###### Abstract  
Embarked from the practical conditions of small samples in time-invariant and time-variant uncertainties, a complete non-probabilistic analysis procedure containing uncertainty quantification, uncertainty propagation, and reliability evaluation is presented in this paper. Firstly, the Grey systematic approach is proposed to determine the boundary laws of static intervals and dynamic interval processes. Through a combination of the policies of the second-order Taylor expansion and the smallest parametric interval set, the structural response histories via quantitative uncertainty results are further confirmed. Additionally, according to the first-passage idea from classical random process theory, the study on the time-dependent reliability measurement on the basis of the interval process model is carried out to achieve a more elaborate estimation for structural safety during its whole life cycle. A numerical example and one experimental application are eventually discussed for demonstration of the usage and reasonability of the methodology developed. uncertainty quantification and propagation; the time-dependent reliability; the Grey systematic approach; the second-order Taylor expansion; the smallest parametric interval set; the first-passage idea +
Footnote †: journal: Applied  
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Footnote 1: institutetext: Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University,  
Beijing 100083, China; [EMAIL_ADDRESS]  
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mation. Several strategies based on statistical theory, including point estimation [14], maximum likelihood estimation [15; 16], hypothesis testing [17; 18], Bayesian method [19; 20; 21], etc. are presented to deal with the above UQ problems, but they generally require adequate data from experiments to determine the statistical characteristics of uncertain parameters so that a weak practicability is inevitable. Under such circumstances, the structural uncertainty analysis based on insufficient samples, particularly the set-theoretical UQ analysis, has gradually attracted more and more attention from scientists and engineers [22]. In late 1960s, [PERSON] [23] first proposed an integral concept of ellipsoidal modeling, quantifying uncertain parameters within an ellipsoid. [PERSON] et al. [6; 24] introduced a minimum-volume parallelepiped for uncertainty buckling analysis. [PERSON] et al. [25] characterized uncertainty by defining two measurements of the ellipsoid, namely, the volume and the sum of the squares of its semi-axes when dealing with parametric identification problems. [PERSON] et al. [26] improved the minimum volume criterion in [24], and further developed the convexity method as well as the interval method for static mechanics. [PERSON] et al. [27] presented a unified method with regard to the correlation analysis as implementation of the classical set theory, and thus the difficulties in multidimensional uncertainty quantification could be tackled to some extent. However, the uncertain structure in practical applications always exhibits obvious time-varying characteristics because of some comprehensive reasons, including material property degeneration, environmental variations, and dynamic load processes, etc. How to guarantee the high reliability level during a system life-cycle under static and dynamic mixed uncertainties remains a big challenge [28]. In view of this, some progresses with regard to the fields of structural time-dependent reliability assessment have been made and reported. For example, [PERSON] et al. proposed a PHI2 method with random parameters and stochastic processes for solving [PERSON]'s classical formulas [29]; [PERSON] and [PERSON] established a probability density evolution model for dynamic response calculation and risk evaluation [30]; a reliability assessment method based on upcrossing rates was proposed by [PERSON] et al. [31] for the function generator mechanism; and the importance sampling method for the reliability evaluation was conducted by [PERSON] and [PERSON] [32] and [PERSON] and [PERSON] [33].