Civil Engineering Reference
In-Depth Information
XX
XXXX
XXX
XXXX
XXX
XX
P
∆
=−
FEPM
(5.36)
The matrix is reduced by the appropriate reduction method. The degrees of
freedom or joints used to represent the stiffness should be placed at the outer
ends of the matrix. These will be the degrees of freedom in the outer col-
umns of Equation 5.37. All of the values above and below the main diagonal
of the matrix will be reduced to zero except those retained in the outer col-
umns. The resulting matrix is shown in Equation 5.37 and is the N-matrix.
K
K
∆
P
S
A
g
g
g
g
XX
X
d
d
d
d
X
X
X
=
X
X
X
V
E
X
X
X
K
K
∆
P
g
g
g
g
(5.37)
The information contained in the N-matrix is used in two ways. The values
in the extreme boundaries of the matrices represent the stiffness, deforma-
tion, or force (
K
g
, ∆
g
, and
P
g
) of the structure related to the retained degrees
of freedom or joints. These values can be combined with other members
in a stiffness analysis. The interior portions of the matrices represent the
values of the stiffness, deformation, or force of the degrees of freedom or
joints eliminated in the reduction. These values are used to find the deforma-
tions and forces of these degrees of freedom or joints once the main analysis
results are known. The N-matrix can be compressed as shown in Equation
5.38. The interior stiffness values can be inverted for later processing.
K K
XX X
XX X
XX X
XX X
K
∆
∆
∆
∆
∆
∆
P
S
A
g
g
g
g
−
1
−
1
=
−
1
V
E
−
1
K
P
g
g
g
g
(5.38)
