<p>A new numerical-analytical method was developed to solve physically nonlinear deformation problems for axisymmetrically loaded irregular bodies of rotation with stress state-dependent characteristics. The problem was linearized by the parameter-based continuous extension method. For the variational formulation of the linearized problem, the Lagrangian functional was constructed, given by kinematically admissible displacement rates. For finding the basic unknowns of the nonlinear deformation problem, the Cauchy problem for the system of ordinary differential equations was formulated and solved by the Runge-Kutta–Merson method with the automatic step choice. The initial conditions were established by solving the linear elastic deformation problem. The right-hand sides of the differential equations at the fixed load parameters corresponding to the Runge-Kutta–Merson scheme were calculated from the solution of the variational problem for the Lagrangian functional. The variational problems were solved by the Ritz method in combination with the method of Rfunctions. The latter can present an approximate solution in the form of a formula, viz a solution structure that exactly satisfies all (general structure) or part (partial structure) of the boundary conditions. The nonlinear elastic deformation of a thick-walled straight cylinder and an irregular rotation body was investigated. The geometry effect on the stress-strain state was studied. Neglect of the different material behaviors in tension and compression was shown to involve tangible errors in the results of calculating the stress-strain state parameters.</p>