92 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



LMarch, 



Al'here w is the weight (in pounds) acting upon the bar, e t}ie ex- 

 tension, and d the compression (in inches). 



To express the relation between w and the corresponding exten- 

 sion and compression, when tlie length of the bar is reduced from 

 10 feet to 1 foot, we assume that tlie extension and compression are 

 uniform throughout the length of the bar. Therefore the exten- 

 sion or compression of one-tentli its length will be reduced in the 

 ratio 1 : 10. t'onseiiuently, in order that the value of w may re- 

 main unaltered in the formula?, the co-efficients of e and d must be 

 increased in the ratio of 10 : 1, and the co-efficients of <r and d- in 

 the ratio of 10- : 1. These modifications being effected, the for- 

 mula; for a bar 1 foot long become 



w = 1 1611706 — 20190o00e^ for extension, 

 u- — WnS'iOd — 8631800d- for compression. 



If the bars were 1 inch only in length, those to which the first 

 formultE applied would be reduced in length in the ratio of 1 : 120. 



Consequently the extensions and compressions would be reduced 

 in the same ratio; and in order that w might remain unaltered, the 

 co-efficients of e and d must be increased in the ratio of 120 : 1, 

 and the co-efficients of e- and d- in the ratio of 120: 1. These 

 clianges being made, the formulae for bars 1 inch long and 1 inch 

 siiuare become 



w= 116117Xl20e— 201905x120^ 



= 1393+04'Oe— 29074320006- for extension, 

 w = 107763Xl20d— 36318X120- (/= 



= 12931560rf— 522979200rf- for compression, 

 where, as before, w is expressed in pounds, d and e in inches. 



Lastly, if the length of the bar be / inches, the corresponding 

 f irmuhe for w may be deduced from those last given, by consider- 

 ing the bar to wliich those formulse apply, increased in length in 

 the ratio / : 1. Consequently as before, to adapt the formulie to 

 the present case the co-efficients of e and d must he diminished in 

 the ratio 1 : /. and the co-efficients oft- and d- in the ratio of 1 : /-. 



The formulse for a bar 1 inch square and / inches in length are, 

 therefore, 



• • (A) 



1393101-0 



— 2907432000-^3- for extension 

 d d- , 



= 12931560--— 522979200 -^ for compression. 



(B) 



I I- 



The mean tensile strength per square inch of section in the irons 

 r.rperimentcd iijjon viis 15711 lb. — 7-OU tons, and the mean ulti- 

 mnte ejctensum for Iciiythx of 10 feet was -1997 inch, or ^ inch nearly. 



The mean romprcssion if bars of the same metal and dimensions, by 

 the wciyht 15711 lb. (the hreakiny ireight by extension , as above stated) 

 was found from the e^tperiments to be 15488 inch, or yfith part of the 

 length. 



To find the values of (e) and {d) in terms of («■), in the preceding 

 equations for Cast-iron. 



w = ae — b(r ; whence be" — «e = — w ; 



e- — e = 



w 

 "b' 



'"--b' + 



a- 



a- 



w 



a 

 Yb 



+ 



Va- w 

 Tb'~ T 



(C) 



Extension of a bar / inches long and 1 inch square in terms of the 



weight stretching it : — 



From equation (A) for elongation of bars of cast-iron, we have, 



in equation (C), 



13934040 , 2907 432000 

 a = , b = ^^— , 



substituting these values of a and h in (C), we have, 

 13934040 



5814864000 

 = •00235628; + 



Xl +y 



13934040 

 ^5814864000 



)>- 



wl- 



2907432000 



00000574215/= 



- •000000000343946/^to- 

 •000000000343946w}, (D), 



:. e= I {-00239628— V00000'5'''l215 



where w is in lbs. and e in parts of an inch, the negative sign being 



tliat alone which is api)licable in tlie quantity under the root. 



If / = 1, the extension is that produced by a length of bar = 1 in. 



lf/= 12, „ " " ZiJft 



If/ = 120, „ „ " -ion. 



Example 1. Suppose w = 11591'48, and /= 10 feet or 120 inches, 

 then substituting for w and / their values in equation (D), we have 



e = /(•00239628 — V00000574215 — -00000398684} 

 = /{-00239628 — -00132488} = 120 X -0010714 = -128568 inch. 

 Comparing this with the result in Table IX., on extension of 

 bars, from the same pressure 15491 -48 lb., we have a defect of 

 :rfj;t'h, the real extension being -1283 inch. 



Example's. Suppose w = 2107-54 lb. and /= 10 feet; substituting 

 for w and / in equation (D) we have 



c= /{ -00239628 — V00000574215 — -00000072488} 



= I (-00239628 — -00223993) = 120 X -00015635 = -01876 



It should be -0186, .-. error = yi^. 



Compression of a bar / inches long and 1 in. square in terms of the 



H eight producing it : — 



The relation between the weight and the compression being ex- 

 pressed by an equation of a similar form to that between the weight 

 and the extension, or w = ad — bd-, we obtain in the same manner 

 as before, — 



" = ^1-^ V^A=" 



substituting the values of a and b derived from the equation (B) 



12931560 ^ , 522979200 . , , . .^ 



for cast-iron, or for a, and jr, for b, we obtain— 



d = 



12931560 

 1045958400 



_ / / 12931560 \ - wl- 



X ^ + \/ (l045958400/ '' 522179200 



-00000000191212 w} 



= / {-012363359 — ^-000152853 — -00000000191212 w}. . (E) 



■Where w is in pounds and d in inches, the quantity under the 

 root is affected by the negative sign, which alone is applicable in 

 this instance. 



Example 1. If «> = 8258-98 lb., and the length 10 feet = 120 

 inches, we obtain by substituting the value of w and I in equa- 

 tion (E),— 



d — I {-012363359 —V 000152853 — -00001579216} 



= (-012363359 — -0117073) / = -07872 inch. 



Comparing this with the experimental result for this pressure in 

 Table VI. on compression of bars, or -07879, we find the deviation 

 or error equal -j-nv^ of the latter. 



Example 2. If «; = 6194-24 lb. and I = 120 inches as before, 

 d=Z {-012363359 — v'-000152853 — -0000118441} 

 = I (-0004S90) = -05868 inch. It should be -05978 .-. error = ^'^th . 



The first example is the case of least deviation of the formula 

 from the results of experiments in Table VI.; and the second is 

 that of greatest deviation for pressures between 2 and 14 tons per 

 square inch, the range between which the results are most trust- 

 worthy. 



Example 3. If w = 15711 lb.— the weight which would tear asun- 

 der an inch bar of these irons— to find the compression of a bar 

 10 feet long and 1 inch square from the same weight. 



Substituting in equation (E) the va lues of w and /, 



d = I {-012363359 - V-000152853 - -00000000191212^} 



= / (-012363359 - -0110820) = / (-0012813) = -15376 inch. 



The decrement, as obtained from the results of experiment in 



Table VI., on compression of bars, was -15488 iuch. 



Extension, 



(Computed from Uie Formula obtaintd.) 



