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

Bending of Thin Plates

Bending of Thin Plates

OutlineIntroductionElementary Beam TheoryAssumptions Formulation in terms of DeflectionInternal Force per Unit Length Relations between Internal Force and Stress Differential Element Equilibrium - Alternative ApproachBoundary ConditionsBoundaryEquationSchemeFourier Method Summary2

Outline • Introduction • Elementary Beam Theory • Assumptions • Formulation in terms of Deflection • Internal Force per Unit Length • Relations between Internal Force and Stress • Differential Element Equilibrium – Alternative Approach • Boundary Conditions • Boundary Equation Scheme • Fourier Method • Summary 2

Introduction: One dimension (thexthickness) is significantlysmaller than the other twot/2t/2(1/8-1/5) > t/b > (1/80-1Middle Surface1/100)Middle Surface: z = O0: Only subjected to transversloads.: If a plate is only subjected to longitudinal loads, theproblem is reduced to plane stress state. The bending problem of thin plates is analyzed withstrategies similar to those of elastic beams.3

• One dimension (the thickness) is significantly smaller than the other two. (1/8-1/5) > t/b > (1/80- 1/100) • Middle Surface: z = 0. • Only subjected to transvers loads. Introduction • If a plate is only subjected to longitudinal loads, the problem is reduced to plane stress state. • The bending problem of thin plates is analyzed with strategies similar to those of elastic beams. 3 t/2 t/2 x y z O Middle Surface b

Review of the Elementary Beam Theory: Plane sections normal to the longitudinal axis of thebeam remain planar.Only uniaxial longitudinal stress is assumed.3MdWEIEIMqdx?2dxdx4

Review of the Elementary Beam Theory • Plane sections normal to the longitudinal axis of the beam remain planar. • Only uniaxial longitudinal stress is assumed. 2 2 2 2 2 2 d d d , d d d w w EI M EI q x x x         4

Assumptions2Straight lines normal to the middle surface112Oremain straight and the same lengthB11/2AStress components acting on planesparallel to the middle surface aresignificantly smaller than othercomponents. The corresponding strain cantherefore be neglectedow0=w = w(x, y)OOzuowOwOu10=U2OzOxaxOzowavavOw-0=I8zy2Ozazdyya.-v(ax1+0Discard:8=E2G2G5

Assumptions 0 ( , ) 1 0 2 1 0 2 ( ) 1 1 Discard: , , . 2 2 z z x z y z x y z zx zx zy zy w w w x y z u w u w z x z x w v v w y z z y E G G                                                                5 t/2 t/2 t/2 • Straight lines normal to the middle surface remain straight and the same length. • Stress components acting on planes parallel to the middle surface are significantly smaller than other components. The corresponding strain can therefore be neglected

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