Contact Problem for a Circular Simply Supported Plate and a Rigid Base

Free-boundary contact problems occupy a special place in plate and shell mechanics; as such problems are inherently nonlinear, linearization would lose physical meaning. A generalized reaction method has been developed at Pitirim Sorokin Syktyvkar State University. It uses an iterative scheme based on an equation linking displacements and reactions. The Kármán–Timoshenko–Naghdi plate theory, avoiding Kirchhoff's hypothesis, has also been developed for contact problems.

This paper solves a contact problem for a circular, axisymmetric simply supported plate on an absolutely rigid and smooth base. The problem is addressed using two distinct approaches, both of which employ the Kármán–Timoshenko–Naghdi equations together with the generalized reaction method.

The first approach exploits axisymmetry, reducing the biharmonic operator to a single-variable operator in a polar radius. A Green's function is then used to formulate the iterative scheme, which is solved via finite differences.

The second approach neglects axisymmetry and approximates second derivatives in Cartesian coordinates. The main difficulty lies in computing numerical derivatives on the circular boundary. A four-variable Green's function is constructed, and the iterative scheme is defined in a similar manner.

The numerical experiments yield deflections of the middle and lower surfaces, contact reactions, and bending moments. The error is computed as the norm of the difference between the solutions, divided by the norm of one of them and multiplied by 100%. For deflections, the error remains below 5%, which confirms the validity of the proposed approach and makes it acceptable for circular plates even when symmetry is not taken into account.