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

2006 ASEE NE Section Conference

•
•
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
2006 ASEE NE Section Conference

•
•
Structured Problem Solving Format
1.
2.
3.
Model
Free-Body Diagrams
Equilibrium Equations
Blue Steps for Statics
4.
5.
6.
7.
8.
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.
2006 ASEE NE Section Conference

7. Solve a) b)
–
Traditional

F x w/ Values and/or

0 ; R
A
R
A


P
B
P
C


P
C
P
B
Unknown

Knowns
–
–
Symbolic
Ours
R
A

P
B

P
C
Do Not Isolate Known and Unknown Variables
–
–
No Algebraic Manipulation – Reduces Errors!
Engineering Tool – Student Choice c)
No Textbook Does This!
2006 ASEE NE Section Conference

•
Question and Test to Verify the “Answers”
• w(0) Suggested Questions
–
–
–
A A Hand Calculation?
Comparison w/ a Known Problem Solution?
x
Examination of Limiting Cases w/ Known Solutions?
–
–
–
Examination of Obvious Known Solutions?
Your Best Judgment?
Comparison w/ Experimentation? – Not done in course.
2006 ASEE NE Section Conference
L w(L)
B
X

•
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.
2006 ASEE NE Section Conference

• 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
2006 ASEE NE Section Conference

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

• Problem Defined & Figure Labeled
Symbolically
• Identify Loading Model
– Axial, Torsion and/or Transverse
• State Assumptions
• Define Coordinate Set y d
1
L
1
L
2 d
2
A
(1) (2)
B
P
B
2006 ASEE NE Section Conference
C
P
C
X

• Complete and/or Parts of Structure
Assumed Deformation y u
B u
C
L
1
L
2
(b)
(c)
R
A
(d)
(1) (2)
B
P
B
C
P
C
A
A
(1)
FBD I
B
F
(1)
B
F B
P
B
(2)
FBD II
B
C
Very Thin IMAGINARY slice shown for clarity of solution only.
P
C
2006 ASEE NE Section Conference x

• Symbolic Equations
• Check Dimensional Homogeneity
• Do Not Isolate Unknowns
– Reduces Algebraic Error!
y
L
1
(b)
FBD
FBD
I :
II :
( 1 )
F
F
B
( 1 )
B


R
A
P
B

P
C
(c)
( 1 )
( 2 )
R
A
A
A
Assumed Deformation u
B
L
2 u
C x
(1) (2)
C
P
C
B
(1)
FBD I
B
F
(1)
B
(1)
F
B
F
(1)
B
(2)
FBD II
P
C
B
C
Very Thin IMAGINARY slice shown for clarity of solution only.
x
2006 ASEE NE Section Conference

• Symbolic Equations
• Do Not Isolate Unknowns
– Reduces Algebraic Error!
• Done for Statically
– Determinate (Not Common) and
– Indeterminate Problems
Treat Both
Problems the
Same Way!
• Done for Both Problems in Textbooks by
– Craig
– Crandall, Dahl, Lardner
– Shames
2006 ASEE NE Section Conference

• Compatibility
– Displacement at Identical Points of Segment Equal
• Boundary Condition
– u
A
= 0 for Rigid Support
Assumed Deformation
(a) x y u
B u
C
L
1
L
2
(b)
(1) (2)
B
P
B
C
P
C
A
(c)
R
2006 ASEE NE Section Conference
A
FBD I
B
(d)
F
(1)
B
B
P
B
(2)
FBD II C
Very Thin IMAGINARY slice shown for clarity of solution only.
P
C x

• Symbolic Equations
• Do Not Isolate Unknowns – Reduces Error!
(a)
(b)
• Check Dimensional Homogeneity x y
L u a u(x) u b y F a u
B u
C F b
x
L
1 a L
2 b
A, E Constant
A
(1)
B
P
B u b
 u a
C

F b
L
P
AE x
R
A
(c) (1)
F
(1)
B
(d)
A FBD I
F B
B
P
B
(2)
FBD II
P
C
B
C
Very Thin IMAGINARY slice shown for clarity of solution only.
Segment
Segment
( 1 ) :
( 2 ) : u
B u
C
 u
A

F
B
( 1 )
L
1
 u
B

A
1
E
1
P
C
L
2
A
2
E
2
2006 ASEE NE Section Conference
( 3 )
( 4 )

• Complementary Formulas
– Stress, Strain, Stiffness, etc.
• Supporting Formulas
– Cross-sectional Area
– Polar Moment of Inertia
– Centroid Location
– Moment of Inertia, etc.
A
1

 d
1
2
4
( i ) A
2

 d
2
2
4
2006 ASEE NE Section Conference
( ii )

• # Independent Equations = 4
• # Unknowns = 4
– R
A
F
B
( 1 )
B
• Solution by and u
C
FBD
FBD
Segment
Segment
I :
II :
( 1 ) :
( 2 ) :
( 1 )
F
F
B
( 1 )
B


R
A
P
B

P
C u
B
 u
A

F
B
( 1 )
L
1 u
C
 u
B

A
1
E
1
P
C
L
2
A
2
E
2
– Hand – Requires Algebraic Manipulation
• Coupled Equations – Indeterminate
• Nonlinear Equations
– Engineering Tool
• ABET Criteria 3k
• Not Found in Textbooks
2006 ASEE NE Section Conference
( 1 )
( 2 )
( 3 )
( 4 )

• 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
2006 ASEE NE Section Conference

• Uniform, Homogenous w/ P
B u
C

P
C
( L
1

L
2
) AE
• Uniform, Homogenous w/ P
C u
C
 u
B

P
B
L
1
A
1
E
1
= 0
= 0 y
• E
1

– u
B
∞ Yields
= 0
• E
–
2 u
C

P
C
( L
2
) A
2
E
2
 ∞ Yields u
B
= u
C
= ( P
B
A

P
C
)( L
1
)
• E
1 u
B
• P
B
 ∞ and E
2
= u
C
= 0
= - P
C

Yields u
B
∞ Yields
= 0 & u
C

P
C
( L
2
)
L
1
(1)
A
1
E
1
A
2
E
2
2006 ASEE NE Section Conference
L
2
B
P
B
(2)
C
P
C

Example 2: Statically Indeterminate
Axially Loaded Bar
• All Equations the Same as Example 1
• Determinate Problem – Example 1
– P
C
– u
C
= Known
= Unknown
L
1
L
2
(1)
B
P
B
(2)
• Indeterminate Problem – Example 2
– P
C
– u
C
= Unknown
= Known = 0
– Only Requires Changing
Known and Unknown
A y
L
1
A
(1)
P
B
B
L
2
(2)
C
C
P
C x
2006 ASEE NE Section Conference

y
• Find d
2
-20 μm to limit u
B to
• 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
L
1
L
2
A
(1)
P
B
B
(2) d
2
=?
C
Commonly Found in
Textbooks
• Coupled
• Non-linear Solution
• No Intermediate Analyses x
2006 ASEE NE Section Conference

•
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
2006 ASEE NE Section Conference

Joe Rencis
Department of Mechanical Engineering
University of Arkansas
V-mail: 479-575-3153
FAX: 479-575-6982
E-mail: jjrencis@uark.edu
2006 ASEE NE Section Conference