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1. 1. ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 20: Plates & Shells
2. 2. Plates & Shells Loaded in the transverse direction and may be assumed rigid (plates) or flexible (shells) in their plane. Plate elements are typically used to model flat surface structural components Shells elements are typically used to model curved surface structural components Are typically thin in one dimension
3. 3. Assumptions Based on the proposition that plates and shells are typically thin in one dimension plate and shell bending deformations can be expressed in terms of the deformations of their midsurface
4. 4. Assumptions Stress through the thickness (perpendicular to midsurface) is zero. As a consequence… Material particles that are originally on a straight line perpendicular to the midsurface remain on a straight line after deformation
5. 5. Plate Bending Theories Kirchhfoff Shear deformations are neglected Straight line remains perpendicular to midsurface after deformations Material particles that are originally on a straight line perpendicular to the midsurface remain on a straight line after deformation Reissner/Mindlin Shear deformations are included Straight line does NOT remain perpendicular to midsurface after deformations
6. 6. Kirchhoff Plate Theory First Element developed for thin plates and shells x y z h θy w1 θx Transverse Shear deformations neglected In plane deformations neglected
7. 7. z Strain Tensor Strains x w ∂ ∂ =θ xzu θ−= x 2 2 x w z x u x ∂ ∂ −= ∂ ∂ =ε
8. 8. z Strain Tensor Strains y w ∂ ∂ =θ yzv θ−= y 2 2 y w z y v y ∂ ∂ −= ∂ ∂ =ε
9. 9. Strain Tensor Shear Strains yx w z x v y u xy ∂∂ ∂ −= ∂ ∂ + ∂ ∂ = 2 γ 0≅= zyzx γγ
10. 10. Strain Tensor                   ∂∂ ∂ ∂ ∂ ∂ ∂ −=           yx w y w x w z xy y x 2 2 2 2 2 2 γ ε ε
11. 11. Moments ∫− = 2/ 2/ h h xx zdzM σ ∫− = 2/ 2/ h h yy zdzM σ
12. 12. Moments ∫− = 2/ 2/ h h xyxy zdzM τ
13. 13. Moments ∫−           =           2/ 2/ h h xy y x xy y x zdz M M M τ σ σ
14. 14. Stress-Strain Relationships z At each layer, z, plane stress conditions are assumed h
15. 15.                       −− =           xy y x xy y x E γ ε ε ν ν ν ν τ σ σ 2 1 00 01 01 1 2 ∫−           =           2/ 2/ h h xy y x xy y x zdz M M M τ σ σ                   ∂∂ ∂ ∂ ∂ ∂ ∂ −=           yx w y w x w z xy y x 2 2 2 2 2 2 γ ε ε
16. 16. Stress-Strain Relationships Integrating over the thickness the generalized stress-strain matrix (moment-curvature) is obtained ∫−             −− = 2/ 2/ 2 2 2 1 00 01 01 1 h h dz E z ν ν ν ν D or           =           xy y x xy y x M M M κ κ κ D
17. 17. Generalized stress-strain matrix ( )             −− = 2 1 00 01 01 112 2 3 ν ν ν ν Eh D
18. 18. Formulation of Rectangular Plate Bending Element h x y z θ1 y θ1 x w1 Node 1 Node 4 Node 2 Node 3 12 degrees of freedom
19. 19. Pascal Triangle 1 x y x2 xy y2 x3 x2 y xy2 y3 x4 x3 y x2 y2 xy3 y4 ……. x5 x4 y x3 y2 x2 y3 xy4 y5
20. 20. Assumed displacement Field 3 12 3 11 3 10 2 9 2 8 3 7 2 65 2 4321 xyayxa yaxyayxaxa yaxyaxayaxaaw ++ +++++ ++++++=
21. 21. Formulation of Rectangular Plate Bending Element 3 12 2 11 2 9 8 2 7542 3 232 yayxaya xyaxayaxaa x w x +++ ++++= ∂ ∂ =ϑ 2 12 3 11 2 10 9 2 8653 33 22 xyaxaya xyaxayaxaa y w y +++ ++++= ∂ ∂ =ϑ
22. 22. For Admissible Displacement Field ( )iii yxww ,= θ1 y θ1 x w1 ( ) y yxw iii x ∂ ∂ = , ϑ( ) x yxw iii y ∂ ∂ −= , ϑ i=1,2,3,4 12 equations / 12 unknowns
23. 23. Formulation of Rectangular Plate Bending Element and, thus, generalized coordinates a1-a12 can be evaluated…
24. 24. Formulation of Rectangular Plate Bending Element For plate bending the strain tensor is established in terms of the curvature                   ∂∂ ∂ ∂ ∂ ∂ ∂ =           yx w y w x w xy y x 2 2 2 2 2 2 κ κ κ           =           xy y x xy y x M M M κ κ κ D
25. 25. Formulation of Rectangular Plate Bending Element xyayaxaa x w 118742 2 6262 +++= ∂ ∂ xyayaxaa y w 1210962 2 6622 +++= ∂ ∂
26. 26. Formulation of Rectangular Plate Bending Element yaxayayaa yx w 12 2 11985 2 664422 ++++= ∂∂ ∂
27. 27. Strain Energy ∫= eV T e dVU σDε 2 1 ∫= eA T e dAU Dκκ 2 1 Substitute moments and curvature… Element Stiffness Matrix
28. 28. Shell Elements x y z h θy w θx u v
29. 29. Shell Element by superposition of plate element and plane stress element Five degrees of freedom per node No stiffness for in-plane twisting
30. 30. Stiffness Matrix           = × × 88 1212 2020 ~ 0 0 ~ ~ stressplane plate x shell k k k
31. 31. Kirchhoff Shell Elements Use this element for the analysis of folded plate structure
32. 32. Kirchhoff Shell Elements Use this element for the analysis of slightly curved shells
33. 33. Kirchhoff Shell Elements However in both cases transformation to Global CS is required And a potential problem arises…         = × × 44 2020 2424 * 0 0 ~ ~ 0 k k shell x shell 2020 * 2424 ~ × = TkTk shell T x shell Twisting DOF
34. 34. Kirchhoff Shell Elements … when adjacent elements are coplanar (or almost) Singular Stiffness Matrix (or ill conditioned) Zero Stiffness θz
35. 35. Kirchhoff Shell Elements         = × × 44 2020 2424 * 0 0 ~ ~ I k k k shell x shell Define small twisting stiffness k
36. 36. Comments Plate and Shell elements based on Kirchhoff plate theory do not include transverse shear deformations Such Elements are flat with straight edges and are used for the analysis of flat plates, folded plate structures and slightly curved shells. (Adjacent shell elements should not be co- planar)
37. 37. Comments Elements are defined by four nodes. Elements are typically of constant thickness. Bilinear variation of thickness may be considered by appropriate modifications to the system matrices. Nodal values of thickness need to be specified at nodes.
38. 38. Plate Bending Theories Kirchhfoff Shear deformations are neglected Straight line remains perpendicular to midsurface after deformations Material particles that are originally on a straight line perpendicular to the midsurface remain on a straight line after deformation Reissner/Mindlin Shear deformations are included Straight line does NOT remain perpendicular to midsurface after deformations
39. 39. Reissner/Mindlin Plate Theory x y z h θy w1 θx Transverse Shear deformations ARE INCLUDED In plane deformations neglected
40. 40. Strain Tensor z xzu β−= y x z x u x x ∂ ∂ −= ∂ ∂ = β ε xzx x w γβ − ∂ ∂ = γxz x w ∂ ∂
41. 41. Strain Tensor z yzv β−= y y z y v y y ∂ ∂ −= ∂ ∂ = β ε yzy y w γβ − ∂ ∂ = γyz y w ∂ ∂
42. 42. Strain Tensor Shear Strains       ∂ ∂ + ∂ ∂ −= ∂ ∂ + ∂ ∂ = xy z x v y u yx xy ββ γ Transverse Shear assumed constant through thickness xzx x w γβ − ∂ ∂ = yzy y w γβ − ∂ ∂ = xxz x w βγ − ∂ ∂ = yyz y w βγ − ∂ ∂ =
43. 43. Strain Tensor                   ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ −=           xy y x z yx y x xy yy xx ββ β β γ ε ε             − ∂ ∂ − ∂ ∂ =       y x yz xz y w x w β β γ γ Transverse Shear StrainPlane Strain
44. 44. Stress-Strain Relationships z At each layer, z, plane stress conditions are assumed h Isotropic Material
45. 45. Stress-Strain Relationships                   ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂             −− −=           xy y x E z yx y x xy y x ββ β β ν ν ν ν τ σ σ 2 1 00 01 01 1 2 Plane Stress
46. 46. Stress-Strain Relationships Transverse Shear Stress             − ∂ ∂ − ∂ ∂ + =       y x yz xz y w x w E β β ντ τ )1(2
47. 47. Strain Energy Contributions from Plane Stress [ ] dzdA E U xy y x A h h xyyx ps                       −− = ∫ ∫− γ ε ε ν ν ν ν γεε 2 1 00 01 01 12 1 2 2/ 2/
48. 48. Strain Energy Contributions from Transverse Shear [ ] ( ) dzdA Ek U yz xz A h h yzxz ts       − = ∫ ∫− γ γ ν γγ 122 2/ 2/ k is the correction factor for nonuniform stress (see beam element)
49. 49. Stiffness Matrix Contributions from Plane Stress [ ] dzdA E U xy y x A h h xyyxps                       −− = ∫ ∫− γ ε ε ν ν ν ν γεε 2 1 00 01 01 12 1 2 2/ 2/ [ ]∫                       −− = A xy y x xyyxps dA Eh κ κ κ ν ν ν ν κκκ 2 1 00 01 01 1 2 3 k
50. 50. Stiffness Matrix Contributions from Plane Stress ( )∫             − ∂ ∂ − ∂ ∂ −      − ∂ ∂ − ∂ ∂ = A y x yx ts dA y w x w Ehk y w x w β β ν ββ 12 k [ ] ( ) dzdA Ek U yz xz A h h yzxzts       − = ∫ ∫− γ γ ν γγ 122 2/ 2/
51. 51. Stiffness Matrix ),,(),( yxtsyxps w ββββ kkk += Therefore, field variables to interpolate are yxw ββ ,,
52. 52. Interpolation of Field Variables For Isoparametric Formulation Define the type and order of element e.g. 4,8,9-node quadrilateral 3,6-node triangular etc
53. 53. Interpolation of Field Variables ∑= = q i i yiy N 1 ββ ∑= = q i i xix N 1 ββ ∑= = q i iiwNw 1 Where q is the number of nodes in the element Ni are the appropriate shape functions
54. 54. Interpolation of Field Variables In contrast to Kirchoff element, the same shape functions are used for the interpolation of deflections and rotations (Co continuity)
55. 55. Comments Elements can be used for the analysis of general plates and shells Plates and Shells with curved edges and faces are accommodated The least order of recommended interpolation is cubic i.e., 16-node quadrilateral 10-node triangular Lower order elements show artificial stiffening Due to spurious shear deformation modes Shear Locking
56. 56. Kirchhoff – Reissner/Mindlin Comparison Kirchhoff: Interpolated field variable is the deflection w Reissner/Mindlin: Interpolated field variables are Deflection w Section rotation βx Section rotation βy True Boundary Conditions are better represented In addition to the more general nature of the Reissner/Mindlin plate element note that
57. 57. Shear Locking Reduced integration of system matrices To alleviate shear locking Numerical integration is exact (Gauss) Displacement formulation yields strain energy that is less than the exact and thus the stiffness of the system is overestimated By underestimating numerical integration it is possible to obtain better results.
58. 58. Shear Locking The underestimation of the numerical integration compensates appropriately for the overestimation of the FEM stiffness matrices FE with reduced integration Before adopting the reduced integration element for practical use question its stability and convergence
59. 59. Shear Locking & Reduced Integration Kb correctly evaluated by quadrature (Pure bending or twist) Ks correctly evaluated by 1 point quadrature only.
60. 60. Shear Locking & Reduced Integration Ks shows stiffer behavior =>Shear Locking
61. 61. Shear Locking & Reduced Integration Kb correctly evaluated by quadrature (Pure bending or twist) Ks cannot be evaluated correctly
62. 62. Shear Locking & Reduced Integration
63. 63. Shear Locking – Other Remedies Mixed Interpolation of Tensorial Components MITCn family of elements To alleviate shear locking Reissner/Mindlin formulation Interpolation of w, β, and γ Good mathematical basis, are reliable and efficient Interpolation of w,β and γ is based on different order
64. 64. Mixed Interpolation Elements
65. 65. Mixed Interpolation Elements
66. 66. Mixed Interpolation Elements
67. 67. Mixed Interpolation Elements
68. 68. Mixed Interpolation Elements
69. 69. FETA V2.1.00 ELEMENT LIBRARY
70. 70. Planning an Analysis Understand the Problem Survey of what is known and what is desired Simplifying assumptions Make sketches Gather information Study Physical Behavior Time dependency/Dynamic Temperature-dependent anisotropic materials Nonlinearities (Geometric/Material)
71. 71. Planning an Analysis Devise Mathematical Model Attempt to predict physical behavior Plane stress/strain 2D or 3D Axisymmetric etc Examine loads and Boundary Conditions Concentrated/Distributed Uncertain stiffness of supports or connections etc Data Reliability Geometry, loads BC, material properties etc
72. 72. Planning an Analysis Preliminary Analysis Based on elementary theory, formulas from handbooks, analytical work, or experimental evidence Know what to expect before FEA
73. 73. Planning an Analysis Start with Simple FE models and improve them
74. 74. Planning an Analysis Start with Simple FE models and improve them
75. 75. Planning an Analysis Check model and results
76. 76. Checking the Model • Check Model prior to computation • Undetected mistakes lead to: – execution failure – bizarre results – Look right but are wrong
77. 77. Common Mistakes In general mistakes in modeling result from insufficient familiarity with: a) The physical problem b) Element Behavior c) Analysis Limitations d) Software
78. 78. Common Mistakes Null Element Stiffness Matrix Check for common multiplier (e.g. thickness) Poisson’s ratio = 0.5
79. 79. Common Mistakes Singular Stiffness Matrix • Material properties (e.g. E) are zero in all elements that share a node • Orphan structure nodes • Parts of structure not connected to remainder • Insufficient Boundary Conditions • Mechanism exists because of inadequate connections • Too many releases at a joint • Large stiffness differences
80. 80. Common Mistakes Singular Stiffness Matrix (cont’d) • Part of structure has buckled • In nonlinear analysis, supports or connections have reached zero stiffness
81. 81. Common Mistakes Bizarre Results • Elements are of wrong type • Coarse mesh or limited element capability • Wrong Boundary Condition in location and type • Wrong loads in location type direction or magnitude • Misplaced decimal points or mixed units • Element may have been defined twice • Poor element connections
82. 82. Example 127 127 127 127178 178 178178 178 Unit: mm 74 o 74 o 11 11 11 1717 12.7
83. 83. (c) Instrumentation placement [7] 2440 Strain Gages Survey Prism DWT 25.4 mm = 1 inch CL A B C D interior exterior Survey Prism 11330 center 17530
84. 84. (c) Cross-bracing 1219 mm 3962 mm 1219 mm (e) Loading configuration Mid-Span x z
85. 85. X Z Y (a) Deck and girder (b) Stud pockets (c) Cross-bracing
86. 86. (a) Deformed shape -8 -7 -6 -5 -4 -3 -2 -1 0 0 1 2 3 4 5 6 7 8 9 Distance from the End of Bridge (m) Deflection(mm) FEM Test 1 Test 2 Center Girder Deflection
87. 87. -8 -7 -6 -5 -4 -3 -2 -1 0 0 1 2 3 4 5 6 7 8 9 Distance from the End of Bridge (m) Deflection(mm) FEM Test 1 Test 2 Interior Girder Deflection -8 -7 -6 -5 -4 -3 -2 -1 0 0 1 2 3 4 5 6 7 8 9 Distance from the End of Bridge (m) Deflection(mm) FEM Test 1 Test 2 Exterior Girder Deflection 8 7 6 5 4 3 0 1 2 3 4 5 6 7 8 9 Distance from the End of Bridge (m) Test 2