PRINCIPAL STRAIN
For the strains on an oblique plane we have an oblique we have two equations which are identical in form with the equation defining the direct stress on any inclined plane q .
Since the equations for stress and strains on oblique planes are identical in form, so it is evident that Mohr's stress circle construction can be used equally well to represent strain conditions using the horizontal axis for linear strains and the vertical axis for half the shear strain.
It should be noted, however that the angles given by Mohr's stress circle refer to the directions of the planes on which the stress act and not the direction of the stresses themselves.
The direction of the stresses and therefore associated strains are therefore normal (i.e. at 900) to the directions of the planes. Since angles are doubled in Mohr's stress circle construction it follows therefore that for a true similarity of working a relative rotation of axes of 2 x 900 = 1800 must be introduced. This is achieved by plotting positive sheer strains vertically downwards on the strain circle construction.
The sign convention adopted for the strains is as follows:
Linear Strains : extension - positive
{ Shear of strains are taken positive, when they increase the original right angle of an unstrained element. }
Shear strains : for Mohr's strains circle sheer strain gxy - is +ve referred to x - direction the convention for the shear strains are bit difficult. The first subscript in the symbol gxy usually denotes the shear strains associated with direction. e.g. in gxy– represents the shear strain in x - direction and for gyx– represents the shear strain in y - direction. If under strain the line associated with first subscript moves counter clockwise with respect to the other line, the shearing strain is said to be positive, and if it moves clockwise it is said to be negative.
N.B: The positive shear strain is always to be drown on the top of Îx .If the shear stain gxy is given ]
Moh's strain circle
For the plane strain conditions can we derivate the following relations
A typical point P on the circle given the normal strain and half the sheer strain 1/2gxy associated with a particular plane. We note again that an angle subtended at the centre of Mohr's circle by an arc connecting two points on the circle is twice the physical angle in the material.
Mohr strain circle :
Since the transformation equations for plane strain are similar to those for plane stress, we can employ a similar form of pictorial representation. This is known as Mohr's strain circle.
The main difference between Mohr's stress circle and stress circle is that a factor of half is attached to the shear strains.
Points X' and Y' represents the strains associated with x and y directions with Î and gxy /2 as co-ordiantes
Co-ordinates of X' and Y' points are located as follows :
In x – direction, the strains produced, the strains produced by sx,and - t xy are Îx and - gxy /2
where as in the Y - direction, the strains are produced by Îy and + gxy are produced by sy and + txy
These co-ordinated are consistent with our sign notation ( i.e. + ve shear stresses produces produce +ve shear strain & vice versa )
on the face AB is txy+ve i.e strains are ( Îy, +gxy /2 ) where as on the face BC, txy is negative hence the strains are ( Îx, - gxy /2 )
A typical point P on the circle gives the normal strains and half the shear strain, associated with a particular plane we must measure the angle from x – axis (taken as reference) as the required formulas for Îq , -1/2 gq have been derived with reference to x-axis with angle measuring in the c.c.W direction
CONSTRUCTION :
In this we would like to locate the points x' & y' instead of AB and BC as we have done in the case of Mohr's stress circle.
steps
1. Take normal or linear strains on x-axis, whereas half of shear strains are plotted on y-axis.
2. Locate the points x' and y'
3. Join x' and y' and draw the Mohr's strain circle
4. Measure the required parameter from this construction.
Note: positive shear strains are associated with planes carrying positive shear stresses and negative strains with planes carrying negative shear stresses.
ILLUSTRATIVE EXAMPLES :
1. At a certain point, a material is subjected to the following state of strains:
Determine the magnitudes of the principal strains, the direction of the principal strains axes and the strain on an axis inclined at 300 clockwise to the x – axis.
Solution:
Draw the Mohr's strain circle by locating the points x' and y'
By Measurement the following values may be computed
The angles being measured c.c.w. from the direction of Îx.
PROB 2.
A material is subjected to two mutually perpendicular strains Îx = 350 x10-6 units and Îy = 50 x 10-6 units together with an unknown sheer strain gxy if the principal strain in the material is 420 x 10-6 units Determine the following.
(a) Magnitude of the shear strain
(b) The other principal strain
(c) The direction of principal strains axes
(d) The magnitude of the principal stresses
If E = 200 GN / m2; g = 0.3
Solution :
The Mohr's strain circle can be drawn as per the procedure described earlier. from the graphical construction, the following results may bre obtained :
(i) Shear strain gxy = 324 x 10-6 radians
(ii) other principal strain = -20 x 10-6
(iii) direction of principal strain = 470 / 2 = 230 30'
(iv) direction of other principal strain = 900 +230 30' = 1130 30'
In order to determine the magnitude of principle stresses, the computed values of Î1and Î2 from the graphical construction may be substituted in the following expressions
Use of strain Gauges :
Although we can not measure stresses within a structural member, we can measure strains, and from them the stresses can be computed, Even so, we can only measure strains on the surface. For example, we can mark points and lines on the surface and measure changes in their spacing angles. In doing this we are of course only measuring average strains over the region concerned. Also in view of the very small changes in dimensions, it is difficult to archive accuracy in the measurements
In practice, electrical strain gage provide a more accurate and convenient method of measuring strains.
A typical strain gage is shown below.
The gage shown above can measure normal strain in the local plane of the surface in the direction of line PQ, which is parallel to the folds of paper. This strain is an average value of for the region covered by the gage, rather than a value at any particular point.
The strain gage is not sensitive to normal strain in the direction perpendicular to PQ, nor does it respond to shear strain. therefore, in order to determine the state of strain at a particular small region of the surface, we usually need more than one strain gage.
To define a general two dimensional state of strain, we need to have three pieces of information, such as Îx , Îy and gxy referred to any convenient orthogonal co-ordinates x and y in the plane of the surface. We therefore need to obtain measurements from three strain gages. These three gages must be arranged at different orientations on the surface to from a strain rossett. Typical examples have been shown, where the gages are arranged at either 450 or 600 to each other as shown below :
 
A group of three gages arranged in a particular fashion is called a strain rosette. Because the rosette is mounted on the surface of the body, where the material is in plane stress, therefore, the transformation equations for plane strain to calculate the strains in various directions.
Knowing the orientation of the three gages forming a rosette, together with the in – plane normal strains they record, the state of strain at the region of the surface concerned can be found. Let us consider the general case shown in the figure below, where three strain gages numbered 1, 2, 3, where three strain gages numbered 1, 2, 3 are arranged at an angles of q1 , q2 , q3 measured c.c.w from reference direction, which we take as x – axis.
Now, although the conditions at a surface, on which there are no shear or normal stress components. Are these of plane stress rather than the plane strain, we can still use strain transformation equations to express the three measured normal strains in terms of strain components Îx , Îy , Îz and gxy referred to x and y co-ordiantes as
 
This is a set of three simultaneous linear algebraic equations for the three unknows Îx, Îy , gxy to solve these equation is a laborious one as far as manually is concerned, but with computer it can be readily done.Using these later on, the state of strain can be determined at any point.
Let us consider a 450 degree stain rosette consisting of three electrical – resistance strain gages arranged as shown in the figure below :
The gages A, B,C measure the normal strains Îa , Îb , Îc in the direction of lines OA, OB and OC.
Thus
Thus, substituting the relation (3) in the equation (2) we get
gxy = 2Îb- ( Îa + Îc ) and other equation becomes Îx = Îa ; Îy= Îc
Since the gages A and C are aligned with the x and y axes, they give the strains Îx and Îy directly
Thus, Îx , Îy and gxy can easily be determined from the strain gage readings. Knowing these strains, we can calculate the strains in any other directions by means of Mohr's circle or from the transformation equations.
The 600 Rossett:
For the 600 strain rosette, using the same procedure we can obtain following relation.
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Principal Strain and Mohr Strain Circle
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