Machine Design

Syllabus

Design for static and dynamic loading; failure theories; fatigue strength and the S-N diagram; principles of design of bolted, riveted and welded joints, shafts, gears, rolling and sliding contact bearings, brakes and clutches, and springs.

Key concepts and formulas

Theories of failure (σ_y = yield strength, N = factor of safety):

In pure shear, Tresca predicts yield at τ = 0.5σ_y and von Mises at τ = σ_y/√3 ≈ 0.577σ_y.

Shafts under combined bending and torsion. Equivalent torque T_e = √(M² + T²) (max shear stress theory), equivalent moment M_e = ½(M + √(M² + T²)) (max principal stress theory). Then τ_max = 16T_e/(πd³) and σ_max = 32M_e/(πd³).

Fatigue. For steels, the endurance limit S_e′ ≈ 0.5S_ut, reduced by surface, size, reliability and other factors. With alternating stress σ_a and mean stress σ_m:

• Goodman: σ_a/S_e + σ_m/S_ut = 1/N
• Soderberg: σ_a/S_e + σ_m/S_y = 1/N (more conservative)
• Gerber: σ_a/S_e + (σ_m/S_ut)² = 1 (with N = 1)

Joints. Fillet weld: throat = 0.707 × leg size; shear stress = load/(0.707 s l). Bolts: tensile stress area, preload.

Helical springs. Spring index C = D/d. Shear stress τ = 8PDK/(πd³) (K = Wahl factor); deflection δ = 8PD³n/(Gd⁴); stiffness k = Gd⁴/(8D³n).

Rolling bearings. Rating life L₁₀ = (C/P)ᵖ million revolutions, with p = 3 for ball and 10/3 for roller bearings.

Gears. Lewis bending strength F = σ_bbmY (Y = Lewis form factor).

Brakes and clutches. Single-plate clutch with uniform wear: T = μW(r₁ + r₂)/2 per friction surface; with uniform pressure: T = (2/3)μW(r₁³ − r₂³)/(r₁² − r₂²). Band brake: T₁/T₂ = e^μθ.

Worked example

A ductile shaft is under σ₁ = 120 MPa and σ₂ = −60 MPa, with σ_y = 300 MPa. Find the factor of safety by the Tresca and von Mises theories. Then find the Goodman factor of safety for σ_a = 100 MPa, σ_m = 150 MPa, S_e = 250 MPa and S_ut = 600 MPa, and the life of a ball bearing with C = 30 kN and P = 6 kN.

import math

s1, s2, sy = 120, -60, 300
tresca = sy / (s1 - s2)
vm = sy / math.sqrt(s1 ** 2 - s1 * s2 + s2 ** 2)
print(f"Tresca N = {tresca:.3f}, von Mises N = {vm:.3f}")
sa, sm, se, sut = 100, 150, 250, 600
print(f"Goodman N = {1 / (sa / se + sm / sut):.4f}")
C, P = 30, 6
print(f"ball bearing life = {(C / P) ** 3:.0f} million revolutions")
G, d, D, n, F = 80e3, 8, 48, 10, 500
print(f"spring stiffness = {G * d ** 4 / (8 * D ** 3 * n):.3f} N/mm")

Tresca N = 1.667, von Mises N = 1.890
Goodman N = 1.5385
ball bearing life = 125 million revolutions
spring stiffness = 37.037 N/mm

Practice questions