Civil Engineering Reference
In-Depth Information
which is exactly equal to the applied force of 3
F
o
multiplied by the length of the
column
L
.
The resulting force-displacement relationship of the column (or the pushover curve) is
shown in Figure 2.4(c).
2.3 Nonlinear Structural Analysis of Moment-Resisting Frames
For a multi-degree-of-freedom (MDOF) system, the derivation of the FAM also begins with a
similar concept of inelastic displacements while representing the force and displacement quan-
tities in vector form. Consider a structure having
n
degrees of freedom (DOFs), the displace-
ment can be written in vector form as
<
=
<
=
x
0
1
x
0
2
.
x
0
n
x
0
1
x
0
2
.
x
0
n
x
=
x
0
+
x
00
=
+
ð2
:
35Þ
:
;
:
;
where
x
represents the total displacement vector,
x
0
is the elastic displacement vector, and
x
00
is
the inelastic displacement vector.
Consider a moment-resisting framed structures, the moments at the two ends of the members
are typically the critical points where yielding occurs. Denote these yielding locations as plastic
hinge locations (PHLs). Let the total moment vector
m
at the locations where plastic hinges
may form be described by the equation
<
=
<
=
m
0
1
m
0
2
.
m
0
q
m
0
1
m
0
2
.
m
0
q
m
=
m
0
+
m
00
=
+
ð2
:
36Þ
:
;
:
;
where
m
0
is the elastic moment vector due to elastic displacement
x
0
, and
m
00
is the inelastic
moment vector due to inelastic displacement
x
00
. The value
q
represents the total number of
potential PHLs in the moment-resisting frame.
Consider first the inelastic moment vector,
m
00
. When plastic rotations occur at certain PHLs
in the structure, these plastic rotations are replaced with a set of fictitious forces. Define the
plastic rotation vector,
Θ
00
,as
<
:
=
;
θ
0
1
θ
0
2
.
θ
0
q
Θ
00
=
ð2
:
37Þ
