# Structural Damage Localization by the Principal Eigenvector of Modal Flexibility Change
## 2 Theory
### The Principle of Deflection-Based Damage Localization
Figure 1 presents a Bernoulli-Euler plane beam element with two nodes giving four DOFs, whose stiffness matrix and nodal displacement vector in local co-ordinates are as follows:  
\[\Delta K=K_{u}-K_{d} \tag{2}\]6\[K^{e}=\frac{EI}{L^{3}}\left[ \begin{array}{cccc}12&6L&-12&6L\\ 6L&4L^{2}&-6L&2L^{2}\\ -12&-6L&12&-6L\\ 6L&2L^{2}&-6L&4L^{2}\end{array} \right] \tag{18}\]  
where \(E\) is the elastic modulus, \(I\) is the moment of inertia, and \(L\) is the element length. The four column vectors of \(K^{e}\) in Equation (18) are the characteristic forces for the Bernoulli-Euler plane beam element in local co-ordinates. Figure 2 shows the corresponding load configurations of these characteristic forces. From Figure 2, it is apparent that all the load configurations are the self-equilibrating force systems. This conclusion is also valid for other types of finite elements. As can be seen in Figure 3, the internal  
Figure 1: A Bernoulli-Euler plane beam element  
Figure 2: Load configurations of the four characteristic forces. force (IF) in most parts of the structure under a characteristic force will be zero, except the element associated with this characteristic force. In short, one can conclude from Figures 2 and 3 that: (1) the characteristic force is a self-equilibrating force; (2) the characteristic force acts only on its own element and not on the rest of the structure. When the characteristic force is applied to the structure without redundant constraints (_i.e._, the statically determinate structure), only the element associated with the characteristic force is deformed, whereas the remaining part of the structure is not deformed and only has rigid body motion. That is to say, the elemental characteristic deflection will, due to its characteristic force, consist of several parts of the rigid displacement, but not the element itself. As stated before, structural deflection change due to defects is a linear combination of the characteristic deflections corresponding to those damaged elements. Thus, the deflection change of the structure without redundant constraints under an arbitrary load will consist of several parts of the rigid displacement but not the damaged segments. In other words, the turning points between each segment of the rigid displacement in the shape of the deflection change for the structure are the locations of the defects. For the statically indeterminate structure, the derivation process is the same as that of the statically determinate structure. The only difference is that the IF in some parts of the statically indeterminate structure under a characteristic force will be slightly greater than zero because of the redundant constraint limitation. Correspondingly, the deflection variation due to damage will consist of several parts of the approximate rigid displacement (or the rigid displacement) but not the damaged segments. Then, the turning points between each segment of the approximate rigid displacement (or the rigid displacement) in the deflection change shape are the defect locations. As a result, the sharp peaks in the curvature of structural deflection variation will indicate the locations of structural defects. Figure 3: The internal force (IF) distributions of the structure under characteristic forces.