A promising and attractive way of performing structural health monitoring (SHM) and damage prognosis (DP) of engineering systems is through utilizing a nonlinear finite element (FE) model. Often, FE models contain parameters that are unknown or known with significant level of uncertainty. Such parameters need to be estimated/updated/calibrated using data measured from the physical system. The Bayesian paradigm to model updating/calibration is attractive as it accounts, using a rigorous probabilistic framework, for numerous sources of uncertainties existing in the real-world. However, applying Bayesian methods to nonlinear FE models of large-scale civil structural systems is computationally very prohibitive. Additionally, non-identifiability of FE model parameters poses challenges in the model updating process. This paper presents Bayesian model updating and identifiability analysis of nonlinear FE models with a specific testbed civil structure, Pine Flat concrete gravity dam, as illustration example. Model updating is performed in the recursive mode using the unscented Kalman filter (UKF) and in the batch mode using the transitional Markov chain Monte Carlo (TMCMC) method. Limitations in terms of applicability and computational challenges of each method for model updating of large-scale nonlinear FE models are addressed and discussed. Identifiability and sensitivity analyses of the model are then performed using local and global methods. Local practical identifiability analysis using local sensitivity in conjunction with the Fisher information matrix is used to assess the parameter identifiability in a certain local region in the parameter space. Due to the nonexistence of a method to assess global practical identifiability, variance-based global sensitivity analysis (Sobol's method) is used herein. Identifiability and sensitivity analysis results are used to choose the parameters to be included in the model updating phase.