Stresses in a plate containing an infinite row of holes

Abstract

It is well known that the presence of a hole in a stressed plate has a very great influence upon the maximum stress, and a number of investigations, both mathematical and experimental, have been directed towards determining the exact extent of this influence. When the hole is isolated and the boundaries of the plate are distant the mathematical method of binding the stresses in its neighbourhood is elementary. But the presence of adjacent boundaries introduces great complications. The problem has been solved for a circular hole near to a single straight boundary by effery, and for a hole mid-way between two straight boundaries by the present writer and others. Here appears, however, to have been no theoretical investigationof the stresses in the neighbourhood of a group of holes sufficiently close together to influence each other to a marked extent. Capper, using optical methods, obtained the stress distributions in a narrow plate under tension when there were three holes in line across the plate or six holes in triangular formation. His results will be referred to at a later stage. The simplest case for mathematical investigation is that of an infinite row of equal and equally spaced holes in a plate of indefinite extent. If the applied tractions are of such a kind that all the holes are similar with regard to them, we have a periodic stress distribution. It will be shown that this is always expressible by means of a set of periodic functions all derivable from one fundamental function. The solution so obtained will be applied to the special cases of tensions applied either parallel to the line of centres or perpendicular to this line. Numerical results will be given.