深层平板载荷试验变形模量计算公式推导

Deduction of calculation formula for deformation modulus of deep flat plate load test

基于Mindlin弹性理论位移解,对深层平板载荷试验变形模量的计算公式进行了系统的推导和阐述,得出以下结论:(1)Mindlin课题下半空间体表面的竖向位移解等于Boussinesq课题下深度c处的竖向位移解;(2)当荷载作用深度与荷载圆半径之比m固定时,荷载圆中心点处的计算系数I和边界处的计算系数I均随着泊松比μ的增大先增大后减小,而I/I却正好相反,两者的极值均出现在μ=0.3左右;在固定的μ值下,Ⅰ_(0)、Ⅰ_(a)和Ⅰ_(0)/Ⅰ_(a)均随着m的增大而减小;当m值由10变化至15时,由Mindlin竖向位移解推导出的荷载圆面积中心点处的沉降为边界处沉降的1.44~1.46倍;(3)深层平板载荷试验由于存在开口效应,其理论模型并不严格等同于Mindlin课题,文章推导出来的计算结果可视为理论解析解的下限解;(4)文章推导出的综合计算系数ω值比规范提供值小,当承压板直径与试验深度之比d/z固定时,其偏差随着μ的增大而增大,而当μ固定时,其偏差随着d/z的减小而降低。该文系统地解答了有关深层平板载荷试验中变形模量计算公式的来源问题,对相关学者全面的了解此类问题有一定指导和借鉴意义。

Based on the displacement solution by the Mindlin elastic theory, the calculation formula of the deformation modulus of the deep flat plate load test is systematically deduced and explained.Following conclusions are drawn:(1)The vertical displacement solution of the half-space body surface obtained for the Mindlin problem is equal to the vertical displacement solution at the depth c obtained for the Boussinesq problem;(2)When m, the ratio of the depth of the load action to the radius of the load circle, is fixed, the coefficient Iat the center point of the load circle and the coefficient Iat the boundary first increase and then decrease with the Poisson’s ratio.But I/Iis just the opposite, and their extreme values appear at about μ=0.33.For a fixed μ, I, Iand I/Iall decrease with m.When the value of m increases from 10 to 15, the settlement at the center point of the load circle area derived from the Mindlin vertical displacement solution is 1.44 to 1.46 times the settlement at the boundary.(3)Due to the opening effect of the deep plate load test, its theoretical model is not strictly equivalent to the Mindlin problem.The calculation results derived in this paper can be regarded as the lower limit solution of the theoretical analytical solution;(4)The value of the comprehensive calculation coefficient ω deduced in this paper is smaller than that provided by the specification.When d/z, the ratio between the diameter of the bearing plate and the test depth is fixed, the deviation increases with μ.And when the μ is fixed, the deviation decreases with the d/z.This paper systematically answers the question about the source of the calculation formula of the deformation modulus in the deep flat plate load test, which provides guidance and reference significance for related scholars to fully understand such problems.