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in which [/V] is the matrix of shape functions. In this case, N\ = I — x/L and N2 = x/L, and hence vary linearly over the element, as shown in Fig. 9.25. Note that the shape functions have the value unity at the node corresponding to the nodal displacement being interpolated and zero at all other nodes (in this case at the only other node), and is a property of all shape functions.

Fig. 9.25. Shape functions for the axial force rod element.

Element stiffness matrix in local coordinates

Consider the axial force element shown in Fig. 9.24. The only strain present will be a direct strain in the axial direction and is given by eqn. (9.4) as ex = e = 3 u/dx

Substituting from eqn. (9.32), gives e = d[N]{u}/dx = [B]{m} (9.33) and, taking the virtual strain to have a similar form to the real strain, gives s = [B]{u] (9.34)

where

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