Deflection analysis of micro-scale microstretch thermoelastic beam resonators under harmonic loading

In this paper, the flexural vibrations of homogeneous, isotropic, generalized micropolar microstretch thermoelastic thin Euler–Bernoulli beam resonators, due to time harmonic load have been investigated. The axial ends of the beam are assumed to be at either clamped-clamped, simply supported-simply supported or clamped-free conditions. The governing equations have been solved analytically by using Laplace transforms technique twice with respect to time and space variables respectively. The inversion of Laplace transform in time domain has been performed by using the calculus of residues to obtain deflection. The numerical simulation has been carried out with the help of MATLAB software for magnesium like material. The graphical representations and observations have been discussed for deflection of beam under various boundary conditions and for distinct considered values of time and space as well.     

Influence of homogenization models on size-dependent nonlinear bending and postbuckling of bi-directional functionally graded micro/nano-beams

In this work, different homogenization schemes are employed to analyze both size-dependent postbuckling and nonlinear bending behavior of micro/nano-beams, made of a bi-directional functionally graded material (BDFGM), under external axial compression and distributed load. To such different homogenization models, including Reuss, Voigt, Mori-Tanaka, and Hashin–Shtrikman bounds schemes, together with nonlocal strain gradient elasticity theory are adopted within the framework of refined exponential shear deformation beam theory, to develop a comprehensive size-dependent BDFGM beam model. Deviation of associated physical neutral plane, from mid-plane counterpart, is also considered. Nonlocal strain gradient load-deflection responses of BDFGM micro/nano-beam are obtained by numerical solution methodology for both nonlinear bending and postbuckling behaviors corresponding to different values of the lateral and longitudinal material property indices and various small scale parameters. We observed that by decreasing the values of material property gradient indices, associated with BDFGM, difference between the estimations of various homogenization schemes is raised. We also indicated that increasing maximum deflection, decreasing the significance of nonlocal size effect on the bending strength of BDFGM micro/nano-beams, whereas strain gradient size effect becomes more important. In addition, we found that at lower material property gradient indices, bending strength reduction in BDFGM micro/nano-beams, causes by the axial gradient property is higher than lateral gradient property. At higher values of these indices, however, the trend is opposite.     

Buckling and lateral buckling interaction in thin-walled beam-column elements with mono-symmetric cross sections

Effects of axial forces on beam lateral buckling strength are investigated here in the case of elements with mono-symmetric cross sections. A unique compact closed-form is established for the interaction of lateral buckling moment with axial forces. This new equation is derived from a non-linear stability model. It includes first order bending distribution, load height level and effect of mono-symmetry terms (Wagner’s coefficient and shear point position). Compared to the so-called three-factors (C₁–C₃) formula commonly employed in beam lateral buckling stability, another factor C₄ is added in presence of axial loads. Pre-buckling deflection effects are considered in the study and the case of doubly-symmetric cross sections is easily recovered. The proposed solutions are validated and compared to finite element simulations where 3D beam elements including warping are used. The agreement of the proposed solutions with bifurcations observed on the non-linear equilibrium paths is good. Dimensionless interaction curves are dressed for the beam lateral buckling strength and the applied axial load, where the flexural-torsional buckling axial force is a taken as reference.     

Thermal effects on buckling of shear deformable nanocolumns with von Kármán nonlinearity based on nonlocal stress theory

Thermal buckling of nanocolumns considering nonlocal effect and shear deformation is investigated based on the nonlocal elasticity theory and the Timoshenko beam theory. By expressing the nonlocal stress as nonlinear strain gradients and based on the variational principle and von Kármán nonlinearity, new higher-order differential governing equations with corresponding higher-order nonlocal boundary conditions both in transverse and axial directions for instability of nanocolumns are derived. New analytical solutions for some practical examples on instability of nanocolumns are presented and analyzed in detail. The paper concluded that the critical buckling load is significantly increased in the presence of nonlocal stress and the results confirm that nanocolumn stiffness is enhanced by nanoscale size effect and reduced by shear deformation. The critical temperature change is increased with larger diameter to length ratio and higher nonlocal nanoscale. It is also concluded that at low and room temperatures the buckling load of nanocolumns increases with increasing temperature change, while at high temperature the buckling load decreases with increasing temperature change.     

粘弹性梁的动力响应

本文利用Voigt力学模型对粘弹性简支梁进行动力分析,得到了梁的自由振动与强迫振动的若干解析解的表达式.并与S.Timoshenko给出的弹性简支梁的相应的结论进行了比较,指出了弹性动力分析的局限性.最后给出了二个数值例子.     

基于非局部弹性应力场理论的纳米尺度效应研究:纳米梁的平衡条件、控制方程以及静态挠度

该文成功地解答了3个关于非局部应力理论用于纳米梁的问题:(ⅰ) 在绝大多数研究中,非局部效应增加导致纳米结构体刚度下降,其现象表现为弯曲挠度增加,固有频率减少,屈曲载荷下降,但为什么Eringen 的非局部弹性理论给出了完全相反的结论;(ⅱ) 为什么在某些研究结果中,非局部效应消失或是对研究结果无影响,比如纳米悬臂梁在集中载荷作用下的弯曲挠度; (ⅲ) 在高阶控制方程中,为什么高阶边界条件不存在．通过应用非局部弹性理论和精确变分原理分析纳米梁的弯曲问题,推导出全新的平衡条件、控制方程、边界条件和静态响应．这些方程和条件包含了与之前的相关研究结果符号相反的高阶微分项,这一差别导致了纳米效应对结构体的影响结果完全相反． 还证明之前为大家所公认的纳米梁静态或动态平衡条件实际上没有达到平衡,只有用等效弯矩代替非局部弯矩时,才可达到平衡．这些结论通常是可以被其它方法,比如应变梯度理论、耦合应力模型以及相关实验所证明．     

Assessment of an edge type settlement of above ground liquid storage tanks using a simple beam model

Analysis of deformation and bending moment distributions along sections of the bottom plate of a large unanchored cylindrical liquid storage tank with appreciable out-of-plane localized differential edge settlement is considered. The analysis uses approximate simple slender beam bending theory to model localized edge settlements of the plate and takes into account the effects of foundation compliance, initial settlement shape, shell and hydrostatic loadings and the shell-bottom plate junction stiffness. The obtained model is solved, in the elastic range, using a combined analytical–numerical procedure for the deflection and bending moment distributions along the beam. The obtained approximate solutions were displayed graphically for selected values of system parameters: edge settlement amplitude, plate thickness, foundation stiffness, and hydrostatic load. The maximum allowable edge displacement amplitudes based on the plate yielding stress predicted by the present study are compared for the selected values of system parameters with those recommended in the API standard 653.     

Static analysis of functionally graded beams using higher order shear deformation theory

Displacement field based on higher order shear deformation theory is implemented to study the static behavior of functionally graded metal–ceramic (FGM) beams under ambient temperature. FGM beams with variation of volume fraction of metal or ceramic based on power law exponent are considered. Using the principle of stationary potential energy, the finite element form of static equilibrium equation for FGM beam is presented. Two stiffness matrices are thus derived so that one among them will reflect the influence of rotation of the normal and the other shear rotation. Numerical results on the transverse deflection, axial and shear stresses in a moderately thick FGM beam under uniform distributed load for clamped–clamped and simply supported boundary conditions are discussed in depth. The effect of power law exponent for various combination of metal–ceramic FGM beam on the deflection and stresses are also commented. The studies reveal that, depending on whether the loading is on the ceramic rich face or metal rich face of the beam, the static deflection and the static stresses in the beam do not remain the same.     

Analysis of functionally graded plates using higher order shear deformation theory

This work addresses a static analysis of functionally graded material (FGM) plates using higher order shear deformation theory. In the theory the transverse shear stresses are represented as quadratic through the thickness and hence it requires no shear correction factor. The material property gradient is assumed to vary in the thickness direction. Mori and Tanaka theory (1973) [1] is used to represent the material property of FGM plate at any point. The thermal gradient across the plate thickness is represented accurately by utilizing the thermal properties of the constituent materials. Results have been obtained by employing a C° continuous isoparametric Lagrangian finite element with seven degrees of freedom for each node. The convergence and comparison studies are presented and effects of the different material composition and the plate geometry (side-thickness, side–side) on deflection and temperature are investigated. Effect of skew angle on deflection and axial stress of the plate is also studied. Effects of material constant n on deflection and the temperature distribution are also discussed in detail.     

不可压饱和多孔弹性梁的大挠度非线性数学模型

在孔隙流体仅存在沿梁轴线方向扩散的假定下,建立了微观不可压饱和多孔弹性梁大挠度问题的非线性数学模型．利用Galerkin截断法,研究了固定端不可渗透、自由端可渗透的饱和多孔弹性悬臂梁在自由端突加集中载荷作用下的非线性弯曲,得到了梁骨架的挠度、弯矩以及孔隙流体压力等效力偶等的时间响应和沿轴线的分布．比较了大挠度非线性和小挠度线性理论的结果,揭示了两者间的差异．研究发现大挠度理论的结果小于相应的小挠度理论结果,并且,大挠度理论的结果趋于其稳态值的时间小于相应的小挠度理论结果趋于其稳态值的时间．     

Stability and vibration analysis of axially-loaded shear beam-columns carrying elastically restrained mass

Classical shear beams only consider the deflection resulting from sliding of parallel cross-sections, and do not consider the effect of rotation of cross-sections. Adopting the Kausel beam theory where cross-sectional rotation is considered, this article studies stability and free vibration of axially-loaded shear beams using Engesser’s and Haringx’s approaches. For attached mass at elastically supported ends, we present a unified analytical approach for obtaining a characteristic equation. By setting natural frequencies to be zero in this equation, critical buckling load can be determined. The resulting frequency equation reduces to the classical one when cross-sections do not rotate. The mode shapes at free vibration and buckling are given. The frequency equations for shear beam-columns with special free/pinned/clamped ends and carrying concentrated mass at the end can be obtained from the present. The influences of elastic restraint coefficients, axial loads and moment of inertia on the natural frequencies and buckling loads are expounded. It is found that the Engesser theory is superior to the Haringx theory.     

高阶剪切变形板理论K■rm■n型方程及在热后屈曲分析中的应用

本文基于Reddy高阶剪切变形板理论导出Karman型非线性大挠度方程并用于层合板热后屈曲分析．分析中计及板初始几何缺陷和热效应．给出了四边简支．对称正交铺设层合板在均匀或非均匀抛物型热分布作用下的后屈曲分析．采用摄动-Galerkin混合法确定板的热屈曲载荷与热后屈曲平衡路径．同时讨论了横向剪切变形，板长宽比，铺层数以及初始几何缺陷等各种参数变化的影响．     

Nonlocal effect on axially compressed buckling of triple-walled carbon nanotubes under temperature field

This paper studies the small scale effect on the buckling behaviors of triple-walled carbon nanotubes (TWCNTs) with the initial axial stress under the temperature field. The TWCNTs are modeled as three elastic shells coupled together through vdW interaction between different layers. Buckling governing equations of CNTs are firstly formulated on the basis of nonlocal elastic theory and the small scale effect on CNTs buckling results with the change of temperature are then achieved. The results show that the critical buckling load is dependent on the temperature, scale parameter and wavenumber. Some conclusions are drawn that small scale effect will arise gradually with the increases of wavenumber, and the temperature can influence the ratio between the nonlocal buckling load and the corresponding local load. Furthermore, with or without effects of nonlocal considered, the same results is obtained that the axial buckling load increases as the value of temperature increases at low and room temperature condition, while at high temperature condition the axial buckling load decreases as the value of temperature increases.     

Analytical solution for nonlocal coupled thermoelasticity analysis in a heat-affected MEMS/NEMS beam resonator based on Green–Naghdi theory

In this article, an analytical solution is presented for coupled thermoelasticity analysis (with energy dissipation) in a micro/nano beam resonator, considering small scale effects on the transient behaviors of fields’ variables. The Green–Naghdi (GN) theory of generalized coupled thermoelasticity and nonlocal Rayleigh beam theory (NRBT) are employed to derive the temperature and lateral deflection in the closed forms. The presented analytical solution is based on Laplace transform. To find the dynamic and transient behaviors of fields’ variables in time domain, an inversion Laplace technique is utilized, which is called Talbot method. The effects of some parameters such as small scale parameter and dimensions of the beam on the dynamic behaviors of temperature and lateral deflections are discussed in details. The propagation of wave fronts in both temperature and lateral deflection domains are obtained and graphically illustrated at various time instants.     

Finite element dynamic analysis of unsymmetric composite laminated beams with shear effect and rotary inertia under the action of moving loads

The finite element dynamic response of an unsymmetric composite laminated orthotropic beam, subjected to moving loads, has been studied. One-dimensional finite element based on classical lamination theory, first-order shear deformation theory, and higher-order shear deformation theory having 16, 20 and 24 degrees of freedom, respectively, are developed to study the effects of extension, bending, and transverse shear deformation. The theories also account for the Poisson effect, thus, the lateral strains and curvatures can be expressed in terms of the axial and transverse strains and curvatures and the characteristic couplings (bend–stretch, shear–stretch and bend–twist couplings) are not lost. The dynamic response of symmetric cross-ply and unsymmetric angle-ply laminated beams under the action of a moving load have been compared to the results of an isotropic simple beam. The formulation also has been applied to the static and free vibration analysis.     

Nonlocal coupled photo-thermoelasticity analysis in a semiconducting micro/nano beam resonator subjected to plasma shock loading: A Green-Naghdi-based analytical solution

In this article, coupled photo-thermoelasticity analysis is carried out using an analytical method in a semiconducting micro/nano beam resonator, considering Green – Naghdi theory (with energy dissipation) and small scale effects. The governing equations for temperature and displacement fields are derived using Eringen nonlocal theory combined with Rayleigh beam theory. One end of the assumed semiconducting MEMS/NEMS is excited by three types of suddenly increasing carrier density and temperature as the plasma and thermal shock loading. The transient behaviours of carrier density field are studied and the effects of disturbances in plasma field on other fields including temperature and deflection are obtained using the proposed analytical solution. The presented analytical solution is based on Laplace transform. To find the dynamic and transient behaviours of fields’ variables in time domain, an inversion Laplace technique is utilized, which is called Talbot method. The effects of small scale parameter and dimensions of the semiconducting micro/nano beam on the dynamic behaviours of fields’ variables are discussed in detail. The axial wave propagation and the distribution of fields’ variables along axial direction are studied at various times.     

A simple mathematical model for approximate analysis of tall buildings

The focus of this article is to present a new and simple mathematical model that may be used to determine the optimum location of a belt truss reinforcing system on tall buildings such that the displacements due to lateral loadings would generate the least amounts of stress and strain in building’s structural members. The effect of belt truss and shear core on framed tube is modeled as a concentrated moment applied at belt truss location, this moment acts in a direction opposite to rotation created by lateral loads. The axial deformation functions for flange and web of the frames are considered to be cubic and quadratic functions respectively; developing their stress relations and minimizing the total potential energy of the structure with respect to the lateral deflection, rotation of the plane section, and unknown coefficients of shear lag, the mathematical model is developed. The proposed model shows a good understanding of structural behavior; easy to use, yet reasonably accurate and suitable for quick evaluations during the preliminary design stage which requires less time. Numerical examples are given to demonstrate the ease of application and accuracy of the proposed modeled.     

Effect of random system properties on bending response of thermo-mechanically loaded laminated composite plates

In this paper, effect of random variation in system properties on bending response of geometrically linear laminated composite plates subjected to transverse uniform lateral pressure and thermal loading is examined. System parameters such as the lamina material properties, expansion of thermal coefficients, lamina plate thickness and lateral load are modeled as basic random variables. The basic formulation is based on higher order shear deformation theory to model the system behavior of the composite plate. A C⁰ finite element method in conjunction with the first order perturbation technique procedure developed earlier by authors for the plate subjected to lateral loading is employed to obtain the second order response statistics (mean and variance) of the transverse deflection of the plate. Typical numerical results for the second order statistics of the transverse central deflection of geometrically linear composite plates with temperature independent and dependent material properties subjected to uniform temperature and combination of uniform and linearly varying temperature distribution are obtained for various combinations of geometric parameters, uniform lateral pressures, staking sequences and boundary conditions. The performance of the stochastic laminated composite model is demonstrated through comparison of mean transverse central deflection with those results available in literature and standard deviation of the deflection with an independent Monte Carlo simulation.     

基于修正偶应力理论的Timoshenko微梁模型和尺寸效应研究

基于修正偶应力理论,将Timoshenko微梁的应力、偶应力、应变、曲率等基本变量,描述为位移分量偏导数的表达式.根据最小势能原理,推导了决定Timoshenko微梁位移场的位移场控微分方程.利用级数法求解了任意载荷作用下Timoshenko简支微梁的位移场控微分方程,得到了反映尺寸效应的挠度、转角及应力的偶应力理论解.通过对承受余弦分布载荷Timoshenko简支微梁的数值计算,研究了Timoshenko微梁的挠度、转角和应力的尺寸效应,分析了Poisson比对Timoshenko微梁力学行为及其尺寸效应的影响.结果表明:当截面高度与材料特征长度的比值小于5时,Timoshenko微梁的刚度和强度均随着截面高度的减小而显著提高,表现出明显的尺寸效应;当截面高度与材料特征长度的比值大于10时,Timoshenko微梁的刚度与强度均趋于稳定,尺寸效应可以忽略;材料Poisson比是影响Timoshenko微梁力学行为及尺寸效应的重要因素,Poisson比越大Timoshenko微梁刚度和强度的尺寸效应越显著.该文建立的Timoshenko微梁模型,能有效描述Timoshenko微梁的力学行为及尺寸效应,可为微电子机械系统（MEMS）中的微结构设计与分析提供理论基础和技术参考.