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Drawn circular cutout: boundary-element K_t = 2.997 against Kirsch's 3

ToolDrawn Cutout
SourceKirsch, G. (1898). "Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre." Zeitschrift des Vereines deutscher Ingenieure, Vol. 42, No. 28, pp. 797–807.
LevelUndergraduate

Problem

A round cutout of unit radius is drawn in a plate under unit tension along y. Find the extreme hoop stresses on the edge and the stresses at the points (2a, 0), (0, 2a) and (1.5a, 1.5a), and compare them with the Kirsch solution.

Inputs to Enter

  • Tension along y (N_yy with t such that σ = 1; e.g. N_yy = 10000 lb/in, t = 0.1 in for 100 ksi)
  • Circle template of radius a (test script: 120-point circle, 240 constant elements)
  • Corner radius 0 (a circle has no corners)
Open the Drawn Cutout tool

Answers

  • Max hoop / σ = 2.9972 (Kirsch 3.0), min hoop / σ = −0.9936 (−1.0); the peak lies on the x axis
  • (2a, 0): σ_xx 0.2803 / σ_yy 1.2187 (Kirsch 0.28125 / 1.21875); (0, 2a): 0.0317 / 0.4692 (0.03125 / 0.46875); (1.5a, 1.5a): −0.1479 / 1.1479 / τ −0.1113 (−0.1481 / 1.1481 / −0.1111)
  • General state σ_xx = 1, σ_yy = 0.5, τ_xy = 0.3: extremes 3.0624 and −0.0569 against 3S_1 − S_2 = 3.062 and 3S_2 − S_1 = −0.0620

Notes

Values from test-scripts/test_bem_cutout.php (tolerance 1 % on the edge, 2 % on the minimum). In the tool use the circle template and leave the corner radius at zero or any value, since a circle has no corners.

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