In this study, a finite element model with two nodes and four degrees of freedom per node (horizontal and vertical translations, rotation and curvature) is presented for free vibration analysis of tapered beams with damages. Element stiffness and mass matrices for Bernoulli-Euler beam with variable cross-section are derived by using the Galerkin method. In the finite element formulation, linear Lagrange polynomials for extensional vibrations and fifth-order Hermitian polynomials for bending vibrations are chosen. Damage is introduced by a stiffness loss coefficient to the element stiffness matrix. The element mass matrix is assumed to be unchanged due to the damage effect. The accuracy of the proposed element is shown by comparison with available literature. Some numerical results are presented to show the success of modal curvature change in finding damage (crack) locations.