【正文】 ically equivalent system of forces acting on the same portion of the surface, the redistribution 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 parison with the linear dimension of the surface which the forces are changed. ? 把物體 一小部分 邊界上的面力，變換為分布不同但靜力等效的面力 （主矢相同，對同一點(diǎn)的主矩相同） ， 那末， 離面力變換處 較近 的地方的應(yīng)力將有顯著變化，離面力變換處 較遠(yuǎn)的地方 的應(yīng)力將基本不變。 Saintvenant`s principle1 圣維南原理 ? 把物體一 小 部分邊界上的面力，變換為分布不同但靜力等效的面力，那末，離 面力變換處 較近 的地方的應(yīng)力將有顯著變化，離面力變換處較 遠(yuǎn)的地方的應(yīng)力將基本不變。 Statically equivalent systems 靜力等效力系 ? By “statically equivalent systems” we mean that the two system have the same resultant force and the same resultant moment. ? 靜力等效力系是指兩個力系的 主矢量相同，對同一點(diǎn)的主矩相同。 舉例：如何在局部邊界上應(yīng)用圣維南原理 局部邊界，小邊界或次要邊界。 舉例：圣維南原理的應(yīng)用 P P P P/2 P/2 P/2 P/2 P/2 P/2 P/A P/A P Notes： ? 在應(yīng)力邊界上應(yīng)用圣維南原理，就是在小邊界上，將精確的應(yīng)力邊界條件代之為 主矢量相同 ， 對于同一點(diǎn)的 主矩也相同的靜力等效條件 。 舉例：圣維南原理的應(yīng)用 x y NFSFMdyy2/h2/hl lxfyfxfyfx?xy?x?xy??????????????ylxxyxlxxff||???????????????ylxxyxlxxff||???????????ylxxyxlxxff||??舉例：圣維南原理的應(yīng)用 ? ?? ?? ?? ???? ???? ???????????2/2/2/2/2/2/2/2/2/2/2/2/)()()()()()(hhhhylxxyhhhhxlxxhhhhxlxxdyyfdyy dyyfy dydyyfdy????????????????2/2/2/2/2/2/)()()(hhSlxxyhhlxxhhNlxxFdyMy dyFdy??? x y NFSFMdyy2/h2/hl lxfyfxfyfx?xy?x?xy?舉例：圣維南原理的應(yīng)用 ?????????????2/2/2/2/2/2/)()()(hhSlxxyhhlxxhhNlxxFdyMy dyFdy??? x y NFSFMdyy2/h2/hl lxfyfxfyfx?xy?x?xy??????????ylxxyxlxxff||??精確條件，函數(shù)方程 兩個條件 不易滿足； 近似的積分條件， 簡單的代數(shù)方程； 三個條件； 易于滿足。 Saintvenant`s principle2 圣維南原理 ? If a balanced system of surface forces is applied to any small portion of a body, it will induce significant stresses only in the neighborhood of the surface forces. ? A balanced force system can be regarded as the difference between two statically equivalent force systems. ? 作用在物體小部分邊界上的平衡面力系僅在面力作用的附近產(chǎn)生顯著的應(yīng)力。 ? 靜力等效的兩個力系的差異是個平衡力系。 思考題？？ ? 1. 為什么在大邊界上，不能用圣維南原理？ ? 2. 列出下圖邊界條件。 h/2 h/2 L L qL qL x y O q 歸納： Mathematical expressions for elasticity problem (plane stress ) 彈性力學(xué)平面應(yīng)力問題的數(shù)學(xué)表述 ? 平衡方程 ? 幾何方程 ? 物理方程 ? 邊界條件 00 ?????????????? yxyyxyxx fxyfyx ????xux ???? yvy ???? yuxvxy ???????vvuuuu ii ??? ,:ysxyyxsxyx flmfml ???? )()( ????平面問題的物理方程 ???? ???? 1,1 2EE平面應(yīng)力問題 Plane stress problem 平面應(yīng)變問題 Plane strain problem ??????????1,)1()21(2EExyxyxyxyyyxxEGEE????????????)1(21][1][1???????xyxyxyxyyyxxEGEE??????????????????2221)11(21]1[1]1[1?????????????Three ways for the solution of an elasticity problem: 彈性力學(xué)求解的三種方法 1. Solution in terms of displacementstake the displacement ponents as the basic unknown functions. 2. Solution in terms of stressestake the stress ponents as the basic unknown functions. 3. Solution in terms of displacements and stressestake some of the displacement ponents and also some of the stress ponents as the basic unknown functions. ? 按位移求解 把位移作為基本未知函數(shù)。 ? 按應(yīng)力求解 把應(yīng)力作為基本未知函數(shù)。 ? 按應(yīng)力和位移混合求解 同時把某些位移和某些應(yīng)力作為基本未知函數(shù)。 Chapter 2 Theory of Plane Problems 第二章：平面問題的理論 ? Plane stress and plane strain 平面應(yīng)力問題與平面應(yīng)變問題 ? Differential equations of equilibrium 平衡微分方程 ? Stress at a point. Principal stresses 斜截面上的應(yīng)力。主應(yīng)力 ? Geometrical equations. Rigidbody displacements 幾何方程。剛體位移 ? Physical equations. 物理方程 ? Boundary conditions 邊界條件 ? Saintvenant`s principle1 圣維南原理 ? solution of plane problem in terms of displacements 按位移求解平面問題 ? solution of plane problem in terms of stresses 按應(yīng)力求解 平面問題 ? Case of constant body forces 常體力情況 下的簡化 ? Airy`s stress function. Inverse method and semiinverse method 艾瑞應(yīng)力函數(shù)。逆解法與半 逆解法 solution of plane problem in terms of displacements 按位移求解平面問題 ? Substituting the strains expressed by displacements through geometrical equations into physical equations in which the stresses are in the left side while the strains are in the right side, we get elastic equations . ? 將用位移表示的應(yīng)變（幾何方程）代入應(yīng)力放左邊的物理方程，得用位移表示應(yīng)力的關(guān)系式（彈性方程） 。 )()1(2)(1)(122xvyuExuyvEyvxuExyyx?????????????????????????????Differential equations of equilibrium expressed by displacements 用位移表示的平衡微分方程 ? Substituting the stresses expressed by displacements through elastic equations into differential equations of equilibrium expressed by stresses, we get another differential equations of equilibrium, expressed by displacements. ? 將用位移表示的應(yīng)力 (彈性方程 )代入用應(yīng)力表示的平衡微分方程，得用位移表示的平衡微分方程。 00??????????????yxyyxyxxfxyfyx????0)2121(10)2121(1222222222222????????????????????????????yxfyxuxvyvEfyxvyuxuE??????)()1(2)(1)(122xvyuExuyvEyvxuExyyx???????????????????????