2014
年
5
月
第
30
卷第
3
期
沈阳建筑大学学报(自然科学版)
Journal
of
Shenyang
Jianzhu
University
(Natural
Sci
巳
nce)
岛
1ay
2014
Vo
l.
30
, No.3
文章编号
;2095
-1922(2014)03 -0414
-08
doi;
10.
11717/j.
issn
;2095
-
1922.
2014.
03.05
基于
Mathematica
的蜂窝梁孔口应力
集中问题的复分析
贾连光，谢国辉，侯祥林
(沈阳建筑大学土木工程学院，辽宁沈阳
110168
)
摘
要:目的研究不同孔型的孔口应力集中问题，为蜂窝梁及其他开孔结构的工程设计应用
提供参考.方法利用弹性力学的基本理论，运用复变函数的方法，引入保角变换，以无限大开
孔板为例，借助
Mathematica
的符号计算功能，编制相应的程序对常用的蜂窝梁孔型一六边
形、四边形、圆形孔口进行分析求解.运用孔口纯弯状态下有
FR
元分析与理论计算结果进行对
比和修正.结果计算推导得出了在几种不同应力边界条件下的孔口处的复应力函数，进而得
到对应的孔口应力分布特征，借助有
FR
元分析给出了纯弯状态下孔口应力的理论计算公式.结
论孔口的应力集中程度随孔角曲率半径减小而加剧，圆形孔口能有效降低应力集中的影响，
是一种理想的开孔形式.
关键词:蜂窝梁;复变函数;保角映射;
Mathematica
;应力集中
中图分类号:
TU392
文献标志码
:A
Complex Analysis of Stress Concentrations in Cellular Beam
with a Hole Based on the Mathematica
JIA Lianguang ,XIE Guohui ,
HOU
Xianglin
(School
of
Civil
Engineering
,
Shenyang
Ji
anzhu
University
,
Shenyang
,
China
,
110168)
Abstract:
To offer references in engineering designs for the cellular beam and other structure with opening ,
this paper studied stress concentrations
of
holes with different shapes in beam. Based on the elastic theory ,
complex function analysis and the conformal mapping technique , with the symbolic computational software
Mathematica
, stress concentrations
of
holes such
as
hexagon ,
squ
缸
e
and circle types that are often used in the
cellular beam were calculated. Theoretical calculation results
of
stresses on the hole under pure bending state
were compared and corrected with the finite element analysis. The complex stress functions under axial load-
ing
,pure shearing and pure bending were calculated and the stress distribution behaviors on the hole were
obtained. The theoretical calculational formulas for stress concentrations on the hole under the pure bending
with the aid
of
finite element analysis were buil
t.
It indicates that stress concentrations on the hole increase
rapidly with decrease
of
curvature radius by the hole angle. The circular hole is a kind
of
ideal form
of
open-
ing
, which can effectively reduce the stress concentrations.
Key
words:
cellular beam; complex function; conformal mapping; Mathematica; stress concentration
收稿日期
;2013
-11
-12
基金项目:国家自然科学基金项目
(51108279)
;辽宁省自然科学基金项目
(201102181
)
作者简介:贾连光(1
961-)
，男，教授，主要从事结构工程研究.
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