Mechanical Engineering Session

1. A New Approach to Mechanics of Materials: An Introductory Course with Integration of Theory, Analysis, Verification and Design Hartley T. Grandin, Jr. Worcester Polytechnic Institute Joseph J. Rencis University of Arkansas Mechanical Engineering Session 2006 New England Section American Society of Engineering Education Conference March 18, 2006

2. Outline • Theory • Analysis • Verification • Design • Examples • Conclusion

3. Theory • Typical of a One Semester Course • Topics • Planar Equilibrium Analysis of a Rigid Body • Stress • Strain • Material Properties and Hooke’s Law • Centric Axial Tension and Compression • Torsion • Bending • Combined Analysis • Static Failure Theories • Columns • Commonly Found in Textbooks

4. Analysis • Structured Problem Solving Format • Model • Free-Body Diagrams • Equilibrium Equations • Material Law Formulas • Compatibility and Boundary Conditions • Complementary and Supporting Formulas • Solve • Verification • Textbooks • Headings to Solve Problem Commonly Used • Craig – Closest to us! But does not use structured format. Blue Steps for Statics

5. Analysis ‘Continued’ • Solve • Traditional • w/ Values and/or • Symbolic • Ours • Do Not Isolate Known and Unknown Variables • No Algebraic Manipulation – Reduces Errors! • Engineering Tool – Student Choice • No Textbook Does This!

6. + = Verification • Question and Test to Verify the “Answers” • Suggested Questions • A Hand Calculation? • Comparison w/ a Known Problem Solution? • Examination of Limiting Cases w/ Known Solutions? • Examination of Obvious Known Solutions? • Your Best Judgment? • Comparison w/ Experimentation? – Not done in course.

7. Verification ‘Continued’ • Important Educational Elements • Reflex Suspicion of Program Results • Check Results with Alternative Methods • Expected of Professionals • Expect Student to be Professional • Textbook by Craig • Intuitive Discussion for One Solution • No Numerical Testing • We Do Both Since We Use Engineering Tools! Allows for Multiple Calculations Easily.

8. Design • Design is Where you Search for Optimum Solution • Interchanging Role of Known & Unknown Variables • ABET Criteria 3c & Criteria 4 (now in 3c) • Textbooks – Homework & Computer • Traditional • Typically Single Solution for a Single Set of Specific Requirements • Ours • Multiple Solution for Any Set of Requirements • Easily Change Known & Unknown Variables

9. Determine the displacement at B and C. Solve using the given specifications: PB = - 18.0 kN L1 = 0.508 m d1 = 40 mm E1 = 207 GPa: Steel PC = 6.0 kN L2 = 0.635 m d2 = 30 mm E2 = 69 GPa: Aluminum y d1 d2 Example 1: Statically DeterminateAxially Loaded Bar x

10. y d1 d2 1. Model • Problem Defined & Figure Labeled Symbolically • Identify Loading Model • Axial, Torsion and/or Transverse • State Assumptions • Define Coordinate Set

11. (1) FB 2. Free-Body Diagrams • Complete and/or Parts of Structure • Symbolic Variables – Even Knowns!

12. (1) FB 3. Equilibrium Equations • Symbolic Equations • Check Dimensional Homogeneity • Do Not Isolate Unknowns • Reduces Algebraic Error!

13. 4. Compatibility and Boundary Conditions • Symbolic Equations • Do Not Isolate Unknowns • Reduces Algebraic Error! • Done for Statically • Determinate (Not Common) and • Indeterminate Problems • Done for Both Problems in Textbooks by • Craig • Crandall, Dahl, Lardner • Shames Treat Both Problems the Same Way!

14. 4. Compatibility and Boundary Conditions ‘Continued’ • Compatibility • Displacement at Identical Points of Segment Equal • Boundary Condition • uA = 0 for Rigid Support

15. (1) FB 5. Material Law Formulas • Symbolic Equations • Do Not Isolate Unknowns – Reduces Error! • Check Dimensional Homogeneity A, E Constant

16. 6. Complementary and Supporting Formulas • Complementary Formulas • Stress, Strain, Stiffness, etc. • Supporting Formulas • Cross-sectional Area • Polar Moment of Inertia • Centroid Location • Moment of Inertia, etc.

17. 7. Solve • # Independent Equations = 4 • # Unknowns = 4 • RA, , uB and uC • Solution by • Hand – Requires Algebraic Manipulation • Coupled Equations – Indeterminate • Nonlinear Equations • Engineering Tool • ABET Criteria 3k • Not Found in Textbooks

18. 8. Verification • Comments • May not Yield Absolute Proof • Does Improve the Level of Confidence • Step 7. Solves Problem Once • Step 8. Solves Problem Multiple Times • Need Engineering Tool! • Compare to • Hand Solution • Similar Problems in other Texts

19. 8. Verification ‘Continued’ • Uniform, Homogenous w/ PB = 0 • Uniform, Homogenous w/ PC = 0 • E1 ∞ Yields • uB = 0 • E2 ∞ Yields uB = uC= • E1 ∞ and E2 ∞ Yields uB = uC = 0 • PB = - PC Yields uB = 0 & y x

20. Example 2: Statically IndeterminateAxially Loaded Bar • All Equations the Same as Example 1 • Determinate Problem – Example 1 • PC = Known • uC = Unknown • Indeterminate Problem – Example 2 • PC = Unknown • uC = Known = 0 • Only Requires Changing Known and Unknown x

21. Example 3: Design Application of Example 2 • Find d2 to limit uB to -20 μm • Solution Alternative 1 • Iterate Input d2 • Solve uB • Solution Alternative 2 • Plot d2 versus uB • Solution Alternative 3 • uB = - 20 μm (Known) • d2 = Unknown d2=? Commonly Found in Textbooks • Coupled • Non-linear Solution • No Intermediate Analyses

22. Conclusion • Integrated Approach • Theory • Analysis • Structured Problem Solving Format • Symbolic Equations • Solution by Engineering Tool • Verification • Hand Solution • Known Solution • Limiting Cases • Design • Change Known and Unknown Variables

23. What do you think? Joe Rencis Department of Mechanical Engineering University of Arkansas V-mail: 479-575-3153 FAX: 479-575-6982 E-mail: jjrencis@uark.edu