Involution and Body Force Method for Solution of Half Plane Problems with Sub-Surface Hole

Abstract

Flamant problem plays significance in practical problems such as bridges. It is concentrated load acting on the free boundary of semi-infinite plate. Solution for flamant problem with hole has been obtained by classical method such as involution technique . Body force method, a semi-numerical method based on principle of superposition is used to obtain solutions [1,2 ]. In the the previous works the actual condition is treated as an imaginary condition i.e. the semi-infinite plate with hole is treated as a plate without hole; the actual hole is regarded as imaginary on whose periphery boundary forces are applied. The problem is solved by superimposing the stress fields of the boundary forces and concentrated force acting at an arbitrary point to satisfy the prescribed boundary conditions so that the stress condition of the actual plate is approximately equal to that of the imaginary plate. In the present work, complex potentials of involution technique for the Kelvin-type problem (In plane concentrated load acting in the infinite plate with hole) [3,4 ] is used as fundamental solution.The flamant problem with hole is shown in Fig.1.

As contrast to dividing the imaginary hole into number of divisions as in the previous works, the imaginary free boundary (AB} of infinite plate with hole acted upon by a concentrated load ‘F' at  "P” is divided into number of equal divisions as shown in Fig.2. On each division unit concentrated load in x and y directions are applied and resultant forces along each division in x and y directions are calculated due to these unit concentrated loads. The boundary conditions require that these resultant forces along each segment are nullified due to resultant forces along each segments created by concentrated load acting on the boundary. By this ,the imaginary free boundary of infinte plate becomes stress free and becomes free edge of semi-infinite plate as shown in Fig,2.The body forces are calculated which are to be applied at the midpoint of each divisions. Now the free boundary becomes stress free and it is equivalent to flamant problem with hole. Hoop stresses, radial stresses and tangential stresses are calculated around the hole and compared with involution technique. It is found that the results are more closure to involution technique than the previous work wherein imaginary hole is descretised. In the present method, singularities near the hole boundary is nullified.