PSI - Issue 77

514 Tianyu Wang et al. / Procedia Structural Integrity 77 (2026) 512–520 Wang et al/ Structural Integrity Procedia 00 (2026) 000 – 000 0 , 0 ). These constants are determined by enforcing boundary conditions at the inner and outer surfaces of the pipe and continuity conditions for stress and displacement at the interfaces between layers. 3

Fig. 1. 3D composite pipe model and loading conditions.

Therefore, the continuity conditions can be summarised as follows: the requirement for displacement continuity prevents delamination under loading, whilst stress continuity ensures load transfer between plies follows the principles of elastic equilibrium. The outcome of this level is an accurate, analytical description of the baseline stress profile resulting from all symmetric loads, providing the foundation upon which asymmetric perturbations are superposed. The second level of the model is dedicated the stresses produced by asymmetric bending loads. This analysis is founded on the classical work of Lekhnitskii, employing a stress function approach within the framework of 3D elasticity theory. The mathematical sophistication required here reflects the increased complexity of the asymmetric problem: unlike the axisymmetric case, all six independent stress components must be determined, and they vary with both radial and angular position. The stress state is described using two stress functions, ( , ) and ( , ) , which must satisfy a set of governing partial differential equations derived from strain compatibility conditions. For a pipe subjected to pure bending moment ( ), the solution is obtained by assuming a sinusoidal dependence on the hoop coordinate ( ), a form suggested by both the loading geometry and the principle of minimum potential energy. This assumption, whilst appearing restrictive, captures the essential physics of bending-induced stress distributions and reduces the governing equations to a set of Cauchy-Euler type ordinary differential equations, which can be solved analytically. The mathematical reduction from partial to ordinary differential equations represents a crucial simplification that enables closed-form solutions whilst preserving the 3D nature of the stress field. The final stress expressions for all six components ( , , , , , ) are functions of the radial and hoop coordinates ( , ), the pipe ’ s curvature ( ), and a set of layer-dependent unknown constants ( ). These constants are determined by applying boundary conditions of zero traction on the inner and outer surfaces and enforcing stress and displacement continuity at the interfaces. The physical interpretation of these boundary conditions is particularly important: the traction-free surfaces ensure that no spurious loads are introduced, whilst the interface conditions guarantee that the multi-layered structure behaves as a coherent whole rather than a collection of independent cylinders. Finally, complete stress state at any point within the composite pipe under combined loading is obtained by the direct summation of the stress tensors calculated at both levels. This superposition is mathematically rigorous within the linear elastic regime and physically meaningful as it reflects how different load components contribute independently to the total stress state. The two-level strategy provides a computationally efficient and conceptually intuitive framework for analysing complex, multi-faceted structural problems. The computational efficiency of this approach deserves special emphasis. Traditional FEA of a similar problem requires the solution of systems containing millions of degrees of freedom, necessitating sophisticated solvers and significant computational resources. In contrast, the analytical approach reduces the problem to solving sets of linear algebraic equations with dimensions proportional to the number of layers, typically fewer than 20 equations. This reduction of computational complexity explains the observed two-order-of-magnitude improvement in solution time.