Comparative study of thermal post-buckling analysis of uniform slender & shear flexible columns using rigorous finite element and intuitive formulations

Thermal post-buckling analysis of uniform, isotropic, slender and shear flexible columns is presented using a rigorous finite element formulation and a much simpler intuitive formulation. The ends of the columns are axially restrained to move and consequently any temperature rise above the stress free condition of the column produces an equivalent constant compressive mechanical load that causes the column to buckle at a critical temperature. Further increase in temperature beyond critical temperature results in the thermal post-buckling phenomenon. As a result of constraints imposed on the axial displacement at the ends of the column, the post-buckling phenomenon is governed by the von-Karman strain displacement relation applicable to one dimensional problems. Empirical formula for ratio of nonlinear axial load to critical load (equivalent constant mechanical load for a given temperature rise) as a function of the central deflection are obtained using both the rigorous finite element and intuitive formulations for various boundary conditions. The boundary conditions considered are the classical such as hinged-hinged, clamped-clamped and clamped-hinged conditions and nonclassical boundary conditions like the hinged-guided or the clamped-guided conditions. Post-buckling analysis results pertaining to nonclassical boundary conditions are meagre in the literature. It is observed that results obtained from both the formulations are in excellent agreement for all boundary conditions considered. Also the accuracy and simplicity of the intuitive formulation is aptly demonstrated to slender and shear flexible columns.     

水平井中钻柱屈曲的非线性有限元分析

建立了直井中钻柱屈曲的平衡方程及对应的泛函表达式,使用有限元法对不同约束下水平井中钻柱从稳定到非线性屈曲的整个过程进行了分析。力学模型中考虑了重力和扭矩对屈曲的影响。分析结果表明:钻柱的屈曲是一个从局部屈曲到总体屈曲的过程,屈曲段的井壁约束力、钻柱弯矩和屈曲位移呈周期性变化;重力在水平井中对屈曲有较强的抑制作用;边界约束对屈曲的影响不可忽略;扭矩对屈曲的影响很小,可以忽略。     

An exact finite strip for the calculation of relative post-buckling stiffness of I-section struts

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of I-section struts. The so-called exact strip is developed based on the concept that it is effectively a plate. The presented method, which is designated by the name Full-analytical Finite Strip Method, provides an efficient and extremely accurate buckling solution. In the development process, the Von-Karman's equilibrium equation is solved exactly to obtain the buckling loads and mode shapes for the I-section struts. The investigation of buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. The current post-buckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. Through the solution of the Von-Karman's compatibility equation, the in-plane displacement functions which are themselves related to the Airy stress function are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. All the displacement functions are then substituted in the total strain energy expressions. The theorem of minimum total potential energy is subsequently applied to solve for the unknown coefficient. Finally, the developed method is subsequently applied to analyze the initial post-buckling behavior of some representative I-sections for which the results were also obtained through the application of a Semi-energy Finite Strip Method [Ovesy HR. The development and application of a semi-energy post-local buckling finite strip. PhD dissertation, Cranfield University, UK, 1998]. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.     

大挠度后屈曲倾斜梁结构的非线性力学特性

基于弹性梁的几何非线性大挠度屈曲理论,建立两端固定对称倾斜支撑梁结构的大挠度后屈曲控制微分方程,采用几何非线性隐式变形协调关系来表达强非线性超静定边值问题,得到描述倾斜梁大挠度后屈曲行为的精确解析解.采用数值方法求解含有第一、二类椭圆积分的强非线性微分方程,给出不同倾角梁结构从初始屈曲到后屈曲并发生两态跳转过程中的位形曲线及非线性刚度.根据最小能量原理和挠曲线拐点个数,分析对称屈曲模态与非对称屈曲模态之间相互跳转的内在联系及其对结构非线性刚度突变的影响,得到了屈曲模态之间的转换条件.跳转过程的数值仿真表明,倾斜支撑梁结构发生大挠度后屈曲时具有明显的双稳态特性且只出现低阶(1、2阶)屈曲模态,仿真计算结果与试验结果相一致.     

搅拌摩擦焊接加筋板轴压稳定性研究

运用半经验公式对采用搅拌摩擦焊接技术的加筋板轴向压缩局部屈曲临界应力进行工程计算,并应用有限元软件ABAQUS对该型结构的稳定性进行数值计算,得到结构的失稳屈曲临界载荷;参考工程计算和有限元计算值,对加筋板进行轴向压缩试验,对其屈曲形式、失稳及破坏载荷进行试验研究,并考虑侧边支持条件对结果的影响。试验分析表明,非承载边的约束条件对试件的屈曲载荷有一定的影响,而对试件的承载能力影响较小;搅拌摩擦焊加筋板具有一定的后屈曲承载能力;有限元计算结果与试验结果较吻合,说明模拟方法可以用来对该型结构的稳定性能进行分析,从而为该型结构的优化设计及工程应用提供分析参考。     

纤维增强FGM梁的自由振动和临界屈曲载荷分析

基于经典梁理论（CBT）研究轴向力作用下纤维增强功能梯度材料（FGM）梁的横向自由振动和临界屈曲载荷问题。首先考虑由混合律模型来表征纤维增强FGM梁的材料属性,其次利用Hamilton原理推导轴向力作用下纤维增强FGM梁横向自由振动和临界屈曲载荷的控制微分方程,并应用微分变换法（DTM）对控制微分方程及边界条件进行变换,计算了纤维增强FGM梁在固定-固定（C-C）、固定-简支（C-S）和简支-简支（S-S）3种边界条件下横向自由振动的无量纲固有频率和无量纲临界屈曲载荷。退化为各向同性梁和FGM梁,并与已有文献结果进行对比,验证了本文方法的有效性。最后讨论在不同边界条件下纤维增强FGM梁的刚度比、纤维体积分数和无量纲压载荷对无量纲固有频率的影响以及各参数对无量纲临界屈曲载荷的影响。     

Buckling of columns: Allowance for axial shortening

This paper investigates the effect of axial shortening on (i) the elastic buckling of columns with a continuous elastic restraint, (ii) the elastic buckling of rotating columns and (iii) the free vibration of columns under a static axial load. These column problems can be solved in a unified approach because the resulting energy functional is similar. The field differential equation is derived by minimizing the energy functional with respect to the lateral displacement function via calculus of variations. The buckling load or fundamental frequency may be obtained by analytically solving the two-point boundary-value problem. It was found that the boundary conditions and the restraint parameter or angular velocity parameter affect the influence of axial shortening on the buckling load. In vibrating columns, tensile forces enhance the effect of axial stretching on the fundamental frequency.     

Buckling analysis of laminated plates using the extended Kantorovich method and a system of first-order differential equations

The extended Kantorovich method using multi-term displacement functions is applied to the buckling problem of laminated plates with various boundary conditions. The out-of-plane displacement of the buckled plate is written as a series of products of functions of parameter x and functions of parameter y. With known functions in parameter x or parameter y, a set of governing equations and a set of boundary conditions are obtained after applying the variational principle to the total potential energy of the system. The higher order differential equations are then transformed into a set of first-order differential equations and solved for the buckling load and mode. Since the governing equations are first-order differential equations, solutions can be obtained analytically with the out-of-plane displacement written in the form of an exponential function. The solutions from the proposed technique are verified with solutions from the literature and FEM solutions. The bucking loads correspond very well to other available solutions in most of the comparisons. The buckling modes also compare very well with the finite element solutions. The proposed solution technique transforms higher-order differential equations to first-order differential equations, and they are analytically solved for out-of-plane displacement in the form of an exponential function. Therefore, the proposed solution technique yields a solution which can be considered as an analytical solution.     

Large amplitude free vibration analysis of Timoshenko beams using a relatively simple finite element formulation

Free vibration analysis of uniform isotropic Timoshenko beams with geometric nonlinearity is investigated through a relatively simple finite element formulation, applicable to homogenous cubic nonlinear temporal equation (homogenous Duffing equation). Geometric nonlinearity is considered using von-Karman strain displacement relations. The finite element formulation begins with the assumption of the simple harmonic motion and is subsequently corrected using the harmonic balance method. Empirical formulas for the non-linear to linear radian frequency ratios, for the boundary conditions considered, are presented using the least square fit from the solutions of the same obtained for various central amplitude ratios. Numerical results using the empirical formulas compare very well with the results available from the literature for the classical boundary conditions such as the hinged–hinged, clamped–clamped and clamped–hinged beams. Numerical results for the beams with non-classical boundary conditions such as the hinged-guided and clamped-guided, hitherto not studied, are also presented.     

Nonlocal beam models for buckling of nanobeams using state-space method regarding different boundary conditions

Buckling analysis of nanobeams is investigated using nonlocal continuum beam models of the different classical beam theories namely as Euler-Bernoulli beam theory (EBT), Timoshenko beam theory (TBT), and Levinson beam theory (LBT). To this end, Eringen’s equations of nonlocal elasticity are incorporated into the classical beam theories for buckling of nanobeams with rectangular cross-section. In contrast to the classical theories, the nonlocal elastic beam models developed here have the capability to predict critical buckling loads that allowing for the inclusion of size effects. The values of critical buckling loads corresponding to four commonly used boundary conditions are obtained using state-space method. The results are presented for different geometric parameters, boundary conditions, and values of nonlocal parameter to show the effects of each of them in detail. Then the results are fitted with those of molecular dynamics simulations through a nonlinear least square fitting procedure to find the appropriate values of nonlocal parameter for the buckling analysis of nanobeams relevant to each type of nonlocal beam model and boundary conditions.analysis.     

Post-buckling behavior of carbon fiber epoxy composite plates

In this study, the pre-buckling and post-buckling behaviors of layered composite plates which were made of woven carbon fiber fabric with a circular hole in the middle were investigated experimentally and numerically. Firstly, load-displacement graphs of composite plates with different hole diameters were experimentally obtained under compressive load. Then the numerical load-displacement graphs of the plates were found with the ANSYS package program which used the finite element method. As a result, after linear buckling experimental and numerical results were found to be compatible with each other. In addition, damage behavior of plates after buckling with the aid of Tsai-Wu damage criterion was obtained similar to experimental results. The increase in hole diameter did not change the load-displacement behavior characteristics of the plates after buckling. However, it has reduced maximum damage load and maximum failure displacement. The stress at the perimeter of the hole increased significantly with the increase of the vertical displacement with immediately after the buckling but later was not significantly affected by this increase.

     

神经网络在结构优化中的应用

采用阶梯折算及传递矩阵的基本方法,利用神经网络算法中的模拟退火法,研究了轴向力作用下非均匀变截面柱的屈曲优化问题耐算了给定柱体总体积约束下,使轴向受压柱的屈曲载荷达到最大的问题。研究表明：优化结果使4种边界条件下的屈曲载荷都得到提高。     

Optimum shape for buckling and post-buckling behavior of a laminated composite panel with I-type stiffeners

A shape optimization of stiffener was conducted to increase buckling load or failure load with stiffened laminated composite panel of I-type under compression loading. Design variables are cap length, web length, and/or thickness under the constraint of volume constancy. The objective function is buckling load and failure load of post-buckling based on Tsai-Hill theory using ABAQLJS 5.8 for analysis and Optimizer on Broydon-Fletcher Goldfarb-Sharno Method and Augmented Lagrange Multiplier Method. The effects of relative length of a web and a cap of stiffener on buckling load and failure load of post-buckling were investigated with the results of optimum design.     

碟形封头压力容器在内压作用下的弹塑性屈曲及后屈曲行为研究

基于ABAQUS对受内压碟形封头压力容器的弹塑性屈曲及后屈曲行为进行分析.通过非对称网格剖分技术诱发结构的屈曲行为,采用Riks算法捕捉完整的屈曲及后屈曲路径,分析屈曲载荷和后屈曲形态.计算结果表明,在内压作用下碟形封头压力容器过渡区域出现波状的分叉屈曲形态,随着加载的继续,过渡段将逐渐生成肉眼可视的褶皱,进入后屈曲阶段.后屈曲路径可以分为第一阶段与第二阶段.对引入初始厚度缺陷的结果分析可知,初始缺陷厚度的布置将对后屈曲路径产生影响,并降低结构的分叉屈曲载荷.     

Analytical buckling solutions for mindlin plates involving free edges

This paper investigates the buckling behaviour of rectangular Mindlin plates having two parallel edges simply supported, one edge free and the remaining edge free, simply supported or clamped. The proper boundary conditions at free edges subjected to in-plane loads have been examined. The buckling analysis is performed by applying the concept of state space to the Levy-type solution method to obtain the closed-form critical loads from the governing differential equations. The results, where possible, are compared with existing solutions to verify the validity of the solution method. The differences between buckling factors obtained with the appropriate and inappropriate free edge conditions are reported. Several design charts representing the essential features of the critical load characteristics of rectangular plates with two opposite edges simply supported at least one free edge are obtained. The critical loads can be determined from the design charts without difficulty.     

大展弦比机翼屈曲及后屈曲分析

针对某大展弦比复合材料无人机机翼结构,利用有限元素法进行机翼结构的屈曲及后屈曲分析.首先采用MSC.Patran/Nastran软件进行结构屈曲分析,目的是确定机翼结构易发生屈曲的区域;然后对不同区域进行分析,确定需要进行后屈曲分析的细节模型.对细节模型进行后屈曲分析时采用MSC.Patran/Marc软件,分析方法采用弧长法.通过机翼结构的后屈曲分析,可以分析出结构的失稳路径.结果表明,该机翼存在屈曲失稳问题.该机翼的地面静力试验结果验证分析结果的可信性.     

Zigzag theory for buckling of hybrid piezoelectric beams under electromechanical loads

A new efficient coupled one-dimensional (1D) geometrically nonlinear zigzag theory is developed for buckling analysis of hybrid piezoelectric beams, under electromechanical loads. The potential field is approximated layerwise as piecewise linear. The deflection is approximated to account for the normal strain due to electric field. The axial displacement is approximated as a combination of a global third-order variation and layerwise linear variation. It is expressed in terms of three primary displacement variables and a set of electric potential variables by enforcing exactly the conditions of zero transverse shear stress at the top and bottom and the conditions of its continuity at the layer interfaces. The governing coupled nonlinear field equations and boundary conditions are derived using a variational principle. Analytical solutions for buckling of simply supported beams under electromechanical loads are presented. Comparisons with the exact 2D piezoelasticity solution establish that the present zigzag theory is very accurate for buckling analysis.     

基于 ABAQUS 的角接触球轴承动力学仿真分析

为研究角接触球轴承在不同转速下温度及位移变化,基于 Palmgren 摩擦生热理论,综合考虑轴向载荷、切向摩擦和法向摩擦等边界条件,在 ABAQUS 软件中建立了轴承完全热力耦合模型。对轴承外圈施加固定的轴向载荷,并对内圈加载不同的转速,使用显示求解器求解轴承在不同转速下各部件温度和轴承内圈位移变化。仿真分析表明,该模型能够准确地反映轴承的温升以及轴承内圈位移情况,为轴承选取合适工作转速区间提供理论支撑。     

Buckling analysis of laminated composite rectangular plates reinforced by SWCNTs using analytical and finite element methods

In this paper, the buckling analysis of laminated composite plates reinforced by single-walled carbon nanotubes (SWCNTs) is carried out using an analytical approach as well as the finite element method. The developed model is based on the classical laminated plate theory (CLPT) and the third-order shear deformation theory for moderately thick laminated plates. The critical buckling loads for the symmetrical layup are determined for different support edges. The Mori-Tanaka method is employed to calculate the effective elastic modulus of composites having aligned oriented straight nanotubes. The effect of the agglomeration of the randomly oriented straight nanotubes on the critical buckling load is also analyzed. The results of analytical solution are compared and verified with the FEM calculations The critical buckling loads obtained by the finite element and the analytical methods for different layup and boundary conditions are in good agreement with each other. In this article, the effects of the carbon nanotubes (CNTs) orientation angle, the edge conditions, and the aspect ratio on the critical buckling load are also demonstrated using both the analytical and finite element methods.     

The dynamics of symmetric post-buckling

This work examines the relationship between natural frequency and compressive load for a conservative elastic mechanical system characterised by symmetric buckling. Application of some general theory shows that for a perfect system, exhibiting a stable-symmetric point of bifurcation, the initial post-buckling curve of compressive load against the square of the natural frequency is linear and has a slope of that of the pre-buckling curve. Simple link models and discretised continuous systems are analysed to illustrate the results of the general theory. These models are also used to show that the initial post-buckling curve is independent of axial inertia. The influence of initial imperfections is then considered and compared with experimental evidence.