Anchor Chain Length & Scope Calculator: Catenary and Gust Loads

Windage follows NASA Glenn's drag equation, $H=\rho C_d A v^2/2$, or the user supplies horizontal forces directly. Roller rise is $h=d+b$; unstretched scope is $(L_c+L_e)/h$.

The geometry and energy equations below are derived here from force balance and virtual work, using the physical assumptions discussed by Alain Fraysse. The chain has submerged weight $w$ per unstretched metre and effective axial rigidity $K_c=EA_c$. A weightless, linear elastic upper segment has length $L_e$ and rigidity $K_e=EA_e$.

For suspended chain length $s$, anchor vertical tension $V_a$, and horizontal force $H$, write $V_t=V_a+ws$, $T_a=\sqrt{H^2+V_a^2}$ and $T_t=\sqrt{H^2+V_t^2}$. Chain rise and horizontal reach are

The upper segment adds $z_e=L_eV_t/T_t+L_eV_t/K_e$ and $x_e=L_eH/T_t+L_eH/K_e$. Height closure requires $z_c+z_e=h$. First solve with $V_a=0$ and $0\le s\le L_c$ (seabed contact); if the full chain cannot reach the roller in that branch, set $s=L_c$ and solve for $V_a>0$. Then $X=x_c+x_e$ and $\alpha=\tan^{-1}(V_a/H)$.

Stored energy is gravitational chain energy $U_g=\int_0^s wz(u)\,du$, chain strain energy $[H^2L_c+V_a^2s+V_aw s^2+w^2s^3/3]/(2K_c)$, and upper-line strain energy $L_eT_t^2/(2K_e)$. Closed-form integrals are evaluated in rode_state; the loaded ground chain's strain energy is included.

At the first astern turning point, find the loading-branch root above the static gust force:

This includes work done by the sustained gust as well as initial kinetic energy. The reported roller tension is $T_t(H_p)$; anchor tension is $T_a(H_p)$. Stored energy is returned on recoil, not dissipated. Swing radius is peak roller reach plus the entered roller-to-farthest-hull distance.

The chain-only static benchmark is $L_{req}=\sqrt{h^2+2hH_g/w}$; its corresponding rigid-chain lift-off force is $w(L_c^2-h^2)/(2h)$. Neither includes dynamic amplification.

• One fixed anchor on a level seabed; a planar, frictionless rode with horizontal vessel motion and constant roller height.
• Chain is perfectly flexible with uniform submerged weight (also used for any short section above water); chain and the weightless upper segment have constant linear axial rigidity. Rope self-weight, chain inertia, hydrodynamic drag, damping, hysteresis and slack-to-taut impacts are omitted.
• Nylon/snubber is in series with the loaded chain. A slack parallel backup chain never becomes taut in this model. Bridles need an equivalent axial relation and are not modeled directly.
• The gust is a sudden force step maintained through the first astern turning point. Effective mass only supplies the optional initial kinetic energy; a zero-speed step peak is independent of mass.
• Defaults are an illustrative 40 m chain plus 8 m elastic upper segment in 5 m water with a 1 m roller height. The windage area, drag coefficient and rigidities are assumptions, not catalog ratings.
• The play control sweeps equilibrium geometry; it does not integrate physical time. Height is exaggerated in the side view and the factor is labeled. A 0.001 N tension floor regularizes the zero-wind limit.

Use for comparative loading and geometry in sheltered conditions. This is not a holding guarantee, an anchor-selection method, a recommended scope rule or a certification of tackle strength. A small angle and a working-load ratio below one do not prove safe anchoring.

Use positive total roller height and unstretched rode longer than that height. The zero-height case exists only to verify the horizontal spring limit; its scope and chain-only benchmarks have no anchoring interpretation. The linear upper-line estimate is flagged above 20% strain; chain strain is flagged above 0.2%. These are model screening limits, not material allowables. Use supplier data for wet rope stiffness and the complete tackle's service working loads. If a backup chain can straighten, revise the model rather than relying on the calculated snubber extension.

Waves, current drag, yaw, heave, anchor movement, uneven ground, buried chain friction, multiple anchors, chafe and cyclic-gust resonance can change actual peaks and swing clearance. Known-force mode can include a separate steady horizontal load, but does not simulate current dynamics. The modeled peak envelope is not an all-weather clearance boundary.

Source discrepancies are retained openly. Fraysse's homogeneous example gives approximately 2.5 daN/m and 52 daN lift-off for a 15 m chain at 5 m rise; those inputs give 50 daN exactly. Attached text s3, p. 1 states “52 daN”; the reference tolerance covers that discrepancy without changing the equation. In Static Behavior (2) §3 the author writes “allows a 10 m (33 ft) extra height” for a 12 m nylon addition; the stated inextensible 25 m chain/5 m rise geometry instead gives 4.62 m. That worked claim is not used. In Dynamic Behavior (1) §2.1 the spring force/displacement results are consistent, while its period and velocity are inconsistent with the stated stiffness; only the consistent force and displacement numbers are checked.