Question 9.2: Panel ABC in the slanted side of a water tank is an isoscele...
Panel ABC in the slanted side of a water tank is an isosceles triangle with the vertex at A and the base BC = 2 m, as shown in Figure 9.13. Find the force on the panel due to water pressure and on this force’s line of action.
Given: Dimensions of panel in water tank.
Find: Resultant force on panel; location of center of pressure.
Assume: No relative motion of fluid elements (hydrostatics); water has constant, uniform density, equal to its tabulated value at 20˚C (998 kg/m³).
We first want to understand the geometry of the triangular panel. We are given a side view of the tank, and the height of the triangle ABC. In a head-on view, we would see the panel as sketched in Figure 9.14 at left.
The water pressure has a hydrostatic distribution, as sketched in Figure 9.14 at right, and the resultant force is found by integrating this pressure over the panel area. This is equivalent to the formula F_R = ρgh_cA , where h_c is the depth of the centroid of the submerged surface, the triangular panel ABC.
The depth of point A is zero; the depth of points B and C is 4 m. The depth of the centroid of the triangular gate ABC is 2/3 of the way down. (Note: This is because the submerged surface is a triangle, not because the pressure has a triangular pressure prism.):
h_c =\frac{2}{3} (4 m) =2.67 m.
A=\frac{1}{2}bh=\frac{1}{2} (2 m)(5 m) =5 m² ,
so
F_R =ρ gh_cA= (998 kg/m³)(9.81 m/s² )(2.67 m )(5 m²) F_R = 131,000 N = 131 kN
This force acts at the center of pressure of the submerged panel ABC. Due to the symmetry of the panel, this is on the centerline (x_{_R} = 0), and we are only required to calculate the y coordinate y_{_R} (Figure 9.15).
y_{_R}=y_c+\frac{I_{xc}}{y_cA}= y_c+\frac{I_{xc}\sin \theta}{h_cA}=y_c+\frac{\frac{1}{36}bh^3 \sin \theta }{h_cA}
=3.33 m + \frac{\frac{1}{36}(2 \textrm{ m})(5 \textrm{ m})^3 \sin(53° ) }{(2.67 \textrm{ m})(5 \textrm{ m}^2)}
=3.33 m + 0.417 m
=3.75 m.
Note that this y_{_R} is measured down from A, along the panel surface, as shown in the sketch in Figure 9.16.
