Meriam, Kraige & Bolton — Equilibrium of Rigid Bodies: Problem 4_26

⚡ Mecademy AIENG정역학 · ch4  Problem Statement Determine the force in member of the loaded truss. Assume that the four external reactions at and are equal in magnitude and are directed perpendicular to the local supporting surface. Problem 4/26 (a) Calculation of External Reactions 1. Formula: From vertical equilibrium of the entire truss: where are the reactions at the supports. 2. Substitution: Given that all reactions are equal in magnitude ( ): 3. Calculation: 4. Result: Each external reaction is . ● Final Conclusion: The magnitude of each of the four equal reactions is directed radially inward from the supports. (b) Determination of Force in Member CG 1. Formula: Consider a vertical section through the middle of the truss, cutting members , , , and . By symmetry, the force in member is equal CG A,B,E,F F = ∑ y 0⇒R cos30+ A ∘ R cos30+ B ∘ R cos30+ E ∘ R cos30− F ∘ 2L R ,R ,R ,R ABEF R = A R = B R = E R = F R 4Rcos30= ∘ 2L 4 R = ( 2 3 )2L⇒ 2 R = 32L R= 3 L R= ≈ 3 L 0.577L 3 L CDCGDHHGCG to the force in member (). We apply vertical equilibrium to the left half of the truss. 2. Substitution: Summing vertical forces on the left half (joints ): where and are the vertical components of the internal forces at the cut. 3. Calculation: From symmetry, . Let their magnitude be . The vertical components are (downward at ) and (upward at ), where is the angle of the diagonals with the horizontal. Substituting the reaction values: This confirms the vertical equilibrium is satisfied for