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1、平面问题基本理论现在学习的是第1页，共87页空间问题中共有空间问题中共有1515个未知量个未知量应力分量：应力分量：x x、y y、z z、xyxy、xzxz、yzyz应变分量：应变分量：x x、y y、z z、xyxy、xzxz、yzyz位移分量：位移分量：u u、v v、w w这这1515个未知量应满足个未知量应满足3 3个平衡微分方程；个平衡微分方程；6 6个几何个几何方程；方程；6 6个物理方程；在边界上还要满足边界条件个物理方程；在边界上还要满足边界条件2 2.1 .1 弹性力学问题的提法弹性力学问题的提法现在学习的是第2页，共87页空间问题空间问题 平面问题平面问题 转转 化化 平
2、面应力问题平面应力问题平面应变问题平面应变问题Spatial problemSpatial problemPlane problemPlane problemPlane stress problemPlane stress problemPlane strain problemPlane strain problem转化条件转化条件：构件的形状构件的形状 荷载性质荷载性质Particular shapeParticular shapeParticular forcesParticular forces现在学习的是第3页，共87页平面问题中，基本未知量为：平面问题中，基本未知量为：x x，y y
3、，xyxy，x x，y y，xyxy，u u，v v（八个）（八个）求解平面问题的基本方程：求解平面问题的基本方程：平衡微分方程（平衡微分方程（2 2个）个）Differential Equations of EquilibriumDifferential Equations of Equilibrium 几何方程几何方程 （3 3个）个）Geometrical EquationsGeometrical Equations 物理方程物理方程 （3 3个）个）Physical EquationsPhysical Equations再考虑边界条件再考虑边界条件(Boundary Condition
4、sBoundary Conditions)，即可求出所，即可求出所有未知量。有未知量。现在学习的是第4页，共87页边界条件分为位移边界条件、应力边界条件边界条件分为位移边界条件、应力边界条件和混合边界条件。和混合边界条件。边值问题的求解：边值问题的求解：位移法位移为基本未知量位移法位移为基本未知量应力法应力为基本未知量应力法应力为基本未知量混合法一部分位移和一部分应力为基本未混合法一部分位移和一部分应力为基本未知量。知量。现在学习的是第5页，共87页2 2.2 .2 解的叠加原理及解的惟一性定理解的叠加原理及解的惟一性定理1 1 解的叠加原理解的叠加原理在小变形和线弹性情况下，作用在物体上的在
5、小变形和线弹性情况下，作用在物体上的几组荷载产生的总效应等于每一组荷载单独几组荷载产生的总效应等于每一组荷载单独作用的效应总和。作用的效应总和。现在学习的是第6页，共87页1 1 解的惟一性定理（解的惟一性定理（Kirchhoff)Kirchhoff)定理定理在给定的荷载作用下处于平衡状态的弹性在给定的荷载作用下处于平衡状态的弹性体，其内部各点的应力、应变的解是惟一体，其内部各点的应力、应变的解是惟一的，如果刚体位移受到约束，则位移解也的，如果刚体位移受到约束，则位移解也是惟一的。是惟一的。现在学习的是第7页，共87页2 2.3 .3 平面应力和平面应变问题平面应力和平面应变问题空间问题空间问
6、题 平面问题平面问题 转转 化化 平面应力问题平面应力问题平面应变问题平面应变问题Spatial problemSpatial problemPlane problemPlane problemPlane stress problemPlane stress problemPlane strain problemPlane strain problem转化条件转化条件：构件的形状构件的形状 荷载性质荷载性质Particular shapeParticular shapeParticular forcesParticular forces一、平面应力问题一、平面应力问题Plane Stress
7、Plane Stress 1 1、构件的形状：、构件的形状：薄板：薄板：t t其它两个方向的尺寸其它两个方向的尺寸xyozoytThin Plate of uniform Thin Plate of uniform thickness tthickness t现在学习的是第8页，共87页2 2、荷载的性质：、荷载的性质：面力面力：沿板边，平行于板面，沿厚度不变：沿板边，平行于板面，沿厚度不变体力体力：平行于板面，沿厚度均布：平行于板面，沿厚度均布薄片两侧没有外荷载薄片两侧没有外荷载xyozoytAll the forces being parallel to the All the force
8、s being parallel to the faces of the plate and distributed faces of the plate and distributed uniformly over the thicknessuniformly over the thickness板面不受力，即：板面不受力，即：z z z=+t/2z=+t/2=0=0结论结论：zxzx z=+t/2z=+t/2=0 =0 zyzy z=+t/2z=+t/2=0 =0 现在学习的是第9页，共87页因为板很薄，荷载不沿厚度变化，应力是连续因为板很薄，荷载不沿厚度变化，应力是连续分布的，所以可以认
9、为，在整个薄板：分布的，所以可以认为，在整个薄板：z z=0 =0 zxzx=0 =0 zyzy=0=0 平面应力问题有那些应变分量和位移分量？平面应力问题有那些应变分量和位移分量？薄板的应力为薄板的应力为：x x y y xyxy 且与且与z z无关，无关，为为x x、y y的函数的函数,称为平面应力问题称为平面应力问题The remaining stress components The remaining stress components x x，y y，xyxy，may be considered to be functions of xmay be considered to be
10、 functions of x、y y onlyonly，such a problem is called such a problem is called a plane a plane stress problem.stress problem.现在学习的是第10页，共87页二、平面应变问题二、平面应变问题Plane StrianPlane Strian1 1、构件的形状：、构件的形状：yzx（1 1）足够长柱体，两端光滑刚性约束）足够长柱体，两端光滑刚性约束（2 2）无限长柱体，两端自由）无限长柱体，两端自由Very long cylindrical or Very long cylin
11、drical or prismatical bodyprismatical body2 2、荷载的性质：、荷载的性质：（1 1）平行于横截面）平行于横截面（2 2）沿长度不变）沿长度不变（任意横截面上（任意横截面上的受力是相同的的受力是相同的）All the forces being parallel to a cross section of the All the forces being parallel to a cross section of the body and not varying along the axial direction.body and not varyin
12、g along the axial direction.现在学习的是第11页，共87页称为平面应变问题称为平面应变问题结论：结论：yzx平面应变问题有平面应变问题有那些应力分量？那些应力分量？（1 1）应力、应变只是应力、应变只是x x、y y的函数的函数（）（）w=0w=0（z z），应变分），应变分量只有量只有 x x y y xyxyWith any cross section of the body as xy plane,the With any cross section of the body as xy plane,the components will be function
13、s of xcomponents will be functions of x、y onlyy only，due to due to symmetry,the shearing stresses symmetry,the shearing stresses zxzx=0,=0,zyzy=0,=0,and and w=0w=0,such a problem is called such a problem is called a plane strain problem.a plane strain problem.现在学习的是第12页，共87页归纳：归纳：平面问题中，共有八个未知量：平面问题中
14、，共有八个未知量：x x y y xy xy x x y y xyxy u uv v求解弹性力学平面问题，就是要求解弹性力学平面问题，就是要根据已根据已知条件知条件（荷载，边界条件）（荷载，边界条件）求未知的应求未知的应力分量、应变分量和位移分量。力分量、应变分量和位移分量。现在学习的是第13页，共87页希望大家提出宝贵的意见和建议！谢谢！现在学习的是第14页，共87页xyO取图示微六面取图示微六面体为隔离体，体为隔离体，厚度厚度 t=1t=1Isolate elementIsolate element.4 .4 平面问题的基本方程平面问题的基本方程 y yx xy xdxxxxdxxxyxy
15、dyyyydyyyxyxcXY建立平衡方程建立平衡方程Formulate Equilibrium EquationsFormulate Equilibrium Equations现在学习的是第15页，共87页 y yx xy xdxxxxdxxxyxydyyyydyyyxyxXYxyoXYc M MC C=0 =0 （1 1）xyxy=yxyx X=0 X=0 （2 2）011111dydxXdxdxdyydydydxxyxyxyxxxx0Xyxyxx现在学习的是第16页，共87页 Y=0 Y=0 （3 3）0Yxyxyy0Xyxyxx0Yxyxyy（平面应力（平面应力问题与平面问题与平面应变
16、问题）应变问题）The elasticity problem is statically indeterminate.To solve for The elasticity problem is statically indeterminate.To solve for the unknow stresses,we have to consider the strains and the unknow stresses,we have to consider the strains and displacements.displacements.Differential Equations o
17、f Equilibrium are applicable both to Differential Equations of Equilibrium are applicable both to plane stress problems and plane strain problems.plane stress problems and plane strain problems.现在学习的是第17页，共87页几何方程几何方程Geometrical EquationsGeometrical Equations ：xuxyvyxvyuxy（平面应力（平面应力问题与平面问题与平面应变问题）应变
18、问题）Geometrical EquationsGeometrical Equations are applicable both to plane stress are applicable both to plane stress problems and plane strain problems.problems and plane strain problems.现在学习的是第18页，共87页物理方程物理方程（应力、应变之间的关系）（应力、应变之间的关系）zyxxE1zxyyE1xyzzE1Physical EquationsPhysical EquationsThe relatio
19、ns between stresses and strainsThe relations between stresses and strains完全弹性、各向同性体，完全弹性、各向同性体，HOOKHOOK定理：定理：In an isotropic and perfectly elastic bodyIn an isotropic and perfectly elastic body，the relations the relations between stresses and strains based on Hookebetween stresses and strains based
20、on Hookes laws law：现在学习的是第19页，共87页E E、G G、为常数为常数(three(three elastic constants)elastic constants)，不随，不随坐标、方向而变化坐标、方向而变化xyxyG1xzxzG1yzyzG1)1(2EG其中：E is the modulus of elasticity or YoungE is the modulus of elasticity or Youngs moduluss modulus，is the Poissonis the Poissons ratios ratio，G is the shear
21、 modulus or modulus of rigidity.G is the shear modulus or modulus of rigidity.现在学习的是第20页，共87页在平面应力问题中在平面应力问题中 z z=0 =0 zxzx=0 =0 zyzy=0 =0 0 xz0yz而且而且)(yxzEIn a plane stress problemIn a plane stress problem物理方程物理方程 yxxE1xyyE1xyxyG1(Physical Equations)(Physical Equations)现在学习的是第21页，共87页物理方程另一种形式：物理方程
22、另一种形式：)(12yxxE)(12xyyExyxyG1现在学习的是第22页，共87页物理方程物理方程 yxxE112xyyE112xyxyG1在平面应变问题中在平面应变问题中 z z=0 =0 zxzx=0 =0 zyzy=0 =0 (In a plane strain problem)(In a plane strain problem)(Physical Equations)(Physical Equations)(yxz现在学习的是第23页，共87页将平面应力问题物理方程中的将平面应力问题物理方程中的E E/E E/（1-1-2 2）；）；/（1-1-）就得平面应变问题的物理方程。就得
23、平面应变问题的物理方程。平面应力问题与平面应变问题两者的物平面应力问题与平面应变问题两者的物理方程不同。理方程不同。The Physical Equations of plane stress problems and plane The Physical Equations of plane stress problems and plane strain problems are different.strain problems are different.平面应力问题与平面应变问题两者的物理平面应力问题与平面应变问题两者的物理方程虽然不同，但平衡微分方程和几何方方程虽然不同，但平衡微分
24、方程和几何方程是相同的。程是相同的。Physical Equations are differentPhysical Equations are different，Differential Equations Differential Equations of Equilibrium and Geometrical Equations same.of Equilibrium and Geometrical Equations same.现在学习的是第24页，共87页平面问题中，基本未知量为：平面问题中，基本未知量为：x x，y y，xyxy，x x，y y，xyxy，u u，v v（八个）（
25、八个）求解平面问题的基本方程：求解平面问题的基本方程：平衡微分方程（平衡微分方程（2 2个）个）Differential Equations of EquilibriumDifferential Equations of Equilibrium 几何方程几何方程 （3 3个）个）Geometrical EquationsGeometrical Equations 物理方程物理方程 （3 3个）个）Physical EquationsPhysical Equations再考虑边界条件再考虑边界条件(Boundary ConditionsBoundary Conditions)，即可求出所，即可求
26、出所有未知量。有未知量。现在学习的是第25页，共87页希望大家提出宝贵的意见和建议！谢谢！现在学习的是第26页，共87页2.5 边界条件 圣维南原理弹性力学问题 平衡微分方程+几何协调方程+物理方程+边界条件或初始条件一、边界条件（Boundary ConditionsBoundary Conditions）a、位移边界条件（displacement boundary conditionsdisplacement boundary conditions）：Boundary Conditions.Boundary Conditions.Saint-VenantSaint-Venants Prin
27、ciples PrincipleAccording to Boundary ConditionsAccording to Boundary Conditions，elasticity problems are elasticity problems are classified as classified as displacement boundary problemsdisplacement boundary problems，stress stress boundary problemsboundary problems，mixed boundary problemsmixed boun
28、dary problems.In a displacement boundary problemIn a displacement boundary problem，the surface displacements of the surface displacements of the body are specified.the body are specified.弹性体在边界上的位移是已知的。现在学习的是第27页，共87页x如：四边固定的板或两端简支的板xbaxaby边界条件如下：us=u vs=v现在学习的是第28页，共87页b、应力边界条件(stress boundary cond
29、itions)stress boundary conditions)：xyqAX XN N=l=l x x+m+m xyxyY YN N=m=m y y+l+l xyxy已知BPyyxxyxsYNXN在边界上，若面力已知，则 XN=X YN=Yl l x x+m+m xyxy=X=Xm m y y+l+l xyxy=Y=YIn a stress boundary problemIn a stress boundary problem，the surface forces acting on the body the surface forces acting on the body are s
30、pecified.are specified.弹性体在边界上所受的面力是已知的现在学习的是第29页，共87页2xy3141、3边界：l l=0 m=+1y=Y;xy=X+2、4边界：m=0 l l=+1x=X;xy=Y+l l x x+m+m xyxy=X=Xm m y y+l+l xyxy=Y=Y平面问题的应力边界条件stress boundary conditions of stress boundary conditions of a plane problema plane problem特殊边界的应力边界条件When the boundary is normal to a coord
31、inate axis,the stress boundary conditions are simplified现在学习的是第30页，共87页当边界的外法线沿坐标轴的正向时，两者的正负号相同；当边界的外法线沿坐标轴的负向时，两者的正负号相反。The boundary stress components are equal to the surface The boundary stress components are equal to the surface force components force components（use positive or negatve sign acco
32、rding use positive or negatve sign according as the outward normal is along the positive or negative as the outward normal is along the positive or negative direction of the coordinate axisdirection of the coordinate axis）.c、混合边界条件(mixed boundary conditions)mixed boundary conditions)：一部分边界具有已知的位移，一部
33、分边界具有已知的面力；或同一部分边界既有位移边界条件，又有应力边界条件。In a mixed boundary problemsIn a mixed boundary problems，some portion of the some portion of the boundary is specified with known displacements while the boundary is specified with known displacements while the other portion is subjected to known surface forces.ot
34、her portion is subjected to known surface forces.现在学习的是第31页，共87页解：AB边：y=-q;yx=0BC边：x=0;xy=0CD边：y=0;yx=0AD边：u=0；v=0例1、写出图示悬臂梁的边界条件，板厚为1ABCDyxqxyABCAB边：v=0AC边：x=0；xy=0混合边界条件For instance：Mixed boundary problem现在学习的是第32页，共87页解：BC边：x=0;xy=0CD边：y=0;yx=0AD边：u=0；v=0AB边：y=yx=00qlxl 例2、写出图示悬臂梁的边界条件，板厚为1xABCDy
35、q0l现在学习的是第33页，共87页解：BC边：y=0;yx=0DE边:AB边：x=-gy;xy=0例3、写出图示结构AB、BC、DE的应力边界条 件(水的重度为)ExyABCDNcos(N,x)=cos=l lcos(N,y)=cos(90+)=-sin=ml l x x+m+m xyxy=X=Xm m y y+l l xyxy=Y=Ycoscos x x-sin-sinxyxy=0=0-sin-sin y+cosy+cosxyxy=0=0现在学习的是第34页，共87页二、圣维南原理(Saint-VenantSaint-Venants Principle)s Principle)在求解弹性力
36、学问题时，要使应力分量、形变分量、位移分量满足基本方程并不困难，但要完全、精确地满足边界条件，却往往发生很大困难。In solving an elasticity problem,itIn solving an elasticity problem,its rather s rather easy to obtain the stresses,strains and easy to obtain the stresses,strains and displacements which satisfy all the basic displacements which satisfy all t
37、he basic equations.However we often encounter equations.However we often encounter difficulties in having all the boundary difficulties in having all the boundary conditions completely satisfied.conditions completely satisfied.现在学习的是第35页，共87页在上述情况下，可利用圣维南原理(Saint-VenantSaint-Venants s Principle)Prin
38、ciple)来写出近似的边界条件：圣维南原理：如果把物体一小部分边界上的面力变换成分布不同但静力等效的面力，则近处的应力分布将有显著的变化，但远处所受的影响可以忽略。(P30)另外，在工程计算中经常碰到这样的情况：在物体的部分边界上，只知道物体所受面力的合力，面力的分布方式并不明确。It happens frequently in the stress calculation for a It happens frequently in the stress calculation for a structural or machine element that we know only th
39、e structural or machine element that we know only the resultant of surface forces on a small portion of the resultant of surface forces on a small portion of the element,but not the distribution of the forces.element,but not the distribution of the forces.Under such circumstances,Saint-Venants princ
40、iple may be of much help to us.现在学习的是第36页，共87页The essence of the principle can be stated as follows:If a system of forces acting on a small portion of the surface of an elastic body is replaced by another statically equivalent system of forces acting on the same portion of the surface,the redistribu
41、tion of loading produces substantial changes in the stresses only in the immediate neighborhood of the loading,and the stresses are essentially the same in the parts of the body which are at large distances in comparison with the linear dimension of the surface on which the forces are changed.By“sta
42、tically equivalent systems”,it means that the two systems have the same resultant force and the same resultant moment.现在学习的是第37页，共87页For instance：PPPP/2P/2P/2P/AP/AP现在学习的是第38页，共87页PPPP/2P/2P/2P/AP/APThe solution for stresses in the case d is rather sample,as the stress boundary conditions are very s
43、imple.According to Saint-Venants principle,its solution for the stresses can be applied to the case a,b and c.For case e,we have displacement boundary conditions.However,it is evident that the surface forces at the fixed end must be equilibrium with the force P.According to Saint-Venants principle,t
44、he simple solution may also be applied to this case.abcde现在学习的是第39页，共87页注意：（1）用圣维南原理必须满足静力等效条件，若不满足，则计算结果不能用于不同的情况。现在学习的是第40页，共87页希望大家提出宝贵的意见和建议！谢谢！现在学习的是第41页，共87页For the solution of an elasticity problem,we can proceed in three different ways:1.Take the displacement components as the basic unknown
45、functions,formulate a system of differential equations and boundary conditions containing the displacement components only,solve for these unknown functions and thereby find the strain components by the geometrical equations and then the stress components by the physical equations.2.6 弹性力学问题的解法现在学习的
46、是第42页，共87页2.Take the stress components as the basic unknown functions,formulate a system of differential equations and boundary conditions containing the stress components only,solve for these unknown functions and thereby find the strain components by the physical equations and then the displacemen
47、t components by the geometric equation.3.Take some of the displacement components and also some of the stress components as the basic unknown functions,formulate a system of differential equations and boundary conditions containing the stress components only,solve for these unknown functions and the
48、reby find the other unknown functions.现在学习的是第43页，共87页一一 位移解法（按位移求解平面问题）位移解法（按位移求解平面问题）平面问题的基本未知量有平面问题的基本未知量有 x x，y y，xyxy，x x，y y，xyxy，u u，v v，根据基本方程即可求解。，根据基本方程即可求解。Solution of Plane Problem in Terms of DisplacementsSolution of Plane Problem in Terms of Displacements求解方法有：按位移求解；按应力求解；混合求解 按位移求解：以按位
49、移求解：以位移分量位移分量为基本函数，由为基本函数，由只含只含位移分量的微分方程位移分量的微分方程和和边界条件边界条件求出位移分量求出位移分量后，再求其他的未知量。后，再求其他的未知量。Take the Take the displacement componentsdisplacement components as the as the basic unknown functionbasic unknown function，formulate a system of differential equations and boundary conditions formulate a sy
50、stem of differential equations and boundary conditions containing the displacement components onlycontaining the displacement components only，solve for these unknown solve for these unknown functions and thereby find the strain components by the geometrical functions and thereby find the strain comp