(1) Figure 1 shows a cross section of a beam made of 3 × 12inch dimension lumber. There is a 3 × 1/2in. steel plate screwed into it on the underside. Figure 2 shows a schematic of the loading. Each point load is 5760lbf. The lumber is Douglas fir, and the steel is A325. Draw a load function (you may use my sign convention or that of the book), a shear diagram, and moment diagram. Calculate maximum tensile stress in wood and steel, and determine the implied safety factors of the wood and steel with respect to yield of steel or failure of wood. Use the least strength version of A325. (Yield strength = 120 ksi)
Figure 1. Schematic diagram of a beam made of Douglas fir with a steel plate screwed into the bottom surface.
(2) Figure 2 shows views of one of the three drums of a wire-saw to make silicon wafers. The drum is 1.328mlong, with a cylinder of aluminum with 125mm outer diameter and 75mm inner diameter. The shell of urethane is 25mm thick and rests on the aluminum cylinder with no gap.
Each roller looks very much like a beam. The cutting wires are spaced 1.260mm apart, beginning and ending 4mm from the ends of the urethane, and each carries a tension of 5N. These wiresd spread at a 60 degree angle to pass over three rollers.
Urethane does not possess a true Young's modulus except in the limit of no load. Rather its elstic modulus diminishes with increasing load, and one usually specifies a 10%, 20%, or 30 % modulus. However, at vanishingly small strain the modulus of this urethane is 690MPa. Using this information calculate a load, shear, and moment diagrams for this drum. Presume that the moment at the beam center applies to it uniformly, and estimate both the curvature of the beam and its mid-point deflection from this value of M. Hint: remember that to calculate the value of I for a composite beam you do not have to draw the transformed section, but only multiply the contribution to I of the urethane by a ratio of elastic moduli.
Because the drum rotates there is a cyclic stress that developes as the lower and upper sides of the drum switch places. To cut an ingot requires feeding 500km of wire through the web in a cycle of 1000m forward and 900m reverse. IOn other words each cycle makes use of only 100m of new wire. How many stress reversals are involved?
Figure 2. Two pictures of a wiresaw roller. The urethane-on-aluminum roller has wires that pass over it and spread at 60 degrees in order to pass over two other rollers.