Normal Tension for a Suspended Steel Tape

Stretch the tape harder and it lengthens; let it hang and its own weight shortens the span. At exactly one pull those cancel and the tape reads true with no arithmetic. Setting the two corrections equal and substituting the span weight W = w L removes the length from both sides and leaves P^2 (P - P0) = A E W^2 / 24 -- whose square root is the textbook P = 0.204 W sqrt(A E) / sqrt(P - P0), because 0.204 is just 1 over the square root of 24. A 100 ft tape weighing 0.02 lb/ft, 0.006 sq in in section, standardized at 10 lb, wants 34.4 lb; pull the usual 20 lb instead and the span reads 0.036 ft LONG. Also shows the residual at whatever pull you actually used. Normal tension can exceed a practical or rated pull, and a fully supported tape has no sag to cancel.

set the tension correction equal and opposite to the sag correction, (P - P0) L / (A E) = w^2 L^3 / (24 P^2); substituting the span weight W = w L eliminates L and gives P^2 (P - P0) = A E W^2 / 24, solved by bisection. Square-rooting the same relation gives the textbook implicit form P = 0.204 W sqrt(A E) / sqrt(P - P0), where 0.204 = 1/sqrt(24).

Normal tension as defined in the standard surveying references (Ghilani/Wolf, Elementary Surveying), by name. The component corrections are the same tension and sag expressions the taping-corrections tile uses, so the two tiles cannot drift.

Both corrections and the normal-tension condition are elementary mechanics published in every surveying text and in free course notes; nothing here is tabulated or proprietary.

Estimate. AHJ and licensed professional govern.

Field names used by the API: span_ft, tape_weight_plf, tape_area_in2, standard_pull_lb, applied_pull_lb, e_psi, normal_tension_lb, span_weight_lb, sag_at_normal_ft, pull_at_normal_ft, net_at_applied_ft

• Cancellation condition tension correction = -(sag correction) at the normal tensiondefinition of normal tension
• Governing equation P^2 (P - P0) = A E W^2 / 24, length-independent once W = w Lderived from the two correction formulas
• 0.204 constant 1/sqrt(24) = 0.20412, derived rather than fittedalgebra of the same relation
• Steel modulus 29,000,000 psi default, editablestandard value for steel tapes
• Scope does not address temperature or standardization error; may exceed a practical or rated pullstated scope limit