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RReeaall LLiiffee Examples in MMeecchhaanniiccss ooff Solids Lesson plans and solutions Eann A Patterson, Editor Real Life Examples in Mechanics of Solids Lesson plans and solutions First edition produced September 2006 Second Edition, 2008 Third [Electronic] Edition, 2011 Real Life Examples in Mechanics of Solids Lesson plans and solutions Copyright © 2006, 2008, 2011 Eann A Patterson (editor) This work is licensed under the Creative Commons Attribution-NonCommercial- NoDerivs 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/3.0/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA. The Editor has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Bounded printed copies can be purchased on-line at www.engineeringexamples.org This edition is distributed free of charge by the ENGAGE project (www.EngageEngineering.org), which is supported by the National Science Foundation under Grant No. 0833076. Real Life Examples in Mechanics of Solids Lesson plans and solutions Suggested exemplars within lesson plans for Sophomore Solids Courses. Prepared as part of the NSF-supported project (#0431756) entitled: “Enhancing Diversity in the Undergraduate Mechanical Engineering Population through Curriculum Change” Eann A Patterson, Editor The University of Liverpool, England [email protected] This work was developed during the NSF-supported project (#0431756) entitled: “Enhancing Diversity in the Undergraduate Mechanical Engineering Population through Curriculum Change”. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The following were members of the project and contributed to the many discussions which led to this production of this volume: Ilene Busch-Vishniac Johns Hopkins University (Project leader) Suzanne Gage Brainard University of Washington Patricia B. Campbell Campbell-Kibler Associates Inc Constantin Chassapis Stevens Institute of Technology Glenn Ellis Smith College Ashley Emery University of Washington Darrell Guillaume California State University, Los Angeles Jeff Jarosz Johns Hopkins University Susan Staffin Metz Stevens Institute of Technology Eann Patterson Michigan State University Pradosh K. Ray Tuskegee University Helen Ryaciotaki-Boussalis California State University, Los Angeles Kuei-Wu Tsai California State University, Los Angeles Horace Whitworth Howard University Real Life Examples in Mechanics of Solids CONTENTS page no. Introduction and acknowledgements 5 ELEMENTARY STRESS SYSTEMS 1. Stress and strain in unixial solid and hollow bars 6 iPod, suspension bridge, femur, cello 2. Displacement plus deformation in control cables 9 Derailleur gears, yacht rudder 3. Stress in pressure vessel wall 11 Bike pump STATICALLY INDETERMINATE PROBLEMS 4. Combined use of principles of compatibility and equilibrium 13 iPod, dinosaur display TORSION 5. Stress and strain due to applied torque 17 Bottle closures STRAIN ENERGY 6. Conservation of Energy/Energy Methods 19 Slingshot, bungee jumper 7. Helical springs 21 Bicycle suspension, pogo stick BEAM BENDING 8. Bending moments and shear stress diagrams 24 Skateboarder, unicyclist on plank METHOD OF SUPERPOSITION 9. Eccentric loading 29 Basketball goal 10. Thermal stress and statically indeterminate loads 31 Rail tracks, jewellery pendant TWO-DIMENSIONAL STRESS SYSTEMS 11. Mohr’s circle of stress 34 Sausages 12. Combined bending and torsion 36 Wind-up clock, motor 3 Real Life Examples in Mechanics of Solids INTRODUCTION These real life examples and supporting material are designed to enhance the teaching of a sophomore course in mechanics of solids, increase the accessibility of the principles and raise the appeal of the subject to students from a diverse background. The examples have been embedded in skeletal lesson plans using the principle of the 5Es: Engage, Explore, Explain, Elaborate and Evaluate. Such lesson plans were developed by the Biological Sciences Curriculum Study1 in the 1980s from work by Atkin and Karplus2 in 1962. Today they are considered to form part of the constructivist learning theory and a number of websites provide easy to follow explanations of them3. This booklet is intended to be used by instructors and is written in a style that addresses the instructor, however this is not intended to exclude students who should find the examples interesting, stimulating and hopefully illuminating, particularly when their instructor is not utilizing them. In the interest of brevity and clarity of presentation, standard derivations and definitions are not included since these are readily available in textbooks which this booklet is not intended to replace but rather to supplement. Similarly, it is anticipated that these lessons plans can be used to generate lectures/lessons that supplement those covering the fundamentals of each topic. Acknowledgements Many of these examples have arisen through lively discussion in the consortium supported by the NSF grant (#0431756) on “Enhancing Diversity in the Undergraduate Mechanical Engineering Population through Curriculum Change” and the input of these colleagues is cheerfully acknowledged as is the support of NSF. The influence of the editor’s mentors and peers at the University of Sheffield is substantial and is gratefully acknowledged since many of the ideas for these examples originate from tutorial questions developed and used in the Department of Mechanical Engineering in Sheffield over many years. 1 http://www.bscs.org/library/BSCS_5E_Instructional_Approach_July_06.pdf 2 Atkin, J. M. and Karplus, R. (1962). Discovery of invention? Science Teacher 29(5): 45. 3 e.g. http://www.science.org.au/primaryconnections/constructivist.htm 5 Real Life Examples in Mechanics of Solids ELEMENTARY STRESS SYSTEMS 1. Principle: Stress and strain in uni-axial solid and hollow bars Engage: Take your iPod into class and dangle it by the earphone cable. Cut open the cable on an old set of earphones to expose cable and insulation. Explore: Pass around class lengths of copper wire and lengths of empty hollow insulation and invite students to stretch them. Discuss relative extensions and stiffness. Someone will probably snap one so talk about ultimate tensile stress. Be sure have to enough lengths that every student has at least one to play with while you are talking. Explain: Work through the example below: An iPod, with a mass of 30 grams is dangled from its earplug cable. a) Assuming that the copper wire of diameter 0.40mm inside the cable carries the entire load, evaluate the stress in the wire due to the weight of the iPod. b) If the wire in (a) is 1.50m long, by how much will it stretch? c) Assuming that the plastic insulation, which fits snugly over the wire and has an external diameter of 1mm, carries the entire load, evaluate the stress in the insulation due to the weight of iPoD. d) If the insulation is made from uPVC calculate the extension of the insulation in the circumstances described in (c). Solution: F mg 301039.81 a) Stress,    2.34×106N/m2 A d2 4 0.401032 4 where F, is force applied, A is cross-section area, m is mass, g is gravitational acceleration, and d is the diameter of the cross-section. 6