A geometric approach for establishing dynamic buckling loads of autonomous potential N-degree-of-freedom systems

Non-linear dynamic buckling of autonomous non-dissipative N-degree-of-freedom systems whose static instability is governed either by a limit point or by an unstable symmetric bifurcation is thoroughly discussed using energy and geometric considerations. Characteristic distances associated with the geometry of the zero level total potential energy “hypersurface” in connection with total energy-balance equation lead to dynamic (global) instability criteria. These criteria allow the determination of “exact” dynamic buckling loads without solving the non-linear initial-value problem. The reliability and efficiency of the proposed geometric approach is demonstrated via several dynamic buckling analyses of 3-degree-of-freedom systems which subsequently are compared with corresponding numerical analyses based on the Verner–Runge–Kutta scheme.     

Dynamic buckling of partially-sway frames with varying stiffness using catastrophe theory

This work deals with the static and dynamic stability analysis of imperfect partially-sway frames with non-uniform columns. The examined two-bar frames are elastically supported and subjected to an eccentrically vertical load at their joint. Through a linear stability analysis, the effect of the taper ratio of the column cross-section on the buckling capacity of the partially-sway frame is thoroughly discussed. Using a non-linear method an accurate formula has been established for determining the exact asymmetric bifurcation point associated with the maximum load carrying capacity. These findings have been re-derived more readily using Catastrophe Theory (CT) and considering the frame as a one degree-of-freedom (1-DOF) system through an efficient technique. A local analysis allows us to classify, after reduction, the total potential energy (TPE) function of the system to one of the seven elementary Thom׳s catastrophes (with known properties) and to obtain static and dynamic singularity as well as bifurcational sets. It has been found that geometrical and loading imperfections, which are always present in structural engineering problems, have a significant effect on the dynamic buckling loads. The efficiency of the present approach is illustrated via several examples, while results from finite element analyses are in good agreement with the analytical solution presented herein.     

Influence of localized imperfections on the buckling of cylindrical shells under axial compression

The influence of localized imperfections on the buckling of a long cylindrical shell under axial compression is analysed by using a double scale analysis including interaction modes. This leads to a system of coupled complex non-linear differential equations with discontinuous derivatives. We propose analytical formulas to predict the reduction of the critical buckling load.     

Imperfection sensitivity of optimal symmetric braced frames against buckling

Imperfection sensitivity characteristics of the non-linear buckling load factors of non-optimal and optimal symmetric frames are investigated. The frames are subjected to symmetric proportional vertical loads. The optimization problem is formulated under constraints on linear buckling load factors. Although the buckling load factors corresponding to sway and non-sway modes coincide at the optimum design, the non-sway-type asymmetric bifurcation point disappears as a result of geometrically non-linear analysis. Therefore, the maximum allowable load factors of perfect and imperfect systems should be determined by assigning displacement constraints. It is shown that quantitative evaluation is possible for imperfection sensitivity and mode interaction based on the higher order differential coefficients of the total potential energy even for frames of which the critical points should be numerically obtained. Numerical examples are presented to show that the properties of the non-sway bifurcation point are similar to those of a symmetric bifurcation point, and the interaction between sway and non-sway modes does not always lead to enhancement of imperfection sensitivity.     

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.     

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.     

Imperfection sensitivity of curved panels under combined compression and shear

The initial buckling load of curved panels under compressive loads is substantially reduced by the existence of imperfections, in particular geometric imperfections. It is therefore essential that these imperfections are considered in analysing components which incorporate such panels in order to accurately predict their buckling behaviour. Finite element analysis allows fully non-linear analysis of shells containing geometric imperfections, however, to obtain accurate results information is required on the exact size and shape of the imperfection to be modelled. In most cases this data is not available. It is therefore generally recommended that the imperfections are modelled on the first eigenmode and have an amplitude selected according to the manufacturing procedure. This paper presents the effects of varying imperfection shape and amplitude on the buckling and postbuckling behaviour of one specific case, a curved panel under combined shear and compression, to test the accuracy of such recommendations.     

Post-buckling behaviour of graphite/epoxy stiffened panels with initial imperfections subjected to eccentric biaxial compression loading

The post-buckling behaviour of anisotropic stiffened panels with initial imperfections is investigated. Since buckling of the skin between the stiffeners often occurs first, a non-linear analysis is developed for symmetric panels under biaxial compression in order to obtain the out-of-plane panel deflection in the post-buckling range. The non-linear differential equations are expressed in terms of the out-of-plane displacement and the Airy function. They are solved with the Galerkin method for various boundary conditions by imposing an edge displacement control. The theoretical and experimental results obtained by the present analysis show that the transverse load can greatly influence the buckling loads and halfwave number. Since no experimental results have been found in the literature, several tests have been carried out on graphite/epoxy blade stiffened panels 900 mm long and 620 mm wide applying simultaneously biaxial compression loads with several combined ratios. An eccentricity results between longitudinal and transverse load, because the longitudinal compression is applied along the centroidal axes of the stiffened section while the transverse compression is applied to the skin panel. The correlation between the experimental and analytical results has been quite good; the experimental results demonstrate the influence of eccentricity of the transverse load on panel deflection in the pre- and post-buckling range.     

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.     

Non-linear dynamic stability of a simple floating bridge model

Summary This paper deals with a simple fluid-structure interaction problem of floating bridges under step loading with main emphasis on the non-linear dynamic stability of the structure itself after been simulated by a simple discrete mechanical model. The analysis concerns systems which under the same loading applied statically experience a limit point instability. On the basis of a theoretical discussion of the non-linear response of a single degree-of-freedom model simple conditions for an unbounded motion associated with dynamic buckling have been properly established.According to these conditions one can determine the exact dynamic buckling load without solving the strongly non-linear differential equation of motion. Such a load corresponds to that equilibrium point of the unstable (static) post-buckling path for which the total potential energy of the model becomes zero, while at the same time its second variation is negative definite. This load is also a lower bound in case that damping is included in the analysis. The foregoing conditions of static evaluation of the dynamic buckling load do not hold, in general, for limit point systems of two degres of freedom.The above theoretical predictions have been confirmed by means of numerical integration of the correspending non-linear equation of motion.

---

Nichtlineare dynamische Stabilität eines einfachen Pontonbrückenmodells

Übersicht In dem Beitrag wird am Beispiel einer zweigliedrigen Pontonbrücke das Problem der nichtlinearen dynamischen Stabilität bei sprungförmiger Längsbelastung behandelt. Die Wechselwirkung Struktur—Fluid wird dabei durch eine linearisierte Rückstellkraft und eine Ersatzmasse des Fluids modelliert. Die Analysis betrifft Systeme, welche unter gleicher statischer Belastung eine Grenzwertinstabilität erfahren. Auf der Grundlage der nichtlinearen Antwort eines Modells mit einem Freiheitsgrad werden einfache Bedingungen für die unbeschränkte Bewegung verbunden mit dynamischem Knicken angegeben.Mit diesen Bedingungen kann die genaue dynamische Knicklast gefunden werden, ohne daß man die stark nichtlineare Bewegungsgleichung zu lösen hat. Diese Knicklast entspricht dem Gleichgewichtspunkt für den instabilen (statischen) Nachknickpfad, für den die potentielle Energie des Systems verschwindet, während zugleich ihre zweite Variation negativ definit ist. Diese Last ist ebenfalls eine untere Schranke für den Fall des Systems mit Dämpfung. Diese statische Ermittlung der dynamischen Knicklast kann i. allg. nicht auf ein System von zwei Freiheitsgraden übertragen werden.Die analytischen Ergebnisse wurden durch numerische Integration der zugehörigen nichtlinearen Bewegungsgleichung bestätigt.

     

Dynamic buckling of a single-degree-of-freedom system with variable mass

In this paper dynamic buckling of the single-degree-of-freedom system with variable mass is analyzed. In the system the mass variation is slow and is a function of slow variable time. Due to mass variation the impact force acts. The motion of the system is described with a nonlinear ordinary differential equation with time variable parameters. A new approximate analytic criterion of dynamic buckling for the non-autonomous systems which have the conservation law of energy type is developed. The conservation law is formed applying the Noetherian approach. The suggested method allows the determination of dynamic buckling load without solving the corresponding nonlinear differential equation of motion. For this value of dynamic load the motion of the system becomes unbounded. The obtained analytic value is compared with the numeric one. It shows a good agreement.     

S型功能梯度扁球壳非线性屈曲的渐近分析

本文利用渐近迭代法获得了边界弹性支撑的功能梯度扁球壳的非线性屈曲问题的理论解．假设材料组分体积分数沿壳体厚度方向呈sigmoid幂函数变化,边界上考虑一般的弹性支撑约束．基于经典的薄壳理论和几何非线性关系,导出了S型功能梯度扁球壳的非线性屈曲问题的控制方程．采用渐近迭代法通过两次迭代得到了无量纲挠度和均布荷载之间的非线性特征关系．通过与已有有限元方法和其他数值方法的结果对比,验证了本文解的有效性和高精度．同时,通过计算阐述了缺陷因子、材料参数、边界约束系数及特征几何参数对壳体临界屈曲荷载的影响．     

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.     

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.     

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.     

Dynamic thermal buckling of suddenly heated temperature-dependent FGM cylindrical shells,under combined axial compression and external pressure

FGM components are constructed to sustain high temperature gradients. There are many applications where the FGM components are vulnerable to transient thermal shocks. If a component is already under compressive external loads (e.g. under a combination of axial compression and external pressure), the mentioned thermal shocks will cause the component to exhibit dynamic behavior and in some cases may lead to buckling. On the other hand, a preheated FGM component may undergo dynamic mechanical loads. Only static thermal buckling investigations were developed so far for the FGM shells. In the present paper, dynamic buckling of a pre-stressed, suddenly heated imperfect FGM cylindrical shell and dynamic buckling of a mechanically loaded imperfect FGM cylindrical shell in thermal environment, with temperature-dependent properties are presented. The general form of Green’s strain tensor in curvilinear coordinates and a high order shell theory proposed already by the author are used. Instead of using semi-analytical solutions that rely on the validity of the separation of variables concept, the complicated nonlinear governing equations are solved using the finite element method. Buckling load is detected by a modified Budiansky criterion proposed by the author. The effects of temperature-dependency of the material properties, volume fraction index, load combination, and initial geometric imperfections on the thermo-mechanical post-buckling behavior of a shell with two constituent materials are evaluated. The results reveal that the volume fraction index and especially, the differences between the thermal stresses created in the outer and the inner surfaces may change the buckling behavior. Furthermore, temperature gradient and initial imperfections have less effect on buckling of a shell subjected to a pure external pressure.     

Kármán-type equations for a higher-order shear deformation plate theory and its use in the thermal postbuckling analysis

Kármán-type nonlinear large deflection equations are derived occnrding to the Reddy’s higher-order shear deformation plate theory and used in the thermal postbuckling analysis The effects of initial geometric imperfections of the plate areincluded in the present study which also includes th thermal effects.Simply supported,symmetric cross-ply laminated plates subjected to uniform or nomuniform parabolictemperature distribution are considered. The analysis uses a mixed GalerkinGolerkinperlurbation technique to determine thermal buckling louds and postbucklingequilibrium paths.The effects played by transverse shear deformation plate aspeclraio, total number of plies thermal load ratio and initial geometric imperfections arealso studied.     

STUDY ON AMPLIFICATION FACTOR OF FLEXURAL BUCKLING CRITICAL LOAD OF AXIAL COMPRESSION BARS1)

为了建立一般条件下轴压构件屈曲临界载荷的计算理论,首先对轴心受压构件发生屈曲时的总势能方程进行了推导,然后采用Rayleigh-Ritz法并基于势能驻值原理得到了4种不同端部约束条件下轴压构件的屈曲临界载荷,对比欧拉临界载荷,给出了临界载荷放大系数 的计算式,全面考虑了构件长细比、压缩变形、剪切变形以及截面形状系数对临界载荷的影响,推导的计算式可用于较小长细比轴压构件发生屈曲时临界载荷的计算.圆截面和双轴对称工字形截面轴压构件屈曲临界载荷的分析表明构件长细比是影响放大系数的主导因素。     

A semi-analytical buckling analysis of imperfect cylindrical shells under axial compression

In the framework of the cellular bifurcation theory, we investigate the effect of distributed and/or localized imperfections on the buckling of long cylindrical shells under axial compression. Using a double scale perturbative approach including modes interaction, we establish that the evolution of amplitudes of instability patterns is governed by a non-homogeneous second order system of three non-linear complex equations. The localized imperfections are included by employing jump conditions for their amplitude and permitting discontinuous derivatives. By solving these amplitude equations, we show the influence of distributed and/or localized imperfections on the reduction of the critical load. To assess the validity of the present method, our results are compared to those given by two finite element codes.     

弹性压应力波下直杆动力失稳的机理的判据

基于应力波理论和失稳瞬间能量的转换和守恒,导出了一个直杆动力分岔失稳的准则：（1）直杆在发生分岔失稳的瞬间所释放出的压缩变形能等于屈曲所需变形能与屈曲动能之和;（2）在上述能量转换过程中,能量对时间的变化率服从守恒定律。应用临界条件（1）推导出的直杆动力失稳的控制方程和杆端边界条件以及连续条件,与应用哈密顿原理推导的结果完全相同,但不足以构成求解直杆动力失稳问题的完备定解条件,导出包含两个特征参数的一对特征方程。从而建立了求解直杆动力失稳模态和两个特征参数（临界力参数和失稳惯性项指数参数即动力特征参数）的较严密理论方法。