Kirchhoff Plates: Field Equations

Transcription

1 20 Kirchhoff Plates: Field Equations AFEM Ch 20 Slide 1

2 Plate Structures A plate is a three dimensional bod characterized b Thinness: one of the plate dimensions, the thickness, is much smaller than the other two Flatness: the midsurface of the plate is a plane AFEM Ch 20 Slide 2

3 Plate: Membrane vs Bending (a) z (b) z AFEM Ch 20 Slide 3

4 Reduction to Two Dimensional Problem (b) Plate Midsurface Mathematical Idealization (a) (c) Γ Ω Thickness h Material normal, also called material filament AFEM Ch 20 Slide 4

5 Plate Models Bent membrane geometricall nonlinear global von Karman geometricall nonlinear global * Kirchhoff geometricall linear global * Reissner-Mindlin geometricall linear global High Order Composite geometricall linear local Eact: 3D elasticit geometricall linear local * treated in this course AFEM Ch 20 Slide 5

6 The Kirchhoff Plate Model Behavioral assumptions: o thin plate but w << h o uniform thickness or varies slowl o smmetric fabrication about midplane o transverse loads distributed over areas of char dimension > h o support conditions respect inetensional bending AFEM Ch 20 Slide 6

7 Main Kinematic Assumption for Kirchhoff Plate θ θ (positive as shown if looking toward ) Deformed misurface Γ θ w(,) Original misurface Ω Section = 0 "Material normals remain straight after deformation and normal to the deformed misurface" AFEM Ch 20 Slide 7

8 Kinematic Relations Deflection of plate midsurface along z w = w(,) Rotations of material normal about, θ = w θ, = w Displacement of a material particle P(,,z) u = z w θ = z, u = z w θ = z, u z = w AFEM Ch 20 Slide 8

9 Kinematic Relations (cont'd) Strain-displacement equations e = u = z 2 w 2 e = u = z 2 w 2 e zz = u z z = z 2 w z 2 = 0, = z κ, = z κ, 2e = u + u = 2z 2 w = 2z κ, 2e z = u z + u z = w + w = 0, 2e z = u z + u z = w + w = 0 in which the κ's are the plate midsurface curvatures Advanced FEM κ = 2 w 2, κ = 2 w 2, κ = 2 w AFEM Ch 20 Slide 9

10 Bending Stresses and Moments Showing Positive Sign Conventions Advanced FEM z d Top surface d h Bending stresses (+ as shown) Normal stresses Inplane shear stresses d d d d σ σ σ = σ Bottom surface M Bending moments (+ as shown) M M = M M M M M M M M M 2D view AFEM Ch 20 Slide 10

11 Moment-Curvature Relations Wall fabrication assumptions: o Plate is homogeneous o Each plate lamina z = constant is in plane stress o Material obes Hooke's law in plane stress: σ σ = σ E 11 E 12 E 13 E 12 E 22 E 23 E 13 E 23 E 33 e e = z 2e E 11 E 12 E 13 E 12 E 22 E 23 E 13 E 23 E 33 κ κ 2κ AFEM Ch 20 Slide 11

12 Moment-Curvature Relations (cont'd) Bending moments are obtained b integrating the in-plane wall stresses over the thickness Advanced FEM M d = M d = M d = M d = σ zddz M = σ zddz M = σ zddz M = σ zddz M = σ zdz, σ zdz, σ zdz, σ zdz. Since M = M (from rotational equilibrium) onl 3 independent components need to be calculated AFEM Ch 20 Slide 12

13 Moment-Curvature Relations (cont'd) Carring out the integration over the thickness: M E 11 E 12 E 13 κ D 11 D 12 D 13 κ M = h3 E 12 E 22 E 23 κ = D 12 D 22 D 23 κ 12 M E 13 E 23 E 33 2κ D 13 D 23 D 33 2κ For isotropic material of modulus E and Poisson's ratio ν M M M 1 ν 0 = D ν (1 + ν) 1 2 κ κ 2κ Ma/min stress computation given the moments: Advanced FEM where Eh D = 3 12(1 ν 2 ) is the plate rigidit σ ma,min =± 6M, σ ma,min h 2 =± 6M, σ ma,min h 2 =± 6M h 2 = σ ma,min AFEM Ch 20 Slide 13

14 Transverse Shear Stresses and Forces Advanced FEM z d Top surface d h d σ z d σ z Parabolic distribution across thickness Transverse shear stresses Bottom surface Q Q Q Transverse shear forces (+ as shown) Q 2D view AFEM Ch 20 Slide 14

15 Transverse Shear Stresses and Forces (cont'd) Wall distribution in a homogeneous plate σ z = σz ma ( 4z 2 ) 1, h 2 σz = σz ma ( 4z 2 ) 1. h 2 Integrating over the thickness provides the transverse shear forces Q = σ z dz = 2 3 σ ma z h, Q = σ z dz = 2 3 σ ma z h, If transverse shear forces given, maimum shear stresses are σ ma z = 3 2 Q h, σma z = 3 2 Q h. AFEM Ch 20 Slide 15

16 (a) Q d Q + Q z-force Internal Equilibrium Equations z q d Q d Q + Q d Distributed transverse load (force per unit area) (b) M + -mom M d M M d M + z M M d M z-mom d M + Advanced FEM M M + d M d -mom Q + Q = q M + M = Q M M = M + M = Q AFEM Ch 20 Slide 16

17 Internal Equilibrium Equations (cont'd) Repeating for convenience: Advanced FEM Q + Q M = q + M M = Q M = M + M = Q Eliminating the shear forces and one of the twist moments gives the moment equilibrium equation 2 M M + 2 M 2 = q AFEM Ch 20 Slide 17

18 Matri and Indicial Form of Field Equations Field Matri Indicial Equationname eqn form form for plate problem KE κ = P w κ αβ = w,αβ CE M = D κ M αβ = D αβγ δ κ γδ BE P T M = q M αβ,αβ = q Kinematic equation Moment-curvature equation Internal equilibrium equation Here P T = [ 2 / 2 2 / / ] = [ 2 / / / 1 2 ], M T = [ M M M ] = [ M 11 M 22 M 12 ], κ T = [ κ κ 2κ ] = [ κ 11 κ 22 2κ 12 ]. Greek indices, such as α, run over 1,2 onl. AFEM Ch 20 Slide 18

19 Strong Form Diagram of Field Equations for Kirchhoff Plate Model Advanced FEM Deflection w Transverse load q Kinematic κ = P w in Ω Equilibrium T P M = q in Ω Γ Ω Curvatures κ Constitutive M = D κ in Ω Bending moments M AFEM Ch 20 Slide 19