College of Engineering

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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
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Outline

• Theory
• Analysis
• Verification
• Design
• Examples
• Conclusion

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Theory

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

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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
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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!

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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.

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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.

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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

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Example 1 Statically DeterminateAxially Loaded
Bar

• 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

x

• PC 6.0 kN
• L2 0.635 m
• d2 30 mm
• E2 69 GPa Aluminum

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1. Model

• Problem Defined Figure Labeled Symbolically
• Identify Loading Model
• Axial, Torsion and/or Transverse
• State Assumptions
• Define Coordinate Set

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2. Free-Body Diagrams

• Complete and/or Parts of Structure
• Symbolic Variables Even Knowns!

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3. Equilibrium Equations

• Symbolic Equations
• Check Dimensional Homogeneity
• Do Not Isolate Unknowns
• Reduces Algebraic Error!

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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!
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4. Compatibility and Boundary Conditions
Continued

• Compatibility
• Displacement at Identical Points of Segment Equal
• Boundary Condition
• uA 0 for Rigid Support

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5. Material Law Formulas

• Symbolic Equations
• Do Not Isolate Unknowns Reduces Error!
• Check Dimensional Homogeneity

A, E Constant
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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.

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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

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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

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8. Verification Continued

• Uniform, Homogenous w/ PB 0
• Uniform, Homogenous w/ PC 0
• E1 ? 8 Yields
• uB 0
• E2 ? 8 Yields uB uC
• E1 ? 8 and E2 ? 8 Yields uB uC
  0
• PB - PC Yields uB 0

y
x
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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
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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

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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

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What do you think?
Joe Rencis Department of Mechanical
Engineering University of Arkansas V-mail
479-575-3153 FAX [phone] E-mail
jjrencis_at_uark.edu