Abstract This paper addresses the determination of the closed-form solutions of redundantly constrained stochastic frames in terms of the probability density function (PDF). The full characterization of any response random variable of the structures, that are characterized by spatially random deformability (or its inverse, the stiffness), has been identified through the application of the force method and the application of the probability transformation method (PTM). The first method one allows for solving the redundancy while the PTM gives the explicit relationship between the PDFs of the response and of bending deformability which is assumed to be a mono-axial Gaussian random field. The formulation becomes particularly simply in the case of beams, for which the exact solutions of some benchmark applications are given, confirming the goodness of the present approach.

中文翻译：

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冗余约束随机框架的闭式解

摘要 本文讨论了根据概率密度函数 (PDF) 确定冗余约束随机框架的封闭形式解的问题。通过力法和概率变换法 (PTM) 的应用，已经确定了结构的任何响应随机变量的完整表征，其特征在于空间随机变形性（或其逆，刚度）。第一种方法允许解决冗余问题，而 PTM 给出了响应的 PDF 和弯曲变形能力的 PDF 之间的明确关系，弯曲变形能力假定为单轴高斯随机场。在梁的情况下，公式变得特别简单，为此给出了一些基准应用程序的精确解，