Naval Architecture, Hydrostatic Stability & Maritime Safety
Ship Metacentric Height (GM), Transverse BM & Righting Arm GZ
Calculate Vessel Transverse Metacentric Radius BM = I_T / Disp_Vol, Free-Surface-Corrected Fluid Metacentric Height GM_eff = KB + BM - KG - FSC, Small-Angle Righting Lever GZ = GM_eff*SIN(theta), and IMO Intact Stability Compliance.
Prevent cargo-stacking capsize by deducting tank free-surface loss from Metacentric Height!
Before a container ship or offshore supply vessel departs berth, the Chief Mate verifies Transverse Intact Stability! Hydrostatic buoyancy acts upward through the Transverse Metacenter KM = KB + BM, where KB is vertical center of buoyancy above keel and Metacentric Radius BM = I_T / Nabla divides waterplane transverse moment of inertia I_T (m^4) by submerged displacement volume Nabla (m^3). Subtracting Vertical Center of Gravity KG AND Slack-Tank Free Surface Correction FSC (sloshing ballast/fuel liquid that shifts toward the heeled side!) yields Effective Fluid Metacentric Height GM_eff = KB + (I_T / Nabla) - KG - FSC, generating initial Righting Lever GZ = GM_eff * SIN(theta_rad) (IMO IS Code requires GM_eff >= 0.15 m)!
Verify on a Geared Multipurpose Vessel—Center_of_Buoyancy KB = 4.50 m (B2), Waterplane_Inertia I_T = 36,000 m^4 (B3), Submerged_Displacement_Volume = 12,000 m^3 (B4 -> BM = 36,000 / 12,000 = 3.00 m in E2, giving KM = 7.50 m), Vertical_CG KG = 6.30 m (B5), Slack_Tank_FSC = 0.20 m (B6), and Heel_Angle = 30 deg (B7): Effective Fluid GM is 4.50 + 3.00 - 6.30 - 0.20 = 1.00 m (E3); Small-Angle Righting Lever GZ is 1.00 * SIN(30 deg) = 0.5000 m (E4); and E5 confirms "IMO Compliant Intact Stability"!
| Hydrostatic Stability Metric (KB=4.5m, IT=36,000, Vol=12,000, KG=6.3m, FSC=0.2m) | Formula | Verified Output |
|---|---|---|
| Transverse Metacentric Radius BM | Waterplane_IT / Displacement_Vol | 3.00 m (KM = 7.50 m) |
| Effective Fluid GM_eff (m) | KB + BM - KG - Free_Surface_FSC | 1.00 m (-0.10 m @ KG=7.4m) |
| Righting Arm Lever GZ @ 30 deg | GM_eff * SIN(Heel_Deg * PI() / 180) | 0.5000 m |
| IMO IS Code Stability Gate | GM_eff >= 0.15 m Intact Criterion | IMO Compliant Intact Stability |
Free-Surface-Corrected Metacentric Height (GM_eff) Formula
Validate positive KB, I_T, Displacement_Vol, and KG, non-negative FSC, and 0 <= Heel_Deg < 90!
=LET(kb_m, B2, it_w, B3, vl_d, B4, kg_m, B5, fs_c, B6, th_d, B7, IF(OR(kb_m <= 0, it_w <= 0, vl_d <= 0, kg_m <= 0, fs_c < 0, th_d < 0, th_d >= 90), "Invalid Hydrostatic Input", ROUND(kb_m + (it_w / vl_d) - kg_m - fs_c, 2)))Top-Heavy Deck Cargo Loll / Capsize Check (Set KG = 7.40 m)
Simulate heavy deck containers raising KG B5 to 7.40 m (with 0.20 m slack-tank FSC): Effective GM E3 turns negative (-0.10 m) and E5 immediately flags "Unstable: Negative GM Capsize Risk"!
=IF(ISTEXT(E3), E3, IF(E3 <= 0, "Unstable: Negative GM Capsize Risk", IF(E3 < 0.15, "Below IMO 0.15 m Minimum GM", "IMO Compliant Intact Stability")))Try the example
Use the original values or change them to test your understanding.