Actuator placement in static bending of smart beams utilizing Mohr’s analogy

An extension of Mohr’s analogy to bending of shear-deformable beams with eigenstrain-type actuation, such as a piezoelectric actuation, is presented first. Various refined shear-deformable beam theories are included by means of a theory-dependent parameter. The one-dimensional version of Reissner’s sixth-order plate theory is exemplarily addressed. The Bernoulli–Euler theory of beams rigid in shear, as well as the shear-deformable theory of Timoshenko, are included as special cases. Afterwards, the extended Mohr analogy is applied to the bending of smart beams with piezoelectric patch actuators. The following special problem of static shape control is solved: Seek a placement of single patch actuators, such that the displacement and the cross-sectional rotation vanish at some pre-selected locations of the beam, despite the beam is loaded by external forces. Using Mohr’s analogy, it is shown that the auxiliary loading of the adjoint beam must form a self-equilibrated system of loading in order to achieve the latter goal. The high potential of the proposed actuator placement is demonstrated for the case of a cantilever beam with a single force acting at the tip. Placements of single actuators are represented such that the tip displacement and the tip cross-sectional rotation vanish. The outcomes of shear-deformable theories are compared to the Bernoulli–Euler theory and to a Finite Element computation using piezoelectrically coupled elements.     

Timoshenko悬臂梁自由振动特性的近似分析方法

本文讨论了用模态摄动法分析Timoshenko悬臂梁的自由振动特性。Timoshenko梁是目前较为流行的梁模型之一,Timoshenko梁方程一般较复杂,很难给出解的解析形式。利用模态摄动法,将Timoshenko梁方程转化为一组非线性代数方程组的求解,不仅可以简化计算过程,而且计算结果具有较高的精度,对任意边界条件下的Timoshenko梁都适用。通过算例,讨论了剪切变形和转动惯量对悬臂梁的固有频率的影响。     

基于精细梁模型的向量式有限元分析

精细梁不同于Euler梁和Timoshenko梁,该模型在考虑剪切变形的同时还考虑了横向弯曲时截面转动产生的附加轴向位移及横向剪切变形影响截面抗弯刚度后产生的附加横向位移。推导了适用于向量式有限元分析的精细梁单元应变和内力表达式,采用FORTRAN自编了向量式有限元程序。对悬臂梁、两端固支梁和门式框架进行了算例分析,对比了采用不同梁单元模型下结构的竖向位移。结果表明:当高跨比较小时,3种梁单元的竖向位移相差不大;当高跨比较大时,精细梁单元的竖向位移较Euler梁和Timoshenko梁明显增大,表明剪切变形及刚度折减引起的附加轴向位移、附加横向位移不能忽略。精细梁单元模型对高跨比较大的梁进行分析可望得到更精确的结果。     

任意开口薄壁截面圆弧曲梁弯扭精确分析

现有薄壁曲梁弯扭理论缺乏严密的理论推导。本文基于薄壁构件分析的两个基本假定，导出了任意开口薄壁截面圆弧曲梁的翘曲位移、正应力、剪应力及其各自合力以及平衡微分方程的精确表示式。为便于应用，本文还给出了一套具有良好精度的内力简化公式及相应的平衡微分方程。     

考虑隧道剪切效应的基坑开挖对邻近隧道纵向变形分析

 基坑施工将打破周围地层已平衡的应力场,引起应力释放,对既有下卧地铁隧道产生不利的影响。针对既有研究的不足,提出考虑隧道剪切效应的基坑开挖对下卧隧道影响的解析解。既有隧道简化为搁置于Winkler地基上的Timoshenko梁。通过两阶段分析法,考虑基坑卸荷作用下已建隧道的响应。首先基于Timoshenko梁理论,建立考虑隧道剪切效应的隧道纵向变形微分方程,然后将由Mindlin弹性解计算得到附加荷载施加于既有隧道上,最后通过有限差分法得到在附加荷载作用下隧道的纵向变形解答。收集3个已发表工程实例的实测数据,并与本文方法及Euler-Bernoulli梁法的计算结果进行对比分析,发现实测结果与2种方法得计算结果有较好的一致性。然而,在内力分析上,相对比于本文方法,Euler-Bernoulli梁法明显高估基坑卸荷引起的隧道弯矩与剪力。由于本文方法可有效模拟既有隧道剪切效应,因而可进一步给出1在卸荷作用下隧道管片间错台量。研究成果可为合理预测邻近基坑施工对既有隧道的影响提供一定的理论支持。     

考虑桩基剪切效应的盾构开挖对邻近桩基水平向变形分析

盾构隧道开挖将引起土体卸荷,会对邻近既有桩基产生不利影响,为探究既有桩基在邻近盾构开挖下的水平向响应规律,针对已有研究的不足,提出了一种在邻近盾构开挖下能考虑桩基剪切效应的计算桩基水平向响应的简化算法。首先既有桩基被视为置于Pasternak双参数地基上的铁木辛柯梁,然后由两阶段法建立起考虑桩基剪切效应的桩基水平向变形控制微分方程,利用有限差分法解得附加荷载作用下的桩基水平向变形解析解。通过离心试验、监测数据的对比,验证了该方法的合理性。并在给定的桩土参数下,把同时考虑桩基剪切和弯曲变形的解析解与只考虑桩基弯曲变形的退化解进行对比,发现较之于欧拉–伯努利梁法,盾构对桩基扰动程度的影响,扰动越大,剪切效应体现越明显,由铁木辛柯梁法计算的位移值与弯矩值更接近实测数据。     

轴向荷载作用下梁结构动力特性的随机分析

传统梁结构动力特性分析方法由于忽略结构参数的不确定性而不能满足实际工程设计需要,因此,将响应面法(RSM)、有限元法(FEM)和蒙特卡罗法各自的优点相结合,提出一种新的结构随机动力特性计算方法-混合分析法。该方法的最大特点是有效地利用响应面法(RSM)、有限元法(FEM)和蒙特卡罗法(MCS)各自的优点,并将其充分地结合起来。然后,运用该方法分析一简支梁自振频率的统计值。结果表明基于确定性模型的梁结构频率响应分析仅能给出频率响应的均值,忽略结构随机参数对频率响应的影响。为获得准确的频率响应值,有必要在今后的结构动力特性分析中考虑结构参数随机性的影响。最后,进行梁结构随机动力特性计算的敏感性因素分析,并指明影响梁结构动力特性的主要随机因素。     

Derivation of the exact stiffness matrix for a two-layer Timoshenko beam element with partial interaction

This paper presents the full closed-form solution of the governing equations describing the behaviour of a shear-deformable two-layer beam with partial interaction. Timoshenko’s kinematic assumptions are considered for both layers, and the shear connection is modelled through a continuous relationship between the interface shear flow and the corresponding slip. The limiting cases of perfect bond and no bond are also considered. The effect of possible transversal separation of the two members has been neglected. With the above assumptions, the present work can be considered as a significant development beyond that available from Newmark et al.’s paper [4]. The differential equations derived considering the above key assumptions have been solved in closed form, and the corresponding “exact” stiffness matrix has been derived using the standard procedure basically inspired by the well-known direct stiffness method. This “exact” stiffness matrix has been implemented in a general displacement-based finite element code, and has been used to investigate the behaviour of shear-deformable composite beams. Both a simply supported and a continuous beam are considered in order to validate the proposed model, at least within the linear range. A parametric analysis has been carried out to study the influence of both shear flexibility and partial interaction on the global behaviour of composite beams. It has been found that the effect of shear flexibility on the deflection is generally more important for composite beams characterized by substantial shear interaction.     

Simulation of linear and non-linear propagation effects of a random turbulence field on bridge flutter instability

This paper presents the results of an analytical and numerical investigation on the implications associated with the propagation of an “uncertain” turbulence field on the aeroelastic stability of long-span bridges. Even though the influence of turbulence on flutter has received the attention of the wind engineering community in the recent past, an alternative formulation for simulating turbulence effects on flutter is proposed, in the general context of uncertainty and modeling error simulation. The coupled-mode flutter threshold, limited to the fundamental bending and torsional modes of a bridge, is estimated in the time domain by stochastic calculus techniques. A generalized modal correlation length is used to simulate turbulence modeling errors. Both linear and non-linear random propagation is investigated by means of stochastic stability. A numerical integration scheme is employed for the solution of the dynamic equations and to derive second-moment stability conditions for non-linear case. A set of simplified bridge examples is analyzed. Either a decrement or an increment in the critical velocity is observed for the non-linear random turbulence case, depending on the selected example. Nevertheless, this variation is usually small, of the order of a few percents in comparison with the “turbulence-free” scenario.     

Out-of-plane behaviour of partially composite or sandwich beams by exact and Finite Element Methods

The out-of-plane vibrations of composite beams with interlayer slip or three-layer sandwich beams are theoretically and numerically investigated in this paper for general boundary conditions. The governing dynamics equations are derived by applying the Hamilton's principle. A Finite Element Resolution is presented for general boundary conditions, and compared to the exact solution based on the resolution of a tenth-order differential equation. The Finite Element Method may exhibit slip locking phenomenon for very stiff connection, a phenomenon widely investigated in the past for the in-plane behaviour of partially composite beams or sandwich beams. This slip locking, analogous to the shear locking for Timoshenko beams, can be faced with some relevant interpolation shape functions of the same order for each kinematics variables, namely the deflections and the torsion angle. The numerical results are presented for layered wood beams and laminated glass beams, with particular emphasis on the rate of convergence of the natural frequencies with respect to the number of Finite Elements. It is theoretically and numerically shown that the elastic spectra of the symmetrical composite beam are composed of two independent spectrums. One spectrum is independent of the connection parameter and can be studied using the solution of the non-composite action, whereas the second spectrum can be obtained from the resolution of a third-order polynomial equation using the Cardano's method. We show the phenomenon of cut-on frequency for this out-of-plane problem, a phenomenon already noticed for the in-plane Timoshenko beam vibrations. The exact method associated to a 10 degrees-of-freedom shape function can be formally associated with the dynamics stiffness method. The numerical and the exact approaches lead to the same dimensionless spectra, up to four digits.     

Bending and buckling of inflatable beams: Some new theoretical results

The non-linear and linearized equations are derived for the in-plane stretching and bending of thin-walled cylindrical beams made of a membrane and inflated by an internal pressure. The Timoshenko beam model combined with the finite rotation kinematics enables one to correctly account for the shear effect and all the non-linear terms in the governing equations. The linearization is carried out around a pre-stressed reference configuration which has to be defined as opposed to the so-called natural state. Two examples are then investigated: the bending and the buckling of a cantilever beam. Their analytical solutions show that the inflation has the effect of increasing the material properties in the beam solution. This solution is compared with the three-dimensional finite element analysis, as well as the so-called wrinkling pressure for the bent beam and the crushing force for the buckled beam. New theoretical and numerical results on the buckling of inflatable beams are displayed.     

Exact solutions for thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment

The exact solutions for torsional analysis of thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment are presented for the first time. For this, a general thin-walled composite beam theory with arbitrary lamination is developed by introducing Vlasov's assumption and the equilibrium equations and the force–deformation relations are derived from the energy principle. Applying the displacement state vector consisting of 14 displacement parameters and the nodal displacements at both ends of the beam, the displacement functions are derived exactly. Then, the exact stiffness matrix for torsional analysis is determined using the force–deformation relations. As a special case, the closed-form solutions for symmetrically laminated composite beams with various boundary conditions are derived. Finally, the finite element procedure based on Hermitian interpolation polynomial is developed. To demonstrate the validity and the accuracy of this study, the numerical solutions are presented and compared with the closed-form solutions and the finite element results using the Hermitian beam elements and ABAQUS's shell elements.     

地基上混凝土梁的温度应力

地基上混凝土梁常需控制温度应力以避免裂缝。本文首先简要介绍马斯洛夫应力解,然后用弹性力学方法给出地基对梁的温度变形的抗力系数,从而得到地基上梁的温度应力的一套完整解法。根据地基弹性模量及梁的长高比,对梁所受到的地基约束作用分为4类:(a)弯曲变形完全约束,水平变形部分约束;(b)弯曲变形完全约束,水平变形无约束;(c)弯曲变形部分约束,水平变形无约束;(d)水平变形和弯曲变形均无约束。对于每类约束情况分别给出了简洁的温度应力计算公式,使计算得到简化。给出了混凝土弹性模量和温度随龄期而变化时的计算方法。指出了目前工程界采用的地基上梁温度应力计算方法的不足之处。     

工字形截面圆弧曲梁的非线性理论

从任意开口薄壁截面圆弧曲梁精确的翘曲位移出发 ,针对常见的单轴对称工字形、槽形和无对称轴的H形截面曲梁 ,通过参数分析找到一个统一的数学表达式 ,给出相应的应力、应变计算式。在这个基础上 ,依据大变形理论 ,建立了工字形截面圆弧曲梁非线性分析的基本理论 ,并考虑了横向正应力的影响。推导过程中未就材料性质做任何假定 ,所以该理论同样可以应用于曲梁弹塑性阶段的分析。最后给出单轴对称工字形截面两种常见放置条件下曲梁的总势能表达式。     

复杂条件下Winkler地基梁的解析解

基于梁的弯曲微分方程、有限元理论和最小势能原理(虚功原理),利用梁单元间的位移、转角、弯矩和剪力协调条件,地基梁的整体平衡条件,给出了复杂条件下Winkler地基梁的一种计算方法。此方法能对Winkler地基梁在受集中力、集中力偶和任意形式的分布荷载共同作用时进行解析解求解,而且地基梁的截面和弹性模量、地基的基床系数可以分段不同。此方法能得到各梁单元和整个地基梁的沉降位移表达式,且这一表达式与简单条件下的Winkler地基梁经典解有相同的结构形式;根据与沉降位移表达式之间的导数关系,可以进一步得到转角、弯矩和剪力表达式。算例表明,计算结果与一般有限元方法的计算结果完全吻合。另外,此方法虽然在概念上属于有限元方法的一种,但单元划分方法与一般有限元方法存在本质差别,单元个数也远远少于一般有限元方法的单元个数。     

压实黏土梁弯曲开裂性状试验研究

裂缝的存在显著降低黏土层的防渗性能。研制了土梁弯曲试验装置,研究了不同含水率压实黏土梁张拉开裂性状,分析了裂纹扩展过程,并采用PIV技术揭示了裂纹尖端的应变局部化现象,获得了不同含水率下压实黏土的开裂应变。试验结果表明:三点弯梁拉伸区可能出现多个应变局部化区域,其中靠近跨中的一个发展为宏观裂纹;裂纹宽度随中点挠度近似呈线性增长,且受含水率影响较小;含水率对土梁开裂特性和开裂应变影响很大,随着含水率由32.6%增至37.0%,土梁断裂挠度显著增加,裂纹尖端的应变局部化区域变大,开裂应变从2.4%增大至4.1%,跨中荷载挠度曲线呈现由跳回型向跳过型变化,荷载峰值明显降低,土梁由脆性断裂逐渐转变为延性断裂。     

Zur Vorhersage des Rißfortschritts bei einbetonierten Kopfbolzendübeln in Verbundträgern unter nicht ruhender Belastung

Realistic prediction of crack propagation of headed studs in composite beams under cyclic loading. The article gives appropriate cyclic load‐displacement‐laws and crack‐propagation‐laws in composite gaps of beams with headed studs ∅︁ 22 mm as well as realistic analytical procedures to determine the load‐displacement state of a composite beam under cyclic loading. The given models are based on a series of displacement controlled Push‐Out‐Tests and verified by cyclically loaded beam tests. Using the derived relationships it is now possible to predict rather exactly the fatigue lifetime of headed studs in composite beams. Further a design method in the limit state of fatigue is proposed.     

Timoshenko beam effect in flexural resonant column tests

Current methods for deriving Young’s modulus (E) from flexural resonant column tests do not account for the Timoshenko beam effect. Hence, the extent of the Timoshenko beam effect is investigated in this study using a series of numerical experiments performed by finite element program LS-DYNA. Prior to conducting the numerical experiments, LS-DYNA is verified for a cantilever beam to ensure that it is capable of accounting for the Timoshenko beam effect. The Young’s modulus determined with the existing interpretation method for flexural resonant column tests, which does not consider the Timoshenko beam effect, is underestimated on average by about 11% for a typical resonant column specimen. A correction factor is proposed here to account for the Timoshenko beam effect using the existing interpretation method for flexural resonant column tests.     

Longitudinal structural modelling of shield tunnels considering shearing dislocation between segmental rings

Shield tunnels in soft deposits are often subjected to differential settlement and longitudinal structural deformation during long-term operation. Since the shear stiffness in circumferential joints between rings is relatively low, shearing induced dislocation between segmental rings becomes a significant aspect of longitudinal deformation of tunnels. The longitudinal deformation mode comprises both rigid rotation of segmental rings (bending mode) and dislocation between rings (dislocation mode). The existing modelling method is usually based on Euler–Bernoulli beam theory, which only considers the bending effect and ignores the shearing deformation. This paper proposes a new longitudinal structural model to consider the shearing dislocation between rings. In the proposed model, the tunnel is simplified as a homogenous Timoshenko beam, and an equivalent shear stiffness is proposed to consider the influence of joints. Since Timoshenko beam theory considers both flexural deformation and shear deformation of the beam, it can describe the actual deformation mode of a tunnel reasonably. The proposed model is compared with the traditional Euler–Bernoulli beam model based on a field measured longitudinal deformation of a Shanghai metro tunnel. The results indicate that the traditional Euler–Bernoulli model overestimates the longitudinal internal forces of a tunnel. The proposed model based on Timoshenko beam theory predicts a smaller joint opening and a greater dislocation between rings than the Euler–Bernoulli beam model.     

Reduction of vibration on a cantilever Timoshenko beam subjected to repeated sequence of excitation with magnetorheological outriggers

This paper deals with the statistical effects of an outrigger system on a cantilever beam under seismic excitation. The nonstationary random approach is employed to simulate seismic events. The Timoshenko beam approach is used to model the frame‐core tube linked at a point of its length by the damped outriggers, therefore are connected vertically two magnetorheological damper devices. The peak root‐mean‐square values of displacement responses is employed as a best measure effective to specify the optimal locations of outriggers according to different vibration modes. To evaluate the performance of the control system, the control algorithm based on Lyapunov stability theory is adopted to seek the input voltage leading to the reduction of vibration.