Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_113

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Draw the shear and moment diagrams for the beam subjected to the two point loads. Determine the maximum bending moment and its location. Problem 5/113 (a) Calculation of Support Reactions 1. Formula: We apply the equilibrium equations for the entire beam. Let and be the vertical reactions at supports and , respectively. Summing moments about point : And summing vertical forces: 2. Substitution: From the moment equation: 3. Calculation: Solving for : Solving for using : 4. Result: (acting downwards), (acting upwards). ● Final Conclusion: The reaction at support is downwards, and the reaction at support is upwards. M max R A R B AB A M = ∑ A 0⟹P⋅ − ( 4 l )2P⋅ + ( 4 3l )R ⋅ B l=0 F = ∑ y 0⟹R + A P−2P+R = B 0 − 4 Pl + 4 6Pl R ⋅ B l=0 R B − + 4 5Pl R ⋅ B l=0⟹R = B (upwards) 4 5P R A F =∑ y 0 R − A P+ = 4 5P 0⟹R = A P− = 4 5P − 4 P R = A − 4 P R = B 4 5P A 4 P B 4 5P (b) Shear and Bending Moment Functions 1. Formula: The shear force and bending moment are determined by considering sections along the beam. with . 2. Substitution and Calculation: For : At , . For : At , . For : At , . 3. Result: The shear diagram consists of three constant steps: , , and . The moment diagram is piecewise linear, connecting points , , , and . ● Final Conclusion: The internal force distributions are defined by the piecewise functions derived above, which characterize the shear and moment diagrams. (c) Maximum Bending Moment and Location 1. Formula: Compare