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Bickley (1928): pin-loaded hole in an infinite plate, cosine pressure, side hoop stress 0.81 σ_br

Tool	Pin-Loaded Hole
Source	Bickley, W. G. (1928). "The Distribution of Stress Round a Circular Hole in a Plate." Philosophical Transactions of the Royal Society of London, Series A, Vol. 227, pp. 383–415; the value 0.81 as quoted on p. 7 of Crews, J. H., Hong, C. S., and Raju, I. S. (1981). Stress-Concentration Factors for Finite Orthotropic Laminates with a Pin-Loaded Hole. NASA Technical Paper 1862, Langley Research Center, Hampton, VA.
Level	Graduate

Problem

A frictionless pin in a 0.25 in hole of a 0.1 in plate (ν = 0.3, plane stress) carries 1,000 lb, pressing the plate with a cosine pressure over the half hole facing the load; the load is reacted at infinity. Find the bearing stress, the peak pressure, the hoop stress at the sides of the hole, the crown and the back, and the bearing stress concentration factor.

Inputs to Enter

P = 1000 lb, t = 0.1 in, D = 0.25 in
ν = 0.3, plane stress; no bypass load (N_xx = N_yy = N_xy = 0)
r = 0.125 in (hole edge), θ = 90° from the load axis (the side)

Open the Pin-Loaded Hole tool (loaded with this example)

Answers

σ_br = P/(Dt) = 40 ksi; p_0 = 4σ_br/π = 50.93 ksi; κ = (3 − ν)/(1 + ν) = 2.077
Side hoop stress 32.41 ksi: K_br = 0.8102 (Bickley 0.81, independent of ν)
Crown (under the pin): hoop +15.87 ksi (0.397 σ_br), radial −p_0 = −50.93 ksi; back: −1.95 ksi (−0.049 σ_br)
ν = 0 gives crown 0.492 and back −0.144 σ_br; ν = 0.5 gives 0.333 and +0.015

Notes

The tool's default inputs are this case. K_br < 1 because the infinite plate has no net section (lugs give 2–4 on the net section); TP-1862's own contact solution with an 83° arc gives 0.87, de Jong 0.91. The series uses 1200 Fourier terms (0.8101 at 800).