Formula Sheet: Fatigue & Failure Theories

Fatigue Fundamentals

Stress-Life (S-N) Approach

Basquin's Equation (High-Cycle Fatigue): \[ S_f = a N_f^b \]

• S_f = fatigue strength (stress amplitude) at N_f cycles (psi or MPa)
• a = fatigue strength coefficient (psi or MPa)
• N_f = number of cycles to failure (cycles)
• b = fatigue strength exponent (dimensionless, typically -0.05 to -0.12)

Alternative Log-Log Form: \[ \log S_f = \log a + b \log N_f \] Endurance Limit Estimation (Steel): \[ S_e' = 0.5 S_{ut} \quad \text{(for } S_{ut} \leq 200 \text{ ksi or 1400 MPa)} \] \[ S_e' = 100 \text{ ksi or } 700 \text{ MPa} \quad \text{(for } S_{ut} > 200 \text{ ksi or 1400 MPa)} \]

• S_e' = uncorrected endurance limit (rotary-beam specimen) (psi or MPa)
• S_ut = ultimate tensile strength (psi or MPa)
• Note: Most non-ferrous metals do not have a true endurance limit

Corrected Endurance Limit: \[ S_e = k_a k_b k_c k_d k_e k_f S_e' \]

• S_e = corrected endurance limit (psi or MPa)
• k_a = surface condition modification factor
• k_b = size modification factor
• k_c = load modification factor
• k_d = temperature modification factor
• k_e = reliability modification factor
• k_f = miscellaneous effects modification factor

Modification Factors

Surface Condition Factor: \[ k_a = a S_{ut}^b \]

• For ground surface: a = 1.34 (1.58 SI), b = -0.085
• For machined or cold-drawn: a = 2.70 (4.51 SI), b = -0.265
• For hot-rolled: a = 14.4 (57.7 SI), b = -0.718
• For as-forged: a = 39.9 (272 SI), b = -0.995
• S_ut in ksi (MPa for SI coefficients)

Size Modification Factor (Bending/Torsion): \[ k_b = \begin{cases} 1.0 & \text{for } d \leq 0.3 \text{ in (7.6 mm)} \\ 0.879 d^{-0.107} & \text{for } 0.3 < d="" \leq="" 10="" \text{="" in="" (2.79="" ≤="" d="" ≤="" 254="" mm)}="" \\="" 0.91="" d^{-0.157}="" &="" \text{for="" }="" d=""> 10 \text{ in (d > 254 mm)} \end{cases} \]

• d = effective diameter (in or mm)
• For non-circular sections, use equivalent diameter: \(d = 0.808(hb)^{0.5}\) where h = height, b = width

Size Modification Factor (Axial Loading): \[ k_b = 1.0 \] Load Modification Factor: \[ k_c = \begin{cases} 1.0 & \text{bending} \\ 0.85 & \text{axial} \\ 0.59 & \text{torsion (shear)} \end{cases} \] Temperature Modification Factor: \[ k_d = \begin{cases} 1.0 & \text{for } T \leq 450°\text{F (840 K)} \\ \text{use material-specific data} & \text{for } T > 450°\text{F (840 K)} \end{cases} \] Reliability Modification Factor:

• 50% reliability: k_e = 1.000
• 90% reliability: k_e = 0.897
• 95% reliability: k_e = 0.868
• 99% reliability: k_e = 0.814
• 99.9% reliability: k_e = 0.753
• 99.99% reliability: k_e = 0.702

Stress Concentration Effects

Fatigue Stress Concentration Factor: \[ K_f = 1 + q(K_t - 1) \]

• K_f = fatigue stress concentration factor (dimensionless)
• K_t = theoretical (geometric) stress concentration factor (dimensionless)
• q = notch sensitivity (0 ≤ q ≤ 1, dimensionless)

Notch Sensitivity (Neuber's Equation): \[ q = \frac{1}{1 + \frac{a}{\sqrt{r}}} \]

• a = Neuber constant (material dependent) (in or mm)
• r = notch radius (in or mm)
• For steels: a ≈ 0.01 to 0.10 in (0.25 to 2.5 mm), depends on S_ut

Fatigue Notch Factor for Shear: \[ K_{fs} = 1 + q_s(K_{ts} - 1) \]

• K_fs = fatigue stress concentration factor for shear (dimensionless)
• K_ts = theoretical stress concentration factor for shear (dimensionless)
• q_s = notch sensitivity for shear (dimensionless)

Fluctuating Stresses

Stress Components

Mean Stress: \[ \sigma_m = \frac{\sigma_{max} + \sigma_{min}}{2} \]

• σ_m = mean stress (psi or MPa)
• σ_max = maximum stress in cycle (psi or MPa)
• σ_min = minimum stress in cycle (psi or MPa)

Alternating (Amplitude) Stress: \[ \sigma_a = \frac{\sigma_{max} - \sigma_{min}}{2} \]

• σ_a = alternating stress amplitude (psi or MPa)

Stress Range: \[ \Delta \sigma = \sigma_{max} - \sigma_{min} = 2\sigma_a \]

• Δσ = stress range (psi or MPa)

Stress Ratio: \[ R = \frac{\sigma_{min}}{\sigma_{max}} \]

• R = stress ratio (dimensionless)
• R = -1 for fully reversed loading
• R = 0 for zero-to-tension loading
• R = 1 for static loading

Amplitude Ratio: \[ A = \frac{\sigma_a}{\sigma_m} \]

• A = amplitude ratio (dimensionless)

Mean Stress Correction Theories

Soderberg Criterion (Conservative): \[ \frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_y} = \frac{1}{n} \]

• n = factor of safety (dimensionless)
• S_y = yield strength (psi or MPa)
• Most conservative; uses yield strength

Modified Goodman Criterion: \[ \frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = \frac{1}{n} \]

• S_ut = ultimate tensile strength (psi or MPa)
• Most commonly used in practice
• Moderately conservative

Gerber Criterion (Parabolic): \[ \frac{\sigma_a}{S_e} + \left(\frac{\sigma_m}{S_{ut}}\right)^2 = \frac{1}{n} \]

• Better fits test data for ductile materials
• Less conservative than Goodman

ASME Elliptic Criterion: \[ \left(\frac{\sigma_a}{S_e}\right)^2 + \left(\frac{\sigma_m}{S_y}\right)^2 = \frac{1}{n^2} \]

• Moderately conservative
• Prevents yielding on first cycle

Yielding Check (First Cycle): \[ \sigma_{max} = \sigma_a + \sigma_m \leq \frac{S_y}{n} \]

Combined Stress Fluctuating Loading

Equivalent Alternating Stress (von Mises Combined): \[ \sigma_{a,eq} = \sqrt{\sigma_a^2 + 3\tau_a^2} \]

• σ_a,eq = equivalent alternating stress (psi or MPa)
• σ_a = alternating normal stress (psi or MPa)
• τ_a = alternating shear stress (psi or MPa)

Equivalent Mean Stress (von Mises Combined): \[ \sigma_{m,eq} = \sqrt{\sigma_m^2 + 3\tau_m^2} \]

• σ_m,eq = equivalent mean stress (psi or MPa)
• σ_m = mean normal stress (psi or MPa)
• τ_m = mean shear stress (psi or MPa)

Modified Goodman for Combined Stresses: \[ \frac{\sigma_{a,eq}}{S_e} + \frac{\sigma_{m,eq}}{S_{ut}} = \frac{1}{n} \]

Strain-Life (ε-N) Approach

Coffin-Manson Relationship

Total Strain Amplitude: \[ \frac{\Delta \varepsilon}{2} = \frac{\Delta \varepsilon_e}{2} + \frac{\Delta \varepsilon_p}{2} \]

• Δε/2 = total strain amplitude (in/in or mm/mm)
• Δε_e/2 = elastic strain amplitude (in/in or mm/mm)
• Δε_p/2 = plastic strain amplitude (in/in or mm/mm)

Strain-Life Equation: \[ \frac{\Delta \varepsilon}{2} = \frac{\sigma_f'}{E}(2N_f)^b + \varepsilon_f'(2N_f)^c \]

• σ_f' = fatigue strength coefficient (psi or MPa)
• E = modulus of elasticity (psi or MPa)
• N_f = cycles to failure (cycles)
• b = fatigue strength exponent (typically -0.05 to -0.12, dimensionless)
• ε_f' = fatigue ductility coefficient (in/in or mm/mm)
• c = fatigue ductility exponent (typically -0.5 to -0.7, dimensionless)

Elastic Strain Amplitude: \[ \frac{\Delta \varepsilon_e}{2} = \frac{\sigma_f'}{E}(2N_f)^b \] Plastic Strain Amplitude: \[ \frac{\Delta \varepsilon_p}{2} = \varepsilon_f'(2N_f)^c \] Transition Life (Elastic equals Plastic): \[ 2N_t = \left(\frac{\varepsilon_f' E}{\sigma_f'}\right)^{\frac{1}{b-c}} \]

• N_t = transition life where elastic and plastic strains are equal (cycles)

Cyclic Stress-Strain Behavior

Ramberg-Osgood Cyclic Stress-Strain Curve: \[ \frac{\Delta \varepsilon}{2} = \frac{\Delta \sigma}{2E} + \left(\frac{\Delta \sigma}{2K'}\right)^{\frac{1}{n'}} \]

• Δσ/2 = stress amplitude (psi or MPa)
• K' = cyclic strength coefficient (psi or MPa)
• n' = cyclic strain hardening exponent (dimensionless)

Cyclic Strength Coefficient Estimation: \[ K' \approx \frac{\sigma_f'}{\left(\varepsilon_f'\right)^{n'}} \] Cyclic Strain Hardening Exponent Estimation: \[ n' \approx \frac{b}{c} \]

Cumulative Damage

Palmgren-Miner Linear Damage Rule

Miner's Rule: \[ D = \sum_{i=1}^k \frac{n_i}{N_i} \leq 1.0 \]

• D = total damage fraction (dimensionless)
• n_i = number of cycles applied at stress level i (cycles)
• N_i = number of cycles to failure at stress level i (cycles)
• k = number of different stress levels
• Failure predicted when D ≥ 1.0
• Conservative approach often uses D ≥ 0.7 to 1.0 for failure

Remaining Life: \[ n_{remaining} = N_i\left(1 - \sum_{j=1}^{i-1} \frac{n_j}{N_j}\right) \]

Static Failure Theories

Ductile Materials

Maximum Shear Stress Theory (Tresca): \[ \tau_{max} = \frac{\sigma_1 - \sigma_3}{2} \leq \frac{S_y}{n} \]

• τ_max = maximum shear stress (psi or MPa)
• σ₁ = maximum principal stress (psi or MPa)
• σ₃ = minimum principal stress (psi or MPa)
• S_y = yield strength (psi or MPa)
• n = factor of safety (dimensionless)

Simplified for Uniaxial Tension: \[ \frac{S_y}{2} \leq \frac{S_y}{n} \] Distortion Energy Theory (von Mises): \[ \sigma' = \sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2} \leq \frac{S_y}{n} \]

• σ' = von Mises equivalent stress (psi or MPa)
• σ₁, σ₂ = principal stresses (psi or MPa)
• Most accurate for ductile materials

von Mises for 3D Stress State: \[ \sigma' = \sqrt{\frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2}{2}} \leq \frac{S_y}{n} \] von Mises in Terms of Stress Components: \[ \sigma' = \sqrt{\sigma_x^2 - \sigma_x\sigma_y + \sigma_y^2 + 3\tau_{xy}^2} \leq \frac{S_y}{n} \]

• σ_x, σ_y = normal stresses in x and y directions (psi or MPa)
• τ_xy = shear stress (psi or MPa)
• For plane stress (σ_z = 0)

von Mises for General 3D State: \[ \sigma' = \sqrt{\frac{1}{2}\left[(\sigma_x - \sigma_y)^2 + (\sigma_y - \sigma_z)^2 + (\sigma_z - \sigma_x)^2 + 6(\tau_{xy}^2 + \tau_{yz}^2 + \tau_{zx}^2)\right]} \]

Brittle Materials

Maximum Normal Stress Theory (Rankine): \[ \sigma_1 \leq \frac{S_{ut}}{n} \quad \text{(tension)} \] \[ |\sigma_3| \leq \frac{S_{uc}}{n} \quad \text{(compression)} \]

• S_ut = ultimate tensile strength (psi or MPa)
• S_uc = ultimate compressive strength (psi or MPa)
• Used for brittle materials

Coulomb-Mohr Theory: \[ \frac{\sigma_1}{S_{ut}} - \frac{\sigma_3}{S_{uc}} \leq \frac{1}{n} \]

• Accounts for different tensile and compressive strengths
• For σ₁ ≥ 0 ≥ σ₃

Modified Mohr Theory:

• If σ₁ ≥ σ₂ ≥ 0: \(\sigma_1 \leq \frac{S_{ut}}{n}\)
• If σ₁ ≥ 0 ≥ σ₂: \(\frac{\sigma_1}{S_{ut}} - \frac{\sigma_2}{S_{uc}} \leq \frac{1}{n}\)
• If 0 ≥ σ₁ ≥ σ₂: \(|\sigma_2| \leq \frac{S_{uc}}{n}\)

Fracture Mechanics

Stress Intensity Factor

Mode I Stress Intensity Factor: \[ K_I = \beta \sigma \sqrt{\pi a} \]

• K_I = stress intensity factor for Mode I (opening) (psi√in or MPa√m)
• β = geometry correction factor (dimensionless)
• σ = applied stress (psi or MPa)
• a = crack length (in or m)

For Center Crack in Infinite Plate: \[ K_I = \sigma \sqrt{\pi a} \]

• β = 1.0 for center crack
• a = half crack length

For Edge Crack: \[ K_I = 1.12 \sigma \sqrt{\pi a} \]

• β = 1.12 for edge crack
• a = full crack length

Fracture Toughness Criterion: \[ K_I \leq K_{Ic} \]

• K_Ic = plane strain fracture toughness (psi√in or MPa√m)
• Critical material property

Critical Crack Length: \[ a_c = \frac{1}{\pi}\left(\frac{K_{Ic}}{\beta \sigma}\right)^2 \]

• a_c = critical crack length for fracture (in or m)

Paris Law (Fatigue Crack Growth)

Paris Equation: \[ \frac{da}{dN} = C(\Delta K)^m \]

• da/dN = crack growth rate per cycle (in/cycle or m/cycle)
• C = material constant (depends on units)
• ΔK = stress intensity factor range (psi√in or MPa√m)
• m = material constant (typically 2 to 4, dimensionless)

Stress Intensity Factor Range: \[ \Delta K = K_{max} - K_{min} = \beta \Delta \sigma \sqrt{\pi a} \]

• Δσ = stress range (psi or MPa)

Cycles to Failure (Integration of Paris Law): \[ N_f = \int_{a_i}^{a_f} \frac{da}{C(\Delta K)^m} \]

• a_i = initial crack length (in or m)
• a_f = final (critical) crack length (in or m)

For Constant β and Δσ: \[ N_f = \frac{1}{C(\beta \Delta \sigma \sqrt{\pi})^m} \int_{a_i}^{a_f} a^{-m/2} da \] Simplified Result (m ≠ 2): \[ N_f = \frac{2}{C(\beta \Delta \sigma \sqrt{\pi})^m (2-m)} \left[a_i^{1-m/2} - a_f^{1-m/2}\right] \]

Impact and Dynamic Loading

Impact Factor

Suddenly Applied Load: \[ \sigma_{dynamic} = 2\sigma_{static} \]

• Impact factor = 2 for suddenly applied load

Axial Impact: \[ \sigma_{max} = \frac{P}{A}\left(1 + \sqrt{1 + \frac{2hAE}{PL}}\right) \]

• σ_max = maximum stress (psi or MPa)
• P = weight of falling object (lb or N)
• A = cross-sectional area (in² or m²)
• h = drop height (in or m)
• E = modulus of elasticity (psi or MPa)
• L = length of member (in or m)

Bending Impact: \[ \sigma_{max} = \frac{Mc}{I}\left(1 + \sqrt{1 + \frac{2hEI}{McL}}\right) \]

• M = static moment from weight (lb·in or N·m)
• c = distance from neutral axis to extreme fiber (in or m)
• I = moment of inertia (in⁴ or m⁴)

Safety Factors and Design Criteria

Factor of Safety Definitions

Factor of Safety Based on Yield: \[ n = \frac{S_y}{\sigma_{applied}} \] Factor of Safety Based on Ultimate Strength: \[ n = \frac{S_{ut}}{\sigma_{applied}} \] Factor of Safety for Fatigue: \[ n = \frac{S_e}{\sigma_a} \quad \text{(fully reversed)} \] Factor of Safety with Mean Stress (Goodman): \[ \frac{1}{n} = \frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} \]

Design Stress Limits

Allowable Stress: \[ \sigma_{allow} = \frac{S}{n} \]

• σ_allow = allowable stress (psi or MPa)
• S = strength parameter (S_y or S_ut) (psi or MPa)

Required Area: \[ A_{req} = \frac{P}{\sigma_{allow}} \]

Multiaxial Fatigue

Critical Plane Approaches

Findley Criterion: \[ \frac{\tau_a + k\sigma_{n,max}}{f} \leq 1 \]

• τ_a = shear stress amplitude on critical plane (psi or MPa)
• σ_n,max = maximum normal stress on critical plane (psi or MPa)
• k = material constant (dimensionless)
• f = fatigue limit in pure torsion (psi or MPa)

Smith-Watson-Topper (SWT) Parameter: \[ SWT = \sigma_{max} \varepsilon_a E \]

• σ_max = maximum normal stress (psi or MPa)
• ε_a = strain amplitude (in/in or mm/mm)
• Used for mean stress effects in crack initiation

Fatigue under Torsion

Endurance Limit in Torsion: \[ S_{se} = 0.577 S_e \]

• S_se = endurance limit in pure shear (psi or MPa)
• Based on von Mises criterion

Alternative Estimation: \[ S_{se} = 0.67 S_{ut} \]

Creep and High-Temperature Failure

Larson-Miller Parameter

Larson-Miller Parameter: \[ LMP = T(C + \log t_r) \]

• LMP = Larson-Miller parameter
• T = absolute temperature (°R or K)
• C = material constant (typically 20 for metals)
• t_r = rupture time (hours)

Time to Rupture: \[ t_r = 10^{(LMP/T - C)} \]

Variable Amplitude Loading

Rainflow Counting Method

• Used to convert complex loading histories into equivalent constant amplitude cycles
• Extracts closed stress-strain hysteresis loops
• Each cycle characterized by mean stress and alternating stress
• Results used with Miner's rule for damage calculation

Equivalent Stress Methods

Root-Mean-Square (RMS) Stress: \[ \sigma_{rms} = \sqrt{\frac{1}{n}\sum_{i=1}^n \sigma_i^2} \]

• σ_rms = root-mean-square stress (psi or MPa)
• n = number of stress values

Equivalent Constant Amplitude: \[ \sigma_{eq} = \left(\sum_{i=1}^k n_i \sigma_i^m\right)^{1/m} \]

• σ_eq = equivalent stress amplitude (psi or MPa)
• m = inverse slope of S-N curve (typically 3 to 5)