Thick Circular Cylindrical Shells Research Articles

Non-linear dynamics of cantilevered circular cylindrical shells with thickness stretch, containing quiescent fluid with small-amplitude sloshing

Studies on nonlinear vibrations of circular cylindrical shells containing fluid are mostly limited to thin simply supported shells, such that no results regarding the behavior of thick cantilevered shells with shear and thickness deformations can be found in the literature. In this article, for the first time, thin and thick circular cylindrical shells with clamped-free boundary conditions are modeled according to the third-order shear deformation theory with thickness stretch. The contained liquid satisfies the Laplace equation and linearized boundary conditions are imposed at the free surface and at the contact with the shell. Using Lagrange equations, the governing differential equations of the system in terms of shell-fluid interaction are derived. In linear free vibration analysis, it was found that increasing the fluid level inside the shell and decreasing the shell thickness, cause a significant rise in the fluid free-surface wave elevation. This increase is such that limits the application of linear sloshing theory. It was observed that the fluid presence can change the nonlinear behavior from softening to hardening type and intensifies the shell thickness deformation. In addition, the presence of contained liquid reduces the circumferential dynamic contraction of the shell caused by large amplitude vibrations.

Mesh refinement of finite element method for free vibration analysis of variable geometrical rotating cylindrical shells

PurposeThis study aims to provide a reliable and effective algorithm that is suitable for addressing the problems of continuous orders of frequencies and modes under different boundary conditions, circumferential wave numbers and thickness-to-length ratios of moderately thick circular cylindrical shells. The theory of free vibration of rotating cylindrical shells is of utmost importance in fields such as structural engineering, rock engineering and aerospace engineering. The finite element method is commonly used to study the theory of free vibration of rotating cylindrical shells. The proposed adaptive finite element method can achieve a considerably more reliable high-precision solution than the conventional finite element method.Design/methodology/approachOn a given finite element mesh, the solutions of the frequency mode of the moderately thick circular cylindrical shell were obtained using the conventional finite element method. Subsequently, the superconvergent patch recovery displacement method and high-order shape function interpolation techniques were introduced to obtain the superconvergent solution of the mode (displacement), while the superconvergent solution of the frequency was obtained using the Rayleigh quotient computation. Finally, the superconvergent solution of the mode was used to estimate the errors of the finite element solutions in the energy norm, and the mesh was subdivided to generate a new mesh in accordance with the errors.FindingsIn this study, a high-precision and reliable superconvergent patch recovery solution for the vibration modes of variable geometrical rotating cylindrical shells was developed. Compared with conventional finite element method, under the challenging varying geometrical circumferential wave numbers, and thickness–length ratios, the optimised finite element meshes and high-precision solutions satisfying the preset error limits were obtained successfully to solve the frequency and mode of continuous orders of rotating cylindrical shells with multiple boundary conditions such as simple and fixed supports, demonstrating good solution efficiency. The existing problem on the difficulty of adapting a set of meshes to the changes in vibration modes of different orders is finally overcome by applying the adaptive optimisation.Originality/valueThe approach developed in this study can accurately obtain the superconvergent patch recovery solution of the vibration mode of rotating cylindrical shells. It can potentially be extended to fine numerical models and high-precision computations of vibration modes (displacement field) and solid stress (displacement derivative field) for general structural special value problems, which can be extensively applied in the field of engineering computations in the future. Furthermore, the proposed method has the potential for adaptive analyses of shell structures and three-dimensional structures with crack damage. Compared with conventional finite element methods, significant advantages can be achieved by solving the eigenvalues of structures with high precision and stability.

Adaptive mesh refinement for finite element analysis of the free vibration disturbance of cylindrical shells due to circumferential micro-crack damage

PurposeModerately thick circular cylindrical shells are widely used as supporting structures or storage cavities in structural engineering, rock engineering, and aerospace engineering. In practical engineering, shells often work with micro-cracks or defects. The existence of micro-crack damage may result in the disturbance of dynamic behaviours and even induce accidental dynamic disasters. The free vibration frequency and mode are important parameters for the dynamic performance and damage identification analysis. In particular, stiffness weakening of the local damage region leads to significant changes in the vibration mode, which makes it difficult for the mesh generated in the conventional finite element method to capture a high-precision solution of the local oscillation.Design/methodology/approachIn response to the above problems, this study developed an adaptive finite element method and a crack damage characterisation method for moderately thick circular cylindrical shells. By introducing the inverse power iteration method, error estimation, and mesh subdivision refinement technique for the analysis of finite element eigenvalue problems, an adaptive computation scheme was constructed for the free vibration problem of moderately thick circular cylindrical shells with circumferential crack damage.FindingsBased on typical numerical examples, the established adaptive finite element solution for the free vibration of moderately thick circular cylindrical shells demonstrated its suitability for solving the high-precision free vibration frequency and mode of cylindrical shell structures. The any order frequency and mode shape of cracked cylindrical shells under the conditions of different ring wave numbers, crack locations, crack depths, and multiple cracks were successfully solved. The influences of the location, depth, and number of cracks on the disturbance of dynamic behaviours were analysed.Originality/valueThis study can be used as a reference for the adaptive finite element solution of free vibration of moderately thick circular cylindrical shells with cracks and lays the foundation for further development of a high-performance computation method suitable for the dynamic disturbance and damage identification analysis of general cracked structures.

Second–order models for the bending analysis of thin and moderately thick circular cylindrical shells

The suitability of employing the second-order shear deformation theory to static bending problems of thin and moderately thick isotropic circular cylindrical shells was investigated. Two variant forms of the polynomial second-order displacement models were considered. Both models account for quadratic expansions of the surface displacements along the shell thickness, although the second model (SSODM) was augmented by the initial curvature term. The equilibrium equations were derived by use of the principle of virtual work. Navier analytical solutions were obtained under simply supported boundary conditions. The results of the displacements and stresses revealed that the theory formulated on the SSODM provides a good depiction of the bending response of thin and moderately thick shells and are in close agreement with those of the first and higher-order shear deformation theories (FSDT; HSDT). The ability of the theory formulated on the first model (FSODM) to predict adequate values of displacements and stresses in thin shells was found to be significantly affected by changes in length to radius of curvature (l/a) ratios.

An hp-version adaptive finite element algorithm for eigensolutions of moderately thick circular cylindrical shells via error homogenisation and higher-order interpolation

PurposeThis study presents a novel hp-version adaptive finite element method (FEM) to investigate the high-precision eigensolutions of the free vibration of moderately thick circular cylindrical shells, involving the issues of variable geometrical factors, such as the thickness, circumferential wave number, radius and length.Design/methodology/approachAn hp-version adaptive finite element (FE) algorithm is proposed for determining the eigensolutions of the free vibration of moderately thick circular cylindrical shells via error homogenisation and higher-order interpolation. This algorithm first develops the established h-version mesh refinement method for detecting the non-uniform distributed optimised meshes, where the error estimation and element subdivision approaches based on the superconvergent patch recovery displacement method are introduced to obtain high-precision solutions. The errors in the vibration mode solutions in the global space domain are homogenised and approximately the same. Subsequently, on the refined meshes, the algorithm uses higher-order shape functions for the interpolation of trial displacement functions to reduce the errors quickly, until the solution meets a pre-specified error tolerance condition. In this algorithm, the non-uniform mesh generation and higher-order interpolation of shape functions are suitable for addressing the problem of complex frequencies and modes caused by variable structural geometries.FindingsNumerical results are presented for moderately thick circular cylindrical shells with different geometrical factors (circumferential wave number, thickness-to-radius ratio, thickness-to-length ratio) to demonstrate the effectiveness, accuracy and reliability of the proposed method. The hp-version refinement uses fewer optimised meshes than h-version mesh refinement, and only one-step interpolation of the higher-order shape function yields the eigensolutions satisfying the accuracy requirement.Originality/valueThe proposed combination of methodologies provides a complete hp-version adaptive FEM for analysing the free vibration of moderately thick circular cylindrical shells. This algorithm can be extended to general eigenproblems and geometric forms of structures to solve for the frequency and mode quickly and efficiently.

A new prestressed dynamic stiffness element for vibration analysis of thick circular cylindrical shells

This paper presents a new spectral element based on the dynamic stiffness matrix of a prestressed cylindrical shell. The dynamic stiffness matrix is built using first-order shear deformation theory, and natural frequencies are processed easily. Vibration analyses are performed with numerical examples to determine the performance of this approach and the effect of prestressing on frequency response functions. This element has many advantages over the finite element method in terms of accuracy, model size and computing time.

Mechanical and thermal postbuckling of FGM thick circular cylindrical shells reinforced by FGM stiffener system using higher-order shear deformation theory

The postbuckling of the eccentrically stiffened circular cylindrical shells made of functionally graded materials (FGMs), subjected to the axial compressive load and external uniform pressure and filled inside by the elastic foundations in the thermal envi-ronments, is investigated with an analytical method. The shells are reinforced by FGM stringers and rings. The thermal elements of the shells and stiffeners in the fundamen-tal equations are considered. The equilibrium and nonlinear stability equations in terms of the displacement components for the stiffened shells are derived with the third-order shear deformation theory and Leckhniskii smeared stiffener technique. The closed-form expressions for determining the buckling load and postbuckling load-deflection curves are obtained with the Galerkin method. The effects of the stiffeners, the foundations, the material and dimensional parameters, and the pre-existent axial compressive and thermal load are considered.

Comparison Study on the Exact Dynamic Stiffness Method for Free Vibration of Thin and Moderately Thick Circular Cylindrical Shells

Comparison study on free vibration of circular cylindrical shells between thin and moderately thick shell theories when using the exact dynamic stiffness method (DSM) formulation is presented. Firstly, both the thin and moderately thick dynamic stiffness formulations are examined. Based on the strain and kinetic energy, the vibration governing equations are expressed in the Hamilton form for both thin and moderately thick circular cylindrical shells. The dynamic stiffness is assembled in a similar way as that in classic skeletal theory. With the employment of the Wittrick-Williams algorithm, natural frequencies of circular cylindrical shells can be obtained. A FORTRAN code is written and used to compute the modal characteristics. Numerical examples are presented, verifying the proposed computational framework. Since the DSM is an exact approach, the advantages of high accuracy, no-missing frequencies, and good adaptability to various geometries and boundary conditions are demonstrated. Comprehensive parametric studies on the thickness to radius ratio (h/r) and the length to radius ratio (L/r) are performed. Applicable ranges of h/r are found for both thin and moderately thick DSM formulations, and influences of L/r on frequencies are also investigated. The following conclusions are reached: frequencies of moderately thick shells can be considered as alternatives to those of thin shells with high accuracy where h/r is small and L/r is large, without any observation of shear locking.

Nonlinear dynamic analysis of Sigmoid functionally graded circular cylindrical shells on elastic foundations using the third order shear deformation theory in thermal environments

This paper presents an analytical approach to investigate the nonlinear dynamic response and vibration of imperfect functionally graded materials (S-FGM) thick circular cylindrical shells surrounded on elastic foundation using the third order shear deformation shell theory. Material properties are assumed to be temperature dependent and graded in the thickness direction according to a Sigmoid power law distribution (S-FGM) in terms of the volume fractions of constituents with metal–ceramic–metal layers. The S-FGM shells are subjected to mechanical, damping and thermal loads. The Galerkin method and fourth-order Runge–Kutta method are used to calculate natural frequencies, nonlinear frequency–amplitude relation and dynamic response of the shells. In numerical results, the effects of geometrical parameters, the material properties, imperfections, the elastic foundations and mechanical loads on the nonlinear dynamic response and nonlinear vibration of the shells are investigated. Accuracy of the present formulation is shown by comparing the results of numerical examples with the ones available in literature.

Nonlinear dynamic response and vibration of shear deformable imperfect eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic foundations

This paper presents an analytical approach to investigate the nonlinear dynamic response and vibration of imperfect eccentrically stiffened functionally graded thick circular cylindrical shells surrounded on elastic foundations using both the first order shear deformation theory and stress function with full motion equations (not using Volmir's assumptions). Material properties are graded in the thickness direction according to a sigmoid power law distribution (S-FGM) in terms of the volume fractions of constituents with metal–ceramic–metal layers. The S-FGM shells are subjected to mechanical and damping loads. Numerical results for dynamic response of the shells are obtained by Runge–Kutta method. The results show the influences of geometrical parameters, the volume fractions of metal–ceramic–metal layers, imperfections, the elastic foundations, eccentrically stiffeners, pre-loaded axial compression and damping loads on the nonlinear dynamic response and nonlinear vibration of functionally graded cylindrical shells. The proposed results are validated by comparing with other results reported in the literature.

Nonlinear buckling of higher deformable S-FGM thick circular cylindrical shells with metal–ceramic–metal layers surrounded on elastic foundations in thermal environment

An analytical approach on the nonlinear response of thick functionally graded circular cylindrical shells with temperature independent material property surrounded on elastic foundations subjected to mechanical and thermal loads is presented. Material properties are graded in the thickness direction according to a Sigmoid power law distribution in terms of the volume fractions of constituents (S-FGM). The formulations are based on the third order shear deformation shell theory taking into account von Karman nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation. By applying Galerkin method and using stress function, explicit relations of thermal load–deflection curves of the S-FGM shells are determined. Detailed parametric studies are carried out to investigate effects of volume fraction index, material properties and geometrical shapes, axial compressions and thermal load, foundation stiffness and imperfection on nonlinear buckling behaviors of S-FGM thick circular cylindrical shells. The present analysis is validated by comparing results with other publications.

On numerical analysis of axisymmetric thick circular cylindrical shells based on higher order shell theories by segmentation method

Numerical analysis is carried out in this paper for thick circular cylindrical shells using higher order shell theory. In orthogonal curvilinear coordinate system, all equations are derived with the inclusion of the additional quadratic and cubic terms in the Taylor’s series expansions of the both in-plane as well as the transverse displacement components for the improvement of bending behaviour of the shell. Assuming ( h/ R)2 ≪ 1, a rigorous formulation involving the reduction of a three-dimensional elasticity problem to a two-dimensional one, based partly upon the Reissner’s variational principle, is presented. These equations are algebraically manipulated to be in the form of a coupled system of first-order differential equations in terms of the intrinsic dependent variables. These are then solved by a segmentation method – numerical integration technique for various combinations of material and geometric parameters. The theory is shown to result in a partial differential equation system of sixteenth order.

Varied effects of shear correction on thermal vibration of functionally graded material shells

AbstractThe effects of varied shear correction coefficient on the first-order transverse shear deformation result of functionally graded material (FGM) thick circular cylindrical shells under thermal vibration are investigated and computed by using the generalized differential quadrature method. The computed and varied values of shear correction coefficient are usually functions of FGM power law index and environment temperature. In the thermoelastic stress–strain relations, the simpler form stiffness of FGM shells under linear temperature rise is considered. The equation of shear correction coefficient is derived and obtained by using the total strain energy principle. Two parametric effects: environment temperature and FGM power law index on the thermal stress and center deflection of FGM thick circular cylindrical shells are obtained and investigated.

Free vibration analysis of homogeneous isotropic circular cylindrical shells based on a new three-dimensional refined higher-order theory

Closed-form formulation of three-dimensional (3-D) refined higher-order shear deformation theory (RHOST) for the free vibration analysis of simply supported–simply supported and clamped–clamped homogenous isotropic circular cylindrical shells is presented. The proposed RHOST accounts for the effects of in-plane and rotary inertias as well as the effects of transverse normal and shear strains on the dynamic response of cylindrical shells. Also, the present analysis incorporates trapezoidal shape factor of a shell element that arise due to the fact that the stresses over the thickness of the shell are to be integrated on a trapezoidal-like cross-section of a shell element to obtain the accurate stress-resultants. Therefore, the present theory refines other HOSTs established hitherto for free vibration analysis of thick circular cylindrical shells. The equations of motion are obtained using Hamilton's principle. Solutions are obtained using the Galerkin method. Numerical results, not hitherto reported in the literature, are presented for the natural frequencies of isotropic long/short and hollow/solid cylinders. Numerical results indicate that for thick cylinders with large length-to-radius ratios, approximate series truncation methods are inadequate for calculating the stress-resultants and just exact integration yields accurate stress-resultants leading to a reliable prediction of natural frequencies. Natural frequencies associated to higher-mode numbers of thick isotropic cylinders, never published in the literature before, are also considered in this paper. It is shown that stress-resultants calculated using exact integration over the cylinder cross-section are necessary to obtain accurate results of the free vibration analysis, especially for higher-mode numbers of thick isotropic cylinders. The closed-form solutions presented herein are compared with the available exact 3-D elasticity and analytical solutions in the available literature and excellent agreement is obtained. Also, the validity of the results is established with the aid of commercial finite element software.

The propagation of stress waves in thick circular cylindrical shells

The propagation of stress waves in thick circular cylindrical shells

152 種々の境界条件を有する厚肉円筒殼の自由振動特性(ロボティクス・メカトロニクス/機械力学・計測制御)

152 種々の境界条件を有する厚肉円筒殼の自由振動特性(ロボティクス・メカトロニクス/機械力学・計測制御)

THE SOUND RADIATION EFFICIENCY OF FINITE LENGTH ACOUSTICALLY THICK CIRCULAR CYLINDRICAL SHELLS UNDER MECHANICAL EXCITATION I: THEORETICAL ANALYSIS

The acoustic radiation from circular cylindrical shells is of fundamental and applied interest primarily because cylindrical shells are widely used in industry, and because their acoustic behaviour is different from that of beams and plates due to curvature effects. In previous studies of the subject, cylindrical shells have been categorized into acoustically thin and acoustically thick shells in terms of the ratio between the ring frequency frand the critical frequency fC, i.e., fr/fC<1 for acoustically thin shells, and fr/fC>1 for acoustically thick shells. For acoustically thin shells, it has been found by statistical methods that the radiation efficiency has a peak at the ring frequency. Above the ring frequency, the shells behave like flat plates. For acoustically thick shells, especially with finite length, however, the behavior is not so clear. From the analysis in the wavenumber domain, a formula for calculating the modal radiation efficiency of finite length circular cylindrical shells (immersed in light fluid) under mechanical excitation is obtained analytically. Based on this method, the modal-averaged sound radiation efficiencies of acoustically thick circular cylindrical shells are calculated. It is found that unlike acoustically thin shells, the radiation efficiencies of acoustically thick cylindrical shells very much depend on the acoustic behaviour of each individual vibration mode, and thus on the geometries and the boundary conditions. Results obtained by acoustic boundary element calculations and experiments verify these conclusions.

SEGMENTED SENSORS AND ACTUATORS FOR THICK PLATES AND SHELLS PART I: ANALYSIS USING FSDT

This paper presents modal sensitivity factors, actuation factors and controlled damping ratio for segmented distributed piezoelectric sensor and actuator layers laminated on simply supported thick rectangular plates and circular cylindrical shells made of cross-ply composite laminate. A Flugge-type first order shear deformation theory (FSDT) has been developed including rotary inertia of the elastic plies and inertia, stiffness and piezoelectric effects (direct and converse) of the piezoelectric layers. Plates and shells with any cross-ply lamination scheme (symmetric/unsymmetric) are considered with the piezoelectric sensor and actuator layers rectangularly segmented. The Navier-type solution is obtained for the case of velocity feedback using modal filtering. Analytical expressions for modal membrane and bending sensitivity factors, membrane and bending actuation factors and controlled damping ratio are developed for cylindrical shells using FSDT and classical lamination theory. The contribution of the in-plane displacement components is included. The results for plates are obtained as a particular case of the results for circular cylindrical shells.

SEGMENTED SENSORS AND ACTUATORS FOR THICK PLATES AND SHELLS PART II: PARAMETRIC STUDY

A parametric study is presented here for the modal sensitivity and actuation factors and controlled damping ratio for the rectangularly segmented distributed piezoelectric sensor and actuator layers laminated on simply supported thick rectangular plates and circular cylindrical shells made of cross ply composite laminate. The analytical expressions of these entities, developed in Part I of this two-part study, based on a Flugge-type first order shear deformation theory (FSDT) and classical lamination theory (CLT), are numerically evaluated to illustrate the effect of the thickness parameter. The contribution of the membrane and bending strains to sensitivity and actuation factors is computed for plates and shells for the lowest mode and a few higher modes. It has been concluded that the classical lamination theory is inadequate for the analysis of spatial actuation and control of all modes for thick plates and shells as well as for the higher modes of thin composite plates and shells.

VIBRATION OF THICK CYLINDRICAL SHELLS ON THE BASIS OF THREE-DIMENSIONAL THEORY OF ELASTICITY

In this paper, an approximate analysis using a layerwise approach is presented to study the vibration of thick circular cylindrical shells on the basis of three-dimensional theory of elasticity. The boundary conditions studied are simply supported–simply supported and clamped–clamped boundary conditions. In this approach, the thick cylindrical shell is discretized into an arbitrary number of thin cylindrical layers in the thickness direction and each layer assumes a three-dimensional stress state. The approach is similar in concept to the finite-strip method. The displacements for each layer are approximated by trigonometric functions in the axial and circumferential directions and by some linear-shape functions in the thickness direction. The governing equation is obtained using an energy minimization principle. Extensive frequency parameters, never seen in the literature before, have been presented for a wide range of thickness-to-radius ratios and thickness-to-length ratios. In addition, the frequency characteristics of thin and thick cylindrical shells are also studied and the displacement fields in the thickness direction for various thickness-to-radius ratios are also presented. The analysis has been verified by comparing results with those in the literature and excellent agreement is obtained.