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Moderately 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.

In 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.

Based 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.

This 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.

This 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.

An 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.

Numerical 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.

The 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.

Gives a bibliographical review of the finite element methods (FEMs) applied for the linear and nonlinear, static and dynamic analyses of basic structural elements from the theoretical as well as practical points of view. The range of applications of FEMs in this area is wide and cannot be presented in a single paper; therefore aims to give the reader an encyclopaedic view on the subject. The bibliography at the end of the paper contains 2,025 references to papers, conference proceedings and theses/dissertations dealing with the analysis of beams, columns, rods, bars, cables, discs, blades, shafts, membranes, plates and shells that were published in 1992‐1995.

This 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.

On 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.

In 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.

The 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.

The purpose of this work is to apply the Fourier–Ritz method to study the vibration behavior of the moderately thick functionally graded (FG) parabolic and circular panels and shells of revolution with general boundary conditions.

The modified Fourier series is chosen as the basis function of the admissible functions of the structure to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges, and the vibration behavior is solved by means of the Ritz method. The complete shells of revolution can be achieved by using the coupling spring technique to imitate the kinematic compatibility and physical compatibility conditions of FG parabolic and circular panels at the common meridian of θ = 0 and 2π. The convergence and accuracy of the present method are verified by other contributors.

Some new results of FG panels and shells with elastic restraints, as well as different geometric and material parameters, are presented and the effects of the elastic restraint parameters, power-law exponent, circumference angle and power-law distributions on the free vibration characteristic of the panels are also presented, which can be served as benchmark data for the designers and engineers to avoid the unpleasant, inefficient and structurally damaging resonant.

The paper could provide the reference for the research about the moderately thick FG parabolic and circular panels and shells of revolution with general boundary conditions. In addition, the change of the boundary conditions can be easily achieved by just varying the stiffness of the boundary restraining springs along all the edges of panels without making any changes in the solution procedure.

In the current manuscript, the authors examine the Belleville spring with the variable thickness. The thickness is assumed to be variable along the meridional and parallel coordinates of conical coordinate system. The calculation of the Belleville springs includes the cases of the free gliding edges and the edges on cylindric curbs, which constrain the radial movement. The equations developed here are based on common assumptions and are simple enough to be applied to the industrial calculations.

In the current manuscript, the authors examine the Belleville spring with the variable thickness. The calculation of the Belleville springs investigates the free gliding edges and the edges on cylindric curbs with the constrained radial movement. The equations developed here are based on common assumptions and are simple enough to be applied to the industrial calculations.

The developed equations demonstrate that the shift of the inversion point to the inside edge does not influence the bending of the cone. On the contrary, the character of the extensional deformation (circumferential strain) of the middle surface alternates significantly. The extension of the middle surface of free gliding spring occurs outside the inversion. The middle surface of the free gliding spring squeezes inside the inversion point. Contrarily, the complete middle surface of the disk spring on the cylindric curb extends. This behavior influences considerably the function of the spring.

A slotted disk spring consists of two segments: a disk segment and a number of lever arm segments. Currently, the calculation of slotted disk spring is based on the SAE formula (SAE, 1996). This formula is limited to a straight slotted disk spring with freely gliding inner and outer edges.

The equations developed here are based on common assumptions and are simple enough to be applied to the industrial calculations. The developed method is applicable for disk springs with radially constrained edges. The vertical displacements of a disk spring result from an axial load uniformly distributed on inner and outer edges. The method could be directly applied for calculation of slotted disk springs.

The nonlinear governing equations for the of Belleville spring centres were derived. The equations describe the deformation and stresses of thin and moderately thick washers. The variation method is applicable for the disc springs with free gliding and rigidly constrained edges. The developed method is applicable for Belleville spring with radially constrained edges. The vertical displacements of a disc spring result from an axial load uniformly distributed on inner and outer edges.

The purpose of this paper is to present the optimal design of a simply supported variable curvature laminated angle-ply composite panel under uniaxial compression. The objective is to maximize the failure load which is defined as the minimum of the buckling load and the first-ply failure load.

The numerical results presented are obtained using a shear deformable degenerated shell finite element, a brief formulation of which is given. Some verification problems are solved and a convergence study is conducted in order to assess the accuracy of the element. The design procedure is presented and optimization results are given for a simply supported symmetric eight layer angle-ply panel composed of a flat and two cylindrical sections.

The influences of the stacking sequence and panel thickness on optimization are investigated and the effects of various problem parameters on the optimization procedure are discussed.

The paper shows that the load carrying capacity of thicker panels is considerably reduced when the first-ply failure constraint is taken into account.

A bibliographical review of the finite element methods (FEMs) applied for the linear and nonlinear, static and dynamic analyses of basic structural elements from the theoretical as well as practical points of view is given. The bibliography at the end of the paper contains 1,726 references to papers, conference proceedings and theses/dissertations dealing with the analysis of beams, columns, rods, bars, cables, discs, blades, shafts, membranes, plates and shells that were published in 1996‐1999.

To investigate the large amplitude free flexural vibration of doubly curved shallow shells in the presence of cutouts.

Finite element model using an eight‐noded C⁰ continuity, isoparametric quadrilateral element is employed. Nonlinear strains of von Karman type are incorporated into the first‐order shear deformation theory.

Cylindrical shell shows mostly hard spring behavior whereas spherical shell shows both hard spring and soft spring behavior with the increase of amplitude ratios for different cutout sizes, radii of curvature and thickness parameters. At a particular value of the amplitude ratio, the frequency ratio of shells is governed by the interactive effects of stiffness and mass.

Aircraft, spacecraft and many other structures where shell panels are used, undergo large amplitude nonlinear vibrations.

The paper will assist researchers of vibration behavior of elastic systems.

The authors examined the numerical natural frequency analysis of a 2D functionally graded (FG) truncated thick hollow cone using 3D elasticity theory.

The material properties of the 2D-FGM (two dimensional-functionally graded materials) cone are graded along the radial and axial axes of the cone using a power–law distribution. The eigenvalue problem was solved using finite element analysis (FEA) employing graded hexahedral elements, and the verification of the finite element approach was assessed by comparing the current solution to earlier experimental studies.

The effects of semivertex angle, material distribution and the cone configuration on the natural frequencies have been analyzed. For various semivertex angles, thickness, length and power law exponents, many results in the form of natural frequencies and mode shapes are presented for the 2D-FGM cone. As a result, the effects of the given parameters were addressed, and the results were compared, demonstrating the direct efficiency of raising the power–law exponents and cone thickness on the rise of natural frequencies.

For the first time, the numerical natural frequency analysis of a 2D-FG truncated thick hollow truncated cone based on 3D equations of elasticity has been investigated. The material properties of the truncated cone have been distributed along two directions, which has not been considered before in any research for the truncated thick cone. The reason for using these innovative volume fraction functions is the lack of accurate coverage by functions that are available in the literature (Asemi et al., 2011; Babaei et al. 2021).