Finite Element Formulation Tarun Kant Department of Civil - PowerPoint PPT Presentation

Reissner-Mindlin Plate Bending Finite Element Formulation Tarun Kant Department of Civil Engineering Indian Institute of Technology Bombay E-mail: tkant@civil.iitb.ac.in Website: www.civil.iitb.ac.in/~tkant 1

BEAM z w p(x) x u τ xz dz σ x z Δ x x 2

Euler-Bernoulli Beam  u   ( , ) ( ,0) u x z u x z z   0 z  ( , ) ( ,0) w x z w x or more concisely    ( ) ( ) u u x z x    u z o  ( ) w w x  w o 3

S-D Relations    u u     o z    x x x x    u w w      o    xz z x x   setting 0 gives us xz  w     o x  Causes occurrence of second derivative of w o in PE functional “π”  Involves C 1 continuous interpolation functions in finite element formulation 4

Non-vanishing stresses and strains   , x xz  x 5

Before deformation 6

After deformation 7

Timoshenko Beam Displacement Model Same as that of Euler-Bernoulli Beam Non-Vanishing Stresses/Strains        , , ( ) x x xz x xz S-D Relations  w           u o    o z xz   x x x x 8

Timoshenko Beam  Π contains only first derivatives of w o and θ FE Formulation uses simple C o shape functions  9

Euler-Bernoulli Beam, Poisson-Kirchhoff Plate and Love Shell Theories Thin beam, plate, shell, i.e., (h/L) << 1 or (h/a) << 1 or  (h/R) << 1 Tangential displacements vary linearly through the  thickness. Transverse shear deformation is neglected, γ 1z = γ 2z =0  Transverse normal strain is zero, ε z = 0  Transverse normal stress is small (neglected), σ z ≈ 0  10

After deformation γ or φ θ Reissner-Mindlin theory 11

Kirchhoff Hypothesis Limitations  Transverse shear deformability of structural elements (especially of shear deformable FRC laminates) is not accounted in the formulation  Transverse normal rotations of the cross sections becomes first derivatives of transverse displacement components,  w     o i x i  Transverse displacement field turns C 1 continuous 12

Remarks  Popular displacement – based formulations include – transverse shear deformation/s. ?  This is not only due to the significance of shear deformation effects in moderately thick/multilayered/FR composite elements.  But mainly because the resulting formulation is C o which enables simple interpolations. 13

Remarks (Contd.)  The classical Poisson-Kirchhoff plate theory requires C 1 continuity.  C o – continuous finite element interpolations are easily constructed.  Multi-dimensional C 1 -interpolations are difficult/tedious to construct.  Considerable ingenuity was required to develop compatible C 1 - continuous plate/shell elements; resulting schemes have always been extremely complicated in one way or another. 14

Remarks (Contd.)  Since mid-seventies, there is turning away from Poisson-Kirchhoff theory based elements to those based on Reissner-Mindlin theory which not only requires C o -continuity but also accommodates transverse shear deformations. However, these too, initially, were not without its own inherent difficulties. - Locking in ‘thin’ regime - Spurious ‘zero - energy’ modes 15

Positive set of stress components 16

Positive set of displacement components z w o w y  v u v o y u o  x x 17

Kinematics   ( , , ) ( , ) u x y z z x y y    ( , , ) ( , ) v x y z z x y x z  ( , , ) ( , ) w x y z w x y o w o w  t ( , , ) u u v w y  v u v o y    t ( , , ) u u w o x y o  x x or   ( , , ) w o x y 18

Strains   u      y z z   x x x x   v        x z z   y y y y       u v        y  x  z z     xy xy   y x y x    w u w         o    xz y x z x x    w v w         o    yz x y z y y 19

Middle Surface Strains        0 0   ε t ( , , )  x   b x y xy     0 0      ε t  ( , ) y     w s y x   o         ε Lu 0        ε t t t   x ( , ) x y       b s     y 0 1    x          1 0     0 0    y     x ε L u         b b 0 1 0 0 L      b  x y       ε L u L  s         s s 1 0  0           y x y 20

Total Potential Energy    U W 1      ε σ u b u t t t t = dV dV dS 2 V V S t u : vector of displacement components of int a po in plate space ε : vector of strain components σ : vector of stress components b : vector of body forces t : vector of boundary tractions V : sol ution domain : S part of boundary on which boundary tractions are prescribed t 21

Strain Energy   U U U b s   1           U dV b x x y y xy xy 2 V   1          = z z z dV x x y y xy xy 2 V   1               ( ( ) ( ) ( ) = zdz zdz zdz dA x x y y xy xy 2   A z z z   1       = M M M dA x x y y xy xy 2 A   1  ε σ t . = dA b b 2 A 22

in which  σ t ( , , ) M M M b x y xy     ε t ( , , ) b x y xy 23

Shear Strain Energy   1        U dV s xz xz yz yz 2 V   1       = dAdz xz x yz y 2 V   1           ( ) ( ) = dz dz dA x xz y yz 2   A z z   1     = Q Q dA x x y y 2 A   1  ε σ t . = dA s s 2 A 24

Strain Energy   U U U b s     1 1     ε σ ε σ t t . . dA dA b b s s 2 2 A A     1 1     ε D ε ε D ε t t . . dA dA b b b s s s 2 2 A A 25

Work Done   W w p dxdy    t ( , , ) o z u w o x y A   t  u p dxdy t ( ,0,0) p p z A 26

Equilibrium    U W             0 U W U W Minimum Potential Energy Principle    or U W Virtual Work Principle 27

Constitutive Relations - 3D Define,       σ t , ,  σ E ε b x y xy   b b b    σ t ,  s xz yz σ E ε    σ σ σ s s s t t t , b s σ = Eε       ε t , , b x y xy      ε t , s x y    ε ε ,ε t t t b s 28

Constitutive Relations - 3D (Contd.)         0 E E     x 11 12 x        σ E ε     0 E   22 b y y b b           sym E     33 xy xy             0 E    xz xz 1 σ S E ε         s s s      0  E     yz 2 yz S For isotropic material, E     ; E E G ; E E   1 2 S S 11 22   2 1 E    ; ; G E E 12 11   2(1 )   1   E E G σ = Eε 33 11 2 29

Constitutive Relations - 2D        0 M D D     x 11 12 x       σ D ε     0 M D   b b y 22 y b          sym D M     33 xy xy            Q 0 D   x x 1 S D ε       s  s     Q  0  D     y 2 y S For isotropic material,   3 3 1 Eh h       ; ; ; D D D D D D G   11 22   12 11 33 11 2 2 12 12 1  2 5 E     ; ; D D kGh G k or   1 2 S S 2(1 ) 6 12  σ Dε 30