Ya. M. Grigorenko

National Academy of Sciences of Ukraine | ISP

The stress state of hollow cylinders with oval cross-section made of orthotropic and isotropic materials is analyzed using spatial problem statement and analytical methods of separation of variables, approximation of functions by discrete Fourier series, and numerical discrete-orthogonalization method. The reference surface in the cross-section is...

The review is devoted to the numerical solution of new problems of electroasticity, namely, determination of the dynamical characteristics of inhomogeneous piezoceramic waveguides of circular cross-section and inhomogeneous piezoceramic cylinders of finite length. To solve these problems, an effective numerical–analytical approach is used. The appr...

The bending problem for a cylindrical shell with oblique cuts is solved using the equations of a model based on the straight-line hypothesis. The two-dimensional boundary-value problem obtained is reduced by the spline-approximation method to one-dimensional one solved by the method of discrete orthogonalization. The results calculated are compared...

The fundamentals of three new discrete-continuum approaches to the solution of the stationary problems of shell theory are discussed: the discrete Fourier series approach, the spline-collocation method, and the complete systems method. The general idea of the discussed approaches consists in using some sort of transformation to convert the original...

The stress problem for layered hollow inhomogeneous cylinders with concave semi-corrugations is solved in spatial statement, and their stress state is studied depending on the stiffness of the core layer. To solve the problem, the analytical methods of variable separation, approximation of functions by discrete Fourier series, and the numerical dis...

An approach to solving static problems for ring plates with parameters varying in two coordinate directions is proposed. The system of equations and boundary conditions are formulated for displacements, forces, and moments. The two-dimensional boundary-value problem is reduced to one-dimensional one using the spline-collocation method. This problem...

The effect of change in the curvature parameters of the stress state of concave corrugated hollow cylinders is studied. The change is attributed to variations in the radius of a moving circle and in the distance to its center. The problem is solved in spatial statement using analytical methods of separation of variables, approximation of functions...

The stress–strain state of open and closed variable-thickness elliptic cylindrical shells is studied. To solve the problem, the Mushtari–Donell–Vlasov shell model and numerical-analytical approach based on the spline-collocation and discrete-orthogonalization methods are used. Various types of boundary conditions and variable loadings are considere...

We present the solution of a three-dimensional boundary-value problem of stresses in the theory of elasticity for hollow inhomogeneous orthotropic cylinders with cross sections in the form of convex semicorrugations with zones of large curvature. The boundary conditions at the ends of the cylinder make it possible to separate variables along the le...

The effect of the change in the curvature due to changes in the epicycle radius on the stress state of longitudinally corrugated hollow cylinders is studied using a spatial problem statement, the variable separation method, discrete Fourier series, and the discrete-orthogonalization method. The results presented in the form of graphs of distributio...

The effect of orthotropy on the stress state of longitudinally corrugated orthotropic hollow cylinders is analyzed using a three-dimensional problem formulation, the analytical variable-separation and Fourier-series methods, and the numerical discrete-orthogonalization method. The results obtained are presented in the form of graphs of displacement...

The effect of the changes in the curvature caused by variations in the radius of the epicycle and the distance to its center on the stress state of longitudinally corrugated hollow cylinders is studied using a three-dimensional problem statement, the variable separation method, discrete Fourier series, and the discrete-orthogonalization method. The...

An approach applying Fourier series expansion to functions defined on a discrete set of points is used to determine the stress state of noncircular hollow cylinders depending on the cross-sectional shape and material properties. The results of analysis of the stress state are presented in the form of plots and tables.

Studies on the static and dynamic deformation of isotropic and anisotropic elastic shell-like bodies of complex shape performed using classical and refined problem statements are reviewed. To solve two-dimensional boundary-value problems and eigenvalue problems, use is made of a nontraditional discrete-continuum approach based on the spline-approxi...

A refined Timoshenko-type model based on the straight-line hypothesis is used to develop an approach to analyzing the stress state of longitudinally corrugated cylindrical shells with elliptic cross-section. The approach is to reduce the two-dimensional boundary-value problem that describes the stress–strain state of the shell to a one-dimensional...

The stress state of longitudinally corrugated layered hollow orthotropic elliptic cylinders is determined using discrete Fourier series for different geometrical parameters. The results obtained are presented in the form of plots and tables and analyzed

A developed approach based on discrete Fourier series is used to find the solution for and analyze the stress state of complex-shaped noncircular hollow cylinders depending on the cross-sectional curvature Keywordsnoncircular hollow cylinders-stress state-cross-sectional curvature-discrete Fourier series

The stress state of longitudinally corrugated hollow orthotropic elliptic cylinders is analyzed using discrete Fourier series and discrete orthogonalization

The problem of bending of beveled circular cylindrical shells is solved by parametrizing the shell and reducing the two-dimensional boundary-value problem to a one-dimensional one by the spline-collocation method. This problem is solved by the stable discrete-orthogonalization method. The effect of the variability of the geometrical parameters on t...

The paper deals with some approaches to solving linear and nonlinear boundary-value stress problems for elastic bodies with complex geometry and structure. The problems are described by partial differential equations solved using discrete Fourier series. The results obtained are presented in the form of plots and tables

An approach developed to solve static problems for longitudinally corrugated elliptic cylindrical shells is used to analyze the influence of their geometric parameters and thickness on the stress–strain state. The circumferential distribution of stresses and displacements is analyzed for different values of the aspect ratio and number of corrugatio...

An approach developed to solve boundary-value problems is used to analyze the effect of geometry and orthotropy parameters on the displacement and stress fields in nonthin orthotropic conical shells

An approach developed earlier to solve boundary-value problems is used to analyze the behavior of the stress-strain state of orthotropic elliptic cylindrical shells with variation in the geometric parameters of their cross section at constant volume (weight)

The approach developed to solve two-dimensional static problems for nonthin conical shells of varying thickness is used to examine the effect of the geometrical parameters on the stress-strain state of shells. The approach is based on spline-approximation and a stable numerical method of solving one-dimensional problems

An approach based on the use of discrete Fourier series is employed to analyze the stress state of orthotropic and transversely isotropic elliptical hollow cylinders

An approach to solving two-dimensional stress-strain problems for orthotropic conical shells of variable thickness in a refined formulation is developed. The approach is based on the spline-approximation and the stable discrete-orthogonalization method, which is used to solve the one-dimensional problem. The dependence of the deflection and stresse...

Stress problems for noncircular cylindrical shells in classical, refined, and spatial statements are solved using nonconventional approaches based on discrete Fourier series and spline functions. Solutions for isotropic and orthotropic shells are presented as plots and tables

The paper presents an exact analytic solution to a nonlinear boundary-value problem for a long noncircular cylindrical shell with variable cross-sectional curvature under two concurrent loads: a surface force and bending moments. The behavior of the shell under these loads is analyzed

An approach developed to solve boundary-value problems is used to analyze the influence of orthotropy and other factors on the displacement and stress fields in nonthin orthotropic cylindrical shells with elliptic cross-section

A problem-solving approach based on discrete Fourier series is used to analyze the stress state of corrugated orthotropic and transversely isotropic hollow cylinders

An exact analytical solution is found to a nonlinear boundary-value deformation problem for a long noncircular cylindrical shell of variable curvature. The shell is subject to bending moments at the edges. The dependence of the stress-strain state of the shell on the curvature is analyzed

The paper presents an approach to solve the boundary-value stress-strain problem for circumferentially corrugated elliptic cylindrical shells. The approach employs splines to approximate the solution and the stable discrete-orthogonalization method to solve the resulting one-dimensional problem. The results are presented as plots and a table

A spatial model and a refined model based on the straight-line hypothesis are used to analyze the stress state of nonthin elliptic cylindrical shells with certain end conditions for different thicknesses and aspect ratios. The results obtained are compared, and the validity range of the refined model is established

The paper proposes an approach to solving a spatial stress problem for solid circular cylinders under axisymmetric surface loading. Two types of boundary conditions at the ends are examined: simply supported or clamped. The circumferential variable is separated using Fourier series for the former type of boundary conditions, and spline-approximatio...

An approach is developed to solve the stress-strain problem for noncircular cylindrical shells with complex cross section in the form of connected convex half-corrugations. Numerical results are presented

A method developed for solving two-dimensional problems in the theory of conical shells is used to analyze the stress-strain state of shells with different boundary conditions and thickness varying in two directions at constant mass. Numerical results are given in the form of plots and tables

An analytic nonlinear boundary-value solution is found and used in analysis of the precritical and postcritical stress states of a flexible long cylindrical shell with variable curvature and hinged longitudinal edges under nonuniform loading

The approach to the solution of a three-dimensional boundary-value stress problem for elastic hollow inhomogeneous cylinders of corrugated elliptic cross-section is proposed. The boundary conditions make it possible to separate variables along the length at the cylinder ends. It is proposed to include additional functions into the resolving system...

An approach is proposed to determine the upper and lower bounds of the critical load for flexible noncircular long cylindrical shells with clamped longitudinal edges under nonuniform loading. An exact analytical solution is designed. Its critical points are analyzed

An approach is developed to solve the two-dimensional boundary-value problems of the stress-strain state of conical shells with circumferentially varying thickness. The approach employs discrete Fourier series to separate variables and make the problem one-dimensional. The one-dimensional boundary-value problem is solved by the stable discrete-orth...

The paper proposes an approach to the stress-strain analysis of orthotropic open and closed cylindrical shells of variable thickness and noncircular cross section under various loading and boundary conditions. As an example, the circumferential distribution of deflection and tangential force in shells with elliptic cross section is analyzed for som...

An approach is proposed to solve static problems for corrugated nonthin cylindrical shells applying spline-approximation in the longitudinal direction and a stable numerical method in the circumferential direction. Solutions are presented in the form of plots and tables for isotropic and transversely isotropic shells of constant and variable thickn...

An approach is proposed to solve boundary-value stress-strain problems for cylindrical shells with thickness varying in two coordinate directions. The approach employs discrete Fourier series to separate circumferential variables. This makes it possible to reduce the problem to a one-dimensional one, which can be solved by the stable discrete-ortho...

The bending problem for laminated orthotropic trapezoidal plates of variable thickness is solved by parametrizing the domain of interest and reducing the two-dimensional boundary-value problem to one-dimensional using the spline-collocation method. The resulting problem is solved by the stable discrete-orthogonalization method. The influence of ort...

The stress problem for corrugated hollow transversely isotropic cylinders is solved in three-dimensional formulation for certain end conditions. Discrete Fourier series are used to make the problem one-dimensional, which is then solved by the stable discrete-orthogonalization method. Examples of analysis are given

The stress problem for corrugated hollow transversely isotropic cylinders is solved in three-dimensional formulation for certain end conditions. Discrete Fourier series are used to make the problem one-dimensional, which is then solved by the stable discrete-orthogonalization method. Examples of analysis are given.

The bending problem for laminated orthotropic trapezoidal plates of variable thickness is solved by parametrizing the domain of interest and reducing the two-dimensional boundary-value problem to one-dimensional using the spline-collocation method. The resulting problem is solved by the stable discrete-orthogonalization method. The influence of ort...

The stress-strain state of elliptic cylindrical shells under loads nonuniformly distributed over a portion of the surface along the directrix was analyzed. The two-dimensional boundary-value problem describing the stress-strain state of the class of shell was solved using an approach based on approximating the solution along the generatrix by splin...

The influence of corrugation frequency and amplitude on the displacement and stress fields is analyzed based on an approach to solving stress problems for corrugated elliptical cylinders with certain end conditions.

The influence of varying thickness at constant mass on the stress–strain state of orthotropic cylindrical shells with elliptic cross-section is analyzed by solving two-dimensional boundary-value problems

The paper proposes an approach to solving three-dimensional stress problems for hollow orthotropic cylinders with noncircular cross section and certain end conditions. The approach is based on the method of separation of generatrix and directrix variables, discrete Fourier series, and the stable method of discrete orthogonalization. The results of...

An approach is developed to solve stress–strain problems in a refined formulation for orthotropic cylindrical shells of variable thickness and noncircular cross section. It is shown, as an example, how the distributions of deflections and stresses depend on changes in the shell thickness at constant weight

The paper presents an approach based on three-dimensional elastic equations to solve boundary-value stress problems for hollow cylinders with corrugated elliptical cross section. Discrete Fourier series are used to make the problem one-dimensional and then to solve it by the stable discrete orthogonalization method. Solutions for cylinders of diffe...

Hybrid equilibrium finite elements based on the direct approximation of the domain stress and boundary displacement fields are presented. The structure is divided into a far field, which is considered as an infinite super element, and a near field, which is in turn discretized into finite elements. The displacements in the domains of typical finite...

The stress–strain state of biconvex laminated orthotropic shells is analyzed against the degree of shallowness and the parameters of orthotropy. Numerical values of deflections and stresses are obtained by solving two-dimensional boundary-value problems using spline-functions and the discrete-orthogonalization method. The effect of the rise of shel...

An approach developed earlier to solve two-dimensional static problems for noncircular cylindrical shells is used to analyze the effect of the spatial frequency and amplitude of corrugation on the stress–strain state of shells. The results of the analysis are presented in the form of plots and tables

The approach to the solution of the boundary-value problems of bending of elastic rectangular plates of variable thickness is presented. It is proposed to introduce into the resolving system of partial differential equations additional functions which enables the variables to be formally separated and the problem to be reduced to a unidimensional o...

The exact analytical solution of the deformation problem for a flexible long noncircular cylindrical shell is used to analyze how a geometrical parameter characterizing the wall thickness and curvature of the shell affects its behavior when its longitudinal edges are rigidly clamped. Plots of the deflection-load relationship are presented for vario...

An analysis of the stress-strain state of biconvex layered orthotropic shells is carried out depending on slope rate and orthotropy parameters. The numerical values of deflections and stresses are obtained on the base of 2D boundary value problems solving using spline-functions and the discrete orthogonalization numerical method. An influence of th...

On the base of developed approach to solving the two-dimensional problems of statics of noncircular cylindrical shells, the investigation of influence of the change of frequency and amplitude of corrugation on the stress-strain state of shells is carried out. The results of calculation are given on plots and tables. The maximum bonding moment behav...

On the base of constructed exact analytical solution of the problem on flexible noncircular long cylindrical shell deforming, the analysis of an influence of characterized the thin-wallity and curvature geometrical parameter on a shell behavior over all area at the clamped support of longitudinal edges is carried out. The graphs are given for a dep...

On the basis of the constructed exact analytical solution of a nonlinear boundary value problem, the influence of a characterized the thickness and curvature generalized geometric parameter on deforming in pre-critical and post-critical areas the infinitely long non-circular cylindrical shell is studied for non-uniformly distributed along a cross s...

A number of approaches to the solution of stress problems for anisotropic inhomogeneous shells in the classical formulation are discussed. A review is made of approaches to the solution of one- and two-dimensional static problems for thin shells with variable parameters and to the solution of stress–strain problems for anisotropic shells of revolut...

By solving the boundary-value stress–strain problems for cylindrical shells with an elliptical cross section and a thickness varying along the directrix, the effect of variability in the thickness and load on their deformation is studied. Tables and plots present the deflection and the bending moment calculated under various boundary and loading co...

The stress problem is solved for inhomogeneous laminated hollow elliptical cylinders depending on their ellipticity and the rigidity of the middle layer. Discrete Fourier series are used. Illustrative calculations are carried out.

An approach developed to solve boundary-value stress problems for noncircular cylinders is used to analyze the stress state of elliptical cylinders in the case where their eccentricity and thickness change under uniform and local loads.

The effect of changed curvature and load distribution on the deflection–load relationship is studied based on the exact analytical solutions of the nonlinear problem on the deformation of a flexible long noncircular cylindrical shell with clamped and hinged edges under a nonuniform normal load. Graphs show how changes in the curvature and load para...

The stress–strain state of rectangular plates with a variable thickness and constant weight is analyzed. The laws of variation in the thickness are specified to depend on parameters so that the plate weight remains constant for any of their values. To solve the problem, discrete Fourier series are used, which makes it possible to reduce a two-dimen...

The stress problem for corrugated hollow cylinders is solved in a three-dimensional formulation. Use is made of end conditions that make the problem two-dimensional. By applying discrete Fourier series, the problem is made one-dimensional and then is solved by the stable numerical method of discrete orthogonalization. The stress state of the cylind...

The problem on the stress-strain state of layered cylindrical shells with bottoms of intricate shape under the action of internal pressure is considered. The elastic system examined is formed by spiral-circular winding. Two variants of the shell bottom structure are investigated. In the first variant, one spiral layer is installed, which leads to g...

An approach is proposed to solve three-dimensional stress problems for noncircular hollow cylinders. The end conditions are such that the problem can be reduced to a two-dimensional problem. This problem is reduced to a one-dimensional problem by introducing additional functions into the resolvable system of equations. These functions are determine...

The problem solving is shown and the study of stress state of hollow inhomogeneous layered elliptical cylinders is carried out depending on their ellipticity ratio and rigidity of the middle layer. Solving the problem is linked with using the discrete Fourier series. Calculation examples are given.

The stress-strain of rectangular plates was investigated when their thickness is changing and weight is conserved. The laws of a thickness change are specified. The depend on parameters in such a manner that at different parameter values the weight of plates is conserved. For the problem solving, the discrete Fourier series are used that permits to...

On the base of solving the boundary-value problems on the stress-strain state of cylindrical shells with an elliptical cross-section, thickness of which is changed along the directrix, the study of influence of the thickness change and loading on their deformations is carried out. In the tables and plots, the results of the deflection and bending m...

The study of an influence of the curvature variations and load distribution on the dependence of an deflection on loading is conducted on the basis of exact analytical solutions of a nonlinear problem about the deformation of a flexible long non-circular cylindrical shell under action of the non-uniform normal load and the hinge supported edges. It...

Certain approach to solving the problems of the stress state of noncircular hollow cylinders is suggested in the 3D statement, at certain conditions on edges that allows for reducing the problem to the 2D statement. This problem is reduced to 1D problem by introducing the additional functions into the solvable system of equations that are determine...

A refined theory of flexible layered shells with orthotropic layers of variable thickness is considered. The theory assumes that each layer has a local rotation angle due to lateral shear. This makes it possible to derive equations whose order does not depend on the number of layers. The basic equations and calculation results are presented for a t...

The nonlinear problem on deformation of a hinged flexible long noncircular cylindrical shell under nonuniform loading is solved exactly. The solution consists of two relations in terms of elementary functions. Plots are presented

A stress analysis of the teeth periodontium is carried out. The periodontium has the form of a thin-walled elastic shell whose thickness varies along the generatrix. The problem is solved by invoking the theory of elasticity, theoretical mechanics, and spline functions that describe the lateral surfaces of the periodontium. Plots show how the chang...

The effect of inhomogeneity of elastic properties in the circumferential direction on the distribution of stress and displacement fields in orthotropic cylindrical panels is studied. The mechanical properties of the panels and the load acting on them are constant in the axial direction, which makes it possible to neglect the influence of the curvil...

The exact solution is constructed to a nonlinear problem on subcritical and postcritical deformation of a flexible long cylindrical shell with a noncircular cross-section and fixed periphery under uniform loading. The solution is represented by two relations in terms of elementary functions. The graphs demonstrate how the deflection behaves dependi...

The accurate solution of the nonlinear problem on subcritical and postcritical deformation of an elastic long cylindrical shell with a noncircular cross-section and fixed contours under uniform loading is obtained. The solution is given in the form of two relations expressed in terms of elementary function. The manner in which the value of deflecti...

A certain variant of the refined theory of flexible layered shells of orthotropic layers with variable thickness is considered. Presence in every layer of local rotation angles caused by lateral shear is taken into consideration, what permits to obtain the equations with the order that does not depend on the number of layers. The basic equations an...

The paper deals with analysis of the stress state of a teeth periodont which looks like a thin-walled elastic shell with variable thickness along a generatrix. Solution of the problem is based on theory of elasticity and theoretical mechanics with use of spline-functions which describe the lateral surfaces of the periodont. The effect of change in...

A heterogeneous mathematical model is formulated. It permits us to use simultaneously the equations of the theories of elasticity and Timoshenko-type shells to describe different fragments of a structure. This model can be written as a closed system of differential equations of different dimensions with boundary conditions on the domain boundary an...

Some approaches to the solution of problems on the elastic deformation of thin-walled solids with a complex shape are analyzed on the basis of linear and geometrically nonlinear models. The general characteristic of the classical approaches to the solution of the problems is discussed. Approaches employing new classes of surfaces are considered. So...

An approach to solving the problem of statics of shallow orthotropic shells is suggested. The approach is based on the reduction of the two-dimensional boundary value problem to an one-dimensional one using the method of spline-collocation and the last problem solution with the help of the stable numerical method of discrete orthogonalization. Ther...

An approach to the solution of problems on the deformation of locally loaded orthotropic shells of revolution is proposed. Some analytical transformations are carried out to reduce the initial resolving system of partial differential equations to the form where the constant terms representing the surface loads are continuous functions of the circum...

The analysis of some approaches to solution of the problems on elastic deformation of the complicated shape thin-walled solids on the basis of linear and geometrically non-linear models is given. Basic characteristic of the classical approaches to the problem solutions is discussed. The approaches with the use of new classes of surfaces are conside...

In the three-dimensional formulation we study a class of problems involving the stressed state of an axisymmetrically heated anisotropic cylinder arbitrarily inhomogeneous over the thickness taking account of the dependence of mechanical characteristics on the temperature. The solution of the boundary-value problems is carried out numerically. We s...

The stress-strain state of noncircular shells has been analyzed based upon solution of boundary value static problems of variable shell thickness using the methods of spline-collocation and discrete orthogonization. Shell weight was preserved depending on the degree of change in the thickness and the value of the eccentricity. Three classes of prob...

A procedure is proposed for solving two-dimensional boundary-value problems on the stress-strain state of open and closed noncircular cylindrical shells of variable thickness under surface loads. The solution is based on the use of the spline-collocation method along the directrix and the method of discrete othogonalization along the generatrix. Ex...