A multi-state Semi-Markov model for nuclear power plants piping systems subject to fatigue damage and random shocks under dynamic environments

•A multi-state Semi-Markov model for degraded piping systems in nuclear power plants was proposed.•The effects of the stochastic environment and random shocks were considered.•The analytical solutions of the time-dependent state probability of the system were derived.•The correctness of the analytical results was verified based on the benchmark case in the literature.•Two case studies were presented.