focus on stress when stiffness is the problem
It has been my experience that one should not be stressed out over stresses, but instead should be more mindful of displacements and deformations. I came to this conclusion after visiting over 100 design firms as an application engineer for a company that sold FEA software. (ERA note: Finite Element Analysis is a computer method of calculating stress).
My salary was proportional to the amount of software sold, and I did not receive bonuses for helping customers solve problems using methods other than FEA. Consequently, I did not dissuade customers from using FEA when other methods would have worked better or been less costly.
On one occasion I was at a customer's site pitching my FEA program to the design staff when the manager showed up and declared, "If this program can tell me which of these new deck designs will have the least deflection, III buy your software." I accepted the challenge and used FEA to model four deck configurations. The analysis took the better part of a week, and the presentation lasted 30 min. The manager said our program correctly predicted the best configuration and bought a copy of the software. Keeping my salary in mind, I neglected to tell him that FEA was unnecessary for this job, or that his CAD package would have given the same answer in about 30 see. All he needed was good engineering discipline and a list of the moments of inertia for each deck.
The problem the manager was solving was a stiffness problem, not a stress problem. My experience in structures has proven this to be true for most problems facing engineers. Designs are rarely pushed to the limit of a material's ultimate strength. (This is where aerospace engineers disagree.) Instead, the driving requirement is that designs are stiff enough to prevent excessive flexing under load.
For structures analysts, the two most important properties relating to stiffness are Young's modulus and moment of inertia, also known as material stiffness and geometric stiffness, respectively. Material stiffness measures the deflection of a sample by a given load per unit area. For example, because steel is stiffer than rubber, a rubber object deflects more than similarly shaped steel objects supporting the same load. Although one could design a rubber staircase without exceeding the rubber's ultimate strength, walking on the staircase would be like running through a funhouse. The design would be acceptable from a stress and fatigue standpoint but unacceptable from a stiffness perspective.
Geometric stiffness comes into play because different geometries deflect differently under similar loads. A hollow cylindrical steel pipe deflects significantly less than a round solid steel pipe of the same area carrying the same load. This is because the moment of inertia, or geometric stiffness, is higher for a hollow cylinder than for a solid cylinder of the same area. Recall the basic bending equation from strength of materials. The maximum stress on a cantilever beam is calculated with the following equation:
where M = moment, lb-in.; c = distance from the neutral axis to the extreme fiber, in.; and I = moment of inertia, in.'
The radius of curvature in bending the same beam is
where E = Young's modulus. These two equations can be combined:
Timoshenko refers to the product of IE as flexural rigidity. The concept of flexural rigidity allows us to examine a structure's rigidity and see that it depends on both material and geometric stiffnesses. These values can be obtained without FEA or without "imposing" any load whatsoever. Flexural rigidity is a static property.
The solution to the manager's question about the stiffest design could be answered by looking at the geometric stiffness of each design. Most 3D CAD packages have a utility to calculate this value. Basically, the stiffer the section, the lower the deflection.
I still encounter problems in which flexural rigidity should be examined, but designers are preoccupied by stresses. And there is a story in this regard frequently told. It may or may not be true, but the lesson is valid nevertheless. It concerns a motorcycle company that won two consecutive world championship races. It then spent untold amounts of money to build a cutting-edge titanium frame and shoot for its third straight championship. A high strength-to-weight ratio was the mantra of the designers, and the frame was to be capable of handling a much higher horsepower-to-weight ratio.
Engineers performed stress calculations at each joint, but when the motorcycle was built and raced, the bike could not complete one lap under full power. Designers had built the frame to the same dimensions as the previous steel frame, but nobody looked at the differences in the material modulus. Titanium has a Young's modulus about half that of steel. The titanium frame was significantly lighter, and the ultimate strength was as high as that of the steel frame, so there was no concern about failure under load. The problem was that the motorcycle felt springy, like "trying to ride and drive a noodle."
The designers should have increased the diameter of the frame tubing to increase the moments of inertia and compensate for the lower material stiffness. If you stop by a bike shop, you'll notice that aluminum-frame bikes have larger diameter tubing than steel-frame bikes. This compensates for the lower Young's modulus of the aluminum.
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