In order to evaluate the failure probability corresponding to the Limit State Function (LSF) in structural reliability, the First Order Reliability Method (FORM) linearizes the LSF and directly calculates the failure probability based on the Most Probable Point (MPP). But this method is unable to effectively handle nonlinear problems. The Second Order Reliability Method (SORM), on the other hand, utilizes the Hessian matrix to obtain curvature information at the MPP and computes the failure probability with Breitung's second-order approximation formula. However, SORM is unable to maintain satisfactory computational accuracy in highly nonlinear problems due to its lack of detailed analysis on the shape of the limit state surface. To address this issue, this paper proposes an Improved Approximation Integration Method (IAIM). The initial approximation of the limit state surface is based on the intersection region between the hypersphere and the n-dimensional paraboloid. Subsequently, a statistical analysis of the deviation between the aforementioned region and its projected area is conducted to construct a corresponding fitting function, which is then utilized for dimensionality reduction in the multidimensional integration of the approximation area. Finally, several examples are presented to demonstrate the accuracy and feasibility of the proposed IAIM.

中文翻译：

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非线性可靠性分析的改进近似积分法


为了评估结构可靠性中极限状态函数（LSF）对应的失效概率，一阶可靠性方法（FORM）将LSF线性化，并根据最可能点（MPP）直接计算失效概率。但该方法无法有效处理非线性问题。另一方面，二阶可靠性方法（SORM）利用 Hessian 矩阵获取 MPP 处的曲率信息，并使用 Breitung 的二阶近似公式计算失效概率。然而，SORM由于缺乏对极限状态面形状的详细分析，在高度非线性问题中无法保持令人满意的计算精度。为了解决这个问题，本文提出了一种改进的近似积分方法（IAIM）。极限状态面的初始近似基于超球面和n维抛物面之间的相交区域。随后，对上述区域与其投影面积的偏差进行统计分析，构建相应的拟合函数，用于近似区域多维积分的降维。最后，给出了几个例子来证明所提出的IAIM的准确性和可行性。