Civil Engineering Reference
In-Depth Information
N
(
)
∑
T
T
()
T
T
t
AR
+
AR
+
Ac r
+
Ak
r
n n
n n
n e t
nae t
n
n
n
=
1
(9.73)
N
(
)
∑
T
T
T
=
A mAr A cAr A kAr
+
+
nnn
nnn
nnnt t
n
1
=
Defining the structural properties
T
⎡
⎤
AmA
M
C
⎡⎤
⎢
⎥
N
⎢⎥
=
∑
T
A cA
(9.74)
⎢
⎥
⎢⎥
⎢
⎥
n
1
⎢
⎣⎦
⎢
=
K
T
AkA
⎥
⎣
⎦
n
the motion induced properties
⎡
T
⎤
N
C
A c
A
⎡ ⎤
ae
∑
ae
=
(9.75)
⎢
⎥
⎢ ⎥
K
T
⎢
Ak A
⎥
⎣ ⎦
ae
n
1
⎣
⎦
=
ae
n
and the load vectors
T
⎡
⎤
N
⎡ ⎤
R
R
∑
AR
=
(9.76)
⎢
⎥
⎢ ⎥
()
T
()
t
t
⎢
⎣ ⎦
⎢
AR
⎥
n
1
⎣
⎦
=
n
then Eq. 9.73 may be written
()
()
t
t
Mr
+
Cr
+
Kr
=
R
+
R
+
R
(9.77)
tot
tot
tot
ae
where
()
()
()
t
t
t
R
=
C
r
+
K
r
(9.78)
ae
ae tot
ae tot
Eq. 9.77 may be split into a time invariant (mean static)
equilibrium condition
and
tot
Since
r r
=
r
=
r
tot
(
)
KK r R
(9.79)
−
=
ae
and a purely dynamic equilibrium condition
() (
) () (
) ()
()
t
t
t
t
(9.80)
Mr
+−
C C
r
+−
K K
r
=
R
ae
ae
