Tools › Mechanical Design › Shafts › Shaft Design › Example

Stepped shaft on two bearings: Marin endurance limit and DE-Goodman factor (Shigley's method)

Tool	Shaft Design
Source	Budynas, R. G., and Nisbett, J. K. (2015). Shigley's Mechanical Engineering Design, 10th ed. McGraw-Hill Education, New York. Section 6-9 (Marin factors, Table 6-2, eq. 6-20, Table 6-5) and Section 7-4 (DE-Goodman, eq. 7-8).
Level	Undergraduate

Problem

A machined steel shaft (S_ut = 95 ksi, S_y = 77 ksi) with diameters 1.25 / 1.5 / 1.75 / 1.5 in runs at 1,750 rpm on bearings 20 in apart and carries two gears at x = 6 in and 14 in with radial loads in two planes and ±1,600 lb-in of torque. Find the Marin-corrected endurance limit of the 1.5-in sections at 99 % reliability, the fully reversed bending and steady torsional stresses at the critical station, the DE-Goodman fatigue factor of safety and the first critical speed.

Inputs to Enter

Bearing span 20 in, speed 1,750 rpm; sections start x −2 / 2 / 10 / 18 in, diameters 1.25 / 1.5 / 1.75 / 1.5 in, K_f 1.0 / 1.6 / 1.5 / 1.6, K_fs 1.0 / 1.3 / 1.25 / 1.3
Loads at x = 6 in: F_y −800 lb, F_z 290 lb, T 1,600 lb-in, mass 4 lb; at x = 14 in: F_y 300 lb, F_z −820 lb, T −1,600 lb-in, mass 6 lb
S_ut 95 ksi, S_y 77 ksi, E 30 Msi, density 0.284 lb/in³, surface finish machined, reliability 99 %, design factor 2.0

Open the Shaft Design tool (loaded with this example)

Answers

k_a (machined) = 2.70 · 95^−0.265 = 0.8077; k_b (d = 1.5 in) = 0.879 · 1.5^−0.107 = 0.8417; k_e (99 %) = 0.814 (Shigley Table 6-5)
S_e = 0.8077 · 0.8417 · 0.814 · 47.5 ksi = 26.29 ksi (S_e′ = 0.5 S_ut = 47.5 ksi)
Critical station: M = 2,827 lb-in, T = 1,600 lb-in in d = 1.5 in → σ_a = 32M/(πd³) = 8.531 ksi, τ_m = 16T/(πd³) = 2.414 ksi
DE-Goodman n_f = 1 / (2 · 8.531/26.29 + √3 · 2.414/95) = 1.443; von Mises static n_y = 8.10
Rayleigh first critical speed n_c = 13,400 rpm (tool 13,403 rpm, shaft mass included)

Notes

The tool's default inputs are this shaft; press Submit. The case is the tool's own default checked against Shigley's equations (test_general_tools.php), not a worked example printed in the book.