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Goodier (1936): round hole in a plate under uniaxial bending, K = (5 + 3ν)/(3 + ν)

Tool	Hole in Plate Bending
Source	Goodier, J. N. (1936). "The Influence of Circular and Elliptical Holes on the Transverse Flexure of Elastic Plates." Philosophical Magazine, Series 7, Vol. 22, No. 145, pp. 69–80.
Level	Graduate

Problem

A 0.1 in thick plate with ν = 0.3 carries a uniform remote bending moment of 1,000 lb·in per inch on its x faces and contains a 0.5 in round hole. Find the moment concentration factor on the hole edge, the peak edge moment and surface stress, and the edge moment at 90° from the peak.

Inputs to Enter

M_xx = 1000 lb·in/in (entered as 1000 lb), M_yy = 0, M_xy = 0
t = 0.1 in, D = 0.5 in, ν = 0.3
r = 0.25 in (hole edge), θ = 0

Open the Hole in Plate Bending tool

Answers

K_b = (5 + 3ν)/(3 + ν) = 1.7879; M_edge,max = 1,787.9 lb·in/in at θ = 0
Surface stress 6M/t² = 1,072.7 ksi (nominal 6M_xx/t² = 600 ksi)
At θ = 90°: M_θ = (1 − ν)/(3 + ν) M = 0.2121 M = 212.1 lb·in/in, 127.3 ksi
Equal biaxial moments give 2M all round (build check)

Notes

Change the default M_yy = 200 to 0 and set the probe angle to 0 to land on the peak. Moments per unit width are entered in force units (lb·in/in = lb). The moment field satisfies M_r = 0 and V_r = 0 on the free edge.