Applied Mathematics and Mechanics

, Volume 38, Issue 1, pp 57–72 | Cite as

Matrix description of differential relations of moment functions in structural reliability sensitivity analysis

  • Tianxiao ZhangEmail author


In a structural system reliability analysis that lacks probabilistic information, calculating the numerical characteristics of the state functions, especially the first four moments of the state functions, is necessary. Based on that, the structural system reliability is analyzed with a fourth-order moment method. The reliability sensitivity is required to conduct the differential operation of the numerical characteristic functions. A reliability sensitivity analysis formula is then derived in combination with the relation of the differential operation. Based on the matrix theory and Kronecker algebra, this paper systematically derives a matrix expression of the first four moments of the state functions, and establishes the matrix relation between the first four moments of the state functions and those of the basic random variables. On this basis, a differential operation formula of the first four moments of the state functions is further derived against the first four moments of the basic random variables. The vector relation between the state functions and the multidimensional basic random variables is described by means of the matrix operation to extend the operation method. Finally, a concise and intuitive formula is obtained to explore the inherent essential relation between the numerical characteristics of the state functions and those of the basic random variables, leading to a universal equation for the two kinds of numerical characteristics.


structural system reliability analysis state function numerical characteristic matrix description Kronecker product functional differential reliability sensitivity 

Chinese Library Classification


2010 Mathematics Subject Classification



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Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial EngineeringUniversity of Illinois at ChicagoIllinoisUSA

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