A general three-dimensional L-section beam finite element for elastoplastic large deformation analysis

In this paper, the development of a general three-dimensional L-section beam finite element for elastoplastic large deformation analysis is presented. We propose the generalized interpolation scheme for the isoparametric formulation of three-dimensional beam finite elements and the numerical procedure is developed for elastoplastic large deformation analysis. The formulation is general and effective for other thin-walled section beam finite elements. To show the validity of the formulation proposed, a 2-node three-dimensional L-section beam finite element is implemented in an analysis code. As numerical examples, we first perform elastic small and large deformation analyses of a cantilever beam structure subjected to various tip loadings, and elastoplastic large deformation analysis of the same structure under reversed cyclic tip loading. We then analyze the failures of simply supported beam structures of different lengths and slenderness ratios under elastoplastic large deformation. The same problems are solved using refined shell finite element models of the structures. The numerical results of the L-section beam finite element developed here are compared with the solutions obtained using shell finite element analyses. We also discuss the numerical solutions in detail.     

A mixed finite element for shell model with free edge boundary conditions Part 2. The numerical scheme

From the mixed variational formulation which was suggested in Part 1, we construct a finite element scheme. Then, a mathematical justification concerning the error estimate is developed. The most important point is to justify two compatibility conditions between the approximation of the kinematics and the stresses. Because of the structure of the shell operator they are not obvious at all and several restrictions on the mesh are necessary. Therefore, the classical result known in fluid mechanics for Stokes equations cannot be applied directly.     

An 8-node assumed stress hybrid element for analysis of shells

In this paper an 8-node quadrilateral assumed-stress hybrid shell element is presented. The formulation is based on Hellinger–Reissner variational principle. The element is developed by flat shell approach by combining a membrane element with a Mindlin plate element. The proposed element has six degrees of freedom per node and permits an easy connection to other types of finite elements. Numerical examples are presented to show that the validity and efficiency of the present element for static and free vibration analysis.     

Hybrid strain based three node flat triangular shell elements—II. Numerical investigation of nonlinear problems

In a companion paper [M. L. Liu and C. W. S. To, Comput. Struct. 54, 1031–1056 (1995)] theories and incremental formulation of nonlinear shell structures discretized by the finite element method are discussed. The updated Lagrangian formulation and the incremental Hellinger-Reissner variational principle are adopted. The independently assumed fields employed are the incremental displacements and incremental strains. Based on the theory and incremental formulation explicit element stiffness and mass matrices of three node flat triangular shell finite elements are derived. In the present paper the derived element matrices are applied to nine examples. The latter include static and dynamic response analysis of shell structures with geometrical, material, and geometrical and material nonlinearities. The formulation adopted and element matrices derived are found to be accurate, flexible and applicable to various types of shell structures with geometrical and material nonlinearities.     

The simulation of sheet metal forming processes via integrating solid-shell element with explicit finite element method

Solid-shell elements can be seen as a class of typical double-surfaced shell elements with no rational degrees of freedom, which are more suitable for analyzing double-sided contact problems than conventional shell elements. In this study, a solid-shell finite element model is implemented into the explicit finite element software ABAQUS/Explicit as a user-defined element, through which the sheet metal forming processes are simulated. The main feature of this finite element model is that the solid-shell element formulation is embedded into an explicit finite element procedure, compared to the previous studies on the solid-shell elements under the implicit finite element framework. To obtain a straightforward element, a complete integration scheme is adopted. No loss of generality, a twelve-parameter enhance assumed strain method is employed to improve the element’s behavior. Two benchmarks from the NUMISHEET conference and a U-channel roll-forming process are simulated with this explicit solid-shell finite element model. The calculated results are comparable with experimental and numerical results presented in the literatures.     

Sensitivity of structural joint stiffnesses with respect to beam properties: A hybrid approach

In the optimization of frame structures the sizes of the beam members change, as do the stiffnesses of the joints where such members meet. In this paper a method for calculating the design sensitivities of structural joints to changes in the size of the members, using a finite element formulation, is presented. The method uses the initial joint stiffness, predicted from a more detailed shell finite element model or experimental data, to calculate the design sensitivies for any number of joint members. The method is developed into a computer program that does not require a finite element model of the joint. The formulation of the method and a test case are presented.     

Numerical analysis of coupled thermomechanical frictional contact problems. Computational model and applications

Summary In this paper a numerical model for the analysis of coupled thermomechanical multi-body frictional contact problems at finite deformations is presented. The multi-body frictional contact formulation is fully developed on the continuum setting and then a spatial (Galerkin projection) and temporal (time-stepping algorithm) discretization is applied. A contact pressure and temperature dependent thermal contact model has been used. A fractional step method arising from an operator split of the governing equations has been used to solve the coupled nonlinear system of equations, leading to a staggered solution algorithm. The numerical model has been implemented into an enhanced version of the computational finite element program FEAP. Numerical examples and simulation of industrial metal forming processes show the performance of the numerical model in the analysis of coupled thermomechanical frictional contact problems.     

Geometrically nonlinear analysis of cylindrical shells using surface-parallel quadratic elements

A moderately thick cylindrical shell isoparametric element that is capable of accurately modeling cylindrically curved geometry, while also incorporating appropriate through-thickness kinematic relations is developed. The analysis accounts for fully nonlinear kinematic relations so that stable equilibrium paths in the advanced nonlinear regime can be accurately predicted. The present nonlinear finite element solution methodology is based on the hypothesis of linear displacement distribution through thickness (LDT) and the total Lagrangian formulation. A curvilinear side 16-node element with eight nodes on each of the top and bottom surfaces of a cylindrical shell has been implemented to model the transverse shear/normal deformation behavior represented by the LDT. The BFGS iterative scheme is used to solve the resulting nonlinear equations. A thin-shallow clamped cylindrical panel is investigated to test the convergence of the present element, and also to compare the special case of the present solution based on the KNSA (von Karman strain approximation) with those computed using the available faceted elements, discrete Kirchhoff constraint theory (DKT) and classical shallow shell finite elements, spanning the entire computed equilibrium path.     

Modelization and numerical approximation of piezoelectric thin shells: Part I: The continuous problems

This paper comprises three parts mainly directed to the obtention of a two-dimensional formulation (Part I), to the analysis of an approximation by finite element methods and to some numerical experiments (Part II) and to the use of piezoelectric components in order to realize active structures (Part III).In this first part, the general three-dimensional equations of piezoelectricity are recalled; they use a representation of the three-dimensional body by a system of three curvilinear coordinates. An existence and uniqueness result is proved. Next, under appropriate assumptions on the mechanical and on the electrical behaviour of the shell during the deformation, the integration of the three-dimensional equations through the thickness leads to a set of two-dimensional equations which are themselves simplified by using an energy criterion. Finally, it is proved that these reduced two-dimensional equations have one and only one solution.     

Behaviour of paraboloid of revolution shell using cross-ply and anti-symmetric angle-ply laminates

A generalised formulation for the analysis of a doubly curved laminated composite shell using an eight-noded curved quadratic isoparametric finite element has already been presented. Based on that formulation, the behaviour of the shell characteristics of the paraboloid of revolution shell using eight types of cross-ply laminates and twelve types of anti-symmetric angle-ply laminates are presented in this paper. A detailed study of the variations of the deflections and other shell actions for the above laminates reveals that symmetric cross-ply laminates appear to be superior to anti-symmetric cross-ply laminates while 45/-45 laminates are superior to 30/-30 and 60/-60 laminates. An overall comparison between cross-ply and angle-ply laminates reveals that angle-ply laminates are more efficient than cross-ply laminates. In general, a decrease in the value of the shell actions is observed with an increase in the number of layers.     

Thermal stress analysis of laminated composite plates and shells

In this paper, Semiloof shell finite element formulation has been extended to thermal stress analysis of laminated plates and shells. The accuracy of the formulation has been verified using sample problems available in the literature. Thermal stresses in cross-ply and angle-ply laminated plates and shells subjected to thermal gradients across the thickness are presented for different boundary conditions, taking into account the temperature dependence of the material properties. The behaviour of laminates under thermal load is found to be different from that under mechanical loads in certain respects.     

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.     

A method for shell contact analysis

A relatively general and computationally efficient method of shell contact analysis using the discrete Fourier transform is developed for linear and certain types of nonlinear problems. The method predicts the contact boundary and the interfacial pressure distribution. It is illustrated by calculating the road contact pressure predicted by a finite element toroidal shell model of a pneumatic tire.     

A stress analysis technique for plate and shell built-up structures with junctions and its application to minimum-weight design of stiffened structures

A finite element formulation using the penalty function method to analyse exactly the junctions of plate and shell built-up structures is suggested for an isoparametric shell element. The connectivity condition at the junction is added to the potential energy functional by the penalty parameter and the interpolating function of displacements. This formulation yields an integral-type stiffness matrix of the special junction elements, which can directly evaluate the surface tractions at the junction. For applying the technique suggested here to the optimum design of structures with junction parts, a design sensitivity analysis formulation for the adhesive special element is also developed. The technique is applied to the minimum-weight design problems of isotropic and composite laminated plates with a stiffener subjected to stress constraints.     

Nonlinear finite element analysis of shell structures using the semi-loof element

The Semi-Loof Shell element originally developed by Irons [2] for linear elastic analysis of thin shell structures is formulated to include large deflection and plastic deformation effects. In this paper the details of the finite element formulation of the problem using total Lagrangian coordinate systems are presented and different element matrices are given. For plastic materials following the Prandtl-Reuss flow rule with isotropic strain hardening a multi-layer approach using a subincremental technique is employed. Numerical results on the performance of the element for a variety of applications are presented. These computer studies include complete load-deflection curves into the post-buckling range and comparisons are made with other existing results. Current experience with the element indicates that it is a reliable and competitive element for nonlinear analysis of shells of general geometry.     

The usefulness of static condensation in the finite element analysis of stiffened submersible cylindrical hulls

The usefulness of the static condensation technique in the finite element analysis of stiffened submersible. cylindrical hulls is examined in this paper. The finite element formulation used herein is essentially the same as outlined by the authors in an earlier paper wherein the stiffener is modeled rigorously using axisymmetric thin annular plate elements for the web and axisymmetric thin shell elements for the flange. The static condensation technique has been applied in this paper to reduce these stiffener finite elements so that their effect can be transferred to the shell node at the point of attachment of the stiffener with the shell. The advantage of such condensation of the stiffener elements is the smaller number of equations to be solved without the rigor of the stiffener modeling being lost in any way. The manner of incorporating the condensation in the computer program has been described. Examples of several stiffened submersible cylindrical hulls have been considered as an illustration of the use of the program.     

A high precision triangular laminated anisotropic shallow thin shell finite element

A high precision triangular laminated shallow thin shell finite element has been developed based on the classical lamination theory. The stiffness matrix is obtained explicitly by pre and post multiplying a few basic matrices. The formulation is almost an order of magnitude faster than those available for similar order elements. The numerical results of the example problems presented demonstrate that both displacements and stresses are predicted accurately with moderately coarse grids. A complete listing of FORTRAN subroutines is presented for users, to ease implementation of the algorithm.     

Geometrically non-linear formulation for the three dimensional solid-shell transition finite elements

This paper presents a geometrically non-linear formulation using total lagrangian approach for the solid-shell transition finite elements. Such transition finite elements are necessary in geometrically non-linear analysis of structures modelled with three dimensional solid elements and the curved shell elements. These elements are an essential connecting link between the solid elements and the shell elements. The element formulation presented here is derived using the properties of the three dimensional solid elements and the curved shell elements. No restrictions are imposed on the magnitude of the nodal rotations. Thus the element formulation is capable of handling large rotations between two successive load increments. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate their behavior, accuracy and applications in three dimensional stress analysis.

It is shown that the selection of different stress and strain components at the integration points do not effect the overall linear response of the element. However, in geometrically non-linear applications it may be necessary to select appropriate stress and the strain components at the integration points for stable and converging element behavior. Numerical examples illustrate various characteristics of the element.