Passive bipeds have become an interesting field of research for investigators. Probably all of passive bipeds which have been modeled previously are considered as a rigid model with point-mass. In this paper, 2D planner compass-gait biped with elastic link is modeled and simulated and its period-one gait is investigated. The stance leg of the passive biped is modeled as an Euler-Bernoulli beam and its vibrations are modeled by using Assumed Modes Method and the equations of motion for the swing phase are developed. Then behavior of the elastic biped is simulated by suing numerical methods in MATLAB software and the changes in leg angles and angular velocities of the biped are discussed. Computer simulations showed that when the vibrations of the stance leg are large, angular velocities become oscillating. Vibrations of the stance leg and the effects of Young's Modulus and damping coefficient on the motion of the elastic biped are discussed. Then model is simulated for the small vibrations of the stance leg and the results show that when the vibrations are small the elastic biped behaves like a rigid biped which verifies our simulations. When the vibrations are small, period-one gait can be found for the elastic biped for the ramp slopes of 0<γ≤0.0328 rad. The split in the eigenvalues of the period-one gait happens at the ramp slope of γ=0.029 rad.