ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 20: 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

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 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

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

Kirchhoff Plate Theory First Element developed for thin plates and shells x y z h yy ww xx Transverse Shear deformations neglected In plane deformations neglected

z Strain Tensor Strains x

z Strain Tensor Strains y

Strain Tensor Shear Strains

Strain Tensor

Moments

Stress-Strain Relationships z At each layer, z, plane stress conditions are assumed h

Stress-Strain Relationships Integrating over the thickness the generalized stress-strain matrix (moment-curvature) is obtained or

Generalized stress-strain matrix

Formulation of Rectangular Plate Bending Element h x y z yy xx ww Node 1 Node 4 Node 2 Node 3 12 degrees of freedom

Pascal Triangle 1 xy x2x2 xyxyy2y2 x3x3 x2yx2yxy2xy2 y3y3 x4x4 x3yx3y x2y2x2y2 xy3xy3 y4y4 ……. x5x5 x4yx4y x3y2x3y2 x2y3x2y3 xy 4 y5y5

Assumed displacement Field

Formulation of Rectangular Plate Bending Element

For Admissible Displacement Field yy xx ww i =1,2,3,4 12 equations / 12 unknowns

Formulation of Rectangular Plate Bending Element and, thus, generalized coordinates a 1 -a 12 can be evaluated…

Formulation of Rectangular Plate Bending Element For plate bending the strain tensor is established in terms of the curvature

Formulation of Rectangular Plate Bending Element

Strain Energy Substitute moments and curvature… Element Stiffness Matrix

Shell Elements x y z h yy w xx u v

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

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 And a potential problem arises… Twisting DOF

Kirchhoff Shell Elements … when adjacent elements are coplanar (or almost) Singular Stiffness Matrix (or ill conditioned) Zero Stiffness  z

Kirchhoff Shell Elements Define small twisting stiffness k

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 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.

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

Reissner/Mindlin Plate Theory x y z h yy ww xx Transverse Shear deformations ARE INCLUDED In plane deformations neglected

Strain Tensor z y  xz

Strain Tensor z y  yz

Strain Tensor Shear Strains Transverse Shear assumed constant through thickness

Strain Tensor Transverse Shear Strain Plane Strain

Stress-Strain Relationships z At each layer, z, plane stress conditions are assumed h Isotropic Material

Stress-Strain Relationships Plane Stress

Stress-Strain Relationships Transverse Shear Stress

Strain Energy Contributions from Plane Stress

Strain Energy Contributions from Transverse Shear k is the correction factor for nonuniform stress (see beam element)

Stiffness Matrix Contributions from Plane Stress

Stiffness Matrix Contributions from Plane Stress

Stiffness Matrix Therefore, field variables to interpolate are

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 Where q is the number of nodes in the element N i are the appropriate shape functions

Interpolation of Field Variables In contrast to Kirchoff element, the same shape functions are used for the interpolation of deflections and rotations (C o continuity)

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

Kirchhoff – Reissner/Mindlin Comparison 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 In addition to the more general nature of the Reissner/Mindlin plate element note that

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.

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 K b correctly evaluated by quadrature (Pure bending or twist) K s correctly evaluated by 1 point quadrature only.

Shear Locking & Reduced Integration K s shows stiffer behavior =>Shear Locking

Shear Locking & Reduced Integration K b correctly evaluated by quadrature (Pure bending or twist) K s cannot be evaluated correctly

Shear Locking & Reduced Integration

Shear Locking – Other Remedies Mixed Interpolation of Tensorial Components MITCn family of elements To alleviate shear locking Reissner/Mindlin formulation Interpolation of w, ,  and  Good mathematical basis, are reliable and efficient Interpolation of w,  and  is based on different order

Mixed Interpolation Elements

FETA V 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

(c) Instrumentation placement [7]

(c) Cross-bracing 1219 mm 3962 mm 1219 mm (e) Loading configuration Mid-Span x z

(a) Deck and girder (b) Stud pockets (c) Cross-bracing

(a) Deformed shape