ON THE INFLUENCE OF FORM ON STRENGTH. 425 



suspended bar of uniform sectional area in its body part, and let fig. 8 re- 

 present another suspended bar of the same sectional area at its lower part as 

 the bar of fig. 7, but having an increased section at its upper part, such in- 

 crease being ahvnpthj made ; then, if these two bars be supposed to be sub- 

 jected to equal loads, there will be more strain at the outside of the smaller 

 part of the bar fig. 8, say at A A, than there is on any part of the uniformly 

 small bar fig. 7. 



Assume that the bars are composed of a number of equally elastic pa- 

 rallel columns (and to get rid of any question of what may be called the 

 natural fibre of wrought iron not following the outline of bar fig. 8, let it be 

 supposed that the bars are made of cast iron or cast steel) ; now if these 

 elastic columns are capable within certain limits of equal increase of exten- 

 sion with equal increments of load, it follows that if the bar fig. 7, when 

 unloaded, were to have a horizontal line drawn across it, as at x y, and the 

 bar were then to be loaded, the result would be to lower this line to the 

 position .^■' y' , the line still being horizontal. 



In fig. 8, however, where the wide upper part contains a greater number 

 of such elastic columns than its lower part (or than the bar of fig. 7 con- 

 tains), then if the horizontal line were in the unloaded state of the bar 

 drawn at x y (the part where the dimensions abruptly change), and if the 

 load were afterwards applied, xy could not be drawn down so great a dis- 

 tance as in the case of the bar fig. 7, because there are more elastic columns 

 on the upper part of the bar fig. 8 to uphold the load than there are in the 

 bar fig. 7 ; and, moreover, x y could not be drawn down so as to preserve 

 its straightness, unless it could be assumed that the elastic columns at the 

 sides of the wide upper part were equally extended with those in the middle ; 

 but it is obvious that these outer upper elastic columns, not having any 

 columns below to pull them, can only be brought down by their lateral con- 

 nexion with the neighbouring upper elastic columns ; but this connexion being 

 in itself elastic, the effect can only be to draw the outer parts partially down, 

 and thus to cause the lowered line x y to assume the curved form of x' y'. Now 

 it will be seen that this curved form of the line x' \j involves the outer elastic 

 columns being more extended than the internal columns of the lower part 

 of the bar ; but as the strain on the elastic columns may be ascertained by 

 referring to their extension, this proves that the strain is not uniformly distri- 

 buted, as in the case of the bar fig. 7, and that the outer elastic columns of 

 the lower part of the bar fig. 8 are more strained than the columns of fig. 7, 

 and still more strained than the internal columns of fig. 8. 



An endeavour will be made to explain and establish this proposition by 

 the diagrams figs. 9 & 10. In these diagrams the cross bar B B is taken as 

 the equivalent of the lateral connexion Avhich exists among the elastic columns 

 in the bars figs. 7 & 8. 



Let fig. 9 represent three spiral springs, AAA, suspended at equal distances 

 apart, and attached at the bottom to the bar B hinged in the middle, and let 

 C C C be three other similar spiral springs attached at their upper ends to 

 the bar B, and at their lower to a perfectly rigid bar D carrying the load L. 



Under such an arrangement as this the load would be uniformly distri- 

 buted, and the result would be simply to stretch the springs equally and to 

 lower the jjarts B and D from their original positions indicated by the 

 dotted lines. Now let it be assumed that, as in fig. 10, two other springs, 

 A' A', each equal to one of the springs A, have been placed alongside of 

 them and have been attached to the outer ends of the hinged bar B, and 

 that the load L has been applied to the bottom bar D as before ; the result 



1869. 2 F 



