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Transcript of L20
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ECIV 720 A Advanced Structural
Mechanics and Analysis
Lecture 20: Plates & Shells
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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
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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
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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
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Plate Bending Theories
Kirchhfoff
Shear deformations are neglectedStraight 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 includedStraight line does NOT remain perpendicular to midsurface after deformations
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Kirchhoff Plate Theory
First Element developed for thin plates and shells
x
y
z
h
y
w
x
Transverse Shear deformations neglected
In plane deformations neglected
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z
Strain Tensor
Strains
x
w
xzu
x
2
2
x
wz
x
ux
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z
Strain Tensor
Strains
y
w
yzv
y
2
2
y
wz
y
vy
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Strain Tensor
Shear Strains
yx
wz
x
v
y
uxy
2
0 zyzx
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Strain Tensor
yx
wy
wx
w
z
xy
y
x
2
2
2
2
2
2
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Moments
2/
2/
h
h
xx zdzM
2/
2/
h
h
yy zdzM
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Moments
2/
2/
h
h
xyxy zdzM
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Moments
2/
2/
h
hxy
y
x
xy
y
x
zdz
M
M
M
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Stress-Strain Relationships
z
At each layer, z, plane stress conditions are assumed
h
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xy
y
x
xy
y
xE
2
100
01
01
1 2
2/
2/
h
hxy
y
x
xy
y
x
zdz
M
M
M
yx
wy
wx
w
z
xy
y
x
2
2
2
2
2
2
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Stress-Strain RelationshipsIntegrating over the thickness the generalized stress-strain matrix (moment-curvature) is obtained
2/
2/2
2
2
100
01
01
1
h
h
dzE
z
D
or
xy
y
x
xy
y
x
M
M
M
D
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Generalized stress-strain matrix
2
100
01
01
112 2
3
Eh
D
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Formulation of Rectangular Plate Bending Element
h
x
y
z
y
x
w
Node 1
Node 4
Node 2
Node 3
12 degrees of freedom
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Pascal Triangle
1
x y
x2 xy y2
x3 x2y xy2 y3
x4 x3y x2y2 xy3 y4
…….
x5 x4y x3y2 x2y3 xy4 y5
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Assumed displacement Field
312
311
310
29
28
37
265
24321
xyayxa
yaxyayxaxa
yaxyaxayaxaaw
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Formulation of Rectangular Plate Bending Element
312
211
29
82
7542
3
232
yayxaya
xyaxayaxaax
wx
212
311
210
92
8653
33
22
xyaxaya
xyaxayaxaay
wy
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For Admissible Displacement Field
iii yxww ,
y
xw
y
yxw iiix
, x
yxw iiiy
,
i=1,2,3,4 12 equations / 12 unknowns
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Formulation of Rectangular Plate Bending Element
and, thus, generalized coordinates
a1-a12 can be evaluated…
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Formulation of Rectangular Plate Bending Element
For plate bending the strain tensor is established in terms of the curvature
yx
wy
wx
w
xy
y
x
2
2
2
2
2
2
xy
y
x
xy
y
x
M
M
M
D
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Formulation of Rectangular Plate Bending Element
xyayaxaax
w118742
2
6262
xyayaxaay
w1210962
2
6622
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Formulation of Rectangular Plate Bending Element
yaxayayaayx
w12
211985
2
664422
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Strain Energy
eV
Te dVU σDε
2
1
eA
Te dAU Dκκ
2
1
Substitute moments and curvature…
Element Stiffness Matrix
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Shell Elements
x
y
z
h
y
w
x
u
v
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Shell Element by superposition of plate element and plane stress
element
Five degrees of freedom per node
No stiffness for in-plane twisting
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Stiffness Matrix
88
1212
2020~
0
0~
~
stressplane
plate
xshell k
kk
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Kirchhoff Shell Elements
Use this element for the analysis of folded plate
structure
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Kirchhoff Shell Elements
Use this element for the analysis of slightly curved shells
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Kirchhoff Shell Elements
However in both cases transformation to Global CS is required
And a potential problem arises…
44
20202424
*
0
0~
~
0
kk
shell
xshell
2020
*
2424
~
TkTk shellT
xshell
Twisting DOF
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Kirchhoff Shell Elements
… when adjacent elements are coplanar (or almost)
Singular Stiffness Matrix (or ill conditioned)
Zero Stiffness z
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Kirchhoff Shell Elements
44
20202424
*
0
0~
~
I
kk
k
shell
xshell
Define small twisting stiffness k
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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)
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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.
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Plate Bending Theories
Kirchhfoff
Shear deformations are neglectedStraight 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 includedStraight line does NOT remain perpendicular to midsurface after deformations
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Reissner/Mindlin Plate Theory
x
y
z
h
y
w
x
Transverse Shear deformations ARE INCLUDED
In plane deformations neglected
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Strain Tensor
zxzu
y
xz
x
u xx
xzx x
w
xz
x
w
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Strain Tensor
zyzv
y
yz
y
v yy
yzy y
w
yz
y
w
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Strain Tensor
Shear Strains
xy
zx
v
y
u yxxy
Transverse Shear assumed constant through thickness
xzx x
w
yzy y
w
xxz x
w
yyz y
w
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Strain Tensor
xy
y
x
z
yx
y
x
xy
yy
xx
y
x
yz
xz
y
wx
w
Transverse Shear StrainPlane Strain
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Stress-Strain Relationships
z
At each layer, z, plane stress conditions are assumed
h
Isotropic Material
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Stress-Strain Relationships
xy
y
xE
z
yx
y
x
xy
y
x
2
100
01
01
1 2
Plane Stress
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Stress-Strain Relationships
Transverse Shear Stress
y
x
yz
xz
y
wx
wE
)1(2
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Strain Energy
Contributions from Plane Stress
dzdAE
U
xy
y
x
A
h
h xyyx
ps
2
100
01
01
12
12
2/
2/
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Strain Energy
Contributions from Transverse Shear
dzdAEk
U
yz
xz
A
h
h yzxz
ts
122
2/
2/
k is the correction factor for nonuniform stress
(see beam element)
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Stiffness Matrix
Contributions from Plane Stress
dzdAE
U
xy
y
x
A
h
h xyyxps
2
100
01
01
12
12
2/
2/
A
xy
y
x
xyyxps dAEh
2
100
01
01
1 2
3
k
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Stiffness Matrix
Contributions from Plane Stress
A
y
x
yx
ts
dA
y
wx
wEhk
y
w
x
w
12
k
dzdAEk
Uyz
xz
A
h
h yzxzts
122
2/
2/
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Stiffness Matrix
),,(),( yxtsyxps w kkk
Therefore, field variables to interpolate are
yxw ,,
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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
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Interpolation of Field Variables
q
i
iyiy N
1
q
i
ixix N
1
q
iiiwNw
1Where q is the number
of nodes in the
element
Ni are the appropriate
shape functions
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Interpolation of Field Variables
In contrast to Kirchoff element, the same shape functions are used for the
interpolation of deflections and rotations
(Co continuity)
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Comments
Elements can be used for the analysis of
general plates and shellsPlates 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
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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
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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.
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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
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Shear Locking & Reduced Integration
Kb correctly evaluated by quadrature (Pure bending or twist)
Ks correctly evaluated by 1 point quadrature only.
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Shear Locking & Reduced Integration
Ks shows stiffer behavior =>Shear Locking
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Shear Locking & Reduced Integration
Kb correctly evaluated by quadrature (Pure bending or twist)
Ks cannot be evaluated correctly
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Shear Locking & Reduced Integration
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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
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Mixed Interpolation Elements
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Mixed Interpolation Elements
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Mixed Interpolation Elements
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Mixed Interpolation Elements
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Mixed Interpolation Elements
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FETA V2.1.00
ELEMENT LIBRARY
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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)
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Planning an Analysis
Devise Mathematical Model
Attempt to predict physical behaviorPlane stress/strain
2D or 3D
Axisymmetric
etc
Examine loads and Boundary Conditions
Concentrated/Distributed
Uncertain stiffness of supports or connections
etc
Data ReliabilityGeometry, loads BC, material properties etc
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Planning an Analysis
Preliminary Analysis
Based on elementary theory, formulas from handbooks, analytical work, or experimental evidence
Know what to expect before FEA
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Planning an Analysis
Start with Simple FE models and improve them
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Planning an Analysis
Start with Simple FE models and improve them
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Planning an Analysis
Check model and results
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Checking the Model
• Check Model prior to computation
• Undetected mistakes lead to:– execution failure – bizarre results– Look right but are wrong
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Common Mistakes
In general mistakes in modeling result from insufficient familiarity with:
a) The physical problem
b) Element Behavior
c) Analysis Limitations
d) Software
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Common Mistakes
Null Element Stiffness Matrix
Check for common multiplier (e.g. thickness)
Poisson’s ratio = 0.5
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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
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Common Mistakes
Singular Stiffness Matrix (cont’d)• Part of structure has buckled• In nonlinear analysis, supports or connections
have reached zero stiffness
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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
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Example
127
127 127
127 178 178
178 178
178
Unit: mm
74 o 74 o
11 11
11
17
17
12.7
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(c) Instrumentation placement [7]
2440
Strain Gages
Survey Prism
DWT
25.4 mm = 1 inch
C L
A B C D
interior
exterior
Survey Prism
1133
0
center
17530
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(c) Cross-bracing
1219 mm3962 mm
1219 mm
(e) Loading configuration
Mid-Spanx
z
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X Z
Y
(a) Deck and girder(b) Stud pockets
(c) Cross-bracing
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(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)
Def
lec
tio
n (
mm
)
FEMTest 1Test 2
Mid-Span
Interior Girder
Center Girder Exterior
Girder
Center Girder Deflection
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-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)
Def
lect
ion
(m
m)
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)
Def
lect
ion
(m
m)
FEM
Test 1
Test 2
Exterior Girder Deflection
-8
-7
-6
-5
-4
-3
-2
-1
0
0 1 2 3 4 5 6 7 8 9
Mid-Span
Interior Girder
Center Girder Exterior
Girder
Center Girder Deflection