Naval Architecture

Hydrostatics Tools

Practical hydrostatic calculators for ship design, displacement and buoyancy analysis.

Calculator

Buoyant Force

Calculate buoyant force from displaced fluid volume.

Fluid Density ρ (kg/m³)

Displaced Volume ∇ (m³)

Gravity g (m/s²)

Buoyant Force —

Equivalent Displacement Mass —

Displacement from Volume

Calculate vessel displacement mass from underwater volume and water density.

Underwater Volume ∇ (m³)

Water Density ρ (kg/m³)

Displacement —

Block Coefficient

Calculate block coefficient from underwater volume and principal dimensions.

Length L (m)

Breadth B (m)

Draft T (m)

Underwater Volume ∇ (m³)

Block Coefficient C_B —

Displacement from Principal Dimensions

Estimate displacement using length, breadth, draft, block coefficient and water density.

Length L (m)

Breadth B (m)

Draft T (m)

Block Coefficient C_B

Water Density ρ (kg/m³)

Underwater Volume —

Displacement —

Draft from Displacement

Estimate vessel draft from displacement, principal dimensions and an assumed block coefficient.

Displacement (tonnes)

Length L (m)

Breadth B (m)

Block Coefficient C_B

Water Density ρ (kg/m³)

Estimated Draft — Assumes the entered block coefficient remains representative at the calculated draft.

Tonnes per Centimeter Immersion (TPC)

Estimate the mass required to change mean draft by one centimeter.

Waterplane Area A_WP (m²)

Water Density ρ (kg/m³)

TPC —

Waterplane Coefficient

Calculate waterplane area coefficient from waterplane area, length and breadth.

Waterplane Area A_WP (m²)

Length L (m)

Breadth B (m)

Waterplane Coefficient C_WP —

Hydrostatic Force & Center of Pressure

Calculate resultant hydrostatic force and center of pressure on a vertical rectangular surface.

Plate Width b (m)

Plate Height h (m)

Depth to Top Edge (m)

Fluid Density ρ (kg/m³)

Gravity g (m/s²)

Resultant Force —

Center of Pressure Depth —

Ship Stability Tools

Practical calculators for initial stability, metacentric height, righting arms and free-surface effects.

Calculator

Metacentric Height (GM)

Calculate initial transverse metacentric height from KM and KG.

KM (m)

KG (m)

Metacentric Height GM —

Metacentric Radius (BM)

Calculate transverse metacentric radius from waterplane second moment of area and displacement volume.

Waterplane Second Moment I_T (m⁴)

Displacement Volume ∇ (m³)

Metacentric Radius BM —

Initial Stability from Hydrostatics

Calculate BM, KM and GM directly from KB, waterplane inertia, displacement volume and KG.

KB (m)

Waterplane Second Moment I_T (m⁴)

Displacement Volume ∇ (m³)

KG (m)

BM —

KM —

GM —

Small-Angle GZ

Estimate the righting arm using GZ = GM sin(φ). Intended for initial-stability calculations at relatively small heel angles.

Metacentric Height GM (m)

Heel Angle φ (degrees)

Righting Arm GZ — For larger heel angles, use a full cross-curve or KN/GZ analysis rather than this approximation.

Righting Moment

Calculate righting moment from vessel displacement and righting arm.

Displacement Δ (tonnes)

Righting Arm GZ (m)

Gravity g (m/s²)

Righting Moment —

Free Surface Correction

Estimate the reduction in GM caused by a partially filled tank using the tank free-surface second moment.

Tank Free-Surface Moment I (m⁴)

Tank Liquid Density ρ (kg/m³)

Ship Displacement Δ (tonnes)

Free Surface Correction —

Corrected GM with Free Surface

Apply the free-surface correction to the vessel's uncorrected metacentric height.

Uncorrected GM (m)

Tank Free-Surface Moment I (m⁴)

Tank Liquid Density ρ (kg/m³)

Ship Displacement Δ (tonnes)

Free Surface Correction —

Corrected GM —

Heel from Transverse Weight Shift

Estimate equilibrium heel angle caused by shifting a weight transversely aboard the vessel.

Shifted Weight w (tonnes)

Transverse Shift Distance d (m)

Ship Displacement Δ (tonnes)

GM (m)

Estimated Heel Angle — Based on the initial-stability relation tan(φ) = wd / (ΔGM).

Ship Resistance Tools

Practical calculators for ship resistance, friction coefficients, dimensionless numbers and effective power.

Calculator

Froude Number

Calculate the length-based Froude number used in ship resistance and wave-making analysis.

Ship Speed V (m/s)

Characteristic Length L (m)

Gravity g (m/s²)

Froude Number Fn —

Reynolds Number

Calculate Reynolds number using ship speed, characteristic length and kinematic viscosity.

Ship Speed V (m/s)

Characteristic Length L (m)

Kinematic Viscosity ν (m²/s)

Reynolds Number Re —

ITTC-1957 Friction Coefficient

Calculate the turbulent skin-friction coefficient using the ITTC-1957 model-ship correlation line.

Reynolds Number Re

Friction Coefficient C_F —

Frictional Resistance

Calculate hull frictional resistance from water density, speed, wetted surface area and friction coefficient.

Water Density ρ (kg/m³)

Ship Speed V (m/s)

Wetted Surface Area S (m²)

Friction Coefficient C_F

Frictional Resistance —

Total Resistance from C_T

Calculate total resistance using the total resistance coefficient, wetted surface area and ship speed.

Water Density ρ (kg/m³)

Ship Speed V (m/s)

Wetted Surface Area S (m²)

Total Resistance Coefficient C_T

Total Resistance —

Total Resistance Coefficient

Calculate the dimensionless total resistance coefficient from measured or estimated resistance.

Total Resistance R_T (kN)

Water Density ρ (kg/m³)

Ship Speed V (m/s)

Wetted Surface Area S (m²)

Total Resistance Coefficient C_T —

Residuary Resistance

Calculate residuary resistance as the difference between total and frictional resistance.

Total Resistance R_T (kN)

Frictional Resistance R_F (kN)

Residuary Resistance R_R —

Effective Power

Calculate the effective towing power required to overcome total ship resistance at a given speed.

Total Resistance R_T (kN)

Ship Speed V (m/s)

Effective Power —

Effective power is the power required to tow the hull at the specified speed. It is not the same as delivered or shaft power.

Ship Propulsion Tools

Practical calculators for propeller performance, shaft power, thrust, torque, RPM and propulsion efficiency.

Calculator

Propeller Advance Ratio

Calculate propeller advance ratio from advance speed, propeller rotational speed and diameter.

Advance Speed V_A (m/s)

Propeller Speed (RPM)

Propeller Diameter D (m)

Advance Ratio J — J = V_A / (nD), where n is propeller speed in revolutions per second.

Apparent Propeller Slip

Estimate apparent propeller slip by comparing pitch-based theoretical advance with actual advance speed.

Propeller Pitch P (m)

Propeller Speed (RPM)

Actual Advance Speed V_A (m/s)

Theoretical Pitch Speed —

Apparent Slip —

Thrust Power

Calculate useful thrust power from propeller thrust and advance speed.

Propeller Thrust T (kN)

Advance Speed V_A (m/s)

Thrust Power —

Overall Propulsive Efficiency

Calculate overall propulsive efficiency from effective power and delivered propeller power.

Effective Power P_E (kW)

Delivered Power P_D (kW)

Overall Propulsive Efficiency η_D —

Delivered Power

Estimate required delivered propeller power from effective power and overall propulsive efficiency.

Effective Power P_E (kW)

Overall Propulsive Efficiency η_D (%)

Required Delivered Power —

Shaft Torque

Calculate shaft torque from delivered power and rotational speed.

Delivered Power P_D (kW)

Shaft Speed (RPM)

Shaft Torque —

Required Propeller RPM

Calculate propeller rotational speed required for a specified advance ratio.

Advance Speed V_A (m/s)

Advance Ratio J

Propeller Diameter D (m)

Propeller Speed —

Rotational Frequency —

Propeller Pitch

Estimate geometric pitch from advance speed, rotational speed and assumed apparent slip.

Advance Speed V_A (m/s)

Propeller Speed (RPM)

Apparent Slip (%)

Estimated Propeller Pitch — This is a simplified pitch–speed–slip relationship and does not replace detailed propeller design calculations.

Propeller Tip Speed

Calculate circumferential blade-tip speed and resultant helical tip velocity.

Propeller Diameter D (m)

Propeller Speed (RPM)

Advance Speed V_A (m/s)

Circumferential Tip Speed —

Resultant Helical Tip Speed —

Ship Structures Tools

Practical structural calculators for stress, strain, deformation, bending, beam deflection and strength assessment.

Calculator

Normal Stress

Calculate average axial normal stress from force and cross-sectional area.

Axial Force F (kN)

Cross-Sectional Area A (mm²)

Normal Stress —

Positive force represents tension and negative force represents compression.

Normal Strain

Calculate engineering strain from change in length and original length.

Change in Length ΔL (mm)

Original Length L (mm)

Engineering Strain ε —

Microstrain —

Axial Deformation

Calculate elastic axial deformation of a uniform member using δ = FL/(AE).

Axial Force F (kN)

Member Length L (m)

Cross-Sectional Area A (mm²)

Young's Modulus E (GPa)

Axial Deformation —

Assumes a uniform prismatic member and linear-elastic material behavior.

Bending Stress

Calculate maximum elastic bending stress using σ = Mc/I.

Bending Moment Magnitude M (kN·m)

Distance to Extreme Fiber c (mm)

Second Moment of Area I (mm⁴)

Maximum Bending Stress — Result is based on simple elastic beam theory.

Section Modulus

Calculate elastic section modulus from second moment of area and distance to the extreme fiber.

Second Moment of Area I (mm⁴)

Distance to Extreme Fiber c (mm)

Section Modulus Z —

Simply Supported Beam Deflection

Calculate maximum deflection of a simply supported beam carrying a concentrated load at midspan.

Central Point Load P (kN)

Beam Span L (m)

Young's Modulus E (GPa)

Second Moment of Area I (mm⁴)

Maximum Deflection —

Deflection / Span —

Formula applies to a simply supported, constant-section beam with a point load at midspan.

Average Shear Stress

Calculate average shear stress from shear force and resisting cross-sectional area.

Shear Force V (kN)

Shear Area A (mm²)

Average Shear Stress — Actual shear stress distribution depends on section geometry; this calculator returns average V/A.

Factor of Safety

Calculate factor of safety by comparing material strength with actual working stress.

Material Strength (MPa)

Actual Stress (MPa)

Factor of Safety —

Utilization Ratio —

Use the material strength appropriate to the design criterion, such as yield strength or ultimate strength.

Ship Maneuvering Tools

Practical calculators for turning motion, yaw rate, drift angle, heading response and maneuvering analysis.

Calculator

Turning Radius

Estimate steady kinematic turning radius from vessel speed and rate of turn.

Vessel Speed V (m/s)

Rate of Turn r (deg/s)

Turning Radius —

Turning Diameter —

Assumes approximately steady circular motion.

Rate of Turn

Calculate angular rate of turn from vessel speed and turning radius.

Vessel Speed V (m/s)

Turning Radius R (m)

Rate of Turn —

Turning Circle Time

Calculate the time required to complete one idealized 360-degree circular turn.

Turning Radius R (m)

Vessel Speed V (m/s)

Circle Time —

Centripetal Acceleration

Calculate lateral centripetal acceleration for idealized circular vessel motion.

Vessel Speed V (m/s)

Turning Radius R (m)

Gravity g (m/s²)

Centripetal Acceleration —

Acceleration as Fraction of g —

Drift Angle

Calculate the kinematic drift angle from longitudinal and transverse velocity components.

Longitudinal Velocity u (m/s)

Transverse Velocity v (m/s)

Drift Angle β —

Resultant Speed —

The sign of β follows the sign of the entered transverse velocity.

Heading Change

Calculate heading change for a constant rate of turn over a specified time.

Rate of Turn (deg/min)

Time (s)

Heading Change —

Number of Full Turns —

First-Order Nomoto Response

Estimate yaw-rate response to a constant step rudder input using the first-order Nomoto steering model.

Nomoto Gain K (1/s)

Time Constant T (s)

Rudder Angle δ (degrees)

Elapsed Time t (s)

Yaw Rate at Time t —

Steady Yaw Rate —

Heading Change —

Assumes a step rudder input, zero initial yaw rate and a linear first-order Nomoto model.

Turning Circle Ratios

Normalize measured turning-circle dimensions by ship length for maneuvering comparison.

Ship Length L (m)

Advance (m)

Tactical Diameter (m)

Transfer (m)

Advance / L —

Tactical Diameter / L —

Transfer / L —

Naval Architecture

Hydrostatics Tools

Buoyant Force

Displacement from Volume

Block Coefficient

Displacement from Principal Dimensions

Draft from Displacement

Tonnes per Centimeter Immersion (TPC)

Waterplane Coefficient

Hydrostatic Force & Center of Pressure

Ship Stability Tools

Metacentric Height (GM)

Metacentric Radius (BM)

Initial Stability from Hydrostatics

Small-Angle GZ

Righting Moment

Free Surface Correction

Corrected GM with Free Surface

Heel from Transverse Weight Shift

Ship Resistance Tools

Froude Number

Reynolds Number

ITTC-1957 Friction Coefficient

Frictional Resistance

Total Resistance from CT

Total Resistance Coefficient

Residuary Resistance

Effective Power

Ship Propulsion Tools

Propeller Advance Ratio

Apparent Propeller Slip

Thrust Power

Overall Propulsive Efficiency

Delivered Power

Shaft Torque

Required Propeller RPM

Propeller Pitch

Propeller Tip Speed

Ship Structures Tools

Normal Stress

Normal Strain

Axial Deformation

Bending Stress

Section Modulus

Simply Supported Beam Deflection

Average Shear Stress

Factor of Safety

Ship Maneuvering Tools

Turning Radius

Rate of Turn

Turning Circle Time

Centripetal Acceleration

Drift Angle

Heading Change

First-Order Nomoto Response

Turning Circle Ratios

Ocean Tech Review

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Total Resistance from C_T