If σ₁ and σ₃ are the algebraically largest and smallest principal stresses respectively, the value of the maximum shear stress is

The correct option is represented by the formula:
$$\frac{{\sigma }_1-{\sigma }_3}{2}$$ which is shown in the correct option image containing the variables ${\sigma }_1$ and ${\sigma }_3$ subtracted in the numerator and divided by 2.

Step-by-Step Derivation and Explanation:

1. Understanding Principal Stresses:
In stress analysis, when a body is subjected to complex loading, the stress state at any point can be resolved into normal and shear stresses. The planes on which the shear stresses are zero are called the principal planes, and the normal stresses acting on these planes are the principal stresses.
By convention, the three principal stresses are ordered algebraically as:
$${\sigma }_1\ge {\sigma }_2\ge {\sigma }_3$$ where:
- ${\sigma }_1$ is the algebraically largest principal stress.
- ${\sigma }_2$ is the intermediate principal stress.
- ${\sigma }_3$ is the algebraically smallest principal stress.

2. Mohr's Circle Representation:
Mohr's circle is a graphical representation of the transformation equations for plane stress. For a three-dimensional state of stress, the stress state is represented by three circles plotted on the normal stress ($\sigma$) versus shear stress ($\tau$) coordinate system:
- The first circle is constructed between ${\sigma }_1$ and ${\sigma }_2$.
- The second circle is constructed between ${\sigma }_2$ and ${\sigma }_3$.
- The third, and largest, circle is constructed between the absolute maximum and minimum values, which are ${\sigma }_1$ and ${\sigma }_3$.

3. Determining Maximum Shear Stress:
The maximum shear stress (${\tau }_{\text{max}}$) is represented by the radius of the largest Mohr's circle.
The diameter of the largest circle lies along the horizontal $\sigma$-axis between the points ${\sigma }_1$ and ${\sigma }_3$. The length of this diameter is calculated as:
$$\text{Diameter}={\sigma }_1-{\sigma }_3$$ Since the radius of this circle is half of its diameter, the maximum shear stress is given by:
$${\tau }_{\text{max}}=\frac{\text{Diameter}}{2}=\frac{{\sigma }_1-{\sigma }_3}{2}$$ This matches the mathematical representation in the correct image option.