We discuss application of the finite element method to the solution of problems with initial stresses within the elasticity theory. Based on the incremental theory of deformable solids, the relationships of the finite element method are derived to calculate the stiffness matrix coefficients for a prestressed spatial element of the serendip family with quadratic approximation of displacements. The calculation of the stressed state of an eccentrically compressed beam and a round plate under conditions of longitudinal-transverse bending is carried out. Comparison of the numerical results with analytical solutions is presented. The variation in the compression and shear strains of a cylindrical damper is studied depending on the degree of deformation and the sequence of load application.Keywords: finite element method, stiffness matrix, incremental theory.There exists a large class of structures wherein a preliminary stress changes their stress-strain behavior considerably. Among such structures are rubber-metal joints, rubber-cord products (pneumatic damper cylinders, rubber pneumatic couplings, tires, etc.). Rubber elements extensively used in engineering that work in compression and tension are usually manufactured in the form of cylinders and parallelepipeds with metal plates vulcanized to the ends, which serve for damper fastening). The character and sequence of load application affect the stress-strain behavior of the material and structures.The majority of the problems in a deformable continuum solved by analytical and numerical methods are based on the traditional theory of elasticity. Taking into account the preliminary stresses in the solution of practical problems assumes the use of the incremental theory and presents a considerable mathematical difficulty. Therefore, there exists only a limited class of problems in the elasticity theory for which the analytical solutions have been derived [1][2][3].For a wide class of structures, taking into account the effect of initial stresses makes it possible to reveal their additional strength and stiffness margins.Among the works devoted to the application of the finite element method to solution of practical problems, we note [4][5][6][7][8][9][10]. At the same time, it should be noted that with a vast theoretical basis available, the solutions of practical problems have been elucidated insufficiently.The goal of the present work is to implement the numerical solution of practical problems with initial stresses using the finite element method based on the three-dimensional incremental theory of a deformable solid body.The boundary-value problem for a structure with a preliminary stress is defined by specifying the additional mass forces q i , additional external forces p i on the surface S σ and the displacements u i on the surface S u where the displacements are counted off from the initial state.
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