Problem M22 A cantilever beam is to be made with a length L and circular cross section, radius R (to be specified by the designer), to support an applied moment, M. a) What combination of material properties must be minimized to obtain the highest bending stiffness (i.e. minimum deflection for a given applied moment) beam for a given mass of material? (show your working). The deflection, d, of a cantilever beam of this shape is given by: 2ML2 d = pR4E where E is the Young's modulus. b) Choose between the following material choices to select the most promising material to construct a wing for a small Unmanned Aerial Vehicle (which will carry load as a cantilever beam) Material Density, r,Modulus, E, Poisson’s Yield Stress, Price , p, (Mg/m3) (GPa) Ratio sy, (MPa) ($/Mg) Mild Steel 7.9 203 0.31 220 375 Aluminum alloy (2024) 2.8 71 0.33 350 1650 Titanium alloy Ti-6Al4V 4.5 120 0.34 850 30000 Carbon fiber composite 1.5 230 0.3 1050 70000 Polyethelene (High Density) 0.96 1.1 0.39 30 1000 Wood (pine) 0.6 12 0.35 300 300 Silicon Carbide (SiC) 3.0 410 0.17 300 (Flexural) 30000 c) To a first approximation the human hip joint carries forces like a cantilever beam. In hip replacement surgery it is important to use material with the same total mass and bending stiffness as the bone that is removed. Bone has a modulus of 18 GPa and a density of 1.55�Mg/m3. By reference to the attached materials selection chart suggest two materials that could be used for hip replacements. Include the chart, and your working with your answers. 5From: Material Selection in Mechanical Design, M.F Ashby, Pergamon Press, Oxford, 1992
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