Ship Anchoring Lab: Chain Scope, Catenary, Holding Power and Spread-Mooring Storm Offset

Presets

Live readouts

Graphs

Restoring force versus offset (orange line: environmental load, white dot: equilibrium)

Fairlead tension and anchor pull versus horizontal tension (red: 60 % of breaking load, grey: anchor capacity)

Chain lying on the seabed versus horizontal tension (vertical line: lift-off load)

Numerical checks

running...

How to use

Drag the 3D view to orbit, or use the camera buttons for the bow, the chain, the anchor and the sea-floor overview. Pick a preset then raise the wind or shorten the chain.

In Short scope in a gale the chain is taut and the anchor drags; Long scope, same gale keeps the same load but the anchor holds.
Watch Chain on the seabed fall to zero at the lift-off load: that is when the anchor sees an upward pull.
Switch to the floating platform and compare 2, 4 and 8 lines: the offset falls as more lines share the load.
Use the light old-type anchor to see a drag long before the chain is anywhere near its breaking load.

The science

The chain hangs as a catenary. A heavy chain with submerged weight w per metre, pulled by a horizontal tension H at the seabed, has the shape y = a (cosh(x/a) - 1) with a = H/w. Between the touchdown point and the fairlead, a height D = depth + fairlead height above the sea floor, the suspended length, horizontal reach and fairlead tension are exactly

ℓ = √(D² + 2aD), x_s = a acosh(1 + D/a), T_f = H + w D, tan θ_f = ℓ/a

The rest of the chain, S - ℓ, lies on the seabed. Its weight and friction (coefficient μ) take load off the anchor: anchor pull = H - μ w (S - ℓ). The fairlead-to-anchor distance is X = (S - ℓ) + x_s. For given X the page inverts this relation for H by bisection, so a heavier load simply means a straighter chain with less of it on the seabed. When the load reaches H_lift = w (S² - D²)/(2D) the last link leaves the bottom; beyond that the chain is taut and arrives at the anchor with an uplift angle, found from the closed form m = atanh(D/S), q = asinh(S / (2a cosh m)), X = 2aq. A short scope (S/D of 3 rather than 7) lifts the chain early, tilts the pull upwards and pulls the anchor out.

Chain data. Mass of stud-link chain 0.0219 d² kg/m (d in mm); submerged weight w = 0.0219 d² g (1 - ρ_w/ρ_steel); breaking load MBL = k d² (44 - 0.08 d) kN with k = 0.0148, 0.0223 or 0.0274 for grades 2, 3 and 4. Design practice keeps the intact tension below about 60 % of MBL.

Storm loads. Wind F_w = ½ ρ_a C_w A_w U² (C_w = 0.9), current F_c = ½ ρ_w C_c A_c U_c² (C_c = 0.6) and mean wave drift F_d = ½ ρ_w g (H_s/2)² B R² with an illustrative reflection factor R² = 0.16. All three act in the same direction, which is the worst case. The pull balances the total: for one anchor the ship drifts out until H equals the load; for a spread of N lines at angles (k + 1/2) 360°/N the platform moves by the offset x that makes the sum of the line forces along the load direction equal to the load, solved by bisection. The restoring force curve is not a straight spring: it hardens as the lines lift off the seabed.

Holding power. An anchor's horizontal holding capacity is taken as (holding ratio) x (anchor weight) x g, with illustrative holding ratios of 4 (old stockless), 12 (high-holding-power) and 30 (drag-embedment). The anchor drags when the pull after chain friction exceeds that capacity, or when the chain arrives with an uplift angle above 5° at more than 80 % of the capacity; the animation then lets the vessel drift. Real holding depends strongly on soil, penetration and setting, and must be taken from the anchor maker's tests.

Honest simplifications. Quasi-static: no wave-frequency surge, no dynamic chain loads, no snap loading, no chain stretch or elasticity, no line drag in the water, wind, current and waves all collinear, no yaw or sheer, frictionless contact except the lying-chain friction coefficient, rigid seabed. Platform and ship areas are teaching values. Educational model only, not for anchoring decisions.