Elastic Cylindrical Shell Research Articles

04.01: Buckling of very short elastic cylinders with weld imperfections under uniform bending

ABSTRACTThe length‐dependent domains of behaviour of thin elastic cylindrical shells under uniform bending have recently received significant research attention. Ovalisation is known to affect very long cylinders which undergo significant cross‐sectional flattening before failing by local buckling. This effect is restrained by the end boundary condition in shorter cylinders, which fail instead by local buckling at moments close to the classical analytical prediction. In very short cylinders, however, even this local buckling is restrained by the end boundary, and failure occurs instead through the development of a destabilising meridional fold on the compressed side. Although this is a limit point instability under bending, ovalisation does not play any role at all. This ‘very short’ length domain has only recently been explored for the first time with the aid of finite element modelling.A brief overview of the nonlinear buckling behaviour of very short elastic cylinders under uniform bending is presented in this paper. Two types of edge rotational restraint are used to illustrate the influence of a varying support condition on the stability in this short length range. It is shown that short cylinders under bending do not suffer at all from local short‐wave buckling. Additionally, when the meridional dimension of such cylinders becomes particularly short the resulting numerical models may predict indefinite stiffening without a limit point, even when the shell is modelled using more complete 3D solid continuum finite elements. Idealised weld depressions, which are realistic representations of a systemic manufacturing defect, are used to demonstrate only a very mild sensitivity to geometric imperfections at such short lengths owing to a pre‐buckling stress state dominated by local compatibility bending. The topic should be of interest to researchers studying shell problems dominated by local bending with computational tools and designers of multi‐segment shells with very close segment spacing.

04.04: Buckling of elastic cylindrical shells under symmetric but non‐uniform bending moment distributions

ABSTRACTThe behaviour of cylindrical shells and circular hollow sections under uniform bending has long received significant research attention. This is mainly due to the phenomenon of Brazier cross‐sectional ovalisation which is known to lead to an almost 50% reduction in the buckling moment relative to the classical analytical prediction. However, very little research appears to have been carried out to investigate the ovalisation behaviour under non‐uniform bending moment distributions (BMDs). This appears to be because until only relatively recently, most studies on the topic were analytical in nature which made the buckling analysis of anything but a uniform BMD difficult. With the rise of modern computing it is now possible to undertake extensive parametric studies to investigate any desired load condition.This paper employs the finite element method to explore the elastic stability of perfect simply‐supported cylinders or tubes under load cases that represent symmetric parabolic and triangular BMDs, and to compare them with the behaviour under the hitherto most‐commonly studied uniform BMD. Each non‐uniform BMD is investigated by four different three‐dimensional shell loading arrangements. While these all share the same resultant BMD, they exhibit very different buckling behaviours illustrating the ambiguity inherent in realistically modelling such systems. This is a consequence of the loss of uniqueness when projecting a 1D construct onto a 3D space.In addition to linear bifurcation analyses, nonlinear elastic buckling analyses are presented to explore the effect of ovalisation at long lengths under the proposed loading arrangements. It is found that ovalisation becomes fully developed at approximately the same dimensionless length and leads to a similar ∼50% reduction in the buckling strength of asymptotically long cylinders for almost any representation of the three considered symmetric BMDs.

Buckling of very short elastic cylinders with weld imperfections under uniform bending

The length-dependent behaviour domains of thin elastic cylindrical shells under uniform bending have recently received significant research attention. Ovalization is known to affect very long cylinders that undergo significant cross-sectional flattening before failing by local buckling. This effect is restrained by the end boundary conditions in shorter cylinders, which instead fail by local buckling at moments close to the classical analytical prediction. In very short cylinders, however, even this local buckling is restrained by the end boundary, and failure occurs instead through the development of a destabilizing meridional fold on the compressed side. Although this is a limit point instability under bending, ovalization does not play any role at all. This ‘very short’ length domain has only recently been explored for the first time with the aid of finite element modelling. A brief overview of the non-linear buckling behaviour of very short elastic cylinders under uniform bending is presented in this paper. Two types of edge rotational restraint are used to illustrate the influence of a varying support condition on the stability in this short length range. It is shown that short cylinders under bending do not suffer at all from local short-wave buckling. Additionally, when the meridional dimension of such cylinders becomes particularly short, the resulting numerical models may predict indefinite stiffening without a limit point, even when the shell is modelled using more complete 3D solid continuum finite elements. Idealized weld depressions, which are realistic representations of a systemic manufacturing defect, are used to demonstrate only a very mild sensitivity to geometric imperfections at such short lengths owing to a pre-buckling stress state dominated by local compatibility bending. The topic should be of interest to researchers studying shell problems dominated by local bending with computational tools and designers of multi-segment shells with very close segment spacing.

Characterization and prediction of the backscattered form function of an immersed cylindrical shell using hybrid fuzzy clustering and bio-inspired algorithms

The acoustic scattering of a plane wave by an elastic cylindrical shell is studied. A new approach is developed to predict the form function of an immersed cylindrical shell of the radius ratio b/a (‘b’ is the inner radius and ‘a’ is the outer radius). The prediction of the backscattered form function is investigated by a combined approach between fuzzy clustering algorithms and bio-inspired algorithms. Four famous fuzzy clustering algorithms: the fuzzy c-means (FCM), the Gustafson-Kessel algorithm (GK), the fuzzy c-regression model (FCRM) and the Gath-Geva algorithm (GG) are combined with particle swarm optimization and genetic algorithm. The symmetric and antisymmetric circumferential waves A, S0, A1, S1 and S2 are investigated in a reduced frequency (k1a) range extends over 0.1<k1a<225 (k1 is the wave number). The time-frequency representation of Smoothed Pseudo Wigner-Ville (SPWV) is applied on the predicted and calculated acoustic backscattered form functions. This representation is used as a comparison criterion between the calculated form function by the analytical method and that predicted by the proposed approach on the one hand and is used to extract the predicted cut-off frequencies on the other hand. Moreover, the transverse velocity of the material constituting the cylindrical shell is extracted. The computational results show that the proposed approach is very efficient to predict the form function and consequently, for acoustic characterization purposes.

Nonlinear stability of elastic elliptical cylindrical shells under uniform bending

Cylindrical metal shells with elliptical cross-sections are gaining increasing popularity as hollow sections due to their unique aesthetic appearance and different geometric properties about their two principal axes, with one axis exhibiting properties that are significantly more favourable than the other under flexure. However, in comparison with other hollow geometries, elliptical cross-sections have only recently begun receiving significant research attention. This is partly because even simple analytical treatments inevitably encounter cumbersome elliptical integrals that have no closed-form solutions, a problem now attenuated by powerful modern computing capabilities.A recent computational study investigated the nonlinear buckling resistance of perfect elastic circular cylindrical shells under uniform bending, establishing four distinct length-dependent domains of behaviour and characterising these in compact form using specially chosen dimensionless parameters. The present study extends this work to cylinders with elliptical cross-sections under bending about both principal axes. The same qualitative domains of length-dependent nonlinear elastic behaviour are found as for circular cylinders, but requiring a different algebraic characterisation that takes account of the varying elliptical radii. On the basis of computational results, a reference equation for the moment governing the ‘Brazier’ ovalisation cross-sectional failure mode for long elliptical thin-walled cylinders is deduced and presented for publication for the first time.

Applicability of a Structurally Orthotropic Model in Problems on Tension, Bending, and Stability of Densely Perforated Plates and Shells

The applicability of a structurally orthotropic model to the calculation of perforated plates and cylindrical shells subjected to tension and bending is studied by the finite-element method. The parameters of the orthotropic material are used in the form of coefficients of stiffness reduction. They are determined from the solution to the problem on deformation of a cyclically repeating structural element, with a varying degree of perforation (porosity), in tension, shear, and bending. The structural element is investigated by the methods of continuum mechanics and the theory of Timoshenko-type shells, and the limit of applicability of the theory of shells to such problems is found. The numerical results obtained are compared with the analytical estimates given by E. I. Grigolyuk and L. A. Filshtinskii. Verification of the numerically obtained orthotropic parameters is carried out based on the solution to the problem of bending of one quarter of a cylindrical strip and a plate perforated with one row of holes. It is shown that the approach chosen is applicable to perforated plates and shells in bending problems with waves whose length exceeds the characteristic size of their structural element. The stability of a perforated elastic cylindrical shell under external pressure, with two variants of boundary conditions, is investigated. Values of the critical pressure and the corresponding buckling modes in relation to the length of the shell and the degree of perforation are obtained.

Application of a Polymer Compound to a Plane Surface by an Elastic Plate

A mathematical model of the hydroelasticity of a flow of a non-Newtonian fluid in a wedge-shaped clearance with an elastic wall has been constructed. This wall represents a thin cylindrical elastic shell inextensible along its central axis, for which the Kirchhoff–Love hypotheses are true. A fluid flow in the clearance was simulated in the approximation of the Reynolds lubrication theory with the use of an asymptotic approximation for the second invariant of the strain rate tensor. Results of numerical investigations with the mathematical model proposed are presented.

Calculation method of locking torque based on elastic thin shell theory for hydraulic locking sleeve

The hydraulic locking sleeve is a key component of precision instruments such as five-axis machine tools, giant astronomical telescope, and satellite antenna. This is subjected to the action of pressure load causing large elastic deformation and locking the rotational freedom of feed shaft at any angle. The maximum locking torque is an important parameter for designing the hydraulic locking sleeve. First, the hydraulic locking sleeve is simplified as elastic thin cylindrical shell structure. Neglecting the bending and twisting effects, the calculation equations describing the deformation and stress state between the hydraulic locking sleeve and rotary shaft are derived by applying the theory of elastic thin shell. Then, taking into account that one end of the hydraulic lock sleeve is fixed to the shaft sleeve seat by the end face flange; the calculating formula of the maximum locking torque of the hydraulic locking sleeve is obtained by modifying the deformation equation based on moment model. Finally, a test platform of hydraulic locking sleeve is designed, which can measure the maximum locking torque of the hydraulic locking sleeve. The error between the calculation result of locking torque theoretical calculation model and the experimental measured value is &lt;15%. As a result, the causes of the error are analyzed, and the effects of the shaft sleeve length, wall thickness, and radius on the maximum locking torque are calculated.

Распространение нелинейных волн в соосных физически нелинейных цилиндрических оболочках, заполненных вязкой жидкостью

Investigation of deformation waves behavior in elastic shells is one of the important trends in contemporary wave dynamics. There exist mathematical models of wave motions in infinitely long geometrically non-linear shells, containing viscous incompressible liquid, based on the related hydroelasticity problems, which are derived by the shells dynamics and viscous incompressible liquid equations in the form of generalized KdV equations. Also, mathematical models of the wave process in infinitely long geometrically non-linear coaxial cylindrical elastic shells are obtained by means of disturbances method. These models differ from the known ones by the consideration of incompressible liquid presence between the shells, based on the related hydroelasticity problems. These problems are described by shells dynamics and viscous incompressible liquid equations with corresponding edge conditions in the form of generalized KdV equations system. The paper presents the investigation of wave occurrences of two geometrically non-linear elastic coaxial cylindrical shells model of Kirchhoff-Love type, containing viscous incompressible liquid between them, as well as inside. The difference schemes of Crank-Nicholson type are obtained for the considered equations system by taking into account liquid impact and with the help of Gro¨bner bases construction. To generate these difference schemes, the basic integral difference correlations, approximating initial equations system, were used. The usage of Gro¨bner bases technology provides generating the schemes, for which it becomes possible to obtain discrete analogs of the laws of preserving initial equations system. To do this, equivalent transformations were made. On the basis of computation algorithm the complex of programs, permitting to construct graphs and obtain numerical solutions under exact solutions of coaxial shell dynamics equations system, was made.

Vibro-acoustics of infinite and finite elastic fluid-filled cylindrical shells

The classical model of an elastic fluid-filled cylindrical shell is used for analysis of its vibrations. The model is based on thin shell theory, standard linear acoustics and the heavy fluid-loading coupling concept. First, several important features of performance of a fluid-filled shell not yet fully explored in literature e.g. the difference between kinematic/forcing excitations, acoustic source type identification (monopole/dipole) and energy transfer between fluid and shell are studied. Then the discussion is extended to finite fluid-filled shells by application of the Boundary Integral Equations Method (BIEM). Two techniques for solving the equations deduced from the BIEM are discussed and investigated with respect to convergence, respectively, Boundary Elements (BE) and modal expansion. Successively, the implementation of the BIEM is validated against numerical and experimental results for the simplified case of an empty shell. Finally, impedance boundary conditions for a fluid-filled shell in an assembled piping system and computations of its resonances and forced response using the BIEM are discussed.

Acoustic scattering reduction using layers of elastic materials

Making an object invisible to acoustic waves could prove useful for military applications or measurements in confined space. Different passive methods have been proposed in recent years to avoid acoustic scattering from rigid obstacles. These techniques are exclusively based on acoustic phenomena, and use for instance multiple resonators or scatterers. This paper examines the possibility of designing an acoustic cloak using a bi-layer elastic cylindrical shell to eliminate the acoustic field scattered from a rigid cylinder hit by plane waves. This field depends on the dimensional and mechanical characteristics of the elastic layers. It is computed by a semi-analytical code modelling the vibrations of the coating under plane wave excitation. Optimization by genetic algorithm is performed to determine the characteristics of a bi-layer material minimizing the scattering. Considering an external fluid consisting of air, realistic configurations of elastic coatings emerge, composed of a thick internal orthotopic layer and a thin external isotropic layer. These coatings are shown to enable scattering reduction at a precise frequency or over a larger frequency band.

Simplified Analytical Approach to Evaluate the Nonlinear Dynamics of Elastic Cylindrical Shells Under Lateral Blast Loads

This paper presents an analytical solution to predict the nonlinear forced vibrations of elastic thin-walled cylindrical shells under suddenly applied loads. Interest in this problem is motivated by effects due to explosions on fluid-storage metal tanks. The model is based on the energy criterion due to Lagrange, in which the kinematic nonlinear relations are assumed using Donnell's simplified shell theory. Solution is achieved as a series summation in terms of trigonometric functions in the axial and circumferential directions, whereas the degrees of freedom depend on time. A blast load is assumed to represent effects due to explosions on the shell as time-dependent pressures with a given circumferential distribution (a cosine square distribution in terms of the central angle). The procedure is validated by comparison with a nonlinear finite element model under the same load conditions. The influence of load level and shell geometry on the transient response is investigated by mean of parametric studies. Good accuracy is found in the results for the range of shells which are representative of horizontal, fuel storage tanks in the oil industry.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ВОЛНОВЫХ ЯВЛЕНИЙ В ДВУХ ГЕОМЕТРИЧЕСКИ НЕЛИНЕЙНЫХ УПРУГИХ СООСНЫХ ЦИЛИНДРИЧЕСКИХ ОБОЛОЧКАХ, СОДЕРЖАЩИХ ВЯЗКУЮ НЕСЖИМАЕМУЮ ЖИДКОСТЬ

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ВОЛНОВЫХ ЯВЛЕНИЙ В ДВУХ ГЕОМЕТРИЧЕСКИ НЕЛИНЕЙНЫХ УПРУГИХ СООСНЫХ ЦИЛИНДРИЧЕСКИХ ОБОЛОЧКАХ, СОДЕРЖАЩИХ ВЯЗКУЮ НЕСЖИМАЕМУЮ ЖИДКОСТЬ

A high transmission broadband gradient index lens using elastic shell acoustic metamaterial elements.

The use of cylindrical elastic shells as elements in acoustic metamaterial devices is demonstrated through simulations and underwater measurements of a cylindrical-to-plane wave lens. Transformation acoustics of a circular region to a square dictate that the effective density in the lens remain constant and equal to that of water. Piecewise approximation to the desired effective compressibility is achieved using a square array with elements based on the elastic shell metamaterial concept developed by Titovich and Norris [J. Acoust. Soc. Am. 136(4), 1601-1609 (2014)]. The sizes of the elements are chosen based on availability of shells, minimizing fabrication difficulties. The tested device is neutrally buoyant comprising 48 elements of nine different types of commercial shells made from aluminum, brass, copper, and polymers. Simulations indicate a broadband range in which the device acts as a cylindrical to plane wave lens. The experimental findings confirm the broadband quadropolar response from approximately 20 to 40 kHz, with positive gain of the radiation pattern in the four plane wave directions.

Semi-analytical technique for isolating the pseudo-Rayleigh component of the field induced by a transiently responding submerged cylindrical shell

A semi-analytical technique is proposed for isolating the pseudo-Rayleigh (A0) component of the field radiated into the surrounding fluid by a submerged elastic cylindrical shell. The technique is based on the simultaneous use of two fluid–shell interaction models, one based on the Reissner–Mindlin shell theory, and the other on the Kirchhoff–Love shell theory, and is aimed at offering the possibility of analyzing the pseudo-Rayleigh component in its pure form, an approach that has certain rather important advantages over analyzing the component as a part of the overall radiated hydrodynamic pattern. The technique is applied to the analysis of the radiation by two elastic shells with parameters that are common in industry, and the high computational efficiency of the technique is demonstrated.

Vibrations of an elastic cylindrical shell near the lowest cut-off frequency

A new asymptotic approximation of the dynamic equations in the two-dimensional classical theory of thin-elastic shells is established for a circular cylindrical shell. It governs long wave vibrations in the vicinity of the lowest cut-off frequency. At a fixed circumferential wavenumber, the latter corresponds to the eigenfrequency of in-plane vibrations of a thin almost inextensible ring. It is stressed that the well-known semi-membrane theory of cylindrical shells is not suitable for tackling a near-cut-off behaviour. The dispersion relation within the framework of the developed formulation coincides with the asymptotic expansion of the dispersion relation originating from full two-dimensional shell equations. Asymptotic analysis also enables refining the geometric hypotheses underlying various ad hoc set-ups, including the assumption on vanishing of shear and circumferential mid-surface deformations used in the semi-membrane theory. The obtained results may be of interest for dynamic modelling of elongated cylindrical thin-walled structures, such as carbon nanotubes.

Multi-layered elastic shell metamaterial elements for improved transmission in sonic crystals

A metamaterial-based sonic crystal, comprised of a lattice arrangement of cylindrical elastic shells, can be tuned to provide wave steering or phononic band filtering. The effective acoustic properties of such sonic crystals are determined by the geometric and material properties of each individual shell and the lattice spacing between shells. By matching the effective acoustic properties to those of the surrounding fluid, the sonic crystal can be made acoustically transparent at low frequencies, leading to improved transmission over that of non-matched shells. Bi-layered shells provide additional design parameters to broaden the region of acoustic transparency provided by a simple shell. In this effort, three-layered shells are employed to improve transmission further. Transformation acoustics analysis is applied to determine theoretical density and bulk modulus distributions, and then real-world materials are selected which best match these theoretical distributions. Finally, shell geometry is optimized to minimize scattering while constraining the effective acoustic properties to match those of the external fluid.

Theoretical and experimental study of a thin elastic cylindrical shell subjected to a point source

The aim of this work is to study theoretically and experimentally, the acoustic radiation of a cylindrical elastic shell excited by a point source with no internal loading and immersed in a water tank. We are primarily focused on the theoretical study of a cylindrical elastic shell receiving a point force in the form of a Dirac pulse and the acoustic pressure radiated along the tube into a point at far-field observation. The model used is based on the theory of elasticity. It is therefore interesting, in a second step, to conduct experiments to validate these theoretical calculations and to realize complementary measures corresponding to the case of a focalized observer. The used shell is a cylindrical stainless steel tube characterized by its ratio equal to 0.94. Two experiments are carried out. The first experiment is based on a bistatic method, and the transmission and reception are performed by two focalized transducers, characterized by a center frequency of 1 MHz. For the second experiment, the two transducers are different in nature; the transducer acting as the receiver is replaced by a plane transducer characterized by a center frequency identical to the transmitter focalized transducer

Modeling of offshore pile driving noise using a semi-analytical variational formulation

Underwater noise radiated from offshore pile driving got much attention in recent years due to its threat to the marine environment. This study develops a three-dimensional semi-analytical method, in which the pile is modeled as an elastic thin cylindrical shell, to predict vibration and underwater acoustic radiation caused by hammer impact. The cylindrical shell, subject to the Reissner–Naghdi’s thin shell theory, is decomposed uniformly into shell segments whose motion is governed by a variational equation. The sound pressures in both exterior and interior fluid fields are expanded as analytical functions in frequency domain. The soil is modeled as uncoupled springs and dashpots distributed in three directions. The sound propagation characteristics are investigated based on the dispersion curves. The case study of a model subject to a non-axisymmetric force demonstrates that the radiated sound pressure has dependence on circumferential angle. The case study including an anvil shows that the presence of the anvil tends to lower the frequencies and the amplitudes of the peaks of sound pressure spectrum. A comparison to the measured data shows that the model is capable of predicting the pile driving noise quantitatively. This mechanical model can be used to predict underwater noise of piling and explore potential noise reduction measures to protect marine animals.

Theoretical Study on Fluid Velocity for Viscous Fluid in a Circular Cylindrical Shell

A theoretical algorithm by united Lagrangian-Eulerian method for the problem in dealing with viscous fluid and a circular cylindrical shell is presented. In this approach, each material is described in its preferred reference frame. Fluid flows are given in Eulerian coordinates whereas the elastic circular cylindrical shell is treated in a Lagrangian framework. The fluid velocity in a two-dimensional uniform elastic circular cylindrical shell filled with viscous fluid is studied under the assumption of low Reynolds number. The coupling between the viscous fluid and the elastic circular cylindrical shell shows kinematic conditions at the shell surface. Also, the radial velocity and axial velocity of the fluid are discussed with the help of graphs.