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k_

and by comparison with Eq. (12.1) we see that the stiffness matrix for this spring element is

' k -k . -k k which is a symmetric matrix of order 2x2.

12.3 Stiffness matrix for two elastic springs in line

Bearing in mind the results of the previous section we shall now proceed, initially by a similar process, to obtain the stiffness matrix of the composite two-spring system shown in Fig. 12.2. The notation and sign convention for the forces and nodal displacements are identical to those specified in Section 12.1.

First let us suppose that itj = it\ and u2 = »3 = 0. By comparison with the single spring case we have

Secondly, we put a, = = 0 and i<2 = h2. Clearly, in this case, the movement of node 2 lakes place against the combined spring stiffnesses ka and A"b. Hence fx.2 = (/v;1 + kb)lt2

Hence the reactive force Fx l (=—k.iu2) is not directly affected by the fact that node 2 is connected to node 3. but is determined solely by the displacement of node 2. Similar conclusions are drawn for the reactive force Fx Finally, we set it| = it2 = 0, it^ = itand obtain

-^jLsmMsmsüiSiSLr

Fig. 12.2 Stiffness matrix for a two-spring system.

Superimposing these three displacement states we have, for the condition = uu u2 = u2, «3 = «3

Writing Eqs (12.11) in matrix form gives

Comparison of Eq. (12.12) with Eq. (12.1) shows that the stiffness matrix [K] of this two-spring system is

'Fx, r |

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