2004 march, 4 page 1 finite element analysis basics – part 2/2 johannes steinschaden
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2004 March, 4 Page 1
Finite Element AnalysisBasics – Part 2/2
Johannes Steinschaden
2004 March, 4 Page 2
Finite Element Analysis Procedure
1. Preliminary analysis of the system:Perform an approximate calculation to gain some insights about the system
2. Preparation of the finite element model:a Geometric and material information of the systemb Prescribe how is the system supportedc Determine how the loads are applied to the system
3. Perform the calculation:Solve the system equations and compute displacements, strains and stresses
4. Post-processing of the results:Viewing the stresses and displacementsInterpret the results
2004 March, 4 Page 3
Direct Stiffness MethodTwo-dimensional Truss Elements
F1 F2K=EA/L
N1 N2
x1 x2
2
1
2
1
2
1
2
1
11
11
x
x
L
EA
F
F
x
x
KK
KK
F
F
2004 March, 4 Page 4
ad Two-dimensional Truss Elements
X
Y
F2
F1
K
N1
N2
Y1X1
Y2 X2
2
2
1
1
2
2
1
1
0000
0101
0000
0101
Y
X
Y
X
L
EA
F
F
F
F
Y
X
Y
X
local stiffness matrix
2004 March, 4 Page 5
ad Two-dimensional Truss Elements
Coordinate transformation equation
Yg
Xg
Yg
Xg
Y
X
Y
X
g
g
g
g
g
g
F
F
F
F
cs
sc
cs
sc
F
F
F
F
Y
X
Y
X
cs
sc
cs
sc
Y
X
Y
X
Y
X
cs
sc
Y
X
c
s
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
1
1
1
1
00
00
00
00
00
00
00
00
cos
sin
2004 March, 4 Page 6
ad Two-dimensional Truss Elements
lKlK
XKF
XlKlF
ll
XlKlF
XlKlFll
XlKFl
XKF
tg
gg
gt
t
g
g
g
g
1
1
11
2004 March, 4 Page 7
ad Two-dimensional Truss Elements
22
22
22
22
sscssc
sccscc
sscssc
sccscc
L
EAKg
2004 March, 4 Page 8
ad Two-dimensional Truss ElementsExample
Truss ATruss B
Node 1 Node 3
Node 2
F
2004 March, 4 Page 9
ad Two-dimensional Truss ElementsExample
Element A:
local stiffness matrix
global stiffness matrix
2
2
1
1
22
22
22
22
2
2
1
1
Y
X
Y
X
sscssc
sccscc
sscssc
sccscc
L
EA
F
F
F
F
g
g
g
g
A
A
YAg
XAg
Yg
Xg
2
2
1
1
2
2
1
1
0000
0101
0000
0101
Y
X
Y
X
L
EA
F
F
F
F
A
A
Y
X
Y
X
2004 March, 4 Page 10
ad Two-dimensional Truss ElementsExample
Element B:
local stiffness matrix
global stiffness matrix
3
3
2
2
22
22
22
22
3
3
2
2
Y
X
Y
X
sscssc
sccscc
sscssc
sccscc
L
EA
F
F
F
F
g
g
g
g
B
B
Yg
Xg
YBg
XBg
3
3
2
2
3
3
2
2
0000
0101
0000
0101
Y
X
Y
X
L
EA
F
F
F
F
B
B
Y
X
Y
X
2004 March, 4 Page 11
ad Two-dimensional Truss ElementsExample
Summing the two seta of global force-displacement equations:
3
3
2
2
1
1
22
22
2222
2222
22
22
3
3
2
2
1
1
00
00
00
00
Y
X
YX
Y
X
sL
Asc
L
As
L
Asc
L
A
scL
Ac
L
Asc
L
Ac
L
A
sL
Asc
L
As
L
As
L
Asc
L
Asc
L
As
L
Asc
L
A
scL
Ac
L
Asc
L
Asc
L
Ac
L
Ac
L
Asc
L
Ac
L
A
sL
Asc
L
As
L
Asc
L
A
scL
Ac
L
Asc
L
Ac
L
A
E
F
F
FF
F
F
g
g
g
g
g
g
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
A
A
B
B
A
A
A
A
A
A
B
B
B
B
B
B
A
A
B
B
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
Yg
Xg
Yg
Xg
Yg
Xg
2004 March, 4 Page 12
ad Two-dimensional Truss ElementsExample
Nodes 1 and 3 are fixed and only load on node 2 in global X direction
0
0
0
0
00
00
00
00
2
2
22
22
2222
2222
22
22
3
3
1
1
YX
sL
Asc
L
As
L
Asc
L
A
scL
Ac
L
Asc
L
Ac
L
A
sL
Asc
L
As
L
As
L
Asc
L
Asc
L
As
L
Asc
L
A
scL
Ac
L
Asc
L
Asc
L
Ac
L
Ac
L
Asc
L
Ac
L
A
sL
Asc
L
As
L
Asc
L
A
scL
Ac
L
Asc
L
Ac
L
A
E
F
F
F
F
g
g
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
A
A
B
B
A
A
A
A
A
A
B
B
B
B
B
B
A
A
B
B
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
Yg
Xg
Yg
Xg
0F
2004 March, 4 Page 13
ad Two-dimensional Truss ElementsExample
Nodes 1 and 3 are fixed and only load on node 2 in global X direction
Solve for nodal displacements:
2
2
2
2
22
22
2
2
3
3
1
1
Y
X
sL
Asc
L
A
scL
Ac
L
A
sL
As
L
Asc
L
Asc
L
A
scL
Asc
L
Ac
L
Ac
L
A
sL
Asc
L
A
scL
Ac
L
A
E
F
F
F
F
g
g
B
B
B
B
B
B
B
B
B
B
A
A
B
B
A
A
B
B
A
A
B
B
A
A
A
A
A
A
A
A
A
A
Yg
Xg
Yg
Xg
0F
2
2
22
22
Y
X
sL
As
L
Asc
L
Asc
L
A
scL
Asc
L
Ac
L
Ac
L
A
Eg
g
B
B
A
A
B
B
A
A
B
B
A
A
B
B
A
A
0
F
2004 March, 4 Page 14
ad Two-dimensional Truss ElementsExample
Substitude the known displacements and solve for the reaction forces:
2g
2g
Y
X
2
2
2
2
3
3
1
1
sL
Asc
L
A
scL
Ac
L
A
sL
Asc
L
A
scL
Ac
L
A
E
F
F
F
F
B
B
B
B
B
B
B
B
A
A
A
A
A
A
A
A
Yg
Xg
Yg
Xg
2004 March, 4 Page 15
ad Two-dimensional Truss Elements
Truss element A:
2
2
1
1
2g
2g
Y
X
Y
X
Y
X
0
0
0000
0101
0000
0101
00
00
00
00
2
2
1
1
2
2
1
1
A
A
Y
X
Y
X
L
EA
F
F
F
F
cs
sc
cs
sc
Y
X
Y
X
2004 March, 4 Page 16
Stress and Momentum Balance
15 unknown variables
– 3 displacements – 6 strains– 6 stresses
15 equations
– 6 displacement-strain equations– 6 strain-stress equations– 3 equilibrium equations
zxyzxyzzyyxx
t
zxyzxyzzyyxxt
t wvuu
,,,,,
,,,,,
,,
0
pD
E
uD
t
2004 March, 4 Page 17
Strains
x
w
z
u
yz
v
y
w
y
u
x
v
z
w
y
v
x
u
zxyzxy
zzyyxx
;;
;;
2004 March, 4 Page 18
Material law
zxzx
yzyz
xyxy
yyxxzzzz
xxzzyyyy
zzyyxxxx
E
E
E
E
E
E
12
12
12
1211
1211
1211
2004 March, 4 Page 19
ad Material law
zx
yz
xy
zz
yy
xx
zx
yz
xy
zz
yy
xx
E
2
2100000
02
210000
002
21000
0001
0001
0001
211
2004 March, 4 Page 20
Equilibrium Equations
2004 March, 4 Page 21
ad Equilibrium Equations
0
dzdydxp
dydxdzz
zdydx
dzdxdyy
dzdx
dzdydxx
dzdy
x
zxzx
yxyxyx
xxxxxx
0
0
0
zzzyzxz
yzyyyxy
xzxyxxx
pzyx
pzyx
pzyx
2004 March, 4 Page 22
Plain Stress
yyxxzz
xy
yy
xx
xy
yy
xx
zz
yzzxzz
E
1
2
100
01
01
1
0
0;0;0
2
2004 March, 4 Page 23
Plain Strain
yyxxzz
xy
yy
xx
xy
yy
xx
zz
zxyzzz
E
2
2100
01
01
211
0
0;0;0
2004 March, 4 Page 24
Principle of Virtual WorkPrinciple of Virtual Displacements
dVdOqudVpuFu
dVW
dOqudVpuFuW
WW
V
t
O
t
V
tt
V
ti
O
t
V
tto
io
2004 March, 4 Page 25
ad Principle of Virtual Work
pdk
dOqGdVpGFGddVGDEGD
dOqGddVpGdFGdddVGDEDGd
Gdu
dGu
dOqudVpuFDuudVDEDu
uDEE
Du
O
t
V
tt
V
t
O
tt
V
tttt
V
ttt
ttt
O
t
V
tttt
V
ttt
ˆ
2004 March, 4 Page 26
Basis Function
Example: two-dimensional beam element
Basis function to approximate displacement inside element
62
2
0
3
3
2
210
2
321
32
3
2211
4
1
xc
xcxccw
xcxccw
xccw
cw
wwd
dxgxw
w
t
ii
i
IV
2004 March, 4 Page 27
ad Basis Function
22
12
2332
212133
212122
11
10
2
10
2
10
LLww
Lc
LLwwL
c
c
wc
w
w
ww
ww
Lx
x
Lx
x
2004 March, 4 Page 28
ad Basis Function
LL
x
L
xg
L
x
L
xg
LL
x
L
x
L
xg
L
x
L
xg
LL
x
L
xw
L
x
L
xL
L
x
L
x
L
xw
L
x
L
xxw
3
3
2
2
4
3
3
2
2
3
3
3
2
2
2
3
3
2
2
1
23
3
2
2
23
3
2
2
13
3
2
2
13
3
2
2
23
2
231
232231
2004 March, 4 Page 29
Thank you!