半无限弹性体内作用竖向矩形和条形均布荷载时的应力计算公式

Equations for calculating stresses in a semi-infinite elastic solid subjecting to a vertical rectangular and strip uniform load beneath ground surface

传统土力学处理岩土工程问题时,经常采用弹性力学方法分析土中应力分布。土中应力通常依据较为简单的Boussinesq解计算,该解答假定荷载作用于土体表面;但是,建筑物基础一般都埋置在地表以下一定深度,此时,应依据Mindlin解计算应力分布。在半无限弹性体内作用竖向矩形和条形均布荷载时的应力分布对计算基础沉降以及分析基坑尺寸对稳定性和变形的影响均有重要意义,在其他岩土工程问题中也有诸多应用。尽管文献[1-2]给出了相关计算公式,但存在几处错误。基于Mindlin解,通过积分重新推导了在半无限体内部作用竖向矩形均布荷载时应力分布的解析表达式,进一步得到了条形均布荷载作用时应力分布的解析表达式,这2组公式均与文献[1-2]给出的结果存在不同。最后,采用数值积分方法验证了新给出的2组公式的正确性。

In dealing with geotechnical engineering problems with the traditional soil mechanics, the elastic mechanics approach is often used to calculate the stress distribution in soil. The calculation of the stresses in soil is usually based on simple Boussinesq＇s solutions which are derived on the assumption that the loading acts on the ground surface. However, building foundations are generally buried in a certain depth beneath ground surface. In this case, the Mindlin solution is more suitable to calculate the stress distribution. The equations for calculating stresses in a semi-infinite elastic solid subjecting to a vertical rectangular and strip uniform load are significant for many geotechnical problems, such as the calculation of foundation settlement and the analysis of the influence of excavation size on its stability and deformation. Although all the equations have been presented in Reference[1-2] by Yuan Ju-yun, several errors are found in these equations. Based on the Mindlin＇s formulas, the calculation equations for calculating stress in the soil subjecting to a vertical rectangular uniform load beneath the surface are derived by the integral method again. Furthermore, the equations for strip load in similar case are deduced, both of which are different from those given in Reference[1-2]. Eventually, the correctness of two groups of formulations is verified by the numerical integration.