含裂纹损伤圆弧曲梁弹性屈曲的有限元网格自适应分析

该文建立圆弧形曲梁裂纹的截面损伤缺陷比拟方案,实现裂纹大小(深度)、位置、数目的模拟.引入变截面Euler-Bernoulli梁的h型有限元网格自适应分析方法,求解含裂纹损伤圆弧曲梁弹性屈曲问题,得到优化的网格和满足预设误差限的高精度屈曲荷载和屈曲模态解答.数值算例表明该算法中网格非均匀加密可适应裂纹损伤引起的屈曲模态...     

变截面变曲率梁振型的有限元超收敛拼片恢复解和网格自适应分析

该文提出变截面变曲率梁振型的有限元后处理超收敛拼片恢复方法,建立各阶振型的超收敛解,并基于振型超收敛解进行变截面曲梁面内和面外自由振动的自适应分析。在位移型有限元后处理阶段,引入超收敛拼片恢复方法和高阶形函数插值技术,得到振型(位移)的超收敛解。利用振型超收敛解估计当前网格下振型有限元解的能量模形式下的误差,并指导网格进行自适应细分加密分析,获得优化的网格和满足预设误差限的高精度解答。数值算例表明该算法适于求解不同曲线形态、多类边界条件、变截面、变曲率形式的曲梁面内和面外自由振动连续阶频率和振型,解答精确、分析过程高效可靠。     

土体阻尼对全(部分)埋入单桩基础自振特性的影响研究

自振频率和振型是反映结构动力特性的重要物理量,对结构设计和动力分析具有重要作用。基于回传射线矩阵法(MRRM),首先推导出全埋入单桩基础在桩顶自由、桩底分别为自由和固定条件下发生轴向自由振动时波数、振型和自振频率的解析解;而后研究部分埋入单桩基础横向和轴向自由振动时的自振特性,探讨了4种工况,工况1～3同时考虑土体弹簧和土体阻尼作用,工况4仅考虑土体阻尼作用,分析了土体阻尼对部分埋入单桩基础自振频率和振型、衰减系数的影响规律。结果表明:土体阻尼对低阶自振频率和振型的影响较大,在结构的低频振动分析中不可忽略。     

变截面Euler-Bernoulli梁在轴力作用下固有振动的级数解

建立了截面特性、轴力和质量分布均连续变化的Euler-Bernoulli梁固有振动方程。采用Frobeniu方法求解方程,得到了方程的级数解析解;并计算了结构的振动频率与振型。算例证明了该方法的有效性,并表明轴向力的几何非线性效应对振动的低阶频率和自振振型有一定的影响,级数项数的取值对算法的收敛性有影响。     

非圆弧拱面内自由振动实用解析

针对非圆弧拱面内线性自由振动没有解析的现状,提出了一种变系数平衡微分方程近似解析方法来解决该问题。基于笛卡尔直角坐标系下非圆弧拱线性应变与Hamilton原理,推演了非圆弧拱面内自由振动变系数平衡微分方程;基于陡拱与浅拱面内振型没有显著差异的基本假定,将该变系数平衡微分方程对应的常系数平衡微分方程的通解,代入变系数平衡微分方程,得到该变系数平衡微分方程的不平衡差;当该不平衡差沿全拱积分为零时自振频率误差最小,进而得到非圆弧拱面内自振频率高精度实用解析。基于所提出的变系数平衡微分方程近似解析方法,推演了非圆弧两铰拱与无铰拱面内自振频率实用解析,并阐明了非圆弧拱与同参数直梁面内自振频率的逻辑关系。抛物线、悬索线、悬链线与组合线等常用非圆弧两铰拱与无铰拱自由振动算例结果表明：该研究的基本假定得到了严格检验;自振频率与有限元结果吻合较好,非圆弧拱前十阶自振频率中,两铰拱自振频率最大相对误差为7.71%,无铰拱自振频率最大相对误差为4.34%;非圆弧拱与同参数直梁面内自振频率的比例系数,可为行业规范条文修订提供参考。     

动力刚度法求解平面曲梁面外自由振动问题

该文将动力刚度法应用于平面曲梁面外自由振动的分析。通过建立单元动力刚度所满足的常微分方程边值问题,用具有自适应求解功能的常微分方程求解器COLSYS 进行求解,获得单元动力刚度的数值精确解。以COLSYS 求解单元动力刚度的网格作为单元上固端频率计数求解的子网格,由单元动力刚度的边值问题解答线性组合出该子网格下各子单元的动力刚度,由Wittrick-Williams 算法获得单元固端频率的计数。从而实现整体结构的Wittrick-Williams频率计数。通过建立单元动力刚度对频率的导数所满足的常微分方程边值问题,调用COLSYS求其数值精确解,并将其引入导护型牛顿法,可迅速求得结构精确的频率和振型。数值算例表明,该文方法准确、可靠、有效。     

圆弧曲梁面内自由振动的微分容积解法

用微分容积法求解圆弧曲梁在面内的自由振动问题。通过微分容积法将曲梁自由振动的控制微分方程和边界约束方程离散成为一组线性齐次代数方程组，这是一典型的特征值问题，求解这一特征值问题可以求得其自由振动的圆频率。中采用了考虑轴向变形、剪切变形和转动效应的理论，并采用子空间迭代法求解频率方程。数值算例表明，本方法稳定收敛、精度较高，对圆弧曲梁问题简单、有效。     

平面曲梁面外自由振动有限元分析的p型超收敛算法

该文用p型超收敛算法对平面曲梁面外自由振动问题进行求解。该法基于频率和振型结点位移在有限元解答中的超收敛特性,在单元上建立振型近似满足的线性常微分方程边值问题,用更高次元对该线性边值问题进行有限元求解获得各单元上振型的超收敛解,将振型的超收敛解代入Rayleigh商,得到频率的超收敛解。该法作为后处理法,修复计算分别在各个单元上单独进行,故通过少量计算即能显著提高频率和振型的精度和收敛阶。数值算例显示该法稳定、高效,值得进一步研究和推广。     

弹性边界下圆弧拱的自由振动分析

工程中圆弧拱的边界不能总被简化为理想的简支或固支形式。为了研究弹性支承对圆弧拱自由振动特性的影响规律,将力、位移等变量无量纲化。根据平衡方程和坐标转换推导得出极坐标下圆弧拱在水平、竖直和转动方向支撑条件为弹性时的边界条件方程。并采用考虑弯曲和轴向变形而忽略剪切变形及转动惯量的自由振动的运动控制方程。运动方程在边界条件情况下,其解仅为关于矢跨比f,细长比s和无量纲刚度阵[K]的函数。采用Runge-Kutta法和行列式搜索法求解运动方程的特征值即无量纲频率Ωi以及特征向量即振型。通过计算发现,与理想支撑相比,弹性支承情况下细长比s对拱自振频率的影响要明显下降。理想支撑情况下圆弧拱的自振频率越高,则弹性支承对其自振频率的影响越小。与水平和竖向弹性支承相比,转动方向弹性支承仅对圆弧拱基频有较大影响。     

平面曲梁面内自由振动有限元分析的p型超收敛算法

该文提出一种求解平面曲梁面内自由振动问题的p型超收敛算法。该法基于有限元解答中频率和振型结点位移的固有超收敛特性,在单个单元上建立了振型近似满足的线性常微分方程边值问题,对该局部线性边值问题采用单个高次元进行有限元求解获得该单元上振型的超收敛解,逐单元计算完毕后,将振型的超收敛解代入Rayleigh商,获得频率的超收敛解。该法为后处理法,且后处理计算仅在单个单元上进行,通过少量计算即能显著提高频率和振型的精度和收敛阶。数值算例表明,该法可靠、高效,值得进一步研究和推广。     

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.     

Experimental dynamic analysis on naturally curved honeycomb sandwich beams with different damage patterns

In this study, dynamic analysis of naturally curved honeycomb sandwich beam including two surface cracks and an impact region at the facing skin is presented. Laminates of facing skin and backing skin are known as carbon fiber‐plain weave composite laminates with 1.6 mm in thickness. In the first part, in order to determine mechanical properties of both the skin with no‐crack/crack(s) and the honeycomb core of the composite beam, static tensile tests are conducted with respect to straingage measurement technique. In the second part, drop weight impact tests and vibration tests are performed to present the free vibration characteristics of the clamped‐clamped honeycomb sandwich beam including cracks and an impact‐damaged region. Corresponding to damage patterns of the sandwich beam, experimental dynamic analyses consist of six steps: (1) Vibration analysis with no‐crack and no‐impact region, (2) Vibration analysis with no‐crack and an impact region, (3) Vibration analysis with a surface crack and no‐impact region, (4) Vibration analysis with a surface crack and an impact region, (5) Vibration analysis with two surface cracks and no‐impact region, (6) Vibration analysis with two surface cracks and an impact region. For these purposes, an impact hammer with a force transducer is used to excite the undamaged or damaged naturally curved honeycomb sandwich beam through the selected points. After the excitation, the responses are obtained by an accelerometer. Resonant frequencies for the modal responses of the naturally curved honeycomb sandwich beam with different damage patterns are discussed.     

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.     

Free vibration analysis of uniform and stepped cracked beams with circular cross sections

This paper presents a novel numerical technique applicable to analyse the free vibration analysis of uniform and stepped cracked beams with circular cross section. In this approach in which the finite element and component mode synthesis methods are used together, the beam is detached into parts from the crack section. These substructures are joined by using the flexibility matrices taking into account the interaction forces derived by virtue of fracture mechanics theory as the inverse of the compliance matrix found with the appropriate stress intensity factors and strain energy release rate expressions. To reveal the accuracy and effectiveness of the offered method, a number of numerical examples are given for free vibration analysis of beams with transverse non-propagating open cracks. Numerical results showing good agreement with the results of other available studies, address the effects of the location and depth of the cracks on the natural frequencies and mode shapes of the cracked beams. Modal characteristics of a cracked beam can be employed in the crack recognition process. The outcomes of the study verified that presented method is appropriate for the vibration analysis of uniform and stepped cracked beams with circular cross section.     

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%.     

Probabilistic inverse analysis for predicting the distribution of multiple internal defects

A concept and a method of probabilistic inverse analysis are proposed for predicting the distribution of internal defects from their two-dimensional image on a cross-section. In this paper, circular (penny-shaped) cracks are chosen as a subject for analysis and a discussion is made on their statistical distribution. The spatial distribution of internal cracks, i.e. the number of cracks in a unit volume and the distribution of crack radius, is inversely predicted from the areal distribution of cracks observed on a cross-section, i.e. the number of cracks in a unit area and the distribution of crack length. When enough information on the areal distribution is given, for example, when a number of intercepted cracks can be observed on a cross-section for dense internal cracks, the spatial distribution of internal cracks is deterministically analyzed. Conversely, when the information is limited in the cross-sectional observation, a unique distribution of internal cracks cannot be determined. Then, a method of probabilistic inverse analysis is proposed. After the validity of the method is confirmed by numerical simulation, it is applied to actual internal cracks which initiate inside the specimens under a creep-fatigue condition. The distribution of internal cracks is successfully predicted.     

含裂纹损伤杆系结构的动态特性研究

运用动刚度有限元法,研究了含裂纹损伤杆系结构的动态特性。提出了一种含裂纹的杆单元,基于断裂力学的线弹簧模型,导出了相应的动刚度矩阵。在此基础上,对含裂纹的悬臂梁和平面框架进行了数值计算,并与已有的实验值和解析解进行了比较。结果表明:损伤位置和损伤程度的不同均会导致结构动态特性发生改变,因而在结构分析中应考虑损伤的影响;而该单元能够方便地用于含裂纹损伤杆系结构的动态特性分析,并具有很好的精度。     

Dynamics of anisotropic composite cantilevers weakened by multiple transverse open cracks

In this paper an exact solution methodology, based on Laplace transform technique enabling one to analyze the bending free vibration of cantilevered laminated composite beams weakened by multiple non-propagating part-through surface cracks is presented. Toward determining the local flexibility characteristics induced by the individual cracks, the concept of the massless rotational spring is applied. The governing equations of the composite beam with open cracks as used in this paper have been derived via Hamilton's variational principle in conjunction with Timoshenko's beam model. As a result, transverse shear and rotatory inertia effects are included in the model. The effects of various parameters such as the ply-angle, fiber volume fraction, crack number, position and depth on the beam free vibration are highlighted. The extensive numerical results show that the existence of multiple cracks in anisotropic composite beams affects the free vibration response in a more complex fashion than in the case of beam counterparts weakened by a single crack. It should be mentioned that to the best of the authors' knowledge, with the exception of the present study, the problem of free vibration of composite beams weakened by multiple open cracks was not yet investigated.