Tiêu đề 3 Hydrostatic Forces.pdf ✅

HGE 3: Hydrostatic Forces 1. Hydrostatic Force on a Plane Surface From the definition of pressure, p = dF dA dF = P dA F = ∫ P dA For the location of the resultant force, the center of pressure, using Varignon’s theorem (see PSAD 2: Moment and Couple), Fyp = ∫ y dF yp = ∫ y dF F When the gate is inclined at an angle θ from the horizontal, h = y sin θ, hp = ∫ y sin θ dF F When the gate is vertical, θ = 90° and h = y, hp = ∫ h dF F 1.1 Formula Method When the top of the liquid is open to atmosphere, the gauge pressure at the surface is zero. P − 0 = γh P = γh From earlier, F = ∫ P dA F = γ ∫ h dA Note that from the definition of centroids (see MSTC 79: Centroids), Ah̅ = ∫ h dA F = γh̅A

πab At the center I̅ x = πab3 4 πr 2 2 x̅= 0 y̅ = 4r 3π I̅ x = 0.1098r 4 πr 2 4 x̅= 4r 3π y̅ = 4r 3π I̅ x = 0.0549r 4 1.2. Geometric Method Drawing the pressure diagram of the surface upon which the hydrostatic force is exerted: From the integral of the hydrostatic force, the intensity of the pressure diagram can be treated as altitude to the plane area. Thus, the magnitude of the hydrostatic force is the volume of the pressure diagram. From the integral of the center of pressure, ∫ y dF F is the centroid of volume of the pressure diagram. Thus, the center of pressure is at the centroid of the pressure diagram. 2. Hydrostatic Force on a Curved Surface