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.     

Stress intensity factors for cracked triangular cross‐section thin‐walled tubes

For one kind of finite‐boundary crack problems, the cracked equilateral triangular cross‐section tube, an analytical and very simple method to determine the stress intensity factors has been proposed based on a new concept of crack surface widening energy release rate and the principle of virtual work. Different from the classical crack extension energy release rate, the crack surface widening energy release rate can be defined by the G*‐integral theory and expressed by stress intensity factors. This energy release rate can also be defined easily by the elementary strength theory for slender structures and expressed by axial strains and loads. These two forms of crack surface widening energy release rate constitute the basis of a new analysis method for cracked tubes. From present discussions, a series of stress intensity factors are derived for cracked equilateral triangular cross‐section tubes. Actually, the present method can also be applied to cracked polygonal tubes.     

方形截面管横向裂纹的应力强度因子KI

利用裂纹张开能量释放率建立了一个求解方形截面管横向裂纹应力强度因子的一个方法。给出了方形截面管裂纹张开能量释放率的 G*-积分表征,以及和应力强度因子的关系。同时也给出了 G*-积分与载荷、几何参量以及机械性能参数的关系,进而得到方形截面管横向裂纹的应力强度因子。给出的方法不仅适用于一般箱形结构件的裂纹问题,也适用于其它有限边界多边管状结构的三维裂纹问题,过程极为简单。     

基于遗传神经网络与模态应变能的斜裂缝两阶段诊断方法

基于遗传神经网络与模态应变能,提出了一种斜裂缝两阶段诊断方法,识别梁体中斜裂缝的位置、角度和深度。根据线弹性断裂力学与虚功原理,推导了斜裂缝梁的单元刚度矩阵,得到了其频率与振型。采用遗传算法对BP神经网络的拓扑结构、权值和阈值进行优化,从而建立了遗传神经网络,用于识别梁体中斜裂缝的位置和角度;结合斜裂缝单元的模态应变能,通过对斜裂缝应力强度因子的积分,得到斜裂缝深度的解析表达式,用于识别斜裂缝的深度。数值仿真表明:能够高精度地诊断出梁体中斜裂缝的损伤状态,包括位置、角度和深度;与BP神经网络相比,遗传神经网络具有更强的泛化能力,且对测量噪声具有较好的鲁棒性。     

Crack identification in rotating shafts by coupled response measurements

In this paper a new method is applied in rotating cracked shafts to identify the depth and the location of a transverse surface crack. A local compliance matrix of different degrees of freedom is used to model the transverse crack in a shaft of circular cross section, based on available expressions of the stress intensity factors and the associated expressions for the strain energy release rates. It is known that when a crack exists in a structure, such as a beam, then the excitation in one-direction causes coupled vibrations in other directions. This property is used here to identify the crack. The shaft is modeled as a rotating Timoshenko beam including the gyroscopic effect and the axial vibration due to coupling. The method used here is based on the measurements of the axial vibration response due to different excitation frequencies and shaft revolutions. The figures presented are used to identify the crack.     

Numerical models verification of cracked tubular T, Y and K-joints under combined loads

This paper summarizes the key steps involved in the construction of an accurate and consistent finite element model for general cracked tubular T, Y and K-joints. The joint under consideration contains a surface crack which can be of any length and located at any position along the brace-chord intersection. Welding details along the brace-chord intersection, compatible with the American Welding Society (AWS) specifications [American Welding Society. Structural welding code-steel, ANSI/AWS, 15th ed., Miami, 2000], are included in the geometrical model. In order to develop a systematic and consistent modelling procedure, the whole process is divided into four key steps. They are, namely, (1) construction of a consistent geometrical model of the joint with welding details, (2) determination of cracked surface to define the semi-elliptical surface crack profile, (3) generation of well-graded finite element meshes, and (4) stress intensity factor studies around the crack front. To produce a well-graded finite element mesh, a sub-zone technique is used in the mesh generation whereby the entire structure is divided into several sub-zones with each zone consisting of different types of elements and mesh densities. The stress intensity factors (SIFs) are evaluated using the standard J-integral method. Two full-scale T and K-joint specimens were tested to failure under axial load (AX), in-plane bending (IPB), and out-of-plane bending (OPB). In the tests, the rate of crack propagation was monitored carefully using the alternating current potential drop (ACPD) technique. Using the known material parameters C and m, the experimental SIFs were obtained, and they are found to be in complete agreement with the computed SIFs obtained from the generated models. Hence, the proposed finite element models are both efficient and reliable.     

Averaged strain energy density estimated rapidly from nodal displacements by coarse FE analyses: Cracks under mixed mode loadings

The strain energy density (SED) averaged over a structural volume allows to assess the fracture and fatigue behaviour of cracked components under mixed‐mode loadings. To rapidly estimate the averaged SED, two approaches have been previously proposed: (a) the direct approach; (b) the peak stress method (PSM) and nodal stress (NS) approach. In this paper, the nodal displacement (ND) approach is presented to rapidly estimate the averaged SED from the nodal displacements by FE analyses. This method combines the advantages of all previous approaches, ie, the use of coarse meshes able to capture the contributions of both stress intensity factors (SIFs) and higher order terms, without requiring geometrical modelling of the control volume. To validate the method, cracked plates and cracked bars under mixed‐mode loadings have been analysed. The averaged SED values estimated by the ND approach agree with those calculated by the direct approach, within a range of applicability.     

聚合物梯度材料黏弹性断裂的双控制参数

针对组分材料体积分数任意分布的聚合物功能梯度材料,研究其在蠕变加载条件下Ⅰ型裂纹应力强度因子（SIFs）和应变能释放率的时间相依特征。由Mori-Tanaka方法预测等效松弛模量,在Laplace变换域中采用梯度有限元法和虚拟裂纹闭合方法计算断裂参数,由数值逆变换得到物理空间的对应量。分析边裂纹平行于梯度方向的聚合物功能梯度板条,分别考虑均匀拉伸和三点弯曲蠕变加载。结果表明,聚合物梯度材料应变能释放率随时间增加,其增大的程度与黏弹性组分材料体积分数正相关;材料的非均匀黏弹性性质产生应力重新分布,导致裂纹尖端应力场强度随时间变化,当裂纹位于黏弹性材料含量较低的一边时,应力强度因子随时间增加,反之,随时间减小。而且,材料的应力强度因子与时间相依的变化范围和体积分数分布以及加载方式有关,当体积分数接近线性分布时,变化最明显,三点弯曲比均匀拉伸的变化大。SIFs随时间的延长增加或减小、加剧或减轻裂纹尖端部位的“衰坏”,表明黏弹性功能梯度裂纹体的延迟失稳需要联合采用应力强度因子与应变能释放率作为双控制参数。     

Analytical calculation of stress intensity factor of cracked steel I-beams with experimental analysis and 3D digital image correlation measurements

An analytical method to calculate the stress intensity factor for cracked steel I-beams under both bending moment and axial load is presented. The method is based on the approach of crack surface widening energy release rate. The crack surface widening energy release rate is formulated by a G^∗-integral and elementary strength theory of materials. Comparisons between the analytical results and results available in the literature for specific cases demonstrate the validity of the methodology. Furthermore, the fatigue and fracture behavior of the steel I-beam are experimentally investigated. The fatigue crack growth rate, residual deflection and stiffness reduction of a cracked beam under cyclic loading are studied. A three-dimensional digital image correlation system is used to illustrate the stress evolution pattern and the plasticity zone around the crack tip using image processing technique, thereby providing further verification of the theoretical models.     

Cracks emanating from a square hole in rectangular plate in tension

A numerical analysis of cracks emanating from a square hole in a rectangular plate in tension is performed using a hybrid displacement discontinuity method (a boundary element method). Detailed solutions of the stress intensity factors (SIFs) of the plane elastic crack problem are given, which can reveal the effect of geometric parameters of the cracked body on the SIFs. By comparing the calculated SIFs of the plane elastic crack problem with those of the centre crack in a rectangular plate in tension, in addition, an amplifying effect of the square hole on the SIFs is found. The numerical results reported here also prove that the boundary element method is simple, yet accurate, for calculating the SIFs of complex crack problems in finite plate.     

DBEM and FEM analysis on non-linear multiple crack propagation in an aeronautic doubler-skin assembly

The performance of a riveted patch repair, applied on a cracked panel, is simulated by using both a commercially available boundary element code (BEASY) and a finite element code (ANSYS). A two-dimensional stress analysis on a single-sided repaired configuration is performed by both methodologies; consequently, the occurrence of out-of-plane bending and its effect on the through-thickness stress intensity factor (SIF) variation is neglected. The connection between the two layers (patch and panel) is realised by 32 rivets, with through-cracks initiated on the most loaded holes. Special elements are used to model the crack: discontinuous elements in the dual boundary element method (DBEM) approach or quarter point elements in the finite element method (FEM) approach. Different loading configurations are considered depending on the presence of a biaxial or uniaxial remote load and the non-linear hole/rivet contact is simulated by gap elements. The most stressed skin holes are highlighted, and the effect of a through crack from such holes is analysed in terms of SIFs and stress redistribution. The accuracy in SIFs assessment by DBEM and FEM and the respective computational and pre-processing efforts are determined. Such a two-dimensional analysis allows us a straightforward pre-processing phase, and very short run times are needed. A peculiar arrangement of the pin configuration in the DBEM analysis allows us to take into account the real in-plane plate stiffness and the transversal pin stiffness, even in a 2D analysis (this is straightforward by using FEM).     

基于摄动法的多条裂纹欧拉梁特征模态分析

基于摄动理论推导了带多条开口裂纹的欧拉梁的特征模态参数的理论计算公式。采用最直接的方式将梁开口裂纹模拟成梁微段内的横截面折减并用δ函数表达了带开口裂纹的梁沿轴线的截面惯矩和线质量等物理参数。基于此,建立了裂纹梁动力微分方程,并采用一阶摄动理论推导得到了梁的模态频率和振型计算公式。简支梁及悬臂梁算例研究表明,该方法具有很好的精度,与有限元模拟结果及实验结果都能很好地吻合。并采用此方法分析了裂纹深度和位置对带多条开口裂纹梁的特征模态参数的影响。结果表明,裂纹对各阶模态频率虽然影响有限,但其引起的各阶频率变化有着明显的模式,可用于结构损伤定位;裂纹对模态振型影响不明显,但对模态曲率影响比较大,可用于结构损伤位置和程度的诊断。     

Stress intensity factors for cracked T-sections and dynamic behaviour of T-beams

This paper presents an extension of a simple and convenient method proposed by Kienzler and Herrmann [An elementary theory of defective beams. Acta Mech 1986;62:37-46] to estimate the stress intensity factors of cracked beams and bars. This method is based on an elementary beam theory estimation of the strain energy release as the crack is widened into a fracture band. As an extension, the power of the simple beam theory analysis is demonstrated by application to cracked T-beams subjected to a bending moment, shear forces and a torsion. Moreover, the present work addresses the coupled bending-torsional vibration of cracked T-beams within the context of the dynamic stiffness matrix method of analysing structures.     

基于小波有限元的悬臂梁裂纹识别

研究了悬臂梁裂纹识别中的正反问题．即通过裂纹位置和尺寸求解梁的固有频率以及利用梁的固有频率．识别裂纹位置和尺寸。以矩形截面裂纹悬臂梁为例，利用小波有限元方法建立了梁自由振动的有限元模型．其中裂纹被看作为一刚度已知的扭转线弹簧，求解出了系统的固有频率；通过行列式变换，将反问题求解简化为只含线弹簧刚度一个未知数的一元二次方程求根问题，分别做出了以不同固有频率作为输入值时裂纹位置与弹簧刚度之间的解曲线，曲线交点预测出裂纹的位置与尺寸。数值算例证实了算法的有效性，为工程结构裂纹故障预示与诊断提供了新的方法。     

直斜裂纹转轴的时变刚度特性研究*

本文采用有限元方法建立了裂纹转子系统的动力学方程,利用应变能释放率方法得到了裂纹单元的刚度矩阵,采用应力强度因子为零法模拟裂纹的呼吸效应,详细研究了不同深度的直裂纹和45°斜裂纹转子,在一个稳态旋转周期内,裂纹开闭规律以及转轴刚度时变特性。研究表明裂纹深度的增大使裂纹转轴的刚度变化增大,直裂纹与斜裂纹转轴的刚度变化特性具有明显差异,斜裂纹引起更多与更强的耦合振动,使转子的动力学性能更复杂。     

Evaluation of fracture parameters of composites subjected to thermal shock using the boundary element method and sensitivity analysis techniques

Quasi-static crack extension in fiber-reinforced composites subjected to thermal shock is analyzed using the boundary integral equation method, in combination with sensitivity analysis techniques. Buekner's formulation is employed to evaluate the stress intensity factor in a cracked body. This method eliminates the need for special element types to model the crack tip, as well as the use of a large number of elements near the cracked zone of interest. A numerical procedure involving sensitivity analysis techniques based on the adjoint structure approach has been developed to evaluate the energy integrals in the cracked body. Gradients of the functionals of response quantities with respect to variables such as the crack length, necessary for the evaluation of fracture parameters, are determined directly by this method. The numerical differentiation used in other numerical methods, such as the finite element method, which requires the repeated solution of the equations for different crack sizes is avoided. Results for stress intensity factors as a function of crack length are presented for various composite systems. These results are in good agreement with analytical results and results from the finite element method. The present approach results in significantly improved computational efficiency.     

Stress intensity factors of square crack inclined to interface of transversely isotropic bi-material

This paper analyzes a square crack in a transversely isotropic bi-material solid by using dual boundary element method. The square crack is inclined to the interface of the bi-material. The fundamental solution for the bi-material solid occupying an infinite region is incorporated into the dual boundary integral equations. The square crack can have an arbitrary angle with respect to the plane of isotropy of the bi-material occupying either finite or infinite regions. The stress intensity factor (SIF) values of the modes I, II, and III associated with the square crack are calculated from the crack opening displacements. Numerical results show that the properties of the anisotropic bi-material have evident influences on the values of the three SIFs. The values of the three SIFs are further examined by taking into account the effect of the external boundary of the internally cracked bi-material.     

Cracks emanating from circular hole or square hole in rectangular plate in tension

A numerical analysis of cracks emanating from a circular hole (Fig. 1) or a square hole (Fig. 2) in rectangular plate in tension is performed by means of the displacement discontinuity method with crack-tip elements (a boundary element method) presented recently by the author. Detail solutions of the stress intensity factors (SIFs) of the two plane elastic crack problems are given, which can reveal the effect of geometric parameters of the cracked bodies on the SIFs. By comparing the SIFs of the two crack problems with those of the center crack in rectangular plate in tension (Fig. 3), in addition, an effect of the circular hole or the square hole on the SIFs of the center crack is discussed in detail. The numerical results reported here also illustrate that the boundary element method is simple, yet accurate for calculating the SIFs of complex crack problems in finite plate.     

Finite Element Weight Function Application for a Cracked Disk

A practical application of the weight function method is used in this paper in order to find values of the stress intensity factor for a cracked disk subjected to different loadings. The finite element method is used in order to obtain discrete values of the crack face displacement in a reference loading case, namely inertia forces due to uniform rotation. These values are interpolated and a general expression of the displacements is obtained, which is further used to determine the stress intensity factor in this case. With the weight function equation, stress intensity factors for other loadings are obtained and the results are compared with those reported by other authors. Very good agreement was obtained, showing thus the reliability of this approach.     

Probabilistic fracture mechanics analysis of linear-elastic cracked structures by spline fictitious boundary element method

The inherent uncertainties in crack geometry, material properties and loadings have large influence on fracture response characteristics of cracked structures. This paper presents the probabilistic fracture mechanics analysis of linear-elastic cracked structures subjected to mixed-mode loading conditions using the spline fictitious boundary element method (SFBEM). The response surface method (RSM) is used to predict the fracture probability of the cracked structure. To determine the unknown coefficients of the response surface function, the SFBEM based on the Erdogan fundamental solutions for infinite cracked plates is adopted to perform deterministic analyses of stress intensity factors (SIFs) corresponding to different test points with given parameters. Numerical examples based on mode-I and mixed-mode crack problems are presented to illustrate the present method. The results show that the predicted failure probability obtained by the present approach is accurate in comparison with the Monte Carlo simulation (MCS) results. Since a much lesser number of numerical tests are required in RSM as compared with that needed in MCS, and since the SFBEM based on the Erdogan fundamental solutions has been used to conduct the numerical tests, reliability analysis of cracked structures can be performed efficiently using the present method.