SUBSTANTIATION FOR SIMPLIFICATION OF CALCULATION MODELS FOR ASSESSMENT OF THE STABILITY OF ROOF ROCKS

2021 ◽  
pp. 25-36
Author(s):  
Oleksandr Ahafonov ◽  
◽  
Daria Chepiga ◽  
Anton Polozhiy ◽  
Iryna Bessarab ◽  
...  
Keyword(s):  
Differential Equations ◽  
Coal Seam ◽  
Cross Section ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Unit Width ◽  
Distributed Load ◽  
The Stability ◽  
Roof Rocks ◽  
Calculation Models

Purpose. Substantiation of expediency and admissibility of use of the simplified calculation models of a coal seam roof for an estimation of its stability under the action of external loadings. Methods. To achieve this purpose, the studies have been performed using the basic principles of the theory of elasticity and bending of plates, in which the coal seam roof is represented as a model of a rectangular plate or a beam with a symmetrical cross-section with different support conditions. Results. To substantiate and select methods for studying the bending deformations of the roof in the coal massif containing the maingates, the three-dimensional base plate model and the beam model are compared, taking into account the kinematic boundary conditions and the influence of external distributed load. Using the theory of plate bending, the equations for determining the deflections of the coal seam roof in three-dimensional basic models under certain assumptions have a large dimension. After the conditional division of the plate into beams of unit width and symmetrical section, when describing the normal deflections of the middle surface of the studied models, the transition from the partial derivative equation to the usual differential equations is carried out. In this case, the studies of bending deformations of roof rocks are reduced to solving a flat problem in the cross-section of the beam. A comparison of solutions obtained by the methods of the three-dimensional theory of elasticity and strength of materials was performed. For a beam with a symmetrical section, the deflection lies in a plane whose angle of inclination coincides with the direction of the applied load. The calculations did not take into account the difference between the intensity of the surface load applied to the beam. Differences in determining the magnitude of the deflections of the roof in the model of the plate concerning the model of the beam reach 5%, which is acceptable for mining problems. Scientific novelty. To study the bending deformations and determine the magnitude of the roof deflection in models under external uniform distributed load, placed within the simulated plate, a strip of unit width was selected, which has a symmetrical cross-section and is a characteristic component of the plate structure and it is considered as a separate load-bearing element with supports, the cross-sections of this element is remained flat when bending. The deflection of such a linear element is described by the differential equations of the bent axis of the beam without taking into account the integral stiffness of the model, and the vector of its complete displacement coincides with the vector of the force line. Practical significance. In the laboratory, to study the bending deformations and their impact on the stability of the coal seam roof under external loads, it is advisable to use a model of a single width beam with a symmetrical section with supports, the type of which is determined by rock pressure control and secondary support of the maingate at the extraction layout of the coal mine.

A 3D Autonomous System with Infinitely Many Chaotic Attractors

2019 ◽  
Vol 29 (12) ◽  
pp. 1950166 ◽  
Author(s):  
Ting Yang ◽  
Qigui Yang
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Autonomous System ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Double Zero ◽  
Finite Dimensional ◽  
Periodic Attractors ◽  
The Stability ◽  
Autonomous Dynamical System

Intuitively, a finite-dimensional autonomous system of ordinary differential equations can only generate finitely many chaotic attractors. Amazingly, however, this paper finds a three-dimensional autonomous dynamical system that can generate infinitely many chaotic attractors. Specifically, this system can generate infinitely many coexisting chaotic attractors and infinitely many coexisting periodic attractors in the following three cases: (i) no equilibria, (ii) only infinitely many nonhyperbolic double-zero equilibria, and (iii) both infinitely many hyperbolic saddles and nonhyperbolic pure-imaginary equilibria. By analyzing the stability of double-zero and pure-imaginary equilibria, it is shown that the classic Shil’nikov criteria fail in verifying the existence of chaos in the above three cases.

scholarly journals Rotating 3D Flow of Hybrid Nanofluid on Exponentially Shrinking Sheet: Symmetrical Solution and Duality

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1637
Author(s):  
Liaquat Ali Lund ◽  
Zurni Omar ◽  
Sumera Dero ◽  
Dumitru Baleanu ◽  
Ilyas Khan
Keyword(s):  
Differential Equations ◽  
Transfer Rate ◽  
Three Dimensional ◽  
Rotation Parameter ◽  
Shrinking Sheet ◽  
Hybrid Nanofluid ◽  
3D Flow ◽  
Exponentially Shrinking Sheet ◽  
Symmetrical Solution ◽  
The Stability

This article aims to study numerically the rotating, steady, and three-dimensional (3D) flow of a hybrid nanofluid over an exponentially shrinking sheet with the suction effect. We considered water as base fluid and alumina (Al2O3), and copper (Cu) as solid nanoparticles. The system of governing partial differential equations (PDEs) was transformed by an exponential similarity variable into the equivalent system of ordinary differential equations (ODEs). By applying a three-stage Labatto III-A method that is available in bvp4c solver in the Matlab software, the resultant system of ODEs was solved numerically. In the case of the hybrid nanofluid, the heat transfer rate improves relative to the viscous fluid and regular nanofluid. Two branches were obtained in certain ranges of the involved parameters. The results of the stability analysis revealed that the upper branch is stable. Moreover, the results also indicated that the equations of the hybrid nanofluid have a symmetrical solution for different values of the rotation parameter (Ω).

The Calculation of the Quasi-Three-Dimensional Flow in an Axial Gas Turbine

1975 ◽  
Vol 97 (2) ◽  
pp. 283-294 ◽  
Author(s):  
S. Biniaris
Keyword(s):  
Numerical Solution ◽  
Cross Section ◽  
Three Dimensional ◽  
Entire Region ◽  
Three Dimensional Flow ◽  
Total Enthalpy ◽  
Blade Thickness ◽  
Meridional Cross Section ◽  
The Finite Difference Method ◽  
The Stability

The flow is calculated within the entire region from far upstream to far downstream of the blade rows and this not only between the blade rows but especially within the blade passages. It is assumed that the flow is steady, adiabatic, and inviscid. However, compressibility, blade forces in all directions, blade thickness, and total enthalpy gradients are taken into account. The shape of the meridional cross section can be arbitrary. The blades can be either cylindrical or twisted. The numerical solution is based on the finite-difference method. The discretization error, the stability error, and the iteration error of the numerical solution are determined.

scholarly journals Calculation of wooden beams on the stability of a flat bending shape enhancement

2018 ◽  
Vol 196 ◽  
pp. 01003 ◽  
Author(s):  
Anton Chepurnenko ◽  
Vera Ulianskaya ◽  
Serdar Yazyev ◽  
Ivan Zotov
Keyword(s):  
Differential Equation ◽  
Cross Section ◽  
Stability Problem ◽  
Secular Equation ◽  
Rectangular Cross Section ◽  
Center Of Gravity ◽  
Critical Force ◽  
Distributed Load ◽  
Matlab Package ◽  
The Stability

Flat bending stability problem of constant rectangular cross section wooden beam, loaded by a distributed load is considered. Differential equation is provided for the cases when load is located not in the center of gravity. The solution of the equation is performed numerically by the method of finite differences. For the case of applying a load at the center of gravity, the problem reduces to a generalized secular equation. In other cases, the iterative algorithm developed by the authors is implemented, in the Matlab package. A relationship between the value of the critical force and the position of the load application point is obtained. A linear approximating function is selected for this dependence.

REFINED BEAM THEORIES BASED ON A UNIFIED FORMULATION

2010 ◽  
Vol 02 (01) ◽  
pp. 117-143 ◽  
Author(s):  
ERASMO CARRERA ◽  
GAETANO GIUNTA
Keyword(s):  
Differential Equations ◽  
Cross Section ◽  
Cross Sections ◽  
Order Approximation ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Approximation Order ◽  
Form Solution ◽  
Geometrical Parameters ◽  
Beam Theories

This paper proposes several axiomatic refined theories for the linear static analysis of beams made of isotropic materials. A hierarchical scheme is obtained by extending plates and shells Carrera's Unified Formulation (CUF) to beam structures. An N-order approximation via Mac Laurin's polynomials is assumed on the cross-section for the displacement unknown variables. N is a free parameter of the formulation. Classical beam theories, such as Euler-Bernoulli's and Timoshenko's, are obtained as particular cases. According to CUF, the governing differential equations and the boundary conditions are derived in terms of a fundamental nucleo that does not depend upon the approximation order. The governing differential equations are solved via the Navier type, closed form solution. Rectangular and I-shaped cross-sections are accounted for. Beams undergo bending and torsional loadings. Several values of the span-to-height ratio are considered. Slender as well as deep beams are analysed. Comparisons with reference solutions and three-dimensional FEM models are given. The numerical investigation has shown that the proposed unified formulation yields the complete three-dimensional displacement and stress fields for each cross-section as long as the appropriate approximation order is considered. The accuracy of the solution depends upon the geometrical parameters of the beam and loading conditions.

scholarly journals STABILITY OF A CYLINDER FROM MURNAGHAN MATERIAL UNDER STRETCHING, COMPRESSION AND INFLATION

2019 ◽  
Vol 81 (1) ◽  
pp. 30-39
Author(s):  
M. I. Karyakin ◽  
L. P. Obrezkov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Three Dimensional ◽  
Deformation Characteristic ◽  
Small Strain ◽  
Stability Loss ◽  
Theory Of Elasticity ◽  
Equilibrium Equations ◽  
The Stability ◽  
Loading Parameter

The problem of equilibrium and stability of a hollow cylinder subjected to simultaneous uniaxial tension/compression and inflation is considered within the framework of the three-dimensional nonlinear theory of elasticity. To describe the mechanical properties of the material of the cylinder five-constant Murnaghan model is used. By the semi-inverse method the three-dimensional problem is reduced to the study of a nonlinear boundary value problem for an ordinary second-order differential equation. For most sets of material parameters known from the literature, the presence of a falling section in the stretching/inflation diagram, indicating the possible existence of instability zones even in the area of tensile stresses, has been found numerically. The stability analysis was carried out using a bifurcation approach based on linearization of the equilibrium equations in the neighborhood of the constructed solution by means of the method of imposing a small strain on a finite one. The value of a particular deformation characteristic, for which non-trivial solutions of a homogeneous boundary-value problem exist for the equations of neutral equilibrium obtained in the linearization process, was identified with the critical value of the loading parameter, i.e. value at which the system loses stability. As a rule, the coefficient of stretching/shortening of the cylinder and the coefficient of increase/decrease of its internal or external radius were chosen as such parameters. On the plane of the above-mentioned deformation characteristics the areas of stability under tension and compression, as well as under compression by external force and inflation by internal pressure, are constructed. The forms of possible of stability loss depending on the type of stress state are constructed, and the effect on the stability of material and geometric parameters is studied.

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.

Stability Loss in Thick Transversely Isotropic Cylindrical Shells Under Axial Compression

1993 ◽  
Vol 60 (2) ◽  
pp. 506-513 ◽  
Author(s):  
G. A. Kardomateas
Keyword(s):  
Axial Compression ◽  
Critical Loads ◽  
Shell Theory ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Stability Loss ◽  
Theory Of Elasticity ◽  
Dimensional Theory ◽  
Perfect Agreement ◽  
The Stability

The stability of equilibrium of a transversely isotropic thick cylindrical shell under axial compression is investigated. The problem is treated by making appropriate use of the three-dimensional theory of elasticity. The results are compared with the critical loads furnished by classical shell theories. For the isotropic material cases considered, the elasticity approach predicts a lower critical load than the shell theories, the percentage reduction being larger with increasing thickness. However, both the Flu¨gge and Danielson and Simmonds theories predict critical loads much closer to the elasticity value than the Donnell theory. Moreover, the values of n, m (number of circumferential waves and number of axial half-waves, respectively, at the critical point) for both the elasticity, and the Flu¨gge and the Danielson and Simmonds theories, show perfect agreement, unlike the Donnell shell theory.

scholarly journals DETERMINING THE PATTERNS OF STABILITY OF MINE WORKINGS FOR CALCULATION ROOF BOLTING PARAMETERS

2021 ◽  
pp. 9-16
Author(s):  
S. Barsukov ◽  
А.Т. Batyrkhanova ◽  
Vladimir Dyomin
Keyword(s):  
Coal Seam ◽  
Longwall Mining ◽  
Practical Significance ◽  
Roof Bolting ◽  
Pull Out ◽  
Deformation Processes ◽  
Mine Workings ◽  
The Stability ◽  
Host Rocks ◽  
Roof Rocks

Purpose.  The published studies are aimed at determining the mechanism of deformation of the rocks of the contour around the workings in terms of the parameters of the emerging fracturing and their dependence on the indicators of the strength of the rocks and the depth of occurrence in the massif. The tasks of the study include the installation of fracture indicators, the determination of the parameters of the development of the deformation process around the working, including the effect of longwall mining and taking into account the possibility of reuse of the workings. Methodology. To solve the set tasks, the method of field observations was used together with the use of regression dependencies to determine the dependences of the parameters on the influencing factors. In addition, the method of full-scale pull-out tests of anchor support was used, which made it possible to determine the clamping forces of the anchors. Originality. In the course of the research, the dependences of stresses and deformation along the K7 coal seam in the conditions of the mine named after Kuzembaev CD JSC "ArcelorMittal Temirtau" for the massif around the mine with fastening. Rational parameters for the use of roof bolting in preparatory mine workings have been established. This type of fastening provides direct contact between rocks and lining. Analysis of the results of calculating the parameters of the roof bolting showed that to maintain the roof in the development workings, it is necessary to take into account the parameters of the roof bolting. The main parameters include the length of the anchors, the total resistance of the roof bolting and the density of the anchors. Anchor support forms laminated rock beams in the roof rocks, which ensure the stability of the workings. Conclusions and practical significance. The results of studies devoted to the establishment of the influence of mining-geological and mining-technical factors on the formation of zones of inelastic deformation in the host rocks were considered. Significant dependences of the deformation processes of rocks in the massif around the workings were obtained, and the parabolic zone of destruction of rocks was determined. The practical significance of the research consists in determining the actual indicators of the required bearing capacity of the anchorage at two levels in the conditions of the development of the coal seam k7 of the Kuzembaev mine.  

Vibration of Thick Circular Disks and Shells of Revolution

2003 ◽  
Vol 70 (2) ◽  
pp. 292-298 ◽  
Author(s):  
A. V. Singh ◽  
L. Subramaniam
Keyword(s):  
Cross Section ◽  
Three Dimensional ◽  
Free Vibrations ◽  
Theory Of Elasticity ◽  
Published Data ◽  
Axially Symmetric ◽  
Shells Of Revolution ◽  
Displacement Fields ◽  
Wide Range ◽  
Circular Disks

A fully numerical and consistent method using the three-dimensional theory of elasticity is presented in this paper to study the free vibrations of an axially symmetric solid. The solid is defined in the cylindrical coordinates r,θ,z by a quadrilateral cross section in the r-z plane bounded by four straight and/or curved edges. The cross section is then mapped using the natural coordinates (ξ,η) to simplify the mathematics of the problem. The displacement fields are expressed in terms of the product of two simple algebraic polynomials in ξ and η, respectively. Boundary conditions are enforced in the later part of the solution by simply controlling coefficients of the polynomials. The procedure setup in this paper is such that it was possible to investigate the free axisymmetric and asymmetric vibrations of a wide range of problems, namely; circular disks, cylinders, cones, and spheres with considerable success. The numerical cases include circular disks of uniform as well as varying thickness, conical/cylindrical shells and finally a spherical shell of uniform thickness. Convergence study is also done to examine the accuracy of the results rendered by the present method. The results are compared with the finite element method using the eight-node isoparametric element for the solids of revolution and published data by other researchers.

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