Estimate of strength of anisotropic bars of arbitrary cross-section in the general case of their combined stress

We consider bars of arbitrary shape made of a homogeneous anisotropic material. In the general case, all six internal force factors (three forces and three moments) are simultaneously nonzero in the transverse cross-sections of the bar. We consider the case of small displacements and strains of the bar. Using the rigid-plastic model of a strained rigid body, the associated strain law, and the traditional hypotheses of static and kinematic character for the bars, we derive parametric equations for the limit surface (the strength surface) in the space of internal forces and moments acting in the the transverse cross-section. We present several versions of the obtained equations in specific cases (for orthotropy, transversal isotropy, and isotropy) and some numerical examples.     

On an inverse problem of bending of a physically nonlinear inhomogeneous plate

An elastic plate with a physically nonlinear inclusion of an arbitrary shape is considered. This plate is subjected to pure bending under the action of transverse forces and bending moments applied at the external boundary of the plate. There are no loads distributed over the surface. The problem of finding external actions that provide a necessary uniform moment state in the inclusion, i.e., prescribed constant moments and curvatures, is formulated and solved. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 104–107, September–October, 2007.     

Influences of aerodynamic loads on hunting stability of high-speed railway vehicles and parameter studies

The influences of steady aerodynamic loads on hunting stability of high-speed railway vehicles were investigated in this study.A mechanism is suggested to explain the change of hunting behavior due to actions of aerodynamic loads：the aerodynamic loads can change the position of vehicle system（consequently the contact relations）,the wheel/rail normal contact forces,the gravitational restoring forces/moments and the creep forces/moments.A mathematical model for hunting stability incorporating such influences was developed.A computer program capable of incorporating the effects of aerodynamic loads based on the model was written,and the critical speeds were calculated using this program.The dependences of linear and nonlinear critical speeds on suspension parameters considering aerodynamic loads were analyzed by using the orthogonal test method,the results were also compared with the situations without aerodynamic loads.It is shown that the most dominant factors a ff ecting linear and nonlinear critical speeds are different whether the aerodynamic loads considered or not.The damping of yaw damper is the most dominant influencing factor for linear critical speeds,while the damping of lateral damper is most dominant for nonlinear ones.When the influences of aerodynamic loads are considered,the linear critical speeds decrease with the rise of cross wind velocity,whereas it is not the case for the nonlinear critical speeds.The variation trends of critical speeds with suspension parameters can be significantly changed by aerodynamic loads.Combined actions of aerodynamic loads and suspension parameters also a ff ect the critical speeds.The effects of such joint action are more obvious for nonlinear critical speeds.     

Relationships of the Timoshenko-type theory of thin shells with arbitrary displacements and strains

A new modified version of the Timoshenko theory of thin shells is proposed to describe the process of deformation of thin shells with arbitrary displacements and strains. The new version is based on introducing an unknown function in the form of a rotation vector whose components in the basis fitted to the deformed mid-surface of the shell are the components of the transverse shear vector and the extensibility in the transverse direction according to Chernykh. For the case with the shell mid-surface fitted to an arbitrary non-orthogonal system of curvilinear coordinates, relationships based on the use of true stresses and true strains in accordance with Novozhilov are obtained for internal forces and moments. Based on these relationships, a problem of static instability of an isotropic spherical shell experiencing internal pressure is solved. The shell is considered to be made either of a linear elastic material or of an elastomer (rubber), which is described by Chernykh’s relationships.     

Electrically-excited (electroosmotic) flows in microchannels for mixing applications

We propose an asymptotic model for quite general liquid microchannel flows in the presence of electrical double layers (EDLs). The model provides an “inner” solution for the wall layer, which reflects the dominant balance between electrical forces and viscous forces (tangentially), respectively between electrical forces and pressure and viscous forces (normally). The electrically-neutral core of the flow is governed by the standard Navier–Stokes equations, providing the “outer” solution. The asymptotic matching of both solutions provides a method for the simplified numerical treatment of such EDLs. The superposition of the solutions in both regions then allows to infer an approximate solution, valid within the entire domain.Based on this model, we apply external oscillatory electrical fields to excite secondary flows (i) in microchannels with an internal obstacle or (ii) in folded (meander) microchannels. These secondary flows are demonstrated to greatly enhance the mixing of two liquids flowing in a layered fashion through these microchannels. Thus, electrical excitation has considerable potential if micromixers for ionic liquids are designed within electrically-insulating (e.g. plastics, glass) substrates.     

Couple stress based strain gradient theory for elasticity

The deformation behavior of materials in the micron scale has been experimentally shown to be size dependent. In the absence of stretch and dilatation gradients, the size dependence can be explained using classical couple stress theory in which the full curvature tensor is used as deformation measures in addition to the conventional strain measures. In the couple stress theory formulation, only conventional equilibrium relations of forces and moments of forces are used. The couple's association with position is arbitrary. In this paper, an additional equilibrium relation is developed to govern the behavior of the couples. The relation constrained the couple stress tensor to be symmetric, and the symmetric curvature tensor became the only properly conjugated high order strain measures in the theory to have a real contribution to the total strain energy of the system. On the basis of this modification, a linear elastic model for isotropic materials is developed. The torsion of a cylindrical bar and the pure bending of a flat plate of infinite width are analyzed to illustrate the effect of the modification.     

General Reduced Forms of Constitutive Relations in the Classical Continuum Mechanics

A theory of constitutive relations describing the resistance of bodies to deformation is developed. This theory takes into account the presence of internal body forces and internal kinematic constraints. A number of axioms and a general reduced form of constitutive relations for classical media are proposed. For simple bodies it is proved that the Il'yushin and Noll constitutive relations are equivalent.     

FINITE ELEMENT DISPLACEMENT PERTURBATION METHOD FOR GEOMETRIC NONLINEAR BEHAVIORS OF SHELLS OF REVOLUTION OVERALL BENDING IN A MERIDIONAL PLANE AND APPLICATION TO BELLOWS (Ⅰ)

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Forces and moments acting on bodies rotating about a symmetry axis in a free molecular flow

Analytic expressions are obtained for the forces and moments acting on symmetrically rotating convex figures of revolution moving in a free molecular flow of rarefied gas under the following assumptions: the velocity distribution function of the molecules of the oncoming flow is Maxwellian and the incident molecules have a diffuse—specular interaction with the surface of the body. For bodies with arbitrary piecewise smooth generator, general expressions are found in terms of quadrature for the components of the aerodynamic forces and moments. For a disk, sphere, and cylindrical and conical surfaces, the integration of the forces and the moments, which depend on the rotation of the body, is carried out to the end. For the moments of the forces, graphs are plotted of the errors of the hypothermal approximation as a function of the velocity ratio.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp, 151–155, May–June, 1980.     

FINITE ELEMENT DISPLACEMENT PERTURBATION METHOD FOR GEOMETRIC NONLINEAR BEHAVIORS OF SHELLS OF REVOLUTION OVERALL BENDING IN A MERIDIONAL PLANE AND APPLICATION TO BELLOWS (Ⅰ)

In order to analyze bellows effectively and practically, the finite-element-displacement-perturbation method (FEDPM) is proposed for the geometric nonlinear behaviors of shells of revolution subjected to pure bending moments or lateral forces in one of their meridional planes. The formulations are mainly based upon the idea of perturba-tion that the nodal displacement vector and the nodal force vector of each finite element are expanded by taking root-mean-square value of circumferential strains of the shells as a perturbation parameter. The load steps and the iteration times are not as arbitrary and unpredictable as in usual nonlinear analysis. Instead, there are certain relations between the load steps and the displacement increments, and no need of iteration for each load step. Besides, in the formulations, the shell is idealized into a series of conical frusta for the convenience of practice, Sander’s nonlinear geometric equations of moderate small rotation are used, and the shell made of more than one material ply is also considered.     

VIBRATION AND ACOUSTIC RADIATION FROM SUBMERGED STIFFENED SPHERICAL SHELL WITH DECK-TYPE INTERNAL PLATE

Based on the motion differential equations of vibration and acoustic coupling system for thin elastic spherical shell with an elastic plate attached to its internal surface, in which Dirac-δ functions are employed to introduce the moments and forces applied by the attachment on the surface of shell, by means of expanding field quantities as Legendre series, a semi-analytic solution is derived for the vibration and acoustic radiation from a submerged stiffened spherical shell with a deck-type internal plate, which has a satisfactory computational effectiveness and precision for an arbitrary frequency range. It is easy to analyze the effect of the internal plate on the acoustic radiation field by using the formulas obtained by the method proposed. It is concluded that the internal plate can significantly change the mechanical and acoustic characteristics of shell, and give the coupling system a very rich resonance frequency spectrum. Moreover, the method can be used to study the acoustic radiation mechanism in similar structures as the one studied here.     

Helical Birods: An Elastic Model of Helically Wound Double-Stranded Rods

We consider a geometrically accurate model for a helically wound rope constructed from two intertwined elastic rods. The line of contact has an arbitrary smooth shape which is obtained under the action of an arbitrary set of applied forces and moments. We discuss the general form the theory should take along with an insight into the necessary geometric or constitutive laws which must be detailed in order for the system to be complete. This includes a number of contact laws for the interaction of the two rods, in order to fit various relevant physical scenarios. This discussion also extends to the boundary and how this composite system can be acted upon by a single moment and force pair. A second strand of inquiry concerns the linear response of an initially helical rope to an arbitrary set of forces and moments. In particular we show that if the rope has the dimensions assumed of a rod in the Kirchhoff rod theory then it can be accurately treated as an isotropic inextensible elastic rod. An important consideration in this demonstration is the possible effect of varying the geometric boundary constraints; it is shown the effect of this choice becomes negligible in this limit in which the rope has dimensions similar to those of a Kirchhoff rod. Finally we derive the bending and twisting coefficients of this effective rod.     

Stability and dynamic analysis of partially cracked thin orthotropic microplates under thermal environment: an analytical approach

Abstract

An analytical model is proposed to analyze the vibration and buckling problem of partially cracked thin orthotropic microplate in the presence of thermal environment. The differential governing equation for the cracked plate is derived using the classical plate theory in conjunction with the strain gradient theory of elasticity. The crack is modeled using appropriate crack compliance coefficients based on the simplified line spring model. The influence of thermal environment is incorporated in governing equation in form thermal moments and in-plane compressive forces. The governing equation for cracked plate has been solved analytically to get fundamental frequency and central deflection of plate. To demonstrate the accuracy of the present model, few comparison studies are carried out with the published literature. The stability and dynamic characteristics of the cracked plate are studied considering various parameters such as crack length, plate thickness, change in temperature, and internal length scale of microstructure. It has been concluded that the frequency and deflection are affected by crack length, temperature, and internal length scale of microstructure. Furthermore, to study the buckling behavior of cracked plate, the classical relations for critical buckling load and critical buckling temperature is also proposed considering the effect of crack length, temperature, and internal length scale of microstructure.     

Stefan-Maxwell Relations in Nonequilibrium Polyatomic Reacting Gas Mixtures

The Stefan-Maxwell relations in nonequilibrium (with respect to the internal degrees of freedom) flows of multicomponent gas mixtures are obtained in an arbitrary approximation in Sonine polynomials.     

A nonlinear planar beam formulation with stretch and shear deformations under end forces and moments

A new nonlinear planar beam formulation with stretch and shear deformations is developed in this work to study equilibria of a beam under arbitrary end forces and moments. The slope angle and stretch strain of the centroid line, and shear strain of cross-sections, are chosen as dependent variables in this formulation, and end forces and moments can be either prescribed or resultant forces and moments due to constraints. Static equations of equilibria are derived from the principle of virtual work, which consist of one second-order ordinary differential equation and two algebraic equations. These equations are discretized using the finite difference method, and equilibria of the beam can be accurately calculated. For practical, geometrically nonlinear beam problems, stretch and shear strains are usually small, and a good approximate solution of the equations can be derived from the solution of the corresponding Euler–Bernoulli beam problem. The bending deformation of the beam is the only important one in a slender beam, and stretch and shear strains can be derived from it, which give a theoretical validation of the accuracy and applicability of the nonlinear Euler–Bernoulli beam formulation. Relations between end forces and moments and relative displacements of two ends of the beam can be easily calculated. This formulation is powerful in the study of buckling of beams with various boundary conditions under compression, and can be used to calculate post-buckling equilibria of beams. Higher-order buckling modes of a long slender beam that have complex configurations are also studied using this formulation.     

Development of the Spectral Sturm–Liouville Method for the Solution of a Boundary-Value Problem for the Biharmonic Equation

We generalize the spectral Sturm–Liouville method for the solution of the biharmonic equation. The characteristic equation for the determination of eigenvalues is investigated and eigenfunctions are constructed. We determine the stress-strain state for a rectangular plate loaded by arbitrary forces on its sides. For an arbitrary external load, we obtain a relation for the stress-strain state in the form of a series in eigenfunctions. A method of integral moments for the determination of the coefficients of the series is proposed. The Saint-Venant principle is verified.     

Analytical evaluation of the elastic and plastic resistances of double symmetric rectangular hollow sections under axial force and biaxial bending

In the recent codes for the design of steel structures, the elastic–plastic methods of analysis are recognised to provide an efficient estimation of the ultimate resistance of some of these structures. These methods are usually based on some basic hypotheses, such as the creation of plastic hinges in the most stressed cross-sections, for instance.As the development of these plastic hinges depends on the interaction between the internal forces and on the cross-section shape, specific equations are required for the analysis of different types of cross-sections. However, most frequently, these equations are not available, or they are expressed by means of simplified expressions; this is usually the case when biaxial bending is involved.This paper presents new interaction criteria for the analysis of steel rectangular hollow sections subjected to an axial force and biaxial bending moments, at the elastic or the plastic limit states (as long as buckling phenomena are not involved). The plastic interaction criteria are presented, in a first step, for some particular combinations of the internal forces, such as axial loading with bending about a main axis, and biaxial bending without axial loading. Then, the global solution for the simultaneous combination of an axial force and bending moments about both the main axes of inertia are described in detail. All these plastic interaction criteria are compared with the corresponding plastic criteria adopted in the Eurocode 3 (EC3). Some suggestions are presented in order to improve the results given by these EC3 criteria.     

Relations between the forces on a body moving in a rarefied gas in a light flux and in a hypersonic newtonian stream

For three-dimensional axisymmetric and two-dimensional bodies we derive relations, valid in an arbitrary coordinate system, between the aerodynamic forces for motion of the body when the locality law is valid (i.e., the momentum flux at the surface of the body depends on the local angle between the velocity and the normal to the surface) and we determine the total forces and moments by integration over that part of the surface of the body facing the flow. We obtain equations defining the change in the lift (and, consequently, the frontal drag and lift-drag ratio), depending on the angle of attack, as a function of the area of projection of the surface of the body on the plane perpendicular to the direction of flight. Particular cases of the relations we obtain are, for example, the Newtonian motion of a body at hypersonic velocities, the motion of a body in a rarefied gas, and the effect of light pressure on a body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 89–95, July–August, 1973.     

Discrete element analysis of stone cantilever stairs

Stone cantilever staircases are present in case of both new constructions and reconstructions. The aim of the present paper is to understand the mechanical behaviour of these staircases with the help of discrete element simulations, and to compare the calculated behaviour to the estimations given by the existing manual calculation methods. First a literature review is presented on the statical calculation of cantilevered staircases: manual calculation methods suggested in the 1990s for straight and spiral staircases are introduced, focusing on Heyman’s theory and its improved counterparts. Then the discrete element method is used as a tool to perform virtual experiments, in order to evaluate the mechanical behaviour of the straight and spiral staircases for selfweight, live loads and support movement. The results obtained (internal forces, stresses, deflections) are then compared with the manual calculation results. The most important conclusions are: (1) the term “cantilever stair” is misleading: significant torsion moments occur in the treads, while the bending moments are much smaller than in a free cantilever; (2) the type of the connection between wall and treads (i.e. the end of the tread is simply supported by the wall against translation and torsion, or it is also partly clamped) has a fundamental influence on the internal forces and stress distributions; (3) for simply supported treads the existing manual methods are conservative for straight stairs, but for spiral stairs they dangerously underestimate the torsional moments.