An accurate two-node finite element for shear deformable curved beams

An accurate two-node (three degrees of freedom per node) finite element is developed for curved shear deformable beams. The element formulation is based on shape functions that satisfy the homogeneous form of the partial differential equations of motion which renders it free of shear and membrane locking. The element is demonstrated to converge to the results obtained from a shear deformable straight beam when the beam becomes shallower. Numerical examples were performed to demonstrate the accuracy and efficiency with respect to previously published formulations. © 1998 John Wiley & Sons, Ltd.     

A conforming unified finite element formulation for the vibration of thick beams and frames

Beams and frames are common features in many engineering structures and in this paper an approach is given to model their dynamic behaviour adequately. Whilst the eigen‐frequencies of continuous systems comprising of slender beams can be identified, in most cases of practical interest, by means of Euler or Timoshenko beam theory, for structures comprising of thick beam models this is not necessarily true since such idealizations constrain the cross‐sections to remain planar. This paper suggests an alternative approach by means of a unified fully conforming plane stress rectangular finite element which is believed to allow for more realistic representation of the shear effects and hence the strain field around the joints of such structures. The usefulness and functionality of this improved numerical approach is explored via comparison against a non‐conforming two‐dimensional plate as well as one‐dimensional Euler–Bernoulli and Timoshenko finite element formulations corresponding to a variety of beam aspect ratios representing the structures of a rotor and a portal frame. The idealization is shown to be particularly advantageous for simulating the effects of shear distortion where beams join at right angles and the transverse forces in one member interact with the extensional forces of the adjoining structure. Copyright © 2004 John Wiley & Sons, Ltd.     

Dynamic analysis of thin-walled and variable open section beams with shear flexibility

The equations of motion of thin-walled beams with open cross-section, considering the effects of shear flexibility and rotatory inertia in the stress resultants, as well as variable cross-sectional properties are presented using a state variables approach. These equations are based on Vlasov's theory of thin-walled beams, which is modified to include the effects indicated above. The resulting equations are used in the determination of the natural frequencies of an open cross-section prestressed concrete beam. Comparisons with other formulations are also presented employing the waves dispersion equation.     

A finite element formulation for sliding beams,Part I

We use the updated Lagrangian and the co-rotational finite element methods to obtain solutions for geometrically non-linear flexible sliding beams. Finite element formulations are normally carried out for fixed domains. Since the sliding beam is a system of changing mass, first we discretize the system by introducing a variable-domain beam element and model the sliding beam by a fixed number of elements with changing length. Second, we transform the system governing equations of motion to a fixed domain and use conventional finite elements (fixed size and number) to discretize the system. Then our investigation is followed by a comparison between two formulations. Finally, we use the co-rotational method in conjunction with a variable domain beam element to obtain the discretized system equations. To do so, we consider the beam to slide with respect to a fixed channel and later we consider a formulation in which the beam remains at rest and the channel slides with a prescribed velocity. We show that both formulations end up with identical discretized equations of motion. © 1998 John Wiley & Sons, Ltd.     

On approximate solutions for the deformation of plane anisotropic beams

Plane deformation of anisotropic beams with narrow rectangular cross sections exhibits coupling of stretching, bending and transverse shearing. For anisotropic cantilever beams with a stiff end-cap under end forces and an end couple, assessments were made for approximate solutions by comparing these with numerically exact finite element (FE) solutions. Specific attention is given to point-wise or approximate satisfaction of the end-fixity conditions. As approximate methodologies, (i) the elementary polynomial form of Airy's stress function for the plane stress problem in a rectangular region, (ii) a Timoshenko-type beam theory, and (iii) the Bernoulli-Euler beam theory were selected. Among these, only the polynomial form of Airy's stress function violates the point-wise end-fixity conditions. Both the polynomial Airy stress function and the Timoshenko-type beam theory successfully model the effects of transverse shear deformation and the coupling of stretching and transverse deflection. Analytical solutions demonstrate that the normal shear coupling effect increases linearly with the thickness-to-span ratios in axial normal stress and axial displacement, while the coupling manifests quadratically in transverse displacement. The comparison of end displacements with the numerically exact FE solutions indicates that the polynomial form of Airy's stress function is no better than the Timoshenko-type beam theory. Similar conclusions were reached for the problem of uniformly loaded cantilever beams. It has been found that the accurate prediction of the deformation of thick anisotropic beams with significant normal-shear coupling requires the use of higher order theories.     

Super convergent shear deformable finite elements for stability analysis of composite beams

The super convergent finite beam elements are newly presented for the spatially coupled stability analysis of composite beams. For this, the theoretical model applicable to the thin-walled laminated composite I-beams subjected to the axial force is developed. The present element includes the transverse shear and the warping induced shear deformation by using the first-order shear deformation beam theory. The stability equations and force–displacement relationships are derived from the principle of minimum total potential energy. The explicit expressions for the seven displacement parameters are then presented by applying the power series expansions of displacement components to simultaneous ordinary differential equations. Finally, the element stiffness matrix is determined using the force–displacement relationships. In order to demonstrate the accuracy and the superiority of the beam element developed by this study, the numerical solutions are presented and compared with the results obtained from other researchers, the isoparametric beam elements based on the Lagrangian interpolation polynomial, and the detailed three-dimensional analysis results using the shell elements of ABAQUS. The effects of shear deformation, boundary condition, fiber angle change, and modulus ratios on buckling loads are investigated in the analysis.     

Symmetric Galerkin BEM for shear deformable plates

A symmetric Galerkin boundary element formulation is developed for shear deformable plates. A mixed strategy is used for the integration process, i.e. partial regularization using simple solutions followed by a singularity subtraction technique. For the shear equation, full regularization is achieved using new kernel relationships found through a constant shear mode of deformation. Some of the strong singular integrals are avoided altogether by using a modified traction obtained through a very simple variable change; appropriate boundary conditions are defined. Details of the implementation are given and several example problems solved to verify the accuracy of the proposed formulation. Copyright © 2003 John Wiley & Sons, Ltd.     

A Galerkin formulation for shear deformable plate bending dynamics

A Galerkin boundary element formulation for shear deformable plate bending dynamics is developed. The formulation makes use of the static fundamental solutions for the weighted residual integral equations. The domain integrals carrying the inertia terms and generic static loads are considered as body forces and approximated with boundary values using the dual reciprocity method. The load is modelled as a series of impact loads of time varying intensity and moving in space in a predetermined path. The formulation was implemented and tested solving a benchmark problem. The results are compared with finite element solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

复合材料空间薄壁梁的有限元分析模型

在剪切梁理论的基础上, 采用9 节点平面单元模拟梁任意截面形状; 采用27 节点体单元, 模拟截面出平面外的二次翘曲位移, 从而建立了空间复合材料任意截面薄壁梁考虑二次翘曲的有限元分析模型。根据本文中导出的复合材料有限元模型编制了相应的分析计算程序。算例表明: 本文中建立的复合材料薄壁梁模型正确, 可以用于考虑多种耦合影响因素作用下复杂结构空间薄壁复合材料梁的有限元分析计算。      

On the dynamic behaviour of the Timoshenko beam finite elements

First, various finite element models of the Timoshenko beam theory for static analysis are reviewed, and a novel derivation of the 4 × 4 stiffness matrix (for the pure bending case) of the superconvergent finite element model for static problems is presented using two alternative approaches: (1) assumed-strain finite element model of the conventional Timoshenko beam theory, and (2) assumed-displacement finite element model of a modified Timoshenko beam theory. Next, dynamic versions of various finite element models are discussed. Numerical results for natural frequencies of simply supported beams are presented to evaluate various Timoshenko beam finite elements. It is found that the reduced integration element predicts the natural frequencies accurately, provided a sufficient number of elements is used. The research reported herein is supported by theOscar S. Wyatt Endowed Chair.     

A shear–flexible element with warping for thin-walled open beams

A new finite element for the analysis of thin-walled open beams with an arbitrary cross section is presented. Combining Timoshenko beam theory and Vlasov thin-walled beam theory, the derived element includes both flexural shear deformations and warping deformations caused by the bimoment. By adopting an orthogonal Cartesian co-ordinate system, one can obviate the ad hoc introduction of St. Venant stiffness. The derived block stiffness matrix is comparable but more general than the one given by earlier researchers. The versatility and accuracy of the new element are demonstrated by comparing the numerical results with the classical solutions or other numerical results available in the literature. © 1998 John Wiley & Sons, Ltd.     

A new method of shear stiffness prediction of periodic Timoshenko beams

This article interprets the new implementation of an asymptotic homogenization method for effective bending stiffness of heterogeneous beam structures with periodic microstructure along its axial direction in an intuitionistic way. With this interpretation, the authors then develop a new method of evaluating effective shear stiffness for their Timoshenko beam model. This method can be easily implemented numerically in commercial software. Different kinds of elements and modeling techniques available in commercial software can be applied to model the unit cell. Several examples are given to demonstrate the effectiveness of this new method.     

Sliding beams,Part II: time integration

In this paper we obtain solutions for the discretized incremental system equations, as obtained in Part I, under certain initial and boundary conditions and/or specified applied loads, using the variable domain beam element. As a check on the validity of implementation, we first limit ourselves to linear analysis and obtain results for the axially inextensible sliding beams which we compare with the results reported in the literature. Second we set the axial velocity to zero and solve some special cases when the length of the beam is constant. In this case, we check the formulation and its implementation for non-linearities in the system due to large displacements. Finally, we solve the sliding beam problem for small amplitude oscillations, with a non-linear solver and compare the results with those obtained by the linear solver used for inextensible sliding beams. With these preliminary tests completed, we obtain the transient response of the free and forced large amplitude vibrations of the flexible sliding beam and demonstrate the need for using a non-linear analysis for this complex system. Finally, we consider the stability of the motion of periodically time varying flexible sliding beams and show that in the case of parametric resonance, the unstable regions obtained in the linear analysis, which imply unbounded amplitudes, are indeed stable and bounded when non-linear terms are taken into account. © 1998 John Wiley & Sons, Ltd.     

Advantages of the mixed format in geometrically nonlinear analysis of beams and shells using solid finite elements

The paper deals with two main advantages in the analysis of slender elastic structures both achieved through the mixed (stress and displacement) format with respect to the more commonly used displacement one: (i) the smaller error in the extrapolations usually employed in the solution strategies of nonlinear problems and (ii) the lower polynomial dependence of the problem equations on the finite element degrees of freedom when solid finite elements are used. The smaller extrapolation error produces a lower number of iterations and larger step length in path‐following analysis and a greater accuracy in Koiter asymptotic method. To focus on the origin of the phenomenon, the two formats are derived for the same finite element interpolation. The reduced polynomial dependence improves the Koiter asymptotic strategy in terms of both computational efficiency, accuracy and simplicity. Copyright © 2016 John Wiley & Sons, Ltd.     

An equilibrium-based formulation with nonlinear configuration dependent interpolation for geometrically exact 3D beams

This article describes a novel equilibrium-based geometrically exact beam finite element formulation. First, the spatial position and rotation fields are interpolated by nonlinear configuration-dependent functions that enforce constant strains along the element axis, completely eliminating locking phenomena. Then, the resulting kinematic fields are used to interpolate the spatial sections force and moment fields in order to fulfill equilibrium exactly in the deformed configuration. The internal variables are explicitly solved at the element level and closed-form expressions for the internal force vector and tangent stiffness matrix are obtained, allowing for explicit computation, without numerical integration. The objectivity and absence of locking are verified and some important numerical and theoretical aspects leading to a computationally efficient strategy are highlighted and discussed. The proposed formulation is successfully tested in several numerical application examples.     

Dynamic behaviour of edge-cracked shear deformable functionally graded beams on an elastic foundation under a moving load

This paper studies the dynamic response of functionally graded beams with an open edge crack resting on an elastic foundation subjected to a transverse load moving at a constant speed. It is assumed that the material properties follow an exponential variation through the thickness direction. Theoretical formulations are based on Timoshenko beam theory to account for the transverse shear deformation. The cracked beam is modeled as an assembly of two sub-beams connected through a linear rotational spring. The governing equations of motion are derived by using Hamilton’s principle and transformed into a set of dynamic equations through Galerkin’s procedure. The natural frequencies and dynamic response with different end supports are obtained. Numerical results are presented to investigate the influences of crack location, crack depth, material property gradient, slenderness ratio, foundation stiffness parameters, velocity of the moving load and boundary conditions on both free vibration and dynamic response of cracked functionally graded beams.     

The mixed finite element method for the quasi‐static and dynamic analysis of viscoelastic timoshenko beams

The quasi‐static and dynamic responses of a linear viscoelastic Timoshenko beam on Winkler foundation are studied numerically by using the hybrid Laplace–Carson and finite element method. In this analysis the field equation for viscoelastic material is used. In the transformed Laplace–Carson space two new functionals have been constructed for viscoelastic Timoshenko beams through a systematic procedure based on the Gâteaux differential. These functionals have six and two independent variables respectively. Two mixed finite element formulations are obtained; TB12 and TB4. For the inverse transform Schapery and Fourier methods are used. The numerical results for quasi‐static and dynamic responses of several visco‐elastic models are presented. Copyright © 1999 John Wiley & Sons, Ltd.     

Polygonal finite elements for finite elasticity

Nonlinear elastic materials are of great engineering interest, but challenging to model with standard finite elements. The challenges arise because nonlinear elastic materials are characterized by non‐convex stored‐energy functions as a result of their ability to undergo large reversible deformations, are incompressible or nearly incompressible, and often times possess complex microstructures. In this work, we propose and explore an alternative approach to model finite elasticity problems in two dimensions by using polygonal discretizations. We present both lower order displacement‐based and mixed polygonal finite element approximations, the latter of which consist of a piecewise constant pressure field and a linearly‐complete displacement field at the element level. Through numerical studies, the mixed polygonal finite elements are shown to be stable and convergent. For demonstration purposes, we deploy the proposed polygonal discretization to study the nonlinear elastic response of rubber filled with random and periodic distributions of rigid particles, as well as the development of cavitation instabilities in elastomers containing vacuous defects. These physically‐based examples illustrate the potential of polygonal finite elements in studying and modeling nonlinear elastic materials with complex microstructures under finite deformations. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

MITC finite elements for laminated composite plates

Within the framework of the first‐order shear deformation theory, 4‐ and 9‐node elements for the analysis of laminated composite plates are derived from the MITC family developed by Bathe and coworkers. To this end the bases of the MITC formulation are illustrated and suitably extended to incorporate the laminate theory. The proposed elements are locking‐free, they do not have zero‐energy modes and provide accurate in‐plane deformations. Two consecutive regularizations of the extensional and flexural strain fields and the correction of the resulting out‐of‐plane stress profiles necessary to enforce exact fulfillment of the boundary conditions are shown to yield very satisfactory results in terms of transverse and normal stresses. The features of the proposed elements are assessed through several numerical examples, either for regular and highly distorted meshes. Comparisons with analytical solutions are also shown. Copyright © 2001 John Wiley & Sons, Ltd.