ME 499/699 Materials Selection. Homework -1 Solutions. (a) Using the E-ρ chart identify metals with both E > 100 GPa and E/ρ > 23 GPa/(Mg/m 3 ).

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1 1. Using Materials Selection Charts M 99/699 Materials Selection Homework -1 s (a) Using the - chart identify metals with both > 100 GPa and / > GPa/(Mg/m ). Stage-1: Tree stage to select metals Stage-: Graph stage - chart to isolate metals with >100GPa Stage-: Graph stage - chart Draw line with slope 1 dit Stage-, and modify Selection. nter X1 Mg/m and Y GPa. If the plot has X axis in kg/m then enter X1000 kg/m. Resultant selection is: Name Identity Cast iron, ductile (nodular) Commercially pure titanium High carbon steel ow alloy steel ow carbon steel Medium carbon steel Nickel Nickel-based superalloys Nickel-chromium alloys Stainless steel Titanium alloys Three classes steels, titanium alloys and nickel alloys (b) Using the - chart establish whether woods have a higher specific stiffness / than epoxies. Certain wood products have / greater than epoxies Name Identity Bamboo BA Hardwood: oak, across grain Hardwood: oak, along grain Plywood Softwood: pine, across grain Softwood: pine, along grain (c) Do titanium alloys have a higher or lower specific strength (strength/density, σ f / ) than tungsten alloys? This is important when you want strength at low weight (landing gear of aircraft, mountain bikes). Titanium alloys have a higher specific strength

2 . Translating design requirements: For the problems below, the best material for minimizing cost is to be selected. Translate the design requirements into the four steps function, constraints, objectives and free variables. Assign numerical values for the requirements where possible. (a) A material is required to manufacture office scissors. Paper is an abrasive material, and scissors sometimes encounter hard obstacles like staples. To resist abrasive wear the scissors must have blades of high hardness. In cutting, they will sooner or later encounter a staple or other hard obstruction that would chip a brittle blade some toughness is required. These two parameters help reduce wear, but there are other factors that influence it, so it is sensible to specify good wear resistance. Finally, the scissors must be formed if the handles are integral with the blades, they must be forged or stamped from sheet, requiring the ability to be processed in this way. The design requirements can be classified into the four steps of material selection Scissors High hardness Adequate toughness: K1c > 15MPa.m 0.5 Good wear resistance Able to be forged Minimize material cost (b) A material is required for the windings of an electric air-furnace capable of temperatures up to 1000ºC. Think out what attributes a material must have if it is to be made into windings and function properly in a furnace. If the material is to be used as windings it must be able to be drawn to wire and wound to a coil, requiring ductility. It must conduct electricity and be able to operate at 1000 o C in air. High temperature furnace winding Maximum service temperature, Tmax > 1000º C Able to be rolled or drawn to wire Good electrical conductor Some ductility so that it can be wound, ε f > % Good resistance to oxidation at elevated temperature Minimize material cost

3 . Derivation of material indices: The problems below deal with cantilever beams with a square cross-section and fixed length. Derive the material index M for case. (a) A cantilever beam of given length and fixed square cross-section (of side t) is loaded at its end by a load F. In order to minimize deflection, show that the material index to be maximized is M, where is Young's modulus (neglect self-weight). nd loaded cantilever beam ength is specified Cross-section t x t is specified nd load F is specified Minimize deflection δ The deflection of the end of an end-loaded cantilever beam is (See Appendix A.) F δ I bh t I δ ( F ) t The last equation expresses the deflection as the product of the loading, geometry and material indices. To minimize deflection, the material index to be minimized is (1/) or the material index to be maximized is M (b) A cantilever beam of given length and fixed square cross-section (of side t) deflects under its own weight (w per unit length). In order to minimize deflection, show that the material index to be maximized is M /, where is Young's modulus and is the density.

4 Self loaded cantilever beam ength is specified Cross-section t x t is specified Minimize deflection δ The beam is subject to a self load per unit length of w gt The deflection of the end of a cantilever beam subject to a distributed load of f is w gt δ ( g) ( ) 8I 8 t /1 t In order to minimize deflection δ, the material index to be maximized is M (c) A cantilever beam of given length and square cross-section (i.e. size is not given) deflects under its own weight (w per unit length). Show that for the lightest beam that does not deflect more than a given value δ, the material index to be maximized is M /, where is Young's modulus and is the density. nd loaded cantilever beam ength is specified maximum deflection δ is specified Minimize mass Choice of cross section At The mass of the beam, which is to be minimized, is m t The deflection of a beam under self loading from part (b) above is: δ t ( g) t δ ( g)

5 Substituting for t into the mass equation gives ( ) ( ) g g m 5 δ δ In order to minimize mass m, the material index to be maximized is M