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.     

Thermal buckling and postbuckling of FGM circular plates with in-plane elastic restraints

Based on von Karman's plate theory, the axisymmetric thermal buckling and post-buckling of the functionally graded material(FGM) circular plates with inplane elastic restraints under transversely non-uniform temperature rise are studied. The properties of the FGM media are varied through the thickness based on a simple power law. The governing equations are numerically solved by a shooting method. The results of the critical buckling temperature, post-buckling equilibrium paths, and configurations for the in-plane elastically restrained plates are presented. The effects of the in-plane elastic restraints, material property gradient, and temperature variation on the responses of thermal buckling and post-buckling are examined in detail.     

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.     

细长弹性杆在轴压下的二次分叉

基于[1]的弹性曲杆的平衡方程,本文研究了矩形横截面细长杆在轴压下的后屈曲行为。设横截面的边长比为 1:2δ,使用 Poincare-Keller 的打靶法并引进坐标的伸缩变换,研究了δ在 δ_0=1 附近的情形。当δ≠1 时,发现了杆平衡态的二次分叉。我们也给出了原始后屈曲解支及二次分支的渐近表示并分析了各个解支的稳定性。     

深海保温输油管后屈曲及屈曲传播的有限元分析

本文运用ABAQUS有限元软件模拟深海输油管道的后屈曲及屈曲传播现象．将深海输油管道视为内、外层为合金钢材,夹心为聚氨酯泡沫的输油管道．采用线弹性-线性硬化的本构关系,运用Riks分析法获得后屈曲平衡路径,以此模拟管道中的屈曲传播过程,并获得屈曲传播压力．通过数值算例,综合讨论了不同初始缺陷、不同径厚比、不同夹层与夹心厚度比对、不同的材料弹性模量对屈曲传播过程和压力的影响．最后将输油管退化为单层管,将传播压力的有限元结果与实验结果、 Palmer理论解比较．结果表明有限元值、实验结果、理论值三者比较吻合．验证了有限元分析的正确性．     

Post-buckling of a hinged-fixed beam under uniformly distributed follower forces

Based on geometrically non-linear theory for extensible elastic beams, governing equations of statically post-buckling of a beam with one end hinged and the other fixed, subjected to a uniformly distributed, tangentially compressing follower forces are established. They consist of a boundary-value problem of ordinary differential equations with a strong non-linearity, in which seven unknown functions are contained and the arc length of the deformed axis is considered as one of the basic unknown functions. By using shooting method and in conjunction with analytical continuation, the non-linear governing equations are solved numerically and the equilibrium paths as well as the post-buckled configurations of the deformed beam are presented. A comparison between the results of conservative system and that of the non-conservative systems are given. The results show that the features of the equilibrium paths of the beams under follower loads are evidently different from that under conservative ones.     

Effects of axisymmetric loads on inflated non-linear membranes

This paper describes the effects of various external axisymmetric loads on pressurized hinged spherical membranes taking into account changes in internal pressure, volume, and temperature. “Exact” geometrical non-linearity along with generalized constitutive relations for a highly non-linearly clastic, isotropic, homogeneous, incompressible material are used in the analysis. The specialized case of a Hookean material is also treated.The non-linear equations of membrane equilibrium are derived in terms of additional finite displacements for the case of nonorthogonal curvilinear midsurface coordinates and are then specialized for the problem of an inflated hinged spherical membrane. The resulting two highly non-linear coupled second order differential equations are solved by means of a finite difference and Newton-Raphson iterative procedure. All results are presented in nondimensionalized graphical form.     

Post-buckling analysis of non-prismatic columns under general behaviour of loading

The paper shows the effects of behaviour of loading during the buckling process on the value of critical force and initial stability of post-buckling path for elastic, non-prismatic columns. Perturbation method combined with Croll's manoeuvre makes it possible to derive general formulae for the first or the second correction of the force, and hence to analyze stability in the post-buckling range. The effects of behaviour of active and reactive forces may be essential: in numerous examples the boundary values of structural parameters separating stability and instability are evaluated. Pre-buckling geometry changes are analyzed as well.     

Anti-plane shear waves in a fibre-reinforced composite with a non-linear imperfect interface

The propagation of non-linear elastic anti-plane shear waves in a unidirectional fibre-reinforced composite material is studied. A model of structural non-linearity is considered, for which the non-linear behaviour of the composite solid is caused by imperfect bonding at the “fibre–matrix” interface. A macroscopic wave equation accounting for the effects of non-linearity and dispersion is derived using the higher-order asymptotic homogenisation method. Explicit analytical solutions for stationary non-linear strain waves are obtained. This type of non-linearity has a crucial influence on the wave propagation mode: for soft non-linearity, localised shock (kink) waves are developed, while for hard non-linearity localised bell-shaped waves appear. Numerical results are presented and the areas of practical applicability of linear and non-linear, long- and short-wave approaches are discussed.     

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。     

Bifurcation and post-buckling analysis of bimodal optimum columns

A mathematical formulation of column optimization problems allowing for bimodal optimum buckling loads is developed in this paper. The columns are continuous and linearly elastic, and assumed to have no geometrical imperfections. It is first shown that bimodal solutions exist for columns that rest on a linearly elastic (Winkler) foundation and have clamped-clamped and clamped-simply supported ends. The equilibrium equation for a non-extensible, geometrically nonlinear elastic column is then derived, and the initial post-buckling behaviour of a bimodal optimum column near the bifurcation point is studied using a perturbation method. It is shown that in the general case the post-buckling behaviour is governed by a fourth order polynomial equation, i.e., near the bifurcation point there may be up to four post-buckling equilibrium states emanating from the trivial equilibrium state. Each of these equilibrium states may be either supercritical or subcritical in the vicinity of the bifurcation point. The conditions for stability of these non-trivial post-buckling states are established based on verification of positive semi-definiteness of a two-by-two matrix whose coefficients are integrals of the buckling modes and their derivatives. In the end of the paper we present and discuss numerical results for the post-buckling behaviour of several columns with bimodal optimum buckling loads.     

Thermal buckling and post-buckling response of imperfect temperature-dependent sandwich FGM plates resting on elastic foundation

Post-buckling behaviour of sandwich plates with functionally graded material (FGM) face sheets under uniform temperature rise loading is considered. It is assumed that the plate is in contact with a Pasternak-type elastic foundation during deformation, which acts in both compression and tension. The derivation of equations is based on the first-order shear deformation plate theory. Thermomechanical non-homogeneous properties of FGM layers vary smoothly by the distribution of power law across the thickness, and temperature dependency of material constituents is taken into account. Using the non-linear von-Karman strain-displacement relations, the equilibrium and compatibility equations of imperfect sandwich plates with FGM face sheets are derived. The boundary conditions for the plate are assumed to be simply supported in all edges. The governing equations are reduced to two coupled equation in terms of stress function and lateral deflection. Employing the single mode approach combined with Galerkin technique, an approximate closed-form solution is presented to calculate the critical buckling temperature and post-buckling equilibrium path of the plate. Presented numerical examples contain the influences of power law index, sandwich plate geometry, geometrical imperfection, temperature dependency, and the elastic foundation coefficients.     

锥柱形膜片后屈曲的材料特性预示研究

使用正交表设计数值试验, 分析弹性模量、屈服强度、延伸率等材料特性参数对锥柱形膜片结构后屈曲状态的影响, 预示材料特性对膜片翻转的影响. 正交数值试验在不影响结果科学性的基础上, 极大地降低了试验次数. 数值试验结果表明: 在一定结构情况下, 弹性模量与屈服强度对膜片正向翻转影响非常显著, 而延伸率的影响与前两个因素比较而言, 几乎可以忽略. 并且, 结构顶点位移随材料的弹性模量增大而增大; 随材料的屈服强度减小而增大.因此, 在结构固定的情况下, 为了提高材料的翻转性能, 可以选择弹性模量较大而屈服强度较小的材料, 且延伸率为20%左右最好.     

非保守力下温度场中FGM圆板的非线性力学行为

研究了温度场中非保守功能梯度材料(FGM)圆板的非线性力学行为.基于经典板理论,推导了受非保守力作用的FGM圆板在温度场中的控制微分方程.采用打靶法分析了由陶瓷二氧化锆和金属钛合金两相材料组成的非保守FGM圆板在均匀和非均匀升温场中的非线性力学行为.给出了不同均匀升温和非均匀升温场下,FGM圆板在非保守载荷作用下的平衡...     

Theory of elastic solids reinforced with fibers resistant to extension,flexure and twist

A model of non-linearly elastic solids reinforced by continuously distributed embedded fibers is formulated in which elastic resistance of the fibers to extension, bending and twist is taken into account explicitly. This generalizes the conventional theory in which the solid is modeled as a transversely isotropic simple material.     

Buckling and post-buckling of long pressurized elastic thin-walled tubes under in-plane bending

The present paper focuses on the structural stability of long uniformly pressurized thin elastic tubular shells subjected to in-plane bending. Using a special-purpose non-linear finite element technique, bifurcation on the pre-buckling ovalization equilibrium path is detected, and the post-buckling path is traced. Furthermore, the influence of pressure (internal and/or external) as well as the effects of radius-to-thickness ratio, initial curvature and initial ovality on the bifurcation moment, curvature and the corresponding wavelength, are examined. The local character of buckling in the circumferential direction is also demonstrated, especially for thin-walled tubes. This observation motivates the development of a simplified analytical formulation for tube bifurcation, which considers the presence of pressure, initial curvature and ovality, and results in closed-form expressions of very good accuracy, for tubes with relatively small initial curvature. Finally, aspects of tube bifurcation are illustrated using a simple mechanical model, which considers the ovalized pre-buckling state and the effects of pressure.     

Elastic nonlinear stability analysis of thin rectangular plates through a semi-analytical approach

A semi-analytical approach to the elastic nonlinear stability analysis of rectangular plates is developed. Arbitrary boundary conditions and general out-of-plane and in-plane loads are considered. The geometrically nonlinear formulation for the elastic rectangular plate is derived using the thin plate theory with the nonlinear von Kármán strains and the variational multi-term extended Kantorovich method. Emphasis is placed on the effect of destabilizing loads and on the derivation of the solution methodologies required for tracking a highly nonlinear equilibrium path, namely: parameter continuation and arc-length continuation procedures. These procedures, which are commonly used for the solution of discretized structural systems governed by nonlinear algebraic equations, are augmented and generalized for the direct application to the PDE. The boundary value problem that results from the arc-length continuation scheme and consists of coupled differential, integral, and algebraic equations is re-formulated in a form that allows the use of standard numerical BVP solvers. The performance of the continuation procedures and the convergence of the multi-term extended Kantorovich method are examined through the solution of the two-dimensional Bratu–Gelfand benchmark problem. The applicability of the proposed approach to the tracking of the nonlinear equilibrium path in the post-buckling range is demonstrated through numerical examples of rectangular plates with various boundary conditions.     

Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation

In this study, simple analytical expressions are presented for large amplitude free vibration and post-buckling analysis of functionally graded beams rest on nonlinear elastic foundation subjected to axial force. Euler–Bernoulli assumptions together with Von Karman’s strain–displacement relation are employed to derive the governing partial differential equation of motion. Furthermore, the elastic foundation contains shearing layer and cubic nonlinearity. He’s variational method is employed to obtain the approximate closed form solution of the nonlinear governing equation. Comparison between results of the present work and those available in literature shows the accuracy of this method. Some new results for the nonlinear natural frequencies and buckling load of the FG beams such as the effect of vibration amplitude, elastic coefficients of foundation, axial force, and material inhomogenity are presented for future references.     

Predicting the creep lives of thin-walled cylindrical polymeric pipe linings subject to external pressure

Installation of a close-fitting polymeric thin-walled lining is now standard practice for the rehabilitation of deteriorating gravity pipes. Design of these linings focuses primarily on their ability to resist an external head of groundwater pressure whilst experiencing long-term creep deformations. Existing structural design guidelines are crude and do not provide consistent safety factors. In this paper an existing simple analysis for linear elastic (geometrically non-linear) buckling loads is developed into a time stepping procedure for calculation of the creep lives of time dependent non-linearly elastic systems subject to long-term constant pressure. The results so obtained using different simulations of the creep data obtained for a particular material are then compared with those derived from a set of corresponding physical tests and alternative numerical modelling, and appropriate conclusions are drawn.