APPLIED MECHANICS. 



[ DISPOSAL Or MATERIAL 



\L OF MATERIAL. If we apply these 

 considerations to a beam subjected to a trausvenie strain, 

 as when it is supported at both ends and loaded in the 

 ini'MIe, we see that at the middle of the beam, just 

 before fracture takes place, there most be a portion of 

 the material above the neutral axis compressed, and a 

 portion below it extended by the action of the load. If 

 the resistance of the material to extension and to com- 

 prrsaion under those conditions be equal, the neutral 

 axis would be in tin- middle of the beam ; and its resis- 

 tance to fracture would then be the greatest possible, 

 liocause the total leverage of the compressed and ex- 

 tended portions on each side of it, would be greater when 

 it Is in the middle than when it is elsewhere. But if it 

 happened that the material was more easily compressed 

 than extended, it would be nearer the lower side ; while 

 if it were more easily fractured by extension than com- 

 pression, it would be nearer the upper side. Desiring, 

 however, in either case to secure the advantage of having 

 it in the middle, and thereby giving both the compressed 

 and the extended portions the greatest possible leverage, 

 we should somewhat alter tlie form of transverse section, 

 so as to increase the area of the weaker part, or add to 

 it some fibres, diminishing the area of the stronger, and, 

 upon the whole, not altering the quantity of material or 

 total area of section, but only modifying it in such a 

 way as to bring the neutral axis to the middle of the 

 depth. In a rectangular beam of malleable iron C (Fig. 

 75). the neutral axis N is below the middle of the depth, 

 because the upper portion yields more readily to com- 

 pression tlian the lower to extension. In a beam of cast- 

 Fig. 74. 



K 



ness of the metal in the middle part and in the lower 

 flange is made as nearly equal as possible, because it is 

 found practically that, in making iron castings, unequal 

 thicknesses of metal cause unequal contractions or shniik- 

 ings in the metal as it cools, and thus tend to distort 

 the work. 



The best form of section for malleable iron rails is 

 nearly the opposite of that for cast-iron girders, a T not 

 Fig. 77. Fig. 7S. inverted (Fig. 77)- A, 



however, the upper part of 

 the rail gets worn and un- 

 even by the friction of the 

 traffic, it is sometimes 

 thought desirable to have 

 the opportunity of invert- 

 ing it, and thereby wear- 

 ing both the upper and 

 lower sides before the rail 

 is rejected as worn out. 

 The section is, therefore, 

 made symmetrical above and below, as well as on both 

 sides (Fig. 78). 



As we nave shown above, the strength of any beam to 

 resist transverse strain increases very greatly as the 

 depth is increased. It is, therefore, of great importance 

 in all cases of transverse strain to give as much depth as 

 possible. This is often effected, not by increasing the 

 total depth of the material, and thus adding greatly to 

 its weight, but by introducing ribs or flanges, and thus 

 dispersing a given quantity of material in a better form. 

 If, for instance, we had to provide a square cast-iron 

 plate, of sufficient strength to resist a great weight 



Fig. 79. 



iron B the neutral axis N is above the middle, because the 

 upper portion resists compression more than the lower 

 resists extension. To bring the neutral axis to the 

 middle in both, we should for malleable iron remove 

 portions of thickness from the lower edge, and add them 

 to the upper, as in A ; for cast-iron we should take from 

 the Upper and add to the lower, as in D. We thus find 

 that the modification of form requisite for increasing the 

 transverse strength of cast-iron of certain depth, without 

 adding to the mass of material, is exactly the opposite 

 of tliat suited to malleable iron. 



The usual section of cast-jron girders, supported at both 

 ends and loaded in the middle, is the inverted ~J~ , swelled 

 a little at the upper edge (Fig. 76). The dimensions of a 

 girder one foot deep are nearly those marked in the dia- 

 gram. This may not be precisely the best form for combin- 

 ing the greatest strength with econo- 

 my of materials, but it approaches it ; 

 and, besides, it possesses practical ad- 

 vantages which deserve consideration. 

 The flange, or increased width at 

 the lower side, not only affords the 

 additional strength required there 

 in order to meet the tensive strain, 

 but presents a projecting ledge for 

 receiving the beams or arches which 

 f the girder may have to bear. The 

 increased thickness at the upper 

 side not only provides additional 

 material to resist oompressive strain 

 at the greatest possible distance 

 from the neutral axis, but verves to 

 stiffen the girder so as to prevent 

 it from bending or buckling side- 

 ways, either from contraction in cooling from its molten 

 tote, or from excessive strain in its place. The thick- 



pressing on its middle (Fig. 79), while it is supported 

 on two piers at the sides : instead of making the plate of 

 solid iron, sufficiently thick to sustain the strain, we 

 should probably make the upper part a thin flat plate 

 (Fig. 80) ; and round the edges, as well as across the 



so. 



diagonals, provide projecting ribs of the greatest depth 

 in the middle (Fig. 81) ; and thus attain sufficient 

 strength and stillness, while we should save a consider- 

 able quantity of material. This mode of attaining 

 strength is particularly useful with such a material as 

 cast-iron ; for when it is formed in thick masses, the 

 cooling of the outer crust, 

 while the inner part of 

 the mass remains fluid, 

 sets or fixes the outside, 

 and the subsequent con- 

 traction of the inside 



I':.-. M. 



PLAN. 



causes a spongmess or 

 looseness of texture, 

 which greatly diminishes 

 strength. It is, therefore, 

 of the utmost importance 

 to attain the required 

 strength without great 

 thickness, as well for the 

 sake of securing solidity and firmness of mater i il, as for 



SECTION. 



