第七章 应力应变分析 强度理论

Chapter7 Analysis of Stress and Strain Failure Criteria

(Analysis of stress-state and strain-state)

第七章 应力和应变分析 强度理论 Chapter7 Analysis of Stress and Strain Strength Theories§7-1 应力状态概述 (Concepts of stress-state) §7-2 平面应力状态分析-解析法 (Analysis of plane stress-state) §7-3 平面应力状态分析-图解法 (Analysis of plane stress-state) §7-4 三向应力状态分析

(Analysis of stress-state and strain-state)

§7-5 平面应变状态分析 (Analysis of plane strain-state) §7-6 广义胡克定律 (Generalized Hook’s law) §7-7 复杂应力状态的变形比能 (Strain-energy density in general stress-state ) §7-8 强度理论 ( Failure criteria) §7-9 莫尔强度理论 (Mohr’s failure criterion)

(Analysis of stress-state and strain-state)

§7-1 应力状态概述 (Introduction of stress-state)一、应力状态的概念 (Concepts of stresses-state)请看下面几段动画 1.低碳钢和铸铁的拉伸实验 (A tensile test of low-carbon steel and cast iron) 2.低碳钢和铸铁的扭转实验

(A torsional test of low-carbon steel and cast iron)

(Analysis of stress-state and strain-state)低碳钢和铸铁的拉伸

低碳钢 (low- carbon steel)

铸铁 (cast-iron)

塑性材料拉伸时为什么会出现滑移线？

(Analysis of stress-state and strain-state) 低碳钢和铸铁的扭转

低碳钢 (low- carbon steel)

铸铁 (cast-iron)

为什么脆性材料扭转时沿45°螺旋面断开？

(Analysis of stress-state and strain-state)3.重要结论(Important conclusions)

(1)拉中有剪,剪中有拉;(2)不仅横截面上存在应力,斜截面上也存在应力; (3)同一面上不同点的应力各不相同; (4) 同一点不同方向面上的应力也是各不相同

应 力哪一个面上？ 哪一点？ 哪一点？ 哪个方向面？

4.一点的应力状态(state of stresses of a given point) 过一点不同方向面上应力的情况,称之为这一点的应力状

态(state of stresses of a given point),亦指该点的应力全貌.

(Analysis of stress-state and strain-state)

二、应力状态的研究方法 (The method for investigating the state of stress)1. 单元体(Element body) 2. 单元体特征 (Element characteristic) (1)单元体的尺寸无限小,每个面上应力均匀分布 (2)任意一对平行平面上的应力相等 3.主单元体(Principal body) 各侧面上切应力均为零的单元体

2

3 1

1

3 2

(Analysis of stress-state and strain-state) 4.主平面(Principal plane)切应力为零的截面 5.主应力(Principal stress)

主面上的正应力

说明:一点处必定存在这样的一个单元体, 三个相互垂直的面 均为主平面, 三个互相垂直的主应力分别记为 1 , 2 , 3 且规定按 代数值大小的顺序来排列, 即

1 2 3

(Analysis of stress-state and strain-stat

e)

三、应力状态的分类 (The classification of stresses-state)1.空间应力状态(Triaxial stress-state or three-dimensional stress-state ) 三个主应力 1 , 2 , 3 均不等于零 2.平面应力状态(Biaxial stress-state or plane stress-state) 三个主应力 1 , 2 , 3 中有两个不等于零 3.单向应力状态(Uniaxial stress-state or simple stress-state) 三个主应力 1 , 2 , 3 中只有一个不等于零

2

3

2

1 1

1 3 2

1

1

2

1

(Analysis of stress-state and strain-state)例题 1 画出如图所示梁S截面的应力状态单元体.

F5 S平面

43

l/2 l/2 l/2 l/2

2 1

(Analysis of stress-state and strain-state)5

S平面

5 4

43 2 1

3 21

2 x11

3

x1

x2

2

x2 2

3

3

(Analysis of stress-state and strain-state)例题 2 画出如图所示梁 危 险截面危险点的应力状态单 元体y S 1 2 z 4 l

F

3

x

FS2 4

a

z

Mz

T

3

(Analysis of stress-state and strain-state)yy

FS1 2 z 3 x 4 z 2

43

Mz

T

x

1z

3 2

Mz T 1 x1 Wt Wz

T 4 FS 2 Wt 3 A

x3

T Mz 3 Wt Wz

(Analysis of stress-state and strain-state)例题3 分析薄壁圆筒受内压时的应力状态m n y

z

p

D

m

l2

n

n

(1)沿圆筒轴线作用于筒底的总压力为F

薄壁圆筒的横截面面积

πD F p 4

′

p

A πD

πD 2 F p 4 pD A πD 4

n

D

(Analysis of stress-state and strain-state) (2)假想用一直径平面将圆筒截分为二,并取下半环为研究对象 "p

直径平面

FN

O

FN d

y

D Fy 0 0 pl 2 sin d plD pD 2 l plD 0 2 π

(Analysis of stress-state and strain-state)

§7-2 平面应力状态分析-解析法 (Analysis of plane stress-state)y

y yx

y

xy

yx x

xx

xy

z

平面应力状态的普遍形式如图所示 .单元体上有 x , xy 和 y , yx

(Analysis of stress-state and strain-state)

一、斜截面上的应力(Stresses on an oblique section)1.截面法(Section method) 假想地沿斜截面 e-f 将单元体截开,留下左边部分的单体元 eaf 作为研究对象y n

e

yx xf

e

x

x

x

xy

αα

n α

xy

αf

a

a

yx

y

(Analysis of stress-state and strain-state)y n

e

yx xf

x

e

x

x

xy

αα

n α

xya

αf

a

yx

y

t

2.符号的确定(Sign convention)

(1)由x轴转到外法线n,逆时针转向时 为正(2)正应力仍规定拉应力 为正

(3)切应力对单元体内任一点取矩,顺时针转 为正