The dynamic behavior of structural components can largely change in the presence of damages. Understanding this behavior is of particular importance for critical engineering systems, and in particular bridges. Damage identification methods forms a key objective in structural health monitoring of bridges so many researches have been conducted in this area. In this thesis, damage identification techniques on beam bridges under moving vehicle loads will be presented in order to produce useful conclusions about the assessment of existing bridges by investigating various numerical applications of different scenarios. The first objective of this thesis is to derive the analytical expressions needed to be able to predict the dynamic response of many different cases of bridges so that as many real scenarios as possible can be treated. This means that these expressions would be used to investigate damaged beam bridges that can be modelled as an assembly of beams with any number of different material properties, any type of interface or boundary conditions and any number of cracks. For this reason an approach to analyze the bridge as an assembly of n piecewise homogeneous damaged Euler-Bernoulli beams jointed at their edges, will be presented, using the generalized functions to obtain a single expression of the solution which depends on the 4 integration constants associated with the boundary conditions. The closed-form expressions of these 4 constants will be provided. Furthermore, in the presence of internal or externals springs, translational or rotational, additional constants representing the discontinuities have to be taken into account and are computed by considering one additional condition for each discontinuity. The feasibility of this approach and the corresponding analytical formulations is shown with two numerical applications that include all the different capabilities mentioned. Moreover, the implementation of these expressions in a deterministic approach for damage localization is presented, mainly as another example of the many possibilities of the use of analytical formulations instead of other approaches and as an introduction of the so called Inverse Problem with deterministic and probabilistic methods. The second objective concerns the optimization of damage identification on bridges by comparing different quantities that are evaluated while measuring the response of the bridge (direct monitoring) and the response of the moving vehicle when it passes along the bridge (indirect monitoring). First, the governing equations for the dynamic response of these models are derived, considering the crack(s) as a rotational spring, the bridge as an Euler-Bernoulli beam (or multiple with different properties) and the moving vehicle as a spring-mass system. In this manner, the dynamic response of the bridge is calculated (modal characteristics and displacement) as well as the one of the moving oscillator (displacement and acceleration) and the reaction force acting on the surface of the beam from the moving vehicles. Numerical applications with different beam properties and different number of cracks are performed, using MATLAB for the analytical expressions and SAP2000 for the finite element model, to derive the optimal quantity to be used for damage identification. Lastly, the results are validated by considering and comparing an alternative way of modelling crack, namely as a zone with reduced rigidity, for the same numerical examples, leading to the same conclusions about the crack(s) identification. Last but not least, the third objective of this thesis is to be able deal not only with the widely used time-invariant damages, namely the always-open crack model, but also with time-variant damages and in this case with the switching crack model. To achieve this, the analytical expressions for the closed-form solutions of the mode shapes derived for the always-open crack are modified to be able to tackle the switching crack model by introducing a Boolean switching crack array which identifies open cracks, modelled as rotational springs. These new expressions would still be able to be used for any number of Euler-Bernoulli beams, any type of interface or boundary conditions and any number of switching cracks. Then, the governing equations for the dynamic response of this model are derived, considering the moving vehicles as moving masses in order to validate the approach with numerical examples existing in the literature and then by introducing its new capabilities. Further, as the computational strategy has been validated, a comparison between time-variant and time-invariant damages is performed concerning crack identification, so that the reader would recognize the importance of understanding the dynamic behavior of different ways of modelling damage in complicate engineering systems like bridges.