平面问题基本理论1.ppt

《平面问题基本理论1.ppt》由会员分享，可在线阅读，更多相关《平面问题基本理论1.ppt（29页珍藏版）》请在淘文阁 - 分享文档赚钱的网站上搜索。

1、 Every elastic body is spatial and,in general,Every elastic body is spatial and,in general,every external force system is spatial.every external force system is spatial.Hence,Hence,strictly speaking,any elasticity problem is strictly speaking,any elasticity problem is a spatial problem.a spatial pro

2、blem.For its solution,we have to For its solution,we have to consider all the components of stress,strain and consider all the components of stress,strain and displacement.displacement.However,if the body has aHowever,if the body has a particular shapeparticular shape and the external forces areand

3、the external forces are distributed in a particulardistributed in a particular manner,we may manner,we may consider the spatial problem as a plane one consider the spatial problem as a plane one and neglect some of the components.and neglect some of the components.This This will greatly simplify the

4、 mathematical will greatly simplify the mathematical aspect of solution while the results may aspect of solution while the results may still be applied in engineering design with still be applied in engineering design with sufficient accuracy.sufficient accuracy.2 2 Theory of Plane ProblemsTheory of

5、 Plane Problems21 21 Plane Stress and Plane StrainPlane Stress and Plane Strain22 22 Differential Equations of EquilibriumDifferential Equations of Equilibrium2323 Geometrical Equations.Rigid-body DisplacementGeometrical Equations.Rigid-body Displacement24 24 Physical EquationsPhysical Equations25 2

6、5 Stress at a PointStress at a Point26 26 Boundary Conditions.Boundary Conditions.Saint-Saint-VenantVenants Principles Principle27 27 Solution of Plane Problem in Terms of DisplacementsSolution of Plane Problem in Terms of Displacements28 28 Solution of Plane Problem in Terms of StressesSolution of

7、Plane Problem in Terms of Stresses29 29 Case of Constant Body Forces.Stress FunctionCase of Constant Body Forces.Stress Function 第二章第二章 平面问题的基本理论平面问题的基本理论21 21 平面应力问题与平面应变问题平面应力问题与平面应变问题22 22 平衡微分方程平衡微分方程23 23 几何方程几何方程 刚体位移刚体位移24 24 物理方程物理方程25 25 平面问题中一点的应力状态平面问题中一点的应力状态26 26 边界条件边界条件 圣维南原理圣维南原理27 2

8、7 按位移求解平面问题按位移求解平面问题28 28 按应力求解平面问题按应力求解平面问题 相容方程相容方程29 29 常体力情况下的简化常体力情况下的简化 应力函数应力函数21 21 Plane Stress and Plane StrainPlane Stress and Plane Strain空间问题空间问题 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

9、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 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 Thin Plate of uniform thickness tuniform thickness t2 2、荷载的性质：、荷

10、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xyozoytAll the forces being parallel to the All the forces 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

11、/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 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 compon

12、ents The remaining stress components x x，y y，xyxy，may be considered to be functions of xmay be considered to be functions of x、y y onlyonly，such a problem is called such a problem is called a plane a plane stress problem.stress problem.二、平面应变问题二、平面应变问题Plane Plane StrianStrian1 1、构件的形状：、构件的形状：yzx（1 1

13、）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 cylindrical or prismatial prismatial bodybody2 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 All the forces being parallel to a c

14、ross section of the body and not varying along the axial direction.the body and not varying along the axial direction.称为平面应变问题称为平面应变问题结论：结论：yzx平面应变问题平面应变问题有那些应力分有那些应力分量？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），），应变应变分量只有分量只有 x x y y xyxyWith any cross section of the body as With any cross s

15、ection of the body as xy xy plane,the plane,the components will be functions 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 str

16、ain problem.a plane strain problem.归纳：归纳：平面问题中，共有八个未知量：平面问题中，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，就是要根据根据已知条件已知条件（荷载，边界条件）（荷载，边界条件）求未知求未知的应力分量、应变分量和位移分量。的应力分量、应变分量和位移分量。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 element2 2 平衡微分方程（静力平衡条件）平衡微分方程（静力平衡条件）y

17、yx xy xcXYDifferential Equations of EquilibriumDifferential Equations of Equilibrium建立平衡方程建立平衡方程Formulate Equilibrium EquationsFormulate Equilibrium Equations y yx xy xXYxyoXYc M MC C=0 =0 （1 1）xyxy=yxyx X=0 X=0 （2 2）Y=0 Y=0 （3 3）（平面应力（平面应力问题与平面问题与平面应变问题）应变问题）The elasticity problem is statically ind

18、eterminate.To The elasticity problem is statically indeterminate.To solve for the solve for the unknow unknow stresses,we have to consider the stresses,we have to consider the strains and displacements.strains and displacements.Differential Equations of Equilibrium are applicable Differential Equati

19、ons of Equilibrium are applicable both to plane stress problems and plane strain problems.both to plane stress problems and plane strain problems.3 3 几何方程几何方程 刚体位移刚体位移AB P取如图所示取如图所示隔离体：隔离体：PABxyoPA=dxPB=dyuvP P、A A、B B各点各点的位移如图的位移如图所示所示Geometrical Equations.Rigid-body DisplacementGeometrical Equatio

20、ns.Rigid-body Displacement x x：PAPA的伸长量的伸长量 y y：PBPB的伸长量的伸长量normal strainnormal strainnormal strainnormal strain xyxy或或 yxyx ：PAPA与与PBPB夹角的改变量：夹角的改变量：+ABPuvPABxyo 几何方程几何方程Geometrical EquationsGeometrical Equations ：已知位移分量，就能求出应变分量。已知位移分量，就能求出应变分量。（平面应力（平面应力问题与平面问题与平面应变问题）应变问题）Geometrical EquationsGe

21、ometrical Equations are applicable both to plane are applicable both to plane stress problems and plane strain problems.stress problems and plane strain problems.For a given set of displacement components u and v,the For a given set of displacement components u and v,the strain components are define

22、d by Geometrical Equations.strain components are defined by Geometrical Equations.For a given set of strain components,the displacement For a given set of strain components,the displacement components u and v are not wholly ponents u and v are not wholly determinate.已知应变分量，能否求出位移分量？已知应变分量，能否求出位移分量？刚

23、体位移刚体位移（Rigid-body DisplacementRigid-body Displacement）：设应变分量已知：设应变分量已知：x x=0=0，y y=0=0 ，xyxy=0=0 u=f1（y）v=f2（x）u=fu=f1 1（y y）=u=u0 0+y y v=fv=f2 2（x x）=v=v0 0-x x （变形为零时的位移）（变形为零时的位移）These are the displacement components corresponding to These are the displacement components corresponding to zero t

24、rains and cannot but be the rigid-body zero trains and cannot but be the rigid-body displacement.displacement.称为刚体位移称为刚体位移(the rigid-body displacement)(the rigid-body displacement)：其中：其中若若 u u0 0=0=0 v v0 0=0=0 和位移为：和位移为：弹性体中任意点弹性体中任意点P P（x x，y y）其位移为：其位移为：u=uu=u0 0+y y v=v v=v0 0-x x Pxxyyr rou u0

25、0，v v0 0弹性体沿弹性体沿x x、y y轴方向的平移轴方向的平移 弹性体沿绕弹性体沿绕z z轴的转动，为什么？轴的转动，为什么？the rigid-body translationsthe rigid-body translationsthe rigid-body rotationthe rigid-body rotation和位移的方向和位移的方向：结论结论：和位移的方向垂直于和位移的方向垂直于OPOP，沿切线方向，所沿切线方向，所以以 表示弹性体绕表示弹性体绕z轴轴的转动的转动Pxxyyr ru=yv=yo归纳：归纳：弹性体在变形为零时有刚体位移，所以当物弹性体在变形为零时有刚体位移

26、，所以当物体发生一定的变形时，由于约束条件不同，体发生一定的变形时，由于约束条件不同，可能具有不同的刚体位移，即由变形不能完可能具有不同的刚体位移，即由变形不能完全确定位移全确定位移（已知（已知，不能完全确定不能完全确定u u，v v），），须考虑边界条件。须考虑边界条件。或者说，没有约束的弹性体存在任意的，不或者说，没有约束的弹性体存在任意的，不确定的刚体位移。确定的刚体位移。An elastic body can have any rigid-body displacements An elastic body can have any rigid-body displacements f

27、or zero strains.Hencefor zero strains.Hence，at a given state of strain,at a given state of strain,the body may have different rigid-body displacements the body may have different rigid-body displacements under different conditions of constraint,in order to under different conditions of constraint,in

28、 order to determine the actual displacement of the body,there determine the actual displacement of the body,there must be three proper conditions of constraint for the must be three proper conditions of constraint for the determination of the three constants.determination of the three constants.4 4

29、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）Physical EquationsPhysical EquationsThe relations 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

30、 between stresses and strains based on between stresses and strains based on HookeHookes laws law：E E、G G、为常数为常数(three elastic constants)(three elastic constants)，不随坐标、方，不随坐标、方向而变化向而变化其中：E is the modulus of elasticity or Youngs modulusE is the modulus of elasticity or Youngs modulus，m is the Poisson

31、s ratio is the Poissons ratio，G is the shear modulus or modulus of rigidity.G is the shear modulus or modulus of rigidity.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 而且而且In a plane stress problemIn a plane stress problem物理方程物理方程 (Physical Equations)(Physical Equations)物理方程另一种形式：物理方程另一种形式：物理方程物理方程

32、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)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 s

33、tress 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，但平衡微分方程和几何方程是相同的。程是相同的。Physical Equations are differentPhysical Equations are different，Differen

34、tial Differential Equations of Equilibrium and Geometrical Equations same.Equations of Equilibrium and Geometrical Equations same.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）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 Equilibrium Differential 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 Equations Physical Equations再考虑边界条件再考虑边界条件(Boundary ConditionsBoundary Conditions )，即可，即可求出所有未知量。求出所有未知量。