## Doctoral Dissertations

January 2012

#### Keywords

Mathematics|Engineering, Mechanical

Ph.D.

#### Abstract

This work entails novel methodologies to be used in risk propagation as well as risk allocation in complex engineering systems. Risk propagation is the process of calculating the risk of failure of the systems based on uncertainties in the design variables (forward analysis). Risk allocation is the process of optimally designing acceptable uncertainties of the design variables to render desired reliability of the system (inverse analysis). ^ Failure is defined as the inability of a system to perform its intended functions within specified performance requirement. Therefore, the design variables space can be divided into two regions: a failure region and a non-failure region. The function achieving this division is called the limit state function. The probability of failure is defined as the possibility that the limit state function reaches a desired threshold. The computational complexity in calculating this probability is insurmountable. Thus there is a compelling need to have efficient and reliable approximations for calculating the probability of failure. In this work, three different analytical methodologies are presented to such calculations. These methods in turn are used in the design process to optimally allocate the acceptable risk of failure of the system to individual design variables. ^ In the first method, a geometric programming optimization is used as the foundation for both deterministic system and stochastic risk based design of the complex engineering systems. In these methods, conservative, near optimum risk allocations are readily available with drastically low computational cost. ^ In the second method, an Eigen-decomposition based methodology is proposed for the risk allocation problem. A diagonal surrogate system is built to mimic the worst-case scenario of the original system. The largest eigenvalue of the covariance matrix of the original system is used to build the main diagonal of the surrogate system. The optimum allocation for the whole system is derived for the surrogate system instead. This method provides the readily available optimum risk allocation by using the intrinsic dynamic of the system. Computationally, compared to the first method, this method has one extra step, the eigenvalue calculation. ^ In the third method, a novel framework is proposed for the risk based design optimization. The fundamental development of this method is an analytical upper bound for calculating the probability of failure. This is in contrast with commonly used First Order Reliability Method (FORM), where a lower bound is used in calculating the probability of failure. In this method, we show that FORM results in an optimistic measure of risk, hence it is potentially catastrophic in engineering design. A more accurate measure of failure is proposed by utilizing an analytical upper bound for the distribution of limit state function. This distribution is a function of the eigenvalues of the linearized limit state function in the normal space. This results in a better understanding of failure phenomenon. ^ All three methods developed in this work are superior in applicability, ease of use, as well as efficiency to the existing methods in the literature. Their efficacy is demonstrated and benched marked in several design cases. ^

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