Research Of Structural Damage Detection Based On Generalized Flexibility Matrix

With the development of society and technology, modern engineering structures such as offshore platform, long-span bridges, tall buildings are being developed toward the direction of large-scale and complication. During the period of engineering structural service, on the one hand, engineering structures will be affected by various kinds of unexpected complex environment, such as storm, blizzard, freezing rain, earthquake, etc. On the other hand, after the engineering structures have been used for years, structures will be aging and fatigue. All of these can alter the structural strength and rigidity, and make structures damage. Structural damage seriously reduced structural security. In order to improve the security and reliability of engineering structures, to avoid dangerous accidents and to reduce the economic losses, it is very necessary to strengthen the research of structural damage identification methods.Engineering structural damage altered the structural properties. The purpose of the research on structural damage identification method is to judge structural damage according to the changes of structural characteristic. If there is structural damage, effective damage detection methods will be used to locate damage and further to evaluate the damage extent. According to the different classification standards, there are different classification methods about structural damage identification technology. The main methods may be classified as follows:local damage identification method and global damage identification method based on the different identification ranges; damage identification method based on dynamic test, static test and combination of dynamic test and static test according to the different testing methods; damaged identification method and undamaged method according to the different recognition means. This paper is mainly focused on global undamaged detection method based on dynamic test.Based on the structural vibration characteristics, a method of structural damage detection based on generalized flexibility matrix is studied in this paper. Generalized flexibility matrix is first put forward. Then generalized flexibility matrix and its sensitivity are used to detect the possible damaged elements. Compared with the original flexibility matrix method, the effect of truncating higher-order modes can be considerably reduced in the generalized matrix approach. When the degrees of freedom can be measured partially, dynamic expansion of mode shapes is used to cope with the incomplete modal data. Finally, an iterative algorithm using the changes in the first several natural frequencies only is proposed to update the damage extents of these possible damaged elements determined by the generalized matrix method. The effectiveness of the proposed method is verified by numerical examples.1. Generalized flexibility matrix methodIt is assumed that the structural damage affects the stiffness matrix only and that the mass matrix denoted as M remains constant. It is further assumed that the number of degrees of freedom after damage remains unchanged. Let Ku and Kd represent the n x n undamaged and damaged global structural stiffness matrices, respectively, and Kut be the n×n stiffness matrix for ith element of undamaged structures, AK be the change of the global stiffness matrix. Here, M,Ku,Kui,K.d and AK are all n×n real symmetrical matrix, n is the number of degrees of freedom. Assumptions above are suitable for the full text.The damaged structural stiffness matrix can be expressed as Referred to the finite element method, the change of the global stiffness matrix can be expressed as the sum of changes of the elemental stiffness matrices where N is the number of structural elements, a, is a scalar denoting the damage extent corresponding to the ith element (0≤α1≤1). Ifαj is equal to 0, the ith element is undamaged. Ifαi, is less than 1 or equal to 1, the ith element is partially or completely damaged. Substituting Eq. (2) into Eq. (1) yields Differentiating Eq. (3) with respect to a, results in 1.1 Flexibility matrix methodStructural flexibility matrix is defined as the inverse matrix of structural stiffness matrix. So the flexibility matrix and stiffness matrix for a damaged structure satisfies the following relation: where Fd is n×n flexibility matrix for the damaged structure, I is the n×n unit matrix.Differentiating Eq. (5) with respect toα, yields Post-multiplying Eq. (6) by results in Using Eq. (5) and rearranging Eq.(7) yield Substituting Eq. (4) into Eq.(8) results in Settingαl= 0(i= 1,...,N) yields where Fu is the n x n flexibility matrix for undamaged structure.Making use of Taylor's series expansion for Fd atαl,= 0(i=1,...,N), the first-order approximation to the flexibility matrix Fd can be expressed as Substituting Eq.(10) into Eq.(11) results inOn the other hand, the change of the flexibility matrix after damaged can be expressed as where AF is the change of the flexibility matrix. Thus Eq.(12) can be rearranged asFlexibility matrix can be expressed by frequency and corresponding mode shape which is normalized with respect to mass matrix, the expressions are given bywhereωui andφui are the ith frequency and the corresponding ith mode shape for the undamaged structure,ωdi andφdi are the ith frequency and the corresponding ith mode shape for the damaged structure. In practical engineering structures only lower mode shapes can be measured, higher mode shape is very difficult to obtain. So the change of the flexibility matrix can be approximately expressed as where m is the number of measured modes. The damage parameters can be calculated by manipulating Eqs.(14) and (17) into a system of linear equations with respect to a1. Solving this system of linear equations using the least squares method, both damage location and damage extent can be determined.1.2 Generalized flexibility matrix and the corresponding structural damage detection methodBased on mode shape normalization with respect to mass matrix M, namelyΦ'd MΦd= I, a new generalized flexibility matrix will be introduced in this paper. The generalized flexibility matrix is defined byfd2(α)= Fd(MFd)'=ΦdAd-1ΦdT(MΦdAd-1ΦdT)1-ΦdAd-1-1ΦdT,l-0,1,2,…(18) whereΦd and Ad are the mode shapes matrix and diagonal matrix of natural frequency squared, respectively, for the damaged structure. From the expression of Eq. (18), we can draw the conclusion that the larger l is, the smaller contribution of the higher order mode is. When l=0, it is reduced to the original flexibility matrix. In this paper, we consider the case of l=1 only.For l=1, the generalized flexibility matrix in Eq. (18) becomes Differentiating Eq. (19) with respect toαi yields Substituting Eq.(9) into Eq.(20) results in Thus atαl,= 0(i=1,…,N) the sensitivity of generalized flexibility matrix is given by Based on the first-order Taylor series expansion of fdg atαi=0(i=i,…,N), the first-order approximation to the generalized flexibility matrix fdg can be expressed as Substituting Eq.(22) into Eq.(23) yields The change△fg of the generalized flexibility matrix can be expressed as Utilizing Eq.(18), for l=1,△fg also can be approximately expressed as whereωdj andφdj are the j th frequency and its corresponding j th mode shape for the damaged structure,ωuj andφuj are the j th frequency and its corresponding j th mode shape for the undamaged structure, where m is the number of measured modes. From Eq.(26) we can see that the j th term of△fg is proportional to 1/ωdj4, while the j th term of AF is only proportional to 1/ωdj2,Therefore, compared with the original flexibility matrix based approach, the effect of the truncating higher-order modes can be considerably reduced in the proposed generalized flexibility method. Then damage parameters can be calculated by manipulating the matrix equations (25) and (26) into a system of linear equations onαi, (i=1,…, N). Solving the system of linear equations using the least squares method, both possible damage location and initial estimate of damage extents can be determined.2. Mode shape expansion schemeIn this paper, both measured mode shape and information of finite element model need be used in the structural damage detection method. Thus it requires that the number of degrees of freedom for the measured mode shape is same as the number of degrees of freedom for the analytical mode shape vector in finite element model. However it is very difficult to satisfy this condition. In practice, the number of degrees of freedom for measured mode shape is far smaller than the number of degrees of freedom for analytical mode shape vector due to the limited number of sensors. To solve this problem two approaches are usually adopted. One is to reduce the number of degrees of freedom of the finite element model; the other is to expand the number of degrees of freedom for measured mode shape. These two technologies are the same in essence. In this paper dynamic expansion scheme is introduced. Based on this method, the mode data for unmeasured degrees of freedom can approximately be expressed by those of the measured degrees of freedom. The complete mode shape vectors obtained through the expansion have the same dimension as the (complete) analytical mode shape vectors.Usually, the general eigenproblem of a full model with n degrees of freedom, referred to as full eigenproblem, is given by where K is the n×n stiffness matrix, M is the n×n mass matrix, andλ, andφI are the j th eigenvalue (square of frequency) and the corresponding eigenvector (mode shape), respectively. Assume that the total degrees of freedom of the full model are categorized as the measured degrees of freedom and the unmeasured degrees of freedom. They are simply indicated by m and s, respectively. With this arrangement, the eigenproblem (27) may be partitioned as The lower part of Eq. (28) is given by Utilizing the mode shape datφmj for the measured degrees of freedom, we can derived the mode shape dataφmj at the unmeasured degrees of freedom as follows The transformation matrix T between mode shape data for the measured and unmeasured degrees of freedom is thus given by T=-(Kss-λjMss)-1(Ksm-λjMsm) (31) The complete mode shape data iswhere I is unit matrix. In the following damage detection procedure, the transformation matrix T is composed of undamaged stiffness, mass matrix and corresponding eigenvalues. 3. Iteration procedureBoth possible damage locations and damage extents can be determined by generalized flexibility matrix method even the mode shape contaminated by noise or incomplete measured data. The damage locations are detected very accurately, but there are smaller or larger deviations in the damage extents. Because frequency is obtained much easier and more accurately than mode shape, in this paper an iterative procedure is proposed using several frequencies only to update the damage extents of the possible damaged elements exactly.Assume that the number of all the possible damaged elements is Nr, generally, Nr <