东南大学：《弹性力学》课程教学课件（英文讲稿）12 Bending of Thin Plates

AssumptionsConstitutive relations-voPEE2G The middle surface of the plate is not strained duringbendingOu02=0=OxCuav= 0ay=0(v). 二=0ovOu02Ox=0ay=06

• Constitutive relations • The middle surface of the plate is not strained during bending. 1 1 1 ( ), ( ), . 2 x x y y y x xy xy E E G              Assumptions           0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 x z z z y z z z x y z z u x u v v y v u x y                                                              6

Governing Equation in terms of Deflection w(x, y): Longitudinal displacements formulated in terms of thevertical deflection w = w(x,y)auOwOwOwz+f(x,y)u:uZOzaxOxax二UavawOwOw.yyVZ-+tOzayayay=0(u).-0fi(x, y) = 0((v):=0 = 0f,(x,y)= 0Longitudinal strains in terms of wa?wQuawOvawdyduos88LaxavaxayOxoya7

1 ( , ) u w w u z f x y z x x v w z y                         2 ( , ) w v z f x y y      0 1 0 2 ( ) 0 ( , ) 0 ( ) 0 ( , ) 0 z z w u z x w v z y u f x y v f x y                             Governing Equation in terms of Deflection w(x, y) • Longitudinal displacements formulated in terms of the vertical deflection w = w(x,y) • Longitudinal strains in terms of w 2 2 2 2 2 1 , , 2 x y xy u w v w v u w z z z x x y y x y x y                                   7

Governing Equation in terms of Deflection w(x, y): Longitudinal stresses in terms of wa"wa"wEzEa1-v2ay?+Voax?oS1-1awa?wEEz+V8aS0ax?Qy?1-v21-EawEzt.8xvT(1 +v)xy1+vaxyTransvers shear stresses in terms of wa3watyxataxEzEzaOtxa'waa1 - v21 - v2x3OzOxoy2axOzaxay2Ot.daOt.atya"wa"waEzEzZ1xyOzayaxOzOyox?1-v2Oy3oy1 - v2IntegrateW.r.t z:8

2 2 2 2 2 2 2 2 2 2 2 2 2 ( ) 1 1 ( ) 1 1 (1 ) 1 x x x y y y x y xy xy x y Ez w w E x y E Ez w w y x E Ez w x y                                                                        • Longitudinal stresses in terms of w • Transvers shear stresses in terms of w 3 3 2 2 3 2 2 3 3 2 2 3 2 2 1 1 1 1 Integrate w.r.t zx x y x z x zy y xy z y Ez w w Ez w z x y z x x y x Ez w w Ez w z y x z y y x y z                                                                                        Governing Equation in terms of Deflection w(x, y) 8

Governing Equation in terms of Deflection w(x, y)Transvers shear stresses in terms of wEz?Ez?aaV?w+ F,(x,y)V?w + F(x, y),2(1 -v2) Qy2(1 -v2) αx: Applying the BCs at the top/bottom surface12Ea2WTX(t-x)=±1/2 = 042(1 - vOx(t,)=±/2 = 0Ea14T42(1 - vayTransvers normal stress in terms of watyet2Eot.ag.YVOzax2(1-v)4ayE12V4w+ F(x,y)↓a2(1 - v49

2 2 2 2 1 2 2 2 , 2(1 ) 2(1 , ) ( ) ( , ) zx zy Ez Ez w F x y F x x y   w y               • Transvers shear stresses in terms of w • Applying the BCs at the top/bottom surface 2 2 2 2 2 2 2 2 2 2 ( ) 0 2(1 ) 4 ( ) 0 2(1 ) 4 z x zx z t zy z t z y E t z w x E t z w y                                          • Transvers normal stress in terms of w 2 2 4 2 2 3 4 2 3 2(1 ) 4 2(1 ) 4 3 ( , ) y z z x z z E t z w z F x y x y E t z z w                                      Governing Equation in terms of Deflection w(x, y) 9

Governing Equation in terms of Deflection w(x, y): Applying the BCs at the bottom surface(α.)=/2 = 0E=9.2(1-EIO26(1 - vFurther applying the BCs at the top surfaceEt3(o. ) =-1/2 = -q=12(1 - v2)Et3DV4w=q,D12(1 - v2)D:Flexural Rigidity10

t t/2   2 2 3 3 4 2 2 4 2 ( ) 0 1 2(1 ) 4 2 3 8 6(1 ) 2 z z t z z E t t t z z w E t z z t w                                              • Applying the BCs at the bottom surface • Further applying the BCs at the top surface 3 4 2 2 3 4 2 ( ) 12(1 ) , 12(1 ) z z t E t q w q E t D w q D                Governing Equation in terms of Deflection w(x, y) • D: Flexural Rigidity 10