General Engineering: Materials, Applied Mechanics, Machines and Design, Thermal and Fluids

Most engineering metals crystallise in one of three structures. Body-centred cubic (BCC) — iron below 912 °C, chromium, tungsten, molybdenum — has 2 atoms per cell, coordination number 8 and atomic packing factor 0.68. Face-centred cubic (FCC) — aluminium, copper, nickel, austenitic iron — has 4 atoms per cell, coordination number 12 and packing factor 0.74. Hexagonal close-packed (HCP) — zinc, magnesium, titanium at room temperature — also packs to 0.74 with coordination 12. The packing factor is not trivia: FCC metals have many slip systems on close-packed planes and are ductile down to very low temperatures, BCC metals show a ductile-to-brittle transition as temperature falls, and HCP metals, with few active slip systems, are the hardest of the three to cold work.

For a fibre composite loaded along the fibres, the rule of mixtures gives the longitudinal modulus as the volume-weighted average, E_c = E_f V_f + E_m V_m (the iso-strain case). Loaded across the fibres the phases carry the same stress instead, and 1/E_c = V_f/E_f + V_m/E_m (the iso-stress case), which is always the lower of the two. That gap is why a unidirectional laminate is stiff along its fibres and weak across them, and why real parts stack plies at several angles.

The iron-iron carbide diagram has three invariant reactions, and GATE asks the two lower ones most. The eutectoid at 727 °C and about 0.76 % carbon turns austenite into pearlite, a lamellar mixture of ferrite and cementite: γ → α + Fe3C. The eutectic at about 1147 °C and 4.3 % carbon turns liquid into ledeburite: L → γ + Fe3C. The peritectic near 1493 °C (δ + L → γ) sits at very low carbon. Austenite dissolves up to about 2.1 % carbon, which is the conventional boundary between steels and cast irons; ferrite dissolves only about 0.022 %. Below 0.76 % a steel is hypo-eutectoid (proeutectoid ferrite plus pearlite), above it hyper-eutectoid (proeutectoid cementite plus pearlite).

On the engineering stress-strain curve of a ductile metal, stress is proportional to strain up to the proportional limit (slope E), the material yields (a sharp upper and lower yield point in mild steel, or a 0.2 % offset yield elsewhere), strain-hardens to the ultimate tensile strength, then necks and the engineering stress falls to fracture. True stress σ_T = σ(1 + ε) and true strain ε_T = ln(1 + ε) are valid up to necking; the true curve keeps rising. The area under the elastic part per unit volume is the resilience, σ_y²/(2E); the area under the whole curve is the toughness. Percentage elongation and reduction in area measure ductility.

Any system of forces on a rigid body reduces to an equivalent force-couple system at a chosen point: a single resultant force R = ΣF and a couple M = Σ(r × F). A body is in equilibrium when both vanish, which in a plane gives three scalar equations, ΣFx = 0, ΣFy = 0 and ΣM = 0 — so a planar free-body diagram can yield at most three unknowns. Draw the free body by isolating it completely and replacing every support by the reactions it can supply: a roller one force normal to its surface, a pin two force components, a fixed support two forces and a moment. A two-force member carries its load along the line joining the two points.

Within the elastic range Hooke's law links stress and strain, σ = Eε and τ = Gγ, and a stretched bar contracts laterally by Poisson's ratio ν = −ε_lateral/ε_axial. The three elastic constants of an isotropic material are not independent: E = 2G(1 + ν) = 3K(1 − 2ν). Since ν cannot exceed 0.5 without K going negative, ν = 0.5 is the incompressible limit. In three dimensions ε_x = [σ_x − ν(σ_y + σ_z)]/E, and the volumetric strain of a body is ε_v = ε_x + ε_y + ε_z.

Mohr's circle for plane stress: plot (σx, τxy) and (σy, −τxy); the circle through them has centre C = (σx + σy)/2 and radius R = √[((σx − σy)/2)² + τxy²]. The principal stresses are σ1,2 = C ± R, the maximum in-plane shear stress is R, the principal planes are at tan 2θp = 2τxy/(σx − σy), and a rotation θ of the element is 2θ on the circle. For σx = 80 MPa, σy = 20 MPa, τxy = 40 MPa: C = 50, R = √(30² + 40²) = 50, so σ1 = 100 MPa, σ2 = 0 and the maximum in-plane shear is 50 MPa.

The flexure formula M/I = σ/y = E/R gives the bending stress σ = My/I, zero at the neutral axis and largest at the outer fibre, so σ_max = M/Z with section modulus Z = I/y_max (bd²/6 for a rectangle, πd³/32 for a solid circle). The transverse shear stress is τ = VQ/(Ib), where Q is the first moment of the area beyond the fibre considered; it is zero at the outer fibre and peaks at the neutral axis, at 1.5 times the average for a rectangle and 4/3 times for a solid circle.

A thin cylinder (t < d/20) under internal pressure p carries a hoop stress σ_h = pd/(2t) and a longitudinal stress σ_l = pd/(4t), half the hoop value — so a thin boiler shell splits along its length first. A thin sphere carries pd/(4t) in every direction. In a thick cylinder the stresses vary through the wall and follow Lamé's equations, σ_r = A − B/r² and σ_θ = A + B/r², with A and B fixed by the pressures at the two radii; the hoop stress is greatest at the inner surface. For internal pressure only, σ_θ at the bore is p(r_o² + r_i²)/(r_o² − r_i²).

Torsion of a circular shaft: T/J = τ/r = Gθ/L, with polar moment J = πd⁴/32 (solid) or π(D⁴ − d⁴)/32 (hollow). The surface shear stress of a solid shaft is 16T/(πd³). Torsional stiffness is GJ/L, and shafts in series add angles of twist while shafts in parallel share torque in proportion to their stiffness. Power transmitted is P = 2πNT/60 with N in rpm.

The mobility of a planar mechanism follows Gruebler's criterion, F = 3(n − 1) − 2j − h, where n counts links including the frame, j lower pairs (turning or sliding, one degree of freedom each) and h higher pairs. A four-bar linkage has n = 4, j = 4, so F = 1. A Grashof four-bar (s + l ≤ p + q, shortest plus longest not more than the other two) has at least one link that can rotate fully: fix the link next to the shortest for a crank-rocker, the shortest itself for a double-crank, and the link opposite the shortest for a double-rocker. The slider-crank, with its inversions, gives the reciprocating engine, the Whitworth and the crank-and-slotted-lever quick-return motions.

Velocity analysis uses instantaneous centres: a mechanism of n links has n(n − 1)/2 of them, and Kennedy's theorem says the three instantaneous centres of any three links lie on a straight line. The velocity of any point is ω times its distance from the instantaneous centre of its link with the frame. Acceleration of a point on a link rotating at ω has a centripetal part ω²r towards the centre and a tangential part αr; a slider on a rotating link adds the Coriolis component 2vω.

A cam drives a follower (knife-edge, roller, flat-faced) through rise, dwell and return. Uniform velocity motion has infinite acceleration at the ends and is used only with modifications; simple harmonic motion has finite acceleration but a jump at the ends; uniform acceleration and retardation (parabolic) gives the least maximum acceleration for a given rise; cycloidal motion has zero acceleration at both ends and suits high speeds. The pressure angle — between the follower's direction of motion and the common normal — should stay small (about 30° is a common limit for a translating roller follower) to keep side thrust down; a larger base circle reduces it.

A governor controls mean speed over many cycles by regulating fuel; a flywheel controls fluctuation within one cycle by storing energy, and neither does the other's job. For a Watt governor the height is h = g/ω² (h ≈ 895/N² metres with N in rpm), so height falls as the square of speed. Porter (a loaded central sleeve) and Hartnell (spring-loaded) governors are more powerful. Terms: sensitive — a small speed change moves the sleeve a lot; stable — radius of rotation increases with speed; isochronous — one equilibrium speed for all radii, infinitely sensitive and prone to hunting. For a flywheel, the maximum fluctuation of energy is ΔE = Iω²C_s, where ω is the mean speed and the coefficient of fluctuation of speed is C_s = (ω_max − ω_min)/ω.

• Bolted joints: the bolt is designed on its tensile stress area; an initial preload keeps the joint closed so the bolt sees only a fraction of the external load, the share set by the stiffness of bolt against members.
• Riveted joints fail by tearing of the plate between rivets, P_t = (p − d)tσ_t, shearing of the rivets, P_s = n(π/4)d²τ per shear plane, or crushing, P_c = n d t σ_c. Joint efficiency = the least of the three divided by the strength of the unpunched plate, p t σ_t.
• Welded joints: a fillet weld of leg s fails across its throat, t = s cos 45° = 0.707s, so a parallel fillet of length L carries P = 0.707 s L τ per weld; a butt weld is designed on the plate thickness.
• Interference (shrink) fit: a diametral interference δ between a solid shaft and a hub of the same material (outer diameter D) creates a contact pressure p = (Eδ/2d)(1 − d²/D²); the torque it can transmit by friction is T = μpπd²L/2.
• Shafts under combined bending M and torque T: by maximum shear stress theory the equivalent torque is T_e = √(M² + T²) and d³ = 16T_e/(πτ); by maximum normal stress the equivalent moment is M_e = [M + √(M² + T²)]/2 and d³ = 32M_e/(πσ).
• Keys transmit torque by shear across the width, τ = 2T/(w l d), and by crushing on the side face, σ_c = 4T/(h l d) for a key half-sunk in the shaft. Couplings: rigid (sleeve, flange) for aligned shafts; flexible (bush-pin, Oldham, universal joint) where misalignment exists.
• Spur gears: module m = d/z, circular pitch πm; the pitch-line velocity ratio is ω1/ω2 = z2/z1. The Lewis equation gives the beam strength of a tooth, F_t = σ_b b m Y, with face width b and form factor Y; interference is avoided by a minimum number of pinion teeth for the pressure angle used (larger for 14.5° than for 20°).
• Belt drives: for a flat belt T1/T2 = e^μθ (θ the angle of lap on the smaller pulley, in radians), for a V-belt e^{μθ/sin β} with β the half groove angle. Power = (T1 − T2)v. Centrifugal tension T_c = mv² adds to both sides; power is greatest when T_c = T_max/3.
• Brakes: a band brake obeys the same e^μθ tension ratio, braking torque (T1 − T2)r; a block brake gives friction μN at the drum. Clutches (single plate, per friction surface): uniform wear, T = μW(R1 + R2)/2; uniform pressure, T = (2/3)μW(R1³ − R2³)/(R1² − R2²). A new clutch is closer to uniform pressure, a worn one to uniform wear, which is the safer (lower) design value.

Friction and lubrication: dry (Coulomb) friction gives F = μN independent of the apparent area. A journal bearing moves, as speed rises, from boundary lubrication (asperities in contact, high μ) through mixed to hydrodynamic lubrication, where a full oil film carries the load and μ rises again only slowly with speed. Petroff's equation for a lightly loaded concentric journal, f = 2π²(μN/P)(r/c), shows friction growing with viscosity μ and speed N and falling with bearing pressure P; the dimensionless group μN/P (the Sommerfeld variable) is what the Stribeck curve is plotted against.

In a fluid at rest pressure rises linearly with depth, p = p0 + ρgh, and acts equally in all directions. The resultant force on a plane submerged surface is F = ρg h̄ A (h̄ the depth of the centroid), acting at the centre of pressure, which is always below the centroid by I_G/(A h̄) for a vertical surface. Continuity for incompressible flow is A1V1 = A2V2. Along a streamline of steady, inviscid, incompressible flow, Bernoulli's equation p/ρg + V²/2g + z = constant; with losses, a head-loss term h_L is added on the downstream side. The momentum equation for a control volume, ΣF = ρQ(V_out − V_in), gives forces on bends, nozzles and vanes.

Pipe flow is laminar below a Reynolds number Re = ρVD/μ of about 2000 and turbulent well above it. The Darcy-Weisbach equation, h_f = fLV²/(2gD), gives the friction head loss for both; in laminar flow f = 64/Re, which reproduces the Hagen-Poiseuille result Δp = 32μVL/D², and the velocity profile is parabolic with the maximum twice the mean. In turbulent flow f depends on Re and relative roughness (the Moody chart) and the profile is much flatter. Minor losses at bends, valves and sudden expansions are written K V²/(2g).

Capillary action: in a tube of diameter d, surface tension σ lifts (or depresses) liquid to a height h = 4σ cos θ/(ρgd). Water wets clean glass (θ ≈ 0) and rises; mercury (θ > 90°) is depressed. The rise is inversely proportional to diameter, so halving the bore doubles it. Dimensional analysis by Buckingham's π theorem: n variables in m fundamental dimensions form n − m independent dimensionless groups. Reynolds (inertia/viscous), Froude (inertia/gravity), Weber (inertia/surface tension), Mach (inertia/compressibility) and Euler (pressure/inertia) numbers are the ones models are matched on.

The zeroth law — two bodies each in thermal equilibrium with a third are in equilibrium with each other — is what makes a thermometer meaningful. The first law for a closed system is Q − W = ΔU; for a steady-flow control volume it becomes the steady-flow energy equation, q − w = (h2 − h1) + (V2² − V1²)/2 + g(z2 − z1) per unit mass. The second law in its Kelvin-Planck form forbids a cycle that converts heat from one reservoir wholly into work, and in its Clausius form forbids heat flowing unaided from cold to hot; together they cap every engine at the Carnot efficiency 1 − T_L/T_H and introduce entropy, dS ≥ δQ/T.

Air-standard cycles replace combustion by heat addition to air treated as an ideal gas. Otto (constant-volume heat addition): η = 1 − 1/r^γ−1, depending on the compression ratio r alone — 56.5 % for r = 8 and γ = 1.4. Diesel (constant-pressure heat addition): η = 1 − (1/r^γ−1)[(ρ^γ − 1)/(γ(ρ − 1))], with cut-off ratio ρ; for the same r the Diesel is less efficient than the Otto, but it runs at much higher r. Dual combines the two. Brayton (the gas turbine): η = 1 − 1/r_p^(γ−1)/γ, depending on the pressure ratio r_p.

Conduction follows Fourier's law, Q = −kA dT/dx, so a plane wall conducts Q = kAΔT/L and has a thermal resistance L/(kA); resistances in series add like electrical ones. Convection follows Newton's law of cooling, Q = hAΔT, resistance 1/(hA). Radiation from a black body is σT⁴ per unit area with σ = 5.67 × 10⁻⁸ W/m²K⁴; a grey body emits εσT⁴, and net exchange depends on T1⁴ − T2⁴, never on the difference of temperatures. Insulating a small pipe can increase heat loss until the outer radius passes the critical radius r_c = k/h (2k/h for a sphere).