马蹄形断面隧洞收缩水深的简易算法

摘要：由于标准马蹄形断面分别由不同圆形半径的圆弧连接而成，因此，断面形式与其它常用输水断面比较相对复杂，收缩水深的计算不但需完成较复杂的超越方程求解，而且计算公式通过分段函数给出，采用解析法无法直接获解，而传统算法（图表法、试算法、迭代法）既繁琐误差又大。笔者对该断面收缩水深基本方程变形整理后的超越函数进行优化拟合替代，并以标准剩余差最小为目标函数，在工程适用参数范围内，经拟合计算获得了表达形式简单直观、便于实际工程应用、计算精度满足设计要求（最大误差为1．396％）的近似公式，具有一定的实用推广意义。

Simplified Method for Calculation of Shrinkage Depth in Tunnel with U-shaped Section

as the typical U-shaped section consists of arcs with different radiuses, it is relatively complicated compared with other conven- tional sections. Calculation of the shrinkage depth not only fulfills solution with complicated transcendental equation but also provides the calculation formulas by sectional functions because the calculation cannot be performed directly by application of analysis method. But the conventional methods （ graphology, trial method and iteration method） are complicated and numerous as well as with higher errors. In this paper, the transcendental equation resulted from transformation of the basic equations for the shrinkage depth at the section is replaced by the optimized fitting and the minimum typical remaining difference is considered as the objective function. In range of the applicable engi- neering parameters, the approximate formulas which are simple, direct and convenient for practical application as well as their calculation precision satisfies design requirements （ maximum error, 1. 396% ） are derived from the fitting calculations. These formulas are worth of developing for practical application.