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The behavior of elastoplastic threads of finite stiffness under lateral bending is analyzed. Geometrical and physical nonlinearities are taken into account. The material is assumed to be elastoplastic. The nonlinear equations describing the stress-strain state of threads are derived using the virtual-displacement principle. Numerical results are discussed Keywords: threads of finite stiffness, geometrical and physical nonlinearities, stress-strain state, lateral bendingIntroduction. The basic theoretical and experimental results on threads of finite stiffness have been obtained solving nonlinearly elastic problems for different geometry, boundary conditions, properties of materials, and types of loading. The most complete studies [7-9, 10] are associated with the solution of differential equilibrium equations. In [5] the solution has been obtained by the energy method based on the generalized Castigliano theorem. The possibility of applying the continuity equation to determine the stress-strain state was discussed in [13]. A method based on the virtual-displacement principle is proposed in [13]. There are also solutions to isolated classes of problems for threads of finite stiffness in general (geometrically and physically nonlinear) formulation. The elastoplastic behavior of threads under active and passive loading was analyzed, and various laws governing the state of the material were considered. However, these issues are addressed just in isolated studies such as [1,3,4,11,14], where special cases of thread behavior were analyzed. The geometrical and physical nonlinearities of the problem result in rather awkward resolving equations and, hence, labor-intensive solution.Of importance for engineering practice is to analyze the stress-strain state of threads of finite stiffness for some practically important cases (type of loading, cross section, and the law of state), which are also of independent theoretical interest [2,[15][16][17][18].In the present paper, the problem is solved based on the virtual-displacement principle, the resolving equations are derived, and a method of allowing for nonlinearities is proposed. Because of the essential nonlinearity of the problem, the Lagrange principle is applied to infinitesimal displacements, making it possible to obtain a closed-form solution. In this case, the theoretical dependences have a compact form and are very convenient for practical application. As a numerical example, the results of stress-strain analysis are given for elastoplastic threads of finite stiffness under a point force and under a load uniformly distributed over a portion of the thread span.

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.

The work of finite-stiffness cable cross-sections in transverse bending is examined. The material is assumed to be rigid-and elastoplastic. The equation for limit curves is derived. Calculation results are presented.

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