Application of third order shear deformation theory in buckling analysis of 2D-functionally graded cylindrical shell reinforced by axial stiffeners

This paper presents buckling analysis of a two-dimensional functionally graded cylindrical shell reinforced by axial stiffeners (stringer) under combined compressive axial and transverse uniform distributive load. The shell material properties are graded in the direction of thickness and length according to a simple power law distribution in terms of the volume fractions of the constituents. Primarily, the third order shear deformation theory (TSDT) is used to derive the equilibrium and stability equations. Since there is no closed form solution, the numerical differential quadrature method, (DQM), is applied for solving the stability equations. Initially, the obtained results for an isotropic shell using DQM were verified against those given in the literature for simply supported boundary conditions. The effects of load, geometrical and stringer parameters along with FG power index in the various boundary conditions on the critical buckling load have been studied. The study of results confirms that, stringers have significant effects on critical buckling load.     

Mechanical stability of metal foam cylindrical shells with various porosity distributions

Abstract

The paper deals with the nonlinear buckling analysis of imperfect cylindrical shells made of porous metal foam subjected to axial compression. For the metal foam shells, porosities are dispersed by uniform, symmetric, and asymmetric distributions in the thickness direction. Using Donnell shell theory and von-Karman nonlinear kinematics, nonlinear equilibrium equations are derived. The critical buckling load and buckling equilibrium curves for both perfect and imperfect shells are solved by using the Galerkin's procedure. A comprehensive investigation into the influence of porosity coefficient, imperfections, porosity distribution, and geometry on the buckling behaviors of the cylindrical shell is performed.     

基于精细传递矩阵法的变厚度圆柱壳自由振动分析

结合精细积分和传递矩阵方法,对变厚度圆柱壳的自由振动进行计算分析。该方法基于圆柱壳的基本微分方程,推导得到关于位移内力向量的一阶齐次偏微分方程,采用精细积分求得场传递矩阵,将其进行组装得到总传递方程,根据边界条件求解总传递方程中系数矩阵的行列式,计算得到变厚度圆柱壳的固有频率。将计算结果与有限元结果进行对比,验证方法的准确性及有效性。同时探究了边界条件、厚度变化形式、厚度变化系数及长径比对自由振动的影响规律。     

Hybrid stress finite element model for non-linear shell problems

The present paper describes a hybrid stress finite element formulation for geometrically non-linear analysis of thin shell structures. The element properties are derived from an incremental form of Hellinger-Reissner's variational principle in which all quantities are referred to the current configuration of the shell. From this multi-field variational principle, a hybrid stress finite element model is derived using standard matrix notation. Very simple flat triangular and quadrilateral elements are employed in the present study. The resulting non-linear equations are solved by applying the load in finite increments and restoring equilibrium by Newton-Raphson iteratioin. Numerical examples presented in the paper include complete snap-through buckling of cylindrical and spherical shells. It turns out that the present procedure is computationally efficient and accurate for non-linear shell problems of high complexity.     

Stability analysis of generally laminated conical shells with variable thickness under axial compression

Abstract

The buckling of generally laminated conical shells having thickness variations under axial compression is investigated. This problem usually arises in the filament wound conical shells where the thickness changes through the length of the cone. The thickness may be assumed to change linearly through the length of the cone. The fundamental relations for a conical shell with variable thickness applying thin-walled shallow shell theory of Donnell-type and theorem of minimum potential energy have been derived. Nonlinear terms of Donnell equations are linearized by the use of adjacent-equilibrium criterion. Governing equations are solved using power series method. This procedure enables us to investigate all combinations of classical boundary conditions. The results are verified in comparison with Galerkin method and the available results in the literature. Effects of thickness function coefficient, semi-vertex angle, lamination sequence, length to diameter ratio, and initial thickness of the cone on the buckling load are investigated. It is observed that these parameters have considerable effects on the critical buckling load of a conical shell.     

The buckling of an orthotropic composite truncated conical shell with continuously varying thickness subject to a time dependent external pressure

In this study, the buckling of an orthotropic composite truncated conical shell with continuously varying thickness, subject to a uniform external pressure which is a power function of time, has been considered. At first, the fundamental relations and the Donnell type stability equations of an orthotropic composite truncated conical shell, subject to an external pressure, have been obtained. Then, employing Galerkin method, those equations have been reduced of time dependent differential equation with variable coefficients. Finally, applying the variational method of Ritz method type, the critical static and dynamic loads, the corresponding wave numbers and the dynamic factor have been found analytically. Using those results, the effects of the variations of the power in the thickness expression, the semi-vertex angle, the power of time in the external pressure expression and the ratio of the Young's moduli on the critical parameters are studied numerically, for the case when the thickness of the conical shell varies as a power and exponential function. It is observed, from the computations carried out, that these factors have appreciable effects on the critical parameters of the problem in the heading.     

Bending,buckling and free vibration of non-homogeneous composite laminated cylindrical shells using a refined first-order theory

The bending, buckling and free vibration problems of non-homogeneous composite laminated cylindrical shells are considered. Hamilton–Reissner's mixed variational principle is used to deduce a consistent first-order theory of composite laminated cylindrical shells with non-homogeneous elastic properties. The governing equations with their required boundary conditions are derived without introducing any shear correction factors. Numerical results for the transverse deflections, stresses, natural frequencies and critical buckling loads are presented to show the advantages of this theory. The influences of the non-homogeneity and thickness ratio on the shell structural response are investigated. The study concludes that the inclusion of the non-homogeneity effect is required, even if it is weak, for predicting the actual structural response of the shells.     

Mechanical buckling of functionally graded material cylindrical shells surrounded by Pasternak elastic foundation

In this study, the mechanical buckling of functionally graded material cylindrical shell that is embedded in an outer elastic medium and subjected to combined axial and radial compressive loads is investigated. The material properties are assumed to vary smoothly through the shell thickness according to a power law distribution of the volume fraction of constituent materials. Theoretical formulations are presented based on a higher-order shear deformation shell theory (HSDT) considering the transverse shear strains. Using the nonlinear strain–displacement relations of FGMs cylindrical shells, the governing equations are derived. The elastic foundation is modelled by two parameters Pasternak model, which is obtained by adding a shear layer to the Winkler model. The boundary condition is considered to be simply-supported. The novelty of the present work is to achieve the closed-form solutions for the critical mechanical buckling loads of the FGM cylindrical shells surrounded by elastic medium. The effects of shell geometry, the volume fraction exponent, and the foundation parameters on the critical buckling load are investigated. The numerical results reveal that the elastic foundation has significant effect on the critical buckling load.     

Nonlinear torsional buckling and postbuckling of eccentrically stiffened FGM cylindrical shells in thermal environment

The main aim of this paper is to investigate the nonlinear buckling and post-buckling of functionally graded stiffened thin circular cylindrical shells surrounded by elastic foundations in thermal environments and under torsional load by analytical approach. Shells are reinforced by closely spaced rings and stringers in which material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction. The elastic medium is assumed as two-parameter elastic foundation model proposed by Pasternak. Based on the classical shell theory with von Karman geometrical nonlinearity and smeared stiffeners technique, the governing equations are derived. Using Galerkin method with three-term solution of deflection, the closed form to find critical torsional load and post-buckling load–deflection curves are obtained. The effects of temperature, stiffener, foundation, material and dimensional parameters are analyzed.     

KOITER'S ANALYSIS OF THIN-WALLED STRUCTURES BY A FINITE ELEMENT APPROACH

This paper refers to the analysis of the postbuckling behaviour of thin-walled structures by means of an asymptotic approach based on a finite element implementation of Koiter's non-linear theory of instability. The analysis has been accomplished by using the following assumptions: (i) the structure is described as an assemblage of flat slender rectangular panels; (ii) a non-linear Kirchhoff-type plate theory is used to model each panel; (iii) HC finite elements discretization is used; (iv) linear and quadratic extrapolations are assumed for the fundamental and the postbuckling paths, respectively; (v) multimodal buckling is considered; and (vi) imperfection sensitivity analysis is performed in both multimodal and monomodal form based on the steepest– descent path criterion. Several numerical results are presented and discussed. Comparisons with numerical solution obtained by standard incremental codes are given, which show the accuracy and reliability of the proposed approach.     

Thermal buckling of shallow/nonshallow piezoelectric-composite cylindrical shells

A thermal buckling analysis is presented for laminated cylindrical shells with surface mounted piezoelectric actuators under combined action of thermal and electrical loads. Derivations of the equations are based on the classical laminated shell theory, using the Sanders nonlinear kinematic relations. The analysis uses the Galerkin method to obtain closed form solutions for the buckling loads of shallow and nonshallow piezolaminated cylindrical shells. Temperature dependency of material properties is taken into account. Illustrative examples are presented to verify the accuracy of the proposed formulation. The effects of the various design parameters on thermal buckling loads are investigated.     

Thermal buckling of laminated cylindrical plates

Thermal buckling of a laminated cylindrical plate subjected to a temperature change is studied. The governing differential equations for Donnell-type laminated cylindrical shells are used and Galerkin's method is employed to determine the critical buckling temperature. Clamped and simply supported boundary conditions are both considered. The effects of various parameters on thermal buckling are examined.     

Symmetricability of pressure stiffness matrices for shells with loaded free edges

For the case of free edges which are loaded, follower forces remaining normal to the middle surface of a shell throughout the deformation history do not have a load potential. In finite element analysis, this results in an unsymmetric pressure stiffness matrix. Depending on the structure of the available computer program, implementation of an equation solver permitting solution of unsymmetric simultaneous systems of algebraic equations may be a tedious task. This explains the significance of the topic of symmetricability of pressure stiffness matrices, turning out to be of special importance in the case of static buckling under the assumption of a linear prebuckling path. At first, incremental equations for tracing the nonlinear load–displacement path are derived. Thereafter, the buckling condition is deduced. Then, it is demonstrated that symmetrization of the pressure stiffness matrix is admissible if the so-obtained ‘buckling pressure’ differs ‘very little’ from the ‘buckling pressure’ resulting from an alternative symmetric ‘buckling matrix’, as is shown to be the case for a simply supported cylindrical shell with a free upper edge, subjected to hydrostatic external pressure. The alternative symmetric ‘buckling matrix’ is a consequence of deleting the virtual work term, causing the unsymmetry of the pressure stiffness matrix, in the expression for the external virtual work. A mechanical interpretation of this virtual work term is given. It is shown to be equal to the difference of virtual work of the original pressure load and of a ‘substitute pressure-field’, of the form of a Fourier series of the former. This explains why, normally, the buckling coefficient resulting from the ‘substitute pressure-field’ represents a good approximation to the buckling coefficient stemming from the original pressure load.     

Elasticity Solution of a Laminated Clamped Cylindrical Shell with Piezoelectric Layer under Dynamic Load

Dynamic elasticity solution for a clamped, laminated cylindrical shell with two orthotropic layers bounded with a piezoelectric layer and subjected to impulse load distributed on inner surface is presented. The piezoelectric layer serves as sensor/actuator. The governing elasticity PDE equations are reduced to ordinary differential equations by means of Legendre polynomial expansion for displacement and electric potential in the axial direction. The resulting equations are transferred into state space form and reduced to an eigenvalue problem by using Galerkin's finite element in radial direction. The static and dynamic results are presented for [0/90/Piezo] lamination. The radius to thickness ratio effect on dynamic behavior is studied. The results are compared for different thickness ratios and applied electric loads with simply-supported shell results. Time responses for sensor and actuated shell are presented and natural frequencies are compared with simply-supported shell results.     

夹芯圆柱壳稳定性优化

研究了在轴压载荷作用下圆柱壳结构的失稳模态和结构承载效率,分析了空心圆柱壳厚度对失稳模态和承载效率的影响,以及圆柱壳填充轻质芯体对提高承载效率的作用;研究了圆柱壳结构的基于参数化建模、稳定性分析以及承载效率优化设计的一体化方法,并基于商用软件PATRAN的PCL语言予以实现。针对特定夹芯圆柱壳结构的稳定性分析和优化表明,空心薄壁圆柱壳结构在轴压载荷作用下容易失稳,结构承载效率低。适当增加壳体厚度,不但提高抗屈曲能力,而且也提高了结构承载效率。但厚度增加到一定限度后,进一步增加壳体厚度会提高结构的失稳荷载,但承载效率下降。利用泡沫状材料填充薄壁圆柱壳结构可以提高圆柱壳的结构抗屈曲承载能力和承载效率。通过优化壳体壁厚和芯体材料的相对密度,可有效地提高结构的承载能力和承载效率。     

A computationally efficient method for the buckling analysis of shells with stochastic imperfections

A computationally efficient method is presented for the buckling analysis of shells with random imperfections, based on a linearized buckling approximation of the limit load of the shell. A Stochastic Finite Element Method approach is used for the analysis of the “imperfect” shell structure involving random geometric deviations from its perfect geometry, as well as spatial variability of the modulus of elasticity and thickness of the shell, modeled as random fields. A corresponding eigenproblem for the prediction of the buckling load is solved at each MCS using a Rayleigh quotient-based formulation of the Preconditioned Conjugate Gradient method. It is shown that the use of the proposed method reduces drastically the computational effort involved in each MCS, making the implementation of such stochastic analyses in real-world structures affordable.     

Modeling and simulation of two-dimensional consolidation for thermoset matrix composites

A finite element method is developed to solve two-dimensional consolidation problems for composites manufacturing. The consolidation governing equations, one for solid stress and one for fluid pressure, are derived using a local volume averaging approach, and the two equations are strongly coupled. A special anisotropic, hyperelastic constitutive equation is developed for the solid stress. This equation matches Gutowski's model for consolidation transverse to the fibers, and has a high stiffness parallel to the fibers. An updated Lagrangian method is used to solve the equations, using implicit time integration and a successive substitution method. The code is applied to several case studies to explore two-dimensional consolidation effects. A free edge affects the thickness profile during consolidation, but the final thickness can still be uniform. This effect is substantial in the region close to the edge, and it propagates progressively from the edge toward the center. Simulations were also performed for laminates that bend to form a corner. The corner is thicker than the flat region after consolidation. Wiggles, similar to fiber buckling, arise at low values of shear modulus when using a male mold. Large values of the solid shear modulus cause the corner effect to extend far into the adjacent flat region. The length of the flat region also affects the consolidation of the corner.     

Postbuckling of functionally graded cylindrical shells with tangential edge restraints and temperature-dependent properties

This paper presents an analytical approach to investigate the buckling and postbuckling behavior of functionally graded cylindrical shells subjected to thermal and axial compressive loads. Material properties are assumed to be temperature dependent and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The governing equations are established within the framework of classical thin shallow shell theory taking both geometrical nonlinearity in von Kármán–Donnell sense and initial imperfection into consideration. Thermal stability analysis also incorporates the effects of tangential edge constraints. A Galerkin procedure is applied to derive expressions of load-deflection relations from which the thermal buckling loads and postbuckling curves of the shells are obtained by an iteration. Effects played by material and geometrical properties, tangential stiffness, imperfection and buckling modes are discussed.     

分析结构自由振动的传递矩阵精确形式

本文以微分方程和矩阵分析理论为基础,导出了求解结构自由振动传递矩阵法的精确工,无论是在计算效率和精度上都是对传递矩阵法的一个很好推广。文中运用这种封闭的精确形式求解了分段变厚圆柱壳的自由振动。     

功能梯度材料薄球壳热屈曲问题

研究了由金属和陶瓷组成的功能梯度材料薄球壳热屈曲问题。用张量方法推导得到轴对称球壳稳定性方程。将热本构方程应用到球壳稳定性方程中,得到以位移表示的球壳热屈曲方程组。分别考虑均布外压和温度作用,采用伽辽金法计算分析简支球壳的热屈曲问题,给出薄球壳厚度、物性参数变化、内外表面温差变化引起的临界温度变化趋势和临界压力变化趋势。