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Procedures of Finite Element AnalysisTwo-Dimensional Elasticity Problems
Professor M. H. Sadd
Two Dimensional Elasticity Element EquationOrthotropic Plane Strain/Stress Derivation Using Weak Form – Ritz/Galerin Scheme
Displacement Formulation Orthotropic Case
0
0
221266
661211
y
x
Fy
vC
x
uC
yx
v
y
uC
x
Fx
v
y
uC
yy
vC
x
uC
x
strain plane and stress plane)1(2
,strain plane
)2-)(1(1
E
stress plane1
strain plane)2-)(1(1
)-E(1
stress plane1
66
2
12
2
2211
EC
E
C
E
CC
Material Isotropic
xyxy
yxy
yxx
eC
eCeC
eCeC
66
2212
1211
Law sHooke'
Two Dimensional Elasticity Weak Form
0
0
2212662
6612111
dxdyFy
vC
x
uC
yx
v
y
uC
xwh
dxdyFx
v
y
uC
yy
vC
x
uC
xwh
e
e
ye
xe
Mulitply Each Field Equation by Test Function & Integrate Over Element
Use Divergence Theorem to Trade Differentiation On To Test Function
yxyyxx
yeyee
xexee
ny
vC
x
uCn
x
v
y
uCTn
x
v
y
uCn
y
vC
x
uCT
dsTwhdxdyFwhdxdyy
vC
x
uC
y
w
x
v
y
uC
x
wh
dsTwhdxdyFwhdxdyx
v
y
uC
y
w
y
vC
x
uC
x
wh
eee
eee
221166661211
2222122
662
11661
12111
,
(constant) icknesselement theh
Two Dimensional Elasticity Ritz-Galerkin Method
ee
ee
e
e
e
dsThdxdyFhFdsThdxdyFhF
dxdyyy
Cxx
ChK
dxdyxy
Cyx
ChKK
dxdyyy
Cxx
ChK
FUKF
F
v
u
KK
KK
yieyieixiexiei
jijieij
jijieijij
jijieij
T
21
226622
66122112
661111
2
1
2212
1211
,
where
}{}]{[}{
}{
}{
}{
][][
][][
formsin weak ,,Let 11
21
N
jjj
N
jjji vvuuww
Two Dimensional Elasticity Element EquationTriangular Element N = 3
23
13
22
12
21
11
3
3
2
2
1
1
)(66
)(56
)(55
)(46
)(45
)(44
)(36
)(35
)(34
)(33
)(26
)(25
)(24
)(23
)(22
)(16
)(15
)(14
)(13
)(12
)(11
23
13
22
12
21
11
3
3
2
2
1
1
2233
1233
1133
2223
2123
2222
1223
1123
1222
1122
2213
2113
2212
2112
2211
1213
1113
1212
1112
1211
1111
2
1
2212
1211
}{
}{
}{
}{
][][
][][
F
F
F
F
F
F
v
u
v
u
v
u
K
KK
KKK
KKKK
KKKKK
KKKKKK
F
F
F
F
F
F
v
u
v
u
v
u
K
KK
KKK
KKKK
KKKKK
KKKKKK
F
F
v
u
KK
KK
e
ee
eee
eeee
eeeee
eeeeee
T
12
3
e
(x1,y1)(x2,y2)
(x3,y3)
Two Dimensional Elasticity Element EquationPlane Strain/Stress Derivation Using Virtual Work Statement
dVuFdSuTdVe iV iS in
iijV ijeee
Statement Workrtual Vi
0)()(
)2(
ee
e
dxdyvFuFhdsvTuTh
dxdyeeeh
yxen
yn
xe
xyxyyyxxe
0
2
eee
dxdyF
F
v
uhds
T
T
v
uhdxdy
e
e
e
hy
xT
eny
nx
T
e
xy
y
x
T
xy
y
x
e
12
3
x
y
e
(Element Geometry)
e = 12 + 23 + 31
(x1,y1)(x2,y2)
(x3,y3)
he = thickness
eee hV
eee hS
Two Dimensional Elasticity Element EquationInterpolation Scheme
iii
iii
yxvyxv
yxuyxu
),(),(
),(),(}]{[ d
v
u
}]{[}]{[
//
/0
0/
//
/0
0/
2
}{ dBde
xy
y
x
v
u
xy
y
x
e
e
e
xy
y
x
}]{][[}]{[}{ dBCeC
66
2212
1211
00
0
0
][
C
CC
CC
C
Material cOrthotropi General
strain plane and stress plane)1(2
strain plane)2-)(1(1
E
stress plane1
strain plane)2-)(1(1
)-E(1
stress plane1
66
2
12
2
2211
EC
E
C
E
CC
Material Isotropic
Two Dimensional Elasticity Element Equation
eee
dsT
Thdxdy
F
Fhdxdyh n
y
nxTT
ey
xTTe
TTe ][}{][}{}]){][[]([}{ ψδdψδddBCBδd
Statement Workirtual V
}{}{}]{[ QFdK Equation Element
VectorLoading][}{
VectorForceBody][}{
MatrixStiffness]][[][][
e
e
e
dsT
Th
dxdyF
Fh
dxdyh
ny
nxT
e
y
xTe
Te
ψQ
ψF
BCBK
Triangular Element With Linear Approximation
1
2
3
1
1
1
2
3
1
2
1
2
3
13
Lagrange Interpolation Functions
x
y
(x1,y1)
(x2,y2)
(x3,y3)
u1
u2
1
2
3u3
v1
v2
v3
ycxccyxu 321),(
33321333
23221222
13121111
),(
),(
),(
ycxccuyxu
ycxccuyxu
ycxccuyxu
3
1332211 ),(),(),(),(),(
iii yxuyxuyxuyxuyxu
)(2
1),( yx
Ayx iii
ei
jkkji yxyx kji yy jki xx
1,),(3
1
i
iijjji yx
Triangular Element With Linear Approximation
}]{[000
000
3
3
2
2
1
1
321
321 dψ
v
u
v
u
v
u
v
u
}]{[}]{[
//
/0
0/
//
/0
0/
2
}{
dBdψ
e
xy
y
x
v
u
xy
y
x
e
e
e
xy
y
x
332211
321
321
332211
321
321
000
000
2
1000
000
][eA
xyxyxy
yyy
xxx
B
]][[][][ BCBK Tee Ah
Matrix Stiffness
662322
23
66331233662311
23
6632223266321223662222
22
662312326632113266221222662211
22
66312231663112136621222166211212662122
21
6613123166311131661212216621112166111211662111
21
4][
CC
CCCC
CCCCCC
CCCCCCCC
CCCCCCCCCC
CCCCCCCCCCCC
A
h
e
eK
Loading Terms for Triangular Element With Uniform Distribution
12
3
x
y
e
(Element Geometry)
e = 12 + 23 + 31
(x1,y1)(x2,y2)
(x3,y3)
he = thickness
eee hV
eee hS
Tyxyxyx
ee FFFFFFAh
F }{3
}{
312312
][][][
][}{
dsT
Thds
T
Thds
T
Th
dsT
Th
ny
nxT
eny
nxT
eny
nxT
e
ny
nxT
e
ψψψ
ψQ
12
12
3
3
2
2
1
1
0
0
2][
1212
ny
nx
ny
nx
e
ny
nx
ny
nx
ny
nx
eny
nxT
eT
T
T
T
Lhds
T
T
T
T
T
T
hdsT
Th ψ
31
31
23
23
0
0
2][,
0
0
2][
3123
ny
nx
ny
nx
eny
nxT
e
ny
nx
ny
nxe
ny
nxT
e
T
T
T
T
Lhds
T
Th
T
T
T
TLhds
T
Th ψψ
Rectangular Element Interpolation
44332211
4321
),(),(),(),(
),(
uyxuyxuyxuyx
xycycxccyxu
onpproximatiBilinear A
b
y
a
xb
y
a
x
b
y
a
x
b
y
a
x
1
1
11
4
3
2
1
Functions ionInterpolat
1 2
34
a
b
x
y
Two Dimensional Elasticity Element EquationRectangular Element N = 4
}{
}{
}{
}{
][][
][][2
1
2212
1211
F
F
v
u
KK
KKT
(x3,y3)
1 2
3
e
(x1,y1) (x2,y2)
4 (x4,y4)
24
14
23
13
22
12
21
11
4
4
3
3
2
2
1
1
2244
1244
1144
2234
2134
2233
1234
1134
1233
1133
2224
2124
2223
2123
2222
1224
1124
1223
1123
1222
1122
2214
2114
2213
2113
2212
2112
2211
1214
1114
1213
1113
1212
1112
1211
1111
F
F
F
F
F
F
F
F
v
u
v
u
v
u
v
u
K
KK
KKK
KKKK
KKKKK
KKKKKK
KKKKKKK
KKKKKKKK
Rectangular Element With BiLinear Approximation
}]{[0
0
000
000
4
4
3
3
2
2
1
1
4
4
321
321 dψ
v
u
vu
v
u
v
u
v
u
ab
y
a
x
bab
y
ab
x
b
y
aab
x
b
y
aa
x
b
a
x
bab
x
ab
x
a
x
b
ab
y
ab
y
b
y
ab
y
a
xyxyxyxy
yyyy
xxxx
xy
y
x
11
11
11
11
11
00011
0
00011
011
0000
0000
][
//
/0
0/
][
44332211
4321
4321
ψB
Two Dimensional Elasticity Rectangular Element Equation - Orthotropic Case
(x3,y3)
1 2
3
e
(x1,y1) (x2,y2)
4
(x4,y4)
24
14
23
13
22
12
21
11
4
4
3
3
2
2
1
1
2244
1244
1144
2234
2134
2233
1234
1134
1233
1133
2224
2124
2223
2123
2222
1224
1124
1223
1123
1222
1122
2214
2114
2213
2113
2212
2112
2211
66126611661266116612661166126611 4
12
6
1
4
1
6
1
4
12
6
1
4
1
3
1
F
F
F
F
F
F
F
F
v
u
v
u
v
u
v
u
K
KK
KKK
KKKK
KKKKK
KKKKKK
KKKKKKK
CCCb
aC
a
bCCCC
a
bCCC
b
aC
a
bCCC
b
aC
a
b
66
2212
1211
00
0
0
][
C
CC
CC
C
Material cOrthotropi General
]][[][][ BCBK Tee Ah
Matrix Stiffness
FEA of Elastic 1x1 Plate Under Uniform Tension
x
T
3
21
y
4
33
2
2
11
1
2
)1(3
)1(3
)1(2
)1(2
)1(1
)1(1
)1(3
)1(3
)1(2
)1(2
)1(1
)1(1
2
1
02
1
12
1
2
32
1
2
1
2
3
02
1
2
1
2
1
2
10101
)1(2
y
x
y
x
y
x
T
T
T
T
T
T
v
u
v
u
v
u
E
Element 1: 1 = -1, 2 = 1, 3 = 0, 1 = 0, 2 = -1, 3 = 1, A1 = ½.
Element 2: 1 = 0, 2 = 1, 3 = -1, 1 = -1, 2 = 0, 3 = 1, A1 = ½
)2(3
)2(3
)2(2
)2(2
)2(1
)2(1
)2(3
)2(3
)2(2
)2(2
)2(1
)2(1
2
2
32
1
2
32
1
2
1
2
1101
1012
1
2
1
2
100
2
1
)1(2
y
x
y
x
y
x
T
T
T
T
T
T
v
u
v
u
v
u
E
FEA of Elastic Plate
)2(3
)2(3
)2(2
)1(3
)2(2
)1(3
)1(2
)1(2
)2(1
)1(1
)2(1
)1(1
4
4
3
3
2
2
1
1
)2(66
)2(56
)2(55
)2(46
)2(45
)2(44
)1(66
)2(36
)2(35
)2(34
)1(56
)2(33
)1(55
)1(46
)1(45
)1(44
)1(36
)1(35
)1(34
)1(33
)1(26
)1(25
)2(24
)1(26
)2(23
)1(25
)1(24
)1(23
)2(22
)1(22
)1(16
)1(15
)2(14
)1(16
)2(13
)1(15
)1(14
)1(13
)2(12
)1(12
)2(11
)1(11
00
00
y
x
yy
xx
y
x
yy
xx
T
T
TT
TT
T
T
TT
TT
V
U
V
U
V
U
V
U
K
KK
KKKK
KKKKKK
KKK
KKKK
KKKKKKKKKK
KKKKKKKKKKKK
System Global Assembled
Boundary Conditions U1 = V1 = U4 = V4 = 0 0,2/,0,2/ )2(
2)1(
3)2(
2)1(
3)1(
2)1(
2 yyxxyx TTTTTTTT
Condtions Loading
0
2/
0
2/
3
3
2
2
)2(44
)1(66
)2(34
)1(56
)2(33
)1(55
)1(46
)1(45
)1(44
)1(36
)1(35
)1(34
)1(33
T
T
V
U
V
U
KK
KKKK
KKK
KKKK
System Reduced
T
3
21
4
33
2
2
1
1
2
1
T
3
21
4
33
2
2
1
1
2
1
Solution of Elastic Plate Problem
mT
V
U
V
U
11
3
3
2
2
10
030.0
441.0
081.0
492.0
Choose Material Properties: E = 207GPa and v = 0.25
Note the lack of symmetry in the displacement solution
Axisymmetric Formulation
}]{[0000
0000
0
01
0
0
01
0
2
}
4
4
3
3
2
2
1
1
4321
4321 dBe
v
u
v
u
v
u
v
u
NNNN
NNNN
rz
z
r
r
u
u
rz
z
r
r
e
e
e
e
z
r
rz
z
r
{
ntDisplaceme-Strain
constant plane
z
r
1 2
4 3
44332211
44332211
vNvNvNvNu
uNuNuNuNu
z
r
Element Quad
Axisymmetric Formulation
}]{][[}]{[
22
21000
01
01
01
)21)(1(} dBCeCσ
rz
z
r
rz
z
r
e
e
e
e
E{
Relations Strain-Stress
rdrdze
T
]][[][][ BCBK
Matrix Stiffness
}{}{}]{[ QFdK Equation Element
4
4
3
3
2
2
1
1
44332211
4321
4321
4321
0000
0000
0000
}]{[
2
}
v
u
v
u
v
u
v
u
r
N
z
N
r
N
z
N
r
N
z
N
r
N
z
Nz
N
z
N
z
N
z
Nr
N
r
N
r
N
r
Nr
N
r
N
r
N
r
N
e
e
e
e
rz
z
r
dBe{
ntDisplaceme-Strain
Two-Dimensional FEA Code MATLAB PDE Toolbox
- Simple Application Package For Two-Dimensional Analysis Initiated by Typing “pdetool” in Main MATLAB Window
- Includes a Graphical User Interface (GUI) to: - Select Problem Type - Select Material Constants - Draw Geometry - Input Boundary Conditions - Mesh Domain Under Study - Solve Problem - Output Selected Results
Two-Dimensional FEA ExampleUsing MATLAB PDE Toolbox
Cantilever Beam Problem
L = 2
g1=0
g2=1002c = 0.4
Mesh: 4864 Elements, 2537 Nodes
L/2c = 5
FEA MATLAB PDE Toolbox ExampleCantilever Beam Problem
Stress Results
L = 2
g1=0
g2=100
E = 10x106 , v = 0.3
300012/)4.0)(1(
)2.0)(2)(40(3max
I
Mc
2c = 0.4
Contours of sx
FEA Result: smax = 3200
FEA MATLAB PDE Toolbox ExampleCantilever Beam Problem
Displacement Results
L = 2
g1=0
g2=100
E = 10x106 , v = 0.3Contours of Vertical Displacement v
002.012/)4.0)(1)(10)(3(
)2)(40(
3 37
33
max EI
PLv
FEA Result: vmax = 0.00204
2c = 0.4
Two-Dimensional FEA ExampleUsing MATLAB PDE Toolbox
Plate With Circular Hole
Contours of Horizontal Stress x Stress Concentration Factor: K 2.7
Theoretical Value: K = 3
Contours of Horizontal Stress x Stress Concentration Factor: K 3.5
Theoretical Value: K = 4
Two-Dimensional FEA ExampleUsing MATLAB PDE Toolbox
Plate With Circular Hole
FEA MATLAB ExamplePlate with Elliptical Hole
(Finite Element Mesh: 3488 Elements, 1832 Nodes)
(Contours of Horizontal Stress x)
Stress Concentration Factor K 3.3
Theoretical Value: K = 5
Aspect Ratio b/a = 2
FEA ExampleDiametrical Compression of Circular Disk
(FEM Mesh: 1112 Elements, 539 Nodes) (Contours of Max Shear Stress)
(FEM Mesh: 4448 Elements, 2297 Nodes) (Contours of Max Shear Stress)
Theoretical Contours of Maximum Shear Stress
Experimental Photoelasticity Isochromatic Contours
