This section is from the book "Safe Building", by Louis De Coppet Berg. Also available from Amazon: Code Check: An Illustrated Guide to Building a Safe House.
Use Formula (44) to get the stress at A, where x = K1 M1 = 9", or 3/4 feet; and p = G1 I1, which we find scales but little over 47 tons; and Formula (45) for stress at A. For d the width of joint we have of course the diameter of base, or 10 2/3 feet. Therefore
Stress at A1 = 47/39 +6.47.3/4/39.10 2/3 = 1,71 tons, per square foot, or 1,71.2000/144 = +24. (compression), per square inch, and stress at A = 47/39 - 6. 47.3/4/39.10 2/3 = + 0,7 tons, per square foot, or
0,7.2000/144 = 10 lbs. (compression), per square inch.
To find the pressure on base E E,; draw P G horizontally at half the whole height; make G M = 657 tons (or the whole load)1 and draw M K horizontally, and = 14 1/4 tons (or the whole wind-pressure).
Draw G K and (if necessary) prolong till it intersects E E, at K. From formulas (44) and (45) we get the stresses at E, and E: p being = G K
= 658 tons; and x = MK= 20", or 1 2/3 feet. For d the width of base we have the total diameter, or 16 feet. Therefore stress at
E1 = 658/151 + 6.658.1 2/3/151.16 = +7tons per sq. ft., or 7.2000/144 = +97 lbs.
(compression), per square inch, and stress at E = 658/151 -6.658.2/2/151.16 =
+1 2/3 tons (compression), per square foot, or 1 2/3.2000/144 = + 23 lbs.
(compression) per square inch.
There is, therefore, absolutely no danger from wind.
Corbels carrying overhanging parts of the walls, etc., should be calculated in two ways, first, to see
1 Scale of weights. Fig. 87, applies to G, H,, etc., but not to G M.
whether the corbel itself is strong enough. We consider the corbel as a lever, and use either Formula (25), (26G) or (27); according to how the overhang is distributed on the corbel, usually it will be (25). Secondly, to avoid crushing the wall immediately under the corbel, or possible tipping of the wall. Where there is danger of the latter, long iron beams or stone-blocks must be used on top of the back or wall side of corbel, so as to bring the weight of more of the wall to bear on the back of corbel.
To avoid the former (crushing under corbel) find the neutral axis G H of the whole mass, above corbel, Figure 95; continue G H till it intersects A B at K, and use Formula (44) and (45).
If M be the centre of A B, then use x = K M, and p = weight of corbel and mass above; remembering to use and measure all parts alike; that is, either, all tons and feet, or all pounds and inches.
A brick tower has pilasters 30" wide, projecting 6". On one side the tower is engaged and for reasons of planning the pilasters cannot be carried down, but must be supported on granite corbels at the main roof level, which is 36 feet below top of tower. The wall is 24" under corbel, and averages from there to top 16", offsets being on both sides, so that it is central over 24" wall. What thickness should the granite corbel be?
If we make the pilaster hollow, of 8" walls, we will save weight on the corbel, though we lose the advantage of bonding into wall all way up. Still, we will make it hollow.
First, find the distance of neutral axis M-N (Figure 96), of pilaster from 24" wall, which will give the central point of application of load on corbel. We use rule given in the first article, and have x = 8.30.12 + 8.8.4 + 8.8.4 = 9 1/3". 8.30 + 8.8 + 8.8 The weight of pilaster (overlooking corbel) will be 6624 lbs., or 3 1/3 tons.
Calculation of corbel.
Assuming that only 30" in length of the wall will come on back part of corbel, the weight on this part would be 8640 lbs., or 4 1/3 tons. The bending moment of 6624 lbs. at 9 1/3" from support on a lever is (see Formula 27): m - 6624.9 1/3 = 61824 (lbs. inch). From Formula (18) we know that: m/(k/f) = r
From Table I, Section Number 3, we have: r = b.d2/6 = 30.d2/6 = 5.d2
And from Table V, we have for average granite:
(k/f)=180 Inserting these values in Formula (18) we have:
61824/180 = 5.d2, or d2 = 61824/900 = 68,7, therefore, d = 8 1/3".
We should make the block 10" deep, however, to work better with brickwork. This would give at the wall a shearing area = 10.30 =
300 square inches, or 6624/300 = 22 lbs., per square inch, which granite will certainly stand; still, it would be better to corbel out brickwork under granite, which will materially stiffen and strengthen the block. To find the crushing strain on brick wall under corbel, find the central axis of both loads by same rule as we find centre of gravity; that is, its distance A K1 from rear of wall will be (Figure 97) 4 1/3.12 + 3 1/3.33 1/3 = 22 1/7"
4 1/3 + 3 1/3 or, say, 22", the pressure at this point will, of course, be = 4 1/3 + 3 1/3 = 7 2/3 tons, or p = 15340 lbs.
The area will be = 24.30 = 720 square inches, while K M measures 10"; we have, then, from Formulae (44) and (45) stress at
B = 15340/720 + 6.15340.10/720.24 = +74 lbs. (compression), and stress at A = 15340/720 - 6. 15340.10/720.24 = - 31 lbs. (tension).
There would seem to be, therefore, some tendency to tipping, still we can pass it as safe, particularly as much more than 30" of the wall will bear on the rear of corbel.
Pressure under corbel.
If the wind could play against inside wall of tower, it might help to upset the corbel, but as this is impossible, its only effect could be against the pilaster, which would materially help the corbel against tipping.