Civil Engineering Reference
In-Depth Information
equilibrium in addition to being compatible. Equating
f
a
in both Eqs. (2.43) and (2.44) and
solving for the inelastic displacements
x
00
gives
x
00
=
K
−1
K
0
Θ
00
ð2
:
45Þ
Due to the induced equivalent forces
f
a
which produce inelastic displacements
x
00
in the struc-
ture, additional moments are also induced at the PHLs. Denoting this induced moment vector as
m
p
, it is related to the inelastic displacement
x
00
by the equation
m
p
=
K
0
T
x
00
ð2
:
46Þ
Then substituting Eq. (2.45) into Eq. (2.46) gives
m
p
=
K
0
T
K
−1
K
0
Θ
00
ð2
:
47Þ
Note that the
K
0
T
matrix is used in Eq. (2.46). Recall from Eq. (2.38) that the
K
0
matrix relates
the plastic rotation at the PHLs with the applied force at the DOFs. Therefore, from the theory of
reciprocity, it follows that
K
0
T
relates the inelastic displacement at the DOFs with the moments
at the PHLs, i.e.
<
:
=
;
2
3
<
:
=
;
<
:
=
;
2
3
<
:
=
;
K
0
11
K
0
21
K
0
n
1
K
0
12
K
0
22
K
0
n
2
. .
.
.
. .
K
0
1
q
K
0
2
q
K
nq
K
0
11
K
0
12
K
0
1
q
K
0
21
K
0
22
K
0
2
q
.
θ
0
1
θ
0
2
.
θ
0
q
m
p
1
m
p
2
.
m
pq
x
0
1
x
0
2
.
x
0
n
f
a
1
f
a
2
.
f
an
4
5
4
5
=
and
=
.
.
.
.
.
K
0
n
1
K
0
n
2
K
nq
ð2
:
48Þ
Finally, the inelastic moments
m
00
at the PHLs in Figure 2.5(a) can be determined by
summing the residual moments
m
R
at the PHLs shown in Figure 2.5(b) and the induced
moment
m
p
shown in Figure 2.5(c), i.e.
m
00
=
m
R
+
m
p
ð2
:
49Þ
Substituting Eqs. (2.40) and (2.47) in Eq. (2.49) gives the equation of inelastic moments
m
00
as a
function of plastic rotations
Θ
00
as:
m
00
= −
K
00
−
K
0
T
K
−1
K
0
Θ
00
ð2
:
50Þ
Equation (2.50) represents inelastic moment vector due to the plastic rotations within the struc-
ture with no external applied force. For example, if an earthquake causes plastic rotations
within the structure, then the inelastic moments represent the forces remaining in the members
after the earthquake motion subsides.
Now consider the relationship between the elastic moments
m
0
and the elastic displacements
x
0
due to the external applied static load
F
a
. Similar to Eq. (2.43) where the inelastic
