Appendix A: The Stiffness Matrix
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Fig. A.4 Local and global axis system
Direct Stiffness Method
Even in the trivial example of Fig. A.1, it is seen that for determining K 11 , the contribution from members connected at joint 1 has been taken into account. In this section,
we present a more formalised approach to the formation of the stiffness matrix. It
will be shown that the overall stiffness matrix of the structure will be generated
automatically by the computer, based on the geometric and elastic properties of the
individual element.
For this, a local coordinate system is chosen for a particular member. Global axis
system is the common coordinate system, which is valid for the entire structure.
The two-coordinate systems may be different, depending on the orientation of the
member. The difference between the two is explained in Fig. A.4, for a bar element
which can take up axial force only.
The local and global axes system is related by the transformation matrix. If the
displacements in the local axis system for a particular member are {δ
} and the
displacements in the corresponding global axis system is {δ}, then
{δ
} = [T ]{δ
}
(A.1)
where [T ] is the member transformation matrix.
For the truss, element of Fig. A.4, the stiffness matrix of the member in local
coordinate is
[k] e =
E A
L
1 −1
−1 1
(A.2)
and
[T ] =
l 1 m 1 0 0
0 0 l 1 m 1
(A.3)