Non-uniform rational B-spline based free vibration analysis of axially functionally graded tapered Timoshenko curved beams

doi:10.1007/s10483-020-2594-7

Applied Mathematics and Mechanics (English Edition) ›› 2020, Vol. 41 ›› Issue (4): 567-586.doi: https://doi.org/10.1007/s10483-020-2594-7

Non-uniform rational B-spline based free vibration analysis of axially functionally graded tapered Timoshenko curved beams

Zhiwei ZHOU¹, Meixia CHEN¹, Kun XIE²

1. School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China;
2. College of Engineering, Huazhong Agricultural University, Wuhan 430070, China

收稿日期:2019-10-23 修回日期:2020-01-03 发布日期:2020-03-26
通讯作者: Meixia CHEN E-mail:chenmx26@hust.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Nos. 51779098 and 51909098) and the Natural Science Foundation of Hubei Province of China (No. 2019CFB132)

Non-uniform rational B-spline based free vibration analysis of axially functionally graded tapered Timoshenko curved beams

Zhiwei ZHOU¹, Meixia CHEN¹, Kun XIE²

1. School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China;
2. College of Engineering, Huazhong Agricultural University, Wuhan 430070, China

Received:2019-10-23 Revised:2020-01-03 Published:2020-03-26
Contact: Meixia CHEN E-mail:chenmx26@hust.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Nos. 51779098 and 51909098) and the Natural Science Foundation of Hubei Province of China (No. 2019CFB132)

摘要/Abstract

摘要： The free vibration of axially functionally graded (FG) tapered Timoshenko curved beams is studied with the numerical approach. By using the non-uniform rational B-spline (NURBS) basis functions, the exact geometry and the generalized displacement field are formulated. Variable geometric parameters and material properties, including the curvature, cross-sectional area, area moment of inertia, mass density, and Young’s modulus, are expanded as functions of the coordinate in a parametric domain. Based on Hamilton’s principle, the weak formulation is derived by applying a refined constitutive relation which considers the thickness effect. Natural frequencies and mode shapes are obtained from the eigenvalue equation. Circular, elliptic, and parabolic curved beams are considered in numerical examples. The obtained results are in good agreement with those in the existing studies and those calculated by the finite element software ANSYS. Moreover, the effects of the material gradient, taper ratio, slenderness ratio, and heightspan ratio on vibration behaviors are discussed.

关键词: free vibration, functionally graded (FG) material, non-uniform crosssection, curved beam, variable curvature, thickness effect

Abstract: The free vibration of axially functionally graded (FG) tapered Timoshenko curved beams is studied with the numerical approach. By using the non-uniform rational B-spline (NURBS) basis functions, the exact geometry and the generalized displacement field are formulated. Variable geometric parameters and material properties, including the curvature, cross-sectional area, area moment of inertia, mass density, and Young’s modulus, are expanded as functions of the coordinate in a parametric domain. Based on Hamilton’s principle, the weak formulation is derived by applying a refined constitutive relation which considers the thickness effect. Natural frequencies and mode shapes are obtained from the eigenvalue equation. Circular, elliptic, and parabolic curved beams are considered in numerical examples. The obtained results are in good agreement with those in the existing studies and those calculated by the finite element software ANSYS. Moreover, the effects of the material gradient, taper ratio, slenderness ratio, and heightspan ratio on vibration behaviors are discussed.

Key words: free vibration, functionally graded (FG) material, non-uniform crosssection, curved beam, variable curvature, thickness effect

中图分类号:

70J10

Zhiwei ZHOU, Meixia CHEN, Kun XIE. Non-uniform rational B-spline based free vibration analysis of axially functionally graded tapered Timoshenko curved beams[J]. Applied Mathematics and Mechanics (English Edition), 2020, 41(4): 567-586.

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Non-uniform rational B-spline based free vibration analysis of axially functionally graded tapered Timoshenko curved beams

Non-uniform rational B-spline based free vibration analysis of axially functionally graded tapered Timoshenko curved beams

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