Characteristics of vibration and damping are investigated for a parabolic curved bar having two layers, one of which is elastic and the other viscoelastic. Equations of motion are obtained from the stationary condition of the Lagrangian, and they are solved exactly in the case of two elastic layers by means of the series expansion method. Composite loss factors are calculated by the strain energy method. It is found that the composite loss factors in the case of out-plane vibration depend on the Young's modulus ratio, loss factor ratio and diameter ratio of the two layers, but in the case of inplane vibration, they change complicatedly, especially in the symmetric mode vibration. Experiments are carried out on a parabolic curved pipe of aluminium alloy which is filled with acrylic acid resin. The natural frequencies and composite loss factors calculated theoretically are compared with the experimental results and found to be in good agreement with them.