A new class of algorithms for transient finite element structural dynamical analysis which is amenable to an efficient implementation in parallel computers (especially Massively Parallel Computers) is proposed. The suitability of the method for parallel computation stems from the fact that, given an arbitrary partition of the finite element mesh, each element in the partition can be processed over a time step independently and simultaneously with the rest, and no global equation solving effort is involved. Although the proposed EBE time integration algorithms are shown to have the structure of an explicit scheme, they are unconditionally stable over a certain range of the algorithmic parameter.