Machine Design – Theory Cards & GATE Formula Sheets

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Topic 01

Stress Analysis & Theories of Failure

⚡ GATE Hot

Principal Stresses

2D stress transformation & Mohr's Circle

2D StressMohr's Circle

Principal Stresses (σ₁, σ₂)

Maximum Shear Stress

σₓNormal stress on x-face

σᵧNormal stress on y-face

τₓᵧShear stress on x-y plane

💡Von Mises effective stress: σ′ = √(σ₁² − σ₁σ₂ + σ₂²)

⚠️

Theories of Failure

Yield & fracture criteria for design

Static Loading⚡ GATE

Theory	Failure Condition	Best For
Von Mises (DET)		Ductile, general
Tresca (MSS)	σ₁ − σ₂ ≥ Syt	Ductile, conservative
Rankine (MNS)	σ₁ ≥ Sut	Brittle materials
Coulomb-Mohr	σ₁/Sut − σ₂/Suc = 1	Brittle, Sut ≠ Suc

⚠️DET is more accurate for ductile; MSS is 15% more conservative. GATE prefers Von Mises.

Combined (Bending + Torsion + Axial)

σ′ = √(σ² + 3τ²) ≤ Syt / FOS

ℹ️For shafts under bending M and torsion T: σ = 32M/πd³, τ = 16T/πd³

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Stress Concentration

Notch effects in machine members

Kₜ FactorFatigue

Theoretical Stress Concentration

Kₜ = σ_max / σ_nom

Fatigue Stress Concentration Factor

Kf = 1 + q(Kₜ − 1)

qNotch sensitivity (0 for very ductile, 1 for very brittle)

KₜTheoretical (geometric) SCF from charts/formula

KfFatigue SCF — used in endurance limit correction

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Factor of Safety

Static & fatigue design criteria

FOSDesign

FOS (Static, Ductile)

n = Syt / σ′ (Von Mises)

FOS (Static, Brittle)

n = Sut / σ_max

Typical FOS Values

Static, well-known loads1.25 – 2.0

Dynamic / fatigue loads1.5 – 3.0

Impact loading3.0 – 5.0

Brittle materials3.0 – 6.0

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Topic 02

Shafts — Design & Analysis

⚡ GATE Hot

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Shaft Stress Formulas

Bending, torsion, combined loading

BendingTorsion

Bending Stress (solid circular)

σ_b = 32M / (π·d³)

Torsional Shear Stress

τ = 16T / (π·d³)

Combined (ASME Elliptic — GATE Favourite)

d³ = (16/π·Ssy) · √[(Kb·M)² + (Kt·T)²]

KbShock factor for bending (1.5–2.0 for minor shock)

KtShock factor for torsion (1.0–1.5 for minor shock)

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Critical Speed of Shaft

Whirling speed — Dunkerley's & Rayleigh's

Dynamics⚡ GATE

Rayleigh's Method (single disc)

Nc = (1/2π)·√(g / δ) [rps]

Dunkerley's Equation (multiple loads)

1/Nc² = 1/N₁² + 1/N₂² + ...

⚠️δ = static deflection due to self-weight. Dunkerley gives lower bound — unsafe side.

Simple supported shaft, central load W

δ = WL³ / (48EI)

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Keys & Keyways

Sunk key, Kennedy key design

ShearCrushing

Shear Failure of Key

T = l · b · τ_key · (d/2)

Crushing Failure of Key

T = l · (h/2) · σ_c · (d/2)

Key Proportions (standard)

Width bd/4

Thickness hd/6

Length l≈ 1.5d

✅Sunk key fails in shear first if τ_key ≤ σ_c/2 (matching standard material).

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Topic 03

Springs — Helical & Leaf

GATE Medium

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Helical Compression Spring

Close-coiled spring under axial load W

StiffnessDeflection

Shear Stress (Wahl's Correction)

τ = Kw · (8WD) / (π·d³)

Wahl's Stress Factor

Kw = (4C−1)/(4C−4) + 0.615/C

Deflection & Stiffness

δ = 8WD³n / (Gd⁴) k = Gd⁴ / (8D³n)

CSpring index = D/d (mean coil dia / wire dia). Typical: 4–12

nNumber of active coils

GModulus of rigidity (80 GPa for steel)

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Springs in Series & Parallel

Equivalent stiffness combinations

⚡ GATE

Series (same force, add deflections)

1/k_eq = 1/k₁ + 1/k₂ + ...

Parallel (same deflection, add forces)

k_eq = k₁ + k₂ + ...

💡Series → softer (k_eq < each k). Parallel → stiffer (k_eq > each k). Common GATE trap!

Strain Energy in Spring

U = W²/(2k) = kδ²/2 = Wδ/2

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Semi-Elliptic Leaf Spring

Nipped and graduated leaf design

AutomotiveBending

Bending Stress

σ = 6WL / (nb t²) = 3WL / (2nbt²)

Central Deflection

δ = 3WL³ / (8nbt³E)

nTotal number of leaves

bWidth of each leaf

tThickness of each leaf

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Topic 04

Gears — Spur, Helical & Bevel

⚡ GATE Hot

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Spur Gear Geometry

Fundamental gear terminology & relations

InvoluteModule

Modulem = d/T = p/π

Pitch Circle Diad = mT

Addenduma = 1·m

Dedendumb = 1.25·m

Clearancec = 0.25·m

Centre DistanceC = m(T₁+T₂)/2

Velocity RatioVR = T₂/T₁ = N₁/N₂

ℹ️T = number of teeth. p = circular pitch. For meshing gears, module must be equal.

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Lewis Beam Strength

Static tooth bending strength

⚡ GATEBending

Lewis Equation

Fb = σ_b · b · m · Y

Effective Load on Tooth

Feff = Ft · Cv (velocity factor Cv)

YLewis form factor (depends on tooth number T)

bFace width (typically 8m to 12m)

σ_bPermissible bending stress = Sut/3

⚠️Weaker gear: has smaller product σ_b·Y. Always check both gears.

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Hertz Contact (Pitting) Stress

Surface durability of gear teeth

ContactPitting

Contact Stress (AGMA)

σ_H = −Z_E · √(Ft · K_A / (b · d₁ · Z_H²))

Simplified (Buckingham)

σ_c = 0.418 · √(F_t · E / (b·r₁·r₂/(r₁+r₂)))

💡Pitting is a surface fatigue failure. σ_H must be ≤ [σ_H] = HB × 2.76 − 70 MPa.

Gear Type / Condition	Cv (Barth)
Ordinary cut gears v < 12 m/s	3/(3+v)
Carefully cut gears v < 12 m/s	4.5/(4.5+v)
Precision gears v < 20 m/s	6/(6+v)
Hobbed & shaved v < 25 m/s	5.6/(5.6+√v)

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Topic 05

Rolling Contact Bearings

⚡ GATE Hot

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Bearing Life (L₁₀)

ISO standard bearing selection

⚡ GATEL₁₀ Life

Basic Rating Life

L₁₀ = (C/P)^p [millions of revolutions]

Life in Hours

L_h = 10⁶/(60N) · (C/P)^p

Dynamic Equivalent Load

P = X·V·Fr + Y·Fa

p3 for ball bearings, 10/3 for roller bearings

CDynamic load capacity (from catalogue)

VRotation factor: 1.0 (inner ring), 1.2 (outer ring)

X,YRadial & axial load factors from bearing tables

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Hydrodynamic Journal Bearing

Petroff's law & Sommerfeld number

LubricationFriction

Petroff's Friction Torque (lightly loaded)

f = 2π²μN r² L / (W·c)

Sommerfeld Number

S = (r/c)² · μN/P

μAbsolute viscosity of lubricant (Pa·s)

cRadial clearance = R − r

NShaft speed in rev/s

ℹ️High S → thick film. Low S → metal-to-metal contact (boundary). ZN/P parameter used in Stribeck curve.

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Topic 06

Power Screws & Threaded Joints

GATE Medium

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Power Screw — Torque & Efficiency

Raising and lowering loads with screw

⚡ GATEEfficiency

Torque to Raise Load (Square thread)

T_R = W·d_m/2 · (l + π·μ·d_m) / (π·d_m − μ·l)

Torque to Lower Load

T_L = W·d_m/2 · (π·μ·d_m − l) / (π·d_m + μ·l)

Efficiency of Power Screw

η = tan(λ) / tan(λ + φ)

λLead angle = arctan(l / π·d_m)

φFriction angle = arctan(μ)

💡Self-locking condition: φ ≥ λ, i.e., μ ≥ tan(λ). Max efficiency = (1−sin φ)/(1+sin φ) at λ = 45° − φ/2.

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Bolt/Screw Preload & Joint Stiffness

Eccentric loading on bolt groups

PreloadEccentric

Initial Tightening Torque

T = K · Fi · d (K ≈ 0.2 for lubricated)

Resultant on critical bolt (direct + shear)

F_r = √(F_s² + F_t² + 2·F_s·F_t·cosθ)

ℹ️For eccentrically loaded bolt group: F_s = P/(no. of bolts), F_t = P·e·r / Σr² (turning moment).

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Topic 07

Welded Joints — Fillet & Butt

⚡ GATE Hot

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Fillet Weld Stresses

Direct shear, bending, torsion on welds

⚡ GATEFillet

Throat Area (fillet weld)

A_throat = 0.707 · h · l

Direct Shear Stress

τ = P / (0.707 · h · l)

Bending Stress in Weld Group

σ_b = M / (0.707 · h · Z_w)

⚠️h = weld leg size. l = weld length. Z_w = section modulus of weld group as a line.

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Weld Groups — Throat Properties

Treating welds as lines for analysis

Section Modulus

Weld Configuration	A_w (as line)	I_w (as line)
Single horizontal	l	l³/12
Two vertical sides (H×b)	2b	b·H + b³/6
Rectangle (b × d)	2(b+d)	d²(3b+d)/6
Circle (dia D)	πD	πD³/4

✅Multiply final I_w and Z_w by 0.707h for actual throat values.

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Topic 08

Fatigue Analysis — S-N Curve & Goodman

⚡ GATE Hot

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Endurance Limit (Se)

Modified endurance limit with correction factors

⚡ GATES-N Curve

Endurance Limit Estimate

Se′ ≈ 0.5 Sut (for Sut ≤ 1400 MPa)

Modified Endurance Limit (Marin Equation)

Se = Ka · Kb · Kc · Kd · Ke · Se′

Factor	Name	Accounts for
Ka	Surface factor	Machined, hot-rolled, forged
Kb	Size factor	d ≤ 8mm → 1.0, d > 51mm → 0.7
Kc	Load factor	Bending=1, Axial=0.85, Torsion=0.59
Kd	Temperature factor	≤350°C → 1.0
Ke	Reliability factor	90%=0.897, 99%=0.814

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Fatigue Failure Criteria

Goodman, Gerber, Soderberg diagrams

⚡ GATEGoodman

Modified Goodman (most common in GATE)

σ_a/Se + σ_m/Sut = 1/n

Soderberg (conservative, uses Syt)

σ_a/Se + σ_m/Syt = 1/n

Gerber Parabola (least conservative)

σ_a/Se + (σ_m/Sut)² = 1/n

Alternating & Mean Stress

σ_a = (σ_max − σ_min)/2 σ_m = (σ_max + σ_min)/2

⚠️GATE: When in doubt, use Modified Goodman. Soderberg is most conservative. Gerber is closest to experimental data.

Goodman Criterion

95%

Endurance Limit

88%

Soderberg

60%

Gerber

40%

Stress Concentration

80%

⚡ GATE Machine Design — Master Formula Quick Reference HOT SHEET

Principal Stress

σ₁,₂ = (σx+σy)/2 ± √[((σx−σy)/2)² + τxy²]

Von Mises Stress

σ′ = √(σ₁²−σ₁σ₂+σ₂²) = √(σ²+3τ²)

Max Shear Stress

τmax = (σ₁−σ₂)/2

Shaft (Bending)

σ = 32M / πd³

Shaft (Torsion)

τ = 16T / πd³

Spring Deflection

δ = 8WD³n / Gd⁴

Spring Stiffness

k = Gd⁴ / 8D³n

Bearing Life

L₁₀ = (C/P)^p [ball: p=3]

Lewis Eq (Gear)

Fb = σb · b · m · Y

Gear Module

m = D/T, C = m(T₁+T₂)/2

Weld (Direct τ)

τ = P / 0.707·h·l

Goodman Criterion

σa/Se + σm/Sut = 1/n

Endurance Limit

Se = Ka·Kb·Kc·Kd·Ke·Se′

Power Screw η

η = tanλ / tan(λ+φ)

Critical Speed

Nc = (1/2π)√(g/δ)

Key (Shear)

T = l·b·τ·(d/2)

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