The beam on elastic foundation model is used to simulate the underground tunnels structures; a vibration equation on two-parameter foundation considering the bending moment

axial force

shear deformation

external and internal damping is obtained. An approximate method for calculating of displacement of underground pipes is used to compute the displacement of underground tunnels structures

avoiding the trouble of integral transform. In numerical example involving the damping. it is convenient to use a special separated variable method to obtain natural frequencies and damping ratio

the attenuation of displacement considering damping. It is concluded that for underground structures

the effect of internal damping can be neglected.