Civil Engineering Reference
In-Depth Information
F
o
L
3
EI
0.1231
0.0737
F
o
L
0.0737
F
o
L
F
o
L
3
EI
0.0740
0.0737
F
o
L
0.0737
F
o
L
F
o
L
2
EI
F
o
L
2
EI
0.0123
0.0123
0.0737
F
o
L
0.0737
F
o
L
F
o
L
2
EI
F
o
L
2
EI
0.1023
0.1023
0.0977
F
o
L
0.0977
F
o
L
F
o
L
2
EI
F
o
L
2
EI
0.0860
0.0860
0.0240
F
o
L
0.0240
F
o
L
F
o
L
2
EI
F
o
L
2
EI
0.0620
0.0620
0.0240
F
o
L
0.0240
F
o
L
Figure 2.14
Graphical illustration of the responses of the two-story frame after loads removed.
2.6 Static Condensation
One important aspect of matrix structural analysis is its ability to reduce the size of the problem
through static condensation, which is a method that is commonly used to compress the stiffness
matrix when the applied forces at certain DOFs are zero. Consider Eq. (2.63), which is repeated
here as follows:
K
K
′
x
F
a
=
ð2
:
141Þ
T
K
′
K
″
−
Θ
″
m
Some entries of
F
a
may be zero, such as when no moment is applied at the rotational degrees of
freedom. Let
d
denotes the number of degrees of freedom that have nonzero applied forces, and
r
denotes the number of degrees of freedom that have zero applied forces. This gives
n
=
d
+
r
in
an
n
-DOF system. The matrices and vectors in Eq. (2.141) are now partitioned as follow:
K
′
x
F
K
K
ad
dd
dr
d
d
K
′
x
=
,
F
=
,
K
=
,
K
′
=
ð2
:
142Þ
a
x
0
K
K
rd
rr
r
r
where the subscript '
d
' denotes the quantities associated with the DOFs with nonzero applied
forces, and the subscript '
r
' denotes the quantities associated with the DOFs with zero applied
forces. Substituting Eq. (2.142) into Eq. (2.141) gives
K
K
K
′
x
F
dd
dr
d
d
ad
K
K
K
′
x
=
0
ð2
:
143Þ
rd
rr
r
r
K
′
T
d
K
′
T
K
″
−
Θ
″
m
r
