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A finite element formulation for the time-domain analysis of linear transient elastodynamicproblems is presented based on the weak form obtained by applying the Galerkin's method to the integro-diferential equations which contain the initial conditions implicitly and does not include the inertia terms.The weak form is extended temporally under the assumptions of the constant and linear time variations offield variables, since the time-stepping algorithms such as the Newmark method and the Wilson q-methodare not necessary, obtaining two kinds of implicit finite element equations which are tested for numericalaccuracy and convergency. Three classical examples having finite and infinite domains are solved andnumerical results are compared with the other analytical and numerical solutions to show the versatilityand accuracy of the presented formulation.