ECIV 720 A Advanced Structural Mechanics and Analysis
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Transcript of ECIV 720 A Advanced Structural Mechanics and Analysis
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ECIV 720 A Advanced Structural
Mechanics and Analysis
Lecture 16 & 17: Higher Order Elements (review)3-D Volume ElementsConvergence RequirementsElement Quality
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Higher Order Elements
Quadrilateral Elements
Recall the 4-node
4321, aaaau
Complete Polynomial
4 generalized displacements ai
4 Boundary Conditions for admissible displacements
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Higher Order Elements
Quadrilateral Elements
29
28
227
26
25
432
1,
aaaaa
aaa
au
Assume Complete Quadratic Polynomial
9 generalized displacements ai
9 BC for admissible displacements
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9-node quadrilateral
9-nodes x 2dof/node = 18 dof
BT18x3 D3x3 B3x18
ke 18x18
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9-node element Shape Functions
Following the standard procedure the shape functions are derived as
1 2
34
4,3,2,14
1 iN iii
Corner Nodes
5
6
7
8
8,7,6,5
11
12
1 22
i
N
iiii
iii
Mid-Side Nodes9
Middle Node
911 22 iN i
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N1,2,3,4 Graphical Representation
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N5,6,7,8 Graphical Representation
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N9 Graphical Representation
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Polynomials & the Pascal Triangle n
nxaxyayaxaayx 3210, u
1
x y
x2 xy y2
x3 x2y xy2 y3
x4 x3y x2y2 xy3 y4
…….
x5 x4y x3y2 x2y3 xy4 y5
Degree
1
2
3
4
5
0
Pascal Triangle
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Polynomials & the Pascal Triangle
To construct a complete polynomial
1
x y
x2 xy y2
x3 x2y xy2 y3
x4 x3y x2y2 xy3 y4
…….
x5 x4y x3y2 x2y3 xy4 y5
etc
Q1
xyayaxaayx 3210, u
4-node QuadQ2
39
28
27
36
254
23
21
01
,
yaxyayxaxa
yaxyaxa
yaxa
a
yx
u
9-node Quad
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Incomplete Polynomials
1
x y
x2 xy y2
x3 x2y xy2 y3
x4 x3y x2y2 xy3 y4
…….
x5 x4y x3y2 x2y3 xy4 y5
yaxaayx 210, u
3-node triangular
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Incomplete Polynomials
1
x y
x2 xy y2
x3 x2y xy2 y3
x4 x3y x2y2 xy3 y4
…….
x5 x4y x3y2 x2y3 xy4 y5
27
26
254
23
21
01
,
xyayxa
yaxyaxa
yaxa
a
yx
u
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8-node quadrilateral
Assume interpolation
1 2
34
5
6
7
8
27
26
254
23
21
01
,
xyayxa
yaxyaxa
yaxa
a
yx
u
8 coefficients to determine for admissible displ.
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8-node quadrilateral
8-nodes x 2dof/node = 16 dof
BT16x3 D3x3 B3x16
ke 16x16
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8-node element Shape Functions
Following the standard procedure the shape functions are derived as
1 2
34
4,3,2,1
1114
1
i
N iiiii
Corner Nodes
5
6
7
8
8,7,6,5
112
1 22
i
N iiiii
Mid-Side Nodes
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N1,2,3,4 Graphical Representation
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N5,6,7,8 Graphical Representation
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Incomplete Polynomials
1
x y
x2 xy y2
x3 x2y xy2 y3
x4 x3y x2y2 xy3 y4
…….
x5 x4y x3y2 x2y3 xy4 y5
254
23
21
01
,
yaxyaxa
yaxa
a
yx
u
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6-node Triangular
Assume interpolation
1 2
3
4
56
254
23
21
01
,
yaxyaxa
yaxa
a
yx
u
6 coefficients to determine for admissible displ.
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6-node triangular
6-nodes x 2dof/node = 12 dof
BT12x3 D3x3 B3x12
ke 12x12
1 2
3
4
56
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6-node element Shape Functions
Following the standard procedure the shape functions are derived as
3,2,112 iLLN iii
Corner Nodes
1 2
3
214 4 LLN
Mid-Side Nodes
4
56
325 4 LLN
136 4 LLN Li:Area coordinates
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Other Higher Order Elements
1
x y
x2 xy y2
x3 x2y xy2 y3
x4 x3y x2y2 xy3 y4
…….
x5 x4y x3y2 x2y3 xy4 y5
12-node quad
1 2
34
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Other Higher Order Elements
x5 x4y x3y2 x2y3 xy4 y5
16-node quad1
x y
x2 xy y2
x3 x2y xy2 y3
x4 x3y x2y2 xy3 y4
……. x3y21 2
34
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3-D Stress state
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3-D Stress State
AssumptionSmall Deformations
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Strain Displacement Relationships
xy
xz
yz
z
y
x
xy
xz
yz
z
y
x
Material Matrix
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3-D Finite Element Analysis
Simplest Element (Lowest Order)
Tetrahedral Element
Solution Domain is VOLUME
12
3
45
6
78
91011
12
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3-D Tetrahedral Element
Parent
(Master)
1 (1,0,0)
2 (0,1,0)
4 (0,0,0)
3 (0,0,1)
Can be thought of an extension of the 2D CST
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3-D Tetrahedral
1
2
3
4
Shape Functions
1N
2N
3N
14N
Volume Coordinates
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Geometry – Isoparametric Formulation
44332211 xNxNxNxNx
44332211 yNyNyNyNy
44332211 zNzNzNzNz In view of shape functions
3424144 xxxxx 3424144 yyyyy
3424144 zzzzz
jiij xxx
jiij yyy
jiij zzz
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Jacobian of Transformation
343434
242424
141414
zyx
zyx
zyx
zyx
zyx
zyx
J
JJ det6
1det
1
0
1
0
1
0
dddVe
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Strain-Displacement Matrix
B is CONSTANT
x
N
z
Ny
N
z
Nx
N
y
Nz
Ny
Nx
N
B
ii
ii
ii
i
i
i
i
0
0
0
00
00
00
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Stiffness Matrix
ee V
T
V
Te dVdVU Dεεσε
2
1
2
1
Element Strain Energy
ee V
TT
V
TT dVdV DBqBqDBqBq2
1
2
1
eeTe
Te
TV qkqDBqBq2
1
2
1
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Force Terms
Body Forces
ddddVWP TT
V
Tf
e JfNqfu det
Tzzyxzyxe
e fffffffV
)112(4 f
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Element Forces
Surface Traction
1
2
3
4Applied on FACE of element
ee A
TT
A
TT dAdAWP TNqTu
Tzyxzyxzyxe TTTTTTTTTA
)112(123 0004
T
eg on face 123
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Stress Calculations
= DB qe
Stress Tensor
Constant
Stress Invariants
zyxI 1
2222 xyxzyzxzzyyxI
222
3 2
xyzxzyyzx
xyxzyzzyxI
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Stress Calculations
Principal Stresses
cos31
1 cI
3
2cos
31
2
cI
3
4cos
31
3
cI
32 ac
ac
b3cos
3
1 1
2
21
3I
Ia 3
21
3
1
332 I
IIIb
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Other Low Order Elements
1
2
3
4
5
6
18 dof 5-hedral
1
2
3
5
6
7
8
24 dof 6-hedral
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Degenerate Elements
1
2
3
5
6
7
8
1
2
4
5
6
8
,3
,7
Still has 24 dof
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Degenerate
1
2
3
5
6
7
8
1
2,3
4
5,6,7,8
Still has 24 dof
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Higher Order Elements 10-node 4-hedral
2
Z
X
Y
1
2
3
4
5 6
7
8
9103
4
6
810
1
9
7
1314
15
5
N L L i
N L L N L L N L L
N L L N L L N L L
i i i
2 1 1 3 4
4 4 4
4 4 45 1 2 6 2 3 7 3 1
8 1 4 9 2 4 10 3 4
,2, ,
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15-node 5-hedral
Z
X
Y
L1
L3
L2
1
2
3
4
5
6
7 8
9
10 11
12
13
14
15
3
2
4
6
8
10
1112
13
14
15
19
7
1314
15
5
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15-node 5-hedral Shape Functions
N L L L i
N L L L i p
N L L i p q
N L L i p q
N L i p
i i i i
i j j j
i p q
i p q
i p
1
22 1 1 1 1 3
1
22 1 1 1 4 5 6 1 3
2 1 7 8 9 1 3 2 31
2 1 101112 1 3 2 31
2 1 131415 1 3
2
2
2
,2,
, , ,2,
, , ,2, , ,
, , ,2, , ,
, , ,2,
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20-node 6-hedral
Z
X
Y
1
2
3
4
5
6
7
910
1112
13 14
1516
17
18
19
20
24
23
22
8
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20-node 6-hedral Shape Functions
20,19,18,171114
1
16,14,12,101114
1
15,13,11,91114
1
8,...,2,121118
1
2
2
2
iN
iN
iN
iN
iii
iii
iii
iiiiiii
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Convergence Considerations
For monotonic convergence of solution
Elements (mesh) must be compatible
Elements must be complete
Requirements
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Monotonic Convergence
Exact Solution
FEM Solution
No of Elements
For monotonic convergence the elements must beFor monotonic convergence the elements must be
complete and the mesh must be compatiblecomplete and the mesh must be compatible
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Mixed Order Elements
Consider the following Mesh
8-node
4-node
Incompatible Elements…
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Mixed Order Elements
We can derive a mixed order element for grading
8-node 4-node
By blending shape functions appropriately
7-node
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Convergence Considerations
For monotonic convergence of solution
Elements (mesh) must be compatible
Elements must be complete
Requirements
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Element Completeness
For an element to be complete
Assumption for displacement field
nnxaxyayaxaayx 3210, u
•RIGID BODY MOTION
•CONSTANT STRAIN STATE
must accommodate
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Element Completeness
Consider
yaxaayxu 321,
This is not a complete polynomial
However,
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Element Completeness
yaxaayxu 321, The Computed nodal displacement corresponding to this field
iii yaxaau 321 i=1,…,#of nodes
Test for ELEMENT completeness
Isoparametric Formulation
iiuNuNuNuNu 332211
Assume displacement field
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Element Completeness
iiiii yNaxNaNau 321
Isoparametric Formulation
iixNx ii yNy
Thus, computed displacement field
yaxaNau i 321
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Element Completeness
yaxaayxu 321, Assumed
yaxaNau i 321 Computed
In order for the computed displacements to be the
assumed ones we must satisfy
1 iN Condition for element completeness
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Effects of Element Distortion
Loss of predictive capability of isoparametric element
No Distortion
1
x y
x2 xy y2
x2y xy2
Behavior accurately predicted
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Effects of Element Distortion
Angular Distortion
1
x y
x2 xy y2
x2y xy2
Predictability is lost for all quadratic terms
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Effects of Element Distortion
Quadratic Curved Edge Distortion
1
x y
x2 xy y2
x2y xy2
Predictability is lost for all quadratic terms
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Effects of Element Distortion
The advantage (reduced #of dof)
of using 8-node higher order element
based on an incomplete polynomial is lost
when high element distortions are present
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Effects of Element Distortion
Loss of predictive capability of isoparametric element
No Distortion
1
x y
x2 xy y2
x2y xy2
Behavior accurately predicted
x2y2
9-node
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Effects of Element Distortion
Angular Distortion
1
x y
x2 xy y2
x2y xy2
9-node
Behavior predicted better than 8-node
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Effects of Element Distortion
Quadratic Curved Edge Distortion
1
x y
x2 xy y2
x2y xy2
Predictability is lost for high order terms
9-node
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Effects of Element Distortion
The advantage (reduced #of dof)
of using higher order element
based on an incomplete polynomial is lost
when high element distortions are present
For angular distortion 9-node element shows better behavior
For Curved edge distortion all elements give low order prediction
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Polynomial Element Predictability
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Tests of Element Quality
Eigenvalue TestIdentify Element Deficiencies
Patch TestConvergence of Solutions
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Eigenvalue Test
1
2
3
4
Apply loads –{r} in proportion to displacements
dr
rdk
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Eigenvalue Test
dr
rdk 0dIk
Eigenproblem
As many eigenvalues as dof
For each there is a solution for {d}
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Displacement Modes & Stiffness Matrix
nn
n
n
nn d
d
d
d
d
d
d
d
d
2
1
2
22
12
1
21
11
D
n
2
1
Λ
DΛKD
For all eigenvalues and modes
dKd
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Eigenvalue Test
Scale {d} so that
1dd T
U
TT
2 dddkd
then
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Eigenvalue Test
U
TT
2 dddkd
Rigid Body Motion => System is not strained => U=0
System is strained => U=0
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Rigid Body Motion
Rigid Body Modes
Total Number of Element Displacement Modes
(=number of degrees of freedom)
Element Straining Modes+
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Displacement Modes & Stiffness Matrix
Consider the 2-node axial element
11
11
L
AEK
Identify all possible modes of displacement
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Displacement Modes & Stiffness Matrix
t=1
E=1
v=0.3
8 degrees of freedom 8 modes
1
1
Consider the 4-node plane stress element
Solve Eigenproblem
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Displacement Modes & Stiffness Matrix
01
Rigid Body Mode 02
Rigid Body Mode
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Displacement Modes & Stiffness Matrix
03
Rigid Body Mode
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Displacement Modes & Stiffness Matrix
495.05
Flexural Mode
495.04 Flexural Mode
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Displacement Modes & Stiffness Matrix
769.06 Shear Mode
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Displacement Modes & Stiffness Matrix
769.07 Stretching Mode
43.18 Uniform Extension Mode
(breathing)
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Displacement Modes & Stiffness Matrix
The eigenvalues of the stiffness matrix display directly how stiff the element is in the corresponding displacement mode
U2
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Patch Test
Objective
Examine solution convergence for displacements, stresses and strains in a particular element type with mesh refinement
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Patch Test - Procedure
Build a simple FE model
Consists of a Patch of Elements
At least one internal node
Load by nodal equivalent forces consistent with state of constant stress
Internal Node is unloaded and unsupported
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Patch Test - Procedure
HtF x2
1
Compute results of FE patch
If
(computed x) = (assumed x)
test passed