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(Solved): A PFC Channel2 and a universal beam (UB) Section1 (See Table 1) are welded together to form the co ...
A PFC Channel2 and a universal beam (UB) Section1 (See Table 1) are welded together to form the composite section as shown in Figure (b). The individual sectional properties are given in Figures (c) and (d) where xo and yo indicate centroidal axes. Determine the largest uniformly distributed load w that can be applied, if the allowable stress is in tension and in compression for the cantilever beam of long. (a)Cantilever beam (c) UB Section (d) PFC Section
Table 1. Sectional properties UB Section PFC Channel a) Determine the location of the centroid with respect to the point ;
Determine the moment of inertia with respect to the centroidal axes. C) Compute the radius of gyration d) Determine the largest uniformly distributed load that can be applied, if the allowable stress is in tension and 120 MPa in compression.
Expert Answer
a) To determine the centroid of the composite section, we need to find the coordinates of the centroid of each individual section and then use the formula for the centroid of a composite section. Let's call the centroid of the UB section point A and the centroid of the PFC section point B. Then, we can find their coordinates with respect to point P as follows:Coordinates of A with respect to P: x_A = x_P + x_oA = 30 + 63 = 93 mm y_A = y_P + y_oA = 80 + 150/2 = 155 mmCoordinates of B with respect to P: x_B = x_P + x_oB = 30 + 100 = 130 mm y_B = y_P + y_oB = 80 + 75/2 = 122.5 mmNow, we can use the formula for the centroid of a composite section:x_c = (A1x1 + A2x2) / (A1 + A2) y_c = (A1y1 + A2y2) / (A1 + A2)where A1 and A2 are the areas of the UB and PFC sections, respectively, and x1, y1 and x2, y2 are their respective centroid coordinates.Substituting the values, we get:x_c = (42.2 * 93 + 120 * 130) / (42.2 + 120) = 116.7 mm y_c = (42.2 * 155 + 120 * 122.5) / (42.2 + 120) = 130.8 mmTherefore, the centroid of the composite section is located at a distance of 116.7 mm horizontally and 130.8 mm vertically from point P. , a) To determine the centroid of the composite section, we need to find the coordinates of the centroid of each individual section and then use the formula for the centroid of a composite section. Let's call the centroid of the UB section point A and the centroid of the PFC section point B. Then, we can find their coordinates with respect to point P as follows:Coordinates of A with respect to P: x_A = x_P + x_oA = 30 + 63 = 93 mm y_A = y_P + y_oA = 80 + 150/2 = 155 mmCoordinates of B with respect to P: x_B = x_P + x_oB = 30 + 100 = 130 mm y_B = y_P + y_oB = 80 + 75/2 = 122.5 mmNow, we can use the formula for the centroid of a composite section:x_c = (A1x1 + A2x2) / (A1 + A2) y_c = (A1y1 + A2y2) / (A1 + A2)where A1 and A2 are the areas of the UB and PFC sections, respectively, and x1, y1 and x2, y2 are their respective centroid coordinates.Substituting the values, we get:x_c = (42.2 * 93 + 120 * 130) / (42.2 + 120) = 116.7 mm y_c = (42.2 * 155 + 120 * 122.5) / (42.2 + 120) = 130.8 mmTherefore, the centroid of the composite section is located at a distance of 116.7 mm horizontally and 130.8 mm vertically from point P.