An innovative Importance Sampling (IS) in calculating reliability for a structural engineering problem using multiple spheres is proposed. Radial-based Importance Sampling (RBIS) builds a single sphere with its center at the origin and a radius of β(adistance from the most probable point to the origin) to recognize the safety samples located inside the sphere. Such samples are excluded for function evaluation to reduce the computational cost. Adaptive radial-based importance sampling (ARBIS) extended RBIS with an adaptive scheme to determine the optimal radius β. To maximize the number of safety samples, multiple spheres with various centers and radii are recommended in current study. Two types of spheres are introduced: the “origin” and “non-origin spheres”. It is shown that in addition to “origin sphere”, the “non-origin spheres” can exclude more safety samples. As a results, computational efficiency is significantly enhanced. Similar to RBIS, samples outside the “origin sphere” are generated in the proposed method. However, only part of these samples is evaluated by the limit state function. A simple but robust line search is adopted to determine the radius of each sphere. Effects of sphere number and locations are discussed. Robustness and efficiency of the proposed method are demonstrated by various benchmark problems. Results show that for most cases, the proposed method greatly reduces the number of function evaluation with similar accuracy and uncertainty levels compared to those of Monte Carlo Simulation (MCS), RBIS and ARBIS.

提出了一种创新的重要性抽样（IS），用于计算使用多个球体的结构工程问题的可靠性。基于径向的重要性采样（RBIS）构建一个球体，其中心位于原点，半径为β（从最可能的点到原点的距离），以识别位于球体内的安全样本。排除这些样本以进行功能评估以减少计算成本。自适应的基于径向的重要性抽样（ARBIS）通过自适应方案扩展了RBIS，以确定最佳半径β。为了使安全样品的数量最大化，当前的研究中建议使用多个具有不同中心和半径的球体。引入了两种类型的球体：“原始球体”和“非原始球体”。结果表明，除了“起源球”以外，“非起源球”还可以排除更多的安全样本。结果，大大提高了计算效率。与RBIS相似，在所提出的方法中会生成“原始球体”之外的样本。但是，这些样本中只有一部分由极限状态函数评估。采用简单但健壮的线搜索来确定每个球体的半径。讨论了球数和位置的影响。各种基准测试问题证明了该方法的鲁棒性和有效性。结果表明，在大多数情况下，