The periodic viaduct is assumed to be composed of an infinite number of spans, and each span is supposed to consist of a pier, two longitudinal beams and three linking springs. Based on the Bernoulli–Euler beam vibration theories and Bloch theorem, a transfer matrix for the junction linking the beams and the pier is obtained. The polynomial eigenvalue equation for the energy bands of the periodic viaduct undergoing in-plane motion is also derived. Based on the obtained eigenvalue equation, the energy bands of the periodic viaduct were presented. With the proposed model, the influences of the ratio of Young’s module of the beams to that of the piers and the stiffness of the spring on the energy bands of the periodic viaduct is investigated. Numerical results in this paper demonstrate that when the periodic viaduct with beam-beam and beam-pier spring junction is undergoing in-plane motion, there exist three lattice waves: the first kind of wave is a highly decaying wave and cannot propagate a long distance along the viaduct; the second kind of lattice wave can propagate only at some frequency ranges; and the third kind of lattice wave can propagate at most frequencies. However, within a low frequency range, the lattice wave does not propagate. As a result, to guarantee the dominant frequency of the base for the periodic viaduct not to be located within the low frequency range is crucial for the periodic viaduct design. Otherwise, the wave components carrying most energy of seismic waves will be localized, which is dangerous for the viaduct。Moreover, with increasing the ratio of Young’s modulus of the beams to that of the piers and the stiffness of the beam-beam spring, the attenuation of the lattice waves decreases significantly, implying that the wave can propagate a longer distance along the structures.