Spanned Cable Minimum Sag for a Tension Limit
The inverse of the spanline-sag-tension tile: the least sag a horizontally spanned cable can be pulled to before the support (anchor) tension reaches the allowable, d_min = w L^2 / (8 sqrt(T_allow^2 - (w L / 2)^2)). A 100 ft span at 1 lb/ft held to a 502 lb rope WLL needs at least 2.5 ft of sag; pull it tighter and the tension climbs past the limit, because tension is inversely proportional to the sag. The allowable must exceed the w L / 2 support vertical reaction, or no sag can carry the load. Enter the rope WLL or anchor capacity with the design factor applied. A planning screen; the WLL, the anchors, and the head rigger govern.
Formula and source
d_min = w x L^2 / (8 x sqrt(T_allow^2 - (w x L / 2)^2)); the inverse of T_support = w L^2 / (8 d) sqrt(1 + (4 d / L)^2) solved for the sag.
Shallow-cable parabola statics (by name) with ASME B30.9 / Wire Rope Users Manual rigging practice; first-principles, no edition cycle.
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