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213 FINAL column distribution
These intuitions are borne out by many traditional building forms where columns, or frames, or stiffeners are larger and further apart near the ground, and finer and closer together higher up. Our key picture shows examples. But what is the structural basis for these intuitions!1
Elastic plate theory gives us a formal explanation.
Consider an unstiffened thin wall carrying an axial load. This wall will usually fail in buckling before it fails in pure compression because it is thin. And this means that Lhe material in the wall is not being used efficiently. It is not able to carry the compressive loads which its compressive strength makes possible because it is too thin.
It is therefore natural to design a wall which is either thick enough or stiffened enough so that it can carry loads up to its full compressive capacity withouL buckling. Such a wall, which uses its material to the limits of its compressive capacity, will then also satisfy the demands of efficient structure (206).
The critical factor is the slenderness of the walclass="underline" the ratio of its height to its thickness. For the simple case of an unstiffened concrete wall, the ACl code tells us that the wall will be able to work at 93 per cent efficiency (that is, carry 93 per cent of its potential compressive load without buckling), if it has a slenderness ratio of 10 or less. A wall 10 feet high and 1 foot thick is therefore efficient in this sense.
Suppose now, that we extrapolate to the case of a stiffened wall using elastic plate theory. By using the equation which relates allowable stress to the spacing of stiffeners, we can obtain similar figures for various walls with stiffeners. These figures are presented in the curve below. For example, a wall with a slenderness of 20 needs stiffeners at 0.5H apart (where H is the height) thus creating panels half as wide as they are high. In general, obviously, the thinner the wall is, in relation to its height, the more often it needs to be stiffened along its length.
In every case, the curve gives the spacing of stiffeners which is needed to make the wall work at 93 per cent of its compressive strength. In short, we may say that a wall built according to the principle of efficient structure (206) ought to be stiffened in accordance with this curve.
The gradient of column spacing over different floors follows
CONSTRUCTION
| This curve is derived from |
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| calibrated by setting fc = 93% of the allowable compressive stress I for lightweight concrete and |
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| \ using the ACI value of 77 — — v 6 H 10\ for the unstiffened case, where |
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| 0.2 0.4 0.6 0.8 1.0 1.2 —HThe curve which relates wall slenderness to the s-pacing of stiffeners. |
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to
directly from this curve. We may see this in the following manner. The walls in a four story building carry loads which are very roughly in the ratio 4:3:2:! (only very roughly). In any case, the loads the walls carry get less and less the higher we go in the building. If all the walls arc reaching their full compressive capacity, this means that they must be getting steadily thinner too, the higher one goes in the building. If we assume that the walls all have the same height, then the four walls will therefore have progressively greater and greater slenderness ratios, and will therefore fall further and further to the left on the curve, and will therefore need to lie stiffened at closer and closer intervals.
For example, suppose a four story building has 8 foot high walls on all floors and has wall thicknesses of 12 inches, 9 inches, 6 inches, and 3 inches on its four floors. The slenderness ratios are 8, II, 17, and 33. In this case, reading off the curve, we find the ground floor has no stiffeners at all (they are infinitely far apart), the second floor has stiffeners at about 8 feet apart, the third floor has them about 5 feet apart, and the top floor has them about 2 feet apart.
In another case, where the walls are thinner (because materials are lighter and loads smaller), the spacing will be closer. Suppose, for example, that the necessary wall thicknesses are 8, 6, 4, and
2 inches. Then the slenderness ratios are 12, 16, 24, and 48, and the stiffeners need to be spaced closer together than before: nine feet apart on the ground story, 5 feet apart on the second story, 3 feet apart on the third, and 15 inches apart on the top.
As you can see from these examples, the variation in column spacing is surprisingly great; greater, in fact, than intuition would allow. But the variation is so extreme because we have assumed that ceiling heights are the same on every floor. In fact, in a correctly designed building, the ceiling height will vary from floor to floor; and under these circumstances, as we shall see, the variation in column spacing becomes more reasonable. There are two reasons why the ceiling height needs to vary from floor to floor, one social and one structural.
In most buildings, the spaces and rooms on the first floor will tend to be larger—since communal rooms, meeting rooms, and so on, are generally better located near the entrance to buildings, while private and smaller rooms will be on upper stories, deeper into the building. Since the ceiling heights vary with the size of social spaces—see ceiling height variety (190)—this means that the ceiling heights are higher on the ground floor, getting lower as one goes up. And the roof floor has either very short walls or no wall at all—see sheltering roof (117).
Variation of room sizes.
And there is a second, purely structural explanation of the fact that ceilings need to be lower on upper stories. It is embodied in the drawing of the granary shown below. Suppose that a system of columns is calculated for pure structure. The columns on upper stories will be thinner, because they carry less load than
those on lower stories. But because they are thinner, they have less capacity to resist buckling, and must therefore be shorter if we are to avoid wasting material. As a result, even in a granary, where there are no social reasons for variation in ceiling height, purely structural considerations create the necessity for thick columns and high ceilings on the lower stories and for thinner and thinner columns and lower and lower ceilings the higher one gets in the building.
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| German granary. |
The same conclusion comes from consideration of our curve. We have used the curve, so far, to tell us that stiffeners need to be closer together on upper stories, because the walls are more slender. We may also use the curve to tell us that, for a given load, we should try to keep the slenderness ratio as low as possible. On the upper stories, where walls are most apt to be thin, we should therefore make the walls as low as possible, in order to keep the slenderness ratios low.
IOOO 213 final column distribution
Let us assume now, that the wall heights do vary in a building, in a manner consistent with these arguments. A four story building, with an attic story on top, might then have these wall heights (remember that the vault height, in a vaulted room, is higher than the wall height) : 9 feet on the ground floor, 7 feet on the second, 6 feet on the third, and 4 feet on the fourth, where the pitched roof comes down low over the eaves. And let us assume that the wall thicknesses are 12 inches, 6 inches, 5 inches, and 3 inches, respectively. In this case, the slenderness ratios will be 9, 14, 14, 15. The ground floor needs no stiffeners at all; the second has them 6 feet apart; the third has them 5 feet apart; and the fourth has them 3 feet apart. We show a similar distribution in the drawing opposite.