Differential Equations Of Equilibrium Research Articles

Asymmetric tensor representations in micropolar continuum mechanics theories

Получены новые представления трехмерного асимметричного тензора напряжений и соответствующие им формы дифференциальных уравнений равновесия. Асимметричные теории механики деформируемого твердого тела по прежнему привлекают пристальное внимание в связи с необходимостью математического моделирования механического поведения современных материалов (например, ауксетиков с помощью теорий гемитропной микрополярной упругости). Исследование ограничивается только такими асимметричными тензорами второго ранга, для которых удается сохранить понятие о вещественных собственных значениях, но отказаться от взаимной ортогональности направлений главного триэдра. Обсуждается точная алгебраическая формулировка указанных условий асимметричности. В статье обобщаются тензорные представления симметричного тензора напряжений, основанные на естественном репере асимптотических направлений. Полученные результаты являются ярким свидетельством в пользу алгебраической «гиперболичности» симметричных и асимметричных тензоров второго ранга в трехмерном пространстве.

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.

Tensile Failure Mechanism of Antiseepage System for Slope with Solid Waste Underground Landfill

The antiseepage membrane applied to the slope of solid waste landfill often shows tensile failure in projects, which results in ineffective antiseepage system and serious environmental pollution. In order to ensure the practical performance of the antiseepage membrane, the tensile force of it was studied, and the settlement mechanical model of the interaction between landfill body, antiseepage membrane, and cushion was established after comprehensively considering the effects of dead weight, lateral pressure, settlement, and foundation boundary. The analytical solutions of the tensile force and displacement of the antiseepage membrane was calculated through differential equation of equilibrium. With general slag and ardealite slag as the research objects, the major parameters affecting the internal tensile force of the antiseepage membrane were analyzed and studied by the combination of numerical and theoretical methods. The results show that the internal tensile force of the antiseepage membrane is greatly affected by the parameters such as the membrane-slag interface friction angle, the membrane-cushion interface friction angle, the buried depth, and the single step height. The theoretical slope normal stress and membrane pull-up force are basically consistent with the numerical calculation results, which indicates that the theory is universally applicable to tackle the tensile failure of the antiseepage membrane in the solid waste landfill system.

ON MAXWELL'S STRESS FUNCTIONS FOR SOLVING THREE DIMENSIONAL ELASTICITY PROBLEMS IN THE THEORY OF ELASTICITY

The governing equations of three dimensional elasticity problems include the six Beltrami-Michell stress compatibility equations, the three differential equations of equilibrium, and the six material constitutive relations; and these are usually solved subject to the boundary conditions. The system of fifteen differential equations is usually difficult to solve, and simplified methods are usually used to achieve a solution. Stress-based formulation and displacement-based formulation methods are two common simplified methods for solving elasticity problems.This work adopted a stress-based formulation for a three dimensional elasticity problem. In this work, the Maxwell's stress functions for solving three dimensional problems of elasticity theory were derived from fundamental principles. It was shown that the three Maxwell stress functions identically satisfy all the three differential equations of static equilibrium when body forces were ignored. It was further shown that the three Maxwell stress functions are solutions to the six Beltrami-Michell stress compatibility equations if the Maxwell stress functions are potential functions. It was also shown that the Airy's stress functions for two dimensional elasticity problems are special cases of the Maxwell stress functions.

EFFECTS OF THE INITIAL DEFLECTION SHAPE ON THE MOMENT AMPLIFICATION FACTOR OF BEAM-COLUMNS

In the design of slender steel beam-columns, the moment amplification factor is used to estimate the maximum moment along with the longitudinal direction. While formulas for evaluating the factor have been presented on the basis of elastic or elastic-plastic analysis, the initial deflection of the column is not considered. The effect that the initial deflection on the strength and behavior of the column has been shown only when the initial deflection shape is half sine wave. This paper discusses the effect of the initial deflection shape on the value of the moment amplification factor by performing the analytical work. The analytical model is the hinged-end beam-column subjected to constant axial compressive force and end moments. First of all, the equilibrium differential equation which governs the problem is solved and the formula for calculating the bending moment is presented. In the parametric study, magnitude of initial deflection, initial deflection shape, axial load ratio, slenderness ratio and end moment ratio are selected as the parameters. In this paper, we discuss the effects of the amount of the initial deflection and the initial deflection shape.

GDQ computation for thermal vibration of thick FGM plates by using fully homogeneous equation and TSDT

The third-order shear deformation theory (TSDT) effects on the thick functionally graded material (FGM) plates with fully homogeneous equation under thermal vibration are investigated by using the generalized differential quadrature (GDQ) method. The nonlinear coefficient c1 term of displacement field of TSDT is used to derive the equations of motion for thermal vibration of thick FGM plates. The dynamic equilibrium differential equations with TSDT of FGM plates in terms of partial derivatives of displacements and shear rotations subjected to partial derivatives of thermal loads, mechanical loads and inertia terms are derived in matrix forms. Three parametric effects of varied-modified shear correction factor, environment temperature and FGM power law index on the thermal stress and center deflection of FGM thick plates with and without containing the nonlinear coefficient c1 term are investigated.

Bending of symmetrically sandwich beams and I-beams – Analytical study

The paper is devoted to simply supported sandwich beams and I-beams of symmetrical structure. Two types of loads are taken into account: three-point bending and uniformly distributed load. Moreover, two models of deformation of planar cross sections of these beams are also considered: zig-zag theory – classical “broken line” hypothesis and nonlinear theory – “polynomial” hypothesis. Based on the principle of stationary total potential energy the differential equations of equilibrium are obtained. The system of the equations is analytically solved for two types of loads and the deflections of the beams are calculated for exemplary beams with consideration of the shear effect. The results of the two models of deformation of planar cross sections are compared and specified in Tables.

Analysis of natural fibre composites for aerospace structures

PurposeThe main idea is the comparison between composites including natural fibres (such as the linoleum fibres) and typical composites including carbon fibres or glass fibres. The comparison is proposed for different structures (plates, cylinders, cylindrical and spherical shells), lamination sequences (cross-ply laminates and sandwiches with composite skins) and thickness ratios. The purpose of this paper is to understand if linoleum fibres could be useful for some specific aerospace applications.Design/methodology/approachA general exact three-dimensional shell model is used for the static analysis of the proposed structures to obtain displacements and stresses through the thickness. The shell model is based on a layer-wise approach and the differential equations of equilibrium are solved by means of the exponential matrix method.FindingsIn qualitative terms, composites including linoleum fibres have a mechanical behaviour similar to composites including glass or carbon fibres. In terms of stress and displacement values, composites including linoleum fibres can be used in aerospace applications with limited loads. They are comparable with composites including glass fibres. In general, they are not competitive with respect to composites including carbon fibres. Such conclusions have been verified for different structure geometries, lamination sequences and thickness ratios.Originality/valueThe proposed general exact 3D shell model allows the analysis of different geometries (plates and shells), materials and laminations in a unified manner using the differential equilibrium equations written in general orthogonal curvilinear coordinates. These equations written for spherical shells degenerate in those for cylinders, cylindrical shell panels and plates by means of opportune considerations about the radii of curvature. The proposed shell model allows an exhaustive comparison between different laminated and sandwich composite structures considering the typical zigzag form of displacements and the correct imposition of compatibility conditions for displacements and equilibrium conditions for transverse stresses.

Flexure Analysis of Laminated Plates Using Multiquadratic RBF Based Meshfree Method

The linear and nonlinear flexure analysis of laminated plates with twenty theories with the five variable higher order shear deformation theory (HSDT) is investigated using multiquadratic radial basis function based meshfree method. The mathematical formulation of the actual physical problem of the plate subjected to transverse loading is presented utilizing von Karman nonlinear kinematics. These non-linear governing differential equations of equilibrium are linearized using quadratic extrapolation technique. The different results for deflection and stresses are obtained for thin to a thick plate and compared with some published results. It is observed that some of the theories taken here are well suited for analysis of thin as well as a thick plate while some theories are suited only for thin plates.

Thermo-elastic analysis of multilayered plates and shells based on 1D and 3D heat conduction problems

Abstract The present work shows a generic 3D exact shell solution for the thermo-mechanical analysis of a heterogeneous group of one- and multi-layered isotropic, composite and sandwich structures. Plates, cylinders, cylindrical and spherical shells can be investigated using orthogonal mixed curvilinear coordinates. The 3D equilibrium equations for spherical shells automatically degenerate in those for simpler geometries. The elastic part of the proposed 3D model is based on a consolidated layer-wise exact solution which uses the exponential matrix method to solve the equilibrium differential equations through the thickness direction. The closed-form solution is obtained assuming simply-supported boundary conditions and harmonic forms for displacement and temperature fields. The temperature amplitudes are imposed at the top and bottom external surfaces in steady-state conditions. Therefore, the temperature profile can be evaluated through the thickness direction in three different ways: – calculation of the temperature profile via the steady-state 3D Fourier heat conduction equation; – evaluation of the temperature profile using the steady-state simplified 1D version of the Fourier heat conduction equation; – a priori assumed linear temperature profile through the entire thickness direction ranging from the bottom temperature value to the top temperature value. Once the temperature profile is defined at each thickness coordinate, it is considered as a known term in the 3D differential equilibrium equations. The obtained system consists in a set of non-homogeneous second order differential equilibrium equations which can be solved introducing appropriate mathematical layers. After a reduction to a first order differential equation system, the exponential matrix method is used to calculate both the general and the particular solutions. The effects of the temperature field on the static response of plates and shells are evaluated in terms of displacements and stresses. The proposed solution will be validated using reference results available in the literature. Then, new analyses will be presented for different thickness ratios, geometries, lamination schemes, materials and temperature values at the external surfaces. Results will demonstrate the importance in the 3D shell model of both the correct definition of the elastic part and the appropriate evaluation of the temperature profile through the thickness of the structure. .

Elastic buckling of a cylindrical panel with symmetrically varying mechanical properties – Analytical study

The subject of the paper is a cylindrical panel with symmetrically varying mechanical properties in the thickness direction. The nonlinear hypothesis of deformation of the straight line normal to the panel neutral surface is assumed. Based on the principle of stationary potential energy differential equations of equilibrium are obtained. The system of the equations is analytically solved and the critical loads of cylindrical panel are derived. Moreover, a simplified model of the cylindrical panel without shear effect is formulated. The results of critical loads for example structures are calculated and presented in Tables.

Flexural responses of nanobeams with coupled effects of nonlocality and surface energy

AbstractThis paper proposes a beam model with inclusion of surface and nonlocal effects. Beam‐bulk kinematics is formulated within the framework of Euler‐Bernoulli beam theory; Eringen nonlocal elasticity theory is employed to account for long‐range atomic interactions of a nanoscale beam; and Gurtin‐Murdoch surface elasticity theory is used to represent the size‐dependent effect inherent to a nanoscale beam. Surface‐layer balance equation between the surface and bulk stresses is treated in a consistent manner. Virtual displacement principle is exploited to consistently derive the governing differential equilibrium equation as well as the natural boundary conditions of the problem. The general form of beam bending solution is presented. Two numerical simulations employing the proposed beam model are conducted to study characteristics and behaviors of the nanobeam with various model parameters. The first simulation investigates the coupled effects of nonlocality and surface energy on transverse displacement and bending moment characteristics of nanowires under various support conditions. The second examines the influences of several model parameters as well as the size‐dependent effect on the effective bulk Young's modulus of the nanobeam.

A simplified analysis method for evaluating the horizontal deformation of adjacent single pile under tunneling

To analyze the horizontal response regularity of adjacent pile under tunneling, an analytical method is developed considering the shearing effect of pile to assess the horizontal responses of pile due to adjacent tunneling. Firstly, the pile is considered as a Timoshenko beam resting on the Pasternak foundation model, then, combining with the two-stage method, the additional unloading pressure due to the tunneling excavation is loaded on the pile, thus the equilibrium differential equation considering the pile’s shearing effect is established. Secondly, the specific horizontal deformation of pile is given by the finite difference method. Furthermore, through comparing with the boundary element method, the validity of this theory is verified.Finally, using the theory here, a comparison is made between the results based on the Timoshenko beam and Euler-Bernoulli beam, and find the results taking the shearing effect of pile into account are more close to the classical solution, the parameter analysis also suggests: the pile’s shearing effect is closely related with the disturbance, the stronger the disturbance, the more obvious the pile’s shearing effect.

Mathematical solutions for the flexural analysis of Mindlin’s first order shear deformable circular plates

In this work, the problem of first order shear deformable solid circular plate under transverse load was solved mathematically. The problem considered was assumed axisymmetric. The plate and loading were considered axisymmetric. The problem was defined as a boundary value problem of a system of differential equations of equilibrium in terms of the stress resultants and the stress – resultants – displacement relations. The set of equations were considered simultaneously to express them in variable separable form. The mathematical technique of separation of variables was then used to obtain solutions for the unknown generalised displacements. Specific problems of clamped edge plates and simply supported edge plates under uniformly distributed load and point load at the centre were considered and solved using the same technique of separation of variables. The mathematical expressions obtained showed that in all cases, the deflection was expressible in terms of flexural and shear components. The maximum deflection was found to occur at the plate centre as is expected from the symmetrical nature of the problem. The shear component of the transverse deflection was found to significantly increase with significant increase in the ratio of the plate thickness to the radius (h/r0).

Elastic deformation behavior of freestanding MoS2 films using a continuum approach

In this paper, an approximate solution based on the continuum mechanics is presented to obtain the mechanical response of freestanding single and two layers molybdenum disulfide (MoS2) films. Both circular and rectangular geometries as well as point load and uniform pressure are investigated. The partial differential equilibrium equations are established based on the non-linear Föppl membrane theory for both circular and rectangular films. The Galerkin method is then employed to solve the non-linear partial differential equilibrium equations in case of circular films. Also, the principle of virtual work affords another means of obtaining an approximate solution for rectangular film. Theoretical predictions for the width of membrane regions enable us to estimate the analytical deflection and stress distributions. Finally, the obtained results are compared to the available experimental and theoretical results and a good agreement is seen. It can be concluded that the continuum approach can be used to study the behavior of the MoS2 films.

Support Draft Calculation for a Ramp in the Form of Developable Helicoid

The article is about the analytical method of calculation a ramp in the form of developable helicoid on the support draft. The asymptotic method of small parameter is applied to solve the system of three differential equilibrium equations for developable helicoid stressstrain. The numerical results of displacements and bending moments are verified and coincide with engineering practice. The suggested approach can be extended for calculation of torso-helicoids with other boundary conditions.

Equivalent model of a dually-fed machine for electric drive control systems

The article shows that the mathematical model of a dually-fed machine is complicated because of the presence of a controlled voltage source in the rotor circuit. As a method of obtaining a mathematical model, the method of a generalized two-phase electric machine is applied and a rotating orthogonal coordinate system is chosen that is associated with the representing vector of a stator current. In the chosen coordinate system in the operator form the differential equations of electric equilibrium for the windings of the generalized machine (the Kirchhoff equation) are written together with the expression for the moment, which determines the electromechanical energy transformation in the machine. Equations are transformed so that they connect the currents of the windings, that determine the moment of the machine, and the voltages on these windings. The structural diagram of the machine is assigned to the written equations. Based on the written equations and accepted assumptions, expressions were obtained for the balancing the EMF of windings, and on the basis of these expressions an equivalent mathematical model of a dually-fed machine is proposed, convenient for use in electric drive control systems.

Effect of Initial Mechanical Anisotropy of the Pipe Blank in Expansion by a Conical Punch

The forces involved in the expansion of a pipe blank by means of a conical punch are analyzed by solving the approximate differential equilibrium equation, together with the corresponding fluidity condition. The influence of the initial mechanical anisotropy of the blank on the force and stress in the deformation region is established.

STUDYING THE EFFECT OF AN ADDITIONAL ACTIVE RESISTANCE IN THE FIELD WINDING CIRCUIT ON STARTING CHARACTERISTICS OF SALIENT-POLE SYNCHRONOUS MOTORS

On the basis of the developed method for calculating steady-state asynchronous operation modes of salient-pole synchronous motors, a procedure of mathematical modelling of the starting characteristics is proposed. The problem of calculating the steady-state asynchronous mode is solved as a boundary value one for differential equations of motor circuit electrical equilibrium. Algebraization of the system of differential equations is carried out by approximating the equations of state using cubic spline functions on a grid of period nodes, taking into account the periodic law of variation of the coordinates. This results in the changeover from continual values to nodal ones. The starting static characteristics are calculated using the parameter continuation method. The study of the effect of the starting resistance value on the asynchronous characteristics of the motor relied on a mathematical model of the motor taking into consideration real field circuits, saturation and asymmetry of the magnetic path.

In-plane non-linear elastic stability of parabolic arches with different rise-to-span ratios

Investigations of in-plane nonlinear elastic buckling and postbuckling of parabolic shallow arches have been reported in some literature, where a basic assumption that curve arc differential being equaled to horizontal line differential was used to facilitate the derivation. However, this assumption can be accepted only when the rise-to-span ratio of an arch is quite small, and when the rise-to-span ratio of an arch is greater than 1/12.5, the analytical predictions for buckling and postbuckling responses of the arch based on the assumption have significant deviations from the corresponding finite element results. It is known that parabolic arches having a rise-to-span ratio from 1/8–1/4 are often used in the engineering practice, especially in the arch bridges. This paper aims to improve accuracy of nonlinear analyses for parabolic arches with different rise-to-span ratios. A more precise strain expression that includes an approximate curve arc differential item in the Cartesian coordinate system is derived in this paper. Based on the strain expressions and principle of virtual work, nonlinear differential equilibrium equations are derived and more accurate analytical solutions for the buckling and postbuckling behavior are derived. Parabolic arches with the rise-to-span ratio ranging from 1/12.5–1/3 are taken as examples to investigate the effectiveness of the proposed method for prediction of the buckling and postbuckling behavior by comparisons with the predictions of a finite element method. The comparisons show that the solutions are in good agreements with the finite element results and that the new strain expression can lead to more accurate closed-form analytical solutions for the buckling and postbuckling behavior of parabolic arches with different rise-to-span ratios. It is found that the rise-to-span ratio has a significant influence on the in-plane nonlinear buckling behavior of parabolic arches.