Frame‐indifferent beam finite elements based upon the geometrically exact beam theory

This paper describes a new beam finite element formulation based upon the geometrically exact beam theory. In contrast to many previously proposed beam finite element formulations the present discretization approach retains the frame‐indifference (or objectivity) of the underlying beam theory. The space interpolation of rotational degrees‐of‐freedom is circumvented by the introduction of a reparameterization of the weak form corresponding to the equations of motion of the geometrically exact beam theory. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

A regularized force‐based beam element with a damage–plastic section constitutive law

A new beam finite element is presented, with a generalized section constitutive law based on damage mechanics and plasticity, to analyse the cyclic structural response of plane frames. Both displacement‐based and force‐based (FB) approaches are used and compared, to demonstrate the significant advantages of the FB formulation in the presence of material non‐linearity. In order to overcome the analytical problems and the pathological mesh dependency of the numerical response in the presence of strain‐softening post‐peak behaviour, a classical non‐local regularization procedure is adopted first, based on the integral definition of the associated variable governing the damaging evolution process. Subsequently, for the FB element a new simple regularization technique is proposed based on a selected integration procedure along the element length, which predefines the location of the Gauss points in the beam region, where the localization phenomena take place. As for the other computational aspects, an iterative element state determination is adopted for the FB formulation and a local predictor–corrector algorithm is used to solve the incremental evolution problems of the damage and plastic internal variables. Finally, some examples are shown on simple beams and frames, subjected to monotonically increasing and cyclic loading conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

Implementation of an ANCF beam finite element for dynamic response optimization of elastic manipulators

Theoretical and practical aspects of an absolute nodal coordinate formulation (ANCF) beam finite element implementation are considered in the context of dynamic transient response optimization of elastic manipulators. The proposed implementation is based on the introduction of new nodal degrees of freedom, which is achieved by an adequate nonlinear mapping between the original and new degrees of freedom. This approach preserves the mechanical properties of the ANCF beam, but converts it into a conventional finite element so that its nodal degrees of freedom are initially always equal to zero and never depend explicitly on the design variables. Consequently, the sensitivity analysis formulas can be derived in the usual manner, except that the introduced nonlinear mapping has to be taken into account. Moreover, the adjusted element can also be incorporated into general finite element analysis and optimization software in the conventional way. The introduced design variables are related to the cross-section of the beam, to the shape of the (possibly) skeletal structure of the manipulator and to the drive functions. The layered cross-section approach and the design element technique are utilized to parameterize the shape of individual elements and the whole structure. A family of implicit time integration methods is adopted for the response and sensitivity analysis. Based on this assumption, the corresponding sensitivity formulas are derived. Two numerical examples illustrate the performance of the proposed element implementation.     

Extended rotation‐free plate and beam elements with shear deformation effects

This paper describes a methodology for extending rotation‐free plate and beam elements to accounting for transverse shear deformation effects. The ingredients for the element formulation are a Hu–Washizu‐type mixed functional, a linear interpolation for the deflection and the shear angles over standard finite elements and a finite volume approach for computing the bending moments and the curvatures over a patch of elements. As a first application of the general procedure, we present an extension of the three‐noded rotation‐free basic plate triangle (BPT) originally developed for thin plate analysis to account for shear deformation effects of relevance for thick plates and composite‐laminated plates. The nodal deflection degrees of freedom (DOFs) of the original BPT element are enhanced with the two shear deformation angles. This allows to compute the bending and shear deformation energies leading to a simple triangular plate element with three DOFs per node (termed BPT+ element). For the thin plate case, the shear angles vanish and the element reproduces the good behaviour of the original thin BPT element. As a consequence the element is applicable to thick and thin plate situations without exhibiting shear locking effects. The numerical solution for the thick case can be found iteratively starting from the deflection values for the Kirchhoff theory using the original thin BPT element. A two‐noded rotation‐free beam element termed CCB+ applicable to slender and thick beams is derived as a particular case of the plate formulation. The examples presented show the robustness and accuracy of the BPT+ and the CCB+ elements for thick and thin plate and beam problems. Copyright © 2010 John Wiley & Sons, Ltd.     

A finite‐strain gradient‐inelastic beam theory and a corresponding force‐based frame element formulation

This paper extends the gradient‐inelastic (GI) beam theory, introduced by the authors to simulate material softening phenomena, to further account for geometric nonlinearities and formulates a corresponding force‐based (FB) frame element computational formulation. Geometric nonlinearities are considered via a rigorously derived finite‐strain beam formulation, which is shown to coincide with Reissner's geometrically nonlinear beam formulation. The resulting finite‐strain GI beam theory: (i) accounts for large strains and rotations, unlike the majority of geometrically nonlinear beam formulations used in structural modeling that consider small strains and moderate rotations; (ii) ensures spatial continuity and boundedness of the finite section strain field during material softening via the gradient nonlocality relations, eliminating strain singularities in beams with softening materials; and (iii) decouples the gradient nonlocality relations from the constitutive relations, allowing use of any material model. On the basis of the proposed finite‐strain GI beam theory, an exact FB frame element formulation is derived, which is particularly novel in that it: (a) expresses the compatibility relations in terms of total strains/displacements, as opposed to strain/displacement rates that introduce accumulated computational error during their numerical time integration, and (b) directly integrates the strain‐displacement equations via a composite two‐point integration method derived from a cubic Hermite interpolating polynomial to calculate the displacement field over the element length and, thus, address the coupling between equilibrium and strain‐displacement equations. This approach achieves high accuracy and mesh convergence rate and avoids polynomial interpolations of individual section fields, which often lead to instabilities with mesh refinements. The FB formulation is then integrated into a corotational framework and is used to study the response of structures, simultaneously accounting for geometric nonlinearities and material softening. The FB formulation is further extended to capture member buckling triggered by minor perturbations/imperfections of the initial member geometry.     

Explicit free‐floating beam element

A two‐node free‐floating beam element capable of undergoing arbitrary large displacements and finite rotations is presented in explicit form. The configuration of the beam in three‐dimensional space is represented by the global components of the position of the beam nodes and an associated set of convected base vectors (directors). The local constitutive stiffness is derived from the complementary energy of a set of six independent deformation modes, each corresponding to an equilibrium state of constant internal force or moment. The deformation modes are characterized by generalized strains, formed via scalar products of the element related vectors. This leads to a homogeneous quadratic strain definition in terms of the generalized displacements, whereby the elastic energy becomes at most bi‐quadratic. Additionally, the use of independent equilibrium modes to set up the element stiffness avoids interpolation of kinematic variables, resulting in a locking‐free formulation in terms of three explicit matrices. A set of classic benchmark examples illustrates excellent performance of the explicit beam element. Copyright © 2014 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.     

A nonlocal higher-order curved beam finite model including thickness stretching effect for bending analysis of curved nanobeams

In the present work, a finite element approach is developed for the static analysis of curved nanobeams using nonlocal elasticity beam theory based on Eringen formulation coupled with a higher-order shear deformation accounting for through-thickness stretching. The formulation is general in the sense that it can be used to compare the influence of different structural theories, through static and dynamic analyses of curved nanobeams. The governing equations derived here are solved introducing a 3-nodes beam element. The formulation is validated considering problems for which solutions are available. A comparative study is done here by different theories obtained through the formulation. The effects of various structural parameters such as thickness ratio, beam length, rise of the curved beam, loadings, boundary conditions, and nonlocal scale parameter are brought out on the static bending behaviors of curved nanobeams.     

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 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.     

A mixed finite element method for beam and frame problems

 In this work we consider solutions for the Euler-Bernoulli and Timoshenko theories of beams in which material behavior may be elastic or inelastic. The formulation relies on the integration of the local constitutive equation over the beam cross section to develop the relations for beam resultants. For this case we include axial, bending and shear effects. This permits consideration in a direct manner of elastic and inelastic behavior with or without shear deformation. A finite element solution method is presented from a three-field variational form based on an extension of the Hu–Washizu principle to permit inelastic material behavior. The approximation for beams uses equilibrium satisfying axial force and bending moments in each element combined with discontinuous strain approximations. Shear forces are computed as derivative of bending moment and, thus, also satisfy equilibrium. For quasi-static applications no interpolation is needed for the displacement fields, these are merely expressed in terms of nodal values. The development results in a straight forward, variationally consistent formulation which shares all the properties of so-called flexibility methods. Moreover, the approach leads to a shear deformable formulation which is free of locking effects – identical to the behavior of flexibility based elements. The advantages of the approach are illustrated with a few numerical examples. Dedicated to the memory of Prof. Mike Crisfield, for his cheerfulness and cooperation as a colleague and friend over many years.     

An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy–momentum conserving scheme in dynamics

We present in this paper a new finite element formulation of geometrically exact rod models in the three‐dimensional dynamic elastic range. The proposed formulation leads to an objective (or frame‐indifferent under superposed rigid body motions) approximation of the strain measures of the rod involving finite rotations of the director frame, in contrast with some existing formulations. This goal is accomplished through a direct finite element interpolation of the director fields defining the motion of the rod's cross‐section. Furthermore, the proposed framework allows the development of time‐stepping algorithms that preserve the conservation laws of the underlying continuum Hamiltonian system. The conservation laws of linear and angular momenta are inherited by construction, leading to an improved approximation of the rod's dynamics. Several numerical simulations are presented illustrating these properties. Copyright © 2002 John Wiley & Sons, Ltd.     

Free vibration analysis of arches using curved beam elements

The natural frequencies and mode shapes for the radial (in‐plane) bending vibrations of the uniform circular arches were investigated by means of the finite arch (curved beam) elements. Instead of the complicated explicit shape functions of the arch element given by the existing literature, the simple implicit shape functions associated with the tangential, radial (or normal) and rotational displacements of the arch element were derived and presented in matrix form. Based on the relationship between the nodal forces and the nodal displacements of a two‐node six‐degree‐of‐freedom arch element, the elemental stiffness matrix was derived, and based on the equation of kinetic energy and the implicit shape functions of an arch element the elemental consistent mass matrix with rotary inertia effect considered was obtained. Assembly of the foregoing elemental property matrices yields the overall stiffness and mass matrices of the complete curved beam. The standard techniques were used to determine the natural frequencies and mode shapes for the curved beam with various boundary conditions and subtended angles. In addition to the typical circular arches with constant curvatures, a hybrid beam constructed by using an arch segment connected with a straight beam segment at each of its two ends was also studied. For simplicity, a lumped mass model for the arch element was also presented. All numerical results were compared with the existing literature or those obtained from the finite element method based on the conventional straight beam element and good agreements were achieved. Copyright © 2003 John Wiley & Sons, Ltd.     

Enhanced mixed interpolation XFEM formulations for discontinuous Timoshenko beam and Mindlin‐Reissner plate

Shear locking is a major issue emerging in the computational formulation of beam and plate finite elements of minimal number of degrees of freedom as it leads to artificial overstiffening. In this paper, discontinuous Timoshenko beam and Mindlin‐Reissner plate elements are developed by adopting the Hellinger‐Reissner functional with the displacements and through‐thickness shear strains as degrees of freedom. Heterogeneous beams and plates with weak discontinuity are considered, and the mixed formulation has been combined with the extended finite element method (FEM); thus, mixed enrichment functions are used. Both the displacement and the shear strain fields are enriched as opposed to the traditional extended FEM where only the displacement functions are enriched. The enrichment type is restricted to extrinsic mesh‐based topological local enrichment. The results from the proposed formulation correlate well with analytical solution in the case of the beam and in the case of the Mindlin‐Reissner plate with those of a finite element package (ABAQUS) and classical FEM and show higher rates of convergence. In all cases, the proposed method captures strain discontinuity accurately. Thus, the proposed method provides an accurate and a computationally more efficient way for the formulation of beam and plate finite elements of minimal number of degrees of freedom.     

Mixed‐mode delamination in 2D layered beam finite elements

In this work, a 2D finite element (FE) formulation for a multi‐layer beam with arbitrary number of layers with interconnection that allows for mixed‐mode delamination is presented. The layers are modelled as linear beams, while interface elements with embedded cohesive‐zone model are used for the interconnection. Because the interface elements are sandwiched between beam FEs and attached to their nodes, the only basic unknown functions of the system are two components of the displacement vector and a cross‐sectional rotation per layer. Damage in the interface is modelled via a bi‐linear constitutive law for a single delamination mode and a mixed‐mode damage evolution law. Because in a numerical integration procedure, the damage occurs only in discrete integration points (i.e. not continuously), the solution procedure experiences sharp snap backs in the force‐displacements diagram. A modified arc‐length method is used to solve this problem. The present model is verified against commonly used models, which use 2D plane‐strain FEs for the bulk material. Various numerical examples show that the multi‐layer beam model presented gives accurate results using significantly less degrees of freedom in comparison with standard models from the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

Mixed state-vector finite element analysis for a higher-order box beam theory

If thin-walled closed beams are analyzed by the standard Timoshenko beam elements, their structural behavior, especially near boundaries, cannot be accurately predicted because of the incapability of the Timoskenko theory to predict the sectional warping and distortional deformations. If a higher-order thin-walled box beam theory is used, on the other hand, accurate results comparable to those obtained by plate finite elements can be obtained. However, currently available two-node displacement based higher-order beam elements are not efficient in capturing exponential solution behavior near boundaries. Based on this motivation, we consider developing higher-order mixed finite elements. Instead of using the standard mixed formulation, we propose to develop the mixed formulation based on the state-vector form so that only the field variables that can be prescribed on the boundary are interpolated for finite element analysis. By this formulation, less field variables are used than by the standard mixed formulation, and the interpolated field variables have the physical meaning as the boundary work conjugates. To facilitate the discretization, two-node elements are considered. The effects of interpolation orders for the generalized stresses and displacements on the solution behavior are investigated along with numerical examples.