As an innovative lightweight solution for noise and vibration problems, locally resonant metamaterials are gaining importance; especially for their favorable vibro-acoustic behavior in targeted frequency ranges commonly referred to as stopbands. This paper investigates the effects of curvature on the stopband behavior observed in locally resonant metamaterials by studying wave motion in infinite metamaterial pipes. Following a curved unit cell modelling approach, wave motion in the metamaterial pipe of infinite extent is described through the analysis of dispersion curves. The existence of stopband zones in metamaterial pipes is shown and the effect of the radius of the pipe on the width of the stopband zone is studied. It is observed that the radius can considerably affect the stopband zone in a metamaterial pipe by influencing the ring frequency. While metamaterial pipes of higher radii show a stopband behavior similar to flat panels, the behavior of smaller radii pipes deviates considerably from their flat counterparts.