## DISCUSSION ON THE TEACHING METHOD OF THE CONCEPT OF CORE OF CROSS-SECTION1)

WU Zeyan, LUO Wei, TIAN Dongfang, PENG Hui, YE Yong,2)

College of Hydraulic ＆ Environmental Engineering, China Three Gorges University, Yichang 443002, Hubei, China

 基金资助: 1)三峡大学教学研究一般项目(J2019001)三峡大学高教研究项目(1901)三峡大学课程建设项目(SDKC202002)

Abstract

The core of cross-section is an important concept in the design of the columnar structure of concrete non-tensile materials under compression in the field of civil engineering. In order to deepen students' understanding of the concept of the core of cross-section in the course of material mechanics, it first developed a programing code with MATLAB to calulate the core of cross-section. The concrete numerical examples help students to understand the concept more easily. Then the convexity of the core of the cross-section was discussed. The convexity of the core of the section is explained by the static equivalence principle and the superposition principle from the perspective of mechanics; the convexity is proved mathematically by the neutral axis equation and the definition of convex set. The teaching process is helpful in cultivating students thinking skills.

Keywords： material mechanics; core of cross-section; visible learning; convex set

WU Zeyan, LUO Wei, TIAN Dongfang, PENG Hui, YE Yong. DISCUSSION ON THE TEACHING METHOD OF THE CONCEPT OF CORE OF CROSS-SECTION1). Mechanics in Engineering, 2022, 44(2): 397-403 DOI:10.6052/1000-0879-21-319

## 1 截面核心的概念

### 图1

$\sigma =\frac{F}{A}\left( {1+\frac{y_{F} }{i_{z}^{2} }y+\frac{z_{F} }{i_{y}^{2} }z} \right)$

$1+\frac{y_{F} }{i_{z}^{2} }y+\frac{z_{F} }{i_{y}^{2} }z=0$

## 2 截面核心的计算

### 2.1 平面图形的几何性质的计算

$\int_\varOmega {\nabla \cdot {B}\;{\rm d}\varOmega } =\int_{\partial \varOmega } {{n}\cdot {B}\;{\rm d}\varGamma }$

$T=\nabla \cdot {B}$

$\int_\varOmega {T{\rm d}\varOmega } =\int_\varOmega {\nabla \cdot {B}\;{\rm d}\varOmega } =\int_{\partial \varOmega } {{n}\cdot {B}\;{\rm d}\varGamma }$

## 3 截面核心是凸集

### 图7

$F_{A} =\frac{l_{BC} }{l_{AC} +l_{BC} }F_{C},\ \ {F_{B}=\frac{l_{AC} }{l_{AC} +l_{BC} }F_{C} }$

3.2.1 凸集的定义

### 图9

Fig.1

3.2.2 命题1

$ay+bz+1=0, kay+kbz+1=0$

### 图10

$\left[ {\lambda +\left( {1-\lambda } \right)k} \right]ay+\left[ {\lambda +\left( {1-\lambda } \right)k} \right]bz+1=0$

3.2.3 命题2

$a_{1} y+b_{1} z+1=0,\ \ a_{2} y+b_{2} z+1=0$

$\left[ {\lambda a_{1} +\left( {1-\lambda } \right)a_{2} } \right]y+\left[ {\lambda b_{1} +\left( {1-\lambda } \right)b_{2} } \right]z+1=0$

### 图11

$\left( {a_{1} y+b_{1} z+1} \right)\left( {a_{2} y+b_{2} z+1} \right)=1$

### 图12

$a_{1} y+b_{1} z+1=-\frac{1-\lambda }{\lambda }\left( {a_{2} y+b_{2} z+1} \right)$

$-\frac{1-\lambda }{\lambda }\left( {a_{2} y+b_{2} z+1} \right)^{2}=1$

3.2.4 截面核心是凸集的证明

$\frac{y_{1} }{i_{z}^{2} }y+\frac{z_{1} }{i_{y}^{2} }z+1=0,\ \ \frac{y_{2} }{i_{z}^{2} }y+\frac{z_{2} }{i_{y}^{2} }z+1=0$

$M\left( {\lambda y_{1} +\left( {1-\lambda } \right)y_{2},\ \ \lambda z_{1} +\left( {1-\lambda } \right)z_{2} } \right)$

$\frac{\lambda y_{1} +\left( {1-\lambda } \right)y_{2} }{i_{z}^{2}}y+\frac{\lambda z_{1} +\left( {1-\lambda } \right)z_{2} }{i_{y}^{2} }z+1=0$

## 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

MATLAB/PDE 在弹性力学可视化教学中的应用

Xu Yangjian, Ruan Hongshi, Shen Qianqian, et al.

Programming practice during teaching and learning of material mechanics-stress state analysis

Mechanics in Engineering, 2018, 40(4):446-450 (in Chinese)

He Feng, Ren Tianjiao, Yang Song, et al.

Development and application of stress state theory solver for elasticity

Mechanics in Engineering, 2021, 43(1):139-143 (in Chinese)

Fan Qinshan, Yin Yajun, Tang Jingjing, et al.

Reform and innovation, a decade practice of improvement of the course of the strength of materials

Mechanics in Engineering, 2018, 40(5):543-549 (in Chinese)

Xue Xiuli, Zeng Chaofeng.

How to open cheaching career for young teachers in the university: taking material mechanics as an example

Journal of Architectural Education in Institutions of Higher Learning, 2018, 27(4):140-143 (in Chinese)

Li Junfeng.

Growth dynamics

Mechanics in Engineering, 2020, 42(2):242-247 (in Chinese)

Carmo MP. 曲线和曲面的微分几何学. 田畴, 忻元龙, 姜国英等译. 上海: 上海科学技术出版社, 1988

/

 〈 〉