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

The correct answer is:

Step-by-Step Explanation:

1. Understanding Principal Stresses:
In a three-dimensional state of stress, there exist three mutually perpendicular planes (called principal planes) on which the shear stresses are zero. The normal stresses acting on these planes are the principal stresses. By convention, they are ordered algebraically as:
${\sigma }_1>{\sigma }_2>{\sigma }_3$
Here, ${\sigma }_1$ is the algebraically largest principal normal stress, and ${\sigma }_3$ is the algebraically smallest principal normal stress.

2. Mohr's Circle Representation:
For a three-dimensional stress state, Mohr's stress representation consists of three circles. The diameters of these three circles are defined by the differences between the principal stresses:
• Circle 1 (in the 1-2 plane) has a maximum shear stress of:
${\tau }_{\text{12}}=\frac{{\sigma }_1-{\sigma }_2}{2}$
• Circle 2 (in the 2-3 plane) has a maximum shear stress of:
${\tau }_{\text{23}}=\frac{{\sigma }_2-{\sigma }_3}{2}$
• Circle 3 (in the 1-3 plane) has a maximum shear stress of:
${\tau }_{\text{13}}=\frac{{\sigma }_1-{\sigma }_3}{2}$

3. Determining the Maximum Shear Stress:
Since ${\sigma }_1$ is the largest value and ${\sigma }_3$ is the smallest value, the largest difference between any two principal stresses is ${\sigma }_1-{\sigma }_3$.
Therefore, the absolute maximum shear stress (${\tau }_{\text{max}}$) is given by the radius of the largest Mohr's circle:
${\tau }_{\text{max}}=\frac{{\sigma }_1-{\sigma }_3}{2}$