Memoirs 7B. Analysis of Hyperbolic Paraboloidsby Milo Ketchum
After moving to Connecticut in 1962, my interest in analysis of hypars continued. One of our clients, Paul Rudolph, then at Yale, had a client, a southern black church who wanted a new sanctuary. I had seen a sketch of the church in Engineering News Record, and I asked him if that was his, but he said no, that it was not his style. The interesting thing about the sketch was that the structure was a typical four gabled hypar (like the structure in Colorado Springs) except that the center point was raised to about a third of the length of the center rib. After that episode I began to think about how such a structure could be analyzed. On September 22, 1970, I visited the site of the failures of a number of high school gymnasiums in Richmond, Virginia, at the request of Dick Elstner, of the well known firm of Wiss, Janney and Elstner, of Northbrook Illinois. There were three structures of the same design, 100 feet square, having a rise of about 15 feet, large inclined supporting ribs, with the center ribs essentially concentric with the shell which was 3.5 inches thick for the most part. There had been a complete collapse of one structure, fortunately with no loss of life. On another structure, the roof had sags of 18 inches at the center and on a third the deflection was somewhat less. It was a strange feeling which suggested that we were time travelers going back to see what had happened to the structure.
The conclusion in my report was that it was essentially a material failure due to the use of lightweight concrete and construction in cold weather. There was good deal of discussion on how the center rib should be placed with respect to the shell to avoid progressive failure by deflection. Now I know the answer: camber the center point of the rib so that forces tend to raise the center instead of depressing it. However it took the studies made later to show that this solution did not impose high stresses on the structure.
It was not until 1974 that I was in a position to investigate this problem by the finite element method. There had been a number of studies by Schnobrich and others but they did not address my problem. I had a doctoral candidate, Ahmed Shaaban, an Egyptian, who had received his Master's degree from the University of Massachusetts in 1969, and who had a structural engineering practice in Springfield, with in commuting distance of Storrs. We never had a candidate with the ability and the knowledge of computers that could equal the ability of Shaaban. Also the structural department must have been much stronger in computer analysis at the University of Massachusetts than at the University of Connecticut.
The original purpose of his thesis was to investigate the effect of raising the center of the hypar. I did not wish to try to duplicate the work of Schnobrich, which was very enlightening. Shaaban’s computer program for the finite element method had all of the bells and whistles, including the effects due to eccentricity of the ribs, and he could plot all of the stresses by computer. The structure investigated was that analyzed by Schnobrich and many others; a hypar 80 feet square, with a rise of eight feet at the middle. There were analyses made with edge beam concentric, upstanding, and down standing. In addition, shells with an additional lift of 8 inches, 2 feet, and 4 feet were investigated, an impressive accomplishment.
The results confirmed my hunches. An additional small rise of the shell had little effect on the stresses. The large, 4 foot rise, caused some bending moments in portions of the shell which would have required extra reinforcement. The forces in the center ridge beam changed from compression to tension for the large elevation of the center.
When it came to publishing the results of the thesis, we decided to present general conclusions rather than the problem of the camber, so we called our paper "Design of Hipped Hypar Shells". It gave a series of practical suggestions on the design of these structures on the theory that it is important to have the right structure to analyze, otherwise the analysis will be invalid, and the design flawed, a general rule in structures. The most important conclusion to me was the demonstration that the membrane forces are essentially compressive and the catenary (tensile) forces are small, and therefore the shell should be thickened at the supports. To my knowledge, we never analyzed a shell for this condition. I believe that it would have shown that the forces in the external ribs would be much smaller. It is evident that the arch is the stiffest path for forces, and we should have known this without elaborate analysis. The trouble is that we were hypnotized by the simplicity of the membrane theory. Incidentally, Sid Simmons at the University of Alberta found that if flexible ties are used, then the tensile forces are much larger and may control.
I wanted to publish the results of the camber studies but it never came to pass. Shaaban moved away, and I retired soon afterward, and I have not heard from him since. I still hope to be able to do work in this field. There are many problems needing study.