A locking-free approximation of curved rods by straight beam elements

Summary. We consider an elastic model for a curved rod with arbitrary three-dimensional geometry, incorporating shear and membrane as well as bending and torsion effects. We define an approximation procedure based on a discretization by linear Timoshenko beam elements. Introducing an equivalent mixed problem, we establish optimal error estimates independent of the thickness, thereby proving that shear and membrane locking is avoided. The approximation scheme is tested on specific examples and the numerical results confirm the estimates obtained by theory. Received May 31, 1995 / Revised version received February 26, 1996     

A new element for analyzing large deformation of thin Naghdi shell model. Part 1: Elastic

One of the best approaches for modeling large deformation of shells is the Cosserat surface. However, the finite-element implementation of this model suffers from membrane and shear locking, especially for very thin shells. The basic assumption of this theory is that the mid-surface of the shell is regarded as a Cosserat surface with one inextensible director. In this paper, it is shown that by constraining the director vector normal to the mid-surface, besides very good and accurate results, shear locking is also eliminated. This constraint is in fact a limiting analysis of the Cosserat theory in which Kirichhoff’s hypothesis is enforced. Numerical solution is performed using nine-node isoparametric element. The principal of virtual work is used to obtain the weak form of the governing differential equations and the material and geometric stiffness matrices are derived through a linearization process. The validity and the accuracy of the method are illustrated by numerical examples.     

Isogeometric Reissner-Mindlin shell analysis - employing different control meshes for displacements and rotations

The main purpose of this paper is the development and implementation of a method for the reduction of the so-called locking effect in the isogeometric Reissner-Mindlin shell formulation. In [1] an isogeometric Reissner-Mindlin shell formulation with an exact interpolation of the director vector based on continuum mechanics was introduced. The numerical examples showed that the accuracy and efficiency increased. However, there are only few effective concepts for the prevention of locking effects for low polynomial degrees. In the work of Beirão da Veiga [2], shear locking is prevented for a Reissner-Mindlin plate formulation by using suitable solution spaces. Here, the method is extended to the Reissner-Mindlin shell formulation. Different control meshes are used for displacements and rotations. Furthermore, the basis functions in the direction of the relevant rotation are one degree less than the ones which are chosen for the displacements. That leads to control meshes with different number and location of the control points. The aim is to avoid shear locking due to the coupling of shear strains and curvature since the compatibility requirement for pure bending is then fulfilled. The accuracy and efficiency of this method are investigated for different examples. In addition, the results are compared to the analytical solutions. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of Locally Loaded Multilayer Shells by a Mixed Finite-Element Method. 2. Geometrically Nonlinear Statement

Based on mixed finite-element approximations, a numerical algorithm is developed for solving linear static problems of prestressed multilayer composite shells subjected to large displacements and arbitrarily large rotations. As the sought-for functions, six displacements and eleven strains of the shell faces are chosen, which allows us to use nonlinear deformation relationships exactly representing arbitrarily large displacements of the shell as a rigid body. The stiffness matrix of a shell element has a proper rank and is calculated based on exact analytical integration. The bilinear element developed does not allow false rigid displacements and is not subjected to the membrane, shear, or Poisson locking phenomenon. The results of solving the well-known test problem on a nonsymmetrically fixed circular arch subjected to a concentrated load and the problem on a locally loaded toroidal multilayer rubber-cord shell are presented.     

无锁定的退化的等参壳元

构造了一个无锁定的8节点退化的等参壳元。在自然坐标系中,对横向剪应变项插值修正使单元满足必要的模式(平移、转动和纯弯曲)以消除"剪切锁定";在局部坐标系中,对薄膜应变插值修正以消除"薄膜锁定"。这样构造的8节点单元的刚度矩阵具有正确的秩,同时具有恰当的零特征值和相应的刚体位移模式。这种单元对大跨-厚比情形既无"剪切锁定"又无"薄膜锁定",无虚假的零能模式和机构出现,可用于厚壳和薄壳。     

Comments on Membrane Locking

Membrane locking is a severe issue frequently neglected in the context of low-order (four node isoparametric) shell elements. From a theoretical point of view, the present contribution illustrates the underlying mechanism by means of shallow shell theory. From a numerical point of view, different element formulations based on standard enhanced assumed membrane strains and reduced integration methods have been compared to each other applying standard as well as modified benchmark examples. A modified integration scheme is proposed. According to preliminary results it behaves virtually optimal within the limits of finite element perfectibility. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Formulation and evaluation of a finite element model for the linear analysis of stiffened composite cylindrical panels

A finite element model for linear static and free vibration analysis of composite cylindrical panels with composite stiffeners is presented. The proposed model is based on a cylindrical shell finite element, which uses a first-roder shear deformation theory. The stiffeners are curved beam elements based on Timoshenko and Saint-Venant assumptions for bending and torsion respectively. The two elements are developed in a cylindrical coordinate system and their stiffness matrices result from a hybrid-mixed formulation where the element assumed stress field is such that exact equilibrium equations are satisfied. The elements are free of membrane and shear locking with correct satisfaction of rigid body motions. Several examples dealing with stiffened isotropic and laminated plates and shells with eccentric as well as concentric stiffeners are analyzed showing the validity of the models.     

黏弹性圆柱壳计及切变形和转动惯量时的动力学稳定性

根据修正的Timoshenko理论,在几何非线性中考虑了剪切变形和转动惯量,对黏弹性圆柱壳的动力稳定性进行了研究.根据Bubnov-Galerkin法,结合基于求积公式的数值方法,将问题简化为求解具有松弛奇异核的非线性积分-微分方程的问题.针对物理-力学和几何参数在大范围内的变化,研究壳体的动力特性,显示了材料的黏弹性对圆柱壳动力稳定性的影响.最后,比较了通过不同的理论得到的结果.     

Problem-Oriented Functionals in the Theory of Nonlinearly Elastic Composite Shells

Based on the Kirchhoff-Love or Timoshenko hypotheses and with regard for a possible membrane or shear degeneration, mixed linearized functionals for four variants of shell theory are presented. The convergence of numerical methods is improved by choosing small strain components as additional variable functions. New classes of problems for thin and nonthin shells are solved. The stress-strain state of shells is studied using different variants of this theory.     

Equivalence estimates for a class of singular perturbation problems

We give some equivalence estimates on the solution of a singular perturbation problem that represents, among other models, the Koiter and Naghdi shell models. Two of the estimates apply to intermediate shell problems and the third is for membrane/shear dominated shells. From these equivalences, many known and some new sharp estimates on the solutions of the singular perturbation problems easily follow. To cite this article: S. Zhang, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

A Four-node Finite Element for Piezoelectric Shell Structures in Convective Co-ordinates

In this contribution, an isoparametric piezoelectric shell element is presented which is based on convective coordinates and which allows for the analysis of arbitrary shell geometries. A two-field variation formulation [1, 2] is used in which the displacements and the electric potentials serve as independent variables. Especially, for thin-walled structures under certain boundary conditions and load cases, the displacement based element tend to shear and membrane locking. In order to avoid this poor behaviour, the Assumed Natural Strain (ANS) method [3] is introduced into the piezoelectric shell element. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Construction of beam elements considering von Kármán nonlinear strains using B-spline wavelet on the interval

The current paper proposes the formulation of beam elements using B-spline wavelet on the interval based wavelet finite element method by incorporating von Kármán nonlinear strains. Formulation is proposed for both Euler–Bernoulli beam theory and Timoshenko beam theory. A background cell based Gauss quadrature is proposed for numerical integration. Numerical examples are solved for transverse deflections and stresses in axial direction, and are compared with the existing converged results from finite element method. The issues of membrane and shear locking for the proposed elements are examined and solution techniques are suggested to overcome the issues.     

Free vibration analysis of FGM cylindrical shells surrounded by Pasternak elastic foundation in thermal environment considering fluid-structure interaction

This paper presents an investigation on partially fluid-filled cylindrical shells made of functionally graded materials (FGM) surrounded by elastic foundations (Pasternak elastic foundation) in thermal environment. Material properties are assumed to be temperature dependent and radially variable in terms of volume fraction of ceramic and metal according to a simple power law distribution. The shells are reinforced by stiffeners attached to their inside and outside in which the material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction. The formulations are derived based on smeared stiffeners technique and classical shell theory using higher-order shear deformation theory which accounts for shear flexibility through shell's thickness. Displacements and rotations of the shell middle surface are approximated by combining polynomial functions in the meridian direction and truncated Fourier series with an appropriate number of harmonic terms in the circumferential direction. The governing equations of liquid motion are derived using a finite strip element formulation of incompressible inviscid potential flow. The dynamic pressure of the fluid is expanded as a power series in the radial direction. Moreover, the quiescent liquid free surface is modeled by concentric annular rings. A detailed numerical study is carried out to investigate the effects of power-law index of functional graded material, fluid depth, stiffeners, boundary conditions, temperature and geometry of the shell on the natural frequency of eccentrically stiffened functionally graded shell surrounded by Pasternak foundations.     

Boundary-Value Problems of the Theory of Thin and Nonthin Orthotropic Shells with Account of Nonlinearly Elastic Properties and Low Shear Rigidity of Composite Materials

The basic geometric and physical relations and resolving equations of the theory of thin and nonthin orthotropic composite shells with account of nonlinear properties and low shear rigidity of their materials are presented. They are derived based on two theories, namely the theory of anisotropic shells employing the Timoshenko or Kirchhoff-Love hypothesis and the nonlinear theory of elasticity and plasticity of anisotropic media in combination with the Lagrange variational principle. The procedure and algorithm for the numerical solution of nonlinear (linear) problems are based on the method of successive approximations, the difference-variational method, and the Lagrange multiplier method. Calculations of the stress-strain state for a spherical shell with a circular opening loaded with internal pressure are presented. The effect of transverse shear strains and physical nonlinearity of the material on the distribution of maximum deflections and circumferential stresses in the shell, obtained according to two variants of the shell theories, is studied. A comparison of the results of the problem solution in linear and nonlinear statements with and without account of the shell shear strains is given. The numerical data obtained for thin and nonthin (medium thick) composite shells are analyzed.     

Computations on thin non‐inhibited hyperbolic elastic shells. Bench‐marks for membrane locking

We study the bending limit problem of shells in relation to the membrane locking, encountered in finite element computation of non‐inhibited very thin shells. Using a new approach of the theory of inextensional displacements (or infinitesimal bendings) we solve the bending limit problem in the case of a clamped hyperbolic paraboloid. We then use this solution to validate computations which can be used as bench‐marks for the membrane locking. Such configuration, non‐inhibited hyperbolic very thin shells, usually lacks numerical ‘validation’. Copyright © 1999 John Wiley & Sons, Ltd.     

Vibrations of Timoshenko beams with isogeometric approach

A study on the free vibration analysis of Timoshenko beams is presented here. In order to determine natural frequencies of beams, a thick beam element is developed by using isogeometric approach based on Timoshenko beam theory which allows the transverse shear deformation and rotatory inertia effect. Three refinement schemes such as h-, p- and k-refinement are used in the analysis and the identification of shear locking is also conducted by using numerical examples. From numerical results, the present element can produce very accurate values of natural frequencies and the mode shapes due to exact definition of the geometry. With higher order basis functions, there is no shear locking phenomenon in very thin beam situations. Finally, the benchmark tests described in this study are provided as future reference solutions for Timoshenko beam vibration problem.     

Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method

This paper presents a novel finite element formulation for static, free vibration and buckling analyses of laminated composite plates. The idea relies on a combination of node-based smoothing discrete shear gap method with the higher-order shear deformation plate theory (HSDT) to give a so-called NS-DSG3 element. The higher-order shear deformation plate theory (HSDT) is introduced in the present method to remove the shear correction factors and improve the accuracy of transverse shear stresses. The formulation uses only linear approximations and its implementation into finite element programs is quite simple and efficient. The numerical examples demonstrated that the present element is free of shear locking and shows high reliability and accuracy compared to other published solutions in the literature.     

The problem of membrane locking in finite element analysis of cylindrical shells

Summary We analyse the problem of membrane locking in (h, p) finite element models of a thin hemicylindrical shell roof loaded by a smoothly varying normal pressure distribution. We show that in the standard finite element method, locking occurs especially at low values ofp and when the finite element grid is not aligned with the axis of the cylinder. A general strategy of avoiding locking by using modified bilinear forms is introduced, and a special implementation of this strategy on aligned rectangular grids is considered.     

On a model of a flexural prestressed shell

E. Sanchez‐Palencia We derive a linearized prestressed elastic shell model from a nonlinear Kirchhoff model of elastic plates. The model is given in terms of displacement and micro‐rotation of the cross‐sections. In addition to the standard membrane, transverse shear, and flexural terms, the model also contains a nonstandard prestress term. The prestress is of the same order as flexural effects; hence, the model is appropriate when flexural effects dominate over membrane ones. We prove the existence and uniqueness of the solutions by Lax–Milgram theorem and compare solution with the solution of the standard shell model via numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

A Spatial Geometric Nonlinearity Spline Beam Element With Nodal Parameters Containing Strains北大核心CSCD

Many slender rods in engineering can be modeled as Euler-Bernoulli beams. For the analysis of their dynamic behaviors, it is necessary to establish the dynamic models for the flexible multi-body systems. Geometric nonlinear elements with absolute nodal coordinates help solve a large number of dynamic problems of flexible beams, but they still face such problems as shear locking, nodal stress discontinuity and low computation efficiency. Based on the theory of large deformation beams’ virtual power equations, the functional formulas between displacements and rotation angles at the nodes were established, which can satisfy the deformation coupling relationships. The generalized strains to describe geometric nonlinear effects in this case were derived. Some parameters of boundary nodes were replaced by axial strains and sectional curvatures to obtain a more accurate and concise constraint method for applying external forces. To improve the numerical efficiency and stability of the system’s motion equations, a model-smoothing method was used to filter high frequencies out of the model. The numerical examples verify the rationality and effectiveness of the proposed element. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.