Effect of nonlocal elasticity on the performance of a flexoelectric layer as a distributed actuator of nanobeams

The paper is concerned with the development of finite element model for the static analysis of smart nanobeams integrated with a flexoelectric layer on its top surface, using nonlocal elastic theory. The flexoelectric layer acts as a distributed actuator of the nanobeam. A layerwise displacement theory has been used to derive the element stiffness matrices from variational principles incorporating nonlocal effects. The finite element model for nonlocal response of the beams has been validated with the exact solution for the case of a simply supported standalone flexoelectric layer. Also, the finite element model of the simply supported smart beam has been validated with exact solutions and numerical models for the local elastic case. The performance of the flexoelectric actuator has been compared for different values of nonlocal parameters and different combinations of nonlocal and local elastic substrate and flexoelectric layer. Further, the model developed has been utlized for investigating the performance of the active flexoelectric layer in case of cantilever beam, for which the exact solutions are not available.     

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.     

Conference diary

A variational higher-order theory for bending and stretching of linearly elastic orthotropic beams including the deformations due to transverse shearing and stretching of the transverse normal fibre is presented. The theory assumes a linear distribution for the longitudinal displacement and a parabolic variation of the transverse displacement across the thickness. Additionally, independent expansions are introduced for the through-thickness displacement gradients with the requirement of a least-square compatibility for the transverse strains and the satisfaction of exact stress boundary conditions at the top/bottom beam surfaces. The theory is shown to be well suited for finite element development requiring simple C⁰- and C^?1- continuous displacement interpolation fields. To demonstrate the computational utility of the theory, a simple two-node stretching-bending finite element is formulated. The analytic and finite element results are obtained for a simple bending problem for which an exact elasticity solution is available. It is shown that the inclusion of the transverse normal deformation in the present theory enables improved displacement, strain and stress predictions, particularly, in the analysis of deep beams.     

解析型Winkler弹性地基梁单元构造

该文采用Winkler弹性地基梁理论确定了弹性地基梁的挠度方程解析通解; 根据最小势能原理建立了解析型Winkler弹性地基欧拉梁及铁摩辛柯梁的单元刚度及等效节点荷载; 得到了解析型弹性地基欧拉梁单元AWFB-E及铁摩辛柯梁单元AWFB-T。同时,论文还采用传统里兹法求得了相应的Winkler弹性地基欧拉梁及铁摩辛柯梁单元刚度矩阵,得到了里兹法弹性地基欧拉梁单元RWFB-E及铁摩辛柯梁单元RWFB-T。对该文构建的两类单元与一般梁-基体系有限元分析结果及理论解析解进行了对比。对比结果表明,传统里兹法由于其多项式形函数无法精确模拟弹性地基梁变形,因此其结果与理论解析解有误差,但随着单元数量增多其误差减小; 采用解析型单元进行计算时,无论单元数量多少,得到的均为“真实”解,说明解析试函数法求得的位移形函数比一般的多项式形函数精确,得到的弹性地基梁单元具备解析型、精确性的特点,可应用于解决实际工程问题。     

含模糊参数弹性地基梁及桩的有限元分析

利用模糊有限元方法研究了文克尔地基上的梁和桩在确定性荷载作用下的反应,其中梁和桩身材料的物性参数及地基参数均模拟成模糊变量。在上述模糊变量为小变量的前提下,利用一种基于普通摄动法原理的模糊摄动展开方法求解模糊有限元平衡方程组得到结构反应量的模糊集。导出了用模糊摄动展开方法求解弹性地基上梁和桩的计算公式,并研究了结构中各模糊参变量对结构反应量模糊集的影响。经与数值模拟的结果比较后表明,方法在精度和实用性方面可较好的满足要求。     

Finite differences/elements in classical beam problems: derivation of feasibility conditions under parametric inequality constraints with the help of Reduce and REDLOG

The solution of the classical fourth-order ordinary differential equation for static beam problems by using the finite difference method is reconsidered, but this time for the derivation of feasibility conditions in cases of validity of parametric linear inequality constraints with respect to the loading/geometry of the beam. To this end, the computer algebra system Reduce has been used, but supplemented by its recent REDLOG (REDuce LOGic) package incorporating the efficient Weispfenning computational quantifier elimination algorithms. A particular problem for a finite beam loaded by a triangular loading has been employed as the vehicle for the illustration of the present approach and the derived feasibility conditions are displayed. The finite element method has also been used (instead of the finite difference method) in the same problem. The present results can also be generalized to problems of beams on an elastic foundation, to two-dimensional problems, to optimization problems, etc.     

A general finite element for beams or beam-columns with or without an elastic foundation

A general finite element is derived for beams or beam-columns with or without a continuous Winkler type elastic foundation. The need to discretize members into shorter elements for convergence towards an ‘exact’ solution is eliminated by employing in the derivation of the element exact shape functions obtained from the equation of the elastic line. Inter-nodal values of deflections, bending moments and shear forces are obtained using the exact shape functions and trigonometric series. The effect of heavy compressive or tensile axial forces on bending stiffness is treated as a linear problem by considering the axial force as a constant parameter affecting the stiffness. FORTRAN subroutines to compute the stiffness matrix, equivalent nodal forces, deflected shape, bending moments and shear forces are provided and verified by an example.     

Nonlinear analysis of non-uniform beams on nonlinear elastic foundation

In this paper a boundary integral equation solution to the nonlinear problem of non-uniform beams resting on a nonlinear triparametric elastic foundation is presented, which permits also the treatment of nonlinear boundary conditions. The nonlinear subgrade model which describes the foundation includes the linear and nonlinear Winkler (normal) parameters and the linear Pasternak (shear) foundation parameter. The governing equations are derived in terms of the displacements for nonlinear analysis in the deformed configuration and for linear analysis in the undeformed one. Moreover, as the cross-sectional properties of the beam vary along its axis, the resulting coupled nonlinear differential equations have variable coefficients which complicate the mathematical problem even more. Their solution is achieved using the analog equation method of Katsikadelis. Several beams are analyzed under various boundary conditions and load distributions, which illustrate the method and demonstrate its efficiency and accuracy. Finally, useful conclusions are drawn from the investigation of the nonlinear response of non-uniform beams resting on nonlinear elastic foundation.     

Analysis of local bending effects in sandwich plates with orthotropic face layers subjected to localised loads

This paper presents a method for the approximate analysis of local bending effects in sandwich plates with specially orthotropic face layers subjected to localised external loads. The local bending analysis is based on the assumption that the relative deflection of the loaded face against the deflection of the face not loaded can be modelled by application of an elastic foundation model. This is achieved by introducing a two-parameter elastic foundation model which takes into account the shearing interaction effects between the loaded face and the core material. An approximate solution to the complete problem is achieved by superposition of the local solution and an overall solution derived by application of classical sandwich plate theory. The results obtained are compared with finite element analysis results, and a good match between the solutions is observed. Finally a brief parametric study shows that the local bending effects are strongly influenced by the modular ratio and the thickness of the loaded face.     

A Cosserat point element (CPE) for the numerical solution of transient large planar motions of elastic–plastic and elastic–viscoplastic beams

The objective of this paper is to develop constitutive equations of a Cosserat point element (CPE) for the numerical solution of transient large planar motions of elastic–plastic and elastic–viscoplastic beams with rigid cross-sections. Specifically, attention is limited to response of a material with constant yield strength. A yield function is proposed which couples the inelastic responses of tension and shear. Another yield function is proposed for bending which depends on a hardening variable that models motion of the elastic–plastic boundary in the beam’s cross-section. Evolution equations are proposed for elastic strains and the hardening variable and an overstress-type formulation is used for elastic–viscoplastic response. In contrast, with standard finite element approaches the CPE model needs no integration through the element region. Also, an implicit scheme is developed to integrate the evolution equations without iteration. Examples of transient large motions of beams, which are impulsively loaded, indicate that the CPE produces reasonably accurate response relative results in the literature and full three-dimensional calculations using ABAQUS.     

A semi-analytical finite element model for the analysis of laminated 3D axisymmetric shells: Bending, free vibration and buckling

A general semi-analytical finite element model is developed for bending, free vibration and buckling analysis of shells of revolution made of laminated orthotropic elastic material. The 3D elasticity theory is used and the equations of motion are obtained by expanding the displacement field and load in the Fourier series in terms of the circumferential coordinate, θ. The coefficients of the expansion are functions of (r, z), and they are approximated using the finite element method. This leads to a semi-analytical finite element in the (r, z) plane. The element is validated by comparing the present results with the analytical and numerical solutions available in the literature.     

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.     

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 elastostatic FEM/BEM analysis of vertically loaded raft and piled raft foundation

In this paper, a static analysis of vertically loaded raft and piled raft foundations in smooth and continuous contact with the supporting soil is presented. In this approach the finite element method (FEM) and the boundary element method (BEM) are coupled: the bending plate is assumed to have linear elastic properties and is modelled by FEM while the soil is considered as an elastic half-space in the BEM. The pile is represented by a single element and the shear force along the shaft is interpolated by a quadratic function. The plate–soil interface is divided into triangular boundary elements (soil) also called cells and finite elements (plate) and the subgrade reaction is linearly interpolated across each cell. The subgrade tractions are eliminated from the FEM and BEM algebraic systems of equations, resulting in the governing system of equations for plate–pile–soil interaction problems. Numerical results are presented and they are close to those resulting from much more elaborate analyses.     

A new approach to the finite element modelling of beams with warping effect

As a first step toward developing a finite element formulation that can model coupling among extensional, bending and torsional behaviour of beams, a new method is proposed to properly represent the warping of arbitrary cross-sections. The basic approach is to introduce a small warping displacement superimposed over flat cross-sections of a shear-flexible beam in a deformed configuration. Numerical tests involving simple isotropic beams undergoing a small elastic displacement demonstrate the validity of the new approach. The present approach can be extended to composite beams as well as isotropic beams experiencing a large deflection or finite rotation.     

非均匀热载荷作用下功能梯度梁的非线性静态响应

由于功能梯度材料结构沿厚度方向的非均匀材料特性,使得夹紧和简支条件的功能梯度梁有着相当不同的行为特征。该文给出了热载荷作用下,功能梯度梁非线性静态响应的精确解。基于非线性经典梁理论和物理中面的概念导出了功能梯度梁的非线性控制方程。将两个方程化简为一个四阶积分-微分方程。对于两端夹紧的功能梯度梁,其方程和相应的边界条件构成微分特征值问题;但对于两端简支的功能梯度梁,由于非齐次边界条件,将不会得到一个特征值问题。导致了夹紧与简支的功能梯度梁有着完全不同的行为特征。直接求解该积分-微分方程,得到了梁过屈曲和弯曲变形的闭合形式解。利用这个解可以分析梁的屈曲、过屈曲和非线性弯曲等非线性变形现象。最后,利用数值结果研究了材料梯度性质和热载荷对功能梯度梁非线性静态响应的影响。     

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。     

STATIC AND DYNAMIC APPLICATIONS OF A FIVE NODED HORIZONTALLY CURVED BEAM ELEMENT WITH SHEAR DEFORMATION

A five-noded thirteen DOF horizontally curved beam element with or without an elastic base is presented. One set of fourth-degree Lagrangian polynomials in natural co-ordinates is used for interpolation of beam geometry and vertical displacement while the angles of transverse rotation and twist are interpolated by another set of third-degree polynomials. For elastic subgrade, the reactive forces offered at any point are assumed to be proportional to the corresponding displacements at that point. The effect of shear deformation is accounted for in the stiffness matrix. In mass matrix evaluation, for dynamic problems, translational as well as rotary intertias have been considered and studied separately. For numerical integration of the stiffness matrix, a four-point Gaussian scheme has been found to be adequate. Numerical results for a number of sample problems and their comparison with analytical solutions have been presented for circular as well as for non-circular curved beams. Displacements, bending moment and torque for static loading with or without elastic foundation, as well as natural frequencies and mode shapes are computed for different cases. Examples include the problem of a cantilever beam of spiral geometry with different parametric values of the spiral and the agreement with the analytical results establishes the efficacy of the element. The performance of the element has been found be be excellent in both static and dynamic conditions. Sufficient details are presented so that the formulation may be readily used. It is hoped that the large number of numerical illustrations will elucidate the validity and the range of applicability of the element and will also serve as benchmark for future researchers. © 1997 by John Wiley & Sons, Ltd.     

Vibration analysis of beams with non‐local foundations using the finite element method

In this paper, a non‐local viscoelastic foundation model is proposed and used to analyse the dynamics of beams with different boundary conditions using the finite element method. Unlike local foundation models the reaction of the non‐local model is obtained as a weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure. In the finite element analysis, the interpolating shape functions of the element displacement field are identical to those of standard two‐node beam elements. However, for non‐local elasticity or damping, nodes remote from the element do have an effect on the energy expressions, and hence the damping and stiffness matrices. The expressions of these direct and cross‐matrices for stiffness and damping may be obtained explicitly for some common spatial kernel functions. Alternatively numerical integration may be applied to obtain solutions. Numerical results for eigenvalues and associated eigenmodes of Euler–Bernoulli beams are presented and compared (where possible) with results in literature using exact solutions and Galerkin approximations. The examples demonstrate that the finite element technique is efficient for the dynamic analysis of beams with non‐local viscoelastic foundations. Copyright © 2007 John Wiley & Sons, Ltd.     

基于n阶GBT初始轴向载荷影响下FGM梁的振动特性

研究了初始轴向载荷影响下弹性地基功能梯度材料(FGM)梁的振动特性。基于一种拓展的n阶广义剪切变形梁理论(n-GBT),以轴向位移、剪切变形挠度与弯曲变形挠度为基本未知函数,应用Hamilton原理,建立了该系统自由振动问题力学模型的控制方程。引入边界控制参数,采用一种改进型广义微分求积(MGDQ)法获得了FGM梁的静动态响应。通过算例验证并给出了GBT阶次n的理想取值,丰富梁理论的同时,可供验证或改进其它各种剪切变形梁理论;提供的数值分析方法切实可行,拓展了GDQ法的使用范围。最后,着重讨论并分析了初始轴向载荷、边界条件、梯度指标、地基刚度、跨厚比等参数对FGM梁振动特性的影响。