Civil Engineering Reference
In-Depth Information
Equations (17.10)-(17.12) are expressed in matrix form as
.
/
0
!
"
.
/
0
!
"
F
A
F
B
F
C
k
AB
−
k
AB
0
w
A
w
B
w
C
=
−
k
AB
k
AB
+
k
BC
−
k
BC
(17.13)
0
−
k
BC
k
BC
×
Note that in Eq. (17.13) the stiffness matrix is a symmetric matrix of order 3
3,
which, as can be seen, connects
three
nodal forces to
three
nodal displacements. Also,
in Eq. (17.5), the stiffness matrix is a 2
2 matrix connecting
two
nodal forces to
two
nodal displacements. We deduce, therefore, that a stiffness matrix for a structure in
which
n
nodal forces relate to
n
nodal displacements will be a symmetric matrix of the
order
n
×
×
n
.
In more general terms the matrix in Eq. (17.13) may be written in the form
k
11
k
12
k
13
[
K
]
=
k
21
k
22
k
23
(17.14)
k
31
k
32
k
33
in which the element
k
11
relates the force at node 1 to the displacement at node 1,
k
12
relates the force at node 1 to the displacement at node 2, and so on. Now, for the
member connecting nodes 1 and 2
k
11
k
12
[
K
12
]
=
k
21
k
22
and for the member connecting nodes 2 and 3
k
22
k
23
[
K
23
]
=
k
32
k
33
Therefore we may assemble a stiffness matrix for a complete structure, not by the
procedure used in establishing Eqs (17.10)-(17.12) but by writing down the matrices
for the individual members and then inserting them into the overall stiffness matrix
such as that in Eq. (17.14). The element
k
22
appears in both [
K
12
] and [
K
23
] and will
therefore receive contributions from both matrices. Hence, from Eq. (17.5)
k
AB
−
k
AB
[
K
AB
]
=
−
k
AB
k
AB
and
k
BC
−
k
BC
[
K
BC
]
=
−
k
BC
k
BC
