最大剪应力计算公式 The formula of maximum shear stress
s x s y 2 s max s min s 1 s 2 t max t x 2 2 2
2
极值剪应力面与主平面的关系 The relation between the plane subjected maximum shear stress and the principle plane
规定sign convention
s 1 s max s x s y s x s y 2 t x s 2 s min 2 2
2
s与截面外法线同向为正；绕研究对象顺时针转为正
The normal stress is positive if the direction of the normal stress is the coordinate external normal line . The shear stress is positive if the shear stress tend to clockwise rotation .
第十章强度理论与组合变形的强度计算
theorem of strength and the
calculation of compose deformation strength §7–1 应力状态的概念
§7–2 平面应力状态分析——解析法
§7–3 平面应力状态分析——图解法
10–１ 应力状态分析 To analysis State of Stress at a Given Point
Two characteristics: 1. The magnitude of stresses in parallel planes is equal 2. the distribution of stress is uniform in parallel planes
七、主单元体、主平面、主应力：
sy
y
sz
z
txy
sx
x
三、单元体：
单元体——构件内的点的代表物，是包围被研究点的
无限小的几何体，常用的是正六面体。 单元体的性质——a、平行面上，应力均布； b、平行面上，应力相等。
Per unit body: surrounded the point we obtain a small body with six planes , we called the body “Per unit body”.
P M
2、组合变形杆将怎样破坏？ How to failure of the members subjected to the composite loadings?
二、一点的应力状态概念：the concept of State of Stress at a Given Point
过一点有无数的截面，这一点的各个截面上应力情况的集合，称为这点的应力 状态.
例10-1 圆轴受力偶矩m=700kN· m作用 试求圆轴表面K点的主 应力和最大剪应力。 The shaft is subjected moment as shown in Fig10-1,to determine the principle stress and maximum stress at given K.
此方程曲线为圆—应力圆（或莫尔圆，由德 国工程师：Otto Mohr引入）
s

sx
y
O
sy
x
txy
t
n
A graphical solution to analysis the plane stress transformation (MOHR’s circle)
t
sy s
t
y x t n D( s , t) C O 2 O
一、引言 guiding
1、铸铁与低碳钢的拉、压、扭试验现象是怎样产生的？ How to create the tension, compression and torsion test phenomena of cast iron and steel?
P 铸铁拉 伸 M
低碳钢 P 铸铁
铸铁压缩 P
s 1 s max
s3
s x s y
2
s x s y 2 t x 55.7 2
2
2
s x s y
2
s x s y 2 t x 55.7 2
2t x tan 2 0 s x s y
20 90 0
n
二、应力圆的画法 the procedures to draw Stress Circle
建立应力坐标系，如下图所示
First; Establish a coordinate system such that the abscissa represents the normal stress,
s 1s 2 s 3
三向应力状态（ Three—Dimensional State of Stress）：
三个主应力都不为零的应力状态。 Three Principal Stresses are not zero. 二向应力状态（Plane State of Stress）： 一个主应力为零的应力状态。 There is a principle stress is zero. 单向应力状态（Unidirectional State of Stress）： 一个主应力不为零的应力状态。 Only subjected to one dimensional normal stress 8.纯剪应力状态 shearing state of stress only subjected to shearing stress
Through a given point has incalculability crosses,the gather of the
stress of each section is called State of Stress at a Given Point By choosing different ordinations, we obtain different stresses, but the states of stress are equivalent.
解:solution
（1）取单元体。Take per unit body.
t x t max
T WT
t y t x
3
T m 16 700 10 tx 55.7 3 WT d 3 40 16
（2）求主应力和最大剪应力 to calculate the principle stress and maximum shear stress
45 t max
s x s y 2 t x 55.7 2
2
例2 分析受扭构件的破坏规律。 To analysis the reason of failure the members subjected tensional moments. 确定危险点并画其单元体 解：
Principal body Principal Plane Principal Stress 主单元体(Principal body)： 各侧面上剪应力均为零的单元体。 The shear stresses are zero on side sections of the body
主平面(Principal Plane)： 剪应力为零的截面。 The plane that the shear stress is zero is called Principal Plane. 主应力(Principal Stress ）： 主面上的正应力。 The stresses on the Principal Plane are called Principal Stress 主应力排列规定：按代数值大小 The array convention of Principal Stress Based on the magnitude of algebra
s x s y tg21 0 10 2t xy
低碳钢 : s s 240MPa;t s 200MPa
低碳钢
Cast iron
灰口铸铁 : s Lb 98~280MPa
s yb 640 ~960MPa;t b 198 ~300MPa
铸铁
小节 summarize Through experiment it has been found that materials that are ductile will fail due to shear stress .As a result, if a torque is applied to a bar made from mild steel, the maximum in-plane shear stress will cause it to fail as shown in the adjacent photo. Materials that are brittle fail due to normal stress. That is why when a brittle material, such as cast iron, is subjected to torsion it will fail in tension at a 45 inclination as seen in the adjacent photo.
2 t xy t
s 1t ;s 2 0;s 3 t
s x s y 2 2 t max （ ）t xy t 2 t min
破坏分析 failure analysis mild steel