Dynamic buckling of simple two-bar frames using catastrophe theory

Non-linear static and dynamic elastic buckling of simple imperfect two-bar frames, treated as continuous systems, are analyzed with the aid of catastrophe theory using a comprehensive and readily employed procedure. Static catastrophes are extended to the corresponding dynamic catastrophes of undamped frames under step loading (autonomous systems) by properly determining the dynamic singularity and bifurcational sets. Attention is focused on fold and cusp catastrophes. A local analysis based on Taylor's expansion of the non-linear equilibrium equation of the frame allows us: (a) to classify the total potential energy function of the frames to the canonical form of the corresponding universal unfolding of the seven elementary Thom's catastrophes, and (b) to easily obtain static and dynamic buckling loads, critical points (singularity sets) and related imperfection sensitivities (bifurcational sets). An illustrative example associated with a static and dynamic fold catastrophe demonstrates the efficiency and reliability of the methodology proposed herein.     

Dynamic buckling of discrete structural systems under combined step and static loads

The dynamic instability of discrete, elastic, multiple degree of freedom (d.o.f.) systems under a combination of static and step loads is studied. Conservative, autonomous and holonomic systems are considered, in which the associated static response is a bifurcation under one load parameter, and a limit point under the second parameter. A review of different criteria and algorithms obtained from them for the computation of dynamic buckling loads is first presented, followed by a procedure derived from previous investigations on one d.o.f. systems. The different procedures are applied to a two d.o.f. problem under axial and lateral load, with quadratic and cubic non-linearities. The response in time shows that the system oscillates about the static equilibrium position before dynamic buckling is reached, with the kinetic energy tending to zero as assumed in the static (energy) procedures of stability.     

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.     

Computing the structural buckling limit load by using dynamic relaxation method

The numerical structural analysis schemes are extensively developed by progress of modern computer processing power. One of these approximate approaches is called "dynamic relaxation (DR) method." This technique explicitly solves the simultaneous system of equations. For analyzing the static structures, the DR strategy transfers the governing equations to the dynamic space. By adding the fictitious damping and mass to the static equilibrium equations, the corresponding artificial dynamic system is achieved. The static equilibrium path is required in order to investigate the structural stability behavior. This path shows the relationship between the loads and the displacements. In this way, the critical points and buckling loads of the non-linear structures can be obtained. The corresponding load to the first limit point is known as buckling limit load. For estimating the buckling load, the variable load factor is used in the DR process. A new procedure for finding the load factor is presented by imposing the work increment of the external forces to zero. The proposed formula only requires the fictitious parameters of the DR scheme. To prove the efficiency and robustness of the suggested algorithm, various geometric non-linear analyses are performed. The obtained results demonstrate that the new method can successfully estimate the buckling limit load of structures.     

Dynamic torsional buckling of a double-walled carbon nanotube embedded in an elastic medium

An elastic double-shell model based on continuum mechanics is presented to study the dynamic torsional buckling of an embedded double-walled carbon nanotube. Based on the presented model, a condition is derived to predict the buckling load of the embedded double-walled nanotube, and the effect of the van der Waals forces to the buckling load is discussed when an inner nanotube is inserted into an embedded outer one. In particular, the paper shows that the buckling load of the embedded double-walled nanotube is always between that of the isolated inner nanotube and that of the embedded outer nanotube for both dynamic and static torsional buckling, due to the effect of the van der Waals forces. This result is different from that obtained by the existing analysis neglecting the difference of the radii for the embedded double-walled nanotube, which indicates that disregarding the difference of the radii of multi-walled nanotubes cannot properly describe the effect of the van der Waals forces between interlayer spacing. In particular, for static torsional buckling of a double-walled nanotube, it is shown that the critical buckling load cannot only be enhanced, but also be reduced when inserting an inner nanotube into an isolated single-walled one. Additionally, it is shown that the elastic medium always increases the critical buckling load of double-walled nanotubes. The critical buckling load of embedded double-walled nanotubes for dynamic torsional buckling is proved to be no less than that for static torsional buckling.     

An asymptotic-numerical analysis for the lower bound dynamic buckling estimates

A finite element asymptotic analysis for determining the lower bound dynamic buckling estimates of imperfection-sensitive structures under step load of infinite duration is presented. The lower bound dynamic buckling loads and the corresponding displacements are sought in the form of asymptotic expansions based on the static stability criterion and they can be determined by solving numerically (FEM) several linear problems with a single nonsingular sub-stiffness matrix. The project supported by the State Education Commission of China     

Non-linear buckling and postbuckling behavior of thin-walled beams considering shear deformation

The static stability of thin-walled composite beams, considering shear deformation and geometrical non-linear coupling, subjected to transverse external force has been investigated in this paper. The theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field (accounting for bending and warping shear) considering moderate bending rotations and large twist. This non-linear formulation is used for analyzing the prebuckling and postbuckling behavior of simply supported, cantilever and fixed-end beams subjected to different load condition. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton-Rapshon method. The numerical results show that the beam loses its stability through a stable symmetric bifurcation point and the postbuckling strength is in relation with the buckling load value. Classical predictions of lateral buckling are conservative when the prebuckling displacements are not negligible and the non-linear buckling analysis is required for reliable solutions. The analysis is supplemented by investigating the effects of the variation of load height parameter. In addition, the critical load values and postbuckling response obtained with the present beam model are compared with the results obtained with a shell finite element model (Abaqus).     

DYNAMIC BUCKLING OF DOUBLE-WALLED CARBON NANOTUBES UNDER STEP AXIAL LOAD

An approximate method is presented in this paper for studying the dynamic buckling of double-walled carbon nanotubes （DWNTs） under step axial load. The analysis is based on the continuum mechanics model, which takes into account the van der Waals interaction between the outer and inner nanotubes. A buckling condition is derived, from which the critical buckling load and associated buckling mode can be determined. As examples, numerical results are worked out for DWNTs under fixed boundary conditions. It is shown that, due to the effect of van der Waals forces, the critical buckling load of a DWNT is enhanced when inserting an inner tube into a single-walled one. The paper indicates that the critical buckling load of DWNTs for dynamic buckling is higher than that for static buckling. The effect of the radii is also examined. In addition, some of the results are compared with the previous ones.     

对桁架结构稳定分析经典理论的讨论

通过算例讨论了桁架结构稳定分析的经典理论,指出用该理论算出的临界荷载远远大于屈曲临界荷载,而且压杆的应力远远超过压缩强度极限。文中分析了问题的来源,提出了桁架结构临界荷载的屈曲理论计算方法,通过比较说明了屈曲理论的正确性。     

Study of static and dynamic stability of flexible rods in a geometrically nonlinear statement

We study static and dynamic stability problems for a thin flexible rod subjected to axial compression with the geometric nonlinearity explicitly taken into account. In the case of static action of a force, the critical load and the bending shapes of the rod were determined by Euler. Lavrent’ev and Ishlinsky discovered that, in the case of rod dynamic loading significantly greater than the Euler static critical load, there arise buckling modes with a large number of waves in the longitudinal direction. Lavrent’ev and Ishlinsky referred to the first loading threshold discovered by Euler as the static threshold, and the subsequent ones were called dynamic thresholds; they can be attained under impact loading if the pulse growth time is less than the system relaxation time. Later, the buckling mechanism in this case and the arising parametric resonance were studied in detail by Academician Morozov and his colleagues.In this paper, we complete and develop the approach to studying dynamic rod systems suggested by Morozov; in particular, we construct exact and approximate analytic solutions by using a system of special functions generalizing the Jacobi elliptic functions. We obtain approximate analytic solutions of the nonlinear dynamic problem of flexible rod deformation under longitudinal loading with regard to the boundary conditions and show that the analytic solution of static rod system stability problems in a geometrically nonlinear statement permits exactly determining all possible shapes of the bent rod and the complete system of buckling thresholds. The study of approximate analytic solutions of dynamic problems of nonlinear vibrations of rod systems loaded by lumped forces after buckling in the deformed state allows one to determine the vibration frequencies and then the parametric resonance thresholds.     

Optimal design of steel frames accounting for buckling

A formulation of a special design problem devoted to elastic perfectly plastic steel frame structures subjected to different combinations of static and dynamic loads is presented. In particular, a minimum volume design problem formulation is presented and the structure is designed to be able to elastically behave for the assigned fixed loads, to elastically shakedown in presence of serviceability load conditions and to prevent the instantaneous collapse for suitably chosen combinations of fixed and ultimate seismic loadings as well as of fixed and wind actions. The actions that the structure must suffer are evaluated by making reference to the actual Italian seismic code. The dynamic response of the structure is performed by utilizing a modal technique. In order to prevent other undesired collapse modes further constraints are introduced in the relevant optimization problem taking into account the risk of element buckling. Different applications devoted to flexural frames conclude the paper. The sensitivity of the structural response has been investigated on the grounds of the determination and interpretation of the Bree diagrams of the obtained optimal structures.     

Global dynamics and integrity of a two-dof model of?a?parametrically excited cylindrical shell

In this paper the global dynamics and topological integrity of the basins of attraction of a parametrically excited cylindrical shell are investigated through a two-degree-of-freedom reduced order model. This model, as shown in previous authors?? works, is capable of describing qualitatively the complex nonlinear static and dynamic buckling behavior of the shell. The discretized model is obtained by employing Donnell shallow shell theory and the Galerkin method. The shell is subjected to an axial static pre-loading and then to a harmonic axial load. When the static load is between the buckling load and the minimum post-critical load, a three potential well is obtained. Under these circumstances the shell may exhibit pre- and post-buckling solutions confined to each of the potential wells as well as large cross-well motions. The aim of the paper is to analyze in a systematic way the bifurcation sequences arising from each of the three stable static solutions, obtaining in this way the parametric instability and escape boundaries. The global dynamics of the system is analyzed through the evolution of the various basins of attraction in the four-dimensional phase space. The concepts of safe basin and integrity measures quantifying its magnitude are used to obtain the erosion profile of the various solutions. A detailed parametric analysis shows how the basins of the various solutions interfere with each other and how this influences the integrity measures. Special attention is dedicated to the topological integrity of the various solutions confined to the pre-buckling well. This allows one to evaluate the safety and dynamic integrity of the mechanical system. Two characteristic cases, one associated with a sub-critical parametric bifurcation and another with a super-critical parametric bifurcation, are considered in the analysis.     

柱壳在轴向冲击载荷作用下动态塑性屈曲的实验研究

圆柱壳的动态塑性屈曲问题的研究主要是集中在屈曲模态方面，至于屈曲的临介载荷与临介时时间则研究的较少，ＳＨＰＢ用于研究材料在高应变率下动态力学性能已为大家所熟悉，但用于研究结构的动态屈曲则未见报道，本文利用一装置对柱壳的动态塑性屈曲进行了实验研究，测出了壳体屈曲过程的载荷，轴向缩短量与时间的关系曲线，得到屈曲时的临介载荷与临介时间，同时发现壳体屈曲变形的一些规律并与静态实验的结果进行了比较，为理论分     

压电薄板屈曲有限元分析及DKQ单元

在机电耦合本构方程基础上,利用Hamilton原理推导了压电薄板屈曲分析的有限元特征方程和机电耦合的内力计算公式,在有限元实现中选择了基于Kirchhoff薄板假定的四边形薄板单元（DKQ单元）,并给出该单元的几何刚度阵及其数值积分方法。在大型通用有限元分析和优化设计软件系统JIFEX中实现了该方法。给出的数值验证了DKQ单元在屈曲分析和压电薄板静力分析中具有较高精度和收敛性,通过机械荷载和电荷载联合作用下的临界荷载计算,表明压电耦合效应能够影响结构的稳定性,可以通过改变外加电压对结构稳定性进行控制。     

连续梁在周期荷载下的动力稳定性简化分析方法

对于一般任意支撑的连续梁结构动力稳定性问题,已有的计算方法求解过程都很复杂,给工程设计带来极大的不便．本文提出了一个简化的分析方法,利用现有的商业软件,只需求得连续梁的自然频率及静力屈曲（失稳）荷载,就可容易得到结构的动力失稳区域,当考虑结构阻尼对不稳定区域的影响时,可将阻尼矩阵表达为Rayleigh阻尼的形式．研究结果表明：采用本文计算方法与已有的理论计算方法得到的连续梁主参数共振的不稳定边界非常吻合,而本文计算方法更为简单,计算结果可靠,计算精度高,可满足工程设计的需要．     

Static and dynamic buckling modes of a cylindrical shell under external pressure

We consider the problem of static and dynamic buckling modes of thin shells under external hydrostatic pressure. If the statement of the problem uses the linearized equations of motion obtained in the moderately large bending theory of shells according to the classical or refined model, then part of terms related to the external load in these equations are assumed to be conservative, and the other terms are assumed to be nonconservative. In this connection, we study four statements of the elastic stability problem for a cylindrical shell with hinged faces. The first of them is the statement of the static boundary value problem in the sense of Euler, where the action of external pressure is assumed to be conservative. The second statement is used to study small vibrations near the static equilibrium by a dynamic method for the same conservative load. The third and fourth statements of the problem correspond to the action of a nonconservative load and are similar to the first and second statements, respectively. They use the linearized equations of equilibrium and motion constructed earlier in a consistent version on the basis of a Timoshenko type model and allowing one to reveal all classical and nonclassical shell buckling modes.     

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。     

Plastic collapse of general frames

A structural theory is presented for the large static plastic deformation of space frames composed of thin walled members. Displacements comparable to the overall structural dimensions are contemplated. The frame is considered to consist of an arbitrary number of beam elements connected at node points. The analysis assumes that plastic deformation is confined to idealized hinges located at the node points. As a basis for a general frame computer program, the equations for a beam element are derived as a relationship between appropriate generalized force and deformation rates. The structural constitutive theory employed for the plastic hinge includes biaxial bending, torsion, and axial extension. It accounts for reduction in the load carrying capacity of the hinge due to local deformation. Predicted force-deformation curves for a space frame are in good agreement with experimental results.     

The influence of multiple cracks on tensile and compressive buckling of shear deformable beams

In this work the static stability of the uniform Timoshenko column in presence of multiple cracks, subjected to tensile or compressive loads, is analyzed. The governing differential equations are formulated by modeling the cracks as concentrated reductions of the flexural stiffness, accomplished by the use of Dirac’s delta distributions. The adopted model has allowed the derivation of the exact buckling modes and the corresponding buckling load equations of the Timoshenko multi-cracked column, as a function of four integration constant only, which are derived simply by enforcing the end boundary conditions, irrespective of the number of concentrated damage. Since shear deformability has been taken into account, the buckling load equation allows capturing both compressive and tensile buckling. The latter phenomenon has been recently investigated with reference to rubber bearing isolators, modeled as short beams, but it has been shown to occur also in slender beams characterized by high distributed shear deformation, like composite and layered beams. The influence of multiple concentrated cracks on the stability of shear deformable beams, particularly under the action of tensile loads, has never been assessed in the literature and is here addressed on the basis of an extensive parametric analysis. All the reported results have been compared with the Euler multi-cracked column in order to highlight its limits of applicability.     

Out-of-plane dynamic stability analysis of curved beams subjected to uniformly distributed radial loading

In this study, out-of-plane stability analysis of tapered cross-sectioned thin curved beams under uniformly distributed radial loading is performed by using the finite-element method. Solutions referred to as Bolotin’s approach are analysed for dynamic stability, and the first unstable regions are examined. Out-of-plane vibration and out-of-plane buckling analyses are also studied. In addition, the results obtained in this study are compared with the published results of other researchers for the fundamental frequency and critical lateral buckling load. The effects of subtended angle, variations of cross-section, and dynamic load parameter on the stability regions are shown in graphics