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where MF1 and MF2 are the modified end fixing moments at ends 1 and 2 of the beam. Where the end rotations of the beam, 01 and 02, are zero, equations (12.48) and (12.49) will give the end moments in the beam.

The above are the general relationships applicable to prismatic members with rotational springs at each end resting on unyielding supports. If the fixing detail between the connection members and the beam end is rigid, the associated value of k will be infinity and the rigidity factor y will be unity. If it functions as a pinned connection, both k and the rigidity factor will be zero.

In Table 12.4 the area of the free bending moment diagram and the position of its centroid from the left hand support are given for commonly occurring load cases.

An example of a beam designed at the ULS condition, with end connections analysed as semi-rigid and also as fully fixed, is given in Example 12.8.2.

With the semi-rigid analysis, it is shown that for a relatively small end rotation the end moment in a semi-rigid connection can be well below the fully fixed value, resulting in a more economic connection design. A more realistic assessment of the deflection of the beam will be obtained and the strength of the member at mid-span can be checked against the increased span moment. It should also be noted that even if the connection is designed to resist the fully fixed moment, there will still be rotation at the connection, and although it will be less than that obtained had the connection been designed as semi-rigid, additional moment will nevertheless be transferred to the beam span and there will also be an increase in its mid-span deflection. These effects will not be taken into account in the rigid analysis.

See Examples 12.8.2 and 12.8.3.

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