This document discusses plate and shell elements for structural analysis. Plate elements are used to model flat surfaces, while shell elements model curved surfaces. Kirchhoff plate theory and Reissner-Mindlin plate theory are described for modeling plate bending, with the latter including transverse shear deformations. The derivation of a rectangular plate bending element is shown, involving assumed displacement fields and strain-curvature relationships. Shell elements can be formulated by combining plate and plane stress elements. Limitations of Kirchhoff shell elements for nearly coplanar or folded plate structures are noted.
1. ECIV 720 A
Advanced Structural Mechanics
and Analysis
Lecture 20:
Plates & Shells
2. 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
3. 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
4. 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
5. 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
6. Kirchhoff Plate Theory
First Element developed for thin plates and shells
x
y
z
h
θy
w1
θx
Transverse Shear deformations neglected
In plane deformations neglected
16. Stress-Strain Relationships
Integrating over the thickness the generalized
stress-strain matrix (moment-curvature) is obtained
∫−
−−
=
2/
2/
2
2
2
1
00
01
01
1
h
h
dz
E
z
ν
ν
ν
ν
D
or
=
xy
y
x
xy
y
x
M
M
M
κ
κ
κ
D
21. Formulation of Rectangular Plate Bending
Element
3
12
2
11
2
9
8
2
7542
3
232
yayxaya
xyaxayaxaa
x
w
x
+++
++++=
∂
∂
=ϑ
2
12
3
11
2
10
9
2
8653
33
22
xyaxaya
xyaxayaxaa
y
w
y
+++
++++=
∂
∂
=ϑ
22. For Admissible Displacement Field
( )iii yxww ,=
θ1
y
θ1
x
w1
( )
y
yxw iii
x
∂
∂
=
,
ϑ( )
x
yxw iii
y
∂
∂
−=
,
ϑ
i=1,2,3,4 12 equations / 12 unknowns
23. Formulation of Rectangular Plate Bending
Element
and, thus, generalized coordinates
a1-a12 can be evaluated…
24. Formulation of Rectangular Plate Bending
Element
For plate bending the strain tensor is
established in terms of the curvature
∂∂
∂
∂
∂
∂
∂
=
yx
w
y
w
x
w
xy
y
x
2
2
2
2
2
2
κ
κ
κ
=
xy
y
x
xy
y
x
M
M
M
κ
κ
κ
D
25. Formulation of Rectangular Plate Bending
Element
xyayaxaa
x
w
118742
2
6262 +++=
∂
∂
xyayaxaa
y
w
1210962
2
6622 +++=
∂
∂
33. Kirchhoff Shell Elements
However in both cases transformation to
Global CS is required
And a potential problem arises…
=
×
×
44
2020
2424
*
0
0
~
~
0
k
k
shell
x
shell
2020
*
2424
~
×
= TkTk shell
T
x
shell
Twisting DOF
34. Kirchhoff Shell Elements
… when adjacent elements are coplanar (or almost)
Singular Stiffness Matrix (or ill conditioned)
Zero Stiffness θz
36. 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)
37. 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.
38. 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
47. Strain Energy
Contributions from Plane Stress
[ ] dzdA
E
U
xy
y
x
A
h
h
xyyx
ps
−−
=
∫ ∫−
γ
ε
ε
ν
ν
ν
ν
γεε
2
1
00
01
01
12
1
2
2/
2/
48. Strain Energy
Contributions from Transverse Shear
[ ] ( )
dzdA
Ek
U
yz
xz
A
h
h
yzxz
ts
−
=
∫ ∫− γ
γ
ν
γγ
122
2/
2/
k is the correction factor for nonuniform stress
(see beam element)
49. Stiffness Matrix
Contributions from Plane Stress
[ ] dzdA
E
U
xy
y
x
A
h
h
xyyxps
−−
= ∫ ∫−
γ
ε
ε
ν
ν
ν
ν
γεε
2
1
00
01
01
12
1
2
2/
2/
[ ]∫
−−
=
A
xy
y
x
xyyxps dA
Eh
κ
κ
κ
ν
ν
ν
ν
κκκ
2
1
00
01
01
1 2
3
k
50. Stiffness Matrix
Contributions from Plane Stress
( )∫
−
∂
∂
−
∂
∂
−
−
∂
∂
−
∂
∂
=
A
y
x
yx
ts
dA
y
w
x
w
Ehk
y
w
x
w
β
β
ν
ββ
12
k
[ ] ( )
dzdA
Ek
U
yz
xz
A
h
h
yzxzts
−
= ∫ ∫− γ
γ
ν
γγ
122
2/
2/
52. 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
53. Interpolation of Field Variables
∑=
=
q
i
i
yiy N
1
ββ
∑=
=
q
i
i
xix N
1
ββ
∑=
=
q
i
iiwNw
1
Where q is the number
of nodes in the
element
Ni are the appropriate
shape functions
54.
55. Interpolation of Field Variables
In contrast to Kirchoff element, the same
shape functions are used for the
interpolation of deflections and rotations
(Co
continuity)
56. 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
57. Kirchhoff – Reissner/Mindlin Comparison
Kirchhoff:
Interpolated field variable is the deflection w
Reissner/Mindlin:
Interpolated field variables are
Deflection w
Section rotation βx
Section rotation βy
True Boundary Conditions are better represented
In addition to the more general nature of the
Reissner/Mindlin plate element note that
58. 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.
59. 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
60. Shear Locking & Reduced Integration
Kb correctly evaluated by quadrature
(Pure bending or twist)
Ks correctly evaluated by 1 point
quadrature only.
64. Shear Locking – Other Remedies
Mixed Interpolation of Tensorial Components
MITCn family of elements
To alleviate shear locking
Reissner/Mindlin formulation
Interpolation of w, β, and γ
Good mathematical basis, are reliable and efficient
Interpolation of w,β and γ is based on different order
71. 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)
72. 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
73. Planning an Analysis
Preliminary Analysis
Based on elementary theory, formulas from
handbooks, analytical work, or
experimental evidence
Know what to expect before FEA
77. Checking the Model
• Check Model prior to computation
• Undetected mistakes lead to:
– execution failure
– bizarre results
– Look right but are wrong
78. Common Mistakes
In general mistakes in modeling result from
insufficient familiarity with:
a) The physical problem
b) Element Behavior
c) Analysis Limitations
d) Software
79. Common Mistakes
Null Element Stiffness Matrix
Check for common multiplier (e.g. thickness)
Poisson’s ratio = 0.5
80. 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
81. Common Mistakes
Singular Stiffness Matrix (cont’d)
• Part of structure has buckled
• In nonlinear analysis, supports or connections
have reached zero stiffness
82. 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