Civil Engineering Reference
In-Depth Information
sEI
=
L
2
−
sEI
=
L
2
sEI
=
L
scEI
=
L
K
0
i
=
ð7
:
38Þ
sEI
=
L
2
−
sEI
=
L
2
scEI
=
L
sEI
=
L
Once the
K
0
T
i
matrix is derived, the
K
0
i
matrix can be written as:
2
4
3
5
2
4
3
5
2
4
3
5
sEI
=
L
2
sEI
=
L
2
M
a
1
M
b
1
M
a
2
M
b
2
M
a
3
M
b
3
M
a
4
M
b
4
V
1
a
V
1
b
M
1
a
M
1
b
V
2
a
V
2
b
M
2
a
M
2
b
sEI
=
L
scEI
=
L
K
0
i
=
=
=
ð7
:
39Þ
−
sEI
=
L
2
−
sEI
=
L
2
scEI
=
L
sEI
=
L
and it represents the fixed-end forces due to unit plastic rotations at PHL '
a
' and PHL '
b
' as
shown in Eq. (7.39) based on Maxwell's reciprocity theorem. Note that if the axial force is zero,
the stability coefficients reduce to
s
=4,
c
= 0.5,
s
= 6, and
s
0
= 12, and the stiffness matrix
K
0
T
i
in Eq. (7.38) becomes
6
EI
=
L
2
−6
EI
=
L
2
4
EI
=
L
2
EI
=
L
K
0
i
=
ð7
:
40Þ
6
EI
=
L
2
−6
EI
=
L
2
2
EI
=
L
4
EI
=
L
7.2.3 Stiffness Matrix
[
K
0
i
]
The stiffness matrix
K
0
i
relates the moments at the PHLs with a corresponding unit plastic rota-
tion at the PHL. To determine the
K
0
i
matrix, the goal is to compute
m
aa
,
m
ab
,
m
ba
, and
m
bb
as
shown in Figure 7.4.
The fixed-end forces derived in the
K
0
i
matrix can be used to calculate the stiffness matrix
K
0
i
. For example, the first column of the 4 × 2
K
0
i
matrix in Eq. (7.39),
V
1
a
=
sEI
=
L
2
,
m
1
a
=
sEI
/
L
,
V
2
a
= −
sEI
=
L
2
, and
m
2
a
=
scEI
/
L
, represents the shear and moment at the ends of the
beam member due to a unit plastic rotation at the left plastic hinge (i.e. PHL '
a
'), as shown
in Figure 7.4. Similarly, the second column of 4 × 2
K
0
i
matrix in Eq. (7.39),
V
1
b
=
sEI
=
L
2
,
m
1
b
=
scEI
/
L
,
V
2
b
= −
sEI
=
L
2
, and
m
2
b
=
sEI
/
L
, represents the shear and moment at the ends
of the beam member due to a unit plastic rotation at the right plastic hinge (i.e. PHL '
b
'), as
shown in Figure 7.4. Then moments at the two plastic hinges for each of the two cases become:
Case '
a
': Imposing a unit plastic rotation
θ
0
a
= 1 and
θ
0
b
= 0 gives
m
aa
=
m
1
a
=
sEI
L
,
m
ba
=
m
2
a
=
scEI
ð7
:
41Þ
L
y
y
θ
a
=1
m
2
b
m
2
a
Case
b
EI
m
ba
m
bb
P
x
P
m
ab
P
x
P
m
aa
Case
a
m
1
a
m
1
b
EI
θ
b
=1
V
1
a
V
2
a
V
1
b
V
2
b
Figure 7.4
Displacement patterns for computation of moments due to unit plastic rotations.
