Post-buckling behaviour of slender structures with a bi-linear bending moment-curvature relationship

Certain classes of slender structures of complex cross-section or fabricated from specialised materials can exhibit a bi-linear bending moment-curvature relationship that has a strong influence on their global structural behaviour. This condition may be encountered, for instance, in (a) non-linear elastic or inelastic post-buckling problems if the cross-section stiffness may be well approximated by a bi-linear model; (b) multi-layered structures such as stranded cables, power transmission lines, umbilical cables and flexible pipes where the drop in the bending stiffness is associated with an internal friction mechanism. This paper presents a mathematical formulation and an analytical solution for such slender structures with a bi-linear bending moment versus curvature constitutive behaviour and subjected to axial terminal forces. A set of five first-order non-linear ordinary differential equations are derived from considering geometrical compatibility, equilibrium of forces and moments and constitutive equations, with hinged boundary conditions prescribed at both ends, resulting a complex two-point boundary value problem. The variables are non-dimensionalised and solutions are developed for monotonic and unloading conditions. The results are presented in non-dimensional graphs for a range of critical curvatures and reductions in bending stiffness, and it is shown how these parameters affect the structure's post-buckling behaviour.     

Fast approximations of dynamic stability boundaries of slender curved structures

Curved beams and panels can often be found as structural components in aerospace, mechanical and civil engineering systems. When curved structures are subjected to dynamic loads, they are susceptible to dynamic instabilities especially dynamic snap-through buckling. The identification of the dynamic stability boundary that separate the non-snap and post-snap responses is hence necessary for the safe design of such structures, but typically requires extensive transient simulations that may lead to high computation cost. This paper proposes a scaling approach that reveals the similarities between dynamic snap-through boundaries of different structures. Such identified features can be directly used for fast approximations of dynamic stability boundaries of slender curved structures when their geometric parameters or boundary conditions are varied. The scaled dynamic stability boundaries of half-sine arches, parabolic arches and cylindrical panels are studied.     

A numerical model for non-linear dynamic analysis of slender masonry structures

A numerical model is presented to enable performing non-linear dynamic analysis of slender masonry structures and elements, such as towers and columns or masonry walls in out-of-plane flexure. Such structures are represented via a continuous one-dimensional model. The main mechanical characteristics of the material in all sections along the height of such structures are taken into account by means of a non-linear elastic constitutive law formulated in terms of generalized stress and strain, under the assumption that the material has no resistance to tension and limited compressive strength. The relations defined herein for the general case of hollow rectangular cross-sections are also aimed at enabling study of towers, bell-towers and similar slender structures.     

On the dynamic behaviour of a piezoelectric laminate with multiple interfacial collinear cracks

This paper investigates the dynamic behaviour of a piezoelectric laminate containing multiple interfacial collinear cracks subjected to steady-state electro-mechanical loads. Both the permeable and impermeable boundary conditions are examined and discussed. Based on the use of integral transform techniques, the problem is reduced to a set of singular integral equations, which can be solved using Chebyshev polynomial expansions. Numerical results are provided to show the effect of the geometry of interacting collinear cracks, the applied electric fields, the electric boundary conditions along the crack faces and the loading frequency on the resulting dynamic stress intensity and electric displacement intensity factors.     

The prediction on in-line vortex-induced vibration of slender marine structures

The in-line (IL) vortex-induced vibration (VIV) that occurs frequently in ocean engineering may cause severe fatigue damage in slender marine structures. To the best knowledge of the authors, in existing literatures, there is no efficient analytical model for predicting pure IL VIV. In this paper, a wake oscillator model capable of analyzing the IL VIV of slender marine structures has been developed. Two different kinds of van der Pol equations are used to describe the near wake dynamics related to the fluctuating nature of symmetric vortex shedding in the first excitation region and alternate vortex shedding in the second one. Some comparisons are carried out between the present model results and experimental data. It is found that many phenomena observed in experiments could be reproduced by the present wake oscillator model.     

桥梁结构静动力性能分析及其CAD微机系统

本文介绍了桥梁结构的力学计算方法，对桥梁结构ＣＡＤ系统的研制和设计方法进行了初步探讨，并列举应用实例说明了平面连续梁模型和空间有限元模型计算结果的区别，表明对桥梁结构进行全面的力学性能计算分析是十分必要的．     

First- and second-order sensitivities of beams with respect to cross-sectional cracks

This paper deals with a variational formulation for the sensitivity problem of beam systems in the context of deformable solids with cracks. Natural frequencies are defined as state variables involved in the energy functional of the system, while the cracks length and position are treated as system parameters. The hierarchical equation system is formed and solved for the first and second derivatives of the natural frequency functions with respect to the cracks length and position. An analytical procedure for calculations of the second-order sensitivities of natural frequencies is proposed for the non-symmetrical equation system operator. Numerical algorithms are worked out and implemented computationally. Analytical and numerical aspects of the problem are discussed in detail through a number of illustrative results.The support of this work by the State Committee for Scientific Research (KBN) under Grant No. 4-050-0148/17-98-00 is gratefully acknowledged.     

The dynamic behaviour of stiffened strings

Summary The small oscillation of a stiffened string about its static equilibrium configuration is studied. The solution requires both the static and dynamic shape of the rod to be determined and since the governing equations do not lend themselves to an analytical solution a Runge-Kutta integration technique was used. Experimental results for the natural frequencies are compared to the numerical solution for a particular stiffened string.

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Übersicht Es werden die kleinen Schwingungen eines versteiften Bandes um die statische Gleichgewichtslage untersucht. Dabei muß sowohl die statische wie auch die dynamische Form des Bandes bestimmt werden. Da eine analytische Lösung der Bewegungsgleichungen nicht möglich ist, wurde numerisch nach Runge-Kutta integriert. Für verschieden versteifte Bänder werden Versuchsergebnisse für die Eigenfrequenzen mit numerischen Lösungen verglichen.

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The authors would like to acknowledge the work of David Stredulinsky in the initial stages of this project and to thank the National Research Council for support under operating grant NRC A7514 (Faulkner).     

A coupled Timoshenko model for smart slender structures

In this paper, a generalized Timoshenko model has been developed for prismatic, beam-like slender structures with embedded or surface mounted piezoelectric type smart materials. Starting from a geometrically exact formulation of the original, three-dimensional electromechanical problem, we apply the variational asymptotic method to carry out a systematic dimensional reduction. In the process, the three-dimensional electromechanical enthalpy functional is approximated asymptotically using the slenderness as the small parameter to find out an equivalent one-dimensional electromechanical enthalpy functional. For Timoshenko-like refinement over the Euler–Bernoulli beam model, terms up to the second order of the slenderness are kept in the enthalpy expression. As an unified analysis tool, the present model can analyze embedded or surface mounted active layer with arbitrary cross-sectional geometry as two cases of a general one, no special assumptions or modifications need to be made for these two separate types of active inclusions.     

细长结构几何非线性分析的子结构方法

将细长结构沿长度方向划分为多个子结构,并在每个子结构上建立一个随结构一起运动的连体基,则结构内任意点的位移可分解为连体基的转动和相对于连体基的小位移。利用细长结构这样的变形特征,本文详细讨论了连体基的转动,给出了与连体基选择方式相协调的节点位移及其虚变分表达式,并将子结构内部位移凝聚到了边界节点上。在此基础上,提出了一种细长结构几何非线性分析的子结构方法,可在不损失计算精度的前提下大幅度降低求解规模,从而提高了计算效率。数值算例验证了所提方法的有效性。     

Flow-induced vibration of elastic slender structures in a cylinder wake

Flow-induced vibration of an elastic airfoil due to the wake propagating from an upstream cylinder at a Reynolds number of 10 000 based on cylinder diameter D was investigated. A laser vibrometer was employed to measure the bending and torsional vibration displacements at the mid-span of the airfoil and the cylinder. The dimensionless gap size S/D between the two structures was selected as the governing parameter of the flow-induced vibration problem. It is found that the vibration amplitudes of the elastic airfoil and the vortex shedding frequency of the coupled cylinder–airfoil system are strongly dependent on S/D, due to the different fluid–structure interaction experienced by the airfoil at various S/D. Strong vortex-induced vibration of the airfoil appears to be excited by the organized Karman-vortex-street (KVS) vortices in the cylinder wake for S/D>3 and becomes stabilized for S/D3. However, as a result of the shear-layer-induced vibration at an appropriate frequency, structural resonance is also found to occur even though the airfoil is located in the stabilizing range. The occurrence of structural resonance is further supported by a complementary experiment where the slender structure is an elastic flat plate. This phenomenon indicates that assuming the structures in any fluid–structure interaction problem to be rigid is not appropriate, even though they might appear to be highly stiff. The experimental results were used to validate a numerical model previously developed to estimate the structural responses in complicated fluid–structure interaction problems.     

Application of the NSCD method to analyse the dynamic behaviour of stone arched structures

In this paper, a masonry arch is simulated in order to assess both its structural and seismic vulnerability. The non-smooth contact dynamics (NSCD) computational method is used to simulate this type of structure as a collection of bodies under the hypothesis of unilateral constraints and frictional contact, with or without cohesion. Sinusoidal oscillations in three dimensions and real earthquake data have been applied to the supporting base element of the arch model.The primary aim of this study is to better understand the dynamic behaviour of the masonry arch, a typical component of historic unreinforced masonry (URM) structures. This study also assesses the influences of the input parameters on the mechanical and dynamic behaviour of the arch structure. Its collapse mechanism is studied for both cohesive and non-cohesive contact.In addition, we examine the behaviour under seismic loading of the Arles aqueduct, a real historical arched structure located in the south-east of France. Significant information can be obtained from the comparison of the results of advanced numerical analysis, taking into account the precise geometry of the model, the mechanical characteristics of the materials and the observation of the in situ monuments after their collapse.     

Finite element formulation of slender structures with shear deformation based on the Cosserat theory

This paper addresses the derivation of finite element modelling for nonlinear dynamics of Cosserat rods with general deformation of flexure, extension, torsion, and shear. A deformed configuration of the Cosserat rod is described by the displacement vector of the deformed centroid curve and an orthogonal moving frame, rigidly attached to the cross-section of the rod. The position of the moving frame relative to the inertial frame is specified by the rotation matrix, parameterised by a rotational vector. The shape functions with up to third order nonlinear terms of generic nodal displacements are obtained by solving the nonlinear partial differential equations of motion in a quasi-static sense. Based on the Lagrangian constructed by the Cosserat kinetic energy and strain energy expressions, the principle of virtual work is employed to derive the ordinary differential equations of motion with third order nonlinear generic nodal displacements. A cantilever is presented as a simple example to illustrate the use of the formulation developed here to obtain the lower order nonlinear ordinary differential equations of motion of a given structure. The corresponding nonlinear dynamical responses of the structures are presented through numerical simulations using the MATLAB software. In addition, a MicroElectroMechanical System (MEMS) device is presented. The developed equations of motion have furthermore been implemented in a VHDL-AMS beam model. Together with available models of the other components, a netlist of the device is formed and simulated within an electrical circuit simulator. Simulation results are verified against Finite Element Analysis (FEA) results for this device.     

沉痛悼念许兵先生

通过对胞壁随机移除的蜂窝结构动态变形过程的有限元模拟,分析了随机缺陷对蜂窝 结构变形模式的影响,得到蜂窝结构在两个加载方向上的变形模式图及不同模式间转换的临 界速度. 对含缺陷蜂窝结构平台应力的研究发现,当变形模式为过渡模式或动态模式时结构 平台应力与冲击速度的平方成线性关系. 相同密度下,低缺陷蜂窝结构的平台应力在由过渡 模式向动态模式转变的临界速度附近高于规则蜂窝结构,较高的随机缺陷则使蜂窝结构的平 台应力在由准静态模式向过渡模式转变的临界速度附近显著下降. 关键词：多孔材料,蜂窝,缺陷,平台应力,有限元分析     

The non-linear dynamic response of the Euler-Bernoulli beam with an arbitrary number of switching cracks

In this study the non-linear dynamic response of the Euler-Bernoulli beam in presence of multiple concentrated switching cracks (i.e. cracks that are either fully open or fully closed) is addressed. The overall behaviour of such a beam is non-linear due to the opening and closing of the cracks during the dynamic response; however, it can be regarded as a sequence of linear phases each of them characterised by different number and positions of the cracks in open state. In the paper the non-linear response of the beam with switching cracks is evaluated by determining the exact modal properties of the beam in each linear phase and evaluating the corresponding time history linear response through modal superposition analysis. Appropriate initial conditions at the instant of transition between two successive linear phases have been considered and an energy control has been enforced with the aim of establishing the minimum number of linear modes that must be taken into account in order to obtain accurate results. Some numerical applications are presented in order to illustrate the efficiency of the proposed approach for the evaluation of the non-linear dynamic response of beams with multiple switching cracks. In particular, the behaviour under different boundary conditions both for harmonic loading and free vibrations has been investigated.     

The dynamic behavior of two collinear interface cracks in magneto-electro-elastic materials

In this paper, the dynamic behavior of two collinear symmetric interface cracks between two dissimilar magneto-electro-elastic material half planes under the harmonic anti-plane shear waves loading is investigated by Schmidt method. By using the Fourier transform, the problem can be solved with a set of triple integral equations in which the unknown variable is the jump of the displacements across the crack surfaces. To solve the triple integral equations, the jump of the displacements across the crack surface is expanded in a series of Jacobi polynomials. Numerical solutions of the stress intensity factor, the electric displacement intensity factor and the magnetic flux intensity factor are given. The relations among the electric filed, the magnetic flux field and the stress field are obtained.     

带裂纹旋转柔性梁系统的刚柔耦合动力学研究

对具有刚柔耦合效应的带裂纹旋转柔性梁进行建模和动力学特性分析研究。采用晶格弹簧离散模型,利用无质量弹簧模拟梁上裂纹,通过考虑梁变形的二阶耦合项建立了带裂纹旋转柔性梁系统的一次近似耦合动力学控制方程。数值计算结果表明,裂纹的存在会使旋转柔性梁的固有频率降低,并且随着梁转速的增大,这种降低效应呈减弱趋势;值得注意的是,裂纹梁的固有频率与裂纹处的弯矩具有正相关关系。此外,裂纹的存在不仅会使转速变化阶段梁的末端位移响应增大,还会对转速稳定后梁的末端振荡产生显著的影响。     

切槽孔爆破动态力学特征的动焦散线实验

应用爆炸加载的透射式动焦散线测试系统,分析了有机玻璃切槽孔爆破模型的裂纹动态特征变化规律。比较了不同切槽角度、切槽深度的定向断裂裂纹尖端的扩展长度、扩展速度和动态应力强度因子。初步探讨了切槽爆破的动态效应,研究表明切槽孔爆破早期裂纹破坏模式为爆炸拉应力波作用下的I型快速扩展裂纹,裂纹尖端拉应力集中积聚的较大应变能维持了爆炸裂纹进一步扩展,裂纹尖端扩展后期表现为P波、S波共同作用下的复合型扩展特征。切槽角为60时获得的定向断裂效果最好,合理切槽深度为炮孔半径的1/4～1/2。