1846.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



165 



bi» the same beam under different circumstances. If a beam ZT betioa of the breadth a 6 ; that is , the breadth a i , ia one position, is repre- 



strained, with parallel lines drawn as above directed, at m" m" p" q", the 

 very fact, that the line b s being bent, shows that its particles are not in a 

 state of quiescenre, although it maj' be the same length as m' n', or that 

 one of the subdivisions of 6 s is equal to one of those in m'n', (he position 

 of the beam before it is bent ; as Z T becomes more and more bent, the 

 divisions will open in the upper regions near m" n", and more contracted 

 towards the lower near q" p", so that the position of the neutral axis must 

 shift towards the lower part of the beam from the top, unless the curves 

 m" n" and p" q" be arcs of concentric circles, then the arc a o in the centre 

 will always be equal to m' n'. But it is not evident that the particles in 

 the arc ao or b s are not strained at all, because either of them happen to 

 be of the same length as m n, m'n', or m" n", which were all equal before 

 the forces were applied 



The conclusion we come to here is, that the whole turns round an axis, 

 sometimes outside of the body and sometimes inside, according to the 

 position of the centre of the circle of curvature of the curve where the 

 greatest strain is applied. Let c' be the centre of the circle of curvature 

 of the point in the centre of the arc q" p", then that arc is supposed to be 

 strained round r.' as an axis. Let c", c'", and C be respectively the cen- 

 tres of the circles of curvature for the points in the middles of the arcs 

 b$, ao, m" n"; then the arc m" n" is supposed to be bent round the centre 

 C; ao, round c'" ; and 4s round c". When the filaments that are the 

 most expanded — that is, those near m" n'' in Z T, and near, 7 p in X Y, — 

 become flatted at the centre, between m", n", or between q p, which is 

 generally the case before fracture ensues, then the centres of the circles of 

 curvature at these centre points, after changing from C to c'" to c", &c., as 

 the beam Z T becomes more bent, now returns in a contrary direction, 

 c', c", c'", &c. ; and when the fibres in m" n" become straight, the radius 

 of curvature becomes infinite, as in the case of the beam V W at rest, io 

 which the particles or fibres at m" n'' are supposed to be straight. 



We shall next consider the nature of the forces exerted by the filaments 

 at different points of the cross section, in the region where fracture would 

 ensue, when the strain exceeds the elastic limit of the body. 



The beams or bars, the nature of the cross sections of which we are in- 

 vestigating, are supposed to be supported at the ends and loaded in the 

 middle. In small beams, the change in the particles that we are about to 

 describe is not perceptible ; yet it will be found very considerable in large 

 girders, such as the tubular bridge about to be constructed by Mr. Stepbeo- 

 (OD, or in small girders of a flexible nature. 



H^-.. 



"---.A 



TTT 



c,pq 



7"L 





The figure used in elucidating this matter is distorted, in order that the 

 change under consideration, near the centre of the beam, may be more ap- 

 parent. A very simple mode of illustrating what we shall describe rela- 

 tive to the molecular action of the particles in a cross section, near the 

 centre of the beam, may be obtained by taking a rectangular piece of 

 caoutchouc, whose cross section would be represented by a 6 ^p ; but, it is 

 to be understood, that in point of structure we do not compare caoutchouc 

 or india rubber with iron, brass, or wood ; — but, merely to show the man- 

 ner in which the particles in the cross sections of bodies, under the cir- 

 cumstances we have just described, endeavour to exert themselves. Let 

 ECDF be the position of a beam before the weight W is applied, — 

 H G A B K L its position after ; the cross sections in the two positions 

 will be represented by the figures a fcyp and cdnm. The action of the 

 weight or force W compels the point s to move to (, and the point r to move 

 to g, and has a tendency to lengthen the whole beam ; while at the same 

 time, the filaments in the upper part of the beam, near the middle, become 

 compressed in the direction of the length A B, and extended in the direc- 



sented by c d in the other. But the fibres in the lower part near r, in 

 changing from r to g, become expanded in the direction of the length A B, 

 and contracted in the direction of the breadth p q, so that p 9 in the cross 

 section becomes mn. From the rigidity of materials, this change may not 

 have place, or may not be perceptible ; but, in all cases, a force acting in 

 the direction of the arrow will have the tendency to change the cross sec- 

 tion a i 9 p into one like cdmn, which if it be not able ultimately to effect, 

 fracture must ensue. 



As we have before observed, what we have just described will become 

 clear by bending a rectangular piece of india-rubber. If the section of 

 the rod or beam be circular, as Z, the change will differ materially from 

 the one already described, for the circle will become, or endeavour to 

 become, a figure like an oval. The change in the molecular particles in 

 the cross section will arrange themselves differently, according to the ex- 

 ternal forms of the beams and the positions in which they are placed : 

 something like the change of form in triangular beams are given in the 

 figures X and Y. When the strain is about to exceed the limit of the 



Fig. 8. 



Fig. 9. 



10. 



elastic power of the material, the fibres which are in a state of quiescence, 

 compared with those in the extreme upper and lower regions of the beam, 

 will arrange themselves in a curve resembling n f in the figures X, Y, and 

 Z, the equation to which will be given hereafter. The behaviour of the force 

 and filaments here stated are in themselves sufficiently simple and explicit, 

 but, in order that none of our readers may enter npon the mathematical 

 investigation of this important subject with incorrect notions respecting 

 the different circumstances under which the beam may be placed, we have 

 thought it expedient to add the following expositions and illustrations : — 



Let E i F be the upper or lower 

 surface of the beam before the 

 weight W is applied ; H < L will 

 represent the upper, and G §■ K 

 the lower, after its application. 

 The same parts are marked with 

 the same letters as the figure pre- 

 ceding, for these surfaces are sup- 

 posed to have reference to that 

 i figure. If the particles at a « i in 



the bending process were such that they would merely become more 

 dense, then the breadth at a j fc would not be changed; but it is not the 

 case, for the harder parts of the material merely obtrude themselves into 

 the softer, and partly become compressed and partly swell the breadth of 

 the beam near these parts, as at ctdia the upper section H t L. But 

 in the lower surface Gg-K, or near it, the pariicles at rag- n become se- 

 parated, and the breadth becomes contracted from a b, which is equal to 

 pqiomn. 



We shall in the next place proceed to the mathematical investigation. 

 Let W be the weight in pounds that would be borne by a beam of wood, 

 iron, or any other material, whose cross section is an inch square, when 

 the strain is as great as it will bear without destroying the elastic force of 

 the body, and the direction of the force coincident with the length or axis 

 of the bar; and let W' be any other weight to be supported under the 

 same circumstances. Suppose the cross section of the piece to support it 

 to be a rectangle, whose breadth = j: and thickness = y inches. 

 W' 

 Then W ; W':: X'.xy; or, ^^=zxy. 



Strictly speaking, the lengths must be the same, or W and W' must in- 

 clude the weights of their respective beams. This proportion has place 

 from the well-known principle— abundantly proved by experiments— 

 that " the strength of a bar or rod to resist a given strain, when drawn in 

 the direction of its length, is directly proportional to the area of its cross 

 section ; while its elastic power remains perfect, and the direction of the 

 force coincides with the axis." 



