弹性力学第一章ppt课件

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 1弹性力学 第一章弹性力学及有限元 Elasticity and FiniGive 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。2弹性力学 第一章Give me a fish and I will eat Textbook:Applied Elasticity 徐芝纶徐芝纶中文教材：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 3弹性力学 第一章Textbook:Applied Elasticity Chapter 1.Introduction 第一章第一章 绪论绪论4弹性力学 第一章Chapter 1.Introduction 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.它是固体力学的一个分枝它是固体力学的一个分枝5弹性力学 第一章 1.1 Contents of Theory ofThree 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6弹性力学 第一章Three branches of solid mechanWhat 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。7弹性力学 第一章What does the Elasticity deal Comparison 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 analysis 研究方法研究方法8弹性力学 第一章Comparison among the three co 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 9弹性力学 第一章 to deal(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：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 10弹性力学 第一章(2)objects studied-differenceMechanics of materials deals essentially with the stresses and displacements 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。11弹性力学 第一章11弹性力学 第一章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12弹性力学 第一章12弹性力学 第一章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 precisely.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13弹性力学 第一章13弹性力学 第一章2.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14弹性力学 第一章2.methods of analysis:研究方法 equilibrium 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 geometrical 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15弹性力学 第一章equilibrium aspects -equi(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 condition 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16弹性力学 第一章(2)methods of analysis:-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 degree of accuracy of the results obtained.17弹性力学 第一章Mechanics of materials:17弹性力学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.18弹性力学 第一章18弹性力学 第一章The 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 of bending stresses.No assumption,that a plane section of the beam remains plane after bending,is made in Elasticity.19弹性力学 第一章The problem of bending of a stA 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 near the hole.20弹性力学 第一章A prismatical tension member w1.2 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21弹性力学 第一章1.2 some important concepts inA.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22弹性力学 第一章A.external forces 外力22弹性力学 第一1.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 body in motion.例如例如：重力，重力，惯性力惯性力23弹性力学 第一章1.Body forces Body force Fig.体力定义图体力定义图。24弹性力学 第一章 Body force Fig.体力F=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上的体力的合力上的体力的合力25弹性力学 第一章F=lim Q/V v 0F-bodyBody 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26弹性力学 第一章Body force components 体力分量F=X 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.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3for example:27弹性力学 第一章Sign Conventions,dimension 2.Surface forces 面力面力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28弹性力学 第一章2.Surface forces 面 Surface force Fig.面力定义图面力定义图。29弹性力学 第一章 Surface force Fig.F=lim Q/SF-surface force vector at P.the vector quantity is the intensity 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30弹性力学 第一章F=lim Q/S30弹性力学 第一章Surface force components 面面力分量力分量F=X i+Y j+Z k=(X,Y,Z)Definition:the projections of F on the x,y and z axes are called the surface force components at P.面面力在坐标轴上的投影叫面力分量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 面面力分量用力分量用X,Y,Z表示表示31弹性力学 第一章Surface force components 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.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2for example:32弹性力学 第一章 Sign Conventions,dimension 写面力分量 3-9 3-1033弹性力学 第一章写面力分量 3-9 3-1033弹性力学 第一章 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34弹性力学 第一章 1.Internal forces:under the 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)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。35弹性力学 第一章 3.s=lim Q/A Stress Fig.应力定义图应力定义图。36弹性力学 第一章 Stress Fig.应力定义图。3Stress 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37弹性力学 第一章Stress components 应力分量s=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致为正 Its dimension is force/length2 因次：力因次：力/长度长度238弹性力学 第一章 Sign Conventions,dimension Another 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 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39弹性力学 第一章Another Stress components另一应力分Normal 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.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 应应力分量用力分量用 N N 表示表示,因次：力因次：力/长度长度240弹性力学 第一章Normal component and tangentiCoordinate plane 坐标面坐标面Coordinate plane-a plane with the outward normal parallel 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41弹性力学 第一章Coordinate plane 坐标面CoordStress 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，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。42弹性力学 第一章Stress Notation 1.坐标面上应Stress Notation 1 坐标面上应力记号坐标面上应力记号1 x-direction y-direction z-directionX plane:xx xy xzy plane:yx yy yzz plane:zx zy zz43弹性力学 第一章Stress Notation 1 坐标面Stress 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 the stress is acting and the direction in which the stress is acting.正应力用正应力用 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，第一个下标表示应力用加二个下标表示，第一个下标表示应力的作用面，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。44弹性力学 第一章Stress Notation 2.坐标面上应力记号2.NStress Notation 2 坐标面上应力记号坐标面上应力记号2 x-direction y-direction z-directionX plane:x xy xzy plane:yx y yzz plane:zx zy z45弹性力学 第一章Stress Notation 2 坐标面上应力记号2Sign 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 axis.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46弹性力学 第一章Sign Conventions 正The fig.of stress notation 坐标面上应力记号图坐标面上应力记号图47弹性力学 第一章The fig.of stress notation 坐标The Equality of Shearing Stresses 剪应力互等剪应力互等 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。48弹性力学 第一章The Equality of Shearing StresTo 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 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 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，可求得经过该点的任意，可求得经过该点的任意截面上的应力。截面上的应力。-称为称为 一点的六个坐标面上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的应力分量可完全确定该点的应力状态。49弹性力学 第一章To precisely define the stress1.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?50弹性力学 第一章1.体力分量、面力分量、应力分量的正负号规定？50弹性力C.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 lengths and angles of its parts.形变形变-物体形状（各部分长度和角度）物体形状（各部分长度和角度）的改变的改变51弹性力学 第一章C.Deformation 形变By deformaTo 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点的三个正点的三个正向微段向微段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 52弹性力学 第一章To study deformation conditionFig.PA PB PC 图 PA PB PC z C P B A y x53弹性力学 第一章Fig.PA PB PC 图 PA PB PNormal 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 PCpositive(negative)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54弹性力学 第一章Normal strain-a change in lenShearing 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 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55弹性力学 第一章Shearing strain-the change ofD.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 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 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56弹性力学 第一章D.Displacements 位移By displace1.研究一点的变形，考虑通过研究一点的变形，考虑通过P点的三个正点的三个正向微段向微段PA,PB,PC，为何要正向微段，为何要正向微段?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？57弹性力学 第一章1.研究一点的变形，考虑通过P点的三个正向微段PA,PB,1.3 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 small.位移和应变是微小的。位移和应变是微小的。58弹性力学 第一章1.3 Basic assumptions 基 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,strains 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。59弹性力学 第一章 The body is continuous The 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 the 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。60弹性力学 第一章The body is perfectly elasticThe body is homogeneous.物体是均质的物体是均质的The elastic constants will be independent of the location in the body.弹性常数与位置无关。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。61弹性力学 第一章The body is homogeneous.物 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62弹性力学 第一章 The body is isotropic 物体是各向 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 original dimensions-We may use the lengths and angles of the body before deformation when we formulate the equilibrium equations.位移远位移远小于物体尺寸小于物体尺寸-可用变形前的尺寸代替变形后的尺可用变形前的尺寸代替变形后的尺寸寸。The strain components and the rotations of all line elements 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应变分量和转角远小于1，其乘积及二次，其乘积及二次幂可忽略幂可忽略。Linear equations and problems 线性方程线性方程,线性问题线性问题63弹性力学 第一章 The displacements and strain作业：作业：1.写图写图3-9、图、图3-10的面力分量的面力分量 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？试述坐标何为坐标面？何为正（负）坐标面？试述坐标面上的应力记号。面上的应力记号。4.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？5.研究一点的变形，考虑通过研究一点的变形，考虑通过P点的三个正向微点的三个正向微段段PA,PB,PC，为何要正向微段，为何要正向微段?6.设设x=-10为负为负x面面,该面上沿该面上沿x正向作用正向作用100KN/m2的荷载，沿的荷载，沿y负向作用负向作用200KN/m2的荷载的荷载,设该面为物体表面，写出该面上面力分量；设设该面为物体表面，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。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？试述应力和面力的异同点？64弹性力学 第一章作业：1.写图3-9、图3-10的面力分量 3-9 3