Analytic solutions of axisymmetric problems in elastic cylinders

Summary A general scheme encompassing expansion methods for the sleeve is presented: one homogeneous condition for each contour portion establishes real eigenvalues ensuring orthogonality to Fourier-Bessel expansions. Solutions by integral equations may be regarded as limiting cases, with summations replaced by integrations. Special attention is devoted to the alternative use of complex eigensolutions, subjected to two homogeneous conditions on the cylindrical boundary.

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Sommario Si presenta uno schema generale che inquadra i metodi di risoluzione attraverso sviluppi, per cilindri cavi: imponendo una condizione omogenea su ciascuna parte, piana o cilindrica, di contorno, si definiscono le autosoluzioni reali, ortogonali, per gli sviluppi in serie di Fourier-Bessel. Rientrano come casi particolari quelli di lunghezza o raggio infinito. Si esamina inoltre l'uso di autosoluzioni complesse, definite da due condizioni omogenee sul contorno cilindrico.

     

The axisymmetric end problem for transversely isotropic circular cylinders

Energy-decay inequalities are applied in investigating the decay of end effects in a transversely isotropic circular cylinder subject to torsionless axisymmetric end loads. A lower bound (in terms of the elastic constants) is obtained for the rate of exponential decay of stresses and this is compared with results of other authors. For a highly anisotropic medium, a slow decay rate is predicted thus anticipating disagreement with Saint-Venant's principle in this case.     

Load-induced stress singularities in the bending of cosserat plates

Summary Presented in this paper are the solutions to several plate bending problems as governed by a recent theory developed by Green and Naghdi, into which couple-stress is incorporated. Specifically, each problem considered is subjected to boundary conditions emanating from a singular load distribution acting on the free edge of a semi-infinite plate. The method of integral transforms is applied in the solutions.In general, it is found that the singularities in the shear and moment resultants are of the same order as those given in Reissner's plate theory, however the detailed structures of these singular functions are altered. The present theory also suggests that, in most cases, the maximum magnitudes of the moment resultants are diminished as compared to the corresponding results given in Reissner's theory.Also discussed is the exact relationship between the Green-Naghdi theory and Reissner's theory.

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Sommario In questa memoria sono presentate le soluzioni di alcuni problemi relativi a piastre inflesse studiati in base ad una teoria recentemente sviluppata da Green e Naghdi, in cui si considerano distribuzioni di coppie di volume.In ogni problema considerato le condizioni al contorno presentano una singolarità nella distribuzione di carico agente su un bordo libero di una piastra semi-indefinita. Per la risoluzione è applicato il metodo delle trasformate integrali.In generale risulta che le singolarità nel taglio e nel momento sono dello stesso ordine di quelle date dalla teoria delle piastre di Reissner, benchè gli andamenti di queste funzioni singolari siano diversi. La teoria sviluppata porta inoltre a valutare, nella maggior parte dei casi, momenti massimi più piccoli di quelli corrispondenti ottenuti in base alla teoria di Reissner.Si discute inoltre l'esatta relazione tra le teorie di Green-Naghdi e di Reissner.

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This research is sponsored by the U. S. Navy under Contract Nonr-610(06) with the Office of Naval Research in Washington, D. C.     

Using spline-approximation to solve problems of axisymmetric free vibration of thick-walled orthotropic cylinders

The three-dimensional theory of elasticity is used to study the free vibrations of an anisotropic hollow cylinder with different boundary conditions at the ends. The relevant problem is solved by a numerical-and-analytic method. Spline approximation and collocation is used to reduce the partial differential equations of elasticity to a boundary-value problem for a system of ordinary differential equations of high order for the radial coordinate, which is solved using the stable discrete-orthogonalization and incremental-search methods. The calculated results for an orthotropic inhomogeneous cylinder with boundary conditions of several types are presented Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 74–85, October 2008.     

Interaction of crack-tip and notch-tip stress singularities for circular cylinder in torsion

Singular integral equations are used to formulate the torsion problem of a circular cylinder containing a polygonal opening and a line crack. The formulation is based on degenerating a system of connecting line cracks to that of a polygon, the sides of which coincides with the cracks. Considered, in particular, is the torsion of a circular cylinder with a rectangular hole and a nearby slanted line crack. Mode III stress intensity factors are computed at both ends of the crack to reflect their relative position to the rectangular hole in addition to change in the dimensions of the crack relative to the other geometric variables. Recognizing that the singular behavior of the stresses near a reentrant's corner differs from that of the crack tip, intensification of the local stresses at the corners of the rectangular hole is also examined. The results show the influence of the crack size and position.     

Elastic compensation simulation of elastic/plastic axisymmetric circular plate bending using a deformation model

A deformation plasticity model is used to implement the elastic compensation (iterative use of linearly elastic models to approximate elastic/plastic response) simulation of small axisymmetric circular plate deformations by the finite element method. Representative numerical results are presented graphically and used to demonstrate the utility of the approach.     

Nonlinear bending of size-dependent circular microplates based on the modified couple stress theory

The present study proposes a nonclassical Kirchhoff plate model for the axisymmetrically nonlinear bending analysis of circular microplates under uniformly distributed transverse loads. The governing differential equations are derived from the principle of minimum total potential energy based on the modified couple stress theory and von Kármán geometrically nonlinear theory in terms of the deflection and radial membrane force, with only one material length scale parameter to capture the size-dependent behavior. The governing equations are firstly discretized to a set of nonlinear algebraic equations by the orthogonal collocation point method, and then solved numerically by the Newton–Raphson iteration method to obtain the size-dependent solutions for deflections and radial membrane forces. The influences of length scale parameter on the bending behaviors of microplates are discussed in detail for immovable clamped and simply supported edge conditions. The numerical results indicate that the microplates modeled by the modified couple stress theory causes more stiffness than modeled by the classical continuum plate theory, such that for plates with small thickness to material length scale ratio, the difference between the results of these two theories is significantly large, but it becomes decreasing or even diminishing with increasing thickness to length scale ratio.     

Load-carrying capacity of stiffened non-circular cylinders in bending

Cylindrical shell beams of mono-symmetrical closed profile, stiffened by stringers and frames, are under consideration for a limit state of bending. Alternative systems of internal forces acting on the frame are determined for the cases:

1. (i) simultaneous deformations of the frame and the shell profile, provided that the sheet remains elastic, smooth and continuous with the frame,

2. (ii) when a second-order effect arises when a plastic hinge of the sheet is formed around the frame and the panel forces are no longer normal to the plane of the frame.

Comments on the buckling of the frame and collapse of the cylinder are added.     

Nonlinear bending of circular membranes

International Applied Mechanics -     

Effect of thermal singularities on stress intensity factors: edge crack in rectangular and circular plate

Thermoelastic solutions for cracked bodies under some types of thermal loading exhibit a thermal singularity at the crack tip as well as the familiar stress singularities. A quarter-point crack tip-element is presented which permits the accurate evaluation of these thermal singularities. They are important, since they affect the values of the stress intensity factors. These effects are accounted for by developing a new formulation for the calculation of the stress intensity factors. Several crack problems are solved and the results are compared where possible, with existing solutions.