The world's largest cable dome, to be completed for the 1992 football season in Atlanta, will be the centerpiece of the 1996 Olympic Games. Spanning 766 ft x 610 ft (233.5 m x 186 m), it will be the first Hypar-Tensegrity Dome. This new cable supported teflon-coated fabric roof is based on the tensegrity principles first enunciated by Buckminster Fuller and Kenneth Snelson. Because of the large deformation characteristics of this type of structures, special geometric nonlinear analysis is required. This paper describes the modelling and the behavior of the roof for the Georgia Dome under different loadings.
The Georgia Dome is the first Hypar-Tensegrity Dome to be built. In a Hypar-Tensegrity Dome, shown in Fig. 1, hyperbolic paraboloid fabric panels are attached to a cable net that is rigidized by the use of tensegrity principles.
The plan configuration of the Georgia Dome is an oval defined by two radii. A ring beam along the outer edge of the roof is supported, on radially sliding bearing pots, by 52 columns projecting up from the seating structure below. Twenty six attachment points spaced about 82 ft (25m) on center around this compression ring serve as the springing points for the cable dome. The top surface of the dome consists of a triangulated network of cables attached together at nodes equally spaced along smaller and smaller meridians located 68, 154 and 250 ft (20, 46 and 75m) from the attachment points on the ring beam. Figure 3 shows that in section the structure appears like a truss in which the bottom chord is discontinuous and is replaced by a series of hoops in plan that link bottom chord nodes.
These tension hoops are connected to the upper cable net by compression posts and diagonal back-stay cables. A center cable truss ties the two circular ends of the cable net together. The upper cable net is deformed by raising the nodes of alternate meridians to achieve a hyperbolic paraboloid geometry for each of the fabric panels.
Apart from a low selfweight of about 6 psf (0.3kN/sq.m), the roof is subject to live load, snow load, wind load, seismic load, temperature load, and loads imposed on the continuous hoops by catwalks. In addition, the structure must be analyzed for construction loads taking into account the erection sequence. Each node is also capable of supporting a suspended load of 1000 lbs (455kg). The minimum roof live load is 20 psf (1 kN/sq.m) which can be reduced to 16 psf (0.8 kN/sq.m) for the design of the fabric and 12 psf (0.6 kN/sq.m) for the design of the cable net. Wind tunnel tests were conducted by RWDI of Guelph, Ontario, on a rigid model under conditions simulating the full scale atmospheric conditions pertinent to the Georgia Dome site. The tests revealed that no resonance would occur for the range of natural frequencies of the structure. Fig. 3 shows the first four vibration modes, which correspond to frequencies of 0.441, 0.682, 0.716, and 0.725 Hz for the prestress and dead load condition. The mean wind loads for a 50-year wind speed, shown in Fig. 4, indicate suction over the entire surface of the roof because of its flat profile.
If built based on the initially defined geometry, the cable dome would seem like a limp noodle even under its own weight. Therefore, a cable dome needs to be prestressed to compensate for the tendency of some cables to go slack. In order to simulate the prestress condition computationally, the simple device of introducing a temperature change in members was employed. A nonlinear analysis using LARSA was performed on the complete cable dome. If the results indicated that some cables were in compression, a local temperature change was applied to those members and the problem was rerun. The runs were repeated to iterate to a condition where no compression cables existed. It was determined that an initial prestress averaging 30% of cable capacity was needed to rigidize the structure. Since the deformation of the structure was well within normally accepted criteria, it was not necessary to use additional prestress.
The results of the analysis for the first loading condition, prestress plus dead load, are shown on Fig. 5. From these results, it is apparent that much of the load on the roof gravitates toward the four corners as can be seen by the ridge and diagonal cable forces. Forces in the hoop cables remain relatively constant which allows the clamping detail to be relatively small. On the other hand, forces in the ridge cables decrease markedly toward the center. This implies large changes in the force at a top node and therefore the need for large clamping forces to absorb the change in tension in the continuous cables. The shape of the moment diagram for the compression ring confirms the fact that the four corners of the roof structure attract much of the roof load. Obviously, in alternate configurations, as the oval shape tends toward a circle, moments would disappear as the circle becomes the funicular for the loads.
The design of the top cable net, ridge cables, was controlled by the wind load, while the design of the diagonal and hoop cables was controlled by the live load. Since the dead load of the roof is very low, seismic loads do not affect the cable design and must be considered only for the supporting column design. The bearing posts that support the compression ring allow for free radial displacements so that the effect of the temperature variations is minimized. Maximum displacements in the cable net were found to be 2.5 ft (0.75 m) down and 2.4 ft. (0.72 m) u pfor live load and wind load, respectively.
The design of the cables and the connections must consider the construction sequence, that in the case of the Hypar-Tensegrity Dome results in large changes in forces and in geometry requiring a complex 3-D nonlinear analysis. The first step is to hang the ridge net from the compression ring and then install a diagonal cable at a time with its corresponding post and hoop cable. Results from a simplified 2-D model using LARSA are shown in Figs 6 and 7 for the variation in roof geometry and in tension force in the diagonal cables. These results assume that the first diagonal to be installed is the outermost, D4, and the last one is the innermost, D1. The sequence can be altered in order to minimize the maximum required jacking force.
In order to verify the choices made in the initial stages of the project, when time constraints did not allow for geometry optimization, and to provide guidelines for future development of the Hypar-Tensegrity Dome concept, an optimization study was prepared with the objective of defining the most economical geometric configuration. Three factors were considered: sector width, the post height, and the number of rings. These are obviously interdependent as exemplified by considering increasing sector width, which requires the membrane curvature to be increased, as shown in Fig. 8, to remain within permissible stresses. An optimal configuration was sought using cost as the ultimate criterion. A parametric nonlinear analysis using LARSA was performed using varying post heights and ring spacing. The results shown in Figs. 9 and 10 suggest that an increase in post height over that used for the Georgia Dome may result in a lower cost configuration. It is also apparent that a two ring solution may be more economical than the three ring configuration used.
Structures such as the Hypar-Tensegrity Dome require special analysis and could not have been realized without the availability of computers and nonlinear programs. Without computers, only rough approximations of the forces in such a highly indeterminate structure could be obtained and furthermore only solutions for symmetrical or other simple loading conditions could be studied. A software package like LARSA provides not only the answer to the every day analysis and design problem but also the nonlinear solutions that large deformation structures require. This leads to the ability to economically construct complex structures which would not otherwise be feasible, opening the door to an exciting range of possibilities limited only by the creativity of the engineer.
James Blaut envisioned "Eight Eurocentric Historians" as the second installment in a trilogy on Eurocentrism, beginning with "Colonizer's Model of the World" and culminating in a study that would be titled "Decolonizing the Past
"Eight Eurocentric Historians" was the second installment in a trilogy on Eurocentrism. First came "The Colonizer's Model of the World" (Guilford, 1993), a broad statement of the problem. The final installment would have been "Decolonizing the Past." Offering an alternative model of history, it would give what Hegel called the "people without history" their proper due. While cancer last year robbed James Blaut of his life and halted completion of this important project, the first two volumes stand on their own as correctives to Eurocentric history.
The eight Eurocentric historians are Max Weber, Lynn White Jr., Robert Brenner, Eric L. Jones, Michael Mann, John A. Hall, Jared Diamond and David Landes. With his emphasis on European rationality, Weber is a forerunner. Although he enjoys enormous prestige as a founder of modern sociology, there is an racist streak in some of his work. In "The Religion of China," we learn that the Chinese have a "strong attachment to the habitual", "absolute docility" and a "lack of genuine sympathy and warmth."
For socialists, the presence of Robert Brenner among this pro-capitalist and occasionally racist group might come as a surprise. In a 1977 NLR article titled "The Origins of Capitalist Development: A Critique of Neo-Smithian Marxism," he recapitulated arguments made earlier in the journal "Past and Present." Among them was the proposal that capitalism and modernity were generated by conditions internal to and arising from the class struggle in late middle ages Europe, with England the most advanced example. Rather than looking to the pillage of the New World, Brenner gave little weight to the plantation system and Spanish forced labor mining. He specifically attacked the "dependency theory" of Paul Sweezy, Andre G. Frank and Immanuel Wallerstein that supposedly had failed to understand the role of the class struggle in sparking the rise of capitalism. They were following Adam Smith rather than Karl Marx. Blaut tweaks Brenner by suggesting that he is a neo-Weberian Marxist.
John Browett wrote in 1980 that "the age of the radical-liberal dependency formulations have come to an end" because of the Brenner thesis. Despite Brenner's best intentions, Blaut argues that his thesis served a backward agenda as the mood in the 1970s shifted dramatically away from support of the colonial revolution. This is something that touched him deeply as a long-time activist in the movement for Puerto Rican independence.
Blaut, a geographer by profession, makes his point through a telling graphic titled "maps of the world before and after 1500AD." It contains dots representing "dated place-name mentions" in Brenner's articles. No region outside of Europe is ever mentioned before 1500AD. After 1500AD, references occur more frequently but tend only to reflect what Blaut regards as "Brenner's view that capitalism began to diffuse outward to the rest of the world after its birth in northwestern Europe."
Given the more enlightened racial framework of today, Eurocentric historians today support their views with "hard" evidence drawn from agronomy, climatology, demographics, etc. rather than openly racialist claims. Few today would argue that the Europeans were genetically endowed with gifts for invention or rationality, or chosen by god. Rather, fortuitous historical circumstances moved them to the head of the class.
It is exactly these pretensions to hard, scientific evidence that Blaut succeeds in demolishing. Barely disguising his contempt, he answers one false claim after another. When Eric J. Jones asserts that Europeans were solely destined to become capitalists after the Middle Ages, Blaut cites Tomé Pires, the 17th century Portuguese chronicler, who described Indian merchants thusly: "They are men who understand merchandise; they are . . . properly steeped in the sound and harmony of it." He adds, "[T]hose of our people who want to be clerks and factors ought to go there and learn, because the business of trade is a science."
Among these Eurocentric historians farming practices loom larger than any other supposedly objective criterion underpinning the rise of the West. The West is the world of the plucky, inventive yeoman farmer, while the despotic East employed unproductive farming techniques. Benefiting from his early training and fieldwork in agronomy, Blaut presents an alternative interpretation. For example, while Michael Mann considers soil fertility in Europe to be the key to its rise, Blaut points out that until the arrival of the potato from South America, a vast swath of land across Europe remained unproductive because of excess rainfall, conditions beneficial only to potato growth. Meanwhile, crop rotation--supposedly unique to the West--was found in the rest of the world.
Combining scholarly erudition and a passionate commitment to the "people without history," James Blaut leaves a legacy for a new generation of scholars and activist committed to internationalism and working class solidarity.