Cable Laying Catenary

What is modelled

A heavy flexible cable hanging in water between the ship and the seabed takes the shape of a catenary. The tension is smallest at the touchdown point where it is purely horizontal (H), and grows with depth because the cable below carries its own submerged weight.

Equations

• Catenary y = a (cosh(x / a) − 1), parameter a = H / w, w the weight per metre in water.
• Depth h fixes the layback x = a acosh(1 + h / a) and the free-span length s = a sinh(x / a); identity s² = h² + 2 a h.
• Tension T(y) = H + w y, so at the ship T = H + w h.
• Top angle from horizontal tanθ = w s / H; slack ratio s / √(x² + h²) − 1; payout speed is ship speed times (1 + slack).

Checks run in the page

• T = H + w h against H cosh(x / a).
• The identity s² = h² + 2 a h.
• Arc length by numerical integration against a sinh(x / a).
• Vertical component of the top tension against the submerged weight w s.

Honest simplifications

The model ignores ocean currents, waves and ship motion, bending stiffness, seabed friction, the stern sheave geometry and the dynamic loads that dominate real operations. Cable weight and breaking load are generic numbers, not those of a real product, and the model does not predict a safe operating window.

• Choose Slack lay: lowering the touchdown tension lets more cable pay out for the same route.
• Choose Near breaking and watch the safety factor fall below 2.
• Raise the depth with the tension fixed and see the top tension grow by w times the extra depth.