A propeller shaft of length L, loaded in torsion, must support a specified torque T without...

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A propeller shaft of length L, loaded in torsion, must support a specified torque T without failing and be as cheap as possible. Section shape is a variable and failure means the first onset of plasticity. The table below summarizes the requirements. Function Constraints Objective Free variables Cheap shaft Specified failure torque T Length L specified Minimize material cost C Choice of material Section shape and scale (a) Derive the material index. O (b) The table below shows the dimensions of 5 candidate materials: Use the index you derived to rank these candidate materials for this application using data from Appendix A. Material R (mm) t (mm) High carbon steel 70 Titanium alloys 50 Aluminum alloy 90 10 Magnesium alloy 80 10 CFRP (carbon fiber reinforced epoxy) 100 86007 HINT: Use the torsional formula: JT R T= = QT where T is the applied torque, is the shear stress, J is the polar moment of area, R is the outer radius of the shaft, and Q = J/R. At onset of yielding, 7 = y/2, where by is the yield strength. The shape factor for strength-limited in torsion can be expressed as D = Q Qo where Qo is the section moment of a square solid shaft. Use Table 10.2 to express Qo in terms of its cross-section area Ao. Do not assume THIN-walled cylinder. A propeller shaft of length L, loaded in torsion, must support a specified torque T without failing and be as cheap as possible. Section shape is a variable and failure means the first onset of plasticity. The table below summarizes the requirements. Function Constraints Objective Free variables Cheap shaft Specified failure torque T Length L specified Minimize material cost C Choice of material Section shape and scale (a) Derive the material index. O (b) The table below shows the dimensions of 5 candidate materials: Use the index you derived to rank these candidate materials for this application using data from Appendix A. Material R (mm) t (mm) High carbon steel 70 Titanium alloys 50 Aluminum alloy 90 10 Magnesium alloy 80 10 CFRP (carbon fiber reinforced epoxy) 100 86007 HINT: Use the torsional formula: JT R T= = QT where T is the applied torque, is the shear stress, J is the polar moment of area, R is the outer radius of the shaft, and Q = J/R. At onset of yielding, 7 = y/2, where by is the yield strength. The shape factor for strength-limited in torsion can be expressed as D = Q Qo where Qo is the section moment of a square solid shaft. Use Table 10.2 to express Qo in terms of its cross-section area Ao. Do not assume THIN-walled cylinder.