Twisted Rope Lab

What is modelled

A rope is made of n strands that coil round the rope axis on a ring of radius R. The strands are only as long as their fibres, so the rope holds together because twist presses the fibres against each other and friction keeps them from sliding. Tighter twist grips better, but the strands lean away from the load direction and carry less of it.

Equations

• Touching strands R = r / sin(π/n), lay tan α = 2πR / pitch.
• Strain of a helical strand εs = ε cos2α, so the rope stiffness is n E A cos3α.
• Fibre retention φ = min(1, μ Lf sin2α / (2R)), from the pressure T sin2α / R a curved strand puts on its own fibres.
• Rope strength Fmax = n Tf φ cos α, best at sin2α = 2R / (μ Lf).
• Untwisting torque M = F R tan α.

Checks run in the page

• Touching-strand geometry and the helix length by 20000 short steps.
• Strand strain and rope stiffness by differencing a stretched helix.
• Virtual work: the untwisting torque from the length change at constant strand length.
• The best lay angle found by scanning against the closed form.

Honest simplifications

Fibres are treated as equal and parallel inside a strand, the strand bending stiffness and the pressure between strands are ignored, and the fibre pull-out model is the simple shear-lag rule above. Real grass rope varies a lot from batch to batch; use the plots for trends, not for crossing a gorge.

• Press Too slack a lay: the fibres slip and the strength fraction collapses.
• Press Too hard a lay: the fibres hold, but the strands lean and the strand tension rises.
• Shorten the fibres and watch the best lay move to a steeper angle.