PE Naval Architecture cheat sheet

Displacement Δ = ρ·L·B·T·C_B, with C_B = ∇/(L·B·T) and prismatic coefficient C_P = C_B/C_M.

Transverse BM = I_T/∇ where I_T is the waterplane second moment about the centreline; longitudinal BM_L = I_L/∇ is far larger (L²/(12T) for a box-shaped hull).

GM = KB + BM − KG; for small heel GZ = GM·sinθ, so initial transverse stability requires positive GM.

Wall-sided formula GZ = sinθ·(GM + ½·BM·tan²θ) extends the GZ curve beyond the small-angle range up to deck-edge immersion.

Free surface correction FSC = ρ_liquid·i/Δ where i is the free-surface second moment; it reduces effective GM but does not change KG.

A wall-sided ship with negative GM rests at an angle of loll given by tan²θ = −2GM/BM; it is unstable upright but stable at ±loll.

Trim t = Δ·(LCG − LCB)/(100·MCTC) in metres, with MCTC = Δ·GM_L/(100L) and TPC = A_w·ρ/100.

IMO intact stability: GZ area to 30° ≥ 0.055 m·rad, to 40° ≥ 0.09 m·rad, 30°–40° ≥ 0.03 m·rad, GZ_max ≥ 0.20 m and initial GM ≥ 0.15 m.

ITTC-57 friction line C_F = 0.075/(log₁₀Re − 2)² with Re = V·L/ν; ν ≈ 1.19×10⁻⁶ m²/s for seawater at 15 °C.

ITTC-78 correlation: C_Ts = C_Fs + C_R + ΔC_F, where ΔC_F = [105·(k_s/L)^(1/3) − 0.64]×10⁻³.

Froude number Fn = V/√(gL) controls wave-making resistance; a resistance hump typically appears near Fn ≈ 0.4–0.5.

Effective power P_E = R_T·V; delivered power P_D = 2πnQ; quasi-propulsive coefficient QPC = P_E/P_D.

Propeller open-water coefficients: J = V_A/(nD), K_T = T/(ρn²D⁴), K_Q = Q/(ρn²D⁵), η_O = J·K_T/(2πK_Q).

Hull efficiency η_H = (1 − t)/(1 − w); propulsive efficiency η_D = η_O·η_H·η_R with η_R typically 1.0–1.05.

Wake fraction w = (V − V_A)/V and thrust deduction t = (T − R)/T; both grow with hull fullness and single-screw stern form.

Cavitation number σ = (p₀ − p_v)/(½ρV²); cavitation risk increases as σ falls and is assessed on the Burrill chart against blade area ratio.

Admiralty coefficient P = Δ^(2/3)·V³/C allows power scaling between geometrically similar ships at preliminary design stage.

Hull girder bending stress σ = M/SM with section modulus SM = I/y, y measured from the neutral axis to the strength deck or bottom.

Along the hull dM/dx = Q and dQ/dx = −w(x): bending moment is the integral of shear force, which is the integral of the load curve.

Total bending moment = still-water moment from loading plus wave-induced moment; design uses the envelope of hogging and sagging.

IACS wave bending moment: hogging M_wh = +0.19·C·L²·B·C_B and sagging M_ws = −0.11·C·L²·B·(C_B + 0.7) kN·m, with C = 10.75 − ((300 − L)/100)^1.5 for 90 < L < 300 m.

IACS minimum midship section modulus W_min = C·L²·B·(C_B + 0.7)×10⁻⁶ m³ using the same coefficient C.

Plate buckling stress σ_E = k·π²E/(12(1 − ν²))·(t/b)², so stiffener spacing b and plate thickness t dominate buckling capacity.

Longitudinal framing with closely spaced longitudinals in deck and bottom resists hull girder bending efficiently; transverse framing suits shorter, locally loaded vessels.

Miner's rule linear fatigue damage D = Σ n_i/N_i with S-N curves; D ≥ 1 predicts fatigue failure at the detail under the assumed loading.

Local load cases - slamming at the bow bottom, panting at the bow flare, whipping and springing - are checked separately from hull girder bending.

Encounter frequency ω_e = ω − (ω²V/g)·cos μ; head seas (μ = 180°) raise ω_e while following seas lower it and can create resonance.

Deep-water dispersion ω² = g·k gives λ = 2πg/ω² and celerity c = √(gλ/2π).

Roll natural period T_R = 2π·k_x/√(g·GM) with k_x ≈ 0.35B; heave and pitch follow the same form using GM_L and the pitch gyradius.

Irregular-sea response spectrum S_r(ω) = |RAO(ω)|²·S_w(ω); significant amplitude equals 4√m₀ of the response spectrum.

Wave spectra: Pierson–Moskowitz for fully developed seas and JONSWAP with peak enhancement γ ≈ 3.3 for fetch-limited seas.

Roll damping is dominated by viscous and eddy effects rather than wave radiation; bilge keels and anti-roll tanks add damping at resonance.

IMO second-generation intact stability criteria cover parametric roll, pure loss of stability, surf-riding/broaching, dead ship condition and excessive acceleration.

Added resistance in waves raises required power and causes speed loss in head seas, a standard contractual performance check.

Parametric roll is excited in head or following seas when the encounter frequency is about twice the roll natural frequency, driven by periodic GM variation.

For a fixed-pitch propeller, power ∝ n³ and thrust ∝ n²; the engine layout must respect the heavy-running limit and torque-rich operation.

Large two-stroke diesels achieve roughly 170–190 g/kWh brake specific fuel consumption, where SFOC = fuel mass flow / brake power.

MARPOL Annex VI sulphur limits are 0.50% m/m globally (since 2020) and 0.10% in ECAs; NOx Tier III applies to new engines in NOx ECAs.

Pump affinity laws: Q ∝ n, H ∝ n², P ∝ n³; throttling and variable speed are the two common capacity control methods.

Oily water separator discharge must not exceed 15 ppm oil content under MARPOL Annex I, with mandatory oil record book entries.

EEDI and EEXI express CO₂ per unit transport work; CII rates operational carbon intensity annually on an A–E scale.

Exhaust-gas economisers and waste-heat recovery from jacket water improve plant efficiency by generating steam or electrical power from otherwise lost heat.

The shafting line comprises propeller shaft, stern tube, intermediate shaft and thrust bearing; torsional vibration analysis must show adequate barred-speed margins.

Central cooling uses a fresh-water low-temperature circuit with sea-water heat exchangers, reducing corrosion and fouling inside machinery.

The design spiral iterates displacement, principal dimensions, power and cost until weight, volume and speed requirements converge.

C_B = ∇/(L·B·T), C_P = ∇/(A_M·L) and C_M = A_M/(B·T), related by C_B = C_P·C_M.

Lightweight plus deadweight equals displacement; deadweight covers cargo, fuel, fresh water, stores and crew at the assigned load line.

Freeboard is governed by ICLL 1966, tonnage by ITC 1969, and subdivision and damage stability by SOLAS Chapter II-1.

Container capacity in TEU follows from hold and deck slot geometry, with stack weights limited by hatch cover and lashing strength.

Block construction with pre-outfitting assembles the hull from prefabricated units, cutting dock time and improving dimensional control.

Preliminary steel weight is estimated from power laws such as W_s ∝ L^1.6·B^0.4·T^0.2 or Admiralty-type coefficients calibrated on similar ships.

Required freight rate = annual capital plus operating cost divided by annual cargo carried, the standard economic comparison between design alternatives.

Escape routes, means of access and machinery space arrangements must satisfy SOLAS II-2 and classification society rules.