Introduction: Usually, to analyze statically indeterminate rod systems, the classical displacement method and preprepared tables for two types of rods of the main system are used. A mathematically correct representation of local loads with the use of generalized functions makes it possible to find an accurate solution of the differential equation for the equilibrium of a beam exposed to an arbitrary transverse load. Purpose of the study: We aimed to obtain analytical expressions for functions of deflection, rotation angles, transverse forces, and bending moments depending on four types of local loads for beams with different boundary conditions, so as to apply accurate solutions in the displacement method. Methods: We propose an analytical form of the displacement method to analyze rod structural models. For beams exposed to different types of transverse load (uniformly distributed force, concentrated force, or a couple of forces), accurate analytical solutions were obtained for functions of deflection, bending moments, and transverse forces at different types of beam ends’ restraint. This is possible due to the fact that concentrated load and load in the form of the moment of force can be specified by using unit column functions. By transforming Mohr’s integrals, using integration by parts, we show that the system of canonical equations of the displacement method was obtained based on the Lagrange principle. Results: Based on the analysis of a statically indeterminate frame, the effectiveness of the proposed analytical method is shown as compared with the classical displacement method.

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The article presents the study results of geometrically nonlinear deformation of elastic shells of arbitrary type with consideration of transverse shifts. There is constructed a new mathematical model of nonlinear deformation of thin-walled elastic isotropic three-layer shells. Each layer of the shell is made of different materials, but with similar shear modules. The thickness of the layers can be different. Averaging of all three layers becomes possible, and deformation of a three-layer shell as a single-layer shell with the given characteristics of the modulus of elasticity and the Poisson's ratio can be considered.

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