Large-amplitude free vibrations of functionally graded beams by means of a finite element formulation

The large-amplitude free vibration analysis of functionally graded beams is investigated by means of a finite element formulation. The Von-Karman type nonlinear strain–displacement relationships are employed where the ends of the beam are constrained to move axially. The effects of the transverse shear deformation and rotary inertia are included based upon the Timoshenko beam theory. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. A statically exact beam element which devoid the shear locking effect with displacement fields based on the first order shear deformation theory is used to study the geometric nonlinear effects on the vibrational characteristics of functionally graded beams. The finite element method is employed to discretize the nonlinear governing equations, which are then solved by the direct numerical integration technique in order to obtain the nonlinear vibration frequencies of functionally graded beams with different boundary conditions. The influences of power-law exponent, vibration amplitude, beam geometrical parameters and end supports on the free vibration frequencies are studied. The present numerical results compare very well with the results available from the literature where possible. Some new results for the nonlinear natural frequencies are presented in both tabular and graphical forms which can be used for future references.     

A spatial shear deformable beam finite element based on the absolute nodal coordinate formulation

In the present paper a three-dimensional beam finite element undergoing large deformations is proposed. Since the definition of the proposed finite element is based on the absolute nodal coordinate formulation (ANCF), no rotational coordinates occur in the formulation. In the current approach, the orientation of the cross section is parameterized by means of slope vectors. Since those are no unit vectors, the cross-section can deform, similar to existing thick beam and shell elements. The nodal displacements and the directional derivatives of the displacements are chosen as nodal coordinates, but in contrast to standard ANCF elements, the proposed formulation is based on the two transversal slope vectors per node only. Different approaches for the virtual work of elastic forces are presented: a continuum mechanics based formulation, as well as a structural mechanics based formulation, which is in accordance with classical nonlinear beam finite elements. Since different interpolation functions as in standard ANCF elements are used, a much better convergence rate (up to order four) can be obtained. Therefore, the present element has high potential for application in geometrically nonlinear problems. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Finite Element Formulation for the Nonlinear Analysis of Piezoelectric Three-Dimensional Beam Structures

A finite element formulation for a three-dimensional piezoelectric beam which includes geometrical and material nonlinearities is presented. To account for the piezoelectric effect, the coupling between the mechanical stress and the electrical displacement is considered. Based on the Timoshenko theory, an eccentric beam formulation is introduced which provides an efficient model to analyze piezoelectric structures. The geometrically nonlinear assumption allows the calculation of large deformations including buckling analysis. A quadratic approximation of the electric potential through the cross section of the beam ensures the fulfilment of the charge conservation law exactly. This assumption leads to a finite element formulation with six mechanical and five electrical degrees of freedom per node. To take into account the typical ferroelectric hysteresis phenomena, a nonlinear material model is essential. For this purpose, the phenomenological Preisach model is implemented into the beam formulation which provides an efficient determination of the remanent part of the polarization. The applicability of the introduced beam formulation is discussed with respect to available data from literature. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Free vibration analysis of spatial Bernoulli–Euler and Rayleigh curved beams using isogeometric approach

This paper deals with the linear free vibration analysis of Bernoulli–Euler and Rayleigh curved beams using isogeometric approach. The geometry of the beam as well as the displacement field are defined using the NURBS basis functions which present the basic concept of the isogeometric analysis. A novel approach based on the fundamental relations of the differential geometry and Cauchy continuum beam model is presented and applied to derive the stiffness and consistent mass matrices of the corresponding spatial curved beam element. In the Bernoulli–Euler beam element only translational and torsional inertia are taken into account, while the Rayleigh beam element takes all inertial terms into consideration. Due to their formulation, isogeometric beam elements can be used for the dynamic analysis of spatial curved beams. Several illustrative examples have been chosen in order to check the convergence and accuracy of the proposed method. The results have been compared with the available data from the literature as well as with the finite element solutions.     

A smart displacement based (SDB) beam element with distributed plasticity

The standard displacement based inelastic beam element suffers of approximations related to the inability of the cubic polynomial interpolation functions to properly describe the displacement response of the beam when exhibiting inelastic behaviour. The increase of the number of finite elements, or the use of higher order functions with additional internal degrees of freedom, are common remedies suggested to improve the approximation leading to an unavoidable reduction of the computational efficiency. Alternatively, it has been shown that the development of force based finite elements, based on the adoption of exact force shape functions, lead to more accurate results, although requiring different and more complicated iterative solution strategies. Within this scenario, this paper proposes a new inelastic beam element, within the context of the displacement based approach, based on variable displacement shape functions, whose analytic expressions are related to the plastic deformation evolution in the beam element. The adaptive generalised displacement shape are obtained by identifying, at each step, an equivalent tangent beam, characterised by abrupt variations of flexural stiffness, as a suitable representation of the current inelastic state of the beam. The presented approach leads to the formulation of a Smart Displacement Based (SDB) beam element whose accuracy appears to be comparable to those obtained through a force based approach but requiring a reduced implementation effort and a more straightforward approach. The term ‘smart’ aims at emphasizing the ability of the element to upgrade the displacement field according to the current inelastic state.     

Nonlinear dynamic analysis of a complex dual rotor-bearing system based on a novel model reduction method

In this paper, a dynamic model of a complex dual rotor-bearing system of an aero-engine is established based on the finite element method with three types of beam elements (rigid disc, cylindrical beam element and conical beam element), as well as taking into account the nonlinearities of all of the supporting rolling element bearings. To rapidly and accurately analyze dynamic behaviors of the complex dual rotor-bearing system, a two-level model order reduction (MOR) method is proposed by combining component mode synthesis (CMS) method and proper orthogonal decomposition (POD) technique. The first-level reduced-order model (ROM) of the dual rotors is obtained by CMS method with a high precision for the original system. Then, the POD method is applied to second-level model order reduction to further decrease the degrees of freedom (DOFs) of first-level ROM. Second-level ROM with mode expansion and direct second-level ROM are obtained, and the nonlinear displacement responses of the two ROMs are compared with the first-level ROM. The numerical results demonstrate that the proposed method has a higher computational efficiency and accuracy in terms of mode expansion than the direct model reduction by using POD method. In addition, the nonlinear vibration responses of the dual rotor-bearing system are studied by this second-level ROM in the case of different clearances of the inter-shaft bearing. The results indicate that the dynamic characteristics of the dual rotor-bearing system are very complicated for a large clearance.     

Shape sensitivity analysis in the extended finite element method

The contribution is concerned with the derivation and implementation of shape sensitivity analysis in the context of the extended finite element method (xfem). Here, the displacement approximation is enhanced in order to model strong and weak discontinuities along the interfaces and cracks. Consequently, sensitivity analysis must be accomplished for the extended formulation based on a sub-integration technique on sub-domains of the cut elements. This cumbersome and error-prone work can be circumvent using a standard treatment of sub-elements on an appropriately defined sub-mesh. These modifications guarantee exact analytical sensitivities even for distorted finite element domains. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite element methods for Timoshenko beam,circular arch and Reissner-Mindlin plate problems

In this paper some finite element methods for Timoshenko beam, circular arch and Reissner-Mindlin plate problems are discussed. To avoid locking phenomenon, the reduced integration technique is used and a bubble function space is added to increase the solution accuracy. The method for Timoshenko beam is aligned with the Petrov-Galerkin formulation derived in Loula et al. (1987) and can be naturally extended to solve the circular arch and the Reissner-Mindlin plate problems. Optimal order error estimates are proved, uniform with respect to the small parameters. Numerical examples for the circular arch problem shows that the proposed method compares favorably with the conventional reduced integration method.     

Construction of beam elements considering von Kármán nonlinear strains using B-spline wavelet on the interval

The current paper proposes the formulation of beam elements using B-spline wavelet on the interval based wavelet finite element method by incorporating von Kármán nonlinear strains. Formulation is proposed for both Euler–Bernoulli beam theory and Timoshenko beam theory. A background cell based Gauss quadrature is proposed for numerical integration. Numerical examples are solved for transverse deflections and stresses in axial direction, and are compared with the existing converged results from finite element method. The issues of membrane and shear locking for the proposed elements are examined and solution techniques are suggested to overcome the issues.     

Follower loads for high order finite elements

Finite element modelling of hydrostatic compaction where the applied pressure acts normal to the deformed surface requires a geometric nonlinear formulation and follower load terms [1, 5, 7]. These concepts are applied to high order [6] (p-FEM) elements with hierarchic shape functions. Applying the blending function method allows to precisely describe curved boundaries on coarse meshes. High order elements exhibit good performance even for high aspect ratios and strong distortion and therefore allow an efficient discretization of thin-walled structures. Since high order finite elements are less prone to locking effects a pure displacement-based formulation can be chosen. After introducing the basic concept of the p-version the application of follower loads to geometrically nonlinear high order elements is presented. For the numerical solution the displacement based formulation is linearized yielding the basis for a Newton-Raphson iteration. The accuracy and performance of the high order finite element scheme is demonstrated by a numerical example. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基面力单元法在空间几何非线性问题中的应用

基于基面力的概念,并结合Euler角的位移描述方法,提出了适用于几何非线性计算的空间6结点余能基面力单元.使用MATLAB语言编程并对典型梁、板结构进行弹性大变形数值模拟.由计算结果可以看出,基于余能原理的基面力元法(BFEM)在计算构件的空间大变形时有较好的计算精度,对比传统有限元计算方法具有网格尺寸影响小和抗畸变能力强的特点,有良好的计算性能.     

An efficient mixed interpolated curved beam element for geometrically nonlinear analysis

In this study, a curved beam element is developed for geometrically nonlinear analysis of planar structures. The main contribution of this research is to use high-performance formulation to alleviate locking phenomena and consider finite rotation. This scheme is based on the mixed interpolation of the strain fields. In this study, special tying points are found and utilized. One of the interesting advantages of the proposed element is the ability to model tapered structures. Moreover, the First-order Shear Deformation Theory (FSDT) and the Green-Lagrange strain are included. Several complicated and applicable nonlinear problems are solved to depict the efficiency and high accuracy of the proposed element, especially by fewer numbers of elements.     

A Piezoelectric Finite Shell Element for the Geometrically Nonlinear Analysis of Sensors

A geometrically nonlinear finite element formulation to analyze piezoelectric shell structures is presented. The formulation is based on the mixed field variational functional of Hu–Washizu. Within this variational principle the independent fields are displacements, electric potential, strains, electric field, stresses and dielectric displacements. The mixed formulation allows an interpolation of the strains and the electric field through the shell thickness, which is an essential advantage when using a three dimensional material law. It is remarked that no simplification regarding the constitutive relation is assumed. The normal zero stress condition and the normal zero dielectric displacement condition are enforced by the independent resultant stress and resultant dielectric displacement fields. The shell structure is modeled by a reference surface with a four node element. Each node possesses six mechanical degrees of freedom, three displacements and three rotations, and one electrical degree of freedom, which is the difference of the electric potential through the shell thickness. The developed mixed hybrid shell element fulfills the in–plane, bending and shear patch tests, which have been adopted for coupled field problems. A numerical investigation of a smart antenna demonstrates the applicability of the piezoelectric shell element under the consideration of geometrical nonlinearity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational differential quadrature: A technique to simplify numerical analysis of structures

A new solution method in the area of computational mechanics is developed in this article, which is called variational differential quadrature (VDQ). The main idea of this method is based on the accurate and direct discretization of the energy functional in the structural mechanics. In the VDQ method, through developing an efficient matrix formulation and using an accurate integral operator, the discretized governing equations are derived directly from the weak form of the equations with no need for the analytical derivation of the strong form. This technique provides an alternative way to discretize the energy functional, which avoids the local interpolation and the assembly process of the methods of this kind. We first implement the VDQ method for the nonlinear elasticity theory considering the Green-St. Venant strain tensor; then we simplify the formulation further for the first-order shear deformable beam and plate theories. The final formulation of these cases demonstrates the simplicity of the implementation for the VDQ method in the numerical analysis of the structures, which is a major goal for this article. Using these examples, one can easily learn and apply this technique to other structures. To assess the performance of the VDQ method, we compare it with the generalized differential quadrature (GDQ) method and finite element method (FEM) in the case of bending analysis of Mindlin plates. It is indicated that computational cost of VDQ is less than that of GDQ, and the convergence rate of VDQ is faster than that of FEM.     

Flexure Hinge Mechanisms Modeled by Nonlinear Euler-Bernoulli-Beams

A flexure hinge is an innovative engineering solution for providing relative motion between two adjacent stiff members by the elastic deformation of an arbitrary shaped flexible connector. In the literature, modeling of compliant mechanisms incorporating flexure hinges is mainly focused on linear methods. However, geometrically nonlinear effects cannot be ignored generally. This study presents a nonlinear modeling technique for flexure hinges based on the Euler-Bernoulli beam theory, in contrast to the predominant linear modeling approaches. Higher order beam elements of variable cross-section are employed to model the flexure hinge region. A Newton-Raphson scheme is applied to solve the resulting nonlinear system equations. The proposed approach reduces the overall degrees of freedom and is computationally efficient compared to commonly applied 3D finite element methods. A compliant displacement amplification mechanism is studied by means of the proposed method, where an excellent agreement with results of a reference solution is achieved. The modeling approach is suitable for the structural optimization of compliant mechanisms towards a less intuitive design process. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite element based nonlinear dynamic response of elastically supported piezoelectric functionally graded beam subjected to moving load in thermal environment with random system properties

The second order statistics in terms of mean and standard deviation (SD) of normalized nonlinear transverse dynamic central deflection (NTDCD) response of un-damped elastically supported functionally graded materials (FGMs) beam with surface-bonded piezoelectric layers under the action of moving load are investigated in this paper. The random system properties such as Young's modulus, Poisson's ratio, density, thermal expansion coefficients, piezoelectric materials, volume fraction exponent and external loading are modeled as uncorrelated random variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics combined with Newton–Raphson technique through Newmark's time integrating scheme using finite element method (FEM). The non-uniform temperature distribution with temperature dependent material properties is taken into consideration for consideration of thermal loading. The one parameter Pasternak elastic foundation with Winkler cubic nonlinearity is considered as an elastic foundation. The stochastic based second order perturbation technique (SOPT) and direct Monte Carlo simulation (MCS) are adopted for the solution of nonlinear dynamic governing equation. The influences of volume fraction exponents, temperature increments, moving loads and velocity, nonlinearity, slenderness ratios, foundation parameters and external loadings with random system properties on the NTDCD are examined. The capability of present stochastic model in predicting the NTDCD statistics are compared by studying their convergence with the existing results those available in the literature.     

Vibrations of spatially suspended flexible pipe-beam with moving fluid

The nonlinear dynamic model of flexible pipe–beam suspended by spatial system of cables is proposed for vibration analysis of pipeline suspension bridges. The model, based on substructure technique, is considered as an assemblage of the following substructures: cables, hangers and pipe–beam. Equation of motion of pipe–beam is derived by Galerkin's FEM with the original finite element formulated in order to include moving mass of transported fluid. To describe cable vibrations, general continuum approach proposed in Ref.[1] is adopted with application into 3D model. Cable model takes into account initial sag, pre–tension force, large displacements and hangers' point reactions. Equation governing the motion of pipe–beam with cables and hangers is obtained regarding equilibrium conditions of interaction forces and compatibility of displacements at connection points between substructures. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Postbuckling characteristics of angle-ply laminated truncated circular conical shells

The postbuckling characteristics of the angle-ply laminated composite conical shells subjected to the torsion, the external pressure, the axial compression, and the thermal loading considering uniform temperature change are studied using the semi-analytical finite element approach. The finite element formulation is based on the first-order shear deformation theory and the field consistency principle. The variations in the stiffness coefficients along the meridional direction due to the changes in the ply-angle and the ply-thickness of the filament wound conical shells are incorporated in the finite element formulation. The nonlinear governing equations are solved using the Newton–Raphson iteration procedure coupled with the displacement control method to trace the prebuckling followed by the postbuckling equilibrium path. The presence of asymmetric perturbation in the form of a small magnitude load spatially proportional to the linear buckling mode shape is considered to initiate the bifurcation of the shell deformation. The influence of semi-cone angle, ply-angle and number of circumferential waves on the prebuckling/postbuckling response of the anti-symmetric angle-ply laminated circular conical shells is investigated.     

A partitioned finite element scheme based on Gauge-Uzawa method for time-dependent MHD equations

In this paper, we mainly introduce a partitioned scheme based on Gauge-Uzawa finite element method for the 2D time-dependent incompressible magnetohydrodynamics (MHD) equations. It is a fully decoupled projection method which combines the Gauge and Uzawa methods within a variational formulation. Firstly, the temporal discretization is based on backward Euler technique for the linear term and semi-implicit scheme for the nonlinear term. Secondly, the spatial approximation of fluid velocity, hydrodynamic pressure, and magnetic field apply the mixed element method. Finally, the validity, reliability, and accuracy of the proposed algorithms are supported by numerical experiments.