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CLC number: O343.1

On-line Access: 2015-10-01

Received: 2015-03-07

Revision Accepted: 2015-07-16

Crosschecked: 2015-09-15

Cited: 0

Clicked: 1356

Citations:  Bibtex RefMan EndNote GB/T7714

 ORCID:

Chun-xiao Zhan

http://orcid.org/0000-0002-7805-4557

Yi-hua Liu

http://orcid.org/0000-0002-5916-3563

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Journal of Zhejiang University SCIENCE A 2015 Vol.16 No.10 P.805-819

10.1631/jzus.A1500043


Plane elasticity solutions for beams with fixed ends


Author(s):  Chun-xiao Zhan, Yi-hua Liu

Affiliation(s):  School of Civil and Hydraulic Engineering, Hefei University of Technology, Hefei 230009, China

Corresponding email(s):   liuyihua@hfut.edu.cn

Key Words:  Beam, Fixed end, Boundary condition, Plane stress, Elasticity solution


Chun-xiao Zhan, Yi-hua Liu. Plane elasticity solutions for beams with fixed ends[J]. Journal of Zhejiang University Science A, 2015, 16(4): 805-819.

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Abstract: 
The plane stress problem of beams is a typical one in elasticity theory. In this paper a new set of boundary conditions for the fixed end is proposed to improve the accuracy of the plane elasticity solution for beams with fixed end(s). Plane elasticity solutions are then derived for the cantilever beam, propped cantilever beam, and fixed-fixed beam. The new set of boundary conditions is constructed by combining two conventional ones with a parameter. The parameters for different kinds of beams are determined by minimizing the square sum of the longitudinal displacements through the thickness of the fixed end. Comparison with the results obtained by the finite element method (FEM) shows the efficiency of the new type of boundary conditions. When the beam is a deep one, it is found that different boundary conditions yield different errors, and the elasticity solution obtained by the new boundary conditions best approaches the FEM results.

This is a quite interesting and complete work on the seemingly old but important problem in elasticity. The paper suggests a new mathematical form to express the fixed boundary of a beam, which combines the two existing ones in Timoshenko and Goodier by introducing a parameter which is determined on a reasonable ground. Numerical comparison with FEM shows that the new form enables more accurate results.

含固支端梁的弹性力学解

目的:在用应力函数法求含固支端梁的应力和位移时,对固支端只能采用简化的固支边界条件。为此,拟提出一种更好的简化固支边界条件。
创新点:在已有固支边界条件的基础上,提出新的固支边界条件,由此得到的含固支端梁的弹性力学解的精度有很大提高。
方法:1. 综合Timoshenko和Goodier提出的两种固支边界条件,构造出一种新的固支边界条件,并应用Airy应力函数法推导出三种含固支端梁的解析解;2. 对由不同固支边界条件得到的解析解与有限元解进行比较。
结论:1. 与已有的固支边界条件相比,本文提出的固支边界条件更佳,尤其是对短梁;2. 理论与数值结果均表明,对超静定短梁,位移u不再保持线性分布,经典梁理论中的平截面假设不再适用。

关键词:梁;固支端;边界条件;平面应力;弹性力学解

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

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