饱和地基动力响应及高架高速铁路环境振动研究
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摘要
随着我国大量高速铁路的兴建和对既有铁路干线的多次提速,列车高速运行引起线路周边环境振动污染问题日益显著。特别是在广泛分布着饱和软土地基的东南沿海地区,高速列车较易超过地基的Rayleigh波速而产生剧烈的地基振动。为节约用地和控制工后沉降,我国已建300公里级以上高铁干线中,如京津、京沪、沪杭、沪宁、杭甬等,高架桥结构占线路总长的80%以上,高架高速铁路引起饱和地基振动的预测研究在国内外尚属空白。本文以Biot饱和土体波动理论为基础,主要开展了饱和地基时域和频域三维动力响应计算以及高架高速铁路环境振动的理论研究工作,具体如下：
1.以移动谐振荷载下均质饱和半空间土骨架位移和孔压为表征量,通过与完整u-1w格式对比,提出了无量纲参数——最大频率比RΩ来定量判定u-p格式、完全不透水格式和完全透水格式等三种简化控制方程在移动荷载下饱和地基动力响应模拟中的适用性。
2.根据完整u-w格式的Biot波动控制方程,采用Galerkin法推导了三维饱和土体单元控制方程,编制了相应二维和三维饱和土体单元程序,并嵌入到自主开发的三维动力有限元分析程序中。分析了一维饱和土柱的固结、波动问题以及二维地基的动力响应问题,并与解析解和已有研究成果进行了对比验证了程序的正确性。同时求解了移动点荷载作用下的平面应变饱和地基动力响应,研究了荷载速度和渗透系数对地表土骨架位移以及孔隙流量的影响。
3.将基于单相介质提出的多次透射人工边界(MTF)推广应用于饱和地基动力响应的有限元模拟中。首先,基于饱和土体的Biot u-w格式动力方程,在频域中解析求解了平面P1、P2和SV波入射时,有限元离散网格中二阶MTF人工边界稳态反射系数的表达式,并分析了不同无量纲频率比、无量纲网格步距和无量纲时间步距下的MTF人工边界反射系数和反射角。继而,在自主开发的有限元动力分析程序中实现了上述二阶MTF人工边界,通过与解析解和边界远置参考解对比,说明了二阶MTF人工边界在一维、二维和三维饱和地基动力响应有限元模拟中的有效性以及在移动荷载下饱和地基动力响应有限元模拟中的适用性。
4.建立了三维层状饱和地基模型并在模型底边施加频域无反射人工边界(考虑地基下卧基岩时则模型底边固定),采用薄层单元法求解了三维层状饱和地基的频域动力Green函数。通过与已有研究成果对比,说明了所求Green函数能准确计算不同振动频率、不同埋深位置处简谐荷载激振下的三维饱和地基的动力响应,所求Green函数无需关于Hankel函数进行数值积分逆变换且能直接计入地基的成层性,提高了计算效率。
5.采用容积法建立了三维饱和地基—群桩基础频域耦合动力分析模型,其中桩体离散为三维Euler-Bernoulli梁单元,饱和地基刚度矩阵则由基于薄层单元法推导的Green函数在各桩土相互作用节点间形成的柔度矩阵求逆而得。通过与已有研究成果对比说明了所建模型的正确性,在此基础上,分析了不同地基渗透系数和激振频率下饱和地基中群桩基础的水平、竖向、摇摆和扭转阻抗函数以及承台中心施加单位力和弯矩简谐荷载时,饱和地基自由场的位移和孔压响应。
6.建立了移动谐振荷载下、等跨简支箱梁上覆无限长板式轨道的竖向动力分析模型,并利用周期结构动力响应的性质进行求解。分析了不同荷载自振频率、移动速度和初始位置下,钢轨、轨道板和箱梁的位移响应频谱、钢轨的时域位移响应和箱梁支座的反力时程。研究表明,当钢轨准柔度低于扣件弹簧柔度时,振动变形和能量集中在钢轨和扣件中,递到箱梁支座的荷载比例会明显降低；带支座箱梁的一阶挠曲自振频率以及簧上轨道板(弹簧模拟轨道板下的CA砂浆垫层)的共振频率对结构动力响应有显著贡献。
7.针对我国高架高速铁路工程常见的上覆板式轨道、下设群桩基础的双线整体式简支箱梁结构形式,提出了一种能详细考虑列车荷载经由钢轨、扣件、轨道板和CA砂浆垫层等细观轨道构件传递至箱梁、橡胶支座、桥墩和群桩基础,并最终传递至饱和地基全过程的耦合分析模型。在频域和时域中进行了模型关键参数分析,并详细研究了群桩基础承台荷载、饱和地基位移和孔压响应随荷载移动速度、自振频率和初始位置的变化,并考察了地表速度响应随观察点横向距离的衰减规律。研究首次表明带支座箱梁的一阶挠曲自振频率、簧上轨道板的共振频率和群桩基础上桥墩一阶挠曲自振频率对饱和地基动力响应有明显影响。
本论文研究工作丰富了饱和土体动力响应分析方法,为高架高速铁路引起的环境振动预测提供了理论指导和科学依据。
With a large number of high speed railway lines being built and the running speed on existed main railway lines being increased in China, the train-induced environmental vibration pollutions have been increasingly significant. Especially in southeastern coastal areas, where the ground soil is soft and saturated, the train operation speed can easily exceed the Rayleigh wave speed of ground to generate excessive ground vibrations. Among the high speed railways lines with an operation speed of more than300km/h in China, such as the Jing-Jin, Jing-Hu, Hu-Hang, Hu-Ning and Hang-Yong railway lines, the elevated railway lines constitutes80%of the total railway length. However, little research has been conducted on the environmental vibrations caused by elevated high-speed railway line on saturated ground. Based on Biot's theory, the following researches concerning calculations of the responses of three-dimensional saturated ground both in time and frequency domains as well as predictions of the environmental vibrations caused by elevated high-speed railway lines are conducted:
1. Under the action of a moving harmonic point load, the soil skeleton displacements and pore pressures are solved for a saturated half-space governed by the simplified u-p formulations, the fully undrained formulations and the fully drained formulations, and then compared with Biot's complete u-w formulation to determine in which circumstances the simplified formulation can be used to replace to the complete formulation. To do so, a dimensionless parameter RΩ, named maximum frequency ratio, is proposed to quantify the application scopes of the various formations.
2. Based on the complete u-w formulation of Biot's theory, the governing equations of the three-dimensional finite element of the saturated soil are derived by using Galerkin method. Accordingly, the two-and three-dimensional finite elements of the saturated soil are programed and inserted into a self-developed finite element solver. The consolidations and wave-propagations in a saturated soil column as well as the dynamic responses of a two-dimensional saturated ground are studied and then compared with analytical and published results to testify the correctness of the developed saturated-soil finite elements. Meanwhile, the effects of the load speed and the soil permeability on displacements of the soil skeleton and the pore fluids are studied for a plan-strain saturated ground under the action of a moving point load.
3. The multi-transmitting formula (MTF), which is originally proposed for the single-phase medium, is generalized and applied to the finite-element modellings of the saturated-ground dynamic responses. Firstly, expressions of the refection coefficients of the second-order MTF boundary condition, which is applied at boundaries of a finite element grid discretizing a saturated ground governed by the u-w formulation, are analytically derived in frequency domain for incidences of the plain P1, P2and SV waves, respectively. The effects of the dimensionless frequency ratio, grid size and time step length on the boundary reflection coefficients and the reflection angles are investigated. Then, the second-order MTF boundary condition is implemented into the self-developed finite element solver. By comparing with the analytical solutions and the reference results obtained from setting the boundary far away from the studied area, the effectiveness of the second-order MTF boundary condition are testified for the finite-element modellings of the one-dimensional, two-dimensional and three-dimensional saturated ground and of the dynamic responses of saturated ground under the actions of moving loads.
4. A three-dimensional layered-saturated ground model is established with non-reflection boundary conditions applied at the ground bottom to simulate the ground's infinity. Note that the ground bottom should be fixed if there were overlying rigid rocks. The model is solved by using the thin layer element method to obtain the dynamic Green function of the three-dimensional layered saturated ground in frequency domain. By comparing with published results, the obtained Green function is proved to be correct in reproducing the dynamic responses of the three-dimensional saturated ground under harmonic loadings with different oscillating frequency and at different depths. It is noted that the obtained Green function involves no numerical integral inversions of the Hankel functions and can take the soil stratification into account in an easy way.
5. An analytical model incorporating the pile groups and the three-dimensional saturated ground is established basing on the flexible volume method. The piles are discretized by three-dimensional Euler-Bernoulli beam elements. The stiffness matrix of the saturated ground is obtained by inversing the receptance matrix, which is formed among the pile-soil interaction nodes by using the obtained Green function. By comparing with published results, the correctness of the established model is justified. Then the horizontal, vertical, rocking and torsional impedances of the pile groups embedded in three-dimensional saturated ground are studied for different combinations of the soil permeability and the excitation frequency. Specially, the free-field displacements and pore pressures of the saturated ground are investigated for unit forces and unit bending moments applied at the pile-cap center, respectively.
6. A vertically-coupled analytical model incorporating the moving oscillating point load, the multi-span elastically supported bridge girders and the slab track carrying an infinite rail beam is established and solved by using the periodic structure theory. The displacement spectrums of the rail, the slab and the box girder are studied for different sets of load oscillating frequencies and load velocities. Meanwhile, both the rail displacements and the reaction forces of the girder supports are investigated for different combinations of the load velocities, oscillating frequencies and initial positions in time-domain. It is concluded that the deformations and the vibration energy will be confined in the rail and the railpads when the quasi-receptance of the rail is lower than that of the railpad, then the loads being transferred into the girder supports will be substantially reduced. Two characteristic frequencies of the elevated railway bridge are identified:(i) the first flexural natural frequency of the girder on springs representing the elastic bearings;(ii) the resonance frequency of the slab on springs account for the under-slab CA cushion layer.
7. As a common type of elevated railway bridges in China, a structure pattern of a single-box girder carrying double slab tracks, which is supported by pile foundations, is often utilized. For studying the environmental vibrations caused by the above-mentioned elevated bridges, a vertically-coupled analytical model is proposed to reproduce the transmission processes of the moving train load to the girder supports by the rail, the railpads, the slabs, the CA cushion layer and the box girder, and then to the saturated ground via the piers and the pile foundations. Three key parameters of the proposed model are studied firstly in both frequency and time domains. Then the loads on pile cap center, the soil skeleton displacements and the pore pressures are investigated for different combinations of the load velocities, oscillating frequencies and initial positions. Also the decreasing of the velocities at the ground surface with the increasing of the transverse distance between the observation point and the railway central line is studied. Three characteristic frequencies are revealed, for the first time, for the dynamic responses of the saturated ground:(i) the first flexural natural frequency of the girder on springs representing the elastic bearings;(ii) the resonance frequency of the slab on springs account for the under-slab CA cushion layer;(iii) the first flexural natural frequency of the pier on springs representing impedances of the pile foundation.
The present work enriches the analysis methods for the dynamic responses of the saturated soil, and is of theoretical significance and application prospect in solving the environmental vibration problems caused by the elevated high-speed railway lines.
1.以移动谐振荷载下均质饱和半空间土骨架位移和孔压为表征量,通过与完整u-1w格式对比,提出了无量纲参数——最大频率比RΩ来定量判定u-p格式、完全不透水格式和完全透水格式等三种简化控制方程在移动荷载下饱和地基动力响应模拟中的适用性。
2.根据完整u-w格式的Biot波动控制方程,采用Galerkin法推导了三维饱和土体单元控制方程,编制了相应二维和三维饱和土体单元程序,并嵌入到自主开发的三维动力有限元分析程序中。分析了一维饱和土柱的固结、波动问题以及二维地基的动力响应问题,并与解析解和已有研究成果进行了对比验证了程序的正确性。同时求解了移动点荷载作用下的平面应变饱和地基动力响应,研究了荷载速度和渗透系数对地表土骨架位移以及孔隙流量的影响。
3.将基于单相介质提出的多次透射人工边界(MTF)推广应用于饱和地基动力响应的有限元模拟中。首先,基于饱和土体的Biot u-w格式动力方程,在频域中解析求解了平面P1、P2和SV波入射时,有限元离散网格中二阶MTF人工边界稳态反射系数的表达式,并分析了不同无量纲频率比、无量纲网格步距和无量纲时间步距下的MTF人工边界反射系数和反射角。继而,在自主开发的有限元动力分析程序中实现了上述二阶MTF人工边界,通过与解析解和边界远置参考解对比,说明了二阶MTF人工边界在一维、二维和三维饱和地基动力响应有限元模拟中的有效性以及在移动荷载下饱和地基动力响应有限元模拟中的适用性。
4.建立了三维层状饱和地基模型并在模型底边施加频域无反射人工边界(考虑地基下卧基岩时则模型底边固定),采用薄层单元法求解了三维层状饱和地基的频域动力Green函数。通过与已有研究成果对比,说明了所求Green函数能准确计算不同振动频率、不同埋深位置处简谐荷载激振下的三维饱和地基的动力响应,所求Green函数无需关于Hankel函数进行数值积分逆变换且能直接计入地基的成层性,提高了计算效率。
5.采用容积法建立了三维饱和地基—群桩基础频域耦合动力分析模型,其中桩体离散为三维Euler-Bernoulli梁单元,饱和地基刚度矩阵则由基于薄层单元法推导的Green函数在各桩土相互作用节点间形成的柔度矩阵求逆而得。通过与已有研究成果对比说明了所建模型的正确性,在此基础上,分析了不同地基渗透系数和激振频率下饱和地基中群桩基础的水平、竖向、摇摆和扭转阻抗函数以及承台中心施加单位力和弯矩简谐荷载时,饱和地基自由场的位移和孔压响应。
6.建立了移动谐振荷载下、等跨简支箱梁上覆无限长板式轨道的竖向动力分析模型,并利用周期结构动力响应的性质进行求解。分析了不同荷载自振频率、移动速度和初始位置下,钢轨、轨道板和箱梁的位移响应频谱、钢轨的时域位移响应和箱梁支座的反力时程。研究表明,当钢轨准柔度低于扣件弹簧柔度时,振动变形和能量集中在钢轨和扣件中,递到箱梁支座的荷载比例会明显降低；带支座箱梁的一阶挠曲自振频率以及簧上轨道板(弹簧模拟轨道板下的CA砂浆垫层)的共振频率对结构动力响应有显著贡献。
7.针对我国高架高速铁路工程常见的上覆板式轨道、下设群桩基础的双线整体式简支箱梁结构形式,提出了一种能详细考虑列车荷载经由钢轨、扣件、轨道板和CA砂浆垫层等细观轨道构件传递至箱梁、橡胶支座、桥墩和群桩基础,并最终传递至饱和地基全过程的耦合分析模型。在频域和时域中进行了模型关键参数分析,并详细研究了群桩基础承台荷载、饱和地基位移和孔压响应随荷载移动速度、自振频率和初始位置的变化,并考察了地表速度响应随观察点横向距离的衰减规律。研究首次表明带支座箱梁的一阶挠曲自振频率、簧上轨道板的共振频率和群桩基础上桥墩一阶挠曲自振频率对饱和地基动力响应有明显影响。
本论文研究工作丰富了饱和土体动力响应分析方法,为高架高速铁路引起的环境振动预测提供了理论指导和科学依据。
With a large number of high speed railway lines being built and the running speed on existed main railway lines being increased in China, the train-induced environmental vibration pollutions have been increasingly significant. Especially in southeastern coastal areas, where the ground soil is soft and saturated, the train operation speed can easily exceed the Rayleigh wave speed of ground to generate excessive ground vibrations. Among the high speed railways lines with an operation speed of more than300km/h in China, such as the Jing-Jin, Jing-Hu, Hu-Hang, Hu-Ning and Hang-Yong railway lines, the elevated railway lines constitutes80%of the total railway length. However, little research has been conducted on the environmental vibrations caused by elevated high-speed railway line on saturated ground. Based on Biot's theory, the following researches concerning calculations of the responses of three-dimensional saturated ground both in time and frequency domains as well as predictions of the environmental vibrations caused by elevated high-speed railway lines are conducted:
1. Under the action of a moving harmonic point load, the soil skeleton displacements and pore pressures are solved for a saturated half-space governed by the simplified u-p formulations, the fully undrained formulations and the fully drained formulations, and then compared with Biot's complete u-w formulation to determine in which circumstances the simplified formulation can be used to replace to the complete formulation. To do so, a dimensionless parameter RΩ, named maximum frequency ratio, is proposed to quantify the application scopes of the various formations.
2. Based on the complete u-w formulation of Biot's theory, the governing equations of the three-dimensional finite element of the saturated soil are derived by using Galerkin method. Accordingly, the two-and three-dimensional finite elements of the saturated soil are programed and inserted into a self-developed finite element solver. The consolidations and wave-propagations in a saturated soil column as well as the dynamic responses of a two-dimensional saturated ground are studied and then compared with analytical and published results to testify the correctness of the developed saturated-soil finite elements. Meanwhile, the effects of the load speed and the soil permeability on displacements of the soil skeleton and the pore fluids are studied for a plan-strain saturated ground under the action of a moving point load.
3. The multi-transmitting formula (MTF), which is originally proposed for the single-phase medium, is generalized and applied to the finite-element modellings of the saturated-ground dynamic responses. Firstly, expressions of the refection coefficients of the second-order MTF boundary condition, which is applied at boundaries of a finite element grid discretizing a saturated ground governed by the u-w formulation, are analytically derived in frequency domain for incidences of the plain P1, P2and SV waves, respectively. The effects of the dimensionless frequency ratio, grid size and time step length on the boundary reflection coefficients and the reflection angles are investigated. Then, the second-order MTF boundary condition is implemented into the self-developed finite element solver. By comparing with the analytical solutions and the reference results obtained from setting the boundary far away from the studied area, the effectiveness of the second-order MTF boundary condition are testified for the finite-element modellings of the one-dimensional, two-dimensional and three-dimensional saturated ground and of the dynamic responses of saturated ground under the actions of moving loads.
4. A three-dimensional layered-saturated ground model is established with non-reflection boundary conditions applied at the ground bottom to simulate the ground's infinity. Note that the ground bottom should be fixed if there were overlying rigid rocks. The model is solved by using the thin layer element method to obtain the dynamic Green function of the three-dimensional layered saturated ground in frequency domain. By comparing with published results, the obtained Green function is proved to be correct in reproducing the dynamic responses of the three-dimensional saturated ground under harmonic loadings with different oscillating frequency and at different depths. It is noted that the obtained Green function involves no numerical integral inversions of the Hankel functions and can take the soil stratification into account in an easy way.
5. An analytical model incorporating the pile groups and the three-dimensional saturated ground is established basing on the flexible volume method. The piles are discretized by three-dimensional Euler-Bernoulli beam elements. The stiffness matrix of the saturated ground is obtained by inversing the receptance matrix, which is formed among the pile-soil interaction nodes by using the obtained Green function. By comparing with published results, the correctness of the established model is justified. Then the horizontal, vertical, rocking and torsional impedances of the pile groups embedded in three-dimensional saturated ground are studied for different combinations of the soil permeability and the excitation frequency. Specially, the free-field displacements and pore pressures of the saturated ground are investigated for unit forces and unit bending moments applied at the pile-cap center, respectively.
6. A vertically-coupled analytical model incorporating the moving oscillating point load, the multi-span elastically supported bridge girders and the slab track carrying an infinite rail beam is established and solved by using the periodic structure theory. The displacement spectrums of the rail, the slab and the box girder are studied for different sets of load oscillating frequencies and load velocities. Meanwhile, both the rail displacements and the reaction forces of the girder supports are investigated for different combinations of the load velocities, oscillating frequencies and initial positions in time-domain. It is concluded that the deformations and the vibration energy will be confined in the rail and the railpads when the quasi-receptance of the rail is lower than that of the railpad, then the loads being transferred into the girder supports will be substantially reduced. Two characteristic frequencies of the elevated railway bridge are identified:(i) the first flexural natural frequency of the girder on springs representing the elastic bearings;(ii) the resonance frequency of the slab on springs account for the under-slab CA cushion layer.
7. As a common type of elevated railway bridges in China, a structure pattern of a single-box girder carrying double slab tracks, which is supported by pile foundations, is often utilized. For studying the environmental vibrations caused by the above-mentioned elevated bridges, a vertically-coupled analytical model is proposed to reproduce the transmission processes of the moving train load to the girder supports by the rail, the railpads, the slabs, the CA cushion layer and the box girder, and then to the saturated ground via the piers and the pile foundations. Three key parameters of the proposed model are studied firstly in both frequency and time domains. Then the loads on pile cap center, the soil skeleton displacements and the pore pressures are investigated for different combinations of the load velocities, oscillating frequencies and initial positions. Also the decreasing of the velocities at the ground surface with the increasing of the transverse distance between the observation point and the railway central line is studied. Three characteristic frequencies are revealed, for the first time, for the dynamic responses of the saturated ground:(i) the first flexural natural frequency of the girder on springs representing the elastic bearings;(ii) the resonance frequency of the slab on springs account for the under-slab CA cushion layer;(iii) the first flexural natural frequency of the pier on springs representing impedances of the pile foundation.
The present work enriches the analysis methods for the dynamic responses of the saturated soil, and is of theoretical significance and application prospect in solving the environmental vibration problems caused by the elevated high-speed railway lines.
引文
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