1818.1 



THE CIVIL ENGINEER AND ARCHITECTS JOURNTAL. 



137 



left to ri^ht is equal to the weia;ht of 42^° of the arch, and that 

 from rig;lit to left, equal to the weight of 41° ; so that the pres- 

 sures are very nearly in equilibrium ; therefore, the first point of 

 rupture is very near this trial point ; and the pressure of the ri^^ht 

 side of the arch preponderates : therefore, take for the next trial, 

 a point a few degrees nearer the crown. 



Thus the first i)oint of rupture will soon be arrived at, which, in 

 this case, is at tlie extrados, 10° from the crown, for when the 

 voussoirs are about to turn on their edges, at this j>oiiit, the pres- 

 sure from the lel't equals the pressure from the right ; each being 

 equal to the weight of 41° of the arch. 



The second point of rujiture on the right side, is at the intrados, 

 55° from the crown, and the pressure there is equal to the weight 

 of 71° of the arch ; then Problem 1 can be applied, to trace the 

 line of resistance through the rest of this side of the structure, 

 and it will be found, that for the arch to be in the condition of 

 unstable equilibrium, about the springing, it is necessary that the 

 voussoirs should be deepened about 12 inches, at 86° 40' from the 

 crown. 



The second point of rupture on the left side, is at the intrados 

 62° from the crown, and the pressure there is equal to the weight 

 of 90° of the arch, and Problem 1 being applied to trace tlie line 

 of resistance through the rest of the structure, it will be found, 

 that for it to be in the condition of unstable equilibrium, at the 

 s]u-inging, the voussoir must be notched at the extrados, to the 

 depth of about 6 inches. 



Art. 17. — Theprinciplesand themethod described in Secti<in III. 

 may be applied to the arch sustaining pressures, as described in the 

 heading of this section, and of unsymmetricical form ; as well as 

 to that arch whose pressures and form are similar on both sides of 

 the crown, as described in Section II 



It is also evident, that the above method will apply to any irre- 

 gular form of arch, and that the principles and method described 

 in Section III. might also be applied. 



ON A GENERAL THEOREM TO CALCULATE THE AREA 



OF A CROSS-SECTION OF A RAILWAY ON 



SIDELONG GROUND. 



By R. G. Clark. 



The intention of this paper is to investigate a formula, free from 

 surds or any approximation, to compute the area of a cross-section, 

 without having regard to the side stakes. We have given (fig. 1 ) 



But(T-T')x 



sin e 



2 m sin 6 



sin 9 



Fig. 1 



the breadth of formation level A B ; the depth O F from centre 

 stake ; the difference of heights q d, taken by the spirit-level ; the 

 corresponding hypothenusal length q r ; and the ratio of the slopes. 

 On referring to page 68, present volume, article " Railway Sections 

 in Sidelong Ground," to formula (3), which is 



(2 6 -f m u) 1 + \(b-\- m a) {x-x') sin e = area ; 

 where h = ^ formation level ; a = depth or height from centre 

 stake to centre of formation level ; x and x' equal the distances 

 O D, O H ; 8 = angle of inclination of ground ; and m base of 

 slope to one perpendicular. Also let h = difference of heights by 

 level r d ; and / = O rf, the hypothenusal length on surface. 

 From (1) and (2), page 67, in the article above referred to, put 



T = ^ .— , and T' = ^ • 



cos e — m sin e cos B -\-m sin 6 ' 



then the above formula becomes, by substituting T and T', 



(2 6-1- »n a) a + (* + m a)- (T — T') X 



sin e 



:= area (4) 



2 cos' 



m- sin^ e 



-m' sin' 

 m tan' 



e 



cos^ — tn - sin' e 1 — rri- tan a' 



Now consider, in the right-angle triangle q d r, the height qd = 



h to be a tangent, the horizontal distance d r the radius ; therefore, 



h-- 

 tan- fl = 



„ , J- Substitute this value in last expression, and then 



in (4) ; we have for the required general formula : 



m A' 



-^, = area H A B D (5) 



{2 b -{- m a) a + {b -\- may 



Let the slope be 1 to I ; then m ^ 1 ; 



(2 6-1- a) a - 



Ib + aYh^ 



(«) 



The general formula is more simple than it appears : when these 

 numbers are large, we shall only require a table of square numbers 

 to work out any question. Two examples are subjoined : — 



1. Given lengtli on slojie of ground = 20 feet ; the difference of 

 heights, 6 feet; slope to be 2 to 1 ; depth of cutting, 20 feet; and 

 breadth of formation level, 30 feet. Find area by the formula (5). 



Substitute the above values in (5) : 



(33 -I- 40) 20 -[- (16-5-1-40)=-^-— ^=2504-7, area required. 



4-00 — 180 



2. Given the length 20 feet on descent of ground ; difference of 

 heights by level, 6 feet ; intended slopes, 1 to 1 ; depth of cutting, 

 45 feet ; and breadth of formation level, 33 feet. Find the are a. 

 {See fig. 1 ; and for embankment, see fig. 2.) 



By substituting the above values in (o) : 



(33 + 45) 45 + il^^±^^l^ = 3925, area required. 



Fig. 2. 



From the above it may be observed, that when a centre line of 

 railway is ranged and staked out, and the depths known on in- 

 clined ground, we can always find most expeditiously the area of 

 any vertical cross- section, by means of the spirit-level, without re- 

 quiring the distances of the side stakes from the centre. By way 

 of comparison, I have to refer to an article for a similar purpose, 

 page 263, vol. VII. of this Journal. 



SOCIETY FOR THE PUBLICATION OF ARCHITEC- 

 TURAL KNOWLEDGE. 



From what has been said of it, the main purpose of this Society 

 — which, perhaps, may not mean to call itself exactly by the title 

 for the present assigned to it — seems to be to bring out what shall 

 be a complete Dictionary or Encyclopaedia of Architecture — a very 

 meritorious undertaking, and one which, as the want of such a 

 work is felt, might have reasonably enough been expected on the 

 part of the Institute. That the present Dictionaries which we have 

 of the kind are all more or less defective and unsatisfactory, even 

 considered with reference to the time when they were produced, 

 is universally admitted. Even had they no other deficiencies, they 

 have one and all lagged very much behind the actual time when 

 they made their appearance, whereas every work of the kind 

 ought to bring down its information to the latest possible moment. 

 Let us hope this important point will be attended to in the one 

 now promised, and that it will duly notice all those improvements, 

 both in matters of construction and those of embellishment, which 



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