Time-domain analytical solution of the vibration response of a large-diameter pipe pile subjected to transient concentrated load

For a large-diameter pipe pile subjected to transient concentrated load in low strain testing, the wave propagation should be explained by three-dimensional wave equation. The effects of the soils on the pile are modeled by Winkler spring. The exciting force is simulated by a semisinusoidal impulse. A mechanical model for the vibration response of the large-diameter pipe pile subjected to transient concentrated load is established. The time-domain analytical solution is obtained by the separation of variables and the variation of constants. The results of the proposed analytical solution are compared with those calculated by numerical Fourier inverse transformation. The validity is also verified by the test results. The time-domain velocity responses on the pile top are analyzed by means of the analytical solution. The calculated results indicate that the arrival and terminative time of the incident waves is proportional to the propagation distance and it varies linearly from the point of 0° to that of 180°. At the exciting point, the terminative time of the incident waves is equal to the width of the input impulses, and at the other points, the difference between the terminative and arrival time of the incident waves is equal to the width of the input impulses. The peak time of the incident waves at the exciting point is the earliest and that from the point of 0° to that of 135° increases gradually. However, the peak time does not increase any more from the point of 135° to that of 180°.