In finite element analysis, nonlinear time-history analysis is a realistic and accurate analysis type for dynamic or seismic analysis due to its solutions contain wealthy data and complete response time-history. The most commonly used method, probably the only practical procedure, in nonlinear time-history analysis is the direct time integration method. In general, every direct time integration method could be classified as either an implicit method or an explicit method. Understanding the differences between the two categories in both theoretical and practical aspects is very important for engineers to make the best analysis strategy for a specific dynamic or seismic analysis. In this treatise, the most popular method in each category, i.e., implicit Newmark method and explicit central difference method, will be introduced and used in transient analyses and results comparisons. In total, five cases studies are included in this thesis, including three cases with linear elastic materials and two cases with quasi-brittle masonry material. These five cases are studied to answer the main research questions of this research: What differences can be observed in comparisons of solutions obtained from implicit and explicit methods for linear elastic material in transient analysis and for quasi-brittle material under seismic load? Also, how are the performances of both methods with respect to the stability and accuracy aspects? The results comparisons show that for linear elastic material, both methods could generate accurate solutions with proper time steps. However, the explicit method shows a better representation of the high-frequency response of the structure. For quasi-brittle masonry material, both methods are possible to generate good results. However, the implicit method may have convergence problems during the iterations, which could lead to inaccurate predictions of nonlinear behavior of the structure. The explicit method shows very good predictions for nonlinear behavior, but the computation time is quite long and the critical time step is highly influenced by many factors in the finite element model. Moreover, the mass scaling technique, which is generally used in the explicit method, is also investigated in this thesis.