Post-buckling analysis of slender elastic rods subjected to terminal forces

This paper presents formulation and solutions for the elastica of slender rods subjected to axial terminal forces and boundary conditions assumed hinged and elastically restrained with a rotational spring. The set of five first-order non-linear ordinary differential equations with boundary conditions specified at both ends constitutes a complex two-point boundary value problem. Solutions for buckling, initial post-buckling (perturbation), large loads (asymptotic) and numerical integration are developed. Results are presented in non-dimensional graphs for a range of rotational spring stiffness, tuning the analysis from double-hinged to hinged-built-in rods.     

Post-buckling analysis of slender elastic vertical rods subjected to terminal forces and self-weight

This article presents the behavior of slender elastic rods subjected to axial terminal forces and self-weight. The mathematical formulation is presented, a solution is sought for a double-hinged boundary condition and the analysis is carried out for different values of non-dimensional weight. The formulation derives from geometrical compatibility, equilibrium of forces and moments and constitutive relations yielding a set of six first order non-linear ordinary differential equations with boundary conditions specified at both ends, which characterizes a complex two-point boundary value problem. Furthermore, a perturbation method is used to find the critical buckling loads and initial post-buckling solutions. A numerical integration scheme based on a three parameter shooting method is employed in the post-buckling solutions.     

Large-deflection and post-buckling behavior of slender beam-columns with non-linear end-restraints

The large-deflection analysis and post-buckling behavior of laterally braced or unbraced slender beam-columns of symmetrical cross section subjected to end loads (forces and moments) with both ends partially restrained against rotation, including the effects of out-of-plumbness, are developed in a classical manner. The classical theory of the “Elastica” and the corresponding elliptical functions utilized herein are those presented previously by Aristizabal-Ochoa [1]. The proposed method can be used in the large-deflection analysis and post-buckling behavior of elastic slender beam-columns with rigid, semi-rigid, and simple flexural connections at both ends including linear and non-linear inelastic connections like those that suffer from flexural degradation (such as flexural cracking and elasto-plastic connections) or flexural stiffening. Only bending strains are considered in the proposed analysis. Results from the proposed method are theoretically exact from small to very large curvatures and transverse and longitudinal displacements for laterally braced or unbraced slender beam-columns under bending caused by end loads. The large-deflection analysis and post-buckling behavior of slender beam-columns with both supports partially restrained against rotation and with sway inhibited or uninhibited are complex problems requiring the simultaneous solution of two coupled non-linear equations with elliptical integrals whose unknowns are the limits of the integrals. The validity of the proposed method and equations are verified against solutions available in the technical literature. Three comprehensive examples are included that show the effects of linear and non-linear connections at both ends on the large-deflection analysis and post-buckling behavior of slender beam-columns.     

Nonlinear strain gradient elastic thin shallow shells

The governing equilibrium equations for strain gradient elastic thin shallow shells are derived, considering nonlinear strains and linear constitutive strain gradient elastic relations. Adopting Kirchhoff’s theory of thin shallow structures, the equilibrium equations, along with the boundary conditions, are formulated through a variational procedure. It turns out that new terms are introduced, indicating the importance of the cross-section area in bending of thin plates. Those terms are missing from the existing strain gradient shallow thin shell theories. Those terms highly increase the stiffness of the structures. When the curvature of the shallow shell becomes zero, the governing equilibrium for the plates is derived.     

Large deflection and post-buckling analysis of non-linearly elastic rods by wavelet method

We propose a wavelet method in the present study to analyze the large deflection bending and post-buckling problems of rods composed of non-linearly elastic materials, which are governed by a class of strong non-linear differential equations. This wavelet method is established based on a modified wavelet approximation of an interval bounded L²-function, which provides a new method for the large deflection bending and post-buckling problems of engineering structures. As an example, in this study, we considered the rod structures of non-linear materials that obey the Ludwick and the modified Ludwick constitutive laws. The numerical results for both large deflection bending and post-buckling problems are presented, illustrating the convergence and accuracy of the wavelet method. For the former, the wavelet solutions are more accurate than the finite element method and the shooting method embedded with the Euler method. For the latter, both bifurcation and limit loads can be easily and directly obtained by solving the extended systems. On the other hand, for the shooting method embedded with Runge–Kutta method, to obtain these values usually needs to choose a good starting value and repeat trial solutions many times, which can be a tough task.     

Thermal buckling and post-buckling of pinned–fixed Euler–Bernoulli beams on an elastic foundation

In this article, both thermal buckling and post-buckling of pinned–fixed beams resting on an elastic foundation are investigated. Based on the accurate geometrically non-linear theory for Euler–Bernoulli beams, considering both linear and non-linear elastic foundation effects, governing equations for large static deformations of the beam subjected to uniform temperature rise are derived. Due to the large deformation of the beam, the constraint forces of elastic foundation in both longitudinal and transverse directions are taken into account. The boundary value problem for the non-linear ordinary differential equations is solved effectively by using the shooting method. Characteristic curves of critical buckling temperature versus elastic foundation stiffness parameter corresponding to the first, the second, and the third buckling mode shapes are plotted. From the numerical results it can be found that the buckling load-elastic foundation stiffness curves have no intersection when the value of linear foundation stiffness parameter is less than 3000, which is different from the behaviors of symmetrically supported (pinned–pinned and fixed–fixed) beams. As we expect that the non-linear foundation stiffness parameter has no sharp influence on the critical buckling temperature and it has a slight effect on the post-buckling temperature compared with the linear one.     

纵横载荷作用下功能梯度梁的弯曲和过屈曲

基于一阶非线性梁理论和物理中面概念,导出了纵横向载荷作用下功能梯度材料(FGM)梁非线性弯曲和过屈曲问题的控制方程,并获得了该问题的精确解;据此解研究了梯度材料性质、外载荷、横向剪切变形以及边界条件等因素对功能梯度材料梁非线性力学行为的影响,分析中假设功能梯度材料性质只沿梁厚度方向,并按成分含量的幂指数函数形式变化。结果表明:纵横载荷共同作用下,功能梯度梁的弯曲构形将有无限多个;随着梯度指数的增大,梁的变形减小,临界载荷升高;随着长高比的增大,横向剪切变形的影响减小。     

热载荷作用下梯度多孔材料梁弯曲及过屈曲力学行为的研究

基于Bernoulli-Euler梁理论,引入物理中面解耦了复合材料结构的面内变形与横向弯曲特性,研究了梯度多孔材料矩形截面梁在热载荷作用下的弯曲及过屈曲力学行为．假设沿梁厚度方向材料的性质是连续变化的,利用能量法推导了矩形截面梁的控制微分方程和边界条件,并用打靶法对无量纲化的控制方程进行数值求解．利用计算得到的结果分析了材料的性质、热载荷、边界条件对矩形截面梁非线性力学行为的影响．结果表明,对称材料模型下,固支梁与简支梁均显示出了典型的分支屈曲行为特征,而其临界屈曲热载荷值均会随着孔隙率系数的增加而单调增加．非对称材料模型下,固支梁仍显示出分支屈曲行为特征,但其临界屈曲热载荷不再随着孔隙率系数的变化而单调变化;而对于两端简支梁,发生了弯曲变形,弯曲挠度随载荷的增大而增大．     

THIRD-ORDER GENERALISED BEAM THEORY AND CALCULATION METHOD

一阶广义梁理论描述通过运用加入弯曲、扭转和畸变函数的普通非耦合微分方程组解决棱柱状结构行为.二阶广义梁理论,是添加上偏离力效果的微分方程. 通过引入纵向膜弯矩和膜剪应变虚功到广义梁理论系统当中,完全展开的三阶广义梁方程组将以一串大型离散迭代函数且能转化为可用于数值分析的若干切线刚度矩阵形式出现. 通过膜应力派生出三阶分项^ijrkv_σ和^ijrkv_τ并结合先进数值技术寻求全解,三阶广义梁理论提供了一种严谨和高效的数值工具用于调查薄壁结构后屈曲大变形行为.     

Post-buckling analysis of FGM annular sector plates based on three dimensional elasticity graded finite elements

In this article, post-buckling and non-linear bending analysis of functionally graded annular sector plates based on three dimensional theory of elasticity in conjunction with non-linear Green strain tensor is considered. In-plane normal compressive loads have been applied to either radial, circumferential, or all edges of annular sector plates. Material properties are graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents while Poisson׳s ratio is assumed to be constant. The governing equations are developed based on the principle of minimum total potential energy and solved based on graded finite element method. Non-linear equilibrium equations are solved based on iterative Newton–Raphson method. The effects of material gradient exponent, different sector angles, thickness ratio, loading condition and two different boundary conditions on the post-buckling behavior of FGM annular sector plates have been investigated. Results denote that due to the stretching–bending coupling effects of the FGMs, the post-buckling behavior of movable simply supported FGM plates is not of the bifurcation-type buckling. Moreover, FGM annular sector plates subjected to uniaxial compression at radial edges show a non-linear bending behavior with unique and stable equilibrium paths following a flattening feature.     

Large-deflection and postbuckling of beam-columns with non-linear semi-rigid connections including shear and axial effects

The non-linear large deflection-small strain analysis and post-buckling behavior of an out-of-plumb Timoshenko beam-column of symmetrical cross section subjected to end loads (forces and moments) with non-linear bending connections at both ends, and its top end partially restrained against transverse and longitudinal translations are developed in a classical manner. A set of non-linear equations based on the “modified shear equation” that includes the effects of (1) shear deformation and the shear component of the applied axial forces; and (2) shortening of the beam-column due to both axial forces and “bowing” are presented. The proposed method and corresponding equations can be used in the large deflection-small strain analysis of Timoshenko beam-columns with non-linear bending connections, as well as lateral and longitudinal non-linear restraints at the top end. This paper is an extension of previous work presented by the senior author on the large deflection and post-buckling behavior of Timoshenko beam-columns with linear elastic semi-rigid connections and linear elastic lateral bracing. Three comprehensive examples are included that show the effectiveness of the proposed method and corresponding equations. Results obtained in the three examples are verified against analytical solutions available in the technical literature and against results from models using the FEM program ABAQUS.     

A nonlinear planar beam formulation with stretch and shear deformations under end forces and moments

A new nonlinear planar beam formulation with stretch and shear deformations is developed in this work to study equilibria of a beam under arbitrary end forces and moments. The slope angle and stretch strain of the centroid line, and shear strain of cross-sections, are chosen as dependent variables in this formulation, and end forces and moments can be either prescribed or resultant forces and moments due to constraints. Static equations of equilibria are derived from the principle of virtual work, which consist of one second-order ordinary differential equation and two algebraic equations. These equations are discretized using the finite difference method, and equilibria of the beam can be accurately calculated. For practical, geometrically nonlinear beam problems, stretch and shear strains are usually small, and a good approximate solution of the equations can be derived from the solution of the corresponding Euler–Bernoulli beam problem. The bending deformation of the beam is the only important one in a slender beam, and stretch and shear strains can be derived from it, which give a theoretical validation of the accuracy and applicability of the nonlinear Euler–Bernoulli beam formulation. Relations between end forces and moments and relative displacements of two ends of the beam can be easily calculated. This formulation is powerful in the study of buckling of beams with various boundary conditions under compression, and can be used to calculate post-buckling equilibria of beams. Higher-order buckling modes of a long slender beam that have complex configurations are also studied using this formulation.     

三阶广义梁理论及其计算方法

一阶广义梁理论描述通过运用加入弯曲、扭转和畸变函数的普通非耦合微分方程组解决棱柱状结构行为.二阶广义梁理论,是添加上偏离力效果的微分方程. 通过引入纵向膜弯矩和膜剪应变虚功到广义梁理论系统当中,完全展开的三阶广义梁方程组将以一串大型离散迭代函数且能转化为可用于数值分析的若干切线刚度矩阵形式出现. 通过膜应力派生出三阶分项ijrkv_σ和ijrkv_τ并结合先进数值技术寻求全解,三阶广义梁理论提供了一种严谨和高效的数值工具用于调查薄壁结构后屈曲大变形行为.      

大跨度悬索桥索股抗弯刚度对锚跨张力测试精度的影响研究

为研究索股抗弯刚度对张力测试精度的影响,基于能量法推导了计入索股抗弯刚度和忽略索股抗弯刚度影响的悬索桥锚跨索股在两端铰接时索股张力和自振频率的关系,考虑到悬索桥索股边界条件的复杂性,结合动平衡法得出了索股在一端铰接一端弹性支承时索股张力与自振频率之间的相关表达式。通过算例,对所得表达式的精度及有效性进行了验证,结果发现索股的抗弯刚度对索股张力控制精度的影响较大。工程应用研究则表明,索股抗弯刚度对测试精度的影响,主要由索股张力产生的应力刚度与索股抗弯刚度之间的比值决定;锚跨索股边界条件的不确定性,对控制精度也产生较大影响,施工锚跨张力控制过程中,结合提出的最小二乘法,能有效地解决该问题,并能大幅提高索股张力控制精度。     

Initial thermo-mechanical post-buckling of beams with temperature-dependent physical properties

The objective of this work is to analyze the elastic buckling and initial post-buckling behavior of slender beams subjected to uniform heating. The beams are assumed to be double-hinged with fixed ends, preventing thermal expansion. Consequently, destabilizing compressive forces arise that may lead to beam buckling. When the temperature is further increased, the beam experiences finite displacements, with the result that the analysis is geometrically non-linear. The modulus of elasticity and the thermal induced strain, key material properties for this problem, are temperature-dependent. Thus, the coefficients of the governing equations are not constant. This suggests the physical non-linearity of the mathematical model. Hence, the analysis is geometrically and physically non-linear. The analysis is sensitive to the beam initial temperature, as the thermal strain is a function of the initial and final temperatures. The material is considered to be linear elastic, and consequently viscoelastic and plastic effects are not taken into account. Furthermore, the beam cross-section properties are assumed to be constant, which is consistent with the small strain formulation. A perturbation method is applied to the governing non-linear differential equations so that the initial post-buckling behavior may be analytically determined when temperatures above the critical temperature are applied to the beam. To illustrate the application of the formulation we present a case study for the aluminum 7075-T6 alloy, a material commonly used in aerospace and naval industries. Nonetheless, it is expected similar behavior for other metallic materials. The curves that define the variation of the modulus of elasticity, the thermal strain and the yield stress with temperature are considered in our analysis. The change in length, reaction forces at the supports and geometric configurations are obtained as a function of temperature and the beam slenderness ratio. The critical buckling loads and temperatures and the initial post-buckling analysis are also calculated in the context of the temperature-independent physical properties. Our results emphasize the importance of modeling the material's non-linearity if accuracy is required. However, from a practical application point of view results are acceptable if temperature-independent physical properties are employed, especially for large slenderness ratios.     

Buckling and post-buckling of extensible rods revisited: A multiple-scale solution

An exact non-linear formulation of the equilibrium of elastic prismatic rods subjected to compression and planar bending is presented, electing as primary displacement variable the cross-section rotations and taking into account the axis extensibility. Such a formulation proves to be sufficiently general to encompass any boundary condition. The evaluation of critical loads for the five classical Euler buckling cases is pursued, allowing for the assessment of the axis extensibility effect. From the quantitative viewpoint, it is seen that such an influence is negligible for very slender bars, but it dramatically increases as the slenderness ratio decreases. From the qualitative viewpoint, its effect is that there are not infinite critical loads, as foreseen by the classical inextensible theory. The method of multiple (spatial) scales is used to survey the post-buckling regime for the five classical Euler buckling cases, with remarkable success, since very small deviations were observed with respect to results obtained via numerical integration of the exact equation of equilibrium, even when loads much higher than the critical ones were considered. Although known beforehand that such classical Euler buckling cases are imperfection insensitive, the effect of load offsets were also looked at, thus showing that the formulation is sufficiently general to accommodate this sort of analysis.     

Interactive buckling in long thin-walled rectangular hollow section struts

An analytical model describing the nonlinear interaction between global and local buckling modes in long thin-walled rectangular hollow section struts under pure compression founded on variational principles is presented. A system of nonlinear differential and integral equations subject to boundary conditions is formulated and solved using numerical continuation techniques. For the first time, the equilibrium behaviour of such struts with different cross-section joint rigidities is highlighted with characteristically unstable interactive buckling paths and a progressive change in the local buckling wavelength. With increasing joint rigidity within the cross-section, the severity of the unstable post-buckling behaviour is shown to be mollified. The results from the analytical model are validated using a nonlinear finite element model developed within the commercial package Abaqus and show excellent comparisons. A simplified method to calculate the local buckling load of the more compressed web undergoing global buckling and the corresponding global mode amplitude at the secondary bifurcation is also developed. Parametric studies on the effect of varying the length and cross-section aspect ratio are also presented that demonstrate the effectiveness of the currently developed models.     

含芯拧绞绳非线性弯曲动力学特性分析与研究

建立细长缆索大柔性多体动力学模型时,现实存在的复杂捻制几何构型多不予考虑,而是将柔索简化为材料均匀梁进行描述,致使运动仿真模型与物理实际存在一定差距. 为此,研究一种典型非线性拧绞绳股的大变形等效动力学建模方法,考虑准静态与大范围运动情况下绳股内的线接触,计算了受摩擦力及弯曲曲率影响的绳股可变弯曲刚度,通过等效梁模型避免了绳股精细建模时的大规模计算消耗. 基于连续介质力学与绝对节点坐标方法,建立了拧绞绳惯性广义坐标下的多柔体动力学模型. 为了验证等效模型的可行性,与基于有限段方法建立的精细模型进行对比仿真分析,通过位形验证了等效模型的精度. 进一步地,根据力载作用下的准静态构型,研究了特定构型绳股弯曲刚度沿轴向的分布规律;通过自重力下一端固定柔性绳摆自由运动仿真并与传统均匀梁模型相比,研究了模型弯曲特性的差异. 最后,根据能量守恒原理分析了摩擦耗散系统内各种能量间的相互转化. 拧绞绳大变形等效动力学模型能够提高绳索动力系统运动预测的仿真计算效率,还能为钢丝绳参数与构型设计提供依据.      

简支梁大挠度弹性后屈曲载荷与变形的优化算法

针对简支梁结构大挠度后屈曲载荷与变形的计算问题,本文提出了一种直接求解其后屈曲载荷和变形的优化算法。在简支梁处于大挠度屈曲平衡状态下,将梁结构划分为有限子段,以待求后屈曲载荷为设计变量,根据起点的边界条件和每个子段满足的弯矩变形公式,累积计算出其他各个节点的坐标,以得到的终点坐标满足的边界条件构建目标函数模型。在此基础上,通过MATLAB编制优化程序分析了两个典型算例,并将理论结果与相关软件的计算结果进行对比,从而证明了本文算法的正确性。本文算法求解过程简单、快速,具有一定的实用性,为变截面结构大挠度弹性屈曲稳定性问题的研究提供了参考。