Civil Engineering Reference
In-Depth Information
The resulting member stiffness matrix is 14×14 in size with the terms
T
1
,
T
2
,
T
3
, and
T
4
defined as follows:
(
)
l
sinh
l
L
TG
=
κ
(
)
−
(
)
1
T
2
cosh
l
L
1
−
l
Lsinh
l
L
(
)
−
cosh
l
L
1
TG
=
κ
(
)
−
(
)
2
T
2
cosh
l
L
1
−
l
Lsinh
l
L
(
)
−
(
)
Gκ
sinh L cosh
l
l
l
L
T
=
T
(
)
−
(
)
l
2
cosh
l
L
1
−
l
Lsinh
l
L
(
)
ll l
l
LsinhL
−
Gκ
T
=
T
(
)
−
(
)
4
l
2
cosh
L
1
−
l
L
sinh
l
L
Gκ
l
=
T
EI
w
In the equation for
l
,
k
T
is the St. Venant torsion constant, which is typi-
cally the polar moment of inertia,
I
x
. The value of
I
w
is known as the warp-
ing constant. Both of these values are normally tabulated in handbooks or
specifications.
5.13 Sub-StRuctuRing
When a structure is of large enough size that the contents for the global
joint stiffness matrix cannot be contained in the RAM of a computer, the
matrix can be transformed into segments by reduction or decomposition.
The resulting transformed matrix can take many forms depending on the
process used. One of the common transformations is the N-matrix, due to
the configuration of resulting values. The following is a general descrip-
tion of the operation used to solve large systems using the N-matrix. This
method can be used by operating on the individual degrees of freedom or
on the entire joint as matrix operations.
The original equation set is normally a sparse matrix with most of
the values near the main diagonal. Equation 4.34 represents the original
stiffness solution set. For clarity, the zero values are left out of the matrices
and X indicates where values exist.
=
K
∆
P
g
g
g
