# Structural Damage Localization by the Principal Eigenvector of Modal Flexibility Change
## 2 Theory  
### The Principle of Deflection-Based Damage Localization  
In this section, the explicit relation between the defect and defect-induced deformation variation has been derived firstly from the theoretical investigations. Then, the basic principle of deflection-based algorithm for structural damage localization has been illustrated. For a \(n\)-DOFs structure, let \(K_{u}\) and \(K_{d}\) are the stiffness matrices of the intact structure, \(F_{u}\) and \(F_{d}\) are the flexibility matrices of the damaged structures. Then they will satisfy the following relationship:  
\[F_{u}\cdot K_{u}=F_{d}\cdot K_{d}=I_{n\times n} \tag{1}\]  
where \(I_{n\times n}\) is the identity matrix. Generally, structural defects reduce the stiffness and increase the flexibility. If \(\Delta K\) and \(\Delta F\) are the changes of stiffness and flexibility matrices, then one has:  
\[\Delta K=K_{u}-K_{d} \tag{2}\]  
\[\Delta F=F_{d}-F_{u} \tag{3}\]  
From Equations (1)-(3), one has  
\[F_{u}K_{u}=(F_{u}+\Delta F)(K_{u}-\Delta K) \tag{4}\]  
Equation (4) can be expanded as  
\[F_{u}K_{u}=F_{u}K_{u}-F_{u}\Delta K+\Delta F(K_{u}-\Delta K) \tag{5}\]  
From Equation (5), one has  
\[F_{u}\Delta K=\Delta F(K_{u}-\Delta K) \tag{6}\]  
Substituting Equation (3) into (6) yields  
\[F_{u}\Delta K=\Delta FK_{d} \tag{7}\]According to Equation (1), Equation (7) can be rewritten as  
\[\Delta F=F_{u}\Delta KF_{d} \tag{8}\]  
Multiplying Equation (8) by a load vector \(l\), the defect-induced deformation variation vector (DV) can be defined as  
\[dv=\Delta Fl=F_{u}\Delta KF_{d}l \tag{9}\]  
According to the FEM theory, the global stiffness change \(\Delta K\) can be expressed as the summation of all the elemental stiffness changes, _i.e._,  
\[\Delta K=\sum_{i=1}^{N_{e}}\alpha_{i}^{e}K_{i}^{e},\ (0\leq\alpha_{i}^{e}\leq 1) \tag{10}\]  
where \(\alpha_{i}^{e}\) and \(K_{i}^{e}\) are the damage coefficient and stiffness matrix of the \(i\)th element, \(N_{e}\) is the number of elements in the structural FEM. \(\alpha_{i}^{e}=\ 0\) denotes the corresponding element is intact, otherwise the element is damaged. According to Equations (9) and (10), we have  
\[\Delta K=K_{u}-K_{d} \tag{2}\]0  
According to Equation (11), the DV may be due to a single defect or multiple defects. For brevity, the single defect case is studied firstly in the following discussion. If only \(\alpha_{i}^{e}\
eq\ 0\), Equation (11) can be reduced to  
\[\Delta K=K_{u}-K_{d} \tag{2}\]1  
Let  
\[\Delta K=K_{u}-K_{d} \tag{2}\]2  
\[\Delta K=K_{u}-K_{d} \tag{2}\]3  
Then Equation (12) simplifies to  
\[\Delta K=K_{u}-K_{d} \tag{2}\]4  
Using the Linear algebra theory [32; 33], Equation (15) has important implications that \(dv\) is the linear combination of the column vectors in \(E_{i}\). From Equation (13), \(E_{i}\) is a sparse matrix because the elemental stiffness matrix \(K_{i}^{e}\) is very sparse. Assuming \(e_{i}^{j}\) and \(k_{i}^{j}\) are the \(j\)th nonzero column vector of \(E_{i}\) and \(K_{i}^{e}\), respectively, one has  
\[\Delta K=K_{u}-K_{d} \tag{2}\]5  
According to Equation (16), the physical meaning of the vector \(e_{i}^{j}\) is the deflection of the intact structure by considering \(k_{i}^{j}\) as a load vector. Thus, \(e_{i}^{j}\) is defined as the characteristic deflection and \(k_{i}^{j}\) is defined as the characteristic force of the \(i\)th element. It has been shown from Equation (15) that the deflection variation in a structure due to defect is a linear combination of the characteristic deflections for the damaged element. For the multiple-damage case, a similar conclusion can be obtained using the derivation as before. That is to say, structural deflection variation before and after damage is always the linear combination of the characteristic deflections for those damaged elements. Now we begin to discuss the properties of the characteristic force and deflection for structural elements. For convenience, the beam element is employed to illustrate these substantive features.