Nonlinear shell analysis via mixed isoparametric elements

Mixed isoparametric elements are presented for the geometrically nonlinear analysis of laminated composite shells. The analytical formulation is based on a form of the nonlinear shallow shell theory with the effects of shear deformation, material anisotropy and bending-extensional coupling included. The fundamental unknowns consist of the 13 stress resultants and generalized displacements of the shell. The generalized stiffness matrix is obtained by using a modified form of the Hellinger-Reissner mixed variational principle. Both triangular and quadrilateral elements are considered. The accuracy of the mixed isoparametric elements developed is demonstrated by means of numerical examples and their advantages over commonly-used displacement elements are discussed. Also, computational procedures are presented for the efficient evaluation of the elemental matrices and for overcoming the difficulties associated with the large, sparse system of equations of the mixed models thus making them competitive with displacement models.     

Nonlinear analysis of an axisymmetric shell using three noded degenerated isoparametric shell elements

An updated Lagrangian formulation of a quadratic degenerated isoparametric shell element is presented for geometrically nonlinear elasto-plastic shell problems. A finite rotation effect is included in the formulation by adopting a co-rotational scheme. The load stiffness matrix has been derived for the treatment of a pressure load. For elasto-plastic behavior, the layered element model is used. The Newton-Raphson iteration method is employed to solve incremental nonlinear equations. For tracking of post-buckling behavior, the work control method is taken into account. Verification of the present technique is obtained by analyzing the available reference problems. Good correlations between the computed results and referenced data can be drawn.     

Nonlinear analysis of reinforced concrete plate and shell structures using 20-noded isoparametric brick elements

This paper describes an efficient and accurate, 3-D finite element model which may be adopted in the nonlinear analysis of reinforced concerete plate and shell structures. Benchmark tests are undertaken to check the computational model.     

A conoidal shell analysis by modified isoparametric element

The quadratic isoparametric element is modified for applications in thin shell analysis. Four extra nonconforming displacement modes are added to transverse displacement then, later, they are eliminated through static condensation. A conoidal shell under uniform pressure is analyzed using the new element and the results compared with previous works.     

Coordinate-free isoparametric elements

Formulations for general solid elements are presented which do not rely explicitly on an arbitrary global coordinate system, and which are independent of the dimension of the embedding space. Abstract data types corresponding to points, vectors and tensors replace matrix representations, allowing the coordinate-free nature of the derivation to be carried over into the implementation. The formulations are oriented primarily toward applications involving general geometric and material nonlinearities, iterative solution algorithms and element-by-element implementations, but traditional linear applications are considered as well.     

Affine approximation of isoparametric elements

The use of rectangular isoparametric elements in finite element analysis of second-order boundary-value problems requires evaluating integrals of rational polynomial functions. Gaussian quadrature formulas are currently the most popular method of obtaining approximations to the exact integrals. A new method is described in which the isoparametric finite element function spaces are approximated. The resulting integrals can be evaluated exactly, avoiding the computational expense of the Gaussian quadrature schemes, particularly the 27 point formula used in three-dimensional elements.     

Basis for isoparametric stress elements

The family of the so-called ‘isoparametric strain (displacement) elements’ is restricted to membranes and solids. The reason for this restriction has led to the development of a new family based on stress assumptions; these elements will be referred to as ‘isoparametric stress elements’. This family contains plates and solids but no membranes. The omission of a particular element in each family is consistent with the plate-membrane analogies. The basic flexibility matrix of an isoparametric stress element is singular since the zero stress state is directly included. The rank technique is adopted to automatically extract the zero stress modes such that the element can be completely interchangeable between any finite element system. The theory for stress assumed isoparametric “quadrilateral” plate bending elements with curved boundaries is given. A brief presentation of the theory for isoparametric stress solid elements is also included.     

A simple isoparametric three-node shell finite element

A three-node isoparametric shell finite element including membrane and bending effects is proposed. The element is based on the degenerated solid approach and uses an assumed strain method to avoid shear locking. An intermediate convected covariant frame is used in order to construct the modified shear strain interpolation matrix. Validation tests show that shear locking is avoided and that a reduced integration procedure can be used without any loss of accuracy which is useful for the numerical efficiency.     

Mixed isoparametric elements for saint-venant torsion

Mixed isoparametric elements are presented for the Saint-Venant torsion problem of laminated and anisotropic bars. Both triangular and quadrilateral elements are considered. The “generalized” element stiffness matrix is obtained by using a modified form of the Hellinger-Reissner mixed variational principle. Group-theoretic techniques are used in conjunction with computerized symbolic integration to obtain analytic expressions for the stiffness coefficients. The accuracy of the mixed isoparametric elements developed is demonstrated by means of numerical examples, and their advantages over commonly used stress and displacement elements are discussed.     

Design of shell shape using finite elements

A finite triangular facet element for the analysis of doubly curved thin shells is presented, the principal feature of which is a particularly simple resolution process. A simple iterative design procedure is developed, the optimality criteria of which are the elimination of bending and the minimization of the surface integral of the membrane stresses. The procedure is used to obtain numerical predictions of the optimal shapes for constant thickness arches and shells that are in good agreement with those expected. Finally, the procedure is extended to provide an optimal shape for a uniform thickness arch dam and an iterative procedure used to provide an optimal thickness variation.     

Cubic isoparametric rolling/traveling finite elements

In this paper, the development and formulation of a new cubic isoparametric rolling/travelling finite element is described. The element can be used to simulate the dyanamic response of steadily rolling/traveling viscoelastic structures without the necessity of numerical time integration of the governing equations. Due to its higher order shape function, the element can more accurately simulate both the inertia and viscoelastic effects associated with rolling/traveling structures than currently available lower order moving elements. To illustrate the capabilities of the new cubic element, comparisons with exact and lower order generated results are presented.     

On a thin shell element for non-linear analysis, based on the isoparametric concept

This paper discusses a finite element method model for the large displacement, moderate strain analysis of thin shells. The model is based on an ‘adapted’ reference configuration for a displaced element, separating the displacements into rigid body displacements and strain-producing deformations. A strategy is developed, making use of the isoparametric concept for both the choice of reference configuration and in the element formulation. This makes the use of arbitrarily shaped elements possible. The model is shown to give accurate results for a range of relevant problems. Some problems in the general application of this type of model are discussed.     

Applications of isoparametric three-dimensional hybrid-stress finite elements with least-order stress fields

This paper investigates the factors relating to element stability, coordinate invariance and optimality in 8- and 20-noded three-dimensional brick elements in the context of hybrid-stress formulations with compatible boundary displacements and both a priori and a posteriori equilibrated assumed stresses. Symmetry group theory is used to guarantee the essential non-orthogonality of the stress and strain fields, resulting in a set of least-order selections of stable invariant stress polynomials. The performance of these elements is examined in numerical examples providing a broad range of analytical stress distributions, and the results are favourably compared to those of the displacement formulation and a stress-function based complete stress approach with regard to displacements, stresses, sampling, convergence and distortion sensitivity.     

An isoparametric finite element with decreased sensitivity to midside node location

A modified isoparametric finite element formulation is developed which is insensitive to midside node placement. An eight-node plane element is developed using the modified approach. Numerical results are presented for both the standard serendipity and modified isoparametric eight-node elements. These results demonstrate that decreased sensitivity to midside node placement occur when the modified formulation is used.     

Comments on the accuracy of nine-noded isoparametric finite elements

The reduction in accuracy in stress of nine-noded quadratic isoparametric elements related to the locations of the mid-side and centre nodes is shown to be due to erroneous relationships between strains and nodal displacements. The effect of nodal position on the calculated strains is discussed and criteria for the proper positioning of mid-side and centre nodes are provided.Errors associated with the use of curved sides are also examined and examples involving body force loading are used to illustrate the errors involved in the non-optimum location of nodes.The possibility of degenerating quadrilateral nine-noded isoparametric elements into triangular shaped elements is also studied. Recommendations to the user of finite element programs, in which nine-noded elements are incorporated are included.     

Anisotropic interpolation error estimates for isoparametric quadrilateral finite elements

Anisotropic local interpolation error estimates are derived for quadrilateral and hexahedral Lagrangian finite elements with straight edges. These elements are allowed to have diameters with different asymptotic behaviour in different space directions. The case of affine elements (parallel-epipeds) with arbitrarily high degree of the shape functions is considered first. Then, a careful examination of the multi-linear map leads to estimates for certain classes of more general, isoparametric elements. As an application, the Galerkin finite element method for a reaction diffusion problem in a polygonal domain is considered. The boundary layers are resolved using anisotropic trapezoidal elements.     

Hybrid-stress isoparametric elements for moderately thick and thin multilayer plates

A hybrid-stress formulation of isoparametric elements for the analysis of moderately thick and thin multilayer laminated composite plates is presented. The element displacement behavior is characterized by laminate reference-surface inplane and transverse displacements and laminate nonnormal crosssection rotations; as a result, the number of degrees of freedom is independent of the number of layers. All components of stress are included and are related to a set of laminate stress parameters, the number of which is independent of the number of layers. By isolating and analytically integrating all through-thickness contributions to the element matrices, the computation time for the element stiffness generation becomes nearly independent of the number of layers, and thus a computationally efficient element is produced. The formulation is used to develop an 8-node isoparametric multilayer plate element which is naturally invariant, of correct rank, and nonlocking in the thin-plate limit. Results for selected example problems show the range of applicability and convergence behavior of the element.     

Direct evaluation of reaction forces and moments using isoparametric elements

The use of isoparametric finite elements for plates, shells, solids and other structures is by now widespread. The quadratic family of elements, in general, gives the optimum results. Very good displacements are obtained and these are used to evaluate stresses (or stress resultants). When it comes to reactions at the boundaries, the generalized nodal forces are often treated as useless since it is usually not obvious how to relate them to the distributed boundary forces and moments. They usually show sharp variation and even reversal of sign between neighboring nodes. In this paper it is shown how to derive the distributed reaction forces and moments directly from the corresponding generalized forces. Several numerical examples are given to show the excellent agreement with exact and other known solutions.     

One point quadrature shell elements for sheet metal forming analysis

Summary Numerical simulation of sheet metal forming processes is overviewed in this work. Accurate and efficient elements, material modelling and contact procedures are three major considerations for a reliable numerical analysis of plastic forming processes. Two new quadrilaterals with reduced integration scheme are introduced for shell analysis in order to improve computational efficiency without sacryfying accuracy: the first one is formulated for plane stress condition and the second designed to include through-thickness effects with the consideration of the normal stress along thickness direction. Barlat’s yield criterion, which was reported to be adequate to model anisotropy of aluminum alloy sheets, is used together with a multi-stage return mapping method to account for plastic anisotropy of the rolled sheet. A brief revision of contact algorithms is included, specially the computational aspects related to their numerical implementation within sheet metal forming context. Various examples are given to demonstrate the accuracy and robustness of the proposed formulations.     

Application of penalty functions to a curved isoparametric axisymmetric thick shell element

A curved axisymmetric shell element with three nodes is developed. Quadratic interpolation is used and as the transverse shearing strain is included only first derivatives are required in the calculation of the strains. The element is found to yield accurate solutions for thick circular plates but a penalty factor must be used when the ratio of plate radius to thickness is of the order of 100. With appropriate values of the penalty factor, though, thin plate behaviour is reproduced with reasonable accuracy. Further, it is shown that for all practical purposes the penalty factor need only be based on the plate thickness. This is a useful conclusion in relation to shell analysis where different penalty factor values would otherwise need to be evaluated on the basis of the radius to thickness ratio. Finally the element is shown to give good results for cylindrical and spherical shells.