An explicit velocity-based Lax-Wendroff/Taylor-Galerkin methodology of computation with emphasis on applicability to the dynamics of structures is proposed. The proposed formulations are general and applicable to the areas of linear/nonlinear computational structural dynamics (CSD). The concepts are based on the philosophy of an improved rationale for treating both the spatial and temporal variations in direct integration methods. As a consequence, the approach is based on first expressing the transient time-derivative terms in conservation form in terms of a Taylor series expansion including higher order time-derivatives, which are then evaluated from the governing dynamic equations of motion also expressed in conservation form. An updating scheme is proposed for the necessary conservation variables for obtaining the dynamic response. The basic methodology is described and developed in technical detail with emphasis on applications to beam-type structural models. Results which are of a comparative nature are presented to validate and therein demonstrate the applicability to linear/nonlinear dynamics of structures.