Dynamic Stability Problems of Anisotropic Cylindrical Shells via a Simplified Analysis

An analytical–numerical method involving a small number of generalized coordinates is presented for the analysis of the nonlinear vibration and dynamic stability behaviour of imperfect anisotropic cylindrical shells. Donnell-type governing equations are used and classical lamination theory is employed. The assumed deflection modes approximately satisfy simply supported boundary conditions. The axisymmetric mode satisfying a relevant coupling condition with the linear, asymmetric mode is included in the assumed deflection function. The shell is statically loaded by axial compression, radial pressure and torsion. A two-mode imperfection model, consisting of an axisymmetric and an asymmetric mode, is used. The static-state response is assumed to be affine to the given imperfection. In order to find approximate solutions for the dynamic-state equations, Hamiltons principle is applied to derive a set of modal amplitude equations. The dynamic response is obtained via numerical time-integration of the set of nonlinear ordinary differential equations. The nonlinear behaviour under axial parametric excitation and the dynamic buckling under axial step loading of specific imperfect isotropic and anisotropic shells are simulated using this approach. Characteristic results are discussed. The softening behaviour of shells under parametric excitation and the decrease of the buckling load under step loading, as compared with the static case, are illustrated.     

Multiple geometric defects: Their effect on stability of cylindrical shells

Thin isotropic circular cylindrical shells of near perfect quality were tested in axial compression with multiple local geometric defects imposed on them. The defects were in the form of diamond-shaped Yoshimura-facet-type dimples, with large amplitudes, of the same order of magnitude as that of the facets in a cylinder buckled in axial compression. This investigation was performed in continuation of a previous study on the effect of single local defects on the stability of axially loaded cylinders. Apart from establishing the changes in load-carrying capacity with increasing number of defects, the present set of experiments reveals certain behavioral aspects of the defects, providing insight into how they affect the overall shell stiffness and its stability. Based on the test results, an empirical formula relating the reduction in buckling load to the number and size of the local defects is developed.     

Dynamic Buckling of Autonomous Systems Having Potential Energy Universal Unfoldings of Cuspoid Catastrophe

This work deals with dynamic buckling universal solutions of discrete nondissipative systems under step loading of infinite duration. Attention is focused on total potential energy functions associated with universal unfoldings of cuspoid type catastrophes with one active coordinate. The fold, dual cusp and tilted cusp catastrophes under statically applied loading occurring via limit points, asymmetric/symmetric bifurcations and nondegenerate hysteresis points are extended to the case of dynamic loading. Catastrophe manifolds of these types showing imperfection sensitivity under both types of loading are fully assessed. Important findings regarding dynamic buckling of imperfect systems generated by perfect systems associated with imperfect bifurcations are explored. The analysis is supplemented by a numerical application of a system exhibiting imperfect bifurcation when it is perfect as well as a hysteresis point associated with a tilted cusp catastrophe, when it becomes imperfect.     

Dynamic and buckling analysis of a thin elastic-plastic square plate in a uniform temperature field

The nonlinear models of the elastic and elasticlinear strain-hardening square plates with four immovably simply-supported edges are established by employing Hamilton‘s Variational Principle in a uniform temperature field. The unilateral equilibrium equations satisfied by the plastically buckled equilibria are also established. Dynamics and stability of the elastic and plastic plates are investigated analytically and the buckled equilibria are investigated by employing Galerkin-Ritz‘s method. The vibration frequencies, the first critical temperature differences of instability or buckling, the elastically buckled equilibria and the extremes depending on the final loading temperature difference of the plastically buckled equillibria of the plate are obtained. The results indicate that the critical buckling value of the plastic plate is lower than its critical instability value and the critical value of its buckled equilibria turning back to the trivial equilibrium are higher than the value. However, three critical values of the elastic plate are equal. The unidirectional snap-through may occur both at the stress-strain boundary of elasticity and plasticity and at the initial stage of unloading of the plastic plate.     

Numerische Untersuchung des elastisch-plastischen Verhaltens eines Fadenverbundwerkstoffes mit metallischer Matrix Eine Fallstudie

Übersicht Die Fehlervorhersage eines einachsigen Fadenverbundwerkstoffes mit metallischer Matrix erfordert die Kenntnis des nichtlinearen elastisch-plastischen Verhaltens in mikroskopischer Abmessung. Als Fallstudie wurde ein spezieller Verbundwerkstoff mit einem FEM-Programm untersucht. Im plastischen Bereich wurde das klassische, v. Mises Potential' mit dem neuen Übergangsfließpotential unter Berücksichtigung der plastischen Volumendehnung verglichen. Unter transversaler Normalbelastung zeigte der Verbund deutliche Unterschiede in der Zunahme und der Ausdehnung der lokalen plastischen Zonen. Alle kritischen Verformungsstadien wurden von dem Übergangsfließpotential bei geringerer Belastung erreicht.

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Numerical investigation of the elastic-plastic behaviour of a fibre-reinforced composite with a metallic matrix

Summary Failure prediction of unidirectional fibre-reinforced composite with a metallic matrix needs knowledge about the nonlinear elastic-plastic behaviour in a microscopic scale. A specific composite was investigated using a FEM-program as a case study. In the plastic range the classical v. Mises Potential was compared with the new Transition Flow Potential, taking into account the plastic volume dilatation. Subjected to transverse normal loading the composite showed evident differences in the increase and the spread of locally plastic regions. All critical deformation states were reached by the Transition Flow Potential at lower loading.

     

Displacement type unified equation for bending, buckling and vibration of conical shells and its applications

By using Donnell's simplication and starting from the displacement type equations of conical shells, and introducing a displacement functionU(s,,) (In the limit case, it will be reduced to cylindrical shell displacement function introduced by V. S. Vlasov) and a generalized loadq,(s,,),the equations of conical shells are changed into an eighth—order solvable partial differential equation about the displacement functionU(s,,). As a special case, the general bending problem of conical shells on Winkler foundation has been studied. Detailed numerical results and boundary coefficients for edge unit loads are obtained.The project supported by the National Natural Science Foundation of China.     

Development of a test machine and method for galling studies

A test method for evaluating galling resistance in tubular connections by using the established evaluation technique known as the up and down or Bruceton method was studied. A unique test machine that would closely simulate the actual field conditions was designed and constructed. By using the Taguchi technique, the relative effects of rotational speed, roughness, and axial load on the galling resistance were also studied and discussed.Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 8–11.     

Effect of electric current on the evolution of plastic strain near a crack tip

The effect of direct current on the evolution of plastic strain near the tip of a crack in a crystal in tension is studied. The plastic strain near the crack tip in the loaded specimen is the result of motion of dislocations in the active slip planes of the crystal under the action of shear stresses caused by external loading and electric current. The Joule heat, Thomson effect, and electron wind (electroplastic effect) are taken into account in calculations. The plastic strain and stress distributions near the crack tip are determined at different moments of time for a given magnitude ofelectric current. The effect of the plastic zone on the stressintensity factor of the crack is studied. It is found that the plastic strain is affected largely by the Joule heat released upon passage of the electric current. A numerical analysis is performed for an Fe crystal.     

Stability of pretwisted beams in cross bracings

New systems of pretwisted bars are suggested, and their stability under a conservative load is investigated. In this paper, closedform relationships are obtained for the direct evaluation of critical loads of pretwisted cross bracings with different end connections and an arbitrary ratio of the dimensionless parameters of tension and compression elements. The equation for the critical loads is derived for tension and compression braces with different section properties, lengths, and axial loading. It is found that the critical load of a twisted brace is higher than that of a nontwisted brace. Parametric solutions are graphically displayed to clarify the distinct behavior, including the boundary separating symmetric and antisymmetric modes of buckling.     

Numerical analysis of axisymmetric buckling of conical shells

Nonlinear boundaryvalue problems of axisymmetric buckling of conical shells under a uniformly distributed normal pressure are solved by the shooting method. The problems are formulated for a system of six firstorder ordinary differential equations with independent rotation and displacement fields. Simply supported and clamped cases are considered. Branching solutions of the boundaryvalue problems are studied for different pressures and geometrical parameters of the shells. The nonmonotonic and discontinuous curves of equilibrium states obtained show that collapse, i.e., snapthrough instability is possible. For a simply supported shell, multivalued solutions are obtained for both external and internal pressure. For a clamped thinwalled shell, theoretical results are compared with experimental data.     

Analysis of the Dynamic Stability of Electrical Graded Piezoelectric Circular Cylindrical Shells

I. INTRODUCTION In recent years, piezoelectric materials have been widely used as sensors and actuators in the areaof smart structures, and shell-type piezoelectric smart structures have become the focus of research.Various analytical and ?nite element models for piezoelectric shells have been developed[1??3]. Inaddition, a series of exact studies on the bending and buckling of piezoelectric circular cylindrical shellhave been presented by Chen et al.[4??7]. Recently, the use of functi…     

Indentation of the semi-infinite elastic solid by a concave rigid punch

A solution is obtained for the relationship between load, displacement and inner contact radius for an axisymmetric, spherically concave, rigid punch, indenting an elastic half-space. Analytic approximations are developed for the limiting cases in which the ratio of the inner and outer radii of the annular contact region is respectively small and close to unity. These approximations overlap well at intermediate values. The same method is applied to the conically concave punch and to a punch with a central hole. , , . , . . .     

Finite Elastic Deformations of a Tyre Modelled as an Ideal Fibre-Reinforced Shell

Vehicle tyres are anisotropic inhomogeneous fibre-reinforced shells which undergo finite elastic deformations. Calculation of their stress and deformation fields is a difficult task and is normally performed using the finite element technique. In this paper an attempt is made to provide an approximate analysis of the deformation field modelling the tyre as an ideal fibre-reinforced material. Radial-ply tyres are reinforced by a belt of fibres running around the wheel in the circumferential direction under the tread of the tyre. A second set of fibres lies in each radial cross-section, of the tyre and runs from the bead wire which seats against one wheel rim to the bead wire at the other wheel rim. We shall assume each radial cross-section of the tyre is in a state of plane strain and is formed from an arch of fibre-reinforced composite material which is reinforced in the hoop direction. This composite is assumed to be an ideal material which is inextensible in the fibre-direction and is incompressible. The plane-strain deformations of this section are examined and then used to analyse the deformation of the tyre as a whole.     

The extension of Beltrami's ellipsoid to anisotropic bodies

Summary The spectral decomposition of the compliance, stiffness, and failure tensors for transversely isotropic materials was studied and their characteristic values were calculated using the components of these fourth-rank tensors in a Cartesian frame defining the principal material directions. The spectrally decomposed compliance and stiffness or failure tensors for a transversely isotropic body (fiber-reinforced composite), and the eigenvalues derived from them define in a simple and efficient way the respective elastic eigenstates of the loading of the material. It has been shown that, for the general orthotropic or transversely isotropic body, these eigenstates consist of two double components, ₁ and ₂ which are shears (₂ being a simple shear and ₁, a superposition of simple and pure shears), and that they are associated with distortional components of energy. The remaining two eigenstates, with stress components ₃, and ₄, are the orthogonal supplements to the shear subspace of ₁ and ₂ and consist of an equilateral stress in the plane of isotropy, on which is superimposed a prescribed tension or compression along the symmetry axis of the material. The relationship between these superimposed loading modes is governed by another eigenquantity, the eigenangle .The spectral type of decomposition of the elastic stiffness or compliance tensors in elementary fourth-rank tensors thus serves as a means for the energy-orthogonal decomposition of the energy function. The advantage of this type of decomposition is that the elementary idempotent tensors to which the fourth-rank tensors are decomposed have the interesting property of defining energy-orthogonal stress states. That is, the stress-idempotent tensors are mutually orthogonal and at the same time collinear with their respective strain tensors, and therefore correspond to energy-orthogonal stress states, which are therefore independent of each other. Since the failure tensor is the limiting case for the respective _x, which are eigenstates of the compliance tensor S, this tensor also possesses the same remarkable property.An interesting geometric interpretation arises for the energy-orthogonal stress states if we consider the projections of _x in the principal₃D stress space. Then, the characteristic state ₂ vanishes, whereas stress states ₁, ₃ and ₄ are represented by three mutually orthogonal vectors, oriented as follows: The ₃ and ₄ lie on the principal diagonal plane (₃₁₂) with subtending angles equaling (–/2) and (-), respectively. On the positive principal ₃-axis, is the eigenangle of the orthotropic material, whereas the ₁-vector is normal to the (₃₁₂)-plane and lies on the deviatoric -plane. Vector ₂ is equal to zero.It was additionally conclusively proved that the four eigenvalues of the compliance, stiffness, and failure tensors for a transversely isotropic body, together with value of the eigenangle , constitute the five necessary and simplest parameters with which invariantly to describe either the elastic or the failure behavior of the body. The expressions for the _x-vector thus established represent an ellipsoid centered at the origin of the Cartesian frame, whose principal axes are the directions of the ₁-, ₃- and ₄-vectors. This ellipsoid is a generalization of the Beltrami ellipsoid for isotropic materials.Furthermore, in combination with extensive experimental evidence, this theory indicates that the eigenangle alone monoparametrically characterizes the degree of anisotropy for each transversely isotropic material. Thus, while the angle for isotropic materials is always equal to _i = 125.26° and constitutes a minimum, the angle || progressively increases within the interval 90–180° as the anisotropy of the material is increased. The anisotropy of the various materials, exemplified by their ratiosE _L/2G_L of the longitudinal elastic modulus to the double of the longitudinal shear modulus, increases rapidly tending asymptotically to very high values as the angle approaches its limits of 90 or 180°.     

On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

Stress-intensity factors for a semi-infinite plane crack with a wavy front

The stress-intensity factors for a semi-infinite plane crack with a wavy front are determined when the crack faces are subjected to normal and shearing tractions. The results are derived using asymptotic methods and are valid to O(²) where =A/1; A is the amplitude and is the wavelength of the wavy front. The normal and shearing tractions are in the form of line loads parallel to the crack front.The results are then used to evaluate, in a qualitative manner, the growth characteristics of a semi-infinite plance crack with a wavy front under combined mode loading. This provides a possible explanation of crack front segmentation observed experimentally.     

The wedge subjected to tractions: a paradox re-examined

The classical two-dimensional solution for the stress distribution in an elastic wedge loaded by a uniform pressure on one side of the wedge becomes infinite when the wedge angle 2 satisfies the equation tan 235-1. This paradox was resolved recently by Dempsey who obtained a solution which is bounded at 235-2. However, for not equal but very close to 235-3, the classical solution can still be very large as approaches 235-4. In this paper we re-examine the paradox. We obtain a solution which remains bounded as approaches 235-5 and reproduces Dempsey's solution in the limit 235-6. At 235-7 the stress distribution contains a (ln r) term for general loadings. The (ln r) term disappears under a special loading and the stresses are bounded for all r. Moreover, the solution is not unique. In other words, for the wedge angle 235-8 under a special loading, infinitely many solutions exist for which the stresses are bounded for all r. We also obtain solutions which are bounded and approach Dempsey's solutions when 2= and 2. Again, under a special loading infinitely many solutions exist for which the stresses are bounded for all r. Care has been exercised in this paper to present the solutions in a form in which the terms r ^- and ln r are replaced by R ^-gl and ln R where R=r/r ₀is the dimensionless radial distance and r ₀ is an arbitrary constant having the dimension of length.     

Hydrodynamic lubrication of ball and socket joints

Summary As a step towards formation of design rules for ball and socket joints, a solution of the lubrication equations for an ideal joint (in which the socket completely surrounds the ball) under steady loading and rotation is presented. The results are compared with those for an infinitely wide plain bearing. The friction for the ball is found to be higher, and the load capacity lower, than that of the plain bearing under similar conditions.Notation h lubricant film thickness - p pressure - absolute viscosity - average of v, w, over film thickness h - v, w components of fluid velocity in , directions - U , U components of relative velocity of surfaces in , directions - R mean radius of spheres - H difference between ball and socket radii - c eccentricity ratio - angular speed of ball - B =pH ²/6cR ² - P load component along Oy - N load component along Oz - M driving moment to spin around Ox - moment coefficient - _p , P _p suffix p refers to unit width of plain bearing     

Flexible shell-theory and buckling of toroidal shells and tubes

Summary Flexible shells, designed to sustain elastic displacements, have a semimembrane type of deformation, described by simplified equations. In what follows, flexible shell equations are derived and used to analyse bending of curved tubes and buckling of tubes and toroidal shells under external pressure, as interdependent problems. Design formulas and curves are presented.

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Übersicht Flexible Schalen, die elastischen Verformungen ausgesetzt sind, genügen einem Semi-Membranspannungszustand. Vereinfachte Gleichungen für diesen Zustand werden hergeleitet. Damit werden die Probleme des Beulens von Torusschalen unter Außendruck, ein Spezialfall des allgemeineren Problems der Biegung und der Stabilität von Rohren, untersucht. Spannungen, Deformationen und kritischer Außendruck werden in Abhängigkeit von der Rohrkrümmung und von dem Abstand zwischen Querrippen oder Flanschen bestimmt.