Stress Transformation

It is defined as, "the state of plane stress at the point is uniquely represented by two normal stress components and one shear component acting on an element (face). To be equivalent, these three components will be different for each specific orientation Ө of the element at that point".

• If no load applied  ⇔  normal and shear stresses are zero.
• Stress Tensor (has magnitude & multiple directions)  ⇔  convenient way of expressing the stress state inside a material.

Plane-Stress Transformation Equations

A method of transforming the normal and shear stress components from x and y to x' and y'.

Max. In-Plane Principal Stresses

Corresponding planes on which stresses act are called the Principal Planes. 

2ӨP1  - 2ӨP2  =  180° 

Max. In-Plane Shear Stress

Mohr's Stress Circle

It is defined as., "A graphical method for determining normal and shear Shear stresses without using the stress transformation equations".

• Mohr's circle can be plotted using equations of Mohr's circle and Stress Transformation.
• While considering the circle CCW  ⇔  Shear stress positive downward (can rotate element) & Normal stress positive towards right (tensile).
• It can be constructed using different ways depending upon which stresses are known and which are to be found.

The construction of Mohr's circle (with normal and shear stresses are known) is quite easy which include following steps:

1. Draw a set of coordinate axes with σX (+ve right) and τY (+ve downward).
2. Locate the center C of the circle at points σX1 = σavg and τX1Y1= 0.
3. Locate point A which represents stress concentration on X-face corresponds to θ = 0° and σX1 = σx and τX1Y1= τXY.
4. Locate point B which represents stress concentration on Y-face corresponds to θ = 90° and σX1 = σy and τX1Y1= -τXY.
5. Draw a line from point A to point B which represents shear on planes at 90° to each other.
6. Using point C as center, draw Mohr's circle through point A and B.

Some key observations are:

• If max. normal stress on X-face  ⇔  Min. normal stress on Y-face.
• On min. and max. normal stress values  ⇔  shear stresses are zero called Principal Planes (separated by 90°).
• Principal Stresses occur when stress element is rotated such that the shear stresses are zero. Important for predicting failure.

References:

• Material from Class Lectures + Book named Engineering Mechanics of Materials by R.C. Hibbeler (6/9th Edition) + my knowledge. 
• Pics and GIF from Google Images.  
• Videos from YouTube (Engineering Sights).