DQEM for free vibration analysis of Timoshenko beams on elastic foundations

 A differential quadrature element method (DQEM) based on first order shear deformation theory is developed for free vibration analysis of non-uniform beams on elastic foundations. By decomposing the system into a series of sub-domains or elements, any discontinuity in loading, geometry, material properties, and even elastic foundations can be considered conveniently. Using this method, the vibration analysis of general beam-like structures is to be studied. The governing equations of each element, natural compatibility conditions at the interface of two adjacent elements and the external boundary conditions are developed in a systematic manner, using Hamilton's principle. The present DQEM is to be implemented to Timoshenko beams resting on partially supported elastic foundations with various types of boundary conditions under the action of axial loading. The general versality, accuracy, and efficiency of the presented DQEM are demonstrated having solved different examples and compared to the exact or other numerical procedure solutions. Received: 11 October 2002/Accepted: 26 November 2002     

Interpolation functions in control volume finite element method

 The main contribution of this paper is the study of interpolation functions in control volume finite element method used in equal order and applied to an incompressible two-dimensional fluid flow. Especially, the exponential interpolation function expressed in the elemental local coordinate system is compared to the classic linear interpolation function expressed in the global coordinate system. A quantitative comparison is achieved by the application of these two schemes to four flows that we know the analytical solutions. These flows are classified in two groups: flows with privileged direction and flows without. The two interpolation functions are applied to a triangular element of the domain then; a direct comparison of the results given by each interpolation function to the exact value is easily realized. The two functions are also compared when used to solve the discretized equations over the entire domain. Stability of the numerical process and accuracy of solutions are compared. Received: 20 October 2002 / Accepted: 2 December 2002     

Analysis of laminated composite beams and plates with piezoelectric patches using the element-free Galerkin method

 An efficient meshfree formulation based on the first-order shear deformation theory (FSDT) is presented for the static analysis of laminated composite beams and plates with integrated piezoelectric layers. This meshfree model is constructed based on the element-free Galerkin (EFG) method. The formulation is derived from the variational principle and the piezoelectric stiffness is taken into account in the model. In numerical test problems, bending control of piezoelectric bimorph beams was shown to have the efficiency and accuracy of the present EFG formulation for this class of problems. It is demonstrated that the different boundary conditions and applied actuate voltages affects the shape control of piezolaminated composite beams. The meshfree model is further extended to study the shape control of piezo-laminated composite plates. From the investigation, it is found that actuator patches bonded on high strain regions are significant in deflection control of laminated composite plates. Received: 23 October 2001 / Accepted: 29 July 2002     

A point interpolation mesh free method for static and frequency analysis of two-dimensional piezoelectric structures

 A mesh free method called point interpolation method (PIM) is presented for static and mode-frequency analysis of two-dimensional piezoelectric structures. In the present method, the problem domain and its boundaries are represented by a set of properly scattered nodes. The displacements and the electric potential of a point are interpolated by the values of nodes in its local support domain using shape functions derived based on a point interpolation scheme. Techniques are discussed to surmount the singularity of the moment matrix. Variational principle together with linear constitutive piezoelectric equations is used to establish a set of system equations for arbitrary-shaped piezoelectric structures. These equations are assembled for all quadrature points and solved for displacements and electric potentials. A polynomial PIM program has been developed in MATLAB with matrix triangularization algorithm (MTA), which automatically performs a proper node enclosure and a proper basis selection. Examples are also presented to demonstrate the accuracy and stability of the present method and their results are compared with the conventional FEM results from ABAQUS as well as the analytical or experimental ones. Received: 6 February 2002 / Accepted: 5 August 2002     

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.     

A semi-analytical approach to three-dimensional normal contact problems with friction

 We present a semi-analytical approach for three-dimensional elastostatic normal contact problems with friction. The numerical approach to iteration on contact area and stick zone size is supported by an underlying analytical solution relating normal and tangential surface tractions to surface displacements in three coordinate directions. The governing equations are fully coupled. The analytical surface displacement solutions for a basic loading element have been derived elsewhere (Li and Berger 2001), and the total surface displacements are constructed as a superposition of deflections due to overlapping pyramid load segments. This approach requires no interpolation scheme for the field variables, which distinguishes it from other numerical techniques such as the FEM, BEM, and meshless methods. A background grid is defined only on the contact surfaces, and iteration approaches are used to determine a convergent configuration for contact domain and stick zone size. The approach is exercised on several normal contact problems, with and without friction, and the results compare favorably to existing analytical and numerical solutions. Received: 10 July 2002 / Accepted: 3 December 2002 The authors appreciate the support of the UC Department of Mechanical Engineering and the UC Office of the Vice President for Research, who jointly provided funds for this work.     

梁杆结构二阶效应分析的一种新型梁单元

推导了一种计及梁杆二阶效应的新型两结点梁单元。首先依据插值理论构造了三结点Euler-Bernoulli梁单元的位移场:使用五次Hermite插值函数建立梁单元的侧向位移场,二次Lagrange插值函数建立梁单元的轴向位移场,进而由非线性有限元理论推导了单元的线性刚度矩阵和几何刚度矩阵,然后使用静力凝聚方法消除三结点梁单元中间结点的自由度,从而得到一种考虑轴力效应的新型两结点梁单元。实例分析表明,此新型梁单元具有很高的计算精度,使用此单元进行梁杆结构分析可获得相当准确的二阶位移和内力。     

A mixed finite element formulation for timoshenko beam on winkler foundation

 A mixed formulation for Timoshenko beam element on Winkler foundation has been derived by defining the total curvature in terms of the bending moment and its second order derivation. Displacement and moment have been chosen as primary variables, while slope and first derivation of moment have been chosen as secondary variables. The behaviour matrix for Timoshenko beam element has been obtained in mixed form by using weak formulation with equilibrium and compatibility equations. The presented formulation makes the analysis of beams free of shear locking. Received: 10 July 2002 / Accepted: 14 January 2003     

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.     

Spatial stability and free vibration of shear flexible thin-walled elastic beams. II: Numerical approach

In a companion paper,¹ equations of motion and closed-form solutions for spatial stability and free vibration analysis of shear flexible thin-walled elastic beams were analytically derived from the linearized Hellinger–Reissner principle. In this paper, elastic and geometric stiffness matrices and consistent mass matrix for finite element analysis are evaluated by using isoparametric and Hermitian interpolation polynomials. Isoparametric interpolation functions with 2, 3 and 4 nodes per element are utilized in isoparametric beam elements, and in Hermitian beam elements, the third- and fifth-order Hermitian polynomials including shear deformation effects are newly derived and applied for the calculation of element matrices. In order to verify the validity of the finite element formulation, both analytic and numerical solutions for spatial buckling and free vibration problems including shear effects are presented and compared.     

A theoretical and computational setting for a geometrically nonlinear gradient damage modelling framework

 The present work deals with the extension to the geometrically nonlinear case of recently proposed ideas on elastic- and elastoplastic-damage modelling frameworks within the infinitesimal theory. The particularity of these models is that the damage part of the modelling involves the gradient of damage quantity which, together with the equations of motion, are ensuing from a new formulation of the principle of virtual power. It is shown how the thermodynamics of irreversible processes is crucial in the characterization of the dissipative phenomena and in setting the convenient forms for the constitutive relations. On the numerical side, we discuss the problem of numerically integrating these equations and the implementation within the context of the finite element method is described in detail. And finally, we present a set of representative numerical simulations to illustrate the effectiveness of the proposed framework. Received: 13 May 2002 / Accepted: 16 September 2002     

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.     

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 complex variable boundary collocation method for plane elastic problems

 Lagrange interpolation is extended to the complex plane in this paper. It turns out to be composed of two parts: polynomial and rational interpolations of an analytical function. Based on Lagrange interpolation in the complex plane, a complex variable boundary collocation approach is constructed. The method is truly meshless and singularity free. It features high accuracy obtained by use of a small number of nodes as well as dimensionality advantage, that is, a two-dimensional problem is reduced to a one-dimensional one. The method is applied to two-dimensional problems in the theory of plane elasticity. Numerical examples are in very good agreement with analytical ones. The method is easy to be implemented and capable to be able to give the stress states at any point within the solution domain. Received: 20 August 2002 / Accepted: 31 January 2003     

Probabilistic analysis of multi-site damage in aircraft fuselages

 Most aircraft fleets nowadays are operating under the concept of damage tolerance, which requires an aircraft to have sufficient residual strength in the presence of damage in one of its principal structural elements (PSE) during the interval of service inspections. The residual strength however is significantly reduced due to multi site damage (MSD). In the present paper, a probabilistic framework for the computation of the failure probability is developed. The MSD problem of a PSE is considered, where the uncertainties in crack initiation and crack growth as well as yield stress and fracture toughness are described by random variables. For the crack growth calculations the finite element alternating method [1], which avoids a remeshing of the finite element problem, is used. After specifying link up and failure criteria, importance sampling is employed to obtain the probability of failure of the PSE due to MSD. Received: 29 July 2002 / Accepted: 18 December 2002     

On material forces and finite element discretizations

 The idea of using material forces also termed configurational forces in a computational setting is presented. The theory of material forces is briefly recast in the terms of a non-linear elastic solid. It is shown, how in a computational setting with finite elements (FE) the discrete configurational forces are calculated once the classical field equations are solved. This post-process calculation is performed in a way, which is consistent with the approximation of the classical field equations. Possible physical meanings of this configurational forces are discussed. A purely computational aspect of material forces is pointed out, where material forces act as an indicator to obtain softer discretizations. Received 12 December 2001 / Accepted 18 March 2002     

Analysis of thin beams, using the meshless local Petrov–Galerkin method, with generalized moving least squares interpolations

In this paper, the conventional moving least squares interpolation scheme is generalized, to incorporate the information concerning the derivative of the field variable into the interpolation scheme. By using this generalized moving least squares interpolation, along with the MLPG (Meshless Local Petrov–Galerkin) paradigm, a new numerical approach is proposed to deal with 4th order problems of thin beams. Through numerical examples, convergence tests are performed; and problems of thin beams under various loading and boundary conditions are analyzed by the proposed method, and the numerical results are compared with analytical solutions. Received 7 February 1999     

Stiffened plate bending analysis by the boundary element method

 In this work, the plate bending formulation of the boundary element method (BEM) based on the Kirchhoff's hypothesis, is extended to the analysis of stiffened elements usually present in building floor structures. Particular integral representations are derived to take directly into account the interactions between the beams forming grid and surface elements. Equilibrium and compatibility conditions are automatically imposed by the integral equations, which treat this composite structure as a single body. Two possible procedures are shown for dealing with plate domain stiffened by beams. In the first, the beam element is considered as a stiffer region requiring therefore the discretization of two internal lines with two unknowns per node. In the second scheme, the number of degrees of freedom along the interface is reduced by two by assuming that the cross-section motion is defined by three independent components only. Received 6 November 2000