Let's **analyze the formula** and answer whether \[ \tau_{max} = \frac{T C}{J} \] is **valid for non-circular cross sections** such as triangles, as questioned in your image. --- ## Step 1: **Understand the Formula** - **\(\tau_{max}\)**: Maximum shear stress in the cross-section - **\(T\)**: Applied torque - **\(C\)**: Distance from the center to the outermost fiber (radius for circles) - **\(J\)**: Polar moment of inertia, \( J = \int_A \rho^2 dA \) For a **circular shaft under torsion**: \[ \tau_{max} = \frac{T r}{J} \] where \(r\) is the outer radius. --- ## Step 2: **Applicability to Non-Circular Shapes** ### **For Circular Cross Sections:** - The formula is **exact** for solid and thin-walled circular shafts. - Shear stress varies linearly from the center to the outer surface. ### **For Non-Circular Cross Sections (like Triangles):** - Torsion theory is **different**. - **Prandtl's membrane analogy** and **Saint-Venant’s theory** show that: - Shear stress distribution is **not linear**. - Cross-section does **not remain plane** after twisting. - The formula \(\tau_{max} = \frac{T C}{J}\) **does NOT apply** as it does for circles. #### **For Triangles:** - Maximum shear stress and its distribution must be found by **advanced methods** (e.g., warping functions, numerical or series solutions). - For a **thin-walled non-circular section**, modifications or different formulas are used. --- ## Step 3: **Why Doesn't the Formula Apply?** - **Assumptions of the formula**: - Purely circular cross section - Uniform stress distribution on periphery - For **triangles (or rectangles, ellipses, etc.)**: - The stress distribution is **not radial**. - Warping occurs, invalidating the direct use of \( \frac{T C}{J} \). --- ## Step 4: **Correct Approach for Triangles (and other shapes)** - Use **torsion formulas** specifically derived for the shape. - For **rectangles**: There are empirical and analytical solutions involving series. - For **triangles**: You need numerical methods or specialized charts. --- ## **Final Answer** > **No, the formula \( \tau_{max} = \frac{T C}{J} \) is NOT valid for cross sections other than circles (like triangles).** > > This formula is derived for circular shafts under pure torsion, where the shear stress varies linearly from the center. For non-circular sections (such as triangles), the stress distribution is more complex and requires different methods to determine \(\tau_{max}\). --- ### **Summary Table** | Shape | Formula for \(\tau_{max}\) | Notes | |------------------|-----------------------------------------|-------------------------------------------| | Circle | \(\frac{T r}{J}\) | Exact | | Thin-walled tube | \(\frac{T}{2A_m t}\) | Approximate, \(A_m =\) mean area, \(t =\) thickness | | Rectangle/Triangle | **Not \(\frac{T C}{J}\)** | Use specialized solutions | --- **In conclusion:** The formula shown in your image is **only for circles**. For triangles, you need other approaches!