A new efficient mixed formulation for thin shell finite element models

A nine node shell element is developed by a new and more efficient mixed formulation. The new shell element formulation is based on the Hellinger–Reissner principle with independent strain and the concept of a degenerate solid shell. The new formulation is made more efficient in terms of computing time than the conventional mixed formulation by dividing the assumed strain fields into a lower order part and a higher order part. Numerical results demonstrate that the present nine node element is free of locking even for very thin plates and shells and is also kinematically stable. In fact the stiffness matrix associated with the higher order assumed strain plays the role of a stabilization matrix.     

A finite element formulation with stabilization matrix for geometrically non-linear shells

An assumed strain finite element formulation with a stabilization matrix is developed for analysis of geometrically non-linear problems of isotropic and laminated composite shells. The present formulation utilizes the degenerate solid shell concept and assumes an independent strain as well as displacement. The assumed independent strain field is divided into a lower order part and a higher order part. Subsequently, the lower order part is set equal to the displacement-dependent strain evaluated at the lower order integration points and the remaining higher order part leads to a stabilization matrix. The strains and the determinant of the Jacobian matrix are assumed to vary linearly in the thickness direction. This assumption allows analytical integration through thickness, independent of the number of plies. A nine-node element with a judiciously chosen set of higher order assumed strain field is developed. Numerical tests involving isotropic and composite shells undergoing large deflections demonstrate the validity of the present formulation.     

An eighteen-node solid element for thin shell analysis

An eighteen-node, three-dimensional, solid element with 54 degrees of freedom is presented for the finite element analysis of thin plates and shells. The element is based on the Hellinger-Reissner principle with independent strain. The assumed independent strain is divided into higher and lower terms. The stiffness matrix associated with the higher order independent strain plays the role of stabilization matrix. A modified stress-strain relation decoupling inplane and normal strain is used to incorporate thin shell behaviour. Numerical results demonstrate that, with a properly chosen set of assumed strain, this element is effectively free of locking even for very thin plates and shells.     

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。     

A new efficient approach to the formulation of mixed finite element models for structural analysis

A new mixed finite element formulation is developed based on the Hellinger-Reissner principle with independent strain. By dividing the assumed strain into its lower order and higher order parts, the new formulation can be made much more efficient than the conventional mixed formulation. In addition the present new approach provides an alternative way of introducing a stabilization matrix to suppress undesirable kinematic modes.     

An efficient co‐rotational formulation for curved triangular shell element

A 6‐node curved triangular shell element formulation based on a co‐rotational framework is proposed to solve large‐displacement and large‐rotation problems, in which part of the rigid‐body translations and all rigid‐body rotations in the global co‐ordinate system are excluded in calculating the element strain energy. Thus, an element‐independent formulation is achieved. Besides three translational displacement variables, two components of the mid‐surface normal vector at each node are defined as vectorial rotational variables; these two additional variables render all nodal variables additive in an incremental solution procedure. To alleviate the membrane and shear locking phenomena, the membrane strains and the out‐of‐plane shear strains are replaced with assumed strains in calculating the element strain energy. The strategy used in the mixed interpolation of tensorial components approach is employed in defining the assumed strains. The internal force vector and the element tangent stiffness matrix are obtained from calculating directly the first derivative and second derivative of the element strain energy with respect to the nodal variables, respectively. Different from most other existing co‐rotational element formulations, all nodal variables in the present curved triangular shell formulation are commutative in calculating the second derivative of the strain energy; as a result, the element tangent stiffness matrix is symmetric and is updated by using the total values of the nodal variables in an incremental solution procedure. Such update procedure is advantageous in solving dynamic problems. Finally, several elastic plate and shell problems are solved to demonstrate the reliability, efficiency, and convergence of the present formulation. Copyright © 2007 John Wiley & Sons, Ltd.     

An eight‐node hybrid‐stress solid‐shell element for geometric non‐linear analysis of elastic shells

This paper presents eight‐node solid‐shell elements for geometric non‐linear analysis of elastic shells. To subdue shear, trapezoidal and thickness locking, the assumed natural strain method and an ad hoc modified generalized laminate stiffness matrix are employed. A selectively reduced integrated element is formulated with its membrane and bending shear strain components taken to be constant and equal to the ones evaluated at the element centroid. With the generalized stresses arising from the modified generalized laminate stiffness matrix assumed to be independent from the ones obtained from the displacement, an extended Hellinger–Reissner functional can be derived. By choosing the assumed generalized stresses similar to the assumed stresses of a previous solid element, a hybrid‐stress solid‐shell element is formulated. Commonly employed geometric non‐linear homogeneous and laminated shell problems are attempted and our results are close to those of other state‐of‐the‐art elements. Moreover, the hybrid‐stress element converges more readily than the selectively reduced integrated element in all benchmark problems. Copyright © 2002 John Wiley & Sons, Ltd.     

A stabilized one‐point integrated quadrilateral Reissner–Mindlin plate element

A new quadrilateral Reissner–Mindlin plate element with 12 element degrees of freedom is presented. For linear isotropic elasticity a Hellinger–Reissner functional with independent displacements, rotations and stress resultants is used. Within the mixed formulation the stress resultants are interpolated using five parameters for the bending moments and four parameters for the shear forces. The hybrid element stiffness matrix resulting from the stationary condition can be integrated analytically. This leads to a part obtained by one‐point integration and a stabilization matrix. The element possesses a correct rank, does not show shear locking and is applicable for the evaluation of displacements and stress resultants within the whole range of thin and thick plates. The bending patch test is fulfilled and the computed numerical examples show that the convergence behaviour is better than comparable quadrilateral assumed strain elements. Copyright © 2004 John Wiley & Sons, Ltd.     

基于共旋三角形厚薄通用壳元的几何非线性分析

基于共旋列式方法发展了一种用于复合材料层合板结构几何非线性分析的简单高效的三结点三角形平板壳元。该壳元由具有面内转动自由度的广义协调膜元GT9与假设剪切应变场和假设单元转角场的广义协调厚薄通用板元TMT组合而成。为避免薄膜闭锁而采用单点积分计算与薄膜应变有关的项, 同时增加一个稳定化矩阵以消除单点积分导致的零能模式。基于层合板一阶剪切变形理论, 给出了考虑层合板具体铺层顺序的修正的横向剪切刚度, 使该壳元可用于中厚层合板结构的分析。由于共旋列式大转动小应变的假设, 共旋列式内核的几何线性的单元刚阵可仅计算一次而保存下来用于整个几何非线性求解的过程以提高计算效率。数值算例表明提出的壳元进行包括复合材料层合板结构的厚薄壳结构的几何非线性分析的精度高且效率高。     

A new reduced integration solid‐shell element based on EAS and ANS with hourglass stabilization

In this paper, a novel reduced integration eight‐node solid‐shell finite element formulation with hourglass stabilization is proposed. The enhanced assumed strain method is adopted to eliminate the well‐known volumetric and Poisson thickness locking phenomena with only one internal variable required. In order to alleviate the transverse shear and trapezoidal locking and correct rank deficiency simultaneously, the assumed natural strain method is implemented in conjunction with the Taylor expansion of the inverse Jacobian matrix. The projection of the hourglass strain‐displacement matrix and reconstruction of its transverse shear components are further employed to avoid excessive hourglass stiffness. The proposed solid‐shell element formulation successfully passes both the membrane and bending patch tests. Several typical examples are presented to demonstrate the excellent performance and extensive applicability of the proposed element. Copyright © 2015 John Wiley & Sons, Ltd.     

Simple and accurate four-node axisymmetric element

Based on the assumed strain method, a simple four-node axisymmetric solid element is introduced. The assumed strain field is carefully selected to preserve the correct rank of the element stiffness matrix and to achieve high accuracy. The strain field is developed in conjunction with orthogonal projections and no matrix inversions are needed. The coarse mesh accuracy in bending and in typical axisymmetric load cases is excellent even for nearly incompressible materials. The strain-driven format obtained is well suited for materials with non-linear stress–strain relations. Several numerical examples are presented where the excellent performance of the proposed simple element is verified. Copyright © 2006 John Wiley & Sons, Ltd.     

Stiffness matrix for a non-prismatic beam-column element

An algorithm is presented which generates an element stiffness matrix for non-prismatic beam-column members using Newmark's numerical procedure of successive approximations. The resulting element stiffness matrix on element co-ordinates is fond to be in excellent agreement with those available for limiting cases. An approximate stability analysis conducted on the covergence of the numerical scheme shows that the proposed algorithm is stable. The critical buckling loads for various and conditions are computed as part of the computational scheme. A computer program listing in FORTRAN IV is included which will handle essentially and order and/or kind of non-prismaticity of beam elements. The results are presented in the conventional and convenient coefficient format.     

A PARTIAL HYBRID DEGENERATED PLATE/SHELL ELEMENT FOR THE ANALYSIS OF LAMINATED COMPOSITES

A degenerated plate/shell element based on the combined energy variational principle and the equivalent single-layer model is proposed. It is derived from the 3-D continuum equation by imposing one constraint on the 3-D isoparametric solid element: a straight line normal to the mid-surface before deformation remains straight but not normal after deformation. The continuities of interlaminar stresses are satisfied at the interlaminar surface and the number of degrees of freedom per node is independent of the number of layers. In this work, the combined energy variational principle is used to overcome transverse stress continuity limitations of single-layer models. The traction-free conditions are satisfied on the upper and lower surfaces of a laminate by assuming the transverse stress components independently. The transverse normal strain is taken into account in order to consider the full 3-D effect in a laminated composite. The iso-function method and the classification method of the stress modes are used to construct the assumed stress field which contains a minimum number of stress modes and guarantees no zero energy mode in the element stiffness matrix. Three examples are presented to illustrate the efficiency and accuracy of the element.     

Equivalent homogeneous finite element for composite materials via reissner principle. Part I: Finite element for plates

The stiffness matrix in the finite element method for multi-layered materials is generally computed by expressing the strain energy in each layer and adding them together. In order to lower the computing time, which may be prohibitive if the number of layers is high, and to get accurate information on the stresses, especially on transverse shear stresses, we present a new finite element using the Reissner principle. In the first part the case of plates will be detailed: extensions to shell problems will be presented in the second part. The efficiency of the method is tested on a special analytic solution, and some examples are given.     

Large reference displacement analysis of composite plates part I: Finite element formulation

This investigation concerns itself with the dynamic analysis of thin, laminated composite plates consisting of layers of orthotropic laminae that undergo large arbitrary rigid body displacements and small elastic deformations. A non-linear finite element formulation is developed which utilizes the assumption that the bonds between the laminae are infinitesimally thin and shear non-deformable. Using the expressions for the kinetic and strain energies, the lamina mass and stiffness matrices are identified. The non-linear mass matrix of the lamina is expressed in terms of a set of invariants that depend on the assumed displacement field. By summing the kinetic and strain energies of the laminae of an element, the element mass and stiffness matrix can be defined in terms of the set of element invariants. It is shown that the element invariants can be expressed explicitly in terms of the invariants of its laminae. By assembling the finite elements of the deformable body, the body invariants can be identified and expressed explicitly in terms of the invariants of the laminae of its elements. In the dynamic formulation presented in this paper, the shape functions of the laminae are assumed to have rigid body modes that need to describe only large rigid body translations. The computer implementation and the use of the formulation developed in this investigation in multibody dynamics are discussed in the second part of this paper.     

Linear smoothed polygonal and polyhedral finite elements

The strain smoothing technique over higher order elements and arbitrary polytopes yields less accurate solutions than other techniques such as the conventional polygonal finite element method. In this work, we propose a linear strain smoothing scheme that improves the accuracy of linear and quadratic approximations over convex polytopes. The main idea is to subdivide the polytope into simplicial subcells and use a linear smoothing function in each subcell to compute the strain. This new strain is then used in the computation of the stiffness matrix. The convergence properties and accuracy of the proposed scheme are discussed by solving a few benchmark problems. Numerical results show that the proposed linear strain smoothing scheme makes the approximation based on polytopes able to deliver the same optimal convergence rate as traditional quadrilateral and hexahedral approximations. The accuracy is also improved, and all the methods tested pass the patch test to machine precision. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite elements based on energy orthogonal functions

It is shown how the convergence requirements for a finite element may be written as a set of linear constraints on the stiffness matrix. It is then attempted to construct a best possible stiffness matrix. The constraint equations restrict the way in which these stiffness terms may be chosen; however, there is normally still room for improving or optimizing an element. It is demonstrated how an element stiffness matrix may be found using rigid body, constant strain and higher order deformation modes. Further, it is shown how the constraint equations may be exploited in deriving an ‘energy orthogonality theorem’. This theorem opens the door to a whole new class of simple finite elements which automatically satisfy the convergence requirements. Examples of deriving plane stress and plate bending elements are given.     

Buckling Analysis of Angle-ply Composite and Sandwich Plates by Combination of Geometric Stiffness Matrix

Buckling response of angle-ply laminated composite and sandwich plates are analyzed using the global-local higher order theory with combination of geometric stiffness matrix in this paper. This global-local theory completely fulfills the free surface conditions and the displacement and stress continuity conditions at interfaces. Moreover, the number of unknowns in this theory is independent of the number of layers in the laminate. Based on this global-local theory, a three-noded triangular element satisfying C¹ continuity conditions has also been proposed. The bending part of this element is constructed from the concept of DKT element. In order to improve the accuracy of the analysis, a method of modified geometric stiffness matrix has been introduced. Numerical results show that the present theory not only computes accurately the buckling response of general laminated composite plates but also predicts the critical buckling loads of soft-core sandwiches. However, the global higher-order theories as well as first order theories might encounter some difficulties and overestimate the critical buckling loads for soft-core sandwich plates.     

Solid elements with rotational degrees of freedom: Part II—tetrahedron elements

This is the second part of a two part paper on three-dimensional finite elements with rotational degrees of freedom (DOF). Part II introduces a solid tetrahedron element having 3 translational and 3 rotational DOF per node. The corner rotations are introduced by transformation of the midside translational DOF of a 10-node tetrahedron element. To further enhance the element performance a least squares strain extraction technique is also implemented to develop the stiffness matrix with a desired field. The strain smoothing improves performance without causing a loss in generality. As with the hexahedron in Part I, the element stiffness is augmented with a small penalty stiffness to eliminate any possible spurious zero energy modes. The new tetrahedron element passes the patch test and demonstrates much improved performance over the 4-node translational DOF only (constant strain) tetrahedron element.     

A resultant 8-node solid-shell element for geometrically nonlinear analysis

A new resultant force formulation of 8-node solid element is presented for the linear and nonlinear analysis of thin-walled structures. The global, local and natural coordinate systems were used to accurately model the shell geometry. The assumed natural strain methods with plane stress concept were implemented to remove the various locking problems appearing in thin plates and shells. The correct warping behavior in the very thin twisted beam test was obtained by using an improved Jacobian transformation matrix. The 2 × 2 Gauss integration scheme was used for the calculation of the element stiffness matrix. From the computational viewpoint, the present solid element is very efficient for a large scale of nonlinear modeling. A lot of numerical tests were carried out for the validation of the present 8-node solid-shell element and the results are in good agreement with references.An erratum to this article can be found at