Title: Developments of Two Methodologies in Uncertainty Quantification in Civil Engineering and Engineering Mechanics
Description: Uncertainty is ubiquitous in civil infrastructure systems and has a major impact on decision-making for structural safety and reliability, and for assessing and managing risk. This dissertation explores the field both from the theoretical-mathematical modeling point of view, as well as from the practical applications perspective. Two problems addressed in the dissertation: 1) how to quantify the randomness in material properties and 2) what’s the impact for the uncertainties in material properties on the response/safety of the civil structures. There are two distinct parts in the dissertation. Part I focuses on the strength evaluation of main cables for long span suspension bridges. A novel methodology is proposed to evaluate the ultimate strength of the main cables of suspension bridges using information obtained from site inspections and from tensile strength tests on selected wire samples extracted from the bridge’s main cables. A new model is proposed accounting for the spatial variation of individual wires’ strength along their length, an important physical attribute of corroded wires considered here for the first time.

This model includes: (1) mapping the corrosion stage variation along one-panel-long wires that are visible during an inspection, (2) establishing probability distribution functions for the ultimate tensile strength of 18″ long wire segments in each corrosion stage group, (3) generating random realizations of the ultimate strength of all the wires in the cable’s cross section, accounting for their strength variation along the entire panel length, and (4) accounting for the effect of broken wires in the evaluation panel as well as in adjacent panels. A Monte Carlo simulation approach is finally proposed to generate random realizations of the ultimate overall strength of the cable, using an incremental loading procedure. The final outcome is the probability distribution of the ultimate strength of the entire cable. The methodology is demonstrated through the cable strength evaluation of the FDR Mid-Hudson Bridge and Bear Mountain Bridge in New York state, and compared with corresponding results obtained using the current guidelines of NCHRP Report 534.

Part II is more theoretical in nature and focuses on estimating the stochastic response variability of structures with uncertain material properties modeled by stochastic fields. The concept of Variability Response Function (VRF) is applied in the dissertation to quantify the response variability (e.g., mean and variance) of statically indeterminate structures. Two types of response for statically indeterminate beams at a specific location x are studied: bending moment M(x) and displacement/deflection w(x). By solving the governing equations of the statically indeterminate structure, the responses along the length of the beam, M(x) and w(x), are expressed as a function of its (random) zero-moment location denoted by h. For bending moment M(x), combined with a second-order Taylor series expansion of the random zero-moment location h, novel Variability Response Function-based integral expressions for the variance of the response bending moment, Var[M(x)], are established. Extensive numerical examples are provided where the accuracy of the results obtained using the proposed formulation is validated using Monte Carlo simulations involving stochastic fields that follow truncated Gaussian and shifted lognormal probability distribution functions.

These Monte Carlo simulation results indicate that the proposed Variability Response Functions are probability-distribution-independent. For deflection w(x), by introducing hinges at zero-moment location h, a statically indeterminate structure can be transformed into its equivalent statically determinate structure. An approximate close-form analytical expression of VRF is therefore built based on the transformed statically determinate structure with all (probabilistically) possible hinge locations. An ensemble average is taken to get the overall variability response function of the system, which can be replaced by the VRF with the hinge located at the same zero-moment point with the deterministic system without any randomness. This Variability Response Function can provide approximate estimates of the stochastic response variability with reasonable accuracy. Moreover, to get more accurate estimates of the statically indeterminate system, the results from the approximate variability response function can be further refined by introducing a correction term Dw.

Finally, the corrected response variability of the original statically indeterminate structure can be obtained with almost perfect accuracy (compared to brute force Monte Carlo simulations). To sum up, the proposed VRFs of statically indeterminate beams, for both bending moment and deflection, have numerous desirable attributes including the capability to perform a full sensitivity analysis of the response variability with respect to the spectral characteristics of the random field modeling the uncertain material/system properties and establishing realizable upper bounds of the response variability.