NUMERICAL MODELING OF DYNAMIC DEFORMATION OF A THREE-LAYER RIBBED CYLINDRICAL SHELL UNDER AXISYMMETRIC IMPULSE LOADS

The non-stationary vibrations of three-layer composite cylindrical shell under axisymmetric impulse loads have been researched. The mathematical model and method for calculating the dynamic deformation of the shell system under consideration is based on the geometrically nonlinear theory of Timoshenko’s type shells and ribs. The results of numerical analysis of the problem are presented.

Introduction.Thin-walled multilayer shell structures reinforced with ribs are widely used in mechanical engineering and other branches of modern technology.However, the use of ribs and various constituent layers represent complex elastic structures that are inhomogeneous in thickness and leads to inhomogeneity of the stress-strain state (SSS) of the shells during non-stationary vibrations.When considering the dynamic problems of multilayer shells, taking into account the discreteness of the reinforcing ribs, one of the features is the wave nature of the processes under study.Therefore, to solve these problems, the mathematical model of vibrations must be considered in a refined formulation.From the review papers [1,2,5,7,13] and the results given [3,4,11,12,14] for shells of canonical form reinforced with stiffeners, it follows that the problem of their dynamic stability and increasing load-bearing capacity under the action of impulse loads, taking into account the influence of structural non-homogenous and mechanical characteristics, is currently insufficiently studied.This requires further development and refinement of existing methods for calculating non-stationary vibrations of multilayer composite cylindrical shells under impulse loading, taking into account various physical and mechanical characteristics of the constituent layers and new structural materials.The formulation of initial problems that describe dynamic processes in inhomogeneous shell structures necessitates the use of numerical methods to solve them.
The aim of this paper is to create a refined mathematical model of nonstationary vibrations and a numerical study of dynamic processes in ribbed multilayer cylindrical shells under axisymmetric impulse loading, taking into account the discrete placement of ribs.

Method of solution.
The problem of unsteady oscillations and dynamic deformation of a three-layer cylindrical shell is considered, taking into account the discrete placement of longitudinal-transverse reinforcing ribs.The influence of structural inhomogeneities on its dynamic stability and stressstrain state under impulse loading is investigated.The shell of thickness h consists of external load-bearing layers and an inner layerpiecewise homogeneous filler.Elongations, shifts and angles of rotation are small compared to unity, but at the same time deformations of elongations and shifts are magnitudes of a higher order of smallness than the angles of rotation.
The reinforced shell is considered as an elastic non-heterogeneous structural system consisting of orthotropic layers and rigidly connected to it along the contact lines by stringers and transverse ribs.The stress-strain state of the shell and reinforcing ribs is determined on the basis of the geometrically nonlinear theory of elastic shells and rods of the .Timoshenko's type, taking into account transverse shear deformations.
The mathematical model of the process of dynamic deformation of the considered structure is a hyperbolic system of nonlinear differential equations.The deformed state of the shell is determined through the components of the generalized vector of surface displacements ) , , , , ( in the mating layers, and the reinforcing ribs are determined through the components of similar generalized vectors Expressions for shell and ribs strain components have the form: , where j R R, radii of the median surface of the shell and transverse ribs; eccentricities of the ribs; h -thickness of the shell; We assume that the initial deflections of the shell do not cause initial stresses.Therefore, at the moment of time 0  t , under the initial conditions for displacement and the absence of external influences on the shell, all stresses should be equal to zero.In this case, the equilibrium equations are satisfied identically.
The Ostrogradsky -Hamilton variational principle is used to derive the equations of motion of a discretely reinforced shell.After standard transformations, taking into account the conditions of contact of the shell with the reinforcing ribs, we obtain the following equations of vibrations of the shell system: The relationship between the forces-moments and the corresponding values of deformations of the shell and ribs is expressed by the following dependencies: k -thickness shear coefficients in the theory of shells and ribs of the .Timoshenko's type [4];  The developed mathematical model of the vibration process is a hyperbolic nonlinear system of partial differential equations, which are supplemented by appropriate boundary and initial conditions.The main feature of the equations of motion ( 2) is their geometric nonlinearity and the presence of discontinuous coefficients in spatial coordinates.This is due to the discreteness of the arrangement of the longitudinal and transverse reinforcing ribs and their variable stiffness of the jump-like character.The lines of discontinuities in these equations are the projections of the lines of the centers of gravity of the cross-sections of the reinforcing ribs on the median surface of the shell.
The numerical algorithm for solving the problem under consideration is based on a finite-difference approximation [8][9][10] of the vibration equations (2) by spatial coordinates and an explicit difference scheme by time coordinate.This entails restrictions on the steps of the difference grid [8].The solution is in a smooth region and on the lines of spatial discontinuities.This approach allows us to preserve the divergent form of the difference representation of the resolving equations and to ensure the fulfillment of the law of conservation of the total mechanical energy of the shell system at the difference level.
In the matrix-vector form, the difference equations (2) are represented by the dependence where   M and   C -are the mass and stiffness matrices of a discrete dif- ference system; U and ) (t P are the vectors of discrete displacements and external load.
The Mises yield criterion is used as a condition for the loss of bearing capacity by the shell [1,6].It characterizes the intensity of stresses at the moment of the appearance of plastic deformations.
Numerical investigation.As a numerical example, let us consider the dynamic behavior of a rigidly fixed cylindrical shell of a finite length of  L 0,4 m with external discrete longitudinal-transverse reinforcement by ribs under normally distributed impulse loading.Stringers and frames are evenly spaced along the coordinate axes . Geometric and physical -mechanical parameters of the shell and reinforcing ribs: ;   T 310 МPа.The boundary conditions on the rigidly fixed ends of the shell for 0  x and L x  have the form: t -current time.The load parameters are: The effect of transverse reinforcement on the SSS is considered on the example of a shell reinforced with three external transverse ribs along the longitudinal coordinate axis in sections As a result of solving the problem, the parameters of the stress- strain state of the shell and ribs are obtained at discrete time points in the interval 0 30  tT .Based on the analysis of the dynamic behavior of the structure, it is established that there is no material fluidity at this time interval.With a uniformly distributed normal load times less than circumferential stresses.At the rigidly fixed ends of the shell, longitudinal stresses play a crucial role, and with a hinged supports, these stresses are close to zero.The circumferential stresses are weakly dependent on the boundary conditions.At the same time, the longitudinal stresses change very much when the boundary conditions change.The longer the shell, the less the stress depends on the boundary conditions at the ends.Moving the transverse ribs closer to the center of the shell leads to an im-provement in its dynamic properties.Depending on the distribution of the transverse ribs along the length and their rigidity, the general shape of the deformation of the shell changes.When placing the first and third frames in cross-sections , the magnitude of the greatest deflection 3 u of the shell system decreased by 1,19 times, and the circum- ferential stresses by 1,12 times.Of considerable interest is the study of the dynamic behavior of the shell with combined longitudinal-transverse reinforcement by ribs.For a shell reinforced by four stringers and three frames evenly spaced along the coordinate axes ), loss of dynamic stability can occur only between the ribs.The values of deflections 3 u along the axis of symmetry between the stringers are about 2.5 times greater than deflections along the axis of the stringer.This is due to an increase in local stiffness along the reinforcing longitudinal rib.
It should be noted that with the above load, geometric and physicalmechanical parameters of the shell, the structure operates in a purely elastic region.Plastic deformations appear when the load increases by 3,2 times.
Conclusions.The analysis of nonstationary vibrations of structurally inhomogeneous composite shells is carried out on the basis of a refined mathematical model.The model takes into account shear and normal deformations according to the applied theory of the Timoshenko's type in a geometrically nonlinear formulation.The results of the analysis of the stressstrain state of a rib-reinforced three-layer shell with a composite filler showed the high efficiency of the developed calculation method in a wide waves range.
components of the generalized displacement vector.
of the normal to the median surface of the shell relative to the coordinate axes.
the axes of stringers and frames;  -Dirac delta function.Equations (1)-(3) are supplemented by the corresponding boundary conditions.

3 u 9  3 u and circumferential stresses 22  11 
that have the greatest effect on the stress-strain state of the shell.It should be noted that the maximum values of and transverse stresses 22  are observed in the initial period of vibra- along the longitudinal coordinate at various points in time is submitted in Fig.1aand Fig.1b, which show the contact zones of the shell with the transverse ribs.Curves 1 characterizes e the distribution of these quantities at time points T t , curves 2at 14  tT , curves 3 -at 19  tT .These figures show a significant decrease in radial displacements and an increase in stresses in the reinforcement zone of the shell with transverse ribs.coordinate of the shell with three transverse ribsThe nature of the behavior of radial displacement curves and transverse stresses is determined by the wave nature of the problem under study.The local maxima and minima on the charts are the result of the interaction of deformation wave fronts.The graphs clearly show the influence of transverse ribs on the magnitude of deflections during dynamic deformation of ribbed shells.It should be noted that the influence of frames on the SSS of the shell has a local expressed character.With an increase in the number of transverse ribs, the maximum values of radial displacements of the shell system decrease at various points in time.At the same time, the nature of the distribution of these quantities changes and their localization in the locations of the ribs is observed.The stresses are observed in the crossin these sections are2,7