# Solution to Pressure Cylinder Bolting Problem Let's break down the problem into **2 steps** as requested, including example calculations and final answer. We'll also use a graph to illustrate the bolt load versus pressure. --- ## **Step 1: Calculate the Required Bolt Preload and Number of Bolts (Static Failure at Pmax)** **Given:** - Internal Pressure Range: \(P_{\text{min}} = 2\) MPa, \(P_{\text{max}} = 16\) MPa - Gasket modulus: \(E_g = 93\) GPa - Cover (cast iron) modulus: \(E_1 = 100\) GPa - Cylinder (steel) modulus: \(E_2 = 207\) GPa - Bolt size/grade: M14 × 2 mm, grade 8.8 - Load safety factor: 1.3 - Strength safety factor: 1.2 - Thread type: Rolled - Reliability: 50% - Life requirement: Infinite - **Cylinder dimensions (all in mm):** - \(R_i = 70\), \(R_b = 140\), \(R_o = 205\) ### **Step 1A: Find the Load on the Bolts at \(P_{\text{max}}\)** #### **Area Pressurized:** The force exerted by the internal pressure: \[ F = P \cdot A \] \[ A = \pi (R_b^2 - R_i^2) \] \[ A = \pi (140^2 - 70^2) = \pi (19600 - 490) = \pi \times 14700 \approx 46181 \text{ mm}^2 \] \[ F_{\text{max}} = 16 \times 10^6 \times 46181 \times 10^{-6} = 738.9 \text{ kN} \] (Here, we've multiplied by \(10^{-6}\) to convert mm² to m².) ### **Step 1B: Bolt Strength** For **M14, grade 8.8 bolts**: - Tensile area, \(A_t\), for M14: ≈ 115 mm² (standard value) - Proof strength, \(\sigma_{\text{proof}}\): 640 MPa (for 8.8 grade) \[ F_{b,\text{max}} = A_t \cdot \sigma_{\text{proof}} = 115 \times 640 = 73600 \text{ N} = 73.6 \text{ kN} \] Including safety factors: \[ F_{b,\text{allowable}} = \frac{73.6}{1.3 \times 1.2} = \frac{73.6}{1.56} = 47.18 \text{ kN} \] ### **Step 1C: Number of Bolts Needed** \[ n = \frac{F_{\text{max}}}{F_{b,\text{allowable}}} = \frac{738.9}{47.18} \approx 15.7 \] **Round up:** \[ \boxed{16 \text{ bolts}} \] --- ## **Step 2: Verify Fatigue (Fluctuating Load) and Preload** ### **Bolt Preload (for Infinite Life):** **To avoid fatigue, preload should be high enough so the fluctuation in bolt load is minimized.** - **Preload typically set at 75-90% of proof load:** \[ F_{\text{preload}} \approx .8 \cdot F_{b,\text{max}} = .8 \times 73.6 = 58.88 \text{ kN} \] **This preload ensures the joint remains tight and bolt load fluctuation is minimized.** --- ## **Graph: Bolt Load vs. Internal Pressure** Below is a conceptual graph showing how bolt load changes with internal pressure (the actual numbers would depend on joint stiffness and more precise calculations): ```plaintext Bolt Load (kN) ^ | / | / | / |----/----------------------> Internal Pressure (MPa) Pmin Pmax ``` - The slope represents the fraction of pressure load transferred to the bolts (very small if preload is high). --- ## **Final Answers** ### **Number of Bolts Required:** \[ \boxed{16 \text{ M14 × 2 mm, grade 8.8 bolts}} \] ### **Proper Preload per Bolt:** \[ \boxed{58.9 \text{ kN (80% of proof load per bolt)}} \] --- **Example Recap:** - **Given:** Cylinder, M14 bolts, \(P_{\text{max}} = 16\) MP - **Step 1:** Calculate total load and allowable load per bolt, then number of bolts. - **Step 2:** Set bolt preload to 80% of proof load to minimize fatigue. If you need a plotted graph or further calculations (like joint stiffness ratio), let me know!