102 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[Aprii., 



making the molecular forces act at a great distance from tlie neutral is 

 illustrated in a very striking way in the floor-joists of a house, which are 

 uniformly laid on their thinnest edges, in order that their upper and lower 

 surfaces may be separated by the greatest possible interval. Again, if a 

 tlat slip of wood, such as a fiat drawing rule, be pressed on its broad side, 

 there is no difficulty in bending it; if, however, it be pressed on its thin 

 edge it will be almost impossible to produce a deflection. Now, it 13 

 carefully to be noted that in both experiments the nature of the forces 

 called into action are precisely the same ; the material remaining un- 

 changed, of course the cohesion or elastic force is unchanged also ; the 

 only dilTerence is that, in the second case it acts with much greater 

 advautase than in the first, simply because its leverage is increased. 



It is but a very slight extension of Ibis idea to conclude, that the form 

 of the greatest possible strength is that in which all the elastic force acts 

 at the greatest possible distance from the neutral line, or in which all the 

 material is collected in the upper and lower flanges, except what is abso- 

 lutely necessary for the due connection of them by the rib or ribs. 



This brings us to consider the second oflice of the rib — the establishing 

 a rigid connection between the flanges. It is not sufiicient that the flanges 

 should merely be kept asunder: for this purpose an open railing of vertical 

 bars would be sufficient. But it is easy to see that if such a contrivance 

 ■were substituted for the solid rib, the "antagonism" of the molecular 

 forces would not be maintained. It is absolutely necessary that the upper 

 and lower flanges should be in opposite states of elasticity and that they 

 •hould mutually counteract each other. These requisites cannot be an- 

 swered unless the web be rigid — not only vertically, to prevent the flanges 

 approaching each other, but also laterally, so as to act in every part as a 

 rigid lever, of which the fulcrum is in the neutral line, the molecular 

 actions of the flanges constitute the balanced forces. 



Before concluding these preliminary remarks, it may be as well to notice 

 one passage from Mr. Stephenson's Report, published in the last number 

 of this Journal. He says — 



" Another instructive lesson which the experiments have disclosed i«, 

 that the rectangular tube is by far the strongest: that the circular and 

 elliptical should be discarded altogether." 



It may, however, be fairly asked, whether it were necessary to make 

 that a matter of erperimfnt which might be unhesitatingly predicted by 

 the ordinary laws of mechanics ? It is clear that, comparing a curvilinear 

 and a rectangular tube of given depth and containing a given quantity of 

 material, the latter is that in which the greatest proportion of the material 

 has the maximum leverage, and confequently that the rectangular form is 

 that of the greatest strength. 



1. Practical Limits to the length of the Girder. 

 It will be found in the following methods of calculation that the particu- 

 lar form of the transverse section adopted by Mr. Stephenson, aiTords peculiar 

 facilities for the determination of each problem without incurring the diffi- 

 culties which are usually opposed to the application of the theory adopted 

 by Bernoulli. It will be assumed in all that follows that the ribs are only 

 sufficiently strong to bear their own weight, and to maintain the necessary 

 rigid connection between the upper and lower flanges— that is that the 

 whole of the available strength of the material is applied where it may 

 have the most useful eff'ect. It will remain to show hereafter how the 

 flanges may be made to satisfy this assumption, or how far they will mo- 

 dify it. 



It is proposed for the Menai Bridge that the plates of iron shall be one 

 inch thick ; this construction very nearly satisfies the conditions of the 

 greatest strength as laid down in the extrnct from Moseley's Engineering 

 given above. The flrst problem which will be the determination of the* 

 greatest possible length of a girder of the depth proposed (30 feet) so that 

 it may bear its own weight. 



It is found by experiment that wrought iron will bear with safety a strain 

 of nine tons to the square inch, and if that amount be much exceeded, 

 the material begins to stretch. Now as the beam cannot be deflected 

 without some part of its material stretching, the point to be determined is 

 this— what is the length of the beam when by its own weight a strain of 

 nine tons to the square inch, is applied to the metal. It is obvious that if 

 the beam be of uniform depth, the longitudinal strains will be greatest in 

 the miildle. 



Let A B C D, fig. S, represent a longitudinal section of one-half the 

 girder, which is supposed to be cut in half by a vertical plane at C D. If 

 we suppose the half beam to be acted upon at C D, by forces similar to the 



Fig. 3. 



I 





molecular actions which actually exist at C D in the undivided beam, it 

 is clear that the conditions of equilibrium will not be affected. 



The forcesacting on A B C D are— lst,P the upward pressure of theabut. 

 ment (the beam being supposed uniform P = ^ the weight of the beam, br 

 the ordinary conditions of equilibrium). 2nd, a downward force W equal 

 to the weight of A B C D, and acting at the centre of gravity half way 

 between A and D. 3rd, the molecular actions at C D. 



Respecting these molecular actions it is to be observed that they are 

 wholly horizontal ; for P and W being both equal to half the weight of the 

 beam P=W, and therefore if a third vertical force were introduced the 

 equation of vertical forces could not hold. The molecular forces are there- 

 fore horizontal ; they are also equal and opposite, for otherwise the equa- 

 tion of horizontal forces could not hold. As therefore we have supposed 

 the plates A C, B D to be of comparatively small thickness, we may sup- 

 pose the molecular actions to be represented by two forces H, M, in the 

 directions indicated by the arrow beads. The only eff'ect of representing 

 all the forces of compression by one single force, and all the forces of ten- 

 sion by another single force, is the assumption of that which is practically 

 true, that all parts of the section C exert equal pressures, and all parts of 

 the section D equal tension, and that all the forces at C and at D, act so 

 near each other that they may in each case be represented by a single 

 force. 



Taking moments about B, W ) A Dr:M. A B. (1-) 



Now we suppose the tension at D to be 9 tons or 20,160 lb. to the square 

 inch ; consequently if we call the area of the section D, a inches, M = 

 20,100 a. 



W is the weight of the plates B C and AD : if the length of each of 

 them be I inches, its solid content is a I cubic inches, and since the weight 

 of a cubic inch of wrought iron is about -28 of a Ih. the weight of each 

 plate is a ix '28, and W is double this or 2 a ix-28. Substitating in (1.) 

 a(ADx-28 = 2O,lC0aAB (2.) 



j5_2oj^i A B = 72 000 A B. 



•28 



A B the depth of the girder is in the proposed bridge 30 feet or 360 inches. 

 Therefore multiplying 300 by 72,000, and extracting the square root, we 

 get the value of ( in inches : this value will be found equivalent to 421-26 

 feet. Hence we arrive at the following conclusion, I being half the length 



of the girder ; 



The greatest length of a girder 30 feet deep, which trill support its own 

 weight safely is Si&fcet. 



It will beobsened that this conclusion is independent of the arch of the 

 cross sections C and D. or of the width of the girder. This circumstance 

 arises from the tension and weight being both proportional to the cross 

 section. 



2. Tension at the centre of a Girder iSO feet long. 

 The length proposed by Mr. Stephenson falls far within the limits of 

 length determined by the last proposition. The next point to determine is 

 the actual tension per square inch when the length is that of the Menai 

 Bridge— namely, 450 feet. 



Using the figure and notation of the last proposition we have putting io 

 2), the value of / or A C = 225 feet (-2700 inches), and the value of A B 

 c 300 inches ; and putting also t for the tension per inch at D. 

 ax(2700)2x28 = <a300, 

 ,290,000 X '28 



* = - 



360 



Efl'ecting the operations indicated by this equation, we find the value of 

 i to be 5(.70 lb., or 253 tons. Hence we come to this conclusion— 



When the girder is i^O feet long the tension prodwcfd at its centre by itt 

 weight is rather more than 2J tons to the square inch. 



1 his conclusion like the last is independent of the area of the section 



CorD. 



3. Vertical strain on any part of the Girder. 

 It has been demonstrated in the first proposition Uiat the molecular ac- 



