成层土中楔形桩的竖向和扭转振动理论及应用研究
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摘要
由于楔形侧面的存在,楔形桩在承受竖向荷载时,桩侧土不仅可以提供侧摩阻力,还会给楔形侧面施加一个斜向挤压的土压力,从而充分发挥桩与土的相互作用,是一种具有良好应用前景的新型桩型。随着楔形桩在工程上逐渐推广使用,它与土体的动力相互作用机理也逐渐引起众多学者的关注。但由于桩土动力相互作用问题本身的复杂特性,以及楔形桩楔形侧面的特殊性会导致桩侧土体性质发生变化,使桩侧土体形成一定范围的扰动圈,楔形桩与土体动力相互作用问题显得十分复杂。因此,开展楔形桩在竖向和扭转动力荷载作用下桩土耦合振动问题的研究不仅是桩基工程自身发展的需要,同时也可以为桩基动态测试、桩基础抗震和防震设计提供更加完善的理论支撑,具有重要的理论研究价值和工程应用意义。
本文从平面应变理论及(粘)弹性杆件竖向和扭转振动理论出发,采用解析研究方法较系统地建立了复杂条件下成层土中楔形桩与土体的耦合振动理论,基于所得理论解,深入地分析了桩土参数对楔形桩竖向和扭转振动特性的影响机理。论文具体研究内容和成果如下：
(1)考虑楔形桩的均匀变截面特性和桩侧土的层状特性,基于土体竖向振动平面应变假定和Rayleigh-Love杆模型,建立了桩土体系竖向耦合振动的定解问题。采用积分变换技术和阻抗函数传递技术,先后推导得到了竖向动力荷载作用下楔形桩桩顶频域响应的解析解和相应的时域响应半解析解。由参数分析法的结果可知,在一定频率范围内,楔形桩其他设计参数不变时,楔形桩桩顶动刚度和动阻尼均会分别随着楔角、桩长、桩端截面半径和桩身弹性纵波波速的增大而增大。在一定低频范围内,当上层桩侧土或下层桩侧土由软到硬变化时,楔形桩桩顶动刚度和动阻尼均会逐渐增大,且上层桩侧土性质变化对桩顶动刚度和动阻尼的影响更为敏感。当桩侧土为上下两层时,如果某一层桩侧土逐渐变软,土层界面会出现同向反射,如果某一层桩侧土逐渐变硬,土层界面会出现反向反射,从而给桩基无损检测信号的判断带来一定的干扰。
(2)综合考虑楔形桩成桩效应引起的桩侧土体施工硬化和施工软化效应,提出竖向剪切复刚度传递模型来描述桩侧扰动土体的径向非均匀特性,建立了考虑成桩效应时桩土系统竖向耦合振动的定解问题。通过竖向剪切复刚度传递方法求得了桩侧土作用在桩身的竖向剪切复刚度的解析式,进一步,结合积分变换技术和阻抗函数传递技术,先后推导得到了考虑成桩扰动效应时楔形桩桩顶频域响应解析解和相应的时域响应半解析解。由参数分析法的结果可知,在一定频率范围内,可以在一定径向范围内加固桩侧土来增强桩土系统抵抗竖向变形和竖向振动的能力。如果成桩效应引起桩侧土体的施工软化,则会导致桩土系统抵抗竖向变形和竖向振动的能力减弱。桩顶速度频域响应曲线共振峰的幅值会随看扰动十体硬化范围和硬化程度的增大而逐渐减小,随着扰动土体软化范围和软化程度的增大而逐渐增大。桩顶速度时域响应曲线一次桩尖反射信号幅值会随着内部区域硬化土体硬化范围和硬化程度的增大而逐渐减小,随着内部区域软化土体的软化范围和软化程度的增大而逐渐增大。
(3)基于微分思想将楔形桩沿竖向离散为一系列的微元段,对桩侧土采用平面应变模型,对楔形桩采用弹性杆件扭转振动模型,建立了楔形桩与桩侧土耦合扭转振动的定解问题。采用阻抗函数传递方法,通过严格求解得到了扭转激振作用下频域内楔形桩桩顶扭转复刚度和桩顶角速度响应的解析解和相应的时域内桩顶角速度响应的半解析解。由参数分析法的结果可知,在一定低频范围内,随着楔角、桩长、桩端截面半径和桩身剪切波速的增大,楔形桩桩顶扭转动刚度和动阻尼均逐渐增大。当桩侧土具有软弱夹层或坚硬夹层时,桩顶角速度频域响应曲线会出现大峰夹小峰的振荡现象,且软弱土夹层性质越差时,共振峰幅值会越大,坚硬土夹层性质越好时,共振峰幅值会越小。当桩侧土存在软弱夹层时,桩顶角速度时域响应曲线相应位置处会出现类似于缩颈的反射信号,当桩侧土存在坚硬夹层时,桩顶角速度时域响应曲线相应位置处会出现类似于扩颈的反射信号。
(4)采用环向剪切复刚度传递模型来描述桩侧扰动土体的径向非均匀特性,建立.了考虑成桩效应时楔形桩扭转振动的定解问题。采用环向剪切复刚度传递方法和阻抗函数递推方法,通过严格推导得到了频域内楔形桩桩顶角速度响应的解析解和时域内桩顶角速度响应的半解析解。由参数分析法的结果可知,在一定低频范围内,楔形桩桩顶扭转动刚度和动阻尼均会随着扰动土体硬化范围和硬化程度的增大而有一定程度的增大。随着扰动土体软化范围和软化程度的增大,楔形桩桩顶扭转动刚度和动阻尼均有一定程度的减小。(?)形桩桩顶角速度频域响应曲线共振峰的幅值随着扰动土体硬化范围和硬化程度的增大而逐渐减小,随着扰动土体软化范围和软化程度的增大而逐渐增大。楔形桩桩顶角速度时域响应曲线一次桩尖反射信号的幅值随着扰动土体硬化范围和硬化程度的增大而逐渐减小,随着扰动土体软化范围和软化程度的增大而逐渐增大。
(5)当楔形桩桩身存在变截面或变模量桩段时,桩顶(角)速度频域响应曲线和桩顶(角)速度时域响应曲线均会与正常楔形桩存在较大的差异,在楔形桩成桩质量分析盯需仔细考虑变截面或变模量桩段的影响。
(6)综合来说,利用桩土系统扭转振动理论得到的桩顶角速度频域响应曲线和时域响应曲线变化规律与利用桩土系统竖向振动理论得到的桩顶速度频域响应曲线和时域响应曲线变化规律一致,只是变化的幅度会相对较小。因此,利用桩土系统竖向振动理论作为桩基无损检测的理论依据更为可靠。
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.
本文从平面应变理论及(粘)弹性杆件竖向和扭转振动理论出发,采用解析研究方法较系统地建立了复杂条件下成层土中楔形桩与土体的耦合振动理论,基于所得理论解,深入地分析了桩土参数对楔形桩竖向和扭转振动特性的影响机理。论文具体研究内容和成果如下：
(1)考虑楔形桩的均匀变截面特性和桩侧土的层状特性,基于土体竖向振动平面应变假定和Rayleigh-Love杆模型,建立了桩土体系竖向耦合振动的定解问题。采用积分变换技术和阻抗函数传递技术,先后推导得到了竖向动力荷载作用下楔形桩桩顶频域响应的解析解和相应的时域响应半解析解。由参数分析法的结果可知,在一定频率范围内,楔形桩其他设计参数不变时,楔形桩桩顶动刚度和动阻尼均会分别随着楔角、桩长、桩端截面半径和桩身弹性纵波波速的增大而增大。在一定低频范围内,当上层桩侧土或下层桩侧土由软到硬变化时,楔形桩桩顶动刚度和动阻尼均会逐渐增大,且上层桩侧土性质变化对桩顶动刚度和动阻尼的影响更为敏感。当桩侧土为上下两层时,如果某一层桩侧土逐渐变软,土层界面会出现同向反射,如果某一层桩侧土逐渐变硬,土层界面会出现反向反射,从而给桩基无损检测信号的判断带来一定的干扰。
(2)综合考虑楔形桩成桩效应引起的桩侧土体施工硬化和施工软化效应,提出竖向剪切复刚度传递模型来描述桩侧扰动土体的径向非均匀特性,建立了考虑成桩效应时桩土系统竖向耦合振动的定解问题。通过竖向剪切复刚度传递方法求得了桩侧土作用在桩身的竖向剪切复刚度的解析式,进一步,结合积分变换技术和阻抗函数传递技术,先后推导得到了考虑成桩扰动效应时楔形桩桩顶频域响应解析解和相应的时域响应半解析解。由参数分析法的结果可知,在一定频率范围内,可以在一定径向范围内加固桩侧土来增强桩土系统抵抗竖向变形和竖向振动的能力。如果成桩效应引起桩侧土体的施工软化,则会导致桩土系统抵抗竖向变形和竖向振动的能力减弱。桩顶速度频域响应曲线共振峰的幅值会随看扰动十体硬化范围和硬化程度的增大而逐渐减小,随着扰动土体软化范围和软化程度的增大而逐渐增大。桩顶速度时域响应曲线一次桩尖反射信号幅值会随着内部区域硬化土体硬化范围和硬化程度的增大而逐渐减小,随着内部区域软化土体的软化范围和软化程度的增大而逐渐增大。
(3)基于微分思想将楔形桩沿竖向离散为一系列的微元段,对桩侧土采用平面应变模型,对楔形桩采用弹性杆件扭转振动模型,建立了楔形桩与桩侧土耦合扭转振动的定解问题。采用阻抗函数传递方法,通过严格求解得到了扭转激振作用下频域内楔形桩桩顶扭转复刚度和桩顶角速度响应的解析解和相应的时域内桩顶角速度响应的半解析解。由参数分析法的结果可知,在一定低频范围内,随着楔角、桩长、桩端截面半径和桩身剪切波速的增大,楔形桩桩顶扭转动刚度和动阻尼均逐渐增大。当桩侧土具有软弱夹层或坚硬夹层时,桩顶角速度频域响应曲线会出现大峰夹小峰的振荡现象,且软弱土夹层性质越差时,共振峰幅值会越大,坚硬土夹层性质越好时,共振峰幅值会越小。当桩侧土存在软弱夹层时,桩顶角速度时域响应曲线相应位置处会出现类似于缩颈的反射信号,当桩侧土存在坚硬夹层时,桩顶角速度时域响应曲线相应位置处会出现类似于扩颈的反射信号。
(4)采用环向剪切复刚度传递模型来描述桩侧扰动土体的径向非均匀特性,建立.了考虑成桩效应时楔形桩扭转振动的定解问题。采用环向剪切复刚度传递方法和阻抗函数递推方法,通过严格推导得到了频域内楔形桩桩顶角速度响应的解析解和时域内桩顶角速度响应的半解析解。由参数分析法的结果可知,在一定低频范围内,楔形桩桩顶扭转动刚度和动阻尼均会随着扰动土体硬化范围和硬化程度的增大而有一定程度的增大。随着扰动土体软化范围和软化程度的增大,楔形桩桩顶扭转动刚度和动阻尼均有一定程度的减小。(?)形桩桩顶角速度频域响应曲线共振峰的幅值随着扰动土体硬化范围和硬化程度的增大而逐渐减小,随着扰动土体软化范围和软化程度的增大而逐渐增大。楔形桩桩顶角速度时域响应曲线一次桩尖反射信号的幅值随着扰动土体硬化范围和硬化程度的增大而逐渐减小,随着扰动土体软化范围和软化程度的增大而逐渐增大。
(5)当楔形桩桩身存在变截面或变模量桩段时,桩顶(角)速度频域响应曲线和桩顶(角)速度时域响应曲线均会与正常楔形桩存在较大的差异,在楔形桩成桩质量分析盯需仔细考虑变截面或变模量桩段的影响。
(6)综合来说,利用桩土系统扭转振动理论得到的桩顶角速度频域响应曲线和时域响应曲线变化规律与利用桩土系统竖向振动理论得到的桩顶速度频域响应曲线和时域响应曲线变化规律一致,只是变化的幅度会相对较小。因此,利用桩土系统竖向振动理论作为桩基无损检测的理论依据更为可靠。
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.
引文
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