Stability and vibration analyses of carbon nanotube-reinforced composite beams with elastic boundary conditions: Chebyshev collocation method

This article aims to investigate stability and vibration behavior of carbon nanotube-reinforced composite beams supported by classical and nonclassical boundary conditions. To include significant effects of shear deformation and rotary inertia, Timoshenko beam theory is used to formulate the coupled equations of motion governing buckling and vibration analyses of the beams. An effective mathematical technique, namely Chebyshev collocation method, is employed to solve the coupled equations of motion for determining critical buckling loads and natural frequencies of the beams with different boundary conditions. The accuracy and reliability of the proposed mathematical models are verified numerically by comparing with the existing results in the literature for the cases of classical boundary conditions. New results of critical buckling loads and natural frequencies of the beams with nonclassical boundary conditions including translational and rotational springs are presented and discussed in detail associated with many important parametric studies.     

Experimental study on scaling of RC beams under close-in blast loading

Damage effects analysis and assessment of buildings under blast loading is an important problem concerned by the area of explosion accident analysis, blast-resistant design, anti-terrorist and military weapon design.The damage character of RC beam under close-in blast loading is investigated through experiments. The damage modes and damage levels of RC beams are studied under different blast loads. The results show that the spallation area increases with the decrease of the scaled distance. The concrete beams are prone to be damaged in flexure mode with concrete crushed on the front face, concrete spallation on the back surface and concrete flake off on the side surface. The scaling of the dynamic response of reinforced concrete beams subjected to close-in blast loadings is also studied. The test results showed a similar macrostructure damage and fracture in all experiment conditions. But the local damage degree of RC beams with smaller size has been reduced a little as compared with that of beams with larger size. Based on the results, empirical equations of the center deflection to height ratio are proposed to correct scaling model considering size effects.     

弹性支撑旋转Timoshenko梁动力学特性

为研究弹性支撑旋转梁动力学特性随转速及弹性支撑参数变化规律,考虑剪切效应、转动惯量和陀螺效应,采用Hamilton原理推导旋转Timoshenko梁动力学方程,应用Chebyshev谱方法获得系统涡动频率与模态振型数值解。结果表明,在高速转动状态下陀螺效应、支撑结构刚度对Timoshenko梁动力学特性有显著影响;各阶固有频率随着转速增加而分成正向涡动频率与反向涡动频率,高阶频率变化幅度更大;涡动频率随支撑结构直线刚度增加而呈阶梯状变化,当直线刚度增加到一定值后系统涡动频率将保持稳定;随着支撑结构转动刚度增加,涡动频率出现一个最小值与最大值,前者低于自由边界条件下频率值,后者高于固定边界条件下频率值。相关结果可用于各类旋转梁机构的设计与优化。     

Analytical solutions of coupled-mode equations for multiwaveguide systems, obtained by use of Chebyshev and generalized Chebyshev polynomials

A novel approach is proposed for obtaining the analytical solutions of the coupled-mode equations (CMEs); the method is applicable for an arbitrary number of coupled waveguides. The mathematical aspects of the CMEs and their solution by use of Chebyshev polynomials are discussed. When mode coupling between only adjacent waveguides is considered (denoted weak coupling), the first and second kinds of the usual Chebyshev polynomials are appropriate for evaluating the CMEs for linearly distributed and circularly distributed multiwaveguide systems, respectively. However, when one is considering the coupling effects between nonadjacent waveguides also (denoted strong coupling), it is necessary to use redefined generalized Chebyshev polynomials to express general solutions in a form similar to those for the weak-coupling case. As concrete examples, analytical solutions for 2 x 2, 3 x 3, and 4 x 4 linearly distributed directional couplers are obtained by the proposed approach, which treats the calculation as a nondegenerate eigenvalue problem. In addition, for the 3 x 3 circularly distributed directional coupler, which gives rise to a degenerate eigenvalue problem, an analytical solution is obtained in an improved way. Also, for comparison and without loss of generality, to clarify the difference between the two coupling cases, analytical solutions for a 5 x 5 circularly distributed directional coupler are obtained by use of the usual and the redefined generalized Chebyshev polynomials.     

Effects of local damage on vibration characteristics of composite pyramidal truss core sandwich structure

The effects of local damage on the natural frequencies and the corresponding vibration modes of composite pyramidal truss core sandwich structures are studied in the present paper. Hot press molding method is used to fabricate intact and damaged pyramidal truss core sandwich structures, and modal testing is carried out to obtain their natural frequencies. A FEM model is also constructed to investigate their vibration characteristics numerically. It is found that the calculated natural frequencies are in relatively good agreement with the measured results. By using the experimentally validated FEM model, a series of numerical analyses are conducted to further explore the effects of damage extent, damage location, damage form on the vibration characteristics of composite pyramidal truss core sandwich structures as well as the influence of boundary conditions. The conclusion derived from this study is expected to be useful for analyzing practical problems related to structural health monitoring of composite lattice sandwich structures.     

Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory

The natural frequencies of the flapwise bending vibrations of a nonuniform rotating nanocantilever has been calculated, considering the true spatial variation of the axial force due to the rotation. The area of the nanobeam cross-section is assumed to change linearly. The problem has been formulated using the nonlocal Eringen elasticity theory and it was solved by a pseudo-spectral collocation method based on Chebyshev polynomials. The effect of the nonlocal small-scale, angular speed, nonuniformity of the section and hub radius on the vibration behavior of the nanocantilever is discussed.     

损伤定位向量法原理及应用

该文中证明了损伤定位向量存在的条件,推导了梁、柱和支撑单元的特征应力计算公式,改进了损伤后振型质量归一化系数的简化求解。完成了一个空间钢支撑框架模型损伤定位的数值模拟分析,模拟了包括支撑损伤和连接损伤的7种损伤模式。分别采用力锤激励和环境激励两种激励方式;使用自然激励技术和特征系统实现算法进行模态参数识别,得到了模态频率和振型;使用损伤定位向量法判定损伤单元,并对两种损伤模式进行了误差分析。结果表明:当损伤达到一定程度时,损伤定位向量法能有效判定损伤单元。     

Numerical convergence of simple and orthogonal polynomials for the unilateral plate buckling problem using the Rayleigh–Ritz method

Unilateral buckling is a contact problem whereby buckling is confined to take place in only one lateral direction. For plate structures, this can occur when a thin steel plate is juxtaposed with a rigid concrete medium and the steel may only buckle locally away from the concrete core. This paper investigates the use of simple and orthogonal polynomials in the Rayleigh–Ritz method for unilateral plate buckling. The orthogonal polynomials used are the classical Chebyshev types 1 and 2, Legrende, Hermite and Laguerre. The study presents a comparison between the efficiency of the polynomial‐based displacement functions with regard to elastic bilateral and unilateral plate buckling, where efficiency is measured as a function of their convergence characteristics. Some buckling solutions for plates with varying boundary conditions and in‐plane shear loads are also provided as an illustration. Copyright © 1999 John Wiley & Sons, Ltd.     

Modal parameter identification and vibration based damage detection of a multiple cracked cantilever beam

This paper presents a detailed investigation on the modal parameter identification and vibration based damage detection of a multiple cracked cantilever beam with hollow circular cross-section. To consider multiple crack effects, a cantilever beam including cracks is considered for six damage scenarios. Finite element models are constituted in ANSYS software for numerical solutions. The results are validated by experimental measurements. Ambient vibration tests are performed to extract the dynamic characteristics using Enhanced Frequency Domain Decomposition (EFDD) and Stochastic Subspace Identification (SSI) methods. Calculated and measured natural frequencies and mode shapes for undamaged and damaged beams are compared with each other. Automated model updating is carried out using the modal sensitivity method based on Bayesian parameter estimation to minimize the differences for damage detection. In addition, modal assurance criterion (MAC) and coordinated modal assurance criterion (COMAC) factors are obtained from the mode shapes and two set of measurements to establish the correlation between the measured and calculated values for damage location identification.     

Stability analysis of delaminated composite beams

This study describes the dynamic stability of composite cantilever beams subjected to periodic axial loading with delaminations at pre-set locations. A computer code based on the finite element method is developed to calculate the natural frequencies, critical buckling loads and dynamic instability regions of the woven and laminated composite beams with different stacking sequences ([0]₄, [0/90]_s and [90]₄), corresponding to this peculiar delamination case. The results of the developed code for the natural frequencies are compared with the natural frequencies obtained experimentally and numerically with commercial FEA (ANSYS). The critical buckling loads are also compared with the ones obtained from ANSYS simulations.     

Spectral collocation methods for the primary two-point boundary value problem in modelling viscoelastic flows

Expansions in terms of beam functions and Chebyshev polynomials are used to compute solutions to the primary two-point boundary value problem within a spectral collocation formulation. The performance of the methods is analysed in terms of accuracy and robustness relative to the level of non-linearity. Accurate results are obtained with Chebyshev polynomials and the performance of these trial functions is insensitive to the level of non-linearity whereas the behaviour of the beam functions shows great sensitivity to the level of non-linearity. The use of Newton's method to solve the mixed linear-non-linear system for the Chebyshev coefficients is successful for highly non-linear problems without the need for parameter continuation.     

Detection of multiple cracks using frequency measurements

A method for detection of multiple open cracks in a slender Euler-Bernoulli beams is presented based on frequency measurements. The method is based on the approach given by Hu and Liang [J. Franklin Inst. 330 (5) (1993) 841], transverse vibration modelling through transfer matrix method and representation of a crack by rotational spring. The beam is virtually divided into a number of segments, which can be decided by the analyst, and each of them is considered to be associated with a damage parameter. The procedure gives a linear relationship explicitly between the changes in natural frequencies of the beam and the damage parameters. These parameters are determined from the knowledge of changes in the natural frequencies. After obtaining them, each is treated in turn to exactly pinpoint the crack location in the segment and determine its size. The forward, or natural frequency determination, problems are examined in the passing. The method is approximate, but it can handle segmented beams, any boundary conditions, intermediate spring or rigid supports, etc. It eliminates the need for any symbolic computation which is envisaged by Hu and Liang [J. Franklin Inst. 330 (5) (1993) 841] to obtain mode shapes of the corresponding uncracked beams. The proposed method gives a clear insight into the whole analysis. Case studies (numerical) are presented to demonstrate the method effectiveness for two simultaneous cracks of size 10% and more of section depth. The differences between the actual and predicted crack locations and sizes are less than 10% and 15% respectively. The numbers of segments into which the beam is virtually divided limits the maximum number of cracks that can be handled. The difference in the forward problem is less than 5%.     

初应力拱侧向振动的固有频率

在初应力位形上附加变形场论的框架下,引入自然坐标系,采用平截面假设,建立对数值方法和解析方法十分有用的、初应力曲拱位移变分方程。给出了平面曲拱在平面初始变形状态下附加变形变分方程,将拱的初始内力嵌入其中。根据所得的方程,讨论了面内载荷对圆截面初应力圆弧形拱侧向振动固有频率的影响,给出了侧向振动固有频率与初载荷的解析关系。结果表明,面内初始载荷的存在减小了侧向挠曲的固有频率;减少的数值与用频率方法诊断构件缺陷通常可得到的灵敏度属同量级。     

Free vibration of axially loaded rectangular composite beams using refined shear deformation theory

Free vibration of axially loaded rectangular composite beams with arbitrary lay-ups using refined shear deformation theory is presented. It accounts for the parabolical variation of shear strains through the depth of beam. Three governing equations of motion are derived from the Hamilton’s principle. The resulting coupling is referred to as triply axial-flexural coupled vibration. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results are obtained for rectangular composite beams to investigate effects of fiber orientation and modulus ratio on the natural frequencies, critical buckling loads and load–frequency curves as well as corresponding mode shapes.     

受边缘非线性分布荷载作用矩形薄板的面内应力分析

矩形薄板边缘受非线性分布面内载荷作用的情况在工程中经常遇到,精确的应力分析也是薄板稳定性分析的基础,由于问题的复杂性目前还没有精确解.该文根据弹性力学理论,采用里兹法求解矩形薄板边缘受非线性分布载荷作用的面内应力,应力函数采用切比雪夫多项式并满足所有应力边界条件.结合数学计算软件Mathematica,分析了不同长宽比...     

Overall moduli and natural frequencies of composite laminates containing multiple interlaminar transverse cracks

In this paper, the overall tensile modulus of a composite laminate containing embedded multiple interlaminar transverse cracks is studied. The modulus is calculated based upon an energy method and the crack opening displacement which is obtained by solving a boundary-value problem. Numerical computation is applied to fibre-reinforced composite laminates following the theoretical analysis. The theoretical prediction is compared with the experimental data, and good agreement is found. The solution is then used to examine the natural frequencies of two representative cross-ply beams with multiple matrix cracks in some of the outer transverse layers. The difference between the natural frequencies of the intact and the damaged cross-ply beams is presented. It is found that for a graphite/epoxy composite, the multiple transverse cracks only have a minor influence on the frequency, whereas for a glass/epoxy composite, the multiple cracks may have a significant influence on the frequency when the cracks reach the saturation level in a relatively large area of a beam.     

Compact computation of natural frequencies and buckling loads for plane frames

Existing theory is assembled to give a method which needs only the core of a mini-computer to calculate the eigenvalues of large rigidly jointed plane frames with certainty, the eigenvalues being natural frequencies and critical load factors in free vibration and buckling problems, respectively. The method is illustrated by annotated listings of vibration and buckling programs, each involving under two hundred Fortran statements and with low number storage requirements (see Table I). The use of the programs as ‘black boxes’ is fully explained, with illustrative examples. The member theory used is the ‘exact’ classical Bernoulli-Euler uniform member theory. Possible applications include: evaluation of answers from approximate methods; calculation of critical loads for substitution in the modified Merchant-Rankine formula to estimate collapse loads of frames; and calculation of shifts in natural frequencies caused by structural damage, in connection with structural integrity monitoring of inaccessible structures.     

Vibration analysis of thin-walled composite beams with I-shaped cross-sections

A general analytical model applicable to the vibration analysis of thin-walled composite I-beams with arbitrary lay-ups is developed. Based on the classical lamination theory, this model has been applied to the investigation of load–frequency interaction curves of thin-walled composite beams under various loads. The governing differential equations are derived from the Hamilton’s principle. A finite element model with seven degrees of freedoms per node is developed to solve the problem. Numerical results are obtained for thin-walled composite I-beams under uniformly distributed load, combined axial force and bending loads. The effects of fiber orientation, location of applied load, and types of loads on the natural frequencies and load–frequency interaction curves as well as vibration mode shapes are parametrically studied.     

A new sampling scheme for developing metamodels with the zeros of Chebyshev polynomials

The accuracy of metamodelling is determined by both the sampling and approximation. This article proposes a new sampling method based on the zeros of Chebyshev polynomials to capture the sampling information effectively. First, the zeros of one-dimensional Chebyshev polynomials are applied to construct Chebyshev tensor product (CTP) sampling, and the CTP is then used to construct high-order multi-dimensional metamodels using the ‘hypercube’ polynomials. Secondly, the CTP sampling is further enhanced to develop Chebyshev collocation method (CCM) sampling, to construct the ‘simplex’ polynomials. The samples of CCM are randomly and directly chosen from the CTP samples. Two widely studied sampling methods, namely the Smolyak sparse grid and Hammersley, are used to demonstrate the effectiveness of the proposed sampling method. Several numerical examples are utilized to validate the approximation accuracy of the proposed metamodel under different dimensions.     

Assessment of CFRP strengthened RC beams through dynamic tests

The use of Fiber Reinforced Polymers (FRPs) for strengthening damaged RC beams has become common practice over recent years. Two methods adopted for repairing or strengthening such beams are FRP plates/sheets glued onto their concrete surface and FRP rods or strips inserted into grooves.This paper investigates experimental vibration monitoring of strengthening according to the two aforementioned methods through dynamic tests on six RC beam models strengthened using carbon FRP. Three beams were strengthened applying CFRP sheets on the tensile cracked surface after loading and three beams were strengthened by near surface mounted (NSM) CFRP rods. The experimental results include both the static tests to create damage and the dynamic tests of strengthened, measuring natural vibration modes and frequency values for free end beams.Comparison between experimental dynamic response and static behavior established that vibration monitoring is a convenient, non-destructive method for assessing strengthened beams under service loads. Further studies and tests must be developed in order to solve the issues that emerged following the analysis of the experimental data obtained.