System Of Differential Equilibrium Equations Research Articles

On the Existence of Solutions of Nonlinear Boundary Value Problems for a System of Differential Equilibrium Equations for Timoshenko-Type Shells in Isometric Coordinates

We prove the existence of solutions of a boundary value problem for a system of five nonlinear second-order partial differential equations with given nonlinear boundary conditions, which describes the equilibrium state of elastic shallow inhomogeneous isotropic shells with free edges in the Timoshenko shear model referred to isometric coordinates. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in the Sobolev space, the solvability of which is established using the contraction mapping principle.

On the Existence of Solutions of Nonlinear Boundary Value Problems for a System of Differential Equilibrium Equations for Timoshenko-Type Shells in Isometric Coordinates

On the Existence of Solutions of Nonlinear Boundary Value Problems for a System of Differential Equilibrium Equations for Timoshenko-Type Shells in Isometric Coordinates

Analytical, numerical and experimental study of the finite inflation of circular membranes

In the present work we derive an analytical expression for the pressure–deflection curve of circular membranes subjected to inflation. This problem has been studied mostly from a numerical point of view and there is still a lack of accurate closed-form solutions in nonlinear elasticity. The analytical formulation is developed with a semi-inverse method by setting a priori the kinematics of deformation of the membrane. A compressible Mooney–Rivlin material model is considered and a pressure–deflection relation is derived from the equilibrium. The kinematics is approximated and therefore the obtained solution is not exact. Consequently, the formulation is adjusted by introducing an additional polynomial function in the pressure–deflection equation. The polynomial is calibrated by fitting numerical solutions of the exact system of differential equilibrium equations. The calibration is done over a wide range of constitutive parameters that covers the response of all rubber materials for technological applications. As a result, a definitive and accurate expression of the applied pressure as a function of the deflection of the membrane is obtained. The formula is validated with finite element (FE) simulations and compared with other solutions available in the literature. The comparison shows that the present model is more accurate. In addition, unlike the other models, it can be applied to compressible materials. Experimental uniaxial and bulge tests are carried out on rubber materials and the model proposed is used to characterize the Mooney–Rivlin constitutive parameters. Since the pressure–deflection formula is accurate and easy-to-use, it is an innovative tool in engineering applications of inflated membranes.

Modeling and calculation of a pipeline section repaired on the basis of composite-coupling technology

The issues of modeling and calculation of a section of a main pipeline reconstructed by a composite coupling are considered. Such a section is a two-layer composite cylindrical structure. At the heart of the construction а model are provisions and the hypothesis of the classical theory of shells (variant of L. I. Balabukh — V. I. Novozhilov) and the theory of deformation of composite structures (A. R. Rzhanitsyn). The aim of the work is mathematical modeling and study of the tensely deformed state (TDS) in the field of installation of the coupling. The problem of calculating a section of a pipeline repaired in accordance with the composite-coupling technology (CCT), which is under the action of internal hydrostatic pressure, is considered. A mathematical model of deformation of a two-layer circular cylindrical shell is presented in the form of a system of differential equilibrium equations and boundary conditions. Shear tension in the interlayer seam are taken into account.

Numerical calculation of elements of thin-walled structures under alternating loading taking into account damage

The statement of the problem and the algorithms for calculating thin-walled rods of arbitrary section under spatially variable loading based on the theory of small elastoplastic deformations and the refined theory of rods are presented. Using the variational Lagrange principle, mathematical models are developed for the deformation and damage of thin-walled rods in a cylindrical coordinate system. A system of differential equilibrium equations for a rod with spatially variable loading in vector form is obtained. To solve the boundary-value problem, the central difference scheme of the second order of accuracy and the matrix sweep method are used. An example of calculation is given.

Deformation of a Three-Layer Rod with a Compressible Core in a Neutron Flow*

We studied the deformation of a three-layer elastoplastic rod with a compressible core in a neutron flow. To describe the kinematics of the sandwich asymmetric across the thickness, the broken line hypothesis are adopted: Bernoulli’s hypothesis for the thin face layers and Tymoshenko’s hypothesis with linear approximation of displacements over the thickness for the compressible core. The resistance of the core in the tangential direction is taken into account. The constitutive stress–strain relations correspond to the theory of small elastoplastic deformations. The system of differential equilibrium equations is derived using the variational method. The kinematic boundary conditions of the rod is free support of the ends by fixed rigid supports. The boundary-value problem is solved by determining four functions of deflections and longitudinal displacements of the midsurfaces of the face layers. The analytical solution is obtained on the basis of the method of elastic solutions. Its numerical analysis is carried out in the case of an evenly distributed load.

Mathematical model of the stress-strain state of belt with the load of tubular belt conveyer

The article developed a mathematical model of the stress-strain state of a tubular conveyor belt filled with bulk load. In this case, the belt is considered as a thin elastic inextensible cylindrical shell, and the bulk load in the belt is in the limit state. A system of differential equilibrium equations for a tubular belt with a bulk load with respect to forces and bending moments in a belt was obtained, which, when simplified, was reduced to a fourth-order differential equation for belt deflections. Based on this mathematical model, analytical dependencies of the deflections of the tubular conveyor belt on the parameters of the conveyor, the radius and properties of the belt, as well as the properties of the bulk load are obtained and analyzed. As a result, the maximum allowable distance between the roller supports of the tubular conveyor is determined. It was found that the allowable distance between the roller bearings is directly proportional to the tension of the belt and inversely proportional to the square of the radius of the belt and the bulk weight of the load. The research results can be used in the design of tubular belt conveyors transporting bulk load.

SOLUTION OF THE MAIN DIFFERENTIAL EQUATION OF DEFORMATIONS OF THE CASING IN A CURVED WELL

In a curvilinear well, the casing functions as a long continuous rod. It is installed on the supports-centralizers and replicates the complex profile of the well, as a result of which it receives large deformations. To describe them, a system of differential equilibrium equations of internal and external forces and moments was composed, which was supplemented to a closed form with a differential equation of curvature. It is non-uniform, because it takes into account the own distributed weight of the rod. Two ways are proposed to solve the problem: by the method of mathematical compression of the system equations into a complex inhomogeneous differential equation or by projecting the equilibrium equations of forces on the global (vertical-horizontal) and on the local (tangent-normal) coordinate systems. It is shown that the first integral of the system can also be found from the equilibrium equations of a portion of a curved rod of finite length. This integral has the form of a second-order inhomogeneous differential equation with variable coefficients and is the main equation that describes the deformation of a long elastic rod under the action of the longitudinal and transverse components of the forces of distributed weight. The main requirement of the technology is the installation of a pipes column on the centering supports, the purpose of which is to ensure the coaxiality of the pipes and the borehole walls and the creation between them a cement ring of the same thickness and strength. Accounting for this requirement allowed us to linearize the main equation. Its solution is the clue to the formulas of deflections, angular slopes, internal bending moments and transverse forces in the rod with the arbitrary arrangement of supports and boundary conditions in their intersections. The solution of the main differential equation of angular deformations of a long bar is found in the form of a linear combination of Airy and Scorer’s functions and in the form of three linearly independent polynomial series in the sum with a partial solution. The obtained formulas of flexure and power parameters allow us to calculate stress and strain in the pipes column during the process of casing the borehole of an arbitrary profile which increases the reliability and durability of the well.

CALCULATION OF STRETCHING AND BENDING STRESS IN A CASING STRING INSTALLED IN A WELL WITH A COMPLEX PROFILE

The casing column in the curvilinear well is represented as a long solid elastic rod. It has a vertically actingweight, which is evenly distributed along the length and creates variable axial tensile forces in the column body. At the same time, it is influenced by the reaction forces of the borehole walls, which, together with the weight, bend the column of initially straight pipes. It is assumed that the casing axis replicates the axis of the bent borehole, and the walls reaction is continuously distributed along the length according to a certain law, which, together with the weight, bends the column of the initially straight pipes. A system of differential equilibrium equations of internal and external forces and moments was composed. This system was supplemented to a closed form with the differential equation of curvature. This system describes large deformations of a long elastic rod in one plane. The introduction of the distributed weight, wall reactions and resistance forces into the calculation makes it non-uniform. Its feature is the need to solve the inverse problem. In this case the external load and rod deformations defined by the well shape in the form of an inclinometric data table are known. The unknown internal forces and the function of the walls reaction which creates its predetermined shape must be determined. It is established that this function depends on the distribution of axial forces caused by weight and resistance forces. As a result the system was reduced to a linear inhomogeneous differential equation with variable coefficients of the first order as to the axialforce and its solution was obtained as a sum of integrals. It is shown that one of them can be found in quadratures only in the case of a constant radius of curvature of the well. This necessitated the use of numerical integration methods. Formulas for the distribution of axial forces and bending moments in the body of the column, as well as the reactions of the walls leading the column to the actual well profile are obtained from the solution of the basic equation. To calculate these force factors, a method for numerical integration of inclinometric measurements data and software for numerical analysis of a real well are developed. This technique allows to detect the areas of local increase of the curvature and difficult passage of the curvilinear wellbore and to calculate the main parameters of the stress-strain state of the casing column in it.

НАПРЯЖЕННО-ДЕФОРМИРОВАННОЕ СОСТОЯНИЕ СФЕРИЧЕСКОЙ ОБОЛОЧКИ ПОД ДЕЙСТВИЕМ ПРОИЗВОЛЬНОЙ НАГРУЗКИ НА ОСНОВЕ НЕКЛАССИЧЕСКОЙ ТЕОРИИ

Considered is the stress state of an isotropic spherical shell exposed to an arbitrary load based on a non-classical theory. When building a mathematical model of the shell, three-dimensional equations of the theory of elasticity are applied. Displacements are represented in the form of polynomials along the coordinate normal to the middle surface two degrees higher relative to the classical theory of the Kirchhoff-Love type. As a result of minimization of the refined value of the Lagrange energy functional, a system of differential equilibrium equations in displacements and natural boundary conditions are obtained. The task of reducing two-dimensional equations to ordinary differential equations is carried out by decomposing the components of displacements and external loads into trigonometric series in the circumferential coordinate. Displacements are represented in form of polynomials along the coordinate normal to the middle surface by two degrees higher relative to the classical theory of the Kirchhoff-Love type. Resulting from minimization of the refined value of the Lagrange energy functional, a system of differential equilibrium equations in displacements and natural boundary conditions are received. The task of reducing two-dimensional equations to ordinary differential equations is carried out by decomposing the components of the displacements and the external loads into trigonometric series as per the circumferential coordinate. The formulated boundary problem is solved by the methods of finite differences and matrix sweep. As a result, displacements are obtained in the grid nodes, for approximation of which splines are used. The shell deformations are found using geometric relationship; tangential stresses are received from the correlations of Hooke's law. One of the features of this paper lies in the fact that the transverse stresses are determined by the direct integration of the equilibrium equations of the three-dimensional theory of elasticity. An example of the calculation of a hemispherical shell rigidly restrained along the lower base contour is brought. The shell is exposed to the wind load. Comparison of the results received by the refined theory with the data of the classical theory has shown that in the zone of distortion of the stressed state, the normal tangential stresses are substantially revised and the transverse normal stresses, which are neglected in the classical theory, are of the same magnitude with the maximum values of the main bending stress. Considered is the influence of the relative thickness on the stress state of the shell. It was discovered that the shell thickness significantly increases the error of the classical theory, while determining the stresses and assessing the strength of the elements of the aircraft structures.

Determination of the maximum allowable distance between the roller conveyors of a tubular belt conveyor

The article developed a mathematical model of the stress-strain state of a tubular conveyor belt filled with bulk load. In this case, the belt is considered as a thin elastic inextensible cylindrical shell, and the bulk load in the belt is in the limit state. A system of differential equilibrium equations for a tubular belt with a bulk load with respect to forces and bending moments in a belt was obtained, which, when simplified, was reduced to a fourth-order differential equation for belt deflections. Based on this mathematical model, analytical dependencies of the deflections of the tubular conveyor belt on the parameters of the conveyor, the radius and properties of the belt, as well as the properties of the bulk load are obtained and analyzed. As a result, the maximum allowable distance between the roller supports of the tubular conveyor is determined. It was found that the allowable distance between the roller bearings is directly proportional to the tension of the belt and inversely proportional to the square of the radius of the belt and the bulk weight of the load. The research results can be used in the design of tubular belt conveyors transporting bulk load.

Variable Thermal-Force Bending of a Three-Layer Bar with a Compressible Filler

Deformation of a three-layer elastoplastic bar with a compressible filler in a temperature field is considered. To describe the kinematics of a pack asymmetric across its thickness, the hypothesis of broken line is accepted, according to which the Bernoulli hypothesis is true in thin bearing layers, and the Timoshenko hypothesis is valid for a filler compressible across the its thickness, with a linear approximation of displacements across the layer thickness. The work of filler in the tangential direction is taken into account. The physical stress-strain relations correspond to the theory of small elastoplastic deformations. Temperature variations are calculated from a formula obtained by averaging the thermophysical properties of layer materials across the bar thickness. Using the variational method, a system of differential equilibrium equations is derived. On the boundary, the kinematic conditions of simply supported ends of the bar are assumed. The solution of the boundary problem is reduced to the search for four functions, namely, deflections and longitudinal displacements of median surfaces of the bearing layers. An analytical solution is derived by the method of elastic solutions with the use of the Moskvitin theorem on variable loadings. Its numerical analysis is performed for the cases of continuous and local loads.

Non-classical effects in FRP composite tubes

This paper presents a formulation of Generalised Beam Theory (GBT) developed to analyse the influence of non-classical effects on the structural behaviour of FRP composite circular hollow section members. Unlike other existing beam theories, the present GBT formulation incorporates non-classical effects comprising (i) elastic material couplings, (ii) deformation of cross-section contour, (iii) warping deformation and (iv) shear deformation. With the purpose of solving the GBT system of differential equilibrium equations, a finite element formulation is briefly presented. In order to clarify the concepts involved in the formulated GBT and illustrate its application and capabilities, the linear ( 1 st order) behaviour of two composite members exhibiting elastic couplings is analysed and the results obtained are thoroughly discussed and compared with estimates available in the literature. It is shown that accurate solutions can only be achieved if these non-classical effects are incorporated in the analysis.

ON THE INFLUENCE OF MATERIAL COUPLINGS ON THE LINEAR AND BUCKLING BEHAVIOR OF I-SECTION COMPOSITE COLUMNS

This paper presents the incorporation of shear deformation effects into a Generalized Beam Theory (GBT) developed to analyze the structural behavior of composite thin-walled columns made of laminated plates and displaying arbitrary orthotropy. Unlike other existing beam theories, the present GBT formulation incorporates in a unified fashion (i) elastic coupling effects, (ii) warping effects, (iii) cross-section in-plane deformation and (iv) shear deformation. The main concepts and procedures involved in the available GBT are adapted/modified to account for the specific aspects related to the member shear deformation. In particular, the GBT fundamental equilibrium equations are presented and their terms are physically interpreted. An I-section is used to illustrate the performance of GBT cross-section analysis and the mechanical properties are explained in detail. With the purpose of solving the GBT system of differential equilibrium equations, a finite element formulation is briefly presented. Finally, in order to clarify the concepts involved in the formulated GBT and illustrate its application and capabilities, the linear (first-order) and stability behavior of three composite I-section members displaying non-aligned orthotropy are analyzed and the results obtained are thoroughly discussed and compared with estimates available in the literature.

GBT buckling analysis of pultruded FRP lipped channel members

The main aspects concerning an orthotropic second order generalised beam theory (GBT), previously developed by the authors to enables the structural analysis of thin-walled members displaying special orthotropy, are briefly presented and discussed in the first part of the paper. Special attention is paid to the symbolic and numerical computational aspects related to the implementation of the GBT equations to perform member linear stability analyses. For this purpose, a finite element formulation is developed, validated and applied to solve the eigenvalue problem associated with the GBT system of differential equilibrium equations. Then, the second order orthotropic GBT is employed to investigate the buckling behaviour of pultruded fiber reinforced plastic lipped channel members, namely columns, beams and beam-columns. In particular, with the objective of illustrating the concepts and procedures involved in the performance of a GBT analysis, a detailed in-depth study of simply supported lipped channel columns is presented, in which the relevant (local and global) buckling modes are identified and the corresponding bifurcation stress values are determined. In addition, the results of an investigation concerning the influence of the applied stress distribution, cross-section geometry, material properties and end support conditions on the member buckling behaviour are presented, which required the completion of a number of parametric studies. In these investigations, careful consideration was given to the occurrence and characterisation of local-plate, distortional and mixed flexural–distortional buckling modes.

Integration of the Equilibrium Equations for Inhomogeneous, Transversely Isotropic Plates

A system of differential equilibrium equations for inhomogeneous transversely isotropic plates is derived based on the Fourier series in terms of Legendre polynomials. It is assumed that Poisson's ratios are constant and the elastic moduli are linear functions of the transverse coordinate. A method of finding the general solution to the system of equations derived is set forth