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#### The is a preliminary materials selection for the “down tube” of a bicycle frame based on bending strength and torsional stiffness

###### Math

The is a preliminary materials selection for the “down tube” of a bicycle frame based on bending strength and torsional stiffness.  Determine the Merit Indices for the following cases:

Constraints for all cases:  Cross-section shape, length of the downtube

Free Variable:  Cross-section area

## Objective #1:  Minimize Mass

Merit Index #1a: Strength-limited bending, circular tube cross-section

Merit Index #1b: Strength-limited bending, elliptical tube cross-section

Merit Index #2a: Stiffness-limited torsion, circular tube cross-section

Merit Index #2b: Stiffness-limited torsion, elliptical tube cross-section

Merit Index #3:          Weighted combination of 60% bend strength and 40%         torsional stiffness for minimum mass

## Objective #2:  Minimize Cost

Merit Index #4a: Strength-limited bending, circular tube cross-section

Merit Index #4b: Strength-limited bending, elliptical tube cross-section

Merit Index #5a: Stiffness-limited torsion, circular tube cross-section

Merit Index #5b: Stiffness-limited torsion, elliptical tube cross-section

Merit Index #6:     Weighted combination of 60% bend strength and 40%

torsional stiffness for minimum cost

Note that each of the merit indices must include a shape factor because we want to directly compare materials being used as circular tubes and elliptical tubes.  Therefore, the indices above ending in “a” should include the circular tube shape factor raised to the appropriate exponent when combined with the material shape factor.  Likewise, the “b” indices should include the shape factor for an elliptical tube cross-section.

• Do not derive the Merit Indices!  Look up the appropriate indices in the Ashby Merit Index tables & write them down.
• I do not advise doing all the algebra required to combine all of the various quantities involved into a single shape factor formula for this assignment.
• Show how the shape factors are combined with the merit indices.  Again, you don’t need to do all the algebraic substitutions.
• Use the shape factor definitions from the notes.  Ashby reformulated the shape factors a couple of times before settling on the ones in the 3rd edition of his “Materials Selection” textbook.  The new shape factor formulations combine with the merit indices in a much more intuitive way, as discussed in class.
• No calculations are required for this assignment … just a collection of the appropriate formulas.  For example, the combined merit index and shape factor for the stiffness limited design of a shaft with specified wall thickness, loaded in torsion, optimized for minimum mass would be:

Ashby, “Materials Selection in Mechanical Design”, 3™ edition, 2005

Structures in Bending:

Strength Limited: oh =6 aa

Suffness Limited: PR = 12 ma

Structures in Torsion:

Strength Limited: ol. = 48 =

A

Suffness Limited: Or = 7.14=

A

Columns in Compression:

Elastic Buckling: Qs = 12

where:

A = Area, units = [L’]

I = Second Moment of Area (aka Area Moment of Inertia), units = [L*]

Tax = Maximum of I,, and I,

Tain = Minimum of I,, and I,,

Z = Section Modulus = I/y,,4,, units = [L?]

K = Torsional Moment of Area, units = [L*]

(= J, polar moment of area, for circular sections)

Q = Torsional Section Modulus ( = J/tuq, for circular sections), units = [L*]