As an example of the use of the above, take an arch of 100 feet span, six feet thick at the crown and semi circular. The horizontal thrust is 6X50=300 cubic feet; and let us take the weight of the half arch as =1200 at a medium, since, on account of the open spandrel, it may be considerably varied. Suppose the arch sprung at 2 ht 2. 18. 300 18 feet high, then h+c=7 4 = 146, h+c 74 9 .3 1-001 3 a 2 also 4.74 c) and ?146 + 147.93=17.14, .3 from which subtract a4 (h+ we have 4.97, or 5 feet nearly, for the thickness of the pier, which is not one-twentieth of the span. In an example nearly the same as this, 13 feet has been given by an eminent ma thematician for the thickness of the pier ; but the rea son is, that the stability which the pier derives from the superincumbent arch, has not been taken into considera tion ; an oversight the more extraordinary, since it is evident, that unless this weight did bear completely on the pier, it could have no tendency whatever to over turn it.
Suppose that c in the above formula is =0, or, what is the same thing, that the pier is carried no higher than the springing, 3 a we have b= t + 4h And in an arch of the above dimensions, 3a 3 x 1200 4 h 21=600, 4x 18 =50, when squared=2500 ?3100-50=55.68-50=5.68 nearly, or about a se venth part more than the former. We see therefore how little the stability may depend on the mere weight of the pier.
We may have a proof of the accuracy of this deter mination, by comparing it with the formula first given for the thickness of piers, viz. b= h t, ht, or the + overturning force, will be 300x18=5400. The pier in the first case, taking it at 5 feet, will be 5 x74=370, and :}p+,1-a will be 185+900 or 1085 ; multiply this by 5, we have 5425, a little more only than the overturning force, as the thickness was taken at 5 feet, which is a little in excess. The reader, if he chooses to go through the calculation for himself, will find 4.97 agree exactly.
In the second case, the pier =5.68 nearly, x 8=102.24, and its half =51.12, which added to 900, and multiplied by 5.68, gives 5402.3. A trifle in excess, because 5.68, like the former, is only an approximate number.
The weight of the pier in this case making so small a part of the whole resisting force, we may readily be lieve, that its total immersion in water would make no great addition to the requisite thickness. Stone, when so immersed, loses about 3 of its weight, being in spe cific gravity about 21 times that of water ; and, in the above example, were the whole pier under water, it ought to be about a fiftieth part thicker.
We have hitherto supposed the arch equilibrated, at least as far as is conveniently practicable, in which case the horizontal thrust is represented by the rectangle un der the radius and thickness at crown. But if the equili bration of the arch has not been attended to, we must consider whether any uncommon weight about the shoul ders may not produce, by the help of friction, a thrust in the arch fully equivalent to what would arise from a greater thickness at the crown ; and our calculations are to be regulated accordingly.
On the other hand, we have given the arch a weight in the above example which is nearly that of solidity. But in general the arch weighs much less. The most common case, where the stability of the pier is any way doubtful, is when it carries no more than the ring of arch-stones, and before it is assisted by the weight of the superincumbent backing. The weight keeping the pier steady, is now much diminished; while the horizontal thrust is unaltered ; for, if not propagated by weight, it is by means of the friction of the sections propagated to the pier, so as to act against it in the same manner as if completed.
Now, as it is by no means likely that the arch will be made thinner at the spring-courses than at the crown, while any additional thickness of the former is always in favour of the piers, we shall proceed upon the supposition, that a regular annulus, or ring of stones, is laid on them every where of equal thick ness. Suppose this thickness, as before, to be six feet. In that case the semi-arch of the above dimensions measures 499.5, or 500 feet, and ? 2 t + s a + 1590 ------= 32.13-20.83, or 4h 4.18 4.18 11 3 feet for the breadth of the pier. But it is by no means likely that the arch would have 6 feet thickness of crown in these circumstances ; 2, or at most 3 feet, would, in all probability be thought sufficient for a depth of keystone ; and a ring of arch-stones 2 feet deep will require a pier of 9 feet only. If we build up the pier behind the springing for about 6 feet, this thickness may be reduced to 8 feet ; and it will be absolutely ne cessary to do so in a case of this kind, to prevent the lower sections of the arch from sliding away.
The above example is taken for a semi-circular arch ; and though the reader must see, that the thickness of the pier is in no certain proportion to the span, it is ne vertheless obvious, that those writers who derive it from that, have hitherto erred considerably in excess. It is usually stated at 1,- for semi-circles ; but we see, that in the most unfavourable circumstances, it need not exceed of the span, and may often be made much less. This, however, we state with limitation, referring to the height of pier above given ; for were the pier much higher, it must be made thicker ; if the pier be infi nitely high, the weight of the arch sinks into insignifi cance, and the thickness, =?2 t, which in the above arch 6 feet thick is =24A feet nearly, and in general, if the thickness at crown =— of radius, then ?2 t = 71 3 R . /2 —= –, that is, taking s the span, nz=2n, the Il 71: thickness —s az whence this rule for the thickness of a pier of infinite height. Find what part the thick ness at crown is of the span, extract the square root, and multiply it by the span for the thickness ; or thus, multii ly the diameter by the thickness at crown, and ex tract the square root.