Ship Calculator

Calculate the righting arm (GZ) and initial metacentric height (GM) for a ship to assess its stability characteristics.

Understanding Ship Stability: The Role of GM and GZ

Ship stability is a critical factor in maritime safety, determining a vessel's ability to resist overturning when subjected to external forces such as wind, waves, or cargo shifting. Two key parameters used to quantify initial stability are the Metacentric Height (GM) and the Righting Arm (GZ).

Metacentric Height (GM)

The metacentric height (GM) is a measure of the initial stability of a floating body. It represents the vertical distance between the ship's center of gravity (G) and its metacenter (M).

• G (Center of Gravity): The point where the total weight of the ship is considered to act. Its position depends on the distribution of weight onboard.
• B (Center of Buoyancy): The centroid of the underwater volume of the ship. It's the point where the buoyant force acts.
• M (Metacenter): The point where the vertical line of action of the buoyant force intersects the ship's centerline when the ship is heeled slightly.

The metacenter (M) can be calculated using the following formula:

KM = KB + BM

Where:

• KM is the height of the metacenter above the keel.
• KB is the height of the center of buoyancy above the keel.
• BM is the distance between the center of buoyancy and the metacenter.

The BM value is calculated using the ship's waterplane properties:

BM = I / V

Where:

• I is the second moment of area of the ship's waterplane about its longitudinal centerline (often approximated for rectangular or simple hull forms). For a rectangular waterplane, I = (Length * Beam³) / 12. For this simplified calculator, we'll assume an input for waterplane area and use a simplified approach or assume a relationship if beam is not directly given. For a general hull shape and simplicity here, we'll assume a more direct calculation for BM or a related value. A common approximation for BM related to ship length and beam might be BM = I / V where I is the transverse moment of inertia of the waterplane and V is the underwater volume. A common simplified BM calculation for a rectangular waterplane is (L*B^2)/12V. However, given the inputs, a more practical approach is to calculate BM using the provided waterplaneArea and shipWeight (which relates to volume and density). V = Displacement / Density. If Density is 1 tonne/m³, then V = Displacement. A common engineering approximation for BM, considering a ship shape, can be linked to the waterplane radius of gyration. For this calculator's simplified inputs, we'll use a direct estimation if possible or infer it. A very simplified approach sometimes used is BM ≈ (I_wl / V_submerged). Given the inputs, and the need to calculate GZ, we'll focus on the direct calculation of GM and GZ. A more practical approach for this calculator is to provide the necessary components for GM. We'll simplify BM calculation using common naval architecture approximations, or by relating it directly to the waterplane area and ship dimensions if beam can be inferred or is typically related. For this calculator, let's consider a typical relationship where BM can be estimated if we know the beam or can approximate it from length and displacement, or it's directly provided or calculable. However, for a direct input calculation, we often relate BM to the waterplane's second moment of area. A common approach is BM = I / V. Given we have waterplaneArea and shipWeight (Volume), we need I. For a simplified rectangular waterplane of length L and beam B, I = (L * B^3) / 12. Without beam, it's hard to calculate I precisely. Let's re-evaluate the inputs. We have shipWeight, KG, KB, waterplaneArea, shipLength, and heelAngleDegrees. The core calculation for stability involves GM and GZ. GM = KM - KG KM = KB + BM So, GM = KB + BM - KG. We need BM. A common approximation for BM for a hull is BM = I / V where I is the second moment of area of the waterplane about the centerline and V is the underwater volume. If we assume water density is 1.025 tonnes/m³ (seawater), then V = shipWeight / 1.025. Calculating I accurately requires hull form. However, if we consider a simplified hull form where I is proportional to WaterplaneArea * (CharacteristicLength)^2, and a characteristic length might be related to beam. Let's simplify by using a direct formula for GZ. The righting arm (GZ) is calculated as: GZ = GM * sin(θ) for small angles of heel (θ). For larger angles, the formula is more complex and considers the movement of the center of buoyancy. A more precise GZ formula is GZ = (KB + BM * sin²(θ)) * tan(θ) - KG * tan(θ) OR GZ = v * (h_b + z_m) - KG * tan(θ) where v is submerged volume, h_b is distance of CB from keel, and z_m is distance of M from CB. Let's stick to the most common simplified calculation using GM. First, we must calculate BM. A common approximation related to the waterplane area and ship dimensions is BM = I / V. For a simplified rectangular waterplane, I = (Length * Beam³) / 12. We don't have beam. Let's use a common formula for GZ directly that requires KM. KM = KB + BM. Let's assume we can directly calculate BM based on waterplane area and ship volume. V (underwater volume in m³) = Ship Displacement (tonnes) / Water Density (tonnes/m³). Let's assume water density = 1.025 tonnes/m³. So, V = shipWeight / 1.025. A common approximation for BM for a ship (especially for slender hulls) is related to the waterplane area and the beam. If we cannot get beam, we must rely on empirical formulas or relationships. However, the prompt implies calculating GZ given heel angle. The standard formula for GZ is: GZ = GM * sin(heel_angle_radians) for small angles. Or, a more general formula considering the curve of stability: GZ = v * (h_b + BM sin²(θ)) * tan(θ) - KG * tan(θ) where v is submerged volume. This is too complex without more inputs. Let's use a direct GZ approximation formula if we assume a hull form, or we stick to the most fundamental: GM = KM - KG KM = KB + BM BM = I / V Here, I is the second moment of area of the waterplane about the longitudinal axis passing through the center of buoyancy. For a ship, this is often approximated using length and beam. Let's re-examine the given inputs: shipWeight, KG, waterplaneArea, KB, shipLength, heelAngleDegrees. We are missing the ship's beam or the second moment of area (I) of the waterplane. Without these, calculating BM and subsequently GM accurately is difficult. Let's try a different approach common in simplified calculators: Calculate GM directly if we can estimate BM. V = shipWeight / 1.025 (assuming seawater density of 1.025 t/m³) BM calculation is the bottleneck. For a ship, BM is typically proportional to (Length * Beam^2) / V or WaterplaneAreaMomentOfInertia / V. If we make a very strong assumption that BM is proportional to WaterplaneArea / ShipLength or similar, it would be highly inaccurate. Let's assume the user is meant to provide inputs that lead to BM. If we don't have BM, but we have waterplaneArea, and shipLength, we can *estimate* a relationship for BM. For example, if we assume the beam (B) is roughly shipLength / 6 (a common waterline beam-to-length ratio for many ship types), then I = (shipLength * B³) / 12. Then BM = I / V. Let's try this approximation: Assume Beam B = shipLength / 6. Then I = (shipLength * (shipLength/6)³) / 12. V = shipWeight / 1.025. BM = I / V. KM = KB + BM. GM = KM - KG. Now, for GZ: For small angles, GZ = GM * sin(θ). For larger angles, the formula becomes more complex. A common simplified approximation for GZ at a given angle θ is: GZ = GM * sin(θ) + (BM * (sin(θ)³)) / 2. This is still an approximation. A more common formula used for stability curves is GZ = v * (KM - KG) * sin(θ) + v * BM * sin(θ) * cos(θ)… No, that's not right. Let's simplify based on the most fundamental understanding: 1. Calculate underwater volume V. 2. Calculate BM using an approximation that involves waterplaneArea and potentially ship dimensions. The most robust way without beam is to assume some relationship. If waterplane area is given, and length, and we assume a beam, we can calculate I. Let's *assume* that waterplaneArea and shipLength are sufficient to imply BM, perhaps through an empirical factor or a simplified hull shape assumption. A common simplification in exercises is to assume a rectangular waterplane. If so, Waterplane Area = L * B. If we have L and Waterplane Area, we can find B. Then I = (L * B³) / 12. V = shipWeight / 1.025. BM = I / V. Okay, let's use this assumption: Assume shipLength and waterplaneArea imply a beam B such that waterplaneArea = shipLength * B. This is a simplification and assumes a rectangular waterplane for calculating I, even if the ship isn't. B = waterplaneArea / shipLength. Then, I = (shipLength * B³) / 12. V = shipWeight / 1.025 (assuming seawater density of 1.025 t/m³) BM = I / V. KM = KB + BM. GM = KM - KG. Now for GZ. The most widely used formula for GZ (the righting arm) for small to moderate angles is: GZ = GM * sin(θ) where θ is the angle of heel in radians. A more accurate GZ equation for larger angles of heel is: GZ = v * (KM - KG) * sin(θ) + v * BM * sin(θ) * cos(θ). This is also complex. Let's use the standard, commonly accepted formula for GZ for practical stability calculations, which involves the "curve of stability": GZ = v * (KM - KG) * sin(θ) + v * BM * sin(θ) * cos(θ) <– This seems incorrect or specific. Let's revert to the simpler, most fundamental definition for GZ when initial stability is discussed: GZ = GM * sin(θ) where θ is the angle of heel in radians. This is valid for small angles. For the purpose of this calculator, we will calculate GM and then use the small-angle approximation for GZ. Let's refine the calculation steps: 1. Convert heel angle from degrees to radians: heelAngleRad = heelAngleDegrees * (Math.PI / 180). 2. Estimate Beam (B) from Waterplane Area and Length: B = waterplaneArea / shipLength. This is a strong simplification. 3. Calculate Moment of Inertia of Waterplane (I) assuming rectangular shape: I = (shipLength * Math.pow(B, 3)) / 12. 4. Calculate Underwater Volume (V) assuming seawater density (1.025 t/m³): V = shipWeight / 1.025. 5. Calculate BM: BM = I / V. 6. Calculate KM: KM = KB + BM. 7. Calculate GM: GM = KM - KG. 8. Calculate GZ (Righting Arm) using small angle approximation: GZ = GM * Math.sin(heelAngleRad). Edge cases: – Division by zero if shipLength is 0. – Negative values for inputs. – Ensure calculated B is reasonable. If waterplaneArea is very small relative to length, B could be tiny. Refined calculation logic: – Check if inputs are valid numbers and positive. – If shipLength is 0, warn. – If waterplaneArea / shipLength results in a non-positive beam, warn. Units: – Ship Displacement: tonnes – KG, KB, BM, KM, GM: meters (m) – Waterplane Area: square meters (m²) – Heel Angle: degrees – GZ: meters (m) */ function calculateStability() { var shipWeight = parseFloat(document.getElementById("shipWeight").value); var centerOfGravityHeight = parseFloat(document.getElementById("centerOfGravityHeight").value); // KG var waterplaneArea = parseFloat(document.getElementById("waterplaneArea").value); var centerOfBuoyancyHeight = parseFloat(document.getElementById("centerOfBuoyancyHeight").value); // KB var shipLength = parseFloat(document.getElementById("shipLength").value); var heelAngleDegrees = parseFloat(document.getElementById("heelAngleDegrees").value); var resultDiv = document.getElementById("result"); resultDiv.innerHTML = ""; // Clear previous results // Input validation if (isNaN(shipWeight) || shipWeight <= 0 || isNaN(centerOfGravityHeight) || centerOfGravityHeight < 0 || // KG can be 0 for keel, but usually positive isNaN(waterplaneArea) || waterplaneArea <= 0 || isNaN(centerOfBuoyancyHeight) || centerOfBuoyancyHeight < 0 || // KB must be positive isNaN(shipLength) || shipLength <= 0 || isNaN(heelAngleDegrees) || heelAngleDegrees < 0) { resultDiv.innerHTML = "Please enter valid positive numbers for all fields."; return; } // Constants var waterDensity = 1.025; // tonnes/m³ for seawater // Step 1: Calculate Beam (B) assuming a rectangular waterplane for simplicity in calculating I // This is a significant simplification. Real ship waterplanes are complex. var beam = waterplaneArea / shipLength; if (isNaN(beam) || beam <= 0) { resultDiv.innerHTML = "Error: Could not calculate a valid ship beam from provided length and waterplane area. Please check inputs."; return; } // Step 2: Calculate Moment of Inertia of Waterplane (I) about longitudinal axis // Assuming a rectangular waterplane for calculation of I var momentOfInertia = (shipLength * Math.pow(beam, 3)) / 12; if (isNaN(momentOfInertia) || momentOfInertia <= 0) { resultDiv.innerHTML = "Error: Could not calculate a valid moment of inertia for the waterplane."; return; } // Step 3: Calculate Underwater Volume (V) var underwaterVolume = shipWeight / waterDensity; if (isNaN(underwaterVolume) || underwaterVolume <= 0) { resultDiv.innerHTML = "Error: Could not calculate a valid underwater volume."; return; } // Step 4: Calculate BM (Distance between Center of Buoyancy and Metacenter) var bm = momentOfInertia / underwaterVolume; if (isNaN(bm) || bm <= 0) { resultDiv.innerHTML = "Error: Could not calculate BM (Metacentric Radius)."; return; } // Step 5: Calculate KM (Height of Metacenter above Keel) var km = centerOfBuoyancyHeight + bm; // KM = KB + BM if (isNaN(km)) { resultDiv.innerHTML = "Error calculating KM."; return; } // Step 6: Calculate GM (Initial Metacentric Height) var gm = km – centerOfGravityHeight; // GM = KM – KG if (isNaN(gm)) { resultDiv.innerHTML = "Error calculating GM."; return; } // Step 7: Calculate GZ (Righting Arm) for the given heel angle // Convert heel angle to radians var heelAngleRad = heelAngleDegrees * (Math.PI / 180); // Using the fundamental GZ = GM * sin(theta) for small angles. // This is an approximation for initial stability assessment. var gz = gm * Math.sin(heelAngleRad); if (isNaN(gz)) { resultDiv.innerHTML = "Error calculating GZ."; return; } // Display Results var stabilityResult = ""; stabilityResult += "

  Stability Results

  "; stabilityResult += "Initial Metacentric Height (GM): " + gm.toFixed(3) + " m"; stabilityResult += "Righting Arm (GZ) at " + heelAngleDegrees + "°: " + gz.toFixed(3) + " m"; if (gm > 0.15) { // A common threshold for positive stability stabilityResult += "Interpretation: The ship has positive initial stability. A higher GM generally indicates greater initial stability, but too high a GM can lead to uncomfortable 'stiff' rolling."; } else if (gm > 0 && gm 0) { stabilityResult += "A positive GZ at this heel angle indicates the ship will tend to return to its upright position."; } else { stabilityResult += "A non-positive GZ at this heel angle indicates the ship may not return to its upright position and could be in danger of capsizing."; } resultDiv.innerHTML = stabilityResult; }

  The righting arm (GZ) is the horizontal distance between the line of action of the buoyant force and the line of action of the gravitational force. It creates a righting moment (Righting Moment = Displacement * GZ) that opposes the heeling moment and tends to return the ship to its upright position.

  For small angles of heel (θ), the GZ can be approximated as:

  GZ ≈ GM * sin(θ)

  Where:

  • GM is the initial metacentric height.
  • sin(θ) is the sine of the angle of heel (θ), which must be in radians for this formula if GM is in meters and GZ is in meters. If θ is in degrees, sin(θ_degrees) can be used directly with GM for GZ in meters. The calculator uses radians internally for accuracy.

  Calculation Assumptions and Simplifications in this Calculator:

  • Water Density: Assumes seawater density of 1.025 tonnes/m³.
  • Hull Form: Assumes a rectangular waterplane shape for calculating the moment of inertia (I) and subsequent BM. This is a significant simplification for real ship hulls.
  • Beam Estimation: Estimates the ship's beam (B) from the waterplane area and length (B = Waterplane Area / Ship Length). This is an approximation.
  • Small Angle Approximation for GZ: Uses the formula GZ ≈ GM * sin(θ), which is most accurate for small angles of heel. For large angles, the actual GZ curve deviates from this linear relationship.

  When to Use this Calculator:

  • Initial Stability Assessment: To get a quick estimate of a ship's initial stability characteristics (GM).
  • Educational Purposes: To understand the fundamental relationships between KG, KB, BM, and GM.
  • Preliminary Design: In the very early stages of ship design when detailed hull form data is not yet available.

  Disclaimer: This calculator provides simplified estimates for educational and preliminary assessment purposes only. Actual ship stability calculations require detailed hydrostatics data from naval architects and adherence to strict maritime regulations (e.g., IMO stability code). Never rely solely on this calculator for critical safety decisions.