A new boundary-type finite element for 2-D- and 3-D-elastic structures

In this paper we discuss the theoretical and numerical formulation of 3-D Trefitz elements. Starting from the variational principle with the so-called hybrid stress method, the trial functions for the stresses have to fulfil the Beltrami equations, which means also the compatibility equations for the strains. The divergence theorem can be applied, and one arrives at a pure boundary formulation in the sense of the Trefftz method. Besides the resulting variational formulation, different regularizations of the interelement conditions are investigated by numerical tests. Two examples show the numerical efficiency of the derived elements. First, a geometric linear 3-D example is presented to show the effects on distorted element meshes. The third example shows the geometrically non-linear analysis of a shallow cylindrical shell segment under a singe load.     

Stress resultant geometrically non-linear shell theory with drilling rotations. Part III: Linearized kinematics

A consistent formulation of the geometrically linear shell theory with drilling rotations is obtained by the consistent linearization of the geometrically non-linear shell theory considered in Parts I and II of this work. It was also shown that the same formulation can be recovered by linearizing the governing variational principle for the three-dimensional geometrically non-linear continuum with independent rotation field. In the finite element implementation of the presented shell theory, relying on the modified method of incompatible modes, we were able to construct a four-node shell element which delivers a very high-level performance. In order to simplify finite element implementation, a shallow reference configuration is assumed over each shell finite element. This approach does not impair the element performance for the present four-node element. The results obtained herein match those obtained with the state-of-the-art implementations based on the classical shell theory, over the complete set of standard benchmark problems.     

A robust non‐linear mixed hybrid quadrilateral shell element

In the paper a non‐linear quadrilateral shell element for the analysis of thin structures is presented. The variational formulation is based on a Hu–Washizu functional with independent displacement, stress and strain fields. The interpolation matrices for the mid‐surface displacements and rotations as well as for the stress resultants and strains are specified. Restrictions on the interpolation functions concerning fulfillment of the patch test and stability are derived. The developed mixed hybrid shell element possesses the correct rank and fulfills the in‐plane and bending patch test. Using Newton's method the finite element approximation of the stationary condition is iteratively solved. Our formulation can accommodate arbitrary non‐linear material models for finite deformations. In the examples we present results for isotropic plasticity at finite rotations and small strains as well as bifurcation problems and post‐buckling response. The essential feature of the new element is the robustness in the equilibrium iterations. It allows very large load steps in comparison to other element formulations. Copyright © 2005 John Wiley & Sons, Ltd.     

基于几何非线性不协调元变分原理的圆柱壳非线性初始稳定性分析

本文根据几何非线性不协调元增量变分原理,按严格的壳体方程,建立了高精度的圆柱壳几何非线性20参数矩形精化不协调元RCSR4,并用于圆柱壳非线性初始稳定性分析。计算结果表明,该方法收敛性良好。     

Numerical analysis of viscoplastic axisymmetric shells based on a hybrid strain finite element

The viscoplastic deformation behavior of the shell is governed by any of the more recently proposed unified constitutive models with internal state variables and with the assumption that the total strain rate tensor can be decomposed additively into an elastic and an inelastic part. For the numerical analysis of viscoplastically deformed shells we use a hybrid strain finite element based on a geometrically linear theory of inelastic shells proposed by Kollmann and Mukherjee (1985). This theory gives the reduction of a two-field variational principle originally proposed by Oden and Reddy (1974) for elastic shells to the shell midsurface. It contains strain and displacement rates as variables to be independently varied. The shell formulation of this variational principle is the basis for the present work. First, a general hybrid finite element model is derived in which the shape functions for the strain and displacement rates can be polynomials of different order. Here we use the term hybrid in the sense of Pian (1988), i.e. in our two-field finite element the strain rates are condensed statically on the element level, leaving nodal displacements as the only unknowns in the final matrix equation. Then the finite element model is specialized for an axisymmetrically loaded conical shell with linear approximation of the strain rate field and quadratic interpolation of the displacement rates. Special emphasis is given to the derivation of the inelastic pseudo-forces and pseudo-moments. Numerical results for elastically and viscoplastically deformed shells are presented, where viscoplastic deformations are described by Hart's (1976) constitutive model.     

Alternate hybrid stress finite element models

Alternate hybrid stress finite element models in which the internal equilibrium equations are satisfied on the average only, while the equilibrium equations along the interelement boundaries and the static boundary conditions are adhered to exactly a priori, are developed. The variational principle and the corresponding finite element formulation, which allows the standard direct stiffness method of structural analysis to be used, are discussed. Triangular elements for a moderately thick plate and a doubly-curved shallow thin shell are developed. Kinematic displacement modes, convergence criteria and bounds for the direct flexibility-influence coefficient are examined.     

INFLUENCE OF GEOMETRIC NON-LINEARITIES ON THE FREE VIBRATIONS OF ORTHOTROPIC OPEN CYLINDRICAL SHELLS

This paper presents a general approach to predict the influence of geometric non-linearities on the free vibration of elastic, thin, orthotropic and non-uniform open cylindrical shells. The open shells are assumed to be freely simply supported along their curved edges and to have arbitrary straight edge boundary conditions. The method is a hybrid of finite element and classical thin shell theories. The solution is divided into two parts. In part one, the displacement functions are obtained from Sanders' linear shell theory and the mass and linear stiffness matrices are obtained by the finite element procedure. In part two, the modal coefficients derived from the Sanders–Koiter non-linear theory of thin shells are obtained for these displacement functions. Expressions for the second- and third-order non-linear stiffness matrices are then determined through the finite element method. The non-linear equation of motion is solved by the fourth-order Runge–Kutta numerical method. The linear and non-linear natural frequency variations are determined as a function of shell amplitudes for different cases. The results obtained reveal that the frequencies calculated by this method are in good agreement with those obtained by other authors. © 1997 by John Wiley & Sons, Ltd.     

压电材料修正后的H-R混合变分原理及其层合板的精确法

将三维弹性体的Hellinger-Reissner(H-R)混合变分原理引入到具有机-电耦合效应的压电材料静力学问题中,建立了压电材料修正后的H-R混合变分原理,通过变分运算和分部积分得到了压电材料的状态向量方程。给出了四边简支的压电材料层合板静力学状态向量方程的精确求解方法,数值实例的结果证明了方法是正确性的。这里的理论和求解方法同样适应于纯弹性材料板和压电材料板混合的层合板静力学问题的分析。变分原理将有利于压电材料问题相应的半解析法或有限元法的推导。     

几何非线性复合材料层合固体壳单元

通过定义广义应力,提出了一个改进的刚度矩阵,以克服固体壳元的厚度自锁问题,并能保证沿复合材料层合结构厚度方向上的连续应力分布;将应力插值函数分为低阶和高阶两部分,建议了一个新的非线性变分泛函,推导了一个用于几何非线性分析的九节点固体壳单元,该单元的计算精度和效率基本上与九节点减缩积分单元相当,与同类型其他单元相比,该单元显著提高了计算效率。     

Finite deformation formulation of a shell element for problems of sheet metal forming

An elastic-plastic thin shell finite element suitable for problems of finite deformation in sheet metal forming is formulated. Hill's yield criterion for sheet materials of normal anisotropy is applied. A nonlinear shell theory in a form of an incremental variational principle and a quasi-conforming element technique are employed in the Lagrangian formulation. The shell element fulfills the inter-element C ¹ continuity condition in a variational sense and has a sufficient rank to allow finite stretching, rotation and bending of the shell element. The accuracy and efficiency of the finite element formulation are illustrated by numerical examples.     

Refined hybrid degenerated shell element for geometrically non-linear analysis

Based on a variational principle with relaxed inter-element continuity requirements, a refined hybrid quadrilateral degenerated shell element GNRH6, which is a non-conforming model with six internal displacements, is proposed for the geometrically non-linear analysis. The orthogonal approach and non-conforming modes are incorporated into the geometrically non-linear formulation. Numerical results show that the orthogonal approach can improve computational efficiency while the non-conforming modes can eliminate the shear/membrane locking phenomenon and improve the accuracy. © 1998 John Wiley & Sons, Ltd.     

Higher‐order hybrid‐mixed axisymmetric thick shell element for vibration analysis

In this study, we present free vibration analysis of shells of revolution using the hybrid‐mixed finite element. The present hybrid‐mixed element, which is based on the modified Hellinger–Reissner variational principle, employs consistent stress parameters corresponding to cubic displacement polynomials with additional nodeless degrees to resolve the numerical difficulties due to the spurious constraints. The stress parameters are eliminated and the nodeless degrees are condensed out by the Guyan reduction. Several numerical examples show that the present element with cubic displacement interpolation functions and consistent quadratic stress functions is highly accurate for the free vibration analysis of shells of revolution, especially for higher vibration modes. Copyright © 2001 John Wiley & Sons, Ltd.     

一种中厚扁壳杂交应力单元

本文建立了一种任意形状四节点杂交应力扁壳单元,考虑了横向■切的效应,可用于中厚壳及薄壳。基于Hellinger-Reissner变分原理,以单元内不相容位移为Lagrange乘子放松了单元应力的平衡约束,并以应力在不相容应变上做功为零做为补充条件,消去了单元内多余的应力参数。此单元具有最少的应力参数,具有坐标不变性。例题计算结果表明,此单元的性能是相当好的。     

Multilayered shell finite element with interlaminar continuous shear stresses: a refinement of the Reissner–Mindlin formulation

A finite element formulation for refined linear analysis of multilayered shell structures of moderate thickness is presented. An underlying shell model is a direct extension of the first‐order shear‐deformation theory of Reissner–Mindlin type. A refined theory with seven unknown kinematic fields is developed: (i) by introducing an assumption of a zig‐zag (i.e. layer‐wise linear) variation of displacement field through the thickness, and (ii) by assuming an independent transverse shear stress fields in each layer in the framework of Reissner's mixed variational principle. The introduced transverse shear stress unknowns are eliminated on the cross‐section level. At this process, the interlaminar equilibrium conditions (i.e. the interlaminar shear stress continuity conditions) are imposed. As a result, the weak form of constitutive equations (the so‐called weak form of Hooke's law) is obtained for the transverse strains–transverse stress resultants relation. A finite element approximation is based on the four‐noded isoparametric element. To eliminate the shear locking effect, the assumed strain variational concept is used. Performance of the derived finite element is illustrated with some numerical examples. The results are compared with the exact three‐dimensional solutions, as well as with the analytical and numerical solutions obtained by the classical, the first‐order and some representative refined models. Copyright © 2000 John Wiley & Sons, Ltd.     

A mixed shell formulation accounting for thickness strains and finite strain 3d material models

A non‐linear quadrilateral shell element for the analysis of thin structures is presented. The Reissner–Mindlin theory with inextensible director vector is used to develop a three‐field variational formulation with independent displacements, stress resultants and shell strains. The interpolation of the independent shell strains consists of two parts. The first part corresponds to the interpolation of the stress resultants. Within the second part independent thickness strains are considered. This allows incorporation of arbitrary non‐linear 3d constitutive equations without further modifications. The developed mixed hybrid shell element possesses the correct rank and fulfills the in‐plane and bending patch test. The essential feature of the new element is the robustness in the equilibrium iterations. It allows very large load steps in comparison with other element formulations. We present results for finite strain elasticity, inelasticity, bifurcation and post‐buckling problems. Copyright © 2007 John Wiley & Sons, Ltd.     

On stress interpolation for hybrid models

A technique is proposed for the selection of stress interpolations for hybrid models. The present paper applies this approach to plane problems. The stiffness matrix is derived using the Hellinger–Reissner variational principle. This formulation uses infinitesimal equilibrium relationships and divides the assumed stress into its lower-order and higher-order parts. The patch test can be passed and the resulting elements are generally invariant. A plane four-node quadrilateral element is described and compared with existing elements. Numerical studies show that the accuracy of the element is generally good.     

Finite Rotation Piezoelectric Exact Geometry Solid-Shell Element with Nine Degrees of Freedom per Node

This paper presents a robust non-linear piezoelectric exact geometry (EG) four-node solid-shell element based on the higher-order 9-parameter equivalent single-layer (ESL) theory, which permits one to utilize 3D constitutive equations. The term EG reflects the fact that coefficients of the first and second fundamental forms of the reference surface are taken exactly at each element node. The finite element formulation developed is based on a new concept of interpolation surfaces (I-surfaces) inside the shell body. We introduce three I-surfaces and choose nine displacements of these surfaces as fundamental shell unknowns. Such choice allows us to represent the finite rotation piezoelectric higher-order EG solid-shell element formulation in a very compact form and to utilize in curvilinear reference surface coordinates the strain-displacement relationships, which are objective, that is, invariant under arbitrarily large rigid-body shell motions. To avoid shear and membrane locking and have no spurious zero energy modes, the assumed displacement-independent strain and stress resultant fields are introduced. In this connection, the Hu-Washizu variational equation is invoked. To implement the analytical integration throughout the element, the modified ANS method is applied. As a result, the present finite rotation piezoelectric EG solid-shell element formulation permits the use of coarse meshes and very large load increments.     

A SIMPLE AND EFFICIENT MIXED FINITE ELEMENT FOR AXISYMMETRIC SHELL ANALYSIS

An efficient, perhaps simplest, three-noded mixed finite element is proposed for axisymmetric shell analysis. The key feature in the present formulation is to start with a better variational principle in which the independent unknowns are only the quantities that can be prescribed at the shell edges. If the consistency for field approximations is satisfied, no other numerical consideration is necessary in the present element. Several examples confirm the satisfactory numerical behaviour of the present mixed element.     

An efficient hybrid / mixed element for geometrically nonlinear analysis of plate and shell structures

In this paper, a geometrically nonlinear hybrid/mixed curved quadrilateral shell element (HMSHEL4N) with four nodes is developed based on the modified Hellinger/Reissner variational principles. The performance of element is investigated and tested using some benchmark problems. A number of numerical examples of plate and shell nonlinear deflection problems are included. The results are compared with theoretical solutions and other numerical results. It is shown that HMSHEL4N does not possess spurious zero energy modes and any locking phenomenon, and is convergent and insensitive to the distorted mesh. A good agreement of the results with theoretical solutions, and better performance compared with displacement finite element method, are observed. It is seen that an efficient shell element based on stress and displacement field assumptions in solution and time is obtained.     

Partial hybrid stress element for the analysis of thick laminated composite plates

A new element—a partial hybrid stress element—is proposed in this paper for the analysis of thick laminated composite plates. The variational principle of this element can be derived from the Hellinger–Reissner principle through dividing six stress components into a flexural part (σ_x, σ_y, σ_xy, σ_z) and a transverse shear part (τ_xy, τ_yz). The element stiffness matrix can be formulated by assuming a stress field only for transverse shear stresses, while all the others are obtained from an assumed displacement field. Consequently, this new element combines the benefits of the conventional displacement method and the hybrid stress method. A twenty-node hexahedron element is employed in each layer for the displacement field. For the assumed transverse shear stress field, only the traction-free boundary conditions and interface traction continuity are satisfied. The equilibrium equation is enforced by the variational principle. Hence, the complicated work of searching an equilibrating stress field for all the six stress components in the hybrid stress method can be avoided. Furthermore, the interlaminar traction discontinuity, especially transverse shear, encountered by the conventional displacement method and higher-order plate element for laminated plate analysis can also be overcome. Examples are illustrated to demonstrate the accuracy and efficiency of this proposed partial hybrid stress element.