lHi6.] 



THE CIVIL ENGINEEERAND ARCHITECT'S JOURNAL 



173 



When X is between 70 and 60 feet, y will be reduced to 15 feet, and 

 inii»t not be farther lessened. By adopting then the form here indicated, 

 »he tension will be for 160 feet on each side of the centre uniformly equal 

 to 841 tons, and for the 05 feet near either extremity will be less. Taking 

 it however at 841 tons, we observe tiiat, since the plates are 15 feet wide 

 and 1 inch thick, and the cross section consequently contains 180 square 

 inches, the strain per square inch on the metal plates is about 4| toni. 

 Iron will bear a strain of 29 tons per square inch without breaking, and 

 D tons without permanent injury ; but the diminished strain here as- 

 Mgned to it is not too great an excess on the side of safety, when the dimi- 

 nation of strength at the places where the several pieces composing the 



■ pper and lower plates are welded or rivetted together is taken into con- 

 tideratioQ. 



6. Necessary strength of the vertical rihs. 



The beam of greatest strength for a given quantity of material is that in 

 which the material is collected in two wide horizontal flanges, separated 

 bj a thin vertical rib. It has been usual to consider that no more strength 

 ii requisite in this rib than will suffice to keep the flanges apart; and the 

 consideration seems to have been hitherto neglected in all philosophical 

 idiestigations of the subject, that the longitudinal strains which the flanges 

 «ert upon the rib, render it necessary that the latter should have much 

 more strength than what is required for the mere separation of the flanges. 

 A very simple method may, however, be given for determining the exact 

 amount of strength actually required in the rib by its connection with the 

 other parts of the beam. As the subject appears to be an entirely new 

 oor, no apology is necessary for discussing it fully. 



The most convenient method of determining the strains upon the vertical 

 web or webs is to consider the vertical strains and the horizontal strains 

 <)uite separately. This plan of keeping the two kind of forces perfectly 

 distinct has been observed throughout these papers, and is by far the best 

 for getting clear and precise notions of the subject. 



We will first consider the Inngitudinal strains which the flanges exert 



■ pon the rib. Now it is to be remembered that the upper flange is in a 

 »tate of more or less compression throughout its length ; and the lower 

 flange in a state of more or less tension. We know, however, that a string 

 stretched by two forces only, one at each end, is in a state of uniform ten- 

 tion throughout its length ; but if forces be applied to the string at inter- 

 mediate points along it, the tension will vary in various parts. For instance, 

 suppose that a string, A B, is fastened at B to a staple firmly fix°d in a 



wall, and at A, passes over a pulley, and is stretched by a weight P,, it 



a clear that if no weight but this act on the string, it will have a uniform 







-VI, 





AS 



Pi 



"T3I 



ff^ 





tenaioD = P,. Suppose, however, that at the points Aj, A„ A^, other 



weights are hooked on to the string and made to act upon it horizontally 

 by means of pulleys, it is obvious that the tension of the string will now 

 vary in various parts ; between A^ and B all the weights are acting on the 

 tiring, and consequently the tension of this part = P, + P^ + Pg + Pj ; 

 between A3 and A4 the tension = P, -f" Pj + P3 i between Aj and A, 

 the tension = P, + Pj ; between Aj and A, the tension = P,. 



Now, to apply this consideration to the lower flange of our girder — we 

 cbserve that the tension of it varies in every part ; this variation, therefore, 

 wises from the action of horizontal forces at every point throughout its 

 length, and it is certain that these horizontal forces have been communi- 

 •ated by no other means than by the vertical ribs. 



In order to estimate the amount of the strains communicated by the bot- 

 tom flange to the ribs, we must find out the law b; which the tension of the 



' fh» 



former varies. Let us first for simplicity suppose the depth of the girder 

 uniform throughout and = a. Also let the whole length be uniformly 

 loaded, and let one foot of the length of the girder with the load upon it 

 = m ; then if 2 i be the length of the girder, 2 ml will be its weight, and 

 the reaction on the abutment A = ml. Also, if a section be made at any 

 part C D and A D = X the weight of the part A B C D = m r. Let the 

 tension at D = M. Then taking moments about C 

 M a = m i X — m X . J X 

 =: m{lx - 4x2). 

 Similary, if we had supposed a section made in the girder at a distance x-' 

 from the end, and that the corresponding tension was M', we should have 

 the equation 



M'a = m (i x'— J i' ). 

 Consequently 



(M-M') a = ,n{t (x-x')-i (x'-x'=} 

 This last equation gives the law of the variation of the tension. 

 We have now to see how this variation is produced by the vertical ribs. 

 Let us first of all suppose that the connection between the upper and lower 

 flanges is maintained— not by continuous plates — but by a lattice consist- 

 ing of vertical and horizontal bars crossing each other. Now the upper 

 horizontal bars will exert thrusts, and the lower, tensions. In the follow- 

 ing figure, let A B represent a portion of the lower flange resting on the 



abutment at A. Above A B are represented some of the horizontal bars 

 of the lattice, but those only which exert tension ; those which exert thrust 

 as well as the upper flange being omitted in the figure. Now by this 

 arrangement the tension of the portion of the bottom flange, between any 

 two vertical bars is constant. For instance, the tension of that part of A B 

 which lies between a and b is uniform, and of the part between * and c 

 the tension is also uniform, only it is greater in amount than the tension 

 between a and b. Call the tension between a and b, M and between b 

 and c, M,, then it is clear that the vertical rod b b' is solicited at the point 

 6 by two diflerent horizontal forces, M^, at the left of b tending to pull it 

 towards the abutment, M^ to the right of b tending to pull it further /roni 

 the abutment. But M^ is the greatest: therefore on the whole the rod 

 b b' is acted on by a force M, — M^ tending to pull it towards the rigbu 

 For the equilibrium of this rod therefore we must have equal forces tend- 

 ing to pull it towards the abutment. These forces are supplied by the 

 tension of the horizontal bars above a b. Hence this conclusion— the 

 aggregate tension of the horizontal bars above a b=M M 



Now if we suppose B the tniddle point of the beam, we know that the 

 tension is equal and opposite on both sides of that point ; that is, the ten- 

 sion of c Bis equal to the tension to the right of B. Consequently the 

 vertical bar at B will stand of itself without any tension of the horizontal 

 bars above c B. The tension of these bars is therefore zero. The bars 

 however above 6 c are in a state of actual tension, the amount of which 

 equals the diflTerence between the tension of the portion B c of the flange 

 and of the portion b c. 



Let the successive dilTerences of the successive portions of the flange be 



represented by P,, P^, P3 P„,then the bars above A c are stretched 



by a force P, at c ; the bars above a b are stretched by a force P at c and 

 a force P^ at 6 ; the bars next to the left are stretched by a force P at c, 

 P^atft, aidPj at a, and so on. Consequently the tensions of the hori- 

 zontal bars are, to the left of the first upright rod, P, ; to the left of the 

 second, l\ -1- P„ ; of the third, P, + Pj + P, and so on. The tension to 

 the left of the rth vertical rod =:P, + P„ + P + + p. 



But the forces P,, P^ P^are respectively equivalent to M — M 



M.-Mj M„ _ , — M^. HenceP,-hP2 + P,+ .... +P^ = 



M - M , + M . — M2 -H + M^ . ^ — M,=M — M^, 



or the aggregate tension of the hori zontal bars at any point is equal to Ibe 



