Strength of Materials : Principal Stress and Strain

PRINCIPAL STRESS AND STRAINS:

Stress tensor:

A tensor is a multi-dimensional array of numerical values that can be used to describe the physical state or properties of a material. A simple example of a geo-physically relevant tensor is stress.

Sign conventions:

While analysing a stress system, the general conventions have been taken as follows:

A tensile stress is positive and compressive stress, negative.
A pair of shear stresses on parallel planes forming a clockwise couple is positive and a pair with counterclockwise couple, negative.
Clockwise angle is taken as positive and counterclockwise as negative.

Methods to find stresses at a point:

The stresses on a point are determined by the following methods:

Sum of Direct stresses of two mutually perpendicular stresses:

PRINCIPAL STRESSES

In general, a body may be acted upon by direct stresses and shear stresses.
However, it will be seen that even in such complex systems of loading, there exist three mutually perpendicular planes, on each of which the resultant stress is wholly normal. These are known as principal planes and the normal stress across these planes, as principal stresses.
The larger of the two stresses σ₁ is called the major principal stress, and the smaller one σ₂ as the minor principal stress. The corresponding planes are known as major and minor principal planes.
In two-dimensional problems, the third principal stress is taken to be zero.

As shear stress is zero in principal planes:

MOHR'S STRESS CIRCLE

The stress components on any inclined plane can easily be found with the help of a geometrical construction known as Mohr’s stress circle.

Some points about Mohr's Circle:

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