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The project: I'm given a 1.2m long cylindrical cantilever with forces acting on it; one axial tensile force, and two forces producing torque/shear along the outside edge of the beam. I can choose either a solid, hollow, or composite rod, and can choose from steel, aluminum, or titanium. I can also vary my outer radius (and inner/boundary radii in the cases of hollow or composite rods). I need to find the rod of the least mass that can still resist the forces without YIELDING (not failing) given a 1.5 factor of safety.

My approach: I wrote up the equations for axial, shear, torsional, and bending stresses in excel. I just dragged and dropped the formula for each down the columns with varying radii in the first column (only for solid rods so far; once I get past this hiccup I'll be working on composites).

My problem: Using a type of aluminum with a ~250 MPa tensile yield strength and 2.7 g/cm^3 density, my equations show that a 2-cm radius aluminum rod can resist a force of 120 kN (diagram shows 80 kN --> 120 kN accounting for the FoS). This definitely sounds wrong, obviously, and the equations for just the tensile force is exceedingly simple. I still, for some reason, can't figure out where I've gone wrong.

My Calculations: Without going into all of the different radii I used, I'll show what I did for the 2-cm radius rod. Axial stress = F/A; the force is 120 kN. Area is (obviously) pi*r^2 --> pi*(0.02)^2 --> 0.001256637. F/A = 120,000/0.001256637 = 95.5 x 10^6 N/m^2 = 95.5 MPa. So the rod is experiencing 95.5 MPa of stress, but its tensile yield strength is 250 MPa, so this 2-cm radius aluminum rod can hold 120 kN (or roughly 27,000 lbs) of force (according to my surely-wrong calculations).

I know this should be very easy, but I honestly can't figure out where I'm going wrong. Any help would be greatly appreciated.