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On Calculating the Slope and Deflection of a Stepped and Tapered Shaft

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2014 ASEE Annual Conference & Exposition


Indianapolis, Indiana

Publication Date

June 15, 2014

Start Date

June 15, 2014

End Date

June 18, 2014



Conference Session

Teaching Mechanics of Materials

Tagged Division


Page Count


Page Numbers

24.946.1 - 24.946.13



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Paper Authors


Carla Egelhoff Montana Tech of the University of Montana

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Dr. Egelhoff teaches courses that include petroleum production engineering, oil property evaluation, and capstone senior design within the petroleum engineering program at Montana Tech of the University of Montana.

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Edwin M. Odom University of Idaho, Moscow

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Dr. Odom teaches courses that include introductory CAD, advanced CAD, mechanics of materials, machine design, experimental stress analysis, and manufacturing technical electives within the mechanical engineering program at the University of Idaho.

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Beam Deflection Calculated a New Way: an Example Using the Stepped and Tapered ShaftRegardless of specific discipline, most engineering students take a first course in the study ofMechanics of Materials wherein beam deflection is a challenging skill for students to learn. Forany beam with more than a simple load, deflection determination by any method (Integration,area-moment, superposition and singularity functions) can present mathematical difficulties,even if students can write correct governing and compatibility equations.Before computers were ubiquitous, engineers designed stepped or tapered shafts or beams usingelegant but time-consuming graphical techniques [3-4]. With the availability of scientificcalculators and mainframe computers, semi-graphical and computer-programmed techniquesappeared [5-7]. In today’s undergraduate Machine Design textbooks, we see few generalapproaches to the solution of deflection for stepped or tapered shafts; one approach is graphicaland other approaches use some form of discontinuity equations [8 -12].Recently a new and novel approach for the deflection calculation has emerged; the approachrequires merely undergraduate-level problem formulation skills to produce analytical results forgraduate-level beam geometry. Founded on the use of basic Statics, writing a beam’s momentequation, incorporating Castigliano’s Second Theorem and a Heaviside Step Function, thisapproach uses the student’s favorite equation solver. Even the novice can find deflectionvirtually “everywhere” along a beam; the results are obtained in about the same amount of timeas an ordinary (and easy) beam deflection problem is solved using paper and pencil.The positive aspects of thisprocess include (a) thegoverning equation writing issystematic and straight-forwardsince it relies primarily onknowledge of statics andmoment equations, (b) thegeneralized form allowsmaximum usefulness and (c)since students select theirpersonal equation-solvingsoftware, the coding proceedsquickly to obtain results.Overall, the process consists ofthe following steps: (1) apply a dummy load, Figure 1. Example geometry and loading. and solving for static support reactions, (2) write a moment equation in Macaulay form augmented with Heaviside step function variables, (3) take a partial derivative of the moment equation with respect to the dummy load, (4) re-write the moment equation to eliminate the dummy load and finally, (5) use the results of steps 3 and 4 to develop the deflection calculation via Castigliano's Theorem applied parametrically to create a deflection curve for the entire length of the beam.In this paper we present the method by way of example, an analysis of distributed vs point loadmodeling for supports and applied loads. Assessment of the new approach over several years inmultiple institutions has shown that virtually every student can determine deflection“everywhere” along a beam regardless of the complexity of loading or changing cross-section.Selected References3. Shroeder, Walter, “Beam Deflections,” Machine Design, p. 85-90, January 1947.4. Cowie, Alexander, “A tabular method for Calculating Deflections of Stepped and Tapered Shafts,” Machine Design, p. 111-118, August 9, 1956.5. Margolin, Lawrence L., “Calculating Deflections in Variable-Section Beams,” Machine Design, p.173-181, March 16, 1961.6. Halasz, Sandor T., “Minicalcs find Stepped-Shaft Deflections,” Machine Design, p. 78-82, June 27, 1974.7. Hopkins, Bruce R., Design Analysis of Shafts and Beams, McGraw-Hill Book Company, New York, NY, 1970.8. Collins, Jack A., Henry Busby and George Staab, Mechanical Design of Machine Elements and Machines, 2nd Edition, John Wiley & Sons, Hoboken, NJ, 2010.9. Spotts, Merhyle F., Terry E. Shoup and Lee E. Hornberger, Design of Machine Elements, 8th Edition, Pearson Prentice Hall, Upper Saddle River, NJ, 2003.10. Mott, Robert L., Machine Elements in Mechanical Design, 4th Edition, Pearson Prentice Hall, Upper Saddle River, NJ, 2004.11. Juvinall, Robert C. and Kurt M. Marshek, Fundamentals of Machine Component Design, 4th Edition, John Wiley and Sons, Inc, Hoboken, NJ, 2006.12. Budynas, Richard G. and J. Keith Nisbett, Shigley’s Mechanical Engineering Design, 8th Edition, McGraw-Hill, New York, NY, 2008.

Egelhoff, C., & Odom, E. M. (2014, June), On Calculating the Slope and Deflection of a Stepped and Tapered Shaft Paper presented at 2014 ASEE Annual Conference & Exposition, Indianapolis, Indiana. 10.18260/1-2--22879

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