Effect of the geometry on the non-linear vibration of circular cylindrical shells

The non-linear vibration of simply supported, circular cylindrical shells is analysed. Geometric non-linearities due to finite-amplitude shell motion are considered by using Donnell's non-linear shallow-shell theory; the effect of viscous structural damping is taken into account. A discretization method based on a series expansion of an unlimited number of linear modes, including axisymmetric and asymmetric modes, following the Galerkin procedure, is developed. Both driven and companion modes are included, allowing for travelling-wave response of the shell. Axisymmetric modes are included because they are essential in simulating the inward mean deflection of the oscillation with respect to the equilibrium position. The fundamental role of the axisymmetric modes is confirmed and the role of higher order asymmetric modes is clarified in order to obtain the correct character of the circular cylindrical shell non-linearity. The effect of the geometric shell characteristics, i.e., radius, length and thickness, on the non-linear behaviour is analysed: very short or thick shells display a hardening non-linearity; conversely, a softening type non-linearity is found in a wide range of shell geometries.     

On the accuracy of the multiple scales method for non-linear vibrations of doubly curved shallow shells

Non-linear free and forced vibrations of doubly curved isotropic shallow shells are investigated via multi-modal Galerkin discretization and the method of multiple scales. Donnell’s non-linear shallow shell theory is used and it is assumed that the shell is simply supported with movable edges. By deriving two different forms of the stress function, the equations of motion are reduced to a system of infinite non-linear ordinary differential equations with quadratic and cubic non-linearities. A quadratic relation between the excitation and the fundamental frequency is considered and it is shown that, although in case of hardening non-linearities the results resemble those found via numerical integration or continuation softwares, in case of softening non-linearity the solution breaks down as the amplitude becomes larger than the thickness. Results reveal that, expressing the relation between the excitation and fundamental frequency in this form, which was considered by many researchers as a useful tool in analyzing strong non-linear oscillators, yields in spurious results when the non-linearity becomes of softening type.     

Effect of foundation nonlinearity on the nonlinear transient response of orthotropic shallow spherical shells

Summary Interaction of clamped orthotropic shallow spherical shells with nonlinear elastic foundations is studied under transient loads. The effect of softening and hardening foundation nonlinearities on the response behaviour of shallow shells has been investigated. Detailed analysis depicting the influence of hardening type foundation nonlinearity on the maximum response of orthotropic shallow spherical shells has been conducted. The numerical results suggest that for the shell-foundation interaction problems undergoing moderately large deformations, the nonlinear model for the foundation must be considered.

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Einfluß der Bettungsnichtlinearität auf die nichtlineare transiente Antwort von orthotropen, leicht gekrümmten Schalen

übersicht Es wird die Wechselwirkung eingespannter orthotroper, leicht gekrümmter Schalen mit nichtnichtlinearer elastischer Bettung unter transienter Belastung behandelt. Dabei wird der Einfluß der erweichenden und der verfestigenden Nichtlinearität der Bettung auf das Antwortverhalten der schwach gekrümmten Schalen untersucht. Speziell wird der Einfluß der verfestigenden Bettungsnichtlinearität auf extreme Antwortverhalten von orthotropen, schwach sphärischen Schalen im einzelnen analysiert. Die numerischen Ergebnisse machen deutlich, daß bei Wechselwirkungsproblemen von Schale und Bettung mit mäßig großen Deformationen das nichtlineare Modell der Einbettung betrachtet werden muß.

     

Nonlinear free vibration of reticulated shallow spherical shells taking into account transverse shear deformation

This paper deals with nonlinear free vibration of reticulated shallow spherical shells taking into account the effect of transverse shear deformation. The shell is formed by beam members placed in two orthogonal directions. The nondimensional fundamental governing equations in terms of the deflection, rotational angle, and force function are presented, and the solution for the nonlinear free frequency is derived by using the asymptotic iteration method. The asymptotic solution can be used readily to perform the parameter analysis of such space structures with numerous geometrical and material parameters. Numerical examples are given to illustrate the characteristic amplitude-frequency relation and softening and hardening nonlinear behaviors as well as the effect of transverse shear on the linear and nonlinear frequencies of reticulated shells and plates.     

Koiter circles in the buckling of axially compressed conical shells

As is well known, the elastic stability of shell structures under certain loading conditions is characterised by a severely unstable postbuckling behaviour. The presence of simultaneous buckling modes (‘competing’ modes corresponding to the same critical buckling load) is deemed to be largely responsible for such a behaviour. In the present paper, within the framework of the so-called classical theory (linear bifurcation eigenvalue analysis), the buckling behaviour of axially compressed cylindrical shells is firstly reviewed. Accordingly, doubly periodic eigenvectors (buckling modes) corresponding to the same eigenvalue (critical buckling load) can be determined, and their locus in a dimensionless meridional and circumferential buckling wavenumber space is described by a circle (known as the Koiter circle). In the case of axially compressed conical shells, no clear evidence of the existence of simultaneous buckling modes can be found in the literature. Then, such a problem is studied here via linear eigenvalue finite element analyses, showing that simultaneous doubly periodic modes do also occur for cones, and that their locus in a specifically defined dimensionless wavenumber space can be described by an ellipse (hereafter termed as the Koiter ellipse) whose aspect ratio is dependent on the tapering angle of the cone.     

Non-local effects on the non-linear modes of vibration of carbon nanotubes under electrostatic actuation

Carbon nanotubes (CNTs) based NEMS with electrostatic sensing/actuation may be employed as sensors, in situations where it is fundamental to understand their dynamic behaviour. Due to displacements that are large in comparison with the thickness and to the non-linearity of the electrostatic force, these CNT based NEMS operate in the non-linear regime. The knowledge of the modes of vibration of a CNT provides a picture of what one may expect from its dynamic behaviour not only in free, but also in forced vibrations. In this paper, the non-linear modes of vibration of CNTs actuated by electrostatic forces are investigated. For that purpose, a p-version finite element type formulation is implemented, leading to ordinary differential equations of motion in the time domain. The formulation takes into account non-local effects, which influence the inertia and the stiffness of CNTs, as well as the electrostatic actuation. The ordinary differential equations of motion are transformed into algebraic equations of motion via the harmonic balance method (HBM) and then solved by an arc-length continuation method. Several harmonics are considered in the HBM. The importance of non-local effects, combined with the geometrical non-linearity and with the action of the electrostatic force, is analysed. It is found that different combinations of these effects can result in alterations of the natural frequencies, variations in the degrees of softening or hardening, changes in the frequency content of the free vibrations, and alterations in the mode shapes of vibration. It is furthermore found that the small scale, here represented by the non-local theory, has an effect on interactions between the first and higher order modes which are induced by the geometrical and material non-linearities of the system.     

VIBRATIONS OF CIRCULAR CYLINDRICAL SHELLS WITH NONUNIFORM CONSTRAINTS,ELASTIC BED AND ADDED MASS; PART I: EMPTY AND FLUID-FILLED SHELLS

Circular cylindrical shells with non uniform edge constraints (with zero radial and circumferential displacement) are investigated, including riveted shells. The linear modes of simply supported shells vibrating in vacuo are used as admissible functions, and the solution is obtained with the artificial spring method. The Flügge theory of shells is used and in-plane inertia is retained. Any shell constraint other than simple supports can be studied with the proposed method. Complicating effects due to the contained inviscid fluid, elastic bed of partial axial and angular dimensions, intermediate constraints and added mass are considered. The convergence of the method is numerically investigated and the effect of the number of rivets (clamped arcs) on shell modes is studied.     

A study of different modes of collapse in metallic hemispherical shells resting on flat platen and compressed with hemispherical nosed indenter

The paper presents experimental and analytical studies on axial compression of aluminium spherical shells having Radius/wall thickness (R/t) ratios between 23 and 135. Quasi-static compressive load was applied centrally and with offset through a indenter having diameter of 22 mm. Testing was carried out on an INSTRON machine having 250 T capacity. Shells having different radius and wall thickness were tested, to classify their modes of collapse and their corresponding energy absorption mechanism. In experiments shells of lower R/t values were found to collapse due to formation of an inward dimple associated with a rolling plastic hinge in central as well as in offset loading. On the other hand, shells of higher R/t values were collapsed initially with formation of an axisymmetric inward dimple, but in later stage of compression showed buckling of non-symmetric shape consisting of integral number of lobes and stationary plastic hinges. The stationary hinges were formed between consecutive lobes. Experimental observations are used to propose an analytical model for prediction of load–compression and energy–compression curves. The results obtained from analytical model compared with the experimental results and found match fairly well.     

Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects

This paper presents an analytical approach to investigate the non-linear axisymmetric response of functionally graded shallow spherical shells subjected to uniform external pressure incorporating the effects of temperature. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and compatibility equations for shallow spherical shells are derived by using the classical shell theory and specialized for axisymmetric deformation with both geometrical non-linearity and initial geometrical imperfection are taken into consideration. One-term deflection mode is assumed and explicit expressions of buckling loads and load-deflection curves are determined due to Galerkin method. Stability analysis for a clamped spherical shell shows the effects of material and geometric parameters, edge restraint and temperature conditions, and imperfection on the behavior of the shells.     

Nonlinear modes of cylindrical panels with complex boundaries. <Emphasis Type="Italic">R</Emphasis>-function method

Nonlinear vibrations of cylindrical panels with complex base are analyzed. The Donnell-Mushtari-Vlasov equations with respect to displacements are used to study vibrations of shallow shell with geometrical nonlinearity. R-function method is applied to satisfy the panel boundary conditions. The Rayleigh-Ritz method is used to obtain the linear vibrations eigenmodes, which contain R-function. The nonlinear vibrations of panel are expanded by using these eigenmodes. The harmonic balance method and nonlinear normal modes are used to study the free nonlinear vibrations.     

Stability of Axially Compressed Noncircular Cylindrical Shells Consisting of Panels of Constant Curvature

The stability problem is solved for an axially compressed cylindrical shell. Its cross section is formed by circular arcs of radius r with ends supported on a closed circle of radius R. The solution is based on the Flügge equations of the classic theory of deep cylindrical shells. It is shown that the critical axial load for shells of medium length and appropriately chosen cross-sectional profile can be increased by a factor of R/r approximately, compared with the circular shell. The shells length affects considerably the efficiency of noncircular shells of this type. This design model allows us to find out how the local properties of the shell and its stiffness are related     

Non-linear static and dynamic response of spherical shells

In the present investigation an analytical technique using Chebyshev series expansion has been presented and used to study the non-linear transient behavior of shallow spherical shells with and without damping. The two coupled non-linear differential equation governing the shallow shell behavior are initially linearised using Taylor's series expansion. Results have been presented for three types of transient loadings, namely, step function, N-shaped pulse and sine wave pulse. It is shown that accurate results can be obtained using a five terms Chebyshev series expansion which is unlikely in the conventional methods.     

Nonlinear vibrations of cylindrical shells with initial imperfections in a supersonic flow

The paper studies the dynamics of nonlinear elastic cylindrical shells using the theory of shallow shells. The aerodynamic pressure on the shell in a supersonic flow is found using piston theory. The effect of the flow and initial deflections on the vibrations of the shell is analyzed in the flutter range. The normal modes of both perfect shells in a flow and shells with initial imperfections are studied. In the latter case, the trajectories of normal modes in the configuration space are nearly rectilinear, only one mode determined by the initial imperfections being stable __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 63–73, September 2007.     

Study of vibration characteristics for orthotropic circular cylindrical shells using wave propagation approach and multivariate analysis

A study of free vibration of orthotropic circular cylindrical shells is presented. The vibration control equations of shells are based on Flügge classical thin shell theory. Wave approach is used in the analysis, in which the boundary conditions of shells can be simplified according to the associated beam. The free vibration frequencies of shells can be obtained from a frequency polynomial equation of order 6. The parametric analysis of the free vibration of orthotropic cylindrical shells is investigated using a statistical method. The effects of geometrical parameters and material characteristics upon frequencies are investigated here. Multivariate analysis (MVA) can be a useful tool for this parametric study. Some statistical characteristics, including correlation analysis and ANOVA are applied. ANOVA has been conducted to predict the statistical significance of the various factors. Calculations are performed in the Minitab statistical software. The results show that the L/R, h/R and m have larger effects on the lowest frequency. The importance of input parameters is ranked according to their contributions to the total variance. A knowledge and data visualization approach, Self-organizing mapping (SOM) is also adopted here for mining some intrinsic characteristics of shells.     

Nonlinear long-term behaviour of spherical shallow thin-walled concrete shells of revolution

The nonlinear long-term buckling behaviour (creep buckling) of spherical shallow, thin-walled concrete shells of revolution (including domes) subjected to sustained loads is investigated herein. A thorough understanding of their nonlinear time-dependent behaviour, as well as the development of comprehensive analytical models for their analysis, has hitherto not been fully established and further studies are required. A nonlinear axisymmetric theoretical model, which accounts for the effects of creep and shrinkage, and which considers the ageing of the concrete material and the variation of the internal stresses and geometry in time, is developed for this purpose. The governing field equations are derived using variational principles, equilibrium requirements, and integral-type constitutive relations. A systematic step-by-step procedure is used for the solution of the integral-type governing equations. First, the nonlinear short-term behaviour is studied to provide a benchmark for the long-term analysis. Different theories for the analysis of the shell structure are examined for this purpose and compared with results obtained by the finite element method. A numerical study, which highlights the capabilities of the nonlinear theoretical model and which provides insight into the nonlinear long-term behaviour of shallow concrete domes, is presented. The results show that long-term effects are critical for the design and structural safety of shallow, thin-walled concrete domes, and so these effects need to be fully understood and quantifiable.     

The effect of material and geometry on the non-linear vibrations of orthotropic circular cylindrical shells

The extensive use of circular cylindrical shells in modern industrial applications has made their analysis an important research area in applied mechanics. In spite of a large number of papers on cylindrical shells, just a small number of these works is related to the analysis of orthotropic shells. However several modern and natural materials display orthotropic properties and also densely stiffened cylindrical shells can be treated as equivalent uniform orthotropic shells. In this work, the influence of both material properties and geometry on the non-linear vibrations and dynamic instability of an empty simply supported orthotropic circular cylindrical shell subjected to lateral time-dependent load is studied. Donnell׳s non-linear shallow shell theory is used to model the shell and a modal solution with six degrees of freedom is used to describe the lateral displacements of the shell. The Galerkin method is applied to derive the set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge–Kutta method. The obtained results show that the material properties and geometric relations have a significant influence on the instability loads and resonance curves of the orthotropic shell.     

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.     

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.     

Experimental and numerical investigations into collapse behaviour of thin spherical shells under drop hammer impact

A study of the collapse behaviour of hemi spherical and shallow spherical shells and their modes of deformation under impact loading are presented in this paper. Aluminium spherical shells of various radii and thicknesses were made by spinning. These were subjected to impact loading under a drop hammer and the load histories were obtained in all the cases. Three-dimensional numerical simulations were carried out for all the tested specimen geometries using LS-DYNA^®. Material, geometric and contact nonlinearities were incorporated in the analysis. The uni-axial stress–strain curve for the material was obtained experimentally and was assumed to be piecewise linear in the plastic region. The results from impact experiments are used for the validation of the numerical simulations. Three distinct modes of deformation, namely local flattening, inward dimpling and formation of multiple numbers of lobes were analysed and influence of various parameters on these modes is discussed.     

Further experimental studies on buckling of integrally ring-stiffened cylindrical shells under axial compression

An experimental study of the buckling of closely spaced integrally stiffened cylindrical shells under axial compression was carried out to determine the influence of shell and ring geometry on the applicability of linear theory. Twenty-nine specimens fabricated from 7075-T6 aluminum alloy with different geometries were tested. Test specimens were designed to fail in general instability and under low critical stresses to assure elastic buckling. Agreement of experimental results of the present study, and of those obtained in other studies with linear theory was found to be governed primarily by the ring area parameter, (A ₂ /ah). Values of “linearity,” ? (ratio of experimental buckling load to the predicted one), higher than 80 percent were obtained for (A ₂ /ah)>0.3 and a clear trend towards ?=1 was observed with increasing values of this parameter. Correlation with linear theory was also found to be influenced by ring spacing, (a/h), or rather the combination (a/h) [1+(A ₂ /ah]^?1/2. No significant effect of shell and other ring parameters on the correlation with linear theory could be discerned for the shells tested. By a conservative structural-efficiency criterion it was observed that only for low values of the area parameter, (A ₂ /ah)<≈0.5 ring-stiffened shells are more efficient than equivalent-weight isotropic ones. Highest efficiencies are obtained for (A ₂ /ah)≈0.2.