A general finite element model for shells of arbitrary geometry

A finite element formulation is developed for the large displacement analysis of arbitrary shells. Formulation is based on a convected coordinate system and a tensorial approach is followed. The strain-displacement relationships used do not reflect the Kirchhoff hypothesis and Love's approximations. Isoparametric interpolation is used for the discretization of the problem, and the number of nodal points is variable. The numerical examples include the buckling analysis of cylindrical shells as well as two problems to test the convergence and accuracy of the algorithm.     

Analyses of large quasistatic deformations of inelastic bodies by a new hybrid-stress finite element algorithm: Applications

In an earlier paper [1] the authors described a new hybrid-stress finite element algorithm suitable for the analysis of large quasistatic deformations of inelastic bodies. However, that paper focused on the subtleties of the algorithm, and only finite homogeneous deformation problems were presented as examples. The present paper is concerned with the algorithm's implementation and application to problems of technological interest.     

Transient response of inelastic shells of revolution

The paper presents a transient response finite difference solution in space and time for the shell of revolution equations derived by Sanders. Inelastic constitutive relationships and nonlinear kinematics are included. The spatial differencing technique is based on a checkerboard half-spacing mesh configuration that allows the use of lower order difference expressions than those normally used for accuracy. Several example problems are considered which illustrate convergence for elastic and elastic-plastic problems. Experimental results for cylindrical shells are compared with predictions based on linear and nonlinear kinematics, various mesh spacings, and constitutive differences.     

Optimization of inelastic conical shells with cracks

Inelastic conical shells loaded by the central rigid boss with vertical load are studied. The thickness of the conical part of the structure is piecewise constant. The connection between the shell and the boss is weakened with a stable crack. The designs with the maximal load-carrying capacity are established under a given material consumption of the shell. Material of the shell wall obeys the von Mises yield condition.     

Instability analysis of thin plates and arbitrary shells using a faceted shell element with loof nodes

The faceted representation is employed in the paper to derive a 24-dof triangular shell element for the instability analysis of shell structures. This element, without the deficiencies of displacement incompatibility, singularity with coplanar elements, inability to model intersections, and low-order membrane strain representation, which are normally associated with existing flat elements, has previously been found by the authors to perform well in linear static shell analyses. The total Lagrangian approach is used in the nonlinear formulation, and the results of the various numerical examples indicate that its performance is comparable to existing nonlinear shell elements. An extrapolation stiffness procedure, which will improve the convergence characteristics of the constant arc length solution algorithm used here, is also presented.     

Optimization of inelastic cylindrical shells with internal supports

A non-linear programming method is developed for optimization of inelastic cylindrical shells with internal ring supports. The shells under consideration are subjected to internal pressure loading and axial tension. The material of shells is a composite which is considered as an anisotropic inelastic material obeying the yield condition suggested by Lance and Robinson. Taking geometrical non/linearity of the structure into account optimal locations of internal ring supports are determined so that the cost function attains its minimum value. A particular problem of minimization of the mean deflection of the shell with weakened singular cross sections is treated in a greater detail.     

Finite element analysis for time-dependent inelastic material behavior

A representation of inelastic, time-dependent material behavior, due to Bodner and Partom, in which both elastic and inelastic deformations are considered to be present at all stages of loading and unloading, is shown to be well-suited to structural analyses using finite element modeling techniques and high speed digital computation methods. The formulation considerably simplifies the computational logic for non-monotonic and cyclic loading problems since no special unloading criteria or yield conditions are required. Examples demonstrating strain-rate sensitivity, work-hardening, and reversed loading behavior are given for problems in the small strain range. Experimental results for a titanium tensile specimen subjected to changes in crosshead velocity are compared with predictions based on a plane stress finite element model. Numerical analyses using axisymmetric, solid of revolution finite element models are presented for unstiffened and ring-stiffened cyclindrical shells subjected to time-dependent external pressure.     

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.     

On the Willmore energy of shells under infinitesimal deformations

In this paper we apply tensor calculus and differential geometry to consider shell structure. Using tensor calculus we examine the stationarity of the Willmore energy under infinitesimal deformations of the surfaces in ${?}^3$. We obtain the class of the surfaces which does not change its Willmore energy under infinitesimal deformations. In particular, a special kind of deformation is considered—infinitesimal bending which preserves the arc length. The change of the Willmore energy under such deformations is determined. Also, we give a new proof of a well-known theorem (that reads that the total mean curvature of a surface is stationary under an infinitesimal bending), applying tensor calculus.     

Natural structural shapes for shells of revolution in the membrane theory of shells

Natural Structural Shapes are derived for axisymmetrically loaded shells of revolution within the membrane theory of shells. The concept of natural structural shapes is based on the simultaneous minimization of the mass and the strain energy of the loaded structure, a multicriteria optimization problem with Edgeworth-Pareto optimality as the basic optimality concept. The problem is formulated as a multicriteria control problem and necessary conditions for arbitrary loading and boundary are derived. Exact and numerical results are obtained for both the case of uniform pressure and that of a ring load with zero surface loads.Presented at the IUTAM Symposium on Structural Optimization in Melbourne, Feb. 1988.     

A mindlin element for thick and deep shells

A strain-displacement relation of the Reissner-Mindlin type is derived for a general shell element which may be used with equal confidence and ease in either plate or shell configurations over a wide range of thicknesses. The usual assumptions pertaining to thin, shallow shell geometry have not been employed, resulting in strain-displacement relations which are consistent with the Reissner-Mindlin hypotheses for thick shell configurations. Test results show that this element has excellent accuracy and convergence characteristics and is free of shear locking.     

Non-linear analysis of shells of revolution under arbitrary loads

A new finite element formulation is presented for the non-linear analysis of elastic doubly curved segmented and branched shells of revolution subject to arbitrary loads. The circumferential variations of all quantities are described by truncated Fourier series with an appropriate number of harmonic terms. A coupled harmonics approach is employed, in which coupling between different harmonics is dealt with directly rather than by the use of pseudo-loads. Key issues in the formulation, such as non-linear coupling and growth of harmonic modes, are carefully and systematically explained. This coupled harmonics approach allows an easy implementation of the arc-length method. As a result, post-buckling load–deflection paths can be traced efficiently and accurately. The formulation also employs a non-linear shell theory more complete than existing classical theories. The results from the present study are independently verified using ABAQUS, while those from other studies are found to be inaccurate in general.     

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.     

Stabilized finite element formulation for elastic–plastic finite deformations

This paper presents a stabilized finite element formulation for nearly incompressible finite deformations in hyperelastic–plastic solids, such as metals. An updated Lagrangian finite element formulation is developed where mesh dependent terms are added to enhance the stability of the mixed finite element formulation. This formulation circumvents the restriction on the displacement and pressure fields due to the Babuška–Brezzi condition and provides freedom in choosing interpolation functions in the incompressible or nearly incompressible limit, typical in metal forming applications. Moreover, it facilitates the use of low order simplex elements (i.e. P1/P1), reducing the degrees of freedom required for the solution in the incompressible limit when stable elements are necessary. Linearization of the weak form is derived for implementation into a finite element code. Numerical experiments with P1/P1 elements show that the method is effective in incompressible conditions and can be advantageous in metal forming analysis.     

A finite element method for plane strain deformations of incompressible solids

A finite element method for incompressible deformation is formulated from the virtual work equation based on deviatoric quantities. Particular emphasis is given to nonlinear material behavior. The incompressibility constraint is imposed on the admissible displacement field by direct elimination of nodal displacements. The resulting stiffness matrix is symmetric and positive definite. Once a convergent solution for the displacement is obtained, the hydrostatic stress is determined from the principle of virtual work. A variety of illustrative examples are presented. The efficiency, economy and limitations of the method are discussed.     

An arbitrary lagrangian-eulerian finite element method for path-dependent materials

The conservation laws, the constitutive equations, and the equation of state for path-dependent materials are formulated for an arbitrary Lagrangian-Eulerian finite element method. Both the geometrical and material nonlinearities are included in this setting. Computer implementations are presented and an elastic-plastic wave propagation problem is used to examine some features of the proposed method.     

Numerically integrated triangular element for doubly curved thin shells

A doubly curved triangular finite element for the analysis of thin shells with nonzero Gaussian curvature is developed by numerical integration. The element, though nonconforming in bending, is found to give good results when applied to cylindrical and spherical shell problems.     

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.     

Nonlinear finite element analysis of shells: Part I. three-dimensional shells

A nonlinear finite element formulation is presented for the three-dimensional quasistatic analysis of shells which accounts for large strain and rotation effects, and accommodates a fairly general class of nonlinear, finite-deformation constitutive equations. Several features of the developments are noteworthy, namely: the extension of the selective integration procedure to the general nonlinear case which, in particular, facilitates the development of a ‘heterosis-type’ nonlinear shell element; the presentation of a nonlinear constitutive algorithm which is ‘incrementally objective’ for large rotation increments, and maintains the zero normal-stress condition in the rotating stress coordinate system; and a simple treatment of finite-rotational nodal degrees-of-freedom which precludes the appearance of zero-energy in-plane rotational modes. Numerical results indicate the good behavior of the elements studied.