Practice Problems: Fatigue & Failure Theories
Question 1
A mechanical engineer is analyzing a steel shaft subjected to cyclic loading in a manufacturing plant. The shaft experiences a completely reversed bending stress. Given the following information:
Ultimate tensile strength (Su) = 620 MPa
Yield strength (Sy) = 450 MPa
Endurance limit (Se') = 310 MPa (uncorrected)
Surface finish factor (ka) = 0.78
Size factor (kb) = 0.85
Load factor (kc) = 1.0
Temperature factor (kd) = 1.0
Reliability factor (ke) = 0.897 (for 95% reliability)
What is the corrected endurance limit of the shaft?
(a) 183 MPa
(b) 205 MPa
(c) 228 MPa
(d) 310 MPa
Question 2
A design engineer is evaluating a connecting rod that experiences fluctuating axial stress. The rod is made of aluminum alloy with the following properties:
Maximum stress (σmax) = 180 MPa
Minimum stress (σmin) = 40 MPa
What is the mean stress (σm) and alternating stress (σa) for this loading condition?
(a) σm = 110 MPa, σa = 70 MPa
(b) σm = 140 MPa, σa = 140 MPa
(c) σm = 70 MPa, σa = 110 MPa
(d) σm = 90 MPa, σa = 90 MPa
Question 3
A structural engineer is analyzing a steel component under biaxial loading conditions. The principal stresses at a critical point are:
σ1 = 280 MPa
σ2 = 140 MPa
σ3 = 0 MPa
Yield strength (Sy) = 350 MPa
Using the maximum shear stress theory (Tresca), what is the factor of safety?
(a) 1.25
(b) 1.50
(c) 1.75
(d) 2.00
Question 4
A reliability engineer is testing a component that failed after 2.5 × 10⁶ cycles. The component was subjected to a fully reversed stress of 220 MPa. The material properties are:
Ultimate tensile strength (Su) = 750 MPa
Endurance limit at 10⁶ cycles (Se) = 375 MPa
Using the Basquin equation with b = -0.085, what is the fatigue strength coefficient (σf')?
(a) 890 MPa
(b) 925 MPa
(c) 1050 MPa
(d) 1125 MPa
Question 5
A consulting engineer is evaluating a machine component subjected to combined fluctuating stresses. The component is made of AISI 1045 steel with:
Alternating stress (σa) = 120 MPa
Mean stress (σm) = 180 MPa
Endurance limit (Se) = 250 MPa
Ultimate tensile strength (Su) = 620 MPa
Using the Goodman failure criterion, what is the factor of safety?
(a) 1.35
(b) 1.52
(c) 1.68
(d) 1.84
Question 6
A manufacturing engineer is designing a bolt that will experience tensile loading. The bolt material has the following properties under triaxial stress state:
σ1 = 350 MPa
σ2 = 210 MPa
σ3 = 70 MPa
Yield strength (Sy) = 450 MPa
Using the von Mises (distortion energy) theory, what is the equivalent stress?
(a) 268 MPa
(b) 289 MPa
(c) 305 MPa
(d) 321 MPa
Question 7
A project engineer is analyzing fatigue life of a gear tooth subjected to pulsating stress. The stress varies from 50 MPa to 350 MPa. The material properties are:
Yield strength (Sy) = 550 MPa
Ultimate strength (Su) = 750 MPa
Endurance limit (Se) = 340 MPa
Using the Soderberg criterion, what is the factor of safety against fatigue failure?
(a) 1.42
(b) 1.68
(c) 1.85
(d) 2.12
Question 8
A failure analysis engineer is investigating a shaft failure. The shaft was subjected to combined bending and torsion with:
Bending stress amplitude (σa) = 90 MPa
Bending mean stress (σm) = 60 MPa
Torsional shear stress amplitude (τa) = 50 MPa
Torsional mean shear stress (τm) = 30 MPa
Endurance limit in bending (Se) = 280 MPa
Ultimate tensile strength (Su) = 600 MPa
Using the modified Goodman criterion with von Mises combination, what is the equivalent alternating stress (assuming Se,shear = 0.577×Se)?
(a) 122 MPa
(b) 138 MPa
(c) 154 MPa
(d) 170 MPa
Question 9
A materials engineer is conducting a stress-life (S-N) fatigue test on a steel specimen. The following data points were obtained:
At 10³ cycles: σf = 520 MPa
At 10⁶ cycles: σf = 280 MPa
Assuming a log-log linear relationship, what is the fatigue strength at 5 × 10⁴ cycles?
(a) 375 MPa
(b) 395 MPa
(c) 415 MPa
(d) 435 MPa
Question 10
A power transmission engineer is designing a shaft with a stress concentration at a fillet. The shaft properties are:
Nominal alternating stress (σnom) = 85 MPa
Theoretical stress concentration factor (Kt) = 2.4
Notch sensitivity (q) = 0.85
Corrected endurance limit (Se) = 240 MPa
What is the fatigue stress concentration factor (Kf) and the actual alternating stress at the notch?
(a) Kf = 2.19, σa = 186 MPa
(b) Kf = 2.40, σa = 204 MPa
(c) Kf = 1.85, σa = 157 MPa
(d) Kf = 2.04, σa = 173 MPa
Question 11
A quality control engineer is testing a component that experiences variable amplitude loading. Using Miner's rule for cumulative damage, the following loading history is recorded:
n1 = 50,000 cycles at stress level σ1 (N1 = 200,000 cycles to failure)
n2 = 80,000 cycles at stress level σ2 (N2 = 400,000 cycles to failure)
n3 = 30,000 cycles at stress level σ3 (N3 = 150,000 cycles to failure)
What is the cumulative damage index, and has failure occurred?
(a) D = 0.75, no failure
(b) D = 0.85, no failure
(c) D = 0.95, no failure
(d) D = 1.05, failure occurred
Question 12
A design engineer is evaluating a pressure vessel subjected to internal pressure. The vessel wall experiences the following stresses:
Hoop stress (σ1) = 180 MPa
Longitudinal stress (σ2) = 90 MPa
Radial stress (σ3) = 0 MPa
Yield strength (Sy) = 250 MPa
Using the maximum distortion energy theory, what is the factor of safety against yielding?
(a) 1.45
(b) 1.60
(c) 1.75
(d) 1.90
Question 13
A maintenance engineer is analyzing a crankshaft that failed due to fatigue. The material has the following properties:
Fully corrected endurance limit (Se) = 200 MPa
Ultimate tensile strength (Su) = 550 MPa
Applied alternating stress (σa) = 110 MPa
Applied mean stress (σm) = 150 MPa
Using the Gerber parabolic relationship, what is the factor of safety?
(a) 1.28
(b) 1.45
(c) 1.62
(d) 1.78
Question 14
A research engineer is studying low-cycle fatigue in a turbine blade. The strain-life approach is being used with:
Fatigue ductility coefficient (εf') = 0.65
Fatigue ductility exponent (c) = -0.6
Expected life (N) = 5000 cycles
What is the plastic strain amplitude at this life?
(a) 0.0125
(b) 0.0158
(c) 0.0192
(d) 0.0225
Question 15
A structural engineer is analyzing a welded joint in a bridge structure. The joint experiences cyclic loading with:
Maximum load (Pmax) = 450 kN
Minimum load (Pmin) = 150 kN
Cross-sectional area (A) = 2500 mm²
What is the stress ratio (R) and stress range (Δσ) for this loading condition?
(a) R = 0.33, Δσ = 120 MPa
(b) R = 0.40, Δσ = 108 MPa
(c) R = 0.33, Δσ = 108 MPa
(d) R = 0.40, Δσ = 120 MPa
Question 16
A reliability engineer needs to determine the required diameter of a rotating shaft subjected to completely reversed bending. Given:
Bending moment amplitude (Ma) = 850 N·m
Corrected endurance limit (Se) = 180 MPa
Desired factor of safety (n) = 2.0
Fatigue stress concentration factor (Kf) = 1.8
What is the minimum required shaft diameter?
(a) 45 mm
(b) 52 mm
(c) 58 mm
(d) 64 mm
Question 17
A consulting engineer is evaluating a component that operates in the finite-life region. The material has:
Fatigue strength at 10³ cycles (Sf at 10³) = 0.9Su = 630 MPa
Endurance limit at 10⁶ cycles (Se) = 350 MPa
Ultimate tensile strength (Su) = 700 MPa
Using the log-log relationship, what is the fatigue strength at 2 × 10⁴ cycles?
(a) 465 MPa
(b) 485 MPa
(c) 505 MPa
(d) 525 MPa
Question 18
A product development engineer is designing a leaf spring for automotive application. The spring experiences:
Maximum stress (σmax) = 550 MPa
Minimum stress (σmin) = -150 MPa
Endurance limit (Se) = 320 MPa
Yield strength (Sy) = 900 MPa
Ultimate strength (Su) = 1100 MPa
Using the modified Goodman criterion, what is the factor of safety against fatigue failure?
(a) 0.82
(b) 0.94
(c) 1.06
(d) 1.18
Question 19
A failure analysis engineer is investigating a shaft with a keyway that failed under torsional fatigue. The shaft parameters are:
Nominal shear stress amplitude (τnom) = 65 MPa
Theoretical stress concentration for keyway (Kt) = 2.0
Notch radius (r) = 0.5 mm
Notch sensitivity index for this material/geometry (q) = 0.90
Endurance limit in shear (Sse) = 195 MPa
What is the factor of safety against torsional fatigue failure?
(a) 1.42
(b) 1.58
(c) 1.74
(d) 1.90
Question 20
A test engineer is conducting a fatigue test on aluminum alloy specimens. Two specimens with different surface finishes show the following results at infinite life:
Ground finish: Endurance limit = 165 MPa (ka = 0.89)
Machined finish: Endurance limit = 145 MPa (ka = 0.78)
Both specimens have the same size factor kb = 0.85
Ultimate tensile strength (Su) = 410 MPa
What was the theoretical (uncorrected) endurance limit Se' before surface and size corrections were applied?
(a) 205 MPa
(b) 218 MPa
(c) 231 MPa
(d) 244 MPa
