Study on solution of limit state function’s gradients in reliability analysis

TAN Xiao-hui; LIU Xin-rong

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Study on solution of limit state function’s gradients in reliability analysis

更新时间：2026-06-25

- Study on solution of limit state function’s gradients in reliability analysis
- Vol. 27, Issue 6, (2006)
- 作者机构：
  
  1. 合肥工业大学资源与环境工程学院,合肥,230009
  2. 重庆大学土木工程学院,重庆,400030
- 作者简介：
- 基金信息：
- DOI：
  CLC： O24
- Online First：14 November 2013，
  
  Published：2006
- 稿件说明：
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TAN Xiao-hui, LIU Xin-rong. Study on solution of limit state function’s gradients in reliability analysis[J]. 2006, 27(6).
DOI：

TAN Xiao-hui, LIU Xin-rong. Study on solution of limit state function’s gradients in reliability analysis[J]. 2006, 27(6). DOI：

摘要

在岩土工程的可靠度分析中，功能函数的形式非常复杂，甚至是隐式的。而对于常用的可靠度分析方法，如一阶可靠性方法（FORM）、二阶可靠性方法（SORM）等，一个重要的问题是求功能函数对基本随机变量的梯度。因此，对于隐式的或复杂的显式功能函数，必须采用数值微分方法来求解功能函数的梯度。对于可靠度分析中常用的有限差分法（FDM）及有理多项式法（RPT）这两种数值微分方法，本文详细研究了二者间的异同及其求导结果与步长的关系，指出了对于相同的步长控制系数及取样方式，FDM是RPT在线性情况下的特例；对于连续的线性功能函数，可直接用3点FDM求导；对于连续的非线性功能函数，可用5点RPT求导；对于非连续的功能函数，应采用RPT求导。建议取步长控制系数等于1。

Abstract

The limit state function in the reliability analysis of geotechnical engineering is often very complicated or even implicit

so it’s difficult to calculate the limit state function’s gradients analytically. However

the solution of the gradients is necessary and important in some general used reliability analysis methods

such as the first order reliability method(FORM) and the second order reliability method(SORM). Therefore

it’s necessary to solve the gradients of the limit state function by numerical methods. The authors studied carefully two numerical differential methods

finite difference method(FDM) and rational polynomial technique(RPT)

and pointed that FDM is the special case of RPT when the limit state function is linear. Accordingly

FDM3 can be used for the continuous linear limit state function and RPT5 for the continuous nonlinear limit state function. For the discontinuous limit state function

RPT should be used because of the large error of FDM. In each case

the control coefficient of step size is suggested to be 1.

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