Analysis of the stress-strain state of a cantilever perforated layer with a cylindrical embedded support

Abstract

The article presents an analysis of the stress-strain state of a cantilever perforated layer in contact with a rigid cylindrical embedded support. Such structural elements are widely used in aviation, aerospace, and general engineering to reduce weight, provide access, or perform specific functional tasks. The presence of holes and supports significantly changes the stress-strain state, leading to local stress concentration, which can cause premature failure. A mathematical model based on three-dimensional equations of elasticity theory has been developed for the study. To solve the boundary value problem, a generalized Fourier method was used, which allows the boundary conditions on all surfaces (layer planes, hole surfaces, and the contact boundary with the support) to be satisfied with high accuracy. Satisfying mixed boundary conditions leads to an infinite system of integro-algebraic equations with unknown coefficients. Through analytical transformations, this system is reduced to an infinite system of linear algebraic equations of the second kind, to which the reduction method is applied, ensuring high accuracy of results. As part of the work, a detailed numerical study was carried out for a specific configuration: a layer of ABS plastic with five parallel cylindrical cavities, one of which acts as a rigid fastening. The case of a localized normal load on the upper surface of the layer is considered. Detailed distribution patterns of all components of the stress tensor were analyzed. The results demonstrate the complex three-dimensional nature of the stress distribution arising from the superposition of bending, tension/compression effects, and significant mutual influence of stress concentrators. It has been established that the stress distribution is significantly asymmetrical both relative to the surfaces of the layer and for each cavity individually, which emphasizes the impossibility of using simplified models. Areas of maximum stress concentration have been identified, which are critically important in terms of strength assessment. The developed analytical and numerical approach allows obtaining reliable data on stress and displacement distribution with high accuracy, which is important for verifying the results obtained using numerical methods. The work has significant practical value for design engineers, providing an effective tool for calculating strength, optimizing structures, and ensuring their operational reliability.