Effect of a Quadrate Hole on the Stress Distribution in a Deep Beam under Uniform Bending Moment

Abstract

Effect of a Quadrate Hole on the Stress Distribution in a Beam under Uniform Bending Moment, By D. Sadahide Ogawa, Member. In the structure such as aeroplanes, ships, steel cars which are regarded to receive bending action, the quadrate holes are frequently perforated for the purpose of saving of weight, side ports, windows etc. So far as the author is aware, however, this practical problem has not yet been solved theoretically, though the effect of circular hole was attempted mathematically and experimentally by Z. Tuzi.(1) Therefore I have given theoretically the stresses in a deep beam having a quadrate hole of various roundness of corners or a crack of star shaped with its center on the neutral axis and subjected to uniform bending moment in the plane of beam.The results obtained by author will be applied well to a beam, the depth of which is great enough compared with the size of hole.First, the stresses due to the quadrate hole with its four corners on the neutral and vertical axes of beam have been examined. The quadrate boundary of hole is given by α=αi of the series of curves, which are introduced by the relationx= a (3eα cos β+re-3α cos 3β), y =a (3eα sin β-re-3α sin 3β), x and y axes being the neutral and vertical axes of beam. The stress-components αα, ββ, and αβ in the orthogonal-curvilinear co-ordinates α and β are determined by the equations (11), (12), (13) and (16), (17).Then, the case of the quadrate hole with its centers of sides on the neutral and vertical axes of beam has been deduced by substituting r = -r and β=β-π/4 (=β' say) in the foregoing results.In these results, α varies from 0 to ∞.