# ECIV 720 A Advanced Structural Mechanics and Analysis by 6Vt6e4J6

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```									          ECIV 720 A
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
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 

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)

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.

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)

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
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

element for practical use question its stability
and convergence
Shear Locking & Reduced Integration

(Pure bending or twist)

Ks correctly evaluated by 1 point
Shear Locking & Reduced Integration

Ks shows stiffer behavior =>Shear Locking
Shear Locking & Reduced Integration

(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
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

Planning an Analysis

Planning an Analysis

Check model and results
Checking the Model

• Check Model prior to computation

– 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

(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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