Thin-walled Circular Cylinders Research Articles

Analysis of Stress-Strain State of a Cylinder with Variable Elasticity Moduli Based on Three-Dimensional Equations of Elasticity Theory

Introduction. Functionally graded materials are of great use, because heterogeneity of properties enables to control the strength and rigidity of structures. This has caused great interest in the topic in the world scientific literature. The construction of solutions to such problems depends significantly on the type of boundary conditions. In this paper, we consider the equilibrium of a thin-walled circular cylinder whose mechanical properties change along the radius. Homogeneous boundary conditions were set on cylindrical surfaces that had not been considered before, the effect was on the ends. The mathematical formulation of the problem was carried out in the linear theory of elasticity in the framework of axisymmetric deformation. Expressions were constructed for the components of the stress-strain state of the cylinder, in which some coefficients were found from the solution to the resulting system of linear algebraic equations.Materials and Methods. The material of the cylinder was linearly elastic, the elastic modulus of which depended linearly on the radial coordinate. The basic research method was the asymptotic method, in which half the logarithm of the ratio of the outer and inner radii acted as a small parameter. Iterative processes were used to construct the characteristics of the stress-strain state of the cylinder.Results. Homogeneous solutions to the boundary value problem were obtained for a linearly elastic functionally gradient hollow thin-walled cylinder. An analysis of these solutions made it possible to reveal the nature of the stress-strain state in the cylinder wall. For this purpose, an asymptotic analysis of the solutions was carried out, relations for displacements and stresses were obtained. It was determined that those solutions corresponded to the boundary layer, while their first terms determined Saint-Venant edge effect similar to the plate theory.Discussion and Conclusion. The analytical solution to the equilibrium problem of a thin-walled cylinder inhomogeneous in radius constructed by asymptotic expansion can be used for numerical solution to a specific problem. For this, it is required to solve the obtained systems of linear algebraic equations and determine the corresponding coefficients. The resulting asymptotic representations provide analyzing the three-dimensional stress-strain state. The selection of the number of expansion terms makes it possible to calculate displacements and stresses with a given degree of accuracy. This analysis can be useful in assessing the adequacy of applied calculation methods used in engineering practice.

Localization and snaking in axially compressed and internally pressurized thin cylindrical shells

Abstract This paper uncovers new manifestations of the homoclinic snaking mechanism in the post-buckling regime of a pressurized thin cylindrical shell under axial load. These new forms tend to propagate either wholly or partially in a direction that is orthogonal to the direction of the applied load and so, unlike earlier forms in Woods & Champneys (1999, Heteroclinic tangles in the unfolding of a degenerate Hamiltonian Hopf bifurcation. Phys. D, 129, 147–170), are fundamentally 2D in nature. The main effect of internal pressurization on the snaking mechanism is firstly to transition the circumferential multiplication of buckles from a one-tier pattern to a three-tier pattern. Secondly, internal pressurization can induce oblique snaking, whereby the sequential multiplication of buckles occurs in a helical pattern across the cylinder domain. For low levels of internal pressure, the single dimple remains—as in the unpressurized case—the unstable edge state that forms the smallest energy barrier around the stable pre-buckling equilibrium. For greater levels of pressure, the edge state changes to a single dimple surrounded by four smaller dimples. By tracing the limit point that denotes the onset of these edge states in the parameter space of internal pressure and axial load, we justify and validate the empirically derived design guideline for buckling of pressurized cylinders proposed by Fung & Sechler (1957, Buckling of thin-walled circular cylinders under axial compression and internal pressure. J. Aeronaut. Sci., 24, 351–356).

Derivation of reliability formula design for cylinder-elliptical head structure

Based on the JC method (reliability calculation method) and the fourth strength theory proposed by Lacvitz et al.[1] and recommended by the Joint Committee on International Security (JCSS), the reliability design formula for pressure vessel elliptical head and thin-walled circular cylinder is deduced. Monte Carlo reliability calculation method of ANSYS is used to verify the correctness of the formula, and the factors that have the greatest impact on the structural reliability are analyzed, which can provide some reference for the design of pressure vessel reliability.

Mechanical behaviour of a granular solid and its contacting deformable structure under uni-axial compression – Part I: Joint DEM–FEM modelling and experimental validation

This paper proposes an exquisite confined compression experiment to validate a recently developed numerical method (Joint DEM–FEM approach). A uni-axial compression apparatus was designed to investigate the mechanical behaviour of a granular solid under vertical loading and the load transmission to the containing wall. This tester consists of an instrumented thin-walled circular cylinder, two platens, a commercial dynamic material test machine (MTS 810), a bottom load cell and twelve bi-axial strain gage rosettes. Experiments were conducted on polystyrene beads. The delicately-designed confined compression test provides abundant experimental data for proving the validity of the proposed modelling technique.The corresponding simulations were performed by the developed numerical approach with the linking of discrete element method (DEM) and finite element method (FEM). To attain a high level of quantitative validation, the important particle properties required for DEM simulation were not simply assumed but measured directly in laboratory tests. Characteristics of the compression system, such as load–displacement response, force transmission to boundary surfaces, lateral pressure ratio, bulk wall friction, and stress distribution in the cylindrical wall, were evaluated. A comparison between the numerical and experimental results was made and discussed in this paper. The majority of compared physical quantities show reasonable to good agreement, thus giving a convincingly quantitative validation for the developed Joint DEM–FEM modelling technique. This proposed linking methodology is shown to give plausible results and can be used as a first step to solve the particulate solid–structure interaction problems.

An Analytical Method for Cylindrical Shells With Nozzles Due to Internal Pressure and External Loads—Part I: Theoretical Foundation

An improved version of the analytical solutions by Xue, Hwang and co-workers (1991, “Some Results on Analytical Solution of Cylindrical Shells With Large Opening,” ASME J. Pressure Vessel Technol., 113, 297–307; 1991, “The Stress Analysis of Cylindrical Shells With Rigid Inclusions Having a Large Ratio of Radii,” SMiRT 11 Transactions F, F05/2, 85–90; 1995, “The Thin Theoretical Solution for Cylindrical Shells With Large Openings,” Acta Mech. Sin., 27(4), pp. 482–488; 1995, “Stresses at the Intersection of Two Cylindrical Shells,” Nucl. Eng. Des., 154, 231–238; 1996, “A Reinforcement Design Method Based on Analysis of Large Openings in Cylindrical Pressure Vessels,” ASME J. Pressure Vessel Technol., 118, 502–506; 1999, “Analytical Solution for Cylindrical Thin Shells With Normally Intersecting Nozzles Due to External Moments on the Ends of Shells,” Sci. China, Ser. A: Math., Phys., Astron., 42(3), 293–304; 2000, “Stress Analysis of Cylindrical Shells With Nozzles Due to External Run Pipe Moments,” J. Strain Anal. Eng. Des., 35, 159–170; 2004, “Analytical Solution of Two Intersecting Cylindrical Shells Subjected to Transverse Moment on Nozzle,” Int. J. Solids Struct., 41(24–25), 6949–6962; 2005, “A Thin Shell Theoretical Solution for Two Intersecting Cylindrical Shells Due to External Branch Pipe Moments,” ASME J. Pressure Vessel Technol., 127(4), 357–368; 2005, “Theoretical Stress Analysis of Two Intersecting Cylindrical Shells Subjected to External Loads Transmitted Through Branch Pipes,” Int. J. Solids Struct., 42, 3299–3319) for two normally intersecting cylindrical shells is presented, and the applicable ranges of the theoretical solutions are successfully extended from d/D≤0.8 and λ=d/(DT)1/2≤8 to d/D≤0.9 and λ≤12. The thin shell theoretical solution is obtained by solving a complex boundary value problem for a pair of fourth-order complex-valued partial differential equations (exact Morley equations (Morley, 1959, “An Improvement on Donnell’s Approximation for Thin Walled Circular Cylinders,” Q. J. Mech. Appl. Math. 12, 89–91; Simmonds, 1966, “A Set of Simple, Accurate Equations for Circular Cylindrical Elastic Shells,” Int. J. Solids Struct., 2, 525–541)) for the shell and the nozzle. The accuracy of results is improved by some additional terms to the expressions for resultant forces and moments in terms of complex-valued displacement-stress function. The theoretical stress concentration factors due to internal pressure obtained by the improved expressions are in agreement with previously published test results. The theoretical results discussed and presented herein are in sufficient agreement with those obtained from three dimensional finite element analyses for all the seven load cases, i.e., internal pressure and six external branch pipe load components involving three orthogonal forces and the respective three orthogonal moments.

Dynamic Visco-elastic Buckling Analysis for Airway Model

In order to clarify the mechanism by which the lung airway narrows during an asthma attack, dynamic buckling analysis of the wall was conducted. The wall was modeled using a visco-elastic thin-walled circular cylinder of the Voigt model for the planestress state. A governing equation for dynamic buckling was derived, and in the equation, the contraction of smooth muscle was replaced by uniform inward transmural pressure. The non-dimensional parameters for the buckling wave number n were nondimensional retardation time τ, non-dimensional increasing velocity of inward transmural pressure β, thickness radius ratio α2, radius length ratio η, density ratio ζ, and Poisson's ratio ν. The validity of the theoretical model was confirmed by comparing the calculated wave number with that obtained from the experiment, in which a silicone rubber tube blended with silicone potting gel was used as the in vitro airway model. In addition, the wave number n increased with β. It was necessary to consider the damping effect of the tube model or the airway wall, and n increased by 1.5 to 2 due to the additional mass effect of surrounding tissues of the basement membrane in the airway wall.

Near-Field and Plane-Wave Electromagnetic Coupling into a Slotted Circular Cylinder: Hard or TE Polarization

An analytically tractable model is proposed in this initial study of the electromagnetic phenomena that control our ability to synthesize, by using a near-field source, the effect of plane-wave coupling through an aperture into the interior of a vehicle under test. An integral equation for the tangential electric field in the slot aperture of a perfectly conducting, infinitesimally thin-walled circular cylinder is solved using a basis set of Chebyshev polynomials that are properly weighted according to the static edge condition. The resulting matrix elements from a Galerkin procedure are computed to high precision upon extracting the logarithmic singularity of the kernel of the integral operator. Exact expressions for the matrix elements, in the form of rapidly convergent series of elementary terms, are constructed by isolating another logarithmic function of the aperture width. A minimization of the mean-square error between the true plane-wave response and that due to a near-field line-source establishes the optimal complex source strength of the near-field source

Inelastic shell instability of thin-walled circular cylinders under external hydrostatic pressure

This paper reports on experimental work carried out on nine thin-walled circular cylinders which were tested to destruction under external hydrostatic pressure. Seven of the cylinders failed through non-symmetric bifurcation buckling and two failed through axisymmetric collapse. The results were used from these tests, together with the results from other experiments, to produce a design chart which could be used for designing against the occurence of elastic and inelastic shell instability.

MACH NUMBER DEPENDENCE OF THE COMPRESSION WAVE GENERATED BY A HIGH-SPEED TRAIN ENTERING A TUNNEL

An analysis is made of the compression wave generated when a high speed train enters a tunnel. Operations in the near future are expected to be at Mach numbersMranging in value up to 0·4. Non-linear steepening of the wave in a very long tunnel exacerbates the environmental damage caused by the subsequent radiation of an impulsivemicro-pressure wavefrom the far end of the tunnel. The compression wave profile depends on train speed and the area ratio of the train and tunnel, and forM<0·2 can be estimated to a good approximation by regarding the local flow near the tunnel mouth during train entry asincompressible. The influence of higher train Mach numbers is investigated for the special case in which the tunnel is modelled by a thin-walled circular cylinder, of the type frequently used in model scale tests. This is done by representing the nose of the train by a distribution of monopole sources, and calculating their interaction with the mouth of the tunnel by using the exact acoustic Green's function for a semi-infinite, circular cylindrical tunnel. Empirical formulae valid up toM=0·6 are obtained for the compression wave amplitude and for the maximuminitialpressure gradient (which determines the amplitude of the micro-pressure wave). Predictions of the theory are found to be within 5% of measurements made in recent experiments.

Plastic axisymmetric collapse of thin-walled circular cylinders and cones under uniform external pressure

This paper reports on two theoretical investigations and an experimental investigation into the plastic axisymmetric buckling of two thin-walled conical shells and several thin-walled circular cylindrical shells, under uniform external pressure.One of the theoretical investigations was a non-linear finite element solution for plastic axisymmetric buckling, which gave good results.The other theoretical solution was for the plastic non-symmetric bifurcation buckling of thin-walled circular cylinders. This second solution was a very simple one and, although it was based on plastic lobar buckling, it gave very good predictions for plastic axisymmetric collapse. This latter observation prompted the conclusion that there may be a link between plastic lobar buckling and plastic axisymmetric buckling.

Energy-absorption abilities of CFRP cylinders during impact crushing

This paper describes the energy-absorption abilities of thin-walled circular cylinders made from carbon-fibre-reinforced plastic (CFRP) during crushing under impact loading. The tests are carried out with an impact testing machine, the crosshead mass of which is 21.5 kg, at a velocity of 8 m s −1 under the temperature range 20 to 130°C. The ply angles of the specimens are 15 and 75°, their dimensions are 1.2 mm in thickness, 30 mm in inner diameter and 100 mm in length. Load-time and displacement-time profiles are acquired precisely by using a newly designed load cell and an optical displacement transducer. The acquired force and displacement profiles are converted to the force and deformation relationship, which manifests the energy-absorption abilities. The specimens produce two different types of crushing mode and definite constant load levels which keep their energy-absorption ability high. The energy-absorption abilities and the crushing modes of the specimens change at temperatures above 60°C.

Bending and Twisting of 60Sn40Pb Solder Interconnects With Creep

An exact analysis is presented for the creep deformation of a thin-walled circular solder cylinder under the actions of bending and twisting moments. Dimensionless interaction curves and charts which relate the variables, interconnect geometry, solder material properties, bending moment, twisting moment, curvature rate, and twist rate are also provided for engineering practice convenience.

Experimental Measurements for Energy Absorption Abilities of Thin-Walled Circular Cylinders by Crushing Due to Impact Loading. On the Method of Measurement and Evaluation.

In this paper, axial crushing tests on thin-walled circular cylinders of aluminium were conducted on an impact testing machine accelerated by air pressure, which is installed in the authors' laboratory at Tokyo Institute of Technology, and measurements for the variations of the loads applied to the specimens and the crushing deformations with time during impact were made with considerable accuracy. The dynamic load-deformation diagrams were extracted from these measurements to estimate the energy absorption abilities of the specimens, thus determining the effects of dimension or material of the specimens and crushing mode. The above results were compared with those of static crushing tests and the differences were discussed.

A theoretical perspective on in-plane forming of sheet metal

The theory of in-plane homogeneous deformation of sheet metal is a classical topic. Nevertheless some practically important aspects remain to be investigated in depth. Among those treated here are (i) connexions between the local geometries of associated paths of stress and strain; (ii) the influence of non-isotropic hardening; (iii) intrinsic eigenstates under coupled loads and deformations: (iv) conditions for the initiation of diffuse instability. The analysis is applicable also to thin-walled circular cylinders; the effects of combined loading by internal pressure and longitudinal tension are examined in detail.

Plastic torsional buckling of thin-walled circular cylinders

Plastic torsional buckling of thin-walled circular cylinders

On partial contact of a thin-walled circular cylinder and a rigid half-space

A long circular cylinder with radius R and thickness t transmits a longitudinal eccentrically applied force to a rigid half-space. We study the influence of the eccentricity of the load and the ratio R t on the stress state of the cylinder using the theory of thin shells in bending. The elastic displacements are presented in the form of a Fourier expansion, the coefficients of which are found by the variational formulation of the problem combined with the penalty approach for a numerical implementation of contact conditions. Results are applied to certain problems of buckling of a circular bar and compared with previous approximate solutions.

Bending and Twisting of Pipes With Strain Hardening

A closed-form solution is presented for the small deformation analysis of a straight thin-walled circular cylinder subjected to the simultaneous action of bending and twisting moments. Dimensionless interaction curves and charts which relate the variables, bending moment, curvature, maximum effective strain, twisting moment, and shear strain are also provided for engineering practice convenience. The average stress-strain diagram of the cylinder is described by two straight lines. The result presented herein is not only a good approximation of a wide class of piping materials, but also provides a standard tool for estimating the accuracy of different direct schemes such as numerical integration, finite-difference, and finite-element methods.

Creep of Thin-Walled Circular Cylinder Under Axial Force, Bending, and Twisting Moments

An exact analysis is presented for the creep deformation of a thin-walled circular cylinder subjected to the simultaneous actions of axial force, bending, and twisting moments. Dimensionless interaction curves and charts which relate the variables, axial force, location of neutral-axis, bending moment, maximum effective strain rate, twisting moment, and shear strain rate are also provided for engineering practice convenience. The constitutive relationship of the cylinder is described by Prandtl-Nadai creep law.

The quasi-static crumpling of thin-walled circular cylinders and frusta under axial compression

Thin-walled circular cylinders, and frusta (truncated circular cones) of aluminium alloy were subjected to axial static loading. Their initial axial length and the outside diameter of the cylinders and frusta (the larger top end) were kept constant whilst their wall thickness was varied. The load-deformation or compression behaviour of the cylinder and frusta of various thicknesses and semi-apical angles were recorded and the modes of collapse were observed and studied. The main purpose of results for circular cylinders is to give a basis of comparison for the frusta. Relatively thick tubes fail by collapsing into circumferential axisymmetric rings and thin ones by folding progressively into diamond-shaped lobes after assuming an initially axisymmetric ring mode of deformation. In the latter case the initial axisymmetric rings developed into nonsymmetric diamond patterns (elliptic, triangular and square, etc.) as loading progressed. Also, initially non-symmetric diamond buckle patterns were observed to be characteristic modes of frustum collapse. Initial peak failure loads and mean post-buckling loads for the various modes of deformation were obtained experimentally. It is shown that buckling loads for both cylinders and frusta increase with increasing slenderness ‡ ratio ( t D for cylinders and t D ̄ for frusta) following, broadly, a parabolic law. Empirical expressions for the post-buckling load required to effect crumpling have been obtained.

Large strain inelastic behaviour of cylindrical tubes

The finite strain behaviour of thin-walled circular cylinders fixed at their ends and subjected to uniform internal pressure is examined. General transversely isotropic inelastic properties are assumed, and both time-independent plastic (elastoplastic) and rate-sensitive inelastic (viscoplastic) material response are considered. Flow theory and deformation theory-type constitutive laws are employed. The influence of constitutive properties and tube geometry on maximum pressure and localized necking instabilities is determined through finite element computations for selected types of anisotropy. Simple relations are developed for the pressure-deformation response and instability of general transversely isotropic materials, and these are shown to be very accurate for long tubes. The accuracy of elementary formulae commonly used in experimental investigations of constitutive relations is also discussed.