Field- and edge-consistency synthesis of A 4-noded quadrilateral plate bending element

In this paper, we demonstrate the use of two conceptual principles, the field-consistency requirement and the edge-consistency requirement, as the basis for deriving a 4-noded quadrilateral plate bending element based on Mindlin plate theory using Jacobean transformations only. The derivation is now free of the use of such devices as strain-interpolation points and Hrennik off strain reference lines, etc., which have been the basis for many recent formulations of this element. The shear strain constraints are now consistently defined within the element domain, and ‘tangential’ shear strains are consistently matched at element boundaries so that there is no locking even under extreme distortion—e.g. even when two nodes are collapsed so that the quadrilateral becomes a triangle. Numerical experiments show that this synthesis produces an element that should be identical to other recent formulations of this element based on tensorial transformations or on shear constraint condensation on the edges, but now given a more complete and formal logical basis.     

Stress oscillations in plane stress modelling of flexure—a field-consistency interpretation

Exactly integrated isoparametric plane stress elements behave poorly in flexure. The 4-noded element ‘locks’, with errors that progress indefinitely as element aspect ratio increases. Reduced integration of the shear strain energy eliminates this locking entirely. The 8-noded element does not lock, but improves in performance with reduced integration of shear strain energy. Both elements, with their original shape functions, show severe shear stress oscillations in flexure. In this paper we attribute these oscillations to the lack of ‘consistency’ of shear strain fields derived directly from independent field-variable interpolations. We derive error models for specific tractable examples which can confirm the accuracy of this conceptual scheme through digital computation using the finite element models. A field-consistent redistribution strategy for the shear strain field is offered as an elegant procedure to free the elements of spurious oscillations and give a ‘lock’-free performance.     

Shear flexible field‐consistent curved spline beam element for vibration analysis

In this paper, an efficient curved cubic B‐spline beam element is developed based on the field consistency principle, for vibration analysis. The formulation is general in the sense that it includes anisotropy, transverse shear deformation, in‐plane and rotary inertia effects. The element is based on laminated refined beam theory, which satisfies the interface transverse shear stress and displacement continuity, and has a vanishing shear stress on the top and bottom surfaces of the beam. The lack of consistency in the shear and membrane strain field interpolations in their constrained physical limits causes poor convergence and unacceptable results due to locking. Hence, numerical experimentation is conducted to check these deficiencies with a series of assumed shear/membrane strain functions, redistributed in a field‐consistent manner. The performance of the element is assessed by studying the free vibration behaviour of a variety of problems ranging from a straight beam to a circular ring. Copyright © 1999 John Wiley & Sons, Ltd.     

A new shear flexible cubic spline plate element for vibration analysis

Here, a new cubic B‐spline plate element is developed using field consistency principle, for vibration analysis. The formulation includes anisotropy, transverse shear deformation, in‐plane and rotary inertia effects. The element is based on a laminated refined plate theory, which satisfies the interface transverse shear stress and displacement continuity, and has a vanishing shear stress on the top and bottom surfaces of the plates. The lack of consistency in the shear strain field interpolations in its constrained physical limits produces poor convergence and results in unacceptable solutions due to locking phenomenon. Hence, numerical experimentation for the evaluation of natural frequencies of plates is carried out to check this deficiency with a series of assumed shear strain functions, redistributed in a field consistent manner. Copyright © 2002 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。     

A pure bending exact nodal-averaged shear strain method for finite element plate analysis

An averaged shear strain method, based on a nodal integration approach, is presented for the finite element analysis of Reissner–Mindlin plates. In this work, we combine the shear interpolation method from the MITC4 plate element with an area-weighted averaging technique for the nodal integration of shear energy to relieve shear locking in the thin plate analysis as well as to pass the pure bending patch test. In order to resolve the numerical instability caused by the direct nodal integration, the bending strain field is computed by a sub-domain nodal integration approach based on the Sub-domain Stabilized Conforming Integration and a modified curvature smoothing scheme. The resulting nodally integrated smoothed strain formulation is shown to contain only the primitive variables and thus can be easily implemented in the existing displacement-based finite element plate formulation. Several numerical examples are presented to demonstrate the accuracy of the present method.     

Assumed strain formulation for the four-node quadrilateral with improved in-plane bending behaviour

A new assumed strain quadrilateral element with highly accurate in-plane bending behaviour is presented for plane stress and plane strain analysis. The basic idea of the formulation consists in identification of various modes of deformation and then in proper modification of the strain field in some of these modes. In particular, the strain operator corresponding to the in-plane bending modes is modified to simulate the strain field resulting from the assumptions usually made in structural mechanics. The modification of the strain field leads to the assumed strain operator on the element level. As a result, the so-called shear and membrane locking phenomena are alleviated. The element exhibits remarkable success in bending- dominated problems even when severely distorted and high aspect ratio meshes are used. Another advantage of the present assumed strain element is that locking for nearly incompressible materials is also mitigated. While this assumed strain element passes the patch test only for the parallelogram shapes, the element provides convergent solutions as long as the initially general form of the element approaches a parallelogram shape with the refinement of the mesh.     

Two-scale method for shear bands: thermal effects and variable bandwidth

A method for the analysis of shear bands using local partition of unity is developed in the framework of the extended finite element method (XFEM). Enrichments are introduced for both the displacement field and the thermal field. The shear band width is determined by minimizing the plastic work. A coupled finite strain thermo-elastoplastic constitutive law is used. The enrichment is injected into the mesh when the material law becomes unstable. The criterion based on a complete stability analysis for materials in the finite strain regime including heat conduction, strain hardening, strain rate hardening and thermal softening is presented. A mixed continuous quadrilateral element is employed. The method is applied to the Nesterenko experiments, which exhibit multiple propagating shear bands and other problems. Copyright © 2007 John Wiley & Sons, Ltd.     

A new drilling quadrilateral membrane element with high coarse-mesh accuracy using a modified Hu-Washizu principle

By utilizing a modified Hu-Washizu principle, a new mixed variational framework and a corresponding high-performing four-node membrane element with drilling degrees of freedom, named as GCMQ element, are proposed. In this work, the generalized conforming concept, which is originally proposed within a displacement-based formulation, is extended to a mixed formulation. The new element is able to handle higher-order displacement, strain, and stress distributions. The interpolations are complete up to second order for stress and strain. The enhanced strain field is optimized so that a complete cubic displacement field can be represented. For numerical integration, a five-point scheme is proposed to minimize computational cost. Compared to other four-node elements in existing literature, numerical examples show that the proposed element has a better performance regarding predictions of both displacements and internal forces, particularly with coarse meshes. The new element is also free from shear locking and volumetric locking. Due to the nature of the mixed framework, GCMQ can be directly used in elastoplastic applications.     

QUADRATIC TRIANGULAR C0 PLATE BENDING ELEMENT

In this paper, a six-node triangular C⁰ plate bending element is developed by the assumed strain formulation. The sampled transverse shear strains in the element are chosen such that the latter has a favourable constraint index of shear locking and the strains are optimized with respect to a linear pure bending displacement/rotation field. It happens that the optimal strains are the mean strains along the element edges and medians. Numerical examples reveal that the element is free from shear locking and passes all the patch tests for plate bending elements. Moreover, the element accuracy is close to that of a state-of-the-art seven-node assumed strain element. © 1997 by John Wiley & Sons, Ltd.     

Influence of Boundary Conditions on the Finite Element Creep Stress and Strain Analysis of a Double Shear Specimen for Isotropic Creep Conditions

The present study investigates the influence of boundary conditions on creep finite element stress and strain results for a double shear creep specimen which was developed by Mayr et al. [1]. The results of the study show that a preliminary analysis [2], which was performed using the finite element code ABAQUS using RIGID SURFACE as a boundary condition for loading is conservative, because it predicts that a homogeneous state of shear stress is only maintained up to shear strains of the order of 10%. Using a more realistic boundary condition for loading (ABAQUS option: CONTACT PAIR) it can be shown that the homogeneous state of shear stress in the specimen is maintained throughout creep up to much higher shear strains than originally estimated. Further anisotropic finite element creep stress analysis of our double shear creep specimen will therefore be based on this more realistic loading condition.     

基于一阶剪切变形理论的新型复合材料层合板单元

基于一阶剪切变形理论(FSDT),本文构造一种新型的20自由度(每结点5个自由度),四边形复合材料层合板单元,适合于任意铺设情形的层合板的计算。它是按如下方式构造的:(1) 单元每边的转角和剪应变由Timoshenko层合厚梁理论来确定;(2) 对单元域内的转角场和剪应变场进行合理的插值;(3) 引入平面内双线性位移场来体现层合板面内与弯曲的耦合作用。本文单元,记为TMQ20,不存在剪切闭锁现象,在计算单层的各向同性板时可以退化为文[1]中优质的中厚板单元TMQ。在文[2]中将给出本文单元对于层合板问题的详细数值算例。     

Higher-order MITC general shell elements

Two mixed-interpolated general shell finite elements for non-linear analysis-a 9-node element and a 16-node element-are presented. The elements are based on the Mixed Interpolation of Tensorial Components (MITC) approach in which the covariant strain component fields for the in-plane and shear actions are interpolated and tied to the also interpolated displacement field. Both the 9-node element, referred to as the MITC9 element, and the 16-node element, referred to as the MITC16 element, are tested numerically and found to have high predictive capabilities.     

Effective shear modulus approach for two dimensional solids and plate bending problems by meshless point collocation method

For the analysis of material nonlinearity, an effective shear modulus approach based on the strain control method is proposed in this paper by using point collocation method. Hencky's total deformation theory is used to evaluate the effective shear modulus, Young's modulus and Poisson's ratio, which are treated as spatial field variables. These effective properties are obtained by the strain controlled projection method in an iterative manner. To evaluate the second order derivatives of shape function at the field point, the radial basis function (RBF) in the local support domain is used. Several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method and comparisons have been made with analytical solutions and the finite element method (ABAQUS).     

Failure analysis of RC beams subjected to shear through different numerical approaches

This work discusses and compares different numerical approaches that can be adopted for the analysis up to failure of reinforced concrete beams, also when they experience a brittle shear collapse. Since the development of inclined shear cracks causes a variation of the strain field normal to the element axis, as well as of the shear strains in the beam depth, this type of problem is often dealt with refined bi-dimensional nonlinear finite element analyses. The effectiveness of this type of simulations is in turn mainly related to the adoption of a sound constitutive law for the material. This work highlights that, given the same material model, also more “traditional” approaches, based on sectional analysis or on 1D finite element simulations, can be satisfactorily applied to study the problem, if their kinematic assumptions are improved and extended. In these cases, a subdivision of beam depth into several layers is also recommended. In fact, this allows to both simulate the actual position of steel reinforcement, and to widen the applicability of the method to elements characterized by a generic cross-section shape.     

Development of shear locking‐free shell elements using an enhanced assumed strain formulation

The degenerated approach for shell elements of Ahmad and co‐workers is revisited in this paper. To avoid transverse shear locking effects in four‐node bilinear elements, an alternative formulation based on the enhanced assumed strain (EAS) method of Simo and Rifai is proposed directed towards the transverse shear terms of the strain field. In the first part of the work the analysis of the null transverse shear strain subspace for the degenerated element and also for the selective reduced integration (SRI) and assumed natural strain (ANS) formulations is carried out. Locking effects are then justified by the inability of the null transverse shear strain subspace, implicitly defined by a given finite element, to properly reproduce the required displacement patterns. Illustrating the proposed approach, a remarkably simple single‐element test is described where ANS formulation fails to converge to the correct results, being characterized by the same performance as the degenerated shell element. The adequate enhancement of the null transverse shear strain subspace is provided by the EAS method, enforcing Kirchhoff hypothesis for low thickness values and leading to a framework for the development of shear‐locking‐free shell elements. Numerical linear elastic tests show improved results obtained with the proposed formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

Two‐scale shear band evolution by local partition of unity

We introduce a methodology to model shear band evolution in the quasi‐static regime using the extended finite element method. We enrich the finite element polynomial displacement field with a fine scale function, which models the high displacement gradient in the shear band. For this purpose we use a local partition of unity and a parameterized displacement enrichment based on closed form solutions for one‐dimensional shear bands. A stabilized consistent penalty method is used to circumvent locking in the regularized elasto‐viscoplastic plane‐strain regime and to guarantee element stability. The loss of stability of the boundary value problem is used as an indicator of shear band initiation point and direction. Shear band development examples are shown, illustrating the capabilities of the method to track shear band evolution and strains as high as 1000%. Copyright © 2005 John Wiley & Sons, Ltd.     

An enhanced cell‐based smoothed finite element method for the analysis of Reissner–Mindlin plate bending problems involving distorted mesh

In this work, an enhanced cell‐based smoothed finite element method (FEM) is presented for the Reissner–Mindlin plate bending analysis. The smoothed curvature computed by a boundary integral along the boundaries of smoothing cells in original smoothed FEM is reformulated, and the relationship between the original approach and the present method in curvature smoothing is established. To improve the accuracy of shear strain in a distorted mesh, we span the shear strain space over the adjacent element. This is performed by employing an edge‐based smoothing technique through a simple area‐weighted smoothing procedure on MITC4 assumed shear strain field. A three‐field variational principle is utilized to develop the mixed formulation. The resultant element formulation is further reduced to a displacement‐based formulation via an assumed strain method defined by the edge‐smoothing technique. As the result, a new formulation consisting of smoothed curvature and smoothed shear strain interpolated by the standard transverse displacement/rotation fields and smoothing operators can be shown to improve the solution accuracy in cell‐based smoothed FEM for Reissner–Mindlin plate bending analysis. Several numerical examples are presented to demonstrate the accuracy of the proposed formulation.Copyright © 2013 John Wiley & Sons, Ltd.     

AN UPPER BOUND FINITE ELEMENT PROCEDURE FOR SOLVING LARGE PLANE STRAIN DEFORMATION

A unique and robust upper bound finite element procedure is developed for the analysis of large plastic deformation problems under plane strain condition. It can consistently treat problems with isotropic strain varying materials. It can also effectively solve problems with any initial ‘guessed’ velocity field, even from an random number generator. To explore and demonstrate the capability of this new approach, strip tension and plane strain compression problems are solved. For validation, the computed results are compared with existing analytical or experimental solutions in good agreement. The phenomenon of shear band formation can be simulated and, as expected, is found to develop more distinctly in strain softening materials than in perfectly plastic and strain hardening materials. © 1997 by John Wiley & Sons, Ltd.     

A simple triangular element for thick and thin plate and shell analysis

A new plate triangle based on Reissner–Mindlin plate theory is proposed. The element has a standard linear deflection field and an incompatible linear rotation field expressed in terms of the mid-side rotations. Locking is avoided by introducing an assumed linear shear strain field based on the tangential shear strains at the mid-sides. The element is free of spurious modes, satisfies the patch test and behaves correctly for thick and thin plate and shell situations. The element degenerates in an explicit manner to a simple discrete Kirchhoff form.