Least-squares finite element formulation for shear-deformable shells

We present a least-squares based finite element formulation for the numerical analysis of shear-deformable shell structures. The variational problem is obtained by minimizing the least-squares functional, defined as the sum of the squares of the shell equilibrium equations residuals measured in suitable norms of Hilbert spaces. The use of least-squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise, i.e. stability requirements such as inf–sup conditions never arise. The proposed formulation retains the generalized displacements and stress resultants as independent variables and, in view of the nature of the variational setting upon which the finite element model is built, allows for equal-order interpolation. A p-type hierarchical basis is used to construct the discrete finite element model based on the least-squares formulation. Exponentially fast decay of the least-squares functional is verified for increasing order of the modal expansions. Several well established benchmark problems are solved to demonstrate the predictive capability of the least-squares based shell elements. Shell elements based on this formulation are shown to be effective in both membrane- and bending-dominated states.     

Quasi-analytical finite element procedures for axisymmetric anisotropic shells and solids

Using complex series representations, a quasi-analytical finite element procedure is developed which can analyze the static and dynamic mechanical fields of anisotropic axisymmetric shells and bodies. Due to its generality the procedure can handle arbitrary laminate construction with possible meridional and radial variations in locally or globally mechanically anisotropic materials. In this respect, in contrast to traditional quasi-analytical procedures which are limited to the ‘specially’ orthotropic case, the present treatment reveals several important effects of material and/or structural anisotropy. To illustrate the procedure as well as the significant effects of material anisotropy, several numerical examples are given along with comparisons with known analytical treatments.     

A finite element formulation for shells of negative Gaussian curvature

An expression for the strain energy of a shell of negative Gaussian curvature, including thickness shear deformations and without neglecting z/R in comparison with unity, is derived. Then a curved trapezoidal finite element formulation based on the principle of minimum potential energy is obtained. The shell element has eight nodes with 40 degrees of freedom and at each node there are three displacements and two rotations. The formulation is applicable for both thin and moderately thick shell analysis. The performance of this finite element is verified by applying it to some problems existing in the literature.     

Geometrically nonlinear finite element behaviour using buckling mode superposition

Within the context of finite element analysis a method is presented for analysing the geometrically nonlinear static behaviour of thin-type structures under one-parametric, conservative loading. The method is applicable in cases of mild geometrical nonlinearities, which are characterized by the existence of a linear bifurcation situation which is “close” to the considered situation. In this case the true nonlinear behaviour in the initial loading range can be quite well approximated by a linear combination of the eigenvectors of the linear buckling problem where the coefficients of combination are nonlinear functions of the load parameter and are easily computable from the eigenvalues, eigenvectors and the applied loading. A comprehensive treatment of this method of linear buckling mode superposition is given, including theoretical derivations, physical interpretation, software implementation, computational considerations and selected application examples which illustrate the basic nature of the method.     

A semi-analytical coupled finite element formulation for shells conveying fluids

A new formulation, based on the semi-analytical finite element method, is proposed for elastic shells conveying fluids. The structural equations are based on the shell element proposed by Ramasamy and Ganesan [Comput Struct 70 (1998) 363] while the fluid model is based on velocity potential. Dynamic pressure acting on the walls is derived from Bernoulli's equation. Imposing the requirement that the normal components of velocity of the solid and fluid be equal, introduces fluid–structure coupling. The proposed technique has been validated using results available in the literature. This study shows that instability occurs at a critical fluid velocity corresponding to the shell circumferential mode with the lowest natural frequency and this phenomenon is also independent of the type of structural boundary conditions imposed.     

On the finite element creep rupture analysis of structures

The analysis of problems involving creep rupture is considered. Two creep material failure criteria are employed, i.e. the Kachanov-Rabotnov damage relations and a new, in conjunction with the finite element method, energy criterion.Calculations are reported for titanium notched tensile specimens, where the plastic strains are evaluated by the Ramberg-Osgood formulas.     

Coons' surface method for formulation of finite element of plates and shells

In this paper, we apply the Coons' surface method and fitted function interpolation to fit boundary conditions of the finite element, and obtain different displacement functions of the plate and shell rectangular element, as well as the sectorial element, parallelogram element, ring element and quadrilateral element. The method is easily implemented and its geometric significance and mechanical conception are quite clear. Both computing technique and operational procedure are relatively unified. It is convenient to formulate conforming elements and high-precision elements, and also may be applied to formulate mixed and hybrid elements.     

Vibration of thin cylindrical shells based on a mixed finite element formulation

A Hellinger-Reissner functional for thin circular cylindrical shells is presented. A mixed finite element formulation is developed from this functional, which is free from line integrals and relaxed continuity terms. This element is applied to the problem of vibration of rectangular cylindrical shells. Bilinear trial functions are used for all field variables. The element satisfies the compatibility and completeness requirements. The numerical results obtained in this work show that convergence is quite rapid and monotonic for a much smaller number of degrees of freedom than other finite element formulations.     

Geometrically nonlinear finite element analysis of beams, frames, arches and axisymmetric shells

The geometrically nonlinear analysis of elastic inplane oriented bodies, e.g. beams, frames and arches, is presented in a total Lagrangian co-ordinate system. By adopting a continuum approach, employing a paralinear isoparametric element, the formulation is applicable to structures consisting of straight or curved members. Displacements and rotations are unrestricted in magnitude. The nonlinear equilibrium equations are solved using the Newton-Raphson method for which a number of examples are given. The derivations are extended to include axisymmetric structures.     

Vibration of thin shells of revolution based on a mixed finite element formulation

A modified Hellinger-Reissner functional for thin shells of revolution is presented. A mixed finite element formulation is developed from this functional which is free from line integrals and relaxed continuity terms. This formulation is applied to the problem of free vibration of spherical and conical shells. Bilinear trial functions are used for all field variables. The quadrilateral curved elements here presented satisfy the C⁰ continuity requirement of the functional. In all the results obtained the accuracy is quite good even for a reasonable     

Study of a nine-node mixed formulation finite element for thin plates and shells

A nine-node element, designated as SHEL9, has been developed for analysis of thin plates and shells. The element formulation is based on the degenerate solid shell concept and a modified Hellinger-Reissner principle with independent in-plane and transverse shear strains. Numerical tests indicate that the present SHEL9 element with uniform 3 × 3 point integration rule is free of locking, and it gives reliable solutions even for thin plates and shells.     

On the finite element solution of the elastoplastic axisymmetric Hertz contact problem

This paper includes a dimensionless treatment of the Hertz contact problem in the elastoplastic range, assuming the material in the bodies involved to be linear elastic and linear strain hardening. The bodies treated in this paper are assumed to be axisymmetric. Based upon the theoretical considerations in this work conversion formulas for engineering use are presented. The use of these formulas makes it possible to convert results from elastic plastic contact analysis for different yield stress and geometries. This will considerably reduce the number of calculations needed.     

Stochastic finite element analysis of shells

In the present paper the stochastic formulation of the triangular composite (TRIC) facet shell element is presented using the weighted integral and local average methods. The elastic modulus of the structure is considered to be a two-dimensional homogeneous stochastic field which is represented via the spectral representation method. As a result of the proposed derivation and the special features of the element, the stochastic stiffness matrix is calculated in terms of a minimum number of random variables of the stochastic field giving a cost-effective stochastic matrix. Under the assumption of a pre-specified power spectral density function of the stochastic field, it is possible to compute the response variability of the shell structure. Numerical tests are provided to demonstrate the applicability of the proposed methodologies.     

On creep buckling analysis of structures

Analysis of creep buckling using ADINA is described and evaluated. The main emphasis is on the selection of solution time steps. An energy method is presented to find an approximate time to buckle.

The evaluation of ADINA results is based on another numerical solution of creep buckling columns. We also give some practical guidelines for creep buckling analysis in general.     

On the role of mass operators in the group finite element formulation

The influence of the mass operators on the accuracy, economy and computational efficiency of the time-split group finite element formulation is investigated for the viscous flow over a backward-facing step. On a coarse grid it is found that mass operators must be retained adjacent to the computational boundaries to obtain the correct steady-state solution. The present time-split (or approximate factorisation) group finite element formulation is only 18% less economical than a fully lumped (equivalent to a finite difference) formulation. If mass lumping is introduced in the interior only, the solution is almost as accurate as if the full mass operators are retained, but there is little gain in economy in two dimensions. An operation-count estimate indicates that interior mass lumping would be about 40% more economical than retaining the mass operators in three dimensions.     

High-precision finite element analysis of cylindrical shells

This paper presents the finite element formulation to study the free vibration of cylindrical shells. The displacement function for the high-precision shell element with 16 degrees of freedom is approximated by a Hermitian polynomial of beam function type. The explicit formulation for the high-precision element is extremely efficient. For the purpose of comparison, the subject element is used to study the sample case of free vibration of a shell structure. The results are in good agreement with those published. The study shows that solution accuracy with fewer elements is assured and that accurate solutions are obtainable in the high-frequency range.     

On a physically stabilized one point finite element formulation for three-dimensional finite elasto-plasticity

In the case of linear elasticity, a direct connection between the concept of reduced integration with hourglass stabilization and a mixed method can usually be established. In the non-linear case, this is in general not possible. To overcome this difficulty we suggest in this paper a new concept based on a Taylor expansion of the constitutively dependent quantities with respect to the centre of the element. The push-forward of the second (linear) term of the Taylor series for the first Piola–Kirchhoff stress tensor to the current configuration determines the so-called hourglass stabilization part of the residual force vector. Due to the fact that the element uses only one Gauss point and the hourglass stabilization part is computed by means of a simple functional evaluation, the present element technology is very efficient from the computational point of view.In contrast to the 2D case the computation of the Jacobi determinant only in the centre of the 3D element does not yield the correct volume, if the element shape deviates from being a parallelipiped. It is shown in the paper that the error becomes negligibly small for a relatively coarse discretization. The formulation is free of volumetric locking and can compete with shell formulations up to an aspect ratio of about hundred. For bending-dominated problems, at least two elements over the thickness are needed in order to compute the onset of plastification correctly. The element behaves very robustly in finite elasticity and inelasticity, also when large element distortions occur.     

Finite element analysis of steady nonlinear harmonic oscillations of axisymmetric shells

A finite element method is presented for the analysis of nonlinear harmonic oscillations of axisymmetric shells. More specifically, a computational approach is described which can be used to calculate the constants which arise due to the geometrically nonlinear effects in the amplitudefrequency equations for shells.     

An axisymmetric formulation of the rigid-plastic domain-boundary element method and indentation analyses

The rigid-plastic domain-boundary element method has been formulated with mixed variables (velocities and velocity's derivatives). This method possesses a merit in that compatibilities of the velocity and the velocity's derivative can be met. On the other hand, the rigid-plastic domain-boundary element method possesses another merit in that it does not need iterative calculations in any computing step and then there is no possibility of divergence with the iterative calculations. In this paper, an axisymmetric rigid-plastic domain-boundary element method is formulated, and axisymmetric indentation processes of three material constants and two friction factors are analyzed by this method.