Frank Lloyd Wright (1867-1959) was a prominent American architect, writer and teacher. His design philosophy was based on the idea that structures should harmonize with their environment. This approach was best exemplified by Fallingwater (1935) and his many Prairie Homes. However, he also designed numerous lesser known structures. One of which stands in Wauwatosa, a city in Milwaukee County, WI, USA, only a few miles from my hometown. The Annunciation Greek Orthodox Church was designed and proposed in 1956, near the end of Wright’s life. In fact, it was not constructed until 1959, after his death.
For the design of this thin shell dome, Wright was inspired by the Hagia Sophia (Istanbul, Turkey). The reinforced concrete dome is unique. First, it is extremely thin for its size. The dome has a 16.2m radius (the Hagia Sophia’s radius is 15.5m) and a height of about 4m (Hagia Sophia’s height is 15m). However, the average thickness of the concrete shell is only 9cm whereas in the masonry Hagia Sophia dome the average thickness is 76cm. Second, the dome rests along its perimeter on greased steel ball bearings. Milwaukee has a wide range of seasonal temperatures and the ball bearings allow the dome to expand and contract about 2 cm of movement due to temperature variations. Finally, the dome is detailed along its perimeter with a series of glass orbs which let light in and give the illusion that the dome is levitating above the building. This resembles the design of the Hagia Sophia, interior pictured below, which has windows allowing streams of light to pass around the perimeter of its dome as well.
Besides its striking lighting effects, I would like to find out what kind and level of stresses occur in the Annunciation dome which has a thickness of only 9cm. Since the radius of the dome is 16.2m and the height is 4m, the aspect angle of the dome is 27.8⁰. Assuming, density of reinforced concrete as 2500kg/m^{3}, the approximate hoop and meridional stresses within the dome of the Annunciation Greek Orthodox Church under its own weight can be found using analytical formulae.
Hoop stress and meridional stresses are membrane stresses in the latitudinal and longitudinal directions, respectively, of the dome as shown in figure 1. In particular, I am interested in establishing the stresses present at the top and bottom of the dome.
In order to solve for these stresses, we need a few equations. For a full derivation of these equations please refer to The Stone Skeleton: structural engineering of masonry architecture (Jacques Heyman). Rather than working in terms of stresses, we will use the equations for resultant forces in the hoop and meridional directions. At the end, we will divide by the dome thickness to solve for internal stresses.
My results show that the Annunciation dome experiences meridional compressive stresses throughout the structure. Compression can be well taken by concrete. Similarly, the hoop stress is compressive throughout because the dome is quite shallow. The following table gives the values of the stress resultants and actual stresses within the dome at the crown and at the base.
Φ (⁰) |
NΦ
(kN/m) |
fΦ
(N/mm2) |
Nθ
(kN/m) |
fθ
(N/mm2) |
Comments |
0 |
-38.3 | -0.43 | -38.3 |
-0.43 |
Nθ = NΦ |
27.9 | -40.7 | -0.45 | -27.1 | -0.45 |
Notice, at the crown of the dome, the hoop and meridional stresses are equal and both in compression. The compressive stress in the meridional direction increases toward the base while the compressive stress in the circumferential (hoop) direction decreases.
These simple calculations show the efficacy of the dome; the stresses in the dome turn out to be rather low under its own weight. Little is known about this shallow dome in Wauwatosa. Because of its efficiency and elegance, it merits more attention from curved surface enthusiasts.
Author: Kendall Schmidt
Sources:
[1] Heyman, Jacques. The stone skeleton: structural engineering of masonry architecture. Cambridge University Press, 1997.
This could be a masterpiece if built in 6th century
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