The German Tradition of vault forms

 

Vault Forms

Introduction Domes and Barrels

The arena roof was designed by Anton Tedesko, who is credited for bringing the German tradition of vaulted forums to the United States.

In addition, I'm going to describe classical forms for vaulted shell structures, such as domes and barrels. And, finally, I'll end the lecture with some examples of the importance of the form of a vault, meaning the shape of the vault.

Before we get into 3D vaulted behavior, let's examine the 2D behavior of an arch. The 2D behavior of an arch is not the same as a 3D response of a vault, but it will help us to move our thoughts in that direction.

What is the ideal form for a concrete arch?

Well, if we hang weights from a string, we get pure tension in that string. If we can freeze it and turn it over, we get pure compression, which is perfect for concrete, because concrete has almost no strength tension. If the load is uniform, the shape is, essentially, a parabola.

 

But, if the load is not symmetrical, due to trucks on a bridge, or wind on an arched roof, the arch cannot adjust the shape as a string can. The arch will deflect, and in this sketch, the dashed line, is the original form, and this solid line is a deflected shape.

 

This creates bending stresses in the structure, something that we want to avoid, in thin concrete walls, in particular, since it does not have much strength and bending.

 

The most efficient way to carry loads in concrete walls is through pure compression. To reduce and, or, control the bending stresses in arches, We can introduce what is called hinges.

 

A hinge is defined as part of the structure that does not resist rotation. Think of the hinge on a door. That hinge will allow the door to swing freely without any resistance to rotation.

 

Bridges are often designed with hinges. For example, Robert Maillart's Salginatobel Bridge is a 3 hinged arch bridge. This concept of 3 hinged arches can be extended to long-spanning roofs. It was applied as early as 1889 for Machinery Hall in the ParisExpo.

 

Now they supply this very general idea of arches to domes. The aspherical dome is the simplest type of dome, but it is not the ideal shape of an arch, meaning a dome is spherical, or circular, it is not parabolic.

 

The loads in a spherical dome are carried by meridians and hoops. The meridians can be thought of, essentially, as arches. It carries the gravity loads to the supports.

 

But the loads in a dome are transferred to the supports much more efficiently than a series of 2D arches, due to the 3D effects of the hoops, which interact with the meridians, tied them together, and keep them from bending.

 

We've begun our scientific study of vaults through domes, we started to look at hoops and meridians. I'm going to continue explaining the stresses in the hoops and meridians in a moment, and then continue with the study of Franz Dischinger, who published the first mathematical formulation for domes in1928.

 

A year later, he designed the Leipzig Market Hall in 1929, and I will go over that structure, as well. Going back to meridians and hoops, notice, in the sketch, the dashed lines.

 

Below the 52 degrees mark, we have tension in the hoops, which can lead to cracking in the concrete. The meridians are always in compression, hence the lines are always drawn solid.

 

The hoops go into tension as you go down towards the support. They're in compression towards the top of the dome and, then, transfer into tension below that 52-degree mark. In ancient times, this was not fully understood.

 

The Pantheon, for example, is the oldest major concrete structure. It's a hemispherical dome, and it's offered, meaning you see these square indentations inside of the dome, and that makes the dome lighter.

 

The dome of the Pantheon spans about 43 meters, and it was built in the 2nd century. Pictures were taken by David Billington, many years ago, clearly show the cracks in locations below 52 degrees.

 

I visited recently, and although the cracks aren't as clear as they were many years ago, they are still there, if you look for them. Concrete walls, today, have steel bars embedded in the concrete to enable the structure to carry a mild amount of tension.

 

This is called reinforced concrete design. That is, concrete reinforced with steel. The Pantheon does not have steel reinforcements, so it can be assumed that these cracks were there from the beginning.

 

The Romans probably saw the cracks develop and consequently loaded the sides of the dome, near the supports, with more stones and concrete to eliminate some of that tension.

 

To understand the meaning of this, take a piece of string, and load it uniformly with weights. That is weights of equal amounts and equal distances. The form is, essentially, a parabola, as I noted earlier.

 

If you want the shape of that string to take the form of a circle, you'll need to load more heavily the edges, near the support. Flip this physical concept to understand the concept of circular arches and domes.

 

If we look at a cross-section of the Pantheon, you'll notice that there are more weights, more concrete, and stones, on the outside near the supports, compared to towards the center.

 

Now, let's trace the origins of modern reinforced concrete domes. This image is of a dome in Breslau, Germany [Please note that this town was placed under Polish administration in 1945, and changed its name to Wrocław.  Notice how the meridians and hoops are expressed as arches and rings.

 

It isn't really acting as a 3D reinforced concrete thin-shell vault in its purest meaning. In comparison to a slightly larger dome, in Leipzig, Germany, This table shows that Breslau weighs about 3times more. And the dome of St. Peters in Rome weighs about 5 times more than the dome of Leipzig.

 

Here's an image of Leipzig Market Hall in Germany, and it was designed by Franz Dischingerin 1929, one year after he developed the mathematical equations that defined a dome.

 

The Leipzig Market Halls were, perhaps, the first modern reinforced concrete thin shell, or vaulted dome, and they began the German tradition. They're About 4 inches thick, and they're polygonal, they have 8 sides.

 

The shape is close to a circle, close to a spherical dome, but not exactly. The major difference between Leipzig and Breslau is that Breslauis conceived of as a series of arches and rings supporting a roof, whereas Leipzig is a self-supporting thin shell roof stiffened at its polygonal folds by thin ribs.

 

A modern view of Leipzig shows that it's still standing in good condition, today. If you would like to know more about Leipzig, a group of students at Princeton made a study of these shells, and their scientific, social, and symbolic analyses are described in a website called Shells.Princeton.edu.

 

The designer of Leipzig, Franz Dischinger, recognized that there were few opportunities for spaces with circular floor plans. More practical use of vaulted forms was one that covered rectangular spaces. He made an experiment of a rectangular floor plan dome.

 

This is a square plan with rigid boundary conditions, which followed the mathematical formulation, and as a demonstration of strength, by loading it with engineers.

 

Now we're going to move to the scientific analysis of another very common geometric form for vaults called barrels. Barrels are, essentially, slices of cylinders.

 

The person who published the mathematical formulation for barrels was Ulrich Finsterwalder, and he did this in 1933. Generally speaking, the behavior of barrels falls into two categories.

 

There's the long barrel and the short barrel. In the short barrel, the length, l, is small in comparison to the radius of curvature of that barrel, and therefore this barrel will behave, essentially, like an arch, the forces flow down to the edges, where it's carried to the supports.

 

In the long barrel, the length, l, is relatively large, and therefore it behaves, essentially, like a beam, with compression on the top of the shell, and tension towards the bottom.

 

This is a very simple way of looking at barrel analysis. Of course, a more refined examination requires higher-order mathematics.

 

On the left is a paper by Finsterwalder, with the handwritten notes of Felix Candela, whom we'll study later. Even the simplest of shells requires significant mathematics, like 8th order partial differential equations, as seen on the right. These formulas assume a boundary condition, that is, a certain behavior at the edges and at the supports.

 

The Germans constructed their vaults so that the boundary condition matched the assumptions of the formula, which makes sense, especially since these very thin vaults of reinforced concrete were an innovation, and they were being as possible.

 

For Finsterwalder and Dischinger, the form follows the formula. The one who starts to break away from this is another engineer, Anton Tedesko, whom we'll study next.

 

Anton Tedesko

As I said at the beginning of the lecture, he came to the united states and pioneered thin-shelled concrete construction. Before I talk about his structures, let me give you a brief bio and introduction to Anton Tedesko. He graduated in 1926 from the Technical Institute in Vienna as a civil engineer.

 

He had European education and practice meaning he studied completed works and his practice was a combination of design and construction.In 1930, he joined Dyckerhoff and Widmann.This is where Finsterwalder and Dischinger worked.

 

And this is where his thin-shelled concrete practice begins. After the successful completion of several large-scale domes and barrels in Europe, Dyckerhoffand Widmann decided to expand their operations to the United States.

 

They sent Tedesko who had prior experience living and working in the United States and they sent him over here in 1932. The tradition begins in New York City, the Hayden Planetarium which was torn down in 1997 but not for structural reasons.

 

It's the first shell that Tedesko worked on in the United States and perhaps the first full-scale reinforced concrete thin-shell structures in the U.S. It was a pure dome.

The construction of these thin-shell concrete structures involved what's called scaffolding formwork.

 

Remember concrete when you mix it is in a fluid form and you need a mold to support it until it hardens. The scaffolding supports the formwork. The formwork is essentially that mold that hardens the concrete into the proper shape.

 

For the construction of this Hayden Planetarium, the formwork was built on scaffolding and this formwork and scaffolding could not be repeated and reused on this project. It all needed to be built all at once.

 

This process was expensive and required intensive labor. The Hersheypark Arena is the first economical Tedesko structure in the United States.

 

Hershey is a famous chocolate brand in the United States and according

to its website, it is "The leading North American manufacturer of quality chocolates."

 

Hershey chocolates were founded by Milton Hershey who, according to David Billington, decided to construct a sports arena with the labor of chocolate workers to provide jobs and boost morale during the Depression of the '30s.

 

It was built in 1936 with Anton Tedesko as the designer and builder of the vaulted roof.

It's used mostly for ice hockey today in both college and high school and in the past, it was used for basketball and special events as well. As you can see, it's still in good condition today more than 80years later.

 

The structure consists of a curved slab of reinforced concrete that is three and a half inches or about nine centimeters thick. It gets thicker at the supports and at the top, the top of the vault is called a crown and at this place, you have large loads such as speakers and lights.

 

The crown sits about 100 feet or about 30 and a half meters above Iceland; the thin shell spans 39 feet 2 inches or about 12 meters between the arches.

 

The shape approximates that of an upper part of a pipe with a radius of 132 feet or 40 meters. The thin slab helps the arches carry some of the load and also provides lateral support for the arches against buckling. The arches are 22 inches wide by 60 inches deep approximately56 centimeters by 152 centimeters.

 

At the supports, the arch depth increases and sits on hinges to allow free rotation. The depth of the arch is reduced at the crown at the very top so that it approximates a three-hinged arch behavior. The arch spans 222 feet approximately 68 meters and rises 81feetor 25 meters above the hinge supports.

 

The scaffolding was reused many times in this structure which led to the economy. It looks expensive but it wasn't because it could be reused for every segment meaning every slice of the building. The roof was built in five segments separated by four expansion joints.

 

The three center segments were 78 feet long, 24 meters approximately and the two end segments were 52 feet long or 16 meters. Here's Tedesko examining some wires that inform the construction procedure. Each wire is connected to a jack in a specific location in the scaffolding.

 

When the concrete hardens it's time to lower those forms and you have to lower them very carefully. This process of lowering is called decentering. You have to lower these in a calculated way very carefully because if you don't do it correctly and evenly the vault might bend and might break.

 

The calculations for the construction procedure are very important and Tedesko emphasizes that these structures can be economical only if you consider construction carefully. In addition, Tedesko designed many warehouses that were built around the time of the war.

 

He needed to make an assumption in the calculations of the warehouses that the ends of the thin shell are rigid and these ribs as shown in the warehouse in Columbus force that rigid boundary condition that was assumed into mathematics.

 

The equations had an immense influence on the design and this is a mathematically driven form. Tedesko didn't like to rely on these heavy ribs. It made the project more expensive and it also made the appearance bulkier.

 

So he carried out a series of experiments. Here's a test that he organized in 1950. These are barrel shells and he loaded them with people to demonstrate their strength. Here you see that he did away with the ribs and showed that it could indeed work.

 

He also performed some simple mathematical calculations. In this warehouse in Middletown, Pennsylvania, in 1959 we have a pure shell with a slight thickening over the supports but you see that there are no ribs.

 

He removes this rigid mathematical influence on form but uses experiments as justification. This is simpler and more economic. In addition to warehouses, Tedesko found a market for hangers in the United States.

 

He arrived in this country at a time when air transportation grew rapidly and a large number of hangers were needed.

 

This hangar in San Diego was designed by Tedesko and you can see that it has a heavy appearance because it was based on the theory of rigid boundary conditions. In this hangar in Maine, the arches are more slender than they were in the previous design.

 

One can see the scaffolding used for construction in the background which follows a similar procedure that he used for the Hersheypark Arena.

 

For all vaulted structures, the form is the most important parameter. If one has the wrong format best the vault will be expensive to build and maintain and at worst it can collapse. Next, we'll examine what I mean by proper form.

Importance of Form

Here I'm going to highlight three items. One is, it will affect the visual impact of the shell structure of the vault itself and I will show you an example of the St. Louis Airport Terminalversus a Bacardi Rum Factory's an example of how the form will affect that visual impact.

 

More important than anything else is safety and I'll illustrate that through the example of the Tucker High School roof and then finally, as I mentioned earlier, the economy plays a major role and I'll illustrate that with an example of the Montreal Stadium versus the Kingdome in Seattle.

 

Going back to Anton Tedesko, one of his major designs was the St. Louis Airport Terminal. He designed this working with the architect Minoru Yamasaki. The architect's vision for this airport terminal as you see here in the model was a series of intersecting barrels where the barrels intersect at 90 degrees these valleys formed in the vaults.

 

This form of intersecting barrels is not the best form for a vault and to the architect's dismay, Tedesko concluded that the vault needed ribs at the valleys and at the edges. This was essentially the wrong form for the vaulted design because it needed ribs.

 

Visually the ribs aren't so bad but they conceal the thinness of the vault and make it look heavier than it actually is. I believe it would have looked more elegant without the ribs but would it have been possible to design this type of vaulted structure without ribs?

 

The answer to that question is both yes and no. If we use intersecting barrels as done here, the ribs were needed, but what if we used intersecting hyperbolic paraboloids instead of barrels?

 

It's a different shape, the hyperbolic paraboloid that has curvature in two directions and adds stiffness and that is indeed what Félix Candela did. Félix Candela designed the Bacardi Rum Factory as a critique of the St. Louis Airport Terminal.

 

By using the hyperbolic paraboloid geometric form, he was able to give the vault enough stiffness to eliminate the need for deep ribs and in my opinion, arrived at a more elegant structure.

 

We're going to learn more about Candela and his structures later on in this course but the visual impact is certainly not as important as safety.

 

Safety is the most important consideration when choosing a form for your vaults. The example of safety begins with a thin shell concrete structure designed by Anton Tedesko in Denver, USA.

 

This is a hyperbolic paraboloid shell that was taken down in the'90s because a town wanted to build a high-rise building in its place.

Another designer with less experience copied the general form of the Denver shells, a gymnasium for kids in Virginia.

 

Tedesko said that the building should not be built because the shell was too flat and improperly reinforced. The shell was completed in 1963 and it collapsed in 1970. This was the roof of a gym of a school which was full of kids a bit earlier.

 

Fortunately, the roof sagged noticeably before crashing to the ground allowing people to leave thus resulting in only minor injuries and no deaths. Three other similar shells after this failure were observed to be flattening out.

 

A fairly recent study indicates that creep was likely the culprit for this collapse. Creep is a phenomenon of concrete where the concrete under constant compressive load will continue to deform over time and this is an important consideration for reinforced concrete thin-shell roofs.

 

In addition to visual appearance and safety, the four most significantly affect the cost of the vault. Let's compare the cost of two domes, one with proper form and one without.

 

Neither of these was designed by Anton Tedesko. The Kingdome was completed in 1976 with a span of approximately 660 feet or approximately 201 meters. To generate the shape the engineer, Jack Christiansen, combined 40 triangular hyperbolic paraboloids.

Unfortunately, the dome was torn down in the year 2000 to make way for two new stadiums. At the time that it was torn down, it was still in excellent condition.

 

Let's compare the Kingdome to the Montreal Stadium in Canada. The Kingdom is a dome as implied in its name and if you take the cost of the Kingdom And divide it by the number of seats in the dome, it comes to about $600 per seat.

 

The Montreal Stadium structurally is a series of horizontal cantilevers. It does not behave like a dome and if you take the cost of the stadium itself divided by the number of seats it comes to $13,000 per seat.

 

You'll note that the Montreal Stadium costs more than 20 times per seat compared to the Kingdom.

 

Why is that?

 

Let's start by examining the form. The Kingdom is a true dome that takes advantage of the 3D behavior of domes to reduce stresses. The Montreal Stadium is a series of horizontal cantilever beams which is not so efficient for carrying loads.

 

The Montreal Stadium is elliptical meaning that every cantilever was different. Standardizing the pieces would have helped to reduce the cost. The Montreal Stadium included many unique pieces and details and didn't have the benefit

like the Kingdom of having few standard pieces. Under a deadline to have the stadium completed in time for the Olympics, lots of equipment and people were brought onto site and the project was rushed leading to costly mistakes.

 

One of which was the misalignment of ring sections. Tedesko commented on the Montreal Stadium in a world congress on structures in Montreal in 1976. He said this: "My reaction is one of disappointment.

 

I find the final product, the details, ill-conceived, ponderous, lacking in simplicity, lacking in grace, and without finesse.

 

I believe that there was no capable knowledgeable engineer experienced in construction active from the very beginning to guide the architect.

 

I suspect a lack of partnership, no early joint effort between designer and constructor.I feel, for instance, that the 600-foot span Velodrome should have had more construction depth in which case forces would have been smaller and details would have been less heavy.

 

I would have preferred to make use of a three-dimensional action.I would like to have seen some advantage taken of dome action in the roof of the stadium with resulting savings and materials and cost.

 

I can think of men like Nervi, Esquillan, Finsterwalder, Candela,if any one of these men had been enduring their early design stage of these structures would have been more elegant, proportions would have been better, costs would have been smaller."

 

Tedesko mentioned Finsterwalder, the German who published the mathematical formulations for barrels. Although I didn't have time to discuss, Finsterwalder designed many structures including bridges.Tedesko also mentioned Nervi and Candela.

 

In my opinion, these two men surpassed Finsterwalder and Tedesko in expressing creative forms for reinforced concrete vaults.

 

Who were these men and what did they design?

 

Keep watching the lectures to find out. Next, we'll travel to Rome to study Pier Luigi Nervi.