带旋转自由度C^0类任意四边形板（壳）单元

基于Ｒｅｉｓｓｎｅｒ－Ｍｉｎｄｉｌｉｎ板弯曲理论和Ｖｏｎ－Ｋａｒｍａｎ大挠度理论,采用单元域内和边界位移插值一致性的概念,将四节点等参弯曲单元与Ａｌｌｍａｎ膜变形二次插值模式相结合,对层合板壳的大挠度分析提供了一种实用的带旋转自由度的四节点Ｃ＾０类板单元。大量算例表明：该单元对板壳结构的线性强度、稳定性和后屈曲分析都表现出良好的收敛性和足够的工程精度。     

Geometrically non-linear analysis of arbitrary elastic supported plates and shells using an element-based Lagrangian shell element

In the present paper, the ELF (element-based Lagrangian formulation) 9-node ANS (assumed natural strain) shell element was combined with the spring element for geometrically non-linear analysis of plates and shells sustained by arbitrary elastic edge supports that are subjected to variation in loading.This particular spring element serves as tool for modeling an arbitrary elastic edge support with 6 DOF (degrees of freedom). The elastic edge support was modeled by combining different spring models. The ANS method was used to overcome shear and membrane locking problems inherent in some thin plate and shell problems. In the formulation of the ELF characteristic arrays, the expression of element strains was adopted in the framework of the element natural coordinates. The non-linear analysis results of idealized edge supports were validated against the reference solutions available in the literature. As a result of the numerical test, the combination of the ELF 9-node shell element and spring element shows an exceptional performance for non-linear analysis of plates and shells under elastic edge supports.     

A geometrically exact Cosserat shell-model including size effects,avoiding degeneracy in the thin shell limit. Part I: Formal dimensional reduction for elastic plates and existence of minimizers for positive Cosserat couple modulus

This contribution is concerned with a consistent formal dimensional reduction of a previously introduced finite-strain three-dimensional Cosserat micropolar elasticity model to the two-dimensional situation of thin plates and shells. Contrary to the direct modelling of a shell as a Cosserat surface with additional directors, we obtain the shell model from the Cosserat bulk model which already includes a triad of rigid directors. The reduction is achieved by assumed kinematics, quadratic through the thickness. The three-dimensional transverse boundary conditions can be evaluated analytically in terms of the assumed kinematics and determines exactly two appearing coefficients in the chosen ansatz. Further simplifications with subsequent analytical integration through the thickness determine the reduced model in a variational setting. The resulting membrane energy turns out to be a quadratic, elliptic, first order, non degenerate energy in contrast to classical approaches. The bending contribution is augmented by a curvature term representing an additional stiffness of the Cosserat model and the corresponding system of balance equations remains of second order. The lateral boundary conditions for simple support are non-standard. The model includes size-effects, transverse shear resistance, drilling degrees of freedom and accounts implicitly for thickness extension and asymmetric shift of the midsurface. The formal thin shell membrane limit without classical h ³-bending term is non-degenerate due to the additional Cosserat curvature stiffness and control of drill rotations. In our formulation, the drill-rotations are strictly related to the size-effects of the bulk model and not introduced artificially for numerical convenience. Upon linearization with zero Cosserat couple modulus we recover the well known infinitesimal-displacement Reissner-Mindlin model without size-effects and without drill-rotations. It is shown that the dimensionally reduced Cosserat formulation is well-posed for positive Cosserat couple modulus by means of the direct methods of variations along the same line of argument which showed the well-posedness of the three-dimensional Cosserat bulk model [72].Received: 16 April 2004, Accepted: 3 May 2004, Published online: 17 September 2004     

Static and free vibration analysis of graphene platelets reinforced composite truncated conical shell,cylindrical shell,and annular plate using theory of elasticity and DQM

Abstract

In this paper, three-dimensional static and free vibration analysis of functionally graded graphene platelets-reinforced composite (FG-GPLRC) truncated conical shells, cylindrical shells and annular plates with various boundary conditions is carried out within the framework of elasticity theory. The main contribution of the present work is that formulation for free vibration and bending behavior of the FG-GPLRC truncated conical shell based on theory of elasticity has not yet been reported. Additionally, formulation and solution for cylindrical shell and annular plate are derived by changing the semi vertex angle in formulation and solution of FG-GPLRC truncated conical shell. A semi-analytical solution is proposed base on employing differential quadrature method (DQM) together with state-space technique. Validity of current approach is assessed by comparing its numerical results with those available in the literature. An especial attention is drawn to the role of GPLs weight fraction, patterns of GPLs distribution through the thickness direction, geometrical parameters such as semi-vertex angle, length to mid-radius ratio on natural frequencies and bending characteristics. Numerical results reveal that desirable static and free vibration response (such as lower radial deflection and higher natural frequencies) can be achieved by locating more square shaped GPLs near inner and outer surfaces.     

Phenomenological invariants and their application to geometrically nonlinear formulation of triangular finite elements of shear deformable shells

A phenomenological definition of classical invariants of strain and stress tensors is considered. Based on this definition, the strain and stress invariants of a shell obeying the assumptions of the Reissner–Mindlin plate theory are determined using only three normal components of the corresponding tensors associated with three independent directions at the shell middle surface. The relations obtained for the invariants are employed to formulate a 15-dof curved triangular finite element for geometrically nonlinear analysis of thin and moderately thick elastic transversely isotropic shells undergoing arbitrarily large displacements and rotations. The question of improving nonlinear capabilities of the finite element without increasing the number of degrees of freedom is solved by assuming that the element sides are extensible planar nearly circular arcs. The shear locking is eliminated by approximating the curvature changes and transverse shear strains based on the solution of the Timoshenko beam equations. The performance of the finite element is studied using geometrically linear and nonlinear benchmark problems of plates and shells.     

Higher order zig-zag theory for smart composite shells under mechanical-thermo-electric loading

A higher order zig-zag shell theory based on general tensor formulation is developed to refine the predictions of the mechanical, thermal, and electric behaviors. All the complicated curvatures of surface including twisting curvatures can be described in a geometrically exact manner in the present shell theory because the present theory is based on the geometrically exact surface representation. The in-surface displacement fields are constructed by superimposing the linear zig-zag field to the smooth globally cubic varying field through the thickness. Smooth parabolic distribution through the thickness is assumed in the out-of-plane displacement in order to consider transverse normal deformation and stress. The layer-dependent degrees of freedom of displacement fields are expressed in terms of reference primary degrees of freedom by applying interface continuity conditions as well as bounding surface free conditions of transverse shear stresses. Thus the proposed theory has only seven primary displacement unknowns and they do not depend upon the number of layers. To assess the validity of present theory, the developed theory is evaluated under the thermal and electric load as well as under the mechanical load of composite cylindrical shells. Through the numerical examples, it is demonstrated that the proposed smart shell theory is efficient because it has the minimal degrees of freedom. The present theory is suitable in the predictions of deformation and stresses of thick smart composite shells under the mechanical, thermal, and electric loads combined.     

Three dimensional degenerated shell theory and assumed strain shell elements

In this paper Reissner-Mindlin plate theory is extended to cater for curved shell structures. It can be considered as Reissner-Mindlin type shell theory. From this theory, the C(O) continuity formulation of shell elements of taking account the transverse shear deformation could be derived directly. These degenerated shell elements have been widely employed. To overcome the locking of shear and membrane and avoid zero energy modes the author proposed the formulation of the new elements with assumed strains. A wide range of numerical tests was conducted and the results illustrate that the assumed strain elements possess high accuracy and good performance.     

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The fiat shell element is constructed by combining a quadrilateral area co- ordinate method （QAC） based membrane element AGQ6- II, and a Timoshenko beam function （TBF） method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory （FSDT）, for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical re- sults show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.     

PARAMETRIC VARIATIONAL PRINCIPLE BASED ELASTIC-PLASTIC ANALYSIS OF COSSERAT CONTINUUM

A new algorithm is developed based on the parametric variational principle for elastic-plastic analysis of Cosserat continuum. The governing equations of the classic elastic-plastic problem are regularized by adding rotational degrees of freedom to the conventional translational degrees of freedom in conventional continuum mechanics. The parametric potential energy principle of the Cosserat theory is developed, from which the finite element formulation of the Cosserat theory and the corresponding parametric quadratic programming model are constructed. Strain localization problems are computed and the mesh independent results are obtained.     

基于参数变分原理的Cosserat连续体弹塑性分析

基于参数变分原理,提出了Cosserat模型弹塑性计算的算法,给出了基于Cosserat理论的参数最小势能原理,基于所提出的变分方程,建立了Cosserat理论弹塑性分析的参数二次规划模型,进一步将算法应用于平面应变软化问题计算中,获得的结果具有良好的非网格依赖性.     

中厚板的耦合多项式基径向点插值无网格法

采用Mindlin平板理论,通过最小位能原理建立了各向同性中厚板的伽辽金整体弱式方程,形函数采用耦合多项式基的径向点插值法构造,可以直接施加本质边界条件. 算例表明,用耦合多项式基的径向点插值无网格法分析中厚板问题,具有效率高、精度高和易于实现等优点,可以避免薄板弯曲时的剪切自锁现象.     

平面广义四节点等参元GQ4及其性能探讨

广义节点有限元是将传统有限元方法中的节点广义化,在不增加节点个数的前提下,仅通过提高广义节点的插值函数的阶次,从而达到提高有限元解精度的目的.与现有的p型和hp型有限元不同,在这种新的有限元中,节点自由度全部定义在节点处,在理论与程序实现上与传统有限元方法具有很好的相容性,传统有限元方法是这种新方法的广义节点退化为0阶时的特殊情形.文中主要讨论了这一新方法的四节点等参元（记为GQ4）的形式.对GQ4进行的各种数值试验表明,所发展的广义四节点等参单元具有精度高且无剪切自锁与体积自锁等的特点.     

广义协调平板型矩形壳元

本文根据修正势能原理通过广义协调方法提出了一种列式简单的平板型矩形壳元ＧＣＲ２４。它在四个角点处各有六个自由度，总共二十四个自由度。作为一种极限协调元，单元的收敛性得到保证，并且不发生薄膜闭锁现象。通过标准问题的数值检验，表明本文提出的平板型矩形薄壳元是性能可靠、计算精度高的优质单元。     

Locking-Free Degenerated Isoparametric Shell Element

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｄｅｇｅｎｅｒａｔｅｄｉｓｏｐａｒａｍｅｔｒｉｃｓｈｅｌｌｅｌｅｍｅｎｔｈａｓｂｅｃｏｍｅｉｎｃｒｅａｓｉｎｇｌｙｐｏｐｕｌａｒｉｎｐｒａｃｔｉｃａｌｅｎｇｉｎｅｅｒｉｎｇｉｎａｗｉｄｅｒａｎｇｅ.ＴｈｉｓｇｅｎｅｒａｌｓｈｅｌｌｅｌｅｍｅｎｔｗａｓｏｒｉｇｉｎａｌｌｙｉｎｔｒｏｄｕｃｅｄｂｙＡｈｍａｄｅｔａｌ[1]ｆｏｒｔｈｅｌｉｎｅａｒａｎａｌｙｓｉｓｏｆｍｏｄｅｒａｔｅｌｙｔｈｉｃｋｓｈｅｌｌｓ .Ｉｔｗａｓｄｅｇｅｎｅｒａｔｅｄｆｒｏｍａｔｈｒｅｅ＿ｄｉｍｅｎｓｉ…     

Refined geometrically nonlinear formulation of a thin-shell triangular finite element

A refined geometrically nonlinear formulation of a thin-shell finite element based on the Kirchhoff-Love hypotheses is considered. Strain relations, which adequately describe the deformation of the element with finite bending of its middle surface, are obtained by integrating the differential equation of a planar curve. For a triangular element with 15 degrees of freedom, a cost-effective algorithm is developed for calculating the coefficients of the first and second variations of the strain energy, which are used to formulate the conditions of equilibrium and stability of the discrete model of the shell. Accuracy and convergence of the finite-element solutions are studied using test problems of nonlinear deformation of elastic plates and shells. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 160–172, September–October, 2007.     

Vibroacoustic comparisons of composite laminated cylindrical shells according to three shear deformation shell theories

Whether the first-order and Reddy third-order shear deformation shell theories are able to evaluate the vibroacoustic responses of laminated cylindrical shells with normal deformation in the high frequency range or not is examined by comparison with a 3D higher-order shear deformation shell theory. The implicit governing equations of arbitrary angle-ply laminated cylindrical shells are derived from the 3D higher-order and Reddy third-order shell theories, and solved on the basis of the Fourier transform. The Reddy third-order shell theory can be obtained as a special case from the 3D higher-order shell theory. The first-order and Reddy third-order shell theories almost give rise to the same vibrational and acoustic results. These two simple shear deformation shell theories can be used to study far-field acoustic radiation from laminated cylindrical shells from the low to high frequency range, but they show some differences from the 3D higher-order shell theory in high frequency vibration of shells. Nevertheless, the differences of vibrational responses seem not to be distinct. The helical wave spectra of the higher-order radial displacements are nearly separate from those of the low-order radial displacement and play a minor role in far-field acoustic radiation, which makes the two simple shell theories applicable in prediction of acoustic power of the cylindrical shells in the much higher frequency range. Moreover, it also results in the fact that far-field sound is least sensitive in comparison with near-field sound and vibration of shells.     

Model optimization of hybrid stress general shell element

A 20 DOF (degree of freedom) hybrid stress element based upon Mindlin plate bending theory is developed using the optimizing design method for thin and moderately thick shell. It is of quadrilateral shape with only four corner nodes and can be used for both shallow and deep shells with excellent performance and the shear locking problem no longer exists.The project supported by National Science Foundation of China.     

Nonlinear vibration of a rotating laminated composite circular cylindrical shell: traveling wave vibration

In this paper, the large-amplitude (geometrically nonlinear) vibrations of rotating, laminated composite circular cylindrical shells subjected to radial harmonic excitation in the neighborhood of the lowest resonances are investigated. Nonlinearities due to large-amplitude shell motion are considered using the Donnell’s nonlinear shallow-shell theory, with account taken of the effect of viscous structure damping. The dynamic Young’s modulus which varies with vibrational frequency of the laminated composite shell is considered. An improved nonlinear model, which needs not to introduce the Airy stress function, is employed to study the nonlinear forced vibrations of the present shells. The system is discretized by Galerkin’s method while a model involving two degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the forced vibration responses of the two-degrees-of-freedom system. The stability of analytical steady-state solutions is analyzed. Results obtained with analytical method are compared with numerical simulation. The agreement between them bespeaks the validity of the method developed in this paper. The effects of rotating speed and some other parameters on the nonlinear dynamic response of the system are also investigated.     

Novel shear deformation model for moderately thick and deep laminated composite conoidal shell

This article presents a novel mathematical model for moderately thick and deep laminated composite conoidal shell. The zero transverse shear stress at top and bottom of conoidal shell conditions is applied. Novelty in the present formulation is the inclusion of curvature effect in displacement field and cross curvature effect in strain field. This present model is suitable for deep and moderately thick conoidal shell. The peculiarity in the conoidal shell is that due to its complex geometry, its peak value of transverse deflection is not at its center like other shells. The C¹ continuity requirement associated with the present model has been suitably circumvented. A nine-node curved quadratic isoparametric element with seven nodal unknowns per node is used in finite element formulation of the proposed mathematical model. The present model results are compared with experimental, elasticity, and numerical results available in the literature. This is the first effort to solve the problem of moderately thick and deep laminated composite conoidal shell using parabolic transverse shear strain deformation across the thickness of conoidal shell. Many new numerical problems are solved for the static study of moderately thick and deep laminated composite conoidal shell considering 10 different practical boundary conditions, four types of loadings, six different hl/hh (minimum rise/maximum rise) ratios, and four different laminations.