496 



BRIDGE. 



Theory. 



PLATE 

 LXXX. 



Fig. 2. 



stone and its load, lay off also abln, bc=:ux, ace. 

 ' and these divisions will represent the weight of the 

 several sections, the superincumbent matter being in- 

 cluded. 



This method is evidently only an approximation ; 

 we consider the principal load as arising from the mass 

 incumbent on each section, or at least that the weights 

 of the sections are proportional to these masses. It 

 becomes pretty accurate, by taking iv in the mean circle 

 drawn between the soffit and back of the arch; and 

 we might render )t still more accurate, by giving the 

 determination a fluxionary form, but we write at pre- 

 sent for the practical builder, to whom the calculus 

 is seldom known ; besides, as the reader will see here- 

 after, we do not think the rigid determination of this 

 matter as yet of much consequence. 



Having thus discovered the weights of the sections, 

 and laid them off on the horizontal line, as if for a 

 flat arch, and having, either from the given form of 

 the keystone, or the horizontal thrust, drawn the 

 anglen of abutment which a flat arch would require, 

 the joints of the arch in question are to be drawn pa- 

 rallel to these, and through the extremities of the 

 proper section.-., previously marked out, as above men- 

 tioned. If ther-.' be intermediate joints, they may 

 either be drawn properly related to the others, or 

 separately, discovered by a repetition of the con- 

 struction. For example, let C be the given centre 

 for the keystone ; draw C Cb, Cc, &c. ; and through 

 1 draw the j< int 1R parallel to Co, also 2T parallel 

 to Cb, and 3W to Cc, &c. : the arch would then be 

 in equilibration. 



Thus we find, that, by the proper adjustment of 

 the joints to the weight of the section, ive may form 

 equilibrated arches, luivitig soffits of any Jtgiire that 

 may be thought proper, and with any proportion of 

 dead weight over them that circumstances may re- 

 quire. Let us now look at the converse of this pro- 

 blem ; where the inclinations of the joints being given, 

 it is required to discover the mass or weight which 

 must be allotted to each section, so as to preserve 

 the whole in equilibrium. 



Pursuing the mode already employed, it is evident, 

 that if we lay off from one centre the angles to be 

 formed by the successive joints, or abutments, with 

 the vertical line, a horizontal line drawn to cut them 

 will represent, by its successive segments, the weights 

 of the several sections ; while, at the same time, the 

 perpendicular let fall from the centre on this line will 

 exhibit the horizontal thrust. If the arch, therefore, 

 must throughout be of equal thickness, we have only 

 to mark off upon the soffit, or rather upon the mean 

 curve, segments proportional to those of the hori- 

 zontal line. If the upper and lower outline of the 

 arch be determined, we must divide it into trapezoids, 

 having the same proportions ; then draw the joints 

 parallel to the lines expressing the given angles of 

 inclination. Such joints will run to several different 

 centres, thereby shewing us, that their union in one 

 is not at all necessary to the security of the arch, 

 even should that be a portion of a circle. 



The position of the joints is usually given in a dif- 

 ferent way from that which we have just considered. 

 In circular arches they are generally formed by pro- 

 ducing the radii from the centre ; and in others they 



are commonly drawn perpendicular to the curve. 

 Now, though we have just shewn, that this is by no 

 means necessary to the equilibrium, yet, as it is in rea- 

 lity the most convenient in practice, it may be of im- 

 portance to attend to the effects likely to be produ- 

 ced by tliib modification. 



We see, in Fig. 10. that the tangents on the ho- 

 rizontal line rapidly increase as we pass outward, 

 and we should therefore increase, in the same pro- 

 portion, the weight of our sections. We cannot in- 

 crease the base as proposed above, for that is neces- 

 sarily given by the position of the joints, but, as we 

 arc still able either to increase the height or the 

 breadth of the sections, we may consider the effect 

 of both these modes. 



Let it be required, then, to equilibrate a circular 

 arch, where the stones being all of i qtial thickness, 

 with joints equally distant, and drawn all to one 

 centre, we are only at liberty to increase the width 

 of the roadway, or length ot the horizontal courses. 



Considering each course of arch stones as a prism 

 of a given base, a supposition sufficiently accurate, 

 it is evident, that its magnitude or weight increases 

 with the length only. But this weight must, from 

 the principles already laid down, be as the difference 

 of the tangents of its abutments ; the length therefore 

 must be in that ratio. Accordingly we find the 

 breadth at different distances from the vertex in the 

 same way with the weights of the sections : the 

 breadth at 4.5 must be double, and at 55 must be 

 about triple of that at the crown, and will increase 

 still more rapidly afterwards. Proportions such as 

 these may answer well in the short flight of steps for a 

 flying staircase, but are quite ui fit tor our present pur- 

 pose. When we recollect, however, that in a bridge, 

 the extraordinary expansion towards the haunches 

 is materially corrected by the increased pressure of 

 the incumbent mass in that part, we are encouraged 

 to proceed a little farther, and consider the effect of 

 the second mode of effecting the equilibrium. 



The pressure of matter upon each section has al- 

 ready been stated as proportional to tvxsiv; but Iv is 

 the difference of the sines of the angular distances of 

 the successive abutments from the vertex, and sw is 

 the mean versed sine added to the given thickness at 

 the crown, when the roadway is horizontal. We have 

 therefore the pressure as the difference of the sines X 

 (mean versed sine + thickness at vertex.) But these 

 pressures are also, from the theory, as the difference 

 of the tangents of these angular distances. In the 

 I case, where the angles of abutment, and, 

 consequently, where the difference of their sines and 

 tangents are known, and where the mean versed sine 

 may also be readily formed, it will not be difficult to 

 state the conditions ot equilibrium for an arch of any 

 dimensions. 



In the common mode of building, we must give the 

 arch a sufficient thickness at the keystone, to resist 

 the horizontal thrust, ensure stability, and bear the 

 loads likelyto come upon it. We must also COM 

 part with a certain thickness of gravel, or other mat- 

 ter, so as to form a roadway. The varying pr. 

 of the wheels of a loaded carriage, when it is propo- 

 gated through this stratum of gravel, will be so far 

 diffused as not to disturb the stone immediately bc- 

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