## 1 1 1

so that for a member aligned with the x axis, joining nodes i and j subjected to nodal forces Fxi and Fxj, we have

The solution proceeds in a similar manner to that given in the previous section for a spring or spring assembly. However, some modification is necessary since frameworks consist of members set at various angles to one another. Figure 12.3 shows a member of a framework inclined at an angle 6 to a set of arbitrary reference axes x, y. We shall refer every member of the framework to this global coordinate system, as it is known, when we are considering the complete structure but we shall use a member or local coordinate system x, y when considering individual members. Nodal forces and displacements referred to local coordinates are written as F, u etc so that Eq. (12.21) becomes, in terms of local coordinates

where the element stiffness matrix is written

In Fig. 12.3 external forces Fxi and Fxj are applied to nodes i and j. It should be noted that Fyj and Fvj do not exist since the member can only support axial forces. However, Fx i and Fx j have components Fxh f>v- and Fx j, Fyj respectively,

Fig. 12.3 Local and global coordinate systems for a member of a plane pin-jointed framework.

Fig. 12.3 Local and global coordinate systems for a member of a plane pin-jointed framework.

so that, whereas only two force components appear for the member in terms of local coordinates, four components are present when global coordinates are used. Therefore, if we are to transfer from local to global coordinates, Eq. (12.22) must be expanded to an order consistent with the use of global coordinates, i.e.

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