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.     

Formulation of reference surface element and its applications in dynamic analysis of delaminated composite beams

This paper presents a new finite element formulation, referred to as reference surface element (RSE) model, for numerical prediction of dynamic behaviour of delaminated composite beams and plates using the finite element method. The RSE formulation can be readily incorporated into all elements based on the Timoshenko beam theory and the Reissner–Mindlin plate theory taking into account the transverse shear deformations. The ‘free model' and ‘constrained model' for dynamic analysis of delaminated composite beams and/or plates have been unified in this RSE formulation. The RSE formulation has been applied to an existing 2-node Timoshenko beam element taking into account the transverse shear deformations and the bending–extension coupling. Frequencies and vibration mode shapes are determined through solving an eigenvalue problem. Numerical results show that the present RSE model is reliable and practical when used to predict frequencies and mode shapes of delaminated composite beams. The RSE formulation has also been used to investigate the effects of the number, size and interfacial loci of delaminations on frequencies and mode shapes of composite beams.     

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.     

Static analysis of Timoshenko beam resting on elastic half-plane based on the coupling of locking-free finite elements and boundary integral

Making use of a mixed variational formulation including the Green function of the soil and assuming as independent fields both the structure displacements and the contact pressure, a finite element (FE) model is derived for the static analysis of a foundation beam resting on elastic half-plane. Timoshenko beam model is adopted to describe structural foundations with low slenderness and to impose displacement compatibility between beam and half-plane without requiring the continuity of the first order derivative of the surface displacements enforced by Euler–Bernoulli beam. Numerical results are obtained by using locking-free Hermite polynomials for the Timoshenko beam and constant reaction over the soil. Foundation beams loaded by many load configurations illustrate accuracy and convergence properties of the proposed formulation. Moreover, the different behaviour of the Euler–Bernoulli and Timoshenko beam models is thoroughly discussed. Rectangular pipe loaded by a force in the upper beam exemplifies the straightforward coupling of the foundation FE with a structure described by usual FEs.     

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.     

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.     

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.     

Winkler地基上修正Timoshenko梁振动分析

利用Bernoulli-Euler梁理论建立的弹性地基梁模型应用广泛,但其在高阶频率及深梁计算中误差较大,利用修正的Timoshenko梁理论建立新的弹性地基梁振动微分方程,由于其在Timoshenko梁的基础上考虑了剪切变形所引起的转动惯量,因而具有更好的精确度。利用ANAYS beam54梁单元进行振动模态的有限元计算,所求结果与理论基本无误差,从而验证了该理论的正确性。基于修正Timoshenko梁振动理论推导出了弹性地基梁双端自由-自由、简支-简支、简支-自由、固支-固支等多种边界条件下的频率超越方程及模态函数。分析了弹性地基梁在不同理论下不同约束条件及不同高跨比情况下的计算结果,从而论证了该理论计算弹性地基梁的适用性。分析了不同弹性地基梁理论下波速、群速度与波数的关系。得到了约束条件和梁长对振动模态及地基刚度对振动频率有重要影响等结论。     

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.     

改进共旋坐标法的Timoshenko梁单元非线性分析

为提高空间Timoshenko梁单元非线性问题的计算精度,在共旋坐标法的基础上,提出了一种改进的Timoshenko梁单元几何非线性分析方法。利用虚功原理得到改进空间梁单元的刚度矩阵;使用有限质点法中的逆向运动思路计算单元局部坐标系下的刚体旋转矩阵;根据整体坐标系与局部坐标系之间旋转角度的转化以及微分关系,求得空间梁单元的切线刚度矩阵;编制了相应的有限元程序,对多个经典的大变形结构进行几何非线性分析。计算结果印证了该文所提出改进方法的正确性,同时与传统共旋坐标法相比,具有更高的精度。     

An accurate laminated element for piezoelectric smart beams including peel stress

This paper presents a laminated element for piezoelectric (PZT) smart beams in taking into account peel stresses. In the finite element analysis (FEA) formulation, a coupled electrical and mechanical beam element is used to model PZT patches, and a conventional structural element is used to model a host beam. A continuous adhesive element with shear and peel stiffness is derived to form a PZT laminated element. For a smart beam with a partially bonded PZT patch or distributed PZTs, the laminated element is applied to an area of the host beam with PZTs and the conventional element is used in the host beam where no PZT is bonded. A novel PZT laminated element is firstly derived based on the Timoshenko beam theory, in which the FEA formulation based on the Euler-Bernoulli beam theory can be considered as its special case. FEA numerical results of static and dynamic analyses based on the Euler-Bernoulli beam theory are compared with the exact static and dynamic solutions to validate the present FEA formulation. The present FEA framework based on the Timoshenko beam theory is then used to investigate the effects of PZT debondings on static behaviors and dynamic responses, and an original and effective procedure for detecting debondings in PZT actuators or sensors is proposed.The authors are grateful to the support of the Australian Research Council through a Large Grant Scheme (Grant No. A10009074).     

FREE VIBRATION ANALYSIS OF KIRCHHOFF PLATES RESTING ON ELASTIC FOUNDATION BY MIXED FINITE ELEMENT FORMULATION BASED ON GÂTEAUX DIFFERENTIAL

The main objective of the present work is to give the systematic way for derivation of Kirchhoff plate-elastic foundation interaction by mixed-type formulation using the Gâteaux differential instead of well-known variational principles of Hellinger–Reissner and Hu–Washizu. Foundation is a Pasternak foundation, and as a special case if shear layer is neglected, it converges to Winkler foundation in the formulation. Uniform variation of the thickness of the plate is also included into the mixed finite element formulation of the plate element PLTVE4 which is an isoparametric C⁰ class conforming element discretization. In the dynamic analysis, the problem reduces to solution of the standard eigenvalue problem and the mixed element is based upon a consistent mass matrix formulation. The element has four nodes and at each node transverse displacement two bending and one torsional moment is the basic unknowns. Proper geometric and dynamic boundary conditions corresponding to the plate and the foundation is given by the functional. Performance of the element for bending and free vibration analysis is verified with a good accuracy on the numerical examples and analytical solutions present in the literature. © 1997 by John Wiley & Sons, Ltd.     

On finite element formulations for nearly incompressible linear elasticity

In this paper we present a mixed stabilized finite element formulation that does not lock and also does not exhibit unphysical oscillations near the incompressible limit. The new mixed formulation is based on a multiscale variational principle and is presented in two different forms. In the first form the displacement field is decomposed into two scales, coarse-scale and fine-scale, and the fine-scale variables are eliminated at the element level by the static condensation technique. The second form is obtained by simplifying the first form, and eliminating the fine-scale variables analytically yet retaining their effect that results with additional (stabilization) terms. We also derive, in a consistent manner, an expression for the stabilization parameter. This derivation also proves the equivalence between the classical mixed formulation with bubbles and the Galerkin least-squares type formulations for the equations of linear elasticity. We also compare the performance of this new mixed stabilized formulation with other popular finite element formulations by performing numerical simulations on three well known test problems.     

An advanced finite element formulation for piezoelectric beam structures

Since many piezoelectric components are thin rod-like structures, a piezoelectric finite beam element can be utilized to analyse a wide range of piezoelectric devices effectively. The mechanical strains and the electric field are coupled by the constitutive relations. Finite element formulations using lower order functions to interpolate mechanical and electrical fields lead to unbalances within the numerical approximation. As a consequence incorrect computational results occur, especially for bending dominated problems. The present contribution proposes a concept to avoid these errors. Therefore, a mixed multi-field variational approach is introduced. The element employs the Timoshenko beam theory and considers strains throughout the width and the thickness enabling to directly use 3D constitutive relations. By means of several numerical examples it is shown that the element formulation allows to analyse piezoelectric beam structures for all typical load cases without parasitically affected results.     

基于Green函数法的Timoshenko曲梁强迫振动分析

该文运用Green函数法求解了Timoshenko曲梁在强迫振动下的解析解,通过分析曲梁截面的力学平衡,建立了Timoshenko曲梁的振动方程。依次采用分离变量法和Laplace变换法,对不同边界的Timoshenko曲梁求解出了相应的Green函数。并且通过引入两个特征参数来考虑阻尼对强迫振动的影响。数值计算中,验证了该解析解的有效性,并对其中涉及的各种重要物理参数的影响进行了研究。研究结果表明:通过将半径R设置为无穷大,可以简化为Timoshenko直梁振动模型,在此基础上,将剪切修正因子κ设置为无穷大,可以退化为Prescott直梁振动模型,最后再把转动惯量γ设置为0,可退化为Euler-Bernoulli直梁振动模型。该文给出的数值结果验证了所得解的有效性。     

A simple finite element for beams on elastic foundations

A simple "routine" beam on elastic foundation finite element using a polynomial displacement function has been developed which yields acceptably accurate deflection, shear and bending moment values for prismatic or non-prismatic beams of elastic material resting on foundations with varying or nonlinear subgrade reactions. Limited extension of the formulation to an "exact" finite element using the exact displacement function of a beam on elastic foundation has also been carried out. The subgrade is represented by a non-homogeneous solid medium to include nonlinear parameters if required. The iterative solution is extended to cases where the beam may uplift because the foundation is a no tension material. The model is also suitable for calculating the elastic deflections, membrane. and bending stress resultants for axisymmetrically loaded variable thickness shells of revolution. A computer program called FEBEF [finite element: beam on elastic foundation] incorporating the routine finite element has been prepared for the solution of beams on elastic foundations and axi symmetrically loaded shells of revolution.     

Effect of shear and rotary inertia on dynamic fracture of a beam or plate in tensile loading

The dynamic fracture response of a long beam of brittle material subjected to tensile loading is studied. If the magnitude of the applied tensile loading is increased to a critical value, a crack will propagate from one of the longitudinal surfaces of the beam. As an extension of previous work, the effect of shear and of rotary inertia on the tensile loading and the induced bending moment at the fracturing section is included in the analysis. Thus an improved formulation is presented by means of which the crack length, crack tip velocity, bending moment and axial force at the fracture section are determined as functions of time after crack initiation. It is found that the rotary effect diminishes the bending moment effect and retards total fracture time whereas the shear has an opposite effect. Thus by combining the two effects (to simulate to first order the Timoshenko beam) overall fracture is retarded. The results also apply for plane strain fracture of a plate in tensile loading provided the value of the elastic modulus is appropriately modified.     

Consistency aspects of out-of-plane bending,torsion and shear in a quadratic curved beam element

Curved beams in civil engineering applications call for out-of-plane bending and torsion under the action of out-of-plane transverse shear loads. The design of a quadratic displacement curved beam element capable of representing shear deformation as in the Timoshenko beam theory requires special attention to the manner in which the shear strain is represented. Field-inconsistent representations of the out-of-plane transverse shear strain will result in a loss of efficiency and introduce spurious oscillations in the bending moment, torsional moment and shear force. The optimal field-consistent assumed strain interpolation for shear is derived and it is demonstrated to posses very high accuracy which is free from spurious force and moment oscillations.