BEM PCE Part B Mechanical cheat sheet

Static equilibrium requires ΣF = 0 and ΣM = 0 about any point; for a 2D body this gives three independent equations, so a determinate free body yields at most three unknowns.

Second moment of area for a rectangle about its centroidal axis is I = b·h³/12; parallel-axis transfer adds A·d² where d is the offset from the centroid.

Bending stress follows σ = M·y/I, with maximum stress at the extreme fibre y = c; section modulus Z = I/c gives σ_max = M/Z.

Euler buckling load P_cr = π²EI/(K·L)²; effective length factors: K = 1.0 pinned-pinned, 0.5 fixed-fixed, 0.7 fixed-pinned, 2.0 fixed-free.

Torsion of a solid circular shaft: τ_max = T·r/J with J = π·d⁴/32; angle of twist θ = T·L/(G·J).

Thin-walled pressure vessel hoop stress σ_h = p·r/t and longitudinal stress σ_l = p·r/(2t); hoop stress is twice the longitudinal value.

S-N fatigue curve: infinite life design uses the endurance limit; below roughly 10³ cycles is low-cycle fatigue governed by strain, not stress.

Stress concentration factor K_t multiplies nominal stress at geometric discontinuities; it applies to both ductile and brittle materials under fatigue loading.

First law for a steady-flow open system: q − w = Δh + Δke + Δpe per unit mass; for an adiabatic turbine w = h₁ − h₂.

Carnot efficiency sets the ceiling: η = 1 − T_L/T_H with absolute temperatures in kelvin; no real cycle between the same reservoirs exceeds it.

Rankine cycle efficiency improves with higher boiler pressure, lower condenser pressure, and reheating; regeneration raises mean temperature of heat addition.

Otto cycle thermal efficiency η = 1 − r^(1−k) where r is compression ratio and k = c_p/c_v; Diesel cycle efficiency additionally depends on cut-off ratio.

Conduction through a plane wall: q = k·A·ΔT/L; for a composite wall the resistances R = L/(kA) add in series.

Newton's law of cooling for convection q = h·A·(T_s − T_∞); radiation exchange q = ε·σ·A·(T_s⁴ − T_surr⁴) with σ = 5.67×10⁻⁸ W/(m²·K⁴).

Log-mean temperature difference for a heat exchanger: ΔT_lm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂); apply a correction factor F for multipass and cross-flow units.

Reynolds number Re = ρ·V·D/μ; pipe flow is laminar below about 2300, transitional to 4000, and turbulent above 4000.

Shaft power transmission: P = 2π·N·T, so torque T = P/(2π·N) with N in revolutions per second; check consistent units between watts, N·m, and rad/s.

Gear ratio equals the inverse ratio of tooth numbers: n₁/n₂ = N₂/N₁; torque scales with the ratio and speed scales inversely.

Belt drives: ratio of tensions T₁/T₂ = e^(μθ) for a flat belt, with θ the arc of contact in radians; slipping limits transmitted power.

Four-bar Grashof condition: shortest + longest ≤ sum of the other two links gives at least one full rotation of the shortest link.

Clutch or brake torque capacity for a uniform-wear disc: T = μ·F·r_mean with r_mean = (r_o + r_i)/2; uniform-pressure theory uses (2/3)·(r_o³ − r_i³)/(r_o² − r_i²).

Screw thread mechanical advantage: torque to raise load T = F·d_m/2 · (μ·secα + tanλ)/(1 − μ·secα·tanλ), where λ is the lead angle.

Journal bearing Sommerfeld number S = (μN/P)(r/c)² governs the friction coefficient and minimum film thickness in hydrodynamic lubrication.

Flywheel energy fluctuation: ΔE = I·ω_mean²·C_s where C_s is the coefficient of speed fluctuation (ω_max − ω_min)/ω_mean.

Tolerance stack-up: worst-case variation is the arithmetic sum of individual tolerances; statistical (root-sum-square) stack-up assumes independent, normally distributed dimensions.

Clearance fit requires the minimum hole size to exceed the maximum shaft size; interference fit is the reverse, with negative clearance throughout.

Geometric tolerance symbols per ISO/ASME: flatness and straightness are form controls needing no datum; position and profile typically reference datums.

Surface roughness Ra is the arithmetic mean deviation of the profile; Rz is the maximum height of profile irregularities over the sampling length.

Limits and fits designation H7/g6: capital letter is the hole tolerance zone, lowercase the shaft; H marks a hole whose lower deviation is zero.

Measurement uncertainty of a vernier or micrometer must be considered against the tolerance band; a gauge should be roughly 10 times more precise than the tolerance.

Hole-basis system is preferred in production because holes are made with fixed-size tooling such as reamers and drills.

Mohr's circle: principal stresses σ₁,₂ = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²]; maximum shear τ_max = (σ₁ − σ₂)/2.

Von Mises equivalent stress for plane stress: σ_vm = √(σx² − σx·σy + σy² + 3τxy²); compare with yield strength for ductile design.

Tresca (maximum shear) criterion predicts yielding when τ_max reaches σ_y/2, a more conservative criterion than von Mises for ductile metals.

Strain gauge rosette: three measured strains solve for εx, εy, and γxy using the strain transformation equation at the three gauge angles.

Hooke's law in 3D: εx = [σx − ν(σy + σz)]/E; the shear modulus relates as G = E/[2(1 + ν)].

Thermal stress in a fully constrained bar: σ = −E·α·ΔT; no stress develops if the bar is free to expand.

Stress intensity factor K = Y·σ·√(π·a) governs linear-elastic fracture; crack growth is characterized by the Paris law da/dN = C·(ΔK)^m.

Fatigue design uses modified Goodman or Gerber diagrams to combine mean and alternating stresses against the endurance limit and ultimate strength.

ISO 286 fit calculations require the fundamental deviation plus the tolerance grade IT number; IT grades 5–7 are typical for precision fits.

Shaft design for combined bending and torsion uses equivalent stress σ_eq = √(σ_b² + 3τ²) for ductile material under von Mises.

Keyway sizing: shear stress τ = 2T/(d·L·w) and bearing stress σ = 4T/(d·L·h) for a square key of width w and height h.

Bolted joint preload F_i must exceed the external separating load divided by the joint stiffness ratio to prevent joint separation.

Welded joint throat thickness for a fillet weld is 0.707 times the leg length; shear stress acts on the throat area.

Spring rate for a helical compression spring: k = G·d⁴/(8·D³·N_a), where d is wire diameter, D is mean coil diameter, and N_a active coils.

Bearing life rating L₁₀ = (C/P)^p in millions of revolutions, with p = 3 for ball bearings and p = 10/3 for roller bearings.

S-N diagram: fatigue strength decreases with increasing mean stress; surface finish, size, and notch effects are applied as modifying factors.