Civil Engineering Reference
In-Depth Information
9.2 The element mechanical properties
A free body diagram of an arbitrary beam (line-like) type element
n
, with local axis
x
,
y
and
z
is illustrated in Fig. 9.3. It is taken for granted that all displacements as well as
forces comprise a time-invariant mean value (the static part) and a stationary fluctuating
(dynamic) part. At position
x
along its span the cross sectional displacements and
rotation (torsion) are defined by
T
(
)
()
( )
x t
,
⎡
r
r
r
r
⎤
x
xt
,
r
=
=
r
+
r
(9.1)
⎣
⎦
el
x
y
z
θ
el
el
tot
el
tot
where index
el
indicates quantities at element level. At ends
1
and
2
it has the element
nodal forces
T
tot
⎧
[
]
F
=
FF F F F F
⎡⎤
F
⎪
1
1
2
3
4
5
6
()
1
()
tot
F
t
=
=
F
+
F
t
where
(9.2)
⎨
⎢
⎣⎦
tot
F
T
[
]
⎪
⎩
F
=
FFFF F F
2
tot
2
7
8
9
10
11
12
tot
tot
and corresponding local displacements
⎧
T
tot
[
]
d
=
dddddd
d
⎡⎤
⎪
1
1
2
3
4
5
6
()
1
()
tot
d
t
=
=
d
+
d
t
where
(9.3)
⎨
⎢
⎣⎦
tot
d
[
]
T
⎪
⎩
d
=
dddd d d
2
tot
2
7
8
9
10
11
12
tot
tot
It is assumed that the cross sectional displacement vector
(
)
r
x t
,
with sufficient
el
tot
accuracy may be described by the product of a shape function matrix
N
00000
N
00 0 0 0
⎡
⎤
1
7
⎢
⎥
0
N
000
N
0
N
0 0 0
N
()
⎢
2
6
8
12
⎥
x
N
=
⎢
0
0
N
0
N
0
0
0
N
0
N
0
⎥
3
5
9
11
⎢
⎥
000
N
00000
N
0 0
⎣
⎦
4
10
(9.4)
and the nodal displacement vector
()
tot
t
d
, i.e. that
(
)
( )
( )
x t
,
x
t
r
=
N
⋅
d
(9.5)
el
tot
tot
where the twelve shape function
N
,
i
112
, are given in Fig. 9.4. These are
identical to the shape functions commonly used elsewhere in structural mechanics. Since
=−
