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A reliability and reliability-based sensitivity analysis on mechanical parts with multiple failure modes is carried out in the proposed paper. Due to the unknown distributions of random variables, the first order second moment method are utilized to calculate the reliability of each failure mode. According to the characteristic that only a few failure modes occur when the parts fail, only the first three main failure modes are considered here. Based on the theory of probability and copula theory, the reliability with three failure modes are then obtained based on the built joint distribution function with selected copula function. Reliability sensitivity refers to the partial derivative of the reliability with respect to basic random variables. According to the matrix differential method, the parameters sensitivity analysis is performed. The proposed method is proved to be practical by the numerical example.