When a body is subjected to bi-axial stress, i.e. direct stress (P1) and (P2) in two mutually perpendicular planes accompanied by a simple shear stress (q), then maximum normal stress is 

Concept:

When a body is subjected to bi-axial stresses (direct stresses P₁ and P₂ ) along with a shear stress q , the maximum and minimum normal stresses can be determined using stress transformation equations. The maximum normal stress is particularly important for failure analysis.

Given:

• Direct stress in the first direction, \( P_1 \)
• Direct stress in the second direction, \( P_2 \)
• Shear stress, \( q \)

Step 1: Understand the Stress State

The given stress state consists of two normal stresses P₁ and P₂ acting in mutually perpendicular directions, along with a shear stress q .

Step 2: Use the Formula for Principal Stresses

The maximum and minimum normal stresses (principal stresses) are given by:

\[ \sigma_{\text{max}}, \sigma_{\text{min}} = \frac{P_1 + P_2}{2} \pm \frac{1}{2} \sqrt{(P_1 - P_2)^2 + 4q^2} \]

Step 3: Identify the Maximum Normal Stress

The maximum normal stress corresponds to the positive root of the expression:

\[ \sigma_{\text{max}} = \frac{P_1 + P_2}{2} + \frac{1}{2} \sqrt{(P_1 - P_2)^2 + 4q^2} \]