## METHODS FOR DETERMINING THE NEUTRAL AXIS POSITION AND STRESS ANALYSIS OF COMPOSITE BEAM1)

ZHAO Junqing,2), ZHEN Yubao, ZHOU Peng, WANG Jun, HU Hengshan

Department of Astronautic Science and Mechanics, Harbin Institute of Technology, Harbin 150001, China

 基金资助: 1)黑龙江省高等教育教学改革项目(SJGY20190170)哈尔滨工业大学教学发展基金(XSZ2020001)

Received: 2021-03-11   Revised: 2021-03-22

Abstract

The key to analyze a composite beam composed of different material is to find out the position of the neutral axis. The position is determined by the method of equivalent section and weighted average. Then the normal and shear stresses of the composite beam are analyzed based the equivalent and weighted counterparts. In term of finding out the position of the neutral axis, the equivalent section method is conceptually clear while the weighted average method is simple and efficient. The equivalent section and weighted average methods can simplify the stress analysis of composite beam. The process can help students to understand the related concepts of beam bending, and to think comprehensively on bending problems.

Keywords： composite beam; neutral axis position; equivalent section method; weighted average method; stress analysis

ZHAO Junqing, ZHEN Yubao, ZHOU Peng, WANG Jun, HU Hengshan. METHODS FOR DETERMINING THE NEUTRAL AXIS POSITION AND STRESS ANALYSIS OF COMPOSITE BEAM1). Mechanics in Engineering, 2022, 44(1): 184-187 DOI:10.6052/1000-0879-21-096

## 1 中性轴位置确定方法

### 图1

$b_{2} =\frac{E_{2} }{E_{1} }b=0.5b$

$Ay_{\rm c} =\sum\limits_{i=1}^2 {A_{i} y_{ci} }$

### 图2

$y_{\rm c}=7h/8$

### 1.2 加权平均法

$y_{\rm c} =\frac{E_{1} S_{Z1} +E_{2} S_{Z2} }{E_{1} A_{1} +E_{2} A_{2} }=\frac{7h}{8}$

$y=\frac{\sum\limits_{i=1}^n {E_{i} } S_{Zi} }{\sum\limits_{i=1}^n {E_{i} A_{i} } }$

## 2 应力分析

### 2.1 正应力分析

$\sigma_{1}=E_{1}\varepsilon = E_{1} \frac{y}{\rho },\ \sigma_{2} =E_{2}\varepsilon =E_{2} \frac{y}{\rho}$

$\frac{1}{\rho }=\frac{M_{Z} }{E_{1} I_{Z1} +E_{2} I_{Z2} }$

### 2.2 切应力分析

$\tau_{1}=\frac{k_{1}F_{{\rm s}} S_{Z1}^{\ast } }{I_{Z} b},\ \tau_{2} =\frac{k_{2} F_{{\rm s}} S_{Z2}^{\ast } }{I_{Z} b}$

${{I}_{Z}}=({{E}_{1}}{{I}_{Z1}}+{{E}_{2}}{{I}_{Z2}})/{{E}_{1}}$

$S_{Zi}^{\ast }$为材料$i$所求应力的点横线以外的所有面积对中性轴$z_{\rm c}$轴的静矩。$k_{i}$为材料$i$的弹性模量与材料1的弹性模量之比。

$k_{1} S_{Z1}^{\ast } =k_{2} S_{Z2}^{\ast }$

$k_{1} S_{Z1}^{\ast } =S_{Z1}^{\ast } =bh_{1} \left(\frac{h_{1} }{2}+a-h_{1}\right)=\frac{3bh^{2}}{16}$

### 图3

$k_{2} S_{Z2}^{\ast } =0.5S_{Z2}^{\ast } =0.5bh_{2} \left(y_{0} -\frac{h_{2} }{2}\right)=\frac{3bh^{2}}{16}$

## 3 组合梁附加分析

$b_{2}=k_{2}b,\ b_{3}=k_{3}b$

$y_{\rm c}=45h/26$

### 图4

$y_{\rm c} =\frac{E_{1} S_{Z1} +E_{2} S_{Z2} +E_{3} S_{Z3} }{E_{1} A_{1} +E_{2} A_{2} +E_{3} A_{3} }=\frac{45h}{26}$

$I_{Z} =\sum\limits_{i=1}^n {E_{i} I_{Zi} }$

$\sigma_{1} =E_{1} \varepsilon,\ \sigma_{2} =E_{2} \varepsilon,\ \sigma_{3}=E_{3} \varepsilon$

$\frac{1}{\rho }=\frac{M_{Z} }{E_{1} I_{Z1} +E_{2} I_{Z2} +E_{3}I_{Z3} }$

$\left.\begin{array}{rl}\tau_{1} & =\frac{k_{1} F_{\mathrm{s}} S_{Z 1}^{*}}{I_{Z} b} \\ \tau_{2} & =\frac{k_{1} F_{\mathrm{s}} S_{Z 1}+k_{2} F_{\mathrm{s}} S_{Z 2}^{*}}{I_{Z} b} \\ \tau_{3} & =\frac{k_{3} F_{\mathrm{s}} S_{Z 3}^{*}}{I_{Z} b}\end{array}\right\}$

${{I}_{Z}}=({{E}_{1}}{{I}_{Z1}}+{{E}_{2}}{{I}_{Z2}}+{{E}_{3}}{{I}_{Z3}})/{{E}_{1}}$

3种材料组合梁的分析可比照本例进行。

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Gao Yunfeng, Jiang Chiping. Detailed Explanations and Comments of Questions of the National Zhou Peiyuan College Student Mechanics Competition. Beijing: Machinery Industry Press, 2015 (in Chinese)

Timoshenko S, Gere J. Mechanics of Materials. Hu Renli Transl. Beijing: Science Press, 1978 (in Chinese)

Sun Xunfang, Fang Xiaoshu, Guan Laitai. Mechanics of Materials (II). Beijing: Higher Education Press, 2009 (in Chinese)

Zhou Qian, Yan Weiming.

Bending analysis on composite beam and combination beam of Chinese ancient wooden buildings

Building Structure, 2012, 42(4): 157-161 (in Chinese)

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