166 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



[Junk, 



F.g. 12. 

 1— A 



— d 



— b 



Let d be the quantity a bar of iron, or other material, an 

 inch square, and a foot in leni;th, represented by a h, would be 

 extended by the force of the weight W in pounds, which, as in 

 the last, is supposed to be the greatest, it would bear without 

 destroying its elastic force ; and let L be the length of any 

 other portion of the same rod in feet. 



Then I : L; -. d : Ld = the extension of the rod, whose 

 length is L. This is evident, for if we suppose the weight W 

 to be attached at «, and to stretch a h a quantity d it will stretch 

 ac,2d; ad,Zd; kc, because \V, after stretching «fc, will 

 apply the same force to he, and, strictly speaking, a little more, 

 for it will have an additional weight in the length ub, a. foot of 

 the material ; the same reasoning will hold with respect to any 

 other foot, as e A for the weight W, and the weight of 4 feet of 

 the material may be supposed to be applied at e, and act in ex- 

 pounding cA, but the weight of the rod is so small, tliat it may 

 be neglected in moderate lengths. Strictly speaking, then W , 

 the ultimate force must involve the weight of the bar to which 

 •• ■ ,nnr,Pd The weight W would ultimately be reduced to zero, for the 

 ;Vr: only be able to support its own weight without destroying its 

 elastTc rce.-Example: cast iron will bear, without permanent al.era- 

 tion 5 300 lb. upon a square inch ; consequently, the length of a rod of 

 cast'i n^^a' would be just able to support itself without P-^-n al.er- 

 aUon would be 4,896 feet; a cubic foot of cast iron weighs «0 lb no 

 Ttter what the cross sectional area ma, be. Again, suppose W to be 

 any other weight less than \V,— 



The number of pounds on each square inch of cross section _ -^, 



tosether with the weight of the bar up to that section. Then just at the 



point where the beam is suspended, there will be -^ +p L pounds on the 



square inch which must not exceed w, the elastic limit in pounds for each 

 AV . . . W r 



square inch of section. 



W 



Av : w" : : d : 



M'd 

 W 



'■ - — = the elongation of each foot 

 Alt' 



W 



in length by the weight -^ on the square inch ; but as each square inch is 

 strained by the same weight, each foot of the bar will be increased by the 

 same length. . • • ^^ ^ = the elongation of the length L, by the weight 



W, 



L + ^L2 =L- 



4 W 



P 

 A + 2 



lU 



E, the elonga- 



= the extension produced by the weight W, sup- 



posin.. the length to'be one foot, and the cross section an inch square ; and 

 aTo that the extension of a bar or rod by a force acting in "e d.rea.on of 

 its length is proportional to the straining force; the area o the cross sec- 

 t!oi remainin'g the same, and the strain not to exceed the elastic power of 

 the beam, bar, or rod. j^^^„ ^ 



The extension for any length L, by the weight W, =— ^y- S for, 



VV d . . LW'd 



W is supposed to involve the weight of the bar, as well as V. . 



Let it be required to the elongation E of a bar suspended vertical y, and 

 suia n ug a given strain or weight W, in the direction of its length equal 

 Lfrethe influence of its own weight being taken into account, the sec- 

 . lalVrea A square inches. Without destroying the elasticity, or surpass- 

 .heetaiic limit, suppose . to be the length that w pounds will elon- 

 'il a bar we^gS; pounds, one foot long and one square inch sectional 



^^^^' .u loc. f„„t nf a bar L feet long to be suspended to the second 



laslr :ktt^fll lu:^::ga.e the sLond last will be found by the 



following proportion :-„, : p : : . : ^ = the elongation of the second foot. 

 Suppose these two feet to be attached to the third foo.,^we have 

 „- • 2p • • € : — = elongation of the third foot ; ir : 3;, : : . : -^' = elon- 

 gation of the fourth foot ; therefore the sum of the following series will 

 ^wTthe whole elongation, the length of the bar L representing the number 



Or 



W. And hence, ^ 1- -r ^ ^, " „ 



tion by the weight W and L A p the weight of the bar. 



We shall next explain certain numbers introduced by writers on this 

 subject and called by them ynoduli. There is the modulus of elasticity, 

 the modulus of resilience, the modulus of fragility, and the modulus of rup- 



"The ,nodulus of elasticily is the strain in pounds which would be required 

 to extend a bar, one foot long and one square inch sectional area, to double 

 its length without altering its sectional area; or it is the trust in pounds 

 that would compress the same unit bar into half its length, that is, till the 

 foot becomes G inches, the sectional area remdining, as before, one square 

 inch We never could see the real use of introducing these imaginary 

 numbers, unless for the purpose of mystifying the subject or to make it 

 assume a very scientific appearance. 



It would be a stretch of the imagination to suppose a brick to be pulled 

 till it would become the length of two bricks. He must be a very clever 

 man indeed who determined the modulus of elasticity of pipeclay. Mere 

 book-makers like Hall and Moseley, of King's College, cannot be offended ; 

 but men like Barlow and Hodgkinson, who have lost their tune experi- 

 menting to find them, may be a little indignant to find their favourite num- 

 bers spoken so lightly of. ,,.,., 



Let us suppose, for argument sake, that Tredgold is right with respect 

 to the modulus of malleable iron ; he found the modulus of elasticity to 

 be 24,920,000 lb., or 7,550,000 feet of the same matter ; sectional area one 

 square inch. Those who have made experiments will give but little credit 

 to one who finds him wrong. Tredgold found that 17,800 lb. on a square 

 inch of good English iron would cause no permanent alteration, and would 

 extend a foot or any other length, not taking into account the weight of 

 the bar, the ^S^, part of its length. ,U i° ""' ''^^^ ^™"''l ^^ '' 17«»0= «> 

 and p = \]i, for a cubic foot of malleable iron weighs 475 lb. nearly. Now 

 it isevideut',ifthe elastic limit of malleable iron were such as to allow it, and 

 that the elongation was in direct proportion to its strain, that it would require 

 1400 times 17,800 lb. to extend this foot of iron till it becomes two feet long, 

 or which is the same thing, till it becomes a foot elongated. Now 1,400 



' "' .u 



times 17,800 =24,920,000 lb., or, 17,800 -^i,U = 24,929,000 =- i this, 



to muke the matter assume a learned appearance, or rather, a more college 

 appearance, we shall represent by M^; the modulus of elasticity and its 



reciprocal -^ by jj ; so that E = L -^ [ ^+ f L | may be written 



of terms, we have^l 1+2 + 3 4-4-1- 



1 L L>£ 



2 w 



the elongation may be determined thus :-Let :r be^the length in feet mea- 

 sured from the lower extremity, then ,„ ; ;, ^ : : e : ^"= the elongation of 

 a foot in length from the strain of the weight ;<.r. And let d x, as wri- 



,Z ON THF. CaLCDLUS SAY, BE THE LENGTH WHICH IS NEXT TO NOTHING. 



foot . feet . . iP^ .IE xdx =the elongation of the length which was 



■.•—-- j^ .. u' ■ ic 



» . „.^tv,incr Intezrating between the limits .r = o and 

 supposed to be next to nothing, inicgia & 



L ^W 



e=m:( A+2 



"t ' 



A bar of malleable iron, one square inch of se«- 



tional Irea and 7,550,000 feet-nearly 1,430 miles long = 24,920,0001b ; 

 from which circumstance 7,550,000 feet is called the modulus in feet ; 



That is, — -i-p — 



pe 



-^e^P-> 



■ PhiE 



J _ 9 ,n — L2. So that a bar 1 inch square 



I =L; that 



and L feet long wiU be elongated by its own weight ^ X U . i indeed, let 

 the sectional area be what it may, the bar will be elongated by its own 

 weight the same length, because the body is uniform, and each loch of 

 sectional area is circumstanced in the same manner. 



24,920,000 -r fi5 = 7,550,000. 

 this modulus Hodgkinson dilTers 400 miles of iron from Tredgold, and 

 Barlow about 250 miles of iron from Hodgkinson. So much for the mo- 

 dulus of elasticity. 



The elongation of a bar suspended vertically, and sustaining a strain of 

 W lb. —the influence of its own weight not being taken into account— was 



c W L , „ „ , 



found to be— • -^, which we shall call I :— 



by substituting for -^ its value -rjr- . Before we eiplaia 



.1- 



what is meant by the modulus of resilience or fragility, it is necessary to eaj 

 what is meant by a unit or work. A pound weight raised vertically 



