Método de distribución de momentos

Leonard K. Eaton

Hardy Cross and the “Moment Distribution Method”
Leonard K. Eaton resurrects the reputation of Hardy Cross, developer of the “moment distribution method” and one of America’s most brilliant engineers. The structural calculation of a large reinforced concrete building in the nineteen fifties was a complicated affair. It is a tribute to the engineering profession, and to HardyCross, that there were so few failures. When architects and engineers had to figure out what was happening in a statically indeterminate frame, they inevitably turned to what was generally known as the “moment distribution” or “Hardy Cross” method. Although the Cross method has been superseded by more powerful procedures such as the Finite Element Method, the “moment distribution method” madepossible the efficient and safe design of many reinforced concrete buildings during an entire generation.

Introduction In his paper “The Influence of Mathematics on the Development of Structural Form” in Nexus II: Architecture and Mathematics, Holger Falter comments on the difficulty of calculating statically indeterminate frames of reinforced concrete. He remarks, “Forming hinges was simple withiron, but difficult with reinforced concrete. It created statically indeterminate frames continuing through several spans, a difficulty surmountable only by using analytical methods” [Falter 1998:61]. He emphasizes the change from graphic to analytical statics and concludes that, “Only a limited number of statically indeterminates could be solved with these new procedures, which is probably thereason for the mostly simple structures, causing as little calculating effort as possible” [Falter 1998: 61]. Whi1e I am in general agreement with Falter’s account, I would like to offer an emendation and in so doing resurrect the reputation of Hardy Cross, one of America’s most brilliant engineers. When I started teaching at the University of Michigan, Ann Arbor, in 1950, architects wereunderstandably cautious about designing large buildings with reinforced concrete frames, because they were uncertain about the location of moments (movements) within a frame of reinforced concrete. Nobody likes building failures. They lead to lawsuits and shattered careers. It happened that I was recruited by an architectural school which has always taken the structural side of the profession seriously.Unlike many places, which have traditionally “farmed out” this end of architecture to schools of civil engineering, the architectural school at University of Michigan has always taken structures as part of its domain. I knew some of my colleagues in the structural program rather well, and occasionally asked them for help in understanding problems of structure in

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Gothic, Renaissance, and Baroque buildings. Sometimes I heard them talking about the behavior of reinforced concrete structures. The name of Hardy Cross would be invoked with awe. The attitude of my fellow faculty members has to be understood in the context of the decade. In the nineteen fifties the post-World War II building boom was well under way, and a number of Americanarchitects were confronted with demands for multistory structures which had, by their very nature, statically indeterminate frames. Many of these were done in reinforced concrete, especially in the years of the Korean war when steel was generally unavailable. Even Mies Van Der Rohe, the apostle of steel construction, did one reinforced concrete building; it was far from his best performance. The oldcolumn and slab system, developed by C. A. P. Turner, and the more refined version of Maillart, were alright for industrial buildings but deemed unsuitable for first class office buildings and apartment houses.1 But even in the fifties concrete was seen as a tricky material. I had one colleague who was always talking about “the dreaded creep,” a movement of concrete after it has hardened. Over in...