Formula sheet

Marine Vessel Performance Formula Sheet

Marine vessel formulas for buoyancy, displacement, draft, stability, resistance, power, propulsion, cavitation, scaling, pressure, stress, fatigue, and corrosion.

Branch: Naval and Marine Engineering
Content: Formula sheet
Updated: May 29, 2026
Revision: v1.0.11 · reviewed

This formula sheet collects common first-pass relationships used in naval and marine engineering. It is intended for screening, design review, sea-trial interpretation, and consistency checks. Detailed design must follow the relevant classification rules, stability codes, structural standards, model-test methods, operating profile, and vessel-specific data.

State the loading condition before using any hydrostatic or performance value. Displacement, draft, trim, center of gravity, free-surface effect, water density, fouling, propeller condition, and sea state can change the result.

Buoyancy and displacement

Buoyant force:

F_B=\rho g \nabla

Floating equilibrium:

W=F_B

Displacement mass:

\Delta=\rho \nabla

where $\Delta$ is displacement mass and $\nabla$ is displaced volume.

Weight from displacement mass:

W=\Delta g

Change in displacement mass from added load:

\Delta_2=\Delta_1+m_{added}-m_{removed}

Use water density appropriate to the operating condition. Freshwater and seawater produce different draft for the same vessel mass.

Hydrostatic pressure

Pressure at depth:

p=p_0+\rho gh

Gauge pressure below a vented free surface:

p_g=\rho gh

Pressure head:

\displaystyle h=\frac{p}{\rho g}

Force on a horizontal surface at uniform pressure:

F=pA

Hydrostatic pressure is static. Waves, sloshing, acceleration, impact, water hammer, and motion-induced loads require additional dynamic checks.

Draft and waterplane estimate

For a small load change near a given waterline, approximate draft change:

\displaystyle \Delta T \approx \frac{\Delta m}{\rho A_{WP}}

where $\Delta T$ is draft change, $\Delta m$ is added mass, and $A_{WP}$ is waterplane area.

Tons per centimeter immersion can be expressed conceptually as:

\displaystyle TPC=\frac{\rho A_{WP}(0.01)}{1000}

when $\rho$ is in kg/m3 and the result is metric tonnes per centimeter.

This is a local approximation. Large loading changes require updated hydrostatic curves.

Initial stability

Small-angle righting arm:

GZ \approx GM\sin\phi

Righting moment:

M_R=\Delta g GZ

Metacentric height:

GM=KB+BM-KG

Waterplane contribution:

\displaystyle BM=\frac{I_{WP}}{\nabla}

where $KB$ is vertical center of buoyancy above keel, $KG$ is vertical center of gravity above keel, $I_{WP}$ is second moment of waterplane area, and $\nabla$ is displaced volume.

These formulas apply to initial intact stability. Large-angle stability, downflooding, damaged stability, free-surface effect, and regulatory criteria require more detailed calculations.

Free-surface correction

A simplified free-surface correction for one slack tank is:

\displaystyle FSC=\frac{\rho_l I_t}{\Delta}

Corrected metacentric height:

GM_{corr}=GM-FSC

where $\rho_l$ is liquid density in the tank, $I_t$ is free-surface second moment of the tank plan, and $\Delta$ is displacement mass.

Multiple slack tanks add corrections:

\displaystyle GM_{corr}=GM-\sum FSC_i

Tank geometry, filling level, permeability, baffles, and damage condition can change the correction method required by rules.

Roll period estimate

A rough roll natural period estimate is:

\displaystyle T_\phi \approx 2\pi\sqrt{\frac{k^2}{gGM}}

where $k$ is radius of gyration in roll.

This is a screening relation. Real roll response depends on damping, bilge keels, hull form, speed, loading condition, wave spectrum, free-surface effects, and nonlinear stability.

Reynolds and Froude numbers

Reynolds number:

\displaystyle Re=\frac{\rho VL}{\mu}

Froude number:

\displaystyle Fn=\frac{V}{\sqrt{gL}}

where $V$ is vessel speed and $L$ is characteristic length.

Reynolds number is tied to viscous effects and boundary layers. Froude number is tied to gravity waves and wave-making. Ship model tests usually match Froude number and correct viscous effects separately.

Resistance and effective power

Total resistance:

R_T=R_F+R_R+R_A+R_{appendage}+R_{added}

where $R_F$ is frictional resistance, $R_R$ is residuary or wave-related resistance, $R_A$ is air resistance, and $R_{added}$ may include added resistance in waves or service allowances.

Effective power:

P_E=R_TV

Delivered power estimate:

\displaystyle P_D=\frac{P_E}{\eta_D}

where $\eta_D$ is overall propulsive efficiency from delivered power to effective power.

Fuel and range calculations should use service power, not only clean calm-water trial power.

Propulsion and thrust

Simplified momentum thrust:

T\approx \dot{m}(V_e-V_0)

Effective power from thrust and vessel speed:

P_T=TV

Propeller advance ratio:

\displaystyle J=\frac{V_A}{nD}

where $V_A$ is advance speed into the propeller, $n$ is revolutions per second, and $D$ is propeller diameter.

Propeller coefficients are commonly expressed as:

\displaystyle K_T=\frac{T}{\rho n^2D^4}

\displaystyle K_Q=\frac{Q_p}{\rho n^2D^5}

where $Q_p$ is propeller torque.

Open-water propeller efficiency:

\displaystyle \eta_0=\frac{J K_T}{2\pi K_Q}

Installed efficiency differs from open-water efficiency because of wake, thrust deduction, hull interaction, shafting, appendages, and operating condition.

Cavitation checks

A common cavitation number form is:

\displaystyle \sigma=\frac{p_\infty-p_v}{0.5\rho V^2}

where $p_\infty$ is reference absolute pressure, $p_v$ is vapor pressure, and $V$ is reference speed.

Cavitation risk increases when local pressure approaches vapor pressure:

p_{local}\leq p_v(T)

Propeller cavitation checks must consider blade loading, immersion, wake nonuniformity, rpm, speed, maneuvering, water temperature, and surface condition.

Wake and hull-propulsor interaction

Wake fraction:

\displaystyle w=1-\frac{V_A}{V}

where $V_A$ is propeller advance speed and $V$ is vessel speed.

Thrust deduction fraction can be expressed as:

\displaystyle t=1-\frac{R_T}{T}

Hull efficiency:

\displaystyle \eta_H=\frac{1-t}{1-w}

These simplified relations depend on the chosen convention and measurement method. State whether values come from model tests, CFD, empirical prediction, or sea trials.

Structural stress

Nominal axial stress:

\displaystyle \sigma=\frac{F}{A}

Elastic bending stress:

\displaystyle \sigma=\frac{My}{I}

Shear stress from simple area average:

\displaystyle \tau_{avg}=\frac{V}{A}

Hull structures require global and local checks. Still-water bending, wave bending, slamming, tank pressure, local buckling, fatigue, corrosion wastage, and classification-rule load cases must be considered.

Fatigue screening

Stress range:

\Delta \sigma=\sigma_{max}-\sigma_{min}

Stress amplitude:

\displaystyle S_a=\frac{\Delta \sigma}{2}

Miner damage rule:

\displaystyle D=\sum_i\frac{n_i}{N_i}

where $n_i$ is applied cycles at stress range $i$ and $N_i$ is cycles to failure at that range.

Fatigue is strongly affected by weld detail, stress concentration, corrosion, residual stress, inspection access, load spectrum, and sea state.

Corrosion allowance

Remaining plate thickness:

t_{rem}=t_0-rt

where $t_0$ is initial thickness, $r$ is corrosion rate, and $t$ is exposure time.

Required initial thickness with corrosion allowance:

t_0=t_{req}+t_{corr}

Corrosion rates are not universal. They depend on material, coating, cathodic protection, seawater chemistry, oxygen, temperature, flow, biological activity, crevices, galvanic couples, and maintenance.

Water hammer screening

Pressure surge from rapid velocity change:

\Delta p=\rho a\Delta V

where $a$ is pressure-wave speed and $\Delta V$ is velocity change.

Marine piping, ballast systems, fire mains, cooling-water systems, hydraulic circuits, and cargo systems can experience damaging pressure transients during rapid valve closure, pump trip, or emergency operation.

Practical checklist

Use these formulas with a short marine review checklist:

State vessel condition: loading, draft, trim, tank status, water density, and operating mode.
Check buoyancy, displacement, freeboard, stability, and free-surface correction.
Separate calm-water, wave, maneuvering, shallow-water, and emergency cases.
Estimate resistance, power, propeller loading, wake, and cavitation risk.
Check hydrostatic, dynamic, tank, slamming, and structural load cases.
Include fatigue, corrosion, coating condition, inspection access, and maintenance.
Compare predictions with class rules, model tests, CFD, sea trials, or operating data.

The formulas are only useful when the vessel condition is explicit. A marine performance value without loading condition, speed, water density, hull condition, and sea state is incomplete.

REF

Disciplines