饱和非均质土中桩土耦合扭转振动理论研究
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摘要
本文基于饱和两相介质的本构关系及波动方程,将桩视为一维弹性杆件,采用解析的方法对弹性桩与非均质饱和土的动力扭转相互作用问题进行了较系统和深入的研究。主要工作和创新成果包括:
1.针对受谐和扭转荷载作用的饱和土体中端承桩扭转振动问题,采用分离变量法,得到了土体扭转振动位移形式解。进一步根据桩土接触面上的位移、应力连续条件,求解了桩身动力平衡方程,在频域内得到了桩顶动力响应的解析解。在此基础上,利用傅立叶逆变换和卷积定理,求得了适用性较广的半正弦脉冲激振扭矩作用下桩顶速度时域响应的半解析解。此外,还研究了谐和扭转荷载作用下弹性支承桩与饱和土的动力相互作用问题。
2.基于饱和土体的运动方程以及横观各向同性介质的本构方程,给出了柱坐标下横观各向同性饱和土体的动力控制方程。采用分离变量法,结合桩土系统的边界条件和衔接条件,求得了端承桩在横观各向同性饱和土中扭转振动频域响应的解析解。进一步地利用傅立叶逆变换和卷积定理,求得了半正弦脉冲激振扭矩作用下桩顶速度时域响应的半解析解。
3.研究了成层饱和土体中受谐和扭转荷载作用的任意段变阻抗桩的动力响应问题,提出了土层层间相互作用简化模型,利用单层土中给出的求解方法,通过阻抗函数的递推得到了成层饱和土中任意段变阻抗桩桩顶频域响应的解析解和半正弦脉冲激振扭矩作用下桩顶速度时域响应的半解析解,并通过与单层严格解的对比论证了该简化模型的精确性和适用性。
4.研究了径向非均匀饱和土体中弹性支承桩受谐和扭转荷载作用的动力响应问题,利用径向非均匀土模型,将桩周径向非均匀土体分为内部环形扰动区域和外部半无限大未受扰动区域,并提出了适用性更强的扰动区域土体复剪切模量变化模式。进一步地将内部扰动区域划分为很多薄层同心环区域,结合土层边界和衔接条件采用土层剪切刚度递推的方法,求得了土体对桩身的扭转阻抗,进而推导得到弹性支承桩桩顶频域响应的解析解以及桩顶速度时域响应的半解析解。
5.利用以上解,通过编程和大量计算,对桩的扭转振动特性进行了较全面和细致的分析,获得了对桩基工程有实际参考价值的结论。
本文给出的弹性桩与非均质饱和土耦合扭转振动响应问题的分析理论和解答,进一步丰富和完善了桩基扭转振动理论,为桩基抗震、防震设计以及剪切波在桩基动态检测中的应用提供了更加合理和严格的理论支持。
Based on wave equation of fluid-saturated porous medium and one-dimensional equation of motion of an elastic pile, the torsional dynamic interactions between elastic pile and heterogeneous soils are systematically investigated by the analytical methods. The principal contents and original work are as follows:
1. For an end bearing pile embedded in saturated soil and subjected to harmonic torsional loading, the equations of equilibrium of saturated soil undergoing axisymmetric torsional deformations in cylindrical coordinates are solved by using the separation of variables technique. Then, the vibration displacement solution with an undetermined constant of the soil is obtained. Based on perfect contact between the pile and soil, the solution is coupled into an one-dimensional governing equation of a pile, and the dynamic response of the pile top is obtained in a closed form in the frequency domain. By virtue of inverse Fourier transform and convolution theorem, a semi-analytical solution for the velocity response of a pile subjected to a semi-sine wave exciting torque is obtained in the time domain. In addition, the dynamic response of an elastic bearing pile embedded in saturated soil and subjected to harmonic torsional loading is investigated.
2. On the basis of the dynamic equations of saturated soil and the stress-strain relationships of transversely isotropic medium, the dynamic governing equations of the transversely isotropic saturated soil are derived in cylindrical coordinates. In combination with boundary and continuous conditions of the pile-soil system, the analytical solutions for the dynamic response of an end bearing pile are gained in the frequency domain by utilizing the separation of variables technique. Furthermore, by virtue of inverse Fourier transform and convolution theorem, a semi-analytical solution for the velocity response of an end bearing pile subjected to a semi-sine wave exciting torque is obtained in the time domain.
3. To study the dynamic response of a pile with variable impedance embedded in a multi-layered saturated soil and subjected to harmonic torsional loading, a simplified and practical mathematical model for simulating dynamic interaction of the adjacent soil layers is developed. By using the same solving methods employed in single layer saturated soil and through the recursion of the impedance function, the analytical and semi-analytical solutions for the dynamic response of the pile top are derived in the frequency and time domain, respectively. Moreover, the accuracy and feasibility of the simplified model is verified by comparing with the single layer rigorous solution.
4. The dynamic response of an elastic bearing pile embedded in radially inhomogeneous saturated soil and subjected to harmonic torsional loading is investigated by using the radially inhomogeneous soil model theory. The radially inhomogeneous soil is divided into two regions, the inner annular disturbed region and outer semi-infinite undisturbed region. Furthermore, a more feasible function that describes the variation of the complex-valued shear modulus is developed. Then, the inner disturbed region is subdivided into many thin concentric annular zones. Combined with the boundary and continuous conditions of the soil layers, the torsional impedance of the radially inhomogeneous soil is derived through the recursion of the shearing rigidity of soil layer. Then, the analytical and semi-analytical solutions for the dynamic response of the pile top are obtained in the frequency and time domain, respectively.
5. Based on these solutions, the corresponding computer programs are developed and parameter study is made to investigate the torsional vibration characteristics of pile, and valuable conclusions are obtained for pile foundation.
The torsional pile-soil dynamic interaction models and corresponding solutions developed in this dissertation are more rigorous and feasible than the previous theories. Therefore, the present dissertation provides more reasonable and rigorous theoretical support for seismic design of pile foundation and pile dynamic testing.
1.针对受谐和扭转荷载作用的饱和土体中端承桩扭转振动问题,采用分离变量法,得到了土体扭转振动位移形式解。进一步根据桩土接触面上的位移、应力连续条件,求解了桩身动力平衡方程,在频域内得到了桩顶动力响应的解析解。在此基础上,利用傅立叶逆变换和卷积定理,求得了适用性较广的半正弦脉冲激振扭矩作用下桩顶速度时域响应的半解析解。此外,还研究了谐和扭转荷载作用下弹性支承桩与饱和土的动力相互作用问题。
2.基于饱和土体的运动方程以及横观各向同性介质的本构方程,给出了柱坐标下横观各向同性饱和土体的动力控制方程。采用分离变量法,结合桩土系统的边界条件和衔接条件,求得了端承桩在横观各向同性饱和土中扭转振动频域响应的解析解。进一步地利用傅立叶逆变换和卷积定理,求得了半正弦脉冲激振扭矩作用下桩顶速度时域响应的半解析解。
3.研究了成层饱和土体中受谐和扭转荷载作用的任意段变阻抗桩的动力响应问题,提出了土层层间相互作用简化模型,利用单层土中给出的求解方法,通过阻抗函数的递推得到了成层饱和土中任意段变阻抗桩桩顶频域响应的解析解和半正弦脉冲激振扭矩作用下桩顶速度时域响应的半解析解,并通过与单层严格解的对比论证了该简化模型的精确性和适用性。
4.研究了径向非均匀饱和土体中弹性支承桩受谐和扭转荷载作用的动力响应问题,利用径向非均匀土模型,将桩周径向非均匀土体分为内部环形扰动区域和外部半无限大未受扰动区域,并提出了适用性更强的扰动区域土体复剪切模量变化模式。进一步地将内部扰动区域划分为很多薄层同心环区域,结合土层边界和衔接条件采用土层剪切刚度递推的方法,求得了土体对桩身的扭转阻抗,进而推导得到弹性支承桩桩顶频域响应的解析解以及桩顶速度时域响应的半解析解。
5.利用以上解,通过编程和大量计算,对桩的扭转振动特性进行了较全面和细致的分析,获得了对桩基工程有实际参考价值的结论。
本文给出的弹性桩与非均质饱和土耦合扭转振动响应问题的分析理论和解答,进一步丰富和完善了桩基扭转振动理论,为桩基抗震、防震设计以及剪切波在桩基动态检测中的应用提供了更加合理和严格的理论支持。
Based on wave equation of fluid-saturated porous medium and one-dimensional equation of motion of an elastic pile, the torsional dynamic interactions between elastic pile and heterogeneous soils are systematically investigated by the analytical methods. The principal contents and original work are as follows:
1. For an end bearing pile embedded in saturated soil and subjected to harmonic torsional loading, the equations of equilibrium of saturated soil undergoing axisymmetric torsional deformations in cylindrical coordinates are solved by using the separation of variables technique. Then, the vibration displacement solution with an undetermined constant of the soil is obtained. Based on perfect contact between the pile and soil, the solution is coupled into an one-dimensional governing equation of a pile, and the dynamic response of the pile top is obtained in a closed form in the frequency domain. By virtue of inverse Fourier transform and convolution theorem, a semi-analytical solution for the velocity response of a pile subjected to a semi-sine wave exciting torque is obtained in the time domain. In addition, the dynamic response of an elastic bearing pile embedded in saturated soil and subjected to harmonic torsional loading is investigated.
2. On the basis of the dynamic equations of saturated soil and the stress-strain relationships of transversely isotropic medium, the dynamic governing equations of the transversely isotropic saturated soil are derived in cylindrical coordinates. In combination with boundary and continuous conditions of the pile-soil system, the analytical solutions for the dynamic response of an end bearing pile are gained in the frequency domain by utilizing the separation of variables technique. Furthermore, by virtue of inverse Fourier transform and convolution theorem, a semi-analytical solution for the velocity response of an end bearing pile subjected to a semi-sine wave exciting torque is obtained in the time domain.
3. To study the dynamic response of a pile with variable impedance embedded in a multi-layered saturated soil and subjected to harmonic torsional loading, a simplified and practical mathematical model for simulating dynamic interaction of the adjacent soil layers is developed. By using the same solving methods employed in single layer saturated soil and through the recursion of the impedance function, the analytical and semi-analytical solutions for the dynamic response of the pile top are derived in the frequency and time domain, respectively. Moreover, the accuracy and feasibility of the simplified model is verified by comparing with the single layer rigorous solution.
4. The dynamic response of an elastic bearing pile embedded in radially inhomogeneous saturated soil and subjected to harmonic torsional loading is investigated by using the radially inhomogeneous soil model theory. The radially inhomogeneous soil is divided into two regions, the inner annular disturbed region and outer semi-infinite undisturbed region. Furthermore, a more feasible function that describes the variation of the complex-valued shear modulus is developed. Then, the inner disturbed region is subdivided into many thin concentric annular zones. Combined with the boundary and continuous conditions of the soil layers, the torsional impedance of the radially inhomogeneous soil is derived through the recursion of the shearing rigidity of soil layer. Then, the analytical and semi-analytical solutions for the dynamic response of the pile top are obtained in the frequency and time domain, respectively.
5. Based on these solutions, the corresponding computer programs are developed and parameter study is made to investigate the torsional vibration characteristics of pile, and valuable conclusions are obtained for pile foundation.
The torsional pile-soil dynamic interaction models and corresponding solutions developed in this dissertation are more rigorous and feasible than the previous theories. Therefore, the present dissertation provides more reasonable and rigorous theoretical support for seismic design of pile foundation and pile dynamic testing.
引文
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