The Effects of Holes and Semicircular Notches on the Distribution of Stress in Tension Members

Abstract

The necessities of practical construction lead to a number of interesting cases of stress problems in which discontinuities like holes and notches occur in great variety both as regards form and arrangement. For the experimental determination of stresses in loaded members an optical examination of a model shaped in transparent material has many advantages. Two cases of primary importance are examined in this way, and the results are compared with those obtained by analysis. The first example relates to the case of a hole in a tension member subjected to a uniformly applied stress, p. The values of (px - py) the difference between the principal stresses are readily obtained optically, and they show a fair agreement with the calculated values if the diameter of the hole is not greater than one-quarter of the width of the plate, but beyond this the agreement is not so good. For practical purposes it is important to be able to estimate the maximum stress from the value obtained by assuming that the total load on a tension member is uniformly distributed over the cross-section. A formula based on the relationship found in the experiments takes the form pmax = [6c3/(2c3 + 2c2 + c + 1)]pmean, where c is the ratio of the width of the member to the diameter of the hole; if c is large compared with unity this reduces to the simple form pmax = 3c/(c + 1)mean In the case of two semicircular notches, arranged symmetrically with regard to the centre line and to the cross-section, there appears to be no exact mathematical solution, but an approximate one has been obtained by Leon, resulting in expressions for px and py at the minimum section of the form px = (p/2)(2 + a2/r2 + a4/r4), py = (p/2)(a2/r2 - a4/r4) provided that the radius of the notch is small compared with the breadth of the plate. Experimental determinations of px - py show that the maximum values agree very well with those of the formulae for notches having a maximum radius of about one-quarter of the breadth of the member, but the minimum values do not show a very good agreement if the notch has a radius greater than one-eighth of the breadth. The results appear to indicate that the radial stress for large notches is greater than that given by the formula. For determining the maximum stress from the applied mean stress a formula is proposed of the form pmax = [12c3/(6c3 + 4c2 + c + 1)]pmean, and this shows a fair agreement with the experimental values.