A pattern language

Again, the basic problem is to maintain the integrity of the social spaces in the plan.

We know, from structure follows social spaces (205), that floor and ceiling vaults must correspond to the important social spaces in the plan. But there are a great number of social spaces, and they range in size from spaces like window place (180), perhaps five feet across, to spaces like farmhouse kitchen (¹ 39), perhaps 15 feet across, to collections of spaces, like common areas at the heart (129), perhaps 35 feet across.

Where vaults of different width are near each other, you must remember to pay attention to the level of the floor above. Either you can level out the floor by making the smaller vaults have proportionately higher arches, or you can put extra material in between to keep the small vaults low—see ceiling height variety (190), or you can make steps in the floor above to correspond to changes in the vault sizes below.

Vaults on different floors do not have to line up perfectly with one another. In this sense they are far more flexible than column-beam structures, and for this reason also better adapted to structure follows social spaces (205). However, there are limits. If one vault is placed so that its loads come down over

the arch of the vault below, this will put undue stress on the lower vault. Instead, we make use of the fact that vertical forces, passing through a continuous compressive medium, spread out downward in a 45 degree angle cone. If the lower columns are always within this cone, the upper vault will do no structural damage to the vault below it.

The angle at which a vertical force sfreads downward.

To maintain reasonable structural integrity in the system of vaults as a whole, we therefore suggest that every vault be placed so that its loads come down in a position from which the forces can go to the columns which support the next vault down, by following a 45 degree diagonal.

Good ... . . . 110 good.

With all this in mind then, work out a vault plan for your building. We suggest that you try to keep the vaults aligned with the rooms, with occasional adjustments to suit a very big room, or a very small nook or alcove. The drawing on the next page shows a floor and ceiling layout for a simple building.

Each space that you single out for a vault may have either a two-way vault (a domical ceiling on a rectangular base) or a one-way vault (a barrel vault). The tw'o-way vaults are the most efficient structurally; but when a space is long and narrow, the domical shape begins to act like a barrel vault. We therefore

CONSTRUCTION
A version of floor-ceiling layout} shown in flan and section, for a simfle ultra-lightweight concrete building.

suggest domical vaults for spaces where the long side is not more than twice the short side and barrel vaults for the spaces which are narrower.

We also suggest that you use barrel vaults for the rooms immediately under the roof. The roof itself is generally a barrel vault— see roof vault (220)—so it is most natural to give the ceiling of the space just under the roof a barrel vault as well.

The vaults described in floor-ceiling vaults (219) may

980 210 FLOOR AND CEILING LAYOUT

span from 5 to 30 feet. And they require a rise of at least 13 per cent of the short span.

Therefore:

Draw a vault plan, for every floor. Use two-way vaults most often; and one-way barrel vaults for any spaces which are more than twice as long as they are wide. Draw sections through the building as you plan the vaults, and bear the following facts in mind:

1. Generally speaking, the vaults should correspond to rooms.

2. There will have to be a support under the sides of each vault: this will usually be the top of a wall. Under exceptional circumstances, it can be a beam or arch.

3. A vault may span as little as 5 feet and as much as 30 feet. However, it must have a rise equal to at least 13 per cent of its shorter span.

4. If the edge of one vault is more than a couple of feet (in plan) from the edge of the vault below it—then the lower vault will have to contain an arch to support the load from the upper vault.

Put a perimeter beam (21 7) on all four sides of every vault, along the top of the bearing wall, or spanning openings. Get the shape of the vaults from floor-cf.iling vaults (219) and as you lay out the sections through the vaults, bear in mind that the perimeter beams get lower and lower on higher floors, because the

981

columns on upper stories must be shorter (top floor columns about 4 feet, one below top 6 feet, two below top 6 to 7 feet, three below top 8 feet)—final column distribution (213). Make sure that variations in floor level coincide with the distinctions between quiet and more public areas—floor surface (233). Complete the definition of the individual spaces which the vaults create with columns at the corners (212). Include the smallest vaults of all, around the building edge, in thickening

THE OUTER WALLS (21 1) . . . .

982

21 I THICKENING THE OUTER WALLS*

983

. . . the arrangement of roof and floor vaults will generate horizontal outward thrust, which needs to be buttressed—cascade of roofs (116). It also happens, that in a sensibly made building every floor is surrounded, at various places, by small alcoves, window seats, niches, and counters which form “thick walls” around the outside edge of rooms—window place (180),

THICK WALLS ( I 97) j SUNNY COUNTER ( I 99) , BUILT-IN SEATS

(202), child caves (203), secret place (204). The beauty of a natural building is that these thick walls—since they need lower ceilings, always, than the rooms they come from—can work as buttresses.

Once the roof layout (209), and the floor and ceiling layout (210) are clear these thick walls can be laid out in such a way as to form the most effective butresses, against the horizontal thrust developed by the vaults.

We have established in THICK walls (197), how important it is for the walls of a building to have “depth” and “volume,” so that character accumulates in them, with time. But when it comes to laying out a building and constructing it, this turns out to be quite hard to do.

The walls will not usually be thick in the literal sense, except in certain special cases where mud construction, for example, lends itself to the making of walls. More often, the thickness of the wall has to be built up from foam, plaster, columns, struts, and membranes. In this case columns, above all, play the major role, because they do the most to encourage people to develop the walls. For instance, if the framework of a wall is made of columns standing away from the back face of the wall, then the wall invites modification—it becomes natural and easy to nail planks to the columns, and so make seats, and shelves, and changes there. But a pure, flat, blank wall does not give this kind of encouragement. Even though, theoretically, a person can always add things which stick out from the wall, the very smoothness of the

wall makes it much less likely to happen. Let us assume then, that a thick wall becomes effective when it is a volume defined by columns.

Thick walls made effective by columns.

How is it possible for a wall of this kind to justify its expense by helping the structure of the building? The fact that the building is conceived as a compressive structure, whose floors and roofs are vaults—efficient structure (206), means that there are horizontal thrusts developed on the outside of the building, where the vaults do not counterbalance one another.

To some extent this horizontal thrust can be avoided by arranging the overall shape of the building as an upside down catenary—see cascade of roofs ( i 16). If it were a perfect catenary, there would be no outward thrust at all. Obviously, though, most buildings are narrower and steeper than the ideal structural catenary, so there are horizontal thrusts remaining. Although these thrusts can be resolved by tensile reinforcing in the perimeter beams—see perimeter beams (217)—it is simplest, and most natural, and stable to use the building itself to buttress the horizontal thrusts.

This possibility occurs naturally wherever there are “thick walls”—alcoves, window seats, or any other small spaces at the outside edge of rooms, which can have lower ceilings than the main room and can therefore have their roofs shaped as continuations of the ceiling vault inside. This requires that thick walls be outside the structure of the main room, so that their roofs and walls come close to forming a catenary with the main vault.

Alcoves within the catenary.

. . . next to the mosaic of subcultures (8), perhaps the most important structural feature of a city is the pattern of those centers where the city life is most intense. These centers can help to form the mosaic of subcultures by their variety; and they can also help to form city country fingers (3), if each of the centers is at a natural meeting point of several fingers. This pattern was first written by Luis Racionero, under the name “Downtowns of 300,000.”

* ❖ ❖

There are few people who do not enjoy the magic of a great city. But urban sprawl takes it away from everyone except the few who are lucky enough, or rich enough, to live close to the largest centers.

This is bound to happen in any urban region with a single high density core. Land near the core is expensive; few people can live near enough to it to give them genuine access to the city’s life; most people live far out from the core. To all intents and purposes, they are in the suburbs and have no more than occasional access to the city’s life. This problem can only be solved by decentralizing the core to form a multitude of smaller cores, each devoted to some special way of life, so that, even though decentralized, each one is still intense and still a center for the region as a whole.

The mechanism which creates a single isolated core is simple. Urban services tend to agglomerate. Restaurants, theaters, shops, carnivals, cafes, hotels, night clubs, entertainment, special services, tend to cluster. They do so because each one wants to locate in that position where the most people are. As soon as one nucleus has formed in a city, each of the interesting services—especially those which are most interesting and therefore require the largest catch basin—locate themselves in this one nucleus. The one nucleus keeps growing. The downtown becomes enormous. It becomes rich, various, fascinating. But gradually, as the metropolitan area grows, the average distance from an individual house

CONSTRUCTION

It is of course rare to be able to have the alcove or thick walls approach a true catenary section—we hardly ever want them that deep or that low. But even when the thick walls and alcoves are inside the line of the catenary, they are still helping to counter outward thrusts. And their buttressing effect can be improved still more by making their roofs heavy. The extra weight will tend to redirect the forces coming from the main vault slightly more toward the ground.

The drawing below' show's the way this pattern works, and the kind of effect it has on a building.

The effect of thickening the outer walls, shown m flan nnd section.

986

21 I THICKENING THE OUTER WALLS

Therefore:

Mark all those places in the plan where seats and closets are to be. These places are given individually by alcoves (179), WINDOW PLACES (l8o), THICK WALLS (197), SUNNY COUNTER (199), WAIST-HIGH SHELF (20l), BUILT-IN SEATS (202), and so on. Lay out a wide swath on the plan to correspond to these positions. Make it two or three feet deep; recognize that it will be outside the main space of the room; your seats, niches, shelves, will feel attached to the main space of rooms but not inside them. Then, when you lay out columns and minor columns, place the columns in such a way that they surround and define these thick volumes of wall, as if they were rooms or alcoves.

For shelves and counters less than 2 feet deep, there is no need to go to these lengths. The thickening can be built simply by deepening columns and placing shelves between them.

(//pi

1 to 3 feet of thickness

A

outside the room

In order to make an alcove or thick wall work as a buttress, build its roof as near as possible to a continuation of the curve of the floor vault immediately inside. Load the roof of the buttress with extra mass to help change the direction of the forces—roof vaults (220). Recognize that these thick walls must be outside

987

the main space of the room, below the main vault of the room— floor-ceiling vaults (219), so that they help to buttress the horizontal forces generated by the main vault of the ceiling. When you lay out columns and minor columns, put a column at the corner of every thick wall, so that the wall space, like other social spaces, becomes a recognizable part of the building structure-COLUMNS AT THE CORNERS (2 I 2) . . . .

2 12 COLUMNS AT THE

CORNERS**

989

. . . assume that you have worked out the roof plan, and laid out ceiling vaults for every room on every floor—roof layout (209), floor and ceiling layout (210). These vaults are not only the basis of the structure, but also define the social spaces underneath them. Now it is time to put columns at the corners of the vaults. This will both complete them as clearly defined social spaces—structure follows social spaces (205)—and also be the first constructive step in the erection of the building —gradual stiffening (208).

•£•

We have already established the idea that the structural components of a building should be congruent with its social spaces.

In structure follows social spaces (205) we have established that the columns need to be at corners of social spaces for psychological reasons. In efficient structure (206) we have established that there needs to be a thickening of material at the corners of a space for purely structural reasons.

Now we give yet a third still different derivation of the same pattern—not based on psychological arguments or structural arguments, but on the process by which a person can communicate a complex design to the builder, and ensure that it can be built in an organic manner.

We begin with the problem of measurement and working drawings. For the last few decades it has been common practice to specify a building plan by means of working drawings. These measured drawings are then taken to the site; the builder transfers the measurements to the site, and every detail of the drawings is built in the flesh, on site.

This process criffiles buildings. It is not possible to make such a drawing without a T-square. The necessities of the drawing itself change the plan, make it more rigid, turn it into the kind of plan which can be drawn and can be measured.

But the kind of plans which you can make by using the pattern

2 12 COLUMNS AT THE CORNERS

language are much freer than that—and not so easy to draw and measure. Whether you conceive these plans out on the site—and mark them on the site with sticks and stones and chalk marks— or draw them roughly on the back of envelopes or scraps of tracing paper—in all events, the richness which you want to build into the plan can only be preserved if the builder is able to generate a living building, with all its slightly uneven lines and imperfect angles.

Chalk marks on the ground.

In order to achieve this aim, the building must be generated in an entirely different manner. It cannot be made by following a working drawing slavishly. What must be done, essentially, is to fix those points which generate the spaces—as few of them as fossible—and then let these points generate the walls, right out on the building site, during the very process of construction.

You may proceed like this: first fix the corner of every major space by putting a stake in the ground. There are no more than a few dozen of these corners in a building, so this is possible, even if the measurements are intricate and irregular. Place these corner markers where they seem right, without regard for the

exact distances between them. There is no reason whatever to try and make modular distances between them. If angles are slightly off, as they often will be, the modular dimensions are impossible anyway,

“Staking out”

These simple marks are all you need to build the building. Once construction starts, you can start very simply, by building a column, over each of these marks. These columns will then generate the rest of the building, by their mere presence, without

212 COLUMNS AT THE CORNERS

any further need for detailed measurements or drawings, because the walls will simply be built along the lines which connect adjacent columns: and everything else follows.

For the upper storys, you can make drawings of the column positions and once again transfer them to the actual building while it is being built. As you will see from final column distribution (213), upper story columns do not need to line up perfectly with downstairs columns.

With this procedure, it becomes possible to transfer a rather complex building from your mind, or from a scrap of paper, to the site—and regenerate it in a way which makes it live out there.

The method hinges on the fact that you can fix the corners of the spaces first—and that these corners may then play a significant role in the construction of the building. It is interesting that although it is based on entirely different arguments from structure follows social spaces (203), it leads to almost exactly the same conclusion.

Therefore:

On your rough building plan, draw a dot to represent a column at the corner of every room and in the corners formed by lesser spaces like thick walls and alcoves. Then transfer these dots onto the ground out on the site with stakes.

columns at corners

Once you have the columns for each floor on your vault plan, reconcile them from floor to floor and put in intermediate col-

CONSTRUCTION

umns—final column distribution (213). Note, especially, that it is not necessary for the corner columns to fall on a grid. The floor vaults and roof vaults can be made to fit any arrangement of columns, and still make a coherent structure—thus allowing the social spaces to determine the building shape without undue constraint from purely structural considerations—floorceiling VAULTS (219), ROOF VAULTS (220).

These columns will not only guide your mental image of the building, they will also guide construction: first put the columns and the column foundations in place; then, to make the frame complete, tie the columns together around each room with the perimeter beam—root foundations (214), box columns

(216), perimeter beams (217). Give special emphasis to all free-standing columns with the idea that when you build them, you will make them very thick—column place (226). . . .

994

213 final column

DISTRIBUTION**

995

to this one center increases; and land values around the center rise so high that houses are driven out from there by shops and offices—until soon no one, or almost no one, is any longer genuinely in touch with the magic which is created day and night within this solitary center.

The problem is clear. On the one hand people will only expend so much effort to get goods and services and attend cultural events, even the very best ones. On the other hand, real variety and choice can only occur where there is concentrated, centralized activity; and when the concentration and centralization become too great, then people are no longer willing to take the time to go to it.

If we are to resolve the problem by decentralizing centers, we must ask what the minimum population is that can support a central business district with the magic of the city. Otis D. Duncan in “The Optimum Size of Cities” (Cities and Society, P. K. Hatt and A. J. Reiss, eds., New York: The Free Press, 1967, pp. 759—72), shows that cities with more than 50,000 people have a big enough market to sustain 61 different kinds of retail shops and that cities with over 100,000 people can support sophisticated jewelry, fur, and fashion stores. He shows that cities of 100,000 can support a university, a museum, a library, a zoo, a symphony orchestra, a daily newspaper, AM and FM radio, but that it takes a population of 250,000 to 500,000 to support a specialized professional school like a medical school, an opera, or all of the TV networks.

In a study of regional shopping centers in metropolitan Chicago, Brian K. Berry found that centers with 70 kinds of retail shops serve a population base of about 350,000 people (Geography of Market Centers and Retail Distribution, New Jersey: Prentice-Hall, 1967, p. 47). T. R. Lakshmanan and Walter G. Hansen, in “A Retail Potential Model” (American Institute of Planners Journal, May 1965, pp. 134—43), showed that full-scale centers with a variety of retail and professional services, as well as recreational and cultural activities, are feasible for groups of 100,000 to 200,000 population.

It seems quite possible, then, to get very complex and rich urban functions at the heart of a catch basin which serves no more than 300,000 people. Since, for the reasons given earlier, it is

. . . assume that you have placed the corner columns which define the spaces—columns at the corners (212). It is now necessary to fill in the gaps between the columns with intermediate stiffener columns as required by efficient structure (206). This pattern gives the spacing of these intermediate stiffener columns, and helps to generate the kind of walls which efficient structure (206) requires. It also helps to generate ceiling height variety (190).

How should the spacing of the secondary columns which stiffen the walls, vary with ceiling height, number of stories and the size of rooms?

In some very gross intuitive way we know the answer to this question. Roughly, if we imagine a building with the walls stiffened at intervals along their length, we can see that the texture of these stiffeners needs to be largest near the ground, where social spaces are largest and where loads are largest, and smallest near the roof, where rooms are smallest and where loads are least. In its gross intuitive form this is the same as the intuition which tells us to expect the finest texture in the ribbing at the fine end of a leaf where everything is smallest, and to expect the grosser, cruder structure to be near the large part of the leaf.

Leaf.

996

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 wall: 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
calibrated by setting fc = 93% of the allowable compressive stress I for lightweight concrete and

t

90

80

70

\ using the ACI value of 77 — — v 6 H 10\ for the unstiffened case, where
0.2 0.4 0.6 0.8 1.0 1.2 —HThe curve which relates wall slenderness to the s-pacing of stiffeners.

60

50

40

30

20

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.

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.

When you try to apply this pattern to floor plan, you will find a certain type of difficulty. Since the corners of rooms may already be fixed by columns at the corners (212), it is not always possible to space the stiffeners correctly within the wall of any given room. Naturally this does not matter a great deal; the stiffeners only need to be about right; the spacing can comfortably vary from room to room to fit the dimensions of the walls. However, on the whole, you must try and put the stiffeners closer together where the rooms are small and further apart where rooms are large. If you do not, the building will seem odd, because it defies one’s structural intuitions.

Consider two rooms on the same floor, one twice as large as the other. The larger room has twice the perimeter, but its ceiling generates four times the load; it therefore carries a greater load per unit length of wall. In an ideal efficient structure, this means that the wall must be thicker; and therefore, by the arguments already given, it will need stiffeners spaced further apart than the smaller room which carries less load and has thinner walls.

We recognize that few builders will take the trouble to make wall thicknesses vary from room to room on one floor of the building. However, even if the wall is uniformly thick, we believe that the stiffeners must at least not contradict this rule. If, for reasons of layout, it is necessary that the spacing of stiffeners varies from room to room, then it is essential that the larger spacings of the stiffeners fall on those walls which enclose the

IOOI

CONSTRUCTION
The final column distribution in a jour story building, built according to our -patterns for columns, walls and vaults.

larger rooms. If the greater spacing of stiffeners were to coincide with smaller rooms, the eye would be so deceived that people might misunderstand the building.

One important note. All of the preceding analysis is based on the assumption that walls and stiffeners are behaving as elastic plates. This is roughly true, and helps to explain the general

1002

213 FINAL COLUMN DISTRIBUTION

phenomenon we are trying to describe. However, no wall behaves perfectly as an elastic plate—least of all the kind of lightweight concrete walls we are advocating in the rest of the construction patterns. We have therefore used a modified form of the elastic plate theory, calibrated according to the AC1 code, so that the numbers in our analysis are based on the elastic behavior of concrete (and fall within the limits of its tension and compression). However, when the plate goes out of the elastic range and cracks, as it almost certainly will in a concrete design, other factors will enter in. We therefore caution the reader most strongly not to take the actual numbers presented in our analysis as more than illustrations. The numbers reflect the general mathematical behavior of such a system, but they are not reliable enough to use in structural computations.

Therefore:

Make column stiffeners furthest apart on the ground floor and closer and closer together as you go higher in the building. The exact column spacings for a particular building will depend on heights and loads and wall thicknesses. The numbers in the following table are for illustration only, but they show roughly what is needed.

in stories ground floor 2nd floor 3rd floor 4th floor

²''5'

1

2

3

4

Mark in these extra stiffening columns as dots between the corner columns on the drawings you have made for different floors. Adjust them so they are evenly spaced between each pair of corner columns; but on any one floor, make sure that they are closer together along the walls of small rooms and further apart along the walls of large rooms.

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CONSTRUCTION

v w w~

floor by floor variation

s/

To the extent consistent with ceiling height variety (190), make walls and columns progressively shorter the higher you go in the building to keep slenderness ratios low.

And make wall thicknesses and column thicknesses vary with the height—-see wall membrane (218). Our calculations, for a typical lightweight concrete building of the kind we have been discussing, suggest the following orders of magnitude for wall thicknesses: Top story—2 inches thick; one below top story—3 inches; two below top story—4 inches; three storys below top (ground floor on a four story building)—5 inches. Of course these numbers will change for different loads, or for different materials, but they show the type of variation you can expect.

Column thicknesses must be proportional to wall thicknesses, so that the thinnest walls have the thinnest columns. If they are very thin, it will be possible to make them simply by placing boards, or one thickness of material, outside the outer skins which form the wall membrane—see wall membrane (218). If the walls are thick, they will need to be full columns, twice as thick as the walls, and roughly square in section, built before the walls, but made in such a way that they can be poured integrally with the walls—box columns (216). . . .

fut stakes in the ground to mark the columns on the site_} and start erecting the main jrame according to the layout oj these stakes;

10 MAGIC OF THE CITY

desirable to have as many centers as possible, we propose that the city region should have one center for each 300,000 people, with the centers spaced out widely among the population, so that every person in the region is reasonably close to at least one of these maj or centers.

To make this more concrete, it is interesting to get some idea of the range of distances between these centers in a typical urban region. At a density of 5000 persons per square mile (the density of the less populated parts of Los Angeles) the area occupied by 300,000 will have a diameter of about nine miles; at a higher density of 80,000 persons per square mile (the density of central Paris) the area occupied by 300,000 people has a diameter of about two miles. Other patterns in this language suggest a city much more dense than Los Angeles, yet somewhat less dense than central Paris—four-story limit (21), density rings (29). We therefore take these crude estimates as upper and lower bounds. If each center serves 300,000 people, they will be at least two miles apart and probably no more than nine miles apart.

One final point must be discussed. The magic of a great city comes from the enormous specialization of human effort there. Only a city such as New York can support a restaurant where you can eat chocolate-covered ants, or buy three-hundred-year-old books of poems, or find a Caribbean steel band playing with American folk singers. By comparison, a city of 300,000 with a second-rate opera, a couple of large department stores, and half a dozen good restaurants is a hick town. It would be absurd if the new downtowns, each serving 300,000 people, in an effort to capture the magic of the city, ended up as a multitude of second-class hick towns.

This problem can only be solved if each of the cores not only serves a-catch basin of 300,000 people but also offers some kind of special quality which none of the other centers have, so that each core, though small, serves several million people and can therefore generate all the excitement and uniqueness which become possible in such a vast city.

Thus, as it is in Tokyo or London, the pattern must be implemented in such a way that one core has the best hotels, another the best antique shops, another the music, still another has the fish and sailing boats. Then we can be sure that every person is

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