A sixteen node shell element with a matrix stabilization scheme

A sixteen node shell element is developed using a matrix stabilization scheme based on the Hellinger-Reissner principle with independent strain. Initially the assumed independent strain is divided into a lower order part and a higher order part. The stiffness matrix corresponding to the lower order assumed strain is equivalent to the stiffness matrix of the assumed displacement model element with the reduced integration scheme. The spurious kinematic modes of the element are suppressed by introducing a stabilization matrix associated with a judiciously chosen set of higher order assumed strain fields. Numerical results show that this element is free of locking even for very thin plates and shells.     

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.     

Refined discrete quadrilateral degenerated shell element by using Timoshenko's beam function

A refined discrete degenerated 20‐DOF quadrilateral shell element RQS20 is proposed. The exact displacement function of the Timoshenko's beam is used as the displacement on the element boundary. The re‐constitute method for shear strain matrix is adopted. The proposed element can be used for the analysis of both moderately thick and thin plates/shells, and the convergence for the very thin case can be ensured theoretically. Numerical examples presented show that the new model indeed possesses higher accuracy in the analysis of thin and thick plates/shells, and that it can pass the patch test required for the Kirchhoff thin plate elements. Most important of all, it is free from the membrane and shear locking phenomena for extremely thin plates/shells, on the one hand, and it can also avoid the phenomenon of oscillatory solutions for thick plates/shells case on the other. Copyright © 2005 John Wiley & Sons, Ltd.     

Refined 15‐DOF triangular discrete degenerated shell element with high performances

A refined discrete degenerated 15‐DOF triangular shell element RDTS15 with high performances is proposed. For constructing the element displacement function, the exact displacement function of the Timoshenko's beam is used as the displacement on the element boundary, and the re‐constitute method for shear strain matrix is adopted. The proposed element can be used in the analysis of both moderate thick and thin plates/shells. Numerical examples presented show that the new model indeed possesses higher accuracy in the analysis of thin and thick plates/shells, and that it can pass the patch test required for the Kirchhoff thin plate elements, and also passed the inf–sup test for free cylindrical shell problems and satisfied both the bending‐ and membrane‐dominated test. Copyright © 2004 John Wiley Sons, Ltd.     

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     

实体退化板单元及其在板的振动分析中的应用

经典板壳单元是由板壳理论构造出来的,而经典的板壳理论是在空间弹性理论的基础上考虑板壳的基本假定得来的。在空间等参数单元的基础上,直接引入板壳的基本假定,修改空间等参数单元的弹性矩阵,从而构造出适合于厚薄板壳分析的20结点实体退化板单元,并将其应用于开口圆柱薄壳的静力分析和厚薄板的固有振动分析。数值算例表明,该单元收敛快,稳定性好,具有较高的精度。此外,该单元还可以用于曲边变厚度板、壳体及层合板的振动分析。     

A co-rotational 8-node assumed strain shell element for postbuckling analysis of laminated composite plates and shells

 The formulation of a nonlinear composite shell element is presented for the solution of stability problems of composite plates and shells. The formulation of the geometrical stiffness presented here is exactly defined on the midsurface and is efficient for analyzing stability problems of thin and thick laminated plates and shells by incorporating bending moment and transverse shear resultant forces. The composite element is free of both membrane and shear locking behaviour by using the assumed natural strain method such that the element performs very well as thin shells. The transverse shear stiffness is defined by an equilibrium approach instead of using the shear correction factor. The proposed formulation is computationally efficient and the test results showed good agreement. In addition the effect of the viscoelastic material is investigated on the postbuckling behaviour of laminated composite shells. Received: 6 February 2002 / Accecpted: 6 January 2003 ID=" Present address: School of Civil Engineering, Asian Institute of Technology     

Transient analysis of FGM and laminated composite structures using a refined 8-node ANS shell element

In this paper, we investigate the vibration analysis of functionally graded material (FGM) and laminated composite structures, using a refined 8-node shell element that allows for the effects of transverse shear deformation and rotary inertia. The properties of FGM vary continuously through the thickness direction according to the volume fraction of constituents defined by sigmoid function, but in this method, their Poisson’s ratios of the FGM plates and shells are assumed to be constant. The finite element, based on a first-order shear deformation theory, is further improved by the combined use of assumed natural strains and different sets of collocation points for interpolation the different strain components. We analyze the influence of the shell element with the various location and number of enhanced membrane and shear interpolation. Using the assumed natural strain method with proper interpolation functions the present shell element generates neither membrane nor shear locking behavior even when full integration is used in the formulation. The natural frequencies of plates and shells are presented, and the forced vibration analysis of FGM and laminated composite plates and shells subjected to arbitrary loading is carried out. In order to overcome membrane and shear locking phenomena, the assumed natural strain method is used. To validate and compare the finite element numerical solutions, the reference solutions of plates based on the Navier’s method, the series solutions of sigmoid FGM (S-FGM) plates are obtained. Results of the present theory show good agreement with the reference solutions. In addition the effect of damping is investigated on the forced vibration analysis of FGM plates and shells.     

复合材料四结点四边形多层退化壳单元

本文提出了一个复合材料四结点四边形多层退化壳单元。单元从修正的Hu-Washizu变分原理导出,独立假设位移场、每层内部应变场和应力场。文中给出了几个复合材料板和壳静力线性问题实例。计算结果表明本文的单元是准确和有效的。     

A mixed formulation of C0-linear triangular plate/shell element—the role of edge shear constraints

A simple and effective linear C⁰-triangular element for plates and shells is developed on the basis of the Hellinger-Reissner mixed variational principle with independently assumed stress and displacement fields. Two main features are emphasized in this development concerning the assumed transverse shear stress field. Firstly, this assumption results in edge-type penalty constraints in the thin plate/shell regimes. Secondly, the element is used in the form of a four-triangle, cross-diagonal macroelement mesh. As a result, the element is shown to be free from shear locking in thin plate/shell applications, and it satisfied all the appropriate patch tests required for Kirchhoff plate models. In addition, the element exhibits good overall convergence properties in a variety of test problems for plates and shells. Finally, from the computational standpoint, the element is very efficient since all stiffness component matrices are derived in explicit forms.     

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.     

Free and forced vibration analysis of laminated composite plates and shells using a 9-node assumed strain shell element

The natural frequencies of isotropic and composite laminates are presented. The forced vibration analysis of laminated composite plates and shells subjected to arbitrary loading is investigated. In order to overcome membrane and shear locking phenomena, the assumed natural strain method is used. To develop a laminated shell element for free and forced vibration analysis, the equivalent constitutive equation that makes the computation of composite structures efficient was applied. The Mindlin-Reissner theory which allows the shear deformation and rotary inertia effect to be considered is adopted for development of nine-node assumed strain shell element. The present shell element offers significant advantages since it consistently uses the natural co-ordinate system. Results of the present theory show good agreement with the 3-D elasticity and analytical solutions. In addition the effect of damping is investigated on the forced vibration analysis of laminated composite plates and shells.     

Refined triangular discrete Mindlin flat shell elements

In this paper flat shell elements are formed by the assemblage of discrete Mindlin plate elements RDKTM and either the constant strain membrane element CST or the Allmans membrane element with drilling degrees of freedom LST. The element RDKTM is a robust Mindlin plate element, which can perform uniformly thick and thin plate bending analysis. It also passes the patch test for thin plate bending, and its convergence for very thin plates can be ensured theoretically. The singularity of the stiffness matrix and membrane locking are studied for the present elements. Numerical examples are presented to show that the present models indeed possess properties of simple formulations, high accuracy for thin and thick shells, and it is free from shear locking for thin plate/shell analysis.     

Triangular C0 bending elements based on the Hu-Washizu principle and orthogonality conditions

This paper deals with simple and effective C⁰ linear triangular bending elements based on the Hu-Washizu variational principle. The following key features are inherent in the proposed formulation: extensional, flexural and transverse shear strains and the corresponding stresses are referred to the directions of the triangle sides. The independent variation of displacements, strains and stresses yields discrete approximations consistent with the continuum theory. Furthermore, a proper identification of the deformational modes and the orthogonality requirements between the assumed transverse shear forces and the undesirable strain fields lead to element formulations with important attributes: minimization of superfluous energy, properly justified assumptions, simplicity of formulation and straightforward extension to non-linear problems. Alternative approximations of strains and stresses, also presented, enhance the element performance in the case of both thick and thin plates and shells. Test examples demonstrate the characteristic properties of the proposed element formulations.     

Ultimate load behaviour of reinforced concrete plates and shells under dynamic transient loading

This paper considers a layered thick shell finite element procedure for determining the dynamic transient nonlinear response of plates and shells. The degenerated three-dimensional isoparametric shell element with independent rotational and translational degrees-of-freedom is employed and a layered formulation is adopted to represent the steel reinforcement and to simulate progressive concrete cracking through the thickness. The dynamic yielding function is assumed to be a function of the current strain rate, in addition to being total plastic strain or work dependent. The concrete model also simulates both compressive crushing and tensile cracking behaviours and an implicit Newmark algorithm is employed for time integration of the equations of motion. Several numerical examples are presented for both slab and shell structures and the results obtained compared with those from other sources wherever available.     

Anisotropic elasto-plastic finite element analysis of thick and thin plates and shells

An elasto-plastic analysis of anisotropic plates and shells is undertaken by means of the finite element displacement method. A thick shell formulation accounting for shear deformation is considered, which is based on a degenerate three-dimensional continuum element. The accommodation of variable material properties, not only along the surface of the structure but also through the thickness, is made possible by a discrete layered approach. Although isoparametric elements of the Serendipity family give satisfactory solutions for thick and moderately thin shells the results exhibit ‘locking’ for an increasing ratio of span to thickness. To develop a numerical model which is applicable to thick or thin plates and shells, the nine-node Lagrangian element and the Heterosis element are also introduced into the present model. Plastic yielding is based on the Huber-Mises yield surface extended by Hill for anisotropic materials. The yield function is generalized by introducing anisotropic parameters of plasticity which are updated during the material strain hardening history. Numerical examples are presented and compared with available solutions. The effects of anisotropy on these solutions are also discussed.     

Geometrically nonlinear analysis of laminate composite plates and shells using the eight-node hexahedral element with one-point integration

An eight-node hexahedral isoparametric element with one-point quadrature for the geometrically nonlinear static and dynamic analysis of plates and shells of laminate composite materials is formulated and implemented in this work. The element is free of volumetric and shear locking and spurious modes are not detected. Shear locking is avoided using a corotational system for strain and stress components, whereas volumetric locking is controlled using one-point quadrature for the dilatational (or spherical) part of the gradient matrix. The efficiency and potential of the three-dimensional element in the analysis of plates and shells of laminate composite materials undergoing large displacements and rotations are shown solving several numerical examples and comparing with results obtained by other authors using different plate and shells elements.     

A stabilized co‐rotational curved quadrilateral composite shell element

A new curved quadrilateral composite shell element using vectorial rotational variables is presented. An advanced co‐rotational framework defined by the two vectors generated by the four corner nodes is employed to extract pure element deformation from large displacement/rotation problems, and thus an element‐independent formulation is obtained. The present line of formulation differs from other co‐rotational formulations in that (i) all nodal variables are additive in an incremental solution procedure, (ii) the resulting element tangent stiffness is symmetric, and (iii) is updated using the total values of the nodal variables, making solving dynamic problems highly efficient. To overcome locking problems, uniformly reduced integration is used to compute the internal force vector and the element tangent stiffness matrix. A stabilized assumed strain procedure is employed to avoid spurious zero‐energy modes. Several examples involving composite plates and shells with large displacements and large rotations are presented to testify to the reliability, computational efficiency, and accuracy of the present formulation. Copyright © 2011 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.