In great horizontal thrusts, where the segment is flat, the immersion of the pier in water comes to have an important effect. On the weight of the pier, in those cases, the stability chiefly depends, and a deduction from that of two-fifths must be compensated by enlarg ing the thickness. For example, in the arch of 100 feet span, with 25 feet rise, and piers 20 feet high, the ring of stones of 3 feet at the crown may be set on a pier of 14 feet broad, taking the half arch at 180 feet. But lithe pier be set in water to the springing, it will lose 3 of its weight; and its breadth must be increased nearly to 16-k feet ere it has the same degree of stability as before. The truth is, that in this case the stability derived from the pier itself is nearly as much as that derived from the arch, (conceiving this always concentrated in the middl e of the half of the pier,) a diminution of 3 from the pier, therefore, is x of the whole, and must be provided for by an increase of breadth, not just equal to 1; for we must observe, that the stability derived From the arch is also increased thereby.
But indeed the immersion of the pier, if it be very tall, that is, if the depth of water be great in proportion to the span, will demand attention, although the arch should not be very flat. In such a case, the stability arising from the pier is often as great as that which is derived from the weight of the arch. It can seldom be greater, and consequently can seldom require an addi tion of more than one fifth of that breadth, which would be sufficient were there no immersion.
We might easily give a theorem for this in rectangu lar piers ; but it is hardly worth while ; the effect of any addition is easily determined in the first formula, which we think, on the whole, although only tentative, the most convenient rule for the practitioner.
But although the total immersion, even of a lofty pier, will seldom require any great alteration in the thick ness, there is yet another circumstance which well de serves attention. Bridges are often built, especially in a tide-way, with the arches springing below the high waters ; we have in that case a diminution from the weight of the arch itself, but unless the keystone be under water, the horizontal thrust is unchanged ; we must, accordingly, in our calculation, make the same diminution for that part of the arch which is thus im mersed, as we did in the above example for the piers. The result will oblige us still more to increase the thick ness of pier.
On the whole, we may conclude from this investiga tion respecting the piers, that the increase of breadth which may be, and usually is given to the pier, is of much less importance, on account of the weight that is thereby gained, than by its increasing the length of that arm of the lever, whereby the weight of the whole re sists the effect of the horizontal thrust oversetting it.
Instead, therefore, of building up the pier with per pendicular sides, we should think it more advisable to begin the foundation of the pier on a base much wider than usual, and from thence, by regular recesses, or otherwise, gradually to diminish it, until, at the spring ing of the arch, it does not exceed the depth of the two archstones, while the outline of the pier may be a curve of any shape that is most pleasing. Many advantages would, in our opinion, be obtained by this construction : the water way will be enlarged ; the pier equally strong ; the stability equally great, nay, much greater than usual ; and the chance of the foundations being hurt in floods will be greatly diminished : and all this with a smaller quantity of materials.
Before we take leave of the stability of piers, it will be proper to request the reader's attention a little longer to a case which we have hitherto but slightly noticed, we mean when the waters come to encroach on the crown of the arch. In this event, the stability arising from the arch is diminished by the loss of weight in all that part which is immersed. The horizontal force acts as be fore ; it will be propagated through the immersed arch stones. The weight of the pier is diminished by the immersion. All this must be compensated by an in crease of breadth in the pier.
Suppose the waters to rise to the key-stone, the hori zontal thrust is still unaltered, and is propagated as be fore ; the intermediate archstones, however, lose two fifths of their weight, and, supposing them jointed to equilibration, they will all have a tendency to rise and slide up. This is particularly the case with the lower stones of an arch with radial joints, for we know that these have such a tendency independent of this. What therefore is there to prevent them ? Their mutual friction, and the back or lateral pressure only. Their friction, however, is now much diminished, and so is the weight of the backing, on account of the immersion.
In drawing the limit of position for the joints to be equilibrated by friction, therefore, in Plate LX XX I. Fig.
5. we ought to diminish the lengths on the line, the key section only excepted, and observe the effect on the po sition of the joints ; the general effect will be, to make these joints approach nearer to the vertical, or, in other words, to draw them to lower centres ; and, if we are so inclined to admit of the arches being flatter segments, this observation is of use, and should be attended to in the formation of culverts, &c. which are often glutted.