Relaxation methods applied to engineering problems XIII. The flexure and extension of perforated elastic plates

Abstract

The theory given in Part VIIA (1941) for a flat plate having a single external boundary is here extended to allow for internal boundaries (holes) at which, as well as at the external boundary, either tractions or displacements may be specified. Such holes entail appreciable complication, more especially when boundary tractions are specified and do not form an equilibrating system at each hole severally: ‘orthodox’ studies (e.g. by Bickley, Green, Howland, Jeffery) have been restricted to simple shapes (e.g. one or more circular holes in a plate having infinite extension). But any shape can be treated by Relaxation Methods, without excessive labour and with all the accuracy that has practical value. A worked example (stresses in a loaded tie-bar) shows that holes may entail a cyclic form for Airy’s stress-function X .