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Transcript of ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric...
![Page 1: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/1.jpg)
ECIV 720 A Advanced Structural
Mechanics and Analysis
Lecture 15: Quadrilateral Isoparametric Elements
(cont’d)Force VectorsModeling Issues
Higher Order Elements
![Page 2: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/2.jpg)
Integration of Stiffness Matrix
1
1
1
1
det dJdt Te DBBk
B (3x8)
D (3x3)
BT(8x3)
ke (8x8)
![Page 3: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/3.jpg)
Integration of Stiffness Matrix
Each term kij in ke is expressed as
1
1
1
1
1
1
1
1
3
1
3
1
,
,det
ddgt
ddJBDBtkm l
ljmlTimij
Linear Shape Functions is each Direction
Gaussian Quadrature is accurate if
We use 2 Points in each direction
![Page 4: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/4.jpg)
Integration of Stiffness Matrix
311 312
311
311
2222212112121111 ,,,, gwwgwwgwwgww
1
1
1
1
, ddgt
11 w
12 w
22211211 ,,,, gggg
![Page 5: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/5.jpg)
Choices in Numerical Integration
• Numerical Integration cannot produce exact results
• Accuracy of Integration is increased by using more integration points.
• Accuracy of computed FE solution DOES NOT necessarily increase by using more integration points.
![Page 6: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/6.jpg)
FULL Integration
• A quadrature rule of sufficient accuracy to exactly integrate all stiffness coefficients kij
• e.g. 2-point Gauss ruleexact for
polynomials
up to 2nd order
311 312
311
311
311 312
311
311
![Page 7: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/7.jpg)
Reduced Integration, Underintegration
Use of an integration rule of less than full order
Advantages
• Reduced Computation Times
• May improve accuracy of FE results
• Stabilization
Disadvantages
Spurious Modes (No resistance to nodal loads that tend to activate the mode)
![Page 8: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/8.jpg)
Spurious Modes
t=1
E=1
v=0.3
8 degrees of freedom 8 modes
1
1
Consider the 4-node plane stress element
Solve Eigenproblem
![Page 9: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/9.jpg)
Spurious Modes
01
Rigid Body Mode 02
Rigid Body Mode
![Page 10: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/10.jpg)
Spurious Modes
03
Rigid Body Mode
![Page 11: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/11.jpg)
Spurious Modes
495.05
Flexural Mode
495.04 Flexural Mode
![Page 12: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/12.jpg)
Spurious Modes
769.06 Shear Mode
![Page 13: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/13.jpg)
Spurious Modes
769.07 Stretching Mode
43.18 Uniform Extension Mode
(breathing)
![Page 14: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/14.jpg)
Element Body Forces
ii
Ti
el
T
eA
T
eA
T
e
e
e
tdl
tdA
tdA
Pu
Tu
fu
Dεε2
1
ii
Ti
eA
T
eV
T
eV
T
e
e
e
dA
dV
dV
Pφ
Tφ
fφ
φεσ 0
Total Potential Galerkin
![Page 15: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/15.jpg)
Body Forces
eA
T dAd detfu
Integral of the form
8
1
4321
4321
0000
0000
q
q
NNNN
NNNN
v
u
8
1
4321
4321
0000
0000
NNNN
NNNN
y
x
eA
T dAd detfφ
![Page 16: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/16.jpg)
Body Forces
In both approaches
e
e
e
e
Ay
Ax
Ay
Ax
dAdNf
dAdNf
dAdNf
dAdNf
qqWP
det
det
det
det
4
2
1
1
81
Linear Shape Functions
Use same quadrature as stiffness maitrx
![Page 17: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/17.jpg)
Element Traction
ii
Ti
el
T
eA
T
eA
T
e
e
e
tdl
tdA
tdA
Pu
Tu
fu
Dεε2
1
ii
Ti
eA
T
eV
T
eV
T
e
e
e
dA
dV
dV
Pφ
Tφ
fφ
φεσ 0
Total Potential Galerkin
![Page 18: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/18.jpg)
Element Traction
Similarly to triangles, traction is applied along sides of element
4
2
3
Tx
Ty
u
v
1
4
0
12
1
12
1
0
4
3
2
1
N
N
N
N
el
TT tdlWP Tu
32
![Page 19: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/19.jpg)
Traction
8
1
32
32
000000
000000
q
q
NN
NN
v
u
0
0
0
0
232
8132
y
x
eT T
Tlt
qqWP
For constant traction along side 2-3
Traction
components
along 2-3
![Page 20: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/20.jpg)
Stresses
311 312
311
311
DBqσ
12221121
1121
1222
00
00
det
1
JJJJ
JJ
JJ
JA
10101010
10101010
01010101
01010101
4
1G
More Accurate at
Integration points
Stresses are calculated at any
![Page 21: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/21.jpg)
Modeling Issues: Nodal Forces
ii
Ti
el
T
eA
T
eA
T
e
e
e
tdl
tdA
tdA
Pu
Tu
fu
Dεε2
1
A node should be
placed at the location
of nodal forces
In view of…
Or virtual potential energy
![Page 22: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/22.jpg)
Modeling Issues: Element Shape
Square : Optimum Shape
Not always possible to use
Rectangles:
Rule of Thumb
Ratio of sides <2
Angular Distortion
Internal Angle < 180o
Larger ratios
may be used
with caution
![Page 23: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/23.jpg)
Modeling Issues: Degenerate Quadrilaterals
Coincident Corner Nodes
1
2
3
4
32
1
4
xx
xx
x
xx
x
Integration Bias
Less accurate
![Page 24: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/24.jpg)
Modeling Issues: Degenerate Quadrilaterals
Three nodes collinear
1
2
3
4
xx
xx
1
2
3
4 x
xx
x
Less accurate
Integration Bias
![Page 25: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/25.jpg)
Modeling Issues: Degenerate Quadrilaterals
Use only as necessary to improve representation of geometry
2 nodes
Do not use in place of triangular elements
![Page 26: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/26.jpg)
A NoNo Situation
3
4
1 2
x
y
(3,2) (9,2)
(7,9)
(6,4)
Parent
All interior angles < 180
J singular
![Page 27: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/27.jpg)
Another NoNo Situation
x, y
not uniquely
defined
![Page 28: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/28.jpg)
FEM at a glance
It should be clear by now that the cornerstone in FEM procedures is the interpolation of the displacement field from discrete values
i
m
ni zyxzyx uNu
1
,,,,
Where m is the number of nodes that define the interpolation and the finite element and N is a set of Shape Functions
![Page 29: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/29.jpg)
FEM at a glance
1=-1 2=1
m=2
1=-1
1
3
2=1
2m=3
![Page 30: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/30.jpg)
FEM at a glance
12
3q6
q5
q4
q3
q2
q1
vu
m=3
4
1
2
3
m=4
![Page 31: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/31.jpg)
FEM at a glance
In order to derive the shape functions it was assumed that the displacement field is a polynomial of any degree, for all cases considered
nn xaxaxaax 2
210u
..., 3210 xyayaxaayxu
Coefficients ai represent generalized coordinates
1-D
2-D
![Page 32: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/32.jpg)
FEM at a glance
For the assumed displacement field to be admissible we must enforce as many boundary conditions as the number of polynomial coefficients
1=-1
1
3
2=1
2e.g.
12121101 uxaxaaxx u
22222102 uxaxaaxx u
32323103 uxaxaaxx u
![Page 33: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/33.jpg)
FEM at a glance
This yields a system of as many equations as the number of generalized displacements
nn u
u
a
a
zyx
Matrix
tCeofficien
10
),,(
nn u
u
C
a
a
1
10
that can be solved for ai
![Page 34: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/34.jpg)
FEM at a glance
nn xaxaxaax 2
210u
Substituting ai in the assumed displacement field
and rearranging terms…
i
m
ni uxNxu
1
![Page 35: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/35.jpg)
FEM at a glance
113N
12
11N 1
2
12N
u(-1)=a0 -a1 +a2 =u1
u(1)=a0 +a1 +a2 =u2
u(0)=a0 =u3
33
22
11
uN
uN
uNu
…
u()=a0+a1 +a2 2
1=-1
1
2=1
3 2
![Page 36: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/36.jpg)
Let’s go through the exercise
x1
1
x2
2
220 xaaxu
Assume an incomplete form of quadratic variation
![Page 37: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/37.jpg)
Incomplete form of quadratic variation
We must satisfy
12
1201 uxaaxu
22
2202 uxaaxu
2
1
1
0
22
21
1
1
u
u
a
a
x
x
x1
1
x2
2
![Page 38: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/38.jpg)
Incomplete form of quadratic variation
2
1
1
0
22
21
1
1
u
u
a
a
x
x
And thus,
2
121
22
21
221
0
11
1
u
uxx
xxa
a
![Page 39: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/39.jpg)
Incomplete form of quadratic variation
21
22
2211
22
0 xx
uxuxa
21
22
211 xx
uua
220 xaaxu
And substituting in
221
22
2121
22
2211
22 x
xx
uu
xx
uxuxxu
![Page 40: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/40.jpg)
Incomplete form of quadratic variation
221
22
2121
22
2211
22 x
xx
uu
xx
uxuxxu
2
1
21
22
221
21
22
222
u
u
xx
xx
xx
xxxu
Which can be cast in matrix form as
![Page 41: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/41.jpg)
Isoparametric Formulation
The shape functions derived for the interpolation of the displacement field are used to interpolate geometry
2
1
21
22
221
21
22
222
x
x
xx
xx
xx
xxx
x1
1
x2
2
![Page 42: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/42.jpg)
Intrinsic Coordinate Systems
Intrinsic coordinate systems are introduced to eliminate dependency of Shape functions from geometry
1 (-1,-1) 2 (1,-1)
4 (-1,1) 3 (1,1)
The price?
Jacobian of transformation
iiiN 114
1
Great Advantage for the money!
![Page 43: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/43.jpg)
Field Variables in Discrete Form
nNxx
Geometry
Displacement
nNuu
= DB un
Stress Tensor
= B un
Strain Tensor
)(intrinsic
)(cartesianJ
![Page 44: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/44.jpg)
FEM at a glance
eeTeV
Te
e
dVU ukuσDε2
1
2
1
Element Strain Energy
ee V
TTeV
Tf dVdVW fNufu
Work Potential of Body Force
ee S
TTeS
Tf dSdSW TNuTu
Work Potential of Surface Traction
etc
![Page 45: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/45.jpg)
Higher Order Elements
Quadrilateral Elements
Recall the 4-node
4321, aaaau
Complete Polynomial
4 generalized displacements ai
4 Boundary Conditions for admissible displacements
![Page 46: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/46.jpg)
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
![Page 47: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/47.jpg)
9-node quadrilateral
9-nodes x 2dof/node = 18 dof
BT18x3 D3x3 B3x18
ke 18x18
![Page 48: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/48.jpg)
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
![Page 49: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/49.jpg)
9-node element – Shape Functions
113N
12
11N 1
2
12N
Can also be derived from the 3-node axial element
1=-1
1
2=1
3 2
![Page 50: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/50.jpg)
Construction of Lagrange Shape Functions
(1,)(1,1)
1 (-1,-1)
12
11N
12
11N
1
2
11
2
1, 111 NNN
4,3,2,14
1 iN iii
![Page 51: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/51.jpg)
N1,2,3,4 Graphical Representation
![Page 52: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/52.jpg)
N5,6,7,8 Graphical Representation
![Page 53: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/53.jpg)
N9 Graphical Representation
![Page 54: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/54.jpg)
Polynomials & the Pascal Triangle n
n xaxyayaxaayx 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
![Page 55: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/55.jpg)
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
![Page 56: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/56.jpg)
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
![Page 57: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/57.jpg)
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
![Page 58: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/58.jpg)
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.
![Page 59: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/59.jpg)
8-node quadrilateral
8-nodes x 2dof/node = 16 dof
BT16x3 D3x3 B3x16
ke 16x16
![Page 60: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/60.jpg)
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
![Page 61: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/61.jpg)
N1,2,3,4 Graphical Representation
![Page 62: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/62.jpg)
N5,6,7,8 Graphical Representation
![Page 63: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/63.jpg)
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
![Page 64: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/64.jpg)
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.
![Page 65: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/65.jpg)
6-node triangular
6-nodes x 2dof/node = 12 dof
BT12x3 D3x3 B3x12
ke 12x12
1 2
3
4
56
![Page 66: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/66.jpg)
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
![Page 67: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/67.jpg)
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
![Page 68: ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order.](https://reader035.fdocuments.in/reader035/viewer/2022062216/56649d1b5503460f949f1151/html5/thumbnails/68.jpg)
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