Bi-Harmonic Analysis in a Perforated Strip

A method of solving the biharmonic equation in a region bounded externally by two parallel straight lines and internally by a circle was given by one of the authors in a recent paper. General formulae were developed, but these were restricted to solutions symmetrical about both co-ordinate axes, and were applied to only one special problem of elasticity. In the present paper the analysis is generalized to include unsymmetrical solutions, and the formulae are developed to a point at which it becomes possible to solve any problem of stress within the specified boundaries. Two important special stress-systems—that corresponding to pure bending-moment, and that giving bending-moment with shear—are worked out in detail. A number of other interesting systems may be discussed by the aid of the results given. In addition, only slight modifications are needed to make the equations applicable to the slow motion of a viscous fluid.