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This paper presents a novel method to construct a plate element enveloping crack tip in the extended finite element method (XFEM) frame for crack growth simulation. The element enveloping a crack tip is sub-partitioned into several triangular quarter-point singular elements around the crack tip. The sub-triangular singular element cut by the crack, like the ordinary elements separated by the crack in other regions, is enriched with the Heaviside function. As the sub-partition scheme retains the original boundary of the crack tip element and the discontinuity of the entire crack is reproduced by the Heaviside enrichment, the proposed method can model cracks independent of the background mesh. Using the singular elements instead of analytical asymptotic functions employed in the conventional XFEM allows to reflect the crack tip singularity for more fracture scenarios. The additional nodes created by the sub-partition act as internal nodes of the crack tip element, thus the classical and enriched degrees of freedom (DOFs) on these nodes can be eliminated through an element-level condensation, which could reduce the condition number of the system matrix. Crack growth in plates and shells under fatigue loading is described by using Paris' law. The performance of the proposed method is investigated through stress intensity factors (SIFs) prediction, convergence study, and fatigue crack growth analysis in several examples of cracked plates and cylindrical shells. • A novel method for constructing crack tip element is developed based on XFEM. • The crack tip element is sub-partitioned into several triangular singular elements. • The sub-triangular singular element cut by crack is enriched with the Heaviside function. • All the DOFs on the sub-partition-induced nodes are eliminated through element-level condensation. • Several examples including fatigue crack growth in cylindrical shells show effectiveness and accuracy of the method. [ABSTRACT FROM AUTHOR]