Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_44

⚡ Mecademy AIENG정역학 · ch6  Problem Statement The uniform slender bar of length has an ideal roller at its upper end . The coefficient of static friction along the horizontal surface varies according to , where is in meters and . Determine the minimum angle for which equilibrium is possible. Problem 6/44 (a) Determination of the minimum angle ● Calculation Process 1. Present Final Formula: Establish the equilibrium equations from the Free Body Diagram (FBD) of the bar. Let be the weight of the uniform bar acting at its midpoint. At end (vertical wall), there is a horizontal normal force due to the ideal roller. At end (horizontal floor), there is a vertical normal force and a horizontal friction force . Vertical equilibrium: Horizontal equilibrium: Moment equilibrium about : From the moment equation, we find the required normal force at : For equilibrium, the required friction force must not exceed the maximum available static friction : 2. Substitute Values: Substitute the given expressions , , , and . L=1.8 mA μ = s μ (1− 0 e) −x xμ = 0 0.50θ θ W=mg AN A BN B f B F =∑ y 0⇒N − B mg=0⇒N = B mg F =∑ x 0⇒N − A f = B 0⇒f = B N A BM =∑ B 0⇒mg cosθ−( 2 L ) N (Lsinθ)= A 0 A N = A = 2sinθ mgcosθ 2tanθ mg f = B N A f = max μ N sB ≤ 2tanθ mg μ (mg)⇒ s ≤ 2tanθ 1 μ s μ = s μ (1− 0 e) −x x= LcosθL=1.8 mμ = 0 0.50 ≤ 2tanθ 1 0.50(1−e ) −1.8cosθ This simplifies to the transcendental equation at the verge of slipping (minimum ): 3. Partial Operations: We solve the equation numerically: Try :