Transient vibrations of flexible structures at mid- and high-frequencies have important applications in aerospace, civil, auto and ship engineering. In this paper, a new method is developed for the determination of the transient vibration solutions of two-dimensional beam frames in mid- and high-frequency regions. In the development, the governing equations of a beam frame structure are formulated by an augmented Distributed Transfer Function Method (DTFM), without the need for discretization and approximation. The augmented DTFM differs from the traditional DTFM in that it does not contain the singularities of subsystem transfer functions, which is crucially important in a mid- or high-frequency analysis. The proposed method delivers exact eigensolutions of a beam structure from low- to high-frequencies without numerical instability. With the platform provided by the augmented DTFM, the transient response of a beam structure can be conveniently estimated by either modal expansion or the residue formula for inverse Laplace transform. A highlight of the augmented DTFM lies in that detailed information at mid- and high-frequencies, such as local displacement, slope, bending moment and shear force at any point, can be obtained, which otherwise may be difficult with conventional methods for mid- and high-frequency analysis. The proposed method is illustrated on several examples and is computationally efficient and stable from low- to high-frequency regions. In the numerical simulation, the augmented DTFM is shown to produce more accurate results than traditional finite element analysis (FEA). The proposed method is extensible to three-dimensional beam structures.