The beeline which is perpendicular to middle plane before deformation is still a beeline after deformation and is still perpendicular to the bent middle plane. Namely Yx=O y 0 (3) the board surface is neutrosphere Na amely )20=0,()2。=0 From the geometric functions =0)=0)=0 (4) the stress 0 has very small effect on deformation, thus can be ignore ed Scilicet we think: 0.=0
11 The beeline which is perpendicular to middle plane before deformation is still a beeline after deformation, and is still perpendicular to the bent middle plane. Namely : xz = 0, yz = 0 （3）the board surface is neutrosphere Namely ( ) 0, ( ) 0 0 0 = = z= z= u v From the geometric functions: ( ) 0, ( ) 0, ( ) 0 0 0 0 = = = = = z= xy z x z y （4）the stress has very small effect on deformation, thus can be ignored. Scilicet we think: = 0 z z
薄城曲 在变形前垂直于中面的直线,变形后仍为直线,并垂 直于弯曲后的中面。即 0 0 (3)中面为中性层假设 即()2=0,()2=0 由几何方程得 6)=0)=0()=0 (4)应力0:对变形的影响很小,可以略去不计。亦即认为 =0 12
12 在变形前垂直于中面的直线，变形后仍为直线，并垂 直于弯曲后的中面。即 xz = 0, yz = 0 （3）中面为中性层假设 即 ( ) 0, ( ) 0 0 0 = = z= z= u v 由几何方程得 ( ) 0, ( ) 0, ( ) 0 0 0 0 = = = = = z= xy z x z y （4）应力 对变形的影响很小，可以略去不计。亦即认为 = 0 z z
§ 12-2 Basic Equations Solving sheet bending problem in terms of displacement Choose the sheet bending w as the basic unknown quantity and express the other physical quantities with w (1) Geometric Function Fetch a small rectangle ABCD on the middle plane, as shown in fig. Its side B lengths are dx and dy. Under action of the load, the rectangle is bent to flexural z plane ABCdo suppose the bending atD point a is w, the obliquities of stretch flexural plane along direction x and y are and ow 13
13 § 12－2 Basic Equations Solving sheet bending problem in terms of displacement. Choose the sheet bending as the basic unknown quantity， and express the other physical quantities with . w w （1）Geometric Function Fetch a small rectangle ABCD on the middle plane, as shown in fig.. Its side lengths are dx and dy. Under action of the load, the rectangle is bent to flexural plane A ’B ’C’D’。suppose the bending at point A is ,the obliquities of stretch flexural plane along direction x and y are and . w y w x w dx x w y w dy y z D C A B A B x D C
薄城曲 §12-2基本方程 按位移求解薄板弯曲问题。取薄板挠度v为基本未知 量,把所有其它物理量都用来表示。 (1)几何方程 在薄板的中面上取一微 小矩形ABCD如图所示。它的 a 边长为x和dy,载荷作用后, 弯成曲面ABCD。设A点的 挠度为,弹性曲面沿利∥大 方向的倾角分别为和, 则 Ow Ow O 14
14 § 12－2 基本方程 按位移求解薄板弯曲问题。取薄板挠度 为基本未知 量，把所有其它物理量都用 来表示。 w w （1）几何方程 在薄板的中面上取一微 小矩形ABCD如图所示。它的 边长为dx和dy，载荷作用后， 弯成曲面A’B’C’D’。设A点的 挠度为 ，弹性曲面沿x和y 方向的倾角分别为 和 ， 则 w y w x w dx x w y w dy y z D C A B A B x D C
The bendi ont B is wx gr dx The bending at point d is w+o-dy Ou aw Ov aw Fromy =Andy,=0 we know 0 z Ox Or can be written as ax az After integral by z, and using u)-=0, )-=0 we get 2 Ox au aw Then the strain components 8 2 can be expressed byW as aw 8.= 2 au e Oy ax Oxay 15
15 The bending at point B is dx x w w + The bending at point D is dy y w w + From and we know xz = 0 yz = 0 0, = 0 + = + y w z v x w z u Or can be written as y w z v x w z u = − = − , After integral by z, and using ( ) 0, ( ) ， 0 we get 0 0 = = z= z= u v z y w z v x w u = − = − , z x y w x v y u xy = − + = 2 2 Then the strain components can be expressed by as z y w y v z x w x u y x 2 2 2 2 = − = = − = W