Secondary buckling of a flat plate under uniaxial compression : Part 1: theoretical analysis of simply supported flat plate

While studies of post-buckling behavior and load-carrying capacities of thin plates subjected to uniaxial compression have been limited to stable conditions, further post-buckling loading generates an unstable condition. The secondary buckling which occurs with snap-through to higher-order deflections under such unstable conditions has not been analyzed in detail as yet. In the first part of this paper, a thin square plate under uniaxial compression, which is simply supported along four edges, is considered. A method based on the second variation of the total potential energy is then proposed for evaluating the stability of the post-buckling equilibrium state and inevitable secondary buckling is derived analytically. The effects of various factors, such as initial imperfections, assumed virtual displacement pattern, post-buckling deflection pattern and in-plane boundary conditions, on the secondary buckling values are discussed. In part 2, secondary buckling of clamped plates is analyzed by use of the finite element method and the resultant numerical results are compared with experimental results.     

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.     

Experimental techniques for testing unstiffened plates in compression and bending

We present details of a dual-actuator rig developed for testing rectangular plates supported on three sides, with the remaining (longitudinal) edge free, under combined uniaxial compression and in-plane bending. Particular attention is given to ensuring a constant strain gradient at the loaded ends, as opposed to a constant load eccentricity, in order to determine the post-buckling behavior and ultimate load and moment capacities of unstiffened thin-walled elements. Strain gradients varying from pure compression to pure bending are facilitated. Attention is also given to ensuring simply supported boundary conditions, and the methods used for anchoring the tensile stresses that develop at the loaded edges as a result of large plate deflections. Details of the methods for controlling the applied displacements are given, for which a system of four laser displacement devices was employed in order to achieve the required strain gradient. The operation of the rig is verified against established theoretical solutions.     

联合载荷作用下简支矩形板的屈曲和过屈曲

本文研究了简支正交各向异性矩形板在两对板受中面压力作用下的屈曲和过屈曲性态,得到了载荷的稳定性区域,证明了临界载荷最多为二重的。利用多参数摄动方法求得临界载荷附近板的过屈曲状态的渐近解,分析了在二重临界载荷附近,当载荷按比例变化时,板的可能的过屈曲状态及其与参数的依赖关系。     

Imperfection sensitivity of the post-buckling behavior of higher-order shear deformable functionally graded plates

This paper investigates the sensitivity of the post-buckling behavior of shear deformable functionally graded plates to initial geometrical imperfections in general modes. A generic imperfection function that takes the form of the product of trigonometric and hyperbolic functions is used to model various possible initial geometrical imperfections such as sine type, local type, and global type imperfections. The formulations are based on Reddy’s higher-order shear deformation plate theory and von Karman-type geometric nonlinearity. A semi-analytical method that makes use of the one-dimensional differential quadrature method, the Galerkin technique, and an iteration process is used to obtain the post-buckling equilibrium paths of plates with various boundary conditions that are subjected to edge compressive loading together with a uniform temperature change. Special attention is given to the effects of imperfection parameters, which include half-wave number, amplitude, and location, on the post-buckling response of plates. Numerical results presented in graphical form for zirconia/aluminum (ZrO₂/Al) graded plates reveal that the post-buckling behavior is very sensitive to the L2-mode local type imperfection. The influences of the volume fraction index, edge compression, temperature change, boundary condition, side-to-thickness ratio and plate aspect ratio are also discussed.     

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...     

Thermal post-buckling analysis of imperfect laminated plates using a higher-order shear deformation theory

Thermal post-buckling analysis is presented for a simply supported, composite laminated plate subjected to uniform or non-uniform tent-like temperature loading. The initial geometrical imperfection of the plate is taken into account. The formulations are based on the Reddy's higher-order shear deformation plate theory, and include thermal effects. The analysis uses a mixed Galerkin-perturbation technique to determine thermal buckling loads and post-buckling equilibrium paths. Numerical examples cover the performances of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated plates. The effects played by transverse shear deformation, thermal load ratio, plate aspect ratio, total number of plies, fiber orientation and initial geometrical imperfections are studied. Typical results are presented in dimensionless graphical form.     

Creep buckling and post-buckling analysis of the laminated piezoelectric viscoelastic functionally graded plates

The creep buckling and post-buckling of the laminated piezoelectric viscoelastic functionally graded material (FGM) plates are studied in this research. Considering the transverse shear deformation and geometric nonlinearity, the Von Karman geometric relation of the laminated piezoelectric viscoelastic FGM plates with initial deflection is established. And then nonlinear creep governing equations of the laminated piezoelectric viscoelastic FGM plates subjected to an in-plane compressive load are derived on the basis of the elastic piezoelectric theory and Boltzmann superposition principle. Applying the finite difference method and the Newmark scheme, the whole problem is solved by the iterative method. In numerical examples, the effects of geometric nonlinearity, transverse shear deformation, the applied electric load, the volume fraction and the geometric parameters on the creep buckling and post-buckling of laminated piezoelectric viscoelastic FGM plates with initial deflection are investigated.     

Thermomechanical Postbuckling of Imperfect Moderately Thick Plates on Nonlinear Elastic Foundations*

ABSTRACT

A postbuckling analysis is presented for a moderately thick rectangular plate subjected to combined axial compression and uniform temperature loading, and resting on a softening nonlinear elastic foundation. The cases of (1) thermal postbuckling of initially compressed plates and (2) compressive post-buckling of initially heated plates are considered. The initial geometrical imperfections of the plates are taken into account. Formulations are based on Reissner-Mindlin plate theory, considering first-order shear deformation effects, and including plate-foundation interaction and thermal effects. The analysis uses a deflection-type perturbation technique to determine buckling loads and postbuckling equilibrium paths. Numerical examples include the performance of perfect and imperfect, moderately thick plates resting on softening nonlinear elastic foundations. Typical results are presented in dimensionless graphical form.     

均匀受压矩形管的屈曲后板组效应研究

本文根据势能驻值原理,建立了均匀受压矩形管的大挠度弹性屈曲理论,用于分析屈曲后相邻板件间的相互约束作用,即屈曲后板组效应。该方法通过引入拉格朗日乘子,使系统总势能取驻值的同时,形函数能满足一系列基本的,但非完备的边界条件;在此基础上采用增量法将非线性方程组线性化,从而确定给定荷载下矩形管的受力状态及变形。由于系统总势能是以解析式的形式给出,且仅需取有限的级数项就可达到满意的精度,故该半解析法具有计算效率高的特点。目前许多国家的钢结构设计规范中,在计算受压板件的有效宽度时,常采用板组约束系数来考虑板组效应。由于有效宽度法本身考虑了板的屈曲后强度,而约束系数多数却是建立在小挠度弹性屈曲理论上的,故有必要分析板件的屈曲后板组效应。本文详细讨论了矩形管的屈曲后板组效应,并与小挠度理论给出的计算结果进行比较,得出了均匀受压矩形管屈曲后板组效应减弱的结论。     

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.     

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。     

Elastic stability of inhomogeneous thin plates on an elastic foundation

We study the elastic stability of infinite inhomogeneous thin plates on an elastic foundation under in-plane compression. The elastic stiffness constants depend on the coordinate variable in the thickness direction of the plate. The elastic foundation is represented as a Winkler-type model characterized by linear and nonlinear spring constants. First we derive the Föppl–von Kármán equations by taking variations of the elastic strain energy. Next we develop the linear stability analysis of the plate under uniform in-plane compression and explicitly derive the critical loads and wave numbers for particular three cases. The effects of the material inhomogeneity, material orthotropy and loading orthotropy on the critical states are examined independently. Finally, we perform a weakly nonlinear analysis of the plate at the onset of the buckling instability. With the multiple scales method, the amplitude equations for the unstable modes that provide insight into the mode type and its amplitude are derived and then the effect of the material inhomogeneity on buckling modes are evaluated qualitatively.     

复合材料层合板蠕变屈曲与变形的优化问题

将瞬时弹性失稳荷载、持久临界荷载和后屈曲持久变形刚度作为复合材料层合板蠕变屈曲与变形的优化指标,对几组纤维铺设方式进行讨论,构造多目标优化模型,就纤维铺设角进行优化分析.考虑横向剪切变形效应,分别利用Lap lace变换和准弹性方法,导出了这三个优化指标的计算公式.     

Post-buckling analysis of a thin stiffened plate with uncertain initial deflection via interval analysis

The post-buckling behavior of the thin stiffened plates with uncertain initial deflection under uni-direction compression load is investigated based on Von-Karman large deflection theory. The uncertain initial deflections are considered to be unknown except that they belong to a given set in the interval range. Interval analysis model for computing the bounds of curves of the post-buckling deflection versus load of the plate is presented. Interval analysis model is compared with the stochastic model, which is taken as the benchmarks of accuracy for judgment. The results indicate that the non-probabilistic and stochastic methods will produce similar results for a large deviation of uncertainty. If the probabilistic information is unavailable, one should not propose the probabilistic method based on an arbitrary assumption on the distribution of the deflection coefficients. Rather, one should use the non-probabilistic method to uncertainty.     

Thermo-mechanical post-buckling of FGM cylindrical panels with temperature-dependent properties

This paper presents thermo-mechanical post-buckling analysis of cylindrical panels that are made of functionally graded materials (FGMs) with temperature-dependent thermo-elastic properties that are graded in the direction of thickness according to a simple power law distribution in terms of the volume fractions of the constituents. The panel is initially stressed by an axial load, and is then subjected to a uniform temperature change. The theoretical formulations are based on the classical shell theory with von-Karman–Donnell-type nonlinearity. The effect of initial geometric imperfection is also included. A differential quadrature (DQ) based semi-analytical method combined with an iteration process is employed to predict the critical buckling load (where it is applicable) and to trace the post-buckling equilibrium path of FGM cylindrical panels under thermo-mechanical loading. Numerical results are presented for panels with silicon nitride and nickel as the ceramic and metal constituents. The effects of temperature-dependent properties, volume fraction index, axial load, initial imperfection, panel geometry and boundary conditions on the thermo-mechanical post-buckling behavior are evaluated in detail through parametric studies.     

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.     

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。     

安全壳中钢衬壳热屈曲问题理论与实验研究(I)———理论分析部分

钢衬壳热屈曲问题是核工程安全壳设计中的主要问题把铆固之间的钢衬壳视为钢衬板的特殊缺陷形式，利用Ｋｏｉｔｅｒ初始后屈曲理论分析了完善和具有初始缺陷钢衬壳的弹性热后屈曲性态给出了用挠度－温度载荷表示的钢衬壳的后屈曲平衡路径表达式和屈曲临界载荷表达式具体分析了三种钢衬壳模型：四点铆固钢衬壳、四边固支钢衬壳和五点铆固钢衬壳给出了钢衬的初始缺陷、锚钉间距、钢衬厚度等参数对钢衬热屈曲载荷的影响结果对安全壳中钢衬壳的设计有很好的参考价值     

On the minimal mass reinforcement of masonry structures with arbitrary shapes

This paper presents a tensegrity approach to the minimal mass design of tensile reinforcements of masonry structures with arbitrary shapes. The proposed strengthening methodology allows for the design of minimal mass resisting mechanisms of systems formed by a network of masonry rods, mainly working in compression, and grids of tensile reinforcements. Assuming a perfectly plastic response by each member, the existence of such resisting mechanisms ensures that the reinforced structure is stable under the examined loading conditions, due to the safe theorem of the limit analysis of elastic-plastic bodies. The approach proposed in this paper includes an explicit determination of the state of prestress to be applied to tensile reinforcements, in order that they are effective under pre-existing loading conditions. Several benchmark examples illustrate the potential of this approach when dealing with minimal mass reinforcements of 3D models of masonry walls under in-plane and out-of-plane forces, and a structural complex formed by a cloister vault resting on supporting walls.