悬链线形断面临界水深的直接计算公式

Direct Formula for Critical Depth in Catenary Cross Sections

悬链线形断面临界水深的计算需求解含反双曲余弦函数的超越方程,数学上无解析解。传统的试算法或图表法计算过程复杂,且精度难以得到保证。由于该超越方程的复杂性,目前仅有的两套公式计算精度均不够高。运用逐次优化拟合原理提出新的直接计算公式,误差分析及实例计算结果表明,在工程适用参数范围内,临界水深计算值的最大相对误差绝对值小于0.10%,平均相对误差绝对值小于0.021%。该公式简明直观,能够满足较高的计算精度要求,且适用范围广,为工程设计及水工设计手册的编制提供了有益的参考。

Transcendental equations involving inverse hyperbolic functions have to be solved for computing critical water depth in catenary cross sections, which have no analytic solutions mathematically. Traditional methods of trial-and-error and chart look-up have complicated computational procedures and can hardly ensure requirements of accuracy. The two existing formulae are without high e nough accuracy due to complexity of the transcendental equations. A new direct formula is established based on the theory of gradual optimal fitting. Results of error analysis and application example indicate that maximum absolute relative error of the computed val- ues is less than 0.10% and average absolute relative error is less than 0. 021% within the practical scope of project. The proposed formula is concise, intuitional and with wide application range, meeting requirements of high accuracy, which will be valuable for en- gineering practice and in the course of compiling handbook of hydraulic structure design.