弹性力学第一章ppt课件.ppt

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1、弹性力学 第一章,1,弹性力学及有限元 Elasticity and Finite Element Method1,The lectures will be given both in English and Chinese采用中英文双语讲授,弹性力学 第一章,2,Give me a fish and I will eat today, Teach me to fish and I will eat for a life time.,授人以鱼，不如授人以渔。,弹性力学 第一章,3,Textbook:Applied Elasticity 徐芝纶,中文教材：弹性力学简明教程徐芝纶,弹性力学 第一章

2、,4,Chapter 1. Introduction 第一章 绪论,弹性力学 第一章,5,1.1 Contents of Theory of Elasticity 1.1 弹性力学的内容NAME Theory of elasticity is often called elasticity for short. It is the branch of solid mechanics. 弹性力学的理论简称为弹性理论或弹性力学. 它是固体力学的一个分枝,弹性力学 第一章,6,Three branches of solid mechanics固体力学的三个分枝,Mechanics of materi

3、als 材料力学, Structural Mechanics 结构力学 Elasticity 弹性力学,弹性力学 第一章,7,What does the Elasticity deal with? It deals with the stresses, deformations and displacements in elastic solids produced by external forces or changes in temperature. 研究弹性体由于外力和温度改变而引起的应力，形变和位移。It analyzes the stresses, deformations and

4、 displacements of structural elements within the elastic range and thereby to check the sufficiency of their strength, stiffness and stability. 分析结构的应力，形变和位移，检查是否满足强度，刚度和稳定性条件。,弹性力学 第一章,8,Comparison among the three courses in solid mechanics 固体力学三门学科的比较,Three branches have the same purpose and do di

5、ffer from one another both in objects studied and the methods of analysis used. 1. Objects studied 研究对象 2. Methods of analysis 研究方法,弹性力学 第一章,9,to deal with the elastic solids 都是研究弹性体,1. objects studied:-研究对象: (1) Similarity-相同点,弹性力学 第一章,10,(2)objects studied-difference研究对象-不同点,Mechanics of materials

6、 : bar element材料力学 单根杆件Structural bar systems:-Mechanics : truss, rigid frame结构力学 杆件系统： 桁架，刚架。 Elasticity: 1. plates and shells 板，壳弹性力学 2.blocks: 块体 e.g. dams,foundations 坝，基础 3.analyze bar element precisely 对杆件作精确分析,弹性力学 第一章,11,Mechanics of materials deals essentially with the stresses and displace

7、ments of structural element in the shape of a bar, straight or curved, which is subjected to tension, compression, shear, bending, or torsion. 材料力学研究受到拉、压、剪、弯或扭的直杆或曲杆的应力和位移。,弹性力学 第一章,12,Structural Mechanics deals with the stresses and displacements of a structural in the form of a bar system, such a

8、s a truss or a rigid frame.结构力学研究杆件系统（例：桁架或刚架）的应力和位移,弹性力学 第一章,13,Elasticity deals with the stresses and displacements of the structural elements such as blocks, plates and shells, which are not in the form of a bar.弹性力学研究块体、板和壳体的应力和位移。Elasticity also analyze a bar element thoroughly and precisely.弹性

9、力学对杆件作更精确分析,弹性力学 第一章,14,2. methods of analysis:研究方法 (1) Similarity- 相同点 :,equilibrium aspects 静力学方面 geometrical aspects 几何学方面 physical aspects 物理学方面,弹性力学 第一章,15,equilibrium aspects -equilibrium of forces of an isolated body静力学方面-脱离体力的平衡 geometrical aspects -the relations between displacements and st

10、rains. 几何学方面-位移和应变的关系 physical aspects- -the relations between stresses and strains 物理学方面-应力和应变的关系,弹性力学 第一章,16,(2) methods of analysis:- difference研究方法- 不同点:,Mechanics of materials: some assumptions on the strain condition or the stress condition are made材料力学： 对应变或应力情况作某些假定Elasticity：no assumptions

11、on the strain condition or the stress condition are made. 弹性力学 : 对应变或应力情况不作假定,弹性力学 第一章,17,Mechanics of materials: some assumptions on the strain condition or the stress condition are madeThe assumptions simplify the mathematical derivation to a certain extent.The assumptions inevitably reduce the de

12、gree of accuracy of the results obtained.,弹性力学 第一章,18,Elasticity：no assumptions on the strain condition or the stress condition are made. The results obtained in elasticity are more accurate and may be used to check the approximate results obtained in Mechanics of materials.,弹性力学 第一章,19,The problem

13、of bending of a straight beam under transverse loads.It is assumed in mechanics of materials that a plane section of the beam remains plane after bending, which leads to the linear distribution of bending stresses. No assumption, that a plane section of the beam remains plane after bending, is made

14、in Elasticity.,弹性力学 第一章,20,A prismatical tension member with a small holeIt is assumed in mechanics of materials that the tensile stresses are uniformly distributed across the net section of the member.The analysis in elasticity shows that the stresses are by no means uniform, but are concentrated n

15、ear the hole.,弹性力学 第一章,21,1.2 some important concepts in theory of elasticity 1.2 弹性力学中的几个重要概念,A. External Forces 外力B. Stress 应力 C. Deformation(Strain) 形变(应变)D. Displacement 位移,弹性力学 第一章,22,A. external forces 外力,1. Body forces 体积力，体力 2. Surface forces 表面力，面力,弹性力学 第一章,23,1. Body forces 体力。,External fo

16、rces or the loads,distributed over the volume of the body,are called body forces.分布在物体体内的外力叫体力E.g. gravitational forces, or inertia forces in the case of a body in motion.例如： 重力， 惯性力,弹性力学 第一章,24,Body force Fig. 体力定义图。,弹性力学 第一章,25,F=lim Q/ V v 0F-body force vector at p, The vector quantity F is the i

17、ntensity of body force at PF- P点的体力矢量 V-an elementary volume of the body around point p V-包含P点的小体积 Q-body force acting on V Q-作用在 V上的体力的合力,弹性力学 第一章,26,Body force components 体力分量,F=X i+Y j+Z k=(X,Y,Z) The projections of F on the x,y,and z axes are called the body force components at P. 体力在坐标轴上的投影叫体力分

18、量。The body force components will be denoted by X,Y and Z 体力分量用X,Y,Z表示,弹性力学 第一章,27,Sign Conventions, dimension 正负约定， 因次,It is considered positive (negative) when it acts in the positive (negative) direction of the corresponding coordinate. 与坐标轴正向一致为正。Its dimension is force/length3 因次：力/长度3for example

19、:,弹性力学 第一章,28,2. Surface forces 面力,Definition: external forces, or the loads, distributed over the surface of a body, are called surface forces. 分布在物体表面的外力叫面力e.g. hydrostatic pressure, the pressure of one body on another 例：水压力，接触力,弹性力学 第一章,29,Surface force Fig. 面力定义图。,弹性力学 第一章,30,F=lim Q/ SF-surface

20、 force vector at P. the vector quantity is the intensity of surface force at P. F- P点的面力矢量。 S-an elementary area of the surface around point P. S -包含P点的小面积 Q-the surface force acting on S 作用在S上的面力矢量,弹性力学 第一章,31,Surface force components 面力分量,F=X i+Y j+Z k=(X,Y,Z)Definition: the projections of F on th

21、e x,y and z axes are called the surface force components at P. 面力在坐标轴上的投影叫面力分量The surface force components will be denoted by X,Y and Z 面力分量用X,Y,Z表示,弹性力学 第一章,32,Sign Conventions, dimension 正负约定， 因次,It is considered positive (negative) when it acts in the positive (negative) direction of the correspo

22、nding coordinate. 与坐标轴正向一致为正。Its dimension is force/length2 因次：力/长度2for example:,弹性力学 第一章,33,写面力分量,3-9 3-10,弹性力学 第一章,34,1. Internal forces: under the action of external forces,internal forces will be produced between the parts of a body.内力：在外力作用下，物体各部分间产生相互作用的力叫内力2. Stresses are the internal forces

23、acting on the per unit area 应力：作用在单位面积上的内力,B. Stress 应力,弹性力学 第一章,35,3. s= lim Q/A （ A 0） (1) s-the stress at point P on the section mPn s- 截面mPn上P点的应力。(2) A-an elementary area on the section mPn around P. A- mPn面上包含 P点的微小面积(3) Q-the internal force acted by part B on part A across A Q-B 部分作用在A部分上的 A上

24、的内力。,弹性力学 第一章,36,Stress Fig. 应力定义图。,弹性力学 第一章,37,Stress components 应力分量,s=XN i+YN j+ZN k=(XN,YN,ZN) The projections of s on the x,y,and z axes are called the stress components at P. 应力在坐标轴上的投影叫应力分量the stress components will be denoted by XN,YN and ZN 应力在坐标轴上分量用 XN,YN and ZN 表示,弹性力学 第一章,38,Sign Conven

25、tions, dimension 正负约定， 因次,It is considered positive (negative) when it acts in the positive (negative) direction of the corresponding coordinate. XN,YN and ZN 与坐标轴正向一致为正 Its dimension is force/length2 因次：力/长度2,弹性力学 第一章,39,Another Stress components另一应力分量,s=N N+N T=(N, N)N is the outward normal to the

26、 plane mPn, T is on the plane mPn, axes N,T and s are on the same plane. N 是 mPn 的外法线， T 在 mPn 面上， N,T 和 s在同一面上The projections of s on the N and T axes are called the stress components at P. s 在 N 和 T 上投影是 另一应力分量,弹性力学 第一章,40,Normal component and tangential component法向和切向分量.,The stress is resolved in

27、to a normal component and a tangential component. 应力分解为法向和切向分量The normal component is called the normal stress. The tangential component is called the shearing stress. 法向分量叫法向 应力，切向分量 叫剪应力the stress components will be denoted by N and N ，Its dimension is force/length2 应力分量用 N N 表示,因次：力/长度2,弹性力学 第一章,

28、41,Coordinate plane 坐标面,Coordinate plane-a plane with the outward normal parallel to the coordinate axes. 坐标面-截面的外法线平行于坐标轴的面Coordinate plane-x plane,y plane,z plane 坐标面-x面,y面,z面positive(negative) coordinate plane- a plane with the outward normal in the positive(negative) direction of the coordinate

29、axis. 正（负）面-外法线为坐标轴 正（负）向 的面,弹性力学 第一章,42,Stress Notation 1. 坐标面上应力记号1.,We associate the stress with two coordinate subscripts, first coordinate subscript indicates the coordinate plane the stress is acting, the second subscript indicates the direction in which the stress is acting. 应力用加二个下标表示，第一个下标表

30、示应力的作用面， 第二个下标表示应力的作用方向。,弹性力学 第一章,43,Stress Notation 1 坐标面上应力记号1,x-direction y-direction z-directionX plane: xx xy xzy plane: yx yy yzz plane: zx zy zz,弹性力学 第一章,44,Stress Notation 2. 坐标面上应力记号2.,Normal stress- with a coordinate subscript which indicates the coordinate plane the stress is acting and t

31、he direction in which the stress is acting.正应力用加一个下标表示，该下标既表示应力的作用面，又表示应力的作用方向。Shearing stress- with two coordinate subscripts剪应力用加二个下标表示，第一个下标表示应力的作用面， 第二个下标表示应力的作用方向。,弹性力学 第一章,45,Stress Notation 2 坐标面上应力记号2,x-direction y-direction z-directionX plane: x xy xzy plane: yx y yzz plane: zx zy z,弹性力学 第一

32、章,46,Sign Conventions 正负约定,A stress component on positive (negative) coordinate plane will be considered positive as it acts in the positive (negative) direction of the corresponding axis. 正（负）面上沿坐标轴正（负）向的应力为正。- 正面正向为正， 负面负向为正A normal stress is positive (negative) for tension (compression) 正应力拉为正，压为

33、负,弹性力学 第一章,47,The fig. of stress notation 坐标面上应力记号图,弹性力学 第一章,48,The Equality of Shearing Stresses 剪应力互等,xy=yx xz= zx yz= zyThe six shearing stresses are mutually equal in pairs, hence, the subscript letters of the notation of shearing stress may be interchanged at will.剪应力互等, 当只考虑大小时，下标顺序可随心所欲。,弹性力学

34、 第一章,49,To precisely define the stress condition完全确定一点的应力状态,The stresses on any section through point P can be evaluated if the stress components x y z xy= yx xz=zx yz=zy at that point are known.Consequently,the six stress components x y z xy= yx xz=zx yz=zy at any point P precisely define the stres

35、s condition at that point.一点的六个坐标面上的应力分量 x y z xy= yx xz=zx yz=zy 已知后，可求得经过该点的任意截面上的应力。-称为 一点的六个坐标面上的应力分量可完全确定该点的应力状态。,弹性力学 第一章,50,1. 体力分量、面力分量、应力分量的正负号规定？2. 坐标面上应力记号?,弹性力学 第一章,51,C. Deformation 形变,By deformation we mean the change of the shape of a body, which may be expressed by the changes in len

36、gths and angles of its parts.形变-物体形状（各部分长度和角度）的改变,弹性力学 第一章,52,To study deformation condition at a certain point P, we consider line segments PA, PB, PC 研究一点的变形，考虑通过P点的三个正向微段PA,PB,PCPA/x PA=dx P A -positive x direction PB/y PB=dy P B -positive y direction PC/z PC=dz P C -positive z direction,弹性力学 第一章

37、,53,Fig. PA PB PC 图 PA PB PC,z C P B A y x,弹性力学 第一章,54,Normal strain-a change in length per unit length正应变-单位长度的长度改变,x-change in length per unit length of PAy-change in length per unit length of PBz-change in length per unit length of PCpositive (negative) for elongation (contraction) x-x向微段PA的相对伸缩y

38、-y向微段PB的相对伸缩 伸长为正z- z向微段PC的相对伸缩 缩短为负,弹性力学 第一章,55,Shearing strain-the change of a right angle（radian)剪应变-直角的改变量(弧度),rxy-the change of a right angle APBryz-the change of a right angle BPCrzx-the change of a right angle APCpositive (negative) for a decrease(increase) of the right anglerxy- 直角APB的改变量 ry

39、z- 直角BPC的改变量 直角 变小为正rzx- 直角APC的改变量 直角 变大为负,弹性力学 第一章,56,D. Displacements 位移,By displacement, we mean the change of position. 位置的移动叫位移Displacement components u,v,w-the projections of the displacement on the x,y and z axes.位移在坐标轴上投影叫位移分量 u,v,wIt is considered positive as it is in the positive direction

40、 of the corresponding coordinate axis. 沿坐标正向的位移分量为正。The dimension is length. 因次为长度,弹性力学 第一章,57,1. 研究一点的变形，考虑通过P点的三个正向微段PA,PB,PC，为何要正向微段?2. 正应变、剪应变的定义和正负号规定？,弹性力学 第一章,58,1.3 Basic assumptions 基本假定,The body is continuous 物体是连续的The body is perfectly elastic. 物体是完全弹性的The body is homogeneous. 物体是均质的The b

41、ody is isotropic 物体是各向同性的the displacements and strains are small. 位移和应变是微小的。,弹性力学 第一章,59,The body is continuous 物体是连续的,The whole volume of the body is filled with continuous matter without any void.假定整个物体的体积都被组成这个物体的介质所充满，不留下任何孔隙。Under this assumption, the physical quantities in the body, such as st

42、resses, strains and displacements, can be expressed by continuous functions of coordinates in the space.物理量（例：应力，应变，位移）能用坐表的连续函数表示。,弹性力学 第一章,60,The body is perfectly elastic. 物体是完全弹性的,The body wholly obeys Hooks law of elasticity. -The relations between the stress components and the strain component

43、s are linear. 物体遵守虎克定律-应力分量和应变分量是线性关系-线性本构关系-物理线性。 The elastic constants will be independent of the stress or strain components under this assumption.弹性常数与应力和应变的大小无关。,弹性力学 第一章,61,The body is homogeneous. 物体是均质的,The elastic constants will be independent of the location in the body. 弹性常数与位置无关。物体由同一种材料

44、组成。物体由多种材料组成，但每一种材料的颗粒远小于物体且在物体内均匀分布。,弹性力学 第一章,62,The body is isotropic 物体是各向同性的,The elastic constants will be independent of the orientation of the coordinate axes. 弹性常数与坐标轴的方向无关。Steel structure-isotropic 钢-各向同性wooden structure-not isotropic 木-各向异性,弹性力学 第一章,63,The displacements and strains are smal

45、l. 位移和应变是微小的-几何线性。,The displacement components are very small in comparison with its original dimensions-We may use the lengths and angles of the body before deformation when we formulate the equilibrium equations.位移远小于物体尺寸-可用变形前的尺寸代替变形后的尺寸。The strain components and the rotations of all line element

46、s are much smaller than unity.-we may neglect the squares and products of the small quantities. 应变分量和转角远小于1，其乘积及二次幂可忽略。 Linear equations and problems 线性方程,线性问题,弹性力学 第一章,64,作业： 1.写图3-9、图3-10的面力分量 3-9 3-102. 体力分量、面力分量、应力分量的正负号规定？3. 何为坐标面？何为正（负）坐标面？试述坐标面上的应力记号。4. 正应变、剪应变的定义和正负号规定？5. 研究一点的变形，考虑通过P点的三个正向微段PA,PB,PC，为何要正向微段?6. 设x=-10为负x面,该面上沿x正向作用100KN/m2的荷载，沿y负向作用200KN/m2的荷载,设该面为物体表面，写出该面上面力分量；设该面原位于物体内部，写出该面上应力分量。试述应力和面力的异同点？,