1846.] 



THE CIVIL ENGINEEERAND ARCHITECT'S JOURNAL 



101 



was extended and the concave compressed, and that consequently on the 

 one side the strains vpere of the nature of tensions, and on the other of 

 pressures. The simple deduction from this view of the case was that there 

 is some place between the two sides of the beam where the extension of 

 the particles ceases, and the compression begins, — that is, there is a line 

 in the beam which is neither lengthened nor diminished in length by the 

 deflection. To this line Bernoulli gave the name of the Neutral Line : 

 his views are now exclusively adopted ; the only difficulty still existing re- 

 fers not to their accuracy, but the means of applying them. 



A very simple illustration of the fact that a deflected beam is partly com- 

 pressed and partly extended, may be obtained by bending a twig of hazel or 

 willow ; if the bark be tender and flexible, it will be found to have a cor- 

 rugated appearance — to be gathered up in folds — on the concave side> 

 and on the convex side to be tightly stretched or actually burst asunder. 



The chief difliculty in the above familiar instance, and one which indeed 

 Bernoulli's theory is generally subject to, is tlie determination of the pre- 

 cise place where the material is neither stretched or compressed. But 

 there is also another objection which will frequently hold — Bernoulli 

 adopted so much of Leibnitz's theory as assumed that the tension of any 

 part of the beam varied as the distance of a fixed line — that line being ac- 

 cording to Leibnitz in the upper or lower surface — according to Bernoulli 

 the neirfra; (in< between the two surfaces. Bernouilli's law of the mole- 

 cular forces was in fact this, that if at any purl of the beam, a plane be 

 drawn perpendicular to the neutral lines, the elastic forces perpendicular 

 to that plane (whether of the nature of pressures or tensions) are propor- 

 tional to the distances of the molecules from tlie neutral line. This law 

 was founded on Hooke's principle that the restituiive forces of a body are 

 proportional to the amount by which its natural length is increased or 

 diminished. But it is found in practice that it is possible to apply so great 

 a strain to a body that Hooke's law ceases to be true ; in^other words, that 

 there is a degree beyond which if the extension or compression of a body 

 be carried, its restitutive powers will no longer continue to increase with 

 the strain, but will actually diminish. The point is called the "elastic 

 limit ;" the nature of it may be illustrated by bending a flexible strip of 

 metal or wood to such an extent that the restitutive forces are no longer 

 able to restore the original form of the body, which remains permanently 

 bent. 



The difficulty certainly is often opposed to the application of Bernoulli's 

 laws, because it is a matter of common experience that beams may be so 

 much strained and bent as to permanently retain their deflection when the 

 load is removed. 



The most celebrated of the modern writers on the theory of the strength 

 of girders are M. Poncelet and Prof. Moseley. The elaborate researches 

 of the latter contained in the fifth part of the Mechanical Principles of En- 

 gineering, are well known to the English reader. In these researches 

 however, the weight of the girder itself is neglected as small compared 

 with the load to be supported : but as in the case of the Menai bridge, a 

 great proportion of the strength of the material is required to support its 

 own weight, a separate investigation seem to be required. 



Distinctions between girders, arches, and suspension bridges. 

 It is very necessary that the effect en girders by which the molecular 

 action on their upper and lower sides have opposite tendencies should be 

 clearly conceived in the mind : for it is this opposite tendency which con- 

 stitutes the essential strength of girders. It is also their characteristic, by 

 which they are distinguished from the other structures for supporting loads 

 between piers or abutments. In the arch the material is entirely in a state 

 of compression, in the suspension bridge, of tension — in the girder alone 

 part of the material resists compression, and part resists extension. Did 

 not this property exist, the girder must, like the arch, and the suspension 

 chain be more and more stretched, the more nearly it was horizontal. It 

 is known that no finite tension will make a suspension chain quite horizon- 

 tal, and that when the deflection is small the strain and tendency to rup- 

 ture is very great. In the arch'also it is known that where the curvature 

 is small, the lateral thrust is greatly increased, and that when the height 

 of the arch is very great compared with its span, the strain on the material 

 i« not much more than that of the superincumbent mass. In the girder, how- 

 ever, it is found that no advantage is gained by giving the surfaces a great 

 curvature ; the strongest and most useful form which can be devised is 

 that for which the girder is equally strong in every part, and for the at- 

 tainment of this object, the curvilinear surfaces of the girder may be made 

 very much flatter than would be the curves of an arch or suspension chain, 

 eabjected to equal strain under similar circumstances. 



Another important characteristic of the girder, due to the opposite ten- 

 dency of the molecular action on its upper and lower sides, is that unless 

 it be positively bent by its load, it exerts no lateral force on its abutments 

 A beam supported on two props will exert upon them a vertical pressure 

 only, and however much it may be loaded will exert no horizontal force at 

 its extremities unless the load be suflScient to alter its geometrical form. 

 In an arch or suspension-chain, however, the smallest load requires a cor- 

 responding lateral force, this force being in the former of the nature of a 

 thrust, in the latter of a tension. The distinguishing property of a girder in 

 this respect is due as has been said to the antagonism of its molecular ac- 

 tions, for it is demonstrable that whatever lateral forces its particles exert 

 on the lower side are counterbalanced by equal and opposite forces on the 

 upper side, so that ultimately the resultant horizontal force is zero. 



Form of section of greatest strength. 

 Before proceeding to determine the amount of the strains actually ex- 

 erted in the proposed Menai tubular bridge, it is necessary to consider 

 what form of the transverse section of the girder gives the greatest 

 strength. These considerations cannot be expressed more clearly than in 

 the following quotation from the fifth part of Professor iMoseley's work 

 already referred to. The extract deserves careful perusal, as the correct 

 comprehension of it will clear up all difficulties as to the means by %vhich 

 the strength of girders is obtained :— 



"Since the extension and the compression ofthe material are the greatest 

 at those points which are most disiant from the neutral axes of the^ection, 

 it is evident that the material cannot be in the slate bordering upon rup- 

 ture at every point of the section at the same instant, unless all the material 

 ofthe compressed side be collected at the same distance from the neutral 

 axis, and likewise all the material of the extended side, or unless the ma- 

 terial of the extended side and the material of the compressed side be re- 

 spectively collected into two geometrical lines parallel to the neutral axis ; 

 a distribution manifestly impossible, since it would produce an entire sepa- 

 ration ofthe two sides of the beam. 



" The nearest practicable approach to this form of section is that repre- 

 sented in the accompanying ligure (tig. 1), where the material is shown 

 collected in two thin but wide flanges, but united by a narrow rib. 

 Fig. 1. "" 



[:>^«ps»->^ 



That which constitutes the strength of the beam being the 

 resistance of its material to compression on the one side of 

 its neutral axis, and its resistance to extension on the other 

 side, it is evidently a second condition of the strongest form 

 of any given section that when the beam is about to break 

 across that section by extension on the one side, it may be 

 about to break by compression on the other. So long, there- 

 ^j gi fi>re, as the distribution of the material is not such as that 



1-''''^'' '^^' 1 the compressed and extended sides would yield together, the 

 strongest form of section is not attained. Hence it is apparent that the 

 strongest form of the section collects the greater quantity of the material 

 on the compressed or the extended side of the beam, according as the re- 

 sistance ofthe material to compression or to extension is the less. M'here 

 the material of the beam is cast iron, whose resistance to extension is greatly 

 less than its resistance to compression, it is evident that the greater portion 

 ofthe material must be collected on the extended side. 



" Thus then it follows, from the preceding condition and this, that the 

 strongest form of section in a cast iron beam is that by which the material 

 is collected into two unequal flanges joined by a rib, the greater flange 

 being on the extended side; and the proportion of this inequality of the 

 flange being just such as to make up for the inequality of the resistances 

 of the material to rupture by extension and compression respectively. 



" iMr. Hodgkinson, to whom this suggestion is due, has directed a series 

 of experiments to the determination of that proportion of the flanges by 

 which the strongest form of section is obtained." 



Effect ofthe vertical ribs. 



Now it will be seen from the representation of the trans- 

 verse srction (fig. 2) of the Tubular Bridge that it is in fact 

 a girder of the required form, except in that it has two 

 lateral ribs instead of one central rib. These ribs, it will be 

 seen from the quotation, contribute to the strength of the 

 r ^. ^^x^Njig girder, not so much directly by their own strength, as 

 Fig. 2. indirectly (I) by separating the upper and lower flanges, 



and (2) by establishing that rigid connection by which the opposite 

 tendency of the molecular action already spoken of is maintained.. 

 These two offices of the vertical ribs ought to be rightly understood 

 Respecting the first, it may be observed that for a given quantity 

 of material, the strength of the beam increases with the distance by 

 which the upper and lower flanges are separated. This increase 

 of strength does not arise from the increased mass of the rib, so mach 

 as from the circumstance that the further the flanges are apart, the 

 further are they from the neutral line within the girder, and, consequently, 

 the greater U the leverage of the molecular forces. The advantages of 



