Meriam, Kraige & Bolton — Equilibrium of Rigid Bodies: Problem 4_113

⚡ Mecademy AIENG정역학 · ch4  Problem Statement Determine the components of all forces acting on each member of the loaded frame. Problem 4/113 (a) Analysis of External Reactions and Pin Forces ● Calculation Process 1. Present Final Formula: We first consider the entire frame as a single rigid body to find the external reactions at supports and . Let the radius of the semicircular arch be . We define a coordinate system with the origin at pin . Then, pin is at and pin is at . Both applied moments are clockwise. The equilibrium equations for the whole frame are: 2. Substitute Values: From the moment equation: From the vertical force sum: 3. Partial Operations (Member-by-Member Analysis): Now, consider member . It is pinned at and , and has a clockwise moment applied. Taking moments about pin for member : AB RA(0,0)C (R,R)B(2R,0)M F = ∑ x A + x B = x 0 F = ∑ y A + y B = y 0 M = ∑ A B (2R)− y M−M=0 B (2R)− y 2M=0 B = y = 2R 2M (upwards) R M A + y = R M 0⇒A = y − (downwards) R M CB C(R,R)B(2R,0)M CCB M = ∑ C B (R)+ y B (R)− x M=0 Substituting : Then, from the horizontal equilibrium of the whole frame: Finally, find the internal pin forces at . For member : By Newton's Third Law, the force on member at pin is equal and opposite: 4. Final Calculation: All force components are found as a function of and . Member : - At : - At : Member : - At : - At : ● Final Conclusion: The force components acting on each member are summarized as follows (assuming positive axes to the right and upward