Principal stress, Principal plane & Mohr's circle analysis

Summary of Lecture

• 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.
• 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).
• 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).
• 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

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



Fig.4 Connecting end of shear stress lines

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



Fig.5 Mohr circle construction

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.6 Determination on normal and shear stress using Mohr cirlce

• 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.



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

• 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.