Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_42

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the height above the base of the centroid of the cross-sectional area of the beam. Neglect the fillets. Problem 5/42 (a) Determination of the centroidal height from the base ● Calculation Process 1. Present Final Formula: The centroidal height of a composite area is determined by the formula: where is the area of each component and is the vertical distance of its individual centroid from the reference base. 2. Substitute Values: We divide the cross-section into three rectangular parts: the bottom flange (1), the vertical web (2), and the top flange (3). Part 1 (Bottom Flange): Part 2 (Web): Height of web Part 3 (Top Flange): Substituting these into the formula: 3. Partial Operations: Numerator sum: yˉ yˉ =yˉ A ∑ i A y ∑ ii A i y i A = 1 12.48×1.40=17.472 in 2 y = 1 = 2 1.40 0.70 in h = w 13.71−1.40−1.40=10.91 in A = 2 0.87×10.91=9.4917 in 2 y = 2 1.40+ = 2 10.91 6.855 in A = 3 6.24×1.40=8.736 in 2 y = 3 13.71− = 2 1.40 13.01 in =yˉ 17.472+9.4917+8.736 (17.472×0.70)+(9.4917×6.855)+(8.736×13.01) A y =∑ ii 12.2304+65.0656+113.6554= 190.9514 in 3 Denominator sum: 4. Final Calculation: ● Final Conclusion: The height of the centroid above the base is approximately . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/42 A =∑ i 17.472+9.4917+8.736=35.6997 in 2 =yˉ ≈ 35.6997 190.9514 5.3488 in 5.35 in =yˉ5.35 in