A CO-ROTATIONAL FORMULATION FOR 2-D CONTINUA INCLUDING INCOMPATIBLE MODES

The co-rotational finite element method is well known for decomposing the motion of an element into a rigid body motion and a strain-producing deformation. A key feature is the definition of the rotating frame and, partially for this reason, the method has almost exclusively been applied to beams and shells. A novel feature of the current paper is that the method is applied to continua—in the current case, two-dimensional. The main motivation is to allow the analyst to quickly introduce the latest and best linear elements into a non-linear context. To this end, in the current work, a set of ‘incompatible modes’ or ‘enhanced strains’ is added to the conventional four-noded elements. While the main body of the paper considers small strains, as a further novel aspect, it later applies the co-rotational method to problems with large strains (here via hyperelasticity) and, to this end, establishes a link between the co-rotational technique and a Biot stress formulation.     

A geodesic beam finite element for use in the-static analysis of reinforced circular cylindrical shells

The development of a geodesic beam finite element for use with a specific circular cylindrical shell finite element in the analysis of reinforced circular cylinders is described. The basic strain and curvature change equations are given and, from these, three versions of the geodesic beam element are developed. Two of the beams have nodal degress-of-freedom identical with the shell element. They differ in the treatment of the terms relating to rotation about the principal normal. The first version ignores this parameter but, under certain circumstances, the stiffness matrix contains terms which can contribute to the strain energy under arbitrary rigid body movement of the beam. This deficiency is removed by applying an aspect of Koiter's theory which is used to transform the curvature change equations. The introduction of additional rotational degress-of-freedom, at the end nodes of the beam element, produces a variation which is capable of accurately representing and transmitting in-surface bending effects to an adjoining beam element. Numerical evaluation successfully compares finite element solutions to basic problems for straight, circular and helical beams with theoretical strain energy solutions. Finally the beam is used in conjunction with a shell element to analyse an infinitely long circular cylinder, reinforced with equispaced rings, subjected to internal pressure.     

摇摆结构刚体模型研究综述

合理的理论模型对于结构分析至关重要,摇摆结构理论研究中应用最为广泛的模型是摇摆刚体模型。该文对摇摆结构刚体模型进行了较为系统考察与总结,对摇摆刚体模型的研究起源与研究现状进行了介绍,针对经典摇摆刚体模型、其他典型摇摆刚体模型、相关试验研究、有限元模拟及其在结构体系中的应用进行了探讨,指出了已有模型的优点与局限性,提出摇摆结构刚体模型未来研究的关键问题,为建立更完善实用的摇摆结构刚体模型及应用于摇摆结构分析提供了参考。     

Elastic cylinder with helicoidal orthotropy: Theory and applications

The elasticity equations are derived for a helicoidally symmetric cylinder. An isotropic cylinder in the center is wrapped by a thick layer of a material with helicoidal anisotropy formed by spiraling of fibers around the core. Such structures are seen in springs, armored cables, trusses, composites, and biostructures. Structure under axial loading and pure bending are considered. The analytical expressions for displacements are obtained from equations of three-dimensional elasticity. The results are verified numerically via the finite element method. The stiffness matrix for an equivalent simplified model that links the displacements of the ends with the applied force and momentum is formulated. A coupling effect is revealed: the bar twists and elongates when axially loaded.The developed technique is used to explain an evolution of structure in nature. A pine trunk with spiralling grain is investigated from an optimization of a mechanical construction viewpoint. We model the trunk as an anisotropic cylinder with helicoidal symmetry and compute the displacements and stresses using nonlinear finite element model. An optimized spiralling angle of the grains accounts for a composite failure, transverse deflection, and fluid transportation. We suggest a combined criterion that could explain this spiralling morphology.     

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.     

Simple Efficient Smart Finite Elements for the Analysis of Smart Composite Beams

This paper is concerned with the development of new simple 4-noded locking-alleviated smart finite elements for modeling the smart composite beams. The exact solutions for the static responses of the overall smart composite beams are also derived for authenticating the new smart finite elements. The overall smart composite beam is composed of a laminated substrate conventional composite beam, and a piezoelectric layer attached at the top surface of the substrate beam. The piezoelectric layer acts as the actuator layer of the smart beam. Alternate finite element models of the beams, based on an “equivalent single layer high order shear deformation theory”, and a “layer-wise high order shear deformation theory”, are also derived for the purpose of investigating the required number of elements across the thickness of the overall smart composite beams. Several cross-ply substrate beams are considered for presenting the results. The responses computed by the present new “smart finite element model” excellently match with those obtained by the exact solutions. The new smart finite elements developed here reveal that the development of finite element models of smart composite beams does not require the use of conventional first order or high order or layer-wise shear deformation theories of beams. Instead, the use of the presently developed locking-free 4-node elements based on conventional linear piezo-elasticity is sufficient.     

A Cosserat point element (CPE) for the numerical solution of transient large planar motions of elastic–plastic and elastic–viscoplastic beams

The objective of this paper is to develop constitutive equations of a Cosserat point element (CPE) for the numerical solution of transient large planar motions of elastic–plastic and elastic–viscoplastic beams with rigid cross-sections. Specifically, attention is limited to response of a material with constant yield strength. A yield function is proposed which couples the inelastic responses of tension and shear. Another yield function is proposed for bending which depends on a hardening variable that models motion of the elastic–plastic boundary in the beam’s cross-section. Evolution equations are proposed for elastic strains and the hardening variable and an overstress-type formulation is used for elastic–viscoplastic response. In contrast, with standard finite element approaches the CPE model needs no integration through the element region. Also, an implicit scheme is developed to integrate the evolution equations without iteration. Examples of transient large motions of beams, which are impulsively loaded, indicate that the CPE produces reasonably accurate response relative results in the literature and full three-dimensional calculations using ABAQUS.     

Large reference displacement analysis of composite plates part I: Finite element formulation

This investigation concerns itself with the dynamic analysis of thin, laminated composite plates consisting of layers of orthotropic laminae that undergo large arbitrary rigid body displacements and small elastic deformations. A non-linear finite element formulation is developed which utilizes the assumption that the bonds between the laminae are infinitesimally thin and shear non-deformable. Using the expressions for the kinetic and strain energies, the lamina mass and stiffness matrices are identified. The non-linear mass matrix of the lamina is expressed in terms of a set of invariants that depend on the assumed displacement field. By summing the kinetic and strain energies of the laminae of an element, the element mass and stiffness matrix can be defined in terms of the set of element invariants. It is shown that the element invariants can be expressed explicitly in terms of the invariants of its laminae. By assembling the finite elements of the deformable body, the body invariants can be identified and expressed explicitly in terms of the invariants of the laminae of its elements. In the dynamic formulation presented in this paper, the shape functions of the laminae are assumed to have rigid body modes that need to describe only large rigid body translations. The computer implementation and the use of the formulation developed in this investigation in multibody dynamics are discussed in the second part of this paper.     

Explicit finite element perfectly matched layer for transient three‐dimensional elastic waves

The use of a perfectly matched layer (PML) model is an efficient approach toward the bounded‐domain modelling of wave propagation on unbounded domains. This paper formulates a three‐dimensional PML for elastic waves by building upon previous work by the author and implements it in a displacement‐based finite element setting. The novel contribution of this paper over the previous work is in making this finite element implementation suitable for explicit time integration, thus making it practicable for use in large‐scale three‐dimensional dynamic analyses. An efficient method of calculating the strain terms in the PML is developed in order to take advantage of the lack of the overhead of solving equations at each time step. The PML formulation is studied and validated first for a semi‐infinite bar and then for the classical soil–structure interaction problems of a square flexible footing on a (i) half‐space, (ii) layer on a half‐space and (iii) layer on a rigid base. Numerical results for these problems demonstrate that the PML models produce highly accurate results with small bounded domains and at low computational cost and that these models are long‐time stable, with critical time step sizes similar to those of corresponding fully elastic models. Copyright © 2008 John Wiley & Sons, Ltd.     

Three-dimensional absolute nodal co-ordinate formulation: plate problem

In this investigation, an absolute nodal co-ordinate dynamic formulation is developed for the large deformations and rotations of three-dimensional plate elements. In this formulation, no infinitesimal or finite rotations are used as nodal co-ordinates, instead global displacements and slopes are used as the plate coordinates. Using this interpretation of the plate coordinates the new method does not require the use of co-ordinate transformation to define the global inertia properties of the plates. The resulting mass matrix is the same constant matrix that appears in linear structural dynamics. The stiffness matrix, on the other hand, is a non-linear function of the nodal co-ordinates of the plate even in the case of a linear elastic problem. It is demonstrated in this paper that, unlike the incremental finite element formulations, the proposed method leads to an exact modelling of the rigid body inertia when the plate element moves as a rigid body. It is also demonstrated that by using the proposed method the conventional plate element shape function has a complete set of rigid body modes that can describe an exact arbitrary rigid body displacement. Using this fact, plate elements in the proposed new formulation can be considered as isoparametric elements. As a consequence, an arbitrary rigid body motion of the element results in zero strain. © 1997 John Wiley & Sons, Ltd.     

Improvement of PUFEM for the numerical solution of high‐frequency elastic wave scattering on unstructured triangular mesh grids

The purpose of this paper, which builds on previous work (Int. J. Numer. Meth. Engng 2009; 77 :1646–1669), is to improve a numerical scheme based on the partition of unity finite element method (PUFEM) for the solution of the time harmonic elastic wave equations. The approach consists to approximate the displacement field by the standard finite element shape functions, enriched locally by superimposing pressure (P) and shear (S) plane waves. The aim is to accurately model two‐dimensional elastic wave problems on relatively coarse mesh grids, capable of containing many wavelengths per nodal spacing, for wide ranges of frequencies. This allows us to relax the traditional requirement of about 10 nodal points per S wavelength. In this work, an exact integration scheme for the linear triangular finite element is developed to evaluate the oscillatory integrals arising from the use of the PUFEM. The main contribution here consists in developing an explicit closed‐form solution for two‐dimensional wave‐based integrals, when the phase variation is linear in the local coordinate element system. The evaluation of the element mass matrix is performed from appropriate edge integrals. All other element matrices, obtained by adequate splitting of the element stress tensor matrix, are simply deduced from the element mass matrix entries. The results show clearly that the proposed integration scheme evaluates accurately the entries of the global matrix with drastic reduction of the computational time. Numerical tests dealing with the scattering of S elastic plane waves by a circular rigid body show that, for the same discretization level, it is possible to improve the accuracy by using large elements associated with high numbers of approximating plane waves rather than using small elements with less plane waves. However, this increases the conditioning and the fill‐in of the global matrix. At high frequency, it is even possible to push the number of degrees of freedom per S wavelength under 2 and still achieve good accuracy. Finally, some remarks on the choice of the numbers of P and S plane waves leading to better accuracy and conditioning are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

Partitioned analysis of flexible multibody systems using filtered linear finite element deformational modes

This work presents a partitioned finite element formulation for flexible multibody systems, based on the floating frame (FF) approach, under the assumption of small deformations but arbitrarily large rotations of the bodies. In classical FF of reference methods, deformational modes are normally computed by modal analysis. In this approach, free‐floating modes are eliminated from the linear model using projection techniques and substituted by a complete set of non‐linear finite rotations. In this way, all deformational modes are retained, and no modal selection is needed. The main difference between this work and a classical FF of reference formulation is an algebraic separation of pure deformational modes from rigid‐body motions. The proposed methodology presents the following advantages. First, the position and orientation of the FF has no restriction and can be freely located in the body with identical results. Second, the formulation uses only the linear finite element matrices of a classical vibration problem; hence, they can be easily obtained from linear FEM packages. Third, no selection of modes is needed, all deformational modes are retained through the filtering process. And finally, thanks to the use of localized Lagrangian multipliers (LLM), a partitioned system is obtained that can be solved iteratively and in a distributed manner by available scalable solvers. Copyright © 2014 John Wiley & Sons, Ltd.     

Finite element of slender beams in finite transformations: a geometrically exact approach

This article is devoted to the modelling of thin beams undergoing finite deformations essentially due to bending and torsion and to their numerical resolution by the finite element method. The solution proposed here differs from the approaches usually implemented to treat thin beams, as it can be qualified as ‘geometrically exact’. Two numerical models are proposed. The first one is a non‐linear Euler–Bernoulli model while the second one is a non‐linear Rayleigh model. The finite element method is tested on several numerical examples in statics and dynamics, and validated through comparison with analytical solutions, experimental observations and the geometrically exact approach of the Reissner beam theory initiated by Simo. The numerical result shows that this approach is a good alternative to the modelling of non‐linear beams, especially in statics. Copyright © 2003 John Wiley & Sons, Ltd.     

组合梁界面滑移的计算分析

钢-混凝土组合梁(以下简称组合梁)的界面滑移总是存在的,滑移的存在会降低组合梁的组合作用和刚度,增大挠度,要计算组合梁界面的滑移及挠度,对于简支梁在简单荷载情况下,还可得到解析解,但对于连续梁要得到解析解是十分困难的,另外简支梁的解析解十分冗长,实际运用十分不便。用有限元法计算组合梁的滑移和挠度将是很有效的,不受荷载及支撑条件限制,而有限元法的关键是单元刚度矩阵,该文用虚功原理推导了组合梁的单元刚度矩阵,并用自编的有限元程序对组合梁的滑移和挠度进行了计算,在简支情况下与解析解进行了对比和验证,误差很小,在1%以内。该文推导的单元刚度矩阵可用于小型的自编有限元软件,为快速经济地解决相关的大量实际工程问题奠定了基础。     

A BEM formulation based on Reissner's hypothesis for analysing the coupled stretching–bending problem of building floor structures

In this work, a plate bending formulation of the boundary element method (BEM) based on the Reissner's hypothesis to perform linear analysis of plates reinforced by rectangular beams is extended to consider the beams not displayed over their middle surface. Therefore eccentricity effects are taken into account. The building floor structure is modelled as a stiffened plate which is treated as a single body without dividing it into beam and plate elements. Moreover the equilibrium and compatibility conditions are automatically imposed by the integral equations. In the proposed model the final system of equation is obtained by coupling the bending problem to the stretching problem. Besides, in order to reduce the number of degrees of freedom, both the displacements and tractions are approximated along the beam width, leading to a model where the values are defined on the beams axis. In order to validate the proposed formulation, the numerical results are compared to a well know finite element code.     

Formulation of a triangular finite element with an embedded interface via isoparametric mapping

 This paper presents the formulation of a triangular finite element with an embedded interface, designed for the simulation of discrete crack propagation processes. The element is developed within a displacement-based framework. Linear interpolation of the displacement discontinuities along the internal interface is assumed in order to ensure compatibility across inter-element boundaries. The proper representation of the rigid body motions and the solvability of the discretised version of the mechanical problem in point are specifically addressed. Finally, the element performance is illustrated through comparison with some alternative proposals. Received 24 January 2001     

压电复合梁热机电耦合有限元模型

压电材料应用于航天结构形状或振动控制时,可能会受到热场、力场和电场的共同作用。为分析处于热场、力场和电场共同作用下的压电复合结构,文中基于高阶剪切变形理论、高阶电势模型和线性温度分布假设,利用虚功原理建立了压电复合梁结构的热-机-电耦合有限元模型。该模型可应用于热机电耦合压电复合结构的形状与振动控制研究。利用本文模型对压电双晶片梁、压电复合悬臂梁进行了数值仿真,仿真结果与文献给出的理论结果和实验值吻合良好,表明本文模型是正确有效的。     

A finite element formulation of three-dimensional contact problems with slip and friction

The paper describes a special finite element for three-dimensional, large displacement analysis of contact problems with slip and friction. This element may be used to model contact between several finite element bodies or contact between a finite element body and a flexible or rigid geometrical surface fixed in space or moving with time. The contact formulation is based on the concept of a spring-supported, moving disk that transfers normal contact forces and Coulomb friction forces. The contact surface has a finite, prescribed boundary.The contact element has been incorporated into the general-purpose, nonlinear, finite element program FENRIS. Three examples of its application are described in the paper.J. W. Simons was previously NTNF Fellow, Division of Technology, Trondheim, Norway     

A BEM formulation for linear bending analysis of plates reinforced by beams considering different materials

In this work, a boundary element method (BEM) formulation to perform linear bending analysis of building floor structures where slabs and beams can be defined with different materials is presented. The proposed formulation is based on Kirchhoff's hypothesis, the building floor being modelled by a zoned plate, where the beams are treated as thin sub-regions with larger rigidities. This composed structure is treated as a single body, the equilibrium and compatibility conditions being automatically taken into account. In the final integral equation, the tractions are eliminated along the interfaces, therefore reducing the number of degrees of freedom. The displacements are approximated along the beam cross-section, leading to a model where the values remain defined on the beam skeleton line instead of their boundaries. The accuracy of the proposed model is shown by comparing the numerical results with a well-known finite element code.