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The deformation of a circular elastic plate of variable stiffness is studied in the present paper. The problem is considered in the framework of Timoshenko’s hypotheses with various conditions of support at the boundary, including the presence of elastic bonds. One of the applications of this model is the problem of modeling a lamina cribrosa sclerae (LC) of an eyeball. Timoshenko’s hypotheses are used in view of the necessity to take into account in modeling shear deformations of a LC. Elastic constraints in boundary conditions are characterized by two coefficients of subgrade resistance. An energy functional is presented with the use of the variational Lagrange principle for an inhomogeneous plate. It also takes into account the potential energy of bonds at the edge. Deflection of the plate and rotation angle of the normal are found using the Ritz method, which makes it possible to derive a solution based on the energy functional. The influence of the number of coordinate functions on the accuracy of the obtained solution is investigated. A comparison is made with the results obtained earlier in the framework of Kirchhoff’s plates.