STRESS-DEFORMED STATE OF THE SHELL WITH A SMALL INITIAL DEFLECTION UNDER THE ACTION OF THE END LOAD

Purpose of work. Construction of method for calculating the stress-strain state of cylindrical shell with small initial deflection, to which an end load is applied, using the method of characteristics. Comparison of the calculation results of the obtained model with the works of other authors in this area. Research methods. For the calculation, the equations of motion of the Timoshenko type shell were used, taking into account both the shear deformation and inertia of rotation, and some nonlinear terms, to which the method of characteristics was applied. To obtain the equations of shell motion, the Hamilton-Ostrogradsky variational principle was used. Results method is proposed for calculating the stress-strain state of a cylindrical shell with a small initial deflection using characteristics. Comparative analysis of the calculation results with research in this area by other authors, which showed the effectiveness of the proposed method. Scientific novelty. The equations of the classical theory of shells, based on the Kirchhoff-Love hypotheses, which do not take into account the shear deformation and inertia of rotation, as well as linear equations of the Timoshenko type, have become widespread. In this work, a model of the stress-strain state of an axisymmetric shell with small initial deflections is constructed, taking into account both shear deformation and rotational inertia, and some nonlinear terms. Practical value. The proposed method can be used to calculate the stress-strain state of structures in which thin shells are present as elements, taking into account small initial deflection. This method makes it possible to study the influence of the characteristics of the initial deflection on the stress-strain state of the entire structure.


Introduction
Questions related to the determination of the deformed and stressed state of elastic shells are urgent problems of mechanics. In particular, dynamic problems for various types of shell loading are of interest. By now, the equations of the classical theory of shells, based on the Kirchhoff-Love hypotheses, which do not take into account the shear deformation and rotational inertia have become widespread, as well as linear equations of the Timoshenko type. In this work, a model of the stressstrain state of an axisymmetric shell with small initial deflections is constructed, taking into account both shear deformation and rotational inertia, and some nonlinear terms.
In a linear formulation, unsteady waves in homogeneous shell structures were investigated in [1-2, etc.]. Nonlinear problems of deformation of shell systems with geometric imperfections were considered by V.S. Gudramovich [4]. Composite constructions in a nonlinear formulation were solved in [3, etc.].

Mathematical formulation of the problem and research results
Consider a semi-infinite cylindrical shell of circular cross-section with constant thickness h. The Ox axis is directed along the generating line by the middle of the shell surface, and the Oy axis is orthogonal to the Ox axis. We place the origin of coordinates at the end of the shell (Fig. 1).
-full deflection. In this case, in the expressions for deformations, only the components caused by the displacement of the dynamic deflection change, and they take the form: -displacement along the normal to the shell; R -middle surface radius.
We will look for displacements in the shell in this form: can be considered as the displacement of some cylindrical surface R h y 12 2 = , and ψ is the angle of rotation of the normal to the middle surface. Assuming (as is customary for thin shells) 0 = σ yy , introducing the notation: ( ρ is the shell density) and using (1)-(3), we get: To derive the equations of motion for the shell, we use the Hamilton-Ostrogradsky variational principle: where K is the kinetic energy of the shell; П is the potential energy of deformation. For a cylindrical shell, using formulas (4), we get: , 144 12 ISSN 1607-6885 Нові матеріали і технології в металургії та машинобудуванні № 1 (2021) The last term in the resulting equality is multiplied by a correcting multiplier k 2 .
Substituting the obtained expressions for the potential and kinetic energy in (5), and taking into account that the variations of the functions δψ δ δ , , 1 v u are independent quantities, we find three equations describing the axisymmetric motion of a cylindrical shell with an initial deflection.
When deriving these relations, it was assumed that it was possible to neglect nonlinear terms containing the function u and its derivatives, as well as nonlinear terms that include the displacements v1 and v0 or their derivatives if the degree of these terms is higher than two.
Next, we expand the coefficients in the resulting system (8) by degrees of h / R and retain only the senior terms in the expansion, as a result we obtain the following Timoshenko-type equations for the shell under consideration: x  They represent a hyperbolic system of equations for the dynamic state of the shell. It is of interest to construct a solution to the problem considered here using characteristic equations. To begin with, we pass in equations (9) to dimensionless variables by the formulas: Then system (9) is reduced to the following: ( ) where for convenience the following notation is introduced: In addition, the continuity conditions are satisfied along any direction: Getting rid of the mixed derivative in equations (13), they can be rewritten as: Let us consider the second equations of system (11), (14) separately, since they do not depend on . Without them, if we substitute tt U and tt V 1 from (14) into equations (11) and introduce the following notation: we get a system of two equations for xx U and xx V 1 : For this system to be linearly dependent, the following relationships must be met: from which we find the characteristics and ratios on them: Let us find the characteristics from the equality to zero of the determinant of the system -∆, and the relations on them from the equality to zero ∆1: Therefore, on the characteristics dt dx ± = , we obtain, in addition to (18), the following relations: The initial conditions were assumed to be zero. The mechanical effect on the shell was modeled by setting the particle velocity at the end

Conclusions
Analysis was carried out for various geometrical and physical parameters of the shell with a small initial deflection, as well as for various types and durations of end loading. In general, the results of this study are in good agreement with studies in this area by other authors, they show that the linear theory is quite acceptable when studying the transfer of a load impulse in a homogeneous shell and in a shell with a small initial deflection, the figures show the effect of nonzero deflection on the deformed state of the shell. In both cases, a rapid decay damping when moving away from the loading edge.