تحلیل ارتعاشات و بهینه‌سازی تنش دینامیکی در تیرهای جدار نازک با مقطع متغیر

Considering the growing trend toward designing lightweight, robust, and efficient structures in advanced industries such as aerospace, renewable energy, and automotive engineering, accurate dynamic analysis of members like thin-walled beams with variable cross-sections is of high importance. In this study, vibration analysis and dynamic stress assessment of thin-walled beams with varying cross-sections are performed using the Rayleigh–Ritz method. Polynomial functions are employed to model more accurately the geometric variations of thickness and diameter along the beam length. By applying the Rayleigh–Ritz approach, the mass and stiffness matrices are derived in a semi-analytical form and the governing equations for free and forced vibration are obtained. The dynamic response of the beam under a spatially distributed sinusoidal (harmonic) load is then computed, and the distribution of bending stress along the beam is evaluated. To achieve an optimal design, the beam geometry is optimized with the simultaneous objective of reducing peak dynamic stress and structural mass, using the multi-objective genetic algorithm NSGA-II. A set of optimal designs on the Pareto front is extracted and analyzed. The results indicate that an appropriate choice of thickness and diameter variation functions, coupled with multi-objective optimization, can substantially reduce dynamic stresses while effectively lowering structural weight. Moreover, comparison of the numerical responses with ABAQUS results shows very good agreement, corroborating the accuracy and efficiency of the proposed model. The presented method can serve as an effective tool for the design and optimization of lightweight, high-strength structures in engineering applications, particularly in the aerospace, energy, and automotive sectors.