Reinforced thermoplastic pipes (RTPs) are flexible composite pipes that are acknowledged as a good alternative to conventional submarine pipes. This study develops analytical models to estimate deformation and stresses in RTPs with any number of layers under internal pressure based on their nonlinear mechanical behaviors. The Hashin–Yeh failure criterion and the damage evolution model of composite materials are combined, and changes due to deformation are introduced to the winding angle to analyze the progressive failure of RTPs. The modeling covered stress analysis, failure evaluation, and material degradation. The stress distribution and failure pressure calculated by them were in good agreement with those obtained by the FEM and the experiments. On this basis of proposed model, this study investigating the impact of the uncertainties involved in the production process of the RTPs on the failure pressure under combined loading conditions. For analyzing the influence of randomness of the related parameters on the first-ply failure (FPF) pressure as well as the final burst failure (FF) pressure of RTPs, a sufficient number of samples were generated using the Monte-Carlo method. Results of the stochastic analysis were validated using the hydraulic burst test data. The statistical evaluation of the stochastic analysis results was performed by using the Weibull probability density distribution function.