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1、21 平面应力和平面应变问题,2、荷载的性质：,All the forces being parallel to the faces of the plate and distributed uniformly over the thickness,平面应力问题有那些应变分量和位移分量？,称为平面应变问题,结论：,平面应变问题有那些应力分量？,归 纳：,平面问题中，共有八个未知量： x y xy x y xy u v,求解弹性力学平面问题，就是要根据已知条件（荷载，边界条件）求未知的应力分量、应变分量和位移分量。,2 平衡微分方程（静力平衡条件）,Differential Equations
2、of Equilibrium,X=0 （2）,Y=0 （3）,The elasticity problem is statically indeterminate.To solve for the unknow stresses,we have to consider the strains and displacements.,Differential Equations of Equilibrium are applicable both to plane stress problems and plane strain problems.,3 几何方程 刚体位移,PA=dx PB=dy,
3、Geometrical Equations. Rigid-body Displacement,normal strain,normal strain,几何方程Geometrical Equations ：,已知位移分量，就能求出应变分量。,Geometrical Equations are applicable both to plane stress problems and plane strain problems.,For a given set of displacement components u and v ,the strain components are defined
4、by Geometrical Equations.,For a given set of strain components,the displacement components u and v are not wholly determinate.,已知应变分量，能否求出位移分量？,刚体位移（ Rigid-body Displacement）：,设应变分量已知：x=0 ，y=0 ，xy=0,These are the displacement components corresponding to zero trains and cannot but be the rigid-body d
5、isplacement.,称为刚体位移(the rigid-body displacement)：其中,若 u0=0 v0=0,和位移为：,和位移的方向：,归纳：,弹性体在变形为零时有刚体位移，所以当物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，即由变形不能完全确定位移（已知，不能完全确定u，v），须考虑边界条件。,或者说，没有约束的弹性体存在任意的，不确定的刚体位移。,An elastic body can have any rigid-body displacements for zero strains.Hence，at a given state of strain,
6、 the body may have different rigid-body displacements under different conditions of constraint,in order to determine the actual displacement of the body,there must be three proper conditions of constraint for the determination of the three constants .,Physical Equations,The relations between stresse
7、s and strains,E、G、为常数(three elastic constants) ，不随坐标、方向而变化,其中：,E is the modulus of elasticity or Youngs modulus， is the Poissons ratio， G is the shear modulus or modulus of rigidity.,而且,In a plane stress problem,物理方程另一种形式：,物理方程,(In a plane strain problem),(Physical Equations),平面问题中，基本未知量为：x，y，xy，x，y，xy，u，v（八个）,求解平面问题的基本方程： 平衡微分方程（2个） Differential Equations of Equilibrium 几何方程 （3个） Geometrical Equations 物理方程 （3个） Physical Equations,再考虑边界条件(Boundary Conditions )，即可求出所有未知量。,再 见,