Geometrically nonlinear formulation for the curved shell elements

This paper presents a geometrically nonlinear formulation using total lagrangian approach for the three-dimensional curved shell elements. The basic element geometry is constructed using the coordinates of the middle surface nodes and the mid-surface nodal point normals. The element displacement field is described using three translations of the mid-surface nodes and the two rotations about the local axes. The existing shell element formulations are restricted to small nodal rotations between two successive load increments. The element formulation presented here removes such restrictions. This is accomplished by retaining nonlinear nodal rotation terms in the definition of the displacement field and the consistent derivation of the element properties. The formulation presented here is very general and yet can be made specific by selecting proper nonlinear functions representing the effects of nodal rotations. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate the behaviour and the accuracy of the elements.     

Geometrically non-linear formulation for three dimensional curved beam elements with large rotations

This paper presents a geometrically non-linear formulation (GNL) for the three dimensional curved beam elements using the total Lagrangian approach. The element geometry is constructed using co-ordinates of the nodes on the centroidal or reference axis and the orthogonal nodal vectors representing the principal bending directions. The element displacement field is described using three translations at the element nodes and three rotations about the local axes

1 The element displacement field has also been described in the literature using Euler parameters, Milenkovic parameters, or Rodriges parameters representing the effects of large rotations.

. The GNL three dimensional beam element formulations based on these element approximations are restricted to small nodal rotations between two successive load increments. The element formulation presented here removes such restrictions. This is accomplished by retaining non-linear nodal terms in the definition of the element displacement field, and the consistent derivation of the element properties. The formulation presented here is very general and yet can be made specific by selecting proper non-linear functions representing the effects of nodal rotations. The details of the element properties are presented and discussed. Numerical examples are also presented to demonstrate the behaviour and the accuracy of the elements. A comparison of the results obtained from the present formulation with those available in the literature using a linearized element approximation clearly demonstrate the superiority of the formulation in terms of large load steps, large rotations between two load steps and extremely good convergence characteristics during equilibrium iterations. The displacement approximation of these elements is fully compatible with the isoparametric curved shell elements (with large rotations), and since the elements possess offset capability, these elements can also serve as stiffeners for the curved shells.     

A 9-node co-rotational quadrilateral shell element

A new 9-node co-rotational curved quadrilateral shell element formulation is presented in this paper. Different from other existing co-rotational element formulations: (1) Additive rotational nodal variables are utilized in the present formulation, they are two well-chosen components of the mid-surface normal vector at each node, and are additive in an incremental solution procedure; (2) the internal force vector and the element tangent stiffness matrix are respectively the first derivative and the second derivative of the element strain energy with respect to the nodal variables, furthermore, all nodal variables are commutative in calculating the second derivatives, resulting in symmetric element tangent stiffness matrices in the local and global coordinate systems; (3) the element tangent stiffness matrix is updated using the total values of the nodal variables in an incremental solution procedure, making it advantageous for solving dynamic problems. Finally, several examples are solved to verify the reliability and computational efficiency of the proposed element formulation.     

A HYBRID STRESS QUADRILATERAL SHELL ELEMENT WITH FULL ROTATIONAL D.O.F.S

A 4-node hybrid stress quadrilateral shell element with 3 rotational d.o.f.s per node is presented. The mid-surface displacement of the element is founded on Allman's rotation. The equal-rotation mode intrinsic to the rotation is suppressed by a stabilization vector. The assumed stress field and the stabilization scalar is chosen such that membrane locking can be avoided. Computational efficiency of the element is improved by employing orthogonal stress modes and admissible matrix formulation. Popular benchmark tests are attempted and the results are found to be satisfactory. © 1997 by John Wiley & Sons, Ltd.     

p-Version plate and curved shell element for geometrically non-linear analysis

This paper presents a p-version geometrically non-linear formulation based on the total Lagrangian approach for a nine node three dimensional curved shell element. The element geometry is defined by the coordinates of the nodes located on its middle surface and nodal vectors describing the bottom and top surfaces of the element. The element displacement approximation can be of arbitrary and different polynomial orders in the plane of the element and in the transverse direction. The element approximation functions and the corresponding nodal variables are derived from the Lagrange family of interpolation functions. The resulting approximation functions and the nodal variables are hierarchical and the element displacement approximation ensures C° continuity. The element properties are established using the principle of virtual work and the hierarchical element approximation. In formulating the properties of the element complete three dimensional stresses and strains are considered, hence the element is equally effective for very thin as well as extremely thick shells and plates. Incremental equations of equilibrium are derived and solved using the standard Newton–Raphson method. The total load is divided into increments, and for each increment of load, equilibrium iterations are performed until each component of the residuals is within a preset tolerance. Numerical examples are presented to show the accuracy, efficiency and advantages of the present formulation. The results obtained from the present formulation are compared with those available in the literature.     

Influence of neutral surface position on the nonlinear stability behavior of functionally graded plates

Nonlinear behavior of functionally graded material (FGM) skew plates under in-plane load is investigated here using a shear deformable finite element method. The material is graded in the thickness direction and a simple power law based on the rule of mixture is used to estimate the effective material properties. The neutral surface position for such FGM plates is determined and the first order shear deformation theory based on exact neutral surface position is employed here. The present model is compared with the conventional mid-surface based formulation, which uses extension-bending coupling matrix to include the noncoincidence of neutral surface with the geometric mid-surface for unsymmetric plates. The nonlinear governing equations are solved through Newton–Raphson technique. The nonlinear behavior of FGM skew plates under compressive and tensile in-plane load are examined considering different system parameters such as constituent gradient index, boundary condition, thickness-to-span ratio and skew angle. An erratum to this article can be found at     

Modeling of functionally graded sandwich shells accounting for variation in transverse displacement

In this study, a simple C₀ isoparametric finite element formulation based on higher-order shear deformation theory is presented for static analysis of functionally graded material sandwich shells (FGMSS). To characterize the membrane-flexure behavior observed in a functionally graded shell, a displacement field involving higher-order terms in in-plane and transverse fields is considered. The proposed kinematics field incorporates for transverse normal deformation, transverse shear deformation, and nonlinear variation of the in-plane displacement field through the thickness to predict the overall response of the shell in an accurate sense. To develop the efficient C₀ formulation, the derivatives of transverse displacement are treated as independent field variables (nodal unknowns). Voigt's rule of mixture is employed to ascertain the mechanical properties of each layer's constituents along the thickness direction. A wide range of numerical problems are solved assuming various parameters: side-thickness ratio, curvature-side ratio, and gradation parameter, and their interactions with regard to static analysis of FGMSS are discussed in brief. Deflection and stresses incorporating different thickness schemes of sandwich shells are presented in the form of figures. To validate the results, a functionally graded shell without sandwich arrangement is considered. Since no results are available on static analysis of FGMSS, the present 2D model based on the finite element method might be helpful in assessing the applicability of other analytical and numerical models in this area in the future.     

p-version hierarchical three dimensional curved shell element for elastostatics

This paper presents a hierarchical three dimensional curved shell finite element formulation based on the p-approximation concept. The element displacement approximation can be of arbitrary and different polynomial orders in the plane of the shell (ξ, η) and the transverse direction (ξ). The curved shell element approximation functions and the corresponding nodal variables are derived by first constructing the approximation functions of orders p_ξ, p_η and p_ξ and the corresponding nodal variable operators for each of the three directions ξ, η and ξ and then taking their products (sometimes also known as tensor product). This procedure gives the approximation functions and the corresponding nodal variables corresponding to the polynomial orders p_ξ, p_η and p_ξ. Both the element displacement functions and the nodal variables are hierarchical; therefore, the resulting element matrices and the equivalent nodal load vectors are hierarchical also, i.e. the element properties corresponding to the polynomial orders p_ξ, p_η and p_ξ are a subset of those corresponding to the orders (p_ξ + 1), (p_η +1) and (p_ξ +1). The formulation guarantees C° continuity or smoothness of the displacement field across the interelement boundaries. The geometry of the element is described by the co-ordinates of the nodes on its middle surface (ξ = 0) and the nodal vectors describing its bottom (ξ = ?1) and top (ξ = +1) surfaces. The element properties are derived using the principle of virtual work and the hierarchical element approximation. The formulation is equally effective for very thin as well as very thick plates and curved shells. In fact, in many three dimensional applications the element can be used to replace the hierarchical three dimensional solid element without loss of accuracy but significant gain in modelling convenience. Numerical examples are presented to demonstrate the accuracy, efficiency and overall superiority of the present formulation. The results obtained from the present formulation are compared with those available in the literature as well as analytical solutions.     

考虑剪切效应三维纤维梁单元的几何非线性增量有限元分析

该文以三维连续介质力学和虚功原理为基础,推导了增量U.L.有限元列式,该列式保留了大位移刚度矩阵项,并对该刚度矩阵进行修正使其成为对称矩阵。根据增量U.L.列式,推导了三维纤维梁单元的刚度矩阵。该单元采用平截面假定,以轴向节点位移表示截面上任意一点的位移,并结合Timoshenko梁理论来考虑剪切效应。以上原理编制分析程序,通过几个算例分析,证明了该方法的精确性、通用性。     

A STRAIN-AND-DISPLACEMENT-BASED VARIATIONAL METHOD APPLIED TO GEOMETRICALLY NON-LINEAR SHELLS

A geometrically non-linear hybrid nine-node finite 2D-shell element is presented. The theoretical formulation is based on a Reissner functional in strains and displacements. The increments of which are interpolated with respect to different spatially fixed triads: both the displacement and rotation increments in the material frame (global rectangular Cartesian) and the Green–Lagrange-strain increments in a suitably chosen local rectangular Cartesian in the centroid of the considered element in the reference configuration. Corresponding transformations then deliver the components on the shell mid-surface. Although a single element possesses one spurious zero-energy mode, an assemblage performs excellently (also in comparison with a full-rank element).     

A least-squares approach for uniform strain triangular and tetrahedral finite elements

A least-squares approach is presented for implementing uniform strain triangular and tetrahedral finite elements. The basis for the method is a weighted least-squares formulation in which a linear displacement field is fit to an element's nodal displacements. By including a greater number of nodes on the element boundary than is required to define the linear displacement field, it is possible to eliminate volumetric locking common to fully integrated lower-order elements. Such results can also be obtained using selective or reduced integration schemes, but the present approach is fundamentally different from those. The method is computationally efficient and can be used to distribute surface loads on an element edge or face in a continuously varying manner between vertex, mid-edge and mid-face nodes. Example problems in two- and three-dimensional linear elasticity are presented. Element types considered in the examples include a six-node triangle, eight-node tetrahedron, and ten-node tetrahedron. © 1998 John Wiley & Sons, Ltd.     

Locking-free curved beam element based on curvature

The formulation of a curved beam element with 3 nodes for curvature to eliminate the shear/membrane locking phenomenon is presented. The element is based on curvature so that it may represent the bending energy fully, and the shear/membrane strain energy is incorporated into the formulation by the equilibrium equations. To deal with general boundary conditions, a transformation matrix between nodal curvature and nodal displacement vector is introduced. Several examples are presented in order to verify the element formulation and its analytical capability. The solutions obtained reveal that the element describes the curved beam behaviour quite correctly and efficiently, showing no locking phenomena, and that it is also applicable to the analysis of both thin and thick curved beams.     

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.     

Large displacement analysis of three-dimensional beam structures

An updated Lagrangian and a total Lagrangian formulation of a three-dimensional beam element are presented for large displacement and large rotation analysis. It is shown that the two formulations yield identical element stiffness matrices and nodal point force vectors, and that the updated Lagragian formulation is computationally more effective. This formulation has been implemented and the resulted of some sample analyses are given.     

Finite-element analysis of hyperelastic thin shells with large strains

The objective of this contribution is the development of theoretical and numerical models applicable to large strain analysis of hyperelastic shells confining particular attention to incompressible materials. The theoretical model is developed on the basis of a quadratic displacement approximation in thickness coordinate by neglecting transverse shear strains. In the case of incompressible materials this leads to a three-parametric theory governed solely by mid-surface displacements. The material incompressiblity is expressed by two equivalent equation sets considered at the element level as subsidiary conditions. For the simulation of nonlinear material behaviour the Mooney-Rivlin model is adopted including neo-Hookean materials as a special case. After transformation of nonlinear relations into incremental formulation doubly curved triangular and quadrilateral elements are developed via the displacement method. Finally, examples are given to demonstrate the ability of these models in dealing with large strain as well as finite rotation shell problems.The present study is supported by a research grant of the German National Science Foundation (DFG) under Ba 969/3-1.dedicated to Prof. Dr. Dr. Erwin Stein for his 65th birthday anniversary     

新型协同转动3节点三边形壳单元

发展了一种新型3节点三边形壳单元。计算单元在局部坐标系下的节点变量时,通过采用协同转动法,预先扣除节点整体变量中的刚体转动成分,从而简化了单元的计算公式。不同于现有的其他协同转动单元,在该单元中采用了增量可以直接累加的矢量型转动变量,单元的切线刚度矩阵可以通过直接计算能量泛函对节点变量的二阶偏微分得到,且对节点变量的偏微分次序是可以互换的,因而在局部和整体坐标系下都得到了对称的单元切线刚度矩阵。为消除单元中可能出现的闭锁现象,引入了MacNeal提出的线积分法,分别用沿单元边线方向的膜应变和剪切应变构造新的假定应变场。最后,通过对几个产生了大位移与大转角变形的板壳问题进行分析,检验了该单元的可靠性、计算精度和计算效率。     

A simple finite element formulation of a higher-order theory for unsymmetrically laminated composite plates

A higher-order theory which satisfies zero transverse shear stress conditions on the bounding planes of a generally laminated fibre-reinforced composite plate subjected to transverse loads is developed. The displacement model accounts for non-linear distribution of inplane displacement components through the plate thickness and the theory requires no shear correction coefficients. A C∘ continuous displacement finite element formulation is presented and the coupled membrane-flexure behaviour of laminated plates is investigated. The nodal unknowns are the three displacements, two rotations and two higher-order functions as the generalized degrees of freedom. The simple isoparametric formulation developed here is capable of evaluating transverse shears and transverse normal stress accurately by using the equilibrium equations. The accuracy of the nine-noded Lagrangian quadrilateral element is then established by comparing the present results with the closed-form, three-dimensional elasticity and other finite element available solutions.     

A new discrete Kirchhoff quadrilateral element based on the third-order theory for composite plates

A new discrete Kirchhoff quadrilateral element based on the refined third-order theory is developed for the analysis of composite plates. The element has seven degrees of freedom per node, namely, the three displacements, two rotations and two transverse shear strain components at the mid-surface. The inplane displacements and the shear strains are interpolated using bilinear interpolation functions and the mid-surface rotations are interpolated using bi-quadratic functions based on the discrete Kirchhoff technique. The element stiffness matrix and the consistent load vector are developed using the principle of virtual work. The finite element formulation is validated by comparing the results for simply-supported plate with the analytical Navier solution. Comparison of the present results with those using other available elements based on the TOT establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The element is free from shear locking     

Finite element analysis of anisotropic non-linear incompressible elastic solids by a mixed model

A mixed finite element method is presented for geometrically and materially non-linear analysis of anisotropic incompressible hyperelastic materials. An incremental iteractive total Lagrangian formulation is adopted. The nodal displacements and the hydrostatic pressure are independently interpolated leading to a mixed system of equations, with characteristic zero diagonal terms. Computations are carried out using a three-dimensional linear displacement, constant pressure element. A mixed penalty approximation is then employed to eliminate the pressure variables at the element level. The anisotropic material handling capability of the formulation is tested through a number of transversely isotropic problems and the results compared to analytical solutions. To demonstrate the applicability of this formulation to model complex anisotropic problems, the inflation of a cut toroidal tube with helical fibre orientation is analysed.     

A novel isoparametric finite element displacement formulation for axisymmetric analysis of nearly incompressible materials

It is well accepted that severe numerical difficulties arise when using the conventional displacement method to analyse incompressible or nearly incompressible solids. These effects are caused by the kinematic constraints imposed on the nodal velocities by the constant volume condition. In elastic-plastic analysis, these effects are due to a conflict between the plastic flow rule and the finite element discretization. Although several methods have been proposed to cope with this problem, none has been based on the appropriate choice of displacement interpolation functions to minimize the constraints. The theoretical formulation of a new six-noded isoparametric displacement finite element, which is well suited for elastic-plastic analysis of axisymmetric constrained solids by using a rational displacement interpolation function, is presented in this paper. The proposed displacement interpolation function implies that the displacement in the axial direction and the product of the displacement in the radial direction and the radius should be treated as two independent basic variables. Alternatively, the proposed displacement interpolation function can also be implemented in a conventional displacement formulation simply by using a modified shape function matrix. The suitability of the proposed formulations is first studied theoretically by assessing the number of degrees of freedom per constraint and then verified by performing numerical experiments on typical boundary value problems which involve incompressible behaviour.