Lecture Problem 20 D5

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Lecture Problem 20 D5

• David Schimon
• Friday, March 23 2007
• E106

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Problem Statement (overview)

• A client needs you to build a structure to
  support a uniform distributed load in the manner
  shown below. Determine what material ought to be
  used.

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E4 Once More Functions, Objectives, and
Constraints

• Functions Support Something, whatever the
  client wants.
• Constraints Length, Width, Force, Deflection.
• Thickness is left as a free variable for
  optimization
• Objectives Cost, mass

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Material Selection Criteria

• Different qualities can be optimized using
  different relationships between the mass of the
  object and its material properties
• Deflection equation for stiffness optimization
• Stress equation for strength optimization
• Use the free variable (thickness) to relate mass
  and secondary equations

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Relevant Formulae

• Deflection at L/2
• Tensile stress at underside at L/2
• Mass equation of slab

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Stiffness Index 1

• Solve out t
• Insert into mass relationship

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Stiffness Index 2

• Regroup
• Only material properties are open to variation.
  Dub this term the Performance Index

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Strength Index 1

• Solve out t
• Insert into mass relationship

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Strength Index 2

• Regroup
• Strength Performance Index

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Answer to Part A

• Strength PI
• Stiffness PI

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Part B

• Task Select Metal alloys with Stiffness PIs
  greater than 1.5.
• Two methods
• Using the book and a calculator
• Using CES Selector with axes
• X density
• Y Youngs modulus

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Setting Up to use CES Selector

• Plots produced on a logscale, so work equation
  into log-log form
• y-intercept 3.375
• slope 3

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Part B Results

• Magnesium alloys and steel alloys both above
  curve, and so are acceptable.
• The further above the curve, the better.
• Less dense, higher modulus, higher PI

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Part C Like Part B

• Use CES selector again, selecting different axes
  
• X Costdensity
• Y Youngs Modulus
• Results
• Aluminum has a higher price index than Magnesium.

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Part D Strength Index Analysis

• Similar to Part B, use CES Selector with below
  axes settings
• X density
• Y Tensile strength
• Set up for CES formula

Y intercept 36 Slope 2
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Part D Results

• Aluminum, Magnesium and Titanium all within
  acceptable range.
• Magnesium alloys have highest performance index

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Part E Strength Performance Cost Analysis

• Similar to part C, use CES Selector with axes
• X pricedensity
• Y tensile strength
• Results
• Aluminum alloy has a better price index

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Part F The Best

• Though Magnesium alloys give a better strength
  index, aluminum alloys have better stiffness
  performance indices.
• For the cost, in both cases, aluminum was best
• Interestingly, Wood performed better than all
  alloys across the board.
• Aluminum Alloys The Winner

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Conclusions

• CES Selector is a tremendously flexible analysis
  tool
• Much easier than Calculating each performance
  index by hand from Appendicies
• If it is possible to isolate a set of system
  parameters and ID at least one free variable, a
  concise list of acceptable materials can be
  gathered and compared for effectiveness.