Non‐linear spatial Timoshenko beam element with curvature interpolation

The paper presents a spatial Timoshenko beam element with a total Lagrangian formulation. The element is based on curvature interpolation that is independent of the rigid‐body motion of the beam element and simplifies the formulation. The section response is derived from plane section kinematics. A two‐node beam element with constant curvature is relatively simple to formulate and exhibits excellent numerical convergence. The formulation is extended to N‐node elements with polynomial curvature interpolation. Models with moderate discretization yield results of sufficient accuracy with a small number of iterations at each load step. Generalized second‐order stress resultants are identified and the section response takes into account non‐linear material behaviour. Green–Lagrange strains are expressed in terms of section curvature and shear distortion, whose first and second variations are functions of node displacements and rotations. A symmetric tangent stiffness matrix is derived by consistent linearization and an iterative acceleration method is used to improve numerical convergence for hyperelastic materials. The comparison of analytical results with numerical simulations in the literature demonstrates the consistency, accuracy and superior numerical performance of the proposed element. Copyright © 2001 John Wiley & Sons, Ltd.     

Thin‐walled closed box beam element for static and dynamic analysis

A new displacement‐based finite element is developed for thin‐walled box beams. Unlike the existing elements, dealing with either static problems alone or dynamic problems only with the additional consideration of warping, the present element is useful for both static and dynamic analyses with the consideration of coupled deformation of torsion, warping and distortion. We propose to use a statically admissible in‐plane displacement field for the element stiffness matrix and a kinematically compatible displacement field for the mass matrix so that the present element is useful for a wide range of beam width‐to‐height ratios. The axial variation of cross‐sectional deformation measures is approximated by C⁰ continuous interpolation functions. Numerical examples are considered to confirm the validity of the present element. Copyright © 1999 John Wiley & Sons, Ltd.     

A 3D beam‐to‐beam contact finite element for coupled electric–mechanical fields

In this paper the formulation of an electric–mechanical beam‐to‐beam contact element is presented. Beams with circular cross‐sections are assumed to get in contact in a point‐wise manner and with clean metallic surfaces. The voltage distribution is influenced by the contact mechanics, since the current flow is constricted to small contacting spots. Therefore, the solution is governed by the contacting areas and hence by the contact forces. As a consequence the problem is semi‐coupled with the mechanical field influencing the electric one. The electric–mechanical contact constraints are enforced with the penalty method within the finite element technique. The virtual work equations for the mechanical and electric fields are written and consistently linearized to achieve a good level of computational efficiency with the finite element method. The set of equations is solved with a monolithic approach. Copyright © 2005 John Wiley & Sons, Ltd.     

A three‐noded shear‐flexible curved beam element based on coupled displacement field interpolations

An efficient shear‐flexible three‐noded curved beam element is proposed herein. The shear flexibility is based on Timoshenko beam theory and the element has three degrees of freedom, viz., tangential displacement (u), radial displacement (w) and the section‐rotation (θ). A quartic polynomial interpolation for flexural rotation ψ is assumed a priori. Making use of the physical composition of θ in terms of ψ and u, a novel way of deriving the polynomial interpolations for u and w is presented, by solving force‐moment and moment‐shear equilibrium equations simultaneously. The field interpolation for θ is then constructed from that of ψ and u. The procedure leads to high‐order polynomial field interpolations which share some of the generalized degrees of freedom, by means of coefficients involving material and geometric properties of the element. When applied to a straight Euler–Bernoulli beam, all the coupled coefficients vanish and the formulation reduces to classical quintic‐in‐w and quadratic‐in‐u element, with u, w, and ?w/?x as degrees of freedom. The element is totally devoid of membrane and shear locking phenomena. The formulation presents an efficient utilization of the nine generalized degrees of freedom available for the polynomial interpolation of field variables for a three‐noded curved beam element. Numerical examples on static and free vibration analyses demonstrate the efficacy and locking‐free property of the element. Copyright © 2001 John Wiley & Sons, Ltd.     

A spatially curved‐beam element with warping and Wagner effects

This paper presents a new spatially curved‐beam element with warping and Wagner effects that can be used for the non‐linear large displacement analysis of members that are curved in space. The non‐linear behaviour of members curved in space shows that the Wagner effects are substantial in the large twist rotation analysis. Most existing finite beam element models, such as ABAQUS and ANSYS cannot predict the non‐linear large displacement response of members curved in space correctly because the Wagner effects, viz. the Wagner moment and the corresponding finite strain terms, have not been considered in these finite beam elements. As a consequence, these finite beam elements do not provide correct predictions for the out‐of‐plane buckling and postbuckling behaviour of arches as well. In this paper, the symmetric tangent stiffness matrix has been derived based on the finite rotations parameterized by the conventional displacements. The warping and Wagner effects: both the Wagner moment and the corresponding finite strain terms and their constitutive relationship, are included in the spatially curved‐beam element. Two components of the initial curvature, the initial twist and their interactions with the displacements are also considered in the spatially curved‐beam element. This ensures that the large twist rotation analysis for the members curved in space is accurate. Comparisons with existing experimental, analytical and numerical results show that the spatially curved‐beam element is accurate and efficient for the non‐linear elastic analysis of curved members, buckling and postbuckling analysis of arches, and in its ability to predict large deflections and twist rotations in more arbitrarily curved members. Copyright © 2005 John Wiley & Sons, Ltd.     

Explicit finite element artificial boundary scheme for transient scalar waves in two‐dimensional unbounded waveguide

To simulate the transient scalar wave propagation in a two‐dimensional unbounded waveguide, an explicit finite element artificial boundary scheme is proposed, which couples the standard dynamic finite element method for complex near field and a high‐order accurate artificial boundary condition (ABC) for simple far field. An exact dynamic‐stiffness ABC that is global in space and time is constructed. A temporal localization method is developed, which consists of the rational function approximation in the frequency domain and the auxiliary variable realization into time domain. This method is applied to the dynamic‐stiffness ABC to result in a high‐order accurate ABC that is local in time but global in space. By discretizing the high‐order accurate ABC along artificial boundary and coupling the result with the standard lumped‐mass finite element equation of near field, a coupled dynamic equation is obtained, which is a symmetric system of purely second‐order ordinary differential equations in time with the diagonal mass and non‐diagonal damping matrices. A new explicit time integration algorithm in structural dynamics is used to solve this equation. Numerical examples are given to demonstrate the effectiveness of the proposed scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

Eight‐node Reissner–Mindlin plate element based on boundary interpolation using Timoshenko beam function

A new eight‐node Reissner–Mindlin plate element is developed with a special interpolation within the element. This special interpolation is an extension of the element boundary interpolation that employs Timoshenko beam function for the boundary segment interpolation. The element function can effectively capture the structural behaviours of thick plates and achieve high precision in the analysis of thick plates. Patch tests and numerical investigations are conducted. It can be seen that the proposed element successfully passes all the patch tests. The results of the numerical investigation show that the proposed element is free of the shear locking phenomenon and possesses a higher accuracy in the analyses, as compared to the earlier research in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

The spectral element method for elastic wave equations—application to 2‐D and 3‐D seismic problems

A spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two‐ and three‐dimensional geometries. The computational domain is discretized into quadrangles, or hexahedra, defined with respect to a reference unit domain by an invertible local mapping. Inside each reference element, the numerical integration is based on the tensor‐product of a Gauss–Lobatto–Legendre 1‐D quadrature and the solution is expanded onto a discrete polynomial basis using Lagrange interpolants. As a result, the mass matrix is always diagonal, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy‐momentum conserving scheme that can be put into a classical explicit‐implicit predictor/multicorrector format. Long term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two‐dimensional problems obtained by other methods. The potentiality of the method is then illustrated by studying a simple three‐dimensional model. Very accurate modelling of Rayleigh wave propagation and surface diffraction is obtained at a low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three‐dimensional surface topographies. Copyright © 1999 John Wiley & Sons, Ltd.     

A rational elasto‐plastic spatially curved thin‐walled beam element

Torsion is one of the primary actions in members curved in space, and so an accurate spatially curved‐beam element needs to be able to predict the elasto‐plastic torsional behaviour of such members correctly. However, there are two major difficulties in most existing finite thin‐walled beam elements, such as in ABAQUS and ANSYS, which may lead to incorrect predictions of the elasto‐plastic behaviour of members curved in space. Firstly, the integration sample point scheme cannot capture the shear strain and stress information resulting from uniform torsion. Secondly, the higher‐order twists are ignored which leads to loss of the significant effects of Wagner moments on the large twist torsional behaviour. In addition, the initial geometric imperfections and residual stresses are significant for the elasto‐plastic behaviour of members curved in space. Many existing finite thin‐walled beam element models do not provide facilities to deal with initial geometric imperfections. Although ABAQUS and ANSYS have facilities for the input of residual stresses as initial stresses, they cannot describe the complicated distribution patterns of residual stresses in thin‐walled members. Furthermore, external loads and elastic restraints may be applied remote from shear centres or centroids. The effects of the load (and restraint) positions are important, but are not considered in many beam elements. This paper presents an elasto‐plastic spatially curved element with arbitrary thin‐walled cross‐sections that can correctly capture the uniform shear strain and stress information for integration, and includes initial geometric imperfections, residual stresses and the effects of the load and restraint positions. The element also includes elastic restraints and supports, which have to be modelled separately as spring elements in some other finite thin‐walled beam elements. Comparisons with existing experimental and analytical results show that the elasto‐plastic spatially curved‐beam element is accurate and efficient. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

Dynamic response of intrinsic continua for use in biological and molecular modeling: Explicit finite element formulation

An intrinsic beam formulation has recently appeared (AIAA J. 2003; 41 (6):1131–1137), which presents the three‐dimensional equations of motion governing spatial and temporal changes in a one‐dimensional continua's curvature, strain, rectilinear velocity, and angular velocity. The formulation would suggest several computational advantages over more‐traditional displacement‐based continua approaches: low‐order interpolation functions can describe generally curved and twisted continua configurations; inter‐element displacements, slopes, strains, and curvatures can be matched; and finite rotational variables and their complexities are absent. Here, we present a completed intrinsic continua finite element development and critical analysis, which follows from an earlier preliminary treatment as applied to carbon nanotubes (Int. J. Solids Struct. 2007; 44 :874–894). Modeling of nodal displacements and rotations are included. Explicit time stepping, with desired high‐frequency damping, is accomplished using an implementation of the generalized‐α method. Zero‐energy modes inherent in the formulation are also identified and rectified. Finally, we document very good agreement between results predicted with the intrinsic continua finite element simulator and results generated using more‐traditional finite element simulations. Copyright © 2009 John Wiley & Sons, Ltd.     

Quadratic NMS Mindlin‐plate‐bending element

The development of a robust and efficient quadratic Mindlin‐plate‐bending elements mainly by the use of non‐conforming displacement modes is presented in this paper. A brief review on the previous efforts to develop efficient non‐conforming Mindlin plate bending elements is also given. The behaviour of the newly proposed plate element is further improved by the combined use of nonconforming displacement modes, the selectively reduced integration scheme, and the assumed shear strain fields. Thus, the newly developed element has been designated as ‘NMS‐8P’. The improvement achieved may be attributable to the fact that the merits of these improvement techniques are merged in the formation of the new element in a complementary manner. The proposed 8‐node element passes the patch tests, does not show spurious mechanism, and does not produce shear locking phenomena even with distorted meshes. It is also shown that the element produces reliable solutions through numerical tests for standard benchmark problems. Copypright © 1999 John Wiley & Sons, Ltd.     

A rotation‐free corotational plane beam element for non‐linear analyses

This paper presents a plane beam element without rotational degrees of freedom that can be used for the analysis of non‐linear problems. The element is based on two main ideas. First, a corotational approach is adopted, which means that the kinematics of the element is decomposed into a rigid body motion part and a deformational part. Next, in the deformational part, the local nodal rotations are extrapolated as a function of the local displacements of the two nodes of the element and the first nodes to the left and right of the element. Six numerical applications are presented in order to assess the performance of the formulation. Copyright © 2007 John Wiley & Sons, Ltd.     

Director‐based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates

In the present work, a new director‐based finite element formulation for geometrically exact beams is proposed. The new beam finite element exhibits drastically improved numerical performance when compared with the previously developed director‐based formulations. This improvement is accomplished by adjusting the underlying variational beam formulation to the specific features of the director interpolation. In particular, the present approach does not rely on the assumption of an orthonormal director frame. The excellent performance of the new approach is illustrated with representative numerical examples. Copyright © 2013 John Wiley & Sons, Ltd.     

A new higher-order hybrid-mixed curved beam element

The purpose of this work is to show the successful use of nodeless degrees of freedom in developing a highly accurate, locking free hybrid-mixed C⁰ curved beam element. In the performance evaluation process of the present field-consistent higher-order element, the effect of field consistency and the role of higher-order interpolation on both displacement-type and hybrid-mixed-type elements are carefully examined. Several benchmark tests confirm the superior behaviour of the present element. © 1998 John Wiley & Sons, Ltd.     

A two‐noded locking–free shear flexible curved beam element

A new two‐noded shear flexible curved beam element which is impervious to membrane and shear locking is proposed herein. The element with three degrees of freedom at each node is based on curvilinear deep shell theory. Starting with a cubic polynomial representation for radial displacement (w), the displacement field for tangential displacement (u) and section rotation (θ) are determined by employing force‐moment and moment‐shear equilibrium equations. This results in polynomial displacement field whose coefficients are coupled by generalized degrees of freedom and material and geometric properties of the element. The procedure facilitates quartic polynomial representation for both u and θ for curved element configurations, which reduces to linear and quadratic polynomials for u and θ, respectively, for straight element configuration. These coupled polynomial coefficients do not give rise to any spurious constraints even in the extreme thin regimes, in which case, the present element exhibits excellent convergence to the classical thin beam solutions. This simple C⁰ element is validated for beam having straight/curved geometries over a wide range of slenderness ratios. The results indicates that performance of the element is much superior to other elements of the same class. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite element solution of free‐surface ship‐wave problems

An unstructured finite element solver to evaluate the ship‐wave problem is presented. The scheme uses a non‐structured finite element algorithm for the Euler or Navier–Stokes flow as for the free‐surface boundary problem. The incompressible flow equations are solved via a fractional step method whereas the non‐linear free‐surface equation is solved via a reference surface which allows fixed and moving meshes. A new non‐structured stabilized approximation is used to eliminate spurious numerical oscillations of the free surface. Copyright © 1999 John Wiley & Sons, Ltd.     

An energy‐momentum method for in‐plane geometrically exact Euler–Bernoulli beam dynamics

Nonlinear geometrically exact rod dynamics is of great interest in many areas of engineering. In recent years, the research was focused towards Timoshenko‐type rod theories where shearing is of importance. However, in many general model of mechanisms and spatial deformations, it is desirable to have a displacement‐only formulation, which brings us back to the classical Bernoulli beam. While it is well known for linear analysis, the Bernoulli beam is not as common in geometrically exact models of dynamics, especially when we want to incorporate the rotational inertia into the model. This paper is about the development of an energy‐momentum integration scheme for the geometrically exact Bernoulli‐type rod. We will show that the task is achievable and devise a general framework to do so. Copyright © 2014 John Wiley & Sons, Ltd.     

High‐order layerwise finite element for the damped free‐vibration response of thick composite and sandwich composite plates

A high‐order layerwise finite element methodology is presented, which enables prediction of the damped dynamic characteristics of thick composite and sandwich composite plates. The through‐thickness displacement field in each discrete layer of the laminate includes quadratic and cubic polynomial distributions of the in‐plane displacements, in addition to the linear approximations assumed by linear layerwise theories. Stiffness, mass and damping matrices are formulated from ply to structural level. Interlaminar shear stress compatibility conditions are imposed on the discrete layer matrices, leading to both size reduction and prediction of interlaminar shear stresses at the laminate interfaces. The C¹ continuous finite element implemented yields an element damping matrix in addition to element stiffness and mass matrices. Application cases include thick [0/90/0], [±θ]_S and [±θ] composite plates with interlaminar damping layers and sandwich plates with composite faces and foam core. In the latter case, modal frequencies and damping were also experimentally determined and compared with the finite element predictions. Copyright © 2008 John Wiley & Sons, Ltd.