Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_40

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the distance from the upper surface of the symmetric double-T beam cross section to the location of the centroid. Dimensions are in millimeters. Problem 5/40 (a) Determination of the centroid distance ● Calculation Process 1. Present Final Formula: The distance of the centroid from the top surface for a composite area is given by: where is the area of each component and is the distance of its individual centroid from the reference top surface. 2. Substitute Values: We divide the cross-section into three rectangles: one top flange and two vertical webs. Part 1 (Top Flange): Width , Thickness Part 2 (Two Webs): Height , Thickness (for each web) The individual centroid of the webs is at the middle of their height, plus the thickness of the top flange: H ˉ H ˉ H ˉ =H ˉ A ∑ i A y ∑ ii A i y i w = 1 200 mmt = 1 15 mm A = 1 200×15=3000 mm 2 y = 1 = 2 15 7.5 mm h = 2 120 mmt= 2 15 mm A = 2 2×(120×15)=2×1800=3600 mm 2 y = 2 15+ = 2 120 15+60=75 mm Substituting into the centroid formula: 3. Partial Operations: Calculate the first moment of area (numerator) and the total area (denominator): 4. Final Calculation: ● Final Conclusion: The distance from the upper surface to the centroid is (rounded to one decimal place). ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/40 =H ˉ 3000+3600 3000(7.5)+3600(75) Ay=∑22,500+270,000=292,500 mm 3 A=∑3,000+3,600=6,600 mm 2 =H ˉ ≈ 6,600 292,500 44.318 mm =H ˉ 44.3 mm =H ˉ 44.3 mm