Coupled bending-torsion vibration of a homogeneous beam with a single delamination subjected to axial loads and static end moments

Delaminations in structures may significantly reduce the stiffness and strength of the structures and may affect their vibration characteristics. As structural components, beams have been used for various purposes, in many of which beams are often subjected to axial loads and static end moments. In the present study, an analytical solution is developed to study the coupled bending-torsion vibration of a homogeneous beam with a single delamination subjected to axial loads and static end moments. Euler–Bernoulli beam theory and the "free mode" assumption in delamination vibration are adopted. This is the first study of the influences of static end moments upon the effects of delaminations on natural frequencies, critical buckling loads and critical moments for lateral instability. The results show that the effects of delamination on reducing natural frequencies, critical buckling load and critical moment for lateral instability are aggravated by the presence of static end moment. In turn, the effects of static end moments on vibration and instability characteristics are affected by the presence of delamination. The analytical results of this study can serve as a benchmark for finite element method and other numerical solutions.     

Surface effect on the biaxial buckling and free vibration of FGM nanoplate embedded in visco-Pasternak standard linear solid-type of foundation

In this study, nonlocal elasticity theory in conjunction with Gurtin–Murdoch elasticity theory is employed to investigate biaxial buckling and free vibration behavior of nanoplate made of functionally graded material (FGM) and resting on a visco-Pasternak standard linear solid-type of the foundation. The material characteristics of simply supported FGM nanoplates are assumed to be varied continuously as a power law function of the plate thickness. Hamilton’s principle is implemented to derive the non-classical governing equations of motion and related boundary conditions, which analytically solved to obtain the explicit closed-form expression for complex natural frequencies and buckling loads. Finally, attention is focused on considering the influences of various parameters on variation of damped natural frequency and buckling load ratio such as nonlocal parameter, surface effects, geometric parameters, power law index and properties of visco-Pasternak foundation and it is clearly demonstrated that these factors highly affect on vibration and buckling behavior.     

梯度多孔金属材料梁的屈曲和屈曲基础上的振动

为研究梯度多孔金属材料梁的屈曲以及屈曲附近的振动特性,首先建立了随从分布压力下梯度多孔材料梁的动力学控制方程,得到了描述后屈曲的静态控制微分方程和描述屈曲前后振动响应的控制方程。通过打靶法数值求解两组强非线性方程,获得了简支-固支梯度多孔梁的屈曲临界载荷以及屈曲前后振动频率与载荷之间的关系曲线。分析了孔隙率系数和孔隙分布方式对屈曲临界载荷和屈曲前后振动频率的影响。结果表明,随着孔隙率系数e₀的增加,发生屈曲时的临界载荷减小;各阶固有频率也减小。屈曲前,各阶振动频率随载荷增大而减小,屈曲后,除三阶频率外,一阶和二阶频率随载荷增大而增大。     

Free vibration analyses of axially loaded laminated composite beams based on higher-order shear deformation theory

The dynamic stiffness matrix method is introduced to solve exactly the free vibration and buckling problems of axially loaded laminated composite beams with arbitrary lay-ups. The Poisson effect, axial force, extensional deformation, shear deformation and rotary inertia are included in the mathematical formulation. The exact dynamic stiffness matrix is derived from the analytical solutions of the governing differential equations of the composite beams based on third-order shear deformation beam theory. The application of the present method is illustrated by two numerical examples, in which the effects of axial force and boundary condition on the natural frequencies, mode shapes and buckling loads are examined. Comparison of the current results to the existing solutions in the literature demonstrates the accuracy and effectiveness of the present method.     

Axial-flexural coupled vibration and buckling of composite beams using sinusoidal shear deformation theory

A finite element model based on sinusoidal shear deformation theory is developed to study vibration and buckling analysis of composite beams with arbitrary lay-ups. This theory satisfies the zero traction boundary conditions on the top and bottom surfaces of beam without using shear correction factors. Besides, it has strong similarity with Euler–Bernoulli beam theory in some aspects such as governing equations, boundary conditions, and stress resultant expressions. By using Hamilton’s principle, governing equations of motion are derived. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results for cross-ply and angle-ply composite beams are obtained as special cases and are compared with other solutions available in the literature. A variety of parametric studies are conducted to demonstrate the effect of fiber orientation and modulus ratio on the natural frequencies, critical buckling loads, and load-frequency curves as well as corresponding mode shapes of composite beams.     

First applications of a novel unified model for global and local buckling of sandwich columns

Due to the intrinsic heterogeneity of sandwich structures, phenomena at various scales can co-exist in these layered-like assembly of thick-soft and thin-stiff materials. Especially under in-plane compression loadings, geometrical instabilities can occur at both global (structure) and local (skins) scales. Therefore, the in-plane compressive response of sandwich structures is of major concern in designing structural applications. In the present paper, the first applications of a novel unified model for sandwiches are presented, with closed-form solutions for both global and local buckling. For the perfect structure, analytical critical loads are extracted for a simply supported beam, through the calculation of two eigenvalues leading to three buckling modes: it appears that the eigenvalue associated with the antisymmetrical mode can correspond to the occurrence of either global or local (wrinkling) buckling. These global and local loads from the present unified model are shown to compare very well with the predictions given by the most complete specific models from the literature. Moreover, it is shown that conversely to the classical models, our approach yields critical loads that depend only on rigorous well-founded mechanical hypotheses. The simple but general analytical expressions from the unified model permit to select quickly configurations against local and global buckling. In this simplified framework, conclusions can be drawn from this unified model capable of properly predicting the phenomena at both scales. This simplified study is essential in getting an insight in the role played by each geometrical and material parameter, the combination of which is of importance for subsequent non-linear interactive post-buckling analyses (Léotoing et al., 2001).     

弹性地基上多孔功能梯度材料矩形板的自由振动与临界屈曲载荷分析

多孔功能梯度材料(FGM)构件的特性与孔隙率和孔隙分布形式有密切关系。本文基于经典板理论,考虑不同孔隙分布形式时修正的混合率模型,研究Winkler弹性地基上四边受压多孔FGM矩形板的自由振动与临界屈曲载荷特性。首先利用Hamilton原理和物理中面的定义推导Winkler弹性地基上四边受压多孔FGM矩形板自由振动的控制微分方程并进行无量纲化,然后应用微分变换法(DTM)对无量纲控制微分方程和边界条件进行变换,得到计算无量纲固有频率和临界屈曲载荷的代数特征方程。将问题退化为孔隙率为零时的FGM矩形板并与已有文献进行对比以验证其有效性。最后计算并分析了梯度指数、孔隙率、地基刚度系数、长宽比、四边受压载荷及边界条件对多孔FGM矩形板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 and vibration of non-ideal simply supported rectangular isotropic plates

Buckling and vibration of a rectangular isotropic plate which has non-ideal simply supported boundary conditions along one of its edges are investigated. It is assumed that one of the edges of the plate allows a small non-zero deflection and a small non-zero moment. Externally applied in-plane loads are considered to be parallel and perpendicular to the edge where non-ideal boundary conditions are present in the buckling problem. Analytical solutions of the buckling and vibration problems are obtained by using the Linshtead–Poincare perturbation technique. Improved buckling loads and natural frequencies are determined for various values of the aspect ratio of the plate.     

Predicting buckling loads from vibration data

There are analytical methods for predicting the buckling loads of columns with the boundaries ideally fixed, i.e., simply supported or built-in, or partially fixed. Vibration-test results may furnish a practical method of measuring the fixity. In this investigation a beam, that may or may not be loaded as a column, is assumed to have a torsional spring at each end such that a zero torsional stiffness corresponds to a simply supported end and an infinite torsional stiffness corresponds to a built-in end. From a Rayleigh-Ritz analysis, the buckling load and the fundamental frequency of the beam are each computed as a function of the torsional stiffness. This procedure leads to a one-to-one nondimensional relationship between the buckling load and the natural frequency. From these calculations, it is seen that regardless of the degree of clamping of one end relative to the other end, all that is needed to predict the buckling load within a 15-percent range is a knowledge of the theoretical buckling load of the simply supported column; the theoretical fundamental frequency of the simply supported beam; and the experimental fundamental frequency. Experimental results are presented to support the theory.     

CLASSICAL AND HOMOGENIZED EXPRESSIONS FOR BUCKLING SOLUTIONS OF FUNCTIONALLY GRADED MATERIAL LEVINSON BEAMS

The relationship between the critical buckling loads of functionally graded material(FGM) Levinson beams(LBs) and those of the corresponding homogeneous Euler-Bernoulli beams(HEBBs) is investigated. Properties of the beam are assumed to vary continuously in the depth direction. The governing equations of the FGM beam are derived based on the Levinson beam theory, in which a quadratic variation of the transverse shear strain through the depth is included.By eliminating the axial displacement as well as the rotational angle in the governing equations,an ordinary differential equation in terms of the deflection of the FGM LBs is derived, the form of which is the same as that of HEBBs except for the definition of the load parameter. By solving the eigenvalue problem of ordinary differential equations under different boundary conditions clamped(C), simply-supported(S), roller(R) and free(F) edges combined, a uniform analytical formulation of buckling loads of FGM LBs with S-S, C-C, C-F, C-R and S-R edges is presented for those of HEBBs with the same boundary conditions. For the C-S beam the above-mentioned equation does not hold. Instead, a transcendental equation is derived to find the critical buckling load for the FGM LB which is similar to that for HEBB with the same ends. The significance of this work lies in that the solution of the critical buckling load of a FGM LB can be reduced to that of the HEBB and calculation of three constants whose values only depend upon the throughthe-depth gradient of the material properties and the geometry of the beam. So, a homogeneous and classical expression for the buckling solution of FGM LBs is accomplished.     

Lower bounds to fundamental frequencies and buckling loads of columns and plates

A general derivation of expressions for lower bounds to fundamental frequencies and buckling loads is given for the class of structures governed by linear elastic theory in the prebuckling state. These expressions involve two Rayleigh quotients both of which are upper bounds for the fundamental frequency under a prescribed load. The displacement trial functions must satisfy force and kinematic continuity but no other conditions are required. Thus, if appropriate high order base functions are used, the finite element procedure can be used to systematically narrow the difference between the upper and lower bounds.The theory is illustrated with several column and plate problems. The finite element method is applied to uniform and nonuniform columns with a representative set of boundary conditions. Elementary trial functions are used to show that reasonable bounds can also be obtained for plates subjected to known states of stress. Since the lower bound is obtained with a variation of the classical technique of Rayleigh, these results indicate that the method may be suitable for conservatively estimating buckling loads and fundamental frequencies of engineering structures.     

Dynamic stability analysis of composite plates including delaminations using a higher order theory and transformation matrix approach

A refined higher order shear deformation theory is used to investigate the dynamic instability associated with composite plates with delamination that are subject to dynamic compressive loads. Both transverse shear and rotary inertia effects are taken into account. The theory is capable of modeling the independent displacement field above and below the delamination. All stress free boundary conditions at free surfaces as well as delamination interfaces are satisfied by this theory. The procedure is implemented using the finite element method. Delamination is modeled through the multi-point constraint approach using the transformation matrix technique. For validation purposes, the natural frequencies and the critical buckling loads are computed and compared with three-dimensional NASTRAN results and available experimental data. The effect of delamination on the critical buckling load and the first two instability regions is investigated for various loading conditions and plate thickness. As expected, the natural frequencies and the critical buckling load of the plates with delaminations decrease with increase in delamination length. Increase in delamination length also causes instability regions to be shifted to lower parametric resonance frequencies. The effect of edge delamination on the static and dynamic stability as well as of delamination growth is investigated.     

弹性圆柱壳在轴向冲击载荷和温度耦合作用下的屈曲

通过引入哈密顿体系,将临界载荷和临界温度及它们所对应的屈曲模态归结为辛体系下的广义本征值和本征解问题。根据辛本征解的正交性和完备性,给出了全部的且独立存在的屈曲模态。数值结果表明,在轴向冲击载荷和温度耦合作用下,弹性圆柱壳的屈曲呈现出复杂的模式,温度直接影响冲击临界载荷的大小。随着温度的增加,冲击临界荷载降低,最后,文中给出各种条件下的屈曲模态。     

Continuum-based stability of anisotropic and auxetic beams

The critical buckling loads of pinned-pinned and cantilever beams are computed using the equations of three-dimensional elasticity rather than typical beam theories. These loads are influenced both by the nature of the assumed displacement field over the beam cross-section and by the inclusion of the terms from the full constitutive tensor. Of special interest are beams that are either anisotropic or auxetic. For anisotropic beams, an increased ratio of longitudinal to shear modulus for cantilevered beams increases the generation of shear buckling rather than flexural buckling. For isotropic auxetic beams, the values of Poisson ratio that define the limit between buckling loads that approach the classical buckling load from above or below are discussed.     

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．     

轴向冲击载荷作用下弹性圆柱壳横向弯曲动态屈曲

研究在轴向冲击载荷下弹性圆柱壳动态屈曲问题.通过构造哈密顿体系,在辛空间中将临界载荷和动态屈曲模态归结为辛本征值和本征解问题.辛本征解反映了局部的压缩屈曲模态和整体的弯曲屈曲模态,特别是在冲击端为自由支承边界时的特殊屈曲方式.数值结果给出了具体的临界载荷和屈曲模态规律.     

Mode Finding in Nonlinear Structural Eigenvalue Calculations

Abstract

The eigenvalue problems resulting from stiffness matrix formulations of structural vibration and buckling problems are nonlinear if substructures are analyzed exactly, or if classical frequency (vibration problems) or load factor (buckling problems) dependent member equations are used. This makes rapid calculation of accurate free vibration or buckling modes difficult. This paper presents several techniques which might overcome this difficulty, examines them theoretically and experimentally, and gives some of the ways in which the more successful techniques can be incorporated in mode finding methods. Coincident eigenvalues (i.e., natural frequencies or critical load factors) are included.     

Torsional buckling of spherical shells under circumferential shear loads

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A genetic algorithm optimization of ring-stiffened cylindrical shells for axial and radial buckling loads

In this research, the general axial and radial buckling optimization of ring-stiffened cylindrical shells is implemented by the genetic algorithm (GA). The stiffened shell is subjected to four constraints including the fundamental frequency, the structural weight, the axial buckling load, and the radial buckling load. In addition, six design variables including shell thickness, number of stiffeners, stiffeners width and height, stiffeners eccentricity distribution order, and stiffeners spacing distribution order are considered. In analytical solution, the Ritz method is applied and stiffeners are treated as discrete elements. The effect of the weighting coefficients of the objective functions on the optimum solution is studied. The results show that optimized stiffening a cylindrical shell leads to a lower structural weight, higher natural frequencies, and larger axial and radial buckling loads, simultaneously. In addition, the upper and lower bounds of the design variables influence the optimum results considerably. It is also found that the distributions of eccentricity and spacing of the stiffeners influence the magnitudes of the axial and radial buckling loads considerably.     

复合材料层合梁的屈曲

本文在铁摩辛柯梁理论基础上,利用迭合刚度方法及Hamilton原理建立了层合梁屈曲问题控制方程,并用此控制方程求解了在具体边界条件下层合梁的屈曲问题,得出了无论在什么边界条件下层合梁的最小屈曲载荷不会大于等效剪切刚度系数C的结论.