On the nonlinear dynamic buckling mechanism of autonomous dissipative/nondissipative discrete structural systems

Summary Nonlinear dynamic buckling of nonlinearly elastic dissipative/nondissipative multi-mass systems, mainly under step load of infinite duration, is studied in detail. These systems, under the same loading applied statically, experience a limit point instability. The analysis can be readily extended to the case of dynamic buckling under impact loading. Energy, topological and geometrical aspects for the total potential energyV, which is constrained to lie in a region of phase-space whereV0, allow conclusions to be drawn directly regarding dynamic buckling. Criteria leading to very good, approximate and lower/upper bound dynamic buckling estimates are readily established without solving the highly nonlinear set of equations of motion. The theory is illustrated with several analyses of a two-degree-of-freedom model.     

Post-buckling dynamic behavior of periodically supported imperfect shells

A circular cylindrical shell, periodically supported and subjected to step-loading in the form of lateral or hydrostatic pressure, is studied. Using the time-dependent von Kármán-Donnell equations, its imperfection sensitivity is examined and a simple asymptotic expression for the dynamic buckling load, valid for small imperfections, is obtained. There is a simple relation, independent of the imperfection, between the dynamic and static buckling loads.     

Finite element analysis of post-buckling dynamics in plates—Part I: An asymptotic approach

Various static and dynamic aspects of post-buckled thin plates, including the transition of buckled patterns, post-buckling dynamics, secondary bifurcation, and dynamic snapping (mode jumping phenomenon), are investigated systematically using asymptotical and non-stationary finite element methods. In part I, the secondary dynamic instability and the local post-secondary buckling behavior of thin rectangular plates under generalized (mechanical and thermal) loading is investigated using an asymptotic numerical method which combines Koiter’s nonlinear instability theory with the finite element technique. A dynamic multi-mode reduction method—similar to its static single-mode counterpart: Liapunov–Schmidt reduction—is developed in this perturbation approach. Post-secondary buckling equilibrium branches are obtained by solving the reduced low-dimensional parametric equations and their stability properties are determined directly by checking the eigenvalues of the resulting Jacobian matrix. Typical post-secondary buckling forms—transcritical, supercritical and subcritical bifurcations are observed according to different combinations of boundary conditions and load types. Geometric imperfection analysis shows that not only the secondary bifurcation load but also changes in the fundamental post-secondary buckling behavior are affected. The post-buckling dynamics and the global analysis of mode jumping of the plates are addressed in part II.     

Qualitative criteria in non-linear dynamic buckling and stability of autonomous dissipative systems

A general qualitative approach for dynamic buckling and stability of autonomous dissipative structural systems is comprehensively presented. Attention is focused on systems which under the same statically applied loading exhibit a limit point instability or an unstable branching point instability with a non-linear fundamental path. Using the total energy equation, the theory of point and periodic attractors of the basin of attraction of a stable equilibrium point, of local and global bifurcations, of the inset and outset manifolds of a saddle and of the geometry of the channel of motion, the stability of the fundamental equilibrium path and the mechanism of dynamic buckling are thoroughly discussed. This allows us to establish useful qualitative criteria leading to exact, approximate and upper/lower bound buckling estimates without integrating the highly non-linear initial-value problem. The individual and coupling effect of geometric and material non-linearities of damping and mass distribution on the dynamic buckling load are also examined. A comparison of the results of the above qualitative analysis with those obtained via numerical simulation is performed on several two- and three-degree-of-freedom models of engineering importance.     

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.     

Buckling load of non-linear systems with multiple eigenvalues

For structural systems with a coincident lowest eigenvalue λ_c, the influence of imperfections on the buckling of the systems depends to a very large extent upon the distribution of the imperfections. Moreover, the system may buckle either at a limit point or at a bifurcation point before this limit point is reached. Considering both possibilities, a lower bound to the buckling load of the system, for a given root mean square of the imperfections, is obtained. Furthermore, with reference to a set of particular, normalized co-ordinates, it was found that the absolute minimum buckling load is given by an imperfection vector parallel to the steepest of all post-buckling paths intersecting at λ_c. At this absolute minimum buckling load the critical point is a limit point. As an example, the lower bound to the buckling load of an imperfect cylindrical shell under axial compression was calculated.     

Reduced-Stiffness Method in the Theory of Shells

The fundamentals of the reduced-stiffness method, which is used in buckling analysis of reinforced and perfect and imperfect nonreinforced shells, are set out. The method is validated analytically and experimentally. The lower bound determined by this method is very close to the experimental lower bound. Some aspects of the current state and prospects for development and generalization of the method are discussed     

Dynamic torsional buckling of multi-walled carbon nanotubes embedded in an elastic medium

In this paper the dynamic torsional buckling of multi-walled carbon nanotubes （MWNTs） embedded in an elastic medium is studied by using a continuum mechanics model. By introducing initial imperfections for MWNTs and applying the preferred mode analytical method, a buckling condition is derived for the buckling load and associated buckling mode. In particular, explicit expressions are obtained for embedded double-walled carbon nanotubes （DWNTs）. Numerical results show that, for both the DWNTs and embedded DWNTs, the buckling form shifts from the lower buckling mode to the higher buckling mode with increasing the buckling load, but the buckling mode is invari- able for a certain domain of the buckling load. It is also indicated that, the surrounding elastic medium generally has effect on the lower buckling mode of DWNTs only when compared with the corresponding one for individual DWNTs.     

Nonlinear dynamic buckling of a viscoelastic model under impact loading

A simple nonlinear buckling analysis is applied to a one-degree-of-freedom arch under impact loading in which viscous damping may also be included. Such a loading consists of a falling body striking centrally the joint mass of the arch in such a way that a completely plastic impact can be postulated. When there is no damping the exact dynamic buckling load for such a kind of loading-associated with an unbounded motion can be established by using a static criterion (approach). More specifically, it was shown that the dynamic buckling load corresponds to that unstable equilibrium state where the total potential energy of the system is zero. Furthermore, it was proved that the second variation of the total potential energy at the foregoing unstable equilibrium state is negative definite. This implies that the curve loading versus displacement resulting by the vanishing of the total potential energy has always a maximum on the afore mentioned unstable state. It was also found that the system may become sensitive to initial conditions. If damping is included the foregoing static criterion yields lower bound buckling estimates. These findings were verified by employing a highly efficient approximate technique as well as the numerical scheme of Runge-Kutta for solving any nonlinear initial-value problem.     

复合材料层合开顶扁球壳的非线性动态屈曲

研究了复合材料层合开顶扁球壳的非线性动态屈曲问题。建立了对称层合圆柱正交异性开顶扁球壳考虑横向剪切的非线性振动微分方程,根据突变理论建立了该壳体动态屈曲的突变模型,得到了动态屈曲的临界方程。     

薄壳失稳机理浅析

对薄壳失稳问题研究的理论与实验成果进行了总结和讨论，对薄壳后屈曲理论研究结果提出了不同的看法，同时应用动力学原理对薄壳失稳问题进行了探讨，并建立了计算模型。文中应用动力学原理描述了从加载初期的一个呈现静力学特征的薄壳随荷载的增加而逐渐成为一个呈现动力学特征的薄壳的过程，从薄壳受扰振动乃至共振的角度解释了失稳临界荷载实验数据值及其离散并低于失稳临界荷载理论值的原因。     

INVESTIGATION OF THE DYNAMIC BUCKLING OF DOUBLEWALLED CARBON NANOTUBE SUBJECTED TO AXIAL PERIODIC DISTURBING FORCES

IntroductionThediscoveryofthefirstcarbonnanotubes[1]hasattractedwideattentionandstimulatedextensivestudies[2 - 5 ].Thestudiesshowedthatthecarbonnanotubesexhibitsuperiormechanical,electronicandchemicalproperties.Onthemechanicalbehavior,thecarbonnanotubespossessexceptionallyhighstrength ,stiffnessandelasticmodulus.Theestimatemodulusofthecarbonnanotubemayreachashighas 1TPa.Itisthelargestofallknownmaterials.Thestrengthorstiffnessishigherthananyknownfiber[3].Thecarbonnanotubeareusedascompositemat…     

On local buckling of thin shells

A class of problems is considered where the buckling initially starts only in a part of the shell—locally. The stability analysis is focused on the zone of initial buckling. This leads to radical simplification. First the basic hypothesis and stability equations are formulated. Closed-form stability criteria asymptotically exact for very thin shells are discussed. This gives sufficient conditions for the local character of buckling and for the adequacy of the asymptotic approximation. The analysis taking into account the variation of stresses and shape inside the buckling zone results in a check of stability by hand calculations or by simple coding of a desk-top computer. The adequacy of the simplest representation of the stress and strain variation in the buckling zone is tested.     

Dynamic, multimode buckling of thin-walled columns subjected to in-plane pulse loading

The present paper deals with dynamic, coupled buckling of long, prismatic columns simply supported at the ends. This investigation concerns thin-walled structures of a square cross-section with or without intermediate stiffeners under in-plane pulse loading. The dynamic load of a rectangular shape has been assumed in the analysis. The structures are composed of rectangular plates interconnected along longitudinal edges. A plate model is adopted in the analysis. The material of the structure is isotropic. The problem has been investigated on the basis of the disturbance theory. The dynamic critical load factor DLF has been determined using the Budiansky and Hutchinson criterion. The results obtained with the analytical-numerical method (ANM), which employs the asymptotic perturbation theory, have been compared with the finite element method (FEM).     

环形板的非轴对称屈曲分析

本文利用打靶法研究了内外边界固支且在外边界受均匀径向压力作用下的极正交各向异性环形板的非轴对称屈曲和过屈曲,计算了临界载荷,讨论了分支解的存在性,得到了分支解的渐近表达式,分析了环形板的屈曲性态。     

环形板的非轴对称屈曲分析

本文利用打靶法研究了内外边界固支且在外边界受均匀径向压力作用下的极正交各向异性环形板的非轴对称屈曲和过屈曲,计算了临界载荷,讨论了分支解的存在性,得到了分支解的渐近表达式,分析了环形板的屈曲性态。     

含脱层损伤复合材料层合梁的冲击动力屈曲

针对含初始缺陷和脱层损伤的复合材料层合梁的轴向冲击动力屈曲问题进行了分析。基于Hamilton原理导出了考虑初始缺陷、轴向和横向惯性、横向剪切变形以及转动惯性影响时含脱层损伤复合材料梁的非线性动力屈曲控制方程;基于B-R准则,采用有限差分方法求解了受轴向冲击载荷作用下含脱层损伤复合材料梁的动力屈曲问题;讨论了冲击速度、初始几何缺陷、铺层角度以及脱层长度等因素对复合材料层合梁动力屈曲的影响。     

Influence of the connecting condition on the dynamic buckling of longitudinal impact for an elastic rod

The stress wave propagation law and dynamic buckling critical velocity are formulated and solved by considering a general axial connecting boundary for a slender elastic straight rod impacted by a rigid body. The influence of connecting stiffness on the critical velocity is investigated with varied impactor mass and buckling time. The influences of rod length and rod mass on the critical velocity are also discussed. It is found that greater connecting stiffness leads to larger stress amplitude, and further results in lower critical velocity. It is particularly noteworthy that when the connecting stiffness is less than a certain value,dynamic buckling only occurs before stress wave reflects off the connecting end. It is also shown that longer rod with larger slenderness ratio is easier to buckle, and the critical velocity for a larger-mass rod is higher than that for a lighter rod with the same geometry.     

Dynamic buckling analysis of composite cylindrical shells using a finite element based perturbation method

In this paper a finite element formulation of a reduction method for dynamic buckling analysis of imperfection-sensitive shell structures is presented. The reduction method makes use of a perturbation approach, initially developed for static buckling and later extended to dynamic buckling analysis. The implementation of a single-mode dynamic buckling analysis in a general purpose finite element code is described. The effectiveness of the approach is illustrated by application to the dynamic buckling of composite cylindrical shells under axial and radial step loads. Results of the reduction method are compared with results available in the literature. The results are also compared with full model finite element explicit dynamic analysis, and a reasonable agreement is obtained.     

Influence of general imperfections in axially loaded cylindrical shells

The buckling of an axially loaded cylindrical shell is considered when imperfection components corresponding to all of the classical buckling modes are taken into consideration. The analysis represents an extension of Koiter's axisymmetric solution and in the asymptotic sense due to Koiter the imperfections considered are as general as possible. The results obtained reveal many interesting aspects of shell buckling which arize for various imperfection forms. The buckling behaviour which results is associated with both bifurcation and limit point critical states.