The equation for major principal stress is

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Option C:

σ_{n 1} = \frac{(σ_{1} + σ_{2})}{2} + \sqrt{{(\frac{σ_{1} - σ_{2}}{2})}^{2} + τ^{2}}

Explanation & Concepts: The major principal stress formula represents the maximum normal stress acting on a plane when a 2D stress state undergoes rotation. It is derived from stress transformation equations and Mohr's circle theory, where sigma1 and sigma2 are normal stresses and tau is shear stress on the reference plane. Working Principle: When a stress element is subjected to normal stresses (sigma1, sigma2) and shear stress (tau), the actual stresses vary with the orientation of the plane. Using Cauchy's stress transformation equations, the principal stresses are obtained by finding the extreme values of normal stress. These occur when shear stress becomes zero and are calculated using the average stress plus/minus a term involving half the stress difference and all shear components. The plus sign gives the maximum (major) principal stress. Key Formulas: sigma_n1 = (sigma1 + sigma2)/2 + sqrt[((sigma1 - sigma2)/2)^2 + tau^2] — Major Principal Stress; sigma_n2 = (sigma1 + sigma2)/2 - sqrt[((sigma1 - sigma2)/2)^2 + tau^2] — Minor Principal Stress; Average stress = (sigma1 + sigma2)/2 — Hydrostatic component of stress Key Points: Principal stresses occur on planes where shear stress is zero, representing extreme normal stresses in any direction; The formula uses the average of applied stresses as baseline and adds/subtracts a combined effect of stress difference and shear stress; The sqrt term represents the radius of Mohr's circle and accounts for both normal stress variation and shear stress magnitude; Major principal stress is always greater than or equal to minor principal stress, regardless of original stress orientation; The sign inside the sqrt must be PLUS (not minus) because both squared terms are inherently positive and contribute to stress magnitude Advantages: Identifies maximum and minimum stresses for failure analysis and safety assessments; Works for any 2D stress state regardless of plane orientation; Directly applicable to Mohr's circle graphical method for visualization; Essential for determining critical planes in material strength analysis Disadvantages: Applicable only to 2D plane stress problems; 3D requires extended tensor analysis; Requires accurate determination of all three stress components (sigma1, sigma2, tau); Does not account for material properties or time-dependent behavior; Shear stress measurement on reference plane must be precise for accurate results Applications: Beam bending and shear stress analysis in structural members; Pressure vessel design where hoop and longitudinal stresses exist together; Foundation design and soil mechanics to determine critical failure planes; Machine component design (gears, shafts) under combined loading conditions; Geotechnical engineering for slope stability and earth pressure analysis Additional Info: Also Known As: Maximum Principal Stress, sigma_max, first principal stress (sigma_1 in transformed coordinates). Mathematical Basis: Derived from Cauchy's stress transformation equations and eigenvalue analysis of 2D stress tensor. Mohr's Circle Relation: The major principal stress is the rightmost point on Mohr's circle; the formula gives its coordinates directly. Sign Convention: Tensile stress positive, compressive negative; the formula works for both based on input stress signs. Wrong Options Explained: Option A: Minus sign and wrong stress difference order—gives minimum stress. Option B: Wrong stress difference in first term and incorrect sqrt structure. Option D: Minus sign inside sqrt creates impossible expression and gives minimum instead of maximum stress.. Critical Exam Point: The PLUS sign before the sqrt is the key distinguishing feature; this ensures the maximum (not minimum) principal stress is obtained Answer: C is correct — The major principal stress formula is sigma_n1 = (sigma1 + sigma2)/2 + sqrt[((sigma1 - sigma2)/2)^2 + tau^2], where the plus sign before the square root term ensures the maximum principal stress is calculated by adding the stress circle radius to the average stress. Exam Tip: This formula directly connects to Mohr's circle (visualized as a circle with center at average stress and radius as the sqrt term), which is frequently combined with failure theories (Von Mises, Tresca) in design problems. Mastering this equation and its geometric interpretation is essential for sections on combined stresses, pressure vessels, and shaft torsion problems.

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Option A: σn1=σ1+σ22-σ1-σ222-τ2

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Option B: σn1=σ1-σ22+σ1+σ222-τ2

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Option D: σn1=σ1+σ22-σ1-σ222+τ2

Feature	Gives Minor Principal Stress	Gives Major Principal Stress
Formula Structure	Average minus sqrt (Option A)	Average plus sqrt (Option C)
Stress Difference Term	(sigma1 - sigma2)/2	(sigma1 - sigma2)/2
Shear Stress Contribution	Minus sign inside sqrt (Options A, D)	Plus sign inside sqrt (Option C)

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Parameter	Detail
Also Known As	Maximum Principal Stress, $sigma_{max}$ , first principal stress ( $sigma_{1}$ in transformed coordinates)
Mathematical Basis	Derived from Cauchy's stress transformation equations and eigenvalue analysis of 2D stress tensor
Mohr's Circle Relation	The major principal stress is the rightmost point on Mohr's circle; the formula gives its coordinates directly
Sign Convention	Tensile stress positive, compressive negative; the formula works for both based on input stress signs
Wrong Options Explained	Option A: Minus sign and wrong stress difference order—gives minimum stress. Option B: Wrong stress difference in first term and incorrect sqrt structure. Option D: Minus sign inside sqrt creates impossible expression and gives minimum instead of maximum stress.
Critical Exam Point	The PLUS sign before the sqrt is the key distinguishing feature; this ensures the maximum (not minimum) principal stress is obtained

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