For nonlinear problems in structural mechanics, finite element analysis based on the incremental theory has been well established. But a large computing time and cost make it difficult to use this method effectively in stractural analysis and design. In this paper the authors propose the application of classical Rayleigh-Ritz procedure to structural nonlinear problems including material and geometrical nonlinearities. By using this method, the size of the final stiffness matrix to be handled can be considerably reduced and therefore substantial cut of computing time may be expected, although it depends upon the selection of trial functions, a scheme of numerical energy integrals, and types of problems. As numerical examples ultimate strength analysis of uniaxially or biaxially loaded steel H-columns are shown. The present analysis is based on the incremental theory by the Lagrangian approach, and the most general displacement functions of beam-columns, derived on the assumptions that cross sections are rigid and shearing deformations can be neglected, are used as trial functions. Some initial imperfections, i. e. load eccentricities, initial deflections, and residual stresses are taken into account. The authors believe that the same procedure may be effectively applied to nonlinear dynamic problems of beam-columns.