Advanced FEMKirchhoff Plates: Field Equations 20 AFEM Ch 20 – Slide 1 is much smaller than the other two Flatness: the midsurface of the plate is a plane AFEM Ch 20 – Slide 2 .Advanced FEM Plate Structures A plate is a three dimensional body characterized by Thinness: one of the plate dimensions. the thickness. Advanced FEM Plate: Membrane vs Bending z (a) (b) z y x x y AFEM Ch 20 – Slide 3 . also called material filament x Ω AFEM Ch 20 – Slide 4 .Advanced FEM Reduction to Two Dimensional Problem Midsurface y Mathematical Idealization (b) Plate Γ (c) (a) Thickness h Material normal. Advanced FEM Plate Models Bent membrane von Karman * Kirchhoff * Reissner-Mindlin High Order Composite Exact: 3D elasticity geometrically nonlinear geometrically nonlinear geometrically linear geometrically linear geometrically linear geometrically linear global global global global local local * treated in this course AFEM Ch 20 – Slide 5 . Advanced FEM The Kirchhoff Plate Model Behavioral assumptions: o thin plate but w << h o uniform thickness or varies slowly o symmetric fabrication about midplane o transverse loads distributed over areas of char dimension > h o support conditions respect inextensional bending AFEM Ch 20 – Slide 6 . y) Section y = 0 Original misurface x "Material normals remain straight after deformation and normal to the deformed misurface" AFEM Ch 20 – Slide 7 .Advanced FEM Main Kinematic Assumption for Kirchhoff Plate θy y θ y (positive as shown if looking toward −y) Deformed misurface θx x Γ Ω w(x. Advanced FEM Kinematic Relations Deflection of plate midsurface along z w = w(x. y ∂w θx = ∂w . θy = − ∂y ∂x Displacement of a material particle P(x.y) Rotations of material normal about x.z) u x = −z ∂w = zθy . ∂y uz = w AFEM Ch 20 – Slide 8 . ∂x u y = −z ∂w = −z θx .y. ∂x ∂x ∂x ∂uz ∂w ∂w + =− + =0 ∂y ∂y ∂y = −z in which the κ's are the plate midsurface curvatures κx y = ∂ 2w . ∂x2 ∂ 2w = −z 2 = −z κ yy . ∂x ∂x∂y ∂uz ∂w ∂w + =− + = 0. ∂ y2 κx y = ∂ 2w ∂x∂y AFEM Ch 20 – Slide 9 . ∂x2 κ yy = ∂ 2w . ∂z ∂uy ∂ 2w + = −2z = −2z κx y .Advanced FEM Kinematic Relations (cont'd) Strain-displacement equations ex x = e yy = ezz = 2ex y = 2ex z = 2e yz = ∂ux ∂x ∂uy ∂y ∂uz ∂z ∂ux ∂y ∂ux ∂z ∂uy ∂z ∂ 2w = −z κx x . ∂y ∂ 2w = −z 2 = 0. Bending Stresses and Moments Showing Positive Sign Conventions Advanced FEM Bending stresses (+ as shown) Inplane shear stresses Normal stresses dx dy dx dy z dx Top surface h y dy y x σxy = σyx x y σxx σyy y x Bottom surface M yy Bending moments (+ as shown) Mxx Mxx Myx M yy Mxy M xx M yx y M xy Mxy = M yx Myy x x 2D view AFEM Ch 20 – Slide 10 . Advanced FEM Moment-Curvature Relations Wall fabrication assumptions: o Plate is homogeneous o Each plate lamina z = constant is in plane stress o Material obeys Hooke's law in plane stress: σx x E 11 σ yy = E 12 σx y E 13 ex x E 11 E 13 E 23 e yy = −z E 12 E 33 2ex y E 13 κx x E 13 E 23 κ yy E 33 2κx y E 12 E 22 E 23 E 12 E 22 E 23 AFEM Ch 20 – Slide 11 . σ yx z dz . σ yy z dz .Advanced FEM Moment-Curvature Relations (cont'd) Bending moments are obtained by integrating the in-plane wall stresses over the thickness Mx x dy = M yy d x = Mx y dy = M yx d x = h /2 − h /2 h /2 − h /2 h /2 − h /2 h /2 − h /2 −σx x z dy dz −σ yy z d x dz −σx y z dy dz −σ yx z d x dz ⇒ ⇒ ⇒ ⇒ Mx x = − M yy = − Mx y = − M yx = − h /2 −h /2 h /2 −h /2 h /2 − h /2 h /2 −h /2 σx x z dz . σx y z dz . Since Mxy = Myx (from rotational equilibrium) only 3 independent components need to be calculated AFEM Ch 20 – Slide 12 . min σxmax =± y 6 Mx y max . h2 .min = σ yx 2 h AFEM Ch 20 – Slide 13 .min σxmax =± x 6 Mx x .Advanced FEM Moment-Curvature Relations (cont'd) Carrying out the integration over the thickness: E 11 Mx x 3 M yy = h E 12 12 Mx y E 13 E 12 E 22 E 23 κx x D11 E 13 = D12 E 23 κ yy E 33 2κx y D13 D12 D22 D23 κx x D13 D23 κ yy D33 2κx y For isotropic material of modulus E and Poisson's ratio ν Mx x 1 M yy = D ν 0 Mx y ν 1 0 κx x 0 κ yy 0 1 (1 + ν ) 2κx y 2 where D= Eh 3 12(1 − ν 2 ) is the plate rigidity Max/min stress computation given the moments: .min σ yy =± 6 M yy . h2 max . Advanced FEM Transverse Shear Stresses and Forces dx dy Parabolic distribution across thickness Transverse shear stresses z dx Top surface h σxz σ yz y dy x y y Qx Qy Qy x Bottom surface Transverse shear forces (+ as shown) y Qx x x 2D view AFEM Ch 20 – Slide 14 . h σ yz = max σ yz 4z 2 1− 2 .Advanced FEM Transverse Shear Stresses and Forces (cont'd) Wall distribution in a homogeneous plate σx z = σxmax z 4z 2 1− 2 . maximum shear stresses are σxmax = z 3 2 Qx . 3 yz If transverse shear forces given. h max σ yz = 3 2 Qy . h AFEM Ch 20 – Slide 15 . Qy = h /2 − h /2 σ yz dz = 2 σ max h . h Integrating over the thickness provides the transverse shear forces Qx = h /2 − h /2 σx z dz = 2 max σ 3 xz h. Advanced FEM Internal Equilibrium Equations z (a) Qy Qx z (b) M yx Mxy Mxx dy q dx dy Q y+ ∂ Qy dy ∂y Myy y x dx y Myy+ ∂ M yy dy ∂y x Q x+ ∂ Qx dx ∂x Distributed transverse load (force per unit area) z-force ∂ M yx ∂ Mx x dx M yx + dx ∂x ∂x ∂ Mx y Mxy + dy ∂y x-mom y-mom Mxx + z-mom ∂ Qy ∂ Qx + = −q ∂x ∂y ∂ Mx y ∂ Mx x + = −Q x ∂x ∂y ∂ M yy ∂ M yx + = −Qy ∂x ∂y Mx y = M yx AFEM Ch 20 – Slide 16 . Advanced FEM Internal Equilibrium Equations (cont'd) Repeating for convenience: ∂ Qy ∂ Qx + = −q ∂x ∂y ∂ Mx y ∂ M yy ∂ M yx ∂ Mx x + = −Q + = −Q y x ∂x ∂y ∂x ∂y Mx y = M yx Eliminating the shear forces and one of the twist moments gives the moment equilibrium equation ∂ 2 Mx y ∂ 2 M yy ∂ 2 Mx x + 2 =q + ∂x2 ∂x∂y ∂ y2 AFEM Ch 20 – Slide 17 . Advanced FEM Matrix and Indicial Form of Field Equations Field eqn KE CE BE Matrix form κ = Pw Indicial form καβ = w. MT = [ Mx x M yy Mx y ] = [ M11 M22 M12 ].2 only. κT = [ κx x κ yy 2κx y ] = [ κ11 κ22 2κ12 ].αβ = q Equationname for plate problem M = Dκ PT M = q Kinematic equation Moment-curvature equation Internal equilibrium equation Here PT = [ ∂ 2 /∂ x 2 ∂ 2 /∂ y 2 2 ∂ 2 /∂ x ∂ y ] = [ ∂ 2 /∂ x1 ∂ x1 ∂ 2 /∂ x2 ∂ x2 2 ∂ 2 /∂ x1 ∂ x2 ]. AFEM Ch 20 – Slide 18 . such as α . run over 1.αβ Mαβ = Dαβγ δ κγ δ Mαβ. Greek indices. Advanced FEM Strong Form Diagram of Field Equations for Kirchhoff Plate Model Deflection w κ=Pw in Ω Transverse load q Γ Ω Kinematic Equilibrium Constitutive PT M = q in Ω Curvatures κ M=Dκ in Ω Bending moments M AFEM Ch 20 – Slide 19 .