234 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[July, 



Fig. 3. 



parallel fo E'H, the bisection of the interval between these will give 

 tlip horizontal projection of the joint. Similarly, the side and end 

 projections are modifications of those belonging to' the circular arch: — 

 lliey are fully investigated in the appendix. 



Having obtained a tolerable a])proximation to the forms of the arch 

 stones, it is not uncommon for bridge-builders to throw the remaining 

 resjjoiisibility on the abutments, which, besides transmitting the pres- 

 sure, have to continue its distribution among the parts of the pier, hi 

 truth, the principles of equilibrium seem never, even in the ease of the 

 right arcli,^ to have penetrated beyond the facing stones of the piers ; 

 and the etlect of the arrangement in every bridge tvhich I hare ueii, or 

 ilie drawings for which I have inftptcted, is to throw the whole weight 

 of the arch on the outside stones of the pier and on the outer row of 

 piles in the foundation. To see this clearly, let us draw one of the 

 abutment stones of a right bridge. The 

 oblique face AB receives the pressure of 

 the lowest voussoir ; and it ought to receive 

 that pressure perpendicularly. But the 

 stone is prevented from yielding by re- 

 sistances against the surfaces CD, DE : the 

 pressure of the voussoir is thus decomposed 

 into two pressures, one against CD, well 

 known to be the horizontal thrust of the 

 bridge, and the other against DE, equal 

 to the weight of all the mason-work between the crown of the arch 

 :nid the vertical line through B. Now, since all the stones of the piers 

 are squared, no change (except by improper straining) can take place 

 in the directions in which these pressures are propagated. The pres- 

 sure against CD is communicated along the abutment course to the 

 spring of the next arch, or to the corresponding breadth of the final 

 abutment; while the pressure against CD is transmitted through the 

 facing stones of the pier to the outer row of piles. It will, indeed, be 

 said, that the cohesion of the mortar, and the alternate jointing of the 

 courses, render the pier one mass, and that, therefore, such niceties 

 are not worthy of attention. But, indeed ! is the final disposal of the 

 entire strain of a bridge such a trifle ? Then let us fit our arch-stones 

 by guess, and sweep the span in any fancy. It is at this very corner 

 that all the care of the engineer is required; and I do maintain, that 

 the method in common use outrages the doctrines of eqiulibrium, and 

 renders our arches less secure than they ought to be. It is a piece of 

 bad engineering to throw the whole weight of a bridge upon one row 

 of its su])ports, and to give the others scarcely any strain; especially 

 ■when it is considered that that row is most liable to decay. The al- 

 ternate jointing of the stones calls info action that species of resist- 

 ance which ordinary building-material is least capable of exhibiting ; 

 one end of a stoni; is pressed downwards, while its other end is en- 

 gaged between two Ijlocks ; the consequence is a tendency to break 

 the stone over, to dia/uid its upper surface ; and it is notorious that 

 the strength of stones in this way is much inferior to their jiower of 

 resisting a simple crush. The alternate jointing and the mortar are 

 useful enough in correcting the bad effects of unavoidable inaccuracy; 

 there is no need for deliberate error to )mt them to a severer use. 



The best possible arrangement is to give to each square foot of the 

 foundation its fair share of the wdiole burden. In order to do this, it 

 becomes necessary to lay a counter arch, of u parabolic form (its con- 

 vexity downwartls), upon the pier-head. Such an abutment course 

 would carry the horizontal thrust to the spring of the next arch, pre- 

 cisely as a Hat course would ; but it would distribute a uniform down- 

 ward pressure on each horizontal foot : and, in tliis way, the ioun- 

 dation w-ould be pi-essed on exactly as if the whole weight of mason- 

 work, from the crown of the one arch to the crown of the other, were 

 piled upon it in squared courses. 



On investigating the forms of the joint on a parabolic skew, I found 

 its plan to be a line of the third order, the double parabola; that its 



end elevation is a semi-cubic parabola; and that its side elevation is 

 another line of the same order. Students of the higher mathematics 

 will at once recognise the equations of these curves as the results of 

 other inquiries. For the computations of the parts, on account of the 

 regular progression of the different examples, the method explained 

 in my treatise On the Solution of Equations of Jill Orders, will be 

 found to atibrd peculiar facilities. 



Appendix, 



In the preceding part of this paper, I have stated thegeneral prin- 

 ciples which ought to regulate the construction of oblique arches. In 

 this, the second part, I propose to enter more into detail, and to give 

 the demonstrations of the theorems above laid down. 



The general investigation into the stability of a vault would neces- 

 sarily be complicated by the peculiarities of the ultimate abutments, 

 and by the assumed directions of the lines of pressure ; for these di- 

 rections are, within certain limits, arbitrary. For the present pur- 

 pose, it is enough to consider the case of a vault resting on parallel 

 abutments, cylindroid, and having the lines of pressure contained in 

 vertical planes parallel to each other. 



Fig. 5. 



Let AB, CD, represent the two abutments, HN the crowni line, GF 

 and PN the horizontal projections of two of the lines of pressure. 



Of rectangular co-ordinates, let the .?■ be in the direction HG, the y 

 in FM, and the z vertically. For convenience, also assume oblique 

 co-ordinates ji along HN, u along NM, and z as before; put also GHN 

 the angle of the skew =: s. The formulae of conversion will be 



■r ::= 11 cos s, y = V s\n g — u; g ::= z] 

 V := x sec s, u :=. X tan s — y, z =^ z j ' ' 

 If the equation of the generating curve of the vault, of which EF is 

 the projection, be taken 



u — <!>:: — — B 



the same equation will serve as that of the vault itself; or in rectan* 

 gular co-ordinates 



.» tan s — 7/ — tfi z = Q := B, whence 



dB dB , rfB , 



-T— = tan s ; -5— = — 1 ; -r— z^ a' z. 

 d X dy d z 



The equation of the plane containing one of the lines of pressure is, 



X — X =: ^ c ; whence 



d c 



d c _ d c _ 

 dlc-^'J^j-^' 



d z 



= 0; 



so that the equations of the straight line touching B ^, c = are 

 X — X _ Y — y Z — z 







— 1 



(D) 



where X, Y, Z belong to any point in the tangent ; x, y, z to the point 



of contact. 



Again, let ;z — 9 u = = E be the equation of the horizontal pro- 

 jection of a joint, or in rectangular co-ordinates, 



X tan 8 — y — 6 (r sec s) = = E ; then 



d E 

 dx 



= tan 8 — sec s. B' r ; 



d E 



dy 



l;15 = 0. 

 dz 



The equations of the joint are B : 

 line tangent to it are 



X 



■^_ Y-y 



: 0, c = 0, therefore, those of a 

 Z —z 



m 



<t>' z <p' z (tan s — sec s. 6' v) sec s.&v 

 The stability of the structure demands, that the line whose equations 

 are (F; be perpendicular to that whose equations are (D), therefore 

 the condition of stability is contained in this equation, 



{<p' gy (sin s — fl' i>) — 6' V * 



