Detection of crack location using cracked beam element method for structural analysis

The local flexibility introduced by cracks changes the dynamic behavior of the structure and, by examining this change, crack position and magnitude can be identified. In order to model the structure for FEM analysis, a special finite element for a cracked Timoshenko beam is developed. Shape functions for rotational and translational displacements are used to obtain the consistent mass matrix for the cracked beam element. Effect of the crack on the stiffness matrix and consistent mass matrix is investigated. Proposed is a procedure for identifying cracks in structures using modal test data.     

Transient wave analysis of a cantilever Timoshenko beam subjected to impact loading by Laplace transform and normal mode methods

This study applies two analytical approaches, Laplace transform and normal mode methods, to investigate the dynamic transient response of a cantilever Timoshenko beam subjected to impact forces. Explicit solutions for the normal mode method and the Laplace transform method are presented. The Durbin method is used to perform the Laplace inverse transformation, and numerical results based on these two approaches are compared. The comparison indicates that the normal mode method is more efficient than the Laplace transform method in the transient response analysis of a cantilever Timoshenko beam, whereas the Laplace transform method is more appropriate than the normal mode method when analyzing the complicated multi-span Timoshenko beam. Furthermore, a three-dimensional finite element cantilever beam model is implemented. The results are compared with the transient responses for displacement, normal stress, shear stress, and the resonant frequencies of a Timoshenko beam and Bernoulli–Euler beam theories. The transient displacement response for a cantilever beam can be appropriately evaluated using the Timoshenko beam theory if the slender ratio is greater than 10 or using the Bernoulli–Euler beam theory if the slender ratio is greater than 100. Moreover, the resonant frequency of a cantilever beam can be accurately determined by the Timoshenko beam theory if the slender ratio is greater than 100 or by the Bernoulli–Euler beam theory if the slender ratio is greater than 400.     

Investigation of the size effects in Timoshenko beams based on the couple stress theory

In this paper, a size-dependent Timoshenko beam is developed on the basis of the couple stress theory. The couple stress theory is a non-classic continuum theory capable of capturing the small-scale size effects on the mechanical behavior of structures, while the classical continuum theory is unable to predict the mechanical behavior accurately when the characteristic size of structures is close to the material length scale parameter. The governing differential equations of motion are derived for the couple-stress Timoshenko beam using the principles of linear and angular momentum. Then, the general form of boundary conditions and generally valid closed-form analytical solutions are obtained for the axial deformation, bending deflection, and the rotation angle of cross sections in the static cases. As an example, the closed-form analytical results are obtained for the response of a cantilever beam subjected to a static loading with a concentrated force at its free end. The results indicate that modeling on the basis of the couple stress theory causes more stiffness than modeling by the classical beam theory. In addition, the results indicate that the differences between the results of the proposed model and those based on the classical Euler–Bernoulli and classical Timoshenko beam theories are significant when the beam thickness is comparable to its material length scale parameter.     

含双裂纹圆截面悬臂梁的损伤识别研究

裂纹的萌生和扩展直接影响构件的振动响应,对构件的安全可靠性具有重要影响．本文以圆截面悬臂梁为对象,结合转角模态振型和模态频率等高线,研究了一种双裂纹识别技术．首先,基于应力强度因子和卡氏定理推导了无裂纹梁单元和含裂纹梁单元的刚度矩阵;在此基础上,建立了含裂纹圆截面悬臂梁的有限元动力学方程;然后,结合裂纹对梁转角模态振型和模态频率的影响,提出了双裂纹识别策略．最后,通过算例讨论了双裂纹识别策略的可行性．结果表明,圆截面悬臂梁的模态转角在裂纹位置出现突变,裂纹深度越大转角突变值越大;将识别出的裂纹位置作为已知参数,通过模态频率等高线法,可以准确地识别出双裂纹的深度．     

工程结构的随机特征问题研究及其在梁结构中的应用

采用子结构模态综合和摄动随机有限元相结合求解工程结构的随机特征问题。为求出随机特征对的方差，借助于模态截断概念推出诸特征值与特征向量对随机变量的偏导数。以染结构为典型算法，定量研究了子结构动模态的选取个数与随机特征对的计算精度间关系，以梁的长细比首次确定使用Timoshenko梁和Euler-Bernoulli梁两模型求解梁类工程结构随机特征问题的适用范围。     

Exact closed-form solutions for the static analysis of multi-cracked gradient-elastic beams in bending

Cracks and other forms of concentrated damage can significantly affect the performance of slender beams under static and dynamic loads. The computational model for such defects often consists of a localised reduction in the flexural stiffness, which is macroscopically equivalent to a beam where the undamaged parts are hinged at the position of the crack, with a rotational spring taking into account the residual stiffness (“discrete spring” model). It has been recently demonstrated that this model is equivalent to an inhomogeneous Euler–Bernoulli beam in which a Dirac’s delta is added to the bending flexibility at the position of each damage (“flexibility crack” model). Since these models concentrate the increased curvature at a single abscissa, a jump discontinuity appears in the field of rotations. This study presents an improved representation of cracked slender beams, based on a general class of gradient elasticity with both stress and strain gradient, which allows smoothing the singularities in the flexibility crack model. Exact closed-form solutions are derived for the static response of slender gradient-elastic beams in flexure with multiple cracks, and the numerical examples demonstrate the effects of the nonlocal mechanical parameters (i.e. length scales of the gradient elasticity) in this context.     

基于多层模拟方法的双层梁断裂有限元分析

本文提出一种用于含分层的双层梁线弹性断裂分析的有限元方法．将上下子梁均模拟为多个子层,采用只有平动位移自由度的新型梁单元,假设单元内的位移沿纵向和横向均线性变化,推导了该单元的单元刚度矩阵．将开裂部分和未开裂部分的子梁进行单元刚度矩阵组装,施加相应的等效结点力,得到整体平衡方程,并结合边界条件进行求解．为验证该方法的有效性和精度,开展非对称双悬臂梁(Asymmetric Double Cantilever Beam, ADCB)和单臂弯曲梁(Single Leg Bending, SLB)试样的断裂分析,利用虚拟裂纹闭合技术(Virtual Crack Closure Technique, VCCT)得到了试样的能量释放率及其分量,并将求得的结果与解析解和二维有限元解进行对比．计算结果表明,相对于传统双层模拟方法,该多层模拟方法能够精确、高效地计算各类梁试样的能量释放率及其分量,并且无需引入界面连续条件．     

基于等效黏性弹簧的黏弹性Timoshenko裂纹梁弯曲解析解

考虑裂纹的缝隙和黏性效应,将梁中横向裂纹等效为黏弹性扭转弹簧,利用广义Delta函数,给出了Laplace变换域内裂纹梁的等效抗弯刚度,得到了具有任意开闭裂纹数目且满足标准线性固体黏弹性本构的Timoshenko梁在时间域内的弯曲变形显式解析通解．在此基础上,通过两个数值算例,分析了时间、梁跨高比和裂纹深度等参数对黏弹性Timoshenko开裂纹梁弯曲变形的影响．结果表明：裂纹黏性对Timoshenko裂纹梁的弯曲具有显著的影响．相比于裂纹的弹性扭转弹簧模型,考虑裂纹黏性效应的黏弹性Timoshenko裂纹梁在裂纹处挠度尖点和转角跳跃现象十分明显．另外,由于横向剪切引起的附加变形,Timoshenko裂纹梁的稳态挠度与Euler-Bernoulli梁挠度的差值为常数,其大小与裂纹模型、梁跨高比或裂纹深度无关,这些结果对梁裂纹无损检测具有指导意义．     

基于裂纹附加模态Timoshenko梁的裂纹损伤识别

将梁中横向裂纹等效为无质量扭转弹簧，并忽略其对梁剪切变形的影响，得到的具有任意裂纹数目Timoshenko 梁自振模态的统一显示解析表达式．将裂纹梁的自振模态分为基本模态和裂纹附加模态，利用最小二乘拟合，建立了利用裂纹附加模态函数的梁裂纹损伤识别方法．通过数值模拟开展了简支单裂纹梁以及悬臂和固支双裂纹梁等的裂纹损伤识别，考察了测量误差对损伤识别的影响，数值结果表明本文所提出的裂纹损伤识别方法对裂纹位置的识别精度高于对裂纹损伤程度的识别精度；随着测量误差的增加，裂纹位置及裂纹损伤程度的识别误差增加，但仍在可接受的范围内，故该裂纹损伤识别方法在实际工程中具有一定的应用价值．     

The effect of transverse normal strain in contact of an orthotropic beam pressed against a circular surface

The contact problem of a straight orthotropic beam pressed onto a rigid circular surface is considered using beam theories that account for transverse shear and transverse normal deformations. The circular nature of the rigid surface emphasizes the difference between Euler Bernoulli theory behavior, where point loads develop at the edge of contact, and the higher order theories that predict non-singular pressure distributions. While Timoshenko beam theory is the simplest theory that addresses this behavior, the prediction of a maximum value of pressure at the edge of contact contradicts the elasticity theory result that contact pressure must drop to zero. Transverse normal strain is therefore introduced, both to study this fundamental discrepancy and to include an important effect in many contact problems. To investigate this effect, higher order beam theories that account for both constant and linear transverse normal strain through the beam thickness are derived using the principle of virtual work. The resulting orthotropic beam theories depend on the bending stiffness (EI), shear stiffness (GA), axial stiffness (EA₁) and transverse normal stiffness (EA₂), which are independent stiffness parameters that can differ by orders of magnitude. The above mentioned contact problem is then solved analytically for these theories, along with the Timoshenko beam model which assumes zero transverse normal strain. The results for different orthotropic materials show that inclusion of transverse normal deformation has a significant effect on the contact pressure solution. Furthermore, the solution using higher order beam theories encompasses the two extremes of a Hertz-like contact pressure when the half contact length is smaller than the thickness of the beam, and the Timoshenko beam theory case when the half contact length is much larger than the thickness. Concerning the behavior of the pressure at the edge of contact, adherence to the boundary conditions required by the principle of virtual work, shows that while the pressure does tend to zero, it does not become zero unless artificially enforced. In this regard the solution for the case of linear strain is better than that for constant strain. All beam solutions are validated with plane elasticity solutions obtained using the commercial finite element software ABAQUS.     

在线振动监测与故障诊断的一种新途径

文中由应力强度因子导出含裂纹单元的刚度矩阵,从而提出了一种有限元法.将该法用于单边裂纹悬臂梁,计算了应变响应.用时域法识别了模态参数,并考虑了裂纹闭合的影响.分析了应变响应对故障的敏感度,计算结果表明,应变响应相对于位移响应而言,对故障更敏感.本文还提出了一种判别函数,该函数对于损伤检测,比现有几种判别函数更有效.最后,给出了一种在线振动监测与故障诊断新方法.     

A new approach to on-line vibration monitoring and fault diagnosis

This paper presents a finite element method by deriving the stiffness matrix of cracked element from the stress intensity factor. The method can be used to evaluate the strain response of the cantilever beam with a single edge crack. By using the method of identification in time domain, the modal parameters are identified and at the same time the effect of crack closure is taken into consideration. Analysis has also been made of the sensitivity of the strain to the faults. Test results show that the strain response is more sensitive to faults than the displacement response. Then, a distinguishing function is introduced, which is more effective than any other existing ones in failure detection. Finally a new approach to on-line vibration monitoring and fault diagnosis is given.     

AN APPROXIMATE ANALYSIS OF FREQUENCY RESPONSE OF A CRACKED HINGEDHINGED BEAM

A new method based on a modified line-spring model is developed forevaluating the natural frequencies of vibration of a cracked beam.This model inconjunction with the Euler-Bernoulli beam theory,modal analysis and linear elasticfracture mechanics is applied to obtain an approximate characteristic equation of acracked hinged-hinged beam.By solving this equation the natural frequencies aredetermined for different crack lengths in different positions.The results show goodagreement with the solutions through finite element analysis.The present method maybe extended to analyze other cracked complicated structures with various boundaryconditions.     

AN IMPEDANCE ANALYSIS FOR CRACK DETECTION IN THE TIMOSHENKO BEAM BASED ON THE ANTI-RESONANCE TECHNIQUE

An alternative technique for crack detection in a Timoshenko beam based on the first anti-resonant frequency is presented in this paper.Unlike the natural frequency,the anti-resonant frequency is a local parameter rather than a global parameter of structures,thus the proposed technique can be used to locate the structural defects.An impedance analysis of a cracked beam stimulated by a harmonic force based on the Timoshenko beam formulation is investigated.In order to characterize the local discontinuity due to cracks,a rotational spring model based on fracture mechanics is proposed to model the crack.Subsequently,the proposed method is verified by a numerical example of a simply-supported beam with a crack.The effect of the crack size on the anti-resonant frequency is investigated.The position of the crack of the simply-supported beam is also determined by the anti-resonance technique.The proposed technique is further applied to the"contaminated"anti-resonant frequency to detect crack damage,which is obtained by adding 1-3% noise to the calculated data.It is found that the proposed technique is effective and free from the environment noise.Finally,an experimental study is performed,which further verifies the validity of the proposed crack identification technique.     

梁结构中裂纹参数识别方法研究

以等效弹簧模型来模拟裂纹引起的局部软化效应,将该模型同Bernoulli-Euler梁理论、模态分析方法以及断裂力学原理等结合起来,利用传递矩阵法导出含裂纹梁振动的各种边界条件下的特征方程通解。借助于特征方程,提出两种识别裂纹深度和位置参数的数值方法,最后,通过对含裂纹悬臂梁的分析说明文中方法的有效性。     

含裂纹梁参数的谐响应识别

主要研究含裂纹梁在简谐激励作用下的动力特性,提出一种依据幅值变化对裂纹参数进行识别的新方法。首先,在振动过程中考虑裂纹的呼吸特性,以悬臂梁为例建立含裂纹梁的二维有限元模型,指出在一般激励频率下,其对应的幅值均是明显信号,可用来描述裂纹梁的动力特性。其次,当激励频率分别取无裂纹梁一阶频率的1/4和二阶频率的1/4时,幅值随裂纹参数的变化明显不同,可依据响应幅值的变化对裂纹参数进行识别。然后,利用曲面拟合技术绘出幅值变化曲面,对未知参数的裂纹进行识别,验证了该方法的有效性,并归纳出利用幅值变化对任意裂纹参数进行识别的基本步骤。最后,针对无裂纹梁频率计算可能存在误差的情况,分析识别方法的鲁棒性,结果显示即使最大误差为10%,该方法也能对裂纹参数进行准确识别。     

不确定边界条件下悬臂梁裂纹参数识别方法研究

基于Bernoulli-Euler梁振动理论,以等效弹簧来模拟裂纹引起的局部软化效应和由非完全固支边界条件引起的转角效应。推导了悬臂梁在不确定边界条件下确定其振动频率的特征方程,直接利用该特征方程,提出一种有效估计裂纹参数的优化方法,通过计算测量频率和理论频率之间的误差目标函数最小化即可识别裂纹参数-裂纹位置和深度。最后,应用两个实例-理想固支边界条件下和非完全固支边界条件下的悬臂梁实验来说明本文方法的有效性。实验结果表明:只需梁结构前三阶频率即可识别裂纹位置和深度。对于理想边界条件下的裂纹参数识别,在测量频率存在小误差情况下,该方法仍能给出比较满意的结果,对于非完全固支边界条件下的裂纹参数识别,利用本文方法能得到比Narkis的方法更精确的裂纹位置识别结果。同时本文方法还能给出比较满意的裂纹深度识别结果。     

Modelling a micro-cantilever vibrating in vacuum,gas or liquid under thermal base excitation

The dynamic behaviour of a micro-cantilever that is transversely excited at its base is investigated in this paper. The base actuation is provided by thermal cycles via taking the advantage of thermal expansion. The Euler–Bernoulli equation along with corresponding boundary conditions is used to model the continuous cantilever beam. The resultant boundary value problem takes into account the thermal expansion and stiffness of the actuator at the base as well as the effect of the surrounding gas or liquid. A closed-form analytical model is developed to compute natural frequencies, mode shapes, and harmonic response of the vibrating cantilever, in addition to an integral function for quality factor. The model is validated via a finite element (FE) analysis using ANSYS commercial package. This validation shows that the proposed model can properly predict the cantilever's vibrating behaviour.     

Modal analysis of cracked continuous Timoshenko beam made of functionally graded material

Abstract

The article addresses development of the Transfer Matrix Method (TMM) for free vibration of cracked continuous Timoshenko beam made of Functionally Graded Material (FGM). The governing equations of free vibration are established for the beam based on the power law of material grading, actual position of neutral plane and double spring model of crack. There is conducted frequency equation of the beam with intermediate rigid supports using the TMM after the transverse displacements at rigid supports have been disregarded. Therefore, the frequency equation is simplified and becomes more useful to compute natural frequencies of continuous FGM Timoshenko beam with a number of cracks. The obtained numerical results show the essential effect of cracks, material properties and also number of spans on natural frequencies of the beam.     

A three-dimensional edge-crack finite element for fracture mechanics applications

A three-dimensional extension of a previously published two-dimensional cracked finite element [Potirniche, G.P., Hearndon, J., Daniewicz, S.R., Parker, D., Cuevas, P., Wang, P.T., Horstemeyer, M.F., 2008. A two-dimensional damaged finite element for fracture applications. Engineering Fracture Mechanics 17(13), 3895–3908] is presented in this paper. The new element has an embedded edge crack, and was developed to model damage in three-dimensional structures using the finite element method. The element simulates the presence of a crack without physically inserting it in the three-dimensional finite element mesh. The method involves the derivation of a modified stiffness matrix that accounts for the change in the element flexibility due to the crack presence. The cracked element was analytically formulated and implemented in the finite element code ABAQUS Standard as a User-defined Element (UEL) subroutine. Tests of various cracked beam configurations were used to estimate the accuracy of the element by comparing two models: one with a UEL and another with an embedded edge crack. Beam deflections and natural frequencies were analyzed and compared for the two models. The results indicate that the new element has a good potential in modeling cracks in three-dimensional parts. Moreover, the method using this UEL computes the global response of damaged structures, in which cracks can be placed at various locations and in an unlimited number.