Tresca Criterion Analysis for Plane Stress
📌 The problem involves determining yield based on the maximum shearing stress criterion (Tresca criterion) for a steel component with a yield strength ($\sigma_y$) of 325 MPa.
⚙️ The Tresca criterion states yield will not occur if $\sigma_1 - \sigma_3 \le \sigma_y$; the factor of safety (FS) is $\sigma_y / (\sigma_1 - \sigma_3)$.
🔍 For all cases (where $\sigma_0$ is 200, 240, or 280 MPa), the Mohr's circle center is at $-\sigma_0$ and the radius ($\tau_{max}$) is 100 MPa.

Case-Specific Stress Calculations and Yield Prediction
🔵 Case A ($\sigma_0 = 200 \text{ MPa}$): Principal stresses are calculated relative to the center ($-\sigma_0$) and radius ($\tau_{max}=100 \text{ MPa}$); $\sigma_1 - \sigma_3 = |(-200 + 100) - (-200 - 100)| = |-100 - (-300)| = 200 \text{ MPa}$.
🟢 Case B ($\sigma_0 = 240 \text{ MPa}$): Since one principal stress is zero, $\sigma_1 - \sigma_3$ is the absolute value of the remaining non-zero stress component, resulting in a value greater than $\sigma_y$.
🔴 Case C ($\sigma_0 = 280 \text{ MPa}$): Similar to Case B, the difference $\sigma_1 - \sigma_3$ exceeds the yield strength of 325 MPa.

Yield Determination and Factor of Safety
✅ Yield does not occur only when $\sigma_0 = 200 \text{ MPa}$ (Case A), as $\sigma_1 - \sigma_3 = 200 \text{ MPa} < 325 \text{ MPa}$.
📉 Yield occurs for $\sigma_0 = 240 \text{ MPa}$ (Case B) and $\sigma_0 = 280 \text{ MPa}$ (Case C).
🛡️ The calculated Factor of Safety (FS) is 1.08 for Case A, while Cases B and C result in FS values lower than 1.0.

Key Points & Insights
➡️ The Tresca criterion is used to predict yielding under plane stress conditions based on the maximum shear stress difference ($\sigma_1 - \sigma_3$).
➡️ For $\sigma_0 = 200 \text{ MPa}$, the material is safe with an FS of 1.08 against yielding when $\sigma_y = 325 \text{ MPa}$.
➡️ Yielding is predicted when $\sigma_0$ reaches 240 MPa or 280 MPa, indicating the stress state has surpassed the failure envelope defined by the yield strength.

📸 Video summarized with SummaryTube.com on Dec 03, 2025, 20:32 UTC