PRINCIPAL STRESS AND PRINCIPAL STRAIN

1. Introduction

In the study of Strength of Materials, a body subjected to external loads experiences stresses and strains at every point.
At a general point inside a loaded body, stresses act on many planes with different magnitudes and directions. Among these, there exist certain planes on which:

• Normal stress is maximum or minimum
• Shear stress is zero

The normal stresses on these planes are called principal stresses, and the corresponding strains are called principal strains. Understanding principal stress and strain is essential for safe and economical design of machine and structural components.

2. Stress at a Point

At a point in a loaded body, the state of stress is generally represented by:

• Normal stresses: $\sigma_x, \sigma_y$σx​,σy​
• Shear stress: $\tau_{xy}$τxy​

This condition is known as a plane stress condition, commonly encountered in thin plates and machine parts.

3. Principal Stress

3.1 Definition

Principal stresses are the maximum and minimum normal stresses acting on planes at a point where the shear stress is zero.

There are two principal stresses in plane stress:

• Major principal stress (σ1\sigma_1σ1​)
• Minor principal stress (σ2\sigma_2σ2​)

3.2 Derivation of Principal Stress

For a plane stress condition:$$\sigma_1, \sigma_2 = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x – \sigma_y}{2}\right)^2 + \tau_{xy}^2}$$

Where:

• $\sigma_x, \sigma_y$σx​,σy​ = normal stresses
• $\tau_{xy}$τxy​ = shear stress

3.3 Orientation of Principal Planes

The angle $\theta_p$θp​ of principal planes is given by:$$\tan 2\theta_p = \frac{2\tau_{xy}}{\sigma_x – \sigma_y}$$

At these angles:

• Shear stress becomes zero
• Normal stress becomes maximum or minimum

3.4 Importance of Principal Stress

• Failure theories are based on principal stresses
• Used in the design of brittle materials
• Helps predict crack initiation and fracture direction

4. Principal Strain

4.1 Definition

Principal strains are the maximum and minimum normal strains occurring at a point, acting on planes where shear strain is zero.

They are denoted as:

• Maximum principal strain ($\varepsilon_1$ε1​)
• Minimum principal strain ($\varepsilon_2$ε2​)

4.2 Strain Components at a Point

At a point, strain components are:

• Normal strains: $\varepsilon_x, \varepsilon_y$εx​,εy​
• Shear strain: $\gamma_{xy}$γxy​

4.3 Expression for Principal Strain

The principal strains are given by:$$\varepsilon_1, \varepsilon_2 = \frac{\varepsilon_x + \varepsilon_y}{2} \pm \sqrt{\left(\frac{\varepsilon_x – \varepsilon_y}{2}\right)^2 + \left(\frac{\gamma_{xy}}{2}\right)^2}$$

4.4 Orientation of Principal Strain Planes

The angle $\theta_\varepsilon$θε​ of principal strain planes is:$$\tan 2\theta_\varepsilon = \frac{\gamma_{xy}}{\varepsilon_x – \varepsilon_y}$$​​

5. Relation Between Principal Stress and Principal Strain

Using Hooke’s Law for isotropic materials:$$\varepsilon_1 = \frac{1}{E}(\sigma_1 – \nu \sigma_2)$$$$\varepsilon_2 = \frac{1}{E}(\sigma_2 – \nu \sigma_1)$$

Where:

• $E$E = Young’s modulus
• $\nu$ν = Poisson’s ratio

This shows that principal strain depends on both principal stresses.

6. Mohr’s Circle for Stress and Strain

Mohr’s Circle is a graphical method used to determine:

• Principal stresses or strains
• Maximum shear stress or strain
• Orientation of principal planes

Advantages

• Simple and visual
• Reduces complex calculations
• Widely used in engineering analysis

7. Maximum Shear Stress and Strain

• Maximum shear stress occurs on planes at 45° to principal planes
• Maximum shear stress is:

$$\tau_{max} = \frac{\sigma_1 – \sigma_2}{2}$$τmax​=2σ1​−σ2​​

• Maximum shear strain is related to principal strains

8. Engineering Applications

• Design of shafts, beams, pressure vessels
• Failure analysis of mechanical components
• Structural engineering (bridges, frames)
• Fatigue and fracture analysis

9. Comparison Between Principal Stress and Principal Strain

Aspect	Principal Stress	Principal Strain
Quantity	Force per unit area	Deformation per unit length
Symbol	$\sigma_1, \sigma_2$σ1​,σ2​	$\varepsilon_1, \varepsilon_2$ε1​,ε2​
Occurs on	Planes with zero shear stress	Planes with zero shear strain
Used in	Failure theories	Deformation analysis