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CHAPTER 3 TOTAL HYDROSTATIC FORCE ON SURFACES Plane Areas a) Vertical Plane Area gate Liquid Surface h e F hp Hydrostatic Force (F) cg cp P = F/A F Pcg A or Fh A Eccentricity e Igx Ah Center of pressure hp h e h Igx Ah where: Pcg = pressure at the centroid of the plane. (kPa, psi) A = area of the plane surface (perpendicular area) (mm2, in2) Igx = centroidal moment of inertia of the plane along x-axis (mm4, in4) h = vertical distance from free liquid surface to c.g. (mm, in) hp = vertical distance from free liquid surface to c.p. (mm, in) b) Inclined Plane Area Liquid Surface h y F e cg cp or Hydrostatic Force F Pcg A Fh A Eccentricity (e) e Igx Ay where: Pcg = pressure at the centroid of the plane. (kPa, psi) A = area of the plane surface (perpendicular area) (mm2, in2) IgX = centroidal moment of inertia of the plane along x-axis (mm4, in4) h = vertical distance from free liquid surface to c.g. (mm, in) y = inclined distance from free liquid surface to c.g. (mm, in) y h sin = angle of plane makes with the horizontal Center of pressure (yp) yp = inclined distance from free liquid surface to c.p. (mm, in) yp y e y Igx Ay Common Shapes Area y Igx bh h 2 bh3 12 1 bh 2 2 h 3 bh3 36 r r4 4 4r or 0.424r 3 0.11r 4 y cg h b y h cg b r y r cg r cg y 2 r2 2 Example 1 A vertical rectangular plane of height h and base b is submerged in a liquid with its top edge at the liquid surface. Determine the total force F acting on one side and its location from the liquid surface. 2h bh2 hp , F 3 2 Example 2 A vertical circular gate of radius r is submerged in a liquid with its top edge flushed on the liquid surface. Determine the magnitude and the location of the total force acting on one side of the gate. F = r3 Example 3 [Force on a submerged vertical plane area] Find the force on the gate and the center of pressure. Steps: 1. find the centroid of the area 2. find the distance from the top of fluids to centroid = h or hc 3. determine the pressure at the centroid = P 4. determine the force at the centroid = F 5. calculate the moment of inertia of the area = Igx or Ic 6. compute the center of pressure = hp Step 1: Centroid of given rectangle = Step 2: h = 3.0 + 0.6 = 3.6 m Step 3: Step 4: 1.2 m h = = 0.6 m 2 2 kN P h 9.81 3 3.6 m 35.32 kPa m kN F = PA = 35.32 2 2 m 1.2 m 84.77 kN m Step 5: bh3 2 m 1.2 m Igx 0.2880 m4 12 12 Step 6: The center of pressure = hp h e h 3 Igx hA = 3.6 m 0.2880 m4 3.63 m 3.6 m 1.2 m 2 m Example 4 [Force on a submerged inclined plane area] A 60-cm square gate (s = 0.6 m) has its top edge 12 m below the water surface. It is on a 45º angle and its bottom edge is hinged as shown. What force P is needed to just open the gate? Fig. 2a: The hydrostatic force on a gate Fig. 2b: Free-body diagram Solution h 12 0.3 sin 45º 12.21 m F hA 9.8112.21 0.6x0.6 43.12 kN 12.21 17.27 m h sin 45º sin 45º 0.6x0.63 Igx 12 e 0.0017 m 0.6x0.6 17.27 Ay y d 0.3 e 0.3 0.0017 0.2983 m M hinge 0 Clockwise F d P a 0 43.12 0.2983 P 0.6 0 P 21.44 kN Exercises 1. A vertical rectangular gate 1.5 m wide and 3 m high is submerged in water with its top edge 2 m below the water surface. Find the total pressure acting on one side of the gate and its location from the water surface. 2. A vertical triangular gate with top base horizontal and 1.5 m wide is 3 m high. It is submerged in oil having sp. gr. of 0.82 with its top base submerged to a depth of 2 m. determine the magnitude and location of the total hydrostatic force acting on one side of the gate. 3. A vertical plate 4 m x 4 m is immersed in a position such that the center of pressure shall be 6 cm from the center of gravity. (a) How far below the oil surface (s = 0.80) should the horizontal upper plate be placed, (b) Locate the position of the center of pressure from the oil surface, and (c) What is the hydrostatic force acting on the vertical plate. 4. The gate in the figure shown is 1.5 m wide, hinged at point A, and rest against a smooth wall at B. Compute a) the total force on the gate due to seawater (b) the reaction at B, and c) the reaction at hinged A. Seawater B 5m 2m A A 3m