### Continuous Beams - Steel Structures.

The most general method is the use of Clapeyron’s Theorem of Three Moments.

The theorem applies only to any two adjacent spans in a continuous beam and in its simplest form deals with a beam which has all the supports at the same level, and has a constant section throughout its length.

The proof of the theorem results in the following expression:

where MA,MB and MC are the numerical values of the hogging moments at the sup- ports A, B and C respectively, and the remaining terms are illustrated in

**Fig. 10.7**.

In a continuous beam the conditions at the end supports are usually known, and these conditions provide starting points for the solution.

The types of end conditions are three in number:

**(1)**simply supported

**(2)**partially ﬁxed, e.g. a cantilever

**(3)**completely ﬁxed, i.e. the end of the beam is horizontal as in the case of a ﬁxed beam.

The SF at the end of any span is calculated after the support moments have been evaluated, in the same manner as for a ﬁxed beam, each span being treated separately.

It is essential to note the difference between SF and reaction at any support, e.g. with reference to

**Fig. 10.7**the SF at support B due to span AB added to the SF at B due to span BC is equal to the total reaction at the support.

If the section of the beam is not constant over its whole length, but remains constant for each span, the expression for the moments is rewritten as follows:

in which I1 is the second moment of area for span L1 and I2 is the second moment of area for span L2.

Fig. 10.7 Clapeyron’s Theorem of Three Moments |

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