Delamination buckling of FRP layer in laminated wood beams

An analytical solution for predicting delamination buckling and growth of a thin fiber reinforced-plastic (FRP) layer in laminated wood beams under bending is presented. Based on a strength-of-materials approach, displacement functions for a delaminated beam under four-point bending are derived. Using force and displacement compatibility conditions, an explicit form relating the applied transverse load with the delamination buckling load is established. An explicit form of the strain-energy release rate is presented to study the delamination growth in beams under bending. The analytical solution is evaluated using experimental data for glued-laminated timber (glulam) beams reinforced with a thin fiber-reinforced plastic composite on the compression face. The delamination growth in bending is shown to behave differently to that of the in-plane loading case.     

Analytical Solution of Buckling of Beams with Two Delaminations

A stability problem of composite beams with multiple delamination was tackled. A closed-form solution was found and buckling loads of composite beams with two delaminations were determined in order to obtain their compressive load-carrying capacity. Crack-opening mode was assumed for each detached delaminated region. Seven different regions having different transverse deformations resulted from assumed positions of delaminations. Developing the derived continuity condition equations reduced the number of algebraic equations required to solve the problem analytically. The results of the work were validated by comparing them to those reported in the literature. The effects of length, location, and distribution of multiple delaminations were considered in the comparison, and the results showed very good agreement. Buckling load decreases as delamination size increases. Buckling load for a beam with two delaminations is lower than that for the same beam with a single delamination.     

Free vibration analysis of rotating Timoshenko beams with multiple delaminations

Analytical solutions are developed to study the free vibrations of rotating Timoshenko beams with multiple delaminations. The Timoshenko beam theory and both the ‘free mode’ and ‘constrained mode’ assumptions in delamination vibration are adopted. Parametric studies are performed to study the influences of Timoshenko effect and rotating speed on delamination vibration. Results show that the effect of delamination on both modes 1 and 2 natural frequencies is significantly influenced by Timoshenko effect and the rotating speed. Also, the comparison between ‘free mode’ assumption and ‘constrained assumption’ are conducted. Furthermore, the effect of delamination on mode shapes is also influenced by rotating speed and Timoshenko effect. For both Timoshenko effect and rotating speed, the influences on the second vibration mode shape are more significant.     

含脱层单向铺设层合梁非线性后屈曲分析

采用四分区模型,将含脱层单向铺设复合材料层合板梁分为4个子梁,根据复合材料层合理论,考虑后屈曲路径上位于脱层界面上、下子梁之间的局部受力与变形机制,建立了子梁之间接触力与变形之间的非线性定量关系。在此基础上,结合可伸长梁的几何非线性理论,推导出了计及接触效应的各子梁的非线性后屈曲控制方程。设定简支板梁的边界条件以及脱层前沿处各子梁之间力和位移的连续性条件,通过对控制方程和定解条件归一化,采用小参数摄动法求解,并根据梁的平衡微分方程的特点,解析其通解与特解的构造,获得了含脱层单向铺设层合梁受轴向压力作用的临界屈曲荷载及后屈曲平衡路径的理论解。通过对含脱层单向铺设的复合材料层合梁进行数值分析,综合讨论了脱层长度和深度等对层合板梁的临界屈曲载荷及接触性能的影响,并将所得的理论解与ABAQUS有限元分析得到的结果进行对比,结果表明二者高度吻合。研究发现梁的屈曲模态包含宏观的整体失效模态和界面的微观屈曲模态。梁的屈曲荷载和接触性能都是其固有属性,前者受梁的几何参数和材料参数的影响较显著,而后者则主要受脱层的位置和大小影响。     

Dynamic instability of channel-section beams under periodic loading

Abstract

This paper presents an analytical investigation on the free vibration, static buckling and dynamic instability of channel-section beams when subjected to periodic loading. The analysis is carried out by using Bolotin’s method. By assuming the instability modes, the kinetic energy and strain energy of the beam and the loss of the potential of the applied load are evaluated, from which the mass, stiffness and geometric stiffness matrices of the system are derived. These matrices are then used to carry out the analyses of free vibration, static buckling and dynamic instability of the beams. Theoretical formulae are derived for the free vibration frequency, critical buckling moment, and excitation frequency of the beam. The effects of the lateral restraint applied to the flange, the section size of the beam and the static part of the applied load on the variation of dynamic instability zones are also discussed.     

复合材料梁结构应力波传播特性研究

基于铁木辛柯梁理论,研究含半无限大分层复合材料梁结构中波的反射与透射。考虑表面无接触压力(张开分层)及表面完全接触(闭合分层)两种极端分层条件,对单向复合材料梁建模导出反射与透射矩阵,计算能量反射与透射系数。数值结果表明,能量反射与透射系数同随波频率及分层位置变化,能量传输遵循守恒定律。研究各模态能量分配,描述前两阶弯曲波及第一阶膨胀波间模态转换关系;通过有限元仿真验证该理论分析的正确性。     

Vibrations of multiple delaminated beams

The free vibrations of beams with two enveloping delaminations have been solved analytically without resorting to numerical approximation. The multiple delaminated beam is analyzed as several interconnected Euler–Bernoulli beams. The differential stretching and the bending–extension coupling are considered in the formulation. The influence of the sizes and locations of the delaminations on the primary and secondary frequencies and mode shapes of a beam are investigated. For clamped–clamped beams, the primary frequency shows a high sensitivity for the long delamination but a low sensitivity for the second short delamination, while for the secondary frequency, the sensitivity is high for both delaminations. For cantilever beams, the primary and secondary frequencies show a high sensitivity for the long delamination but low sensitivity for the second short delamination. Results are compared with the analytical and experimental data reported in the literature to verify the validity of the present model.     

Vibrations of beams with two overlapping delaminations

Delamination in composite laminates may arise from either imperfect fabrication processes or impact during service. The locations of the delaminations are arbitrary. In this research, an analytical solution to the free vibrations of a beam with two overlapping delaminations is presented. The delaminated beam is analyzed as seven interconnected Euler–Bernoulli beams. The influence of the sizes and the locations of the delaminations on the fundamental frequency and the mode shape of the beam are investigated. Complex vibration behaviors emerge for different sizes and locations of the delaminations. The results based on the present model agree well with the analytical data reported in the literature.     

湿-热-机耦合作用下多孔功能梯度梁的振动及屈曲特性

采用一种拓展的n阶广义梁理论（GBT）,研究了轴向机械载荷作用下多孔功能梯度材料（FGM）梁在湿热环境中的振动及屈曲特性。考虑了材料的物性随温度变化,湿-热沿梁厚按三种不同类型分布,采用含孔隙率的修正Voigt混合率模型描述多孔功能梯度梁的材料属性,在宏-细观力学模型框架下应用Hamilton原理统一建立了系统的自由振动及屈曲方程,采用Navier法求解FGM简支梁的静动态响应。通过算例验证并讨论了GBT阶数n的理想取值,可用于丰富梁理论。探讨了湿热效应、湿-热-机耦合、孔隙率、材料梯度指标、跨厚比对FGM梁振动及屈曲特性的影响。结果表明:湿-热加剧降低了FGM梁的频率和临界载荷,且不同类型的湿热分布对其减小程度有显著差异;随着孔隙率增大,梁结构的整体刚度虽有所弱化,但在湿热环境中频率反而增大,稳定性增强;湿-热效应对多孔FGM细长梁频率和稳定性影响十分显著,但对短粗梁的影响比较有限。      

轴压作用下自由-自由运动梁振动特性研究

针对轴向压力作用下的两端自由运动梁的振动问题,根据Timoshenko梁理论和Hamilton原理建立了梁的横向振动控制方程,通过解析法和微分求积法(DQM)求解了梁的振动特性,分析了轴向压力和运动效应以及轴向力导数和运动加速度对梁固有特性的影响,并对临界载荷、临界速度等的影响因素进行了比较研究。结果表明:轴向压力和运动效应都使得固有频率降低,压力和运动速度的特定组合会导致超临界现象和耦合模态颤振的出现;压力导数和加速度效应都会使得梁的基础频率产生不稳定性;梁的临界载荷随着运动速度增大而变小,临界速度随轴向压力增大而降低。     

基于压弯耦合效应下预应力梁的竖向自由振动研究

基于大位移广义变分原理,考虑梁的压弯耦合、剪切应变能和转动惯量的影响,建立了预应力梁的不完全广义势能泛函,通过对位移变分,推导出预应力梁自由振动微分方程。并以预应力混凝土简支梁和悬臂梁为例,通过引入边界条件,求出了自由振动频率的解答。对比Bernoulli-Eular梁和Timoshenko梁,详细分析了轴向荷载、剪切效应和转动惯量对自振频率的影响,研究发现,轴向压力荷载可使梁的自振频率降低,反之增大。剪切变形的影响约为转动惯量的3倍,随着主模态阶数的增加和长细比L/r的减小,轴向荷载、剪切变形和转动惯量的影响非常显著。因此,对于预应力混凝土梁,当跨高比L/h≤8,或长细比L/r≤28时,必须考虑轴向荷载、剪切变形和转动惯量的影响,通过与Bernoulli-Eular梁和Timoshenko梁的精确解相比较,证明该文的解答是正确的。     

Efficient modelling of delamination buckling in composite cylindrical shells under axial compression

Composite cylindrical shells and panels are widely used in aerospace structures. These are often subjected to defects and damage from both in-service and manufacturing events. Delamination is the most important of these defects. This paper deals with the computational modelling of delamination in isotropic and laminated composite cylindrical shells. The use of three-dimensional finite elements for predicting the delamination buckling of these structures is computationally expensive. Here combined double-layer and single-layer of shell elements are employed to study the effect of delamination on the global load-carrying capacity of such systems under axial compressive load. It is shown that through-the-thickness delamination can be modelled and analysed effectively without requiring a great deal of computing time and memory. A parametric study is carried out to study the influence of the delamination size, orientation and through-the-width position of a series of laminated cylinders. The effect of material properties is also investigated. Some of the results are compared with the corresponding analytical results. It is shown that ignoring the contact between the delaminated layers can result in wrong estimations of the critical buckling loads in cylindrical shells under compressive load.     

湿-热-机-弹耦合FGM梁的稳定性及振动特性

研究了初始轴向机械载荷作用下Winkler-Pasternak弹性地基上功能梯度材料（FGM）梁在湿-热环境中的稳定性及振动特性。假设温度和湿度沿梁厚度方向稳态分布,材料的物性依赖于温度且按Voigt混合幂律模型连续分布。首先,基于一种扩展的n阶广义梁理论,应用Hamilton原理,统一建立了以轴向位移、弯曲变形项挠度及剪切变形项挠度为基本未知函数FGM梁的屈曲及自由振动方程,采用Navier解法获得了FGM简支梁静动态响应的精确解。其次,通过算例验证并给出了该广义梁理论阶次n的理想取值,丰富梁理论的同时,可供验证或改进其他各种剪切变形梁理论。最后,着重探讨了3种湿-热分布下湿度与温度增加、初始轴向机械载荷、跨厚比、地基刚度、梯度指标等诸多参数对FGM梁稳定性和振动特性的影响。     

The stability of delamination growth in compressively loaded composite plates

The stability of growth of internal delaminations in composite plates subjected to compressive loading is investigated. Due to the compressive loading, these structures can undergo buckling of the delaminated layer and subsequently growth of the delamination. The study does not impose any restrictive assumptions regarding the delamination thickness and plate length (as opposed to the usual thin film assumptions). The growth characteristics of the delamination under monotonic compressive loading are obtained on the basis of a combined delamination buckling/postbuckling and fracture mechanics model. The postbuckling solution is derived through a perturbation procedure, which is based on an asymptotic expansion of the load and deformation quantities in terms of the distortion parameter of the delaminated layer, the latter being considered a compressive elastica. The closed form solutions for the energy release rate at the delamination tip versus applied compressive strain during the initial postbuckling phase are used to define the combinations of delamination length and applied strain that lead to unstable growth. This would practically cause either contained jump growth or complete (catastrophic) growth of the delamination. Estimates for the lower and upper bounds of the jump distance (unstable growth) are provided. Moreover, a study of the influence of the mode dependence of interface toughness on the conditions of initiation and extent of delamination growth is performed.     

Exact reconstruction of multiple concentrated damages on beams

In this paper, an exact procedure for the reconstruction of multiple concentrated damages on a straight beam is proposed. The concentrated damages are modelled as Dirac’s delta distributions capturing the effect of concentrated stiffness reduction. The presented procedure requires the knowledge of vibration mode shape displacements together with the relevant natural frequency, for the reconstruction of each damage position and intensity. The exact solution of the inverse problem at hand has been pursued by exploiting the analytical structure of the explicit closed form expressions provided for the vibration mode shapes of beams in the presence of an arbitrary number of cracks. The proposed procedure is first presented under the hypothesis that the displacements of a vibration mode shape are known at the cracked cross-sections. In this case, explicit closed form expressions of the crack severities are formulated. A further simple reconstruction approach allows the evaluation of the exact positions and intensity of the concentrated damages, if displacements of two vibration mode shapes are known at a single cross-section between two consecutive cracks. The proposed reconstruction procedure is applied for the identification of multiple cracks on a free–free beam where measurements have been simulated by means of a finite element analysis.     

带脱层的复合材料层板的屈曲分析

用基于Mindlin 板理论的有限元方法进行了带脱层损伤的复合材料层板的屈曲载荷分析。为了获得物理上可能的屈曲模态, 即避免上下脱层的相互贯穿, 在接触区域引入一些假想弹簧, 并给出了这些假想弹簧刚度系数的计算公式和接触计算的迭代格式, 通过这些弹簧对原始刚度矩阵进行修正可以有效地求解屈曲载荷特征值分析中的接触问题。数值计算结果表明了本算法的有效性和引入接触分析对这类屈曲分析的重要性。同时, 还对脱层的大小、形状、位置和脱层的纤维铺层方向对屈曲载荷的影响进行了研究。      

Buckling and postbuckling behavior of delaminated sandwich beams

An investigation was performed to study the buckling and postbuckling behavior of sandwich beams containing lengthwise and depthwise through-the-width delaminations. An analytical beam model was developed to predict the buckling load of the beam and to describe its postbuckling response for arbitrarily situated delaminations and various combinations of boundary conditions. Griffith's energy release rate model was employed to predict the stability of delamination propagation under external loading and to determine the direction of delamination growth.

Parametric studies over a wide range of beam geometries, damage sizes and locations, composite facings and beam boundary conditions were carried out to study their effects on the overall behavior of the sandwich structure, as well as its damage tolerance. The results demonstrated that a sandwich construction is very ‘sensitive’ to the presence of delaminations situated at the core-faceplate interface. Premature buckling failure occurs at external loads which are significantly lower than the buckling load for a ‘perfect’ sandwich beam; in ‘imperfect’ beams with composite faceplates, the layup sequence affects significantly the load-carrying capacity of the beam; varying either the boundary conditions in a sandwich beam or the lengthwise location of a delamination has a small effect on the postbuckling behavior of the beam. Delaminations located within composite faceplates have less pronounced influence, and as the defect is moved outwards the limit load may reach the buckling load corresponding to that of the ‘perfect’ beam.

The proposed model is capable of analyzing the postbuckling behavior of both sandwich and composite laminated beams for arbitrary locations of the delamination, and various combinations of boundary conditions.     

含分层损伤复合材料层合板非线性动力稳定性

采用Reddy 的板高阶剪切变形简化理论研究了含分层损伤复合材料层合板的非线性动力稳定性问题。建立了分层模型, 推导了考虑几何非线性和阻尼效应的Methieu 方程, 给出了该方程的解析解表达式; 研究了参数振动解的稳定性; 然后通过典型数例讨论了分层损伤对层合板固有频率、屈曲临界力以及动力稳定区域的影响; 研究了保守与非保守体系的外载荷的激励频率对层合板第一参数振动的振幅的影响, 以及线性、非线性阻尼对非保守体系的最大牵引深度的影响。由典型算例讨论可知, 随着复合材料层合板分层损伤的扩大, 其动力稳定性能逐渐减弱, 特别是损伤接近层合板的中面时, 分层损伤对其动力稳定性能的影响最大。      

On the Through-the-Width Multiple Delamination, and Buckling and Postbuckling Behaviors of Symmetric and Unsymmetric Composite Laminates

Multiple delamination causes severe degradation of the stiffness and strength of composites. Interactions between multiple delamination, and buckling and postbuckling under compressive loads add the complexity of mechanical properties of composites. In this paper, the buckling, postbuckling and through-the-width multiple delamination of symmetric and unsymmetric composite laminates are studied using 3D FEA, and the virtual crack closure technique with two delamination failure criteria: B-K law and power law is used to predict the delamination growth and to calculate the mixed-mode energy release rate. The compressive load-strain curves, load-central deflection curves and multiple delamination process for eight composite specimens with different initial delamination sizes and their distributions as well as two angle-ply configurations 0₄//(±θ)₆//0₄ (θ?=?0° and 45°, and “//” denotes the delaminated interface) are comparatively studied. From numerical results, the unsymmetry decreases the local buckling load and initial delamination load, but does not affect the global buckling load compared with the symmetric laminates. Besides, the unsymmetry affects the unstable delamination and buckling behaviors of composite laminates largely when the initial multiple delamination sizes are relatively small.     

A higher order plate theory for dynamic stability analysis of delaminated composite plates

A higher order shear deformation theory is used to investigate the instability associated with delaminated composite plates subject to dynamic loads. Both transverse shear and rotary inertia effects are taken into account. The procedure is implemented using the finite element method. Delamination is modeled using the penalty parameter approach. The natural frequencies are computed and compared with NASTRAN 3D 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, plate thickness and boundary conditions. As expected the natural frequencies and the critical buckling load of the delaminated plate are lower than those of the nondelaminated plate. They decrease with increase in delamination length. Increase in delamination length causes instability regions to be shifted to lower parametric resonance frequencies and the normalized width of the instability regions to increase.