Chapter 6 and some parts of Chapter 4 are accepted as C. Hao, and S. H. Cheung, "Bayesian updating of failure probability curves with multiple performance functions of nonlinear structural dynamic systems", now in press with journal 'Computer Methods in Applied Mechanics and Engineering'.The contributions of the co-authors are as follows:• Prof. CHEUNG Sai Hung provided the research directions, the basic ideas and revised the manuscript.• I wrote the codes, designed and executed the examples, and prepared the manuscript drafts.