<p>New numerical-analytical method for solving problems of physically nonlinear deformation of thin shallow shells has been developed. The shells have a complex-shaped plan and are made of materials with different resistance in tension and compression. The method is based on the combined use of the <i>R</i>-functions, the Ritz method and the Runge–Kutta–Merson method. The <i>R</i>-functions method makes it possible to present an approximate solution in the form that exactly satisfies the boundary conditions and is invariant with respect to the domain shape where the approximate solution is sought. The problem of nonlinear-elastic deformation of the shallow shell of complex shape with combined boundary conditions has been solved. The influence of the direction of external loading, the geometric shape, and the boundary conditions on the stress–strain state has been studied.</p>