elzaki transform method for finding solutions to two-dimensional elasticity problems in polar coordinates formulated using airy stress functions

This paper presents the elzaki transform method for finding solutions to two-dimensional (2d) elasticity problems in plane polar coordinates formulated using airy stress potential functions. airy stress function is used to express the stress compatibility equation as a biharmonic equation. elzaki transform is applied with respect to the radial coordinate to a modified form of the stress compatibility equation, and the biharmonic equation simplified to a fourth order ordinary differential equation (ode). the general solution for the airy stress potential function in the elzaki transform space is obtained by solving the ode. by inversion, the general solution for the airy stress potential function is obtained in the physical domain space variables in terms of four unknown integration constants. normal stresses and shear stress fields are also determined for the general case of 2d elasticity problems. the flamant problem is solved as a particular illustration of 2d elasticity problems. the stress boundary conditions and the requirement of equilibrium of the internal stress resultants and the external forces are used simultaneously to determine the four constants of integration. the airy stress potential function and the normal and shear stress fields were thus completely determined. the principle of superposition is used to obtain the elasticity solutions for the stress fields in the elastic half plane due to strip load of infinite extent, and solutions for horizontal stresses on smooth rigid retaining walls due to strip loads and parallel line load of infinite extent acting on the elastic half plane.