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Transformation of displacements and forces

To enable assembly of contributions from each constituent element meeting at each joint, it is necessary to transform the force/displacement relationships to some global coordinate system, by means of a transformation matrix [7(e)]. This matrix is derived by establishing the relationship between the displacements (or forces), in local coordinates x',y', and those in global coordinates x,y. Note that the element inclination a, is taken to be positive when acting clockwise viewed from the origin along the positive z-axis, and is measured from the positive global Jt-axis. With reference to Fig. 9.23, for node i, u) = Ui cos a + Vj sin a, and v- = — h; sin a + Vj cos a.

Fig. 9.23. Transformation of displacements.

Similarly, for node u j cos a + vj sin a, and v'j = —uj sin a + vj cos a.

Writing in matrix form the above becomes:

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