Meriam, Kraige & Bolton — Rigid Bodies: Equivalent Systems of Forces: Problem 3_69

⚡ Mecademy AIENG정역학 · ch3  Problem Statement The large bracket is constructed of heavy plate which has a mass per unit area. Determine the force and moment reactions at the support bolt at . Problem 3/69 (a) Force and moment reactions at O 1. Formula: The bracket consists of three segments. Let be the origin . Based on the diagram, we assume a coordinate system where is vertical, is along segment , and is along segments and . The mass and weight of each segment are: The total force reaction and moment reaction at are determined from the equilibrium equations: where and is the position vector of the centroid of each segment relative to . 2. Substitution: Identify the parameters for each of the three plate segments (assuming is at the vertical and horizontal center of the first segment): Segment 1 (width ): Area , Centroid , Weight Segment 2 (length ): Area , Centroid , Weight Segment 3 (length ): Area , Centroid , Weight Total weight . 3. Calculation: Force Reaction: ρ O O(0,0,0) zx byac W = i m g= i (ρA )g= i ρ(L h)g i RM O O F= ∑0⟹R+W = ∑ i 0 M = ∑ O 0⟹M + O ( r × ∑ i W )= i 0 W = i −W k i r i O O aA = 1 ahr = 1 (0,0,0) W = 1 −ρahgk bA = 2 bhr = 2 ( , ,0) 2 b 2 a W = 2 −ρbhgk cA = 3 chr = 3 (b, + 2 a ,0) 2 c W = 3 −ρchgk W = tot ρgh(a+b+c) R = z W + 1 W + 2 W = 3 ρgh(a+b+c) Moment Reaction: Calculate the cross products : Sum of moments from weights: The reaction moments at are : 4. Result: ● Final Conclusion: The reaction at bolt consists of a vertical force and a reaction