18i6.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



103 



lions are entirely horizontal at the centre of the girder ; this however is not 

 the case at any other part of it. AVe shall find that if a section be sup- 

 posed to be made at any point but ihe centre, that llie molecular actions 

 have to be replaced by a vertical strain in addition to the horizontal couple 

 of tension and pressure, and it will be found also that this vertical strain 

 increases continuously from the centre to the extremities of the girder, 

 while the horizontal couple on the contrary is greatest at the centre aud 

 zero at the extremities. The efl'ect of this vertical strain, if it were suf- 

 ficient to produce rupture, would be— not to tear the material asunder 



but to make the particles at the surfaces of the section g/iVe upon each 

 other. It may be shown however that the vertical strain is so small that 

 this effect need not be apprehended, and in fact may be neglected without 

 appreciable error. 



Fig. 4. 



Let A B CD', fig. 4, represent the longitudinal section of a larger por- 

 tion than half the beam, the vertical line CD' being now beyond the mid- 

 dle point. Let W be the weight of A B C D', aud let N be the sum of 

 the vertical strains acting at €' and D'. Then we have for the equation 

 of vertical forces^ 



W' = P + N. 



But the value of P is the same as in fig. 3, and is of course unaltered by 

 the vertical section being removed to C D' ; that is, the value of P is, as 

 before, W or half the weight of the girder. Hence 



N = W — W. 

 That is, the vertical strain is the diilerence between the weight of the por- 

 tion of the beam on which it acts and the weight of half the beam. From 

 the last equation it is clear that N increases as W increases, aud if the 

 section be taken close to the further extremity of Ihe beam where W is 

 nearly equal to the weight of the whole beam, N will be equal to 2 W-W 

 or half the weight of the beam. Giving N this its greatest value we may 

 readily ascertain the strain which it produces per square inch of Ihe verti- 

 cal section. Taking as before the area of the section C or D' to be a inches 

 and therefore of the two together 2a, and putting r for Ihe vertical strain 

 per square inch, N =2 a u. Also the solid content of the two plates to- 

 gether is 2 a multiplied by the length 5400 in. (4J0 feet), and sinte Uie 

 weight of the cubic inch of iron is -28 lb., the total weight of the two plates 

 is 2 a X 5400 X "28. Hence since N equals half this weight ; 

 2 a « = a X 5400 X -28. 



It will be found from this equation that the value of ris 7561b., or the 

 greatest vertical straiit per square iuch is rather more than one-third of a 

 ton. 



It is clear that this strain would have so small an elTect to produce rup- 

 tare that it may safely be neglected. 



4. Tension produced in the Girder by a given load. 

 It has been shown that the greatest strain produced in the girder by its 

 weight merely is rather more than 2^ tons to the square inch : so that if 9 

 tons to the square inch be taken to be the degree of strain which may safely 

 be applied to the material, we have raiher less than Oi tons to the square 

 inch, which may be produced by the railway train or other load upon the 

 bridge. It is clear that when the load is at the centre it has ihe greatest 

 efl^-ect or moment about the points of support at Ihe extremities. Let us 

 now esammewhat strain a given load would of itself produce at the centre 

 neglecting the weight of the girder. Kecurring to fig. 3, let us suppose 

 the force marked W no longer to exist, and that a(.the point D a force «• 

 equal to weight of Ihe given load is applied. Let the force marked M now 

 represent the strain produced by w. Also let P instead of its former value 

 take the value now required, namely, ^ „■ : then it will be clear by reason- 

 ing similar to that in Prop. 1, that no vertical force but w acts atD. Tak- 

 ing moments about C, and putting P=^«', 



fit) . AD = M . CD. 

 A D and C B are in the proportion of 225 feet : 30 feet, or 15 : 2, so that 

 we may substitute for the above equation ; 



Hence whatever number of tons w may weigh, the strain produced by w 

 will be 3i times as many tons. To find the number of tons strain per square 

 mcho/the vertical 3ec(«,«, we observe that the width of the plate is supposed 

 to be 15 feet or 180 inches, and its thickness one inch, so that the area of 

 the section is 180 .square inches, and consequently the strain per inch is the 

 180th part of M. Consequently the strain per iuch is equal to 

 15 7.<: 1 



(3.) 



. 15 



4xlH0"'-7^' 



"48 



From this equation we get the following simple rule- 

 Fur every 48 tons load actini; at the centre of the bridge a strain of 1 ton 

 per square is produced on the metal plates. 



It follows from this that since af.er deducting Ihe strain produced by the 

 weight of the bridge an additional strain of ra.her less than Ci tons per 

 inch may safely be produced by the load, the load which could safelu be au 

 pUed at the centre of the bridge is rather less than three hundred and twelve 

 tons. 



In Mr. Stephenson's report it is calculated that the bridge can bear a 

 load of 747 tons at its centre. But this discrepancy may easily be ac 

 counted for, by supposing that Mr. Hodgkinson's experiments had refe- 

 rence to the ir.«A-i«^ ueight, whereas here the load calculated is that 

 which with the weight of the bridge would produce a strain of 9 tons to 

 the square inch. It is to be observed also that in the present calculation 

 the amount of the extreme load is somewhat underrated, because it is sup 

 posed to act at a single point, whereas in the case of a railway train it would 

 be distributed over a considerable portion of the length of the brid<.e and 

 consequently when Ihe train was at the centre of the bridge, the part Jf the 

 load cut oir by the vertical seciion C D, and resting on A D, would not 

 act wholly at D, but the centre of gravity of this portion of the train would 

 be applied at a point somewhat nearer the extremity, and the moment of ihe 

 load would be proportionabiy smaller. 



This however does not make a material difference. Mr. Stephenson 

 says in his report tliat for practical purposes a strength equivalent to 747 

 tons in the centre would be insufficient; it is clear therefore that as 312 tons 

 (which is less than half this load) causes a strain of 9 tons to the inch 

 when the dimensions of the bridge are those here assumed, it is necessary 

 to determine other dimensions by which the strength may be increased 

 There are two ways of effecting this object-Ist, by increasing the depth" 

 of the girder; 2nd, by increasing the area of the transverse section of the 

 plates, (that is, by giving the plates greater width, or greater thickness or 

 both). There is indeed a third method of increasing Ihe strength, namely 

 by increasing the dimensions of the vertical ribs beyond the degree of 

 strength necessary for the rigid connection of the upper and lower plates 

 but this method is so uuphilosophical and involves such a waste of mate- 

 rial that it may fairly be excluded. 



The examination of the means of obtaining the requisite degree of strength 

 by increasing the thickness of the plates, and the depth of Ihe tube sh'all 

 be given in the next number of this Journal. It is proposed also to examine 

 how Ihe dimensions of the tube may be varied in dillerent parts of it so that 

 the strength may be uniform throughout, to examine the ellects of expansion 

 and contraclion of the material by variations of temperature, the form of the 

 vertical ribs so that Ihey may be sufficiently strong to perform their office 

 without adding to the strain on the flanges, aud lastly, the effects of imbed- 

 ding the ends of the tube in solid masonry. 



It may be as well to say one or two words to prevent the purpose of 

 these suggestions being misinterpreted. They are certainly not intended for 

 the guidance or direction of ihe distinguished engineer who has planned the 

 Meuai tubular bridge, and whose scientific knowledge is fully adequate 

 for the calculation of its dimensions ; but to those who have not fully con- 

 sidered the principles of the strength of girders the present investigations 

 may offer an instructive lesson, especially as the conclusions are derived 

 not from gratuitous and dangerous hypotheses, but from the common funda- 

 mental principles of statical equilibrium. 



H. C. 



The syndicate of the Filzwilliam Museum, Cambridge, have received 

 from Mr Cockerell designs for completing the hall and staircase of Ihe 

 new buildmg, for parts of which Mr. Basevi had not left any sett ed 

 designs H orking drawings and estimates were ordered to be prepared 



