岩土工程可靠度分析的拉格朗日乘数法

基于可靠指标标准正态空间计算的数学模型,运用拉格朗日乘数法,提出了计算岩土工程设计验算点和可靠指标的方法.该方法适用于任何概率分布的相关变量,不必计算当量正态均值和方差、相关变量独立变换,也适用于功能函数为高度线性时采用JC法计算可靠指标迭代不能收敛的情况.通过建立构造函数,然后求其对各个变量的一阶偏导并与结构的功能函数联立求解出验算点坐标,之后求出可靠性指标.由于标准正态分布函数的反函数表达式不能给出,在求解其导数时可使用有理多项式法求解.通过二个算例分析表明该方法简单、可靠和有效,值得在工程实践中推广.

Lagrange Multiplier Method for Reliability Analysis

Based on the computational mathematical model of reliability index in the standard normal space,the Lagrange multiplier method is introduced and applied to the test point and the reliability index analysis of geotechnical engineering.The Lagrange multiplier method has many advantages:it is applicable to correlated variables with any probability distributions;it need not compute the equivalent normal means and standard deviations and independence transform of correlated variables and it is applicable to the procedure of the iteration of reliability index is not converged when the limit state function is highly nonlinear.Through establishing the constructor and calculating the first order partial derivatives of the variables of the constructor,combining the limit state function,the test point and the reliability index can be computed.The rational polynomial technique(RPT5) can be used to calculate the derivative of the inverse function of the standard normal distribution function because the expression of the inverse function of the standard normal distribution function can not be given.The results of four examples reliability analysis show that the Lagrange multiplier method is simple、reliable and effective and can be applied tremendously in practice.