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Introduction of some optimization and design problems into undergraduate solid mechanics

Introduction

In previous works 1, 2, 3 examples were given illustrating benefits of introducing modern software, such as MAPLE®I, into undergraduate mechanics courses. The emphasis there was mainly on dynamics. Here the same theme is explored focusing on strength of materials. Three problems are considered. (i) An example involving bending of a beam with rectangular cross- section. The task is to find the cross-section dimensions that minimize the weight and the maximum bending stress. This exposes students to some design issues (such as sizing) as well as to introductory notions of optimization; (ii) The second example involves an axially loaded, fixed-free rod, with a spring attached at some arbitrary location along its length. The problem is to find the spring location that minimizes certain deflections, further exposing students to some notions of optimization. (iii) Finally an example employing the use of “smart” materials to control beam deflections is treated. A particular smart material in which Young’s modulus varies along the beam length (functionally graded material, FGM 4) is considered, making students aware of the potential of using modern materials in design.

Previously these problems were not easily solvable. However with the current wide availability of optimization programs, such as, for example, those in MAPLE®, they can readily be addressed.

The problems treated are targeted at students in introductory courses in strength of materials. Typically the mathematics pre-requisites of these courses involve basic knowledge of differential equations and some exposure to math software. As a refresher, the finite difference example given in one of the references1 is covered in class. MAPLE® is then introduced via examples, i.e., worksheets are shown and discussed for problems solved in the board. Students learn basic operations with the software, and tutorials, as well as a great amount of examples, are made available via the courses / authors web site. In some instances Teaching Assistants are used in the courses for more challenging assignments.

Some of these examples are currently used by the authors in the classroom. The problems are discussed and then numerical solutions are shown and analyzed. In other instances, a problem is given as an assignment and a numerical solution utilizing MAPLE® is required. In the dynamics class at the University of Michigan computer projects are common. For instance, a problem involving friction effects on the velocity of a bead sliding on a circular vertical track is usually given as an assignment. This example was discussed in a previous paper3.

The authors found that the software and examples can be introduced into the courses without compromising time availability for teaching main course topics.

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Mazzei, A., & Scott, R. (2009, June), Introduction Of Some Optimization And Design Problems Into Undergraduate Solid Mechanics Paper presented at 2009 Annual Conference & Exposition, Austin, Texas. 10.18260/1-2--4505