薄板统计分析的随机边界元法

本文建立了分析含随机材料参数的薄板弯曲问题的随机边界元法。基于Ｔａｙｌｏｒ级数展开技术，分别得到了位移的均值和一阶偏差的边界积分方程，发现材料参数的随机性可作为一个等效的随机荷载处理，从而得到边界位移或边界力的均值和协方差，并进一步求出内点位移和力矩的均值和协方差，最后用本文方法计算了两个算例，并对结果进行了必要的分析。     

一种考虑剪切变形的平行四边形厚/薄板通用单元

根据Timoshenko二广义位移梁理论，构造了深梁位移场的插值函数。利用斜坐标系与直角坐标系的变换关系、有限条带思想和深梁位移插值函数，构造了一种考虑剪切变形的平行四边形厚／薄板弯曲通用单元的位移(曲率、剪应变、转角、横向位移)插值函数，导出了刚度矩阵和非结点荷载等效力。并对简支阍支方板、Razzaque斜板、四边简支斜交板弯曲进行了数值计算。算例表明此单元有较好的精度，对于薄板不出现剪切闭锁，可适应于目前桥梁建设中大量采用的斜交板桥结构分析。     

基于广义变分原理的梁板单元分析的数值流形方法

数值流形方法(NMM)是一种基于有限覆盖技术的新型数值方法.以该方法的覆盖位移模式为基础,利用广义变分原理中罚函数理论,详细推导了梁板流形单元的覆盖位移函数,刚度矩阵和应变矩阵,并建立了可应用于梁板单元分析的数值流形方法.最后通过算例分析表明,该方法在对梁板弯曲问题分析是有效的.     

考虑界面滑移效应的组合裂纹梁弯曲解析解

将上部子梁的裂纹等效为线性扭转弹簧,考虑组合梁连接面的滑移位移,建立了以组合裂纹梁挠度和滑移位移为基本未知量的组合裂纹梁弯曲变形一维数学模型．利用Laplace变换及其逆变换,给出了组合裂纹梁弯曲变形一维数学模型的解析通解．在此基础上,研究了均布载荷作用下简支组合裂纹梁的弯曲变形问题,数值分析了连接面剪切刚度、裂纹深度、数目和位置等参数对组合裂纹梁弯曲变形的影响,结果表明：在裂纹处,组合裂纹梁挠度曲线存在尖点,而横截面转角曲线存在跳跃,且随着裂纹数目和深度的增加,挠度和横截面转角跳跃值增大;随着连接面剪切刚度的增加,挠度和横截面转角减小,并最终趋于定值．并且,随着组合梁跨高比的增加,连接面剪切刚度对梁挠度影响逐渐减弱．     

中厚板统计分析的随机边界元法

本文建立了分析含随机材料参数并具厚度不均匀性的中厚板问题的随机边界元法,基于Taylor级数展开技术,分析和到广义位移的均值和一阶偏差的积分方程,其中将材料参数的随机性和厚度的不均匀性作为等效荷载处理,从而得到广义边界位移或面力的均值和协方差,并进一步求出部点广义位移和内力的均值和协方差,最后用本文方法计算了两个数例,并对所得结果进行了分析,探讨。     

考虑轴力二阶效应的损伤梁弯曲摄动解

将损伤梁等效为阶梯型变刚度Euler-Bernoulli梁，利用Heaviside广义函数，给出了阶梯型变刚度梁抗弯刚度的统一表达式．在此基础上，考虑轴向压力二阶效应，并以损伤为摄动参数，得到了均布横向载荷作用下，简支损伤梁弯曲挠度的一阶和二阶摄动解析解，并数值分析了摄动解析解的精度和损伤梁的弯曲变形特性，结果表明：随着轴向压力和刚度损伤参数的增加，挠度一阶和二阶摄动解析解误差增加，挠度二阶摄动解析解误差通常小于其一阶摄动解析解误差，且二阶摄动解的误差很小，满足工程应用的精度．同时，损伤梁的挠度和转角分布与完整梁的挠度和转角分布差异较大，在刚度变化位置处损伤梁转角斜率存在突变．这些结果可为轴力作用下Euler-Bernoulli梁损伤识别提供理论支撑．     

一种考虑剪切变形的矩形板元素刚度矩阵

本文应用应变能分项插值的概念推导了一种具有12个参数的考虑剪切变形的正交各向异性矩形平板元素的刚度矩阵,在计算应变能的近似值时,对不同的项采取了不同的位移函数.基本的位移函数是根据考虑剪切变形的直梁的位移得到的,其中包含了弯曲刚度与剪切刚度的比值D/C,因此得到的刚度矩阵对各种剪切刚度值直至薄板(C→∞)都能应用.刚度矩阵以显式表示,使用方便.对典型问题的静力、动力计算表明结果是良好的.     

Winkler地基上裂纹Timoshenko梁的弯曲解析解

将梁中裂纹等效为无质量线性扭转弹簧,研究了温克勒(Winkler)基础上具有任意开裂纹数目Timoshenko梁的弯曲变形．利用Delta广义函数和Heaviside函数以及Laplace变换,给出了Winkler基础上具有任意裂纹数目Timoshenko梁弯曲变形的解析通解．在此基础上,研究了Winkler基础上受均布荷载作用简支裂纹Timoshenko梁的弯曲变形,数值分析了裂纹数目和位置以及深度、梁剪切刚度和基础反力系数等对裂纹Timoshenko梁弯曲变形的影响．结果表明：在裂纹处,梁挠度存在尖点,转角存在跳跃;梁挠度随着裂纹深度和数目的增加而增加,但横截面弯矩和转角减小;随着基础反力系数的增加,梁挠度、弯矩和转角减小;随着剪切刚度的增加,梁挠度减少,弯矩和转角增大．     

区间五次Hermite样条多小波Euler-Bernoulli梁单元

首先采用区间五次Hermite样条函数,分别构造了三节点梁的边界和中间节点的多小波尺度函数,然后,基于小波多辨分析思想,构建了梁单元位移多尺度近似空间的基函数系;最后,采用最小势能原理,得到弯曲梁的平衡方程,从而构造了区间五次Hermite样条多小波Euler-Bernoulli梁单元。算例结果表明,该小波单元可通过改变尺度来重新划分网格,从而可自由调节单个小波单元的计算精度,其计算精度与在相同网格划分下采用传统三节点Hermite梁单元计算的完全一致;与其它小波单元相比较,该小波单元具有计算简单明了,物理意义明确,易于理解的特点。     

样条有限条塑性铰法分析板梁的极限荷载

用样条有限条塑性铰法分析了板梁的极限荷载。首先对条单元以样条位移函数表达的总势能进行求导而推导了位移-荷载关系式。然后用塑性铰法推导了单元的塑性刚度矩阵。因此该方法兼具二者优点:样条有限条法的位移量少和塑性铰法形成塑性刚度矩阵的便利。它还可以考虑梁的初始缺陷,如残余应力和初弯曲。通过与相关的试验数据比较,证明该方法有效与可靠。     

Experiments and theory in strain gradient elasticity

Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, b_h. To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam's thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, b_h, is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam's bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.     

Bending of Timoshenko beam with effect of crack gap based on equivalent spring model

Considering the effect of crack gap,the bending deformation of the Timoshenko beam with switching cracks is studied.To represent a crack with gap as a nonlinear unidirectional rotational spring,the equivalent flexural rigidity of the cracked beam is derived with the generalized Dirac delta function.A closed-form general solution is obtained for bending of a Timoshenko beam with an arbitrary number of switching cracks.Three examples of bending of the Timoshenko beam are presented.The influence of the beam’s slenderness ratio,the crack’s depth,and the external load on the crack state and bending performances of the cracked beam is analyzed.It is revealed that a cusp exists on the deflection curve,and a jump on the rotation angle curve occurs at a crack location.The relation between the beam’s deflection and load is bilinear,each part corresponding to an open or closed state of crack,respectively.When the crack is open,flexibility of the cracked beam decreases with the increase of the beam’s slenderness ratio and the decrease of the crack depth.The results are useful in identifying non-destructive cracks on a beam.     

基于等效弹簧模型的裂纹Euler-Bernoulli梁弯曲变形分析

考虑裂纹的缝隙效应,研究了开闭裂纹Euler-Bernoulli梁的弯曲变形．首先,将裂纹等效为内部旋转弹簧,利用广义函数,给出了考虑裂纹缝隙影响的Euler-Bernoulli梁的等效抗弯刚度,推导了具有任意数目开闭裂纹梁弯曲变形的显式通解．在此基础上,研究了均布载荷作用下上侧单裂纹简支梁以及裂纹处承受集中力和集中力偶共同作用的固支梁的弯曲变形,分析了梁长细比、裂纹深度和位置以及载荷等对裂纹开闭状态和梁弯曲变形的影响。结果表明：梁挠度分布在裂纹处存在尖点,而转角分布存在跳跃;梁挠度与载荷的响应关系一般为双折线形式,分别对应于裂纹的张开和闭合状态;且裂纹张开时,裂纹梁的柔度随着梁长细比的增加和裂纹深度的减小而减小。这些结果对梁裂纹无损检测具有指导意义．     

基于等效黏性弹簧的黏弹性Timoshenko裂纹梁弯曲解析解

考虑裂纹的缝隙和黏性效应,将梁中横向裂纹等效为黏弹性扭转弹簧,利用广义Delta函数,给出了Laplace变换域内裂纹梁的等效抗弯刚度,得到了具有任意开闭裂纹数目且满足标准线性固体黏弹性本构的Timoshenko梁在时间域内的弯曲变形显式解析通解．在此基础上,通过两个数值算例,分析了时间、梁跨高比和裂纹深度等参数对黏弹性Timoshenko开裂纹梁弯曲变形的影响．结果表明：裂纹黏性对Timoshenko裂纹梁的弯曲具有显著的影响．相比于裂纹的弹性扭转弹簧模型,考虑裂纹黏性效应的黏弹性Timoshenko裂纹梁在裂纹处挠度尖点和转角跳跃现象十分明显．另外,由于横向剪切引起的附加变形,Timoshenko裂纹梁的稳态挠度与Euler-Bernoulli梁挠度的差值为常数,其大小与裂纹模型、梁跨高比或裂纹深度无关,这些结果对梁裂纹无损检测具有指导意义．     

On the static and dynamic stability of thin beam conveying fluid

In this paper, numerical investigation of the statical and dynamical stability of aligned and misaligned viscoelastic cantilevered beam is performed with a terminal nozzle in the presence of gravity in two cases: (1) effect of fluid velocity on the flutter boundary of beam conveying fluid and (2) effect of gravity on the buckling boundary of beam conveying fluid. The beam is assumed to have a large width-to-thickness ratio, so the out-of-plane bending rigidity is far higher than the in-plane bending and torsional rigidities. Gravity vector is considered in the vertical direction. Thus, deflection of the beam because of the gravity effect couples the in-plane bending and torsional equations. The beam is modeled by Euler–Bernoulli beam theory, with the flow-induced inertia, Coriolis and centrifugal forces along the beam considered as a distributed load along the beam. Furthermore, the end nozzle is regarded as a lumped mass and modeled as a follower axial force. The extended Hamilton’s principle and the Galerkin method are utilized to derive the bending–torsional equations of motion. The coupled equations of motion are solved as eigenvalue problems. Also, several cases are examined to study the impact of gravity, beam inclination angle, mass ratio, nozzle aspect ratio, bending-to-bending rigidity ratio and bending-to-torsional rigidity ratio on flutter and buckling margin of the system.

     

Deterministic and Stochastic Response of Nonlinear Coupled Bending-Torsion Modes in a Cantilever Beam

The purpose of this study is to understand the main differences between the deterministic and random response characteristics of an inextensible cantilever beam (with a tip mass) in the neighborhood of combination parametric resonance. The excitation is applied in the plane of largest rigidity such that the bending and torsion modes are cross-coupled through the excitation. In the absence of excitation, the two modes are also coupled due to inertia nonlinearities. For sinusoidal parametric excitation, the beam experiences instability in the neighborhood of the combination parametric resonance of the summed type, i.e., when the excitation frequency is in the neighborhood of the sum of the first bending and torsion natural frequencies. The dependence of the response amplitude on the excitation level reveals three distinct regions: nearly linear behavior, jump phenomena, and energy transfer. In the absence of nonlinear coupling, the stochastic stability boundaries are obtained in terms of sample Lyapunov exponent. The response statistics are estimated using Monte Carlo simulation, and measured experimentally. The excitation center frequency is selected to be close to the sum of the bending and torsion mode frequencies. The beam is found to experience a single response, two possible responses, or non-stationary responses, depending on excitation level. Experimentally, it is possible to obtain two different responses for the same excitation level by providing a small perturbation to the beam during the test.     

Optimization of a metal honeycomb sandwich beam-bar subjected to torsion and bending

In this paper, we analyze a metal honeycomb sandwich beam/torsion bar subjected to combined loading conditions. The cell wall arrangement of the honeycomb core is addressed in the context of maximizing resistance to either bending, torsion, or combined bending and torsion for given dimensions, face sheet thicknesses and core relative density. It is found that the relative contributions of the honeycomb core to torsion and bending resistances are sensitive to the configuration of cell walls and the optimal properties significantly exceed those of stochastic metallic foams as sandwich beam core materials for this configuration.     

Surface effects of adsorption-induced resonance analysis on micro/nanobeams via nonlocal elasticity

The governing differential equation of micro/nanbeams with atom/molecule adsorption is derived in the presence of surface effects using the nonlocal elasticity.The effects of the nonlocal parameter,the adsorption density,and the surface parameter on the resonant frequency of the micro/nanobeams are investigated.It is found that,in addition to the nonlocal parameter and the surface parameter,the bending rigidity and the adsorption-induced mass exhibit different behaviors with the increase in the adsorption density depending on the adatom category and the substrate material.     

Stochastic response of an axially loaded composite Timoshenko beam exhibiting bending–torsion coupling

A spectral finite element method is proposed to investigate the stochastic response of an axially loaded composite Timoshenko beam with solid or thin-walled closed section exhibiting bending–torsion materially coupling under the stochastic excitations with stationary and ergodic properties. The effects of axial force, shear deformation (SD) and rotary inertia (RI) as well as bending–torsion coupling are considered in the present study. First, the damped general governing differential equations of motion of an axially loaded composite Timoshenko beam are derived. Then, the spectral finite element formulation is developed in the frequency domain using the dynamic shape functions based on the exact solutions of the governing equations in undamped free vibration, which is used to compute the mean square displacement response of axially loaded composite Timoshenko beams. Finally, the proposed method is illustrated by its application to a specific example to investigate the effects of bending–torsion coupling, axial force, SD and RI on the stochastic response of the composite beam.     

用切比雪夫多项式研究短梁的弯曲计算

考虑剪切效应,利用切比雪夫多项式构造严格满足表面切应力边界条件的轴向位移表达式,建立了短梁弯曲问题的新理论．利用奇异函数把作用在短梁上的复杂外载荷表示为分布载荷,推导出了短梁弯曲时的截面正应力公式及挠曲线表达式．把采用切比雪夫多项式推导出短梁的弯曲计算公式计算结果与弹性理论计算结果进行比较,可知该方法的计算精度较高．研究结果表明：在复杂外载荷作用下,当长高比小于等于6时,剪切变形对梁的弯曲挠度影响较大,而当长高比小于3时,剪切变形对梁的弯曲应力影响较大;因此建议采用切比雪夫多项式方法给出的挠度表达式、弯曲应力进行计算,因为切比雪夫多项式方法不但给出了复杂外载荷作用下梁截面挠度、弯曲应力的计算通式,而且该方法具有计算过程简便、精度高的优点．