Appendix A: The Stiffness Matrix
611
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
δ
1
δ
2
δ
3
· · ·
δ
4
δ
5
δ
6
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
l 1 m 1 0
. . . 0 0 0
−m 1 l 1 0
. . . 0 0 0
0 0 1
. . . 0 0 0
· · · · · · · · · · · · · · · · · · · · ·
0 0 0
. . . l 1 m 1 0
0 0 0
. . . −m 1 l 1 0
0 0 0
. . . 0 0 1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
δ 1
δ 2
δ 3
· · ·
δ 4
δ 5
δ 6
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(A.6)
or
{δ
} = [T ] {δ}
(A.7)
The stiffness matrix of the member in local axis system is
[k] e =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
E A
L
0 0 −
E A
L
0
0
12E I
L 3
6E I
L 2
0 −
12E I
L 3
6E I
L 2
4E I
L
0 −
6E I
L 2
2E I
L
E A
L
0
0
Symmetrical
12E I
L 3
−
6E I
L 2
4E I
L
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(A.8)
The element stiffness matrix in the global coordinate system can be formed by
using Eq. (A.4).
Overall Stiffness Matrix
The stiffness matrix of an element of the structure forms the basic component. It is
now discussed here, as to how the element stiffness matrix of the individual element
is assembled, to form the overall stiffness matrix of the structure.
It might have been noticed earlier that the stiffness at a joint is obtained by adding
the stiffness of all the members meeting at the joint.
The degrees of freedom of the structure are first numbered starting with 1 and
ending with NP, where NP is the total degrees of freedom. The restraints are then
numbered beyond NP. This numbering is referred to as degrees of freedom corresponding to global numbering or global degrees of freedom. All the restraints can
be given the number (N P + 1). This procedure will save some storage space.
We have already introduced the concept of local axis system and the global axis
system. We shall henceforth deal with two different numberings in the global axis
system for the displacements at member ends. The first set of numbering is referred to