Some closed-form solutions in random vibration of Bresse-Timoshenko beams

Random vibration of simply supported uniform Bresse-Timoshenko beams is considered under ‘rain-on-the-roof’ (stationary space- and time -wise ideal white noise) excitation. An approximate differential equation is used with both shear distortion and rotary inertia included, but with the term which covers the simultaneous action of these effects omitted. A closed-form solution is derived for the displacement and velocity space-time correlation function of the Bresse-Timoshenko beam with transverse damping, generalizing the corresponding result by Eringen for the classical Bernoulli-Euler beam. Closed-form solutions are also derived for beams with structural or Voigt damping mechanisms. The mean-square value of the stress diverges for both the classical Bernoulli-Euler and Bresse-Timoshenko beams with transverse damping, but converges for the beam possessing structural damping. The main finding of this study is identity of the space-time correlation functions of displacement according to the refined Bresse-Timoshenko and classical Bernoulli-Euler theories, when the joint action of rotary inertia and shear deformation is neglected for the beam under the ‘rain-on-the-roof’ excitation. This remarkable coincidence takes place for beams possessing (a) transverse viscous damping, (b) Voigt damping, and (c) combined rotary and transverse viscous damping.     

Deflection of thick beams of multimodular materials

A transfer-matrix analysis is presented for determining the static behaviour of thick beams of ‘multimodular materials’ (i.e. materials which have different elastic behaviour in tension and compression, with nonlinear stress–strain curves approximated as piecewise linear, with four or more segments). To validate the transfer-matrix method results, a closed-form solution is also presented for cases in which the neutral-surface location is constant along the beam axis. Numerical results for axial displacement, transverse deflection, bending slope, bending moment, transverse shear, axial force and location of neutral surface are presented for multimodular and bimodular models of unidirectional aramid cordrubber. The transfer-matrix method results agree very well with the closed-form solutions.     

湿-热-机-弹耦合FGM梁的稳定性及振动特性

研究了初始轴向机械载荷作用下Winkler-Pasternak弹性地基上功能梯度材料（FGM）梁在湿-热环境中的稳定性及振动特性。假设温度和湿度沿梁厚度方向稳态分布,材料的物性依赖于温度且按Voigt混合幂律模型连续分布。首先,基于一种扩展的n阶广义梁理论,应用Hamilton原理,统一建立了以轴向位移、弯曲变形项挠度及剪切变形项挠度为基本未知函数FGM梁的屈曲及自由振动方程,采用Navier解法获得了FGM简支梁静动态响应的精确解。其次,通过算例验证并给出了该广义梁理论阶次n的理想取值,丰富梁理论的同时,可供验证或改进其他各种剪切变形梁理论。最后,着重探讨了3种湿-热分布下湿度与温度增加、初始轴向机械载荷、跨厚比、地基刚度、梯度指标等诸多参数对FGM梁稳定性和振动特性的影响。     

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.     

基于最大耗能增效原则的惯容减震系统解析设计公式

惯容减震系统在高效吸能的同时可放大其内部耗能装置的变形，从而提高耗能减震效率，此耗能增效特性是惯容减震系统的关键机理，可作为物理意义明确的惯容系统设计原则。但目前惯容耗能增效设计方法的计算公式复杂，不便应用。为充分发挥惯容系统的耗能增效特性，提出以“最大耗能增效”为设计原则，以满足性能需求为直接导向，进而推导出解析形式的惯容减震系统设计公式。基于随机振动理论得出了单自由度惯容减震结构在白噪声激励下均方响应的解析式，并结合极值条件及零固有阻尼假定完成了惯容系统最优参数的推导。在解析公式的基础上给出了惯容减震系统基于最大耗能增效原则的设计流程，并编制相应程序进行了单自由度惯容减震结构的算例设计与验证。算例分析结果验证了该文设计原则与解析公式的正确性与有效性。     

考虑自平衡条件槽形曲梁的静力学特性分析

考虑了剪滞翘曲应力自平衡条件、剪力滞后、剪切变形和翘曲扭转等因素的影响,以最小势能原理为基础,建立了薄壁槽形曲梁的弹性控制微分方程和自然边界条件,获得了弯、扭、翘和剪滞效应相耦合广义位移的闭合解。算例中,分析了不同荷载形式、曲梁半径R、宽跨比等因素对曲线槽形梁力学特性的影响。该文解析法更好揭示了曲线槽形梁的力学特性以及各参数之间的内在关系,所得公式是对曲梁剪滞理论的发展。     

Large amplitude vibration analysis of composite beams: Simple closed-form solutions

Large amplitude vibration analysis of laminated composite beam with axially immovable ends is investigated with symmetric and asymmetric layup orientations by using the Rayleigh–Ritz (R–R) method. The displacement fields used in the analytical formulation are coupled by using the homogeneous governing static axial equilibrium equation of the beam. Geometric nonlinearity of von-Karman type is considered which accounts for the membrane stretching action of the beam. The simple closed-form solutions are presented for the nonlinear harmonic radian frequency as function of central amplitude of the beam using the R–R method. The nonlinear harmonic radian frequency results obtained from the closed-form solutions of the R–R method in general show good agreement with the results obtained from simple iterative finite element formulation. Furthermore, the closed-form expressions are corrected for the harmonic motion assumption from the available literature results on the existence of quadratic and cubic nonlinearity. It is interesting to note that the composite beams can result in asymmetric frequency vs. amplitude curves depending upon the nature of direction of displacement in contrast to isotropic beams which exhibit cubic nonlinearity only and leads to symmetric frequency vs. amplitude curves with respect to sign of the amplitude.     

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。     

剪切可变形梁热过屈曲解析解

该文导出了面内热载荷作用下,梁过屈曲问题的精确解。首先基于非线性一阶剪切变形梁理论,推导了控制轴向和横向变形的基本方程。然后,将3 个非线性方程化简为一个关于横向挠度的四阶非线性积分-微分方程。该方程与相应的边界条件构成了微分特征值问题。直接求解该问题,得到了热过屈曲构形的闭合解,这个解是外加热载荷的函数。利用精确解,得到了临界屈曲载荷的一阶结果与经典结果的解析关系。为考察热载荷、横向剪切变形以及边界条件的影响,根据得到的精确解给出了两端固定、两端简支以及一端固定一端简支边界条件下的具体数值算例,讨论了梁在面内热载荷作用下的过屈曲行为,并与经典结果进行了比较。该文得到的精确解可以用于验证或改进各类近似理论和数值方法。     

Thermo-elastic mismatch in nonhomogeneous beams

The problem of thermo-elastic stress analysis in multi-layered nonhomogeneous beams is considered. The proposed analytical approach based on the multi-layered beam theory permits to take into account an arbitrary distribution of the Young’s modulus, of the thermal-expansion coefficient, and of the temperature variation along the beam depth. The effect of shear deformability of the interfaces is also carefully analyzed. Useful closed-form solutions for the normal stresses in the layers and for the interface tangential stresses are provided in the case of nonhomogeneous bi- and tri-layered beams. The obtained results show the effectiveness of using functionally graded materials to relieve stress-concentrations due to the thermo-elastic mismatch typical of laminated beams with homogeneous layers.     

Inelastic bending of beam including transverse shear

An incremental force–displacement relation is derived using the finite element matrix method to study the inelastic bending of beams including transverse shear. Solutions are obtained for the case of a cantilever beam with a point load and a simply supported beam with distributed loading considering and also ignoring the effect of transverse shear. They show good agreement for those cases where solutions are available. Also, the effect of transverse shear is found to be quite significant.     

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.     

Analytical solution for a functionally graded beam with arbitrary graded material properties

The plane stress problem of an orthotropic functionally graded beam with arbitrary graded material properties along the thickness direction is investigated by the displacement function approach for the first time. A general two-dimensional solution is obtained for a functionally graded beam subjected to normal and shear tractions of arbitrary form on the top and bottom surfaces and under various end boundary conditions. For isotropic case explicit solutions are given to some specific through-the-thickness variations of Young’s modulus such as exponential model, linear model and reciprocal model. The influence of different grade models on the stress and displacement fields are illustrated in numerical examples. These analytical solutions can serve as a basis for establishing simplified theories and evaluating numerical solutions of functionally graded beams.     

The modified couple stress model for bending of normal deformable viscoelastic nanobeams resting on visco-Pasternak foundations

ABSTRACT

The modified couple stress theory (MCST) is utilized to investigate the bending of viscoelastic nanobeams laying on visco-Pasternak elastic foundations based on a new shear and normal deformations beam theory. This model consists of the material length scale coefficient that captures the size impact on small-scale beams. The simply supported beam is made of viscoelastic material, subjected to time harmonic transverse load. The nanobeam is presumed to be laying on double layers of foundations. The first layer is modeled as Kelvin–Voigt viscoelastic model and the second is taken as a shear layer. Based on the proposed beam theory and MCST, the differential motion equations are deduced using Hamilton’s principle. To check the validity of the obtained formulations, the predicted results are compared with those available in the open literature. In addition, the influences of various parameters such as the material length scale parameter, length-to-depth ratio, viscoelastic damping structure, the stiffness and damping coefficients of the viscoelastic substrate, and shear and normal strains on the deflection and stresses are illustrated.     

Mesh‐free models for static analysis of smart laminated composite beams

This paper is concerned with the development of mesh‐free models for the static analysis of smart laminated composite beams. The overall smart composite beam is composed of a laminated substrate composite beam and a piezoelectric layer attached partially or fully at the top surface of the substrate beam. The piezoelectric layer acts as the distributed actuator layer of the smart beam. A layer‐wise displacement theory and an equivalent single‐layer theory have been used to derive the models. Several cross‐ply substrate beams are considered for presenting the numerical results. The responses of the smart composite beams computed by the present new mesh‐free model based on the layer‐wise displacement theory excellently match with those obtained by the exact solutions. The mesh‐free model based on the equivalent single‐layer theory cannot accurately compute the responses due to transverse actuation by the piezoelectric actuator. The models derived here suggest that the mesh‐free method can be efficiently used for the numerical analysis of smart structures. Copyright © 2016 John Wiley & Sons, Ltd.     

Transverse Shear and Normal Deformation Theory for Bending Analysis of Laminated and Sandwich Elastic Beams

The influence of transverse normal strain on bending analysis of cross-ply laminated and sandwich beams is presented. A higher-order shear deformation beam theory is developed. Euler-Bernoulli classical, Timoshenko first-order and simple higher-order theories have been also used in the analysis. The governing equations for a beam composed of orthotropic layers and subjected to any given mechanical load distribution are derived. Making use of Navier-like approach, exact solutions are obtained for cross-ply laminated and sandwich beams subjected to arbitrary loadings. Numerical results for beams with the simply-supported boundary conditions are presented. The effects due to transverse normal strain, transverse shear deformation and number of layers on the static response of the beams are investigated.     

黏弹性轴向运动变张力梁非线性动力学

在黏弹性轴向运动梁横向参数振动的非线性动力学行为研究中,首次计入因速度变化引起的、沿梁的径向变化的、轴向变张力的影响。给出描述变张力轴向运动梁横向非线性振动的偏微分—积分控制方程。基于微分求积法给出轴向运动梁横向非线性参数振动的数值解,通过观察梁中点的位移、速度随时间变化的历程,识别轴向运动系统的非线性动力学行为。同时,通过从数值解中提取的相图、Poincaré映射图和频谱分析,考察轴向运动梁横向振动的分岔与混沌特性,揭示了工程应用中的非线性轴向运动系统的混沌动力学行为。     

An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model

This paper is concerned with the free transverse vibration of cracked nanobeams modeled after Eringen's nonlocal elasticity theory and Timoshenko beam theory. The cracked beam is modeled as two segments connected by a rotational spring located at the cracked section. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. The governing equations of cracked nanobeams with two symmetric and asymmetric boundary conditions are derived; then these equations are solved analytically based on concerning basic standard trigonometric and hyperbolic functions. Besides, the frequency parameters and the vibration modes of cracked nanobeams for variant crack positions, crack ratio, and small scale effect parameters are calculated. The vibration solutions obtained provide a better representation of the vibration behavior of short, stubby, micro/nanobeams where the effects of small scale, transverse shear deformation and rotary inertia are significant.     

基于响应的梁损伤识别

采用扭转弹簧模拟悬臂梁的损伤，导出了损伤梁位移模态和转角模态的近似公式，获得了损伤梁在单点激励下，零初始条件的位移和转角响应。发现损伤梁的转角响应在损伤处发生阶跃变化，而损伤梁的转角响应对位置的一阶偏导数在损伤处有脉冲两数型突变的特点，从而提出了基于损伤梁转角响应的损伤判据函数。通过对损伤梁和损伤桁架结构的数值模拟，表明提出的判据函数不仅可以利用简谐激励下的响应识别梁的损伤，电可以利用冲击激励下的响应识别桁架的损伤。实验结果表明利用所提出的判据函数，可以同时判别损伤的位置和损伤的程度。