120 APPLIED MECHANICS 
bending moments at these points, and these are supplied by the method 
of fixing. In Fig. 160 the beam is shown with flanges, which are sup- 
posed to be bolted to the walls, 
and the forces PP shown produce 
the necessary negative bending 
moments just referred to.* Also 
these forces PP produce a uniform 
bending action over the whole of 
the beam. Let aa’ be the bending 
momentat A. Draw a’b’ parallel Ah 
to ab, cutting ac and be at e and g— W----l,--- 
f respectively, then the shaded kK ee 
figure will be the actual bending C Fh 
‘i 1 
moment diagram for the beam ‘ wir : ! 
nen lag it 2 Ww —— I 
AB, with fixed ends and loaded 
at the centre, ab’ being the base 
of the diagram, This diagram Fra. 160. 
shows that the portion EF of the 
beam is subjected to positive bending, and that the parts AE and BF 
are subjected to negative bending; also, that there is no bending 
moment at either E or F. Hence if the beam be cut at E and F, and 
the parts be again connected by pin joints, the axes of the pins being 
perpendicular to the plane of bending, the jointed beam will behave 
exactly as the solid beam. Hence the original beam is equivalent to 
two cantilevers AE and BF loaded at E and F, and a beam EF sup- 
ported at E and F, and loaded at the centre, as shown in the lower 
part of Fig. 160. 
The slope of the cantilever AE at E is represented by the area of the 
triangle aea’ (Art. 126), and the slope of the beam EF at E is repre- 
sented by the area of the triangle cec’. But these two slopes must be 
equal, therefore the triangles aea’ and cec’ are equal in area, and as they 
are also similar, it follows that-a’e=c’e. Therefore the points of inflexion 
and the middle point of the beam divide the span into four equal parts, 
and L, =1L, also L,=4L. 
The cantilever A, E, of length=}L carries a load=$W at E,, hence 
by Art. 120 the deflection at E, = (S\G)+ SET = ae : 
The beam E,F, of length=4L carries a load W at its centre C,, 
hence by Art. 122 the deflection of C, below E, 
ae L\3". 2a 
=w(5) +48EI= 
The total deflection of the whole beam at the centre is therefore 
equal to We + 
192E1 
* In order that the theory developed in this Article and the next may be 
strictly applicable, the method of fixing must not hinder any horizontal move- 
ment of the beams as a whole at the ends, The fixing is only supposed to keep 
the beam horizontal at the ends, 
