352 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL 



iXov 



.-. 3 X 2D9 + A. = 300^ = ,8 = mf<p 1 A. (1) 



2 A , - A„ = 20 1}), or n A , + A J = e (jj (2) 



13 Aa = 2^<p, or qA„ = € <p (3) 



Eliminating A, and Aj, we liaTe 



{mng+ (?+l) leB {5.2.13 + (13 + 1)} 29.317 



vnq 



2400.2.13 



(p = 21} times (p very nearly. Hence the line N determines the 



1323792 

 62400 

 angle aON = 132^27 

 In the expression 



{mny ^ (y + l)}e5 



rnq 



(R) 



substituting the numerals of the first example, then 



{6.3.2s + (2S - 1)} 20.11 



fl = 



108.3.28 



29205 

 2268 



(p = 12-9 times cp nearly, the result before obtained. 



The ambiguous signs of (R) cannot be mistaken or lead to error, if the 

 manner in which it is deduced from (1), (2), (3), be attended to. From (3) 



A, = — ; substituting this value of A^ in (2), 

 J 



i^^ =1 tip ^ ^2 =• «^ + l?; which, when substi- 

 9 

 luted for A, in (1), gives 



-3 mt<p + { tip - ^-^\ ; from which (R) is easily 



a - \ ? / 



found. 



ON MODEL EXPERIMENTS. 

 Until this present era of the " railway and the steam-ship and the 

 thoughts that shake mankind," the studies of the engineer, like those of 

 the lawyer, were confined to the acquiring of details of precedent, while 

 the knowledge of the scientific principles of his profession was neglected as 

 of comparatively little importance. Thus, although the recognised modes 

 of construction were numerous, the laws of structural equilibrium were few 

 and but imperfectly developed ; the builder was satisfied if the edifice he 

 was about to raise were similar in character and magnitude to others which 

 had been raised before ; — this fact at least he knew — they had been found 

 to stand, perhaps for ages; and the most ordinary, pariter paribus, style of 

 reasoning might be sufficient to assure him that his own work would be no 

 exception to the general rule. But with the railway arose a new epoch in 

 the history of engineering ; works were required to be constructed of un- 

 precedented magnitude and solidity, and for the execution of which a 

 higher amount of mechanical science and a wider range of experience were 

 required. To supply the latter of these two desiderata, numerous experi- 

 ments have been conducted of late years on the strength of materials, and 

 on models of the whole or most important component parts of proposed 

 structures. It is to these last class of experiments we would now direct the 

 reader's attention, particularly with reference to the difference of amount 

 of thrusts and strains in the model and its original. We shall first consider 

 the case of a simple horizontal girder, composed of a web and an upper and 

 lower flange, and loaded with a given weight ; — to this case may be referred 

 almost all the cast iron railway bridges now completed, as well as the pro- 

 posed tubular Menai Bridge. 



Lei A B be the girder, supported at A and B, and composed of the lower 

 flange A B nJ, the upper flange CD erf, and the web caid. Let AB = au; 

 AC -= bu; Aa = cu; Cc = du; Ctc =^ hu. 



Let the weight at w = wu' (because the weight varies as the cube of the 

 scale u) ; the weight of beam and girder >= w'v,^ ; CE = /u ; R and R' the 

 reactions at A and I!. 



Then first considering the equilibrium of the whole girder, we shall have 

 R + R' = («,• + «)'). u' 



R'.flu = (whu. + "^-j^l .«'; the girder being sup- 

 posed uniform and symmetrical throughout its mass. 



For the equilibrium of the portion C F, if T be the tension and thrust of 

 the lower and upper flanges, Y the vertical force at F, z the distance between 

 the paints of application of the thrust and tension, we have 



T = T; Y + R = (» + -«>').«'; and 



/ , , u)' .Fu\ 



u — «■/!«+ I . u' — 



\ ■ Za J 



Y/i 



Tr. 



Whence T; 



-{r-^')--r-2^') ■-■}•- 



Now, in applying the results of experiments upon a model girder to its 

 original, all we have to do is to vary the scale u from the scale of the model 

 to the scale of the original. Consequently, we find that Tr varies as the 

 fourth power of the scale or dimensions of the girder. If the web of the 

 girder be very thin compared with the breadth of the flanges and their ver- 

 tical depth, — and if their vertical depth, uc, ud, be small compared with u d, 

 — and if ui, uk' be the width of the upper and lower flanges respectively, — 

 t and t' their thrusts and tensions per square inch respectively — then we 

 shall have 



T = tkcv? = /' A' d «= nearly ; and z = a a nearly. 

 .-. Ti = takcv? — i'k'adv? nearly. 



.• . t and t' both vary as k nearly ; that is, approximately, the teruioa per 

 square inch on the lower flange, and the thrust per square inch on the upper 

 flange, of all similar and similarti/ loaded girders varies as their scale of 

 linear dimension. This we consider so important a fact, that we shall en- 

 deavour to give a proof of it in popular language. 



Suppose a vertical section made of a loaded girder at E F; then supposing 

 F the fulcrum about which the mass AE is turned, — AE will be prevented 

 from turning about F by the opposite action of the tension at F and reaction 

 at A, and the weight w and thrust at E, and the weight of A E collected at 

 the centre of gravity of C F. Now the weight mi, the weight of C F, and 

 the reaction at A, will all vary as the cube of the dimensions of the girder, 

 if we suppose the girder loaded proportioually to its mass. And the leverage 

 of these forces varies as the linear dimensions of the girder ; consequently, 

 their moment about F varies as the fourth power of the dimensions of the 

 girder; therefore, the moments of the tension and thrust at F and E, vary 

 as the fourth power of the dimensions of the girder ; therefore, if we sup- 

 pose Aa and Cc small, the tension and thrust vary as the cube of the scale ; 

 but as the tension and thrust are composed of the sum of all the tensions 

 and thrusts per square inch at a vertical section of the flanges, and a^ the 

 area of this vertical section varies as the square of the scale, — in order to 

 make up the fourth power, we must have the tension per square inch varying 

 as the scale «. 



We next propose to determine the amount of the load V«' which can be 

 supported at the centre of a girder of the dimensions u, in order that / and 

 I' at; the centre may be the same as in a girder of the dimensions m = 1 

 supporting a load to at its centre. We have proved Tr ~ Ctu", where C is 

 some constant independent of u. 



Making / = A =: - we have, therefore, 



/Va v' a\ 



ctu' = (y + "rj" ' 



Iw a w a\ 



= It * TJ' 



(// a f\' a w' a\ 



T- = (t + TJ « 



and Ct 



2V = 



2w— (k — l).tF' 



.•. 2Vu" = ^2ic-(u-l).t//j .«-. 



