Sailboat Stability Calculator: Heel, Reefing and GZ Curve

The framework is transverse hydrostatics as treated by Rawson and Tupper, Basic Ship Theory, 5th edition (2001), chapter 4. The wall-sided expression is also stated explicitly in the SQA/MCA 21 October 2016 Naval Architecture examination, Q2, PDF p.30 at the linked college source. The section geometry and numerical implementation below are derived here, not empirical yacht correlations.

For beam $B$, hull draft $T$, freeboard $F$ and bottom ratio $r$, the underwater section has bottom width $rB$, a chine at half draft, and vertical sides from the chine to the deck. Thus $A_0=BT(3+r)/4$, $KB=T(r+11)/(6(r+3))$, $L_e=(m/\rho_w)/A_0$, $I_{wp}=L_eB^3/12$, $BM=I_{wp}/\nabla$ and $GM=KB+BM-KG$. The datum is the bottom of the buoyant hull, not the bottom of a fin keel.

At each heel, rotate the six section vertices, clip the polygon below a horizontal waterline, and solve for the immersion that preserves $A_0$. Polygon first moments give B; rotating the body-fixed G gives $GZ=x_B-x_G$. The restoring moment is $mgGZ$. A 3° sweep supplies plots and cached animation. Equilibrium search checks intervals in order, locating an interior net-moment maximum even when both sampled endpoints are negative, then refining the lower crossing with direct centroid evaluation. A golden-section maximum search resolves each 3° interval to less than 0.00008°. Strict monotonicity in the wall-sided range and a geometric upper bound skip intervals that cannot hide a root. Bracketed roots also refine AVS. AVS is the first zero after the initial positive-righting range; 180° means no earlier zero was found. The sampled peak is reported at 3° resolution.

The independent comparison $GZ=\sin\theta[GM+(BM/2)\tan^2\theta]$ is restricted to $\theta\le\arctan[2\min(F,T/2)/B]$: neither deck immersion nor chine emergence may have occurred. The analytical GM and wall-sided lever are checked against centroid integration.

NASA's aerodynamic force relation gives $F_0=\rho_a C_H A V^2/2$. We adopt projected sail area and force-lever factors of $\cos\theta$ each, giving $M_H=F_0h\cos^2\theta$. Gust speed is $kV$, so gust moment is $k^2$ times mean moment. Solve this balance at the chosen heel for reefing mean wind, and at the entered wind for maximum retained sail area. Equilibrium search stops at downflooding, model AVS or 80°, whichever comes first.

The historical USYRU/SNAME screen is $CSF=B/(m/\rho_s)^{1/3}$ with the original 64 lb/ft³ convention, explicitly converted to SI. It is independent of the user's water density. Wolfson Unit (1991), section 6.3, documents the screen. One international knot is exactly 1852 m/h. Cards and chart unit labels follow the metric/US switch; the schematic explicitly labels its own metre and knot readouts.

A symmetric, rigid, closed prismatic hull in calm water; no trim, longitudinal variation, hull flare or coachroof reserve buoyancy. The fin keel's mass enters the overall KG; its displaced volume is neglected. The inferred equivalent length is not a prediction of actual waterline length. Draft and mass are independent descriptions of one loading case and must be updated consistently when load changes.

All fluid and load constants have stated inputs. The default hull dimensions, KG, 60° downflooding angle, side-force coefficient 1, force lever and gust factor are illustrative engineering assumptions, not a measured production yacht. Obtain KG and downflooding from the vessel's own loading and stability data. Tanks are assumed pressed full or empty; no free-surface correction is applied.

Wind is apparent wind at sail height, acting through a single effective lateral coefficient referenced to upright working sail area. The cosine-squared heel law, fixed coefficient and fixed force lever are modeling assumptions. Reefing changes area only; enter new coefficient and lever values when the sail plan changes. The play control is a prescribed angle sweep, not a dynamic roll simulation.

Use this as a static, comparative hull and sail-load estimate for an initially stable monohull with a reasonably represented section. The full GZ curve and its AVS belong to the sealed approximation; they do not establish a real yacht's recovery angle. Beyond the entered downflooding angle, continued buoyancy is hypothetical. A reefing target must remain below downflooding, model AVS, and the sampled peak righting lever. Wind projection is used only up to 80°.

The historic screen's often quoted value of 2 is a screening convention, not a pass/fail safety threshold. CSF does not include CG, ballast location, openings, hull shape or dynamic response; it cannot be converted into AVS. Adding weight high up can lower CSF while reducing actual stability.

No breaking waves, roll energy, transient overshoot, gust duration, aerodynamic depowering, rig or sail strength, crew motion, shifted ballast, flooding, free surfaces, trim, multihulls, racing eligibility or regulatory certification are assessed. No ISO 12217 method, table or figure is implemented. Use a naval architect's yacht-specific hydrostatics and loading assessment, and the vessel and sailmaker's operating guidance, before setting real reefing limits.

Numerical references validate upright hydrostatics and the common aerodynamic force law; they are not sea-trial validation of the reefing or AVS approximation. The nonlinear hull is additionally checked against wall-sided theory, symmetry and displacement conservation.

Retained sail fractions down to 0.001 of full working area can be entered. When the required fraction is smaller, it is reported without an upward clamp and flagged as below the supported input range. Such a result requires a different operating or sail-plan assessment; it is not a recommendation for a realizable reef.