《弹性力学》（双语版） 第二章 平面问题的基本理论

Chapter 2 The Basic Theory of the Plane problem

1 Elasticity

弹单性生力学

2

Chapter 2 The basic theory of the Plane problem 2 82-1 Plane stress problem and plane strain problem >82-2 Differential equation of equilibrium D82-3 The stress on the incline. Principalstress [32-4 Geometricalequation The displacement of the rigid body 82-5 Physicalequation D82-6 Boundary conditions [D$2-7 Saint-Venant's principle ID 82-8 Solving the plane problem according to the displacement >82-9 Solving the plane problem according to the stress. Compatible equation 82-10 The simplification under the circumstances of ordinary physical force 82-11 Stress function. Inverse solution method and semi-inverse method I Exercise lesson

3 Chapter 2 The Basic theory of the Plane Problem §2-11 Stress function.Inverse solution method and semi-inverse method §2-1 Plane stress problem and plane strain problem §2-2 Differential equation of equilibrium §2-3 The stress on the incline.Principal stress §2-4 Geometrical equation.The displacement of the rigid body §2-5 Physical equation §2-6 Boundary conditions §2-7 Saint-Venant’s principle §2-8 Solving the plane problem according to the displacement §2-9 Solving the plane problem according to the stress.Compatible equation §2-10 The simplification under the circumstances of ordinary physical force Exercise Lesson

平的签论 第二章平面问题的基本理论 §2-1平面应力问题与平面应变问题 §2-2平衡微分方程 §2-3斜面上的应力主应力 §2-4几何方程刚体位移 §2-5物理方程 习§2-6边界条件 四§2-7圣维南原理 §2-8按位移求解平面问题 §2-9按应力求解平面问题。相容方程 §2-11应力函数逆解法与半逆解法 习题课

4 第二章 平面问题的基本理论 §2-11 应力函数逆解法与半逆解法 §2-1 平面应力问题与平面应变问题 §2-2 平衡微分方程 §2-3 斜面上的应力主应力 §2-4 几何方程刚体位移 §2-5 物理方程 §2-6 边界条件 §2-7 圣维南原理 §2-8 按位移求解平面问题 §2-9 按应力求解平面问题。相容方程 §2-10 常体力情况下的简化 习题课

82-1 Plane stress problem and plane strain problem In actual problem, it is strictly saying that any elastic body whose external force for suffering is a space system of forces is generally the space object. However, when both the shape and force circumstance of the elastic body for investigating have their own certain characteristics. As long as the abstraction of the mechanics is handled together with appropriate simplification, it can be concluded as the elasticity plane problem The plane problem is divided into the plane stress problem and plane strain problem 1. Plane stress problem Equal thickness lamella bears the surface force that parallels with plate face and don t change along the thickness at the same time so does the volumetric force y 02=0x=0x2=0 Fig2-1 5

5 1.Plane stress problem §2-1 Plane stress problem and plane strain problem In actual problem,it is strictly saying that any elastic body whose external force for suffering is a space system of forces is generally the space object.However,when both the shape and force circumstance of the elastic body for investigating have their own certain characteristics.As long as the abstraction of the mechanics is handled together with appropriate simplification,it can be concluded as the elasticity plane problem. The plane problem is divided into the plane stress problem and plane strain problem. Equal thickness lamella bears the surface force that parallels with plate face and don’t change along the thickness.At the same time,so does the volumetric force. σz = 0 τzx = 0 τzy = 0 Fig.2－1

平富的论 §2-1平面应力问题与平面应变问题 在实际问题中,任何一个弹性体严格地说都是空间物体, 它所受的外力一般都是空间力系。但是,当所考察的弹性体 的形状和受力情况具有一定特点时,只要经过适当的简化和 力学的抽象处理,就可以归结为弹性力学平面问题。 平面问题分为平面应力问题和平面应变问题 平面应力问题 等厚度薄板,板边承受平 行于板面并且不沿厚度变化的 面力,同时体力也平行于板面 并且不沿厚度变化 F=0 t=0V 0 y2 图

6 一、平面应力问题 §2-1 平面应力问题与平面应变问题 在实际问题中，任何一个弹性体严格地说都是空间物体， 它所受的外力一般都是空间力系。但是，当所考察的弹性体 的形状和受力情况具有一定特点时，只要经过适当的简化和 力学的抽象处理，就可以归结为弹性力学平面问题。 平面问题分为平面应力问题和平面应变问题。 等厚度薄板，板边承受平 行于板面并且不沿厚度变化的 面力，同时体力也平行于板面 并且不沿厚度变化。 σz = 0 τzx = 0 τzy = 0 图2－1

Characteristics: 1) The dimension of length and breadth is far larger than that of thickness 2) The force along the plate face for suffering is the face force in parallel with plate face, and along the thickness even, the volumetric force is in parallel with plate force and doesnt change along the thickness, and has no external force function on the surface front and back of the flat par Attention: Plane stress problemo, =0,but 8#0, this is contrary to plane strain problem 7

7 x y Characteristics： 1) The dimension of length and breadth is far larger than that of thickness. 2) The force along the plate face for suffering is the face force in parallel with plate face,and along the thickness even,the volumetric force is in parallel with plate force and doesn’t change along the thickness, and has no external force function on the surface front and back of the flat panel. Attention: Plane stress problem   z  0 z =0,but ,this is contrary to plane strain problem

平河的签论 特点 1)长、宽尺寸远大于厚度 2)沿板边受有平行板面的面力,且沿厚度均布,体力 行于板面且不沿厚度变化,在平板的前后表面上 无外力作用。 y 注意:平面应力问题σ=0,但E≠0,这与平面应变 问题相反

8 x y 特点： 1) 长、宽尺寸远大于厚度 2) 沿板边受有平行板面的面力，且沿厚度均布，体力 平行于板面且不沿厚度变化，在平板的前后表面上 无外力作用。 问题相反。 注意：平面应力问题  z  0 z =0，但 ，这与平面应变

2. Plane strain problem change along the length on the column face, at the same time, so does the t Very long column bears the face force in parallel with plate face and does volumetric force e2=0x=0n=0 For example: dam, circular cylinder piping by the internal air pressure and long level laneway etc P Fig. 2-2 Attention: Plane strain probleme=0, buto#0, this is contrary to plane stress problem. 9

9 2.Plane strain problem Very long column bears the face force in parallel with plate face and doesn’t change along the length on the column face,at the same time,so does the volumetric force. εz = 0 τzx = 0 τzy = 0 x Fig. 2－2 For example:dam,circular cylinder piping by the internal air pressure and long level laneway etc. Attention: Plane strain problem  z  0 z = 0，but ,this is contrary to plane stress problem. x y P

平河的签论 二、平面应变问题 很长的柱体,在柱面上承受平行于横截面并且不沿长度 变化的面力,同时体力也平行于横截面并且不沿长度变化。 E=0 0 0 如:水坝、受内压的圆柱管道和长水平巷道等。 P 图2-2 注意平面应变问题E=0,但≠0,这恰与平面应) 问题相反

10 二、平面应变问题 很长的柱体，在柱面上承受平行于横截面并且不沿长度 变化的面力，同时体力也平行于横截面并且不沿长度变化。 ε z = 0 τ zx = 0 τ zy = 0 x 图 2－2 如：水坝、受内压的圆柱管道和长水平巷道等。 注意平面应变问题  z  0 z = 0，但 问题相反。 ，这恰与平面应力 x y P