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 nine node finite element for analysis of geometrically non-linear shells

A nine node finite element is presented for the analysis of thin shell structures undergoing large deflection. The finite element formulation is based on the concept of degenerate solid shell element and the Hellinger-Reissner principle with independent strain. Three versions of assumed independent strain are selected to suppress spurious kinematic modes. One version leads to a finite element model which is kinematically stable at element level while the other two give globally stable models. Numerical tests indicate that the finite element model which is stable at element level may reveal the locking effect in certain cases. However, the other two models are free of locking.     

THE FOUR-NODE C0 SHELL ELEMENT REFORMULATED

A reformulated four-node shell element, based on the analysis of the moment redistribution mechanism development by C⁰ plate bending and shell elements, is presented. The moment redistribution mechanism of a finite shell element model is shown to be predominantly activated by the membrane flexural action of the shell. This action is triggered through the membrane strain components which participate in the moment equilibrium equations of the finite element assembly system. An equivalent elastic foundation action, along with the activation of the in-plane twisting stiffness of the shell, may also contribute to the moment redistribution mechanism of the finite shell element model. The proposed shell element formulation aims at retaining the non-spurious contribution of the transverse shear/membrane strain energy to the flexural behaviour of the shell, through the activation of the moment redistribution mechanism. Yet, any potentially spurious, whether locking or kinematic, mechanism is rejected. In warped configurations, the element activates appropriate coupling mechanisms of the bending terms to nodal translations. The so-obtained reformulated four-node shell element exhibits an excellent behaviour without experiencing any locking phenomena or zero-energy modes, while its formulation is kept simple, based on physical considerations. The proposed formulation performs equally well in flat as well as in warped shell element applications.     

钢质圆柱壳在侧向局部冲击荷载下的变形及失效破坏

基于动力有限元程序LS-DYNA及随动塑性Cowper-Symonds模型,对两端固支钢质薄壁圆柱壳经受半球头弹体侧向局部冲击的非线性动力响应问题进行数值模拟,获得了不同冲击条件下圆柱壳的变形及破坏模态,并研究了弹体在不同周向冲击倾角时壳壁产生穿透性破裂的最小速度（临界破裂速度）。研究表明,圆柱壳破坏模式与弹体冲击倾角θ0、冲击速度V等因素密切相关,将发生局部凹陷、碟形变形及穿透现象,且临界破裂速度随冲击倾角的增大而增大。研究结果可应用于圆柱壳在侧向局部冲击作用下的毁伤预测,从而为圆柱壳结构的安全防护设计提供理论依据。     

An assumed strain formulation of efficient solid triangular element for general shell analysis

An efficient assumed strain triangular solid element is developed for the analysis of plate and shell structures. The finite element formulation is based on the two‐field assumed strain formulation with two independent fields of assumed displacement and assumed strain. The assumed strain field is carefully selected to alleviate the shear locking effect without triggering undesirable spurious kinematic modes. The curvilinear surface of shell structures is modelled with flat facet elements to obviate the membrane locking effect. The patch tests are successfully passed, and numerical test involving various example problems demonstrates the validity and efficiency of the present element. Copyright © 2000 John Wiley & Sons, Ltd.     

An Assumed Strain Triangular Solid Element for Efficient Analysis of Plates and Shells with Finite Rotation

A simple triangular solid shell element formulation is developed for efficient analysis of plates and shells undergoing finite rotations. The kinematics of the present solid shell element formulation is purely vectorial with only three translational degrees of freedom per node. Accordingly, the kinematics of deformation is free of the limitation of small angle increments, and thus the formulation allows large load increments in the analysis of finite rotation. An assumed strain field is carefully selected to alleviate the locking effect without triggering undesirable spurious kinematic modes. In addition, the curved surface of shell structures is modeled with flat facet elements to obviate the membrane locking effect. Various numerical examples demonstrate the efficiency and accuracy of the present element formulation for the analysis of plates and shells undergoing finite rotation. The present formulation is attractive in that only three points are needed for numerical integration over an element.     

A lanczos-based technique for exact vibration analysis of skeletal structures

This paper presents and discusses a Lanczos-based eigensolution technique for evaluating the natural frequencies and modes from frequency-dependent eigenproblems in structural dynamics. The new solution technique is used in conjunction with a mixed finite element modelling procedure which utilizes both the polynomial and frequency-dependent displacement fields in formulating the system matrices. The method is well suited to the solution of large-scale problems. The new solution methodology presented here is based on the ability to evaluate a specific set of parameterized non-linear eigenvalue curves at given values of the parameter through an implicitly restarted Lanczos technique. Numerical examples illustrate that the implicitly restarted Lanczos method with secant interpolation accurately evaluates the exact natural frequencies and modes of the non-linear eigenproblem and verifies that the new eigensolution technique coupled with the mixed finite element modelling procedure is more accurate than the conventional finite element models. In addition, the eigenvalue technique presented here is shown to be far more computationally efficient on large-scale problems than the determinant search techniques traditionally employed for solving exact vibration problems.     

Simple and efficient shear flexible two-node arch/beam and four-node cylindrical shell/plate finite elements

Several simple and accurate C° two-node arch/beam and four-node cylindrical shell/plate finite elements are presented in this paper. The formulation used here is based on the refined theory of thick cylindrical shells and the quasi-conforming element technique. Unlike most C° elements, the element stiffness matrix presented here is given explicitly. In spite of their simplicity, these C° finite elements posseses linear bending strains and are free from the deficiencies existing in curved C° elements such as shear and membrane locking, spurious kinematic modes and numerical ill-conditioning. These finite elements are valid not only for thick/thin beams and plates, but also for arches/straight beams and cylindrical shells/plates. Furthermore, these C° elements can automatically reduce to the corresponding C¹ beam and plate elements and give the C° beam element obtained by the reduced integration as a special case. Several numerical examples indicate that the simple two-node arch/beam and four-node cylindrical shell/plate elements given in this paper are superior to the existing C° elements with the same element degrees of freedom. Only the formulation of the rectangular cylindrical shell and plate element is presented in this paper. The formulation of an arbitrarily quadrilateral plate element will be presented in a follow-up paper³².     

A sixteen node shell element with a matrix stabilization scheme

A sixteen node shell element is developed using a matrix stabilization scheme based on the Hellinger-Reissner principle with independent strain. Initially the assumed independent strain is divided into a lower order part and a higher order part. The stiffness matrix corresponding to the lower order assumed strain is equivalent to the stiffness matrix of the assumed displacement model element with the reduced integration scheme. The spurious kinematic modes of the element are suppressed by introducing a stabilization matrix associated with a judiciously chosen set of higher order assumed strain fields. Numerical results show that this element is free of locking even for very thin plates and shells.     

A generalized layerwise finite element for multi-layer damping treatments

This paper presents a 4-node facet type quadrangular shell finite element, based on a layerwise theory, developed for dynamic modelling of laminated structures with viscoelastic damping layers. The bending stiffness of the facet shell element is based on the Reissner–Mindlin assumptions and the plate theory is enriched with a shear locking protection adopting the MITC approach. The membrane component is corrected by using incompatible quadratic modes and the drilling degrees of freedom are introduced through a fictitious stiffness stabilization matrix. Linear static tests, using several pathological tests, showed good and convergent results. Dynamic analysis evaluation is provided by using two eigenproblems with exact analytical solution, as well as a conical sandwich shell with a closed-form analytical solution and a semi-analytical ring finite element solution. The applicability of the proposed finite element to viscoelastic core sandwich plates is assessed through experimental validation.     

A hybrid stress ANS solid‐shell element and its generalization for smart structure modelling. Part I—solid‐shell element formulation

In the recent years, solid‐shell finite element models which possess no rotational degrees of freedom and applicable to thin plate/shell analyses have attracted considerable attention. Development of these elements are not straightforward. Shear, membrane, trapezoidal, thickness and dilatational lockings must been visioned. In this part of this paper, a novel eight‐node solid‐shell element is proposed. To resolve the shear and trapezoidal lockings, the assumed natural strain (ANS) method is resorted to. The hybrid‐stress formulation is employed to rectify the thickness and dilatational locking. The element is computationally more efficient than the conventional hybrid elements by adopting orthogonal‐assumed stress modes and enforcing admissible sparsity in the flexibility matrix. Popular benchmark tests are exercised to illustrate the efficacy of the elements. In Part II of the paper, the element will be generalized for smart structure modelling by including the piezoelectric effect. Copyright © 2000 John Wiley & Sons, Ltd.     

Investigation of the Embedded Element Technique for ModellingWavy CNT Composites

This paper presents a comparison of different finite element approaches to modelling polymers reinforced with wavy, hollow fibres with the aim of predicting the effective elastic stiffness tensors of the composites. The waviness of the tubes is described by sinusoidal models with different amplitude-to-wavelength parameters. These volume elements are discretized by structured volume meshes onto which fibres in the form of independently meshed beam, shell or volume elements are superimposed. An embedded element technique is used to link the two sets of meshes. Reference solutions are obtained from conventional three-dimensional volume models of the same phase arrangements. Periodicity boundary conditions are applied in all cases and fibre volume fractions of up to a few percent are considered. The results indicate that embedded element techniques using shell elements for discretizing the fibres may provide an attractive combination of accuracy, computational cost and flexibility for modelling composites reinforced by arbitrarily, three-dimensionally curved nanotubes.     

Formulation and solution of the non-linear,damped eigenvalue problem for skeletal systems

This paper presents and discusses an Arnoldi-based eigensolution technique for evaluating the complex natural frequencies and mode shapes from frequency dependent quadratic eigenproblems associated with vibration analysis of damped structures. The new solution technique is used in conjunction with a mixed finite element modelling procedure which utilizes both the polynomial and frequency dependent displacement fields in formulating the system matrices. This modelling provides the ability to represent a frequency dependent damping matrix in vibration analysis of skeletal systems. The eigensolution methodology presented here is based upon the ability to evaluate a specific set of parametrized curves for the non-linear eigenvalue problem at given values of the parameter. Numerical examples illustrate that this method, used in conjunction with a secant interpolation, accurately evaluates the complex natural frequencies and modes of the quadratic non-linear eigenproblem and verifies that the new eigensolution technique coupled with the mixed finite element modelling procedure is more accurate than the conventional finite element models.     

Experimental measurements and finite element analysis of the coupled vibrational characteristics of piezoelectric shells

Piezoelectric plates can provide low-frequency transverse vibrational displacements and high-frequency planar vibrational displacements, which are usually uncoupled. However, piezoelectric shells can induce three-dimensional coupled vibrational displacements over a large frequency range. In this study, three-dimensional coupled vibrational characteristics of piezoelectric shells with free boundary conditions are investigated using three different experimental methods and finite element numerical modeling. For the experimental measurements, amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) is used to obtain resonant frequencies and radial, lateral, and angular mode shapes. This optical technique utilizes a real-time, full-field, non-contact optical system that measures both the natural frequency and corresponding vibration mode shape simultaneously. The second experimental technique used, laser Doppler vibrometry (LDV), is a pointwise displacement measurement method that determines the resonant frequencies of the piezoelectric shell. An impedance analyzer is also used to determine the resonant frequencies of the piezoelectric shell. The experimental results of the resonant frequencies and mode shapes for the piezoelectric shell are verified with a numerical finite element model. Excellent agreement between the experimental and numerical results is found for the three-dimensional coupled vibrational characteristics of the piezoelectric shell. It is noted in this study that there is no coupled phenomenon at low frequencies over which radial modes dominate. However, three-dimensional coupled vibrational modes do occur at high resonant frequencies over which lateral or angular modes dominate.     

Curved beam elements with penalty relaxation

Shallowly curved beam elements, including shear deformation and rotary inertia effects, are derived from Hamilton's variational principle. Different degree polynomials, labelled ‘anisoparametric’, are used to interpolate the kinematic variables, instead of uniform interpolations as in the conventional isoparametric procedure. This approach yields a correct representation of the bending strain and, importantly, the membrane and transverse shear strains. Consequently, the severe shortcomings of the exactly integrated isoparametric elements, characterized by excessively stiff solutions in the thin regime (a phenomenon often referred to as membrane and shear locking), are overcome. Uniform (isoparametric-like) nodal patterns are achieved by explicitly enforcing higher-degree penalty modes in the membrane and shear strains. This procedure preserves the compatibility of the kinematic field and the capability of the element to move rigidly without straining. Exact quadratures are used on all element matrices, producing a correct rank stiffness matrix, a consistent load vector and a consistent mass matrix. The elements suffer no limitations over the entire theoretical range of the slenderness ratio. For further enhancement and, particularly, in coarse-mesh situations, an effective relaxation of penalty constraints at the local element level is introduced. This technique ensures a well-conditioned stiffness matrix. Although the element penalty constraints are relaxed, the corresponding global structure constraints are enforced as is required by the analytic theory. Particular attention is given to the simplest element—a two-node, six degree-of-freedom beam in which all strains are constant. Solutions to static and free vibration arch and ring problems are presented, demonstrating the exceptional modelling capabilities of this element.     

Incompressibility at large strains and finite-element implementation

Summary. This contribution is concerned with the consideration of material incompressibility at large strains and proposes various methods for the enforcement of the corresponding constraint into finite-rotation shell models. The incompressibility condition can be expressed in terms of displacement as well as strain variables and is considered by means of three different procedures in the numerical implementation. As kinematic hypothesis a quadratic assumption with respect to the thickness coordinate is used in which the corresponding directors are decomposed into two stretch parameters and a common inextensible unit vector. Various constitutive laws holding for incompressible isotropic hyperelasticity are considered and directly coupled with shell equations through a numerical thickness integration. A 4-node isoparametric shell element is developed parameterizing the inextensible shell director in terms of rotation variables in the framework of an up-dated rotation formulation. Finally, several examples are analysed to identify the most effective procedure for modelling isochoric deformations in thin-walled structures.In memory of Y. Baar, who passed away on August 30, 2002.     

Coupled mechanics and finite element for non‐linear laminated piezoelectric shallow shells undergoing large displacements and rotations

A theoretical framework is presented for analysing the coupled non‐linear response of shallow doubly curved adaptive laminated piezoelectric shells undergoing large displacements and rotations. The formulated mechanics incorporate coupling between in‐plane and flexural stiffness terms due to geometric curvature, coupling between mechanical and electric fields, and encompass geometric non‐linearity effects due to large displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear co‐ordinates and are combined with the kinematic assumptions of a mixed‐field shear‐layerwise shell laminate theory. Based on the above formulation, a finite element methodology together with an incremental‐iterative technique, based on Newton–Raphson method is formulated. An eight‐node coupled non‐linear shell element is also developed. Various evaluation cases on laminated curved beams and cylindrical panels illustrate the capability of the shell finite element to predict the complex non‐linear behaviour of active shell structures including buckling, which is not captured by linear shell models. The numerical results also show the inherent capability of piezoelectric shell structures to actively induce large displacements through piezoelectric actuators, by jumping between multiple equilibrium states. Copyright © 2005 John Wiley & Sons, Ltd.     

A computational model for analysing interactive buckling and delamination growth in composite structures

In this paper, a unified method is presented: (i) to model delaminated stiffened laminated composite shells; (ii) for synthesising accurate multiple post-buckling solution paths under compressive loading; and (iii) for predicting delamination growth. A multi-domain modelling technique is used for modelling the delaminated stiffened shell structures. Error-free geometrically nonlinear element formulations — a 2-noded curved stiffener element (BEAM2) and a 3-noded shell element (SHELL3) — are used for the finite element analysis. An accurate and simple automated solution strategy based on Newton type iterations is used for predicting the general geometrically nonlinear and postbuckling behaviour of structures. A simple method derived from the 3-dimensionalJ-integral is used for computing the pointwise energy release rate at the delamination front in the plate/shell models. Finally, the influence of post-buckling structural behaviour and the delamination growth on each other has been demonstrated.     

A stabilization procedure for the quadrilateral plate element with one-point quadrature

A stabilization procedure is developed for controlling the kinematic modes of the four-node, bilinear quadrilateral element when single-point quadrature is used. These kinematic modes manifest themselves by spatial oscillations or singularity of the total stiffness. In this stabilization procedure, additional generalized strains are defined which are activated by the kinematic modes; these generalized modes are furthermore not activated by rigid body motions regardless of the shape of the quadrilateral. By using a scaling law developed in an earlier paper, the stabilization parameters are defined so they do not adversely affect the element's performance. Several problems which are subject to kinematic modes are presented to illustrate the performance of this stabilization procedure for linear problems.     

复合材料大变形任意加筋壳单元

构造了用于复合材料偏心加筋壳形结构大变形分析的任意加筋壳单元。在此模型中,肋骨连同壳的整体被视为一个单元偏心加筋壳单元。肋骨可放在壳单元内的任意位置和任意方向。所构造单元的特点是在网格划分时,可不必考虑肋骨的位置,这就给网格划分带来了很大的灵活性。在壳和肋骨的方程中,引用Von-Karman大变形理论计及几何非线性的影响,按照Mindlin-Reissner一阶剪切变形理论考虑横向剪切变形。