Civil Engineering Reference
In-Depth Information
2
x
ðÞ−
ω
0
2
θ
00
ðÞ
a
ðÞ=
ω
ð8
:
13Þ
Ω
ðÞ=
ω
0
2
x
ðÞ−
ω
00
2
θ
00
ðÞ
ð8
:
14Þ
<
<
θ
00
ðÞ=0
Ω
ðÞ≤
Ω
y
if
,
then
ð8
:
15Þ
Ω
ðÞ=
1
θ
00
ðÞ
:
:
m
f
ð
Þ
Ω
ðÞ>
Ω
y
Equations (8.12) to (8.15) are the governing equations of the mass-normalized SDOF system
in the FAM. Assume that the mass-normalized SDOF system is subjected to a continuous
increasing horizontal displacement at the roof and there are also totally
l
steps, as shown in
Eq. (8.2). Structural information at step
k
+ 1 can be predicted by that at step
k
, thus there is
()
k
k
+
1)
∆
(
2
2
( )
k
2
2
x
a
ωω
′
x
ωω
′
=
+
ð8
:
16Þ
(
k
+
1)
2
2
2
2
′
′′
()
k
′
′′
′′
Ω
ωω
′′
ωω
−∆θ
()
k
θ
in which,
a
(
k
+1)
and
Ω
(
k
+1)
are the normalized-mass force and moment at step
k
+ 1. Assume
that the incremental plastic rotation at the plastic hinge location (PHL) equals zero as:
00
ðÞ
Δθ
=0
ð8
:
17Þ
Then, the trial normalized-mass force and moment at step
k
+ 1 is expressed by
( )
k
+
2 2 ()
k
2 2 ()
k
′
′
a
ωω
x
ωω ∆
x
˜
˜
=
+
ð8
:
18Þ
(
k
+
1)
2
2
( )
k
2
2
Ω
ω
′
ω
′′
Ω
ω
′
ω
′′
0
If the moment at step
k
+ 1exceeds the mass-normalized yield moment
Ω
y
, the final mass-
normalized moment equals the yield moment
Ω
y
and the incremental plastic rotation Δ
θ
00
(
k
)
at step
k
+ 1 can be solved by
+
Δ
x
ðÞ
ω
0
2
1
ω
00
2
00
ðÞ
=
2
x
ðÞ
Δθ
−
Ω
y
+
ω
ð8
:
19Þ
Substituting Eq. (8.19) back into Eq. (8.16), the final mass-normalized lateral force and
moment at step
k
+ 1 can be expressed by
( )
k
+
2
( )
k
2
2
()
k
2
′
′
a
ωω
x
ωω ∆
x
=
+
ð8
:
20Þ
( )
k
+
2
2
2
2
Ω
ωω
′
′′
()
k
ωω
−
∆θ
′
′′
()
k
′′
′′
θ
Example 8.1 NSPA for a normalized SDOF system
Consider a mass-normalized SDOF system conversed from Example 3.6. By using Eq. (8.11),
there are
