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1、弹性力学 第一章1弹性力学及有限元弹性力学及有限元 Elasticity and Finite Element Method1 The lectures will be given both in English and Chinese采用中英文双语讲授采用中英文双语讲授 弹性力学 第一章2Give 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。授人以鱼,不如授人以渔。弹性力学 第一章3Textbook:Applied Elasticity 徐芝纶徐芝纶中文
2、教材:中文教材:弹性力学简明教程弹性力学简明教程徐芝纶徐芝纶 弹性力学 第一章4Chapter 1. Introduction 第一章第一章 绪论绪论弹性力学 第一章5 1.1 Contents of Theory of Elasticity 1.1 1.1 弹性力学的内容弹性力学的内容 NAMENAME 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
3、个分枝它是固体力学的一个分枝弹性力学 第一章6Three branches of solid mechanics固体力学的三固体力学的三个分个分枝枝 Mechanics of materials 材料力学材料力学, , Structural Mechanics 结构力学结构力学 Elasticity 弹性力学弹性力学弹性力学 第一章7What does the Elasticity deal with? It deals with the stresses, deformations and displacements in elastic solids produced by externa
4、l forces or changes in temperature. 研究弹性体研究弹性体由于外力和温度改变而引起由于外力和温度改变而引起的应力,的应力,形变和位移。形变和位移。It analyzes the stresses, deformations and 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,检查是否满足强分析结
5、构的应力,形变和位移,检查是否满足强度,刚度和稳定性条件度,刚度和稳定性条件。弹性力学 第一章8Comparison among the three courses in solid mechanics 固体力学三门学科的比较固体力学三门学科的比较 Three branches have the same purpose and do differ from one another both in objects studied and the methods of analysis used. 1. Objects studied 研究对象研究对象 2. Methods of analysi
6、s 研究方法研究方法弹性力学 第一章9 to deal with the elastic solids 都是研究弹性体都是研究弹性体 1. objects studied:-研究对象研究对象: (1) Similarity-相同点相同点 弹性力学 第一章10(2)objects studied-difference研究对象研究对象-不同点不同点Mechanics of materials : bar element材料力学材料力学 单根杆件单根杆件Structural bar systems:-Mechanics : truss, rigid frame结构力学结构力学 杆件系统:杆件系统:
7、桁架,刚架桁架,刚架。 Elasticity: 1. plates and shells 板,壳板,壳弹性力学弹性力学 2.blocks: 块体块体 e.g. dams,foundations 坝,基础坝,基础 3.analyze bar element precisely 对杆件作精确分析对杆件作精确分析 弹性力学 第一章11 Mechanics of materials deals essentially with the stresses and displacements of structural element in the shape of a bar, straight or
8、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。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 as a truss or a rigid frame. 结构力学研究杆
9、件系统(例:桁架或刚架)的应力结构力学研究杆件系统(例:桁架或刚架)的应力和位移和位移弹性力学 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。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 p
10、recisely. 弹性力学对杆件作更精确分析弹性力学对杆件作更精确分析弹性力学 第一章142. methods of analysis:研究方法研究方法 (1) Similarity- 相同点相同点 : equilibrium aspects 静力学方面静力学方面 geometrical aspects 几何学方面几何学方面 physical aspects 物理学方面物理学方面弹性力学 第一章15equilibrium aspects -equilibrium of forces of an isolated body静力学方面静力学方面-脱离体力的平衡脱离体力的平衡 geometrica
11、l aspects -the relations between displacements and strains. 几何学方面几何学方面-位移和应变的关系位移和应变的关系 physical aspects- -the relations between stresses and strains 物理学方面物理学方面-应力和应变的关系应力和应变的关系弹性力学 第一章16(2) methods of analysis:- difference研究方法研究方法- 不同点不同点:Mechanics of materials: some assumptions on the strain condi
12、tion or the stress condition are made材料力学:材料力学: 对应变或应力情况作某些假定对应变或应力情况作某些假定Elasticity:no assumptions on the strain condition or the stress condition are made. 弹性力学弹性力学 : 对应变或应力情况不作假定对应变或应力情况不作假定弹性力学 第一章17Mechanics of materials: some assumptions on the strain condition or the stress condition are made
13、The assumptions simplify the mathematical derivation to a certain extent.The assumptions inevitably reduce the degree of accuracy of the results obtained.弹性力学 第一章18Elasticity:no assumptions on the strain condition or the stress condition are made. The results obtained in elasticity are more accurate
14、 and may be used to check the approximate results obtained in Mechanics of materials.弹性力学 第一章19The problem 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
15、 of bending stresses. No assumption, that a plane section of the beam remains plane after bending, is made in Elasticity.弹性力学 第一章20A 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 m
16、ember.The analysis in elasticity shows that the stresses are by no means uniform, but are concentrated near the hole. 弹性力学 第一章211.2 some important concepts in theory of elasticity 1.2 弹性力学中的几个重要概念弹性力学中的几个重要概念A. External Forces 外力外力B. Stress 应力应力 C. Deformation(Strain) 形变形变(应变应变)D. Displacement 位移位移弹
17、性力学 第一章22A. external forces 外力外力1. Body forces 体积力,体力体积力,体力 2. Surface forces 表面力,面力表面力,面力弹性力学 第一章231. Body forces 体力体力。External forces 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
18、body in motion.例如例如: 重力,重力, 惯性力惯性力弹性力学 第一章24 Body force Fig. 体力定义图体力定义图。弹性力学 第一章25F=lim Q/ V v 0F-body force vector at p, The vector quantity F is the intensity of body force at PF- P点的体力矢量点的体力矢量 V-an elementary volume of the body around point p V-包含包含P点的小体积点的小体积 Q-body force acting on V Q-作用在作用在 V上
19、的体力的合力上的体力的合力弹性力学 第一章26Body force components 体力分量体力分量 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量。体力在坐标轴上的投影叫体力分量。 The body force components will be denoted by X,Y and Z 体力分量用体力分量用X,Y,Z表示表示弹性力学 第一章27Sign Conventions, dim
20、ension 正负约定,正负约定, 因次因次 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。与坐标轴正向一致为正。 Its dimension is force/length3 因次:力因次:力/长度长度3 for example:弹性力学 第一章282. Surface forces 面力面力 Definition: external forces, or the loads,
21、distributed over the surface of a body, are called surface forces. 分布在物体表面的外力叫面力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,接触力例:水压力,接触力弹性力学 第一章29 Surface force Fig. 面力定义图面力定义图。弹性力学 第一章30 F=lim Q/ S F-surface force vector at P. the vector quantity is the intens
22、ity of surface force at P. F- P点的面力矢量点的面力矢量。 S-an elementary area of the surface around point P. S -包含包含P点的小面积点的小面积 Q-the surface force acting on S 作用在作用在 S上的面力矢量上的面力矢量弹性力学 第一章31Surface force components 面力分量面力分量 F=X i+Y j+Z k=(X,Y,Z) Definition: the projections of F on the x,y and z axes are called
23、the surface force components at P. 面力在坐标轴上的投影叫面力分量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 面力分量用面力分量用X,Y,Z表示表示弹性力学 第一章32 Sign Conventions, dimension 正负约定,正负约定, 因次因次 It is considered positive (negative) when it acts in the positive (negative) direction of the correspo
24、nding coordinate. 与坐标轴正向一致为正。与坐标轴正向一致为正。 Its dimension is force/length2 因次:力因次:力/长度长度2 for 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互作用
25、的力叫内力2. Stresses are the internal forces acting on the per unit area 应力:作用在单位面积上的内力应力:作用在单位面积上的内力 B. Stress 应力应力弹性力学 第一章35 3. s= lim Q/ A ( A 0) (1) s-the stress at point P on the section mPn s- 截面截面mPn上上P点的应力点的应力。(2) A-an elementary area on the section mPn around P. A- mPn面上包含面上包含 P点的微小面积点的微小面积(3)
26、Q-the internal force acted by part B on part A across A Q-B 部分作用在部分作用在A部分上的部分上的 A上的内力。上的内力。弹性力学 第一章36 Stress Fig. 应力定义图应力定义图。弹性力学 第一章37Stress components 应力分量应力分量 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. 应应力在坐标轴上的投影叫应力分量力在坐标轴上的投影叫应力
27、分量 the stress components will be denoted by XN,YN and ZN 应应力在坐标轴上分量用力在坐标轴上分量用 XN,YN and ZN 表示表示弹性力学 第一章38 Sign Conventions, dimension 正负约定,正负约定, 因次因次 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与坐标轴正向一
28、致为正 Its dimension is force/length2 因次:力因次:力/长度长度2弹性力学 第一章39Another Stress components另一应力分量另一应力分量 s=N N+N T=(N, N) N is the outward normal to the plane mPn, T is on the plane mPn, axes N,T and s are on the same plane. N 是是 mPn 的外法线,的外法线, T 在在 mPn 面上,面上, N,T 和和 s在同一面上在同一面上 The projections of s on the
29、N and T axes are called the stress components at P. s 在在 N 和和 T 上投影是上投影是 另一应力分量另一应力分量弹性力学 第一章40Normal component and tangential component法向和切向分量法向和切向分量. The stress is resolved into a normal component and a tangential component. 应力分解为法向和切向分量应力分解为法向和切向分量 The normal component is called the normal stress
30、. The tangential component is called the shearing stress. 法向分量叫法向法向分量叫法向 应力,切向分量应力,切向分量 叫剪应力叫剪应力 the stress components will be denoted by N and N ,Its dimension is force/length2 应力分量用应力分量用 N N 表示表示,因次:力因次:力/长度长度2弹性力学 第一章41Coordinate plane 坐标面坐标面 Coordinate plane-a plane with the outward normal paral
31、lel to the coordinate axes. 坐标面坐标面-截面的外法线平行于坐标轴的面截面的外法线平行于坐标轴的面 Coordinate plane-x plane,y plane,z plane 坐标面坐标面-x面面,y面面,z面面 positive(negative) coordinate plane- a plane with the outward normal in the positive(negative) direction of the coordinate axis. 正(负)面正(负)面-外法线为坐标轴外法线为坐标轴 正(负)向正(负)向 的面的面弹性力学 第
32、一章42Stress Notation 1. 坐标面上应力记号坐标面上应力记号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. 应力用应力用 加二个下标表示,第一个下标表示应力加二个下标表示,第一个下标表示应力的作用面,
33、的作用面, 第二个下标表示应力的作用方向第二个下标表示应力的作用方向。弹性力学 第一章43Stress Notation 1 坐标面上应力记号坐标面上应力记号1 x-direction y-direction z-direction X plane: xx xy xz y plane: yx yy yz z plane: zx zy zz弹性力学 第一章44Stress Notation 2. 坐标面上应力记号坐标面上应力记号2. Normal stress- with a coordinate subscript which indicates the coordinate plane th
34、e stress is acting and the direction in which the stress is acting. 正正应力用应力用 加一个下标表示,该下标既表示应力加一个下标表示,该下标既表示应力的作用面,又表示应力的作用方向。的作用面,又表示应力的作用方向。 Shearing stress- with two coordinate subscripts 剪应力剪应力 用加二个下标表示,第一个下标表示应用加二个下标表示,第一个下标表示应力的作用面,力的作用面, 第二个下标表示应力的作用方向。第二个下标表示应力的作用方向。弹性力学 第一章45Stress Notation
35、2 坐标面上应力记号坐标面上应力记号2 x-direction y-direction z-direction X plane: x xy xz y plane: yx y yz z plane: zx zy z弹性力学 第一章46Sign Conventions 正负约定正负约定 A stress component on positive (negative) coordinate plane will be considered positive as it acts in the positive (negative) direction of the corresponding ax
36、is. 正(负)面上沿坐标轴正(负)向的应力为正。正(负)面上沿坐标轴正(负)向的应力为正。- 正面正向为正,正面正向为正, 负面负向为正负面负向为正 A normal stress is positive (negative) for tension (compression) 正应力拉为正,压为负正应力拉为正,压为负弹性力学 第一章47The fig. of stress notation 坐标面上应力记号图坐标面上应力记号图弹性力学 第一章48The Equality of Shearing Stresses 剪应力互等剪应力互等 xy= yx xz= zx yz= zy The six
37、 shearing stresses are mutually equal in pairs, hence, the subscript letters of the notation of shearing stress may be interchanged at will. 剪应力互等剪应力互等, 当只考虑大小时,下标顺序可随当只考虑大小时,下标顺序可随心所欲。心所欲。弹性力学 第一章49To precisely define the stress condition完全确定一点的应力状态完全确定一点的应力状态 The stresses on any section through po
38、int 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 stress condition at that point. 一点的六个坐标面上的应力分量一点的六个坐标面上的应力分量 x y z xy= yx xz= zx yz= zy 已知后已知后,可求得
39、经过该点的任意截,可求得经过该点的任意截面上的应力。面上的应力。-称为称为 一点的六个坐标面上的应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力分量可完全确定该点的应力状态。弹性力学 第一章501. 体力分量、面力分量、应力分量的正负体力分量、面力分量、应力分量的正负号规定?号规定?2. 坐标面上应力记号坐标面上应力记号?弹性力学 第一章51C. Deformation 形变形变 By deformation we mean the change of the shape of a body, which may be expressed by the changes in lengt
40、hs and angles of its parts. 形变形变-物体形状(各部分长度和角度)的物体形状(各部分长度和角度)的改变改变弹性力学 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,考虑通过研究一点的变形,考虑通过P点的三个点的三个正向微段正向微段PA,PB,PC PA/x PA=dx P A -positive x direction PB/y PB=dy P B -positive y direction PC/z PC=
41、dz P C -positive z direction 弹性力学 第一章53Fig. PA PB PC 图 PA PB PC z C P B A y x弹性力学 第一章54Normal strain-a change in length per unit length正应变正应变-单位长度的长度改变单位长度的长度改变 x-change in length per unit length of PA y-change in length per unit length of PB z-change in length per unit length of PC positive (negati
42、ve) for elongation (contraction) x-x向微段向微段PA的相对伸缩的相对伸缩 y-y向微段向微段PB的相对伸缩的相对伸缩 伸长为正伸长为正 z- z向微段向微段PC的相对伸缩的相对伸缩 缩短为负缩短为负弹性力学 第一章55Shearing strain-the change of a right angle(radian)剪应变剪应变-直角的改变量直角的改变量(弧度弧度) rxy-the change of a right angle APB ryz-the change of a right angle BPC rzx-the change of a righ
43、t angle APC positive (negative) for a decrease(increase) of the right angle rxy- 直角直角APB的改变量的改变量 ryz- 直角直角BPC的改变量的改变量 直角直角 变小为正变小为正 rzx- 直角直角APC的改变量的改变量 直角直角 变大为负变大为负弹性力学 第一章56D. Displacements 位移位移 By displacement, we mean the change of position. 位置的移动叫位移位置的移动叫位移 Displacement components u,v,w-the pr
44、ojections of the displacement on the x,y and z axes.位移在坐标轴上投影叫位移在坐标轴上投影叫位移分量位移分量 u,v,w It is considered positive as it is in the positive direction of the corresponding coordinate axis. 沿坐标正向的位移分量为正沿坐标正向的位移分量为正。 The dimension is length. 因次为长度因次为长度弹性力学 第一章571. 研究一点的变形,考虑通过研究一点的变形,考虑通过P点的三个点的三个正向微段正向微
45、段PA,PB,PC,为何要,为何要正向微段正向微段?2. 正应变、剪应变的定义和正负号规定?正应变、剪应变的定义和正负号规定?弹性力学 第一章581.3 Basic assumptions 基本假定基本假定 The body is continuous 物体是连续的物体是连续的 The body is perfectly elastic. 物体是完全弹性的物体是完全弹性的 The body is homogeneous. 物体是均质的物体是均质的 The body is isotropic 物体是各向同性的物体是各向同性的 the displacements and strains are s
46、mall. 位移和应变是微小的。位移和应变是微小的。弹性力学 第一章59 The body is continuous 物体是连续的物体是连续的 The whole volume of the body is filled with continuous matter without any void.假定整假定整个个物体的体积都被组成这个物体的介质所充满,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。不留下任何孔隙。 Under this assumption, the physical quantities in the body, such as stresses, strai
47、ns and displacements, can be expressed by continuous functions of coordinates in the space.物理量物理量(例:应力,应变,位移)能用坐表的连续函(例:应力,应变,位移)能用坐表的连续函数表示数表示。弹性力学 第一章60The body is perfectly elastic. 物体是完全弹性的物体是完全弹性的 The body wholly obeys Hooks law of elasticity. -The relations between the stress components and th
48、e strain components are linear. 物体遵守虎克定律物体遵守虎克定律-应力分量和应变分量是应力分量和应变分量是线性关系线性关系-线性本构关系线性本构关系-物理线性物理线性。 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。弹性常数与应力和应变的大小无关。弹性力学 第一章61The body is homogeneous. 物体是均质的物体是均质的 The elastic constant
49、s will be independent of the location in the body. 弹性常数与位置无关。弹性常数与位置无关。 物体由同一种材料组成。物体由同一种材料组成。 物体由多种材料组成,但每一种材料的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 物体是各向同性的物体是各向同性的 The elastic constants will be independent of the orientation of the coordinate axes. 弹性常
50、数与坐标轴的方向无关。弹性常数与坐标轴的方向无关。 Steel structure-isotropic 钢钢-各向同性各向同性 wooden structure-not isotropic 木木-各向异性各向异性弹性力学 第一章63 The displacements and strains are smallThe displacements and strains are small. . 位移和应变是微小的位移和应变是微小的-几何线性。几何线性。 The displacement components are very small in comparison with its origi