scholarly journals Behavior of solution of the elasticity problem for a radial inhomogeneous cylinder with small thickness

2021 ◽  
Vol 6 (7 (114)) ◽  
Author(s):  
Natik Akhmedov ◽  
Sevda Akbarova
Keyword(s):  
Boundary Layer ◽  
Lateral Surface ◽  
Strain State ◽  
Theory Of Elasticity ◽  
Equilibrium Equations ◽  
Stress Strain ◽  
Small Thickness ◽  
Homogeneous Solutions ◽  
Stress Strain State ◽  
Inhomogeneous Cylinder

A non-axisymmetric problem of the theory of elasticity for a radial inhomogeneous cylinder of small thickness is studied. It is assumed that the elastic moduli are arbitrary positive piecewise continuous functions of a variable along the radius. Using the method of asymptotic integration of the equations of the theory of elasticity, based on three iterative processes, a qualitative analysis of the stress-strain state of a radial inhomogeneous cylinder is carried out. On the basis of the first iterative process of the method of asymptotic integration of the equations of the theory of elasticity, particular solutions of the equilibrium equations are constructed in the case when a smooth load is specified on the lateral surface of the cylinder. An algorithm for constructing partial solutions of the equilibrium equations for special types of loads, the lateral surface of which is loaded by forces polynomially dependent on the axial coordinate, is carried out. Homogeneous solutions are constructed, i.e., any solutions of the equilibrium equations that satisfy the condition of the absence of stresses on the lateral surfaces. It is shown that homogeneous solutions are composed of three types: penetrating solutions, solutions of the simple edge effect type, and boundary layer solutions. The nature of the stress-strain state is established. It is found that the penetrating solution and solutions having the character of the edge effect determine the internal stress-strain state of a radial inhomogeneous cylinder. Solutions that have the character of a boundary layer are localized at the ends of the cylinder and exponentially decrease with distance from the ends. These solutions are absent in applied shell theories. Based on the obtained asymptotic expansions of homogeneous solutions, it is possible to carry out estimates to determine the range of applicability of existing applied theories for cylindrical shells. Based on the constructed solutions, it is possible to propose a new refined applied theory.

scholarly journals STRESS-STRAIN STATE IN BOUNDARY LAYER OF THE CIRCULAR PLATES WITH VARIOUS THICKNESSES BASED ON THE REFINED THEORY

2020 ◽  
Vol 82 (1) ◽  
pp. 32-42
Author(s):  
Val.V. Firsanov ◽  
Q.H. Doan ◽  
N.D. Tran
Keyword(s):  
Boundary Layer ◽  
Stress State ◽  
Strain State ◽  
Three Dimensional ◽  
Refined Theory ◽  
Circular Plates ◽  
Equilibrium Equations ◽  
Stress Strain ◽  
Stress Strain State ◽  
Three Dimensional Elasticity

A variant of the refined theory on calculation of the stress-strain state of circular plates with symmetrically various thicknesses according to an arbitrary law in the radial direction was presented. Equations of the plate state were established by using the three-dimensional elasticity theory. The required displacements were approximately calculated according to upright direction to the middle plane by polynomials with two degrees higher than in the classical Kirchhoff - Love theory. The differential equation at equilibrium in displacements with various coefficients was obtained by using means of the Lagrange variational principle. The direct integration of the equilibrium equations in the three-dimensional elasticity theory was used to determine the transverse normal and shear stresses. Of an isotropic circular plate with changing in thickness by using the analyzing Fourier chain, the obtained differential equilibrium equations in displacements with variable coefficients containing supplement components and taking into account of the effect of thickness on the stress-strain state of the plate. Examples of calculating the stress state of a circular plate with a thickness varying according to linear and parabolic laws under the action of a uniformly distributed load were considered. The limited difference method was employed to solve the boundary value problem. Comparison results of the refined and classical theories were investigated. It is demonstrated that the study on the stress state in the zones of its distortion (compounds, local loading zones, etc.) should use a refined theory, since the additional corresponding stresses of the “boundary layer” type are of the same order with the values of the main (internal) stress state. This is important to increase the reliability of strength calculations of such elements of aircraft-rocket structures as the power housings of aircraft, their various transition zones and connections, as well as objects in various engineering industries.

Determination and prediction of the stress-strain state of pipeline, taking into account soil changes during operation

2020 ◽  
Vol 10 (4) ◽  
pp. 372-378
Author(s):  
Aydar К. Gumerov ◽  
◽  
Rinat M. Karimov ◽  
Robert М. Askarov ◽  
Khiramagomed Sh. Shamilov ◽  
...  
Keyword(s):  
Strain State ◽  
Mathematical Framework ◽  
Equilibrium Equations ◽  
Stress Strain ◽  
Optimal Method ◽  
Initial Stress State ◽  
Soil Response ◽  
Stress Strain State ◽  
Support Reaction ◽  
Key Factor

The key factor determining the strength, reliability, service life and fail-safe operation of the main pipeline is its stress-strain state. The purpose of this article is to develop a mathematical framework and methodology for calculating the stress-strain state of a pipeline section laid in complex geotechnical conditions, taking into account all planned and altitude changes and impacts at various points of operation, as well as during repair and after its completion. The mathematical framework is based on differential equations reflecting the equilibrium state of the pipeline, taking into account the features of the sections (configuration, size, initial stress state, acting forces, temperature conditions, interaction with soil, supports, and pipe layers). The equilibrium equations are drawn up in a curvilinear coordinate system – the same one that is used for in-pipe diagnostics. According to the results of the solution, all stress components are determined at each point both along the length of the pipeline and along the circumference of any section. At the same time, transverse and longitudinal forces, bending moments, shearing forces, pipeline displacements relative to the ground and soil response to displacements are determined. As an example, a solution is given using the developed mathematical framework. During the course of calculation, the places where the lower form of the pipe does not touch the ground and the places where the support reaction becomes higher than a predetermined limit are determined. A comparative analysis was accomplished, and the optimal method for section repair has been selected.

scholarly journals PECULIARITIES OF MODELING OF STRESS-STRAIN STATE OF PIPELINES THROUGH WHICH GAS-LIQUID MIXTURES WITH AGGRESSIVE COMPONENTS ARE TRANSPORTED

2019 ◽  
pp. 128-135
Author(s):  
A. P. Oliinyk ◽  
B. S. Nezamay ◽  
L. I. Feshanych
Keyword(s):  
Internal Pressure ◽  
Cross Sections ◽  
Strain State ◽  
Smoothing Splines ◽  
Liquid Mixtures ◽  
Equilibrium Equations ◽  
Stress Strain ◽  
Stress Strain State ◽  
Current State ◽  
The Given

The task of estimating the stress-strain state of pipelines through which gas-liquid mixtures with aggressive components are transported is considered, the purpose, object and object of research are established. The analysis of the current state of scientific and technical researches on the given subject is carried out, the circle of unresolved problems is revealed. The combined effect on the pipelines through which gas-liquid mixtures with aggressive components are transported stress – strained state change  is estimated by two models - the model for determining the change of the stress-strain state of the pipeline by data on the surface points certain set displacement   taking into account the quasi-stationarity of the process. The device uses interpolation smoothing splines and methods of differential geometry, 6 components of strain and stress tensors are determined. In order to substantiate the method of estimation of annular stresses at the wear of the pipeline walls due to the action of the aggressive components of the transported mixtures, systems of equilibrium equations for pipeline sections and for quasi-rectilinear sections with altered cross-section configuration have been derived. Boundaryt conditions for equilibrium equations are established. Calculation formulas for estimation of annular stresses arising under the action of internal pressure for sections with shape defects caused by the action of aggressive components are established. The results of calculations that allow to quantify the change of the most significant ring stresses arising in the pipeline material under the action of internal pressure in the pipeline cross sections, which were exposed to the aggressive components, are presented. It is assumed that the deformed sections are little different from the shape of the circle.

scholarly journals DETERMINATION OF STRESS-STRAIN STATE OF A THREE-LAYER BEAM WITH APPLICATION OF CONTACT LAYER METHOD

Vestnik MGSU ◽  
2016 ◽  
pp. 17-26 ◽  
Author(s):  
Vladimir Igorevich Andreev ◽  
Robert Alekseevich Turusov ◽  
Nikita Yur’evich Tsybin
Keyword(s):  
Experimental Data ◽  
Edge Effect ◽  
Strain State ◽  
Rectangular Cross Section ◽  
Contact Layer ◽  
Theory Of Elasticity ◽  
Stress Strain ◽  
Tangential Stresses ◽  
Stress Strain State ◽  
Layer Method

The article deals with the solution for the stress-strain state of a multilayer composite beam with rectangular cross-section, which is bended by normally distributed load. The intermolecular interaction between layers is accomplished by the contact layer, in which the substances of adhesive and substrate are mixed. We consider the contact layer as a transversal anisotropic medium with such parameters that it can be represented as a set of short elastic rods, which are not connected to each other. For simplicity, we assume that the rods are normally oriented to the contact surface. The contact layer method allows us to solve the problem of determining the concentration of tangential stresses arising at the boundaries between the layers and the corner points, their changes, as well as to determine the physical properties of the contact layer basing on experimental data. Resolving the equations obtained in this article can be used for the solution of many problems of the theory of layered substances. These equations were derived from the fundamental laws of the theory of elasticity and generally accepted hypotheses of the theory of plates for the general case of the bending problem of a multilayer beam with any number of layers. The article deals with the example of the numerical solution of the problem of bending of a three-layer beam. On the basis of this solution the curves were obtained, which reflect the stress-strain state of one of the layers. All these curves have a narrow area of the edge effect. The edge effect is associated with a large gradient tangential stresses in the contact layer. The experimental data suggest that in this zone the destruction of the samples occurs. This fact allows us to say that the equations obtained in this article can be used to construct a theory of the strength layered beams under bending.

scholarly journals STRESS-STRAIN STATE OF THE CONICAL SHELL OF VARIABLE THICKNESS BASED ON THREE-DIMENSIONAL EQUATIONS OF ELASTICITY THEORY

2020 ◽  
Vol 82 (2) ◽  
pp. 189-200
Author(s):  
Val.V. Firsanov ◽  
V.T. Pham
Keyword(s):  
Differential Equations ◽  
Classical Theory ◽  
Variable Thickness ◽  
Conical Shell ◽  
Strain State ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Median Surface ◽  
Stress Strain ◽  
Stress Strain State

The results of a study of the stress-strain state of a conical shell of variable thickness based on a non-classical theory are presented. The sought-for displacements of the shell are approximated by polynomials in the normal coordinate to the median surface two degrees higher in relation to the classical theory of the Kirchhoff-Love type. When developing the theory, the three-dimensional equations of the theory of elasticity, as well as Lagrange variational principle are used as the equation of the shell state. As the result of minimizing the specified value of the total energy of the shell, a mathematical model is constructed, which is a system of differential equations of equilibrium in the displacements with variable coefficients and the corresponding boundary conditions. Two cases are considered: the shell is under the action of symmetric and asymmetric loads. Two-dimensional equations are transformed to the system of ordinary differential equations by means of trigonometric sequences as per circumferential coordinate. To solve the formulated boundary value problem, finite difference and matrix sweep methods are applied. The calculations have been made by means of a computer program. After having determined the displacements, shell deformations and tangential stresses are found from geometric and physical equations, transverse stresses - from the equilibrium equations of the three-dimensional theory of elasticity. As an example, a conical shell rigidly restrained at the two edges, with asymmetrically varying thickness is considered. Compared are the results of the VAT calculations obtained as per the improved and classical theories. The significant contribution of additional stresses in the boundary zone to the total stress state of the shell is shown. The received results can be used in the strength and durability calculations and tests of machine-building facilities of various purposes.

scholarly journals The influence of pressure changes on the integrity of tanks for storage of petroleum products and toxic substances

2021 ◽  
pp. 31-36
Author(s):  
Taras Hlova ◽  
Mykhailo Semerak ◽  
Bogdanna Hlova ◽  
Mykola Mykhailyshyn
Keyword(s):  
Cylindrical Surface ◽  
Strain State ◽  
Petroleum Products ◽  
Theory Of Elasticity ◽  
Oil Products ◽  
Thin Plates ◽  
Toxic Substances ◽  
Stress Strain ◽  
Stress Strain State ◽  
Analytical Expressions

Tanks for the storage of oil products and toxic substances in warehouses are the main ones. They can be in the form of separate tanks or a group of tanks. The most widespread are vertical steel tanks with a stationary roof that a placed in open areas. The tanks heat up, and the intensity of evaporation of the oil product increases in case of fire. If there is a permanent roof, the pressure in the tank will increase. If the capacity of the breathing valves is less than the intensity of evaporation then there is a risk of explosion. Explosions in the tank often lead to the separation of the bottom, and the side cylindrical surface and the roof fly away instantly, spilling oil on neighboring tanks and the territory of the tank’s park. Then the combustion area increases intensively. The destruction of the integrity of the tank, due to the separation of the bottom, contributes to temperature and power stresses, the value of which increases with increasing temperature of their heating and increasing pressure, respectively. The values of temperature stresses are added to the power stresses caused by pressure, and when the critical value is reached, destruction occurs. We investigated the stress-strain state of a steel vertical tank for the storage of oil products and toxic substances. The analysis of the reasons for the occurrence of admissible pressure in the tank, which is the reason for the loss of its integrity, is carried out. Using the differential equation of a closed cylindrical shell, which is under the action of internal pressure, analytical expressions are obtained to find deformations and stresses in the side cylindrical surface and bottom. Were calculated axial and annular stresses for the tank of RVS-900. Based on the basic relations of the theory of elasticity of thin plates and shells analytical expressions of the stress-strain state of the cylindrical tanks are obtained for conditions for changing of pressure on their structural elements. It is shown that the greatest values of axial stresses are obtained on the surface of the connection of the cylindrical surface with the bottom. The researches results are presented graphically.

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

2019 ◽  
Vol 3 (122) ◽  
pp. 42-54
Author(s):  
Ruslan Vissarionovych Kyriia ◽  
Hryhorii Ivanovych Larionov ◽  
Mykola Hryhorovych Larionov
Keyword(s):  
Mathematical Model ◽  
Strain State ◽  
Order Differential Equation ◽  
Limit State ◽  
Conveyor Belt ◽  
Equilibrium Equations ◽  
Stress Strain ◽  
Stress Strain State ◽  
Bulk Load ◽  
Belt Conveyors

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.

Study of the stress–strain state of spongy bone around an implant under occlusal load

2020 ◽  
Vol 25 (5) ◽  
pp. 1172-1181
Author(s):  
Natela Zirakashvili
Keyword(s):  
Mathematical Model ◽  
Strain State ◽  
Theory Of Elasticity ◽  
Boundary Element Methods ◽  
Porous System ◽  
Spongy Bone ◽  
Stress Strain ◽  
Jaw Bone ◽  
Stress Strain State ◽  
Occlusal Load

The article studies the stress–strain state of the spongy bone of an implanted jaw. A spongy bone can be considered as a multiporous area with its fissures and pores as the most visible components of a double-porous system. The work studies the stress–strain state of the spongy jaw-bone near the implant, which is under occlusal load. A mathematical model of the problem is the contact problem of the theory of elasticity between the implant and the jaw-bone. The problem is solved by using the boundary element methods, which are based on the solutions of Flamant (BEMF) and Boussinesq’s (BEMB) problems. The cases of various lengths of an implant diameter are considered. Stress contours (isolines) in the jaw-bone are drawn and the results obtained by BEMF and BEMB for the different-diameter implants are compared.

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