502 



Profs. John Perry and W. E. Ayrton. [Dec. 11, 



it quite easy to prove that any ordinate, as NK for instance, repre- 

 sents the value of m for the point K', and B"B' is m T .* 



In fig. 6 the diagram for m has been altered in shape by the student, 

 but this extra labour was not necessary. 



7. Any ordinate to the curve AE.EEB, fig. 2, represents the value 



at that place of — , where PJ is the modulus of elasticity of the sub- 

 El 



stance of which the beam is composed, and J is the moment of inertia 

 of. the section about its neutral line, any unit of measurement being 

 employed. 



The value of d at any point P is shown by the ordinate PP f to the 

 ourve AP'B', and BB' equals e^, the curve AP'B' beiug obtained by 

 raising at a number of points, such as P, the ordinate PP', measured 

 to any convenient scale, and representing the area AEE'P. It is 

 evident that is the total area of the figure AP'B'B, the scale of 

 measurement being computed in the manner described below. 



8. To find f. — A E 1 PJ 2 E 3 B, fig. 3, shows the value of — at every 



EI 



point. Take A T any convenient distance, and raise the perpendicular 

 TT' ; make TS equal to BE S , T2 equal to the ordinate at E 2 and Tl 

 equal to the ordinate at JS7 X . Join A with the points, 1, 2, 3, &c, and 

 produce if necessary. Now it is quite evident that any ordinate of the 



figure AA^A^A^A^'B represents — to a known scale ; thus, for 



EI 



example, PA! represents PA divided by JEI at the point P. (This 

 diagram might, of course, have been easily obtained numerically.) 

 Now at any point make the ordinate PP' represent to any suitable 

 scale the area AA^A^AIP, and it is evident that any ordinate of the 

 ourve through all such points as P' , say AP'B', is / to a known scale. 

 It is also evident that g 1 is the total area of the figure AP'B'B.f 



9. To find g. — At any point P we now know (see fig. 1) the value 



of m; we also know the value of — ; we can, therefore, compute the 



EI 



* For we know that the ordinate KK" represents the bending moment at the 

 point K' if the beam is merely supported at the ends. Also NK" represents the 

 bending moment about K' due to the proper supporting force at A, and therefore 

 NK, the difference, is m the bending moment at K' due to the loading merely 

 between A and K' . 



f Generally as to scale : — When at the point P, fig. 4, we raise a perpendicular PP" 

 which represents, according to any scale, by its length the area of PAP ', then if the 

 ordinate PP' was to such a scale that one centimetre represents a units, and if AB 

 is the span to such a scale that one centimetre represents b feet, if the area AP'P is 

 c square centimetres, and if it is represented by a line PP" whose length is e centi- 

 metres, it is evident that the ordinates of our new curve AP"B"B have a scale such 



that one centimetre represents a ~ units. 



