Principal stress, Principal plane & Mohr’s circle analysis

Concepts of principal stress and plane form backbone of material stress analysis. Purpose of this video lecture is to give you a good introduction to concept of Principal stress, Principal plane and Mohr’s circle analysis.

Summary of the video lecture is given below

Summary of Lecture

1.Engineers most often wants to determine maximum normal stress induced at a given point for their design purpose. But there can be infinite number of planes passing through a point, and normal stress on each plane will be different from one another.

2.There will be one plane on which normal stress value is maximum, this plane is known as Principal plane ( more precisely maximum principal plane) and normal stress on this plane is known as principal stress (more precisely maximum principal stress).

3.Similarly there will be one more plane on which normal stress value is minimum, this is also a principal plane (minimum principal plane) and normal stress on this plane is known as Principal stress (minimum principal stress).

4. 2-Dimensional Stress Analysis – Stress acting on a 2D element is shown in figure below

Fig:1 Stress boundary conditions on a 2 dimensional element

1.Mohr’s circle method is the most easy and convenient way to do stress analysis

2.The procedure to draw Mohr’s circle for above case is explained below

Step1 – Draw normal and shear axes with positive axes as shown

Fig:2 Normal and shear axes of a Mohr circle

Step2 – Mark normal stress values with sign convention, tensile stress is positive and compression stress is negative

Fig:3 Marking normal stress values on normal axis

Step3 – Draw shear stress values starting from already marked normal stress points.

Fig:4 Drawing shear stress values

Step4 – Connect end of shear stress lines

Step5 – Draw Mohr’s circle assuming the connection line as diameter of the circle

Step6 – Stress Analysis on Mohr circle – To get normal and shear stress values at any plane theta, take angle 2theta in Mohr circle starting from diagonal of the circle and locate a peripheral point as as shown. Shear stress value will be Y axis value and normal stress value will be X axis value.

Fig:7 Stress boundary conditions in a 3 dimensional body and normal stress values induced in it

3-Dimensional Stress Analysis – Stress boundary condition of a 3 dimensional case is shown in left side of Fig.7. There will be 3 normal stress values induced in a 3 dimensional case, this is shown in right size of the figure.

There is no graphical method for 3 Dimensional stress analysis, instead we have to use analytical method for this. Values of Principal stress in a 3 dimensional systems are given by solution of following equation.

Where values of stress invariants I1,I2 and I3 are given by

Application of Principal Stresses

Values of principal stresses at a given point is vital design information. Material failure theories extensively use this data to predict whether the design will withstand given load at a specified location.