1. Mechanical Engineering Session March 18, 2006 2006 New England Section American Society of Engineering Education Conference 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

2. 2006 ASEE NE Section Conference Outline 1.Theory 2.Analysis 3.Verification 4.Design 5.Examples 6.Conclusion

3. 2006 ASEE NE Section Conference Theory Typical of a One Semester Course Topics 1.Planar Equilibrium Analysis of a Rigid Body 2.Stress 3.Strain 4.Material Properties and Hooke’s Law 5.Centric Axial Tension and Compression 6.Torsion 7.Bending 8.Combined Analysis 9.Static Failure Theories 10.Columns Commonly Found in Textbooks

4. 2006 ASEE NE Section Conference Analysis Structured Problem Solving Format 1. Model 2. Free-Body Diagrams 3. Equilibrium Equations 4. Material Law Formulas 5. Compatibility and Boundary Conditions 6. Complementary and Supporting Formulas 7. Solve 8. Verification Textbooks Headings to Solve Problem Commonly Used Craig – Closest to us! But does not use structured format. Blue Steps for Statics

5. 2006 ASEE NE Section Conference Analysis ‘Continued’ 7.Solve a) Traditional – w/ Values and/or – Symbolic b) Ours – Do Not Isolate Known and Unknown Variables – No Algebraic Manipulation – Reduces Errors! – Engineering Tool – Student Choice c) No Textbook Does This!

6. 2006 ASEE NE Section Conference 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. 2006 ASEE NE Section Conference 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. 2006 ASEE NE Section Conference 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. 2006 ASEE NE Section Conference Example 1: Statically Determinate Axially Loaded Bar Determine the displacement at B and C. Solve using the given specifications: P B = - 18.0 kN L 1 = 0.508 m d 1 = 40 mm E 1 = 207 GPa: Steel P C = 6.0 kN L 2 = 0.635 m d 2 = 30 mm E 2 = 69 GPa: Aluminum y d2d2 d1d1 x

10. 2006 ASEE NE Section Conference 1. Model Problem Defined & Figure Labeled Symbolically Identify Loading Model –Axial, Torsion and/or Transverse State Assumptions Define Coordinate Set y d2d2 d1d1

11. 2006 ASEE NE Section Conference 2. Free-Body Diagrams Complete and/or Parts of Structure Symbolic Variables – Even Knowns! FBFB (1)

12. 2006 ASEE NE Section Conference 3. Equilibrium Equations Symbolic Equations Check Dimensional Homogeneity Do Not Isolate Unknowns –Reduces Algebraic Error! FBFB (1)

13. 2006 ASEE NE Section Conference 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. 2006 ASEE NE Section Conference 4. Compatibility and Boundary Conditions ‘Continued’ Compatibility –Displacement at Identical Points of Segment Equal Boundary Condition –u A = 0 for Rigid Support

15. 2006 ASEE NE Section Conference 5. Material Law Formulas Symbolic Equations Do Not Isolate Unknowns – Reduces Error! Check Dimensional Homogeneity A, E Constant FBFB (1)

16. 2006 ASEE NE Section Conference 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. 2006 ASEE NE Section Conference 7. Solve # Independent Equations = 4 # Unknowns = 4 –R A,, u B and u C Solution by –Hand – Requires Algebraic Manipulation Coupled Equations – Indeterminate Nonlinear Equations –Engineering Tool ABET Criteria 3k Not Found in Textbooks

18. 2006 ASEE NE Section Conference 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. 2006 ASEE NE Section Conference 8. Verification ‘Continued’ Uniform, Homogenous w/ P B = 0 Uniform, Homogenous w/ P C = 0 E 1  ∞ Yields –u B = 0 – E 2  ∞ Yields u B = u C = E 1  ∞ and E 2  ∞ Yields u B = u C = 0 P B = - P C Yields u B = 0 & x y

20. 2006 ASEE NE Section Conference Example 2: Statically Indeterminate Axially Loaded Bar All Equations the Same as Example 1 Determinate Problem – Example 1 –P C = Known –u C = Unknown Indeterminate Problem – Example 2 –P C = Unknown –u C = Known = 0 –Only Requires Changing Known and Unknown x

21. 2006 ASEE NE Section Conference Example 3: Design Application of Example 2 Find d 2 to limit u B to -20 μm Solution Alternative 1 –Iterate Input d 2 –Solve u B Solution Alternative 2 –Plot d 2 versus u B Solution Alternative 3 –u B = - 20 μm (Known) –d 2 = Unknown Commonly Found in Textbooks Coupled Non-linear Solution No Intermediate Analyses d 2 =?

22. 2006 ASEE NE Section Conference 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. 2006 ASEE NE Section Conference 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