Nonlinear elastic buckling and postbuckling of axially compressed functionally graded cylindrical shells

Based on the nonlinear large deflection theory of cylindrical shells as well as the Donnell assumptions, this paper presents nonlinear buckling and postbuckling analyses for axially compressed functionally graded cylindrical shells by using the Ritz energy method and the nonlinear strain-displacement relations of large deformation. The material properties of the shells vary smoothly through the shell thickness according to a power law distribution of the volume fraction of constituent materials. Meanwhile, by taking into account the temperature-dependent material properties, various effects of external thermal environment are also investigated. Numerical results show various effects of the inhomogeneous parameter, dimensional parameters and external thermal environments on nonlinear buckling and postbuckling behaviors. There is a mode-jumping observed after buckling. The present theoretical results are verified by those in the literature.     

Vibration of functionally graded cylindrical shells based on different shear deformation shell theories with ring support under various boundary conditions

In the present work, study of the vibration of thin cylindrical shells with ring supports made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented. Material properties are graded in the thickness direction of the shell according to volume fraction power law distribution. Effects of boundary conditions and ring support on the natural frequencies of the FGM cylindrical shell are studied. The cylindrical shells have ring supports which are arbitrarily placed along the shell and which imposed a zero lateral deflection. The study is carried out using different shear deformation shell theories. The analysis is carried out using Hamilton’s principle. The governing equations of motion of a FGM cylindrical shells are derived based on various shear deformation theories. Results are presented on the frequency characteristics, influence of ring support position and the influence of boundary conditions. The present analysis is validated by comparing results with those available in the literature. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin M. M. Najafizadeh received his BS degree in 1995 from Azad University (Arak) and the Ms Degree in 1997 from Azad University (Arak), and his Ph.D. degree in 2003 from Science and Research Branch Islamic Azad University (Tehran, Iran), all in mechanical Engineering. He is member of faculty in Islamic Azad University (Arak) since 1998. He teaches courses in the areas of dynamics, theory of plates and shells and finite element method. He has published more than 20 articles in journals and conference proceeding. Mohammad Reza Isvandzibaei received his Ms Degree from Azad University (Arak), and now he is the student of Ph.D. in university of Pune, (India) all in mechanical Engineering. He is member of faculty in Islamic Azad University (Andimeshk).     

Thermomechanical postbuckling analysis of moderately thick functionally graded plates and shallow shells

In this paper, an analytical solution is provided for the postbuckling behaviour of moderately thick plates and shallow shells made of functionally graded materials (FGMs) under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell, according to a power law distribution of the volume fraction of the constituents. The fundamental equations for moderately thick rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection and high-order shear deformation theory for moderately thick plates. The solution is obtained in terms of mixed Fourier series and the obtained results are compared with those of the Reissner–Mindlin's theory for moderately thick plates and the classical theory ignoring transverse shear deformation. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behaviour and the associated stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.     

Vibration of circular membrane backed by cylindrical cavity

The natural vibration of a circular membrane backed by a cylindrical air cavity is investigated using the multimodal approach. The cavity-backed membrane is modeled as a dynamical system composed of two subsystems, and their modal receptance or “inverse receptance” characteristics are used to study the system vibration. The natural frequencies of the system are determined for four typical cases and the results show that the resulting modes can be categorized into three groups depending on the strength of the sybsystem interaction.     

Dynamic stability of cross-ply laminated composite cylindrical shells

The dynamic stability of thin, laminated cylindrical shells under combined static and periodic axial forces is studied using Love’s classical theory of thin shells. A normal-mode expansion of the equations of motion yields a system of Mathieu–Hill equations. Bolotin’s method is then employed to obtain the dynamic instability regions. The present study examines the dynamic stability of antisymmetric cross-ply circular, cylindrical shells of different lamination schemes. The effect of the magnitude of the axial load on the instability regions is also examined.     

Vibration of completely free composite plates and cylindrical shell panels by a higher-order theory

This paper deals with the free vibration of open, laminated composite, circular cylindrical panels having a rectangular plan-form and all their edges free of external tractions. The material arrangement of the shell panels considered may vary from this of the single isotropic (or special orthotropic) layer to that of a general angle-ply lay-up. The analysis is based on the application of the Ritz approach on the energy functional of the Love-type version of a unified shear deformable shell theory. A through-thickness parabolic distribution of the transverse shear deformation is mainly assumed but, for comparison purposes, numerical results that are based on the assumptions of the classical Love-type shell theory are also presented. The Ritz method is a powerful analytical technique since, provided that a complete set of trial functions is employed, it can provide the exact solution of the problem considered in infinite series forms. The mathematical formulation is therefore presented in a general form, appropriate for any set of basis functions. The variational approach is, however, finally applied in conjunction with a complete functional basis made of the appropriate admissible orthonormal polynomials.     

Free vibration of a laminated composite shell of revolution: Effects of shear non-linearity

Effects of shear non-linearity on free vibration of a laminated composite shell of revolution are investigated using a semi-analytical method based on the Reissner–Mindlin shell theory. The coupling between symmetric and anti-symmetric vibration modes of the shell is considered in the shear deformable shell element employed in this study. The Hahn–Tsai non-linearly elastic shear stress–shear strain relation is adopted. Numerical examples are given for laminated composite circular cylindrical and conical shells with various boundary conditions. The numerical results indicate that shear non-linearity may reduce significantly the fundamental frequencies of cross-ply composite shells of revolution.     

Forced axisymmetric response of an annular plate of parabolically varying thickness

Forced axisymmetric response of an annular plate of parabolically varying thickness, based on classical theory, is analyzed by the eigenfunction method. Plate subjected to constant and half-sine pulse loads uniformly distributed over an annular portion of the plate for various combination of boundary conditions are taken as example problems. Numerical results computed for the transverse deflection are plotted in graphs.     

统一强度理论在厚壁圆筒自增强中的应用

采用统一强度理论分析了厚壁圆筒自增强中的一些关键问题,得出了非自增强厚壁圆筒弹性极限载荷和塑性极限载荷的统一解的形式,以及弹塑性界面上当量应力最小时的弹塑性界面半径,并导出了当材料拉、压强度不同,及考虑中间主应力的情况下,自增强处理不发生反向屈服时的圆筒径比。另外,利用统一强度理论公式比较了现有的几种强度理论在自增强分析中所得的结果。     

On the eigenfrequencies of cantilevered beams carrying a tip mass and spring-mass in-span

The present study is concerned with the derivation of the eigenfrequencies and their sensitivity of a cantilevered Bernoulli-Euler beam carrying a tip mass (primary system) to which a spring-mass (secondary system) is attached in-span. After establishing the exact frequency equation of the combined system, a Dunkerley-based approximate formula is given for the fundamental frequency. Using the normal mode method, a second approximate frequency equation is established which is then used for the derivation of a sensitivity formula for the eigenfrequencies. The frequency equations of some simpler systems are obtained from the general equation as special cases. These frequency equations are then numerically solved for various combinations of physical parameters. The comparison of the numerical results with those from exact frequency equations indicate clearly that the eigenfrequencies of the combined system described above can be accurately determined by the present method.     

Transverse vibration of a circular plate with arbitrary thickness variation

Rayleigh-Ritz method has been employed to obtain approximations to frequencies and mode shapes of circular plates with variable thickness. The boundary is either clamped, simply supported or completely free. The main distinguishing feature of the present investigations is that the thickness approximation is done by measuring thickness at a suitable set of sample points and then using interpolation to get the approximating polynomial. Thus, unlike other methods already available in literature where either linear or quadratic variation of thickness has been examined, here one can have a polynomial of arbitrary degree depending upon the number and locations of the sample points. The results have been tabulated in a large number of cases and three-dimensional mode shapes have been plotted for some selected cases. Comparison has been made with available results. A short tables are also given to depict the rate of convergence with the order of approximation.