The governing theory for most structural beam calculations. Key assumptions: plane sections remain plane; shear deformation is negligible (slender beams); material is linear elastic and homogeneous. Breaks down for deep beams (span/depth < 5), where Timoshenko theory should be used.

SFD shows internal shear along the span; BMD shows bending moment. Key relationships: dV/dx = −w(x) (distributed load intensity), dM/dx = V(x). Where V crosses zero, M is at maximum or minimum — this is where maximum bending stress occurs in a simply supported beam.

Memorising the key deflection formulae is essential for rapid checking. Note the strong dependence on span (L³ or L⁴) — doubling the span increases deflection 8× or 16×. EI appears in the denominator; increasing either E or I reduces deflection proportionally.

Deflection limits are set by serviceability requirements, not strength. Excessive deflection cracks ceilings, misaligns machinery, or creates ponding. AISC/BS5950 typical limits: L/360 for floors with brittle finishes, L/240 for roofs, L/180 for flexible roofs. Machinery supports: often L/500 or stricter.

Unrestrained beams in bending can fail by lateral-torsional buckling (LTB) — the compression flange buckles sideways while the tension flange resists. Risk increases with span and reduces with lateral bracing, bigger I_y, and torsional stiffness J. LTB governs for most unbraced steel beams over 3–4 m.

Many real structural members carry both axial load and bending moment simultaneously — roof beams carrying gravity and wind, columns with eccentric connections. Interaction equations account for the amplification of bending moment by axial load (P-Δ effect).

Shear stress in a beam cross-section is not uniform — it varies with the first moment of area Q above the point of interest. Maximum shear stress occurs at the neutral axis. For a rectangle: τ_max = 1.5 × V/A. For I-beams, the web carries most of the shear.

If a component is constrained from expanding when heated, thermal stress develops. Free thermal expansion produces no stress. Stress develops only when expansion is prevented. Critical in pipework, pressure vessels, and engine components. α (coefficient of thermal expansion) for steel ≈ 12×10⁻⁶/°C.

Residual stresses exist without external load. Sources: welding (shrinkage), cold working (plastic deformation), heat treatment (differential cooling). Tensile residual stresses near weld toes dramatically accelerate fatigue crack initiation. Shot peening introduces compressive residual stresses to improve fatigue life.

Internal pressure creates biaxial stress in a cylindrical vessel. Hoop (circumferential) stress σ_h acts around the circumference and is the critical stress — it is twice the longitudinal stress. This is why pipes typically split along their length rather than across. Valid when t/D < 0.1 (thin-wall assumption).

A sphere is the most efficient pressure vessel shape — equal biaxial stress, half the hoop stress of a cylinder for the same diameter and pressure. Cylinders are more practical to manufacture and inspect. The sphere stores ~2× more volume per unit mass of material but is harder to fabricate.

When the wall is thick relative to radius, stress varies across the wall thickness. Lamé equations give exact stress distribution. Hoop stress is maximum at the inner surface. Used for gun barrels, hydraulic cylinders, and autofrettage design.

The ASME Code gives specific formulas for minimum required wall thickness. The allowable stress S accounts for material, temperature, and the ASME safety factor of 3.5 on UTS (or 1.5 on 0.2% proof stress for some materials). Joint efficiency E (0.6–1.0) reflects weld inspection level.

A flange joint seals by compressing a gasket between two flange faces. Two conditions must be satisfied: (1) seating condition — minimum bolt load to seat the gasket at assembly; (2) operating condition — bolt load must maintain sealing under pressure and thermal loads. The gasket factor m (residual factor) and y (seating stress) define these requirements per ASME B16.5/ASME VIII Appendix 2.

Every nozzle opening removes material from the shell, reducing its pressure capacity. Area compensation (reinforcement) replaces the missing metal either using a reinforcing pad or excess shell/nozzle wall thickness. The compensation area must equal or exceed the removed area within a defined boundary.

ASME VIII requires hydrostatic testing at 1.3× MAWP (Maximum Allowable Working Pressure) after fabrication. Pneumatic testing at 1.1× MAWP is allowed with restrictions and risk assessment. Leak testing at operating pressure validates sealing. All test pressures adjusted for test temperature vs design temperature.

Pre-loading a bolt in tension (torquing) puts the clamped joint into compression. External tensile loads partially relieve this compression — the bolt sees only a fraction of the external load (depending on joint stiffness ratio). Pre-load is the key to joint integrity: it prevents fatigue, leakage, and joint separation.

Tensile stress area A_t is less than the nominal cross-section area — threads reduce the effective area. ISO 898 bolt grades define proof and ultimate strength. Grade 8.8: Sp=600 MPa, Sut=800 MPa. Grade 10.9: Sp=830 MPa, Sut=1040 MPa. Always design to proof load, not ultimate.

Tightening torque creates bolt pre-load but most torque (≈50%) overcomes thread friction, ≈40% overcomes bearing surface friction, only ≈10% goes into bolt tension. The K factor (nut factor) accounts for all friction. K≈0.2 for unlubricated steel threads, K≈0.15 for lubricated.

Bolts loaded in shear (lap joints, gusset plates) are analysed differently. Friction-grip (preloaded) bolts transfer load in shear by friction before any bolt shear. Bearing-type joints rely on bolt shear and plate bearing. AISC LRFD: φRn must exceed factored shear load Vu.

Fillet welds are the most common. Stress is calculated on the effective throat — 0.707 × leg size for 45° fillet welds. Groove welds can be full-penetration (CJP) or partial-penetration (PJP). CJP welds develop full parent metal strength. Weld metal strength must exceed parent metal (over-matching).

When load is applied eccentrically to a weld group, the weld experiences both direct shear and torsional shear. The torsional component varies with distance from the weld group centroid. The critical point is the weld element farthest from the centroid and most aligned with the combined vector.

Common weld defects: porosity (gas entrapment), undercut (reduction of parent metal at weld toe), lack of fusion (incomplete bonding), incomplete penetration, cracks (most serious — demand rejection). Inspection methods: Visual (VT), Radiographic (RT), Ultrasonic (UT), Magnetic particle (MT), Dye penetrant (PT).

The HAZ is the region of parent metal whose microstructure has been altered by welding heat. In steel, rapid cooling can produce hard, brittle martensite in the HAZ. Preheat and controlled heat input reduce this risk. For high-carbon or alloy steels, calculate carbon equivalent (CE) to determine preheat requirements.

Involute gears maintain a constant velocity ratio regardless of small centre-distance errors. The module m = d/N links pitch circle diameter, number of teeth, and physical size. Standard pressure angle 20° gives a good balance between tooth strength and smooth operation. Interference occurs when the number of teeth is too small.

The transmitted tangential force W_t is the primary driving force. Normal pressure angle φ introduces a radial (separating) force W_r. Helical gears also develop an axial thrust W_a proportional to the helix angle ψ. The resultant W_N acts along the line of action.

AGMA provides the definitive method for gear strength rating. Two failure modes govern: bending fatigue of the tooth root (Lewis stress with AGMA factors) and pitting/contact fatigue of the tooth surface (Hertzian contact stress). Both must be checked. High hardness (>350 HB) allows much higher ratings.

Simple gear pair: ratio = N_gear/N_pinion = ω_in/ω_out. For large ratios, multiple stages (compound trains) are used. Each stage ratio multiplies. A 3-stage gear box can achieve ratios >1000:1 with reasonable gear sizes. Epicyclic (planetary) gears achieve high ratios in compact form.

Most power transmission shafts experience simultaneous bending (from gear/belt forces and self-weight) and torsion (from transmitted torque). The ASME-DE-Goodman criterion is the preferred design method: it accounts for both mean and alternating stress components and includes a fatigue correction.

Every rotating shaft has critical speeds where resonance causes large deflections. Operating below the first critical speed (undercritical) is safest. Above it (supercritical) is possible but requires careful design. The critical speed depends on shaft stiffness and mass distribution — same as a natural frequency problem.

A key transmits torque between shaft and hub. Parallel keys are most common. Design checks: shear stress on key cross-section, and bearing (compressive) stress on the key side. Key dimensions are standardised relative to shaft diameter (width ≈ d/4). Key length is the design variable.

Rolling element bearings are rated by their basic dynamic load capacity C and the L10 life (hours at which 10% of bearings fail). Speed and load directly determine life. Life varies as load cubed (ball bearings) or to the 10/3 power (roller bearings). Always check both radial and axial load components.

Spring rate (stiffness) k depends on wire diameter d, mean coil diameter D, number of active coils Na, and shear modulus G. Hard-drawn wire is cheapest; music wire has highest strength; chrome-vanadium is best for high-temperature and fatigue. Spring index C = D/d should be 4–12 for good design.

The Wahl factor (K_W) corrects for the actual shear stress in a coil spring, which exceeds the simple torsion formula due to curvature and direct shear effects. Stress concentration is higher for low C (tight coils). The corrected maximum shear stress is used for static failure analysis.

Compression springs must not be compressed to solid height (coil clash) under maximum load — always provide a clearance of ≥10-15% of free length. Spring surge (resonance of the spring itself) occurs when the forcing frequency reaches the spring's natural frequency. Surge causes fatigue failure at coil contacts.

Springs are often subjected to repeated cyclic loading. The Goodman-Zimmerli diagram for spring wire gives the allowable stress amplitude vs mean stress. Surface quality is critical — any surface defects initiate fatigue cracks. Shot peening significantly improves fatigue resistance by introducing compressive residual stress in the wire surface.

During bending, the outer surface is in tension and the inner in compression. The neutral axis shifts inward. Springback occurs when the elastic portion of deformation recovers on die removal — the bend angle reduces. Springback increases with higher yield strength and higher bend radius-to-thickness ratio. Overbend to compensate, or use bottoming.

Bending too tightly causes cracking on the outer surface. Minimum bend radius depends on material ductility (elongation %) and grain direction (bends across grain are tighter than with grain). Soft annealed materials can bend tight; work-hardened materials require larger radii.

The developed blank size is calculated from the flat pattern — the sum of straight sections plus bend allowances. This must be calculated accurately to avoid wasted material and rework. CNC press brakes use stored K-factor tables for specific material-thickness combinations from physical trials.

Fatigue failure occurs below yield strength under cyclic loading. The S-N (Wöhler) curve shows stress amplitude vs cycles to failure. For steels, a true endurance limit S_e exists below which fatigue life is theoretically infinite (typically at 10⁶–10⁷ cycles). For aluminium and other non-ferrous metals, there is no true endurance limit — life continues to decrease with stress.

Geometric stress concentrations (notches, holes, keyways, shoulders) are far more damaging in fatigue than in static loading. The fatigue stress concentration factor K_f = 1 + q(K_t − 1), where q is the notch sensitivity (0 for very ductile materials, 1 for fully notch-sensitive). K_f is always applied to the alternating stress amplitude, not the mean.

The modified Goodman line relates alternating and mean stress to the endurance limit and ultimate strength. Design points below the line are safe from fatigue. The Langer yield line (σ_a + σ_m = Sy) must also be checked — if the Goodman line intersects it first, yielding governs.

The Marin equation modifies the polished-specimen endurance limit to real-world conditions. Surface factor k_a (machined > ground > as-rolled) dominates for moderate/high strength. Size factor k_b accounts for larger stressed volumes. Reliability factor k_e: for 99% reliability (vs 50%), multiply by 0.814.

Linear Elastic Fracture Mechanics (LEFM) provides a rigorous framework for crack analysis. The stress intensity factor K_I characterises the stress field ahead of a crack tip. When K_I reaches the material's fracture toughness K_Ic, unstable fracture occurs. K_Ic is a material property; lower-strength steels have higher K_Ic (more tough).

The critical crack size a_c is the crack length at which unstable fracture occurs at a given stress level. This is the basis for damage-tolerant design: ensure that cracks detectable by inspection grow too slowly to reach a_c between inspection intervals. This approach is mandated for aircraft structures and nuclear components.

Under cyclic loading, cracks grow incrementally. Paris Law gives the crack growth rate per cycle as a function of stress intensity range ΔK. Constants C and m are material-dependent. Crack growth is slow for small ΔK (near-threshold region), accelerates in the Paris regime, then accelerates rapidly near K_Ic.

Ductile fracture is preceded by significant plastic deformation (necking, void nucleation). Fracture surface appears rough/fibrous. Brittle fracture is sudden with little or no plastic deformation — fracture surface is flat, granular (cleavage). Temperature, strain rate, and triaxial stress state drive brittle behaviour (ductile-to-brittle transition in BCC steels).

Carbon steel is the structural workhorse. Low carbon (<0.3% C): weldable, ductile, lower strength — S275/S355/A36/A572. Medium carbon (0.3–0.6% C): heat-treatable — shafts, gears. High carbon (>0.6% C): hard but brittle — cutting tools. Alloy additions (Mn, Cr, Mo, Ni, V) improve hardenability, strength, or toughness.

Minimum 10.5% chromium gives passivation (self-healing oxide layer). Austenitic (304, 316): non-magnetic, excellent corrosion resistance, low strength unless cold-worked. 316L (Mo added): superior resistance to chlorides. Duplex (2205): higher strength + better stress corrosion cracking resistance. Martensitic (410, 420): hard, magnetic, moderate corrosion resistance.

Aluminium: density 2700 kg/m³ vs steel 7850 kg/m³ — 3× lighter. Excellent corrosion resistance. No DBTT (safe at cryogenic temps). No true fatigue limit. 6061-T6: general-purpose structural, good weldability. 7075-T6: aerospace, very high strength but lower corrosion resistance and not weldable. 2024-T3: high fatigue resistance for airframes.

Titanium: density 4500 kg/m³, excellent specific strength, outstanding corrosion resistance, biocompatible. Ti-6Al-4V is the most widely used titanium alloy. Difficult to machine — special cutting conditions needed. Nickel superalloys (Inconel 625, 718): retain strength at temperatures up to 1000°C — used in gas turbines, petrochemical reactors.

Engineering plastics are used for bearings, housings, seals, and light structural parts. Key properties: low density, self-lubricating, chemical resistance, electrical insulation. Weakness: low E (1–5 GPa vs 70–200 for metals), creep under sustained load, degradation above glass transition temperature T_g. Choose material and factor in creep, temperature, and UV exposure.

CFRP achieves exceptional specific stiffness and strength — better than any metal. Used in aerospace, motorsport, and high-performance structures. Properties are anisotropic: much stronger and stiffer along fibre direction. Weak in matrix-dominated directions (transverse, interlaminar shear). Design must account for laminate orientation and interlaminar failure.

Annealing: heat above A3, slow cool — maximum softness, stress relief. Normalising: air cool — uniform microstructure, moderate strength. Quench and temper: rapid quench then temper — controls hardness/toughness tradeoff. Case hardening: carburising, nitriding — hard surface, tough core. Induction hardening: local surface hardening for gears and shafts.