Circular solid plate supported along an edge arc and deflected by a central transverse force

A purely flexural structural analysis is carried out for a thin solid circular plate, deflected by a static central transverse concentrated force, and simply supported along an edge arc, the remaining part of its periphery being free. This problem is modelled in terms of a Fredholm integral equation of the first kind, where the kernel is expressed analytically, and where the unknown function is the reaction force along the support. The initial equation is then modified into a new Fredholm integral equation of the first kind, which implicitly respects the condition imposed on the plate edge deflections by the rigidity of the support, but which has still to be coupled with the translational and rotational equilibrium conditions. By showing that a certain operator is a contraction mapping, it is demonstrated that this new integral equation coupled only with the translational equilibrium condition possesses a unique solution expressed in terms of a smooth function with square root singularities at the support ends. It is also shown that this unique solution, when expressed via Chebyshev polynomials, does not fulfil the rotational equilibrium condition, apart from the limit case when the plate is axisymmetrically supported. It is concluded that, in the framework of the purely flexural plate theory, the title problem does not possess any smooth solution with square root singularities at the ends. An approximate solution is nevertheless computed with the collocation method, by accepting limited undulations of the plate periphery.