《弹性力学》（双语版） 第十二章 薄板弯曲

薄城曲 将前面二式对z进行积分,得: E 2 V2I 4丿ax E a 1- 4)a 再由平衡微分方程第三式,得: az:Ox ay 将x,用挠度w表达式代入,并化简得: E a=20-m2)4 26

26 将前面二式对z 进行积分，得： ( ) w x t z E zx 2 2 2 2 2 1 4            − − =   ( ) w y t z E zy 2 2 2 2 2 1 4            − − =   再由平衡微分方程第三式，得： z x y z zx zy   −   = −     将  zx , zy 用挠度 w 表达式代入，并化简得： ( ) z w E t z z 2 4 2 2 2 1 4         − − =     （1）

Because the bending wdoesn't change along z axis, and we have the boundary condition: (0)_1=0 Integrating formula (1) on z, we get E(1 O.= 1+- Supposing the load acts on per unit area on top surface of sheet is including transverse surface force and transverse body force), and the boundary condition on board top surface is 三q Substitute the expression of o -into the above formula, we get the differential equation 4w三 D 27

27 Because the bending doesn’t change along z axis,and we have the boundary condition: w ( ) 0 2 = = t z z  Integrating formula（1）on z, we get: ( ) w t z t Et z z 4 2 2 3 1 2 1 6 1        +      − − = −   Supposing the load acts on per unit area on top surface of sheet is q(including transverse surface force and transverse body force),and the boundary condition on board top surface is ( ) t q z z = − =− 2  Substitute the expression of into the above formula, we get the differential equation:  z D q  w = 4

薄城曲 由于挠度γ不随z变化,且薄板有边界条件: )=0 将(1)式对z积分,得 E(1 0=6-以2 1+- 设在薄板顶面上每单位面积作用的载荷q(包括横向面 力和横向体力),板上面的边界条件为: I==9 将σ的表达式代入该边界条件,得薄板挠曲微分方程: D 28

28 由于挠度 w 不随z 变化，且薄板有边界条件: ( ) 0 2 = = t z z  将（1）式对z 积分，得: ( ) w t z t Et z z 4 2 2 3 1 2 1 6 1        +      − − = −   设在薄板顶面上每单位面积作用的载荷q(包括横向面 力和横向体力），板上面的边界条件为: ( ) t q z z = − =− 2  将  z 的表达式代入该边界条件，得薄板挠曲微分方程: D q  w = 4

Et where D We call d bend rigidity of sheet The sheet bending differential equation is also called stretch flexural plane differential equation, which is the basic differential equation of sheet bending problems 29

29 where ( ) 2 3 12 1−  = Et D We call D bend rigidity of sheet. The sheet bending differential equation is also called stretch flexural plane differential equation, which is the basic differential equation of sheet bending problems

薄城曲 Et 其中D= 称为薄板的弯曲刚度。 薄板挠曲微分方程也称为薄板的弹性曲面微分方 程,它是薄板弯曲问题的基本微分方程

30 其中 ( ) 2 3 12 1−  = Et D 称为薄板的弯曲刚度。 薄板挠曲微分方程也称为薄板的弹性曲面微分方 程，它是薄板弯曲问题的基本微分方程