有限厚度弹性地基支撑梁的屈曲

Buckling of a beam supported by elastic foundation of finite thickness

摘要

柔性基底支撑平板常见于自然界或工程中，如皮肤、沥青道路等，屈曲是其常见的失效形式之一。地基的侧向支撑作用有助于提高平板的稳定性，当地基厚度相对屈曲波长较厚时，现有的弹性模型无法准确揭示地基的厚度效应，临界屈曲应变的解析解还未给出。为此，本研究针对基底支撑平薄板，基于线弹性理论，提出了有限厚度弹性地基模型，研究了地基支撑无限长梁的屈曲。探讨了地基在刚性基底上黏结与滑动两种情况，得出了临界屈曲应变及临界屈曲波长与地基的杨氏模量、泊松比、厚度间的简洁显式关系，分析了地基厚度对临界屈曲波长及临界屈曲应变的影响。开展有限元分析验证此模型，结果显示相对已有的解析模型，本模型预测结果与有限元结果吻合得更好，当地基厚度大于1/π屈曲波长时，其相对误差小于10%。

Abstract

Flat plates supported by flexible substrate are commonly found in nature or in engineering，such as skin and asphalt roads，and buckling is one of their common forms of failure. The lateral support effect of the foundation helps to improve the stability of the flat plates. When the foundation is thick compared to the buckling wavelength，the existing elastic model cannot accurately reveal the thickness effect of the foundation，and the analytical solution of the critical buckling strain is not yet given. Therefore，in this paper，a finite-thickness elastic foundation model is proposed for a flat and thin plate supported by the foundation，and the buckling of a beam with infinite length is studied using the theory of linear elasticity. Two cases of bonded and sliding of the foundation on the rigid substrate are explored，and the concise explicit relationships of the critical buckling strain and critical buckling wavelength with Young's modulus，Poisson's ratio and thickness of the foundation are derived，and the influence of the foundation thickness on the critical buckling wavelength and critical buckling strain is analyzed. The finite element analysis is carried out to verify the model，and the prediction results show that the results of the model are in better agreement with the finite element results than the existing analytical model，and the relative error between the model and the finite element is less than 10% when the foundation thickness is greater than 1/π of the buckling wavelength，and the model can reveal the supporting effect of elastic foundation more accurately.

关键词

弹性地基 / 有限厚度 / 屈曲 / 屈曲应变 / 有限元

Key words

elastic foundation / finite thickness / buckling / buckling strain / finite element

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基金资助

国家自然科学基金资助项目(12272068)

重庆市自然科学基金资助项目(CSTB2022NSCQ-MSX0364)