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Calculate the Centroid of the Section - Solution

   

Added on  2020-03-23

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Solution no 1a.Determine the centroid of the section.Let us consider, X = 924 mm and Y = 541 mm (as per FIN/FRIC number)Therefore, the I-section can be updated as:Let us consider this as three segment and calculate the centroid for each segmentAlso Ai and yi to be the area and distance of centroid (from the datum line) for the individual segment.Calculating for each segment:Segment 1:A1 = 924 × 40 = 36960 mm2
Calculate the Centroid of the Section - Solution_1
y1 = 40 + 300 + 402 = 360 mmSegment 2:A2 = 40 × 300 = 12000 mm2y2 = 40 + 3002 = 190 mmSegment 3:A3 = 541 × 40 = 21640 mm2y3 = 402 = 20 mmNow, Ӯ (Centroid of the I-section) = ∑Aiyi∑Ai = A1y1+A2y2+A3y3A1+A2+A3 =(36960×360)+(12000×190)+(21640×20)36960+12000+21640 = 13305600+2280000+43280070600 = 1601840070600= 226.89 mmb.Determine the second moment of area about a horizontal axis at the centroid of the section. To calculate the second moment of inertia, we need to use the “Parallel Axis Theorem”:Itotal = ∑(Ii¿+Aidi2)¿Where, di = Vertical distance from the centroid of the segment to the neutral axisAi = Area of the segmentIi = Moment of Inertia of the sectionWe also know that, for rectangular Section:I = 112bh3b = width of rectangleh = height of rectangle
Calculate the Centroid of the Section - Solution_2
Now, consider the individual segment:Segment 1:I1= 112¿924 * 403¿ = 5913600/12 = 492800 mm4A1 = 924 × 40 = 36960 mm2d1 = (y1 - ẏ) = 360 – 226.89 = 133.11 mm Segment 2:I2= 112 (40 * 3003) = 1080000000/12 = 90000000 mm4A2 = 40 × 300 = 12000 mm2d2 = (y2 - ẏ) = 190 – 226.89 = 36.89 mm Segment 3:I3= 112 (541 * 403) = 34624000/12 = 2885333 mm4A3 = 541 × 40 = 21640 mm2d3 = (y3 - ẏ) = 20 – 226.89 = 206.89 mm Itotal = ∑(Ii¿+Aidi2)¿Itotal = (I¿¿1+A1d12)¿ + (I¿¿2+A2d22)¿ + (I¿¿3+A3d32)¿Itotal = (492800+(36960∗133.112)) + (90000000+(12000∗36.892)) + ¿Itotal =655360136+ 1508346720 + 929152469.244Itotal = 3092859325.244 mm4 = 3.09 * 109mm4Solution no 2Now, according to the Id, W = 24 KN/m and Y = 19 KN
Calculate the Centroid of the Section - Solution_3

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