Fatigue Notch Sensitivity (q) and Factor K_f (Peterson Formula)

Why ductile shafts survive cyclic stress peaks below static elastic K_t

Elastic FEA or Peterson charts give geometric stress concentration K_t = sigma_peak / sigma_nom, yet localized grain-scale plasticity and steep stress gradients at small fillet/groove radii r_notch blunt full cyclic fatigue damage! Peterson expressed notch sensitivity as q = 1 / (1 + a_pet / r_notch) in (0, 1), mapping K_t to the reduced Fatigue Stress-Concentration Factor K_f = 1 + q * (K_t - 1), where characteristic length a_pet shrinks as ultimate tensile strength rises.

Compare two steels on a shoulder fillet with static K_t = 2.50 (B2) and fillet radius r_notch = 1.50 mm (B3): normalized medium-carbon steel (a_pet = 0.50 mm in B4) has q = 1 / (1 + 0.50 / 1.50) = 0.7500 and K_f = 1 + 0.75 * 1.50 = 2.1250; quenched ultra-high-strength steel (a_pet = 0.06 mm) jumps to q = 0.9615 and K_f = 2.4423 — nearly full brittle notch sensitivity!

Peterson Notch Sensitivity q = 1 / (1 + a_pet / r_notch) Formula

Validate K_t >= 1, notch radius r_notch > 0, and material constant a_pet >= 0 (note a_pet = 0 corresponds to q = 1.0000 brittle behavior where K_f = K_t).

=LET(kt_st, B2, r_nt, B3, a_pt, B4, IF(OR(kt_st < 1, r_nt <= 0, a_pt < 0), "Invalid Notch Input", ROUND(1 / (1 + a_pt / r_nt), 4)))

Reduced Fatigue Stress-Concentration Factor K_f = 1 + q * (K_t - 1) Formula

Evaluate K_f using unrounded q inside LET before feeding K_f into Marin endurance-limit or Goodman alternating-stress equations.

=LET(kt_st, B2, r_nt, B3, a_pt, B4, IF(OR(kt_st < 1, r_nt <= 0, a_pt < 0), "Invalid Notch Input", LET(q_sen, 1 / (1 + a_pt / r_nt), ROUND(1 + q_sen * (kt_st - 1), 4))))

Further reading

• Arte Sheets Lesson — Goodman, Soderberg, and Gerber Mean-Stress Fatigue Safety Factor
• Microsoft Support — LET function