1. PLANE STRAIN TRANSFORMATION
CHAPTER 10
PLANE STRAIN TRANSFORMATION

2. CHAPTER OBJECTIVES
Apply the stress transformation methods derived in Chapter 9 to similarly transform strain
Discuss various ways of measuring strain
Develop important material-property relationships; including generalized form of Hooke’s law
Discuss and use theories to predict the failure of a material

3. CHAPTER OUTLINE Plane Strain
General Equations of Plane-Strain Transformation
Strain Rosettes
Material-Property Relationships

4. 10.1 PLANE STRAIN
As explained in Chapter 2.2, general state of strain in a body is represented by a combination of 3 components of normal strain (x, y, z), and 3 components of shear strain (xy, xz, yz).
Strain components at a pt determined by using strain gauges, which is measured in specified directions.
A plane-strained element is subjected to two components of normal strain (x, y) and one component of shear strain, xy.

5. The deformations are shown graphically below.
10.1 PLANE STRAIN
The deformations are shown graphically below.
Note that the normal strains are produced by changes in length of the element in the x and y directions, while shear strain is produced by the relative rotation of two adjacent sides of the element.

6. Note that plane stress does not always cause plane strain.
In general, unless  = 0, the Poisson effect will prevent the simultaneous occurrence of plane strain and plane stress.
Since shear stress and shear strain not affected by Poisson’s ratio, condition of xz = yz = 0 requires xz = yz = 0.

7. 10.2 GENERAL EQNS OF PLANE-STRAIN TRANSFORMATION
Sign Convention
To use the same convention as defined in Chapter 2.2.
With reference to differential element shown, normal strains xz and yz are positive if they cause elongation along the x and y axes
Shear strain xy is positive if the interior angle AOB becomes smaller than 90.

8. 10.2 GENERAL EQNS OF PLANE-STRAIN TRANSFORMATION
Sign Convention
Similar to plane stress, when measuring the normal and shear strains relative to the x’ and y’ axes, the angle  will be positive provided it follows the curling of the right-hand fingers, counterclockwise.
Normal and shear strains
Before we develop the strain-transformation eqn for determining x;, we must determine the elongation of a line segment dx’ that lies along the x’ axis and subjected to strain components.

9. 10.2 GENERAL EQNS OF PLANE-STRAIN TRANSFORMATION
Normal and shear strains
Components of line dx and dx’ are elongated and we add all elongations together.
From Eqn 2.2, the normal strain along the line dx’ is x’ =x’/dx’. Using Eqn 10-1,

10. 10.2 GENERAL EQNS OF PLANE-STRAIN TRANSFORMATION
Normal and shear strains
To get the transformation equation for x’y’, consider amount of rotation of each of the line segments dx’ and dy’ when subjected to strain components. Thus,

11. 10.2 GENERAL EQNS OF PLANE-STRAIN TRANSFORMATION
Normal and shear strains
Using Eqn 10-1 with  = y’/x’,
As shown, dy’ rotates by an amount .

12. 10.2 GENERAL EQNS OF PLANE-STRAIN TRANSFORMATION
Normal and shear strains
Using identities sin ( + 90) = cos , cos ( + 90) =  sin ,
Thus we get

13. 10.2 GENERAL EQNS OF PLANE-STRAIN TRANSFORMATION
Normal and shear strains
Using trigonometric identities sin 2 = 2 sin cos, cos2 = (1 + cos2 )/2 and sin2 + cos2 = 1, we rewrite Eqns 10-2 and 10-4 as

14. 10.2 GENERAL EQNS OF PLANE-STRAIN TRANSFORMATION
Normal and shear strains
If normal strain in the y direction is required, it can be obtained from Eqn 10-5 by substituting ( + 90) for . The result is

15. 10.2 GENERAL EQNS OF PLANE-STRAIN TRANSFORMATION
Principal strains
We can orientate an element at a pt such that the element’s deformation is only represented by normal strains, with no shear strains.
The material must be isotropic, and the axes along which the strains occur must coincide with the axes that define the principal axes.
Thus from Eqns 9-4 and 9-5,

16. 10.2 GENERAL EQNS OF PLANE-STRAIN TRANSFORMATION
Principal strains
Maximum in-plane shear strain
Using Eqns 9-6, 9-7 and 9-8, we get

17. 10.2 GENERAL EQNS OF PLANE-STRAIN TRANSFORMATION
Maximum in-plane shear strain
Using Eqns 9-6, 9-7 and 9-8, we get

18. 10.2 GENERAL EQNS OF PLANE-STRAIN TRANSFORMATION
IMPORTANT
The state of strain at the pt can also be represented in terms of the maximum in-plane shear strain. In this case, an average normal strain will also act on the element.
The element representing the maximum in-plane shear strain and its associated average normal strains is 45 from the element representing the principal strains.

19. EXAMPLE 10.1
A differential element of material at a pt is subjected to a state of plane strain x = 500(10-6), y =  300(10-6), which tends to distort the element as shown. Determine the equivalent strains acting on an element oriented at the pt, clockwise 30 from the original position.

20. EXAMPLE 10.1 (SOLN)
Since  is clockwise, then  = –30, use strain-transformation Eqns 10-5 and 10-6,

21. EXAMPLE 10.1 (SOLN)
Since  is counterclockwise, then  = –30, use strain-transformation Eqns 10-5 and 10-6,

22. EXAMPLE 10.1 (SOLN)
Strain in the y’ direction can be obtained from Eqn 10-7 with  = –30. However, we can also obtain y’ using Eqn 10-5 with  = 60 ( = –30 + 90), replacing x’ with y’

23. EXAMPLE 10.1 (SOLN)
The results obtained tend to deform the element as shown below.

24. EXAMPLE 10.2
A differential element of material at a pt is subjected to a state of plane strain defined by x = –350(10-6), y = 200(10-6), xy = 80(10-6), which tends to distort the element as shown. Determine the principal strains at the pt and associated orientation of the element.

25. EXAMPLE 10.2 (SOLN) Orientation of the element From Eqn 10-8, we have
Each of these angles is measured positive counterclockwise, from the x axis to the outward normals on each face of the element.

26. EXAMPLE 10.2 (SOLN)
Principal strains
From Eqn 10-9,

27. EXAMPLE 10.2 (SOLN) Principal strains
We can determine which of these two strains deforms the element in the x’ direction by applying Eqn 10-5 with  = –4.14. Thus

28. EXAMPLE 10.2 (SOLN) Principal strains
Hence x’ = 2. When subjected to the principal strains, the element is distorted as shown.

29. EXAMPLE 10.3
A differential element of material at a pt is subjected to a state of plane strain defined by x = –350(10-6), y = 200(10-6), xy = 80(10-6), which tends to distort the element as shown. Determine the maximum in-plane shear strain at the pt and associated orientation of the element.

30. EXAMPLE 10.3 (SOLN) Orientation of the element From Eqn 10-10,
Note that this orientation is 45 from that shown in Example 10.2 as expected.

31. EXAMPLE 10.3 (SOLN) Maximum in-plane shear strain Applying Eqn 10-11,
The proper sign of can be obtained by applying Eqn 10-6 with s = 40.9.

32. EXAMPLE 10.3 (SOLN) Maximum in-plane shear strain
Thus tends to distort the element so that the right angle between dx’ and dy’ is decreased (positive sign convention).

33. EXAMPLE 10.3 (SOLN) Maximum in-plane shear strain
There are associated average normal strains imposed on the element determined from Eqn 10-12:
These strains tend to cause the element to contract.

34. *10.3 MOHR’S CIRCLE: PLANE STRAIN
Advantage of using Mohr’s circle for plane strain transformation is we get to see graphically how the normal and shear strain components at a pt vary from one orientation of the element to the next.
Eliminate parameter  in Eqns 10-5 and 10-6 and rewrite as

35. *10.3 MOHR’S CIRCLE: PLANE STRAIN
Procedure for Analysis
Construction of the circle
Establish a coordinate system such that the abscissa represents the normal strain , with positive to the right, and the ordinate represents half the value of the shear strain, /2, with positive downward.
Using positive sign convention for x, y, and xy, determine the center of the circle C, which is located on the  axis at a distance avg = (x + v)/2 from the origin.