Computational Methods
Bar Elements & Trusses
Małgorzata Stojek
Cracow University of Technology
March 2012
MS
(L-53 CUT)
FEM
03/2012
1 / 26
Bar Member - Nomenclature & Notation
axial rigidity EA
u(x)
q(x)
x
L
cross
section
P
Quantity
Meaning
x
Longitudinal bar axis
u
(
x
)
Axial displacement
q
(
x
)
Distributed axial force, given per unit of bar length
L
Total length of bar member
E
Elastic modulus
A
Cross section area, may vary with x
P
Prescribed end load
MS
(L-53 CUT)
FEM
03/2012
2 / 26
Bar Member - Strong Formulation
Quantity
Meaning
e
=
du/dx
=
u
′
Infinitesimal axial strain
σ
=
Ee
=
Eu
′
Axial stress
F
=
A
σ
=
EAe
=
EAu
′
Internal axial force
Definition
Governing Equations:
Kinematic
e
=
du/dx
=
u
′
Constitutive
σ
=
Ee
Equilibrium
F
′
+
q
=
0
Definition
(S):
− (
EAu
′
)
′
=
q
on
(
0, L
)
u
(
0
) =
0
at x
=
0
EA u
′
(
L
) =
P
at x
=
L
MS
(L-53 CUT)
FEM
03/2012
3 / 26
Bar Member - Weak Formulation
RECALL:
Z
L
0
EAu,
x
w ,
x
dx
=
Z
L
0
qw dx
+
EAu,
x
(
L
)
w
(
L
)
−
EAu,
x
(
0
)
w
(
0
)
P
0
←−
[
bar member
]
P
L
−→
global
coordinate system
x
−→
P
0
−→
[
bar member
]
P
L
−→
Z
L
0
EAu,
x
w ,
x
dx
=
Z
L
0
qw dx
+
P
L
w
(
L
) +
P
0
w
(
0
)
MS
(L-53 CUT)
FEM
03/2012
4 / 26
c
Felippa
Introduction to FEM
FEM Displacement Trial Function
End node 1 assumed fixed
Axial displacement plotted normal to
x
for visualization convenience
(1)
(2)
(3)
(4)
u
= 0
1
u
2
u
3
u
4
u
5
u
1
f
1
u
3
f
3
u
4
f
4
u
5
f
5
u
2
f
2
x
u(x)
u
IFEM Ch 11 Slide 10
2
3
4
1
5
MS
(L-53 CUT)
FEM
03/2012
5 / 26
c
Felippa
Introduction to FEM
Element Shape Functions
1
2
(e)
1
0
0
1
N
e
N
e
L
e
IFEM Ch 11 Slide 12
1
2
=
ξ
1−ξ
ξ
MS
(L-53 CUT)
FEM
03/2012
6 / 26
Element Stiffness Matrix
Localization & Strain-Displacement Matrix
element DOFs and shape functions,
N
e
1
=
x
−
x
2
x
1
−
x
2
=
1
−
ξ,
N
e
2
=
x
−
x
1
x
2
−
x
1
=
ξ
local interpolant
u
e
(
x
) =
N
e
1
N
e
2
u
1
u
2
=
N
e
u
e
strain-displacement
matrix, B
e
=
d N
e
dx
du
e
(
x
)
dx
=
dN
e
1
dx
dN
e
2
dx
u
1
u
2
=
B
e
u
e
MS
(L-53 CUT)
FEM
03/2012
7 / 26
Element Stiffness Matrix
Natural Coordinates & Change of Variables
natural coordinates,
x
∈ (
0, L
)
,
ξ
∈ (
0, 1
)
,
h
=
l
ξ
=
x
−
x
1
h
,
d
ξ
=
1
h
dx
RECALL: change of variables - rules of calculus for any f
(
x
)
Z
h
0
f
(
x
)
dx
=
h
Z
1
0
f
(
ξ
)
d
ξ
;
df
(
x
)
dx
=
df
(
ξ
)
d
ξ
d
ξ
dx
=
1
h
df
(
ξ
)
d
ξ
MS
(L-53 CUT)
FEM
03/2012
8 / 26
Element Stiffness Matrix
Strain-Displacement Matrix
chain rule
B
e
=
d N
e
dx
=
1
h
d
d
ξ
N
e
1
N
e
2
symbolic derivation
B
e
=
1
h
d
d
ξ
1
−
ξ
ξ
results in
B
e
=
1
h
−
1 1
MS
(L-53 CUT)
FEM
03/2012
9 / 26
Bar Element Stiffness Matrix I
Fact
a
(
w
e
,
u
e
) =
Z
h
0
dw
e
dx
EA
du
e
dx
dx ,
w
e
∈ {
N
e
1
,
N
e
2
}
Fact
For B
=
B
1
B
2
B
1
B
2
B
1
B
2
=
B
1
B
1
B
1
B
2
B
2
B
1
B
2
B
2
Definition
Bar Element Stiffness Matrix
K
e
2
×
2
=
Z
h
0
(
EA
)
B
T
B dx
MS
(L-53 CUT)
FEM
03/2012
10 / 26
Bar Element Stiffness Matrix II
EA=const
RECALL:
B
=
1
h
−
1 1
change of variables
K
e
=
Z
h
0
(
EA
)
B
T
B dx
= (
h
)
Z
1
0
(
EA
)
B
T
B d
ξ
prismatic bar, i.e. EA
=
constant
K
e
=
EA
h
1
h
2
Z
1
0
−
1
1
−
1 1
d
ξ
K
e
=
EA
h
1
−
1
−
1
1
MS
(L-53 CUT)
FEM
03/2012
11 / 26
Element Load Vector
RECALL:
(
q, w
e
) =
Z
h
0
qw
e
dx ,
w
e
∈ {
N
e
1
,
N
e
2
}
element load vector due to internal load, q
(
x
)
:
F
e
q
=
Z
h
0
qN
T
dx
change of variables
F
e
q
=
h
Z
1
0
q
1
−
ξ
ξ
d
ξ
for q
(
x
) =
const
F
e
q
=
qh
Z
1
0
1
−
ξ
ξ
!
d
ξ
=
qh
1
2
1
2
!
MS
(L-53 CUT)
FEM
03/2012
12 / 26
Interpretation
c
Felippa
EA
L
1
−
1
−
1
1
u
i
u
j
=
−
F
F
d
=
u
j
−
u
i
F
=
k
s
d
MS
(L-53 CUT)
FEM
03/2012
13 / 26
2D Bar Element
Local 2D Generalization
Fact
1D local interpolant
u
e
(
x
) =
N
e
1
N
e
2
u
1
u
2
=
N
e
u
e
Definition
2D local interpolant
u
e
(
x
) =
N
e
1
0 N
e
2
0
u
1
v
1
u
2
v
2
=
N
e
d
e
MS
(L-53 CUT)
FEM
03/2012
14 / 26
2D Bar Element
Local 2D Generalization
Definition
2D strain-displacement matrix, B
e
=
d N
e
d x
d u
e
(
x
)
d x
=
dN
e
1
d x
0
dN
e
2
d x
0
u
1
v
1
u
2
v
2
=
B
e
d
e
Definition
2D Bar Element Stiffness Matrix
K
e
4
×
4
=
Z
h
0
(
EA
)
B
T
B d x
MS
(L-53 CUT)
FEM
03/2012
15 / 26
2D Bar Element
Local 2D Stiffness Matrix
1D
B
=
1
h
−
1 1
K
e
=
EA
h
Z
1
0
−
1
1
−
1 1
d
ξ
=
EA
h
1
−
1
−
1
1
2D
B
=
1
h
−
1 0 1 0
K
e
=
EA
h
Z
1
0
−
1
0
1
0
−
1 0 1 0
d
ξ
=
EA
h
1
0
−
1
0
0
0
0
0
−
1
0
1
0
0
0
0
0
MS
(L-53 CUT)
FEM
03/2012
16 / 26
2D Bar Element
Local 2D Load Vector
RECALL:
F
e
q
=
Z
h
0
qN
T
dx
1D
N
=
N
e
1
N
e
2
=
1
−
ξ
ξ
F
e
q
=
h
Z
1
0
q
1
−
ξ
ξ
d
ξ
q
=
const
=
qh
Z
1
0
1
−
ξ
ξ
!
d
ξ
=
qh
1
2
1
2
!
2D
N
=
N
e
1
0 N
e
2
0
=
1
−
ξ
0 ξ 0
F
e
q
=
qh
Z
1
0
1
−
ξ
0
ξ
0
d
ξ
=
qh
1
2
0
1
2
0
=
F
e
q
1
0
F
e
q
2
0
MS
(L-53 CUT)
FEM
03/2012
17 / 26
Rotation of Coordinate System
α
— the angle of rotation
(
x , y
)
global
coord.
α
rotated
−→
(
x , y
)
local
coord.
x
y
=
cos α
sin α
−
sin α cos α
x
y
For c
=
cos α, s
=
sin α
A
=
cos α
sin α
−
sin α cos α
=
c
s
−
s
c
RECALL: rotation is an
orthonormal
transformation
A
−
1
=
A
T
MS
(L-53 CUT)
FEM
03/2012
18 / 26
2-D Bar Element
Globalization: Displacement Transformation
u
xi
u
xi
c
u
yi
s
u
yi
u
xi
s
u
yi
c
u
x j
u
x j
c
u
yj
s
u
yj
u
x j
s
u
yj
c
Node displacements transform as
i
x
y
c
cos
s
sin
in which
Globalization: Displacement Transformation
Introduction to FEM
x
y
j
u
xi
u
yi
u
xj
u
yj
u
yi
u
xi
u
xj
u
yj
IFEM Ch 2 Slide 18
α
_
_
_
_
_
_
T
=
c
s
0
0
−
s
c
0
0
0
0
c
s
0
0
−
s
c
u
x
i
u
y
i
u
x
j
u
y
j
=
c
s
0
0
−
s
c
0
0
0
0
c
s
0
0
−
s
c
u
x
i
u
y
i
u
x
j
u
y
j
d
e
=
T
d
e
MS
(L-53 CUT)
FEM
03/2012
19 / 26
2-D Bar Element
Globalization: Load Vector Transformation
Globalization: Force Transformation
Node forces transform as
or
x
y
i
j
f
xi
f
yi
f
x j
f
yj
f
xi
f
yi
f
xj
f
yj
c
s 0
0
s
c
0
0
0
0
c
s
0
0
s
c
Note:
global on LHS,
local on RHS
Introduction to FEM
f
xi
f
yi
f
x j
f
yj
f
yi
f
xi
f
xj
f
yj
f
(
T
)
f
e
e T
e
_
IFEM Ch 2 Slide 20
α
_
_
_
_
T
=
c
s
0
0
−
s
c
0
0
0
0
c
s
0
0
−
s
c
f
x
i
f
y
i
f
x
j
f
y
j
=
c
−
s
0
0
s
c
0
0
0
0
c
−
s
0
0
s
c
f
x
i
f
y
i
f
x
j
f
y
j
f
e
=
T
T
f
e
MS
(L-53 CUT)
FEM
03/2012
20 / 26
2-D Bar Element I
Globalization: Stiffness Matrix Transformation
RECALL:
global
&
local
coordinate systems
K
e
·
d
e
=
f
e
.
formal replacement
d
e
=
T
d
e
K
e
T
d
e
=
f
e
;
multiplication by T
T
T
T
K
e
T
d
e
=
T
T
f
e
;
replacement
f
e
=
T
T
f
e
T
T
K
e
T
d
e
=
f
e
;
leads to
K
e
·
d
e
=
f
e
;
where
K
e
=
T
T
K
e
T
MS
(L-53 CUT)
FEM
03/2012
21 / 26
2-D Bar Element II
Globalization: Stiffness Matrix Transformation
RECALL:
K
e
=
T
T
K
e
T
K
e
=
EA
h
×
c
s
0
0
−
s
c
0
0
0
0
c
s
0
0
−
s
c
T
1
0
−
1 0
0
0
0
0
−
1 0
1
0
0
0
0
0
c
s
0
0
−
s
c
0
0
0
0
c
s
0
0
−
s
c
K
e
=
EA
h
c
2
cs
−
c
2
−
cs
cs
s
2
−
cs
−
s
2
−
c
2
−
cs
c
2
cs
−
cs
−
s
2
cs
s
2
MS
(L-53 CUT)
FEM
03/2012
22 / 26
2-D Bar Element III
Globalization: Load Vector Transformation
RECALL:
F
e
q
=
F
e
q
1
0
F
e
q
2
0
,
f
e
=
T
T
f
e
F
e
q
=
c
−
s
0
0
s
c
0
0
0
0
c
−
s
0
0
s
c
F
e
q
1
0
F
e
q
2
0
=
F
e
q
1
c
F
e
q
1
s
F
e
q
2
c
F
e
q
2
s
MS
(L-53 CUT)
FEM
03/2012
23 / 26
Nonhomogeneous Natural BCs & Concentrated Forces
Global Coordinate System
1D
—
global
coordinate system
x
−→
P
0
−→
[
bar member
]
P
L
−→
R
L
0
EAu,
x
w ,
x
dx
=
R
L
0
qw dx
+
P
L
w
(
L
) +
P
0
w
(
0
)
↓
Kd
=
F
q
added after assembly
+
W
2D
—
global
coordinate system
Kd
=
F
q
added after assembly
+
W
MS
(L-53 CUT)
FEM
03/2012
24 / 26
Postprocessing in Global Coordinate System
Siły przyw ˛ezłowe W
−→
W
x
1
,
↑
W
y
1
α
−→
W
x
2
,
↑
W
y
2
K
e
d
e
=
F
e
=
F
e
q
+
W
e
W
e
=
W
e
x
1
W
e
y
1
W
e
x
2
W
e
y
2
=
K
e
d
e
−
F
e
q
MS
(L-53 CUT)
FEM
03/2012
25 / 26
Finite Element Program
c
Felippa
The Direct Stiffness Method (DSM) Steps
(repeated here for convenience)
Disconnection
Localization
Member (Element) Formation
Globalization
Merge
Application of BCs
Solution
Recovery of Derived Quantities
Breakdown
Assembly &
Solution
Introduction to FE
post-processing
steps
processing
steps
conceptual
steps
IFEM Ch 3 Slide 2
MS
(L-53 CUT)
FEM
03/2012
26 / 26