# OPTIMUM WEIGHT STIFFNESS STRUCTURAL DESIGN

## Description

My adventures with flexible structures began on the IIT campus with an extracurricular undergraduate project to design an “Open House Exhibit” for the Civil Engineering Department. I chose to display a reinforced concrete diving board together with a prestressed concrete diving board. Visitors... Show moreMy adventures with flexible structures began on the IIT campus with an extracurricular undergraduate project to design an “Open House Exhibit” for the Civil Engineering Department. I chose to display a reinforced concrete diving board together with a prestressed concrete diving board. Visitors enthusiastically pounced on the reinforced concrete structure whose rigid response disappointed one and all. Their indignation was transferred to the prestressed cantilever which thrust them upward from six to ten feet into the air. This unexpected response from a diving board became so dangerous that the Exhibit was unceremoniously closed. I still have the display sign, “More Bounce to the Ounce.”While still an undergraduate, I secured a part-time job at Armour Research Foundation where I responded to a bid request from Rock Island Arsenal to design the 26 foot Honest John Rocket Launcher Rail at minimum weight. This tactical weapon was transported by helicopter. I basked in the fantasy that I was Leonardo da Vinci without his artistic proclivity. Rocket launchers that droop during operation are similar in concept to a circular firing squad. So began my research into minimum weight beams based on deflection rather than strength. I searched for the shoulders of Giants. I found them in the form of mathematicians not structural engineers. I achieved a 26.5% weight savings in the 1126 pound rail by optimizing the geometry. When I developed an optimum prestressed and segmented Kentanium cermet rail, the weight savings became 89%. The right material provides a bigger bang for the buck.

When my journey into optimum design began, I was armed only with analysis tools: strength, stability, and stiffness. This thesis begins with an outline of my present toolbox which contains eight design concepts: 1. Establish the Geometry, 2. Select a material from a finite number of candidates, 3. Prestress and Prestrain, 4. Statistical Screening (Proof Testing), 5. Manipulation of Boundary Conditions, 6. Energized Systems, 7. Counterweights, 8. Self-Healing and Self-Reinforcing. Four of these are used through this review which focuses on stiffness.

Beginning with beams, deflection control examples are described where prestraining and prestressing techniques are used to produce both a zero-deflection beam and a method for pushing with a chain. The calculus of variations made it possible to establish optimum tapers for the flanges and webs of I-beams that minimize beam weight for a specified deflection or, because of reciprocity, minimize beam deflection for a specified beam weight. An anomaly is encountered that enables one to achieve an upward, downward, or zero deflection with a set of beams of vanishing weight. In addition, special circumstances are defined where a uniform strength design is identical to the minimum weight design based on a specific deflection. Closed form solutions are obtained for a variety of loading scenarios. One problem is presented for self-weight that leads to a nonlinear integral equation.

The optimum stiffness-weight design of trusses is undertaken where the area distribution of the truss members is optimized using Lagrange’s method of undetermined multipliers. Once again, we obtain a degenerate case where upward, downward, and zero deflection conditions can be met with an infinite set of trusses of vanishing weight. We photograph a simply supported truss under a downward load that leads to an upward deflection at one of the joints. Special loading conditions are identified that lead to uniform stress designs that are identical to the minimum weight designs based on deflections. This study provides a Segway into the world of minimum weight strength design of trusses. The resulting Maxwell and Michell trusses sometimes display the optimum distribution of bar areas from the point of view of stiffness. Many practitioners are under the mistaken impression that Michell structures, when they exist, provide the optimum truss profile for stiffness. Unfortunately, the optimum array of truss joints based on deflection does not exist. For both trusses and beams the optimum distribution of mass is shown to be necessary and sufficient; the sufficiency is established using well-known inequalities.

The role of stiffness in the design of columns is explored in our final chapter. This cringe-worthy history of column analysis begins our study as a warning to practitioners who use analysis as their basis for design and especially optimum design. Conventional elastic and inelastic buckling theories provide little insight into the design of columns. The fundamentals of minimum weight column design are presented to show the power of design theory in contrast to analysis. Both prismatic and tapered columns are studied with one surprise result; the optimum taper gives rise to a uniform bending stress (without axial stresses). It was fun to see that in 1733 Lagrange made a mistake in calculus of variations that led to the incorrect solution for the optimum tapered column. It took 78 years before Clausen obtained the correct solution. The problem has been revisited by William Prager and again by the author who used dynamic programming. Of course, we all got the same result which is a dreadful solid circular tapered column that is heavier than any ordinary waterpipe. The best of a class is not necessarily the best possible design. Under the heading, “Intuition is a good servant but a bad master,” we introduce the notions of tension members that buckle, columns constructed from spherical beads, optimum rigging of crane booms, and deflection reversal of beam-columns. In several places we observe that the weight of optimum columns is proportional to P^α where P is the axial load and α is less than unity. We fail to tell the reader that this implies that minimum weight columns require putting all your eggs in one basket; one column under load P is lighter than two columns each under load P/2. On the other hand, we expose the solid circular column as the least efficient shape among all regular polygons, the equilateral triangle is the best. Indeed, there is a family of rectangles that are superior to the circular cross-section.

Finally, the author’s prestressed tubular column is introduced that is pressurized to eliminate local buckling. Euler’s buckling can always be eliminated with a thin-wall section of sufficient width without a weight penalty. The weight of the balloon-like member is proportional to (PL) which implies that at last we have a compressive member that meets the requirement of a Michell structure. Bundling of pressurized gas columns are possible without a weight penalty. Further, the column is insensitive to most imperfections. It is the lightest known column for small structural indices (P/L^2 ). When coupled with circulating cryogenic liquid as a prestressing system, a limiting column has a vanishing weight. Show less