ECIV 720 A Advanced Structural Mechanics and Analysis by 6Vt6e4J6

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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          xy



                zx   zy  0
     Strain Tensor



            2w 
                   
            2
                  2
x            x   
           w 
 y   z        
               y
                  2
                
 xy        w
                 2

             2
            xy 
                   
                          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
                                                         xy 
                                                               
            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  wxi , 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      
          xy 
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
    xy
              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        
~            1212                   
k shell               ~
20 x 20
             0        k plane stress 
            
                          88
                                      
                                      
         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                    2020

                            ~
                          k shell    0
     ~*
     k shell              2020         
          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      2020          
  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
                                      21    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
                                                  21    yz 

k ts 

                                           w      
                                                x
    w                w      Ehk  x           
                                                    
A  x   x              y            w      dA
                      y      21         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)

								
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