Complete Exam Study Guide · Parts I–III
Advanced Machine Design — Study Guide
Drawn directly from your MEPP 436 slide decks and lecture notes, then elaborated with the reference texts where the slides are thin. Core (examinable) content is plain; reference/beyond-slides enrichment is always in a coloured box so you can tell them apart.
★ Most-recent paper (25 May 2026 internal)
The newest paper leans on Juvinall & Marshek (fracture “similar sheet” scaling; fatigue S-N with modifying factors) and Shukla (Paris-law crack growth). Those exact methods are elaborated in §5, §7 below and worked in the Subjective Bank.
Click a topic heading to expand or collapse it. Use the sidebar to jump between topics; each figure is taken from your course slides.
1 Introduction to Mechanical Engineering Design
DefinitionDesign = applying creativity to plan the optimum solution of a given problem and communicate it to others. Keyword chain: given problem → creativity → planning → optimum solution → communication. Engineering design narrows this to engineering problems; mechanical engineering design to mechanical elements.
1.1 The design process & considerations
Design proceeds through recognised stages, iterating as needed:
identification of need → problem definition → synthesis → analysis & optimisation → evaluation → presentation
Typical design considerations juggled by the engineer: functionality, strength/stress, distortion/deflection, wear, corrosion, safety, reliability, manufacturability, cost, weight, life, noise, styling and environmental impact.
1.2 Standards and codes
| Term | Definition | Purpose |
|---|---|---|
| Standard | Specifications for parts, materials or processes | Uniformity, efficiency, a specified quality |
| Code | Specifications for analysis, design, manufacture & construction | A specified degree of safety, efficiency & performance |
Bodies to know: NBSM (Nepal), NBC 105:2020, BIS, AISI, ASME, ASTM, ASHRAE, SAE.
1.3 Design economics
- Using standard sizes is the first principle of cost reduction (cuts tooling & procurement cost).
- Close (tight) tolerances raise cost — extra processing, extra inspection, slower machines.
- Break-even analysis compares two production methods; below the break-even quantity the low-setup method wins, above it the high-rate method wins.
1.4 Design factor & factor of safety common MCQ
\[ n_d=\frac{\text{loss-of-function parameter}}{\text{max-allowable parameter}}\quad\Longrightarrow\quad n=\frac{\text{Strength}}{\text{Stress}}=\frac{S}{\sigma} \]
- Stress & strength must be the same type, same units, same critical location.
- The design factor n_d is chosen up-front for uncertainty; the factor of safety is the realised margin after rounding to standard sizes.
Exam tip (Feb 2025 · Q1)The purpose of a factor of safety is “to compensate for uncertainties in material properties and loading conditions” — not weight, cost or aesthetics.
1.5 Reliability (introductory)
\[ R=1-p_f,\qquad p_f=\frac{\text{number of failures}}{\text{total instances}},\qquad 0\le R\le 1 \]
Series system (all must work): R = R₁·R₂·…·Rₙ. Example: two bearings 0.95 & 0.98 → R = 0.931. (Full treatment in §9.)
1.6 Dimensions & tolerances
| Term | Meaning |
|---|---|
| Nominal size | The size used when referring to a part (need not equal the actual dimension) |
| Limits | The stated max & min dimensions |
| Tolerance | The difference between the two limits |
| Bilateral / Unilateral | Variation in both directions / in one direction only |
| Clearance / Interference | Internal member smaller / larger than external member |
| Allowance | Minimum clearance (or maximum interference) of mating parts |
Beyond the slides — why “design determines cost”Roughly 70–80% of a product's total lifecycle cost is locked in at the design stage — the same idea that drives DFMA (§8). Standard sizes, tolerances and design factors are really early lessons in cost- and risk-management, not just geometry.
2 Material Properties & Selection
Choosing the material is one of the earliest and most important design decisions — usually made before dimensions are fixed. Properties come from specimen testing (standardised, e.g. ASTM) or, when risk is high, component testing under real service loads.
2.1 The tensile test (ASTM E8) & key properties
A load–elongation curve becomes a stress–strain curve, giving: elastic limit / yield strength, elastic modulus E (Hooke's law), ultimate tensile strength (UTS), ductility (% elongation, % area reduction), tensile toughness, Poisson's ratio ν.
Engineering vs true stress/strainEngineering values use the original gauge dimensions. In the plastic region area & length change, so true stress = load ÷ instantaneous area and true strain εₜ = ln(l/l₀) describe plastic behaviour accurately.
2.2 Other mechanical tests
| Test | Measures | Key fact |
|---|---|---|
| Compression | Behaviour under compressive load | Needed for ceramics, concrete |
| Shear | Shear stress–strain | Shear yield ≈ 0.5–0.75 × tensile yield; G ≈ 0.4E |
| Hardness (Brinell/Rockwell/Vickers) | Resistance to surface penetration | Not fundamental; for steels UTS(MPa) ≈ 3.4 × HB |
| Impact (Izod/Charpy) | Energy absorbed on sudden load | Materials get more brittle at high strain rate |
| Creep | Time-dependent deformation at high T | Governs high-temperature design |
| Fatigue | Strength loss under repeated stress | See §6–§7 |
Exam tip (Feb 2025 · Q5)For high-temperature service the most important property is creep resistance.
2.3 Heat treatment (effect on properties)
| Process | Effect |
|---|---|
| Quenching | Very hard/strong martensite; trades ductility for strength |
| Tempering | After quench: lowers strength a little, restores some ductility |
| Annealing | Soft, relaxed state; removes residual stresses |
| Normalizing | Stronger/harder than fully annealed but close to it |
2.4 Material classes
| Class | Strengths | Weaknesses |
|---|---|---|
| Metals & alloys | High strength/stiffness, ductile, tough, good fatigue & wear, conductive | Heavy, can corrode |
| Polymers | Low density → good specific strength, corrosion-resistant, insulating, easily formed | Low strength, poor at high T |
| Ceramics | Excellent compressive strength, high E, hard, wear/corrosion-resistant, high-T stable | Very brittle; tension ≈ 10% of compression; no plasticity |
| Composites | High specific strength/modulus, good fatigue/creep, tailorable | Cost, anisotropy, harder to recycle |
Exam tip (Feb 2025 · Q4)Main advantage of composites = “high strength-to-weight ratio and tailored properties.”
2.5 Material selection charts (Ashby charts)
An Ashby chart plots one property against another (e.g. Young's modulus vs density). Each material class occupies a “bubble.” Overlay a guideline of constant performance (specific stiffness E/ρ = C, or specific strength σ/ρ = C); materials in the top-left/upper region win for light-and-stiff or light-and-strong designs.
- Metals heaviest; foams lightest; ceramics stiffest.
- Light-and-stiff bike frame: polymers too floppy, ceramics too brittle in tension → composites best; Mg/Al/Ti competitive.
From the reference books — the material index (Norton / Ashby)Behind the guideline is a material index: maximise E1/2/ρ for a light-stiff beam, σf2/3/ρ for a light-strong beam. A line of slope 2 (or 3/2) on the log–log chart is a line of constant index; sliding it up-left finds the winning material. Knowing the index exists lets you explain why the chart guideline has a particular slope.
3 Stresses, Strains & Failure Theories
3.1 Stress & strain as tensors
Stress = internal resistance per unit area developed against external load; strain = the resulting deformation. At a point, three mutually perpendicular planes fully describe the stress state — the “stress cube.”
\[ \sigma_{ij}=\begin{bmatrix}\sigma_{xx}&\tau_{xy}&\tau_{xz}\\ \tau_{yx}&\sigma_{yy}&\tau_{yz}\\ \tau_{zx}&\tau_{zy}&\sigma_{zz}\end{bmatrix}\qquad\text{symmetric 2nd-order tensor — 9 components, 6 independent} \]
- Plane stress: all z-face stresses = 0 (thin plates, pressure-vessel walls, free surfaces). Components σₓₓ, σᵧᵧ, τₓᵧ.
- Plane strain: all z-direction strains = 0 (thick bodies: dams, long shafts).
3.2 Generalized Hooke's law (isotropic, linear elastic)
\[ \varepsilon_x=\frac1E\big[\sigma_x-\nu(\sigma_y+\sigma_z)\big]\ (\text{+ cyclic}),\qquad \gamma_{xy}=\frac{\tau_{xy}}{G} \]
\[ \text{Lamé: }\sigma_{ij}=\lambda\,\varepsilon_{kk}\delta_{ij}+2\mu\,\varepsilon_{ij}\ (\mu=G),\qquad \text{anisotropic: }\sigma_{ij}=C_{ijkl}\,\varepsilon_{kl} \]
3.3 Principal stresses, transformation & Mohr's circle
Principal planes carry no shear, only normal stress; those are the principal stresses σ₁ ≥ σ₂ ≥ σ₃.
\[ \sigma_{1,2}=\frac{\sigma_x+\sigma_y}{2}\pm\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2} \]
\[ \tau_{\max}=\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}=\frac{\sigma_1-\sigma_3}{2},\qquad \tan2\theta_p=\frac{2\tau_{xy}}{\sigma_x-\sigma_y} \]
3.4 Bending & torsion (revision)
\[ \text{Flexure: }\frac{M}{I}=\frac{\sigma_b}{y}=\frac{E}{\rho}\qquad \text{Torsion: }\frac{T}{J}=\frac{\tau}{\rho}=\frac{G\phi}{L}\qquad J=\frac{\pi}{32}(D^4-d^4) \]
Worked example — hollow shaft torsion (July 2025 · Q2)D = 100, d = 60 mm, L = 1 m, τ = 35 MPa, G = 85 GPa. With ρ = R = 50 mm: φ = τL/(ρG) = (35×1000)/(50×85000) = 0.0082 rad ≈ 0.47° per metre.
Worked example — hollow shaft stress (Feb 2025 · Q3)dᵢ = 15, dₒ = 30 mm, T = 100 N·m. J = (π/32)(30⁴−15⁴) = 74 490 mm⁴; τ = T·rₒ/J = 100000·15/74490 ≈ 20.12 MPa.
3.5 Failure — the five classical theories
| Theory (a.k.a.) | Failure criterion | Best for | Yield surface |
|---|---|---|---|
| Max principal stress (Rankine) | σ₁ ≥ σult (or σᵧ) | Brittle | Square |
| Max shear stress (Tresca) | σ₁ − σ₃ ≥ σᵧ | Ductile (conservative) | Hexagon |
| Max principal strain (St. Venant) | ε₁ ≥ εᵧ | Rarely used | Rhombus |
| Max total strain energy (Haigh) | U ≥ U at yield | — | Ellipse |
| Max distortion energy (Von Mises) | \(\sigma'=\sqrt{\tfrac12[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2]}\ge\sigma_y\) | Ductile (accurate) | Ellipse |
\[ \text{von Mises (biaxial, }\sigma_3=0\text{): }\ \sigma'=\sqrt{\sigma_1^2-\sigma_1\sigma_2+\sigma_2^2}\ge\sigma_y \]
The five yield surfaces individually (course slides, all in \(\sigma_1/\sigma_{yp}\)–\(\sigma_2/\sigma_{yp}\) space — a point inside is safe, on the boundary it yields):
max principal stress · brittle
max shear · ductile (conservative)
max principal strain (\(\nu=0.35\))
max total strain energy (\(\nu=0.35\))
max distortion energy · ductile (accurate)
Exam tip (Feb 2025 · Q2)For ductile shafts the “commonly used” criterion is Maximum Shear Stress (Tresca) — the safe, simple choice. Von Mises is more accurate/less conservative. Know both yield-surface shapes.
Beyond the slides — why Von Mises worksTotal strain energy splits into volumetric (size change) + distortional (shape change). Hydrostatic pressure alone doesn't yield metals, so only the distortion energy should count — the physical reason Von Mises predicts ductile yielding better than Tresca. This one-line “why” lifts a 3-mark answer to 4.
4 Introduction to Fracture Mechanics
DefinitionFracture mechanics studies the propagation of cracks. A crack is a sharp void that acts as a stress amplifier — cracks don't create stress, they intensify the far-field stress. The sharper the tip, the higher the local stress.
4.1 Why fracture mechanics? (the design shift)
Traditional design compares stress to strength (2 parameters). Fracture mechanics adds a third: flaw size. The design is safe if the combination of stress and largest expected crack keeps K < Kᴵᴄ. It answers: what is the strength as a function of crack size, and what is the maximum tolerable crack size?
Historical driver: the WWII Liberty Ships broke in cold water — welded (continuous) construction, brittle high-sulphur steel and micro-cracks let a crack run through the whole hull.
4.2 Characteristics of a crack
- Connection to free surface — fully internal / internal-connected-to-surface / surface crack.
- Crack length — longer is more dangerous.
- Crack-tip radius — sharper is more dangerous; plasticity blunts tips in ductile metals.
- Crack orientation relative to loading.
Exam tip (Feb 2025 · Q7)“Crack surface roughness” is NOT a defining characteristic (the four are length, tip radius, orientation, surface connection).
4.3 Modes of loading (memorise cold)
Common mistake (Feb 2025 · Q6)“Rolling mode” is NOT real. The only three are Opening (I), Sliding (II), Tearing (III). Real cracks are often mixed-mode.
4.4 Cohesive stress & why real strength is low
Cohesive stress σᴄ = theoretical stress to break atomic bonds, from the interatomic force–displacement curve (≈ half a sine wave):
\[ \sigma_c=\sqrt{\frac{E\,\gamma_s}{x_0}}\qquad(\gamma_s=\text{surface energy},\ x_0=\text{atomic spacing}) \]
Real materials fracture far below σᴄ because flaws concentrate stress. For an elliptical crack (tip radius ρ = b²/a):
\[ \sigma_{tip}\approx\sigma\!\left(1+\frac{2a}{b}\right)\approx2\sigma\sqrt{\frac{a}{\rho}}\ (\text{sharp flaw})\quad\Longrightarrow\quad \sigma_f\approx\sqrt{\frac{E\,\gamma_s}{4a\,x_0}} \]
A micron crack gives σ_f ≈ 0.01 σᴄ — matching experiment; a “perfect” solid always contains flaws.
5 Linear Elastic Fracture Mechanics (LEFM)
LEFM applies to brittle behaviour — sharp cracks, little tip plasticity. For growth, both criteria hold: the global energy criterion (Griffith) and the local stress criterion (Inglis).
5.1 Inglis (1913) — local, stress-based
Stress concentration at a notch depends on the tip radius. The crack grows when the amplified tip stress reaches the theoretical fracture stress: σmax = σ_A ≥ σ_f.
5.2 Griffith (1920) — global, energy-based
\[ \frac{dU_s}{da}\ge\frac{dU_\gamma}{da},\quad U_s=\frac{\pi a^2\sigma^2}{E}\ (\text{per unit thickness}),\ U_\gamma=4\gamma_s a \]
\[ \sigma_f=\sqrt{\frac{2E\gamma_s}{\pi a}}\ (\text{plane stress}),\qquad \sigma_f=\sqrt{\frac{2E\gamma_s}{\pi a(1-\nu^2)}}\ (\text{plane strain}) \]
Modified Griffith (Irwin–Orowan) — crucial for metalsPure Griffith works only for truly brittle solids (glass, ceramics). In metals, huge energy is dissipated by plastic deformation at the tip; Irwin & Orowan replaced γₛ by an effective fracture energy w_f = γₛ + γₚ, with γₚ ≫ γₛ. This is why tough steel absorbs thousands of times more energy than its bond energy alone.
5.3 Stress Intensity Factor K — the central quantity
\[ \sigma_{ij}=\frac{K}{\sqrt{2\pi r}}\,f_{ij}(\theta)\quad\Longrightarrow\quad K_I=Y\,\sigma\sqrt{\pi a}\qquad(\text{units MPa}\cdot\sqrt{\text{m}}) \]
- a = half-length for a fully internal (central) crack; full length for an edge crack.
- Y = geometry/shape factor. Insight: quadrupling crack length ≡ doubling stress — K captures the combined effect.
| Geometry | Shape factor Y |
|---|---|
| Central crack, infinite plate | 1.0 |
| Single edge crack | 1.12 (12% higher — extra energy at free surface) |
| Embedded penny-shaped crack | 2/π ≈ 0.64 |
| Surface half-penny crack | 0.713 |
5.4 Irwin's fracture criterion & fracture toughness Kᴵᴄ
Fracture occurs in Mode I when Kᵢ ≥ Kᴵᴄ. Kᴵᴄ = fracture toughness (a material property like σᵧ), microstructure-sensitive, measured under plane strain (thick specimen → conservative, lowest value).
\[ \text{Critical crack length: }\ a_c=\frac{1}{\pi}\left(\frac{K_{IC}}{Y\sigma}\right)^2 \]
| Material | Kᴵᴄ (MPa√m) | Material | Kᴵᴄ (MPa√m) |
|---|---|---|---|
| Cast iron | 33 | Al 2024-T3 | 33 |
| Low-carbon steel | 77 | Al 7075-T6 | 28 |
| Stainless steel | 220 | Ti-6Al-4V | 55 |
Worked example — critical crack length (Feb 2025 · Q12)Edge crack, Δσ = 0→260 MPa, Kᴵᴄ = 165 MPa√m, Y = 1.12. aᴄ = (1/π)(165/(1.12·260))² = (1/π)(0.5666)² = 0.1022 m ≈ 102.2 mm.
From the reference books — Griffith scaling (Juvinall Ch. 6 · Shukla) ★ 2026 internalBecause Kᴵᴄ = Yσ_f√(πa) is constant for a given material, a “similar sheet” with a different crack obeys σf1√a₁ = σf2√a₂. E.g. maraging steel, central crack 40 mm at σ_f = 480 MPa → for a 100 mm crack, σ_f2 = 480·√(20/50) = 303.6 MPa. This exact type appeared in the 25 May 2026 internal (Q2).
5.5 Energy release rate G and the K–G relation
\[ G=-\frac{d\Pi}{dA}\ (\text{crack driving force}),\qquad G=\frac{K_I^2}{E'}\ \big(E'=E\text{ plane stress},\ E/(1-\nu^2)\text{ plane strain}\big) \]
\[ \text{Crack grows when }G\ge G_c;\qquad \text{compliance: }G=\frac{P^2}{2}\frac{dC}{da} \]
K is local; G is global; for linear elasticity they are uniquely related.
5.6 Crack-tip plasticity & the LEFM limit
LEFM predicts infinite tip stress (the stress singularity); plasticity intervenes:
- Irwin model: plastic-zone size r_y = (1/2π)(Kᵢ/σ_ys)² (plane stress); use effective crack length a + r_y.
- Strip-yield (Dugdale–Barenblatt): a thin plastic strip length ρ at each tip; total crack = 2(a+ρ).
LEFM vs EPFM — one line that shows masteryK and G belong to Linear-Elastic FM (brittle, small plastic zone). When the plastic zone is large (tough metals, thin sections) LEFM fails and you move to Elastic-Plastic FM using the J-integral (Rice, 1968) or CTOD δ (Wells, 1961). J reduces to G in the elastic limit — that's why it “generalises” the energy-release-rate concept.
5.7 Summary of fracture criteria
| Criterion (year) | Basis | Condition | Key formula |
|---|---|---|---|
| Inglis (1913) | Local stress, tip radius | σ_A ≥ σ_f | σtip = 2σ√(a/ρ) |
| Griffith (1920) | Global energy | dUₛ/da ≥ dUᵧ/da | σ_f = √(2Eγ/πa) |
| Irwin [K] | Stress intensity | Kᵢ ≥ Kᴵᴄ | Kᵢ = Yσ√(πa) |
| Irwin [G] | Energy release rate | G ≥ Gᴄ | G = K²/E' |
| Wells (1961) — CTOD δ | Crack-tip opening | δ ≥ δᴄ | δ = Kᵢ²/(Eσᵧ) |
| Rice (1968) — J-integral | Generalised energy (elastic-plastic) | J ≥ Jᴄ | HRR fields |
6 Material Fatigue — mechanism & concepts
Definition (ASTM)Fatigue = progressive, localised, permanent structural change under fluctuating stress/strain that may end in cracks or fracture after enough cycles — even when the peak load is below the yield or ultimate strength. Fatigue life = number of cycles to failure.
Fatigue gives no warning (little deflection), is sudden (brittle-like), and is only partly understood, so life is estimated empirically. Infamous cases: 1842 Versailles rail crash (locomotive axle); 1980 Alexander L. Kielland platform (fatigue crack in bracing D-6, 123 deaths).
6.1 The five-stage failure mechanism
- Cyclic plastic deformation → dislocations pile at the surface, forming persistent slip bands.
- Micro-crack initiation along slip bands, grain boundaries, inclusions.
- Micro-crack coalescence into a macro-crack.
- Macro-crack propagation — governed by ΔK (LEFM); plotted as da/dN vs ΔK.
- Final failure — remaining section can't carry the load; rapid fracture.
Governing parameters across the process: K_t (stress-concentration) → K_I (stress-intensity) → K_IC (fracture toughness).
6.2 Fatigue loading parameters
\[ \sigma_a=\frac{\sigma_{max}-\sigma_{min}}{2}\ (\text{amplitude}),\qquad \sigma_m=\frac{\sigma_{max}+\sigma_{min}}{2}\ (\text{mean}) \]
\[ \Delta\sigma=\sigma_{max}-\sigma_{min}\ (\text{range}),\qquad R=\frac{\sigma_{min}}{\sigma_{max}}\ (\text{stress ratio}),\qquad A=\frac{\sigma_a}{\sigma_m}\ (\text{amplitude ratio}) \]
| Type | R | Description |
|---|---|---|
| Fully reversed | −1 | σ_m = 0 (rotating-bending) |
| Repeated | 0 | σmin = 0 |
| Fluctuating | 0 < R < 1 | General tension–tension |
Worked example (Feb 2025 · Q10)σmin=50, σmax=250 → σ_a = (250−50)/2 = 100 MPa (mean 150, range 200).
6.3 HCF vs LCF — a guaranteed question
| High-Cycle Fatigue (HCF) | Low-Cycle Fatigue (LCF) | |
|---|---|---|
| Cycles to failure | > 10³ (often 10⁴–10⁵) | < 10³ |
| Stress level | Low (below yield) | High (local yielding) |
| Dominant strain | Mostly elastic | Plastic strain dominates |
| Best analysis | Stress-life (S-N) | Strain-life (ε-N) |
| Typical source | High-frequency loading (valve springs) | Start-up/shut-down thermal cycles |
6.4 Factors affecting fatigue
Cyclic load state (amplitude, mean, sequence, biaxiality) · geometry/stress concentration · surface quality · residual stress (compressive helps, tensile hurts) · microstructure (finer grains → longer life) · temperature · environment (corrosion fatigue).
Exam tip — residual stressShot-peening / cold-rolling put the surface in compression, lowering the effective mean stress there, delaying crack initiation and increasing fatigue life. Tensile residual stress does the opposite.
6.5 Fatigue testing & the S-N curve
Rotating-beam (R.R. Moore) test: the specimen rotates under bending so each point cycles tension↔compression; max stress ≈ 5.09 FL/d³. Data plotted as stress S vs cycles N (usually log N).
- Endurance/fatigue limit: stress below which (ferrous metals) fatigue never occurs (~10⁶–10⁷ cycles).
- Fatigue strength: stress to fail at a specified N. Fatigue life: cycles permitted at a given stress.
- Endurance ratio = endurance limit / UTS ≈ 0.3–0.4 for metals.
Beyond the slides — aluminium has no true endurance limitNon-ferrous metals (Al, Cu) and high-strength steels show a continuously falling S-N curve, so engineers quote a fatigue strength at 5×10⁸ cycles instead — which is why aircraft aluminium is designed by safe-life / damage-tolerance, not infinite-life.
7 Fatigue Analysis Approaches
| Approach | Best for | Idea |
|---|---|---|
| Stress-life (S-N) | HCF | Oldest, most data; assumes little plasticity; least accurate for LCF |
| Strain-life (ε-N) | LCF | Analyses local plastic strain; cyclic stress–strain (Ramberg–Osgood) |
| LEFM (crack growth) | Structures with detectable cracks | Assumes a crack exists; predicts growth vs ΔK; used with inspection |
7.1 Stress-life: Basquin's equation
\[ S_a=a\,N^{\,b}\ (b<0);\qquad \text{two lives: }\frac{S_1}{S_2}=\left(\frac{N_1}{N_2}\right)^{b} \]
Worked example — Basquin (Feb 2025 · Q11)S = a·N^(−0.15), S = 400 MPa at N = 10⁵. At 10⁶: S₂ = 400·10^(−0.15) = 400·0.708 ≈ 283 ≈ 280 MPa.
From the reference books — estimated S-N with modifying factors (Juvinall & Marshek Ch. 8) ★ 2026 internalJuvinall builds the finite-life S-N line between 0.9 Sᵤ at 10³ cycles and the corrected endurance limit at 10⁶ (or 10⁷). The endurance point is scaled by modifying factors: CL (load) · CG (gradient/size) · CS (surface) · CO (other). Fit S = aN^b through the two anchor points. This is exactly internal-2026 Q3 and July-2025 Q3(b) — see the worked solutions in the Subjective Bank.
7.2 Strain-life (ε-N)
For LCF, total strain amplitude = elastic + plastic parts (Basquin + Coffin–Manson), with a cyclic Ramberg–Osgood curve (K′, n′).
Beyond the slides — full strain-life equation
\[ \frac{\Delta\varepsilon}{2}=\frac{\sigma_f'}{E}(2N)^b+\varepsilon_f'(2N)^c \]
First term (Basquin, exponent b) is the elastic line dominating HCF; second (Coffin–Manson, exponent c) is the plastic line dominating LCF. Where the two cross is the transition life separating LCF from HCF.7.3 LEFM crack growth & Paris' law highest-yield numerical
\[ \Delta K_I=Y\,\Delta\sigma\sqrt{\pi a},\qquad \Delta\sigma=\sigma_{max}-\sigma_{min}\quad(\textbf{use }\Delta\sigma,\ \text{NOT }\sigma_{max}) \]
| Region | Behaviour | Controlled by |
|---|---|---|
| I (threshold) | Below ΔK_th no growth (~10⁻¹⁰ m/cycle) | Microstructure, mean stress, environment |
| II (Paris) | Linear log–log: da/dN = C(ΔK)^m | ΔK; insensitive to microstructure |
| III (unstable) | Accelerating as Kmax → Kᴵᴄ | Fracture toughness K_c |
\[ \text{Paris' law: }\frac{da}{dN}=C(\Delta K)^m;\qquad N=\int_{a_i}^{a_f}\frac{da}{C\,(Y\Delta\sigma\sqrt{\pi a})^m} \]
\[ (m=3,\ Y,\Delta\sigma\text{ const}):\quad N=\frac{2}{C(Y\Delta\sigma\sqrt{\pi})^3}\left(a_i^{-1/2}-a_f^{-1/2}\right) \]
Worked example — Paris-law life (Feb 2025 Q3c & internal-2026 Q4) ★Edge crack aᵢ = 30 mm, Δσ = 0→180 MPa, Kᴵᴄ = 150 MPa√m, Y = 1.12, C = 7.2×10⁻¹², m = 3.
aᴄ = (1/π)(150/(1.12·180))² = 0.1762 m → a_f = aᴄ/2 = 0.0881 m.
YΔσ√π = 357.3; cubed = 4.56×10⁷; ×C = 3.28×10⁻⁴. aᵢ⁻½ = 5.774, a_f⁻½ = 3.369 → N ≈ 1.46×10⁴ cycles.
aᴄ = (1/π)(150/(1.12·180))² = 0.1762 m → a_f = aᴄ/2 = 0.0881 m.
YΔσ√π = 357.3; cubed = 4.56×10⁷; ×C = 3.28×10⁻⁴. aᵢ⁻½ = 5.774, a_f⁻½ = 3.369 → N ≈ 1.46×10⁴ cycles.
Common mistakes(1) using σmax instead of Δσ in ΔK; (2) forgetting the m=3 integral gives a⁻¹ᐟ² terms (exponent 1−m/2 = −½). Keep a in metres if C is in metre units.
7.4 Variable amplitude — Palmgren–Miner rule
\[ D=\sum_i\frac{n_i}{N_i};\qquad \text{failure when }D\ge1;\qquad \text{blocks to failure}=1/D_{block} \]
Beyond the slides — Miner is only approximateIt ignores load sequence: high-then-low ≠ low-then-high because of crack-tip residual stresses and overload retardation. Real failures often occur at D between 0.7 and 2.2. Mentioning “sequence effects / D ≠ exactly 1” is a strong extra point. (Shukla states this cumulative-damage rule as ΣN₀/N = 1.)
7.5 Fatigue / damage design strategies
| Strategy | Principle | Example |
|---|---|---|
| Infinite-life | Keep stress below the fatigue limit forever | Engine valve springs |
| Safe-life | Design for a finite life, then retire (with scatter margin) | Bearings, jet-engine parts |
| Fail-safe | System still holds if one part fails; multiple load paths | Aircraft structure |
| Damage-tolerance | Assume cracks; fracture mechanics + NDI before critical | Modern airframes |
Damage-tolerance needs three things: residual strength, fatigue-crack-growth behaviour, and crack detection (NDI).
8 Design for Manufacturing & Assembly (DFMA)
DefinitionDFMA = DFM + DFA: reduce time-to-market and total cost by making parts easy to make and easy to assemble, deciding this at the design stage.
Instructor's note: in the Dieter/Schmidt reference, focus on sections 13.5, 13.6, 13.11, 13.12, 13.14, 13.15.
8.1 Paradigms → concurrent engineering
- Over-the-wall — traditional; almost no design↔manufacturing communication.
- Sign-off — manufacturing must approve drawings.
- Limited collaboration — teams interact only on critical points.
- Concurrent engineering — design & manufacturing work together from concept to launch (basis of DFX).
Key fact: 60–80% of product cost is fixed by design decisions (Feb 2025 Q13 quotes the 70–80% band).
8.2 DFM vs DFA
| DFM | DFA | |
|---|---|---|
| Goal | Reduce part production cost | Reduce assembly cost |
| Method | Optimise material & process, reduce complexity | Reduce part count, simplify, fewer assembly moves |
| When | Detailed design | Early, before prototypes |
Exam tip (Feb 2025 · Q14)DFA's role in DFM is “optimising the assembly process for efficiency.”
8.3 Five principles & guidelines
Five principles of DFMA: Process · Design · Material · Environment · Compliance/Testing.
DFM guidelines
- Minimise number of parts; standardise components
- Use common parts across product lines
- Keep designs simple & functional; make parts multifunctional
- Avoid tight tolerances; minimise finishing operations
DFA guidelines
- Minimise part count & assembly directions
- Use subassemblies; mistake-proof (poka-yoke)
- Avoid separate fasteners; self-aligning & self-locating parts
- Provide unobstructed access; design for symmetry (or clear asymmetry)
Assembly = handling (grasp, orient, position) + insertion & fastening. Automation levels: manual, automatic (feeder + workhead), robotic.
8.4 Classic case — Ford vs GM
Boothroyd's DFA software saved Ford billions on the Taurus (1988). GM traced 41% of its productivity gap to manufacturability: Ford's front bumper had 10 parts vs GM's 100.
8.5 Process-specific DFM rules
| Process | Key design rules |
|---|---|
| Castings | Orderly (directional) solidification; uniform section; avoid shrinkage cavities & hot tears; pattern must draw; add machining allowance |
| Forging | Taper surfaces (draft 5–7° external, 7–10° internal); single-plane parting line; uniform sections; allow for scale & warpage |
| Machining | Machine only functional surfaces; good reference/holding surface; avoid re-clamping; minimise burrs |
| Welding | Straight force flow-lines, fewest welds; equal-thickness parts; low-stress locations; ensure access |
From the reference books — DFX aspects (Ulrich & Eppinger)DFX = “Design for X,” X = any life-cycle concern: Manufacture, Assembly, Reliability, Safety, Serviceability/Maintenance, Environment/Sustainability, Cost, Quality, Ergonomics. Ulrich & Eppinger frame DFM as estimating manufacturing cost, reducing component/assembly/overhead cost, then iterating.
9 Design for Safety & Reliability
DefinitionReliability R(t) = probability that a system performs its intended function, within tolerances, under stated conditions, for a specified time. 0 ≤ R ≤ 1.
Design is a three-way trade-off: performance ↔ reliability ↔ cost. Reliability grows through a test–fix–test–fix prototype cycle.
9.1 The bathtub curve
| Region | Failure rate | Cause |
|---|---|---|
| 1 · Infant mortality (burn-in) | Decreasing | Manufacturing defects |
| 2 · Useful life | Constant (random) | Random overloads: surges, impact, vibration |
| 3 · Wear-out | Increasing | Corrosion, fatigue, wear |
9.2 Reliability functions & indices
\[ F(t)=\int_0^t f\,dx\ (\text{CDF, unreliability}),\qquad R(t)=1-F(t),\qquad \lambda(t)=\frac{f(t)}{R(t)} \]
\[ \lambda=\frac{\#\text{ failures}}{\text{operating time}},\qquad \text{MTTF}=\int_0^\infty R\,dt=\frac1\lambda\ (\text{non-repairable}),\qquad \text{MTBF}=\frac1\lambda\ (\text{repairable}) \]
9.3 Three failure-time distributions
| Distribution | Reliability R(t) | Use |
|---|---|---|
| Exponential | e^(−λt) (constant λ; memoryless) | Useful-life / random failures |
| Normal | 1 − Φ(z), z = (t−μ)/σ | Wear-out / aging |
| Weibull | e^(−(t/θ)^m) | Most versatile — all three regions |
Weibull shape m (≡ β): m<1 → decreasing rate (infant mortality); m=1 → exponential; m>1 → increasing (wear-out); m≈3.5 → ≈ normal. θ = characteristic life.
Worked example — Normal (Feb 2025 · Q4c)μ=6 yr, σ=1 yr. (i) P(fail<7): z=1 → Φ=0.8413. (ii) R(7)=1−0.8413=0.1587. (iii) 10% failure: Φ(z)=0.10 → z=−1.28 → T = 6−1.28 = 4.72 yr.
Worked example — Weibull (Feb 2025 · Q16)m=0.7, θ=7500, t=1450: R = e^(−(1450/7500)^0.7) = e^(−0.314) = 0.728 ≈ 72.8%.
Worked example — Exponential parallel (July 2025 · Q4b)MTTF=650 h, t=850: λt=1.308, R=e^(−1.308)=0.270; F=0.730; two in parallel R=1−0.730²=0.468; need n≥3 for R≥0.6.
9.4 System reliability
\[ \text{Series (all needed): }R_s=\prod R_i=e^{-(\sum\lambda_i)t}\qquad \text{Parallel (any one): }R_s=1-\prod(1-R_i) \]
Worked example — series vs parallelUnits 0.95, 0.85, 0.75. Series R = 0.605; Parallel R = 1−(0.05)(0.15)(0.25) = 0.998. Redundancy dramatically improves reliability.
9.5 Design for Reliability (DFR) & Safety (DFS)
DFR strategies: fail-safe (monitor the weak link), “one-horse-shay” (equal-life components), absolute worst-case. DFR guidelines: margin of safety, derating, redundancy, durability, damage tolerance, ease of inspection, simplicity, specificity.
Exam tipsQ15: increasing complexity does NOT enhance reliability. · Q19: for DFR of electrical appliances, “Safety” is the odd one out among derating/redundancy/simplicity. · Q17: “safety factor” = ratio of actual load to design load (as worded).
Reliability ≠ Safety. An aircraft that never takes off is safe but not reliable; one that flies reliably but kills passengers is reliable but not safe.
Design for Safety — hazard hierarchy: (1) design the hazard out; (2) add protective devices (guards, cut-offs, relief valves); (3) warn the user (labels, lights, sounds). Fail-safe variants: fail-passive (circuit breaker), fail-active (standby redundancy), fail-operational (valve fails open).
Beyond the slides — FMEA & FTAFMEA (Failure Mode & Effects Analysis) is bottom-up: list each component's failure modes, effects and a Risk Priority Number (Severity×Occurrence×Detection). FTA (Fault Tree Analysis) is top-down: start from an undesired top event and trace causes with AND/OR gates. Naming these signals real-world literacy.
10 Human Engineering / Ergonomics
DefinitionErgonomics = designing so the product fits human abilities and limits — “fitting the job to the man,” not the man to the job. Goal: optimise efficiency, health, safety and comfort. (Greek ergos work + nomos law.)
10.1 Four forms of human factors
| Factor | Concerns | Example |
|---|---|---|
| Anthropometric | Physical size of the body (static interaction) | Reach, seat/table height, lever placement |
| Physiological | Human sensations: visual, auditory, tactile | Alarm loudness, display brightness |
| Psychological | Mental: behaviour, strain, fatigue | Digital display for precise values; pointer for trends |
| Ergonomic | Whole working system | Preventing musculoskeletal disorders (MSD) |
Exam tips (Feb 2025 · Q18, Q20)Anthropometric = physical size of the body. · For rapidly changing quantitative info the best display is the analog dial (moving pointer) — pointers show rate/direction; digital is best for precise static values.
10.2 The 10 principles & the man–machine system
(1) Neutral postures · (2) reduce excessive force · (3) everything in easy reach · (4) proper heights · (5) reduce motions · (6) minimise fatigue/static load · (7) minimise pressure points · (8) provide clearance · (9) move/stretch · (10) comfortable environment.
The man is the flexible controller: senses stimuli → perceives → judges → stores/recalls → decides → acts. Design must match task requirements to human capability.
10.3 Guidelines — displays & controls
Displays
- Show only the accuracy needed; no superfluous info
- Scale subdivisions in multiples of 1, 2 or 5
- Sharp single-plane pointer (avoid parallax)
- Letter height (mm) = viewing distance (mm) / 200
Controls
- Locate clearly visible & comfortably operable
- Clockwise → increase; pointer moves with the control
- Mark on/off & levels; colour/shape/symbol coding
- Use conventional standard sizes to avoid errors
Beyond the slides — design for the extremesSize reach for the 5th-percentile user (if the smallest can reach, everyone can) and clearances for the 95th-percentile user (if the largest fits, everyone fits). Designing for the “average person” fits almost no one.
11 Master formula sheet
Stress / strain / failure
\[ \sigma=\frac{F}{A}\quad \varepsilon=\frac{\delta}{L}\quad \tau=\frac{T\rho}{J}\quad \sigma_b=\frac{My}{I}\quad G\approx0.4E\quad \tau_y\approx0.5\text{–}0.75\,\sigma_y \]
\[ \sigma_{1,2}=\frac{\sigma_x+\sigma_y}{2}\pm\sqrt{\left(\tfrac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}\qquad \tau_{\max}=\frac{\sigma_1-\sigma_3}{2} \]
\[ \text{Tresca: }\sigma_1-\sigma_3=\sigma_y\quad \text{von Mises: }\sqrt{\sigma_1^2-\sigma_1\sigma_2+\sigma_2^2}=\sigma_y\quad n=\frac{S}{\sigma}\quad J=\frac{\pi}{32}(D^4-d^4) \]
Fracture
\[ K_I=Y\sigma\sqrt{\pi a}\quad a_c=\frac1\pi\!\left(\frac{K_{IC}}{Y\sigma}\right)^2\quad G=\frac{K_I^2}{E'}\quad \sigma_f=\sqrt{\frac{2E\gamma}{\pi a}}\quad \text{CTOD }\delta=\frac{K_I^2}{E\sigma_y} \]
\[ Y:\ 1.0\text{ central},\ 1.12\text{ edge},\ 0.64\ (2/\pi)\text{ penny};\qquad \text{Irwin }r_y=\frac{1}{2\pi}\!\left(\frac{K_I}{\sigma_{ys}}\right)^2 \]
\[ \text{Similar-sheet scaling: }\sigma_{f1}\sqrt{a_1}=\sigma_{f2}\sqrt{a_2} \]
Fatigue
\[ \sigma_a=\frac{\sigma_{max}-\sigma_{min}}{2}\quad \sigma_m=\frac{\sigma_{max}+\sigma_{min}}{2}\quad R=\frac{\sigma_{min}}{\sigma_{max}}\quad \text{Basquin }S_a=aN^b \]
\[ \Delta K=Y\Delta\sigma\sqrt{\pi a}\quad \text{Paris }\frac{da}{dN}=C(\Delta K)^m\quad \text{Miner }\sum\frac{n_i}{N_i}=1\quad \text{endurance ratio}\approx0.3\text{–}0.4 \]
\[ \text{Paris life }(m=3):\ N=\frac{2\,(a_i^{-1/2}-a_f^{-1/2})}{C(Y\Delta\sigma\sqrt{\pi})^3}\qquad \text{S-N fit: }b=\frac{\log(S_1/S_2)}{\log(N_1/N_2)},\ a=\frac{S_1}{N_1^{\,b}} \]
Reliability
\[ R=1-F\quad \lambda=\frac{f}{R}=\frac{\#\text{fail}}{\text{op. time}}\quad \text{MTTF}=\text{MTBF}=\frac1\lambda\quad \text{Exp }R=e^{-\lambda t} \]
\[ \text{Normal }z=\frac{t-\mu}{\sigma},\ R=1-\Phi(z)\quad \text{Weibull }R=e^{-(t/\theta)^m}\quad \text{Series }R=\prod R_i\quad \text{Parallel }R=1-\prod(1-R_i) \]
Ergonomics
\[ \text{Letter height (mm)}=\frac{\text{viewing distance (mm)}}{200} \]
12 Last-night revision checklist
- ☐ Purpose of FoS; write n = S/σ; design factor vs factor of safety.
- ☐ Draw the Tresca hexagon inside the Von Mises ellipse; state which is conservative.
- ☐ Write generalized Hooke's law and the stress tensor (6 independent components).
- ☐ Sketch a ceramic stress–strain curve; explain plane stress vs plane strain.
- ☐ Name the three crack modes; write K = Yσ√(πa); recall Y = 1.12 (edge).
- ☐ Compute critical crack length aᴄ; do the “similar sheet” σ_f1√a₁ = σ_f2√a₂ scaling.
- ☐ Define G and the K–G relation; Griffith & the Irwin–Orowan modification.
- ☐ Contrast HCF vs LCF; compute σ_a, σ_m, R, A from σmax/σmin.
- ☐ Build an S-N line with modifying factors (C_L, C_G, C_S); use Basquin ratio.
- ☐ Integrate Paris' law (remember Δσ, a in metres); apply Miner's rule; blocks = 1/D.
- ☐ List the four fatigue design strategies; explain damage tolerance.
- ☐ Explain concurrent engineering & give four DFM + four DFA guidelines.
- ☐ Do a Normal, Weibull and exponential (series/parallel) reliability calculation.
- ☐ Sketch the bathtub curve; define λ, MTTF, MTBF; three fail-safe variants.
- ☐ Name the four human-factor types; letter-height rule & clockwise-increase rule.
In the examFor every numerical: write the formula → substitute in consistent SI units → box the answer with units. For “predict the failure mode,” compute BOTH the fracture load and the yield load — the smaller governs. It's an “attempt-all” paper, so write something structured for every part.