## 2 The Expression of the Cracked Beam Deflection  
This section presents a method for determining the deflection at the free end of a cantilever beam with a crack. The challenge faced when attempting to evaluate damages is that the effect of the crack, both on the deflection as well as on the natural frequencies, is different when it is placed in different positions along the beam. However, the crack has the biggest effect when it is in the beam slice in which the mechanical stresses are highest, i.e., where the bending moment reaches its maximum value. In the case of the cantilever beam, this location is the fixed end. In prior research [29], we have determined a method for assessing the severity of transverse cracks, considering the deflection of a cantilever beam in the intact state and when it is altered by a breathing crack of known depth \(a\) that is located at the fixed end. This mathematical relation is:  
\[\gamma(a)=\frac{\sqrt{\delta(a,0)}-\sqrt{\delta_{u}}}{\sqrt{\delta(a,0)}} \tag{1}\]  
In Equation (1) we denoted: \(\gamma(a)\) the severity of a crack with depth \(a\) located at the fixed end; \(\delta(a,0)\) the deflection at the free end of the cantilever beam having a crack with depth \(a\) at the fixed end (index 0 stays for location \(x=0\) mm); \(\delta_{u}\) the deflection of the intact beam at the free end. It is easy to determine the deflection at the free end for the beam with a constant cross-section subjected to dead mass, as:  
\[\delta_{u}=\frac{\rho AgL^{4}}{8 EI} \tag{2}\]  
Here, \(\rho\) is the volumetric mass density, \(A\) is the cross-sectional area, \(g\) is the gravity, \(E\) is [PERSON]'s modulus and \(I\) is the second moment of inertia. This deflection is also easy to be obtained from a finite element analysis (FEA). Regarding the deflection of a beam affected by cracks, there are no analytical relations for calculating the deflection. Thus, it becomes difficult to determine the severity of cracks. Problems also occur when involving FEA. For a crack positioned at the fixed end, the stresses and deformations of the beam can manifest only on one side of the crack, unlike the case where the crack is located elsewhere along the beam. Thus, the rotation in the cracked region is smaller than that achieved for a crack located in the neighborhood. This has as a consequence a smaller deflection than expected at the free end. The phenomenon is explained in detail in [30]. A suggestive representation of transverse displacements for a cantilever beam's extreme segment fixed at the left end, for two crack positions, is given in Figure 1.  
From the color scheme, one can observe in the above figure that, when the crack is positioned exactly at the fixed end (upper image), the deflection in the transverse direction is bigger at the slice located at 6 mm than the deflection of the beam with a crack located on that slice (bottom image). Subsequently, going toward the free end, the deflections increase faster for the beam with the crack located at 6 mm. Without a doubt, at the free end, this latter beam will achieve a bigger deflection. A supplementary proof can be made with the results presented in Table 1. Here, we present the deflection under dead mass for a steel cantilever beam of length \(L=1\) m and cross-section \(A=0.02\times 0.005\) m\({}^{2}\). The simulations were performed using ANSYS, for the crack positions and depths presented in Table 1. The chosen material is _Structural Steel_ and the mesh is made using hexahedral elements of a maximum 1 mm edge size, thus obtaining a mesh of \(\sim\)30,000 elements. From Table 1, it is easy to observe that the deflection caused by a crack located at the fixed end is smaller than that when the crack is located at \(x=20\) mm and even \(x=200\) mm for all analyzed crack depths.