# Analysis of Shaft Design Problem ## Problem Overview The task is to **determine the critical diameter of a shaft** (using the DE-Goodman criteria and a design factor of 1.5) based on loading conditions and to find the factor of safety against yielding. ### Provided Data - **Ultimate strength, $S_{ut}$:** 848 MPa - **Yield strength, $S_y$:** 618 MPa - **Shaft diameter at critical location:** **50 mm** - **Fillet radius:** .50 mm - **Design factor:** 1.5 ## Steps for Solution ### 1. **Identify Critical Section** - The critical section is usually where the maximum moment or torque occurs, or where stress concentration is highest (e.g., at keyseats or fillets). ### 2. **Calculate Stresses** - **Bending Stress:** \[ \sigma_b = \frac{M \cdot c}{I} \] - **Torsional Shear Stress:** \[ \tau_t = \frac{T \cdot c}{J} \] - Where: - \( M \) = Bending moment - \( T \) = Torque - \( c \) = Outer radius = 25 mm = .025 m - \( I \) (moment of inertia for a solid shaft) = \( \frac{\pi d^4}{64} \) - \( J \) (polar moment of inertia for a solid shaft) = \( \frac{\pi d^4}{32} \) ### 3. **Find Combined Stress** Using the maximum shear stress theory (or DE-Goodman for fatigue if fluctuating loads are present): \[ \sigma_{eq} = \sqrt{\sigma_b^2 + 3\tau_t^2} \] ### 4. **Apply Factor of Safety** \[ \text{Allowable stress} = \frac{S_y}{n} \] Where \( n = 1.5 \). ### 5. **Calculate Factor of Safety Against Yielding** \[ n_{actual} = \frac{S_y}{\sigma_{eq}} \] --- ## Sample Calculation (Based on Provided Diameter) ### **Step 1: Calculate Allowable Stress** \[ \text{Allowable Stress} = \frac{618\ \text{MPa}}{1.5} = 412\ \text{MPa} \] ### **Step 2: Factor of Safety Against Yielding** Assuming you calculated an equivalent stress at the critical section (let's denote it as \( \sigma_{eq} \)): \[ n_{actual} = \frac{618\ \text{MPa}}{\sigma_{eq}} \] --- ## **Summary Table** | Parameter | Value | Unit | |---------------------------------|--------------|---------| | Shaft diameter (critical) | 50 | mm | | Allowable stress (with n=1.5) | 412 | MPa | | Yield strength, \(S_y\) | 618 | MPa | | Ultimate strength, \(S_{ut}\) | 848 | MPa | | Fillet radius | .50 | mm | | Factor of safety (against yield)| (Plug your calculated \(\sigma_{eq}\)) | | --- ## **To Complete the Solution** - Plug in the actual values for the bending moment and torque at the critical section (from the diagram and loading). - Calculate combined stress. - Compute the factor of safety against yielding as shown above. --- **Note:** If you provide the actual values for \( M \) and \( T \), I can help you with exact calculations. If you need further help with DE-Goodman fatigue safety factor, please specify if the loading is fluctuating.