OK THE INFLUENCE OF FORM ON STRENGTH. 429 



the difficulty of the fracturing that took place in the ordinary bolts imme- 

 diately at the junction of the screw parts with the shank, by diminishing 

 the area of the shank, so as to be equal to that at the bottom of the thread, 

 and has thus given in his armour-plate bolts a practical instance that the 

 strength of parts may be added to by reducing that of their larger neigh- 

 bours. 



Besides the armour-plating bolts and other bolts, there are no doubt many 

 cases in which both quiescent strain and the strain of impact are exerted in 

 the direction of the length of the object ; but there are probably a still larger 

 number of cases in which the strains are applied transversely, and among 

 them are the important instances of railway axles. 



It may be well therefore to glance briefly at the influences exercised by a 

 sudden alteration of form when that alteration occurs in an object exposed 

 to transverse strain. 



Let fig. 18 represent an elastic bar (A) of uniform section, placed on sup- 

 ports B B, and subjected to the action of the quiescent load L, the result will 

 be simply to deflect it as sketched. 



In this deflection the central parts may be assumed to be extended on 

 the underside, and compressed on the upper, as represented by the con- 

 verging space abed. 



If, now, the depth of the bar be abruptly increased in the part that lies 

 between these lines abed and the two ends, as shown in fig. 19, the result 

 will be to aggravate the strains at the parts a bed in a, similar way to that 

 which was pointed out in respect of a perpendicularly suspended bar under a 

 quiescent load. But if the bars have to resist impact, then a more serious 

 difierence, to the disadvantage of the bar with unequal sections, wHl be 

 found. It is well known that if one bar be double the depth of another, the 

 first bar will, under a quiescent load, deflect only one-eighth part of that 

 which the second would deflect, the deflections being inversely as the cubes 

 of the depths. 



With respect, however, to the resistance ofiered to the flexure of the bars 

 under impact and not under quiescent loads, the writer believes it can be 

 shown that if there be two elastic imponderable bars alike in aU respects 

 except their depth, and if they be exposed within their elastic limits to the 

 impact of equal forces, the result will be that, if the one bar be taken as 

 tmity in depth and as deflecting unity under the force, the other bar, having 



a depth of n, will deflect according to the formula - \ / — . 



n V « 

 Appljing this formula to the case of a weight let fall upon a bar which is 

 double the depth of another, the deflection of this bar of double depth will 



only ^6 2 \ / 75> or about -35 of that which it would have been if of the 



single depth ; but to produce this -35 of deflection, the strain on the whole 

 section of the bar of the double depth wlU be 2-828 times as great as it would 

 be upon the bar of single depth. 



So that if the bars (figs. 18 and 19) be exposed to the impact of similar 

 forces, the bar fig. 18 would deflect through a space of 2-828 with a strain 

 of -35 on the central parts, while before the bar fig. 19 could, so far as 

 the greater part of it (namely its enlarged ends) is concerned, be deflected 

 through a space of one, there must be put upon the middle section of it, 

 equal to one only in area, a strain of 2-828. As in the case of the vertical 

 bars, this extra strain would be aggravated by the fact that it would not 



