Plane-strain & Strain Measurement

1. Transformation of Plane-Strain
and Strain Measurement

2. Transformation of Plane-Strain
Review:
What is normal strain?
And what is shear strain?
• Normal strain in a rod under axial loading is defined as the deformation
per unit length of that rod.
normal strain
normal stress
L
P
A



 
 

3. • Shear strain is defined as the horizontal deformation (displacement) per
unit length of that rod.
• Normally due to the shear stress.
Transformation of Plane-Strain

4. Transformation of plane strain is very similar to the transformation of the
plane stress.
Transformation of Plane-Strain
• Plane strain occurs in a plate subjected
along its edges to a uniformly distributed
load and restrained from expanding or
contracting laterally by smooth, rigid, and
fixed supports
• The deformations of the material take
place in parallel planes and are the same in
each of those planes.
 
x
components of strain:
0
y xy z zx zy
     
  

5. Transformation of Plane Strain
• Global axis & Local axis
2
2
2
,
1
2
2
2
2
cos
2
2
sin
2
2
2
sin
2
2
cos
2
2
2
sin
2
2
cos
2
2
















 






















xy
y
x
y
x
xy
y
x
y
x
xy
y
x
y
x
y
xy
y
x
y
x
x




























• General equation of plane strain transformation

6. Transformation of Plane Strain
Example:
Solution:






















2
cos
2
2
sin
2
2
2
sin
2
2
cos
2
2
2
sin
2
2
cos
2
2
xy
y
x
y
x
xy
y
x
y
x
y
xy
y
x
y
x
x


















• General equation of plane strain transformation
x ' 450 , ' 200 , ' 375
y xy
     
     

7. Example:
Page 520 (7.128)
For the given state of plane strain, use the methods of Sec 7.10 to determine the
state of strain associated with axes x’ and y’ rotated through the given angle 






















2
cos
2
2
sin
2
2
2
sin
2
2
cos
2
2
2
sin
2
2
cos
2
2
xy
y
x
y
x
xy
y
x
y
x
y
xy
y
x
y
x
x


















• General equation of plane strain transformation
x ' 670 , ' 50 , ' 474
y xy
     
     
Solution:

8. EXERCISE:
Page 520 (7.129)
For the given state of plane strain, use the methods of Sec 7.10 to determine the
state of strain associated with axes x’ and y’ rotated through the given angle 






















2
cos
2
2
sin
2
2
2
sin
2
2
cos
2
2
2
sin
2
2
cos
2
2
xy
y
x
y
x
xy
y
x
y
x
y
xy
y
x
y
x
x


















• General equation of plane strain transformation

9. • Similarly, Mohr’s circle for the strain can be obtained,
• Given
y
x
xy
p






2
tan
R
A ave 

 
max
R
B ave 

 
min
2
y
x
ave
C






2
2
2
2 















 

xy
y
x
R



R
2
max 

Transformation of Plane Strain
)
2
,
(
)
2
,
(




















xy
y
xy
x
Y
X




2 2
1,2
2 2 2
x y x y xy
    

 
   

 
   
    
xy
y
x 

 ;
;

10. )
2
,
(
)
2
,
(




















xy
y
xy
x
Y
X




2
y
x
ave
C






2
2
2
2 















 

xy
y
x
R



y
x
xy
p






2
tan

11. )
2
,
(
)
2
,
(




















xy
y
xy
x
Y
X




2
y
x
ave
C






2
2
2
2 















 

xy
y
x
R



y
x
xy
p






2
tan

12. 2
y
x
ave
C






y
x
xy
p






2
tan
2
2
2
2 















 

xy
y
x
R




13. Measurement of Strain
Example:
Page 482 (7.137)

14. Measurement of Strain
Solution:

15. Measurement of Strain
Solution:

16. Strain Gauges / Rosette

17. 1
1
1
2
1
2
1 cos
sin
sin
cos 






 xy
y
x 


2
2
2
2
2
2
2 cos
sin
sin
cos 






 xy
y
x 


3
3
3
2
3
2
3 cos
sin
sin
cos 






 xy
y
x 


x, y, xy can be obtained by solving the above equation simultaneously.
• Normal strain on the surface of structure a structural
element can be measured using strain gages
• Strain gages indicate normal strain through changes in
resistance.
• Normal and shearing strains may be obtained from
normal strains in any three directions.
Measurement of Strain : Strain Rosette

18. Example:
For below orientation and reading of the strain gages, determine:
i) Principle strains
ii) Maximum shear strain
Measurement of Strain : Strain Rosette
1
1
1
2
1
2
1 cos
sin
sin
cos 






 xy
y
x 


2
2
2
2
2
2
2 cos
sin
sin
cos 






 xy
y
x 


3
3
3
2
3
2
3 cos
sin
sin
cos 






 xy
y
x 



19. Solution:
Measurement of Strain : Strain Rosette
i)Principle strains
ii)Maximum shear strain

20. Measurement of Strain : Strain Rosette

21. Measurement of Strain : Strain Rosette
Example:
Page 483 (7.144)
Solution

22. Measurement of Strain : Strain Rosette

23. Measurement of Strain : Strain Rosette
Example:
Page 483 (7.146)
Solution

24. Measurement of Strain : Strain Rosette

25. Example:
Page 484 (7.150)
Measurement of Strain : Strain Rosette

26. Measurement of Strain : Strain Rosette
Solution:

27. Measurement of Strain : Strain Rosette
t

28. Measurement of Strain : Strain Rosette
Example:
Page 484 (7.152)

29. Measurement of Strain : Strain Rosette
Solution: