{"title":"Free vibration of beams using stochastic finite element method considering three-dimensional randomness of material properties","authors":"Diem Dang Nguyen , Hien Duy Ta","doi":"10.1016/j.finmec.2025.100312","DOIUrl":null,"url":null,"abstract":"<div><div>This study presents the development of the stochastic finite element method (SFEM) to investigate the random free vibrations of beams with three-dimensional (3D) random field of material properties. Material properties, such as mass density and Young's modulus, are modeled as 3D stationary univariate random fields, with their correlations considered in the dynamic analysis. The random material fields are discretized into random variables using the weighted integral method combined with the perturbation technique to obtain a first-order approximation of the eigenvalues of free vibration. Statistical quantities, including mean, variance, and coefficient of variation (COV), of the eigenvalues are derived. Monte Carlo simulations (MCs), based on standard FEM and the spectral representation method for stochastic fields, are employed to validate the SFEM solution. The three-dimensional randomness of material properties significantly affects the random dynamic response of the structure. Results reveal that as the correlation distance increases, the dispersion of the eigenvalues around the expected value also increases. A perfect positive correlation between the 3D random fields of Young's modulus and mass density results in a smaller COV, whereas a perfect negative correlation leads to a larger COV. As the correlation distance approaches infinity, the COV approaches the total standard deviation for a negative correlation, while it becomes negligible for a positive correlation. For independent random fields, the COV converges to approximately 70 % of the total standard deviation. The nearly linear relationship between COV and standard deviation enables the prediction of the random response of the structure once the material property randomness is defined.</div></div>","PeriodicalId":93433,"journal":{"name":"Forces in mechanics","volume":"19 ","pages":"Article 100312"},"PeriodicalIF":3.2000,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Forces in mechanics","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2666359725000083","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
This study presents the development of the stochastic finite element method (SFEM) to investigate the random free vibrations of beams with three-dimensional (3D) random field of material properties. Material properties, such as mass density and Young's modulus, are modeled as 3D stationary univariate random fields, with their correlations considered in the dynamic analysis. The random material fields are discretized into random variables using the weighted integral method combined with the perturbation technique to obtain a first-order approximation of the eigenvalues of free vibration. Statistical quantities, including mean, variance, and coefficient of variation (COV), of the eigenvalues are derived. Monte Carlo simulations (MCs), based on standard FEM and the spectral representation method for stochastic fields, are employed to validate the SFEM solution. The three-dimensional randomness of material properties significantly affects the random dynamic response of the structure. Results reveal that as the correlation distance increases, the dispersion of the eigenvalues around the expected value also increases. A perfect positive correlation between the 3D random fields of Young's modulus and mass density results in a smaller COV, whereas a perfect negative correlation leads to a larger COV. As the correlation distance approaches infinity, the COV approaches the total standard deviation for a negative correlation, while it becomes negligible for a positive correlation. For independent random fields, the COV converges to approximately 70 % of the total standard deviation. The nearly linear relationship between COV and standard deviation enables the prediction of the random response of the structure once the material property randomness is defined.
本文介绍了随机有限元法(SFEM)的发展,用于研究具有材料特性三维随机场的梁的随机自由振动。材料性能,如质量密度和杨氏模量,被建模为三维平稳的单变量随机场,并在动态分析中考虑它们的相关性。采用加权积分法结合微扰技术将随机材料场离散为随机变量,得到自由振动特征值的一阶近似。导出了特征值的统计量,包括平均值、方差和变异系数。采用蒙特卡罗模拟(Monte Carlo simulation, MCs),基于标准有限元法和随机场谱表示法,对SFEM解进行了验证。材料性能的三维随机性对结构的随机动力响应有显著影响。结果表明,随着相关距离的增大,特征值在期望值周围的离散度也增大。三维随机场的杨氏模量与质量密度的正相关关系导致COV减小,负相关关系导致COV增大。当相关距离趋近于无穷大时,对于负相关,COV趋近于总标准差,而对于正相关,COV变得可以忽略不计。对于独立随机场,COV收敛到总标准差的约70%。COV与标准差之间的近似线性关系可以在确定材料特性随机性后预测结构的随机响应。