The folded cross-section of the arches allows the elements to act as a composite beam in resisting lateral forces instead of acting like flimsy plane, like a piece of paper. This arch beam, as we will call it, looks something like a channel beam where a large prtion of the material of the beam is taken away from the center axis, therefy increasing the radius of gyration (r). The increase in r can be derived from the equation r equals the square root of the moment of inertia divided by the area. If the moment of inertia is increased (and increase in h) and the area of the material is constant, then r will increase as well.
While it is true that the effect of increasing the r and I would much greater in the above explanation if the whold system was taken off the centroidal axis, this is not possible in an arch form, who's load path must be within the middle third of the elem ent in order to be stable and in complete compression.
The other great element of stabilty against wind loading is that of the weight of the structure itself or its inertia. The inertia of the arches gives the members enough vertical loading that a lateral loading should only shift the load path a small marg in and V will be kept within the middle third of the members. When a wind loads one side of the building, the tendency of that side is to go into tension and the other side will go into compression. With the great weight of the structure however, given the density of concrete, the wind loads will not have agreat effect and if there is a great deal fo wind the steel reinforcing (which acts better in tension) will handle the tensile forces on the windward side.
Another advantage of the arch form against lateral loading is that of its geometry. Wind speeds (and therefore loads) increase as their height from the earth increases, thus the higher one builds, the higher the wind loads will be. With continuous membe rs like in this arch, the farther the load is from the support, the greater the moment or potential for a moment on the supports. In this case, the iarch is fixed witha rigid connection so the support will have to withstand that moment as will the member s have to resist bending as the wind will push in one direction and the reactionary moment will act in the other direction. Juckily, the rounded, decreasing profile of the top of the arch responds to that load by increasing the angle at which the wind hi ts the arches. The lower angle of incidence then, gives less of a load on the structure and less stress to the material.
Vertical Loading of the Structure
Since the airship hangars are located in France, just south of Paris, their only appreciable vertical loading will be that of the structure itself. The snow loads in the region are nearly negligible due to the mild winter clibate. This especially true w hen one looks at the comparable wight of the structure.
For a true parabolic arch to be successful, the load path must be within the center or "middle third" of the arch in relation to both its width and depth. This must be true for the cross sections throughout its length. If the loading does not fall withi n this area, miments will occur in the arch and put tensile forces in the materials which in this case is reinforced concrete and though it performs better than the earlier masonry arches, will be stressed and suffer some external cracking. This configur ation allows the arch to operate in pure compression, which is the goal of the builder.
The cross sectional area of the arches, which make up the vaulted structure of the hangare, increase in size as they travel down either side of the arch. This increase is due to the increased load that is accumulated as the arches approach the ground. T he stress in a member is equal to the force divided by the area, so to retain a constant stress in the arch, the area must increase in step with the increasing load (weight).
The actual transfer of the vertical loading from the top of the structure starts at the narrowed point of the crown of the arch and angles over and down in a mostly lateral direction. With this shallow pitch at the top of the arch, a comparatively great deal of compression is taken up by these members even a small vertical load will result in a much higher load in the downward lateral member. These loads are added to by the weight of the receiving members that are increasingly pitched downward and thus more efficient carriers of the load. wth the added load from above and their own weight, these members translate the downward force with much of the same stress because the area is increasing as is the load. This progression continues down to the spring ing of the vaults.
The springing of the vaults start just above the foundation and its comparitive thickness shows the need of the vertical load to be distributed over the ground. The cross-sections of the arches increase in thickness more dramatically here as they descend the last 16 meters ot the foundation. This is the point of the arches that change from their articulated form into a more massive element that meets the top of the foundation. These elements are still slightly tilted at this point and it is not until t he bottom fo the foundation that the loading will have a completely vertical component.
The foundation for the vaults was dug quite deep to ensure that it would have an adequete compressive strength to withstand the weight of this collosal structure. Horizontal pads of deep concrete were laid with a slight inward angle toward the center of
the hangars to receive the springing elements which were to be laid on top of these foundation pads. The earth below, which must be of a high grade of compressive strength must equal the load of the fondation in order for it to be in equilibrium. The de
pth of the foundation is what was needed for the loading from the arch to be completely vertical. This was a wise and very efficient move for Freyssinet, because if the load had reached verticality earlier, in the springing, a lot of concrete would have
ben wasted in raising the arch up in the air unneccesarily.
Sheldon Berg and Edward Running. ARCH 461 Spring 1995
Do you have questions about adding a case? or a building to suggest??????? send a message to me....... chrisl@aaa.uoregon.edu