Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_174

⚡ Mecademy AIENG정역학 · ch5  Problem Statement The Quonset hut is subjected to a horizontal wind, and the pressure against the circular roof is approximated by . The pressure is positive on the windward side of the hut and is negative on the leeward side. Determine the total horizontal shear force on the foundation per unit length of roof measured normal to the paper. Problem 5/174 (a) Total horizontal shear force Q per unit length ● Calculation Process 1. Present Final Formula: The total horizontal wind force per unit length is obtained by integrating the horizontal component of the pressure force over the entire semicircular roof. Let be the angle measured from the windward horizontal base ( at the windward stagnation point, at the leeward base). The horizontal component of the force exerted by the pressure on an infinitesimal area element (per unit length) is: The total horizontal force per unit length required for equilibrium is equal to the integral of this component: 2. Substitute Values: Substitute the trigonometric identity : 3. Partial Operations: Evaluate the integral: p p=p cosθ 0 Q F x θ θ=0θ=π p dA=rdθ dF = x p⋅cosθ⋅dA=(p cosθ)⋅ 0 (cosθ)⋅(rdθ) Q Q= p r cos θ d θ∫ 0 π 0 2 cosθ= 2 2 1+cos2θ Q=p r d θ 0 ∫ 0 π 2 1+cos2θ d θ = ∫ 0 π 2 1+cos2θ + [ 2 θ 4 sin2θ ] 0 π Substitute the limits: 4. Final Calculation: Multiply by the constants: ● Final Conclusion: The total horizontal shear force on the foundation per unit length is . This force is directed in the direction of t