DEFLECTION OF BEAMS 



303 



Thus when x = a, M => ; when x - 0, M = - ; and 



8 6 



when x =* ^. = 0-577a, M = 0. 



Therefore the bending moment curve is a parabola, and the 

 bending moment increases from - - at the point of fixing to 



2 



+ - - at the centre of the beam, while at a point situated at a 



distance 0-577a from the centre of the beam the bending moment 

 vanishes. 



160. Let A and B be two sections of a beam taken very close 

 together (Fig. 105), x being the distance between these sections 

 and w $x the load on this elementary length of the beam. 



^M 



JL 



o- 



FIG. 105. 



Let M be the bending moment at A and M + 8M the bending 

 moment at B. Also F is the shearing force at A. 

 Taking moments about O, 



F $x - - w 



Jl 



M + 8M - M 

 or 8M = F &c 



taking w &r 2 to be negligibly small in comparison with F 



When Sx is made infinitely small, 



dM. 



F = 



dx 



that is, the shearing force at a section is the rate at which the 

 bending moment is changing with respect to the length. 



161. Let CD and C 1 D 1 be two sections of a beam, &r apart 

 (Fig. 106), the bending moment at CD being M, and at C J D 1 



