Progressive damage analysis of composite structures in the presence of uncertainties is a computationally-expensive and highly-complex process. This work tackles these issues by developing efficient finite element (FE)-based surrogate models that are constructed with artificial neural network (ANN) models and design of experiment (DOE) methods. The proposed framework for building surrogate models (metamodels) can capture various multi-scale uncertainties. In addition, to alleviate the computational burden in non-deterministic analyses, a novel strategy is proposed using the Plackett-Burman method to determine sources of uncertainties that have significant impacts in scattering the response. The response surface methodology (RSM), along with FE analysis, is used to create datasets. The RSM and ANN metamodels are constructed by using these datasets. Key results show that for complex models, ANN metamodels have more accuracy than RSM ones. The surrogate models are used for stochastic, probabilistic and reliability analysis of filament wound composite tubes. As a result, this multi-scale surrogate-based framework significantly decreases the computational time and cost of the analyses. Reliability analysis demonstrates that the statistical correlation between ply properties is significant, and it must be considered for an accurate evaluation.