The problem of elastoplastic deformation of bars is solved taking into account geometrical and physical nonlinearities. An elastoplastic model is used. A method based on the virtual-work principle is proposed. As an example, the problem is solved numerically for various types of loading. Numerical results are discussed Keywords: elastoplastic deformation, bar, nonlinearity Introduction. The behavior of elastoplastic bars was theoretically studied in [4,6,7,10,[12][13][14][15]. The deformation of bars beyond the elastic limit was studied in [5] by analyzing differential equilibrium equations. The energy method based on the generalized Castigliano theorem was used in [2,3]. The possibility of using the strain compatibility equation, which relates the length of the bar before and after loading, to determine the stress-strain state was examined in [1,7]. Because of the geometrical and physical nonlinearity of the problem, the governing equations produced by all these approaches are rather awkward and difficult to solve.The present paper offers a method to solve the problem based on the virtual-work principle [8][9][10][11]. Since the systems under consideration are highly nonlinear, use is made of the rigorous form of Lagrange's principle and the closed-form representation of the solution. In this case, formulas are compact and convenient for practical calculations. As an example, we will solve the problem for various types of loading and plot the load, thrust, and deflection as functions of a length parameter that indicates how far the bar is from the limiting state.1. Problem Formulation. Derivation of the Governing Equations. We will address the geometrically nonlinear deformation of elastoplastic bars and methods of solving the associated problem. Consider a bar [4,7] with cross-sectional area F, moment of inertia I, and elastic modulus E bent by a uniformly distributed load q = q 1 + q 2 (q 1 is the initial load and q 2 is the additional load) and being in equilibrium. The bar is hinged, its material is perfect elastoplastic, and the cross-section has a perfect profile. According to [1], with such a cross-section, only two types of sections, elastic and plastic, occur in the bar during deformation. This fact somewhat facilitates the solution by making it possible to disregard elastoplastic sections without major alteration of the formulas. Thus, it is possible to present governing equations in typical form. Let the cross-sectional area F be an independent variable, and the thrust H be an unknown function.Let the cross-sectional area F be incremented by dF, which is small, yet finite. As a result, the internal forces in the bar and its geometry are changed: the bending moment M is incremented by dM, the angle j between two infinitely close cross sections is incremented by dj, and the bar shifts from the equilibrium position by dy. Moreover, the thrust H is incremented by dH. Let us now describe the elastic and plastic behavior of the bar.