Line elements in 2D and 3D

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15 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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Line elements in 2D and 3D


BAR ELEMENT IN 2D (TRUSS, LINK)


This element can be loaded only in the axial direction, its orientation in 2D space in
relation to local (element) and global coordinate system shows Fig.3
-
1












Fig.3
-
1 Bar element in 2D space


α
x
g
y
g
y
x
u
1
v
1
v
2
u
2
L
In comparison with the simplest element
Fig.3,
there are two additional deformation
parameters
v
1
, v
2
,

not connected to any element stiffness. The stiffness matrix in
local element coordinate system is obtained from
bar
element

in 1D
, adding two
zero rows and columns, corresponding to parameters
v
1
, v
2

:





Matrix of
deformation parameters


The stiffness matrix in global coordinate system is then obtained by transformation

k
g

= T
T
. k . T
, where

T
is a transformation matrix according to Fig.3
-
1




This element is used for solution of pin
-
jointed frame structures, or as a simple
model of cables and springs. In ANSYS it can be found under the name
LINK1
,
other details see

ANSYS Online Manuals
.

Simple illustration of its usage can be
found in the
Example 3.1


.
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
1















L
ES
k
,
2
2
1
1













v
u
v
u
δ







λ
0
0
λ
T















)
cos(
)
cos(
)
cos(
)
cos(
cos
sin
sin
cos
g
g
g
g
yy
yx
xy
xx




λ
BAR ELEMENT IN 3D (TRUSS, LINK)


3D version of the previous element according to Fig.3
-
2 has six degrees of freedom







Fig.3
-
2 Bar element in 3D space




T
w
v
u
w
v
u
2
2
2
1
1
1
,
,
,
,
,

δ
v
1
w
1
u
1
u
2
v
2
w
2
x
y
z
Its stiffness matrix in local coordinate system is created in by a similar procedure






and transformation to global coordinate system is done by the matrix
T





Bar element in 3D is used for solution of pin
-
jointed frame structures, or as a simple model
of cables and springs. In ANSYS it can be found under the name
LINK8

,
other details see

ANSYS Online Manuals
.

An illustration of its usage can be found in the
Example 3.2



,
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1























L
ES
k







λ
0
0
λ
T
,
.
)
cos(
)
cos(
)
cos(
)
cos(
)
cos(
)
cos(
)
cos(
)
cos(
)
cos(











g
g
g
g
g
g
g
g
g
zz
yz
xz
yz
yy
yx
xz
xy
xx
λ
BEAM ELEMENT

Beam is a line element with bending capability. Primary unknown function is the
deflection
w
. To assure convergence, there must be interelement continuity of deflection
and slope φ. Each node then has two degrees of freedom according to Fig. 3
-
3.







Fig.3
-
3 Beam element

Deflection is approximated by , where




are the degrees of freedom


cubic polynomial shape functions.

w
1
w
2
L
φ
1
φ
2
x
δ
N
.
)
(

x
w
δ
T
= | w
1
,
φ
1
,

w
2
,
φ
2
|
N =
| N
1
N
2
N
3
N
4
|
The stiffness matrix of beam element has the explicit form





where
I
is the second moment of area,
L

length and
E

modulus of elasticity.

In this form, the element can be used only to model bending of straight beams. In
practical applications it must be combined with the bar capability as described in
the next paragraph.


GENERAL BEAM OR FRAME ELEMENT IN 2D

This element can be obtained as a combination of previous beam element (
Fig. 3
-
3
)
and the simplest
1D
-
bar
element

-

see Fig.3
-
4. It has six degrees of freedom, three
in each node



δ
T

= │ u
1
, w
1
, φ
1
, u
2
, w
2
, φ
2

│,

2
2
2
3
4
.
6
12
2
6
4
6
12
6
12
L
Sym
L
L
L
L
L
L
L
EI




k
w
1
w
2
L
φ
1
φ
2
u
1
u
2
Fig. 3
-
4 Frame element in 2D

Frame element in 2D is used for solution of frame structures with its members loaded by
bending moments and axial forces. In ANSYS it can be found under the name
BEAM3
,
other
details see

ANSYS Online Manuals
.

and its stiffness matrix is derived from the matrices of the mentioned elements:






where T = ES/L, A = 4EI/L, B = 2EI/L, C = 6EI/L
2
, D = 12EI/L
3
.

A
C
B
C
C
D
C
D
T
T
B
C
A
C
C
D
C
D
T
T









0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
k
GENERAL BEAM OR FRAME ELEMENT IN 3D

Straight element according to Fig.3
-
5 has twelve degrees of freedom, six in each node
-

three displacements and three rotations.








Fig.3
-
5 Frame element in 3D

The parameters
u
1
,
u
2

are linked with the axial loading, with the stiffness matrix
according to simplest
bar
element
. Parameters
v
1
, φ
z1
, v
2
, φ
z2

are linked with the bending
in the
y
direction, parameters
w
1
, φ
y1
, w
2
, φ
y2

are linked with the bending in the
z
direction. Their stiffness matrix is created according to the
beam
element

with
appropriate second area moment
I
z
, resp.

I
y

.

The rest of parameters,

φ
x1
, φ
x2

,
correspond

to torsion and using the analogy with
tension, the stiffness matrix can be written as




T
z
y
x
z
y
x
w
v
u
w
v
u
2
2
2
2
2
2
1
1
1
1
1
1
,
,
,
,
,
,
,
,
,
,
,







δ
v
1
w
1
u
1
φ
x1
φ
y1
φ
z1
u
2
φ
x2
φ
y2
v
2
w
2
φ
z2
x
y
z
where
G
is the shear modulus and
I
p

the polar moment of area.

The final stiffness matrix is completed from the partial ones in the same way as the
frame element matrix in 2D. The element is used for solution of general frame
structures in 3D with its members loaded by bending moments and axial forces and
torsion. In ANSYS it can be found under the name
BEAM4
,
other details see

ANSYS
Online Manuals
.











1
1
1
1
L
GI
p
k