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ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 20: Plates & Shells Plates & Shells Loaded in the transverse direction and may be assumed rigid (plates) or flexible (shells) in their plane. Are typically thin in one dimension Plate elements are typically used to model flat surface structural components Shells elements are typically used to model curved surface structural components 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 Assumptions As a consequence… Stress through the thickness (perpendicular to midsurface) is zero. Material particles that are originally on a straight line perpendicular to the midsurface remain on a straight line after deformation Plate Bending Theories Material particles that are originally on a straight line perpendicular to the midsurface remain on a straight line after deformation Kirchhfoff Reissner/Mindlin Shear deformations Shear deformations are neglected are included Straight line remains Straight line does NOT perpendicular to remain perpendicular midsurface after to midsurface after deformations deformations Kirchhoff Plate Theory First Element developed for thin plates and shells z w1 y qy qx x h In plane deformations neglected Transverse Shear deformations neglected Strain Tensor Strains u zq x z x w q x u w 2 x z x x 2 Strain Tensor Strains v zq y z y w q y v w 2 y z y y 2 Strain Tensor Shear Strains u v w 2 xy z y x xy zx zy 0 Strain Tensor 2w 2 2 x x w y z y 2 xy w 2 2 xy Moments h/2 h/2 Mx x zdz My y zdz h / 2 h / 2 Moments h/2 M xy xy zdz h / 2 Moments Mx x h/2 My y zdz M h / 2 xy xy Stress-Strain Relationships z h At each layer, z, plane stress conditions are assumed Mx x h/2 My y zdz M h / 2 xy xy x 1 0 x E y 1 0 y 1 2 1 2w xy 0 0 xy 2 2 2 x x w y z y 2 xy w 2 2 xy Stress-Strain Relationships Integrating over the thickness the generalized stress-strain matrix (moment-curvature) is obtained Mx x M y D y M xy xy 1 0 h/2 E 1 dz D 2 z 0 1 2 h / 2 1 0 0 or 2 Generalized stress-strain matrix 1 0 3 Eh D 1 0 12 1 2 1 0 0 2 Formulation of Rectangular Plate Bending Element z w1 q1 x y Node 2 x q1 y Node 1 Node 3 Node 4 h 12 degrees of freedom Pascal Triangle 1 x y x2 xy y2 x3 x 2y xy2 y3 x4 x 3y x2y2 xy3 y4 x5 x 4y x3y2 x2y3 xy4 y5 ……. Assumed displacement Field w a1 a2 x a3 y a4 x a5 xy a6 y 2 2 a7 x a8 x y a9 xy a10 y 3 2 2 3 a11 x y a12 xy 3 3 Formulation of Rectangular Plate Bending Element w x a2 2a4 x a5 y 3a7 x 2a8 xy 2 x a9 y 3a11 x y a12 y 2 2 3 w y a3 a5 x 2a6 y a8 x 2a9 xy 2 y 3a10 y a11 x 3a12 xy 2 3 2 For Admissible Displacement Field w1 q1 x q1 y wi wxi , yi w xi , yi w xi , yi x i y i y x i=1,2,3,4 12 equations / 12 unknowns Formulation of Rectangular Plate Bending Element and, thus, generalized coordinates a1-a12 can be evaluated… Formulation of Rectangular Plate Bending Element For plate bending the strain tensor is established in terms of the curvature 2w x x 2 Mx x w 2 y M y D y y 2 M xy 2 w xy xy 2 xy Formulation of Rectangular Plate Bending Element w 2 2a4 6a7 x 2a8 y 6a11 xy x 2 w 2 2a6 2a9 x 6a10 y 6a12 xy y 2 Formulation of Rectangular Plate Bending Element w 2 2a5 4a8 y 4a9 y 6a11 x 6a12 y 2 2 xy Strain Energy 1 Ue T ε σDdV Ve 2 Substitute moments and curvature… 1 Ue T κ DκdA Ae 2 Element Stiffness Matrix Shell Elements z w v y qy qx x u h Shell Element by superposition of plate element and plane stress element Five degrees of freedom per node No stiffness for in-plane twisting Stiffness Matrix ~ k plate 0 ~ 1212 k shell ~ 20 x 20 0 k plane stress 88 Kirchhoff Shell Elements Use this element for the analysis of folded plate structure Kirchhoff Shell Elements Use this element for the analysis of slightly curved shells Kirchhoff Shell Elements However in both cases transformation to Global CS is required ~* T k shellT T k shell 24 x 24 2020 ~ k shell 0 ~* k shell 2020 24 x 24 0 0 Twisting DOF 4 4 And a potential problem arises… Kirchhoff Shell Elements … when adjacent elements are coplanar (or almost) Zero Stiffness qz Singular Stiffness Matrix (or ill conditioned) Kirchhoff Shell Elements Define small twisting stiffness k ~ k shell 0 ~* k shell 2020 24 x 24 0 k I 4 4 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) Comments Elements are typically of constant thickness. Elements are defined by four nodes. Bilinear variation of thickness may be considered by appropriate modifications to the system matrices. Nodal values of thickness need to be specified at nodes. Plate Bending Theories Material particles that are originally on a straight line perpendicular to the midsurface remain on a straight line after deformation Kirchhfoff Reissner/Mindlin Shear deformations Shear deformations are neglected are included Straight line remains Straight line does NOT perpendicular to remain perpendicular midsurface after to midsurface after deformations deformations Reissner/Mindlin Plate Theory z w1 y qy qx x h In plane deformations neglected Transverse Shear deformations ARE INCLUDED Strain Tensor u z x z y xz w x xz x w x u x x z x x Strain Tensor v z y z y yz w y yz y w y v y y z y y Strain Tensor Shear Strains u x v y xy z y x y x Transverse Shear assumed constant through thickness w w x xz y yz x y w w xz x yz y x y Strain Tensor Plane Strain Transverse Shear Strain x w xx x x y xz x yy z w y yz y xy y y x y x Stress-Strain Relationships Isotropic Material z h At each layer, z, plane stress conditions are assumed Stress-Strain Relationships Plane Stress x x 1 0 x y E y z 1 0 1 y 2 1 xy 0 0 y 2 x y x Stress-Strain Relationships Transverse Shear Stress w x xz E x w yz 2(1 ) y y Strain Energy Contributions from Plane Stress U ps 1 0 x xy 1 h/2 E y y dzdA h / 2 x 1 2 1 0 1 A 2 0 0 xy 2 Strain Energy Contributions from Transverse Shear U ts xz yz k h/2 E 2 A h / 2 xz dzdA 21 yz k is the correction factor for nonuniform stress (see beam element) Stiffness Matrix Contributions from Plane Stress 1 0 x xy 1 h/2 E y y dzdA U ps h / 2 x 1 2 1 0 1 A 2 0 0 xy 2 1 0 x 3 xy Eh k ps x y 1 0 y dA 1 2 A 1 0 0 xy 2 Stiffness Matrix Contributions from Plane Stress xz yz k h/2 E U ts 2 A h / 2 xz dzdA 21 yz k ts w x w w Ehk x A x x y w dA y 21 y y Stiffness Matrix k k ps ( x , y ) k ts ( w, x , y ) Therefore, field variables to interpolate are w, x , y 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 Interpolation of Field Variables q w N i wi Where q is the number i 1 of nodes in the q element x Ni x i i 1 q Ni are the appropriate y Ni y shape functions i i 1 Interpolation of Field Variables In contrast to Kirchoff element, the same shape functions are used for the interpolation of deflections and rotations (Co continuity) Comments Elements can be used for the analysis of general plates and shells Plates and Shells with curved edges and faces are accommodated Lower order elements show artificial stiffening Due to spurious shear deformation modes Shear Locking The least order of recommended interpolation is cubic i.e., 16-node quadrilateral 10-node triangular Kirchhoff – Reissner/Mindlin Comparison In addition to the more general nature of the Reissner/Mindlin plate element note that Kirchhoff: Interpolated field variable is the deflection w Reissner/Mindlin: Interpolated field variables are Deflection w Section rotation x Section rotation y True Boundary Conditions are better represented Shear Locking To alleviate shear locking Reduced integration of system matrices 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. 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 Shear Locking & Reduced Integration Kb correctly evaluated by quadrature (Pure bending or twist) Ks correctly evaluated by 1 point quadrature only. Shear Locking & Reduced Integration Ks shows stiffer behavior =>Shear Locking Shear Locking & Reduced Integration Kb correctly evaluated by quadrature (Pure bending or twist) Ks cannot be evaluated correctly Shear Locking & Reduced Integration Shear Locking – Other Remedies To alleviate shear locking Mixed Interpolation of Tensorial Components MITCn family of elements Reissner/Mindlin formulation Interpolation of w, , and Interpolation of w, and is based on different order Good mathematical basis, are reliable and efficient Mixed Interpolation Elements Mixed Interpolation Elements Mixed Interpolation Elements Mixed Interpolation Elements Mixed Interpolation Elements FETA V2.1.00 ELEMENT LIBRARY 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) 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 Planning an Analysis Preliminary Analysis Based on elementary theory, formulas from handbooks, analytical work, or experimental evidence Know what to expect before FEA Planning an Analysis Start with Simple FE models and improve them Planning an Analysis Start with Simple FE models and improve them Planning an Analysis Check model and results Checking the Model • Check Model prior to computation • Undetected mistakes lead to: – execution failure – bizarre results – Look right but are wrong Common Mistakes In general mistakes in modeling result from insufficient familiarity with: a) The physical problem b) Element Behavior c) Analysis Limitations d) Software Common Mistakes Null Element Stiffness Matrix Check for common multiplier (e.g. thickness) Poisson’s ratio = 0.5 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 Common Mistakes Singular Stiffness Matrix (cont’d) • Part of structure has buckled • In nonlinear analysis, supports or connections have reached zero stiffness 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 Example 178 127 127 178 17 11 12.7 11 178 11 o o 74 74 17 127 178 178 127 Unit: mm 17530 S u rv e y P ris m 2440 DW T A B C D S tra in G a g e s 11330 c e n te r in te rio r e x te r i o r 2 5 .4 m m = 1 i n c h CL S u r v e y P r is m (c) Instrumentation placement [7] Mid-Span x z 1219 mm 3962 mm 1219 mm (e) Loading configuration (c) Cross-bracing Y X Z (b) Stud pockets (a) Deck and girder (c) Cross-bracing 0 Center Girder Deflection -1 FEM -2 Test 1 Deflection (mm) Test 2 -3 -4 -5 -6 Mid-Span -7 Center Girder Interior Exterior Girder Girder -8 0 1 2 3 4 5 6 7 8 9 Distance from the End of Bridge (m) (a) Deformed shape Test 2 3 4 5 6 Mid-Span 7 Center Girder Interior Exterior Girder Girder 8 0 0 1 2 3 4 5 6 0 7 8 9 -1 -1 -2 Distance from the End of Bridge (m) -2 Deflection (mm) Deflection (mm) -3 -3 -4 -4 Interior Girder Deflection Exterior Girder Deflection -5 -5 FEM FEM -6 -6 Test 1 Test 1 -7 Test 2 -7 Test 2 -8 -8 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 Distance from the End of Bridge (m) Distance from the End of Bridge (m)