2.9.4 Structure Stiffness Matrix

Equation 2.138 has been expressed in terms of the coordinate system of the individual members. In a structure consisting of many members there would be as many systems of coordinates as the number of members. Before the internal actions in the members of the structure can be related, all forces and deflections must be stated in terms of one single system of axes common to all — the global axes. The transformation from element to global coordinates is carried out separately for each element and the resulting matrices are then combined to form the structure stiffness matrix. A separate transformation matrix [T] is written for each element and a relation of the form

is written in which [T]n defines the matrix relating the element deformations of element n to the structure deformations at the ends of that particular element. The element and structure forces are related in the same way as the corresponding deformations as

where [q]n contains the element forces for element n and [W ]n contains the structure forces at the extremities of the element. The transformation matrix [T ]n can be used to transform element n from its local coordinates to structure coordinates. We know, for an element n, the force-deformation relation is given as

Substituting for [q]n and [d]n from Equations 2.140 and 2.141 one obtains

[K]n is the stiffness matrix that transforms any element n from its local coordinate to structure coordinates. In this way, each element is transformed individually from element coordinate to structure coordinate and the resulting matrices are combined to form the stiffness matrix for the entire structure.

The member stiffness matrix [K] n in global coordinates for a truss member shown in Figure 2.75, for example, is given as

To construct [K]n for a given member it is necessary to have the values of 1 and m for the member. In addition, the structure coordinates i, j, k, and l at the extremities of the member must be known.

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