20pdf&source=web&cd=1&cad=rja&ved=0CCkQFjAA&url=http://www.ignou.ac.in/upload/unit%204.pdf&ei=N7wTUufENsiXrAfn24EQ&usg=AFQjCNEqTMr1qLrKhrtETcfCqS2Fh88vIQ&bvm=bv.50952593,d.bmkhttp://www.google.co.in/url?sa=t&rct=j&q=ignition%20system%20pdf&source=web&cd=1&cad=rja&ved=0CCkQFjAA&url=http://www.ignou.ac.in/upload/unit%204.pdf&ei=N7wTUufENsiXrAfn24EQ&usg=AFQjCNEqTMr1qLrKhrtETcfCqS2Fh88vIQ&bvm=bv.50952593,d.bmkhttp://www.google.co.in/url?sa=t&rct=j&q=ignition%20system%20pdf&source=web&cd=1&cad=rja&ved=0CCkQFjAA&url=http://www.ignou.ac.in/upload/unit%204.pdf&ei=N7wTUufENsiXrAfn24EQ&usg=AFQjCNEqTMr1qLrKhrtETcfCqS2Fh88vIQ&bvm=bv.50952593,d.bmkhttp://en.wikipedia.org/wiki/Ignition_systemhttp://en.wikipedia.org/wiki/Ignition_systemhttp://en.wikipedia.org/wiki/Ignition_systemhttp://en.wikipedia.org/wiki/Ignition_systemhttp://en.wikipedia.org/wiki/Ignition_systemhttp://www.google.co.in/url?sa=t&rct=j&q=ignition%20system%20pdf&source=web&cd=1&cad=rja&ved=0CCkQFjAA&url=http://www.ignou.ac.in/upload/unit%204.pdf&ei=N7wTUufENsiXrAfn24EQ&usg=AFQjCNEqTMr1qLrKhrtETcfCqS2Fh88vIQ&bvm=bv.50952593,d.bmkhttp://www.google.co.in/url?sa=t&rct=j&q=ignition%20system%20pdf&source=web&cd=1&cad=rja&ved=0CCkQFjAA&url=http://www.ignou.ac.in/upload/unit%204.pdf&ei=N7wTUufENsiXrAfn24EQ&usg=AFQjCNEqTMr1qLrKhrtETcfCqS2Fh88vIQ&bvm=bv.50952593,d.bmkhttp://www.google.co.in/url?sa=t&rct=j&q=ignition%20system%20pdf&source=web&cd=1&cad=rja&ved=0CCkQFjAA&url=http://www.ignou.ac.in/upload/unit%204.pdf&ei=N7wTUufENsiXrAfn24EQ&usg=AFQjCNEqTMr1qLrKhrtETcfCqS2Fh88vIQ&bvm=bv.50952593,d.bmkhttp://www.google.co.in/url?sa=t&rct=j&q=ignition%20system%20pdf&source=web&cd=1&cad=rja&ved=0CCkQFjAA&url=http://www.ignou.ac.in/upload/unit%204.pdf&ei=N7wTUufENsiXrAfn24EQ&usg=AFQjCNEqTMr1qLrKhrtETcfCqS2Fh88vIQ&bvm=bv.50952593,d.bmkhttp://www.google.co.in/url?sa=t&rct=j&q=ignition%20system%20pdf&source=web&cd=1&cad=rja&ved=0CCkQFjAA&url=http://www.ignou.ac.in/upload/unit%204.pdf&ei=N7wTUufENsiXrAfn24EQ&usg=AFQjCNEqTMr1qLrKhrtETcfCqS2Fh88vIQ&bvm=bv.50952593,d.bmkhttp://www.google.co.in/url?sa=t&rct=j&q=ignition%20system%20pdf&source=web&cd=1&cad=rja&ved=0CCkQFjAA&url=http://www.ignou.ac.in/upload/unit%204.pdf&ei=N7wTUufENsiXrAfn24EQ&usg=AFQjCNEqTMr1qLrKhrtETcfCqS2Fh88vIQ&bvm=bv.50952593,d.bmkhttp://www.google.co.in/url?sa=t&rct=j&q=ignition%20system%20pdf&source=web&cd=1&cad=rja&ved=0CCkQFjAA&url=http://www.ignou.ac.in/upload/unit%204.pdf&ei=N7wTUufENsiXrAfn24EQ&usg=AFQjCNEqTMr1qLrKhrtETcfCqS2Fh88vIQ&bvm=bv.50952593,d.bmk

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