## 1 Introduction  
Cylindrical shells are widely used in various industries such as storage tanks, pressure vessels, pipelines, boilers, and aircraft fuselages due to their ability to evenly distribute internal pressure. However, cracks caused by external loads, fatigue, or environmental factors can compromise their structural integrity. To address this, composite patch reinforcement has been studied as an effective solution [1]. Composite materials are widely used in patch reinforcement due to their high strength, stiffness, lightweight nature, and cost-effectiveness [2]. Since the late 1970s, research on composite patch reinforcement has primarily focused on flat plates, particularly for crack propagation control and fatigue life extension [3, 4, 5, 6, 7, 8, 9]. More recently, attention has shifted toward applying these techniques to pipes. [PERSON] et al. [10] confirmed the benefits of bonded composite wrap reinforcement in cracked pipes under bending loads, while [PERSON] et al. [11] investigated the influence of patch thickness and placement on crack growth in circumferentially cracked pipes under combined loading conditions. [PERSON] et al. [12] analyzed the failure pressure of defective pipes repaired with composite systems, emphasizing the role of plastic deformation in structural integrity. Additionally, [PERSON] et al. [13] examined the repair of circumferential through-wall cracked pipes using localized composite patches, demonstrating the effectiveness of targeted reinforcement in restoring mechanical performance. [PERSON] et al. [14] evaluated composite repair for large-diameter pipes with extensive metal loss, and [PERSON] [15] assessed composite patch reinforcement in spiral-welded pipes, demonstrating its contribution to crack mitigation and reliability improvement. Various numerical methods have been employed to assess the reliability of these reinforcement techniques, emphasizing the need for more advanced computational models to optimize the structural performance of composite-reinforced pipes. Several theoretical approaches have been developed to analyze composite-repaired structures, including equivalent single-layer (ESL), layer-wise (LW), and zigzag theories [16]. ESL-based methods, implemented in commercial software such as ABAQUS 2024, ANSYS 2024 R2, and LS-DYNA R15.0.2, effectively predict global behavior in thin composite plates and shells, but struggle with local stress distribution in thick laminated composites [17; 18; 19; 20]. Unlike ESL methods, LW theory accounts for interlaminar effects and localized damage mechanisms such as matrix cracking and delamination [21]. However, conventional LW models require a higher number of degrees of freedom, reducing their computational efficiency [22]. The \(p\)-refinement finite element method (FEM) differs from the conventional \(h\)-refinement FEM in that it enhances the solution accuracy by increasing the polynomial order of shape functions rather than refining the mesh by increasing the number of elements. This method is particularly effective for problems with stress singularities and crack-containing structures, providing high accuracy in capturing localized stress distributions [23]. Additionally, it ensures faster convergence with fewer elements, leading to more efficient modeling and reduced computational costs. Due to these advantages, \(p\)-refinement FEM has been widely studied and applied in various engineering fields, particularly in fracture mechanics and structural analysis [24; 25; 26; 27]. To efficiently analyze patch reinforcement for cracked plate structures, a previous study [25] applied the LW theory with \(p\)-refinement, developing a model to investigate composite patch reinforcement for pipes with circumferential cracks and comparing it with a model utilizing the conventional \(h\)-refinement technique. This study demonstrated that the LW model with \(p\)-refinement offers the advantage of simplified modeling compared to conventional finite element methods.