FEM for Plates Shells

<span class='text_page_counter'>(1)</span>The Finite Element Method A Practical Course. CHAPTER 7:. FEM FOR PLATES & SHELLS.

<span class='text_page_counter'>(2)</span> CONTENTS  INTRODUCTION  PLATE. ELEMENTS. – Shape functions – Element matrices.  SHELL. ELEMENTS. – Elements in local coordinate system – Elements in global coordinate system – Remarks.  CASE. STUDY.

<span class='text_page_counter'>(3)</span> INTRODUCTION  FE. equations based on Reissner-Mindlin plate theory will be developed.  FE equations of shells will be formulated by superimposing matrices of plates and that of 2D solids.  Computationally tedious due to more DOFs..

<span class='text_page_counter'>(4)</span> PLATE ELEMENTS  Geometrically. similar to 2D plane stress solids except that it carries only transverse loads. Leads to bending.  2D equivalent of the beam element.  Rectangular plate elements based on ReissnerMindlin plate theory will be developed – conforming element.  Many software like ABAQUS do not offer plate elements, only the general shell element..

<span class='text_page_counter'>(5)</span> PLATE ELEMENTS  Consider z, w. y. a plate structure:. fz. Middle plane. h. x Middle plane. (Reissner-Mindlin plate theory).

<span class='text_page_counter'>(6)</span> PLATE ELEMENTS  Reissner-Mindlin. plate theory:. Middle plane. u ( x, y, z ) z y ( x, y ) v( x, y, z )  z x ( x, y ). In-plane strain: where. ε  zχ     y x   χ Lθ   x y   x   y  y  x . in which.     L     .   x 0.   y.  0      y      x . (Curvature).

<span class='text_page_counter'>(7)</span> PLATE ELEMENTS w      xz   y x  Off-plane shear strain: γ      w   yz    x    y . Potential (strain) energy:. h/2 h/2 1 1 T U e    ε σdAdz    τT γdAdz 2 Ae  h / 2 2 Ae  h / 2. In-plane stress & strain. Off-plane shear stress & strain.  xz  G 0  τ     γ c s γ   0 G  yz    2 /12. or 5/6.

<span class='text_page_counter'>(8)</span> PLATE ELEMENTS Substituting. ε  zχ.  xz  G 0  , τ  yz    0 G  γ c s γ    . 1 h3 T 1 U e   χ cχdA   hγ T c s γdA 2 Ae 12 2 Ae. Kinetic energy: Substituting. 1 Te    (u 2  v 2  w 2 )dV 2 Ve u ( x, y, z )  z y ( x, y ) v( x, y, z )  z x ( x, y ). 1 h3  2 h3  2 1 2  Te    (hw   x   y )dA   (dT I d)dA 2 Ae 12 12 2 Ae.

<span class='text_page_counter'>(9)</span> PLATE ELEMENTS 1 h3  2 h3  2 1 2  Te    (hw   x   y )dA   (d T I d )dA 2 Ae 12 12 2 Ae. where. w   d  x  ,    y.   h  I  0    0 . 0.  h3 12 0.  0   0     h3  12 .

<span class='text_page_counter'>(10)</span> Shape functions . Note that rotation is independent of deflection w 4. 4. 4. i 1. i 1. i 1. w  N i wi ,  x  N i x i ,  y  N i y i. where. N i  14 (1   i  )(1   i ). (Same as rectangular 2D solid).

<span class='text_page_counter'>(11)</span> Shape functions h. 4 ( 1, +1) (w4,x4,y4) 2 2 1 ( 1,  1) (w1,x1,y1). w    x  Nd e    y. z, w. . 3 (1, +1) (w3,x3,y3). where.  2 (1,  1) (w2,x2,y2).  N1 N  0  0. 0 N1 0. 0 0 N1.      Node 1. N2 0 0. 0 N2 0. 0 0 N2.       Node 2. N3 0 0.  w1     x1   y1     w2     x2   y 2  d e    w3   x 3     y 3  w   4  x 4     y 4 e. 0 N3 0. 0 0 N3.      Node 3.                    . displacement at node 1. displacement at node 2. displacement at node 3. displacement at node 4. N4 0 0. 0 N4 0. 0 0  N 4 .      Node 4.

<span class='text_page_counter'>(12)</span> Element matrices h. w 1 T     )dA T  ( d I d Substitute  x  d Nde into e Ae 2    y.  where. 1 T  Te  d e m ed e 2 T. m e   N I N dA Ae. (Can be evaluated analytically but in practice, use Gauss integration). Recall that:   h  I  0    0 . 0.  h3 12 0.  0   0     h3  12 .

<span class='text_page_counter'>(13)</span> Element matrices h. w   Substitute  x  d Nde into potential energy function    y. from which we obtain h3 I T I k e  [B ] cB dA  h[B O ]T c s B O dA Ae 12 Ae I. . I. B  B1. B. I 2. B. I 3. B. I 4. . 0 0  N j x    B Ij  0 N j y 0   0 N j x  N j y   . N j. N j  1    i (1   i ) x  x 4a N j N j  1   (1   i  ) i y  y 4b. Note:  x a ,   y b.

<span class='text_page_counter'>(14)</span> Element matrices. . B O  B O1. B O2. B 3O. B O4. . 0  N j x B   N j y  N j O j. Nj 0 . (me can be solved analytically but practically solved using Gauss integration).  fz    f e   N T  0  dA Ae 0  . For uniformly distributed load, f eT abf z 1 0 0 1 0 0 1 0 0 1 0 0.

<span class='text_page_counter'>(15)</span> 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..

<span class='text_page_counter'>(16)</span> Elements in local coordinate system Consider a flat shell element  d e1  node 1 d  node 2   d e  e 2  d e 3  node 3 d e 4  node 4  ui  displacement in x direction  v  displacement in y direction  i  wi  displacement in z direction d ei   rotation about x-axis  xi   yi  rotation about y -axis   rotation about z -axis  zi . 4 ( 1, +1) (u4, v4, w4, x4,y4,z4) 2 2 1 ( 1,  1) (u1, v1, w1, x1,y1,z1). z, w. . 3 (1, +1) (u3, v3, w3, x3,y3,z3).  2 (1,  1) (u2, v2, w2, x2,y2,z2).

<span class='text_page_counter'>(17)</span> Elements in local coordinate system Membrane stiffness (2D solid element):. (2x2).   k em     . node1 node2 node3 node4 m m m m k 13 k 11 k 12 k 14  node 1 k m23 k m21 k m22 k m24  node 2  m m m m k 31 k 32 k 33 k 34  node 3  k m41 k m42 k m43 k m44  node 4. Bending stiffness (plate element):. (3x3).   k be     . node1 node2 node3 b b b k 13 k 11 k 12 k b21 k b22 k b23 k b33 k b31 k b32 k b41. k b42. k b43. node4 b k 14  node 1 k b24  node 2  b k 34  node 3  k b44  node 4.

<span class='text_page_counter'>(18)</span> Elements in local coordinate system  node  1          k e            . m k 11 0. 0 b k 11.  node  2  m 0 k 12 0 0. 0.  node  3 .  node  4 . 0 b k 12. 0 0. m k 13 0. 0 b k 13. 0 0. m k 14 0. 0 b k 14. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. k m21 0 0. 0 k b21 0. 0 k m22 0 0 0 0. 0 k b22 0. 0 k m23 0 0 0 0. 0 k b23 0. 0 k m24 0 0 0 0. 0 k b24 0. m k 31 0 0. 0 k b31 0. m 0 k 32 0 0 0 0. 0 k b32 0. m 0 k 33 0 0 0 0. 0 k b33 0. m 0 k 34 0 0 0 0. 0 k b34 0. k m41 0. 0 k b41. 0 k m42 0 0. 0 k b42. 0 k m43 0 0. 0 k b43. 0 k m44 0 0. 0 k b44. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0     0   node 1 0    0      node 2 0   0     0   node 3 0    0    0    node 4 0    0  . (24x24). Components related to the DOF z, are zeros in local coordinate system..

<span class='text_page_counter'>(19)</span> Elements in local coordinate system Membrane mass matrix (2D solid element): node1 node2 node3 node4  m em     . m m11. m m12. m m21 m m 31 m m41. m m22 m m32 m m42. m m13 m m23 m m 33 m m43. m m14. m m24 m m 34 m m44.  node 1  node 2   node 3   node 4. Bending mass matrix (plate element): node1 node2 node3 node4  m be     . b m11 m b21. b m12 mb22. b m13. m b23. b m14 m b24. m b31. mb32. m b33. m b34. m b41. mb42. m b43. m b44.  node 1  node 2   node 3   node 4.

<span class='text_page_counter'>(20)</span> Elements in local coordinate system  node  1          m e            . m m11 0 b 0 m11. 0. 0. m m21 0 0 m b21 0. 0. m m 31. 0. 0 0. m b31 0. m m41. 0. 0 0. m b41 0.  node  2  m 0 m 12 0 0. 0. 0. 0 m m22 0 0 0. 0. m 0 m 32. 0 0. 0 0. 0 m m42 0 0. 0 0.  node  3 . 0 b m12. 0 0. m m13 0. 0. 0. 0. 0 m b22 0 0 m b32 0 0 m b42 0. 0 m m23 0 0 0. 0. m 0 m 33. 0 0. 0 0. 0 m m43 0 0. 0 0.  node  4 . 0 b m13. 0 0. m m14 0. 0 b m 14. 0. 0. 0. 0. 0 m b23 0 0 m b33 0 0 m b43 0. 0 m m24 0 0 0. 0. m 0 m 34. 0 0. 0 0. 0 m m44 0 0. 0 0. 0 m b24 0 0 m b34 0 0 m b44 0. 0     0   node 1 0    0      node 2 0   0     0   node 3 0    0    0    node 4 0    0  . (24x24). Components related to the DOF z, are zeros in local coordinate system..

<span class='text_page_counter'>(21)</span> Elements in global coordinate system K e T T k e T T. M e T m e T. where. T. Fe T f e.  lx T3   l y  l z.  T3 0  0  0 T  0  0 0   0. mx my mz. 0. 0. 0. 0. 0. 0. T3 0. 0 T3. 0 0. 0 0. 0 0. 0 0. 0. 0. T3. 0. 0. 0. 0. 0. 0. T3. 0. 0. 0. 0. 0. 0. T3. 0. 0. 0. 0. 0. 0. T3. 0. 0. 0. 0. 0. 0. nx  n y  n z . 0 0  0  0 0  0 0  T3 .

<span class='text_page_counter'>(22)</span> 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..

<span class='text_page_counter'>(23)</span> CASE STUDY  Natural. frequencies of micro-motor.

<span class='text_page_counter'>(24)</span> Natural Frequencies (MHz) 768 triangular elements with 480 nodes. 384 quadrilateral elements with 480 nodes. 1280 quadrilateral elements with 1472 nodes. 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. Mode. CASE STUDY.

<span class='text_page_counter'>(25)</span> CASE STUDY Mode 1:. Mode 2:.

<span class='text_page_counter'>(26)</span> CASE STUDY Mode 3:. Mode 4:.

<span class='text_page_counter'>(27)</span> CASE STUDY Mode 5:. Mode 6:.

<span class='text_page_counter'>(28)</span> CASE STUDY Mode 7:. Mode 8:.

<span class='text_page_counter'>(29)</span> CASE STUDY  Transient. analysis of micro-motor F. Node 210. x x. F. Node 300 F.

<span class='text_page_counter'>(30)</span> CASE STUDY.

<span class='text_page_counter'>(31)</span> CASE STUDY.

<span class='text_page_counter'>(32)</span> CASE STUDY.

<span class='text_page_counter'>(33)</span>