Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_40

⚡ Mecademy AIENG정역학 · ch6  Problem Statement Determine the maximum value of the angle for which the uniform slender rod will remain in equilibrium. The coefficient of static friction at is , and friction associated with the small roller at may be neglected. Problem 6/40 (a) Determination of the maximum angle 1. Formula: First, establish the equilibrium conditions for the rod. For a rod supported at two points where one is a frictionless roller, the total reaction at the other support must be vertical to balance the vertical weight and the vertical reaction of the roller. Sum of horizontal forces: . Since point is a frictionless roller on a horizontal surface, . Thus, the reaction at must be purely vertical: . Sum of vertical forces: . Sum of moments about : . At the limit of slipping, the angle between the normal to the track at and the total vertical reaction is equal to the friction angle : From geometry, the sine of the angle that the normal makes with the vertical is: 2. Substitution: Substitute the given value to find : Set the geometric expression equal to this limit: θ Aμ = A 0.80 B θ F =∑ x R + Ax R = Bx 0B R = Bx 0A R = Ax 0 F =∑ y R + Ay N − B W=0 BM =∑ B W(Rcosθ)−R (2Rcosθ)= Ay 0⟹R = Ay 2 W αA R A β α=β=tan(μ ) −1 s α sinα= = R ∣x ∣ A = R 2 R sinθ−sinθ 2 2 sinθ−sinθ 2 μ = s 0.80α β=tan(0.80)≈ −1 38.66 ∘ sinα=sin(38.66)≈ ∘ 0.6247 2 = sinθ−sinθ 2 0.6247 3. Calculation: Solve the equation for : Square both sides: Rearrange into a quadratic equation: or Apply the quad