1850.] 



THE CIVIL EXGIXEER AND ARCHITECT'S JOURNAL. 



93 



2064-745 



fil94-24 

 10323-73 

 14453-22 

 18582 71 

 2l77!'r'i5 

 33033 80 



Tran/tverxe Flcrure. 



When a beam is bent in any deiri-ee tlie fibres or particles on the 

 convex side are extended, and those on tlie cnncave side are com- 

 pressed; and there is a line within the beam, intermediate between 

 the two sides, in any transverse section where the particles are 

 neither extended nor compressed. This is called the neutral line, 

 and the particles on each side of it are stretched or compressed 

 according to their distance from it; but the forre exerted by these 

 particles is not in proportion to the distance, in cast iron, at least, 

 which we are treatinj; of. It varies as a function composed of the 

 first and second powers of the distance nearl)-. 



Thus, in the longitudinal extensions and compressions of a bar 

 one inch area of section aiul / inches long-, we have from the mean 

 results of experiments on four kinds of cast-iron, eijuations (A) 

 and (B), 



«- = 13934040 — 2907132000 



rf-. 



w = 12931560 - — 522979200 



where w is the weight in pounds producing the extension e or com- 

 pression d in inches. 



To apply this to transverse pressure, suppose the extension e and 

 compression d of a small length of the material at a distance I from 

 the neutral line t<i be represented by jh/, h(7, respectively, then the 

 extension and compression at any other distance .*- of a portion of 

 the material originally of the same length will be mx and m'a\ and 

 the formulce will become — 



mx ( m,rV 



w — 13934C40 2907432000 —-- 



I I' 



{m'x)- 



in .r 

 12931560 — 



522979200 



/-' 



(F) 

 (O) 



where to, w\ are the forces of tension and compression exerted by 

 the fibres at a distance .r from the neutral line, and m, ni co-effi- 

 cients dependent on them. 



In the 'Pixperimental Researches on the Strength of Iron, pub- 

 lished by the author, and forming an additional volume to 'Tred- 

 gold on Cast-iron,' an attempt was made to give a more general 

 computation of the strength of beams than had hitherto been done, 

 the solution depending upon the supposition that the resistance of 

 the particles to tension and compression varied in terms of the 1st 

 and some other constant power of the extension and compression. 

 Thus if X be the distance from the neutral line — 



<c (.r) = J- — — — j; if !i^=2 



)ia na 



x'r' 



(J) 



<f'(x')^=x' — = x — -;- , if y' = 2 . . . (K), 



n a na 



where (/> (.r) and <l>' (.r) n-ould be quantities respectively propor- 

 tional to the forces of extension and compresssion of a particle at 

 a distance x from the neutral line, and n, n', quantities supposed to 

 be constant. 



From the experiments given in this inquiry, it appears that r, ii', 

 are equal to 2; and in the equations (J) and (K) a is the same 

 quantity as / in equations (F) and(G), a = /; and to adapt the 

 formulas (F), (G), for cast-iron, found before, to the forms above, 

 we liave — 



U'l 





13934.040 m ' 

 2907432000 m 



13934.0 to"" 

 In like manner — 

 _ »■"/ _ _ 522979200 m 



r2931560m " ' ~ 



2907432000 mH 

 X -_-- _ — 7^ X X- 



13924010 »h/- 



X -y, for extension 



(L) 



12931560^ ^ y, for compression 



(M) 



M'hence we obtain the values of n. ).', in equations (J), (K), as 

 behiw, — 



_ 13934040 , _ 12931560 



" ~ 2U07432000m' " ~ ^22979200 ni 

 By inserting these values in the formulie given in the work 

 above referred to, the position of the neutral line and the strength 

 of a cast-iron beam of the form considered may be found. 



Abstract No. VI. 



Abstract of Re.sultK on the Transverse Strength of Cast-iron Bars 

 of different *■;'.;■(«, Init mntliemutically similar, or re/atiiK/y propor- 

 tional in all their dimensions. 

 The bars w-ere of Blaenavon iron, No. 2, and were respectively 



cast to be 3, 2, and 1 inches square, and 15, 10, and 5 feet long. 



They were placed on supports 135, 9, and 4J feet asunder, and the 



strength and ultimate deflections of tlie bars, when reduced to 



their exact size, were as below: — 



size of Bars, 



Ft. span. In. 8q. 



4i 1 



13i 



6} 



Vertical Pressures. 



Strength. 



lbs. Mean 

 461 



437 J. 4-10 

 423 



Ultimate 

 Deflection. 



Incii.-s. Mean. 



1-7J0 



1850 !■ 1-779 



I1-C917 



1249 

 1414 

 1121 



1097 

 15520 ] 

 15944-' 



|.1338 



12 996 

 13-180 

 2.527 

 2-!98 

 3-G20 

 2.984 



2f.9S T 4'R63 ~ 

 2671 UgoiiJ-SOOS 

 3389c f-°"' 5-024 

 1-391 _ 



1 

 I 

 "-3 0035 



1 

 J 



Horizontal Pr«ssurescomputed 

 from the Vertical Pressures. 



Strength. 



tt)8. Mean. 

 468] 

 444 1 447 

 430 I 



1303, 

 14C9 



'U51M394 



1610 

 1648 J 



2686(iJ 



6:'.41 

 5795 

 6215 



.6ii; 



1-3319- 

 |1-190 

 11-353 . 



4-067 



1-2916 



2877- 

 ■2854 

 3573 

 2869 



3043 



0431] 

 5865 16207 

 6306 J 



Ultimate 

 Deflection. 



Inches. 

 1-823 

 1-880 

 1-720 



. Mean. 

 1 1-808 



3032 



3-622 



2649 



2-621 



3-746 I 



3-085-' 



!• 3-126 



4 906 



1-351] 



1 208 U-311 



1-373 J 



The results marked with the letters «, b, c, rf, are from the bars 

 which had been previously subjected to 4000 impacts, each bending 

 them through fjrd of their ultimate deflections. 



The sti-engtiis of similar bars 1, 2, 3 inches square, and 45, 9, 

 and 135 feet between the supports are respectively 447, 1894, and 

 30431b. to resist an horizontal pressure. 



If the elasticity of the beams had been perfect, their strengths 



should have been as the square of their lineal dimensions, or as 1, 



■t, 9. Dividing, therefore, the strengths as above by these squares, 



the quotients ought, on this supposition to be equal. We have, 



however, 



447 

 From the smallest bars . . — — =: 447, 



From the next larger bars 



From the largest bars . 



1394 



= 349, 



3043 



= 33S. 



The quotients are unequal; but -ive see that the deviation from 

 theory, on the sii|)position of perfect elasticity, is much greater in 

 the smaller than in the larger bars, and that the strength of the 

 smallest bar is greatly above that derived from others, partly, it is 

 probable, arising from defect of elasticity, but principally from the 

 superior hardness of the smaller castings. 



The ultimate deflections of similar elastic bars from horizontal 

 pressure are as the lineal dimensions of the bars, nearly; and, 

 therefore, similar bars, one, two, and tliree inches square, ought to 

 deflect before fracture in those proportions. The ultimate deflec- 

 tions from experiments, as above, are below. 



In bars, 1 inch square .... 1-808 

 „ 2 inches square . . . 3-126 

 „ 3 „ .... 4-966 



The deviation in the ultimate deflection of the bars, from 1, 2, 

 3, the ratio of their size, is, therefore, larger in the smallest 

 (hardest) bars than in the others. 



U 



