半空间体上无限长轨道梁作用集中力的Fourier变换解

A FOURIER TRANSFORM SOLUTION FOR AN INFINITE BEAM ON A SEMI-INFINITE SPACE LOADED BY A CONCENTRATED FORCE

从弹性力学半空间问题的Boussinesq解出发,建立无限长梁的弯曲微分方程与半空间体上表面的位移表达式,对两个方程进行Fourier变换联合求解,然后通过Fourier逆变换求得半空间体的支承反力与梁的位移表达式。通过理论解与有限元分析比较,求得Fourier变换解中的待定系数C=γ+e≈3.295 5。分析发现,半无限空间上的梁的特征长度是开四次方的量。最后提出梁下承压应力计算的等效计算长度,用于下部半空间材料的强度计算。所得到的梁的弯矩和剪力公式,可用于轨道梁的强度计算。

This paper made a study on the classic problem of an infinite beam on a semi-infinite space acted by a concentrated load on the beam. One of the basic equation to determine the settlement of the semi-space below the beam was based on the well-known Boussinesq＇s solution of a normal load on a semi-infinite space, the other was the beam differential equation. Fourier transform technique was used to solve the equations. Because the Fourier transform was involved with an integration of function with singularity, it was found that a constant could not be determined analytically. So finite element analysis was used. Different values of the constant were tried to get a best fit to the FE results,and the undetermined coefficient was found to be C=y＋e≈3.2955. The characteristic length of the problem was found to be a fourth root expression rather than a third root one in the current available literature. Finally an equivalent bearing length was proposed to simplify the computation of local maximum bearing compressive stess of the beam under the load. The formulae of bending moment and shear force of the beam could be used strength calculation of track beam.