Teaching deflection of stepped shafts: Castigliano's theorem, dummy loads, heaviside step functions and numerical integration

The need for finding the deflections of shafts, many of which are stepped or varying cross-sectional areas is timeless. Each generation of engineers has used that part of mechanics of materials theory that fit the calculating capability available to them. The method presented here is offered in that vein. The method uses an engineer's ability to construct free body diagrams, derive moment equations, and knowledge of energy methods. The problem solution is kept general until the last step which is a digital numerical integration. The digital numerical integration can be performed on a wide variety of software to include TKSolver™, MatLab
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, MathCad
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, EES
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and spread sheets. This method keeps the section properties independent of the moment equations making it straightforward to include scaling and shape factors on the cross-sectional dimensions. This allows an engineer to run any number of "what if" scenarios during a design process. Additionally, this method provides intermediate opportunities to validate the solution path by a) plotting the moment equation and comparing it against shear and moment diagram developed by hand, or b) plotting the cross section and comparing it against the drawings. Thus far, this approach to solving for the deflection of stepped shafts has been presented to nearly 300 junior Mechanical Engineering students.