1850. J 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



319 



21 tons per square inch, its ratio to the extension is 16139:1, 

 which is between one-fourteenth and one-fifteenth of the original 

 value. 



Mr. Tate assumes the extension to be always in the same constant 

 ratio to the tension; and the latter he taltes to be 21 tons per 

 square inch at the bottom of the model tube. It appears, then, 

 that lie assumes the ratio at more than fourteen times its real value! 



This is, of course, decisive of the character of his assumption 

 as to "'perfect elasticity." But the nature of the error may be 

 even more palpably shown. If h be tlie distance from the neutral 

 axis to the under edge of the beam, and a tlie ratio which the tension 

 per square inch bears to the extension of a unit of length of metal, 

 and p be the radius of curvature, we know, by the ordinary laws 

 of simple beams, which Mr. Tate applies here, that the tension per 

 square inch of the under side of the beam is 



ah 



The quantity h Mr. Tate makes equal to 25-55, and the tension 

 := 21 tons = 47040 lb. Also we have stated that the corresponding 

 ratio of the tension to the extension of a bar 10 feet or 120 inclies 

 long, is 16139 : I, consequently, a — 16139 X 120 = 1936680. 

 Therefore, the equation follows 



47040 = ?5:££J1J£?!!L0. 



or, p = 



49482174 

 47040 



According to the hypotheses of the work before us, the con- 

 stant moduli of extension and compression are different; but 

 the equation to the curve of deflection will be of the same form as 

 the ordinary elastic curve. Consequently, by known principles, 



•' ~ VZp' 



where / is the central deflection and / the length of the beam 



between supports. Taking the value of p determined above, and 



/ = 75 feet = 900 inches, 



810000 X 47040 , . , 



/ =: • — = 64-1 inches. 



•' 12 X 49482174 



That is, the deflection is more than five feet four inches. Com- 

 paring this rather startling result with Mr. Fairhairn's statement 

 to the Iron Commission of the observed deflection (which in tlie 

 Report, p. 410, he says was 4-88 inches), we find that Mi-. Tate's 

 hypotheses make the deflection between thirteen and fourteen times 

 its observed amount. 



In addition to the above evidence, that the tension and compres- 

 sion of the lower and upper parts of the tube could not be what Mr. 

 Tate, calculates them to be, if the beam retained perfect elasticity, 

 we have strong corroborative testimony of authorities on this very 

 point. We have already quoted the opinion of Mr. Ilodgkinson, 

 that the metal would be destroyed by far less strain. Mr. Edwin 

 Clark, also, in his evidence before the Iron Commission (Report, 

 p. 361), observes: — 



" We looked upon 12 tons to the inch to be as much as we 

 could safely subject wrought-iron to as regards compression. We 

 took the resistance to compression to wrought-iron as about 10 

 tons per square inch. AVe found, generally speaking, when you 

 get up to 10 tons to the inch, most iron begins then to be percep- 

 tibly altered in shape." 



Again, page 362, he says : — 



" Ws were therefore limited to 12 tons to the inch, but as we 

 were not going anywhere near such a limit as that, nor even half 

 ofit, it hardly came into play. If we made a thin cell it puckered; 

 if we kept the same dimensions, and kept making the plates thicker, 

 we avoided the puckering till at last we arrived at the thickness 

 at which iron no longer puckers, but sustains nearly tlie whole 

 strain of 12 tons per inch.' 



It was a matter of diflSculty then, and of rare occurrence, to 

 carry the strain up to 12 tons, whereas Mr. Tate computes it at 

 one half as much more. 



There is other evidence that the hypothesis of constant propor- 

 tion of the elastic forces to the corresponding extension and com- 

 pression is frequently quite remote from the truth. If the hypo- 

 thesis were true, the deflection of all the experimental tubes 

 should be proportional to the deflecting pressure. This, however, 

 is found not to be the case. AVe here give a few of the deflecting 

 weights, and corresponding deflections, of various rectangular tubes 



experimented upon by Mr. Hodgkinson (Iron Commission Report, 

 pp. 125 et seq.), together with wliat the deflection would be if pro- 

 portional to tlie weight. All tlie columns of the following table, 

 except the last, are taken from the Report. 



It appears manifest from a mere inspection of the preceding 

 table, that Mr. Tute's assumption of the law of elasticity is not in 

 accordance with a large number of observed facts. It is very 

 important, however, to observe that whatever inaccuracies may be 

 involved in the law assumed by him, are quite inadequate to explain 

 the enormous discrepancy between the deflection of the model tube 

 as observed and as computed. The discrepancy may arise in one 

 of three ways — from inaccuracy of theory, from what Mr. Hodgkin- 

 son terms "some error" in the data of the experiment itself, or 

 from combination of mistakes of both kinds. A very careful exa- 

 mination of the question has satisfied us that Mr. Hodgkinson's 

 surmise is indisputably correct, and that the principal source of 

 error is in the data of the experiment itself. 



Without any doubtful assumption of the law of elasticity, we 

 ascertain with sufficient accuracy the limits within whicli the ten- 

 sion at the bottom of the tube certainly must lie. We observe that 

 the compression of the top of the tube is supplied partly by the 

 plates forming its upper side, and partly by plates lower down. 

 Consequently the " centre of compression" is somewhere below 

 the top; similarly, the "centre of tension" is somewhere above the 

 bottom. The moment of the "couple," of which the distance be- 

 tween these two centres is the arm, is etjual to the moment of the 

 pressure of the beam on its abutment or fulcrum. The greater the 

 "arm" the \ess caiteris paribus are the equal forces of compression 

 and tension. Therefore taking the arm equal to the whole depth 

 of the beam, we make the tension less than it can possibly be. 

 Supposing all the tension to be at the bottom of the tube, let t be 

 the tension in tons per square inch. Take the area subject to this 

 tension at 19 square inches, deducting 3^ inches, the area of the 

 rivet holes. Also the depth of the tube is 54 inches, the moment 

 of the tension is therefore 54. x 19" t. The pressure on the ful- 

 crum is half 89-15 tons, the breaking weight, and the distance of 

 the fulcrum is 450 inches. Therefore, 



54 X 19- t must he greater than 89-15 X 225, 

 or the tension must exceed 19^ tons per square inch; and Mr. Tate 

 makes it 21 tons per square inch, a closely corresponding result. 

 The result is not materially aff'ected if we take into account the 

 area of the vertical sides of the tube. 



The deflection above computed is not that which would actually 

 occur in a tube of the dimensions of the model strained to 21 tons 

 per square inch on its under side; but merely the result of Mr. 

 Tate's hypotheses. In the investigation of the ordinary elastic 

 curve a certain quantity is neglected as smaU, which would not be 



