Civil Engineering Reference
In-Depth Information
An example of having plastic rotations developed at two PHLs in the
i
th beam member is
shown in Figure 2.5(a). This state of the structure can never exist because this
i
th beammember
violates the compatibility condition. Without any force applied to the
i
th member, it should
remain straight, yet the plastic rotations at the two ends restrict the member from being straight.
To ensure the structure deforms in a compatible mode, the member with plastic rotations
Θ
00
is
first isolated from the structure and restoring forces are applied to restore this member back to
the original undeformed shape, as shown in Figure 2.5(b). This induces internal restoring forces
f
RF
as the fixed-end forces on the member. At the global degree of freedom level, the restoring
force is an
n
× 1 vector of the form:
<
=
F
RF
1
F
RF
2
.
F
RFn
= −
K
0
Θ
00
f
RF
=
ð2
:
38Þ
:
;
where
K
0
is the assembled stiffnessmatrix that relates the plastic rotation at the PHLswith the applied
forces at the global DOFs. This
K
0
matrix is the collection of individual
K
0
i
matrices for each beam
member appropriately assembled, where
K
0
i
associated with the
i
th beammember is of the form
<
=
2
3
V
ðÞ
RF
1
M
ðÞ
RF
1
V
ðÞ
RF
2
M
ðÞ
RF
2
6
EI
=
L
2
6
EI
=
L
2
()
= −
K
0
i
Θ
0
i
4
5
θ
0
1
i
θ
0
2
i
4
EI
=
L
2
EI
=
L
f
RF
ðÞ
=
= −
ð2
:
39Þ
:
;
−6
EI
=
L
2
−6
EI
=
L
2
2
EI
=
L
4
EI
=
L
where
E
is the Young's modulus,
I
is the moment of inertia, and
L
is the length of the beam
member, subscripts '1' and '2' denote the '1'-end and the '2'-end of the beam member, respec-
tively, and
V
(
i
)
denotes fixed-end shear and
M
(
i
)
denotes fixed-end moment of the
i
th beam
member. In addition to the restoring forces
f
RF
that are applied at the global DOFs, plastic
rotations
Θ
00
induce residual moments
m
R
at the PHLs as shown in Figure 2.5(b). At the local
PHL level, the residual moment is a
q
× 1 vector in the form:
<
=
m
R
,
1
m
R
,
2
.
m
R
,
q
= −
K
00
Θ
00
m
R
=
ð2
:
40Þ
:
;
where
K
00
is a matrix that relates the plastic rotations at the PHLs with the moments at these
PHLs. This
K
00
matrix is the collection of individual
K
0
i
matrices for each beam member appro-
priately assembled, where
K
0
i
associated with the
i
th beam member is
()
= −
= −
K
0
i
Θ
0
i
m
ðÞ
θ
0
1
i
θ
0
2
i
4
EI
=
L
2
EI
=
L
R
,
1
m
ðÞ
m
R
ðÞ
=
ð2
:
41Þ
2
EI
=
L
4
EI
=
L
R
,
2
