124 SUPPLEMENT TO CHAP. VTL [CHAP. HL 



value of this greatest moment is P 1. Hence we have from (11) P I 



2 TI 27 T I 27 



= T or P = , or as great as for the same beam supported at 



n n t lo 



P I 



the ends only. If the weight is in the middle, however, we have 



8 



2TI 16 T I 



or P = - , or twice as much as the same beam supported at the 



ends. 



19. The above ia sufficient to introduce the reader to the theory of 

 flexure. He can now discuss for himself the above case for uniform load, 

 and prove that the maximum deflection is at the centre and equal to 



884 E I' ^ a * *^ e S reatest moment is at the end and equal to pF, 



24 T I 



and that the breaking weight is p I ; . We may also observe that 



h I 



both in the beam fixed at one end and supported at the other, and fixed at 

 both ends, the moment at the fixed end is positive. From this end it de- 

 creases towards the weight, and finally reaches a point where the moment 

 is zero. Past this point the moment becomes negative, and in the case of 

 the beam, free at the other end, increases gradually to a maximum and then 

 decreases to zero. In the beam fixed at both ends, it increases to a maxi- 

 mum, then decreases to zero, then changes sign and becomes positive and 

 increases to the other end. These points at which the moments are zero 

 are points of inflection, because here the curvature changes from convex to 

 concave, or the reverse. 



They can be easily found from the equations for the moments by finding 

 the value of x necessary to make the moments zero. 



Thus, for a beam fixed at one end and supported at the other, uniform 



load, the inflection point is at a distance from the fixed end x = . For 



both ends fixed, we make M = -p [P 6 (I as) x] = 0, and find x = 



12 



- (3 T Vty I = 0.21131 I and 0.7887 I The reader will also do well to 

 6 



discuss the curves of moments. He will find the moments represented by 

 the ordinates to parabolas, and limited by straight lines similarly to Figs. 

 73 and 75, PI. 18. 



We shall give in the Supplement to Chap. XIV. much more general 

 formulae, from which, for one or both ends fixed or free, the moments and 

 reactions at the supports may be found, when any number of spans of nary- 

 ing length intervene, for single load anywhere upon any span, or for load 

 uniformly distributed over any span. 



