| Due to the existence of wedge side, there are not only lateral friction, but also oblique extrusion earth pressure acting on the shaft of tapered pile which are provided by the pile surrounding soil. It can be seen that tapered pile can maximize the interaction between the pile and the pile surrounding soil. Thus tapered pile is a new type pile which can be widely introduced in the engineering applications. With promoting the use of tapered pile, the dynamic interaction between soil and tapered pile has gradually attracted the attention of many investigators. However, owing to the complexity of soil-pile dynamic interaction and the complicated characteristics of surrounding soil caused by the construction effect of tapered pile, the dynamic interaction between soil and tapered pile is extremely sophisticated. Therefore, investigation on the coupled vibration problem of soil and tapered pile system undergoing vertical and torsional dynamic loading is not only the needs of their own development of pile foundation engineering, but also the theoretical support for seismic resistance of pile foundation and pile nondestructive testing, and the research findings have great significance of theory and valuable application of engineering.In this paper, on the basis of the plane strain theory and vertical and torsional vibration theory of elastic and viscoelastic rod, the coupled vibration theory of soil and tapered pile system is proposed when there is vertical dynamic load or torsional dynamic load acting on the head of tapered pile. By means of parametric study method, the influence of the properties of soil and tapered pile on the vertical and torsional dynamic response of tapered pile is systematically investigated. The main results are listed as followings: (1) Considering the variable cross section of tapered pile and the stratification of soil foundation, the governing equations of soil and tapered pile system undergoing vertical dynamic load are established based on the plane strain assumption of soil and Rayleigh-Love rod model. By means of the integral transform technique and impedance function transfer method, the analytical solutions of complex stiffness and velocity response in frequency domain at the head of tapered pile are derived. By utilizing the convolution theorem and the inverse Fourier transform technique, the velocity response in time domain is also obtained when there is half-cycle sine pulse acting on the head of tapered pile. Using the parametric study method, the main analysis results show that:(1) Within the low frequency range, if other parameters of tapered pile remain unchanged, the vertical dynamic stiffness and dynamic damping at the head of tapered pile increase as the cone-angle, pile length, radius of pile end, and shear wave velocity of tapered increase, respectively.(2) Within the low frequency range, the vertical dynamic stiffness and dynamic damping at the head of tapered pile increase as the upper pile surrounding soil or the lower pile surrounding soil change from soft to hard.(3) When the pile surrounding soil is divided into two layers, there is reflected signal with the same direction as the head wave at the interface of soil layer if the upper pile surrounding soil or the lower pile surrounding soil become softer, but there is reflected signal with the inverse direction as the head wave at the interface of soil layer if the upper pile surrounding soil or the lower pile surrounding soil become harder.(2) Considering the hardening effect and softening effect of pile surrounding soil caused by the construction effect of tapered pile, the vertical shear complex stiffness transfer model is presented to simulate the radial inhomogeneity of soil, and the governing equations of soil and tapered pile system undergoing vertical dynamic load are builded. Then, by virtue of the vertical shear complex stiffness transfer method, integral transform technique and impedance function transfer method, the analytical solutions of complex stiffness and velocity response in frequency domain and the semi-analytical solution of velocity response in time domain at the head of tapered pile are obtained. By means of the parametric study method, the main analysis results show that:(1) Within the low frequency range, the ability to resist vertical deformation and vertical vibration of tapered pile can be strengthened if the pile surrounding soil is reinforced within certain radial range. In other words, if the pile surrounding soil becomes softer owing to the construction effect of tapered pile, the ability to resist vertical deformation and vertical vibration of tapered pile can be weakened.(2) With the increase of hardening range and hardening degree of the internal area of soil, the formant amplitude of velocity response in frequency domain will gradually decrease, and the reflected signal amplitude at the pile tip in the curves of velocity response in time domain will also gradually decrease.(3) With the increase of softening range and softening degree of the internal area of soil, the formant amplitude of velocity response in frequency domain will gradually increase, and the reflected signal amplitude at the pile tip in the curves of velocity response in time domain will also gradually increase.(3) Based on the differential thought, the soil-tapered pile system is divided into a series of segments along the vertical direction. By means of the plane strain theory and torsional vibration theory of elastic rod to simulate the pile surrounding soil and tapered pile, the governing equations of soil and tapered pile system undergoing torsional dynamic load are established. Then, by utilizing impedance function transfer method, the analytical solutions of torsional complex stiffness and angular velocity response in frequency domain are derived. By using the convolution theorem and the inverse Fourier transform technique, the angular velocity response in time domain is also proposed when there is torsional half-cycle sine pulse acting on the head of tapered pile. Utilizing the parametric study method, the main analysis results show that:(1) Within the low frequency range, both the torsional dynamic stiffness and torsional dynamic damping at the head of tapered pile increase with the increase of cone-angle, pile length, radius of pile end and shear wave velocity of tapered pile.(2) If there is soft intercalated soil layer or hard intercalated soil layer in the pile surrounding soil, the curves of angular velocity response in frequency domain may oscillate with big peak and small peak. The softer the intercalated soil layer is, the bigger the formant amplitude of angular velocity response in frequency domain is. By contrast, the harder the intercalated soil layer is, the smaller the formant amplitude of angular velocity response in frequency domain is.(3) If there is soft intercalated soil layer in the pile surrounding soil, a reflected signal with the same direction as the head wave will appear at the interface of soil layer in the curves of angular velocity response in time domain. If there is hard intercalated soil layer in the pile surrounding soil, a reflected signal with the inverse direction as the head wave will appear at the interface of soil layer in the curves of angular velocity response in time domain.(4) Utilizing the circumferential shear complex stiffness transfer model to allow for the radial inhomogeneity of soil, the governing equations of soil and tapered pile system undergoing torsional dynamic load are established. Then, by virtue of the circumferential shear complex stiffness transfer method and impedance function transfer method, the analytical solution of angular velocity response in frequency domain and its corresponding semi-analytical solution of angular velocity response in time domain at the head of tapered pile are derived. By virtue of the parametric study method, the main analysis results show that:(1) Within the low frequency range, with the increase of hardening range and hardening degree of the internal area of soil, both the torsional dynamic stiffness and torsional dynamic damping at the head of tapered pile increase. By contrast, with the increase of softening range and softening degree of the internal area of soil, both the torsional dynamic stiffness and torsional dynamic damping at the head of tapered pile decrease.(2) Both the formant amplitude of angular velocity response in frequency domain and the reflected signal amplitude at the pile tip in the curves of angular velocity response in time domain will gradually decrease with the increase of hardening range and hardening degree of the internal area of soil.(3) Both the formant amplitude of angular velocity response in frequency domain and the reflected signal amplitude at the pile tip in the curves of angular velocity response in time domain will gradually increase with the increase of softening range and softening degree of the internal area of soil.(5) If there is pile segment with variable cross-section or modulus, both the curves of (angular) velocity response in frequency domain and the curves of (angular) velocity response in time domain are noticeably different from that of normal taped pile. Therefore, we should take the influence of the pile segment with variable cross-section or modulus into account in the construction quality analysis of tapered pile.(6) In general, both the curves of angular velocity response in frequency domain and the curves of angular velocity response in time domain calculated by the torsional vibration theory of tapered pile have the same changing regularity with the curves of velocity response in frequency domain and the curves of velocity response in time domain calculated by the vertical vibration theory of tapered pile. But the changing magnitude of the curves obtained by the torsional vibration theory of tapered pile is smaller than the changing magnitude of the curves obtained by the vertical vibration theory of tapered pile. Therefore, the vertical vibration theory of tapered pile is more suitable for pile nondestructive testing. |
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