Civil Engineering Reference
In-Depth Information
2
4
3
5
4
EI
c
=
L
c
2
EI
c
=
L
c
0
0
0
0
0
2
EI
c
=
L
c
4
EI
c
=
L
c
0
0
0
0
0
0
0
4
EI
c
=
L
c
2
EI
c
=
L
c
0
0
0
K
00
=
0
0
2
EI
c
=
L
c
4
EI
c
=
L
c
0
0
0
ð5
:
87Þ
0
0
0
0
4
EI
b
=
L
b
2
EI
b
=
L
b
0
0
0
0
0
2
EI
b
=
L
b
4
EI
b
=
L
b
0
0
0
0
0
0
0
EA
b
=
l
b
According to the static condensation by Eq. (2.149), the condensed stiffness matrix is written as
K
=
K
dd
−
K
dr
K
−1
rr
K
rd
=24
EI
c
=
L
c
+
EA
b
cos
2
β
=
l
b
−1
4
EI
c
=
L
c
+4
EI
b
=
L
b
6
EI
c
=
L
c
6
EI
c
=
L
c
2
EI
b
=
L
b
6
EI
c
=
L
c
6
EI
c
=
L
c
−
ð5
:
88Þ
4
EI
c
=
L
c
+4
EI
b
=
L
b
2
EI
b
=
L
b
Thus, the governing equation is written as:
Ft
()
xt
()
′′
mt
()
−θ
()
t
1
1
′′
mt
()
−θ
()
t
2
2
′
KK
KK
′′
mt
()
−θ
()
t
3
3
=
ð5
:
89Þ
′
T
′′
′′
mt
()
−θ
()
t
4
4
′′
mt
()
−θ
()
t
5
5
′′
mt
()
−θ
()
t
6
6
′′
Pt
()
−δ
()
t
Substituting the first equation in Eq. (5.89) in (5.84) gives
2
3
θ
0
1
ðÞ
θ
0
2
ðÞ
θ
0
3
ðÞ
θ
0
4
ðÞ
θ
0
5
ðÞ
θ
0
6
ðÞ
δ
00
ðÞ
4
5
mx
ðÞ+
c
_
x
ðÞ+
K
x
ðÞ= −
mx
g
ðÞ+
K
0
ð5
:
90Þ
Using the state space numerical integration method of analysis, Eq. (5.84) is expressed as
z
ðÞ=
Az
ðÞ+
H
a
g
ðÞ+
F
s
ðÞ
ð5
:
91Þ
where
,
,
H
=
0
1
x
ðÞ
x
0
ðÞ
0
1
z
ðÞ=
A
=
−
K
=
m
−
c
=
m
