mechanical engineering session march 18, 2006 2006 new england section american society of...
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![Page 1: Mechanical Engineering Session March 18, 2006 2006 New England Section American Society of Engineering Education Conference A New Approach to Mechanics](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d475503460f94a22bc0/html5/thumbnails/1.jpg)
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 DesignHartley T. Grandin, Jr.
Worcester Polytechnic InstituteJoseph J. Rencis
University of Arkansas
![Page 2: Mechanical Engineering Session March 18, 2006 2006 New England Section American Society of Engineering Education Conference A New Approach to Mechanics](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d475503460f94a22bc0/html5/thumbnails/2.jpg)
2006 ASEE NE Section Conference
Outline
1. Theory
2. Analysis
3. Verification
4. Design
5. Examples
6. Conclusion
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2006 ASEE NE Section Conference
Theory
• Typical of a One Semester Course• Topics
1. Planar Equilibrium Analysis of a Rigid Body2. Stress3. Strain4. Material Properties and Hooke’s Law5. Centric Axial Tension and Compression6. Torsion7. Bending8. Combined Analysis9. Static Failure Theories10. Columns
• Commonly Found in Textbooks
![Page 4: Mechanical Engineering Session March 18, 2006 2006 New England Section American Society of Engineering Education Conference A New Approach to Mechanics](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d475503460f94a22bc0/html5/thumbnails/4.jpg)
2006 ASEE NE Section Conference
Analysis
• Structured Problem Solving Format1. 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
![Page 5: Mechanical Engineering Session March 18, 2006 2006 New England Section American Society of Engineering Education Conference A New Approach to Mechanics](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d475503460f94a22bc0/html5/thumbnails/5.jpg)
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!
CBAx PPRF =+=∑ ;0
KnownsUnknown
PPR BCA
=−=
CBA PPR =+
![Page 6: Mechanical Engineering Session March 18, 2006 2006 New England Section American Society of Engineering Education Conference A New Approach to Mechanics](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d475503460f94a22bc0/html5/thumbnails/6.jpg)
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.
X
w(0)
w(L)
x
AB
L
+ =
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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.
![Page 8: Mechanical Engineering Session March 18, 2006 2006 New England Section American Society of Engineering Education Conference A New Approach to Mechanics](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d475503460f94a22bc0/html5/thumbnails/8.jpg)
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
![Page 9: Mechanical Engineering Session March 18, 2006 2006 New England Section American Society of Engineering Education Conference A New Approach to Mechanics](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d475503460f94a22bc0/html5/thumbnails/9.jpg)
2006 ASEE NE Section Conference
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
• PC = 6.0 kN• L2 = 0.635 m• d2 = 30 mm• E2 = 69 GPa: Aluminum
X
L L1 2
P P
A B CB C(1) (2)
y
d2d1
x
![Page 10: Mechanical Engineering Session March 18, 2006 2006 New England Section American Society of Engineering Education Conference A New Approach to Mechanics](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d475503460f94a22bc0/html5/thumbnails/10.jpg)
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
X
L L1 2
P P
A B CB C(1) (2)
y
d2d1
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2006 ASEE NE Section Conference
2. Free-Body Diagrams
• Complete and/or Parts of Structure• Symbolic Variables – Even Knowns!
R
F
L L1 2
uu
P P
P
A B C
BC
A
B C
C
PB
A BFBD I
B CFBD II
Very Thin IMAGINARY sliceshown for clarity of solution only.
(1) (2)
FB(1)
(2)
Assumed Deformation
(a)
(b)
(c)
(d)
x
x
y
B
(1)
(1)
FB
(1)
![Page 12: Mechanical Engineering Session March 18, 2006 2006 New England Section American Society of Engineering Education Conference A New Approach to Mechanics](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d475503460f94a22bc0/html5/thumbnails/12.jpg)
2006 ASEE NE Section Conference
3. Equilibrium Equations
• Symbolic Equations• Check Dimensional Homogeneity• Do Not Isolate Unknowns
– Reduces Algebraic Error!
)2(:
)1(:)1(
)1(
CBB
AB
PPFIIFBD
RFIFBD
+==
R
F
L L1 2
uu
P P
P
A B C
BC
A
B C
C
PB
A BFBD I
B CFBD II
Very Thin IMAGINARY sliceshown for clarity of solution only.
(1) (2)
FB(1)
(2)
Assumed Deformation
(a)
(b)
(c)
(d)
x
x
y
B
(1)
(1)
FB
(1)
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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!
![Page 14: Mechanical Engineering Session March 18, 2006 2006 New England Section American Society of Engineering Education Conference A New Approach to Mechanics](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d475503460f94a22bc0/html5/thumbnails/14.jpg)
2006 ASEE NE Section Conference
4. Compatibility and Boundary Conditions ‘Continued’
• Compatibility– Displacement at Identical Points of Segment Equal
• Boundary Condition– uA = 0 for Rigid Support
R
F
L L1 2
uu
P P
P
A B C
BC
A
B C
C
PB
A BFBD I
B CFBD II
Very Thin IMAGINARY sliceshown for clarity of solution only.
(1) (2)
FB(1)
(2)
Assumed Deformation
(a)
(b)
(c)
(d)
x
x
y
B
(1)
(1)
![Page 15: Mechanical Engineering Session March 18, 2006 2006 New England Section American Society of Engineering Education Conference A New Approach to Mechanics](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d475503460f94a22bc0/html5/thumbnails/15.jpg)
2006 ASEE NE Section Conference
5. Material Law Formulas• Symbolic Equations• Do Not Isolate Unknowns – Reduces Error!• Check Dimensional Homogeneity
F
u
y
uL
F x
u(x)
ba
a
b
a b
A, E Constant
AE
LFuu b
ab +=
)3(:)1(11
1)1(
EA
LFuuSegment B
AB +=
)4(:)2(22
2
EA
LPuuSegment C
BC +=
R
F
L L1 2
uu
P P
P
A B C
BC
A
B C
C
PB
A BFBD I
B CFBD II
Very Thin IMAGINARY sliceshown for clarity of solution only.
(1) (2)
FB(1)
(2)
Assumed Deformation
(a)
(b)
(c)
(d)
x
x
y
B
(1)
(1)F
B
(1)
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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.
)(4
21
1 id
Aπ
= )(4
22
2 iid
Aπ
=
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2006 ASEE NE Section Conference
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
)1(BF
)2(:
)1(:)1(
)1(
CBB
AB
PPFIIFBD
RFIFBD
+==
)3(:)1(11
1)1(
EA
LFuuSegment B
AB +=
)4(:)2(22
2
EA
LPuuSegment C
BC +=
![Page 18: Mechanical Engineering Session March 18, 2006 2006 New England Section American Society of Engineering Education Conference A New Approach to Mechanics](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d475503460f94a22bc0/html5/thumbnails/18.jpg)
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
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2006 ASEE NE Section Conference
8. Verification ‘Continued’
X
L L1 2
P P
A B CB C(1) (2)
• 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 &
AELLPu CC )( 21 +=
111 EALPuu BBC ==
222 )( EALPu CC =
111))(( EALPP CB +
x
y
222 )( EALPu CC =
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2006 ASEE NE Section Conference
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
L L1 2
A B C
y
xPB(2)(1)
X
L L1 2
P P
A B CB C(1) (2) x
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2006 ASEE NE Section Conference
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
L L1 2
A B C
y
xPB(2)(1)
Commonly Found in Textbooks
• Coupled • Non-linear Solution
• No Intermediate Analyses
d2=?
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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
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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: [email protected]