基于Betti-Rayleigh动力互易定理求解移动荷载引起的地基土振动

 基于Betti-Rayleigh动力互易定理，论证了地基土Green函数的互易性，推导了移动荷载引起的地基土振动解析解，进而推证出移动常力荷载和移动简谐荷载作用下地基土表面振动在波数–频率域内和空间–时间域内的解析表达式，并采用Matlab对其进行数值计算。通过互易定理的使用将振源的移动问题变换为地面上受振点的移动问题，使移动荷载作用下地基土振动响应的解析表达式大大简化。最后结合某地基土参数对移动的单位常力荷载和单位简谐荷载作用下的地面振动进行算例分析，结果表明：移动常力荷载引起的地面振动属于典型的低频振动；移动简谐荷载作用下地基土表面出现波动的Doppler效应，且地面上受振点的振动频率范围由最上层地基土的Rayleigh波波速控制；地面上的受振点距振源较近时，地基土表面R波对振动的贡献明显高于P波和S波，而当受振点距振源较远时，P波对振动的影响较明显。

SOLUTION OF FOUNDATION SOIL VIBRATIONS INDUCED BY MOVING LOADS BASED ON BETTI-RAYLEIGH DYNAMIC RECIPROCAL THEOREM

Based on the Betti-Rayleigh dynamic reciprocal theorem，the reciprocal property of the Green function is demonstrated；and the analytical solution of soil vibration subjected to moving loads is deduced. By means of double Fourier integral transform，the problem is firstly transformed into wave number-frequency domain，which makes it very convenient to derive the solution and furthermore to analyze the frequency characteristics of soil vibration. By successful application of the dynamic reciprocal theorem，the moving source problem is converted to the fixed source problem with receiver point moving in opposite direction，which greatly simplifies the complex analytical solution of soil/ground vibration induced by a moving load. A computer code for solving the ground responses subjected to the constant moving load and harmonic moving load is developed by using Matlab and is employed to perform a case study. The analytical results show that，for the constant moving load，the ground vibration is of typical low-frequency feature；while for the harmonic moving load，the frequency range of ground vibration is controlled by the Rayleigh wave velocity of the top layer soil；moreover，the Doppler effect appears in the ground vibrations. When the receiver is near the source path，R-wave contributes to ground vibration much more than P- and S-wave；while the receiver is far from the source path，the contribution of P-wave is more obvious.