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MECH4450 Introduction to Finite Element Methods
Chapter 3
FEM of 1-D Problems: Applications
Plane Truss Problems
Example 1: Find forces inside each member. All members have the same length L. E = 10 GPa, A = 1 cm2, L = 1 m, F = 10 kN
F
Arbitrarily Oriented 1-D Bar Element on 2-D Plane
1 1
2 2
1 1
1 1
u PAEu PL
P1 u1 P2 u2
1 1 P u
2 2 P u
1 1
2 2
1 1
1 1
u PAEuL P
1 1 P u
2 2 P u1 1 Q v
2 2 Q v1 1
1 1
2 2
2 2
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
u P
v QAEu PL
v Q
Arbitrarily Oriented 1-D Bar Element on 2-D Plane
1 1
2 2
1 0 1 0
0 0 0 0 0 0
1 0 1 0
0 0 0 0 0 0
u P
AE
uL P
2
2
1
1
2
2
1
1
cossin00
sincos00
00cossin
00sincos
0
0
v
u
v
u
v
u
v
u
2
2
1
1
2
1
cossin00
sincos00
00cossin
00sincos
0
0
Q
P
Q
P
P
P
1 1
2 2
1 1
1 1
u PAEuL P
x
y
xy
q
Stiffness Matrix of 1-D Bar Element on 2-D Plane
2
2
1
1
2
2
1
1
cossin00
sincos00
00cossin
00sincos
cossin00
sincos00
00cossin
00sincos
v
u
v
u
K
Q
P
Q
P
ij
0000
0101
0000
0101
L
AEK ij
Q2 , v2
11 , uP
q
22 u ,P
P2 , u2
Q1 , v1
P1 , u1
2
2
1
1
22
22
22
22
2
2
1
1
sincossinsincossin
cossincoscossincos
sincossinsincossin
cossincoscossincos
v
u
v
u
L
AE
Q
P
Q
P
Matrix Assembly of Multiple Bar Elements
1 1
1 1
2 2
2 2
1 0 1 0
0 0 0 0
1 0 1 0
0 0 0 0
I I
I I
I I
I I
P u
Q vAE
LP u
Q v
Element I
Element II
Element II I
1 1
1 1
2 2
2 2
1 3 1 3
3 3 3 3
4 1 3 1 3
3 3 3 3
II II
II II
II II
II II
P u
Q vAE
LP u
Q v
1 1
1 1
2 2
2 2
1 3 1 3
3 3 3 3
4 1 3 1 3
3 3 3 3
III III
III III
III III
III III
P u
Q vAE
LP u
Q v
Matrix Assembly of Multiple Bar Elements11
11
22
22
33
33
4 0 4 0 0 0
0 0 0 0 0 0
4 0 4 0 0 0
0 0 0 0 0 04
0 0 0 0 0 0
0 0 0 0 0 0
I
I
I
I
I
I
uP
vQ
uP AEvLQ
uP
vQ
Element I
11
11
22
22
33
33
0 0 0 0 0 0
0 0 0 0 0 0
0 0 1 3 1 3
0 0 3 3 3 34
0 0 1 3 1 3
0 0 3 3 3 3
II
II
II
II
II
II
uP
vQ
uP AEvLQ
uP
vQ
11
11
22
22
33
33
1 3 0 0 1 3
3 3 0 0 3 3
0 0 0 0 0 0
0 0 0 0 0 04
1 3 0 0 1 3
3 3 0 0 3 3
III
III
III
III
III
III
uP
vQ
uP AEvLQ
uP
vQ
Element II
Element II I
Matrix Assembly of Multiple Bar Elements
3
3
2
2
1
1
3
3
2
2
1
1
v
u
v
u
v
u
33333333
33113131
33303000
31301404
33003030
31043014
L4
AE
S
R
S
R
S
R
0v
0u
0v
?u
?v
0u
603333
023131
333300
313504
330033
310435
L4
AE
?S
?R
?S
FR
0S
?R
3
3
2
2
1
1
3
3
2
2
1
1
Apply known boundary conditions
Solution Procedures
0v
0u
0v
?u
?v
0u
603333
023131
333300
310435
330033
313504
L4
AE
?S
?R
?S
?R
0S
FR
3
3
2
2
1
1
3
3
2
1
1
2
u2= 4FL/5AE, v1= 0
0v
0u
0vAE5
FL4u
0v
0u
603333
023131
333300
310435
330033
313504
L4
AE
?S
?R
?S
?R
0S
FR
3
3
2
2
1
1
3
3
2
1
1
2
Recovery of Axial Forces1
1
11
22
22
0 41 0 1 0 50
0 0 0 0 04
1 0 1 0 455
0 0 0 0 00
I
I
I
I
uP
vQ AE
FFLL uP
AEQ v
Element I
Element II
Element II I
21
12
23
23
1541 3 1 3
5 33 3 3 3 50
14 1 3 1 3 0 53 3 3 3 0 3
5
II
II
II
II
FLuP
AEQ AE v F
LP uQ v
11
11
32
32
1 3 1 3 0 0
03 3 3 3 0
0 04 1 3 1 30 03 3 3 3
III
III
III
III
uP
vQ AEuLP
vQ
Stresses inside members
Element I
Element II
Element II I
480 MPa
5
F
A
1
4
5I FP 2
4
5I F
P
1
1
5IIP F
1
3
5IIQ F
2
3
5IIQ F
2
1
5IIP F
240 MPa
5
F
A
Governing Equation and Boundary Condition
• Governing Equation
• Boundary Conditions -----
0at , ? &? &? & ?2
2
2
2
x
dx
vdEI
dx
d
dx
vdEI
dx
dvv
Lxdx
vdEI
dx
d
dx
vdEI
dx
dvv
at , ? &? &? & ?
2
2
2
2
,0)()(
2
2
2
2
xq
dx
xvdEI
dx
d 0<x<L
q(x)
x
y
Weak Formulation for Beam Element
• Weighted-Integral Formulation for one element
2
1
)()(
)(02
2
2
2x
x
dxxqdx
xvdEI
dx
dxw
2
1
2
1
2
1
2
2
2
2
2
2
2
2
0
x
x
x
x
x
x dx
vdEI
dx
dw
dx
vdEI
dx
dwdxwq
dx
vdEI
dx
wd
V(x2)
x = x1
M(x2)
q(x)y
x x = x2
V(x1)
M(x1)
L = x2-x1
2
1
2
1
2
2
2
2
0x
x
x
x
Mdx
dwwVdxwq
dx
vdEI
dx
wd
• Weak Form from Integration-by-Parts
Weak Formulation
• Weak Form
24231211 , , , xMQxVQxMQxVQ
42
21
32112
2
2
2
)()(2
1
Qdx
dwQ
dx
dwQxwQxwdxwq
dx
vdEI
dx
wdx
x
2
1
2
1
2
2
2
2
0x
x
x
x
Mdx
dwwVdxwq
dx
vdEI
dx
wd
Q3
x = x1
Q4
q(x)y(v)
x x = x2
Q1
Q2
L = x2-x1
Ritz Method for Approximation
n
jjj nxuxvLet
1
4 and )()(
42
21
32112
24
12
2
)()(2
1
Qdx
dwQ
dx
dwQxwQxwdxwq
dx
duEI
dx
wdx
x
j
jj
Let w(x)= fi (x), i = 1, 2, 3, 4
42
21
32112
24
12
2
)()(2
1
Qdx
dQ
dx
dQxQxdxq
dx
duEI
dx
d iiii
x
x
ij
jj
i
Q3
x = x1
Q4
q(x)y(v)
x x = x2
Q1
Q2
L = x2-x1
where 1 2
1 1 1 2 1 3 2 2 4 2; ; ; ;x x x x
dv dvu v x v u u v x v u
dx dx
Derivation of Shape Function for Beam Element
In the global coordinates:
1 1 2 2 3 3 4 4
1 1 1 2 2 3 2 4
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
v x u x u x u x u x
v x x v x x
2 3
1 1
2 1 2 1
2
111
2 122 3
3 1 1
4 2 1 2 1
2
1 11
2 1 2 1
1 3 2
1
3 2
x x x x
x x x x
x xx x
x x
x x x x
x x x x
x x x xx x
x x x x
Element Equations of 4th Order 1-D Model
u3
x = x1
u4
q(x)y(v)
xx = x2
u1
u2
L = x2-x1
4
3
2
1
44342414
34332313
24232212
14131211
4
3
2
1
4
3
2
1
u
u
u
u
KKKK
KKKK
KKKK
KKKK
q
q
q
q
Q
Q
Q
Q
2
1
2
1
2
2
2
2 x
x
ii
x
x
jiji
ij qdxqandKdxdx
d
dx
dEIKwhere
x=x2 x=x1
1 1 f1
f3 f2
f4
Element Equations of 4th Order 1-D Model
u3
x = x1
u4
q(x)y(v)
x x = x2
u1
u2
L = x2-x1
2
1
x
x
ii qdxqwhere
24
23
12
11
22
22
3
4
3
2
1
4
3
2
1
233
3636
323
3636
2
u
vu
u
vu
LLLL
LL
LLLL
LL
L
EI
q
q
q
q
Q
Q
Q
Q
Finite Element Analysis of 1-D Problems Example 1.
Finite element model:
2
2
1
1
22
22
3
4
3
2
1
233
3636
323
3636
2
v
v
LLLL
LL
LLLL
LL
L
EI
Q
Q
Q
Q
P1 , v1 P2 , v2 P3 , v3 P4 , v4
M1 , q1
M2 , q2
M3 , q3
M4 , q4
I II III
Discretization:
F
LLL
Matrix Assembly of Multiple Beam Elements
Element I
Element II
112 2
12
232 2
243
3
3
4
4
6 3 6 3 0 0 0 0
3 2 3 0 0 0 0
6 3 6 3 0 0 0 0
3 3 2 0 0 0 02
0 0 0 0 0 0 0 00
0 0 0 0 0 0 0 00
0 0 0 0 0 0 0 00
0 0 0 0 0 0 0 00
I
I
I
I
vL LQ
L L L LQ
vL LQ
L L L LQ EIvL
v
1
1
212 2
22
333
2 234
4
4
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 6 3 6 3 0 0
0 0 3 2 3 0 02
0 0 6 3 6 3 0 0
0 0 3 3 2 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
II
II
II
II
v
vQ L L
Q L L L LEIvQ L LL
Q L L L L
v
Matrix Assembly of Multiple Beam Elements
Element II I
1
1
2
2
31 3
2 22 3
3 42 2
4 4
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 02
0 0 0 0 6 3 6 3
0 0 0 0 3 2 3
0 0 0 0 6 3 6 3
0 0 0 0 3 3 2
III
III
III
III
v
v
EIQ vL LL
Q L L L L
Q vL L
Q L L L L
4
4
3
3
2
2
1
1
22
2222
2222
22
3
4
4
3
3
2
2
1
1
2330000
36360000
32233300
3633663600
00322333
0036336636
0000323
00003636
2
v
v
v
v
LLLL
LL
LLLLLLLL
LLLL
LLLLLLLL
LLLL
LLLL
LL
L
EI
M
P
M
P
M
P
M
P
Solution Procedures
?
?
?
0
?
0
0
0
2330000
36360000
340300
360123600
003403
003601236
0000323
00003636
2
0
0
?
0
?
?
?
4
4
3
3
2
2
1
1
22
222
222
22
3
4
4
3
3
2
2
1
1
v
v
v
v
LLLL
LL
LLLLL
LL
LLLLL
LL
LLLL
LL
L
EI
M
FP
M
P
M
P
M
P
Apply known boundary conditions
?
?
?
0
?
0
0
0
360123600
003601236
0000323
00003636
2330000
36360000
340300
003403
2
?
?
?
?
0
0
0
4
4
3
3
2
2
1
1
22
22
222
222
3
3
2
1
1
4
4
3
2
v
v
v
v
LL
LL
LLLL
LL
LLLL
LL
LLLLL
LLLLL
L
EI
P
P
M
P
M
FP
M
M
Solution Procedures
?
?
?
?
230
3630
34
004
2
0
0
0
4
4
3
2
22
222
22
3
4
4
3
2
v
LLL
LL
LLLL
LL
L
EI
M
FP
M
M
4
4
3
22
3
3
2
1
1
3603
0030
000
0003
2
?
?
?
?
vLL
L
L
L
L
EI
P
P
M
P
?
?
?
?
0
0
0
0
360312600
003061236
0000323
00030636
2303000
36306000
340300
004303
2
?
?
?
?
0
0
0
4
4
3
2
3
2
1
1
22
22
222
222
3
3
2
1
1
4
4
3
2
v
v
v
v
LL
LL
LLLL
LL
LLLL
LL
LLLLL
LLLLL
L
EI
P
P
M
P
M
FP
M
M
Shear Resultant & Bending Moment Diagram
FL7
2
FL7
1
FL
F7
9
F7
3
2P
F
Plane Frame
Frame: combination of bar and beam
E, A, I, LQ1 , v1 Q3 , v2
Q2 , q1
P1 , u1
Q4 , q2
P2 , u2
2
2
2
1
1
1
22
2323
22
2323
4
3
2
2
1
1
460
260
6120
6120
0000
260
460
6120
6120
0000
v
u
v
u
L
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EIL
AE
L
AEL
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EIL
AE
L
AE
Q
Q
P
Q
Q
P
Finite Element Model of an Arbitrarily Oriented Frame
q x
y
q x
y
Finite Element Model of an Arbitrarily Oriented Frame
2
2
2
1
1
1
22
2323
22
2323
4
3
2
2
1
1
40
620
6
0000
60
1260
12
20
640
6
0000
60
1260
12
v
u
v
u
L
EI
L
EI
L
EI
L
EIL
AE
L
AEL
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EIL
AE
L
AEL
EI
L
EI
L
EI
L
EI
Q
Q
P
Q
Q
P
local
global
Plane Frame Analysis - Example
Rigid Joint
Element II
Element IF F
Plane Frame Analysis
2
2
2
1
1
1
22
2323
22
2323
4
3
2
2
1
1
40
620
6
0000
60
1260
12
20
640
6
0000
60
1260
12
v
u
v
u
L
EI
L
EI
L
EI
L
EIL
AE
L
AEL
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EIL
AE
L
AEL
EI
L
EI
L
EI
L
EI
Q
Q
P
Q
Q
P
I
I
P1 , u1
P2 , u2
Q2 , q1
Q4 , q2
Q1 , v1
Q3 , v2
Plane Frame Analysis
P1 , u2
Q3 , v3
Q2 , q2Q4 , q3
Q1 , v2
P2 , u3
4
3
3
2
2
2
22
2323
22
2323
4
3
2
2
2
1
460
260
6120
6120
0000
260
460
6120
6120
0000
v
u
v
u
L
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EIL
AE
L
AEL
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EIL
AE
L
AE
Q
Q
P
Q
Q
PII
Plane Frame Analysis
2
2
2
1
1
1
22
2323
22
2323
4
3
2
2
1
1
40
620
6
0000
60
1260
12
20
640
6
0000
60
1260
12
v
u
v
u
L
EI
L
EI
L
EI
L
EIL
AE
L
AEL
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EIL
AE
L
AEL
EI
L
EI
L
EI
L
EI
Q
Q
P
Q
Q
P
I
I
4
3
3
2
2
2
22
2323
22
2323
4
3
2
2
2
1
460
260
6120
6120
0000
260
460
6120
6120
0000
v
u
v
u
L
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EIL
AE
L
AEL
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EIL
AE
L
AE
Q
Q
P
Q
Q
PII
Plane Frame Analysis
FL8
1
F2
1
F2
1
FL8
3
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