Treatment of slip locking for displacement‐based finite element analysis of composite beam–columns

In the conventional displacement‐based finite element analysis of composite beam–columns that consist of two Euler–Bernoulli beams juxtaposed with a deformable shear connection, the coupling of the transverse and longitudinal displacement fields may cause oscillations in slip field and reduction in optimal convergence rate, known as slip locking. This locking phenomenon is typical of multi‐field problems of this type, and is known to produce erroneous results for the displacement‐based finite element analysis of composite beam–columns based on cubic transverse and linear longitudinal interpolation fields. This paper introduces strategies including the assumed strain method, discrete strain gap method, and kinematic interpolatory technique to alleviate the oscillations in slip and curvature, and improve the convergence performance of the displacement‐based finite element analysis of composite beam–columns. A systematic solution of the differential equations of equilibrium is also provided, and a superconvergent element is developed in this paper. Numerical results presented illustrate the accuracy of the proposed modifications. The solutions based on the superconvergent element provide benchmark results for the performance of these proposed formulations. Copyright © 2010 John Wiley & Sons, Ltd.     

Locking-free finite elements for shear deformable orthotropic thin-walled beams

Numerical models for finite element analyses of assemblages of thin-walled open-section profiles are presented. The assumed kinematical model is based on Timoshenko–Reissner theory so as to take shear strain effects of non-uniform bending and torsion into account. Hence, strain elastic-energy coupling terms arise between bending in the two principal planes and between bending and torsion. The adopted model holds for both isotropic and orthotropic beams. Several displacement interpolation fields are compared with the available numerical examples. In particular, some shape functions are obtained from ‘modified’ Hermitian polynomials that produce a locking-free Timoshenko beam element. Analogously, numerical interpolation for torsional rotation and cross-section warping are proposed resorting to one Hermitian and six Lagrangian formulation. Analyses of beams with mono-symmetric and non-symmetric cross-sections are performed to verify convergence rate and accuracy of the proposed formulations, especially in the presence of coupling terms due to shear deformations, pointing out the decay length of end effects. Profiles made of both isotropic and fibre-reinforced plastic materials are considered. The presented beam models are compared with results given by plate-shell models. Copyright © 2007 John Wiley & Sons, Ltd.     

A shape‐free 8‐node plane element unsymmetric analytical trial function method

The unsymmetric FEM is one of the effective techniques for developing finite element models immune to various mesh distortions. However, because of the inherent limitation of the metric shape functions, the resulting element models exhibit rotational frame dependence and interpolation failure under certain conditions. In this paper, by introducing the analytical trial function method used in the hybrid stress‐function element method, an effort was made to naturally eliminate these defects and improve accuracy. The key point of the new strategy is that the monomial terms (the trial functions) in the assumed metric displacement fields are replaced by the fundamental analytical solutions of plane problems. Furthermore, some rational conditions are imposed on the trial functions so that the assumed displacement fields possess fourth‐order completeness in Cartesian coordinates. The resulting element model, denoted by US‐ATFQ8, can still work well when interpolation failure modes for original unsymmetric element occur, and provide the invariance for the coordinate rotation. Numerical results show that the exact solutions for constant strain/stress, pure bending and linear bending problems can be obtained by the new element US‐ATFQ8 using arbitrary severely distorted meshes, and produce more accurate results for other more complicated problems. Copyright © 2012 John Wiley & Sons, Ltd.     

采用面积坐标的四边形厚薄板通用单元

本文采用四边形面积坐标,利用假设剪切应变场方法和广义协调理论构造出一个具有12个自由度的四边形厚薄板通用弯曲单元TACQ。基本思路如下：首先从Mindlin厚板理论出发,独立假设剪应变场和挠度场,而转角场则由挠度场和剪应变场导出;其次,单元剪应变场是先按Timoshenko厚梁理论确定单元各边剪应变,然后在单元内进行合理插值导出;第三,单元挠度场是根据单元角点处挠度的点协调条件以及单元各边挠度和法向转角的平均协调条件导出。这个方法有两个特点,（1）由于满足点协调和边协调的广义协调条件,故能保证收敛;（2）由于在薄板情况剪应变退化为零,故不出现剪切闭锁现象。数值算例表明：该单元具有精度高,收敛性和可靠性好,对网格畸变不敏感,无剪切闭锁现象等优点;适用于从极薄板到厚板较大的范围。     

梁杆结构二阶效应分析的一种新型梁单元

推导了一种计及梁杆二阶效应的新型两结点梁单元。首先依据插值理论构造了三结点Euler-Bernoulli梁单元的位移场:使用五次Hermite插值函数建立梁单元的侧向位移场,二次Lagrange插值函数建立梁单元的轴向位移场,进而由非线性有限元理论推导了单元的线性刚度矩阵和几何刚度矩阵,然后使用静力凝聚方法消除三结点梁单元中间结点的自由度,从而得到一种考虑轴力效应的新型两结点梁单元。实例分析表明,此新型梁单元具有很高的计算精度,使用此单元进行梁杆结构分析可获得相当准确的二阶位移和内力。     

Conference diary

A variational higher-order theory for bending and stretching of linearly elastic orthotropic beams including the deformations due to transverse shearing and stretching of the transverse normal fibre is presented. The theory assumes a linear distribution for the longitudinal displacement and a parabolic variation of the transverse displacement across the thickness. Additionally, independent expansions are introduced for the through-thickness displacement gradients with the requirement of a least-square compatibility for the transverse strains and the satisfaction of exact stress boundary conditions at the top/bottom beam surfaces. The theory is shown to be well suited for finite element development requiring simple C⁰- and C^?1- continuous displacement interpolation fields. To demonstrate the computational utility of the theory, a simple two-node stretching-bending finite element is formulated. The analytic and finite element results are obtained for a simple bending problem for which an exact elasticity solution is available. It is shown that the inclusion of the transverse normal deformation in the present theory enables improved displacement, strain and stress predictions, particularly, in the analysis of deep beams.     

Fully plastic analyses of plane strain single-edge-cracked specimens subject to combined tension and bending

Accurate yield surfaces of plane strain single-edge-cracked specimens having shallow as well as deep cracks are developed using finite element limit analyses and monotonic interpolation functions. Fully plastic shallow crack configurations are classified based on certain aspects of the yield surfaces. Relationships between incremental plastic crack tip and crack mouth opening displacements and incremental load point displacement/rotation are obtained for a wide range of relative crack depths and loading ratios. Fully plastic crack-tip fields for a sufficiently deep crack in a single-edge cracked specimen are examined to provide the stress triaxiality and the angular orientation of flow line at the crack tip in terms of the remotely applied tension-to-bending ratio. Evidence for fully plastic crack-tip stress fields consisting of an incomplete Prandtl fan and a crack plane constant state region is discussed.     

A new approach to the finite element modelling of beams with warping effect

As a first step toward developing a finite element formulation that can model coupling among extensional, bending and torsional behaviour of beams, a new method is proposed to properly represent the warping of arbitrary cross-sections. The basic approach is to introduce a small warping displacement superimposed over flat cross-sections of a shear-flexible beam in a deformed configuration. Numerical tests involving simple isotropic beams undergoing a small elastic displacement demonstrate the validity of the new approach. The present approach can be extended to composite beams as well as isotropic beams experiencing a large deflection or finite rotation.     

On an isoparametric finite-element for composite laminates with finite rotations

For composite laminates consisting of an arbitrary number of orthotropic laminae first a finite-rotation theory is presented as basis of isoparametric finite-element formulations. The derivation is achieved by a Reissner-Mindlin type kinematic assumption which allows a constant shear deformation distribution across the thickness. The constitutive equations are presented in a general form such that orthotropic material behaviour with material axes varying arbitrarily across the thickness may easily be considered in numerical implementation, also when using curvilinear coordinates. Special attention is taken to predict the force distribution in the deformed shell structure. This theory is then transformed into a four-node isoparametric assumed-strain finite element. Unlike in the degeneration approach, interpolation polynomials are introduced directly for rotation variables determining the deformed position of the unit normal vector. The capability of the finite element developed to deal with strongly nonlinear situations is demonstrated by many examples. Also numerical results are presented permitting a systematical comparison of classical and isoparametric approaches concerning the numerical efficiency.     

A mixed finite element for the nonlinear analysis of in-plane loaded masonry walls

A mixed membrane eight-node quadrilateral finite element for the analysis of masonry walls is presented. Assuming that a nonlinear and history-dependent 2D stress-strain constitutive law is used to model masonry material, the element derivation is based on a Hu-Washizu variational statement, involving displacement, strain, and stress fields as primary variables. As the behavior of masonry structures is often characterized by strain localization phenomena, due to strain softening at material level, a discontinuous, piecewise constant interpolation of the strain field is considered at element level, to capture highly nonlinear strain spatial distributions also within finite elements. Newton's method of solution is adopted for the element state determination problem. For avoiding pathological sensitivity to the finite element mesh, a novel algorithm is proposed to perform an integral-type nonlocal regularization of the constitutive equations in the present mixed formulation. By the comparison with competing serendipity displacement-based formulation, numerical simulations prove high performances of the proposed finite element, especially when coarse meshes are adopted.     

Finite element model of efficient zig-zag theory for static analysis of hybrid piezoelectric beams

Finite element model is presented for the analysis of hybrid piezoelectric beams under static electromechanical load, using the one-dimensional (1D) coupled zig-zag theory developed recently by the authors. Two noded elements are used with cubic Hermite interpolation for deflection and electric potentials at the sub-layers and with linear interpolation for axial displacement and shear rotation. The expressions for the variationally consistent stiffness matrix and load vector are derived and evaluated in closed form using exact integration. The formulation is validated by comparison with the analytical solution for simply-supported beam. The finite element model is free of shear locking. The present zig-zag finite element results for cantilever beams are compared with the 2D finite element results using ABAQUS to establish the accuracy of the zig-zag theory for these boundary conditions.S. Kapuria is grateful to Department of Science and Technology, Government of India, for providing financial assistance for this work.     

Patch‐averaged assumed strain finite elements for stress analysis

A finite element model for linear‐elastic small deformation problems is presented. The formulation is based on a weighted residual that requires a priori the satisfaction of the kinematic equation. In this approach, an averaged strain‐displacement matrix is constructed for each node of the mesh by defining an appropriate patch of elements, yielding a smooth representation of strain and stress fields. Connections with traditional and similar procedure are explored. Linear quadrilateral four‐node and linear hexahedral eight‐node elements are derived. Various numerical tests show the accuracy and convergence properties of the proposed elements in comparison with extant finite elements and analytic solutions. Specific examples are also included to illustrate the ability to resist numerical locking in the incompressible limit and insensitive response in the presence of shape distortion. Furthermore, the numerical inf‐sup test is applied to a selection of problems to show the stability of the present formulation. Copyright © 2012 John Wiley & Sons, Ltd.     

A novel isoparametric finite element displacement formulation for axisymmetric analysis of nearly incompressible materials

It is well accepted that severe numerical difficulties arise when using the conventional displacement method to analyse incompressible or nearly incompressible solids. These effects are caused by the kinematic constraints imposed on the nodal velocities by the constant volume condition. In elastic-plastic analysis, these effects are due to a conflict between the plastic flow rule and the finite element discretization. Although several methods have been proposed to cope with this problem, none has been based on the appropriate choice of displacement interpolation functions to minimize the constraints. The theoretical formulation of a new six-noded isoparametric displacement finite element, which is well suited for elastic-plastic analysis of axisymmetric constrained solids by using a rational displacement interpolation function, is presented in this paper. The proposed displacement interpolation function implies that the displacement in the axial direction and the product of the displacement in the radial direction and the radius should be treated as two independent basic variables. Alternatively, the proposed displacement interpolation function can also be implemented in a conventional displacement formulation simply by using a modified shape function matrix. The suitability of the proposed formulations is first studied theoretically by assessing the number of degrees of freedom per constraint and then verified by performing numerical experiments on typical boundary value problems which involve incompressible behaviour.     

Effects of approximations in analyses of beams of open thin‐walled cross‐section—part I: Flexural–torsional stability

In formulating a finite element model for the flexural–torsional stability and 3‐D non‐linear analyses of thin‐walled beams, a rotation matrix is usually used to obtain the non‐linear strain–displacement relationships. Because of the coupling between displacements, twist rotations and their derivatives, the components of the rotation matrix are both lengthy and complicated. To facilitate the formulation, approximations have been used to simplify the rotation matrix. A simplified small rotation matrix is often used in the formulation of finite element models for the flexural–torsional stability analysis of thin‐walled beams of open cross‐section. However, the approximations in the small rotation matrix may lead to the loss of some significant terms in the stability stiffness matrix. Without these terms, a finite element line model may predict the incorrect flexural–torsional buckling load of a beam. This paper investigates the effects of approximations in the elastic flexural–torsional stability analysis of thin‐walled beams, while a companion paper investigates the effects of approximations in the 3‐D non‐linear analysis. It is found that a finite element line model based on a small rotation matrix may predict incorrect elastic flexural–torsional buckling loads of beams. To perform a correct flexural–torsional stability analysis of thin‐walled beams, modification of the model is needed, or a finite element model based on a second‐order rotation matrix can be used. Copyright © 2001 John Wiley & Sons, Ltd.     

对转角场和剪应变场进行合理插值的厚薄板通用四边形单元

本文提出构造厚薄板通用四边形单元的一种合理方法：先按Timoshenko厚梁理论导出单元各边的转角和剪应变公式,然后进行合理插值,导出单元的转角场、曲率场和剪应变场。当板的厚度变小时,厚梁理论自动退化为薄梁理论,各边剪应变以及单元剪应变插值函数自动退化为零,厚板单元自动退化为薄板单元,彻底消除了剪切闭锁现象。此单元对厚板和薄板都给出了高精度的结果。     

Displacement‐based finite elements with nodal integration for Reissner–Mindlin plates

An assumed‐strain finite element technique is presented for shear‐deformable (Reissner–Mindlin) plates. The weighted residual method (reminiscent of the strain–displacement functional) is used to enforce weakly the balance equation with the natural boundary condition and, separately, the kinematic equation (the strain–displacement relationship). The a priori satisfaction of the kinematic weighted residual serves as a condition from which strain–displacement operators are derived via nodal integration, for linear triangles, and quadrilaterals, and also for quadratic triangles. The degrees of freedom are only the primitive variables: transverse displacements and rotations at the nodes. A straightforward constraint count can partially explain the insensitivity of the resulting finite element models to locking in the thin‐plate limit. We also construct an energy‐based argument for the ability of the present formulation to converge to the correct deflections in the limit of the thickness approaching zero. Examples are used to illustrate the performance with particular attention to the sensitivity to element shape and shear locking. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical tests for assessing finite element model convergence

New requirements for finite element model convergence are defined. It is shown that the presence of constant strain states in an element is not sufficient to guarantee convergence. Numerical tests are proposed to assess convergence, direction of convergence, monotonicity of convergence, convergence rate and monotonicity of convergence rate. The proposed tests are mathematically stronger and more general than existing tests.     

Performing RVE calculations under constant stress triaxiality for monotonous and cyclic loading

In the present work the mesoscopic stress, strain rate and strain states of axisymmetric cells under two types of boundary loadings are formulated. Then, the stress triaxiality of axisymmetric cells is expressed in terms of the axial and radial mesoscopic stress components. Based on the formulations of the mesoscopic stress, three strategies for numerical realization of constant stress triaxiality are presented. The advantages and disadvantages of these strategies are discussed. These numerical strategies are implemented on the platform of the general‐purpose finite element programme ABAQUS. They can be applied for representative volume element (RVE) calculations under constant triaxiality, monotonous and cyclic loading controlled by displacement, force, traction and the mesoscopic equivalent strain of the RVE. Several numerical examples are shown to prove the effectivity of these strategies and programme. Copyright © 2005 John Wiley & Sons, Ltd.     

Stress oscillations and spurious load mechanisms in variationally inconsistent assumed strain formulations

Assumed field-consistent strain formulations of the displacement finite element procedure can lead to poor convergence and spurious stress oscillations if the assumed strain fields are not variationally correct, i.e. they do not satisfy an important orthogonality condition emerging from the equivalence sought between assumed strain displacement procedures and mixed procedures based on the Hellinger–Reissner theorem. Failure to ensure variational correctness introduces errors which can be equated to the presence of spurious loading mechanisms that cause stress oscillations. In this paper, we use the Timoshenko beam element to demonstrate that field-consistency and variational' consistency are two complementary but mutually exclusive principles—one does not imply the other and that both are necessary to successfully implement a displacement type finite element for constrained media.     

Variational multiscale approach to enforce perfect bond in multiple-point constraint applications when forming composite beams

Composite laminates that consist of two or more layers find widespread applications in a variety of engineering structures. In the computational modelling of composite laminates, the layers can be stacked together and connected conveniently at the nodes by using multiple-point constraints (MPCs). However, this type of modelling leads to weakening of the kinematic constraint conditions imposed by the bond between the juxtaposed layers and as a consequence, MPCs application at the nodes produces behaviour that is softer than the perfectly bonded composite beam behaviour. The work herein shows that when kinematic conditions for composite action are weakly imposed in the variational form, they can be enforced in the point-wise sense by proper selection of the interpolation field or otherwise reinforced by using variational multiscale approach without modifying the kinematic model. The originality of the approach presented herein is in the interpretation of the MPCs application as the solution in a superfluously extended space because of the weakening in the kinematic constraints. It is shown that the perfect bond between the composite beam layers can be recovered by excluding the identified fine-scale effect from the solution of the multiple point constraint application. The convergence characteristic of the finite element formulation is also improved by using the variational multi-scale approach. It is also shown that the fine-scale effects can be represented by using extra fictitious elements and springs, which offers a direct correction technique in modelling of composite beams that is especially useful when access to the numerical procedure is limited.