FEM FOR PLATES SHELLS

  Finite Element Method FEM FOR PLATES & SHELLS

  1 CONTENTS 

  INTRODUCTION 

PLATE ELEMENTS

• – Shape functions
• – Element matrices

   SHELL ELEMENTS

• – Elements in local coordinate system
• – Elements in global coordinate system
• – Remarks

  INTRODUCTION  FE equations based on Mindlin plate theory will be developed.

   FE equations of shells will be formulated by superimposing matrices of plates and those of 2D solids.

   Computationally tedious due to more DOFs.

PLATE ELEMENTS

   Geometrically similar to 2D plane stress solids except that it carries only transverse loads. Leads to bending.

   2D equilvalent of the beam element.  Rectangular plate elements based on Mindlin plate theory will be developed – conforming element.

   Much software like ABAQUS does not offer plate elements, only the general shell element.

PLATE ELEMENTS

   Consider a plate structure: f z

  Middle plane y z, w h x

  Middle plane (Mindlin plate theory)

PLATE ELEMENTS

  L θ χ (Curvature)

  L

     

     

     y x y x

          

          

   Mindlin plate theory: ( , , ) ( , ) ( , , ) ( , ) ^y _x u x y z z x y v x y z z x y

      

       y x y x y x x y

              

     

  Middle plane where

    In-plane strain:

  χ ε z

     

PLATE ELEMENTS

  1    

      

   

       

    

  

G

G

 



  γ c γ τ s yz xz

  In-plane stress & strain Off-plane shear stress & strain

    γ τ σ ε

  Off-plane shear strain:     

         

  1 d d

  2

  T h e d d

  A z z A U _{e e} A T h A

  γ Potential (strain) energy:

     

       y w x w x y yz xz

  



 

      

  2

PLATE ELEMENTS

  

 

  G xz   c

       Substituting τ γ γ

    s   z

    ε χ ,

   G

yz

   

 

  3 1 h

  1 T T c c

  U d A  h d A  

  χ χ γ γ e s

   A  A _{e e}

  2

  12

  2

  1

  

2

  2

  2 Kinetic energy:

T   ( u   v w )d

  V e

  V  _e

  2 u x y z ( , , ) z ( , ) x y

    _y Substituting v x y z ( , , ) z ( , ) x y

     _x

  3

  

3

1 h h

  1

  2

  2

2 T

  d I d T ( hw )d A ( )d A

         e x y

   A _e  A _e

  2

  12

  

12

  2

PLATE ELEMENTS

  3

  3 1 h h

  1

  2

  2

2 T

  d I d T ( hw )d A ( )d A

         e x y

   A _e  A _e

  2

  12

  12

  2    h 

  



  w

    ₃ h

    

   

  I d  , where x

       

  12   ₃  

   h

   y

     

   

  12   Shape functions 

  Note that rotation is independent of deflection w , ,

  4

  1

  4

  1

  4

  1 i y i i y i x i i x i i i

  N N w N w    

       

     ) 1 )(

  1 (

  4

  1     i i i

  N    where

  (Same as rectangular

  2D solid) Shape functions h x e y w

              

   _{z 3}

  ,

   _{y 2}

  ,

   _{z 2}

  ) 3 (1, +1) (u ³ , v ³ , w ³ ,

   _{x 3}

  ,

   _{y 3}

  ,

  )

  (u ² , v ² , w ² ,

  2a

  4 ( 1, +1)

  (u ⁴ , v ⁴ , w ⁴ ,

   _{x 4}

  ,

   _{y 4}

  ,

   _{z 4}

  )

   _{x 2}

  ) 2 (1, 1)

  Nd ^{1 1} _{1 2}

     d

where

  2 ₂

  3 _{3 3}

  4 _{4 4} displacement at node 1 displacement at node 2 displacement at node 3 displacement at node 4 ^{x y x y e x y x} _{y e} w w w w

         

                

             

       

                

             

  1 ^{2 3 4} _{1 2 3 4}

   _{z 1}

  1 ^{2 3 4} _{Node 1 Node 2 Node 3 Node 4} N N N N

N N N N

N N N N

            

  N

  1 ( 1, 1)

  (u ¹ , v ¹ , w ¹ ,

   _{x 1}

  ,

   _{y 1}

  ,

  2b   z, w

  Element matrices Substitute h x e y w

   m N I N

            

         

    

  12 h h h

  12

  3

  3

  Recall that:

  A 

                d Nd into e e T e e

   where T d _e e A

   d I d

  A T A 

  2 ^e T

e

  1  1 ( )d

  2

   

  T d m d

  I (Can be evaluated analytically but in practice, use Gauss integration) Element matrices A h A h s A A e _{e e}

  ] d [ d ] [

  N y x N N y N x j j ^{j j}

     

      

     

  1    

  4

  1 ) 1 (

  4

    ) 1 (

         

  N x N

  N y N a x

  I j B ^{i i j j i i j j} b y

            

  12 O T O

     

       

  I ₁ B B B B B 

  2 I

  3 I

  4 I

  I

   

                d Nd into potential energy function from which we obtain

     Substitute h x e y w

  3 B c B cB B k  

  I

  I T

   , b y a x     Note: Element matrices  

  O

     

  1 z T e abf

  1

  1

  1

   

  N f For uniformly distributed load,

        

  T 

  4 O

  A f _e A z e d

  (m e can be solved analytically but practically solved using Gauss integration)

  N y N N x N B

   O j j j j j

        

     

  2 O O ₁ B B B B B 

  3 O

   f

SHELL ELEMENTS

   Loads in all directions  Bending, twisting and in-plane deformation

   Combination of 2D solid elements (membrane effects) and plate elements (bending effect).

   Common to use shell elements to model plate structures in commercial software packages.

  Elements in local coordinate system Consider a flat shell element z, w

  4 ( 1, +1)

   d node

  1  _{e 1} (u 4 , v 4 , w 4 ,

   3 (1, +1)

       d x y z

  4 , 4 , 4 ) ₂ node

  2  e  (u 3 , v 3 , w 3 , d _e    d

     _{e 3} node

  3 x y z

  3 , 3 , 3 )  

  2b d

    node 4 _{e 4}

     2a u x

    displacement in direction _i 2 (1,

1 (

  1) 1, 1)   v y

  (u 2 , v 2 , w 2 , _i displacement in direction (u 1 , v 1 , w 1 ,         x , y , z ) x y z

  2

  2

  2  

  1 , 1 , 1 ) w z displacement in direction   i d _ei

     x

   rotation about -axis _xi     y

   rotation about -axis _yi  

   rotation about -axis z   _zi   Elements in local coordinate system Membrane stiffness (2D solid element):

  4 node 3 node 2 node 1 node node4 node3 node2 node1

  44 ^{34 24 14 43 33 23 13 42 32 22 12 41 31 21 11}      

       

   _{m m} ^{m m} _{m m} ^{m m} _{m m} ^{m m} _{m m} ^{m m m} _e k k k k k k k k k k k k k k k k k

  Bending stiffness (plate element): 4 node 3 node 2 node 1 node node4 node3 node2 node1

  44 ^{34 24 14 43 33 23 13 42 32 22 12 41 31 21 11}      

       

   _{b b} ^{b b} _{b b} ^{b b} _{b b} ^{b b} _{b b} ^{b b b} _e k k k k k k k k k k k k k k k k k

  (2x2) (3x3) Elements in local coordinate system

  4 node 3 node 2 node 1 node

  4 node 3 node 2 node 1 node

  44

⁴⁴

³⁴

³⁴

²⁴

²⁴

¹⁴

¹⁴

^{43 43 33 33 23 23 13 13 42 42 32 32 22 22 12 12 41 41 31 31 21 21 11 11}

     

      

      

      

                       

                     

  

  b

^m

^b

^m

^b

^m

^b

^m

^{b m b m b m b m b m b m b m b m b m b m b m b m e} k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k

             

  (24x24) Components related to the DOF

   z

  , are zeros in local coordinate system. Elements in local coordinate system Membrane mass matrix (2D solid element):

  13 ^{11 12}

¹⁴

_{23 21 22}

₂₄

_{33 31 32}

₃₄

  43 ^{41 42}

⁴⁴

node3 node1 node2 node4 node 1 node 2 node 3 node 4 ^{m m m m m m m m m e m m m m m m m m} 

              m m m m m m m m m m m m m m m m m

  Bending mass matrix (plate element):

  13 ^{11 12}

¹⁴

_{23 21 22}

₂₄

_{33 31 32}

₃₄

  43 ^{41 42}

⁴⁴

node3 node1 node2 node4 node 1 node 2 node 3 node 4 ^{b b b b b b b b b e b b b b b b b b} 

              m m m m m m m m m m m m m m m m m Elements in local coordinate system 4 node 3 node 2 node 1 node

  4 node 3 node 2 node 1 node

  44 ^{44 34 34 24 24 14 14 43 43 33 33 23 23 13 13 42 42 32 32 22 22 12 12 41 41 31 31 21 21 11 11}    

      

      

      

                       

                     

   b ^{m b m b m b m b m b m b m b m b m b m b m b m b m b m b m b m e} _m ^{m m m m m m m} _m ^{m m m m m m m} _m ^{m m m m m m m} _m ^{m m m m m m m m}

              Components related to the DOF

   z

  , are zeros in local coordinate system.

  (24x24) Elements in global coordinate system T k T K e T e

   T m T M e T e

   e T e

  T f F 

              

              

  

  3 ^{3 3 3 3 3 3 3} T T T T T T T T T      

     

   z z z y y y x x x n m l n m l n m l

  where

3 T

  Remarks  The membrane effects are assumed to be uncoupled with the bending effects in the element level.

   This implies that the membrane forces will not result in any bending deformation, and vice versa.

   For shell structure in space, membrane and bending effects are actually coupled (especially for large curvature), therefore finer element mesh may have to be used.

CASE STUDY

   Natural frequencies of micro-motor

  Natural Frequencies (MHz) 1280 Mode

  CASE 768 triangular 384 quadrilateral quadrilateral elements with elements with elements with

  480 nodes 480 nodes 1472 nodes STUDY

  1

  7.67

  5.08

  4.86

  2

  7.67

  5.08

  4.86

  3

  7.87

  7.44

  7.41

  4

  10.58

  8.52

  8.30

  5

  10.58

  8.52

  8.30

  6

  13.84

  11.69

  11.44

  7

  13.84

  11.69

  11.44

  8

  14.86

  12.45

  12.17

CASE STUDY

  Mode 1: Mode 2:

CASE STUDY

  Mode 3: Mode 4:

CASE STUDY

  Mode 5: Mode 6:

CASE STUDY

  Mode 7: Mode 8:

CASE STUDY

   Transient analysis of micro-motor F F F x x

  Node 210 Node 300