Training Civil Engineering Hydrostatics — Pressure & Fluid Forces
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Hydrostatics — Pressure & Fluid Forces

24 min Civil Engineering

Hydrostatics — Pressure & Fluid Forces

Civil engineers design dams, retaining walls, and tanks by calculating hydrostatic pressure and the resultant force on surfaces.

Hydrostatic Pressure

$$P = \rho g h$$

where $\rho$ = fluid density (kg/m³), $g = 9.81$ m/s², $h$ = depth below the surface (m). Pressure increases linearly with depth.

Force on a Submerged Vertical Surface

$$F = \frac{1}{2} \rho g h^2 \cdot w$$

where $h$ = height of the fluid and $w$ = width of the surface. The force acts at $\frac{2}{3}h$ from the surface.

Example 1

A dam face is 8 m high and 20 m wide. Water is at full height. Find the total hydrostatic force.

$$F = \frac{1}{2}(1{,}000)(9.81)(8^2)(20) = \frac{1}{2}(1{,}000)(9.81)(64)(20) = 6{,}278 \text{ kN}$$

It acts at $\frac{2}{3}(8) = 5.33$ m below the surface.

Example 2

A rectangular tank is 3 m deep, filled with water. Find the pressure at the bottom.

$$P = 1{,}000 \times 9.81 \times 3 = 29{,}430 \text{ Pa} = 29.4 \text{ kPa}$$

Example 3

A retaining wall holds back water 5 m deep. Width = 1 m (per unit length). Find the overturning moment about the base.

$F = \frac{1}{2}(1{,}000)(9.81)(25)(1) = 122.6$ kN. Acts at $\frac{5}{3} = 1.667$ m above the base.

$$M = 122.6 \times 1.667 = 204.4 \text{ kN·m per meter of wall}$$

Practice Problems

1. Find the pressure at 12 m depth in fresh water.
2. A vertical gate is 4 m wide and 6 m high, submerged with one side holding water. Find the total force.
3. At what depth does the pressure in seawater ($\rho = 1{,}025$ kg/m³) reach 200 kPa?
4. A triangular dam face has height 10 m and base width 15 m. Find the total force.
5. Find the center of pressure for a vertical rectangle of height $h$ submerged from the surface.
6. A swimming pool is 2 m deep at one end. Find the force per meter width on that end wall.
7. If atmospheric pressure is 101.3 kPa, find the absolute pressure at 5 m depth in water.
8. A cylindrical tank has diameter 3 m and is filled to 4 m depth. Find the total force on the curved wall.
9. Compare the hydrostatic force on a 6 m high wall when water is at 3 m vs. full 6 m.
10. A lock gate is 8 m wide. Water is 7 m on one side, 3 m on the other. Find the net horizontal force.
11. An inclined surface at $60°$ from horizontal is 5 m long and 2 m wide, submerged from the surface. Find the force.
12. A dam must resist overturning. If $F = 500$ kN acting at 4 m above the base, and the dam weight is 800 kN acting 3 m from the toe, is it stable?
Show Answer Key

1. $P = 1{,}000 \times 9.81 \times 12 = 117.7$ kPa

2. $F = \frac{1}{2}(1{,}000)(9.81)(36)(4) = 706$ kN

3. $h = P/(\rho g) = 200{,}000/(1{,}025 \times 9.81) = 19.9$ m

4. Average width = 7.5 m; $F = \frac{1}{2}(1{,}000)(9.81)(100)(7.5) = 3{,}679$ kN (using integration for triangular shape)

5. $y_{cp} = \frac{2h}{3}$ from the surface

6. $F = \frac{1}{2}(1{,}000)(9.81)(4)(1) = 19.62$ kN/m

7. $P_{abs} = 101.3 + 49.05 = 150.4$ kPa

8. $F = \rho g \bar{h} \cdot A_{\text{projected}} = 1{,}000(9.81)(2)(3 \times 4) = 235.4$ kN

9. $F \propto h^2$: at 6 m force is $4\times$ the force at 3 m

10. $F_1 = \frac{1}{2}\rho g (49)(8)$, $F_2 = \frac{1}{2}\rho g (9)(8)$; net $= \frac{1}{2}(9{,}810)(40)(8) = 1{,}570$ kN

11. Vertical depth to centroid $= \frac{5\sin 60°}{2} = 2.165$ m; $F = \rho g (2.165)(5 \times 2) = 212.6$ kN

12. Overturning moment $= 500 \times 4 = 2{,}000$ kN·m; Stabilising $= 800 \times 3 = 2{,}400$ kN·m; Factor of safety $= 1.2$ — marginally stable

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