基于几何非线性不协调元变分原理的圆柱壳非线性初始稳定性分析

本文根据几何非线性不协调元增量变分原理,按严格的壳体方程,建立了高精度的圆柱壳几何非线性20参数矩形精化不协调元RCSR4,并用于圆柱壳非线性初始稳定性分析。计算结果表明,该方法收敛性良好。     

THE NONCONFORMING ELEMENT METHOD AND REFINED HYBRID ELEMENT METHOD FOR AXISYMMETRIC SOLID

The general approach for constituting non-conforming displacement function has been developed for axisymmetric finite element analysis and a pure non-conforming quadrilateral axisymmetric element, from a non-conforming displacement function and without any reduced integration technique, is given. Based on a proposed functional for formulating axisymmetric element and the orthogonal approach, a quadrilateral axisymmetric refined hybrid element has been presented which can be used to achieve superior performances such as higher accuracy and free from locking.     

Linear and Geometrically nonlinear analysis of plates and shells by a new refined non-conforming triangular plate/shell element

A refined non-conforming triangular plate/shell element for linear and geometrically nonlinear analysis of plates and shells is developed in this paper based on the refined non-conforming element method (RNEM). A conforming triangle membrane element with drilling degrees of freedom in Cartesian coordinates and the refined non-conforming triangular plate-bending element RT9, in which Kirchhoff kinematic assumption was adopted, are used to construct the present element. The displacement continuity condition along the interelement boundary is satisfied in an average sense for plate analysis, and the coupled displacement continuity requirement at the interelement is satisfied in an average sense, thereby improving the performance of the element for shell analysis. Selectively reduced integration with stabilization scheme is employed in this paper to avoid membrane locking. Numerical examples demonstrate that the present element behaves quite satisfactorily either for the linear analysis of plate bending problems and plane problems or for the geometrically nonlinear analysis of thin plates and shells with large displacement, moderate rotation but small strain.     

Geometrically non-linear analysis of composite structures with integrated piezoelectric sensors and actuators

This paper deals with the geometrically non-linear analysis of thin plate/shell laminated structures with embedded integrated piezoelectric actuators or sensors layers and/or patches. The motivation for the present developments is the lack of studies in the behavior of adaptive structures using geometrically non-linear models, where only very few published works were found in the open literature.

The model is based on the Kirchhoff classical laminated theory and can be applied to plate and shell adaptive structures with arbitrary shape, general mechanical and electrical loadings.

The finite element model is a non-conforming single layer triangular plate/shell element with 18 degrees of freedom for the generalized displacements and one electrical potential degree of freedom for each piezoelectric layer or patch.

An updated Lagrangian formulation associated to Newton–Raphson technique is used to solve incrementally and iteratively the equilibrium equations.

The model is applied in the solution of four illustrative cases, and the results are compared and discussed with alternative solutions when available.     

Stress resultant geometrically non-linear shell theory with drilling rotations. Part III: Linearized kinematics

A consistent formulation of the geometrically linear shell theory with drilling rotations is obtained by the consistent linearization of the geometrically non-linear shell theory considered in Parts I and II of this work. It was also shown that the same formulation can be recovered by linearizing the governing variational principle for the three-dimensional geometrically non-linear continuum with independent rotation field. In the finite element implementation of the presented shell theory, relying on the modified method of incompatible modes, we were able to construct a four-node shell element which delivers a very high-level performance. In order to simplify finite element implementation, a shallow reference configuration is assumed over each shell finite element. This approach does not impair the element performance for the present four-node element. The results obtained herein match those obtained with the state-of-the-art implementations based on the classical shell theory, over the complete set of standard benchmark problems.     

An efficient assumed strain element model with six DOF per node for geometrically non-linear shells

The present paper describes an assumed strain finite element model with six degrees of freedom per node designed for geometrically non-linear shell analysis. An important feature of the present paper is the discussion on the spurious kinematic modes and the assumed strain field in the geometrically non-linear setting. The kinematics of deformation is described by using vector components in contrast to the conventional formulation which requires the use of trigonometric functions of rotational angles. Accordingly, converged solutions can be obtained for load or displacement increments that are much larger than possible with the conventional formulation with rotational angles. In addition, a detailed study of the spurious kinematic modes and the choice of assumed strain field reveals that the same assumed strain field can be used for both geometrically linear and non-linear cases to alleviate element locking while maintaining kinematic stability. It is strongly recommended that the element models, described in the present paper, be used instead of the conventional shell element models that employ rotational angles.     

A geometrical non-linear brick element based on the EAS-method

This paper presents the development of a 3-D brick element with enhanced assumed strains for a geometrically non-linear theory. Some linear and non-linear examples show that this element can be used successfully in the whole range of solid structures. Thin 2-D- and 3-D-beam and shell structures are calculated with few 3-D elements and the results are the same as for shell or beam elements. © 1997 John Wiley & Sons, Ltd.     

Geometrically non-linear method of incompatible modes in application to finite elasticity with independent rotations

We discuss a geometrically non-linear method of incompatible modes. The model problem chosen for the discussion is the finite elasticity with independent rotations. The conditions which ensure the convergence of the method and the methodology to construct incompatible modes are presented. A detailed derivation of variational equations and their linearized form is given for a two-dimensional plane problem. A couple of geometrically non-linear two-dimensional elements with independent rotational freedoms are proposed based on the presented methodology. The elements exhibit a very satisfying performance over a set of problems in finite elasticity.     

A general procedure for deriving symmetric expressions for the secant and tangent stiffness matrices in finite element analysis

The paper presents a general and straightforward procedure based on the use of the strain energy density for deriving symmetric expressions of the secant and tangent stiffness matrices for finite element analysis of geometrically non-linear structural problems. The analogy with previously proposed methods for deriving secant and tangent matrices is detailed. The simplicity of the approach is shown in an example of application. © 1998 John Wiley & Sons, Ltd.     

A locking-free hexahedral element for the geometrically non-linear analysis of arbitrary shells

The development of the formulation for a highly adaptable hexahedral shell finite element is presented in this paper. A basic 18-node isoparametric hexahedral element is adopted as the basis of the formulation. Potential strategies to alleviate transverse shear, trapezoidal, thickness and membrane locking are investigated, in several combinations, using a wide variety of geometrically linear benchmarks. The most promising approach is further assessed using geometrically non-linear shell and plate problems. The recommended ANS-formulation performs well against an extensive range of benchmarks, and continues to be accurate at an aspect ratio of 1:10,000.     

A systematic development of ‘solid-shell’ element formulations for linear and non-linear analyses employing only displacement degrees of freedom

In the present contribution we propose a so-called solid-shell concept which incorporates only displacement degrees of freedom. Thus, some major disadvantages of the usually used degenerated shell concept are overcome. These disadvantages are related to boundary conditions—the handling of soft and hard support, the need for special co-ordinate systems at boundaries, the connection with continuum elements—and, in geometrically non-linear analyses, to a complicated update of the rotation vector. First, the kinematics of the so-called solid-shell concept in analogy to the degenerated shell concept are introduced. Then several modifications of the solid-shell concept are proposed to obtain locking-free solid-shell elements, leading also to formulations which allow the use of general three-dimensional material laws and which are also able to represent the normal stresses and strains in thickness direction. Numerical analyses of geometrically linear and non-linear problems are finally performed using solely assumed natural shear strain elements with a linear approximation in in-plane direction. Although some considerations are needed to get comparable boundary conditions in the examples analysed, the solid-shell elements prove to work as good as the degenerated shell elements. The numerical examples show that neither thickness nor shear locking are present even for distorted element shapes. © 1998 John Wiley & Sons, Ltd.     

Boundary element analysis of two-dimensional elastoplastic contact problems

A general boundary element formulation for contact problems, capable of dealing with local elastoplastic effects and friction, is presented. Both conforming and non-conforming problems may be analysed. The contact problem is solved by means of a direct constraint technique, in which compatibility and equilibrium conditions are directly enforced in the general system of equations. The contact areas are modelled with linear interpolation functions, and quadratic interpolation functions are used everywhere else. Elastoplasticity is solved by a BEM initial strain approach The Von Mises yield criterion with its associated flow rule is adopted. Both perfectly plastic and work hardening materials are studied in the proposed formulation.

An incremental loading technique is proposed, which allows accurate development of the loading history of the problem. The non-linear nature of these problems demands the use of an iterative procedure, to determine the correct frictional conditions at every node of the contact area and the value of the plastic strains at selected points where local yielding may have occurred. Several numerical examples are presented to demonstrate the efficiency of the proposed formulation.     

The use of bubble functions for the post-local buckling of plate assemblies by the finite strip method

The finite strip displacement functions for post-local buckling are augmented with so-called bubble functions, which are extra modes associated with internal or nodeless degrees of freedom. A non-linear finite strip method of analysis including bubble functions is described for the post-local buckling of geometrically imperfect plate assemblies. It is shown that the use of bubble functions improves significantly the convergence of the method. The non-linear finite strip method is then used to study channel and I-section members in compression and bending.     

A 3-D ADAPTIVE MESH REFINEMENT USING VARIABLE-NODE SOLID TRANSITION ELEMENTS

An automated three-dimensional adaptive h-refinement strategy using the solid transition elements with variable midside nodes at edges and faces of the element is presented. The basic behaviour of these transition elements were improved by addition of associated non-conforming modes. By introducing these transition elements, some difficulties associated with imposing displacement constraints on irregular nodes to enforce interelement compatibility in the conventional adaptive h-refinement are easily overcome. A superconvergent patch recovery technique is also extended to three-dimensional problem. Numerical examples show the effectiveness of the proposed adaptive mesh refinement scheme using transition elements.     

Towards automatic adaptive geometrically non-linear shell analysis. Part I: Implementation of an h-adaptive mesh refinement procedure

Effective methods leading to automated adaptive numerical solutions to geometrically non-linear shell-type problems are studied in this work. In particular, procedures for improving the accuracy, the reliability and the computational efficiency of the finite element solutions are of primary interest here. This is addressed using h-adaptive mesh refinement based on a posteriori error estimation, self-adaptive methods in global incremental/iterative processes, as well as smart algorithms and heuristic approaches based on methods of knowledge engineering. Seemless integration of h-adaptive finite element methods with adaptive step-length control makes it possible to maintain a prescribed accuracy while maintaining the solution efficiency without user intervention throughout the process of a non-linear analysis. Several examples illustrate the merit and potential of the approach studied herein and confirm the feasibility of developing an automatic adaptive environment for geometrically non-linear analysis of shell structures.     

An assumed strain finite element model for large deflection composite shells

A nine node finite element model has been developed for analysis of geometrically non-linear laminated composite shells. The formulation is based on the degenerate solid shell concept and utilizes a set of assumed strain fields as well as assumed displacement Two different local orthogonal co-ordinate systems were used to maintain invariance of the element stiffness matrix. The formulation assumes strain and the determinant of the Jacobian matrix to be linear in the thickness direction. This allows analytical integration in the thickness direction regardless of ply layups. The formulation also allows the reference plane to be different from the shell midsurface. The results of numerical tests demonstrate the validity and the effectiveness of the present approach.     

Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes

A class of ‘assumed strain’ mixed finite element methods for fully non-linear problems in solid mechanics is presented which, when restricted to geometrically linear problems, encompasses the classical method of incompatible modes as a particular case. The method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part, formulated in the context of a three-field variational formulation. The resulting class of mixed methods provides a possible extension to the non-linear regime of well-known incompatible mode formulations. In addition, this class of methods includes non-linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements. The good performance of the proposed methodology is illustrated in a number of simulations including 2-D, 3-D and axisymmetric finite deformation problems in elasticity and elastoplasticity. Remarkably, these methods appear to be specially well suited for problems involving localization of the deformation, as illustrated in several numerical examples.     

‘On-line’ G-control chart for attribute data

On the basis of the theory of random processes the concept of a G-chart is elaborated. In this case the observed variable G is a number of conforming units between two consecutive appearances of non-conforming ones. If the process is a Poisson one then G-variables are geometrically distributed (the G-chart is called after the distribution type). Application of a new SPC concept for attribute data makes it possible to improve SPC employment for: high quality processes with ? < 10^?4 (100 ppm); low volume manufacturing, short runs and ‘stepped’ processes. In the paper individual G, G-bar and stabilized G/? charts are presented. The sensitivities of a G-chart and a classical p-chart for the detection of process changes are compared by constructing operating characteristic curves and ARL curves. Depending on the required degree of the detection process changes, an optimal size of subgroup is found. For this size of subgroup the average number of non-conforming units and the average number of observed units between the process change and this change detection are minimal. Compared with the classical p-chart of the same sensitivity, the G-chart requires on the average fewer observed units for process change detection and also on the average fewer non-conforming units are produced.     

Refined non-conforming quadrilateral thin plate bending element

A refined non-conforming quadrilateral thin plate bending element RPQ4 which can satisfy the requirement of convergence is established such that the non-conforming displacement function can be derived directly. A simple explicit expression of a refined constant strain matrix can be introduced into the formulation of the standard displacement element which results in the constraint condition of interelement continuity being satisfied in an average sense. Numerical examples are presented to show that the present model can pass the patch test and possesses high accuracy. © 1997 John Wiley & Sons, Ltd.     

Refined nine-parameter triangular thin plate bending element by using refined direct stiffness method

Based on a new generalized variational principle, a refined direct stiffness method (RDSM) which can be directly used to improve non-conforming elements is proposed. The formulation is similar to that of the direct stiffness method (DSM), but the constraint condition of interelement continuity is satisfied in an average sense and as a result convergence and high accuracy are insured. The well-known BCIZ nine-parameter triangular thin plate bending element is refined by the RDSM to yield a new nine-parameter thin plate bending element RT₉. Numerical examples are presented to show that the present model passes the patch test and possesses high accuracy.