Optimal design of a beam subject to bending: a basic application

The minimisation of both the mass and deflection of a beam in bending is addressed in the paper. To solve the minimisation problem, a multi-objective approach is adopted by imposing the Fritz John conditions for Pareto-optimality. Constraints on the maximum stress and elastic stability (buckling) of the structure are taken into account. Additional constraints are set on the beam cross section dimensions. Three different cross sections of the beam are analysed and compared, namely the hollow square, the I-shaped and the hollow rectangular cross sections. The analytical expressions of the Pareto-optimal sets are derived. As expected, the I-shaped beam exhibits the best compromise in structural performance, which is related on the particular loading considered.     

Vibration analysis of a stepped laminated composite Timoshenko beam

The goal of this study is to investigate the vibration characteristics of a stepped laminated composite Timoshenko beam. Based on the first order shear deformation theory, flexural rigidity and transverse shearing rigidity of a laminated beam are determined. In order to account for the effect of shear deformation and rotary inertia of the stepped beam, Timoshenko beam theory is then used to deduce the frequency function. Graphs of the natural frequencies and mode shapes of a T300/970 laminated stepped beam are given, in order to illustrate the influence of step location parameter exerts on the dynamic behavior of the beam.     

Three-dimensional finite element modeling of a vibrating beam with a breathing crack

This paper develops a full three-dimensional finite element model in order to study the vibrational behavior of a beam with a non-propagating surface crack. In this model, the breathing crack behavior is simulated as a full frictional contact problem between the crack surfaces, while the region around the crack is discretized into three-dimensional solid finite elements. The governing equations of this non-linear dynamic problem are solved by employing an incremental iterative procedure. The extracted response is analyzed utilizing either Fourier or continuous wavelet transforms to reveal the breathing crack effects. This study is applied to a cracked cantilever beam subjected to dynamic loading. The crack has an either uniform or non-uniform depth across the beam cross-section. For both crack cases, the vertical, horizontal, and axial beam vibrations are studied for various values of crack depth and position. Coupling between these beam vibration components is observed. Conclusions are extracted for the influence of crack characteristics such as geometry, depth, and position on the coupling of these beam vibration components. The accuracy of the results is verified through comparisons with results available from the literature.     

Reflection of a beam of elastic waves by a periodic surface profile

Reflection of elastic waves from a traction-free solid-air boundary of periodic saw-tooth profile is investigated analytically and experimentally. For an incident plane wave the surface displacements on the profile are computed as the solution of a singular integral equation. The reflected field is subsequently obtained by using an integral representation. Incident beams of finite width are represented by Fourier superpositions of plane waves. The dependence of the reflected signal spectra on the incident beam width is examined closely near the fundamental surface resonance frequency. Experimental spectra which were obtained using two different diameter transducers, are compared to the corresponding theoretical spectra. It is found that the depth of the spectral minima depends on the incident beam width. Both analytical and experimental results exhibit the splitting of an incident beam of elastic waves into two reflected beams. The beam splitting is more pronounced for a narrower incident beam and for frequencies close to a resonance frequency of the profile.     

The dynamic response of a strain-softening beam

The dynamic response of a strain-softening beam subjected to a transverse impulsive on its tip is investigated. A softening moment-curvature relation is assumed for the beam and a closed form solution is obtained for a special kind of load, which shows that there exists a softening region in the beam and this region propagates along the beam. This result indicates that, except for the possible discrete softening points with rotation discontinuity caused by the deformation localization^[1], the existence of the softening region and its travelling along the beam are the essential features of the dynamic response of a strain-softening beam. The results also show that the failure of the beam should take place under a special load and the critical condition on which the dynamic failure occurs is given. The project supported by National Natural Science Foundation of China     

A microstructure-dependent Timoshenko beam model based on a modified couple stress theory

A microstructure-dependent Timoshenko beam model is developed using a variational formulation. It is based on a modified couple stress theory and Hamilton's principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Timoshenko beam theory. Moreover, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, which differ from existing Timoshenko beam models. The newly developed non-classical beam model recovers the classical Timoshenko beam model when the material length scale parameter and Poisson's ratio are both set to be zero. In addition, the current Timoshenko beam model reduces to a microstructure-dependent Bernoulli-Euler beam model when the normality assumption is reinstated, which also incorporates the Poisson effect and can be further reduced to the classical Bernoulli-Euler beam model. To illustrate the new Timoshenko beam model, the static bending and free vibration problems of a simply supported beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. Also, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that by the classical model, with the difference between them being significantly large only for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally. Finally, the Poisson effect on the beam deflection, rotation and natural frequency is found to be significant, which is especially true when the classical Timoshenko beam model is used. This indicates that the assumption of Poisson's effect being negligible, which is commonly used in existing beam theories, is inadequate and should be individually verified or simply abandoned in order to obtain more accurate and reliable results.     

旋转楔形-回形梁的高次动力学建模和末端变形分析

对四种不同结构中心刚体-柔性Euler Bernoulli梁系统进行刚柔耦合动力学分析．其中以等截面梁、变截面梁、等截面回形梁、变截面回形梁为对象,研究楔形梁及回形梁对系统的末端变形位移影响．变截面梁的宽高尺寸沿着轴向线性变化．梁的变形包含了轴向、横向、耦合变形项（横向弯曲引起的纵向缩短）．采用假设模态法和第二类Lagrange方程建立系统的动力学方程,并用C++编写软件进行动力学仿真．研究表明：在相同条件下,梁的截面尺寸及空心部分对梁末端变形位移影响十分明显,且当梁在较大变形情况下,该高次耦合模型依然能得到正确的结果,因此在针对实际结构建模时,建立符合实际截面的模型至关重要．     

自由梁受集中质量两点撞击的刚塑性动力响应

对矩形截面自由梁在两端同时受到完全相同的集中质量横向撞击问题进行了理论上的研究 ,通过采用刚塑性的材料模型得到了其动力响应完全解。结合数值方法给出了梁的瞬态变形 ,并讨论了输入能量、质量比等参数对梁的最终变形、能量耗散的影响。针对典型算例将完全解的结果与MSC/Dytran的计算结果进行了比较 ,两者具有合理的近似 ,但理论预测的结果略高估计了梁的最终变形。     

Impact of a surfacewave on an elastic hull

In order to model a ship hull’s response to the impact of surface waves, the two-dimensional problem of wave impact on an elastic beam whose ends are connected by springs with a rigid structure uniformly submerged in a fluid is considered. The fluid is assumed to be ideal and incompressible and its flow symmetric; the lateral bending of the beam is described by the Euler equation. The fluid flow and the size of the wetted region are determined simultaneously with the calculation of the the beam deflection within the framework of the Wagner approach which takes into account the reshaping of the free surface of the fluid on interacting with a body. The stresses and strains arising in the beam and at its ends during impact are found. The numerical algorithm developed makes it possible to analyze the elastic effects in fluid impacts on thin-walled structures of finite length. Moreover, as the stiffness of the connecting springs tends to zero, the solution of this problem describes the impact of an elastic beam with free ends on a weakly curved fluid surface.     

Out-of-plane responses of a circular curved Timoshenko beam due to a moving load

For the cases of using the finite curved beam elements and taking the effects of both the shear deformation and rotary inertias into consideration, the literature regarding either free or forced vibration analysis of the curved beams is rare. Thus, this paper tries to determine the dynamic responses of a circular curved Timoshenko beam due to a moving load using the curved beam elements. By taking account of the effect of shear deformation and that of rotary inertias due to bending and torsional vibrations, the stiffness matrix and the mass matrix of the curved beam element were obtained from the force–displacement relations and the kinetic energy equations, respectively. Since all the element property matrices for the curved beam element are derived based on the local polar coordinate system (rather than the local Cartesian one), their coefficients are invariant for any curved beam element with constant radius of curvature and subtended angle and one does not need to transform the property matrices of each curved beam element from the local coordinate system to the global one to achieve the overall property matrices for the entire curved beam structure before they are assembled. The availability of the presented approach has been verified by both the existing analytical solutions for the entire continuum curved beam and the numerical solutions for the entire discretized curved beam composed of the conventional straight beam elements based on either the consistent-mass model or the lumped-mass model. In addition to the typical circular curved beams, a hybrid curved beam composed of one curved-beam segment and two identical straight-beam segments subjected to a moving load was also studied. Influence on the dynamic responses of the curved beams of the slenderness ratio, moving-load speed, shear deformation and rotary inertias was investigated.     

Modeling of system dynamics of a slewing flexible beam with moving payload pendulum

When a tower crane is handling payload via rotation and moving the carriage simultaneously the jib structure and the payload can be modeled as a system consisting of a slewing flexible clamed-free beam with the spherical payload pendulum that moves along the beam. The present work completes the dynamic modeling of the system mentioned above. The clamed-free beam attached to a rotating hub is modeled by Euler–Bernoulli beam theory. The payload is modeled as a sphere pendulum of point mass attached to via massless inextensible cable the carriage moving on the rotating beam. Non-linear coupled equations of motion of the in- and out-of-plane of the beam and the payload pendulum are derived by means of the Hamilton principle. Some remarks are made on the equations of motion.     

An improved dynamic model for a silicone material beam with large deformation

Dynamic modeling for incompressible hyperelastic materials with large deformation is an important issue in biomimetic applications. The previously proposed lower-order fully parameterized absolute nodal coordinate formulation (ANCF) beam element employs cubic interpolation in the longitudinal direction and linear interpolation in the transverse direction, whereas it cannot accurately describe the large bending deformation. On this account, a novel modeling method for studying the dynamic behavior of nonlinear materials is proposed in this paper. In this formulation, a higher-order beam element characterized by quadratic interpolation in the transverse directions is used in this investigation. Based on the Yeoh model and volumetric energy penalty function, the nonlinear elastic force matrices are derived within the ANCF framework. The feasibility and availability of the Yeoh model are verified through static experiment of nonlinear incompressible materials. Furthermore, dynamic simulation of a silicone cantilever beam under the gravity force is implemented to validate the superiority of the higher-order beam element. The simulation results obtained based on the Yeoh model by employing three different ANCF beam elements are compared with the result achieved from a commercial finite element package as the reference result. It is found that the results acquired utilizing a higher-order beam element are in good agreement with the reference results, while the results obtained using a lower-order beam element are different from the reference results. In addition, the stiffening problem caused by volumetric locking can be resolved effectively by applying a higher-order beam element. It is concluded that the proposed higher-order beam element formulation has satisfying accuracy in simulating dynamic motion process of the silicone beam.     

Suppression of multiple modal resonances of a cantilever beam by an impact damper

Impact dampers are usually used to suppress single mode resonance. The goal of this paper is to clarify the difference when the impact damper suppresses the resonances of different modes. A cantilever beam equipped with the impact damper is modeled. The elastic contact of the ball and the cantilever beam is described by using the Hertz contact model. The viscous damper between the ball and the cantilever beam is modeled to consume the vibrational energy of the cantilever beam. A piecewise ordinary differential-partial differential equation of the cantilever beam is established, including equations with and without the impact damper. The vibration responses of the cantilever beam with and without the impact damper are numerically calculated. The effects of the impact absorber parameters on the vibration reduction are examined. The results show that multiple resonance peaks of the cantilever beam can be effectively suppressed by the impact damper. Specifically, all resonance amplitudes can be reduced by a larger weight ball. Moreover, the impacting gap is very effective in suppressing the vibration of the cantilever beam. More importantly, there is an optimal impacting gap for each resonance mode of the cantilever beam, but the optimal gap for each mode is different.     

Wake of a cylinder: a paradigm for energy harvesting with piezoelectric materials

Short-length piezoelectric beams were placed in the wake of a circular cylinder at high Reynolds numbers to evaluate their performance as energy generators. The coherent vortical structures present in this flow generate a periodic forcing on the beam which when tuned to its resonant frequency produces maximum output voltage. There are two mechanisms that contribute to the driving forcing of the beam. The first mechanism is the impingement of induced flow by the passing vortices on one side of the beam, and the second is the low pressure core region of the vortices which is present at the opposite side of the beam. The sequence of these two mechanisms combined with the resonating conditions of the beam generated maximum energy output which was also found to vary with the location in the wake. The maximum power output was measured when the tip of the beam is about two diameters downstream of the cylinder. This power drops off the center line of the wake and decays with downstream distance as (x/D)^−3/2.     

均质简支梁简化为单自由度系统条件研究

简支梁是土木工程中的常见结构形式，工程计算中常将其简化为单自由度系统进行初步分析和计算．本文系统地讨论了简谐荷载作用下将简支梁简化为单自由度系统进行竖向位移计算的限定条件．对限定条件分析发现将梁上质量的一半集中到梁跨中处所得近似结果精度最好．给出公式可以通过单自由度系统的位移得到梁上任意点处的竖向位移．本文对结构力学教材中的集中质量法进行了进一步的分析和讨论，给出的近似计算公式对土木工程中简支梁在简谐力作用下竖向位移的快速估算具有一定的参考价值．     

Nonlinear vibration of a rotating cantilever beam in a surrounding magnetic field

The nonlinear vibrations of a rotating cantilever beam made of magnetoelastic materials surrounded by a uniform magnetic field are investigated. The kinetic energy, potential energy and work done by the electromagnetic force are obtained. A nonlinear dynamic model, based on the Hamilton principle, which includes the stretching vibration and bending vibration is presented. The Galerkin method is adopted to discretize the dynamic equations. The proposed method is validated by comparison with the literature. The nonlinear behaviors of the responses are studied. Then simulations for different kinds of magnetic field are conducted. The effects of magnetic field parameters, including the amplitude, plane angle, spatial angle and time-varying frequency, on the dynamic behaviors of the stretching motion and bending motion are investigated in detail. The results illustrate that the interaction effects between the rotating cantilever beam and the magnetic field will increase the vibration amplitude and fluctuation of the beam. In particular, we found that: collinear magnetic fields with equal amplitude lead to the same dynamic responses; the amplitude of magnetic field intensity increases the dynamic responses remarkably; the response amplitude changes nonlinearly with the plane angle and spatial angle of the magnetic field; and the increase of time-varying frequency enhances dynamic responses of the rotating cantilever beam.     

Load motion along a beam on a viscoelastic half-space

An expression is derived for equivalent foundation of a viscoelastic half-space interacting with an Euler–Bernoulli beam. It is shown that this equivalent viscoelastic foundation depends on frequencies and wave numbers of the waves in the beam. The real and imaginary part of it substantially varies for phase velocities in between the Rayleigh and shear waves velocities. Radiation of elastic waves occurs for velocities larger than some velocity in that interval. The steady-state beam displacements due to a uniformly moving constant load are calculated for different velocities. The maximum displacement under the load takes place for a velocity of order of the Rayleigh waves velocity.     

Limit point buckling of a finite beam on a nonlinear foundation

In this paper, we consider an imperfect finite beam lying on a nonlinear foundation, whose dimensionless stiffness is reduced from 1 to k as the beam deflection increases. Periodic equilibrium solutions are found analytically and are in good agreement with a numerical resolution, suggesting that localized buckling does not appear for a finite beam. The equilibrium paths may exhibit a limit point whose existence is related to the imperfection size and the stiffness parameter k through an explicit condition. The limit point decreases with the imperfection size while it increases with the stiffness parameter. We show that the decay/growth rate is sensitive to the restoring force model. The analytical results on the limit load may be of particular interest for engineers in structural mechanics.