Thermal stresses in cylindrical Cosserat elastic shells

We consider the problem of thermal stresses in cylindrical elastic shells, modelled as Cosserat surfaces. In the theory of Cosserat shells, the thermal effects are described generally by means of two temperature fields. The problem consists in finding the equilibrium of the shell under the action of a given temperature distribution. The temperature fields are assumed to be general polynomial functions in the axial coordinate, whose coefficients depend on the circumferential coordinate.     

Influence of general imperfections in axially loaded cylindrical shells

The buckling of an axially loaded cylindrical shell is considered when imperfection components corresponding to all of the classical buckling modes are taken into consideration. The analysis represents an extension of Koiter's axisymmetric solution and in the asymptotic sense due to Koiter the imperfections considered are as general as possible. The results obtained reveal many interesting aspects of shell buckling which arize for various imperfection forms. The buckling behaviour which results is associated with both bifurcation and limit point critical states.     

A general nonlinear theory of elastic shells

This paper presents a general nonlinear theory of elastic shells for large deflections and finite strains in reference to a certain natural state. By expanding the displacement components into power series in the coordinate θ³ normal to the undeformed middle surface of shells, the expansions of the Cauchy-Green strain tensors are expressed in terms of these expanded displacement components. Through the modified Hellinger-Reissner variational principle for a three-dimensional elastic continuum, a set of the fundamental shell equations is derived in terms of the expanded Cauchy-Green strain tensors and Kirchhoff stress resultants. The Love-Kirchhoff hypothesis is not assumed and higher order stretching and bending are taken into consideration. For elastic shells of isotropic materials, assuming the strain-energy to be an analytic function of the strain measures, general nonlinear constitutive equations are then derived. Thus, a complete and consistent two-dimensional shell theory incorporating the geometrical and physical nonlinearities is established. The classical theories of shells are directly derivable from the present results by proper truncations of the series.     

Vibrations and buckling of axially loaded stiffened cylindrical shells with elastic restraints

A theory is derived for calculation of the influence of elastic edge restraints on the vibrations and buckling of stiffened cylindrical shells. The stiffeners are considered “smeared” and the edge restraints can be axial, radial, circumferential or rotational. Extensive computations are performed for special kinds of stringer-stiffened shells, and the theoretical predictions are compared with experimental results. A method of definition of equivalent elastically restrained boundary conditions by use of vibration tests is discussed. Application of this technique to tests on 10 shells significantly reduces the scatter in the ratio of experimental to predicted buckling loads.     

On consistent first approximations in the general linear theory of thin elastic shells

Summary This paper, as a number of earlier ones, is concerned with the rational establishment of twodimensional differential equations for the approximate analysis of stress and strain in elastic layers with spacecurved middle surface. It has been known for some time that the principal difficulty of this problem is to establish rational two-dimensional constitutive equations which correspond to a given system of constitutive equations for the layer treated as a three-dimensional continuum. — In an earlier publication [18] the point had been made that since two-dimensional shell theory was concerned with stress resultants and stress couples, it ought to be advantageous to derive such a theory from a three-dimensional theory in which force stresses as well as moment stresses were incorporated, even for media which, actually, were incapable of supporting moment stresses. — The earlier work [18] had indicated that, mathematically, the advantages of approaching the derivation of two-dimensional shell theory from three-dimensional moment stress elastically theory had to do with the form of the compatibility equations for strain in such a three-dimensional theory. Briefly, with these three-dimensional compatibility equations it becomes possible to concentrate all three-dimensional aspects of the shell problem in a three-dimensional system of integro-differential constitutive equations, and the task of deriving rational two-dimensional constitutive equations becomes nothing but the task of establishing suitable asymptotic expansions for the solutions of these three-dimensional integro-differential equations. In the work in [18] this task had not actually been carried out. The present paper establishes a significant rearrangement of the system of integro-differential equations, in such a way that the nature of the necessary asymptotic expansions is made evident. — With this, explicit results are obtained which include the system of two-dimensional constitutive equations of Koiter and Sanders for an iotropic homogeneous medium, as well as a system of constitutive equations for a class of shells for which the normals to the middle surface are not directions of elastic symmetry, as well as a system of constitutive equations for shells which are sufficiently soft in transverse shear to make transverse shear deformation a first-order effect.

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Übersicht In dieser Veröffentlichung wird die rationelle Aufstellung der zweidimensionalen Differentialgleichungen für die näherungsweise Bestimmung von Spannungen und Verformungen in elastischen Schichten mit räumlich gekrümmter Mittelfläche behandelt. Es ist bekannt, daß die Hauptschwierigkeit dabei im Aufstellen von zweidimensionalen Stoffgleichungen besteht, die einem gegebenen System von Stoffgleichungen für eine als dreidimensionales Kontinuum behandelten Schicht entsprechen. In einer früheren Veröffentlichung [18] wurde darauf hingewiesen, daß es vorteilhaft sein könnte, eine solche Theorie aus einer dreidimensionalen Theorie abzuleiten, in der sowohl Momentspannungen als auch Kraftspannungen berücksichtigt werden. Das gilt auch für solche Stoffe, die in Wirklichkeit nicht in der Lage sind, Momentenspannungen aufzunehmen. — Es wurde seinerzeit gezeigt, daß die Vorteile einer Ableitung der genäherten zweidimensionalen Schalentheorie aus der dreidimensionalen Elastizitätstheorie mit der Form der Verträglichkeitsbedingungen für die Verformungen in dieser dreidimensionalen Theorie zusammenhängen. Mit Hilfe dieser dreidimensionalen Verträglichkeitsbedingungen wird es möglich, alle dreidimensionalen Aspekte des Schalenproblems in einem dreidimensionalen System von Integro-Differentialgleichungen für das Stoffverhalten zu konzentrieren, so daß die Ableitung zweidimensionaler Stoffgleichungen nichts anderes ist, als das Aufstellen geeigneter asymptotischer Reihenentwicklungen für die Lösungen dieser dreidimensionalen Integro-Differentialgleichungen. Das wurde jedoch in [18] noch nicht ausgeführt. In der vorliegenden Veröffentlichung wird das System der Integro-Diffe-rentialgleichungen so umgeformt, daß die Art der notwendigen asymptotischen Entwicklungen deutlich wird. Auf diese Weise werden explizite Ergebnisse erhalten, die das System der zweidimensionalen Stoffgleichungen von Koiter und Sanders für ein isotropes homogenes Medium einschließen. Desgleichen sind darin enthalten die Stoffgleichungen für eine Klasse von Schalen, für die die Normalen zur Mittelfläche nicht mit den Richtungen der elastischen Symmetrie übereinstimmen, sowie auch die Stoffgleichungen für Schalen, die hinreichend weich gegenüber Querschub sind, so daß Querschubdeformationen als Effekte erster Ordnung auftreten.

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A report on work supported by the Office of Naval Research, Washington, D.C.     

Postbuckling of internal pressure loaded FGM cylindrical shells surrounded by an elastic medium

This paper presents a study on the postbuckling response of a functionally graded cylindrical shell of finite length embedded in a large outer elastic medium and subjected to internal pressure in thermal environments. The surrounding elastic medium is modeled as a tensionless Pasternak foundation that reacts in compression only. The postbuckling analysis is based on a higher order shear deformation shell theory with von Kármán–Donnell-type of kinematic nonlinearity. The thermal effects due to heat conduction are also included and the material properties of functionally graded materials (FGMs) are assumed to be temperature-dependent. The nonlinear prebuckling deformations and the initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the postbuckling response of the shells and an iterative scheme is developed to obtain numerical results without using any assumption on the shape of the contact region between the shell and the elastic medium. Numerical solutions are presented in tabular and graphical forms to study the postbuckling behavior of FGM shells surrounded by an elastic medium of tensionless elastic foundation of the Pasternak-type, from which results for conventional elastic foundations are obtained as comparators. The results reveal that the unilateral constraint has a significant effect on the postbuckling response of shells subjected to internal pressure in thermal environments when the foundation stiffness is sufficiently large.     

On dynamic buckling phenomena in axially loaded elastic-plastic cylindrical shells

Some characteristic features of the dynamic inelastic buckling behaviour of cylindrical shells subjected to axial impact loads are discussed. It is shown that the material properties and their approximations in the plastic range influence the initial instability pattern and the final buckling shape of a shell having a given geometry. The phenomena of dynamic plastic buckling (when the entire length of a cylindrical shell wrinkles before the development of large radial displacements) and dynamic progressive buckling (when the folds in a cylindrical shell form sequentially) are analysed from the viewpoint of stress wave propagation resulting from an axial impact. It is shown that a high velocity impact causes an instantaneously applied load, with a maximum value at t=0 and whether or not this load causes an inelastic collapse depends on the magnitude of the initial kinetic energy.     

Laterally loaded elastic shells of revolution

Summary An alternate form of the Chernin type equations for laterally loaded shells of revolution is obtained. Except for terms of order of the inherent error in shell theory, the final two coupled second order ordinary differential equations are remarkably similar to the Reissner-Meissner type equations for problems involving axi-symmetric stress distributions. Unlike all previous versions, our two equations can be further reduced (just as in the case of axi-symmetric stress distributions) to a single second order equation for a complex stress function without any additional approximation for uniform cylindrical, spherical, conical and toroidal shells. The side force and tilting moment problem for a shell frustum is shown to be the static geometric analogue of the problem of asymmetric bending and twisting of a ring shell sector. An efficient method for the evaluation of the overall influence coefficients is discussed. The stress state of a complete uniform spherical shell subject to concentrated side force and tilting moment at the two poles is analyzed.

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Übersicht Es wird eine neue Form der Cherninschen Gleichung für rotationssymmetrische, seitlich belastete Schalen abgeleitet. Abgesehen von Gliedern, deren Größenordnung dem allgemeinen Fehler der Schalentheorie entspricht, haben die zwei gekoppelten gewöhnlichen Differentialgleichungen zweiter Ordnung eine merkwürdige Ähnlichkeit zu den von Reissner und Meissner abgeleiteten Gleichungen für Probleme mit axialsymmetrischem Spannungszustand. Zum Unterschied zu früheren Fassungen der Cherninschen Gleichungen können die jetzigen zu einer einzigen Gleichung zweiter Ordnung auf eine komplexe Spannungsfunktion zusammengefaßt werden, wie dies im Falle rotationssymmetrischer Spannung auch möglich ist. Dabei sind keine zusätzlichen Näherungen für gleichförmige Zylinder-, Kugel-, Kegel- und Ringflächenschalen notwendig. Es zeigt sich, daß das Problem der Beanspruchung einer Stumpfschale durch Seitenkraft und Momente statisch und geometrisch analog ist zum Problem der Beanspruchung eines Ringschalensektors durch unsymmetrische Biegung und Drillung. Es wird ein Verfahren zur Bestimmung der Einflußkoeffizienten angegeben. Außerdem wird der Spannungszustand in einer gleichförmigen vollständigen Kugelschale analysiert, die durch eine einzelne Seitenkraft und Momente an den zwei Polen beansprucht wird.

     

A general theory of a Cosserat surface

This paper is concerned with a general dynamical theory of a Cosserat surface, i.e., a deformable surface embedded in a Euclidean 3-space to every point of which a deformable vector is assigned. These deformable vectors, called directors, are not necessarily along the normals to the surface and possess the property that they remain invariant in length under rigid body motions. An elastic Cosserat surface and other special cases of the theory which bear directly on the classical theory of elastic shells are also discussed.     

Studies on the torsional buckling of elastic cylindrical shells

The experimental phenomenon and theoretical analysis are given for the torsional buckling of elastic cylindrical shells. From the experiment, it is found that the postbuckling deformation doesn't occupy the whole length when the shell is longer. In the theoretical calculation, only the normal displacement boundary condition is taken into account. By comparing the present calculation results with the accurate result of Yamakis theory and the results of the present experiment, it is shown that the influence of the axial and circumference boundary condition is less important.     

Vibrations and buckling of eccentrically loaded stiffened cylindrical shells

The influence of eccentricity of loading on the vibrations and buckling of stringer-stiffened shells is studied. An established nonlinear theory, which takes into account nonlinear prebuckling, is applied and the predictions are compared with experimental results. Two families of shells, one ‘heavily’ stiffened and the other ‘moderately’ stiffened, were tested but detailed results are presented only for the ‘heavily’ stiffened shells. In each family there are three identical shells, each with different eccentricity of loading. In all cases, different in-plane-boundary conditions are considered and correlated with experimental results.     

Elastoplastic bifurcation and collapse of axially loaded cylindrical shells

In this paper, a shell finite element is designed within the total Lagrangian formulation framework to deal with the plastic buckling and post-buckling of thin structures, such as cylindrical shells. First, the numerical formulation is validated using available analytical results. Then it is shown to be able to provide the bifurcation modes—possibly the secondary ones—and describe the complex advanced post-critical state of a cylinder under axial compression, where the theory is no longer operative.     

On shear correction factors in the non-linear theory of elastic shells

Theoretical values of two correction factors α_s = 5/6 and α_t = 7/10 are established for the respective transverse shear stress resultants and stress couples within the general, dynamically and kinematically exact, six-field theory of elastic shells. These values do not depend on the shell material symmetry, geometry of the base surface, the shell thickness, or any kind of kinematic and/or dynamic constraints. The analysis is based on the complementary energy density following from the transverse shear stresses acting only on the shell cross section. The appropriate quadratic and cubic distributions of the stresses across the thickness allow one to derive the consistent constitutive equations for the transverse shear stress resultants and stress couples with α_s and α_t as the respective correction factors. Four numerical examples of highly non-linear shell structures illustrate the influence of different values of α_s and α_t on the results. In particular, some influence of α_t is noticed on the placement of bifurcation points. In dynamic problem of flight of three intersecting plates analysed with Newmark-type temporal algorithm, the value of α_t influences the moment at which the relative error of total energy of the system begins to grow indefinitely leading to the solution failure.