1) Given the loading for a three point bend by the load P on a beam of length "L" and cross section width and height of "b" and "t", respectively, as shown, calculate the MAXIMUM MOMENT (M) for this configuration. Then show that the MAXIMUM DEFLECTION (δ_max) at which FAILURE occurs = σ_fL²/6tE where σ_f is the stress representing the onset of failure or yielding, t is the height of the beam, and E is the elastic modulus.

2) You work for FICTIONAL SPACE FLIGHT EXCURSIONS, a company that makes various types of space transportation vehicles for commercial space flight. As an engineer, you are asked to pick materials for protective tiles to be used as the outer skin in various regions on one of their new spacecraft designs.

For the several scenarios presented in the three parts of this problem below (I, II, and III), and for our purposes of this fantasy design, assume the tile dimensions are fixed at 10 inches x 4 inches in length (1) and width (b) respectively and that they require a maximum temperature of operation minimum of 600 C. Furthermore, treat the stress on the tile as a panel in bending. EXCLUDE METALS AND ALLOYS from your search, and use the FLEXURAL STRENGTH as the criteria for the onset of failure.

PART I) The first design requires that the tiles be light weight, yet strong. In addition, a further constraint is that during a critical portion of an atmospheric re-entry time that lasts for 15 minutes, the tiles must NOT allow any appreciable temperature change to the spacecraft.

a. List the TRANSLATION details for this particular design requirement including your free variables:

b. List the performance equation(s) in proper FGM form FOR THIS SITUATION, along with the COUPLING CONSTANT (in simplest form) for these constraints below (Assign mass2 = that for the strength constraint):

c. Plot the graph showing the optimum choice of materials for the multiple constraints, placing the STRENGTH consideration (M2) on the y-axis.

d. Using the same plot as in "c" above, place 4 lines on that plot representing loads P of 30 N, 300N, 3000 N, and 30000 N and list the TWO BEST materials that satisfy both constraints for EACH force.

PART II) In the next design problem, consider a situation where the materials chosen for this application have to be STRONG (a constraint), but must also minimize both mass and cost.

a. Show on a plot (with cost considerations on the y-axis) the locus of non-dominated solutions for this condition.

b. State a PENALTY FUNCTION Z for this condition in terms of materials properties: ___________________ and provide a plot of the penalty function value for the following exchange constants (one plot per exchange constant). LIST AT MOST TWO of the best candidates from each of the graphs next to the exchange constants below and include labels on the plots:

i) 0.1

ii) 1

iii) 10

iv) 100

v) 1000

c. Finally, a very common material that has been used for thermal insulation in the space shuttle program is silica. Use FUSED SILICA as the material to replace and provide another plot showing the RELATIVE TRADEOFFS of mass and cost. Is there any material better than fused silica according to your plot? (Circle one: YES / NO)

PART III) Suppose for yet another application you want to replace a MULLITE Al203-SiO2 ALLOY ceramic with another material for which you need to MAXIMIZE the thermal shock performance of the material (R= λ_th*σ*(1-v)/α_thE where λ_th is the thermal conductivity, v is Poisson's ratio, σ is the failure strength, α_th is the thermal expansion coefficient, and E is Young's Modulus.) In addition, you also want to MAXIMIZE THE TIME (t) IT TAKES for any appreciable change in temperature of the spacecraft to occur.

a. Write the PERFORMANCE equations for these two objectives

b. Plot the material parameter for the thermal shock resistance on the y-axis versus the material parameter for the maximum time objective on the x-axis, both relative to the material being replaced (mullite Al203-5i02 alloy).

c. Label the pareto candidates on the graph and list out at least 6 candidates that are better than mullite (Al203-SiO2 alloy) in the spaces below and CIRCLE THE BEST ONE:

d. Write a RELATIVE PENALTY FUNCTION for this situation

e. Show on your graph (the approximate regions) where the ratio of the relative exchange constants (thermal shock / time) equal:

i. 0.1

ii. 1

iii. 10