# Bolted Connection Fatigue Problem (Goodman Criterion) Let's break the problem into parts and solve step by step. --- ## **Given Data** - **Material:** AISI 102 steel (for bracket and beam) - **Bolt:** 1/2 in coarse-thread SAE Grade 8, tightened to recommended preload - **Stiffness (bolt):** \( K_b = 4 \times 10^6 \) lbf/in - **Stiffness (members):** \( K_m = 16 \times 10^6 \) lbf/in - **Fatigue design factor:** \( n = 2 \) - **Failure governed by bolts, not welds** - **Use Goodman criterion for fatigue** --- ## **Step 1: Find Bolt Properties** ### SAE Grade 8, 1/2 in Bolt (Coarse Thread) - **Major diameter (\( d \)):** .5 in - **Tensile stress area (\( A_t \)):** ≈ .142 in² (from standard tables) - **Ultimate tensile strength (\( S_{ut} \)):** 150 ksi (Grade 8) - **Yield strength (\( S_y \)):** 130 ksi (Grade 8) --- ## **Step 2: Preload in the Bolt** Recommended preload for non-permanent assembly is typically \( 75\% \) of proof strength. - **Proof strength (\( S_p \)):** 120 ksi (Grade 8) - **Preload (\( F_i \)):** \[ F_i = .75 S_p A_t = .75 \times 120,000 \times .142 = 12,780 \text{ lbf} \] --- ## **Step 3: Load Distribution** The fluctuating load \( F \) applied to the bracket does **not** all go into the bolt; only a fraction does, depending on the stiffnesses: - **Fraction to bolt:** \[ C = \frac{K_b}{K_b + K_m} = \frac{4}{4 + 16} = .2 \] So, the bolt sees 20% of the external load. If the external load cycles between and \( F \) (fully reversed loading), the **load range** in the bolt is: \[ \Delta F_b = C \Delta F = .2F \] --- ## **Step 4: Stresses in the Bolt** - **Mean load on bolt:** \( F_{m, b} = F_i + .1F \) (since mean of and \( F \) is \( F/2 \); fraction to bolt is .2, so mean change is \( .1F \)) - **Amplitude load on bolt:** \( F_{a, b} = .1F \) (half the load range) So: - Mean stress: \( \sigma_m = \frac{F_i + .1F}{A_t} \) - Stress amplitude: \( \sigma_a = \frac{.1F}{A_t} \) --- ## **Step 5: Goodman Criterion** The Goodman criterion for fatigue is: \[ \frac{\sigma_a}{S_e/n} + \frac{\sigma_m}{S_{ut}} \leq 1 \] Where: - \( S_e \): Endurance limit (corrected) - \( S_{ut} \): Ultimate strength - \( n \): Design factor ### **Find \( S_e \) (Endurance Limit)** For bolts, a common estimate: - Rotating beam endurance limit: \( S_e' = .5 S_{ut} = .5 \times 150 = 75 \) ksi - Corrections for size, surface, etc. For a rolled thread, typical \( S_e \approx .22 S_{ut} \) to \( .3 S_{ut} \) - **We'll use:** \( S_e = .22 \times 150 = 33 \) ksi (conservative) --- ## **Step 6: Plug Values into Goodman Equation** Let \( F \) be the **maximum external load**. The criterion becomes: \[ \frac{\frac{.1 F}{A_t}}{S_e/n} + \frac{\frac{F_i + .1F}{A_t}}{S_{ut}} = 1 \] Plug in values: - \( A_t = .142 \) in² - \( S_e/n = 33,000/2 = 16,500 \) psi - \( S_{ut} = 150,000 \) psi - \( F_i = 12,780 \) lbf So: \[ \frac{.1F}{.142 \times 16,500} + \frac{12,780 + .1F}{.142 \times 150,000} = 1 \] \[ \frac{.1F}{2,343} + \frac{12,780 + .1F}{21,300} = 1 \] --- ## **Step 7: Solve for \( F \) (Safe Repeated Load)** Let: - \( x = F \) Equation: \[ \frac{.1x}{2343} + \frac{12,780 + .1x}{21,300} = 1 \] Multiply both sides by 21,300 to clear denominators: \[ .1x \frac{21,300}{2,343} + 12,780 + .1x = 21,300 \] Calculate \( 21,300/2,343 \approx 9.095 \): \[ .1x \times 9.095 + 12,780 + .1x = 21,300 \] \[ .9095x + .1x + 12,780 = 21,300 \] \[ 1.0095x = 21,300 - 12,780 \] \[ 1.0095x = 8,520 \] \[ x = \frac{8,520}{1.0095} \approx 8,445 \text{ lbf} \] --- ## **Step 8: Static Load Factor** Static load factor is the ratio of the static strength of the bolt to the actual fluctuating load. - **Static strength of bolt:** For proof strength, \( F_{\text{proof}} = S_p \times A_t = 120,000 \times .142 = 17,040 \) lbf - **Factor:** \( \frac{F_{\text{proof}}}{F} = \frac{17,040}{8,445} \approx 2.02 \) --- # **Final Answers** ## **a. Safe Repeated Load Using Goodman Criterion** > **The safe repeated load that can be imposed is approximately \( \boxed{8,450~\text{lbf}} \) (rounded to 3 significant digits).** ## **b. Static Load Factor** > **The static load factor for the load found above is approximately \( \boxed{2.} \).** --- ## **Summary Table** | Item | Value | |----------------------|--------------| | Safe repeated load | 8,450 lbf | | Static load factor | 2. | --- ### **Assumptions** - Only one bolt considered (if multiple, divide load appropriately) - Conservative estimate for endurance limit - Preload based on non-permanent assembly - External load cycled from to \( F \) --- If you have the actual number of bolts, or if the load is distributed differently, adjust the calculation accordingly.