FATIGUE EXAMPLE Cantilever Beam CASE 1: Tip is flexed ±0.075 in. What is life for 95% survival? psiIMcinlbFlMinlbFkStiffnesslbFFEbhFlEIFly076,23)1094.0)(75.0()12)(2/1094.0)(52.34(..52.34)4)(631.8(/1.115075.0631.8631.8)1094.0)(75.0)(1030)(3()12()4(075.031233363333max±==========×===σδ Since Sut = 245 ksi > 200 ksi, we set Se’ not equal to Sut/2 = 122.5 ksi, but limit it to the maximum of 100 ksi. Surface Factor (Figure 7.10a with Sut = 245 ksi) = 0.63 Check: Table 7.3: e = 2.70 ksi, f = -0.265 628.0)233.0)(7.2()245(7.2265.0===−fk Size Factor Area = (0.75)(0.1094) = 0.08205 in2 Area loaded > 95% stress is 5% of Area = (0.05)(0.08205) = 0.0041 in2 Equivalent diameter from Eqn. 7.20: Because this diameter is less than 0.3 in, we don't use 112.0869.0−= dks, but just set ks = 1. Reliability From Table 7.4, kr = 0.87 Ignore Stress Concentration Details 12 Gauge (0.1094” thick) 0.75 in. wide 4 in. long High Strength Steel, with Sut = 245 ksi Machined finish Room Temperature Note: The “e” here is not the exponential, but a variable from Table 7.3. It just happens to be almost equal to e = 2.7183 in this example. .2314.005356.00766.00041.00766.095inAd ====Derated Endurance Strength Se = kf ks kr S’e = (0.63)(1)(0.87)(100) = 54.8 ksi σalt = 23.1 ksi < 54.8 ksi Endurance Strength, ∴∴∴∴ Life is ∞∞∞∞ . Question: How big could the stress concentration at the attachment be and still have 100,000 cycles of life? Draw the S-N Diagram 0501001502002503001 10 100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000No. of Stress Cycles (N)Fatigue Strength (Sf) Sut = 245 ksiSL' = 0.90 Sut = 220.5 ksiSe = kf ks kr S'e = 54.3 ksiaNb ksiSaNSSSbksiSSafbfeLeL16.87)098.0)(2.887()000,100(2.887202.03605.0)024.4(log318.545.220log31log312.8878.545.220202.010101022====−=−=−=−=−====− So the stress concentration would have to be 77.31.2316.87=. 54.8 ksiCASE 2: Tip is flexed between 0.075 in and 0.225 in. What is life for 95% survival? By proportioning, the force now fluctuates between 8.631 lb and 3 x 8.631 = 25.893 lb. Stresses go from +23.1 ksi to +69.3 ksi. ksiksialtmean1.2321.233.6922.4621.233.692minmaxminmax=−=−==+=+=σσσσσσ What is the Factor of Safety? A) If both alternating and mean stresses increase proportionately: 64.1611.01611.0189.0422.02452.468.541.231===+=+=+=nSSnutMeAσσ B) If only alternating stress increases: 92.11.235.445.44)189.01(8.54)2452.461(8.54)1(maxmax====−=−=−=AautMeanksiSSσσσσ σalt 0 σmean 0.075 0.150 0.225 Tip Deflection (in) 0 23.1 46.2 69.3 Stress (ksi) 02040600 20 40 60 80 100 120 140 160 180 200 220 240 260Mean Stress (ksi)Alternating Stress (ksi)Sut Se σσσσM