Mechanics of materials CHAPTER7 AN⊥OF STRESSANDSTRAIN
Mechanics of Materials
树料力
CHAPTER 7 ANALYSS OFTHE STATE OF STRESSAND STRAIN 」§7-1 CONCEPTS OF THE STATE OF STRESS 」§7-2 ANALYSIS OF THE STATE OF PLANE STRESS ANALYTICAL METHOD 87-3 ANALYSIS OF THE STATE OF PLANE § princesa宝 ESAND THEIR TRAJECTORIES OF THE BEAMI 87-5 ANALYSIS OF TRIAXIAL STRESSED STATE--METHOD OF STRESS CIRCLE □§7-6 ANALYSIS OF STRAIN IN A PLANE 」§7-7 RELATION BETWEEN STRESS AND STRAIN UNDER COMPLEX STRESSED STATE-(GENERALIZED HOOKES LAW) D 87-8 STRAIN-ENERGY DENSITY UNDER COMPLEX STRESSED STATE
CHAPTER 7 ANALYSIS OF THE STATE OF STRESS AND STRAIN §7–4 PRINCEPAL STRESSES AND THEIR TRAJECTORIES OF THE BEAM §7–5 ANALYSIS OF TRIAXIAL STRESSED STATE—METHOD OF STRESS CIRCLE §7–6 ANALYSIS OF STRAIN IN A PLANE §7–7 RELATION BETWEEN STRESS AND STRAIN UNDER COMPLEX STRESSED STATE—（GENERALIZED HOOKE’S LAW） §7–8 STRAIN -ENERGY DENSITY UNDER COMPLEX STRESSED STATE §7–1 CONCEPTS OF THE STATE OF STRESS §7–2 ANALYSIS OF THE STATE OF PLANE STRESS — ANALYTICAL METHOD §7–3 ANALYSIS OF THE STATE OF PLANE STRESS — GRAPHYCAL METHOD
第七章应力状态与应变状态分析 §7-1应力状态的概念 D§7-2平面应力状态分析—解析法 回§7-3平面应力状态分析—图解法 §7-4梁的主应力及其主应力迹线 □§7-5三向应力状态研究—应力圆法 回§7-6平面内的应变分析 □§7-7复杂应力状态下的应力--应变关系 (广义虎克定律) §78复杂应力状态下的变形比能
第七章 应力状态与应变状态分析 §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 §7–4 梁的主应力及其主应力迹线 §7–5 三向应力状态研究——应力圆法 §7–6 平面内的应变分析 §7–7 复杂应力状态下的应力 -- 应变关系 ——（广义虎克定律） §7–8 复杂应力状态下的变形比能
ANALYSIS OFSTRESSAND STRAIN 87-1 CONCEPTS OF THE STAFE OF STRISS 1、 Forward 1) Investigation on the tensile, compressive and torsional test of cast iron and low-carbon steel Cast iron in Cast iron in compression tension M Low-carbon steel Cast iron 2) How will the member rupture in combined deformations M
§7–１ CONCEPTS OF THE STAFE OF STRISS 1、Forward 1)、Investigation on the tensile, compressive and torsional test of cast iron and low-carbon steel M Low-carbon steel Cast iron P P Cast iron in tension P Cast iron in compression 2)、How will the member rupture in combined deformations？ M P
§7-1应力状态的概念 、引言 1、铸铁与低碳钢的拉、压、扭试验现象是怎样产生的? 铸铁拉伸 铸铁压缩 M P 低碳钢 铸铁 P P 2、组合变形杆将怎样破坏? M
§7–１ 应力状态的概念 一、引言 1、铸铁与低碳钢的拉、压、扭试验现象是怎样产生的？ M 低碳钢 铸铁 P P 铸铁拉伸 P 铸铁压缩 2、组合变形杆将怎样破坏？ M P
ANALYSIS OFSTRESSAND STRAIN 2 State of stress at a point There are countless sections through a point. The gathering of stresses in all sections is called the state of stress at this point 3. Element: O Element-- Delegate of a point in the member. It is a infinitesimal geometric body enveloping the studied point In common use it is a correctitude cubic 2 Properties of an element-a Stresses are distributed uniformly in the sections b, The stresses in two planes that are Ox parallel to each other are equal 4 Expression ofstressesin general case
4、Expression of stresses in general case 3、Element：Element—Delegate of a point in the member. It is a infinitesimal geometric body enveloping the studied point. In common use it is a correctitude cubic body. Properties of an element—a、Stresses are distributed uniformly in the sections； b、The stresses in two planes that are parallel to each other are equal. 2、State of stress at a point： There are countless sections through a point. The gathering of stresses in all sections is called the state of stress at this point. x y z sx sz s y txy
二、一点的应力状态: 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( State of stress at a given point)。 三、单元体:①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 ②单元体的性质—a、平行面上,应力均布; b、平行面上,应力相等。 十a四、普逾状态下的应力表示
四、普遍状态下的应力表示 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。 二、一点的应力状态： 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（State of Stress at a Given Point）。 x y z sx sz s y txy
ANALYSIS OFSTRESSAND STRAIN 5, Theorem of conjugate shearing stress Shearing stresses on perpendicular planes are equal in magnitude and have directions such that both stresses point toward or both point away form the line of intersection of the faces Provement The element is in equilibrium ∑M=0 (I dydz dx(T dzdx dy=0 Ox x
x y z sx sz s y txy 5、 Theorem of conjugate shearing stress: Shearing stresses on perpendicular planes are equal in magnitude and have directions such that both stresses point toward ,or both point away form ,the line of intersection of the faces . Mz = 0 (t xydydz)dx−(t yxdzdx)dy=0 t xy =t yx Provement : The element is in equilibrium
五、剪应力互等定理( Theorem of Conjugate Shearing Stress): 过一点的两个正交面上,如果有与相交边垂直的剪应力分 量,则两个面上的这两个剪应力分量一定等值、方向相对或相 离 证明单元体平衡∑M2=0 (r dydz)dx(t dzdx)dy=O Ox x
x y z sx sz s y txy 五、剪应力互等定理（Theorem of Conjugate Shearing Stress): 过一点的两个正交面上,如果有与相交边垂直的剪应力分 量,则两个面上的这两个剪应力分量一定等值、方向相对或相 离。 证明: 单元体平衡 Mz =0 (t xydydz)dx−(t yxdzdx)dy=0 t xy =t yx