Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_190

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the vertical distance from the lower edge of the built-up wooden beam to the location of the centroid. Problem 5/190 (a) Determination of the Centroidal Distance ● Calculation Process 1. Present Final Formula: For a composite area, the distance from a reference axis to the centroid is given by: where is the area of each component part and is the distance from the bottom edge to the centroid of each part. 2. Substitute Values: We divide the beam cross-section into three rectangular segments: the bottom flange (1), the vertical web (2), and the top flange (3). The reference axis is the lower edge (). Segment 1 (Bottom Flange): Segment 2 (Web): Segment 3 (Top Flange): H ˉ H ˉ =H ˉ A ∑ i A y ∑ ii A i y i y=0 A = 1 10× ′′ 2= ′′ 20 in 2 y = 1 = 2 2 ′′ 1 ′′ A = 2 2× ′′ 12= ′′ 24 in 2 y = 2 2+ ′′ = 2 12 ′′ 2+ ′′ 6= ′′ 8 ′′ A = 3 16× ′′ 2= ′′ 32 in 2 y = 3 2+ ′′ 12+ ′′ = 2 2 ′′ 14+ ′′ 1= ′′ 15 ′′ Substituting into the centroid formula: 3. Partial Operations: Calculate the sum of the first moments of area () and the total area (): Numerator: Denominator: 4. Final Calculation: ● Final Conclusion: The vertical distance from the lower edge to the centroid of the built- up wooden beam is . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/190 =H ˉ 20 in+24 in+32 in 222 (20 in⋅1)+(24 in⋅8)+(32 in⋅15) 2 ′′ 2 ′′ 2 ′′ A y ∑ ii A ∑ i 20+192+480=692 in 3 20+24+32=76 in 2 =H ˉ ≈ 76 in 2 692 in 3 9.10526 in =H ˉ 9.11 in =H ˉ 9.11