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

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

2006 ASEE NE Section Conference

Outline

1. Theory

2. Analysis

3. Verification

4. Design

5. Examples

6. Conclusion


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


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


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 =+


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

+ =


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.


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


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


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


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)


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


(1) (2)

FB(1)

(2)

Assumed Deformation

(a)

(b)

(c)

(d)

x

x

y

B

(1)

(1)

FB

(1)


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!


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


(1) (2)

FB(1)

(2)

Assumed Deformation

(a)

(b)

(c)

(d)

x

x

y

B

(1)

(1)


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


(1) (2)

FB(1)

(2)

Assumed Deformation

(a)

(b)

(c)

(d)

x

x

y

B

(1)

(1)F

B

(1)


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

=


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 +=


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


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 =


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


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=?


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


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]

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

Documents

Transcript of Mechanical Engineering Session March 18, 2006 2006 New England Section American Society of...