Leonard K. EatonBox 300 Otter Rock, Oregon 97365 USA
When I started teaching at the University of Michigan, Ann Arbor, in 1950, architects were understandably 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 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 American architects 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 old column 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 Jackson, Michigan a building framed in concrete collapsed and four or five men were killed. An investigation showed that the material had not been given sufficient time to cure. The workmen pulled away the forms too quickly, and a failure at the joints was the result. 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 Hardy Cross, that them were so few failures (This may have been one of the reasons for the widespread interest in the unusual reinforced concrete systems developed by Frank Lloyd Wright for his Johnson Wax office building in 1939). 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.
The paper was entitled "Analysis of Continuous Frames by Distributing Fixed-End Moments", and it set forth an entirely new method of analyzing building frames. Discussion was closed in September 1930; the paper was not published in the Transactions until September 1932. At that time space was afforded for 38 commentators who took up 146 pages. This may be a record for comment on a single paper. Cross was immediately hailed as a man who had solved one of the knottiest problems in structural analysis. And he did so in a way that could be adopted by any engineer working in the field. Although Cross was known as "philosophical", his approach here was extremely practical. His view was that engineers lived in a real world with real problems and that it was their job to come up with answers to questions in design even if approximations were involved. In his first paragraph he wrote, 'The essential idea which the writer wishes to present involves no mathematical relations except the simplest arithmetic" [Cross 1949:1].[2] This statement is not quite accurate. The Cross method depended on the solution of three problems for beam constants: the determination of fixed-end moments, of the stiffness at each end of a beam, and of the carry-over factor at each end for every member of the frame under consideration. The author remarks that the determination of these values is not a part of his method. In point of fact, these moments are often determined by calculus. In 1932, however, that branch of mathematics was part of the intellectual equipment of any well-trained engineer. Its application required no contrivance more complicated than a slide rule. In order to understand the achievement of Hardy Cross, it will be helpful to review developments of the theory of elasticity as it applies to statically indeterminate structutes. While there were important additions to the theory of elasticity in the late 18th century, major interest in the analysis of indeterminate structures started in the early 19th century. Most of the attention was directed to the solutions of trusses in timber and iron having redundant (i.e. indeterminate) membors. Clebsch and Castigliano come to mind as pioneers. Indeterminate frame structures, like continuous beams, were of little interest, because in the available materials it was difficult (and perhaps undesirable) to create continuity at the nodes between members. The developed methodology in solving indeterminate trusses was cumbersome, since it required the solution of as many simultaneous equations as there were redundancies (indeterminacies) in the structure. Tedious longhand calculations were required, with an accuracy to many decimals, carried forward at each step. Anyone who has tried to solve only five equations with that many unknowns in them will appreciate the problem. And with each higher order of indeterminacy the amount of longhand calculation work increased geometrically. Clapeyron was the first to offer a practical solation to the problem of continuous beams over supports. His Three Moment Method was widely used well into the 20th century [Timeoshenko 1953]. He still had as many equations as there were indeterminacies (number of supports beyond two) but in any of these there were only a maximum of three unknowns. The development of a new material, reinforced concrete, made it imperative to find solutions for statically indeterminate frames. Monolithic reinforced concrete structures are highly indeterminate. Hence methods which gave reasonably accurate results and did not require an horrendous amount of calculation were a necessity. In this, Professor Falter is quite correct. Axel Bendixen in 1914 offered a procedure known as the slope deflection method [Timoshenko 1983]. It was the first readily practiced way to solve rigid frame structures. His method leads to an easily written series of simultaneous equations. The initial writing of the equations required little work. Each equation is rather "sparse" (i.e. it contains only a few of the unknowns). Thus the effort required for the solution of these simultaneous equations became less as compared to the methods developed earlier. The results from the solution of the simultaneous equations yielded rotations and displacements at the ends of individual members that in turn could be used to find moments and shears. In 1922, K. A. Calisev, writing in Croatian, offered a method of solving the slope deflection equations by successive approximations [Timoshenko 1983; Bulletin 108]. Probably because of the linguistic difficulty, Hardy Cross seems not to have been aware of Calisev's contribution. Though cumbersome, it was a pioneering work. The problem with it was that Calisev still used successively adjusted rotations to establish moment balances at the nodes. It was Hardy Cross's genius that he recognized he could bypass adjusting rotations to get to the moment balance at each and every node. He found that he could accomplish the same task by distributing the unbalanced moments while unlocking one joint at a time and keeping all the others temporarily fixed. By going around from joint to joint, the method converged very fast (at least in most cases) and it had enormous psychological advantages. Many practicing engineers had dubious mathematical skills in handling simultaneous equations, and many had difficulties in visualizing rotations and displacements. Moments are.much more "friendly" for the average eagineer and therefore easier to deal with. Thus the Moment Distribution Method (also known as the Cross Method) became the preferred calculation technique for reinforced concrete structures. The description of the moment distribution method by Hardy Cross is a little masterpiece. He wrote: "Moment Distribution. The method of moment distribution is this: (a) Imagine all joints in the structure held so that they cannot rotate and compute the moments at the ends of the members for this condition; (b) at each joint distribute the unbalanced fixed-end moment among the connecting members in proportion to the constant for each member defined as "stiffness"; (c) multiply the moment distributed to each member at a joint by the carry-over factor at the end of the member and set this product at the other end of the member; (d) distribute these moments just "carried over"; (e) repeat the process until the moments to be carried over are small enough to be neglected; and (f) add all moments - fixed-end moments, distributed moments, moments carried over - at each end of each member to obtain the true moment at the end." [Cross 1949:2] In the next paragraph Cross observed that for the mathematically inclined his method would appear "... as one of solving a series of normal simultaneous equations by successive approximation" [Cross 1949:2]. Indeed it was. Cross supplied an illustration shown in Figure
1. He assumed that the members
would be straight and of uniform section. Siffnesses were proportional
to the moments of inertia (I), divided by the lengths (L), but
the relative volume given for I/L in the problems might as well
be the relative stiffnesses of a series of beams of varying section.
In that case the carry-over factors would not be -½. Figure
1 is entirely academic. It does not represent any particular
type of structure or any probable conditions of loading. It has
the advantage that it involves all the conditions that can occur
in a frame that is made up of straight members and in which the
points are not displaced. Cross's willingness to accept approximation troubled that minority in the engineering profession who insisted on exactitude. Cross himself readily admitted that the values of moments and shears could not be found exactly. He concluded that there was then no point in trying to find them exactly. He was after a method of analysis which combined reasonable precision with speed. That he achieved his end is clear from a survey of the thirty eight commentators to his brief article. These individuals, partly academics, partly practicing engineers, generally saw that his method was a major contribution to structural analysis. It quickly passed into the curricula of the best American schools of engineering and by 1935 was generally taught. It continued to be received doctrine until the nineteen-sixties. In the light of subsequent developments in twentieth century architecture and engineering, it is worth noting that Hardy Cross included a list of seventeen problems for which his method of analysis would not work. Chief among these were (1) Methods of constructing curves of maximum moments and (2) Methods of constructing curves of maximum shears. The Cross system would not therefore be of any assistance to an engineer computing the stresses in an elegant bridge by Maillart or one of the superb hyperbolic paraboloids of Felix Candela. For this kind of building other tools of analysis were required. And since much late twentieth century architecture involves curved forms (one thinks immediately of Frank Gehry), there is no possible application of the moment distribution method. But for anyone building a structure with a reinforced concrete frame between 1932 and 1960, the method of Hardy Cross was a blessing indeed. And there were many of these, perhaps more in Europe than in the United States, which has always had a building economy where multistory structures have been framed in steel. Professor Robert Darvas, an eminent structural engineer who was educated in Budapest after World War II, learned the moment distribution method of Hardy Cross. Bob remarked in conversation that "His reputation was absolutely world wide". Hence it seems to me that Profeasor Falter overlooked an important mathematical contribution to structural theory in the years 1932-1969.
The most creative years of Hardy Cross were spent at the University
of Illinois. There he developed a reputation as a brilliant,
if forbidding, classroom teacher. Like his eminent colleague,
Harald Westergard, Cross suffered from deafness, and he used
this handicap to his own advantage, both in and out of the classroom.
Students soon found out that it was difficult to improvise answers
at the tops of their voices. And they learned either to be explicit
or to admit that they couldn't answer his questions. He lectured
without notes, and his performance was always calculated to produce
an atmosphere. Occasionally he would stomp out of the class early
because no one had attempted a particular problem, and then ask
some one who had observed his exit, "how do you think they
took it?" Cross believed that the classroom was, above all
else, the place to develop ingenuity and self confidence. He
held that a university was a place to make inte1lectual mistakes,
many mistakes, and learn to rectify them. This is not a bad definition
of a university. Cross began his next lecture by saying, "In our last
meeting Mr. Alford raised a serious and unfounded charge against
the author of our text." Staring at Alford, he said, "Have
you reconsidered your accusation?"
[2] In addition to the publications noted in the text,
this classic paper made two other appearances. It was included
in the report of a symposium at Illinois Institute Technology
in Chicago. The symposium was dedicated to Hardy Cross, "whose
simple demonstration of the power of numerical analysis brought
these methods within the horizon of practicing engineers".
It was also published in [Cross 1963]. This extremely valuable
volume contains two of Cross's additional papers on moment design.
[3] This account of Cross at the University of Illinois
is taken from Men and Ideas in Engineering: Twelve Histories
from Illinois (Urbana, 1967). I am much indebted to Professor
Emory Kemp of West Virginia for this reference and for his encouragement.
Cross, Hardy. 1949. Cross, Hardy. 1952. Cross, Hardy. 1963. Eaton, Leonard K. 1989. Eaton, Leonard K. 1998. Frank Lloyd and the
Concrete Slab and Column. Timoshenko, Stephen. 1983.
The Moment Distribution Method -Summary
Landscape Artist
in America: the Life and Work of Jens Jensen (1964), Two
Chicago Architects and their Clients (1969), American
Architecture Comes of Age (1972) and Gateway Cities and
Other Essays (1989). He is best known for his work on Frank
Lloyd Wright, he has presented two papers at Nexus conferences,
"Fractal Geometry in the Late Work of Frank Lloyd Wright:
the Palmer House" (Nexus 98) and "Mathematics and Music
in the Art Glass Windows of Frank Lloyd Wright" (Nexus 2000).
Prof. Eaton is contemplating a longer work on Hardy Cross and
would be glad to hear from any one with information on this remarkable
man.
Copyright ©2001 Kim Williams top of
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