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Steel Designers' Manual - 6th Edition (2003)
330 Beam analysis
Io _J
B
Fig. 10.4 Bending moment diagram for point load on fixed-end beam
It will be seen, therefore, that the fixing moments for any built-in beam on level supports
can be calculated provided that the area of the free BM diagram and the position
of its centre of gravity are known.
For point loads, however, the principle of reciprocal moments provides the
simplest solution.
With reference to Fig. 10.4,
M
M
A
B
Wab b Wab
L L L
Wab
2
= ¥ = 2
2 a Wa b
= ¥ = 2
L L L
i.e. the fixing moments are in reciprocal proportion to the distances of the ends of
the beam from the point load.
In the case of several isolated loads, this principle is applied to each load in turn
and the results summed.
It should be noted that appropriate formulae for built-in beams are given in the
Appendix Bending moment, shear and deflection tables for built-in beams.
10.3.2 Beams with supports at different levels
The ends are assumed, as before, to be horizontal.
The bent form of the unloaded beam as shown in Fig. 10.5 is similar to the bent
form of two simple cantilevers, which can be achieved by cutting the beam at the
centre C, and placing downward and upward loads at the free ends of the cantilevers
such that the deflection at the end of each cantilever is d/2.