几何缺陷浅拱的动力稳定性分析

研究了几何缺陷对粘弹性铰支浅拱动力稳定性能的影响。从达朗贝尔原理和欧拉-贝努利假定出发推导了粘弹性铰支浅拱在正弦分布突加荷载作用下的动力学控制方程,并采用Galerkin截断法得到了可用龙格-库塔法求解的无量纲化非线性微分方程组。同时引入能有效追踪结构动力后屈曲路径的广义位移控制法,对含几何缺陷浅拱的响应曲线进行几何、材料双重非线性有限元分析。用这两种方法分析了前三阶谐波缺陷对浅拱动力稳定性能的影响,其中动力临界荷载由B-R准则判定。主要结论有:材料粘弹性使浅拱动力临界荷载增大且结构响应曲线与弹性情况差别很大;二阶谐波缺陷影响显著,它使动力临界荷载明显下降且使得浅拱粘弹性动力临界荷载可能低于弹性动力临界荷载。     

Analysis of in-plane 1:1:1 internal resonance of a double cable-stayed shallow arch model with cables' external excitations

The nonlinear dynamic behaviors of a double cable-stayed shallow arch model are investigated under the one-to-one-to-one internal resonance among the lowest modes of cables and the shallow arch and external primary resonance of cables. The in-plane governing equations of the system are obtained when the harmonic excitation is applied to cables. The excitation mechanism due to the angle-variation of cable tension during motion is newly introduced. Galerkin's method and the multi-scale method are used to obtain ordinary differential equations(ODEs) of the system and their modulation equations, respectively. Frequency-and force-response curves are used to explore dynamic behaviors of the system when harmonic excitations are symmetrically and asymmetrically applied to cables. More importantly, comparisons of frequency-response curves of the system obtained by two types of trial functions, namely, a common sine function and an exact piecewise function, of the shallow arch in Galerkin's integration are conducted.The analysis shows that the two results have a slight difference; however, they both have sufficient accuracy to solve the proposed dynamic system.     

钢管混凝土圆弧桁架拱平面内徐变稳定分析

核心混凝土的徐变会增加钢管混凝土拱肋的屈曲前变形,降低结构的稳定承载力,因此只有计入屈曲前变形的影响,才能准确得到钢管混凝土拱的徐变稳定承载力。基于圆弧形浅拱的非线性屈曲理论,采用虚功原理,建立了考虑徐变和剪切变形双重效应的管混凝土圆弧桁架拱的平面内非线性平衡方程,求得两铰和无铰桁架拱发生反对称分岔屈曲和对称跳跃屈曲的徐变稳定临界荷载。探讨了钢管混凝土桁架拱核心混凝土徐变随修正长细比、圆心角和加载龄期对该类结构弹性稳定承载力的影响,为钢管混凝土桁架拱长期设计提供理论依据。     

外激励作用下弹性支撑浅拱的非线性动力学分析

研究了外激励下两端采用转动弹簧约束的铰支浅拱在发生1:1内共振时的非线性动力学行为。通过引入基本假定和无量纲化变量得到浅拱的动力学控制方程, 将阻尼项、外荷载项和非线性项去掉后,所得线性方程及对应边界条件即可确定考虑转动弹簧影响的频率和模态, 发现转动约束取不同刚度值时系统存在模态交叉与模态转向两种内共振形式。对动力方程进行Galerkin全离散, 并采用多尺度法对内共振进行了摄动分析, 得到了极坐标和直角坐标两种形式的平均方程, 其中平均方程系数与转动弹簧刚度一一对应。最低两阶模态之间1:1内共振的数值研究结果表明: 外激励能激发内共振模态的非线性相互作用, 参数处于某一范围时系统存在周期解、准周期解和混沌解窗口, 且通过(逆)倍周期分岔方式进入混沌。     

Linear and geometrically nonlinear analysis of non-uniform shallow arches under a central concentrated force

In this paper an integral equation solution to the linear and geometrically nonlinear problem of non-uniform in-plane shallow arches under a central concentrated force is presented. Arches exhibit advantageous behavior over straight beams due to their curvature which increases the overall stiffness of the structure. They can span large areas by resolving forces into mainly compressive stresses and, in turn confining tensile stresses to acceptable limits. Most arches are designed to operate linearly under service loads. However, their slenderness nature makes them susceptible to large deformations especially when the external loads increase beyond the service point. Loss of stability may occur, known also as snap-through buckling, with catastrophic consequences for the structure. Linear analysis cannot predict this type of instability and a geometrically nonlinear analysis is needed to describe efficiently the response of the arch. The aim of this work is to cope with the linear and geometrically nonlinear problem of non-uniform shallow arches under a central concentrated force. The governing equations of the problem are comprised of two nonlinear coupled partial differential equations in terms of the axial (tangential) and transverse (normal) displacements. Moreover, as the cross-sectional properties of the arch vary along its axis, the resulting coupled differential equations have variable coefficients and are solved using a robust integral equation numerical method in conjunction with the arc-length method. The latter method allows following the nonlinear equilibrium path and overcoming bifurcation and limit (turning) points, which usually appear in the nonlinear response of curved structures like shallow arches and shells. Several arches are analyzed not only to validate our proposed model, but also to investigate the nonlinear response of in-plane thin shallow arches.     

Nonlinear in-plane thermal buckling of rotationally restrained functionally graded carbon nanotube reinforced composite shallow arches under uniform radial loading

The nonlinear in-plane instability of functionally graded carbon nanotube reinforced composite (FG-CNTRC) shallow circular arches with rotational constraints subject to a uniform radial load in a thermal environment is investigated. Assuming arches with thickness-graded material properties, four different distribution patterns of carbon nanotubes (CNTs) are considered. The classical arch theory and Donnell’s shallow shell theory assumptions are used to evaluate the arch displacement field, and the analytical solutions of buckling equilibrium equations and buckling loads are obtained by using the principle of virtual work. The critical geometric parameters are introduced to determine the criteria for buckling mode switching. Parametric studies are carried out to demonstrate the effects of temperature variations, material parameters, geometric parameters, and elastic constraints on the stability of the arch. It is found that increasing the volume fraction of CNTs and distributing CNTs away from the neutral axis significantly enhance the bending stiffness of the arch. In addition, the pretension and initial displacement caused by the temperature field have significant effects on the buckling behavior.     

Planar nonlinear dynamic analysis of cable-stayed bridge considering support stiffness

Support stiffness is one of important factors on structure dynamics. Considering the vertical support stiffness, a multi-cable-stayed shallow-arch model of the cable-stayed bridge is established. Its differential equation governing the planar motion of cables and the shallow arch and the boundary conditions are derived by Hamilton’s principle. Firstly, the in-plane free vibration of the system is explored in order to find the modal functions and the possible internal resonances of nonlinear dynamics. Then, the 1:2:2 internal resonance among the different modes of the shallow arch and two cables are investigated by the multiple time scale method and pseudo-arclength algorithm. Meanwhile, the frequency-/force–response curves are used to explore the nonlinear behaviors of the system, especially the influence of vertical support stiffness, excitation frequency and amplitude on the internal resonance of the system is considered. To a certain extent, the support stiffness can reduce the response amplitudes of members by absorbing some energy from excitation.

     

固支浅圆拱受子弹撞击的实验研究

本文报导了铝合金固支浅圆拱在子弹撞击下动力响应的实验研究,试验中采用应变片测量并由高速摄影得到试件变形的瞬态记录。实验结果表明,此类浅拱的动力响应不存在失稳现象;但在某一撞击速度区间内,拱的中心位移增加较快,拱的响应前期轴力可以忽略不计,而在后期则必须考虑。     

Nonlinear analysis and buckling of elastically supported circular shallow arches

Arches are often supported elastically by other structural members. This paper investigates the in-plane nonlinear elastic behaviour and stability of elastically supported shallow circular arches that are subjected to a radial load uniformly distributed around the arch axis. Analytical solutions for the nonlinear behaviour and for the nonlinear buckling load are obtained for shallow arches with equal or unequal elastic supports. It is found that the flexibility of the elastic supports and the shallowness of the arch play important roles in the nonlinear structural response of the arch. The limiting shallownesses that distinguish between the buckling modes are obtained and the relationship of the limiting shallowness with the flexibility of the elastic supports is established, and the critical flexibility of the elastic radial supports is derived. An arch with equal elastic radial supports whose flexibility is larger than the critical value becomes an elastically supported beam curved in elevation, while an arch with one rigid and one elastic radial support whose flexibility is larger than the critical value still behaves as an arch when its shallowness is higher than a limiting shallowness. Comparisons with finite element results demonstrate that the analytical solutions and the values of the critical flexibility of the elastic supports and the limiting shallowness of the arch are valid.     

Post-buckling behavior of a fixed arch for variable geometry structures

The behavior of a bistable strut for variable geometry structures was investigated in this paper. A fixed shallow arch subjected to a central concentrated load was used to study the equilibrium path of the bistable strut. Based on a nonlinear strain–displacement relationship, the critical loads for both the symmetric snap-through and asymmetric bifurcation buckling modes were obtained. Moreover, the principal of virtual work was also used to establish the post-buckling differential equilibrium equations of the arch in the horizontal and vertical directions. Therefore, the whole mechanical behavior before and after the buckling of fixed arches is investigated.     

两端固支复合材料浅拱的动力屈曲分析

本文研究两端固支层合复合材料浅拱在阶跃载荷作用下的动力稳定性问题。通过对浅拱动力响应的数值计算结果,然后利用B-R动力屈曲准则,着重分析了集中阶跃载荷作用下几种铺层顺序及铺层数对浅拱动力临界载荷的影响,并给出了能够产生‘跳跃失稳’的最小的结构参数γ0。此外,在利用伽辽金法求解浅拱动力学控制方程时,通过取梁的自由振动模态和柱的静力屈曲模态作为浅拱的动力屈曲模态,分别进行计算并比较了二者的结果,进而讨论了二级数解的收敛性。     

Non-linear in-plane analysis and buckling of pinned–fixed shallow arches subjected to a central concentrated load

This paper is concerned with an analytical study of the non-linear elastic in-plane behaviour and buckling of pinned–fixed shallow circular arches that are subjected to a central concentrated radial load. Because the boundary conditions provided by the pinned support and fixed support of a pinned–fixed arch are quite different from those of a pinned–pinned or a fixed–fixed arch, the non-linear behaviour of a pinned–fixed arch is more complicated than that of its pinned–pinned or fixed–fixed counterpart. Analytical solutions for the non-linear equilibrium path for shallow pinned–fixed circular arches are derived. The non-linear equilibrium path for a pinned–fixed arch may have one or three unstable equilibrium paths and may include two or four limit points. This is different from pinned–pinned and fixed–fixed arches that have only two limit points. The number of limit points in the non-linear equilibrium path of a pinned–fixed arch depends on the slenderness and the included angle of the arch. The switches in terms of an arch geometry parameter, which is introduced in the paper, are derived for distinguishing between arches with two limit points and those with four limit points and for distinguishing between a pinned–fixed arch and a beam curved in-elevation. It is also shown that a pinned–fixed arch under a central concentrated load can buckle in a limit point mode, but cannot buckle in a bifurcation mode. This contrasts with the buckling behaviour of pinned–pinned or fixed–fixed arches under a central concentrated load, which may buckle both in a bifurcation mode and in a limit point mode. An analytical solution for the limit point buckling load of shallow pinned–fixed circular arches is also derived. Comparisons with finite element results show that the analytical solutions can accurately predict the non-linear buckling and postbuckling behaviour of shallow pinned–fixed arches. Although the solutions are derived for shallow pinned–fixed arches, comparisons with the finite element results demonstrate that they can also provide reasonable predictions for the buckling load of deep pinned–fixed arches under a central concentrated load.     

Nonlinear dynamic buckling of pinned–fixed shallow arches under a sudden central concentrated load

The structural behavior of a shallow arch is highly nonlinear, and so when the amplitude of the oscillation of the arch produced by a suddenly-applied load is sufficiently large, the oscillation of the arch may reach a position on its unstable equilibrium paths that leads the arch to buckle dynamically. This paper uses an energy method to investigate the nonlinear elastic dynamic in-plane buckling of a pinned–fixed shallow circular arch under a central concentrated load that is applied suddenly and with an infinite duration. The principle of conservation of energy is used to establish the criterion for dynamic buckling of the arch, and the analytical solution for the dynamic buckling load is derived. Two methods are proposed to determine the dynamic buckling load. It is shown that under a suddenly-applied central load, a shallow pinned–fixed arch with a high modified slenderness (which is defined in the paper) has a lower dynamic buckling load and an upper dynamic buckling load, while an arch with a low modified slenderness has a unique dynamic buckling load.     

黏弹性复合材料梁动力学实验建模的研究

以ABS树脂为基材, 填充1%~10%的金红石纳米二氧化钛制成纳米复合材料样本系列,搭建了参数激励非线性振动实验系统. 采用实验建模的方法, 基于非线性增量谐波平衡识别理论,建立了黏弹性复合材料屈曲梁的动力学控制方程. 通过数值模拟与实验结果的比较, 验证了理论模型和实验系统在定性定量分析上的一致性, 并且对一类不同配比成分的纳米复合材料也有很好的适用性.      

Differential quadrature method for analyzing nonlinear dynamic characteristics of viscoelastic plates with shear effects

The integro-partial differential equations governing the dynamic behavior of viscoelastic plates taking account of higher-order shear effects and finite deformations are presented. From the matrix formulas of differential quadrature, the special matrix product and the domain decoupled technique presented in this work, the nonlinear governing equations are converted into an explicit matrix form in the spatial domain. The dynamic behaviors of viscoelastic plates are numerically analyzed by introducing new variables in the time domain. The methods in nonlinear dynamics are synthetically applied to reveal plenty and complex dynamical phenomena of viscoelastic plates. The numerical convergence and comparison studies are carried out to validate the present solutions. At the same time, the influences of load and material parameters on dynamic behaviors are investigated. One can see that the system will enter into the chaotic state with a paroxysm form or quasi-periodic bifurcation with changing of parameters.     

Snap-through under unilateral displacement control with constant velocity

When the side of a beverage can or the domed lid of a jar is pushed inward, all or part of the structure may suddenly snap into an inverted configuration. The velocity of the pushing motion affects this instability. Most previous analyses of snap-through have considered force control (increasing the pushing force, e.g., a weight). Snap-through under dynamic, unilateral displacement control is investigated here, with the indentor moving at constant velocity (as in a universal testing machine) until snap-through occurs. Shallow elastic arches with immovable pinned ends are analyzed. Attention is focused on the critical height of the indentor at which snap-through is initiated. The effects of the indentor velocity, indentor location along the span, initial arch height, and damping magnitude are investigated. In addition, experiments are conducted on shallow buckled beams, which behave similarly to arches. Usually, the higher the indentor velocity, the further the indentor must move before snap-through occurs.     

Dynamic buckling of a shallow arch under shock loading considering the effects of the arch shape

In this paper, the influence of the initial curvature of thin shallow arches on the dynamic pulse buckling load is examined. Using numerical means and a multi-dof semi-analytical model, both quasi-static and non-linear transient dynamical analyzes are performed. The influence of various parameters, such as pulse duration, damping and, especially, the arch shape is illustrated. Moreover, the results are numerically validated through a comparison with results obtained using finite element modeling. The main results are firstly that the critical shock level can be significantly increased by optimizing the arch shape and secondly, that geometric imperfections have only a mild influence on these results. Furthermore, by comparing the sensitivities of the static and dynamic buckling loads with respect to the arch shape, non-trivial quantitative correspondences are found.     

Nonlinear Isolator Dynamics at Finite Deformations: An Effective Hyperelastic,Fractional Derivative,Generalized Friction Model

In presenting a nonlinear dynamic model of a rubber vibrationisolator, the quasistatic and dynamic motion influences on theforce response are investigated within the time and frequencydomain. It is found that the dynamic stiffness at the frequency ofa harmonic displacement excitation, superimposed upon the longterm isolator response, is strongly dependent on staticprecompression, dynamic amplitude and frequency. The problems ofsimultaneously modelling the elastic, viscoelastic and frictionforces are removed by additively splitting them, modelling theelastic force response by a nonlinear, shape factor basedapproach, displaying results that agree with those of aneo-Hookean hyperelastic isolator at a long term precompression.The viscoelastic force is modeled by a fractional derivativeelement, while the friction force governs from a generalizedfriction element displaying a smoothed Coulomb force. A harmonicdisplacement excitation is shown to result in a force responsecontaining the excitation frequency and its every otherhigher-order harmonic, while using a linearized elastic forceresponse model, whereas all higher-order harmonics are present forthe fully nonlinear case. It is furthermore found that the dynamicstiffness magnitude increases with static precompression andfrequency, while decreasing with dynamic excitationamplitude – eventually increasing at the highest amplitudes due tononlinear elastic effects – with its loss angle displaying amaximum at an intermediate amplitude. Finally, the dynamicstiffness at a static precompression, using a linearized elasticforce response model, is shown to agree with the fully nonlinearmodel except at the highest dynamic amplitudes.     

变温场中具损伤粘弹性矩形板的非线性动力响应分析

基于热粘弹性理论、Von Karman板理论和连续损伤力学,导出了二维状态下各向同性材料的变温粘弹性本构方程,建立了含损伤效应的各向同性粘弹性矩形板在变温场中的非线性运动控制方程,且应用有限差分法对问题进行求解.算例中,讨论了损伤演化及温度场等因素对粘弹性矩形板非线性动力学行为的影响,得出一些有意义的结论.     

In-plane stability of arches

Classical buckling theory is mostly used to investigate the in-plane stability of arches, which assumes that the pre-buckling behaviour is linear and that the effects of pre-buckling deformations on buckling can be ignored. However, the behaviour of shallow arches becomes non-linear and the deformations are substantial prior to buckling, so that their effects on the buckling of shallow arches need to be considered. Classical buckling theory which does not consider these effects cannot correctly predict the in-plane buckling load of shallow arches. This paper investigates the in-plane buckling of circular arches with an arbitrary cross-section and subjected to a radial load uniformly distributed around the arch axis. An energy method is used to establish both non-linear equilibrium equations and buckling equilibrium equations for shallow arches. Analytical solutions for the in-plane buckling loads of shallow arches subjected to this loading regime are obtained. Approximations to the symmetric buckling of shallow arches and formulae for the in-plane anti-symmetric bifurcation buckling load of non-shallow arches are proposed, and criteria that define shallow and non-shallow arches are also stated. Comparisons with finite element results demonstrate that the solutions and indeed approximations are accurate, and that classical buckling theory can correctly predict the in-plane anti-symmetric bifurcation buckling load of non-shallow arches, but overestimates the in-plane anti-symmetric bifurcation buckling load of shallow arches significantly.