The need for an analytical method that one can apply manually to estimate dynamic impact forces on railway tracks that occur because of varying track stiffness or track profile initiated a study to develop an analytical method named as the Bezgin Method. The advancement of this method presented in this paper includes an extension of a set of equations developed and introduced by the first author earlier as the Bezgin Equations using the proposed method and development of a new equation. In addition to track stiffness taken into consideration in the equations introduced earlier, the Extended Bezgin Equations presented in this paper take into account bogie stiffness, wheel spring stiffness, Hertzian contact stiffness, and a factor for damping. The new equation takes into account the effect of vertical wheel acceleration as a train transitions to a stiffer structure or transitions along an ascending track profile. The paper unites and applies these equations to estimate wheel forces that develop along stiffness transition zones by considering an array of train speeds for an array of track stiffness ratios and representative values for track profile deviations along the transitions. Final section of the paper includes elaborate finite element analyses of structural track models that involve transitions of soil supported ballasted railway tracks with concrete based ballasted tracks along various transition lengths and compares their estimates for dynamic impact force factors with those estimated by the Extended Bezgin Equations. The paper concludes with a discussion of the potential uses, benefits, and the value of the Bezgin Method for railway engineering.