Where, on account of party lines or other buildings, the stepping out of a foundation wall has to be done entirely at one side, the stepping should be even steeper than 60°, if possible; and particular attention must be paid to anchoring the walls together as soon and as thoroughly as possible, in order to avoid all danger of the foundation wall tipping outwardly.

Where a front or other wall is composed of isolated piers, it is well to combine all their foundations into one,and to step the piers down for this purpose, as shown in Figure 41. Where there is not sufficient depth for this purpose, inverted arches must be resorted to. The manner of calculating the strength of inverted arches will be given under the article on arches. Inverted arches are not recommended, however (except where the foundation wall is by necessity very shallow), as it requires great care and good mechanics to build them well.

Two things must particularly be looked out for: 1, That the end arch has sufficient pier or other abutment; otherwise it will throw the pier out, as indicated in Figure 42. (This will form part of calculation of strength of arch.) Where there is danger of this, ironwork should be resorted to, to tie back the last pier.

2. The skew-back of the arch should be sufficiently wide to take its proportionate share of load from the pier (that is, amount of the two skew-backs should be proportioned to balance of pier or centre part of pier, as the width of opening is to width of pier); otherwise the pier would be apt to crack and settle past arch, as shown in Figure 43. An easy way of getting the width of skew-back graphically is given below. In Figure 44, draw A B horizontally at springing-line of inverted arch; bisect A C at F, and C B at E. Draw E O at random to vertical through F; then draw O C, and parallel to 0 C draw G D; then is C D the required skew-back.1

Inverted arches.

Size of Skew-back.

Fig. 41.

Fig. 42.

Fig. 43.

A good way to do is to give the arches wide skew-backs, and then to introduce a thick granite or blue stone pier stone over them, as shown in Figure 45. This will force all down evenly and avoid cracks. The stone must be thick enough not to break at dotted lines, and should be carefully bedded.

Example.

A foundation pier carrying 150000 lbs. is 5' wide and 3' broad The inverted arches are each 24" deep. What thickness should the granite block have?

We have here virtually a granite beam, 60" long and 36" broad, supported at two points (the centre lines of skew-backs) 36" apart. The load is a uniform load of 150000 lbs. The safe modulus of rupture, according to Table V, for average granite is (k/f) = 180 lbs.

Fig. 44.

Fig. 45.

1 In reality C D should be somewhat larger than the amount thus obtained; but this can be overlooked, except in cases where the pier approaches in width the width of opening. In such cases, however, stepping can generally be resorted to in place of inverted arches. Then, too, if the opening were very wide and the line of pressure came very much outside of central third of C D, it might be necessary to still further increase the width of skew-back, C D.

The bending moment on this beam, according to Formula (21) is m = u.l/8 = 150 000.36/8 = 675000

The moment of resistance, r, is, from Formula (18) r = m/(k/f) = 675 000/180 = 3750

From Table I, No. 3, we find r = bd2/6, therefore bd2/6 = 3750; now, as 6 = 36, transpose and we have d2 = 3750.6/36 = 625.

Therefore d = 25" or say 24". The size of granite block would have to be 5' x 3' x 2'.

As this would be a very unwieldy block, it might be split in two lengthwise of pier; that is, two stones, each 5' x 18" x 2' should be used, and clamped together. Before building piers, the arch should be allowed to get thoroughly set and hardened, to avoid any after shrinkage of the joints.

A parabolic arch is best. Next in order is a pointed arch, then a semi-circular, next elliptic, and poorest of all, a segmental arch, if it is very flat. But, as before mentioned, avoid inverted arches, if possible, on account of the difficulty of their proper execution.

A rock foundation makes an excellent one, and needs little treatment, but is apt to be troublesome because of water. Remove all rotten rock and step off all slanting surfaces, to make level beds, filling all crevices with concrete, as shown in Figure 47: In no case build a wall on a slanting foundation. Look out for springs and water in rock foundations. Where soft soils are met in connection with rock, try and dig down to solid rock, or, if this is impossible, on account of the nature of the case or expense, dig as deep as possible and put in as wide a concrete base-course as possible. If the bad spot is but a small one, arch over from rock to rock, as shown in Figure 48:

Rock foundations.

Fig. 46.

Fig. 47.

Fig. 48.

Good hard sand or even quicksand makes an excellent foundation, if it can be kept from shifting and clear of water. To accomplish this purpose it is frequently "sheath-piled" each side of the base course.

Gravel and sand mixed make an excellent, if not the best foundation; it is practically incompressible, and the driest, most easily drained and healthiest soil to build on.