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THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL, 



[March, 



lias been done above ), be reckoned among the external forces of the 

 system. The thr ust of the arch is uniform in every part of its haunches 

 where there are only vertical forces. But were the horizontal exter 

 nal pressures just alluded to exist, the thrust of the arch will be 

 greater at the crown than at the abutments — being greater than we 

 have calculated it at the crown, and less than we have calculated it 

 at the abutments. It seems impossible to c alculate the amount of this 

 alteration, for to ascertain it we must know the mutual pressures of 

 the contiguous portions of loading, the friction of the materials, and 

 the degree of cohesion produced by ramming or se tiling — effects ut- 

 terly beyond calculation. It may be observed, however, the ballast 

 will generally be so firmly compacted that the part of eacli spandril, 

 even if unsupported, would in most cases have little tendency to 

 slide : and therefore where this precaution is used, the above methods 

 will answer all practical purposes. 



Sometimes the voussoirs of two arches, which spring from the same 

 pier, do not rise independently, but are built together, and press 

 upon each other at their extradosses for some distance as they 

 rise together from the pier. It is clear that for the purposes 

 of our calculation, the springing of each arch must be reckon- 

 ed to commence from that point where the adjacent arch ceases to 

 affect it. Wherever the spandrils of the contiguous arches are con- 

 nected near their springing by small inverted arches (as in Blackfriars 

 bridge), the modifying effect of these subsidiary structures must be 

 taken into account. The thrust in such cases can only be reckoned for 

 that portion of the main arch which is not affected by the contiguity 

 of the other arches. 



The general conclusion from the above reasoning is, that the smaller 

 the rise of the arch iii comparison with its width, the greater extern 

 paribus will be the lateral thrust (of course these conclusions cannot 

 be applied to the platebande or flat arch, where the depth of the 

 voussoirs is so great, compared with the other dimensions, that the 

 methods given above are totally inapplicable). As instances of this 

 truth, may be cited the lofty Pointed arches of cathedrals, which fre- 

 quently sustain enormous weights without exerting great lateral 

 thrust. But it may be as well to refer, in passing, to an erroneous no- 

 tion which is frequently entertained, that because high Pointed arches 

 can sustain great weights, they therefore ought to do so for the sake 

 of their stability. An idea of this sort is expressed in Pratt's Princi- 

 ples of Mechanics, and is supported by very confused and perfectly in- 

 applicable reasoning : as has been already said, and vvill be proved 

 hereafter, the form of the arch depends not on the amount of the 

 loading, but on the distribution of it. 



To return from the digression — we easily see that the limiting 

 cases of the general conclusions just stated respecting the thrust of 

 the arch are these. — If the arch were quite flit (the voussoirs not 

 being of appreciable depth), a finite load would produce an infinite 

 lateral thrust: again, were the arch so lofty that its span could be 

 considered inappreciable in comparison with its rise, the greatest 

 load would produce no horizontal thrust at all. In fact, this last hy- 

 pothetical case is equivalent to that of a weight sustained by vertical 

 posts or columns. 



The consideration, that the thrust of the arch depends only on the 

 rise and span — that the form does not affect the thrust (except indi- 

 rectly.by influencing the position of the centre of gravity of the load), 

 forms an appropriate introduction to the investigation of the lateral 

 pressures of groined vaulting. In plain cylindrical arches, the thrust 

 determined above is distributed in lines parallel to the axis of the 

 arch throughout the whole of the springing. Consequently, it acts 

 on the piers of a bridge (to take an instance) along a surface of which 

 one of the dimensions is the breadth of the roadway, and may in gene- 

 ral be considered as uniformly distributed. But if we take the case 

 of double or intersecting groined arches resting, as supposed in the 

 following diagram, on four detached piers, the amount of surface over 

 which the thrust will be distributed is diminished with the diminu- 

 tion of the horizontal dimensions of the piers. In this case also the 

 double arch will exert a double set of thrusts. Supposing the plan a 



rectangle, and that the arches and the distribution of the loading 

 are perfectly symmetrical. Let a/, bg,dh,ce,he the four piers, 

 and K the keystone or boss. Also let P be the thrust arising from 

 the arch of which the axis is parallel to /g ; and let Q be the thrust 



arising from the arch of which the axis is parallel to c d. We may 

 suppose that the forces at a, b, c, d, are all equal to P, and the forces 

 at e,/, g, h, are all equal to Q, since we have supposed the two halves 

 of each arch to be under exactly the same circumstances. 



Now, if we take moments about a c, for the half arch, of which the 

 axis is parallel to a c, no other moments will appear in the equation 

 but that of the pressure at the crown of this arch, and that of the 

 vpeight resting upon it. Hence the strain P is determined by the 

 rules already laid down for single cylindrical arches— that is, it is 

 equal to the moment of the weight divided by the rise of the arch. 

 In the same way is the pressure Q determined. And hence we arrive 

 at the conclusion, that the total pressure on any one of the four piers, 

 may be considered to be made up of two component forces, — the 

 thrust of each of the two arches considered separately and inde- 

 pendently of the other. 



It may so happen that the form of the groining materially affects 

 the position of the centre of gravity of the loading. There will not 

 however be generally much difficulty in estimating the moment of the 

 weight by methods analogous to those already described. 



This is as much as it seems necessary to say at present respecting 

 the thrusts of arches. Of the means of resisting those thrusts, or of 

 fixing the dimensions of the buttresses or piers which sustain them, 

 mention will be made hereafter. In conclusion, it may be observed, 

 that though these methods are confessedly approximative, they ap- 

 pear quite as much entitled to confidence as others of a more elabo- 

 rate nature. M. Garidel has, with wonderful ingenuity and labour, 

 formed tables of the thrusts of arches (Poussees des Voutes), calculated 

 from a long mathematical formula, analogous to that arrived at by 

 Prof. Moseley. Respecting, however, all long mathematical formula 

 applied to practical mechanics, we are convinced, from considerable 

 experience, that the following strictures are correct — first, these 

 formulEB are too difficult to be employed by the engineer ; secondly, 

 if he could employ them, it-would not be worth his while to do so — 

 for they generally neglect some practical circumstance which entirely 

 destroys their accuracy. In the case, for instance, of M. Garidel's 

 tables and Prof. Moseley's formula, both proceed on the supposition 

 that the materials of the voussoirs are perfectly unyielding and ma- 

 thematically adjusted. If, by the slightest settling, the point of appli- 

 cation ol the resultant of the forces at the crown and springing be 

 altered, the whole investigation fails. It is also obviously impossible to 

 estimate rigorously the effect of the cohesive forces between the con- 

 tiguous portions of loading resting on a series of arches ; and the he- 

 terogenity of the materials is an insuperable obstable to any but ap- 

 proximative calculation. 



H. C. 



