Iran Nezam Mechanical Design Exam cheat sheet

Second moment of area of a rectangle about its centroidal axis: I = b·h³/12; about the base: I = b·h³/3.

Parallel-axis theorem: I = I_c + A·d², where d is the perpendicular distance between parallel axes.

Centroid of a composite area: x̄ = Σ(Aᵢ·xᵢ)/ΣAᵢ, ȳ = Σ(Aᵢ·yᵢ)/ΣAᵢ; subtract holes with negative areas.

Taylor series: f(x) = f(a) + f′(a)(x−a) + f″(a)(x−a)²/2! + …; truncation error scales with the first omitted term.

Newton–Raphson iteration: x_{n+1} = x_n − f(x_n)/f′(x_n); quadratic convergence near a simple root.

Trapezoidal rule error is O(h²); Simpson's 1/3 rule requires an even number of intervals and has error O(h⁴).

For a first-order ODE y′ + P(x)y = Q(x), the integrating factor is μ = exp(∫P dx).

Euler's identity and complex form: e^{iθ} = cosθ + i·sinθ; used for phasor and vibration analysis.

Unit conversion anchors: 1 kgf = 9.80665 N; 1 bar = 100 kPa; 1 kcal = 4.1868 kJ; 1 hp (metric) = 735.5 W.

Static equilibrium in 2D: ΣF_x = 0, ΣF_y = 0, ΣM = 0; a three-force body in equilibrium has concurrent or parallel lines of action.

Coulomb friction: F_f ≤ μ_s·N at impending motion; kinetic friction F_k = μ_k·N with μ_k < μ_s.

Beam shear and moment: V = dM/dx; a concentrated load produces a jump in V and a kink in M; a pure couple produces a jump in M.

Bending stress: σ = M·y/I = M/S; maximum at the extreme fibre, S = I/c.

Torsion of a circular shaft: τ = T·r/J; angle of twist φ = T·L/(G·J).

Euler buckling: P_cr = π²·E·I/(K·L)²; K = 1 pinned-pinned, 0.5 fixed-fixed, 0.7 fixed-pinned, 2 fixed-free.

Principal stresses in 2D: σ₁,₂ = (σ_x+σ_y)/2 ± √[((σ_x−σ_y)/2)² + τ_xy²]; τ_max = (σ₁−σ₂)/2.

Von Mises equivalent stress: σ_vm = √(σ₁² − σ₁σ₂ + σ₂²) for plane stress; Tresca uses σ₁ − σ₃.

Work–energy: T₁ + V₁ + U_{1→2} = T₂ + V₂; for constant acceleration, v² = v₀² + 2as.

Reynolds number Re = ρVD/μ = VD/ν; pipe flow is laminar below about 2300 and turbulent above about 4000.

Darcy–Weisbach head loss: h_f = f·(L/D)·(V²/2g); laminar friction factor f = 64/Re.

Energy equation for a pump system: h_pump = (ΔP/ρg) + Δz + h_f + h_m − h_turbine.

Pump affinity laws: Q ∝ N, H ∝ N², P ∝ N³ for geometrically similar machines at constant diameter.

Specific speed (SI, metric): N_s = N·√Q/H^{0.75}; low N_s favors radial (centrifugal), high N_s favors axial flow.

Cavitation margin: NPSH_a = (P_atm − P_v)/(ρg) + z − h_f,suction; require NPSH_a > NPSH_r.

Continuity and momentum: Q = A·V; force on a bend F = ρQ(V₁ − V₂cosθ) in the flow direction for a horizontal reducer-bend.

Bernoulli along a streamline (incompressible, steady, frictionless): P/(ρg) + V²/(2g) + z = constant.

First law for a steady-flow control volume: Q̇ − Ẇ = ṁ[(h₂ − h₁) + (V₂² − V₁²)/2 + g(z₂ − z₁)].

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

Rankine cycle: η = (W_t − W_p)/Q_in; reheating raises the average temperature of heat addition and limits turbine moisture.

Conduction: q = −k·A·dT/dx; plane wall R = L/(kA); cylindrical shell R = ln(r₂/r₁)/(2πkL).

Convection: q = h·A·(T_s − T_∞); Nusselt number Nu = h·L/k, with Nu = f(Re, Pr) for forced convection.

LMTD for a heat exchanger: ΔT_lm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂), multiplied by F for multipass and cross-flow units.

Radiation: q = ε·σ·A·(T_s⁴ − T_sur⁴), σ = 5.67×10⁻⁸ W/(m²·K⁴); view factors satisfy ΣF_ij = 1.

Biot number Bi = h·L_c/k distinguishes lumped capacitance (Bi < 0.1) from internal conduction resistance.

Sensible cooling coil load: Q_s = 1.23·ṁ_air(L/s)·ΔT(°C) in W, or Q_s = 1.08·cfm·ΔT(°F) in Btu/h.

Latent load: Q_l = 3010·ṁ_air(kg/s)·ΔW(kg/kg) in W; total coil load equals sensible plus latent.

Sensible heat ratio SHR = Q_s/(Q_s + Q_l); the apparatus dew point is found where the coil line meets the saturation curve.

Air-side heating/cooling equation: Q = ṁ·c_p·ΔT with c_p ≈ 1.005 kJ/(kg·K) for dry air.

Duct sizing by equal friction selects a constant pressure drop per unit length; velocity limits control noise (e.g., ≤ 5 m/s in main supply ducts near occupied spaces).

Chilled-water flow: Q = ṁ·c_p·ΔT with c_p ≈ 4.19 kJ/(kg·K); a 6 K ΔT roughly gives 0.04 L/s per kW.

Outdoor air for ventilation is set by occupancy rate per person or per floor area, plus exhaust make-up; economizer operation uses free cooling when outdoor enthalpy permits.

Fan laws: Q ∝ N, P ∝ N², power ∝ N³; system curve intersection with the fan curve sets the operating point.

Soderberg criterion: 1/n = σ_a/S_e + σ_m/S_y; Goodman uses S_u in the mean-stress term and is more permissive.

Endurance limit estimate for steel: S_e ≈ 0.5·S_u, then multiplied by surface, size, load, temperature, and reliability factors (Marin).

Stress concentration: σ_max = K_t·σ_nom; fatigue uses K_f = 1 + q(K_t − 1), where q is notch sensitivity.

Ball bearing rating life: L₁₀ = (C/P)^p million revolutions, p = 3 for ball bearings and 10/3 for roller bearings.

Gear geometry: module m = d/z; circular pitch p = π·m; center distance a = m(z₁ + z₂)/2 for external spur gears.

Belt drives: velocity ratio = D₂/D₁; arc of contact and belt length depend on center distance; V-belts rely on wedge friction.

Power screw torque: T_R = (F·d_m/2)·[(l + π·μ·d_m)/(π·d_m − μ·l)] for raising a load.

Shaft design under combined loading: use equivalent moment M_e = √(M² + T²) for bending plus torsion in ductile shafts.

Carbon steel classification: low carbon < 0.30 %C, medium 0.30–0.60 %C, high 0.60–1.0 %C; weldability falls as carbon equivalent rises.

Carbon equivalent CE = %C + %Mn/6 + (%Cr + %Mo + %V)/5 + (%Ni + %Cu)/15; CE above about 0.40 requires preheat.

Heat treatment: annealing softens and relieves stress; normalizing refines grain; quenching plus tempering trades hardness for toughness.

Austenitic stainless steels (300 series) are welded with low heat input and often low-carbon or stabilized grades to avoid sensitization.

Shielded metal arc welding uses a consumable coated electrode; the coating provides shielding gas and slag; electrode E60xx means 60 ksi tensile strength.

Casting: risers feed shrinkage; a riser must solidify after the casting it feeds; chill and padding control directional solidification.

Rolling, forging, and extrusion refine grain along the flow direction, giving anisotropic strength; machining does not.

Hardness conversions and tensile estimates: for steels, S_u (MPa) ≈ 3.45·HB is a rough correlation only.

Fixture units convert to probable flow via a demand curve; the building supply is sized on the total fixture-unit load, not the sum of fixture flows.

Water supply pressure budget: available pressure minus elevation, meter, and friction losses must exceed the minimum fixture pressure.

Drainage: horizontal branches slope 1–2 % (commonly 1/4 in/ft ≈ 2 % for small pipes); venting protects trap seals from siphonage.

Trap seal depth is typically 50–100 mm; a vent stack limits pressure fluctuation in the drainage system.

Hazen–Williams head loss: h_f = 10.67·L·Q^{1.852}/(C^{1.852}·D^{4.87}) in SI units.

Sprinkler design: density (mm/min or L/min·m²) multiplied by the design area gives the required flow; sprinklers are spaced per hazard class.

Standpipe and hose system demand is added to the sprinkler demand when the systems share a supply.

Fire pumps must not be undersized for the combined demand; churn pressure and rated flow define the pump curve.

National Building Regulations (مقررات ملی ساختمان) are issued as numbered mabhath; Mabhas 14 covers HVAC installations and Mabhas 16 covers sanitary plumbing.

Design practice follows Iranian National Building Code volumes plus the relevant standard (e.g., Mabhas 3 for structural fire and load combinations).

Seismic design uses the Iranian seismic code (Standard 2800) with the national loading code; both must be applied together.

Professional engineering responsibility is regulated by the Building Engineering Organization (سازمان نظام مهندسی ساختمان) with supervision and design certificates.

Energy efficiency requirements for buildings derive from Mabhas 19; compliance is checked through envelope and system performance criteria.

Fire safety design references Mabhas 3 and the fire protection requirements for occupancy type, egress, and detection/suppression.

Design documents must be signed and sealed by a licensed engineer; supervision reports are filed at defined construction stages.

Iranian standards (استاندارد ایران) govern product conformity, including pipes, fittings, and mechanical equipment used in projects.