Engineering structures are inevitably affected by a variety of uncertain factors, such as fluctuations in the service environment, which leads to a certain degree of randomness and uncertainty in their service state. With the improvement of engineering safety requirements, traditional failure assessment methods based on deterministic assumptions have been unable to meet the practical needs. Probabilistic analysis methods based on stochastic theory could quantify the structural integrity of cracked structures through reliability degree, while their practical application is limited by the statistical data of random variables. Convex set theory is introduced into the failure assessment diagram (FAD) to propose a non-probabilistic failure assessment method, which can evaluate structural integrity in the case of scarce statistical data. The polygonal convex set (PCS) model is used to characterize the non-probabilistic failure assessment results, and the non-probabilistic reliability degree can be obtained by area ratio. To verify the accuracy of the proposed non-probabilistic failure assessment method, the probabilistic and non-probabilistic reliability degrees of cracked pipes under bending moments are compared. The results show that probabilistic and non-probabilistic reliability degrees have the same variation trend, and the non-probabilistic reliability based on the PCS model is closer to probabilistic reliability compared with the interval model. In practical engineering applications, the proposed non-probabilistic analysis method can be used as a pre-evaluation of structural integrity.