- The magnitude of normal stress and shear stress may vary with respect to the inclination of planes. If we are concerned with the safety of solids under stress, we are required to find on which planes extreme values of normal and shear stress components are present. Hence, it is essential to know :
(a) Maximum tensile stress,
(b) Maximum compressive stress, and
(c) Maximum shear stress.
- The extreme values of normal stresses are called the Principal Stresses and the planes on which the principal stresses act are called the principal planes.
- In two-dimensional problems, there are two principal stresses, namely the major principal stress and the minor principal stress which are defined as the maximum and minimum values of the normal stresses respectively. Here, the maximum or minimum is to be considered algebraically.
- For example, if the principal stresses happen to be 20 N/mm2 tensile and 75 N/mm2 compressive, the tensile stress of 20 N/mm2 is to be taken as the major principal stress denoted by the symbol σ1 and the compressive stress of 75 N/mm2 is to be taken as the minor principal stress (algebraically – 75 N/mm2) and denoted by the symbol σ2. The corresponding planes are defined as major and minor principal planes.
Fig: Principal Stress
Expressions of Principal Planes and Principal Stresses:
In calculus, you have learnt that when a function reaches maximum or minimum its derivative with respect to the independent variable becomes zero. Since the normal stress on an arbitrary plane is a function of the aspect angle θ as given by the expression,
Expression for major and minor principal stresses as follows :
Above equations may be used to readily determine the principal planes and principal stresses.