Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_122

⚡ Mecademy AIENG정역학 · ch6  Problem Statement The uniform slender bar of Prob. 6/40 is repeated here, but now the ideal roller at B is removed. The coefficient of static friction at A is 0.70 and that at B is 0.50. Determine the maximum value of the angle for which equilibrium is possible. [Note: From Prob 6/40, the bar has a length of and rests on a circular surface of radius .] Problem 6/122 (a) Determination of the maximum equilibrium angle ● Calculation Process 1. Geometric Relations: Let the center of the circular arc be . For a bar of length resting on the arc at and the floor at , we define as the angle from the downward vertical to the radius . The height of point from the floor is . Since , we have: 2. Equilibrium Equations: From the free-body diagram of the bar, we establish moment equilibrium about and force equilibrium in and directions. At the state of impending slip, friction forces are and . These lead to the governing transcendental equation: 3. Substitution of Known Values: Given and : 4. Numerical Solution: We solve for using the constraint : Assume : θ 2RR θ O L=2RABβ OAAy = A R(1−cosβ)y = A 2Rsinθ 1−cosβ=2sinθ⟹cosβ=1−2sinθ Bxy F = A μ N AA F = B μ N BB tanθ= 2μ (tanβ−μ ) BA (1−μ μ )tanβ−(μ +μ ) ABAB μ = A 0.70μ = B 0.50 tanθ= 2×0.50×(tanβ−0.70) (1−0.70×0.50)tanβ−(0.70+0.50) tanθ= tanβ−0.70 0.65tanβ−1.2 θcosβ=1−2sinθ θ=21.5 ∘ RHS: LHS: Since LHS RHS, the solution is verified. 5. Result: ● Final Conclusion: The maximum value of the angle for which equilibriu