Civil Engineering Reference
In-Depth Information
by the use of beam or beam-column type of elements as shown in Fig. 7.4. At any level
it is taken for granted that the load and load effect vectors can be split into a mean part
and a fluctuating part, i.e. at a global system level
R
R
R
r
r
r
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎡⎤
⎢⎥
⎢⎥
⎢⎥
1
1
2
2
()
3
()
()
3
()
t
t
and
t
t
(7.6)
R
=
=
R R
+
r
=
=
r
+
r
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢
⎣⎦
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢
⎣⎦
tot
tot
R
R
R
r
r
r
4
4
5
5
6
6
and at the local level for an arbitrary element
m
F
F
F
d
d
d
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎡⎤
⎢⎥
⎢⎥
⎢⎥
1
1
2
2
()
3
()
()
3
()
F
t
=
=
F
+
F
t
and
d
t
=
=
d
+
d
t
(7.7)
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢
⎣⎦
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢
⎣⎦
tot
m
m
tot
m
m
m
F
F
F
m
d
d
d
4
4
5
5
6
6
m
m
The relationship between local forces and displacements is defined by the local stiffness
matrix
k
, i.e.
F
=⋅
k
d
(7.8)
tot
m
tot
m
m
and the relationship between local and global degrees of freedom is defined by the
matrix
A
, i.e.
d Ar
=⋅
(7.9)
tot
m
tot
m
According to standard element method procedures the global stiffness matrix is then
obtained by summation of contributions from all elements
KAk A
(7.10)
T
mm m
=
⋅
⋅
m
