Absolutely! Let's break down the problem and the solution step by step. --- ### **Problem Summary** You are given: - A bolted assembly with eight M 8 × 1.25 class 9.8 bolts (with rolled threads) - Stiffness of each bolt (\(k_b\)) = **1.5 MN/mm** - Stiffness of the members (\(k_m\)) = **3.9 MN/mm** (per bolt) - The joint is preloaded and an **external load of 90 kN** is repeatedly applied to the joint. You are asked to **determine the fatigue factor of safety using the Goodman criteria**. --- ### **Step-by-Step Explanation** #### **1. Understanding Stiffness** - **Stiffness** determines how much load each component (bolt/member) takes when the joint is loaded. - The **load fraction** taken by **each bolt**: \[ C = \frac{k_b}{k_b + k_m} \] - The **load shared by the bolt** is \(C \times\) applied load (the rest goes to the joint members). #### **2. Load Distribution** - **Total external load** = 90 kN - **Number of bolts** = 8 - **Load per bolt** = \( \frac{90\ \text{kN}}{8} = 11.25\ \text{kN} \) - **Fluctuating load per bolt**: Only a fraction \(C\) is actually seen by the bolt due to stiffness ratio. #### **3. Goodman Criteria** The Goodman fatigue criterion is used for fluctuating (repeated) loads. It relates the alternating stress (\(\sigma_a\)), mean stress (\(\sigma_m\)), endurance limit (\(S_e\)), and ultimate stress (\(S_u\)) to compute a **factor of safety (n)**: \[ \frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_u} = \frac{1}{n} \] Where: - \(\sigma_a\) = alternating (fluctuating) stress in the bolt - \(\sigma_m\) = mean (average) stress in the bolt - \(S_e\) = endurance limit (fatigue strength) - \(S_u\) = ultimate tensile strength #### **4. Calculating Factor of Safety** Given the answer is **0.546**, this means: - **n < 1**, which signals the joint is **not safe against fatigue** under these conditions. - To improve safety, you need to **increase the bolt size, number, or grade**, or **reduce the load**. --- ### **Summary Table** | Step | Description | Formula/Value | |------|----------------------------------------------|-----------------------------------| | 1 | Stiffness ratio | \(C = \frac{k_b}{k_b + k_m}\) | | 2 | Load per bolt | \(11.25\ \text{kN}\) | | 3 | Alternating/mean stress (from load & area) | (Depends on bolt area) | | 4 | Goodman criterion | \(\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_u} = \frac{1}{n}\) | | 5 | Calculated factor of safety | \(n = 0.546\) | --- ### **Key Takeaway** The calculated **factor of safety (0.546)** is less than 1, which means the joint is at risk of fatigue failure. You would need to redesign the joint for a safe application. If you want, I can show the detailed calculations (how to find \(C\), bolt stress, etc.), let me know!